COSMOS FloWorks Fundamentals

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Contents

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Solving Engineering Tasks

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-1

Simulating Engineering Tasks with COSMOSFloWorks . . . . . . . . . . . . . . . . . . . . . .1-4

Solving Engineering Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-6

Frequent Errors and Improper Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11

Advanced Knowledge

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1

Mesh Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1

Types of Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-1Mesh Construction Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-3Basic Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-4Control Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-5Resolving Small Features by Using the Control Planes . . . . . . . . . . . . . . . . . . . .2-5Contracting the Basic Mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-6

COSMOSFloWorks 2008 Fundamentals -i

Resolving Small Solid Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-7Curvature Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-7Tolerance Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-9

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Narrow Channel Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9Local Mesh Settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12Recommendations for Creating the Computational Mesh. . . . . . . . . . . . . . . . . 2-13

Mesh-associated Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13Visualizing the Basic Mesh Before Constructing the Initial Mesh . . . . . . . . . . 2-14Enhanced Capabilities of the Results Loading . . . . . . . . . . . . . . . . . . . . . . . . . 2-14Viewing the Initial Computational Mesh Saved in the .cpt Files . . . . . . . . . . . 2-15Viewing the Computational Mesh Cells with the Mesh Option . . . . . . . . . . . . 2-15Visualizing the Real Computational Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 2-16Switching off the Interpolation and Extrapolation of Calculation Results . . . . 2-18Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20

Calculation Control Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21Finishing the Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21Refinement of the Computational Mesh During Calculation . . . . . . . . . . . . . . 2-23

Flow Freezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-25

What is Flow Freezing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-25How It Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-26Flow Freezing in a Permanent Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-26Flow Freezing in a Periodic Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-28

Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-29Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-29Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-30Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-30Examples of use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-31Rotating impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-31Hydrofoil in a tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-32Ball valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-32Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33

Steam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33

Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-33Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-34Example of use. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35Heat exchanger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-35

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Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35

Humidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35

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Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-36Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-36Example of use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-38Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-38Recommendations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-38

Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-39Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-39Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-40Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-40Example of use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42Joule-Thomson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42Recommendations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-43References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-43

Meshing – Additional Insight

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

Initial Mesh Generation Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

Refinements at Interfaces Between Substances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9

Local Mesh Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12

Irregular Cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13

The "Optimize thin walls resolution" option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13

Postamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14

Validation Examples

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

Flow through a Cone Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3

Laminar Flows Between Two Parallel Plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7

Laminar and Turbulent Flows in Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-17

Flows Over Smooth and Rough Flat Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-23

Flow in a 90-degree Bend Square Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27

Flows in 2D Channels with Bilateral and Unilateral Sudden Expansions . . . . . . . . 4-31

Flow over a Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-35

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Supersonic Flow in a 2D Convergent-Divergent Channel . . . . . . . . . . . . . . . . . . . . 4-39

Supersonic Flow over a Segmental Conic Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-43

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Flow over a Heated Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-49

Convection and Radiation in an Annular Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-55

Heat Transfer from a Pin-fin Heat Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-61

Unsteady Heat Conduction in a Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-65

Tube with Hot Laminar Flow and Outer Heat Transfer . . . . . . . . . . . . . . . . . . . . . 4-69

Flow over a Heated Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-73

Natural Convection in a Square Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-77

Particles Trajectories in Uniform Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-83

Porous Screen in a Non-uniform Stream. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-87

Lid-driven Flows in Triangular and Trapezoidal Cavities . . . . . . . . . . . . . . . . . . . 4-93

Flow in a Cylindrical Vessel with a Rotating Cover . . . . . . . . . . . . . . . . . . . . . . . . 4-99

Flow in an Impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-103

Cavitation on a hydrofoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-109

Thermoelectric Cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-113

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4-117

Technical Reference

Physical Capabilities of COSMOSFloWorks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2The Navier-Stokes Equations for Laminar and Turbulent Fluid Flows. . . . . . . . 5-2Laminar/turbulent Boundary Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-6Constitutive Laws and Thermophysical Properties . . . . . . . . . . . . . . . . . . . . . . . .5-6Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-7Compressible Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-10Non-Newtonian Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-10Equilibrium volume condensation of water from steam . . . . . . . . . . . . . . . . . . .5-11Conjugate Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12Thermoelectric Coolers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13Radiation Heat Transfer Between Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-14General Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-14Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-15View Factor Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-17Environment and Solar Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-18

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Radiative Surface Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-18Viewing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-19


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Global Rotating Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19Local rotating regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20Mass Transfer in Fluid Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21Flows in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21Two-phase (fluid + particles) Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-23Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-25Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-26Internal Flow Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-26External Flow Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-27Wall Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28Internal Fluid Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29

Numerical Solution Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29Computational Mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29Spatial Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-31Spatial Approximations at the Solid/fluid Interface . . . . . . . . . . . . . . . . . . . . . . 5-31Temporal Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-32Form of the Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-33Methods to Resolve Linear Algebraic Systems . . . . . . . . . . . . . . . . . . . . . . . . . 5-34Iterative Methods for Nonsymmetrical Problems. . . . . . . . . . . . . . . . . . . . . . . . 5-34Iterative Methods for Symmetric Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34Multigrid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-35

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1

Solving Engineering Tasks

Introduction

Engineering problems are problems connected with designing various objects or systems.

There are three general approaches to solving engineering problems:

• an experimental approach: a hardware rig or prototype, i.e., the full-scale object and/or its model, is manufactured and the experiments needed for designing the object are conducted with this hardware;

• a computational approach: the computations needed for designing the object are performed and their results are directly used for designing the object, without conducting any experiments;

• a computational-experimental approach combines computations and experiments (with the manufactured full-scale object and/or its model) needed for designing the object, their sequence and contents depending on the solved problem, e.g. iterative procedures may be run.

Each of the first two approaches has advantages and disadvantages.

The purely experimental approach, being properly conducted, does not require additional validations of the obtained results, but it is very expensive, even if it is realized on the object models, since testing facilities and hardware are required anyway. Moreover, if the object models are tested, the obtained results must be scaled to the full-scale object, so some computations are still required.

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Chapter Introduction

The purely computational approach, being properly performed, is substantially less expensive than the experimental one, both in finances and in time, but it requires assurance in adequacy of the obtained computational results. Naturally, such assurance must be based on numerous verifications and validations of the used computational codes, both from mathematical and physical viewpoints, i.e., both on the mathematical accuracy of the obtained results (the results’ adequacy to the used mathematical model) and on the adequacy of the used mathematical model to the governing physical processes, that is validated by comparing the computations with the available experimental data.

The third approach, if it reasonably combines experiments and computations, joins the advantages of both of the first two above-mentioned approaches and avoids their disadvantages. Complex engineering problems are solved mainly in this way. A computational code validated on available experimental data allows of quickly selecting the optimal object design and/or its optimal operating mode. Then necessary experiments are conducted to verify the selection.

When selecting from the world market a computational code that is most suitable for solving your problems, it is necessary to take into account the following suggestions. Any computational code is based, firstly, on a mathematical model of the governing physical processes, expressed, as a rule, in the form of a set of differential and/or integral equations derived from physical laws, and include, if necessary, semi-empirical and empirical constants and/or relationships. Secondly, a method of solving these equations is required. Since, as a rule, the equations of the mathematical model cannot be solved analytically, they are solved in a discrete form on a computational mesh, so the solution of the mathematical problem is obtained with a certain degree of accuracy. Naturally, the accuracy of solution of the mathematical problem depends both on the method of discretising the differential and/or integral equations and on the method of solving the obtained discrete equations. Once these methods have been selected, the accuracy of solution of the mathematical problem depends on how well the computational mesh resolves the problem regions of non-linear behavior. To provide good accuracy, the mesh must be rather fine in these regions. Moreover, a usual way of estimating the accuracy of solution of the mathematical problem consists of obtaining solutions on several different meshes, from coarser to finer. So, if beginning from some mesh in this set, the difference in the interesting physical parameters between the solutions obtained on the finer and coarser meshes becomes negligible from the viewpoint of the engineering problem, i.e., the solution flattens, then the accuracy of solution of the mathematical problem required for solving this engineering problem is considered to be attained, since the so-called solution mesh convergence is attained. Naturally, the solution of the mathematical problem can differ from the experimental values (i.e., from the solution of the physical problem, if it is known), and this difference depends, firstly, from the conformity of the mathematical model and the simulated physical processes, and, secondly, on the errors with which these experimental values have been measured and which, as a rule, are known and tend to decrease upon increasing the number of tests in which they are measured.

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Correspondingly, the computational codes presented on the market differ from each other not only in their cost, but also in accuracy of mathematical simulation of the physical problems, as well as in the procedure of specifying the initial data, in the amount of user’s time needed for this specification, in the procedure of solving a problem and the computer memory and CPU time needed for obtaining a solution of the required accuracy, and at last in the procedures of processing and visualization of the obtained results and the user’s time needed for that.

Naturally, a highly accurate solution requires a fine computational mesh, and consequently rather substantial computer memory and CPU time, as well as, in some cases, increased user time and efforts for specifying the initial data for the calculation. As a result, if the time needed to solve an engineering problem with a computational code exceeds some threshold time, then either the engineering problem becomes irrelevant, e.g. because your competitors have out-distanced you by this time, or alternative approaches, which may be not so accurate, but are surely faster, are used instead in order to solve this problem at given time span.

Before getting acquainted with the recommended procedure of obtaining a reliable and rather accurate solution of an engineering problem with COSMOSFloWorks, it is expedient to consider COSMOSFloWorks’ features governing the below-described strategy of solving engineering problems with COSMOSFloWorks.

Since COSMOSFloWorks is based on solving time-dependent Navier-Stokes equations, steady-state problems are solved through a steady-state approach. To more quickly obtain the steady-state solution, a method of local (over the computational domain) time steps is employed. A multigrid method is used for accelerating the solution convergence and suppressing parasitic oscillations. The computational domain is designed as a parallelepiped enveloping the model with planes orthogonal to the axes of the SolidWorks model’s Cartesian Global coordinate system. The computational mesh is built by dividing the computational domain into parallelepiped cells whose sides are orthogonal to the Global coordinate system axes. (The cells lying outside the fluid-filled regions and outside solids with heat conduction inside do not participate in the subsequent calculations). Procedures of the computational mesh refinement (splitting) are used to better resolve the model features, such as high-curvature surfaces in contact with fluid, thin walls surrounded by fluid, narrow flow passages (gaps), and the specified insulators’ boundaries. During the subsequent calculations during the solving of the problem the computational mesh can be refined additionally (if that is allowed by the user-defined settings) to better resolve the high-gradient flow and solid regions revealed in these calculations (Solution-Adaptive Meshing).

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Chapter Simulating Engineering Tasks with COSMOSFloWorks

Since steady-state problems are solved in COSMOSFloWorks through the steady-state approach, it is necessary to properly select the termination moment for the calculation. If the calculation is finished too early, i.e., when the steady state solution has not been attained yet, then the obtained solution can depend on the specified initial conditions and so be not very reliable. On the contrary, if the calculation is finished too late, then some time has been wasted uselessly. To optimize the termination moment for the calculation and to determine more accurately physical parameters of interest which oscillate in iterations (e.g. a force acting on a model surface, or a model hydraulic resistance), you may specify physical parameters of interest as the calculation goals.

The way to simulate an engineering problem with SolidWorks+COSMOSFloWorks correctly and adequately from the physical viewpoint, i.e. to state the corresponding model problem, and to solve this model problem properly and reliably with COSMOSFloWorks, is described in the chapters Simulating Engineering Tasks with COSMOSFloWorks and Solving Engineering Tasks.

1 Simulating Engineering Tasks with COSMOSFloWorks

It is necessary to remember that a fast but inaccurate beginning will cost you much efforts and time spent uselessly not only for specifying the initial data, but, even worse, for the subsequent calculations, until they will finally become reliable. Therefore, we strongly recommend that you carefully read this section.

1.1 Selecting Geometrical and Physical Features of the Task

Before you start to create a SolidWorks model and a COSMOSFloWorks project, it is necessary to select the engineering problem’s geometrical and physical features that most substantially influence this problem’s solution - first of all, those of them which are important for estimating the possibility of solving this problem with COSMOSFloWorks. For example,

• if the problem contains movable parts, then it is necessary to estimate the importance of taking their motions into account when solving the problem, and, if these motions are important, then to estimate a possibility of solving this problem with a quasi-stationary approach, since model parts’ motions during a calculation are not considered in COSMOSFloWorks (however, you may specify a translational and/or rotational motion of the specific wall),

• if the problem includes several fluids, or fluid and solid, then it is necessary to estimate the importance of chemical reactions between them for the problem’s solution, and, if the reactions are important, i.e., the reactions rates are rather high and the reacting fluids are intensely mixed with each other under the problem’s conditions, then to estimate a possibility of introducing the reaction products as an additional fluid when solving this problem, since chemical reactions are not

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considered in COSMOSFloWorks,

• if the problem includes fluids of different types (for example, a gas and a liquid), and there is an interface between them or these fluids are mixing, then it is


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necessary to estimate the importance of taking it into account, since COSMOSFloWorks can not consider a free fluid surface, or mixing of fluids of different types.

We can present other examples of an clear impossibility of solving some engineering problems with COSMOSFloWorks, as well as of simplifying the engineering problems for solving them with COSMOSFloWorks, but it is impossible to envision and describe all the possible situations in the present document, so that on each particular case you will have to make decision by yourself.

1.2 Creating the Model and the COSMOSFloWorks Project

If the SolidWorks model has already been created when designing the object, i.e. it is fully adequate to the object, then, to solve the engineering problem with COSMOSFloWorks, it may be required:

• to simplify the model by removing the parts which do not influence the problem’s solution, but consume computer resources, i.e. memory and CPU time. For example, a corrugated model surface which will result in an exceedingly large number of mesh cells required to resolve it can be specified instead as smooth surface with equivalent wall roughness. If a model has thin solid protrusions or narrow fluid-filled blind holes whose influence on the overall flow pattern is, by rough estimate, barely perceptible, it would be better to remove these features in order to avoid the excessive mesh splitting around them.

• to add auxiliary parts to the model, e.g. inlet and outlet tubes for stabilization of the flow, lids to cover the inlet and outlet openings, and parts to denote rotating regions, local initial meshes or other areas where special conditions are applied.

Both these actions, being executed properly, can be very pivotal in obtaining a reliable and accurate solution. Naturally, adding the auxiliary parts to a model will inevitably cause an increase of the computational mesh cells and, consequently, the required computer memory and CPU time, therefore these parts’ dimensions must be adequate to the stated problem.

If a model has not been created yet, it is expedient to take all the above-mentioned factors into account when creating it.

If all effects of these actions are not clear enough, it may be worthwhile to vary the model parts and/or their dimensions in a series of calculations in order to determine their effects on the obtained solution.

Then, in accordance with the problem’s physical features revealed and adapted to COSMOSFloWorks capabilities, the basic part of the COSMOSFloWorks project is specified, i.e., the problem type (internal or external), fluids and solids involved in the problem, physical features taken into account (e.g. heat conduction in solids, time-dependent analysis, gravitational effects, etc.), boundaries of the calculation domain,

COSMOSFloWorks 2008 Fundamentals 1-5

initial and boundary conditions, and, if necessary, fluid subdomains, rotating regions, volume and/or surface heat sources, fans and other features and conditions.


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Chapter Solving Engineering Tasks

The specified boundary conditions, as well as heat sources, fans, and other conditions and features must correspond to the problem’s physical statement and must not conflict with each other.

Eventually, you specify the physical parameters of interest as the COSMOSFloWorks project goals. They can be local or integral, defined within the whole computational domain or on some model surfaces, or in some volumes (local parameters are determined over some region in the form of their minimum, or maximum, average, or bulk average values). This will allow you to substantially increase reliability and accuracy of determination of these physical parameters, since their behavior is saved on each iteration during the calculation and can be analyzed later. On the contrary, the convergence behavior of all other parameters can not be analyzed afterwards, as they are saved only at the last iteration and, probably, at some user-specified iterations, whose number is restricted by the disk space limit.

2 Solving Engineering Tasks

As soon as you have specified the basic part of the COSMOSFloWorks project that is unlikely to be changed in the subsequent calculations, the next step is to select the strategy of solving the engineering problem with COSMOSFloWorks, i.e., obtaining the reliable and accurate solution of the problem.

2.1 Strategy of Solving the Engineering Tasks

As it has been mentioned in Introduction, by performing a series of calculations on a set of computational meshes ranging from coarser to finer ones, we can estimate the accuracy of solution of the mathematical problem. As soon as the calculation on a finer mesh does not yield a noticeably different (from the engineering problem’s viewpoint) solution, i.e. the solution flattens with respect to the mesh cells’ number, we can conclude that the solution of the mathematical problem has achieved mesh convergence, i.e., the required mathematical solution accuracy is attained. Naturally, first you must determine the threshold for a solution-vs.-mesh change, so that the change smaller than this threshold will be considered as negligible. Since the determination of this threshold is possible only in relation with some physical parameter, it is natural to connect it with the physical parameters of interest of the engineering problem in question, in particular, with the admissible determination errors of these physical parameters. Moreover, since steady-state problems are solved with COSMOSFloWorks through the steady-state approach, the supervision for a behavior of the calculation goals during the calculation (i.e., in iterations) can serve two purposes. Firstly, if these parameters oscillate during the solution, it will allow you to determine their values and observation errors more accurately by averaging them over a number of iterations and determining their deviation from this average value. Secondly, you may want to intervene in the calculation process by finishing the calculation manually if you see that either the calculation is unacceptable for you by some reasons, or, vice versa, if the solution has actually already converged, so that there is

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no reason to calculate any further.


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Therefore, the strategy of solving an engineering problem with COSMOSFloWorks consists, first of all, in performing several calculations on the same basic project (i.e., with the same model, inside the same computational domain, and with similar boundary and initial conditions) varying only the computational mesh. Since the computational mesh is built automatically in COSMOSFloWorks, it may be varied by varying the project parameters that govern its design (the initial computational mesh on which the calculation starts, and maybe its refinement during the calculation): Result Resolution Level, Minimum Gap Size, Minimum Wall Thickness.

An additional item of this strategy of solving an engineering problem with COSMOSFloWorks consists in varying the auxiliary elements added to the model as needed to solve the problem with COSMOSFloWorks (e.g. inlet and outlet tubes attached to the inlet and outlet openings, for internal problems), whose dimensions are questionable from the viewpoint of their necessity and sufficiency. Those physical parameters of the engineering problem whose values are not known exactly and which, in your opinion, can influence the problem solution, must be varied also. When performing these calculations, there is no need to investigate the solution-vs.-mesh convergence again, since it has been already achieved before. It is enough to just perform these calculations with the project mesh settings that provided the solution with satisfactory accuracy during the solution-vs.-mesh convergence investigation. The same applies also to the parametric engineering calculations while you are changing the model parts and/or flow parameters. However, you must keep in mind the potential necessity for checking the solution-vs.-mesh convergence, because in doubtful cases it must be checked again.

In spite of the apparent simplicity of the proposed strategy, its full realization is usually troublesome due to the substantial difficulties including, first of all, a dramatic increase of the requirements for computer memory and CPU time when you are substantially increasing the number of cells in the computational mesh. Since both the computer memory and the time for which the engineering problem must be solved are usually restricted, the specific realization of this strategy eventually governs the accuracy of the problem solution, whether it will be satisfactory or not. Perhaps, a further simplification of the model and/or reducing the computational domain will be required.

Some specific description of this strategy are presented in the next sections of this document.

2.2 Settings for Resolving the Geometrical Features of the Model and for Obtaining the Required Solution Accuracy

The computational mesh variation described in Section 2.1 is the key item of the proposed strategy of solving engineering problems with COSMOSFloWorks.

The result resolution level specified in the Wizard governs the number of basic mesh cells, the criteria for refinement (splitting) of the basic mesh to resolve the model geometry, i.e., creating the initial mesh, as well as the criteria for refinement (splitting) of the initial mesh


during the problem solution. The Result Resolution specified in the Wizard defines the following parameters in the created project: the Level of initial mesh and the Results resolution level. The Level of initial mesh governs only the initial mesh and is accessible


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(after the Wizard has been finished) from the Initial Mesh dialog box. The Results resolution level is accessible from the Calculation Control Options dialog box and controls the refinement of computational mesh during calculation and the calculation finishing conditions. The Geometry Resolution options that also influence the initial mesh may be changed on the Automatic Settings tab of the the Initial Mesh dialog box. Their effects can be altered on the other tabs of the Initial Mesh dialog box or in the Local Initial Mesh dialog box.

Before creating the initial mesh, COSMOSFloWorks automatically determines the minimum gap size and the minimum wall thickness for the walls whose are in contact with a fluid on both sides. That is required for resolving the geometrical features of the model with COSMOSFloWorks computational mesh. So, when creating the initial mesh, it is taken into account that the number of the mesh cells along the normal to the model surface must not be less than a certain number if the distance along this normal from this surface to the opposite wall is not less than the minimum gap size. Depending on the mesh cell arrangement, the model flow passages not resolved with the computational mesh either are automatically replaced with a wall, or increased up to the mesh cell size. In the automatic mode these mesh parameters are determined from dimensions of the surfaces on which boundary conditions have been specified, e.g. the model inlet and outlet openings in an internal analysis, as well as those surfaces and volumes on or in which heat sources, local initial conditions, surface and/or volume goals and some of the other conditions and features have been specified. Before the calculation, you can see the minimum gap size and the minimum wall thickness determined in such a way. If these values cannot provide an adequate resolution of the model geometry, you can specify them manually. At that, it is necessary to take into account that the number of the computational mesh cells generated to resolve the model geometrical features depends on the specified result resolution level.

Evidently, when creating a COSMOSFloWorks project it is necessary to make sure that both the minimum gap size and the minimum wall thickness are relevant to the model geometry. However, if the model geometry is complicated (e.g. there are non-circular flow passages, sharp edges protruding into the stream, etc.), it can be difficult to determine these parameters unambiguously. In this case it may be useful to perform several calculations by varying these parameters within a reasonable range in order to reveal their influence on the problem solution. In accordance with the strategy of solving engineering problems, these calculations must be performed at different result resolution levels.

The initial mesh created at result resolution levels of 3…5 is not changed during the solving of a problem, i.e. is not adapted to the solution being obtained. Result resolution levels of 5…7 yield the same initial mesh, but at result resolution levels of 6 and 7 the mesh is refined during the calculations in the regions of increased physical parameters gradients. At level 8, a finer initial mesh is generated and refinements during calculation takes place.

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It makes sense to perform calculations at the result resolution level of 3 if both the model geometry and the flow field are relatively smooth. For more complex problems we recommend first of all to perform the calculation at the result resolution level of 4 or,


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better, 5 (naturally, with specifying explicitly the minimum gap size and minimum wall thickness). After that, if the calculation at the result resolution level of 5 has been performed, we recommend, in order to ascertain the mesh convergence, to perform the calculation at the result resolution level of 7 and, if the computer resources allow you to do this, at the result resolution level of 8.

2.3 Monitoring the Calculation

Monitoring the calculation, i.e., at least, monitoring behavior of the physical parameters specified by you as the project goals (you can inspect also physical parameters fields at the specified planar cross-sections) is useful for the following reasons:

• you can intervene in the process of calculation, i.e., manually finish the calculation before it finishes automatically, if you see that either the calculation is unacceptable for you for some reasons (e.g. if COSMOSFloWorks has generated warnings making clear that the sequential calculation is senseless), or, vice versa, when solving a steady-state problem (that concerns some time-dependent problems also), the solution has already converged, so that there is no reason to continue the calculation;

• if a steady-state problem is solved, and the physical parameters specified by you as the project goals oscillate during the iterations, then inspecting these parameters’ behavior during the calculation will allow you to determine their values and determination errors more accurately by averaging their values over the iterations and determining their deviations from these average values;

• if the physical parameters of interest do not change substantially during the calculation, you can obtain their intermediate (preliminary) values beforehand, and in the subsequent iterations they will be refined finally;

• if you solve a time-dependent problem, you can immediately see the calculation results before the calculation is finished.

The first above-mentioned reason is especially useful since it allows you to substantially reduce the CPU time in some cases. For example, if you do not specify the high Mach number gas flow in the project settings, whereas in fact the flow becomes supersonic, or if COSMOSFloWorks warns you about a vortex at the model outlet, that substantially reduces the calculation accuracy, making it necessary to change some of the problem settings (i.e. specify high Mach number flow for the first case or lengthen the model outlet tube for the second one). If you solve a steady-state problem at the result resolution level of 7 or 8 and you see that the computational mesh refinements performed during the calculation do not increase the number of cells in the mesh and, therefore, do not noticeably improve the problem solution (the values of the project goals does not change), you can finish the calculation relatively early (say, after 1…2 travels have been performed).


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2.4 Viewing and Analyzing the Obtained Solution

When viewing and analyzing the obtained solution after finishing the calculation, it is recommended first of all, in accordance with the above-mentioned suggestions, to plot the evolutions of the project goals during the calculation, if you did not monitor them directly as the calculation went on. If a steady-state problem is solved, and you have specified the physical parameter of interest as the project goal, then, if this parameter has oscillated during the calculation, you can determine its value more accurately by averaging it over the last iterations interval in which its steady-state oscillation is seen. By that you also determine the variance of this goal, i.e., its deviation from the average value, that characterizes the goal determination error in the obtained solution.

It is also useful to check for vortices at the model outlet, as well as to see the flow pattern in the model and, if heat transfer in solids has been calculated, the temperature distribution over the solid parts of the model. Naturally, first of all it is expedient to see the obtained field of the physical parameter you are interesting in, not only in the region of interest, but also in a broader area, in order to check this field for apparently incosistent results.

It is also worthwhile to examine the obtained fields of other physical parameters related to the one you are interested in. For example, if you are interested in the total pressure loss, you may want to see the velocity field, whereas if you are interested in the temperature of solid, a picture of the fluid-to-solid heat flux field is also useful.

2.5 Estimating the Reliability and Adequacy of the Obtained Solution

In accordance with the general approach to estimating reliability and accuracy of the engineering problem solution obtained with a computational code, this estimation consists of the following two parts: an estimation of how accurate is the solution of the mathematical problem corresponding to the mathematical model of the physical process, and an estimation of accuracy of simulating the physical process with the given mathematical model.

The accuracy of solution of the mathematical problem is determined by mathematical methods, independently of the consistency of the model to the physical process under consideration. In our case, this accuracy estimation is based on analyzing the mesh convergence of the problem solutions obtained on different computational meshes (See Section 2.2). Then, since steady-state problems are solved with COSMOSFloWorks via a steady-state approach by employing local time steps, it is useful to verify additionally the accuracy of the obtained solution by solving the similar time-dependent problem not employing local time steps.

As soon as the mathematical problem solution of a satisfactory accuracy has been obtained, the next step consists of estimating the accuracy of simulating the physical process under consideration with the mathematical model employed in the computational code. To do this, the obtained solution is compared with the available experimental data

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(taking into account their errors which consist of measurement errors and experimental errors arising from possible spurious influences). Naturally, since experimental data are always restricted, for this validation it is desirable to select the data which are as close to


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the engineering problem being solved as possible. To validate the computational code on the available experimental data, you have to solve the corresponding test problem in addition to the engineering problem being solved (preferably before you start to solve the engineering problem following the above-mentioned strategy), but this operation increases the reliability of estimating the obtained solution of the engineering problem so substantially that the required additional time and efforts will be fully paid back later on, in particular when solving similar engineering problems.

If after solving the test problem you see that accuracy of its solution obtained with COSMOSFloWorks is not satisfactory from your viewpoint, check to see that you have properly specified the COSMOSFloWorks project, that all substantial features of the engineering problem have been taken into account, and, finally, that COSMOSFloWorks restrictions do not impede solving this engineering problem.

3 Frequent Errors and Improper Actions

Let us consider errors and improper actions frequently done when solving engineering problems with COSMOSFloWorks.

When Specifying Initial Data:

• not taking into account physical features which are important for the engineering problem under consideration: e.g. high Mach number gas flow (should be taken into account if M>3 for steady-state or M>1 for transient tasks or supersonic flow occurs in about a half of the computational domain or greater), gravitational effects (must be taken into account if either the fluid velocity is small, the fluid density is temperature-dependent, and a heat source is considered, or several fluids having substantially different densities are considered in a gravitational field), necessity of the time-dependent analysis (e.g. at the moderate Reynolds numbers, when unsteady vortices are generated);

• incorrectly specifying symmetry planes as the computational domain boundaries (e.g. at the moderate Reynolds numbers, when unsteady vortices are generated; you should keep in mind that the symmetry of model geometry and initial and boundary conditions does not guarantee you the symmetry of flow field);

• if symmetry planes have been specified and you click Reset at the Size tab of the Computational Domain dialog box, please do not forget to replace Symmetry by Default at the Boundary Condition tab;

• if you have specified symmetry planes and intend to specify mass or volume flow rates at a model inlet or outlet openings, please do not forget to specify only their parts falling into the computational domain instead of the total flow rates at these openings;

• if you specify integral boundary or volume conditions (heat transfer rates, heat


generation rates, etc.), please remember that their values specified in the COSMOSFloWorks dialog boxes correspond to the area or volume's part falling into the computational domain;


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Chapter Frequent Errors and Improper Actions

• if you specify a flow swirl on a model inlet or outlet openings (in the Fans or Boundary Conditions dialog boxes), please do not forget to specify properly their swirl axes and the proper coordinate system for that in the Definition tab;

• if you specify a Unidirectional or Orthotropic porous medium, please do not forget to specify their directions;

• please do not forget that the specified boundary conditions must not conflict with each other. For example, if you deal with gas flows and the model inlet flow is subsonic, whereas the flow inside the model becomes supersonic, it is incorrect to specify flow velocity or volume flow rate as a boundary condition at the model inlet, since they are fully determined by the geometry of the model flow passage and the fluid’s specific heat ratio;

• if you solve a time-dependent problem, and this problem has cyclic-in-time boundary conditions, thus leading to a steady-state cyclic-in-time solution, to obtain which you have to calculate the flow several times in cycle, every time specifying the solution from the previous calculation as the initial condition for the next calculation, there is no need to specify the boundary conditions for several cycles. Instead it is more convenient to specify them for a cycle and perform a series of calculations, running each calculation with selected Take previous results check box in the Run dialog box;

• when specifying Surface Goals, Volume Goals, Equation Goals, it is better to give them sensible names to identify these goals unambiguously, instead of selecting them in the tree and looking for the respective places at the model in the SolidWorks graphics area;

• if you want to monitor the intermediate calculation results at certain sections of the model during the calculation, it is better to determine these sections’ positions in the Global coordinate system beforehand, i.e. before actually running the calculation, since during the calculation it is a bit more difficult and you may be literally ’late’ in terms of the problem’s physical time;

When Monitoring a Calculation:

• when monitoring intermediate calculation results during a calculation, please do not forget the spatial nature of the problem being solved (of course, if the problem itself is not 2D). To take a look at the full pattern it is expedient to see the results at least in 2 or 3 intersecting planes;

When Viewing the Obtained Solution after Finishing a Calculation:

• please take into account that all settings made in the View Settings dialog box concern all Cut Plots, 3D Plots, Surface Plots, Flow Trajectories, Isosurfaces, which are active in the SolidWorks graphics area, therefore:

• your will not be able to open the Flow Trajectories dialog box if a parameter defined only on wall surfaces has been selected on the Contours tab and the

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Use from contours option has been selected at the Flow Trajectories tab of the View Settings dialog box;


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• to view different result features in different panes simultaneously, it is necessary to split the SolidWorks graphics area into 2 or 4 panes and build different result features in different graphical areas through their individual Cut Plots, 3D Plots, Surface Plots, Flow Trajectories, Isosurfaces defined in these areas;

• if you intend to see integral physical parameters (e.g. area, mass or volume flow rates, heat generation rates, forces, etc.) with the Surface Parameters dialog box, please remember that

• their shown values are determined over the parts of the surface that belong to the computational domain;

• their determination errors include errors of representing these surfaces in SolidWorks and COSMOSFloWorks, the latter depends on the computational mesh;

• if you want to see a computational mesh in Cut Plots and/or Surface Plots, please do not forget to select Display mesh under Tools, Options, Third Party Options, otherwise the Mesh button in the Cut Plots and Surface Plots PropertyManagers will be absent, so you can not view a computational mesh.


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Chapter Frequent Errors and Improper Actions


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2

Advanced Knowledge

Introduction

The present document supplies you with our experience of employing the advanced COSMOSFloWorks capabilities, organized in the following topics:

Manual adjustment of the initial computational mesh settings

Mesh-associated tools (viewing the mesh before and after the calculation and advanced post-processing tools)

Calculation control options (refinement of the computational mesh during calculation, conditions of finishing the calculation)

Flow freezing

1 Mesh Introduction

This chapter provides the fundamentals of working with the COSMOSFloWorks computational mesh, describes the mesh generation procedure, and explains the use of parameters governing both automatically and manually controlled meshes.

First, let us introduce a set of definitions.

1.1 Types of Cells

Any COSMOSFloWorks calculation is performed in a rectangular parallelepiped-shaped computational domain whose boundaries are orthogonal to the axes of the Cartesian Global Coordinate System. A computational mesh splits the computational domain with a set of planes orthogonal to the Cartesian Global Coordinate System's axes to form rectangular parallelepipeds called cells. The resulting computational mesh consists of


cells of the following four types:

• Fluid cells are the cells located entirely in the fluid.


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Chapter Mesh Introduction

• Solid cells are the cells located entirely in the solid.

• Partial cells are the cells which are partly in the solid and partly in the fluid. For each partial cells the following information is kept: coordinates of intersections of the cell edges with the solid surface and normal to the solid surface within the cell.

• Irregular cells are partial cells for which the normal to the solid surface cannot be determined. Cells of this type are never generated with the modern version of COSMOSFloWorks, however, such cells may be found in the meshes built with the previous versions of COSMOSFloWorks)

As an illustration let us look at the original model (Fig.1.1) and the generated computational mesh (Fig.1.2).

Fig.1.1 The original model.

Fig.1.2 The computational mesh cells of different types

Zero level cell (basic cell)

Solid cell

Partial cell

First level cellFluid cell

Partial cell


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1.2 Mesh Construction Stages

Refinement is a process of splitting a rectangular computational mesh cell into eight cells by three orthogonal planes that divide the cell's edges in halves. The non-split initial cells that compose the basic mesh are called basic cells or zero level cells. Cells obtained by the first splitting of the basic cells are called first level cells, the next splitting produces second level cells, and so on. The maximum level of splitting is seven. A seventh level cell is 87 times smaller in volume than the basic cell.

The following rule is applied to the processes of refinement and merging: the levels of two neighboring cells can only be the same or differ by one, so that, say, a fifth level cell can have only neighboring cells of fourth, fifth, or sixth level.

The mesh is constructed in the following steps:

Construction of the basic mesh taking into account the Control Planes and the respective values of cells number and cell size ratios.Resolving of the interface between substances, including refinement of the basic mesh at the solid/fluid and solid/solid boundaries to resolve the relatively small solid features and solid/solid interface, tolerance and curvature refinement of the mesh at a solid/fluid, solid/porous and a fluid/porous boundaries to resolve the interface curvature (e.g. small-radius surfaces of revolution, etc).

Narrow channels refinement, that is the refinement of the mesh in narrow channels taking into account the respective user-specified settings.

Refinement of all fluid, and/or solid, and/or partial mesh cells up to the user-specified

Optimize thin walls resolution. In the early versions of COSMOSFloWorks refinement of the mesh within model's walls was needed to resolve thin walls properly, but it could also lead to increase in number of cells in adjacent fluid regions, especially in narrow channels between walls. If this additional mesh refinement is critical for obtaining the proper results and you want to perform calculation on the same mesh as in the earlier version of COSMOSFloWorks, clear the Optimize thin walls resolution check box. In this case the mesh will be almost the same as in that earlier version, with the main difference of absence of irregular cells.

During the solution-adaptive meshing the cells can be refined and merged. See ”Refinement of the Computational Mesh During Calculation’ on page 23.

If you switch on or off heat conduction in solids, or add/move insulators, you should rebuild the mesh.


level.

Mesh conservation, i.e. a set of control procedures, including check for the difference in area of cell facets common for the adjacent cells of different levels.


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After each of these stages is passed, the number of cells is increased to some extent.

In COSMOSFloWorks you can control the following parameters and options which govern the computational mesh:

1 Nx, the number of basic mesh cells (zero level cells) along the X axis of the Global Coordinate System. 1 ≤ Nx ≤ 1000

2 Ny, the number of basic mesh cells (zero level cells) along the Y axis of the Global Coordinate System. 1 ≤ Ny ≤ 1000.

3 Nz, the number of basic mesh cells (zero level cells) along the Z axis of the Global Coordinate System. 1 ≤ Nz ≤ 1000.

4 Control planes. By adding and relocating them you can contract and/or stretch the basic mesh in the specified directions and regions. Six control planes coincident with the computational domain's boundaries are always present in any project.

5 Small solid features refinement level (Lb). 0 ≤ Lb ≤ 7.

6 Curvature refinement level (Lcur). 0 ≤ Lcur ≤ 7.

7 Curvature refinement criterion (Ccur). 0 ≤ Ccur ≤ π.

8 Tolerance refinement level (Ltol). 0 ≤ Ltol ≤ 7.

9 Tolerance refinement criterion (Ctol). 0 ≤ C tol.

10 Narrow channels refinement: Characteristic number of cells across a narrow channel, Narrow channels refinement level, The minimum and maximum height of narrow channels to be refined.

These options are described in more detail below in this chapter.

1.3 Basic Mesh

The basic mesh is a mesh of zero level cells. In case of 2D calculation (i.e. if you select the 2D plane flow option in the Computational Domain dialog box) only one basic mesh cell is generated automatically along the eliminated direction. By default COSMOSFloWorks constructs each cell as close to cubic shape as possible.

The number of basic mesh cells could be one or two less than the user-defined number (Nx, Ny, Nz). There is no limitation on a cell oblongness or aspect ratio, but you should carefully check the calculation results in all cases for the absence of too oblong or stretched cells.


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1.4 Control Planes

The Control Planes option is a powerful tool for creating an optimal computational mesh, and the user should certainly become acquainted with this tool if he is interested in optimal meshes resulting in higher accuracy and decreasing the CPU time and required computer memory. Control planes allow you to resolve small features, contract the basic mesh locally to resolve a particular region by a denser mesh and stretch the basic mesh to avoid excessively dense meshes.

1.5 Resolving Small Features by Using the Control Planes

If the level of splitting is not high enough, small solid features may be not resolved properly. In this case, two methods can be used to improve the mesh:

• increase the level of splitting. However, this may result in unnecessary increase of the number of cells in other regions, creating a non-optimal mesh, or

• set a control plane crossing the relevant small feature (e.g. a solid's sharp edge). This will allow you to resolve this feature better without creating an excessively dense mesh elsewhere. It is especially convenient in cases of sharp edges oriented along the Global Coordinate System axes.

It is recommended that you place a control plane slightly submerged into the solid, and avoid placing it coincident with the solid surface.

Fig.1.3 Basic mesh examples.

a) 10x12x1 b) 40x36x1


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1.6 Contracting the Basic Mesh

Using control planes you may contract the basic mesh in the regions of interest. To do this, you need to set control planes surrounding the region and assign the proper Ratio values to the respective intervals. The cell sizes on the interval are changed gradually so that the proportion between the first and the last cells of the interval is close (but not necessarily equal) to the entered Ratio value. Negative values of the ratio correspond to the reverse order of cell size increase. Alternatively, you may explicitly set the Number of cells for each interval, in which case the Ratio value becomes mandatory. For example, assume that there are two control planes Plane1 and Plane2 (see Fig.1.4) and the ratio on the interval between them is set to 2. Then the basic mesh cells adjacent to the Plane1 will be approximately two times longer than the basic mesh cells adjacent to the Plane2.

Fig.1.4 Specifying custom control planes.


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Use of control planes is especially recommended for external analyses, where the computational domain may be substantially larger than the model.

In the Fig.1.6 two custom control planes are set through the center of the body with the ratio set to 5 and -5, respectively, on the intervals to the both sides of each plane.

1.7 Resolving Small Solid Features

The procedure of resolving small solid features refines only the cells where the solid/fluid (solid/solid, solid/porous as well as fluid/porous) interface curvature is too high: the maximum angle between the normals to a solid surface inside the cell exceeds 120°, i.e. the solid surface has a protrusion within the cell.

Such cells are split until the the Small solid features refinement level of splitting mesh cells is achieved.

1.8 Curvature Refinement

The curvature refinement level is the maximum level to which the cells will be split during refinement of the computational mesh until the curvature of the solid/fluid or fluid/porous interface within the cell becomes lower than the specified curvature criterion (Ccur).

The curvature refinement procedure has the following stages:

1 Each solid surface is triangulated: COSMOSFloWorks gets triangles that make up the SolidWorks surfaces.

2 A local (for each cell) interface curvature is determined as the maximum angle between the normals to the triangles within the cell.

Fig.1.5 Default control planes. Fig.1.6 Two custom control planes.

The performance settings do not govern the triangulation performance.


3 If this angle exceeds the specified Ccur, and the curvature refinement level is not reached then the cell is split.


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The curvature refinement is a powerful tool, so that the competent usage of it allows you to obtain proper and optimal computational mesh. Look at the following illustrations to the curvature refinement by the example of a sphere.

Fig.1.7 Curvature refinement level is 0;Total number of cells is 64.

Fig.1.8 Curvature refinement level is 1;Total number of cells is 120.

Fig.1.9 Curvature refinement level is 2; Curvature criterion is 0.317;Total number of cells is 120.

Fig.1.10 Curvature refinement level is 2; Curvature criterion is 0.1;Total number of cells is 148.


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1.9 Tolerance Refinement

Tolerance refinement allows you to control how well (with what tolerance) mesh polygons approximate the real interface. The tolerance refinement may affect the same cells that were affected by the small solid features refinement and the curvature refinement. It resolves the interface's curvature more effectively than the small solid features refinement, and, in contrast to the curvature refinement, discerns small and large features of equal curvature, thus avoiding refinements in regions of less importance (see images below).

Any surface is approximated by a set of polygons whose vertices are surface's intersection points with the cells' edges. This approach accurately represents flat faces though curvature surfaces are approximated with some deviations (e.g. a circle can be approximated by a polygon). The tolerance refinement criterion controls this deviation. A cell will be split if the distance (h, see below) between the outermost interface's point within the cell and the polygon approximating this interface is larger than the specified criterion value.

1.10 Narrow Channel Refinement

The narrow channel refinement is applied to each flow passage within the computational domain (or a region, in case that local mesh settings are specified) unless you specify for COSMOSFloWorks to ignore passages of a specified height. The Narrow Channels term is conventional and used for the definition of the flow passages of the model in the direction normal to the solid/fluid interface.

The basic concept of narrow channel refinement is to resolve the narrow channels with a sufficient number of cells to provide a reasonable level of solution accuracy. It is especially important to have narrow channels resolved in analyses of low Reynolds numbers or analyses with long channels, i.e. in such analyses where the boundary layer thickness becomes comparable to the size of the partial cells where the layer is developed.

The narrow channel settings available in COSMOSFloWorks are the following:

• Narrow channels refinement level – the maximum level of cells refinement in narrow channels with respect to the basic mesh cell.

• Characteristic number of cell across a narrow channel – the number of cells (including partial cells) that COSMOSFloWorks will attempt to set across the model flow passages in the direction normal to the solid/fluid interface. If possible, the number of cells across narrow channels will be equal to the specified characteristic number, otherwise it will be as close to it as possible. The Characteristic number of cells across a narrow channel (let us denote it as Nc) and the Narrow channels refinement level (let us denote it as L) both influence the mesh in narrow channels in the following manner: the basic mesh in narrow channels will be split to have Nc number per channel, if the resulting cells satisfy the specified L. In other words, whatever the specified Nc, the smallest possible cell in a narrow channel is 8L times


smaller in volume (or 2L times smaller in each linear dimension) than the basic mesh cell. This is necessary to avoid undesirable mesh splitting in very fine channels that may cause the number of cells to increase to an unreasonable value.


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• The minimum height of narrow channels, The maximum height of narrow channels – the minimum and maximum bounds for the height outside of which a flow passage will not be considered as a narrow channel and thus will not be refined by the narrow channel resolution procedure.

For example, if you specify the minimum and maximum height of narrow channels, the cells will be split only in those fluid regions where the distance between the opposite walls of the flow passage in the direction normal to wall lies between the specified minimum and maximum heights.

The narrow channel refinement operates as follows: the normal to the solid surface for each partial cell is extended up to the next solid surface, which will be considered to be the opposite wall of the flow passage. If the number of cells per this normal-to-wall direction is less than the specified Nc, the cells will be split to satisfy the narrow channel settings as described above.

Although the settings that produce an optimal mesh depends on a particular task, here are some ’rule-of-thumb’ recommendations for narrow channel settings:

1 Set the number of cells across narrow channel to a minimum of 5.

2 Use the minimum and maximum heights of narrow channels to concentrate on the regions of interest.

3 If possible, avoid setting high values for the narrow channels refinement level, since it may cause a significant increase in the number of cells where it is not necessary.

Fig.1.11 Curvature refinement level is 3; Small solid features refinement level is 3; Narrow channel

2-10

refinement is disabled; Total number of cells is 6476.


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Fig.1.12 Curvature refinement level is 3; Small solid features refinement level is 3; Narrow channel refinement is on: 5 cells across narrow channels, Narrow channels refinement level is 2; Total number of cells is 8457.

Fig.1.13 Curvature refinement level is 3; Small solid features refinement level is 3; Narrow channel refinement is on: 5 cells across narrow channels, Narrow channels refinement level is 7; Total number of


cells is 33293.


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1.11 Local Mesh Settings

The local mesh settings option is one more tool to help create an optimal mesh. Use of local mesh settings is especially beneficial if you are interested in resolving a particular region within a complex model.

The local mesh settings can be applied to a component, face, edge or vertex. You can apply local mesh settings to fluid regions and solid bodies. To apply the local mesh settings to a fluid region you need to specify this region as a solid part or subassembly and then disable this component in the Component Control dialog box. The local mesh settings are applied to the cells intersected with the selected component, face, edge, or a cell enclosing the selected vertex. However, cells adjacent to the cell of the local region may be also affected due to the refinement rules described in the Mesh Construction Stages chapter.

Fig.1.14 The local mesh settings used: The rhombic channel is refined into 4th level cells, and two narrow channels are refined to have 10 cells across each channel.


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1.12 Recommendations for Creating the Computational Mesh

1 At the beginning create the mesh using the default (automatic) mesh settings. Start with the Level of initial mesh of 3. On this stage it is important to recognize the appropriate values of the minimum gap size and minimum wall thickness which will provide you with the suitable mesh. The default values of the minimum gap size and minimum wall thickness are calculated using information about the overall model dimensions, the Computational Domain size, and area of surfaces where conditions (boundary conditions, sources, etc.) and goals are specified. Don't switch off the Optimize thin walls resolution option, since it allows you to resolve the model's thin walls without the excessive mesh refinement.

2 Closely analyze the obtained automatic mesh, paying attention to the total numbers of cells, resolution of the regions of interest and narrow channels. If the automatic mesh does not satisfy you and changing of the minimum gap size and minimum wall thickness values do not give the desired effect you can proceed with the custom mesh.

3 Start to create your custom mesh with the disabled narrow channel refinement, while the Small solid features refinement level and the Curvature refinement level are both set to 0. This will produce only zero level cells (basic mesh only). Use control planes to optimize the basic mesh.

4 Next, adjust the basic mesh by step-by-step increase of the Small solid features refinement level and the Curvature refinement level. Then, enable the narrow channels refinement.

5 Finally, try to use the local mesh settings.

2 Mesh-associated Tools

2.1 Introduction

Since the mesh settings tool is an indirect way of constructing the computational mesh, to better visualize the resulting mesh various post-processing tools are offered by COSMOSFloWorks. In particular, these tools allow to visualize the mesh in detail before the calculation, substantially reducing the CPU and user time.

The computational mesh constructed by COSMOSFloWorks or other CFD codes cannot resolve the model geometry at the mesh cell level exactly. A discrepancy can lead to prediction errors. To facilitate an analysis of these errors and/or to avoid their appearance, COSMOSFloWorks offers various options for visualizing the real computational geometry corresponding to the computational mesh used in the analysis.


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Chapter Mesh-associated Tools

Since the numerical solution is obtained inevitably in the discrete form, i.e., in the centers of computational mesh cells, it is interpolated and extrapolated by the post-processor to present the results in a smooth form, which is typically more convenient to the user. As a result, some prediction errors can stem from these interpolations and extrapolations. To facilitate an analysis of such errors and/or to prevent their appearance, COSMOSFloWorks offers an option to visualize the physical parameters’ values calculated at the centers of computational mesh cells, so that when presenting results by coloring an area with a palette, the results are considered constant within each cell.

2.2 Visualizing the Basic Mesh Before Constructing the Initial Mesh

Using this option the user can inspect the Basic mesh and its Control planes corresponding to the mesh settings, which can be made manually or retained by default. The plot appears as soon as these settings have been made or changed, so you immediately see the resulting mesh. (See Help or User’s Guide defining the Basic mesh and its Control planes).

To enable this option, select the Show basic mesh option in the FloWorks, Project menu, or in the Initial Mesh dialog box. The option is accessible both before and after the calculation.

Using this option, you may shifting the Control planes to desired positions to assure that certain features of the model geometry are captured by the computational mesh.

2.3 Enhanced Capabilities of the Results Loading

COSMOSFloWorks allows to view not only the calculation results and the current computational mesh on which they have been obtained, but also the initial (i.e., on which the calculation begins) computational mesh separately. The latter can be viewed either before or after the calculation, allowing the user to compare the initial and current (i.e., refined during the calculation) computational meshes.

Fig.2.1 The Basic mesh (left) and the Initial mesh (right).


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To view various meshes, you must open the corresponding file via the Load results dialog box. The calculation results, including the current computational mesh, are saved in the .fld files, whereas the initial computational mesh is saved separately in the .cpt file. All these files are saved in the project folder, whose name (a numeric string) is formed by COSMOSFloWorks and must not be changed. The .cpt files and the final (i.e., with the solution obtained at the last iteration) .fld files have the name similar to that of the project folder, whereas the solutions obtained during the calculation at the previous iterations (corresponding to certain physical time moments, if the problem is time-dependent) are saved in the .fld files with names “r_<iteration number>”, e.g. the project initial data are saved in the r_000000.fld file.

2.4 Viewing the Initial Computational Mesh Saved in the .cpt Files

To optimize the process of solving an engineering problem and to save time, in some cases it may be useful to view the initial computational mesh before performing the calculation, particularly to be sure that the model features are resolved well by this mesh. To view the initial computational mesh after loading the .cpt file, COSMOSFloWorks offers you Cut Plots, Surface Plots, and the Mesh option (see below), which are also used for viewing the calculation results.

2.5 Viewing the Computational Mesh Cells with the Mesh Option

To view fluid cells of the computational mesh cells (i.e. the cells lying fully in the fluid), solid cells (lying fully in the solid), and partial cells lying partly in the fluid and partly in the solid, COSMOSFloWorks offers you the Mesh option.

Different colors can be used to better differentiate between the computational mesh cells of each of the above-mentioned types. To see the cells in a certain parallelepiped region,

Do not try to load the calculation results obtained in another project with a different geometry; the effect is unpredictable.

Fig.2.2 The Load Results dialog box.


the user must specify the coordinates of the region boundaries in the Global Coordinate System.


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Using the Mesh option, you can also save the information concerning the mesh cells, including the physical parameters values obtained in their centers, in ASCII or Excel files.

2.6 Visualizing the Real Computational Geometry

Since the SolidWorks model geometry, especially its high-curvature parts, cannot be resolved exactly at the cell level by the rectangular (parallelepiped) computational mesh, the real computational geometry corresponding to the computational mesh used in the analysis can be viewed after the calculation to avoid or estimate the prediction errors stemming from this discrepancy. If no solution-adaptive meshing occurs during the calculation, the real computational geometry can be viewed just after the mesh generation. This option is employed by clearing the Use CAD geometry check box in Cut Plots, 3D Plots, Surface Plots, Flow Trajectories, Point Parameters and XY Plots. The result is especially clear when colored Contours are used to visualize a physical parameter values (see Fig.2.3).

This capability is especially useful for revealing the model surface regions which are inadequately resolved by the computational mesh. Let us consider Fig.2.4 with the temperature Cut Plots as an example. The white detail in the bottom part is an insulator, so the heat transfer within it is not considered. It contains a small closed cavity which is omitted in the analysis and therefore is invisible on the cut plot, too. However, the mesh resolution of the triple border between the insulator, the cavity, and the heat-conducting solid body leads to the formation of cog-shaped artifacts.

Visualization of a large amount of computational mesh cells (e.g. all fluid cells in the whole computational domain) may be impractical, since it could require substantial time and memory, and even then you might not be able to see all the visualized cells because the majority of them will likely be screened from view by other cells.

Fig.2.3 Surface Plots on a SolidWorks model inner surface (left) and on its computational realization (right).


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On the other hand, this option may be useful when creating Surface Plots for SolidWorks models containing rippled surfaces whose ripples have not been resolved by the computational mesh and are not essential from the problem solution viewpoint, since the coloring of the simplified solid/fluid interface instead of coloring the actual SolidWorks model faces allows you to substantially reduce the CPU time and memory requirements.

If the computational mesh has resolved the SolidWorks model well, so the obtained computational results are adequate, then enable the Use CAD geometry option before performing the final Cut Plots and Surface Plots to obtain smooth pictures which are more convenient for the analysis.When creating a Surface Plot with the Use CAD geometry option switched off, only the solid/fluid interfaces of partial cells within the computational mesh are colored. When a Surface Plot is created in the Use all faces mode, solid/fluid interfaces of all partial cells are colored. However, when a Surface Plot is created on a selected surface, the solid/fluid interfaces are colored only in the partial cells intersected by the SolidWorks model surface approximated by triangles inside SolidWorks, which may differ from the mesh-approximated surface of the model. As a result, there may exist some partial cells which are not intersected by the triangulated surface, and therefore their solid/fluid interfaces would not be colored (see gray strips in Fig.2.5). Naturally, this circumstance concerns the picture only and does not affect the calculation results.

Fig.2.4 Cut plots using CAD geometry (left) and meshed geometry (right).


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2.7 Switching off the Interpolation and Extrapolation of Calculation Results

Since the numerical solution is obtained inevitably in the discrete form, i.e., in the form of values in the centers of the computational mesh cells in COSMOSFloWorks, it is interpolated and extrapolated by the post-processor to present the results in a smooth form, which is typically more convenient to the user. As a result, prediction errors can stem from and/or be hidden by such interpolation and extrapolation that smoothens the calculation results. To facilitate the revealing, analysis, and elimination of such errors, COSMOSFloWorks offers an option to visualize the physical parameter values ’as is’, i.e. without interpolation, when presenting calculation results in Cut Plots and Surface Plots (other result features, namely, isolines, isosurfaces, flow streamlines and particle trajectories can not be built at all without interpolation), so when coloring a surface with a palette, the results are considered constant within the mesh cells (see Fig.2.6).

Since the mesh cells’ centers used in coloring the surface can lie at different distances from the surface, this can introduce an additional variegation into the picture, if the value of the displayed parameter depends noticeably on this distance (see Fig.2.6).

Fig.2.5


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Fig.2.6 The fluid velocity Surface Plots created with (left) and without (right) interpolating and extrapolating the calculation results.


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2.8 Conclusion

The presented mesh-associated tools of COSMOSFloWorks are additional tools for obtaining reliable and accurate results with this code. These tools are summarized in the table:

Option

Application

ReasonBasic mesh

Initial mesh

After the calculation

Visualizing the Basic mesh

+ + + To inspect the Basic mesh and setting its Control planes

Widened capabilities of loading the results

+ + To view the Initial mesh and the calculation results

Viewing the Initial mesh

+ + To analyze the Initial mesh

Viewing mesh cells of different type

+ + To view mesh cells and save the respective physical parameters values

Visualizing the computational geometry

+ + For analysis of inadequate results and quick post-processing of the results of complicated models

Switching off the interpolation of results

+ For analysis of inadequate results


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3 Calculation Control Options

3.1 Introduction

The Calculation Control Options dialog box introduced into COSMOSFloWorks allows you to control:

• conditions of finishing the calculation,

• saving of the results during the calculation,

• refinement of the computational mesh during the calculation,

• freezing the flow calculation,

• time step for a time-dependent analysis,

• number of rays traced from the surface if radiating heat transfer is enabled.

This dialog box is accessible both before the calculation and during the calculation. In the last case the new-made settings are applied to the current calculation starting from the next iteration.

The main information on employing the options of Finishing the calculation and Refining the computational mesh during calculation is presented in this document.

3.2 Finishing the Calculation

COSMOSFloWorks solves the time-dependent set of equations for all problems, including steady-state cases. For such cases it is necessary to recognize the moment when a steady-state solution is attained and therefore the calculation should be finished. A set of independent finishing conditions offered by COSMOSFloWorks allow the user to select the most appropriate conditions and criteria on when to stop the calculation. The following finishing conditions are offered by COSMOSFloWorks:

• maximum number of refinements;

• maximum number of iterations;

• maximum physical time (for time-dependent problems only);

• maximum CPU time;

• maximum number of travels;

• convergence of the Goals.

Travel is the number of iterations required for the propagation of a disturbance over the whole computational domain. Current number of iterations per one travel is presented in the Info box of the Calculation monitor.


In COSMOSFloWorks you can select the finishing conditions that are most appropriate from your viewpoint to solve the problem under consideration, and specify their values. For the latter two conditions (i.e., for the maximum number of travels and the Goals


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Chapter Calculation Control Options

convergence settings) COSMOSFloWorks presents their default values (details are described below). You can also select the superposition mode for multiple finishing conditions in the Finish Conditions value cell: either to finish the calculation when all the selected finishing conditions are satisfied or when at least one of them is satisfied.

In any case, information on the finishing conditions due to which the calculation has finished is shown in the Monitor’s Log box.

The Goals convergence finishing condition is complex since it consists of satisfying all the specified Goals criteria. A specified Goal criterion includes a specified dispersion, which is the difference between the maximum and minimum values of the Goal, and a specified analysis interval over which this difference (i.e., the dispersion) is determined. The interval is taken from the last iteration rearwards and is the same for all specified Goals. The analysis interval is applied after an automatically specified initial calculation period (in travels), and, if refinement of the computational mesh during calculation is enabled, after an automatically or manually specified relaxation period (in travels or in iterations) since the last mesh refinement is reached. As soon as the Goal dispersion obtained in the calculation becomes lower than the specified dispersion, the Goal is considered converged. As soon as all Goals included in the Goals convergence finishing condition (by selecting them in the On/Off column) have converged, this condition is considered satisfied. The Goals not included into the Goals convergence finishing condition are used for information only, i.e., with no influence on the calculation finishing conditions.

Let us consider the COSMOSFloWorks default values for the maximum number of travels and the Goals convergence settings in detail. These default (recommended by COSMOSFloWorks) values depend on the Result resolution level either specified in the Wizard or changed by pressing the Reset button in the Calculation Control Options dialog box. For higher Result resolution levels the finishing conditions are tighter.

The default maximum number of travels depends on

• the type of the specified Goal (i.e., dynamic or diffusive, see below);

• the specified Result resolution level;

• the problem's type (i.e., incompressible liquid or compressible gas, low or high Mach number gas flow, time-dependent or steady-state).

The Dynamic goals are: Static Pressure, Dynamic Pressure, Total Pressure, Mass Flow Rate, Forces, Volume Flow Rate, and Velocity. The Diffusive goals are: Temperature, Density, Mass in Volume, Heat flux, Heat transfer rate, Concentrations, Mass Flow Rate of species, and Volume Flow Rate of species.


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The default Goals convergence settings are the default analysis interval, which is shown in the Finish tab of the Calculation Control Options dialog box, and the default Goals criterion dispersion values, which are not shown in the Calculation Control Options dialog box, but, instead, are shown in the Monitor’s Goal Table or Goal Plot table (in the Criteria column), since they depend on the values of the Goal physical parameter calculated in the computational domain, and therefore are not known before the calculation and, moreover, can change during it. In contrast, the Goals criterion dispersion values specified manually do not change during the calculation.

As for the automatically specified initial calculation period (measured in travels), it depends on the problem type, the Goal type, and the specified Result resolution level.

3.3 Refinement of the Computational Mesh During Calculation

Refinement of the computational mesh during calculation is a process of splitting or merging of the computational mesh cells in high-gradient flow areas. This option has the following governing parameters:

• refinement level,

• splitting/merging criteria (also named refinement/unrefinement criteria, respectively),

• permission to refine cells in fluid and/or solid regions,

• approximate maximum number of cells,

• strategy of refinements during the calculation.

The first four parameters are described in COSMOSFloWorks Help and User’s Guide. Here, let us consider the Refinement Strategy in detail. The following three strategies are

• the manually specified analysis interval for the Goals convergence finishing criteria must be substantially longer than the typical period of the flow field oscillation (if it occurs);

• the Goals determined on solid/fluid interfaces or model openings, as well as the Post-processor Surface Parameters, yield the most accurate and correct numerical information on flow or solid parameters, especially integral ones;

• Global Goals yield the most reliable information on flow or solid parameters, although they may be too general;

• the CPU time depends slightly on the number of the specified Goals, but, in some cases, vary substantially in the case of presence of a Surface Goal;

• Surface and Volume Goals provide exactly the same information that may be obtained via the Surface and Volume Parameters Post-processor features, respectively.


available:

• Periodic refinement;


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Chapter Calculation Control Options

• Tabular Refinement;

• Manual Only refinement.

In the first two strategies the refinement moment is known beforehand. The solution gradients are analyzed over iterations belonging to the Relaxation interval, which is calculated from the current moment rearwards. As a result, only steady-state gradients are taken into account. The default length of the Relaxation interval can be adjusted manually. On the other hand, the analysis must not continue with the same relaxation interval defined from the start of the calculation, in order to avoid taking into account the initial highly unsteady period. Therefore, a period of at least two relaxation intervals is recommended before the first refinement. If the first assigned refinement is scheduled in a shorter term from the beginning, the period over which the gradients are analyzed is shortened accordingly, so that in the extreme case it can be as short as one current iteration. If you initiate a refinement manually within this period, the gradients are analyzed in one current iteration only. Naturally, such a short period give not very reliable gradients and hence may result in an inadequate solution or excessive CPU time and memory requirements.

The figure below illustrates these concepts. Here, the letter r denotes the Relaxation interval. This figure involves both Periodic and Tabular refinements. Case 1 is the recommended normal approach. In the Case 2 the first refinement is too close to the starting point of the calculation, so the gradients are analyzed over the shorter interval (which could even be reduced to only one current iteration in the extreme case). Case 3 is a particular case when a refinement is initiated manually just before a previously assigned refinement. As a result, the manual refinement is well-defined, since the gradients have


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been analyzed over almost the entire relaxation interval, but on the other hand, the previously assigned refinement is performed on the substantially shorter interval, and therefore its action can be incorrect. Case 3 demonstrates the possible error of performing manual and previously assigned refinements concurrently.

4 Flow Freezing

4.1 What is Flow Freezing?

Sometimes it is necessary to solve a problem that deals with different processes developing at substantially different rates. If the difference in rates is substantial (10 times or higher) then the CPU time required to solve the problem is governed almost exclusively by the slower process. To reduce the CPU time, a reasonable approach is to stop the calculation of the fastest process (which is fully developed by that time and does not change further) and use its results to continue the calculation of the slower processes. Such an approach is called “freezing”.

In the case of problems solved with COSMOSFloWorks the processes of convective mass, momentum, and energy transport are the fastest processes to develop and to converge, whereas the processes of mass, momentum, and energy transfer by diffusion are the slowest ones. Accordingly, COSMOSFloWorks offers the “Flow Freezing” option that

Fig.3.1 Refinement strategy.

Collecting of the statistics is prohibited

Statistics are collectedRefinement

Case 1

Auto ref.

r r

rr

0

0

Ref. point 1 Ref. point 2

Case 2 Case 3

Ref. point 1

r r

Manual ref.


allow you to freeze, or fix, the pressure and velocity field while continuing the calculation of temperature and composition. This option is especially useful in solving steady-state


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Chapter Flow Freezing

problems involving diffusion processes that are important from the user’s viewpoint, e.g. species or heat propagation in dead zones of the flow. Time-dependent analyses with nearly steady-state velocity fields and diffusion processes developing with time are also examples of this class of problems. As a result, the CPU time for solving such problems can be substantially reduced by applying the Flow Freezing option.

COSMOSFloWorks treats Flow Freezing for the High Mach number flows differently. All flow parameters are frozen, but the temperature of the solid is calculated using these fixed parameters at the outer of the boundary layer and user defined time step. Temperature change on the solid's surface and relevant variation of the heat flows are accounted in the boundary layer only. It is impossible (and makes no sense) to use Flow Freezing for calculation of concentration propagation in the High Mach number flow. If custom time step is not specified, the steady-state temperature of solid will be reached in one time step assumed to be infinite.

4.2 How It Works

To access the Flow Freezing option, open the Calculation Control Options dialog box, then the Advanced tab. This option has three modes: Disabled (by default), Periodic, and Permanent.

Flow Freezing in a Permanent Mode

As an example of applying the Flow Freezing option, let us consider a plane flow (2D) problem of heating the vortex core in a vessel (Fig.4.1).

At the beginning the entire fluid region is filled with a cold (T=300 K) liquid. A hot (T=400K) liquid enters the vessel through the lower channel (the upper channel is the exit). As a result, a vortex with a cold core is developed in the vessel. The vortex core temperature is changed mainly due to heat diffusion. To measure it, a small body is placed at the vortex center and disabled in the Component Control dialog box, so that it is treated by COSMOSFloWorks as a fluid region. Its minimum temperature (i.e., the minimum

Fig.4.1 Heating the vortex core in a vessel.

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fluid temperature in this region) is the Volume Goal of the calculation.


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First of all, let us consider Flow Freezing operating in the Permanent mode. The only user-specified parameter in Permanent mode is the starting moment of enabling the Flow Freezing option. Until this moment the calculation runs in a usual manner. After this moment the fluid velocity field becomes frozen, i.e., it is no longer calculated, but is taken from the last iteration performed just before the Flow Freezing Start moment. For the remainder of the run only the equations’ terms concerning heat conduction and diffusion are calculated. As a result, the CPU time required per iteration is reduced.

The starting moment of the Flow Freezing option should be set not too early in order to let the flow field to fully develop. As a rule, an initial period of not less than 0.25 travels is required to satisfy this condition. In most problems the 0.5 travel initial period is sufficient, but there are problems that require a longer initial period.

When first solving the problem under consideration we set the maximum number of travels to 10. The calculation performed without applying the Flow Freezing option then required about 10 travels (the CPU time of 13 min. 20 s on a 500 MHz PIII computer) to reach the convergence of the project Goal (the steady-state minimum fluid temperature in the vortex core). However, the steady-state fluid velocity field was reached in about 0.5 travels, i.e., substantially earlier. So, by applying the Flow Freezing option in the Permanent mode (just after 0.5 travels) the same calculation only required a CPU time of 7 min. 25 s on the same computer to reach the convergence of the project Goal. Convergence histories of the both Goal are plotted in Fig.4.2.

If it is necessary to perform several calculations with the same fluid velocity field, but different temperatures and/or species concentrations, it is expedient to first calculate this fluid velocity field without applying the Flow Freezing option. Then, clone the COSMOSFloWorks project into several projects (including copying the calculation results), make the required changes to these projects, and perform the remaining calculations for these projects using the calculated results as initial conditions and applying the Flow Freezing option in the Permanent mode with a zero Start period.

The Flow Freezing Start moment, as well as other parameters of the Calculation Control Options dialog box can be changed during a calculation.

As soon as the Flow Freezing option is invoked, only the slowest processes are calculated. As a result, the convergence and finishing criteria can become non-optimal. Therefore, to avoid obtaining incorrect results when enabling the Flow Freezing option, it is recommended to increase the maximum number of travels specified at the Finish tab of the Calculation Control Options dialog box by 1.5…5 times compared to the number that was set automatically or required for the calculation performed without the Flow Freezing option.


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Chapter Flow Freezing

Flow Freezing in a Periodic Mode

In some problems the flow field depends on temperature (or species concentrations), so both the velocity and the temperature (concentrations) change simultaneously throughout the calculation. Nevertheless, since they change in a different manner, i.e., the velocity field changes faster than the temperature (concentrations) field, therefore approaching its steady state solution earlier, the Flow Freezing option can be used in a Periodic mode to reduce the CPU time required for solving such problems. The Periodic mode of the Flow Freezing option consists of calculating the velocity field not in each of the iterations (time steps), but periodically for a number of iterations specified in No freezing (iterations) after a period of freezing specified in the Freezing (iterations) (see Fig.4.3) The temperatures and concentrations are calculated in each iteration. Examples include channel flows with specified mass flow rates and pressures, so the fluid density and, therefore, velocity depend on the fluid temperature, or flows involving free convection, where due to the buoyancy the hot fluid rises, so the velocity field depends on the fluid temperature.

If you forget to use the calculated results as initial conditions, then the saved fluid velocity field will be lost in the cloned project, so the project must be created again. To use the calculated results as initial conditions for the current project, select the Transferred type of Parameter definition for the initial conditions in the General Settings dialog box.

Fig.4.2 The convergence history of the Goal (which is the minimum fluid temperature in the vortex core) with and without applying the Flow Freezing option.

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Fig.4.3


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As an example, let us consider a 3D external problem of an air jet outflow from a body face into still air (see Fig.4.4, in which the jet outflow face is marked by a red line). Here, the wire frame is the computational domain. The other body seen in this figure is introduced and disabled in the Component Control dialog box (so it is a fluid region) in order to see the air temperature averaged over its face (the project Goal), depending on the air temperature specified at the jet outflow face.

This problem is solved in several stages. At the first stage, the calculation is performed for the cold (T = 300 K, which is equal to the environment temperature) air jet. This calculation requires a CPU time of 12 min. 14 s. Then we clone the project including copying the results. Next, we set the outlet air temperature to T = 400 K, specify the Periodic mode of the Flow Freezing option by its Start moment of 0.25 travels (in order for the heat to have time to propagate along the jet to the measuring face) and under Duration specify 10 as both the Freezing (iterations) and No freezing (iterations) values. Then perform the calculation on the same computational mesh with the Take previous results option in the Run box. As you can notice, the calculation with flow freezing takes less CPU time than the similar calculation without the Flow Freezing option enabled.

5 Cavitation

5.1 Physical model

Cavitation is a common problem for many engineering devices dealing with liquid flows. The deleterious effects of cavitation include: lowered performance, load asymmetry, erosion and pitting of blade surfaces, vibration and noise, and reduction of the overall machine life. Cavitation models used today range from rather crude approximations to sophisticated bubble dynamics models. Details about bubble generation, growth and collapse are important for the prediction of a solid surfaces erosion, but are not necessary to estimate the performance of a pump, valve or other equipment. In COSMOSFloWorks an engineering model of cavitation is employed to predict the extent of cavitation and its

Fig.4.4 Air jet outflow from a body face into a still air.


influence on the performance of the analyzed device.


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Chapter Cavitation

Limitations and Assumptions

The model has the following limitations and/or assumptions:

• Cavitation is currently available only for incompressible water (when defining the project fluids you should select Water from the list of Pre-Defined liquids); cavitation in mixtures of different liquids cannot be calculated.

• The properties of the dissolved non-condensable gas are set to be equal to those of air.

• Thermodynamic parameters in the phase transition areas should be contained within the following bounds:

277.15 < T < 583.15 K, 800 < P < 107 Pa.

• The model does not describe the detailed structure of the cavitation area, i.e parameters of individual vapor bubbles are not considered.

• The parameters of the flow at the inlet boundary conditions must be such that the volume fraction of liquid water in the inlet flow would be at least 0.1.

5.2 Interface

Cavitation option in COSMOSFloWorks is switched on by checking the Cavitation check box either in Wizard or in the General Settings window. Since COSMOSFloWorks may consider cavitation only in incompressible water, the selection of any fluid type other than liquid or of any fluid other than water renders this check box unavailable.

Cavitation option for a fluid subdomain is switched on in a manner similar to that for the whole project.

Once enabled, the Cavitation option requires you to specify the Dissolved gas mass fraction. The default value of this parameter is 0.00001. This value is typical for air dissolved in water under normal conditions and therefore is appropriate for most cases. If needed, you can specify a different value of the Dissolved gas mass fraction in the range of 10-4...10-8.


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Cavitation is represented in the calculation results via the following parameters: Water Mass Fraction, Water Volume Fraction, Vapour Mass Fraction, and Vapour Volume Fraction, which describe the local fraction of the fluid components (water and vapour) by mass and by volume. Note that you may need to check some of those parameters in the Parameter list to enable their selection in the View Settings window.

5.3 Examples of use

Rotating impeller

Water flows through a rotating impeller with five blades of curved shape, as shown on the picture. The aim of simulation is to predict the impeller characteristics.

Due to the pressure drop on suction side of the impeller blades, a cavitation may develop in these areas, which cannot but affect the impeller performance.

The appearance of calculated cavitation area in the form of isosurfaces is shown below on Figure 2.1.

Fig. 2.1. Isosurfaces for vapour volume fraction of 10%.


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Chapter Cavitation

Hydrofoil in a tunnel

A symmetric hydrofoil is placed in a sufficiently wide water-filled tunnel with a non-zero angle of attack. Obviously, water flow develops some pressure drop on the leeward side of the hydrofoil, which at certain conditions can lead to cavitation.

Figure 2.2 contains a representation of the calculated cavitation area visualized in terms of vapour volume fraction.

Ball valve

Water flows inside an about half-opened ball valve (see Figure 2.3) at the relatively low pressure and high velocity producing cavitation.

The results visualized in the form of Cut plot with Vapour volume fraction as displayed parameter are presented on Figure 2.4. It is clearly seen that sudden expansion of the flow produce an area of strong cavitation.

Fig. 2.2. Calculated cavitation area.

Fig. 2.3. Model of the ball valve.

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Fig. 2.4. Distribution of the vapour volume fraction.


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5.4 Recommendations

• If your analyze a flow of water in some points of which the local static pressure may fall below the liquid's vapour pressure at the local temperature causing cavitation or if a localized boiling of water may occur in the water flow due to intense heating, it is recommended to select the Cavitation option in the Fluids dialog box of the Wizard or General Settings.

• Cavitation area growths slowly during calculation and there is a risk that the calculation will stop before the cavitation area develops completely. To avoid this, always specify Global Goal of Average Density and increase the Analysis interval on the Finish tab of the Calculation Control Options dialog box up to 2.5 travels. Also make sure that the other finish conditions do not cause the calculation to stop before goals are converged. The easiest way to ensure that is to select If all are satisfied in the Value cell for the Finish conditions on the Finish tab of the Calculation Control Options dialog box.

• The Cavitation option should not be selected if you calculate a water flow in the model without flow openings (inlet and outlet).

• The fluid region where cavitation occurs should be well resolved by the computational mesh.

• To see the cavitation areas, you may select, for example, Vapour Volume Fraction or Density (the latter one is probably the best choice) as the parameter for visualization.

6 Steam

6.1 Physical model

COSMOSFloWorks allows you to consider water steam among the project fluids. Like Humidity, the Steam option may be used to analyze engineering problems concerning water vapour and its volume condensation, along with the corresponding changes in the physical properties of the project fluid. Steam option in COSMOSFloWorks describes the behavior of pure water steam (for which, say, a description as "air with 100% humidity at 100°C", although nominally correct, would sound a bit weird) or its mixtures with other gases.



• COSMOSFloWorks project may include pure Steam or its mixture with Gases (but not with Real gases).

• Thermodynamic parameters of steam should be contained within the following


bounds:

283 < T < 610 K, P < 107 Pa.

• The volume fraction of condensed water should never exceed 5%.


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Chapter Steam

• Steam option is incompatible with the High Mach number flow option, i.e. the two can not be employed simultaneously.

• The employed model of condensation is fully equilibrium and considers only volume condensation.

6.2 Interface

Steam is treated by COSMOSFloWorks as a special kind of fluid and may be selected from the Engineering Database just like any other fluid.

Steam may be assigned for a fluid subdomain as well as for the whole project.

Steam may be mixed with any regular Gases (but not with Real gases). In this case, its concentration in a form of mass or volume fraction must be specified in Initial conditions, as well as in all boundary conditions.

Steam content in the mixtures of water steam with other gases is represented in the calculation results via Steam Mass Fraction, Steam Volume Fraction (that represent mass and volume fractions of water, respectively) and Relative Humidity (which is the ratio of the local partial density of water to the density of saturated water vapor under current conditions). The content of particular form of water, i.e. vapor or liquid, is represented via Condensate Mass Fraction (that represents mass fraction of condensed steam in the fluid) and Moisture Content (that represents the fraction of condensed steam with respect to the overall content of steam). Note that you may need to check some of those parameters in the Parameter list to enable their selection in the View Settings window.


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6.3 Example of use

Heat exchanger

COSMOSFloWorks calculates the equilibrium condensation in water steam as steam flows through a cooled tube of a heat exchanger. Figure 2.5 shows cut plot of the condensate mass fraction parameter.

6.4 Recommendations

• To avoid the risk of finishing the calculation before the condensation develops completely, always specify some goal strongly dependent on condensation, for example Global Goal of Average Density, and make sure that the calculation will not stop before this goal is converged.

• To see the condensation areas, you may use Relative Humidity or the Condensate Mass Fraction as the parameter for visualization.

7 Humidity

7.1 Physical model

COSMOSFloWorks allows you to consider the relative humidity of the gas or mixture of gases. This allows you to analyze engineering problems where the condensation of water vapor contained in the air (or other gas), or, more generally speaking, where any differences in physical properties of wet and dry air play an important role. Examples may include air conditioning systems (especially in wet climate or in the places where relative humidity is very important, e.g. libraries, art museums, etc.), tank steamers, steam turbines and other kinds of industrial equipment. COSMOSFloWorks can calculate equilibrium volume (but not surface) condensation of steam into water. As a result, the local fractions of gaseous and condensed steam are determined. In addition, the corresponding changes of the fluid temperature, density, enthalpy, specific heat, and sonic velocity are determined

Fig. 2.5. Cut plot showing the condensate mass fraction.


and taken into account.


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Chapter Humidity



• Humidity is currently available only in Gases (both in individual gases and in mixtures), but not in Real gases.

• Thermodynamic parameters in the fluid areas where humidity is considered should be contained within the following bounds:

283 < T < 610 K, P < 107 Pa.

• The volume fraction of condensed water should never exceed 5%.

• Humidity option is incompatible with the High Mach number flow option, i.e. the two can not be employed simultaneously.

• The model does not describe the condensation process in as subtle detail as the parameters of individual liquid droplets.

• Surface condensation, i.e. the formation of dew on solid surfaces, is not considered.

• The condensed steam has no history, since the employed condensation model is fully equilibrium. In other words, the state of condensed steam at given point is governed solely by the local conditions at this point.

7.2 Interface

Humidity option in COSMOSFloWorks is switched on by checking the Humidity check box either in Wizard or in the General Settings window. This check box is present only if the current fluid type is set to Gases.


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Once Humidity is switched on, the relative humidity of the gas becomes available to specify in the Initial conditions window. The relative humidity is defined as the ratio of the current water vapor density to that of saturated water vapor under current conditions.

Humidity can be assigned for a fluid subdomain as well as for the whole project by selecting the check box of the same name, and, once assigned, becomes available to specify in the Humidity Parameters group box.

The relative humidity must be specified within all boundary and initial conditions in contact with the fluid region for which the calculation of relative humidity is performed.

Together with the humidity value for boundary and initial conditions you must also specify the values of Humidity reference pressure and Humidity reference temperature that describe the conditions under which the relative humidity has been determined, since these values may differ from the current pressure and temperature.


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Chapter Humidity

Humidity is represented in the calculation results via the following parameters: Steam Mass Fraction, Steam Volume Fraction (that represent mass and volume fractions of water, respectively) and Relative Humidity (which is the ratio of the local partial density of water to the density of saturated water vapor under current conditions). The content of particular form of water, i.e. vapor or liquid, is represented via Condensate Mass Fraction (that represents mass fraction of water condensate in the fluid) and Moisture Content (that represents the fraction of condensed water with respect to the overall content of water). Note that you may need to check some of those parameters in the Parameter list to enable their selection in the View Settings window.

7.3 Example of use

Aircraft

An air flow around an aircraft model can be simulated with the Humidity option selected. The examination of relative humidity distribution (Figure 2.6) reveals broad areas of more than 80% relative humidity from above of both wings. Naturally, these areas (together with smaller zones near the cockpit and the tail unit) are enriched with water condensate, as it may be seen on Figure 2.7.

7.4 Recommendations

• If your analyze a flow of gas containing some amount of water vapor and the conditions are likely to get over the dew point, it is recommended to consider

Fig. 2.7. Isosurfaces of condensate mass fraction = 0.00015

Fig. 2.6. Flow trajectories colored in accordance with relative humidity.

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humidity in the calculation as described in this chapter.


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• To avoid the risk of finishing the calculation before the condensation develops completely, always specify some goal strongly dependent on condensation, for example Global Goal of Average Density, and make sure that the calculation will not stop before this goal is converged.

• To see the condensation areas, you may use Relative Humidity or the Condensate Mass Fraction as the parameter for visualization.

8 Real Gases

8.1 Physical model

COSMOSFloWorks has an ability to consider real gases. A wide choice of predefined real gases is presented. The user may also create user-defined real gases by specifying their parameters. This option may be useful in the engineering problems concerning gases at nearly-condensation temperatures and/or at nearly-critical and supercritical (that is to say, very high) pressures, i.e. at conditions where the behavior of the gas can no longer be represented adequately by the ideal-gas state equation.

The model of real gas implemented in COSMOSFloWorks employs a custom modification of the Redlich-Kwong state equation. Naturally, the equation unavoidably has certain bounds of applicability, which are explained on the picture below:


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Chapter Real Gases

The area of validity of the model includes zones 10, 11 and 12. (Each predefined real gas has its own values of Pmin, Pmax, Tmin, and Tmax, and those are also to be specified for a user-defined real gas.) If the calculated pressure and/or temperature fall outside of this area, COSMOSFloWorks issues a warning. The warning for zones 1 - 8 is: Real gas parameters (pressure and/or temperature) are outside the definitional domain of substance properties, with comment: P < Pmin, P > Pmax, T < Tmin, or T > Tmax,

depending on what has actually happen. The warning for zone 9 is: Phase transition in the Real gas may occur.



• Real gas may be used in a COSMOSFloWorks project as pure fluid or in mixture with Gases (but not with other Real gases).

• Pressure and temperature of real gas should be contained within certain limits (those are specified individually for each of the predefined real gases).

• Real gas should not be put under conditions that cause its condensation into liquid.

• The use of real gas is incompatible with the High Mach number flow option.

• The precision of calculation of thermodynamic properties at nearly-critical temperatures and supercritical pressures may be lowered to some extent in comparison to other parameter ranges. The calculations involving user-defined real gases at supercritical pressures are not recommended.

• The copying of pre-defined real gases to user-defined folder is impossible since the employed models are not exactly similar.

8.2 Interface

Real gases are a special type of fluids and may be selected from the Engineering Database along with other fluids.

Real gas may be assigned for a fluid subdomain as well as for the whole project.


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Real gases may be mixed with regular Gases (though not with each other). In this case, substance concentrations in a form of mass or volume fractions must be specified in Initial conditions, as well as in all boundary conditions.

To create a user-defined real gas, the user must create a new item in the corresponding folder in the Engineering Database and specify the following parameters:

• Molar mass;

• Critical pressure pc;

• Critical temperature Tc;

• Critical compressibility factor Zc;

• Redlich-Kwong equation type that should be used, i.e. the original one or its modifications by Wilson, Barnes-King, or Soave;

• Acentric factor ω (if applicable);

• Minimum temperature, i.e. the lower margin of validity of the model;

• Maximum temperature, i.e. the corresponding upper margin;

• Order of ideal gas heat capacity polynomial, i.e. the order of polynomial function of temperature that defines the "ideal-gas" constituent of the real gas specific heat at constant pressure;

• Coefficients of ideal gas heat capacity polynomial, i.e. the coefficients of the aforementioned polynomial;

• Polarity (check if the gas in question has polar molecules);

• Vapor viscosity dependence on temperature, i.e. the coefficients a and n in the equation describing vapor viscosity as η = a·Tn;

• Vapor thermal conductivity dependence on temperature, which includes the coefficients a and n and the choice of dependency type between linear λ = a+n·T and power-law λ = a·Tn forms;


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Chapter Real Gases

• Liquid viscosity dependence on temperature, which includes the coefficients a and n and the choice of dependency type between power-law η = a·Tn and exponential η = 10a(1/T-1/n) forms;

• Liquid thermal conductivity dependence on temperature, which includes the coefficients a and n and the choice of dependency type between linear λ = a+n·T and power-law λ = a·Tn forms;

The coefficients of the user-specified dependencies for thermophysical properties should be entered only in SI unit system, except those for the exponential form of dynamic viscosity of the liquid, which should be taken exclusively from Ref. 1.

Note that the foregoing dependencies for the specific heat and transport properties cover only the ’ideal-gas’ constituents of the corresponding properties, i.e. their values at low-pressure limit, and the actual formulae contain pressure-dependent corrections which are calculated automatically.

The post-processor display parameters concerning real gas includes its mass and volume fractions in a mixture (if it is not a sole component of the fluid) and the Real Gas State. The latter parameter represents the local phase state of real gas, which may be Vapor, Liquid, Supercritical, or Out of range. Once selected, it renders inaccessible the Palette and Min/Max settings within the View settings window and replaces the Color bar with the schematic phase diagram that provides an explanation of meaning of particular colors, as shown on the picture.

8.3 Example of use

Joule-Thomson effect

A flow of nitrogen through a tube containing narrow restriction is simulated. To reduce computation time, the tube was split in halves by a symmetry plane and Symmetry condition was applied to the corresponding boundary of the Computational Domain.


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The calculation within ideal gas approximation, i.e. with nitrogen selected from Gases as the project fluid, results in the temperature distribution shown on Figure 2.1. It is clearly seen that the temperature of the gas, after undergoing a noticeable drop while passing through the hole, later reinstates its initial value. This is an expected behavior of an ideal gas, as its enthalpy does not depend on pressure.

The calculation was repeated with fluid changed to nitrogen selected from Real Gases and all other conditions similar. Now the gas temperature at outlet is different from that at inlet (see Figure 2.9).

Hence we may conclude that the real gas reveals a nonzero Joule-Thomson effect, as expected.

8.4 Recommendations

• Minimum temperature for user-defined real gas should be set at least 5...10 K higher than the triple point of the actual substance.

• Maximum temperature for user-defined real gas should be set so as to keep away from the area of dissociation of the gas.

• The user-specified dependencies for the specific heat and transport properties of the user-defined real gases should be valid in the whole temperature range from Tmin to Tmax (or, as for liquid, in the whole temperature range where the liquid exists).

8.5 References

1 R.C. Reid, J.M. Prausnitz, B.E. Poling. The properties of gases and liquids, 4th edition,

Fig. 2.8. Field of temperatures for a flow of ideal gas.

Fig. 2.9. Field of temperatures for a flow of real gas.


McGraw-Hill Inc., NY, USA, 1987.


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Chapter Real Gases


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3

Meshing – Additional Insight

Introduction

COSMOSFloWorks considers the real model designed in SolidWorks and generates a rectangular computational mesh automatically distinguishing the fluid and solid domains. The corresponding computational domain is generated in the form of a rectangular parallelepiped enclosing the model. In the mesh generation process, the computational domain is divided into uniform rectangular parallelepiped-shaped cells, which form a so-called basic mesh. Then, using information about the model geometry, COSMOSFloWorks further constructs the mesh by means of various refinements, i.e. splitting of the basic mesh cells into smaller rectangular parallelepiped-shaped cells, thus better representing the model and fluid regions. The mesh from which the calculation starts, so-called initial mesh, is fully defined by the generated basic mesh and the refinement settings.

Each refinement has its criterion and level. The refinement criterion denotes which cells have to be split, and the refinement level denotes the smallest size to which the cells can be split. Regardless of the refinement considered, the smallest cell size is always defined with respect to the basic mesh cell size so the constructed basic mesh is of great importance for the resulting computational mesh.

The main types of refinements are:

Small Solid Features Refinement

Curvature Refinement

Tolerance Refinement

Narrow Channel Refinement

Square Difference Refinement


In addition, the following two types of refinements can be invoked locally:

Cell Type Refinement


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Chapter Initial Mesh Generation Stages

Solid Boundary Refinement

During the calculation, the initial mesh can be refined further using the

Solution-Adaptive Refinement.

Though it depends on a refinement which criterion or level is available for user control, we will consider all of them (except for the Solution-Adaptive Refinement) to give you a comprehensive understanding of how the COSMOSFloWorks meshing works.

In the chapter below the most important conclusions are marked with the blue italic font. For abbreviation list refer to the Glossary paragraph.

1 Initial Mesh Generation Stages

1.1 Basic Mesh Generation and Resolving the Interface

1.1.1) Create basic mesh cells whose sizes are governed by the computational domain size, the user-specified Control Planes and the number of the basic mesh cells. [Nx, Ny, Nz, Control Planes. Parameters which act on each stage are summarized in square brackets at the end of the stage.]

1.1.2) Analyze triangulation in each basic mesh cell at the interfaces between different substances (such as solid/fluid, solid/porous, solid/solid and porous/fluid interfaces) in order to find the maximum angle between normals to the triangles which compose the interface within the cell.

1.1.3) Depending on the maximum angle found, the decision whether to split the cell or not is made in accordance with the specified Small solid features refinement level (SSFRL), Narrow channel refinement level (NCRL), Curvature refinement level (CRL) and Curvature criterion (CRC), Tolerance refinement level (TRL) and Tolerance Refinement Criterion (TRC) (see the Refinements at Interfaces Between Substances paragraph). [SSFRL, NCRL, CRL and CRC]

1.1.4) If a basic mesh cell is split, the resulting child cells are analyzed as described in 1.1.2 and 1.1.3, and split further, if necessary. The cell splitting will proceed until the interface resolution satisfies the specified SSFR criterion, CRC and TRC, or the corresponding level of splitting reaches its specified value.

1.1.5) The operations 1.1.2 to 1.1.4 are applied for the next basic mesh cell and so on,

If a cell belongs to a local initial mesh area, then the corresponding local refinement levels will be applied (see the Local Mesh Settings paragraph).

The specified levels of splitting denote the maximum admissible splitting, i.e. they show to which level a basic mesh cell can be split if it is required for resolving the solid/fluid interface within the cell.

3-2

taking into account the following Cell Mating rule: two neighboring cells (i.e. cells having a common face) can be only cells whose levels are similar or differ by one. This rule has the highest priority as it is necessary for simplifying numerical algorithm in solver.


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The mesh at this stage is called the primary mesh. The primary mesh implies the complete basic mesh with the resolution of the solid/fluid (as well as solid/solid, solid/porous, etc.) interface by the small solid features refinements and the curvature refinement also taking into account the local mesh settings.

1.2 Narrow Channel Refinement

After the primary mesh has been created, the narrow channel refinement is put in action. The Narrow Channels term is conventional and used for the definition of the model flow passages which are ’narrow’ in the direction normal to the solid/fluid interface.

Regardless of the real solid curvature, the mesh approximation is that the solid boundary is always represented by a set of flat elements, whose nodes are the points where the model intersects with the cell edges. Thus, whatever the model geometry, there is always a flat element within a partial cell and the normal to this element denotes the direction normal to the solid/fluid interface for this partial cell. See the Irregular Cells paragraph for details.

The narrow channel refinement operates as follows:

1.2.1) For each partial cell COSMOSFloWorks calculates the “local” narrow channel width as the distance between this partial cell and the next partial cell found on the line normal to the solid/fluid interface of this cell (i.e. normal to the flat surface element located in the cell).

The fourth-level red cells appearing after resolving the cog cause the neighboring cells to be split up to third level (yellow cells), that, in turn, causes the subsequent refinement producing second level cells (green cells) and first level cells (blue cells). The white zero level cell (basic mesh cell) remains unsplit since it borders on first level cells only, thus satisfying the rule.

The Cell Mating rule is strict and has higher priority than the other cell operations. The rule is also enforced for the cells that are entirely in a solid.

If the line normal to the solid/fluid interface crosses a local initial

Fig.1.1Fluid cell refinement due to the Cell Mating rule.


mesh area, then the corresponding local narrow channel refinement settings is applied to the cells in this direction.


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1.2.2) If the distance value falls within the range defined by the Minimum height of narrow channel (NCHmin) and Maximum height of narrow channel (NCHmax) options, the number of cells per this interval is calculated including both partial cells and taking into account which portion of each partial cell is in fluid. [NCHmin, NCHmax]

1.2.3) More precisely, the number of cells across the channel (i.e. on the interval between the two partial cells) is calculated as N = Nf + np1 + np2, where Nf is the number of fluid cells on the interval, and np1 and np2 are the fluid portions of the both partial cells. This value is compared with the specified Characteristic number of cells across a narrow channel (CNC). If N is less than the specified CNC then the cells on this interval are split. For example, on Fig.1.2 Nf = 2, np1 = np2 = 0.4, and N = 2+0.4+0.4 = 2.8 which is less than the criterion. On Fig.1.3 the partial cells are split, so that the fluid portions of the newly-formed partial cells are np1 = np2 = 9/10, and the criterion is satisfied (N > CNC).

The narrow channel refinement is symmetrical with respect to the midpoint of the interval and proceeds from the both ending partial cells towards the midpoint. [CNC, NCRL].

On Fig.1.4 the specified Characteristic number of cells across a channel is 5 but only two

Fig.1.2NCRL = 2; CNC = 3;

N = 2.8 < CNC

Fig.1.3NCRL = 3; CNC = 3;

N = 3.8 > CNC

Like in the other refinements, the Narrow channel refinement level (NCRL) denotes the maximum level to which the cells can be split to satisfy the CNC criterion. The NCRL has higher priority than the CNC, so the refinement will proceed until the CNC criterion is satisfied or all the cells reach the Narrow channel resolution level.

Fig.1.4CNC = 5; NCRL = 1

Fig.1.5CNC = 5; NCRL = 3

3-4

cells were generated since the maximum refinement level of one allows only basic mesh cells and first-level cells to be generated.


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On Fig.1.5 the specified Narrow channel refinement level is high enough to allow five cells to be placed across the channel.

Note that (see Fig.1.6) the partial cells near the channel’s dead end and the orifice were not split like the other partial cells along the channel. This is due to the fact that the right angle was approximated by the flat element sloping to the both sides of the channel. Therefore the normal to the solid\fluid interface determined in these corner cells, unlike the other partial cells, is not perpendicular to the channel, so the number of cells per this direction satisfies the criterion without further splitting.

1.2.4) Refinement on Openings This refinement is intended to force the splitting of the partial cells which were not refined due to the “chamfer” approximation of the right angles (Fig.1.6), if these cells are at the boundary condition surface.

On Fig.1.7 the boundary condition specified on the wall in the end of the channel causes the Refinement on Opening procedure that splits the red partial cells to the next level.

1.2.5) Next, for all the fluid cells within the entire computational domain the following Fluid Cell Leveling procedure is applied: if a fluid cell is located between two cells of higher level, it is split to be equalized with the level of neighboring smaller cells.

Fig.1.6Normal to the solid\fluid interface direction in

the corner cells.

Fig.1.7CNC = 5; NCRL = 3; Inlet velocity at the

channel end-wall.


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1.3 Thin walls resolution

In contrast to the narrow channels, thin walls can be resolved without the mesh refinement inside the wall, since the both sides of the thin wall may reside in the same cell. Therefore, the amount of cells needed to resolve a thin wall is generally lower than the number of cells needed to properly resolve a channel of the same width. See Fig.1.8 - 1.10 illustrating the thin walls resolution technology and its limitations.

Fig.1.8One mesh cell can contain more than one fluid and/or solid volume; during calculation each volume has an individual set of parameters depending on its type (fluid or solid).

Solid 1

Solid 2 Fluid 1

Fluid 2

Fig.1.9If the wall thickness is greater than the basic mesh cell's size across the wall or if the

wall creates only one fluid volume in the cell, then the opposite sides of the wall will not lay within the same cell. Such walls are resolved with two or more cells across.


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1.4 Square Difference Refinement

The Square Difference Refinement checks the neighboring partial cells of different levels for the difference between their fluid passage areas. If the difference between the fluid passage area of the higher-level cell and the total fluid passage areas of the adjacent lower-lever cells exceeds the Square Difference Refinement Criterion (SDRC) then the greater-level cell is split to the level of adjacent cells in order to equalize the fluid passage areas (see Fig.1.11). The Square Difference Refinement is always enabled and cannot be disabled since it is a strict solver requirement. As with the Cell Mating rule, this is another condition imposed by the solver to provide stability for the convergence processes.

Though you cannot turn off the Square Difference Refinement, you can control its criterion, which is directly proportional to the Curvature refinement criterion. [CRC].

Fig.1.11Two adjacent partial cells of

different levels at the cylinder surface.

Fig.1.12Cut plot of the cylinder. The concerned cells are blue.

SSFRL = 2; CRL = 0; CRC = 3.14; NCRL = 1.

Fig.1.10The edges of thin walls ending within a mesh cell are trimmed. These mesh cells are called

Trimmed cells.

Model geometry Meshed geometry

Trimmed edge

Trimmed cell


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Fig.1.11 shows neighboring partial cells of different levels at the cylinder's solid/fluid interface. The fluid passage area of the higher-level cell is the ABDE polygon. The total fluid passage area of the lower-level cells is the ABCDE polygon, so the difference between the fluid passages is the yellow BCD triangle. In this example we have increased the curvature refinement criterion to π, thereby increasing the Square Difference Refinement Criterion so that the fluid passage difference (BCD) is smaller than the criterion, and thus, there is no need to split the higher-level cell.

Note that the Square Difference Refinement may cause a domino effect when one splitting produces cells which become lower-level cells for the next adjacent cell causing it to split too, and so on, resulting in an increased number of cells.

In the Fig.1.13 the total number of cells is nearly 20% more than in the Fig.1.14 in spite of the fact that the Curvature refinement is disabled (CRL = 0) in the first case. Here, the model geometry is similar and before the Square Difference Refinement the mesh is practically the same in both cases and mostly governed by the Small Solid Features Refinement when the SSFRL exceeds the CRL, i.e. changing the CRL from 0 to 3 would not change substantially the number of cells. However, in the first case the curvature criterion is lower, resulting in a more stringent criterion of the Square Difference Refinement. So the smaller Square Difference Refinement criterion leads to a greater number of cells subject to the Square Difference Refinement. In the Fig.1.13 you can see a stripe of the third level cells along the cylinder. This is the result of the Square Difference Refinement and the domino effect when a cell on the cylinder edge involves the neighboring cell in the refinement procedure and so forth along the cylinder.

Fig.1.13SSFRL = 3, CRL = 0; CRC = 0.45

Total cells = 49391.

Fig.1.14SSFRL = 3, CRL = 2; CRC = 0.50;

Total cells = 41376.

Increase of the curvature criterion will increase the Square Difference Refinement Criterion, and, in turn, decrease the number of cells in both cases.

3-8

If in the first case we specify the same CRC as in the second case (0.5054 rad), the total number of cells decreases to 40963.


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1.5 Mesh Diagnostic

The mesh diagnostic is intended to inspect the resulting initial mesh but not to change the total number of cells.

2 Refinements at Interfaces Between Substances

Different interface types (solid/fluid, solid1/solid2, solid/porous or porous/fluid) are checked on different refinement criteria, namely: small solid features criterion, curvature refinement criterion, tolerance refinement criterion and narrow channel refinement criterion for solid/fluid and solid/porous interfaces; small solid features criterion for solid1/solid2 interfaces; small solid features criterion and curvature refinement criterion for porous/fluid interfaces. Whereas the specified refinement levels are equally applied to any interface type.

2.1 Small Solid Features Refinement

The small solid features refinement acts on the cells where the maximum angle between normals to the surface-forming triangles is strictly greater than 120°. To make this 120-degree criterion easier to understand, let us consider simple small solid features of planar faces only. The normal to triangles that form the planar face is normal to the planar face too. Therefore, instead of considering the normals to the triangles we can consider normals to faces, or better the angle between faces.

In Fig.2.1 the cells with the cogs of 150 and 60 degrees were not split by the small solid features refinement because the maximum angles between the faces (i.e. between normals to the triangles enclosed by the cell) are 30° and 120°, respectively. If the angle between the normals becomes greater than 120° (121° for the 59°-cog) then the cell is split. The cell with the square spike surely has to be split because the lateral faces of the spike have their normals at the angle of 180°, thus satisfying the 120-degree criterion.

Note that rectangular corners (like in the rightmost cell) do not satisfy the criterion and therefore will not be resolved by the small solid features refinement.

Fig.2.1SSFRL = 1, CRL = 0, NCRL = 0


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Chapter Refinements at Interfaces Between Substances

From Fig.2.2 it is clear that the cells are split by the 120-degree criterion up to the first level, as defined by the narrow channel refinement level.

For the information about how the NCRL influences the narrow channel refinement see the Narrow Channel Refinement paragraph.

2.2 Curvature Refinement

The curvature refinement works in the same manner as the small solid features refinement with the difference that the critical angle between the normals can be specified by the user (in radians) as curvature refinement criterion (CRC). Here, the smaller the criterion, the better resolution of the solid curvature. To give more precise and descriptive explanation, the following table presents several CRC values together with the corresponding angles between normals and the angles between planar faces.

Remember that if the Narrow channel refinement is enabled, the maximum level to which the small solid features refinement can split the cells is set as the maximum level from the specified SSFRL and Narrow channel refinement level (NCRL). In other words, if the Narrow channel refinement is enabled, the SSFRL has no effect if it is smaller than the NCRL.

Table 2.1: Influence of the curvature criterion on the solid curvature resolution.

Curvature criterion, rad 0.3176 0.4510 0.5548 0.6435 1.0472 1.5708 2.0944 3.1416

α ' between normals, [degrees]

>19 >25 >31 >36 >60 >90 >120 180

α between faces, [degrees]

<161 <154 <148 <143 <120 <90 <60 0

Fig.2.2 SSFRL = 0, CRL = 0, NCRL = 1


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The table states that if the CRC is equal to 0.4510 rad, then all the cells where the angle between normals to the surface-forming triangles is more than 25 degrees will be split.

You can see that the curvature criterion set to 0.4510 rad splits the cells with the 150-degrees cog.

However, the default curvature criterion values are small enough to resolve obtuse angles and curvature well. Increasing the curvature criterion is reasonable if you want to avoid superfluous refinement but it is recommended that you try different criteria to find the most appropriate one.

The curvature criterion also denotes the criterion of the Square Difference Refinement. The square difference refinement criterion is directly proportional to the CRC, so the smaller CRC, the smaller square difference refinement criterion, resulting in a greater number of cells appearing after the Square Difference Refinement.

2.3 SSFRL or CRL

Why is it necessary to have two criteria? As you can see, the curvature refinement has higher priority than the small solid features refinement if the curvature criterion is smaller than 2/3 π. Note that COSMOSFloWorks-specified values of the curvature criterion are always smaller than 2/3 π.

Nevertheless, the advantage of the small solid features refinement is that being sensitive to relatively small geometry features it does not “notice” the large-scale curvatures, thus avoiding refinements in the entire computational domain but resolving only the areas of small features. At the same time, the curvature refinement can be used to resolve the large-scale curvatures. So both the refinements have their own coverage providing a

Fig.2.3 CRL = 1, CRC = 0.5548,SSFRL = 0, NCRL = 0

Fig.2.4 CRL = 1, CRC = 0.451,SSFRL = 0, NCRL = 0

Note that the curvature refinement works exactly as the small solid features refinement when the curvature criterion is equal to 2.0944 rad (2/3π).

In other words, if you did not set the CRC greater than 2/3 π and if the SSFRL and NCRL are smaller than the CRL, then the small solid feature refinement would be idle.


flexible tool for creating an optimal mesh.


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Chapter Local Mesh Settings

2.4 Tolerance Refinement

Any surface is approximated by a set of polygons whose vertices are the points of intersection of this surface with the cells' edges. This approach accurately represents flat faces though curved surfaces are represented by some approximation (e.g. as a circle can be represented by a polygon). The tolerance refinement criterion controls the precision of this approximation. A cell will be split if the distance between the outermost point of the surface within the cell and the polygon approximating this surface is larger than the specified criterion value.

3 Local Mesh Settings

The local mesh settings influence only the initial mesh and do not affect the basic mesh in the local area, but are basic mesh sensitive in that all refinement levels are set with respect to the basic mesh cell.

The local mesh settings are applied to the cells intersected with the local mesh region which can be represented by a component, face, edge or vertex.

If a cell intersects with different local mesh setting regions, the refinement settings in this cell will be used to achieve the maximum refinement.

Cell Type Refinement. The refinement level of cells of a specific type (all cells, fluid and partial cells, solid and partial cells, or only partial cells) denotes the minimum level to which the corresponding cells must be split if it doesn’t contradict the Cell Mating rule.

The minimum level means the lower bound to which it is obligatory to split cells, though the cells can be split further if it is required to

Tolerance Refinement Small Solid Feature Refinement

Tolerance criterion = 0.1 Tolerance criterion = 0.08

Tolerance Refinement Curvature Refinement

Refines cells taking into account the curvature only.

Tolerance criterion = 0.1

Refines cells only if the solid part cut by the polygon is large enough (h > 0.1)

Tolerance criterion = 0.03

3-12

satisfy the other criteria such as Small solid features refinement, Curvature refinement, Narrow channels refinement or Solid Boundary Refinement.


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If different cell types are to be refined, the refinement level of partial cells is set as the maximum level among all selected levels.

The local mesh settings have higher priority over the initial mesh settings. Therefore, the local mesh cells will be split to the specified local refinement levels regardless of the general SSFRL, CRL and NCRL (specified in the Initial Mesh dialog box). This, however, may cause refinement of cells located outside of the local region due to imposing the Cell Mating rule.

4 Irregular Cells

When analyzing the computational mesh from the results file obtained with the earlier versions of COSMOSFloWorks, you may notice the presence of irregular cells. An irregular cell is a computational mesh cell lying at the solid/fluid interface (or solid/solid interface in cases where two or more different solids are within the cell), partly in one substance and partly in another, and characterized by the impossibility to define the solid/fluid interface position within the cell, given the cell’s nodes positions relative to solid region and the intersections of the solid/fluid interface with the cell. Please note that there are no irregular cells in the newly generated meshes, because the solid/fliud interface is now always resolved properly.

You can use use the Results Summary to find out whether irregular cells are present and use the Mesh Visualization tool to detect where they are located.

5 The "Optimize thin walls resolution" option

In the earlier versions of COSMOSFloWorks refinement of the mesh around model's walls was needed to resolve thin walls properly, but it could also lead to increase in number of cells in adjacent fluid regions, especially in narrow channels between walls. If this additional mesh refinement is critical for obtaining proper results and you want to perform calculation on the same mesh as in the earlier version of COSMOSFloWorks, clear the Optimize thin walls resolution check box. In this case the mesh will be almost the same as in the previous version, the main difference is the absence of irregular cells. (see Fig.5.1).


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Chapter Postamble

6 Postamble

The problem of resolving a model with the computational mesh is always model-specific. In general, a denser mesh will provide better accuracy but you should tend to create an optimal mesh and to avoid redundant refinement.

When performing a calculation, try different mesh settings and analyze the obtained results carefully in order to understand whether it is necessary to refine the mesh or a coarser resolution is acceptable for the desired accuracy.

7 Glossary

Nx, Ny, Nz – Number of basic mesh cells per X, Y and Z directions, respectively.

SSFRL – Small solid features refinement level.

CRL – Curvature refinement level.

CRC – Curvature refinement criterion.

TRL – Tolerance refinement level.

Fig.5.1Mesh refinement around a thin wall: (a) the Optimize thin walls resolution option is switched

off, i.e. the mesh cells are split as in the previous versions of COSMOSFloWorks; (b) the Optimize thin walls resolution option is selected (the default state), i.e. the mesh cells are not split.

(a) (b)

solid/fluidinterfaces

3-14

TRC – Tolerance refinement criterion.

NCRL – Narrow channel refinement level.


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CNC – Characteristic number of cells across a narrow channel.

NCHmin – The minimum height of narrow channels.

NCHmax – The maximum height of narrow channels.

SDRC – Square difference refinement criterion.


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Chapter Glossary


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4

Validation Examples

Introduction

A series of calculation examples presented below validate the ability of COSMOSFloWorks to predict the essential features of various flows, as well as to solve conjugate heat transfer problems (i.e. flow problems with heat transfer in solids). In order to perform the validation accurately and to present clear results which the user can check independently, relatively simple examples have been selected. For each of the following examples, exact analytical expression or well-documented experimental results exist. Each of the examples focus on one or two particular physical phenomena such as: laminar flow with or without heat transfer, turbulent flows including vortex development, boundary layer separation and heat transfer, compressible gas flow with shock and expansion waves. Therefore, these examples validate the ability of COSMOSFloWorks to predict fundamental flow features accurately. The accuracy of predictions can be extrapolated to typical industrial examples (encountered every day by design engineers and solved using COSMOSFloWorks), which may include a combination of the above-mentioned physical phenomena and geometries of arbitrary complexity.

COSMOSFloWorks 2008 Fundamentals 4-1adfamily.com EMail:[email protected] is for study only,if tort to your rights,please inform us,we will delete

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Chapter Introduction


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1 Flow through a Cone Valve

Let us see how COSMOSFloWorks predicts incompressible turbulent 3D flows in a 3D cone valve taken from Ref.14 (the same in Ref.2) and having a complex flow passage geometry combining sudden 3D contractions and expansions at different turning angles ϕ (Fig. 1.1.). Following the Refs.2 and 14 recommendations on determining a valve’s hydraulic resistance correctly, i.e. to avoid any valve-generated flow disturbances at the places of measuring the flow total pressures upstream and downstream of the valve, the inlet and outlet straight pipes of the same diameter D and of enough length (we take 7D and 17D) are connected to the valve, so constituting the experimental rig model (see Fig. 1.2.). As in Ref.14, a water flows through this model. Its temperature of 293.2 K and fully developed turbulent inlet profile (see Ref.1) with mass-average velocity U ≈ 0.5 m/s (to yield the turbulent flow’s Reynolds number based on the pipe diameter ReD = 105) are specified at the model inlet, and static pressure of 1 atm is specified at the model outlet.

The corresponding model used for these predictions is shown in Fig. 1.2.. The valve’s turning angle ϕ is varied in the range of 0…55° (the valve opening diminishes to zero at ϕ = 82°30′).

Fig. 1.1. The cone valve under consideration: D = 0.206 m, Dax = 1.515⋅D, α = 13°40′.

Fig. 1.2. The model for calculating the 3D flow in the cone valve.

Outlet static pressure

P = 101325 Pa

Inlet velocity profile


The flow predictions performed with COSMOSFloWorks are validated by comparing the valve’s hydraulic resistance ζv, and the dimensionless coefficient of torque M (see Fig. 1.1.) acting on the valve, m, to the experimental data of Ref.14 (Ref.2).


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Chapter Flow through a Cone Valve

Since Ref.14 presents the valve’s hydraulic resistance (i.e. the resistance due to the flow obstacle, which is the valve) ζv, whereas the flow calculations in the model (as well as the experiments on the rig) yield the total hydraulic resistance including both ζv and the tubes’ hydraulic resistance due to friction, ζf , i.e. ζ = ζv + ζf , then, to obtain ζv from the flow predictions (as well as from the experiments), ζ f is calculated (measured in the experiments) separately, at the fully open valve (ϕ = 0); then ζv = ζ - ζ f .

In accordance with Ref.14, both ζ and ζf are defined as (Po inlet - Po outlet)/(ρU2/2), where

Po inlet and Po outlet are the flow total pressures at the model’s inlet and outlet, accordingly, ρ is the fluid density. The torque coefficient is defined as m = M/[D3⋅(ρU2/2)⋅(1+ ζv)], where M is the torque trying to slew the valve around its axis (vertical in the left picture in Fig. 1.1.) due to a non-uniform pressure distribution over the valve’s inner passage (naturally, the valve’s outer surface pressure cannot contribute to this torque). M is measured directly in the experiments and is integrated by COSMOSFloWorks over the valve’s inner passage.

The COSMOSFloWorks predictions have been performed at result resolution level of 5 with manual setting of the minimum gap size to the valve’s minimum passage in the Y = 0 plane and the minimum wall thickness to 3 mm (to resolve the valve’s sharp edges).

COSMOSFloWorks has predicted ζ f = 0.455, ζv shown in Fig. 1.3., and m shown in Fig. 1.4. It is seen that the COSMOSFloWorks predictions well agree with the experimental data.

This cone valve's 3D vortex flow pattern at ϕ = 45° is shown in Fig. 1.5. by flow trajectories colored by total pressure. The corresponding velocity contours and vectors at the Y = 0 plane are shown in Fig. 1.6..

The predictions have been performed on a computational mesh consisting of about 150000 cells and have required about 270 MB memory and about 7 hours to run on a 1GHz PIII platform for each specified ϕ .


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0.1

1

10

100

15 20 25 30 35 40 45 50 55ϕ(°)ϕ(°)ϕ(°)ϕ(°)

v

Experimental data

Calculation

Fig. 1.3. Comparison of the COSMOSFloWorks predictions with the Ref.14 experimental data on the cone valve’s hydraulic resistance versus the cone valve turning angle.

Fig. 1.4. Comparison of the COSMOSFloWorks predictions with the Ref.14 experimental data on the cone valve’s torque coefficient versus the cone valve turning angle.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

15 20 25 30 35 40 45 50 55(°)

m

Experimental data

Calculation


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Chapter Flow through a Cone Valve

Fig. 1.5. Flow trajectories colored by total pressure at ϕ = 45°.

Fig. 1.6. The cone valve’s velocity contours and vectors at ϕ = 45°.


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2 Laminar Flows Between Two Parallel Plates

Let us consider two-dimensional (planar) steady-state laminar flows of Newtonian, non-Newtonian, and compressible liquids between two parallel stationary plates spaced at a distance of 2h (see Fig. 2.1.).

In the case of Newtonian and non-Newtonian liquids the channel has a 2h = 0.01 m height and a 0.2 m length, the inlet for these liquids have standard ambient temperature (293.2 K) and a uniform inlet velocity profile of uaverage = 0.01 m/s (entrance disturbances are neglected). The inlet pressure is not known beforehand, since it will be obtained from the calculations in accordance with the specified channel exit pressure of 1 atm. (The fluids pass through the channel due to a pressure gradient.)

Since the Reynolds number based on the channel height is equal to about Re2h=100, the

flow is laminar.

As for the liquids, let us consider water as a Newtonian liquid and four non-Newtonian

liquids having identical density of 1000 kg/m3, identical specific heat of 4200 J/(kgK) and identical thermal conductivity of 10 W/(mK), but obeying different non-Newtonian liquid laws available in COSMOSFloWorks.

The considered non-Newtonian liquids' models and their governing characteristics are presented in Table 4.1. These models are featured by the function connecting the flow

shear stress (τ) with the flow shear rate ( ), i.e. , or, following Newtonian

liquids, the liquid dynamic viscosity (η) with the flow shear rate ( ), i.e. :

1 the Herschel-Bulkley model: , where K is the consistency coefficient, n is the power-law index, and is the yield stress (a special case with n = 1 gives the Bingham model);

2 the power-law model: , i.e., , which is a special case of

Fig. 2.1. Flow between two parallel plates.

uaverage or m

γ& ( )γτ &f=

γ& ( ) γγητ && ⋅=

( ) onK τγτ +⋅= &

oτ

( )nK γτ &⋅= ( ) 1−⋅= nK γη &


Herschel-Bulkley model with = 0;oτ


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Chapter Laminar Flows Between Two Parallel Plates

3 the Carreau model: , , where

is the liquid dynamic viscosity at infinite shear rate, i.e. the minimum dynamic

viscosity, is the liquid dynamic viscosity at zero shear rate, i.e. the maximum

dynamic viscosity, K1 is the time constant, n is the power-law index (this model is a

smooth version of the power-law model).

In accordance with the well-known theory presented in Ref.1, after some entrance length, the flow profile u(y) becomes fully developed and invariable. It can be determined from

the Navier-Stokes x-momentum equation corresponding to

this case in the coordinate system shown in Fig. 2.1. (y = 0 at the channel's center plane,

is the longitudinal pressure gradient along the channel, in the flow under

consideration).

As a result, the fully developed u(y) profile for a Newtonian fluid has the following form:

u(y) = - ,

where η is the fluid dynamic viscosity and η is the half height of the channel,

,

where uaverage is the flow's mass-average velocity defined as the flow's volume flow rate

divided by the area of the flow passage cross section.

For a non-Newtonian liquid described by the power-law model the fully developed u(y)

γητ &⋅= ( ) ( )[ ]( ) 2/1211

−

∞∞ ⋅+⋅−+=n

o K γηηηη & ∞η

oη

Non-Newtonian liquid No. 1 2 3 4

Non-Newtonian liquid model Herschel-Bulkley Bingham Power law Carreau

Consistency coeffic ient, K (Pa⋅sn) 0.001 0.001 0.001 -

Power law index, n 1.5 1 0.6 0.4

Yield stress, o (Pa) 0.001 0.001 - -

Minimum dynamic viscosity, ∞

(Pa⋅s)- - - 10-4

Maximum dynamic viscosity, o

(Pa⋅s)- - - 10-3

Time constant, K 1 (s) - - - 1

hconst

dy

d

dx

dP wττ ===

dx

dP

dy

du=γ&

)(2

1 22 yhdx

dP −η

2

3

h

u

dx

dP average −=

η

4-8

profile and the corresponding pressure gradient can be determined from the following formulae:


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, .

For a non-Newtonian liquid described by the Herschel-Bulkley model the fully developed u(y) profile can be determined from the following formulae:

at ,

at ,

where the unknown wall shear stress is determined numerically by solving the

nonlinear equation

,

e.g. with the Newton method, as described in this validation. The corresponding pressure

gradient is determined as .

For a non-Newtonian liquid described by the Carreau model the fully developed u(y) profile can not be determined analytically in an explicit form, so in this validation example it is obtained by solving the following parametric equation:

,

,

where p is a free parameter varied within the ±pmax range,

( )

−

++=

+n

n

average h

y

n

nuyu

1

11

12n

average

n

n

h

u

h

K

dx

dP

+⋅⋅−= 12

( ) ( ) n

n

oww

n n

n

K

huyu

1

/1max 1

+−

+== ττ

τhy

w

o

ττ<

( )

−

−−⋅=

+n

n

ow

ow h

y

uyu

1

max 1ττ

ττhy

w

o

ττ>

wτ

( )

−+

−⋅−⋅+

⋅=+

w

own

n

oww

naverage n

n

n

n

K

hu

τττττ

τ 121

1

1

/1

hdx

dP wτ=

( )( )2/)1(220 1)(

−∞∞ +−+=

n

w

pph

y λµµµτ

( )( 1) / 22 2 2 20 0max 2 2

( ) ( )11

2 ( 1) ( 1)

n

w w w

h hhu u p p np

n n

µ µ µ µµ λτ τ λ τ λ

−∞ ∞∞ − − = − − + − − + +


,( )

+−+=

−

∞∞

2/)1(2max

20max 1)(

n

w pp λµµµτ


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The p value is varied to satisfy .

The corresponding pressure gradient is equal to .

The SolidWorks model for the 2D calculation is shown in Fig. 2.2.. The boundary conditions are specified as mentioned above and the initial conditions coincide with the inlet boundary conditions. The results of the calculations performed with COSMOSFloWorks at result resolution level 5 are presented in Figs.2.3 - 2.8. The channel exit u(y) profile and the channel P(x) profile were obtained along the sketches shown by green lines in Fig. 2.2..

( )( 1) / 22 2 2 20 0max max max max 2 2

( ) ( )11

2 ( 1) ( 1)

n

w w w

h hhu p p np

n n

µ µ µ µµ λτ τ λ τ λ

−∞ ∞∞ − − = + + − + + +

max

0 0

h p

average

dyhu udy u dp

dp= =∫ ∫

hdx

dP wτ=

Fig. 2.2. The model for calculating 2D flow between two parallel plates with COSMOSFloWorks.

Fig. 2.3. The water and liquid #3 velocity profiles u(y) at the channel outlet.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

-0.005 -0.003 -0.001 0.001 0.003 0.005Y, m

U, m/s

Water, theory

Water, calculation

Liquid #3, theory

Liquid #3, calculation


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From Figs.2.4, 2.6, and 2.8 you can see that for all the liquids under consideration, after some entrance length of about 0.03 m, the pressure gradient governing the channel pressure loss becomes constant and nearly similar to the theoretical predictions. From Figs.2.3, 2.5, and 2.7 you can see that the fluid velocity profiles at the channel exit obtained from the calculations are close to the theoretical profiles.

Fig. 2.4. The water and liquid #3 longitudinal pressure change along the channel, P(x).

101325

101325.05

101325.1

101325.15

101325.2

101325.25

101325.3

0 0.05 0.1 0.15 0.2X, m

P, Pa

Water, theory

Water, calculation

Liquid #3, theory

Liquid #3,calculation

Fig. 2.5. The liquids #1 and #2 velocity profiles u(y) at the channel outlet.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

-0.005 -0.003 -0.001 0.001 0.003 0.005 Y, m

U, m/s

Liquid #1, theory


Liquid #2, theory



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In the case of compressible liquids the channel has the height of 2h = 0.001 m and the length of 0.5 m, the liquids at its inlet had standard ambient temperature (293.2 K) and a uniform inlet velocity profile corresponding to the specified mass flow rate of

= 0.01 kg/s.

Fig. 2.6. The liquids #1 and #2 longitudinal pressure change along the channel, P(x).

101325

101325.1

101325.2

101325.3

101325.4

101325.5

101325.6

0 0.05 0.1 0.15 0.2X, m

P, Pa

Liquid #1, theory


Liquid #2, theory


Fig. 2.7. The liquid #4 velocity profile u(y) at the channel outlet.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

-0.005 -0.003 -0.001 0.001 0.003 0.005 Y, m

U, m/s

Liquid #4, theory


m&

4-12

The inlet pressure is not known beforehand, since it will be obtained from the calculations as providing the specified mass flow rate under the specified channel exit pressure of 1 atm. (The fluids pass through the channel due to the pressure gradient).


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Let us consider two compressible liquids whose density obeys the following laws:

• the power law:

, where ρ0, P0, B and n are specified: ρ0 is the liquid's density under the reference pressure P0, B and n are constants,

• the logarithmic law:

, where ρ0, P0, B and C are specified: ρ0 is the liquid's density under the reference pressure P0, B and C are constants.

In this validation example these law's parameters values have been specified as ρ0=103

kg/m3, P0 = 1 atm, B = 107 Pa, n = 1.4, C = 1, and these liquids have the 1Pa·s dynamic

viscosity.

Since this channel is rather long, the pressure gradient along it can be determined as

, where η is the liquids' dynamic viscosity, is the liquid mass flow rate, S is the channel's width, ρ is the liquid density.

Therefore, by substitution the compressible liquids' ρ(P) functions, we obtain the

Fig. 2.8. The liquid #4 longitudinal pressure change along the channel, P(x).

101325

101325.02

101325.04

101325.06

101325.08

101325.1

101325.12

101325.14

0 0.05 0.1 0.15 0.2 X, m

P, Pa

Liquid #4, theory


0 0

nP B

P B

ρρ

+= +

0

0

1 *lnB P

CB P

ρρ = +−+

2

3P m

x h S

ηρ

∂ = −∂

& m&


following equations for determining P(x) along the channel:

• for the power-law liquid:


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its solution is

,

where C1 is a constant determined from the boundary conditions;

• for the logarithmic-law liquid:

, this equation is solved numerically.

Both the theoretical P(x) distributions and the corresponding distributions computed within COSMOSFloWorks on a 5*500 computational mesh are presented in Figs.2.9 and 2.10. It is seen that the COSMOSFloWorks calculations agree with the theoretical distributions.

1/

02

0

3n

P BP m

x h S P B

µρ

+∂ = − ∂ +

&

( ) 1 11

0 120

3( )

1n nn m

P B P B x Cn h S

µρ

++ = + ++

&

20 0

31 ln

P m P BC

x h S P B

µρ

∂ += − −∂ +

&

Fig. 2.9. The logarithmic-law compressible liquid's longitudinal pressure change along the channel, P(x).

0

2000000

4000000

6000000

8000000

10000000

12000000

14000000

16000000

18000000

0 0.1 0.2 0.3 0.4 0.5 X, m

P, Pa

LN, theory

LN, calculation


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Fig. 2.10. The power-law compressible liquid's longitudinal pressure change along the channel, P(x).

0

5000000

10000000

15000000

20000000

25000000

30000000

35000000

0 0.1 0.2 0.3 0.4 0.5 X, m

P, Pa

Power, theory

Power, calculation


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3 Laminar and Turbulent Flows in Pipes

Having been encouraged by the 2D results presented in the previous example, let us now see how the 3D flow through a straight pipe is predicted. Let us consider water (at standard 293.2 K temperature) flowing through a long straight pipe with circular cross section of d = 0.1 m (see Fig. 3.1.). At the pipe inlet the velocity is uniform and equal to uinlet. At the pipe outlet the static pressure is equal to 1 atm.

The SolidWorks model used for all the 3D pipe flow calculations is shown in Fig. 3.2. The initial conditions have been specified to coincide with the inlet boundary conditions. The computational domain is reduced to domain (Z ≥ 0, Y ≥ 0) with specifying the flow symmetry planes at Z = 0 and Y = 0.

According to theory (Ref.1), the pipe flow velocity profile changes along the pipe until it becomes a constant, fully developed profile at a distance of Linlet from the pipe inlet. According to Ref.1, Linlet is estimated as:

Fig. 3.1. Flow in a pipe.

uinlet

Fig. 3.2. The SolidWorks model for calculating 3D flow in a pipe with COSMOSFloWorks.


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Chapter Laminar and Turbulent Flows in Pipes

where Red = ρ⋅U⋅d/µ is the Reynolds number based on the pipe diameter d, U is the mass-average flow velocity, ρ is the fluid density, and µ is the fluid dynamic viscosity.

Therefore, to provide a fully developed flow in the pipe at Red under consideration, we will study the cases listed in Table 1:. Here, Lpipe is the overall pipe length. All the COSMOSFloWorks predictions concerning the fully developed pipe flow characteristics are referred to the pipe section downstream of the inlet section.

*) the lengths in brackets are for the rough pipes.

The flow regime in a pipe can be laminar, turbulent, or transitional, depending on Red. According to Ref.1, Red = 4000 is approximately the boundary between laminar pipe flow and turbulent one (here, the transitional region is not considered).

Theory (Refs. 1 and 4) states that for laminar fully developed pipe flows (Hagen-Poiseuille flow) the velocity profile u(y) is invariable and given by:

where R is the pipe radius, and dP/dx is the longitudinal pressure gradient along the pipe, which is also invariable and equal to:

The COSMOSFloWorks predictions of dP/dx and u(y) of the laminar fully developed pipe flow at Red = 100 performed at result resolution level 6 are presented in Fig. 3.3. and Fig. 3.4. The presented predictions relate to the smooth pipe, and similar ones not presented

Table 1: Pipe inlet velocities and lengths.

Red uinlet, m/s Linlet, m Lpipe, m

0.1 10-6 0.3 0.45

100 0.001 0.3 0.45

1000 0.01 3 4.5

104 0.1 4 (5)* 6 (10)*

105 1 4 (5)* 6 (10)*

106 10 4 (5)*6 6 (10)*

=⋅=⋅

=⋅≥⋅⋅

610...6000Re,40

6000...2500Re,100

2500...1.0Re,3Re03.0

d

d

dd

d

d

dd

Linlet =

)(41

)( 22 yRdx

dPyu −−=

µ,

2

8

R

u

dx

dP inlet−= µ .

4-18

here have been obtained for the case of the rough tube with relative sand roughness of k/d=0.2…0.4 %, that agrees with the theory (Ref.1).


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From Fig. 3.3. one can see that after an entrance length of about 0.15 m the pressure gradient predicted by COSMOSFloWorks coincides with the one predicted by theory. Therefore, the prediction of pipe pressure loss is excellent. As for local flow features, from Fig. 3.4. one can see that the fluid velocity profiles predicted at the pipe exit are rather close to the theoretical profile.

101325.0000

101325.0002

101325.0004

101325.0006

101325.0008

101325.0010

101325.0012

101325.0014

101325.0016

101325.0018

101325.0020

0 0.1 0.2 0.3 0.4 0.5 X (m)

Pressure (Pa)

Theory

Calculation

Fig. 3.3. The longitudinal pressure change (pressure gradient) along the pipe at Red ≈ 100.

Fig. 3.4. The fluid velocity profile at the pipe exit for Red ≈ 100.

0

0.0005

0.001

0.0015

0.002

0.0025

0 0.01 0.02 0.03 0.04 0.05 Y (m)

Velocity (m/s)

Theory

Calculation


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The velocity profile and longitudinal pressure distribution in a smooth pipe at Red = 105, i.e., in a turbulent pipe flow regime, predicted by COSMOSFloWorks at result resolution level 6 are presented in Figs.3.5 and 3.6 and compared to theory (Ref.1, the Blasius law of pressure loss, the 1/7-power velocity profile).

Then, to stand closer to engineering practice, let us consider the COSMOSFloWorks predictions of the pipe friction factor used commonly and defined as:

where L is length of the pipe section with the fully developed flow, along which pressure loss ∆P is measured.

101200

101400

101600

101800

102000

102200

102400

0 2 4 6 8 10 X (m)

Pressure (Pa)

Theory

Calculation

Fig. 3.5. The longitudinal pressure change (pressure gradient) along the pipe at Red = 105.

L

d

u

Pf

inlet

⋅∆=

2

2

ρ,


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In Figs. 3.7 and 3.8 (scaled up) you can see the COSMOSFloWorks predictions performed at result resolution level 5 for the smooth pipes in the entire Red range (both laminar and turbulent), and compared with the theoretical and empirical values determined from the following formulae which are valid for fully-developed flows in smooth pipes (Refs.1, 2, and 4):

It can be seen that the friction factor values predicted for smooth pipes, especially in the laminar region, are fairly close to the theoretical and empirical curve.

As for the friction factor in rough pipes, the COSMOSFloWorks predictions for the pipes having relative wall roughness of k/d=0.4% (k is the sand roughness) are presented and compared with the empirical curve for such pipes (Refs.1, 2, and 4) in Fig. 3.8.. The underprediction error does not exceed 13%.

Additionally, in the full accordance with theory and experimental data the COSMOSFloWorks predictions show that the wall roughness does not affect the friction factor in laminar pipe flows.

Fig. 3.6. The fluid velocity profile at the pipe exit at Red = 105.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.01 0.02 0.03 0.04 0.05 Y (m)

Velocity (m/s)

Theory

Calculation

1 4 5

25

64, Re 2300

Re

0.316 Re , 4000 Re 10

, Re 10

dd

d d

dd

laminar flows,

f turbulent flows,

Re1.8 log turbulent flows

6.9

−

−

≤ −

= ⋅ < < − ⋅ ≥ −


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Fig. 3.7. The friction factor predicted by COSMOSFloWorks for smooth pipes in comparison with the theoretical and empirical data (Refs.1, 2, and 4).

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 Red

Friction factor

Smooth pipes,theoretical andempirical dataCalculation

0.010

0.100

1.E+03 1.E+04 1.E+05 1.E+06Red

Friction Factor

Smooth wall,theoretical andempirical dataSmooth wall,calculation

Rough wall,k/d=0.4%, empiricaldataRough wall,k/d=0.4%,calculation

Fig. 3.8. The friction factor predicted by COSMOSFloWorks for smooth and rough pipes in comparison with the theoretical and empirical data (Refs.1, 2, and 4).


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4 Flows Over Smooth and Rough Flat Plates

In the previous example we have presented a validation for laminar and turbulent flows in smooth and rough pipes for a wide range of Reynolds numbers. Now let us consider uniform flows over smooth and rough flat plates with laminar and turbulent boundary layers, so that COSMOSFloWorks predictions of a flat plate drag coefficient are validated.

We consider the boundary layer development of incompressible uniform 2D water flow over a flat plate of length L (see Fig. 4.1.). The boundary layer develops from the plate leading edge lying at the upstream computational domain boundary. The boundary layer at the leading edge is considered laminar. Then, at some distance from the plate leading edge the boundary layer automatically becomes turbulent (if this distance does not exceed L).

The SolidWorks model is shown in Fig. 4.2.. The problem is solved as internal in order to avoid a conflict situation in the corner mesh cell where the external flow boundary and the model wall intersect. In the internal flow problem statement, to avoid any influence of the upper model boundary or wall on the flow near the flat plate, the ideal wall boundary condition has been specified on the upper wall. The plate length is equal to 10 m, the channel height is equal to 2 m, the walls’ thickness is equal to 0.5 m.

Water

L

Fig. 4.1. Flow over a flat plate.

Fig. 4.2. The model for calculating the flow over the flat plate with COSMOSFloWorks.

Outlet

Rough or smooth wall

Ideal wallInlet


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Chapter Flows Over Smooth and Rough Flat Plates

To solve the problem, an incoming uniform water flow of a certain velocity (see below), temperature of 293.2 K, turbulence intensity of 1%, and turbulence length of 0.01 m is specified at the channel inlet, whereas the water static pressure of 1 atm is specified at the channel outlet.

The flow computation is aimed at predicting the flat plate drag coefficient, defined as (see Refs. 1 and 4):

where F is the plate drag force, A is the plate surface area, ρ is fluid density, and V is the fluid velocity.

According to Refs.1 and 4, the plate drag coefficient value is governed by the Reynolds number, based on the distance L from the plate leading edge (ReL = ρVL/µ, where ρ is the fluid density, V is the incoming uniform flow velocity, and µ is fluid dynamic viscosity), as well as by the relative wall roughness L/k, where k is the sand roughness. As a result, Refs.1 and 4 give us the semi-empirical flat plate CD (ReL) curves obtained for different L/k from the generalized tubular friction factor curves and presented in Fig. 4.3. (here, ε ≡ k). If the boundary layer is laminar at the plate leading edge, then the wall roughness does not affect CD until the transition from the laminar boundary layer to the turbulent one, i.e., the CD (ReL) curve is the same as for a hydraulically smooth flat plate. The transition region’s boundaries depend on various factors, the wall roughness among them. Here is shown the theoretical transition region for a hydraulically smooth flat plate. The transition region's boundary corresponding to fully turbulent flows (i.e., at the higher ReL) is marked in Fig. 4.3. by a dashed line. At the higher ReL, the semi-empirical theoretical curves have flat parts along which ReL does not affect CD at a fixed wall roughness. These flat parts of the semi-empirical theoretical curves have been obtained by a theoretical scaling of the generalized tubular friction factor curves to the flat plate conditions under the assumption of a turbulent boundary layer beginning from the flat plate leading edge.

To validate the COSMOSFloWorks flat plate CD predictions within a wide ReL range, we have varied the incoming uniform flow velocity at the model inlet to obtain the ReL values of 105, 3⋅105, 106, 3⋅106, 107, 3⋅107, 108, 3⋅108, 109.To validate the wall roughness influence on CD, the wall roughness k values of 0, 50, 200, 103, 5⋅103, 104 µm have been considered. The COSMOSFloWorks calculation results obtained at result resolution level 5 and compared with the semi-empirical curves are presented in Fig. 4.3..

As you can see from Fig. 4.3., CD (ReL) of rough plates is somewhat underpredicted by COSMOSFloWorks in the turbulent region, at L/k ³ 1000 the CD (ReL) prediction error does not exceed about 12%.

AV

FCd

2

2ρ=

,


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Fig. 4.3. The flat plate drag coefficient predicted with COSMOSFloWorks for rough and hydraulically smooth flat plates in comparison with the semi-empirical curves (Refs.1 and 4).

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

1.00E+05 1.00E+06 1.00E+07 1.00E+08 1.00E+09 Re

CD

L/k=1e3

L/k=2e3

L/k=1e4

L/k=5e4

L/k=2e5

Smooth


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Chapter Flows Over Smooth and Rough Flat Plates


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5 Flow in a 90-degree Bend Square Duct

In the previous examples we have considered laminar and turbulent flows over flat plates and in straight pipes. Let us now see how COSMOSFloWorks predicts 3-dimensional

incompressible flow in a 90o-bend square duct.

Following Ref.8, we will consider a steady-state flow of water (at 293.2 K inlet temperature and Uinlet = 0.0198 m/s inlet uniform velocity) in a 40×40 mm square cross-sectional duct having a 90°-angle bend with ri = 72 mm inner radius (ro = 112 mm outer radius accordingly) and attached straight sections of 1.8 m upstream and 1.2 m downstream (see Fig. 5.1.). Since the flow's Reynolds number, based on the duct's hydraulic diameter (D=40 mm), is equal to ReD = 790, the flow is laminar.

The COSMOSFloWorks prediction was performed at result resolution level 7.

The predicted dimensionless (divided by Uinlet) velocity profiles are compared in Figs.5.2, 5.3 with the ones measured with a laser-Doppler anemometry at the following duct cross sections: XH = -5⋅D, -2.5⋅D, 0 (or θ=0°) and at the θ=30°, 60°, 90° bend sections. The z

and r directions are represented by coordinates and , where z1/2 = 20 mm.

The dimensionless velocity isolines (with the 0.1 step) at the duct's θ= 60° and 90° sections, both measured in Ref.8 and predicted with COSMOSFloWorks, are shown in Figs.5.4 and 5.5.

Fig. 5.1. The 90°-bend square duct's configuration indicating the velocity measuring stations and the dimensionless coordinates used for presenting the velocity profiles.

o

i o

r r

r r

−− 1/ 2

z

z


It is seen that the COSMOSFloWorks predictions are close to the Ref.8 experimental data.


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Chapter Flow in a 90-degree Bend Square Duct

4-28

Fig. 5.2. The duct's velocity profiles predicted by COSMOSFloWorks (red lines) in comparison with the Ref.8 experimental data (circles).


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Fig. 5.3. The duct's velocity profiles predicted by COSMOSFloWorks (red lines) in comparison with the Ref.8 experimental data (circles).

z/z1/2=0.5 z/z1/2=0

z/z1/2=0.5 z/z1/2=0


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Chapter Flow in a 90-degree Bend Square Duct

Fig. 5.4. The duct's velocity isolines at the θ = 60° section predicted by COSMOSFloWorks (left) in comparison with the Ref.8 experimental data (right).

Fig. 5.5. The duct's velocity isolines at the θ =90° section predicted by COSMOSFloWorks (left)

4-30

in comparison with the Ref.8 experimental data (right).


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6 Flows in 2D Channels with Bilateral and Unilateral Sudden Expansions

In this example we will consider both turbulent and laminar incompressible steady-state flows through 2D (plane) channels with bilateral and unilateral sudden expansions and parallel walls, as shown in Figs.6.1 and 6.2. At the 10 cm inlet height of the bilateral-sudden-expansion channels a uniform water stream at 293.2 K and 1 m/s is specified. The Reynolds number is based on the inlet height and is equal to Re = 105, therefore (since Re > 104) the flow is turbulent. At the 30 mm height inlet of the unilateral-sudden-expansion channel an experimentally measured water stream at 293.2 K and 8.25 mm/s mean velocity is specified, so the Reynolds number based on the inlet height is equal to Re = 250, therefore the flow is laminar. In both channels, the sudden expansion generates a vortex, which is considered in this validation from the viewpoint of hydraulic loss in the bilateral-expansion channel (compared to Ref.2) and from the viewpoint of the flow velocity field in the unilateral-expansion channel (compared to Ref.13).

In accordance with Ref.2, the local hydraulic loss coefficient of a bilateral sudden expansion (the so-called total pressure loss due to flow) for a turbulent (Re > 104) flow with a uniform inlet velocity profile depends only on the expansion area ratio and is determined from the following formula:

Fig. 6.1. Flow in a 2D (plane) channel with a bilateral sudden expansion.

outlet

X

Y

inletWater

1 m/s

Fig. 6.2. Flow in a 2D (plane) channel with a unilateral sudden expansion.

400 mm20 mm

30 m

m

h =

15

Y

X0

Inlet experimental velocity profile Outlet static pressure

recirculation Lr

2

1

020

10 1

−=−=

A

A

u

PPs ρ

ζ ,


where A0 and A1 are the inlet and outlet cross sectional areas respectively, P0 and P1 are the inlet and outlet total pressures, and ρu0

2/2 is the inlet dynamic head.

2


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Chapter Flows in 2D Channels with Bilateral and Unilateral Sudden Expansions

In a real sudden expansion the flow hydraulic loss coefficient is equal to ζ = ζf + ζs, where ζf is the friction loss coefficient. In order to exclude ζf from our comparative analysis, we have imposed the ideal wall boundary condition on all of the channel walls.

In this validation example the channel expansion area ratios under consideration are: 1.5, 2.0, 3.0, and 6.0. To avoid disturbances at the outlet due to the sudden expansion, the channel length is 10 times longer than its height. The 1 atm static pressure is specified at the channel outlet.

The ζ s values predicted by COSMOSFloWorks at result resolution level 8 for different channel expansion area ratios A0/A1 are compared to theory in Fig. 6.3.

From Fig. 6.3., one can see that COSMOSFloWorks overpredicts ζ s by about 4.5...7.9 %.

The model used for the unilateral-sudden-expansion channel's flow calculation is shown in Fig. 6.4. The channel's inlet section has a 30 mm height and a 20 mm length. The channel's expanded section (downstream of the 15 mm height back step) has a 45 mm height and a 400 mm length (to avoid disturbances of the velocity field compared to the experimental data from the channel's outlet boundary condition). The velocity profile measured in the Ref.13 at the corresponding Reh = 125 (the Reynolds number based on the step height) is specified as a boundary condition at the channel inlet. The 105 Pa static pressure is specified at the channel outlet.

Fig. 6.3. Comparison of COSMOSFloWorks calculations to the theoretical values (Ref.2) for the sudden expansion hydraulic loss coefficient versus the channel expansion area ratio.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1A0/A1

s

Theory

Calculation


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The flow velocity field predicted by COSMOSFloWorks at result resolution level 8 is compared in Figs.6.5, 6.6, and 6.7 to the values measured in Ref.13 with a laser anemometer. The flow X-velocity (u/U, where U = 8.25 mm/s) profiles at several X = const (-20 mm, 0, 12 mm, … 150 mm) cross sections are shown in Fig. 6.5. It is seen that the predicted flow velocity profiles are very close to the experimental values both in the main stream and in the recirculation zone. The recirculation zone's characteristics, i.e. its length LR along the channel's wall, (plotted versus the Reynolds number Reh based on the channel's step height h, where Reh =125 for the case under consideration), the separation streamline, and the vortex center are shown in Figs.6.6 and 6.7. It is seen that they are very close to the experimental data.

Fig. 6.4. The SolidWorks model for calculating the 2D flow in the unilateral-sudden-expansion channel with COSMOSFloWorks.

Inlet velocity profile

Outlet static pressure


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Chapter Flows in 2D Channels with Bilateral and Unilateral Sudden Expansions

As one can see, both the integral characteristics (hydraulic loss coefficient) and local

Fig. 6.5. The unilateral-sudden-expansion channel's velocity profiles predicted by COSMOSFloWorks (red lines) in comparison with the Ref.13 experimental data (black lines with dark circles).

0

2

4

6

8

10

12

0 50 100 150 200 250 Reh

L/h

- r

eci

rcu

latio

n zo

ne le

ngt

h

Fig. 6.6. The unilateral-sudden-expansion channel's recirculation zone length predicted by COSMOSFloWorks (red square) in comparison with the Ref.13 experimental data (black signs).

0

5

10

15

0 0.2 0.4 0.6 0.8 1

separation streamlines,calculation

vortex center, calculation

Fig. 6.7. The unilateral-sudden-expansion channel recirculation zone's separation streamlines and vortex center, both predicted by COSMOSFloWorks (red lines and square) in comparison with the Ref.13 experimental data (black signs).

4-34

values (velocity profiles and recirculation zone geometry) of the turbulent and laminar flow in a 2D sudden expansion channel under consideration are adequately predicted by COSMOSFloWorks.


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7 Flow over a Circular Cylinder

Until now, we have considered only internal flows. Let us now consider an external incompressible flow example. In this example, water at a temperature of 293.2 K and a pressure of 1 atm flows over a cylinder of 0.01 m or 1 m diameter. The flow pattern of this example substantially depends on the Reynolds number which is based on the cylinder diameter. At low Reynolds numbers (4 < Re < 60) two steady vortices are formed on the rear side of the cylinder and remain attached to the cylinder, as it is shown schematically in Fig. 7.1. (see Refs.3).

At higher Reynolds numbers the flow becomes unstable and a von Karman vortex street appears in the wake past the cylinder. Moreover, at Re > 60…100 the eddies attached to the cylinder begin to oscillate and shed from the cylinder (Ref.3). The flow pattern is shown schematically in Fig. 7.2..

To calculate the 2D flow (in the X-Y plane) with COSMOSFloWorks, the model shown in Fig. 7.3. has been created. The cylinder diameter is equal to 0.01 m at Re ≤104 and 1 m at Re>104. The incoming stream turbulence intensity has been specified as 1%. To take the flow’s physical instability into account, the flow has been calculated by COSMOSFloWorks using the time-dependent option. All the calculations have been performed at result resolution level 7.

Fig. 7.1. Flow past a cylinder at low Reynolds numbers (4 < Re < 60).

Fig. 7.2. Flow past a cylinder at Reynolds numbers Re > 60…100.


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Chapter Flow over a Circular Cylinder

In accordance with the theory, steady flow patterns have been obtained in these calculations in the low Re region. An example of such calculation at Re=41 is shown in Fig. 7.4. as flow trajectories over and past the cylinder in comparison with a photo of such flow from Ref.9. It is seen that the steady vortex past the cylinder is predicted correctly.

Fig. 7.3. The SolidWorks model used to calculate 2D flow over a cylinder.

Fig. 7.4. Flow trajectories over and past a cylinder at Re=41 predicted with COSMOSFloWorks (above) in comparison with a photo of such flow from Ref.9 (below).


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The unsteady vortex shedding from a cylinder at Re > 60..100, yields oscillations of both drag and lateral forces acting on the cylinder and a von Karman vortex street is formed past the cylinder. An X-velocity field over and past the cylinder is shown in Fig. 7.5. The COSMOSFloWorks prediction of the cylinder drag and lateral force oscillations' frequency in a form of Strouhal number (Sh = D/(tU), where D is the cylinder diameter, t is the period of oscillations, and U is the incoming stream velocity) in comparison with experimental data for Re≥103 is shown in Fig. 7.6..

Fig. 7.5. Velocity contours of flow over and past the cylinder at Re=140.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Re

Sh

Fig. 7.6. The cylinder flow's Strouhal number predicted with COSMOSFloWorks (red triangles) in comparison with the experimental data (blue line with dashes, Ref.4).


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Chapter Flow over a Circular Cylinder

The time-averaged cylinder drag coefficient is defined as

where FD is the drag force acting on the cylinder, ρU2/2 is the incoming stream dynamic head, D is the cylinder diameter, and L is the cylinder length. The cylinder drag coefficient, predicted by COSMOSFloWorks is compared to the well-known CD(Re) experimental data in Fig. 7.7..

DLU

FC

2

DD

ρ21

=

0.1

1

10

100

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07Re

CD

Fig. 7.7. The cylinder drag coefficient predicted by COSMOSFloWorks (red diamonds) in comparison with the experimental data (black marks, Ref.3)


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8 Supersonic Flow in a 2D Convergent-Divergent Channel

Until now we have only considered incompressible flows, so now we will study a compressible, supersonic flow.

The first example is a supersonic flow of air in a 2D (plane) convergent-divergent channel whose scheme is shown on Fig. 8.1.

A uniform supersonic stream of air, having a Mach number M=3, static temperature of 293.2 K, and static pressure of 1 atm, is specified at the channel inlet between two parallel walls. In the next convergent section (see Fig. 8.2.) the stream decelerates through two oblique shocks shown schematically in Fig. 8.1. as lines separating regions 1, 2, and 3. Since the convergent section has a special shape adjusted to the inlet Mach number, so the shock reflected from the upper plane wall and separating regions 2 and 3 comes to the section 3 lower wall edge, a uniform supersonic flow occurs in the next section 3 between two parallel walls. In the following divergent section the supersonic flow accelerates thus forming an expansion waves fan 4. Finally, the stream decelerates in the exit channel section between two parallel walls when passing through another oblique shock.

The SolidWorks model of this 2D channel is shown in Fig. 8.3.

Fig. 8.1. Supersonic flow in a 2D convergent-divergent channel.

Fig. 8.2. Dimensions (in m) of the 2D convergent-divergent channel including a reference line for comparing the Mach number.


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Chapter Supersonic Flow in a 2D Convergent-Divergent Channel

Since the channel was designed for the inviscid flow of an ideal gas, the ideal wall boundary condition has been specified and the laminar only flow has been considered instead of turbulent. The computed Mach number along the reference line and at the reference points (1-5) are compared with the theoretical values in Fig. 8.4..

To obtain the most accurate results possible with COSMOSFloWorks, the calculations have been performed at result resolution level 8. The predicted Mach number at the selected channel points (1-5) and along the reference line (see Fig. 8.2.), are presented in Table.8.1 and Fig. 8.4. respectively.

Table.8.1 Mach number values predicted with COSMOSFloWorks with comparison to the theoretical values at the reference points.

From Table.8.1 and Fig. 8.4. it can be seen that the COSMOSFloWorks predictions are very close to the theoretical values. In Fig. 8.4. one can see that COSMOSFloWorks properly predicts the abrupt parameter changes when the stream passes through the shock and a fast parameter change in the expansion fan.

Fig. 8.3. The model for calculating the 2D supersonic flow in the 2D convergent-divergent channel with COSMOSFloWorks.

Point 1 2 3 4 5X coordinate of point, m 0.0042 0.047 0.1094 0.155 0.1648Y coordinate of point, m 0.0175 0.0157 0.026 0.026 0.0157

Theoretical M 3.000 2.427 1.957 2.089 2.365COSMOSFloWorks

prediction of M 3.000 2.429 1.965 2.106 2.380Prediction error,% 0.0 0.1 0.4 0.8 0.6


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To show the full flow pattern, the predicted Mach number contours of the channel flow are shown in Fig. 8.5..

This example illustrates that COSMOSFloWorks is capable of capturing shock waves with a high degree of accuracy. This high accuracy is possible due to the COSMOSFloWorks solution adaptive meshing capability. Solution adaptive meshing automatically refines the mesh in regions with high flow gradients such as shocks and expansion fans.

Fig. 8.4. Mach number values predicted with COSMOSFloWorks along the reference line (the reference points on it are marked by square boxes with numbers) in comparison with the theoretical

1.92

2.12.22.32.42.52.62.72.82.9

33.1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18x, m

M

Theory

Calculation

1

2

3

4

5

Fig. 8.5. Mach number contours predicted by COSMOSFloWorks.


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Chapter Supersonic Flow in a 2D Convergent-Divergent Channel


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9 Supersonic Flow over a Segmental Conic Body

Now let us consider an external supersonic flow of air over a segmental conic body shown in Fig. 9.1. The general case is that the body is tilted at an angle of α with respect to the incoming flow direction. The dimensions of the body whose longitudinal (in direction t, see Fig. 9.1.) and lateral (in direction n) aerodynamic drag coefficients, as well as longitudinal (with respect to Z axis) torque coefficient, were investigated in Ref.5 are presented in Fig. 9.2. They were determined from the dimensionless body sizes and the Reynolds number stated in Ref.5.

Fig. 9.1. Supersonic flow over a segmental conic body.

Center of gravity

n

y

xa

t

External air flowM∞ = 1.7

Fig. 9.2. Model sketch dimensioned in centimeters.


The model of this body is shown in Fig. 9.3..


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Chapter Supersonic Flow over a Segmental Conic Body

To compare the COSMOSFloWorks predictions with the experimental data of Ref.5, the calculations have been performed for the case of incoming flow velocity of Mach number 1.7. The undisturbed turbulent incoming flow has a static pressure of 1 atm, static temperature of 660.2 K, and turbulence intensity of 1%. The flow Reynolds number of 1.7×106 (defined with respect to the body frontal diameter) corresponds to these conditions, satisfying the Ref.5 experimental conditions.

To compare the flow prediction with the experimental data of Ref.5, the calculations have been performed for the body tilted at α = 0°, 30°, 60°, 90°, 120°, 150° and 180° angles. To reduce the computational resources, the Z = 0 flow symmetry plane has been specified in all of the calculations. Additionally, the Y = 0 flow symmetry plane has been specified at α = 0° and 180°.

The calculations have been performed at result resolution level 6.

The comparison is performed on the following parameters:

• longitudinal aerodynamic drag coefficient,

where Ft is the aerodynamic drag force acting on the body in the t direction (see Fig. 9.1.), ρU2/2 is the incoming stream dynamic head, S is the body frontal cross section (being

Fig. 9.3. The SolidWorks model for calculating the 3D flow over the 3D segmental conic body with COSMOSFloWorks.

SU

FC

2

tt

ρ21

= ,

4-44

perpendicular to the body axis) area;


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• lateral aerodynamic drag coefficient,

where Fn is the aerodynamic drag force acting on the body in the n direction (see Fig. 9.1.), ρU2/2 is the incoming stream dynamic head, S is the body frontal cross section (being perpendicular to the body axis) area;

• on the longitudinal (with respect to Z axis) aerodynamic torque coefficient,

where Mz is the aerodynamic torque acting on the body with respect to the Z axis (see Fig. 9.1.), ρU2/2 is the incoming stream dynamic head, S is the body frontal cross section (being perpendicular to the body axis) area, L is the reference length.

The calculation results are presented in Figs.9.4 and 9.5.

From Fig. 9.4., it is seen that the COSMOSFloWorks predictions of both Cn and Ct are excellent.

SU

FC

2

nn

ρ21

= ,

SLU

Mm

2

zz

ρ2

1=

,

Fig. 9.4. The longitudinal and lateral aerodynamic drag coefficients predicted with COSMOSFloWorks and measured in the experiments of Ref.5 versus the body tilting angle.

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

11.21.41.6

0 30 60 90 120 150 180Attack angle

(degree)

Ct , Cn

Ct experiment

Ct calculation

Cn experiment

Cn calculation


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As for the longitudinal aerodynamic torque coefficient (mz) prediction, it is also close to the experimental data of Ref.5, especially if we take into account the measurements error.

To illustrate the quantitative predictions with the corresponding flow patterns, the Mach number contours are presented in Figs. 9.6, 9.7, and 9.8. All of the flow patterns presented on the figures include both supersonic and subsonic flow regions. The bow shock consists of normal and oblique shock parts with the subsonic region downstream of the normal shock. In the head subsonic region the flow gradually accelerates up to a supersonic velocity and then further accelerates in the expansion fan of rarefaction waves. The subsonic wake region past the body can also be seen.

Fig. 9.5. The longitudinal aerodynamic torque coefficient predicted with COSMOSFloWorks and measured in the experiments (Ref.5) versus the body tilting angle.

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 30 60 90 120 150 180

Attack angle (degree)

mz

Experiment

Calculation

Fig. 9.6. Mach number contours at α = 0o

.


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As the forward part becomes sharper, the normal part of the bow shock and the

Fig. 9.7. Mach number contours at α = 60°.

Fig. 9.8. Mach number contours at α = 90°.


corresponding subsonic region downstream of it become smaller. In the presented pictures, the smallest nose shock (especially its subsonic region) is observed at a = 60o.


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10 Flow over a Heated Plate

Until now we have only considered flow in or around bodies with adiabatic walls. We will

now consider flows with other thermal boundary conditions.

The first example, is a uniform 2D flows with a laminar boundary layer on a heated flat

plate, see Fig. 10.1.. The incoming uniform air stream has a velocity of 1.5 m/s, a

temperature of 293.2 K, and a static pressure of 1 atm. Thus, the flow Reynolds number

defined on the incoming flow characteristics and on the plate length of 0.31 m is equal to

3.1⋅104, therefore the boundary layer beginning from the plate’s leading edge is laminar

(see Ref.6).

Then, let us consider the following three cases:

Case #1: the plate over its whole length (within the computational domain) is 10°C

warmer than the incoming air (303.2 K), both the hydrodynamic and the thermal boundary

layer begin at the plate's leading edge coinciding with the computational domain

boundary;

Case #2: the upstream half of the plate (i.e. at x ≤ 0.15 m) has a fluid temperature of 293.2

K, and the downstream half of the plate is 10°C warmer than the incoming air (303.2 K),

the hydrodynamic boundary layer begins at the plate's leading edge coinciding with the

computational boundary;

Case #3: plate temperature is the same as in case #1, the thermal boundary layer begins at

the inlet computational domain boundary, whereas the hydrodynamic boundary layer at

the inlet computational domain boundary has a non-zero thickness which is equal to that

in case #2 at the thermal boundary layer starting.


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Chapter Flow over a Heated Plate

The calculation goal is to predict the local coefficient of heat transfer from the wall to the

fluid, as well as the local skin-friction coefficient.

The SolidWorks model used for calculating the 2D flow over the heated flat plate with

COSMOSFloWorks is shown in Fig. 10.2.. The problem is solved as internal in order to

avoid the conflict situation when the external flow boundary with ambient temperature

conditions intersects the wall with a thermal boundary layer.

To avoid any influence of the upper wall on the flow near the heated lower wall, the ideal

wall boundary condition has been specified on the upper wall. To solve the internal

problem, the incoming fluid velocity is specified at the channel inlet, whereas the fluid

static pressure is specified at the channel exit. To specify the external flow features, the

incoming stream's turbulent intensity is set to 1% and the turbulent length is set to 0.01 m,

i.e., these turbulent values are similar to the default values for external flow problems.

Fig. 10.1. Laminar flow over a heated flat plate.

Heated plate

T = 303.2 K

Air flow

V = 1.5 m/s

T = 293.2 K

Computational domain


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The heat transfer coefficient h and the skin-friction coefficient Cf are COSMOSFloWorks

output flow parameters. The theoretical values for laminar flow boundary layer over a flat

plate, in accordance with Ref.6 can be determined from the following equations:

where

k is the thermal conductivity of the fluid,x is the distance along the wall from the start of the hydrodynamic boundary layer,

Nux is the Nusselt number defined on a heated wall as follows:

for a laminar boundary layer if it’s starting point coincides with the thermal

boundary layer starting point, and

Fig. 10.2. The SolidWorks model used for calculating the 2D flow over heated flat plate with COSMOSFloWorks.

Inlet velocity Ux

Ideal wall

Heated wall

Static pressure opening

x

kh xNu

= ,

2/1x

3/1x RePr332.0Nu ⋅=

2/1x

3/1 RePr332.0 ⋅


3 4/30 )x/x(1 −

=xNu


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for a laminar boundary layer if the thermal boundary layer begins at point x0 lying downstream of the hydrodynamic boundary layer starting point, in this case Nux is defined at x>x0 only;

where is the Prandtl number, µ is the fluid dynamic viscosity, Cp is the

fluid specific heat at constant pressure, is the Reynolds number

defined on x, ρ is the fluid density, and V is the fluid velocity;

at , i.e., with a laminar boundary layer.

As for the hydrodynamic boundary layer thickness δ needed for specification at the

computational domain boundary in case #3, in accordance with Ref.6, it has been

determined from the following equation: , so δ = 0.00575 m in this

case. For these calculations all fluid parameters are determined at the outer boundary of

the boundary layer.

The COSMOSFloWorks predictions of h and Cf performed at result resolution level 7, and

the theoretical curves calculated with the formulae presented above are shown in Figs.10.3

and 10.4. It is seen that the COSMOSFloWorks predictions of the heat transfer coefficient

and the skin-friction coefficient are in excellent agreement with the theoretical curves.

PrµCp

k----------=

RexρV x

µ-----------=

0,664

RefxC = 5Re 5 10x ≤ ⋅

0.54.64 / Rexxδ = ⋅


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Fig. 10.3. Heat transfer coefficient change along a heated plate in a laminar boundary layer: COSMOSFloWorks predictions compared to theory.

0

5

10

15

20

25

0 0.05 0.1 0.15 0.2 0.25 0.3 X (m)

h (W/m^2/K)

Case #1, theory

Case #1, calculation

Case #2, theory


Case #3, theory


Fig. 10.4. Skin-friction coefficient change along a heated plate in a laminar boundary layer: COSMOSFloWorks predictions compared to theory.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 0.05 0.1 0.15 0.2 0.25 0.3 X (m)

Cf

Cases #1 and #2,theoryCase #1, calculation


Case #3, theory

Case#3, calculation


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11 Convection and Radiation in an Annular Tube

We will now consider incompressible laminar flow in a portion of an annular tube, whose outer shell is a heat source having constant heat generation rate Q1 with a heat-insulated outer surface, and whose central body fully absorbs the heat generated by the tube’s outer shell (i.e. the negative heat generation rate Q2 is specified in the central body); see Fig. 11.1.. (The tube model is shown in Fig. 11.2.). We will assume that this tube is rather long, so the tube's L=1 m portion under consideration has fully developed fluid velocity and temperature profiles at the inlet, and, since the fluid properties are not temperature-dependent, the velocity profile also will not be temperature-dependent.

Fig. 11.1. Laminar flow in a heated annular tube.

X

Y

∅ 1.2m

∅ 0.4m

∅ 1.4 m

X

Y

1m

Q1 or T1

Q2 or T2

P = 1 atm

U, T

Fig. 11.2. A model created for calculating 3D flow within a heated annular tube using COSMOSFloWorks.


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Chapter Convection and Radiation in an Annular Tube

To validate the COSMOSFloWorks capability for solving conjugate heat transfer problems both with and without radiation, let us solve the following three problems: 1) a conjugate heat transfer problem with convection only, 2) radiation heat transfer only problem, and 3) a conjugate heat transfer problem with both convection and radiation.

In the first problem we specify Q2 = -Q1, so the convective heat fluxes at the tube inner and outer walls are constant along the tube. The corresponding laminar annular pipe flow's fully developed velocity and temperature profiles, according to Ref.6, are expressed analytically as follows:

u(r) = ,

T(r)= ,

where ϕ = ,

u is the fluid velocity,T is the fluid temperature,r is the radial coordinate,r1 and r2 are the tube outer shell’s inner radius and tube’s central body radius, respectively,

is the volume-average velocity, defined as the volume flow rate divided by the tube cross-section area,

q2 is the the heat flux from the fluid to the tube’s central body,

k is the fluid thermal conductivity,

T2 is the surface temperature of the central body.

The heat flux from the fluid to the tube's central body (negative, since the heat comes from the fluid to the solid) is equal to

−

−

−

⋅ 1

)/ln(

)/ln(1

21

2

2

2

1

2

2 rr

rr

r

r

r

rϕ

−

22

22 ln

r

rr

k

qT

1)/ln(/1

22

2

121

2

2

1 −

−

−

r

rrr

r

r

u

u

2rrr

T

=

∂∂

Lr

Q

⋅⋅ 2

2

2πq2 = k =


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Let Q1 = - Q2 = 107.235 W and ≈13.59 m/s (ϕ = -10 m/s), the fluid has the following

properties: k = 0.5 W/(m⋅K), Cp = 500 J/(kg⋅K), µ = 0.002 Pa⋅s, ρ = 0.1 kg/m3. Since the corresponding (defined on the equivalent tube diameter) Reynolds number Red ≈ 815 is rather low, the flow has to be laminar. We specify the corresponding velocity and temperature profiles as boundary conditions at the model inlet and as initial conditions, and Pout = 1 atm as the tube outlet boundary condition.

To reduce the computational domain, let us set Y=0 and X=0 flow symmetry planes (correspondingly, the specified Q1 and Q2 values are referred to the tube section's quarter lying in the computational domain). The calculation have been performed at result resolution level 7.

The fluid temperature profile predicted at 0.75 m from the tube model inlet is shown in Fig. 11.3. together with the theoretical curve.

It is seen that this prediction practically coincides with the theoretical curve.

Before solving the third problem coupling convection and radiation, let us determine the radiation heat fluxes between the tube's outer and inner walls under the previous problem's wall temperatures. In addition to holding the outer shell's temperature at 450 K and the central body's temperature at 300 K as the volume sources, let us specify the emissivity of ε1 = 0.95 for the outer shell and ε2 = 0.25 for the central body. To exclude any convection,

let us specify the liquid velocity of 0.001 m/s and thermal conductivity of 10-20 W/(m⋅K).

u

Fig. 11.3. Fluid temperature profiles across the tube in the case of convection only, predicted with COSMOSFloWorks and compared to the theoretical curve.

250

275

300

325

350

375

400

425

450

475

500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Y, m

T

TheoryCalculation


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Let J2 denote the radiation rate leaving the central body, and G2 denotes the radiation rate coming to the central body, therefore Q2r = J2 - G2 (the net radiation rate from the central body). In the same manner, let J1 denote the radiation rate leaving the outer shell's inner surface, and G1 denote the radiation rate coming to the outer shell's inner surface, therefore Q1r = J1 - G1 (the net radiation rate from the outer shell's inner surface). These radiation rates can be determined by solving the following equations:

J2 = A2σε2T24 + G2(1-ε2),

G2 = J1F1-2 ,

J1 = A1σε1T14 + G1(1-ε1),

G1 = J2F2-1 + J1F1-1 ,

where σ=5.669⋅10-8 W/m2⋅K4 is the Stefan-Boltzmann constant, F1-2, F2-1, F1-1 are these surfaces' radiation shape factors, under the assumption that the leaving and incident radiation fluxes are uniform over these surfaces, Ref.6 gives the following formulas:

F1-2=(1/X) - (1/π/X)arccos(B/A) - (1/2/Y)[(A2 + 4A - 4X2 + 4)1/2arccos(B/X/A) +

+ Barcsin(1/X)-πA/2],

F1-1=1-(1/X)+(2/π/X)arctan[2(X2-1)1/2/Y]-(Y/2/π/X)[(4X2+Y2)1/2/Y]arcsin[4(X2-1)+

+ (Y/X)2(X2-2)]/[Y2+4(X2-1)]-arcsin[(X2-2)/X2]+(π/2)[(4X2+Y2)1/2/Y-1]

F2-1 = F1-2⋅A1/A2, where X=r1/r2, Y=L/r2, A=X2+Y2-1, B=Y2-X2+1.

These net and leaving radiation rates (over the full tube section surface), both calculated by solving the equations analytically and predicted by COSMOSFloWorks at result resolution level 7, are presented in Table 2:.

Table 2: Radiation rates predicted with COSMOSFloWorks with comparison to the theoretical values.

It is seen that the prediction errors are quite small. To validate the COSMOSFloWorks capabilities on the third problem, which couples convection and radiation, let us add the theoretical net radiation rates, Q1 r and Q2 r scaled to the reduced computational domain, i.e., divided by 4, to the Q1 and Q2 values specified in the first problem. Let us specify Q1

Value, W Prediction error,%Q2 r -383.77 -388.30 1.2%J2 r 1728.35 1744.47 0.9%Q1 r 4003.68 3931.87 -1.8%J1 r 8552.98 8596.04 0.5%

Parameter Theory (Ref.6), WCOSMOSFloWorks predictions

4-58

= 1108.15 W and Q2 = -203.18 W, so theoretically we must obtain the same fluid temperature profile as in the first considered problem.


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The fluid temperature profile predicted at 0.75 m from the tube model inlet at the result resolution level 7 is shown in Fig. 11.4. together with the theoretical curve. It is seen that once again this prediction virtually coincides with the theoretical curve.

275

300

325

350

375

400

425

450

475

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Y, m

T, K

TheoryCalculation

Fig. 11.4. Fluid temperature profiles across the tube in the case coupling convection and radiation, predicted with COSMOSFloWorks and compared to the theoretical curve.


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12 Heat Transfer from a Pin-fin Heat Sink

Heat sinks play an important role in electronics cooling. Following the experimental work presented in Ref.19 and numerical study presented in Ref.21, let us consider heat transfer from an electrically heated thermofoil which is mounted flush on a plexiglass substrate, coated by an aluminum pin-fin heat sink with a 9×9 pin fin array, and placed in a closed plexiglass box. In order to create more uniform ambient conditions for this box, it is placed into another, bigger, plexiglass box and attached to the heat-insulated thick wall, see Figs. 12.1, 12.2. Following Ref.19, let us consider the vertical position of these boxes, as it is shown in Fig. 12.1. (c) (here, the gravity acts along the Y axis).

Fig. 12.1. The pin-fin heat sink nestled within two plexiglass boxes: lcp=Ls=25.4 mm, hcp=0.861 mm, Hp=5.5 mm, H =1.75 mm, S =1.5 mm, S = Ls/8, L=127 mm, H=41.3 mm, Hw=6.35 mm (from Ref.19).


b p ps


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Chapter Heat Transfer from a Pin-fin Heat Sink

The corresponding model used in the calculations is shown in Fig. 12.2.. In this model's coordinate system the gravitational acceleration vector is directed along the X axis. The computational domain envelopes the outer surface of the external box, and the Z=0 symmetry plane is used to reduce the required computer resources.

According to Ref.19, both the heat sink and the substrate are coated with a special black paint to provide a surface emissivity of 0.95 (the other plexiglass surfaces are also opaque, diffuse and gray, but have an emissivity of 0.83).

The maximum steady-state temperature Tmax of the thermofoil releasing the heat of known power Q was measured. The constant ambient temperature Ta was measured at the upper corner of the external box. As a result, the value of

Rja = (Tmax - Ta)/Q (12.1)

was determined at various Q (in the 0.1...1 W range).

The ambient temperature is not presented in Ref.19, so, proceeding from the suggestion that the external box in the experiment was placed in a room, we have varied the ambient temperature in the relevant range of 15...22ºC. Since Rja is governed by the temperature difference Tmax - Ta, (i.e. presents the two boxes’ thermal resistance), the ambient temperature range only effects the resistance calculations by 0.6ºC/W at Q = 1W, (i.e. by 1.4% of the experimentally determined Rja value that is 43ºC/W). As for the boundary conditions on the external box’s outer surface, we have specified a heat transfer coefficient of 5.6 W/m2 K estimated from Ref. 20 for the relevant wind-free conditions and an ambient temperature lying in the range of 15...22ºC (additional calculations have shown that the variation of the constant ambient temperature on this boundary yield nearly

Fig. 12.2. A model created for calculating the heat transfer from the pin-fin heat sink through the two nested boxes into the environment: (a) the internal (smaller) box with the heat sink; (b) the whole model.

a b

4-62

identical results). As a result, at Q = 1W (the results obtained at the other Q values are shown in Ref.21) and Ta =20ºC we have obtained Rja = 41ºC/W, i.e. only 5% lower than the experimental value.


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The flow streamlines visualized in Ref.19 using smoke and obtained in the calculations are shown in Fig. 12.3..

Fig. 12.3. Flow streamlines visualized by smoke in the Ref.19 experiments (left) and obtained in the calculations (colored in accordance with the flow velocity values) (right).


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Chapter Heat Transfer from a Pin-fin Heat Sink


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13 Unsteady Heat Conduction in a Solid

Until now, we have studied various flow problems including those with heat flow from the model walls to the fluid, but we have not considered heat conduction in solids (i.e., a conjugate heat transfer). To validate this capability, let us consider unsteady heat conduction in a solid. To compare the COSMOSFloWorks predictions with the analytical solution (Ref.6), we will solve a one-dimensional problem.

A warm solid rod having the specified initial temperature and the heat-insulated side surface suddenly becomes and stays cold (at a constant temperature of T=300 K) at both ends (see Fig. 13.1.). The rod inner temperature evolution is studied. The constant initial temperature distribution along the rod is considered: Tinitial (x)=350K.

The problem is described by the following differential equation:

where ρ, C, and k are the solid material density, specific heat, and thermal conductivity, respectively, and τ is the time, with the following boundary condition: T=T0 at x = 0 and at x = L.

In the general case, i.e., at an arbitrary initial condition, the problem has the following solution:

where coefficients Cn are determined from the initial conditions (see Ref.6).

With the uniform initial temperature profile, according to the initial and boundary conditions, the problem has the following solution:

Fig. 13.1. A warm solid rod cooling down from an initial temperature to the temperature at the ends of the rod.

Tinitial = 350 KT = 300 K

T = 300 K

XL

τρ

∂∂=

∂∂ T

k

C

x

T2

2,

∑∞

=

−+=1

)/()/(0 sin

2

n

CkLnn L

xneCTT

πρτπ ,


[ ]∑∞

=

−+=1

)/(/ )sin(14

503002

n

CkLn

L

xne

nT

ππ

ρτπ (K).


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Chapter Unsteady Heat Conduction in a Solid

To perform the time-dependent analysis with COSMOSFloWorks, a SolidWorks model representing a solid parallelepiped with dimensions 1×0.2×0.1 m has been created (see Fig. 13.2.).

The evolution of maximum rod temperature, predicted with COSMOSFloWorks and compared with theory, is presented in Fig. 13.3.. The COSMOSFloWorks prediction has been performed at result resolution level 5. One can see that it coincide with the theoretical curve.

The temperature profiles along the rod at different time moments, predicted by COSMOSFloWorks, are compared to theory and presented in Fig. 13.4.. One can see that the COSMOSFloWorks predictions are very close to the theoretical profiles. The

Fig. 13.2. The SolidWorks model used for calculating heat conduction in a solid rod with COSMOSFloWorks (the computational domain envelopes the rod).

Fig. 13.3. Evolution of the maximum rod temperature, predicted with COSMOSFloWorks and compared to theory.

305

315

325

335

345

355

0 2000 4000 6000 8000 10000Physical time (s)

Tem

per

atu

re (

K)

Theory

Calculation

4-66

maximum prediction error not exceeding 2K occurs at the ends of the rod and is likely caused by calculation error in the theoretical profile due to the truncation of Fourier series.


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Fig. 13.4. Evolution of the temperature distribution along the rod, predicted with COSMOSFloWorks and compared to theory.

300

305

310

315

320

325

0 0.2 0.4 0.6 0.8 1 X (m)

Tem

per

atu

re (

K)

t=5000s/theory

t=5000s/calculation

t=10000s/theory

t=10000s/calculation


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Chapter Unsteady Heat Conduction in a Solid


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14 Tube with Hot Laminar Flow and Outer Heat Transfer

Let us now consider an incompressible laminar flow of hot fluid through an externally cooled circular tube (Fig. 14.1.). The fluid flow has fully developed velocity and temperature profiles at the tube inlet, whereas the heat transfer conditions specified at the tube outer surface surrounded by a cooling medium sustain the self-consistent fluid temperature profile throughout the tube.

In accordance with Ref.6, a laminar tube flow with a fully developed velocity profile has a self-consistent fully developed temperature profile if the following two conditions are satisfied: the fluid's properties are temperature-independent and the heat flux from the tube inner surface to the fluid (or vise versa) is constant along the tube. These conditions provide the following fully developed tube flow temperature profile:

T(r, z) = T(r=0, z=zinlet) - ,

where

T is the fluid temperature,

r is a radial coordinate (r = 0 corresponds to the tube axis, r = Ri corresponds to the tube inner surface, i.e., Ri is the tube inner radius),

z is an axial coordinate (z = zinlet corresponds to the tube inlet),

qw is a constant heat flux from the fluid to the tube inner surface,

k is the fluid thermal conductivity,

ρ is the fluid density,

Fig. 14.1. Laminar flow in a tube cooled externally.

Liquid

Laminar flow

Polystyrene Te(z)

αe = const

r

( )2 441

4w inletw i

i i p max i

q z zq R r r

k R R C u Rρ

⋅ − − +


Cp is the fluid specific heat under constant pressure,


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Chapter Tube with Hot Laminar Flow and Outer Heat Transfer

umax is the maximum fluid velocity of the fully developed velocity profile

= umax .

Since the tube under consideration has no heat sinks and is cooled by surrounding fluid medium, let us assume that the fluid medium surrounding the tube has certain fixed temperature Te, and the heat transfer between this medium and the tube outer surface is

determined by a specified constant heat transfer coefficient αe.

By assuming a constant thermal conductivity of the tube material, ks, specifying an

arbitrary αe, and omitting intermediate expressions, we can obtain the following expression for Te:

,

where Ro is the tube outer radius.

In the validation example under consideration (Fig. 14.2.) the following tube and fluid characteristics have been specified: Ri = 0.05 m, Ro = 0.07 m, z - zinlet = 0.1 m, the tube material is polystyrene with thermal conductivity ks = 0.082 W/(m⋅K), umax = 0.002 m/s,

T(r=0, z=zi) = 363 K, qw = 147.56 W/m2, k = 0.3 W/(m⋅K), Cp = 1000 J/(kg⋅K), fluid

dynamic viscosity µ = 0.001 Pa⋅s, ρ = 1000 kg/m3 (these fluid properties provide a laminar flow condition since the tube flow Reynolds number based on the tube diameter is equal to Red = 100). The T(r,zinlet) and u(r) profiles at the tube inlet, the Te(z) distribution

along the tube, αe = 5 W/(m2⋅K), and tube outlet static pressure Pout = 1 atm have been

specified as the boundary conditions.

The inlet flow velocity and temperature profiles have been specified as the initial conditions along the tube.

To reduce the computational domain, the calculations have been performed with the Y=0 and X=0 flow symmetry planes. The calculations have been performed at result resolution level 7.

The fluid and solid temperature profiles predicted at z = 0 are shown in Fig. 14.3. together with the theoretical curve. It is seen that the prediction practically coincides with the theoretical curve (the prediction error does not exceed 0.4%).

( )u r 1

2rRi

−

( ) +−+−===

oRiR

skoRekiRwqinletzzrTzeT ln1143),0( α iRmaxupC

inletzzwq

ρ

−⋅

+4


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Outlet static pressure opening

Computational domainInlet velocity

opening

Sketch line for temperature profile determination

Fig. 14.2. The model used for calculating the 3D flow and the conjugate heat transfer in the tube with COSMOSFloWorks.

Fig. 14.3. Fluid and solid temperature profiles across the tube, predicted with COSMOSFloWorks and compared with the theoretical curve.

290

300

310

320

330

340

350

360

370

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 R, m

T, K

Theory liquid

Theory solid

Calculation


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Chapter Tube with Hot Laminar Flow and Outer Heat Transfer


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15 Flow over a Heated Cylinder

Let us now return to the earlier validation example of incompressible flow over a cylinder and modify it by specifying a heat generation source inside the cylinder (see Fig. 15.1.). The cylinder is placed in an incoming air stream and will acquire certain temperature depending on the heat source power and the air stream velocity and temperature.

Based on experimental data for the average coefficient of heat transfer from a heated circular cylinder to air flowing over it (see Ref.6), the corresponding Nusselt number can be determined from the following formula:

where constants C and n are taken from the following table:

Here, the Nusselt number, NuD = (h×D)/k (where h is the heat transfer coefficient averaged over the cylinder, and k is fluid thermal conductivity), the Reynolds number, ReD = (U×D)/µ (where U is the incoming stream velocity, and µ is fluid dynamic viscosity), and the Prandtl number, Pr=µ×Cp/k (where µ is fluid dynamic viscosity, Cp is

ReD C n

0.4 - 4 0.989 0.330

4 - 40 0.911 0.385

40 - 4000 0.683 0.466

4000 - 40000 0.193 0.618

40000 - 400000 0.0266 0.805

Fig. 15.1. 2D flow over a heated cylinder.

External air flow

Heat source q

Y

X

( ) 31PrRe ⋅⋅= nDD CNu ,


fluid specific heat at constant pressure, and k is fluid thermal conductivity) are based on the cylinder diameter D and on the fluid properties taken at the near-wall flow layer. According to Ref.6, Pr = 0.72 for the entire range of ReD.


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Chapter Flow over a Heated Cylinder

To validate the COSMOSFloWorks predictions, the air properties have been specified to provide Pr = 0.72: k = 0.0251375 W/(m×K), µ = 1.8×10-5 Pa×s, specific heat at constant pressure Cp = 1005.5 J/(kg×K). Then, the incoming stream velocity, U, has been specified to obtain ReD = 1, 10, 100, 103, 104, 5×104, 105, 2×105, and 3×105 for a cylinder diameter of D = 0.1 m (see Table 3:).

This validation approach consists of specifying the heat generation source inside the cylinder with a power determined from the desired steady-state cylinder temperature and the average heat transfer coefficient, h = (NuD×k)/D. NuD is determined from the specified ReD using the empirical formula presented above. The final cylinder surface temperature, that is also required for specifying the heat source power Q (see Table 3:) is assumed to be 10°C higher than the incoming air temperature. The initial cylinder temperature and the incoming air temperature are equal to 293.15 K. The cylinder material is aluminum. Here, the heat conduction in the solid is calculated simultaneously with the flow calculation, i.e., the conjugate heat transfer problem is solved.

As a result of the calculation, the cylinder surface has acquired a steady-state temperature differing from the theoretical one corresponding to the heat generation source specified inside the cylinder. Multiplying the theoretical value of the Nusselt number by the ratio of the obtained temperature difference (between the incoming air temperature and the cylinder surface temperature) to the specified temperature difference, we have determined the predicted Nusselt number versus the specified Reynolds number. The values obtained by solving the steady-state and time dependent problems at result resolution level 5 are presented in Fig. 15.2. together with the experimental data taken from Ref.6.

Table 3: The COSMOSFloWorks specifications of U and Q for the problem under consideration.

From Fig. 15.2., it is seen that the predictions made with COSMOSFloWorks, both in the time-dependent approach and in the steady-state one, are excellent within the whole ReD range under consideration.

1 1.5× 10-4

0.007

10 1.5× 10-3

0.016

102

0.015 0.041

103

0.15 0.121

104

1.5 0.405

105

15 1.994

R e D U , m /s Q ,W


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Fig. 15.2. Nusselt number for air flow over a heated cylinder: COSMOSFloWorks predictions and the experimental data taken from Ref.6.

0.1

1

10

100

1000

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 Re D

NuD

Ca lcula tion,steady-sta teCa lcula tion,time-dependent


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Chapter Flow over a Heated Cylinder


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16 Natural Convection in a Square Cavity

In the previous validation examples we have considered forced convection, i.e., heat transfer between a wall and a fluid while the fluid flow is caused by some driving force other than temperature gradient, so that its characteristics are specified via the flow parameters at model openings (internal problems) or at far-field boundaries (external problems). Let us now consider a heat transfer due to a heat-induced natural convection within a closed cavity.

Here we will consider a 2D square cavity with a steady-state natural convection, for which a highly-accurate numerical solution has been proposed in Ref.10 and used as a benchmark for about 40 computer codes in Ref.11, besides it well agrees with the semi-empirical formula proposed in Ref.12 for rectangular cavities. This cavity's configuration and imposed boundary conditions, as well as the used coordinate system, are presented in Fig. 16.1.. Here, the left and right vertical walls are held at the constant temperatures of T1 = 305 K and T2 = 295 K, accordingly, whereas the upper and bottom walls are adiabatic. The cavity is filled with air.

The square cavity's side dimension, L, is varied within the range of 0.0111...0.111 m in

order to vary the cavity's Rayleigh number within the range of 103…106. Rayleigh number descibes the characteristics of the natural convection inside the cavity and is defined as follows:

,

where is the volume expansion coefficient of air,

g is the gravitational acceleration,

Fig. 16.1. An enclosed 2D square cavity with natural convection.

Adiabatic walls

L

305 K

295 K

Y

g

X

µρβ

k

TLCgRa p ∆

=32

1Tβ =


Cp is the air's specific heat at constant pressure,

∆T=T1 - T2 = 10 K is the temperature difference between the walls,


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Chapter Natural Convection in a Square Cavity

k is the thermal conductivity of air,

µ is the dynamic viscosity of air.

The cavity's model is shown in Fig. 16.2..

Due to gravity and different temperatures of the cavity's vertical walls, a steady-state natural convection flow (vortex) with a vertical temperature stratification forms inside the

cavity. The Ra = 105 flow's prediction performed with COSMOSFloWorks is shown in Fig. 16.3..

Fig. 16.2. The model created for calculating the 2D natural convection flow in the 2D square cavity using COSMOSFloWorks.

4-78

Fig. 16.3. The temperature, X-velocity, Y-velocity, the velocity vectors, and the streamlines, predicted by COSMOSFloWorks in the square cavity at Ra = 105.


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A quantitative comparison of the COSMOSFloWorks predictions performed at result resolution level 8 with References.10, 11 (computational benchmark) and 12 (semi-empirical formula) for different Ra values is presented in Figs.16.4 - 16.6. The Nusselt number averaged over the cavity's hot vertical wall (evidently, the same value

must be obtained over the cavity's cold vertical wall) , where

qw av is the heat flux from the wall to the fluid, averaged over the wall, is considered in

Fig. 16.4..

Here, the dash line presents the Ref.12 semi-empirical formula

,

where D is the distance between the vertical walls and L is the cavity height (D=L in the case under consideration). One can see that the COSMOSFloWorks predictions practically

coincide with the benchmark at Ra ≤ 105 and are close to the semi-empirical data.

/( )av wavNu q L T k= ⋅ ∆ ⋅

1/ 4 1/ 40.28 ( / )avNu Ra L D −= ⋅

Fig. 16.4. The average sidewall Nusselt number vs. the Rayleigh number.

0

1

2

3

4

5

6

7

8

9

10

1.E+03 1.E+04 1.E+05 1.E+06Ra

Nuav

Refs.10, 11Ref.12

Calculation


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The dimensionless velocities of the natural convection flow in the X and Y directions,

and (which are maximum along the cavity's

mid-planes, i.e., along the vertical mid-plane and along the horizontal

mid-plane) are considered in Fig. 16.5.. The dimensionless coordinates, and

, of these maximums' locations (i.e., for and for ) are

presented in Fig. 16.6.. One can see that the COSMOSFloWorks predictions of the natural

convection flow's local parameters are fairly close to the benchmark data at Ra ≤ 105.

pU L CU

k

ρ⋅ ⋅ ⋅= pV L C

Vk

ρ⋅ ⋅ ⋅=

maxU maxV

L

xx =

L

yy = y maxU x maxV

Fig. 16.5. Dimensionless maximum velocities vs. Rayleigh number.

1

10

100

1000

1.E+03 1.E+04 1.E+05 1.E+06 Ra

Dim

ensi

on

less

U

max

, V

max

Vmax, Refs.10, 11

Vmax, calculation

Umax, Refs.10, 11

Umax, calculation


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Fig. 16.6. Dimensionless coordinates of the maximum velocities' locations.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.E+03 1.E+04 1.E+05 1.E+06 Ra

Dim

ensi

on

less

X V

max

, Y

Um

ax

Y Umax, Refs.10, 11

Y Umax, calculation

X Vmax, Refs.10, 11

X V max,


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17 Particles Trajectories in Uniform Flows

Let us now consider the COSMOSFloWorks capability to predict particles trajectories in a gas flow (i.e. two-phase flow of fluid + liquid droplets or solid particles).

In accordance with the particles motion model accepted in COSMOSFloWorks, particle trajectories are calculated after completing a fluid flow calculation (which can be either steady or time-dependent). That is, the particles mass and volume flow rates are assumed substantially lower than those of the fluid stream, so that the influence of particles’ motions and temperatures on the fluid flow parameters is negligible, and motion of the particles obeys the following equation:

where m is the particle mass, t is time, Vp and Vf are the particle and fluid velocities (vectors), accordingly, ρf is the fluid density, Cd is the particle drag coefficient, A is the particle frontal surface area, and Fg is the gravitational force.

Particles are treated as non-rotating spheres of constant mass and specified (solid or liquid) material, whose drag coefficient is determined from Henderson’s semi-empirical formula (Ref.7). At very low velocity of particles with respect to carrier fluid (i.e., at the relative velocity’s Mach number M → 0) this formula becomes

where Reynolds number is defined as

d is the diameter of particles, and µ is the fluid dynamic viscosity.

To validate COSMOSFloWorks, let us consider three cases of injecting a particle perpendicularly into an incoming uniform flow, Fig. 17.1.. Since both the fluid flow and the particle motion in these cases are 2D (planar), we will solve a 2D (i.e. in the XY-plane) flow problem.

gd

pfpffp FACVVVV

dt

dVm +

−⋅−−=

2

)(ρ ,

380.Re480Re0301

124.

Re

24Cd +

+⋅++=

..

µρ dVV pff −

=Re ,

Uniform fluid flow

Particle injection


Fig. 17.1. Injection of a particle into a uniform fluid flow.


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Chapter Particles Trajectories in Uniform Flows

Due to the same reason as in the previous validation examples with flow over flat plates, we will solve this validation as an internal problem. The corresponding SolidWorks model is shown in Fig. 17.2.. Both of the walls are ideal, the channel has length of 0.233 m and height of 0.12 m, all the walls have thickness of 0.01 m. We specify the uniform fluid velocity Vinlet, the fluid temperature of 293.2 K, and the default values of turbulent flow parameters with the laminar boundary layer at the channel inlet, and the static pressure of 1 atm at the channel outlet. All the fluid flow calculations are performed at a result resolution level of 5.

To validate calculations of particles trajectories by comparing them with available analytical solutions of the particle motion equation, we consider the following three cases:

a) the low maximum Reynolds number of Re max = 0.1 (air flow with Vinlet = 0.002 m/s, gold particles of d = 0.5 mm, injected at the velocity of 0.002 m/s perpendicularly to the wall),

b) the high maximum Reynolds number of Re max = 105 (water flow with Vinlet = 10 m/s, iron particles of d = 1 cm, injected at the velocities of 1, 2, 3 m/s perpendicularly to the wall),

c) a particle trajectory in the Y-directed gravitational field (gravitational acceleration gy = -9.8 m/s2, air flow with Vinlet = 0.6 m/s, an iron particle of d = 1 cm, injected at the 1.34 m/s velocity at the angle of 63.44o with the wall).

In the first case, due to small Re values, the particle drag coefficient is close to Cd=24/Re (i.e., obeys the Stokes law). Then, neglecting gravity, we obtain the following analytical solution for the particle trajectory:

Fig. 17.2. The model.

Inlet

Outlet

Ideal Wall

Origin and particle injection point

)18

exp()(18

)( 20

2

0t

dVV

dtVXtX

pfxtpx

pfxt ρ

µµρ

−⋅−+⋅+===

,

)18

exp()(18

)( 20

2

0t

dVV

dtVYtY

pfytpy

pfyt ρ

µµρ

−⋅−+⋅+===

,


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where Vfx, Vpx, Vfy, Vpy are the X- and Y-components of the fluid and particle velocities, accordingly, ρp is the particle material density. The COSMOSFloWorks calculation and the analytical solution are shown in Fig. 17.3.. It is seen that they are very close to one another. Special calculations have shown that the difference is due to the CD assumptions only.

In the second case, due to high Re values, the particle drag coefficient is close to Cd=0.38. Then, neglecting the gravity, we obtain the following analytical solution for the particle trajectory:

The COSMOSFloWorks calculations and the analytical solutions for three particle injection velocities, Vpy(t=0) = 1, 2, 3 m/s, are shown in Fig. 17.4.. It is seen that the COSMOSFloWorks calculations coincide with the analytical solutions. Special calculations have shown that the difference is due to the CD assumptions only.

In the third case, the particle trajectory is governed by the action of the gravitational force only, the particle drag coefficient is very close to zero, so the analytical solution is:

Fig. 17.3. Particle trajectories in a uniform fluid flow at Re max = 0.1, predicted by COSMOSFloWorks and obtained from the analytical solution.

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.00 0.05 0.10 0.15 0.20 0.25 X (m)

Y (m)

AnalyticalsolutionCalculation

)285.0

1ln()(285.0

)(00

td

VVd

tVYtYp

fytpyp

fyt ρρ

ρρ

+⋅−+⋅+===

,

).

ln()(.

)( td

28501VV

2850

dtVXtX

pfx0tpx

pfx0t ρ

ρρ

ρ+⋅−+⋅+=

==.

2


0

00 2

1

−⋅++= =

==px

ttpyt V

XXy

gtVYY .


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Chapter Particles Trajectories in Uniform Flows

The COSMOSFloWorks calculation and the analytical solution for this case are presented in Fig. 17.5.. It is seen that the COSMOSFloWorks calculation coincides with the analytical solution.

I

Fig. 17.4. Particle trajectories in a uniform fluid flow at Re max = 105, predicted by COSMOSFloWorks and obtained from the analytical solution.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.00 0.05 0.10 0.15 0.20 X (m)

Y (m)

Vp = 1 m/s,analytical solution

Vp = 1 m/s,Calculation





Fig. 17.5. Particle trajectories in the Y-directed gravity, predicted by COSMOSFloWorks and obtained from the analytical solution.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.00 0.03 0.06 0.09 0.12 0.15 X (m)

Y (m)

Calculation

Theory


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18 Porous Screen in a Non-uniform Stream

Let us now validate the COSMOSFloWorks capability to calculate fluid flows through porous media.

Here, following Ref.2, we consider a plane cold air flow between two parallel plates, through a porous screen installed between them, see Fig. 18.1.. At the channel inlet the air stream velocity profile is step-shaped (specified). The porous screen (gauze) levels this profile to a more uniform profile. This effect depends on the screen drag, see Ref.2.

The SolidWorks model used for calculating the 2D (in XY-plane) flow is shown in Fig. 18.2.. The channel has height of 0.15 m, the inlet (upstream of the porous screen) part of the 0.3 m length, the porous screen of the 0.01 m thickness, and the outlet (downstream of the porous screen) part of the 0.35 m length. All the walls have thickness of 0.01 m.

Air

L

Porous screen

X

Y

Fig. 18.1. Leveling effect of a porous screen (gauze) on a non-uniform stream.

Fig. 18.2. The SolidWorks model used for calculating the 2D flow between two parallel plates and through the porous screen with COSMOSFloWorks.


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Chapter Porous Screen in a Non-uniform Stream

Following Ref.2, we consider porous screens (gauzes) of different drag, ζ:

ζ = 0.95, 1.2, 2.8, and 4.1, defined as:

where ∆P is the pressure difference between the screen sides, ρV2/2 is the dynamic pressure (head) of the incoming stream.

Since in COSMOSFloWorks a porous medium’s resistance to flow is characterized by parameter k = - gradP/ρV, then for the porous screens k = V⋅ζ /(2⋅L), where V is the fluid velocity, L is the porous screen thickness. In COSMOSFloWorks, this form of a porous medium’s resistance to flow is specified as k = (A⋅V+B)/ρ, so A = ρ⋅ζ /(2⋅L), B = 0 for the porous screens under consideration. Therefore, taking L = 0.01 m and ρ = 1.2 kg/m3 into account, we specify A = 57, 72, 168, and 246 kg/m-4 for the porous screens under consideration. In accordance with the screens’ nature, their permeability is specified as isotropic.

According to the experiments presented in Ref.2, the step-shaped velocity profiles V(Y) presented in Fig. 18.3. have been specified at the model inlet. The static pressure of 1 atm has been specified at the model outlet.

The air flow dynamic pressure profiles at the 0.3 m distance downstream from the porous screens, both predicted by COSMOSFloWorks at result resolution level 5 and measured in the Ref.2 experiments, are presented in Fig. 18.4. for the ζ = 0 case (i.e., without screen) and Figs.18.5-18.8 for the porous screens of different ζ .

22

V

P

ρζ ∆= ,

Fig. 18.3. Inlet velocity profiles.

0

5

10

15

20

25

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Y (m)

V (m/s)

ζ=0, 0.95ζ=0, 0.95ζ=0, 0.95ζ=0, 0.95

ζ=1.2ζ=1.2ζ=1.2ζ=1.2

ζ=2.8ζ=2.8ζ=2.8ζ=2.8

ζ=4.1ζ=4.1ζ=4.1ζ=4.1


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It is seen that the COSMOSFloWorks predictions agree well, both qualitatively and quantitatively, with the experimental data both in absence of a screen and for all the porous screens (gauzes) under consideration, demonstrating the leveling effect of the gauze screens on the step-shaped incoming streams. The prediction error in the dynamic pressure maximum does not exceed 30%.

Fig. 18.4. The dynamic pressure profiles at ζ = 0, predicted by COSMOSFloWorks and compared to the Ref.2 experiments.

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Fig. 18.5. The dynamic pressure profiles at ζ = 0.95, predicted by COSMOSFloWorks and compared to the Ref.2 experiments.

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19 Lid-driven Flows in Triangular and Trapezoidal Cavities

In the previous validation examples we have considered COSMOSFloWorks predictions of pressure-driven incompressible flows in various channels with stationary (motionless) walls. Since COSMOSFloWorks allows the motion of walls, let us now see how COSMOSFloWorks predicts lid-driven (i.e., shear-driven) 2D recirculating flows in closed 2D triangular and trapezoidal cavities with one or two moving walls (lids) in comparison with the calculations performed in Refs.15 and 16.

These two cavities are shown in Fig. 19.1.. The triangular cavity has a moving top wall, the trapezoidal cavity has a moving top wall also, whereas its bottom wall is considered in two versions: as motionless and as moving at the top wall velocity. The no-slip conditions are specified on all the walls.

Shown in Refs. 15 and 16, the shear-driven recirculating flows in these cavities are fully governed by their Reynolds numbers Re = ρ·Uwall·h/µ, where ρ is the fluid density, µ is the fluid dynamic viscosity, Uwall is the moving wall velocity, h is the cavity height. So, we can specify the height of the triangular cavity h = 4 m, the height of the trapezoidal cavity h = 1 m, Uwall = 1 m/s for all cases under consideration, the fluid density ρ = 1 kg/m3, the fluid dynamic viscosity µ=0.005 Pa·s in the triangular cavity produces a Re = 800, and µ= 0.01, 0.0025, 0.001 Pa·s in the trapezoidal cavity produces a Re = 100, 400, 1000, respectively.

Fig. 19.1. The 2D triangular (left) and trapezoidal (right) cavities with the moving walls (the motionless walls are shown with dashes).

2

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Chapter Lid-driven Flows in Triangular and Trapezoidal Cavities

The cavities’ models are shown in Fig. 19.2.. The COSMOSFloWorks calculation of flow in the triangular cavity has been performed on the 48×96 computational mesh. The results in comparison with those from Ref.15 are presented in Fig. 19.3. (streamlines) and in Fig. 19.4. (the fluid velocity X-component along the central vertical bisector shown by a green line in Fig. 19.2.). A good agreement of these calculations is clearly seen.

Fig. 19.2. The models for calculating the lid-driven 2D flows in the triangular (left) and trapezoidal (right) cavities with COSMOSFloWorks.

Fig. 19.3. The flow trajectories in the triangular cavity, calculated by COSMOSFloWorks (right) and compared to the Ref.15 calculation (left).


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The COSMOSFloWorks calculations of flows in the trapezoidal cavity with one and two moving walls at different Re values have been performed with the 100×50 computational mesh. Their results in comparison with those from Ref.16 are presented in Fig. 19.5.-19.10 (streamlines) and in Fig. 19.11. (the fluid velocity X-component along the central vertical bisector shown by a green line in Fig. 19.2.). A good agreement of these calculations is seen.

Fig. 19.5. The flow streamlines in the trapezoidal cavity with a top only moving wall at Re = 100, calculated by COSMOSFloWorks (right) and compared to the Ref.16 calculation (left).

Fig. 19.4. The triangular cavity’s flow velocity X-component along the central vertical bisector, calculated by COSMOSFloWorks (red line) and compared to the Ref.15 calculation (black line with circlets).

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Fig. 19.8. The flow streamlines in the trapezoidal cavity with two moving walls at Re = 100, calculated by COSMOSFloWorks (right) and compared to the Ref.16 calculation (left).


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Fig. 19.11. The flow velocity X-component along the central vertical bisector in the trapezoidal cavity with two moving walls at Re = 400, calculated by COSMOSFloWorks (red line) and compared to the Ref.16 calculation (black line).

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20 Flow in a Cylindrical Vessel with a Rotating Cover

In the previous COSMOSFloWorks validation example we considered a lid-driven 2D recirculating flow in a 2D cavity. Since COSMOSFloWorks allows to consider rotating walls, let us now see how COSMOSFloWorks predicts a 3D recirculating flow in a cylindrical vessel closed by a rotating cover (see Fig. 20.1.) in comparison with the experimental data presented in Ref.17 (also in Ref.18). This vessel of R = h = 0.144 m dimensions is filled with a glycerol/water mixture. The upper cover rotates at the angular velocity of Ω. The other walls of this cavity are motionless. The default no-slip boundary condition is specified for all walls.

Due to the cover rotation, a shear-driven recirculating flow forms in this vessel. Such flows are governed by the Reynolds number Re = ρ·Ω·R2/µ, where ρ is the fluid density, µ is the fluid dynamic viscosity, Ω is the angular velocity of the rotating cover, R is the radius of the rotating cover. In the case under consideration the 70/30% glycerol/water mixture has ρ = 1180 kg/m3, µ = 0.02208 Pa·s, the cover rotates at Ω = 15.51 rpm, so Re = 1800.

The COSMOSFloWorks calculation has been performed on the 82×41×82 computational mesh. The formed flow pattern (toroidal vortex) obtained in this calculation is shown in Fig. 20.2. using the flow velocity vectors projected onto the XY-plane. The tangential and radial components of the calculated flow velocity along four vertical lines arranged in the XY-plane at different distances from the vessel axis in comparison with the Ref.17 experimental data are presented in Figs.20.3-7 in the dimensionless form (the Y-coordinate is divided by R, the velocity components are divided by Ω·R). There is good

Fig. 20.1. The cylindrical vessel with the a rotating cover.

rotating coverΩ

R

h


agreement with the calculation results and the experimental data shown.


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Chapter Flow in a Cylindrical Vessel with a Rotating Cover

Fig. 20.2. The vessel's flow velocity vectors projected on the

Fig. 20.3. The vessel's flow tangential and radial velocity components along the X = 0.6 vertical, calculated by COSMOSFloWorks (red) and compared to the Ref.17 experimental data.


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Chapter Flow in a Cylindrical Vessel with a Rotating Cover



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21 Flow in an Impeller

In the previous validation example we had considered COSMOSFloWorks predictions of a recirculating flow in a cylindrical vessel with a rotating cover. The calculation had been performed in the coordinate system related to the stationary object (vessel), and a part (cover) rotated in this coordinate system. Let us now validate the COSMOSFloWorks ability to perform calculations in a rotating coordinate system related to a rotating solid. Following Ref.22, we will consider the flow of water in a 9-bladed centrifugal impeller having blades tilted at a constant 60º angle with respect to the intersecting radii and extending out from the 320 mm inner diameter to the 800 mm outer diameter (see Fig. 21.1.). The water in this impeller flows from its center to its periphery. To compare the calculation with the experimental data presented in Ref.22, the impeller's angular velocity of 32 rpm and volume flow rate of 0.00926 m3/s are specified.

Since the impeller's inlet geometry and disk extension serving as the impeller's vaneless diffuser have no exact descriptions in Ref.22, to perform the validating calculation we arbitrarily specified the annular inlet as 80 mm in diameter with an uniform inlet velocity profile perpendicular to the surface in the stationary coordinate system.The impeller's disks external end was specified as 1.2 m diameter, as shown in Fig. 21.2..

Fig. 21.1. The impeller's blades geometry.

Fig. 21.2. The model used for calculating the 3D flow in the impeller.


The above-mentioned volume flow rate at the annular inlet and the potential pressure of 1 atm at the annular outlet are specified as the problem's flow boundary conditions.


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Chapter Flow in an Impeller

The COSMOSFloWorks 3D flow calculation is performed on the computational mesh using the result resolution level of 5 and the minimum wall thickness of 2 mm (since the blades have constant thickness). To further capture the curvature of the blades a local initial mesh was also used in the area from the annular inlet to the blades' periphery. As a result, the computational mesh has a total number of about 1,000,000 cells.

Following Ref.22, let us compare the passage-wise flow velocities (ws, see their definition in Fig. 21.3., β = 60°) along several radial lines passing through the channels between the blades (lines g, j, m, p in Fig. 21.4.) at the mid-height between the impeller's disks.

The passage-wise flow velocities divided by Ω⋅r2, where Ω is the impeller's angular velocity and r2 = 400 mm is the impeller's outer radius, which were measured in Ref.22 and obtained in the performed COSMOSFloWorks calculations, are shown in Fig. 21.5., 6, 7, and 8. In these figures, the distance along the radial lines is divided by the line's length. The COSMOSFloWorks results are presented in each of these figures by the curve obtained by averaging the corresponding nine curves in all the nine flow passages between the impeller blades. The calculated passage-wise flow velocity's cut plot covering the whole computational domain at the mid-height between the impeller's disks is shown in

Fig. 21.3. Definition of the passage-wise flow velocity.

Fig. 21.4. Definition of the reference radial lines along which the passage-wise flow velocity was measured in Ref.22 (from a to s in the alphabetical order).

4-104

Fig. 21.9.. Here, the g, j, m, p radial lines in each of the impeller's flow passages are shown. A good agreement of these calculation results with the experimental data is seen.


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Fig. 21.5. The impeller's passage-wise flow velocity along the g (see Fig. 21.4.) radial line, calculated by COSMOSFloWorks and compared to the experimental data.

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Fig. 21.6. The impeller's passage-wise flow velocity along the j (see Fig. 21.4.) radial line, calculated by COSMOSFloWorks and compared to the experimental data.

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Fig. 21.7. The impeller's passage-wise flow velocity along the m (see Fig. 21.4.) radial line, calculated by COSMOSFloWorks and compared to the experimental data.

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Fig. 21.8. The impeller's passage-wise flow velocity along the p (see Fig. 21.4.) radial line, calculated by COSMOSFloWorks and compared to the experimental data.

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Fig. 21.9. A cut plot of the impeller's passage-wise flow velocity calculated by COSMOSFloWorks.


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22 Cavitation on a hydrofoil

When the local pressure at some point in the liquid drops below the liquid's vapour pressure at the local temperature, the liquid undergoes phase transition and form cavities filled with the liquid's vapor with an addition of gas that has been dissolved in the liquid. This phenomenon is called cavitation.

In this validation example we consider COSMOSFloWorks abilities to model cavitation on the example of water flow around a symmetric hydrofoil in a water-filled tunnel. The calculated results were compared with the experimental data from Ref. 23.

The problem is solved in the 2D setting. A symmetric hydrofoil with the chord c of 0.305 m is placed in a water-filled tunnel with the angle of attack of 3.5°. The part of the tunnel being modelled has the following dimensions: length l = 2 m and height h = 0.508 m. The calculation is performed four times with different values of the cavitation number σ defined as follows:

where is the inlet pressure, Pv is the saturated water vapor pressure equal to 2340 Pa at given temperature (293.2 K), ρ is the water density at inlet, and is the water velocity at inlet (see Fig. 22.1.).

The inlet boundary condition is set up as Inlet Velocity of 8 m/s. On the tunnel outlet an Environment Pressure is specified so that by varying it one may tune the cavitation number to the needed value. The project fluid is water with the cavitation option switched on, while the other parameters are default. A local initial mesh was created in order to resolve the cavitation area better. The resulting mesh contains about 30000 cells.

σP∞ Pv–

12---ρU∞

2-------------------=

P∞U∞

Fig. 22.1. The model geometry.


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Chapter Cavitation on a hydrofoil

The qualitative comparison in a form of cut plots with Vapor Volume Fraction as the visualization parameter are shown on Fig. 22.2.

Fig. 22.2. A comparison of calculated and experimentally observed cavitation areas for different σ

σ=1.1 σ=0.97

σ=0.9 σ=0.88


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The calculated length of the cavitation area was derived from the distribution of the Vapor Volume Fraction parameter over the hydrofoil’s surface as the distribution’s width at half-height. The results are presented on Fig. 22.3.

According to Ref. 23, the "clear appearance" of the cavity becomes worse for larger cavity lengths. The experimental data also confirm that the amount of uncertainty increases with increasing cavity extent. Taking these factors into account together with the comparison performed above, we can see that the calculated length of the cavitation area agrees well with the experiment for a wide range of cavitation numbers.

Pressure measurements were performed on the hydrofoil surface at x/c = 0.05 in order to calculate the pressure coefficient defined as follows:

A comparison of the calculated and experimental values of this parameter is presented on Fig. 22.4. and also shows a good agreement.

Fig. 22.3. A comparison of calculated and measured cavitation lengths

Cp–P∞ Px c⁄ 0.05=–

12---ρU∞

2--------------------------------------=


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Chapter Cavitation on a hydrofoil

Fig. 22.4. A comparison of calculated and measured pressure coefficient


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23 Thermoelectric Cooling

COSMOSFloWorks has the ability to model the work of a thermoelectric cooler (TEC), also known as Peltier element. The device used in this example has been developed for active cooling of an infrared focal plane array detector used during the Mars space mission (see Ref. 24).

According to the hardware requirements, the cooler (see Fig. 23.1.) has the following dimensions: thickness of 4.8 mm, cold side of 8X8 mm2 and hot side of 12X12 mm2. It was built up of three layers of semiconductor pellets made of (Bi,Sb)2(Se,Te)3-based material. The cooler was designed to work at temperatures of hot surface in the range of 120-180 K and to provide the temperature drop of more than 30 K between its surfaces.

To solve the engineering problem using COSMOSFloWorks, the cooler has been modelled by a truncated pyramidal body with fixed temperature (Temperature boundary condition) on the hot surface and given heat flow (Heat flow boundary condition) on the cold surface (see Fig. 23.3.).

Fig. 23.1. Structure of the thermoelectric cooler. Fig. 23.2. The thermoelectric module test setup. (Image from Ref. 24)


Fig. 23.3. The model geometry.


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Chapter Thermoelectric Cooling

The TEC characteristics necessary for the modelling, i.e. temperature dependencies of the maximum pumped heat, maximum temperature drop, maximum current strength and maximum voltage, were represented in the COSMOSFloWorks Engineering Database as a linear interpolation between the values taken from Ref. 24 (see Fig. 23.4.).

As it can be seen on Fig. 23.5., the temperature drop between the cooler’s hot and cold surfaces in dependence of current agrees well with the experimental data.

Fig. 23.4. The TEC’s characteristics in the Engineering Database.

Fig. 23.5. ∆T as a function of current under various Th.

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The dependency of ∆T against heat flow under various Th (see Fig. 23.6.) is also in a good agreement with the performance data, as well as the coefficient of performance COP (see Fig. 23.7.) defined as follows:

where Pin is the cooler’s power consumption, and Qc and Qh are the heat flows on the cold and hot faces, respectively.

COPQc

Pin-------

Qc

Qh Qc–-------------------= =

Fig. 23.6. ∆T as a function of heat flow under various Th.

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Chapter Thermoelectric Cooling

Finally, we may conclude that COSMOSFloWorks reproduces thermal characteristics of the thermoelectric coolers at various currents and temperatures with good precision.

Fig. 23.7. COP as a function of ∆T under various Th.

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References

1 Schlichting, H., Boundary Layer Theory. 7th ed., McGraw –Hill, New York, 1979.

2 Idelchik, I.E., Handbook of Hydraulic Resistance. 2nd ed., Hemisphere, New York, 1986.

3 Panton, R.L., Incompressible Flow. 2nd ed., John Wiley & Sons, Inc., 1996.

4 White, F.M., Fluid Mechanics. 3rd ed., McGraw-Hill, New York, 1994.

5 Artonkin, V.G., Petrov, K.P., Investigations of aerodynamic characteristics of segmental conic bodies. TsAGI Proceedings, No. 1361, Moscow, 1971 (in Russian).

6 Holman, J.P., Heat Transfer. 8th ed., McGraw-Hill, New York, 1997.

7 Henderson, C.B. Drag Coefficients of Spheres in Continuum and Rarefied Flows. AIAA Journal, v.14, No.6, 1976.

8 Humphrey, J.A.C., Taylor, A.M.K., and Whitelaw, J.H., Laminar Flow in a Square Duct of Strong Curvature. J. Fluid Mech., v.83, part 3, pp.509-527, 1977.

9 Van Dyke, Milton, An Album of Fluid Motion. The Parabolic press, Stanford, California, 1982.

10 Davis, G. De Vahl: Natural Convection of Air in a Square Cavity: a Bench Mark Numerical Solution. Int. J. for Num. Meth. in Fluids, v.3, p.p. 249-264 (1983).

11 Davis, G. De Vahl, and Jones, I.P.: Natural Convection in a Square Cavity: a Comparison Exercise. Int. J. for Num. Meth. in Fluids, v.3, p.p. 227-248 (1983).

12 Emery, A., and Chu, T.Y.: Heat Transfer across Vertical Layers. J. Heat Transfer, v. 87, p. 110 (1965).

13 Denham, M.K., and Patrick, M.A.: Laminar Flow over a Downstream-Facing Step in a Two-Dimensional Flow Channel. Trans. Instn. Chem. Engrs., v.52, p.p. 361-367 (1974).

14 Yanshin, B.I.: Hydrodynamic Characteristics of Pipeline Valves and Elements. Convergent Sections, Divergent Sections, and Valves. “Mashinostroenie”, Moscow, 1965.

15 Jyotsna, R., and Vanka, S.P.: Multigrid Calculation of Steady, Viscous Flow in a Triangular Cavity. J. Comput. Phys., v.122, No.1, p.p. 107-117 (1995).

16 Darr, J.H., and Vanka, S.P.: Separated Flow in a Driven Trapezoidal Cavity. J. Phys. Fluids A, v.3, No.3, p.p. 385-392 (1991).

17 Michelsen, J. A., Modeling of Laminar Incompressible Rotating Fluid Flow, AFM 86-05, Ph.D. Dissertation, Department of Fluid Mechanics, Technical University of Denmark, 1986.

18 Sorensen, J.N., and Ta Phuoc Loc: Higher-Order Axisymmetric Navier-Stokes Code: Description and Evaluation of Boundary Conditions. Int. J. Numerical Methods in


Fluids, v.9, p.p. 1517-1537 (1989).


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Chapter References

19 Enchao Yu, Yogendra Joshi: Heat Transfer Enhancement from Enclosed Discrete Components Using Pin-Fin Heat Sinks. Int. J. of Heat and Mass Transfer, v.45, p.p. 4957-4966 (2002).

20 Kuchling, H., Physik, VEB FachbuchVerlag, Leipzig, 1980.

21 Balakin, V., Churbanov, A., Gavriliouk, V., Makarov, M., and Pavlov, A.: Verification and Validation of EFD.Lab Code for Predicting Heat and Fluid Flow, In: CD-ROM Proc. Int. Symp. on Advances in Computational Heat Transfer “CHT-04”, April 19-24, 2004, Norway, 21 p.

22 Visser, F.C., Brouwers, J.J.H., Jonker, J.B.: Fluid flow in a rotating low-specific-speed centrifugal impeller passage. J. Fluid Dynamics Research, 24, pp. 275-292 (1999).

23 Wesley, H. B., and Spyros, A. K.: Experimental and computational investigation of sheet cavitation on a hydrofoil. Presented at the 2nd Joint ASME/JSME Fluid Engineering Conference & ASME/EALA 6th International Conference on Laser Anemometry. The Westin Resort, Hilton Head Island, SC, USA August 13 - 18, 1995.

24 Yershova, L., Volodin, V., Gromov, T., Kondratiev, D., Gromov, G., Lamartinie, S., Bibring, J-P., and Soufflot, A.: Thermoelectric Cooling for Low Temperature Space Environment. Proceedings of 7th European Workshop on Thermoelectrics, Pamplona, Spain, 2002.


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5

Technical Reference

1 Physical Capabilities of COSMOSFloWorks

With COSMOSFloWorks it is possible to study a wide range of fluid flow and heat transfer phenomena that include the following:

• External and internal fluid flows

• Steady-state and time-dependent fluid flows

• Compressible gas and incompressible fluid flows (either in different projects or simultaneously in different regions not in contact with each other)

• Free, forced, and mixed convection

• Fluid flows with boundary layers, including wall roughness effects

• Laminar and turbulent fluid flows

• Multi-species fluids and multi-component solids

• Heat conduction in fluid, solid and porous media with/without conjugate heat transfer and/or contact heat resistance between solids and/or radiation heat transfer between opaque solids (some solids can be considered transparent for radiation), and/or volume (or surface) heat sources, e.g. due to Peltier effect, etc.

• Various types of thermal conductivity in solid medium, i.e. isotropic, unidirectional, biaxial/axisymmetrical, and orthotropic

• Fluid flows and heat transfer in porous media

• Flows of non-Newtonian liquids

• Flows of compressible liquids

• Real gases


• Two-phase (fluid + particles) flows

• Equilibrium volume condensation of water from steam and its influence on fluid flow and heat transfer


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Chapter Governing Equations

• Relative humidity in gases and mixtures of gases

• Fluid flows in models with moving/rotating surfaces and/or parts

• Cavitation in incompressible water flows

• Periodic boundary conditions.

2 Governing Equations

2.1 The Navier-Stokes Equations for Laminar and Turbulent Fluid Flows

COSMOSFloWorks solves the Navier-Stokes equations, which are formulations of mass, momentum and energy conservation laws for fluid flows. The equations are supplemented by fluid state equations defining the nature of the fluid, and by empirical dependencies of fluid density, viscosity and thermal conductivity on temperature. Inelastic non-Newtonian fluids are considered by introducing a dependency of their dynamic viscosity on flow shear rate and temperature, and compressible liquids are considered by introducing a dependency of their density on pressure. A particular problem is finally specified by the definition of its geometry, boundary and initial conditions.

COSMOSFloWorks is capable of predicting both laminar and turbulent flows. Laminar flows occur at low values of the Reynolds number, which is defined as the product of representative scales of velocity and length divided by the kinematic viscosity. When the Reynolds number exceeds a certain critical value, the flow becomes turbulent, i.e. flow parameters start to fluctuate randomly.

Most of the fluid flows we encounter in engineering practice are turbulent, so COSMOSFloWorks was mainly developed to simulate and study turbulent flows. To predict turbulent flows, we use the Favre-averaged Navier-Stokes equations, where time-averaged effects of the flow turbulence on the flow parameters are considered, whereas the other, i.e. large-scale, time-dependent phenomena are taken into account directly. Through this procedure, extra terms known as the Reynolds stresses appear in the equations for which additional information must be provided. To close this system of equations, COSMOSFloWorks employs transport equations for the turbulent kinetic energy and its dissipation rate, the so-called k-ε model.

COSMOSFloWorks employs one system of equations to describe both laminar and turbulent flows. Moreover, transition from a laminar to turbulent state and/or vice versa is possible.

Flows in models with moving walls (without changing the model geometry) are computed by specifying the corresponding boundary conditions. Flows in models with rotating parts are computed in coordinate systems attached to the models rotating parts, i.e. rotating with them, so the models' stationary parts must be axisymmetric with respect to the rotation axis.


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The conservation laws for mass, angular momentum and energy in the Cartesian coordinate system rotating with angular velocity Ω about an axis passing through the coordinate system's origin can be written in the conservation form as follows:

where u is the fluid velocity, ρ is the fluid density, is a mass-distributed external force

per unit mass due to a porous media resistance (Siporous), a buoyancy (Sigravity = - ρgi, where gi is the gravitational acceleration component along the i-th coordinate direction), and the coordinate system’s rotation (Sirotation), i.e., Si = Siporous + Sigravity + Sirotation, h is

the thermal enthalpy, is a heat source or sink per unit volume, is the viscous

shear stress tensor, is the diffusive heat flux. The subscripts are used to denote

summation over the three coordinate directions.

For calculations with the High Mach number flow option enabled, the following energy equation is used:

where e is the internal energy.

For Newtonian fluids the viscous shear stress tensor is defined as:

(5.1)0) = u(ρx

+ t

ρi

i∂∂

∂∂

(5.2)iRijij

jiji

j

i Sx

=x

p) + uu(ρ

x +

t

uρ ++∂∂

∂∂

∂∂

∂∂

)( ττ 3,2,1=i

(5.3)( ) ,)( Hiij

iRiji

Rijijj

ii

i QuSx

u

t

pqu

x =

x

Hu

t

H +++∂∂

−∂∂+++

∂∂

∂∂

+∂

∂ ρετττρρ

,2

2uhH +=

Si

QH τ ik

qi

(5.4)( ) ,)( Hiij

iRiji

Rijijj

ii

i

QuSx

uqu

x =

x

pEu

t

E +++∂∂

−++∂∂

∂

+∂+

∂∂ ρετττρ

ρρ

,2

2ueE +=

(5.5)

∂−

∂+∂= kji uuu δµτ 2


∂∂∂ k

ijij

ij xxx 3


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Following Boussinesq assumption, the Reynolds-stress tensor has the following form:

Here is the Kronecker delta function (it is equal to unity when i = j, and zero

otherwise), is the dynamic viscosity coefficient, is the turbulent eddy viscosity

coefficient and k is the turbulent kinetic energy. Note that and k are zero for laminar

flows. In the frame of the k-ε turbulence model, is defined using two basic turbulence

properties, namely, the turbulent kinetic energy k and the turbulent dissipation ε,

Here is a turbulent viscosity factor. It is defined by the expression

and y is the distance from the wall. This function allows us to take into account laminar-turbulent transition.

Two additional transport equations are used to describe the turbulent kinetic energy and dissipation,

(5.6)ij

k

kij

i

j

j

it

Rij k

x

u

x

u

x

u δρδµτ3

2

3

2 −

∂∂−

∂∂

+∂∂=

δi j

µ µ t

µ t

µ t

ερ

µ µµ

2kCft = (5.7)

µf

( )[ ]

+⋅−−=

Ty R

Rf5,20

1025.0exp1 2µ

(5.8),

where µερ 2k

RT = ,µ

ρ ykRy =

( ) kik

t

ii

i

Sx

k

xku

xt

k +

∂∂

+

∂∂=

∂∂+

∂∂

σµµρρ

, (5.9)

( ) εε

εσµµερρε

Sxx

uxt i

t

ii

i

+

∂∂

+

∂∂=

∂∂+

∂∂ , (5.10)


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where the source terms and are defined as

Here represents the turbulent generation due to buoyancy forces and can be written as

where is the component of gravitational acceleration in direction , the constant

σB = 0.9, and constant is defined as: CB = 1 when , and 0 otherwise;

The constants , , , , are defined empirically. In COSMOSFloWorks

the following typical values are used:

Cµ = 0.09, Cε1 = 1.44, Cε2 = 1.92, σ ε = 1.3,

Where Lewis number Le=1 the diffusive heat flux is defined as:

Here the constant σ c = 0.9, Pr is the Prandtl number, and h is the thermal enthalpy.

These equations describe both laminar and turbulent flows. Moreover, transitions from

one case to another and back are possible. The parameters k and are zero for purely

laminar flows.

Sk Sε

Btj

iRijk P

x

uS µρετ +−

∂∂= (5.11)

kfCPC

x

uf

kCS BBt

j

iRij

2

2211

ρεµτεεεε −

+

∂∂= . (5.12)

PB

iB

iB x

gP

∂∂−= ρ

ρσ1 (5.13)

gi xi

CB PB 0>

3

105.0

1

+=

µff , ( )2

2 exp1 TRf −−= (5.14)

Cµ Cε1 Cε2 σk σε

σk 1= (5.15)

ic

ti x

hq

∂∂

+=

σµµ

Pr , i = 1, 2, 3. (5.16)

µ t


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Laminar/turbulent Boundary Layer Model

A laminar/turbulent boundary layer model is used to describe flows in near-wall regions. The model is based on the so-called Modified Wall Functions approach. This model is employed to characterize laminar and turbulent flows near the walls, and to describe transitions from laminar to turbulent flow and vice versa. The modified wall function uses a Van Driest's profile instead of a logarithmic profile. If the size of the mesh cell near the wall is more than the boundary layer thickness the integral boundary layer technology is used. The model provides accurate velocity and temperature boundary conditions for the above mentioned conservation equations.

Constitutive Laws and Thermophysical Properties

The system of Navier-Stokes equations is supplemented by definitions of thermophysical properties and state equations for the fluids. COSMOSFloWorks provides simulations of gas and liquid flows with density, viscosity, thermal conductivity, specific heats, and species diffusivities as functions of pressure, temperature and species concentrations in fluid mixtures, as well as equilibrium volume condensation of water from steam can be taken into account when simulating steam flows.

Generally, the state equation of a fluid has the following form:

where y =(y1, ... yM) is the concentration vector of the fluid mixture components.

Excluding special cases (see below subsections concerning Real Gases, Equilibrium volume condensation of water from steam), gases are considered ideal, i.e. having the state equation of the form

where R is the gas constant which is equal to the universal gas constant Runiv divided by

the fluid molecular mass M, or, for the mixtures of ideal gases,

where , m=1, 2, ...,M, are the concentrations of mixture components, and is the

molecular mass of the m-th component.

Specific heat at constant pressure, as well as the thermophysical properties of the gases, i.e. viscosity and thermal conductivity, are specified as functions of temperature. In addition, proceeding from Eq. (5.18), each of such gases has constant specific heat ratio

(5.17),( ),y,,Tpf=ρ

RT

P=ρ (5.18),

∑=m m

muniv

M

yRR (5.19),

ym Mm

5-6

Cp/Cv.


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Excluding special cases (see below subsections Compressible Liquids, Non-Newtonian Liquids), liquds are considered incompressible, i.e. the density of an individual liquid depends only on temperature:

and the state equation for a mixture of liquids is defined as

The specific heat and the thermophysical properties of the liquid (i.e. viscosity and thermal conductivity), are specified as functions of temperature.

Real Gases

The state equation of ideal gas (5.18) become inaccurate at high pressures or in close vicinity of the gas-liquid phase transition curve. Taking this into account, a real gas state equation together with the related equations for thermodynamical and thermophysical properties should be employed in such conditions. At present, this option may be used only for a single gas, probably mixed with ideal gases.

In case of user-defined real gas, COSMOSFloWorks uses a custom modification of the Redlich-Kwong equation that may be expressed in dimensionless form as follows:

where pr = p/pc, Tr = T/Tc, Φr = Vr·Zc, Vr=V/Vc, F=Tr-1.5, pc, Tc, and Vc are the

user-specified critical parameters of the gas, i.e. pressure, temperature, and specific volume at the critical point, and Zc is the gas compressibility factor that also defines the a, b, and c constants. A particular case of equation (5.22) with Zc=1/3 (which in turn means that b=c) is the original Riedlich-Kwong equation as given in Ref. 11.

Alternatively, one of the modifications (Ref. 11) taking into account the dependence of F on temperature and the Pitzer acentricity factor ω may be used: the Wilson modification, the Barnes-King modification, or the Soave modification.

The specific heat of real gas at constant pressure (Cp) is determined as the combination of

the user-specified temperature-dependent "ideal gas" specific heat (Cpideal) and the

automatically calculated correction. The former is a polynomial with user-specified order and coefficients. The specific heat at constant volume (Cv) is calculated from Cp by means of the state equation.

(5.20),( )Tf=ρ

1−

= ∑

m m

my

ρρ (5.21)

+ΦΦ

−−Φ

=)(·

·1·

c

Fa

bTp

rrrrr

(5.22)


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Likewise, the thermophysical properties are defined as a sum of user-specified "basic" temperature dependency (which describes the corresponding property in extreme case of low pressure) and the pressure-dependent correction which is calculated automatically.

The basic dependency for dynamic viscosity η of the gas is specified in a power-law form:

η = a·Tn. The same property for liquid is specified either in a similar power-law form

η = a·Tn or in an exponential form: η = 10a(1/T-1/n). As for the correction, it is given according to the Jossi-Stiel-Thodos equation for non-polar gases or the Stiel-Thodos equations for polar gases (see Ref. 11), judging by the user-specified attribute of polarity. The basic dependencies for thermal conductivities λ of the substance in gaseous and liquid

states are specified by the user either in linear λ = a+n·T or in power-law λ = a·Tn forms, and the correction is determined from the Stiel-Thodos equations (see Ref. 11).

All user-specified coefficients must be entered in SI unit system, except those for the exponential form of dynamic viscosity of the liquid, which should be taken exclusively from Ref. 11.

In case of pre-defined real gas, the custom modification of the Riedlich-Kwong equation of the same form as Eq. (5.22) is used, with the distinction that the coefficients a, b, and c are specified explicitly as dependencies on Tr in order to reproduce the gas-liquid phase transition curve at P < Pc and the critical isochore at P > Pc with higher precision.


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When the calculated (p, T) point drops out of the region bounded by the temperature and pressure limits (zones 1 - 8 on Fig.2.1) or gets beyond the gas-liquid phase transition curve (zone 9 on Fig.2.1), the corresponding warnings are issued and properties of the real gas are extrapolated linearly.

If a real gas mixes with ideal gases (at present, mixtures of multiple real gases are not considered), properties of this mixture are determined as an average weighted with mass or volume fractions:

where ν is the mixture property (i.e., Cp, µ, or λ), N is the total number of the mixture gases (one of which is real and others are ideal), Yi is the mass fraction (when calculating Cp) or the volume fraction (when calculating µ and λ) of the i-th gas in the mixture.

The real gas model has the following limitations and assumptions:

• The precision of calculation of thermodynamic properties at nearly-critical temperatures and supercritical pressures may be lowered to some extent in comparison to other temperature ranges. Generally speaking, the calculations involving user-defined real gases at supercritical pressures are not recommended.

Fig.2.1

(5.23)ν Yiνi

i 1=

N

∑=


• The user-defined dependencies describing the specific heat and transport properties of the user-defined real gases should be applicable in the whole Tmin...Tmax range (or, speaking about liquid, in the whole temperature range where the liquid exists).


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• Tmin for user-defined real gas should be set at least 5...10 K higher than the triple point of the substance in question.

Compressible Liquids

Compressible liquids whose density depends on pressure and temperature can be considered within the following approximations:

• obeying the logarithmic law:

where ρo is the liquid's density under the reference pressure Po, C, B are coefficients, here ρo, C, and B can depend on temperature, P is the calculated pressure;

• obeying the power law:

where, in addition to the above-mentioned designations, n is a power index which can be temperature dependent.

Non-Newtonian Liquids

COSMOSFloWorks is capable of computing laminar flows of inelastic non-Newtonian liquids. In this case the viscous shear stress tensor is defined, instead of Eq. (5.5), as

where shear rate,

and for specifying a viscosity function the following three models of inelastic

non-Newtonian viscous liquids are available in COSMOSFloWorks:

The Herschel-Bulkley model:

++⋅−=

00 ln1/

PB

PBCρρ ,

n

BP

BP/1

00

++⋅= ρρ ,

, (5.24)( )

∂∂

+∂∂

⋅=i

j

j

iij x

u

x

uγµτ &

i

j

j

iijjjiiij x

u

x

udddd

∂∂

+∂∂=⋅−= ,2γ&

( )γµ &

( ) ( )γτγγµ&

&& onK +⋅= −1 ,

5-10

where K is the liquid's consistency coefficient, n is the liquid's power law index, and

is the liquid's yield stress. This model includes the following special cases:oτ


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• n = 1, = 0 describes Newtonian liquids, in this case K is the liquid's dynamic viscosity;

• n = 1, > 0 describes the Bingham model of non-Newtonian liquids, featured by a non-zero threshold yield stress ( ) below of which the liquid behaves as a solid, so to achieve a flow this threshold shear stress must be exceeded. (In COSMOSFloWorksthis threshold is modeled by automatically equating K, named plastic viscosity in this case, to a substantially high value at );

• 0 < n < 1, = 0 describes the power law model of shear-thinning non-Newtonian liquids (see also below).

• n > 1, = 0 describes the power law model of shear-thickening non-Newtonian liquids (see also below).

The power law model: ,

in contrast to the Herschel-Bulkley model's special case, the µ values are restricted: µmin ≤ µ ≤ µmax;

The Carreau model:

where:

is the liquid's dynamic viscosity at infinite shear rate, i.e., the minimum dynamic viscosity,

is the liquid's dynamic viscosity at zero shear rate, i.e., the maximum dynamic

viscosity,

Kt is the time constant,

n is the power law index.

This model is a smooth version of the power law model with the µ restrictions. In these models, all parameters with the exception of the dimensionless power law index are temperature-dependent in a general case.

Equilibrium volume condensation of water from steam

If the gas whose flow is computed includes steam, COSMOSFloWorks can predict an equilibrium volume condensation of water from this steam (without any surface condensation) taking into account the corresponding changes of the steam temperature, density, enthalpy, specific heat, and sonic velocity. In accordance with the equilibrium approach, local mass fraction of the condensed water in the local total mass of the steam and the condensed water is determined from the local temperature of the fluid, pressure,

oτ

oτoτ

oττ <oτ

oτ

( ) ( ) 1−⋅= nK γγµ && ,

( ) ( )[ ]( ) 2/121−

∞∞ ⋅+⋅−+=n

to K γµµµµ &,

∞µ

oµ


and, if a multi-component fluid is considered, the local mass fraction of the steam. Since this model implies an equilibrium conditions, the condensation has no history, i.e. it is a local fluid property only.


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In addition, it is assumed that

• the volume of the condensed water is neglected, i.e. considered zero, so this prediction works properly only if the volume fraction of the condensed water does not exceed 5%,

• the steam temperature falls into the range of 283...610 K and the pressure does not exceed 10 MPa.

2.2 Conjugate Heat Transfer

COSMOSFloWorks allows to predict simultaneous heat transfer in solid and fluid media with energy exchange between them. Heat transfer in fluids is described by the energy conservation equation (5.3) where the heat flux is defined by (5.16). The phenomenon of anisotropic heat conductivity in solid media is described by the following equation:

where e is the specific internal energy, e = cT, c is specific heat, QH is specific heat

release (or absorption) per unit volume, and λi are the eigenvalues of the thermal conductivity tensor. It is supposed that the heat conductivity tensor is diagonal in the considered coordinate system. For isotropic medium λ1 = λ2 = λ3 = λ.If a solid consists of several solids attached to each other, then the thermal contact resistances between them (on their contact surfaces), specified in the Engineering database

in the form of contact conductance (as m2·K/W), can be taken into account when calculating the heat conduction in solids. As a result, a solid temperature step appears on the contact surfaces. In the same manner, i.e. as a thermal contact resistance, a very thin layer of another material between solids or on a solid in contact with fluid can be taken into account when calculating the heat conduction in solids, but it is specified by the material of this layer (its thermal conductivity taken from the Engineering database) and thickness. The surface heat source (sink) due to Peltier effect may also be considered (see "2.3. Thermoelectric Coolers" on page 5-13).

The energy exchange between the fluid and solid media is calculated via the heat flux in the direction normal to the solid/fluid interface taking into account the solid surface temperature and the fluid boundary layer characteristics, if necessary.

Hi

ii

Qx

T

xt

e +

∂∂

∂∂=

∂∂ λρ

(5.25),


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If a solid medium is porous with a fluid flowing through it, then a conjugate heat transfer problem in this porous-solid/fluid medium can be solved also in the manner described below. The equations (5.3) and (5.25) are solved in a usual way, but with addition of energy exchange between the fluid and the porous solid matrix, defined via the volumetric

heat exchange in the Eq. (5.25) in a form of , where

γ is the user-defined volumetric coefficient of heat transfer between fluid and the porous matrix, Tp is the temperature of the porous matrix, T is the fluid temperature, and the same QH with the opposite sign is employed in Eq. (5.25) for the porous matrix. Note that the γ and c of the porous matrix used in Eq. (5.25) can differ from those of the corresponding bulk solid material. Naturally, both the fluid flow equations and the porous matrix heat transfer equation take into account the fluid and solid densities multiplied by the corresponding fluid and solid volume fractions in the porous matrix.

2.3 Thermoelectric Coolers

Thermoelectric cooler (TEC) is a flat sandwich consisting of two plates covering a circuit of p-n semiconductor junctions inside. When a direct electric current (DC) i runs through this circuit, in accordance with the Peltier effect the a·i·Tc heat, where a is the Seebeck

coefficient, Tc is the TEC's "cold" surface temperature, is pumped from the TEC's "cold"

surface to its "hot" surface (the "cold" and "hot" sides are determined from the DC direction). This heat pumping is naturally accompanied by the Joule (ohmic) heat release at both the TEC surfaces and the heat transfer from the hotter side to the colder (reverse to

the Peltier effect). The ohmic heat release is defined as R·i2/2, where R is the TEC's electric resistance, while the heat transfer is defined as k·∆T , where k is the TEC's

thermal conductivity, ∆T = Th - Tc , Th is the TEC's "hot" surface temperature. The net

heat transferred from the TEC's "cold" surface to its "hot" surface, Qc, is equal to

,

Correspondingly, the net heat released at the TEC's "hot" surface, Qh, is equal to

.

In COSMOSFloWorks a TEC is specified by selecting a flat plate (box) in the model, assigning its "hot" face, and applying one of the TECs already defined by user in the Engineering Database. The following characteristics of TEC are specified in the Engineering Database:

• the maximum DC current, imax• the maximum heat Qcmax transferred at this imax at ∆T = 0

QH poros i t y γ Tp T–( )⋅=

TkiRTiaQ cc ∆⋅−⋅−⋅⋅= 2/2

TkiRTiaQ hh ∆⋅−⋅+⋅⋅= 2/2


• the maximum temperature difference ∆Tmax, attained at Qc = 0

• the voltage Vmax corresponding to imax.


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All of these characteristics are specified for two Th values, in accordance with the

information usually provided by the TEC suppliers. Proceeding from these characteristics, the a(T), R(T), and k(T) linear functions are determined. The functional boundary conditions are specified automatically on the TEC's "cold" and "hot" surfaces, which must be free from other boundary conditions.

The temperature solution inside the TEC and on its surfaces is obtained using a special procedure differing from the standard COSMOSFloWorks calculation procedure for heat conduction in solids.

The TEC's "hot" face must be in contact with other solids, i.e it must not be in contact with any fluid. In addition, it is required that the obtained TEC solution, i.e. Th and ∆T, lie

within the TEC's operating range specified by its manufacturer.

2.4 Radiation Heat Transfer Between Solids

In addition to heat transfer in solids, COSMOSFloWorks is capable to calculate radiation heat transfer between solids whose surface emissivity is specified. If necessary, a heat radiation from the computational domain's far-field boundaries or the model's openings to the model surfaces can be defined and considered either as from solid surfaces, i.e. by specifying these boundaries' emissivity and temperature, or as a solar radiation defined by the specified location (on the surface of the Earth) and time (including date) or by constant or time-dependent direction and intensity.

General Assumptions

The radiation heat transfer is analyzed under the following assumptions:

The heat radiation from the solid surfaces, both the emitted and reflected, is assumed diffuse (except for the symmetry radiative surface type), i.e. obeying the Lambert law, according to which the radiation intensity per unit area and per unit solid angle is the same in all directions.

The propagating heat radiation passes through a body specified as radiation transparent without any refraction and/or absorption.

The project fluids neither emit nor absorb heat radiation (i.e., they are transparent to the heat radiation), so the heat radiation concerns solid surfaces only.

The radiative solid surfaces which are not specified as a blackbody or whitebody are assumed an ideal graybody, i.e. having a continuous emissive power spectrum similar to that of blackbody, so their monochromatic emissivity is independent of the emission wavelength. For certain materials with certain surface conditions, the graybody emissivity can depend on the surface temperature.


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Ray Tracing

In a general case, the surfaces participating in the heat radiation (hereinafter radiative surfaces) can emit, absorb, and reflect a heat radiation. Therefore, both the heat radiation L leaving a radiative surface and the net radiation N being the difference between the radiation heat leaving this surface and the radiation heat arriving at it, are calculated for each of these surfaces.

,

,

where ε is this surface emissivity, σ is the Stefan-Boltzmann constant, T is the

temperature of the surface (ε·σ·T4 is the heat radiated by this surface in accordance with

the Stefan-Boltzman law), I is the radiation arriving at this surface, ρ is a reflection coefficient (ρ = 1 - ε for graybody walls and ρ = 0 for openings).

In order to reduce the of memory requirements, the problem of determining the leaving and net heat radiation fluxes is solved using a discrete ray Monte-Carlo approach consisting of the following main elements:

To reduce the number of radiation rays and, therefore, the required calculation time and resources, the computational mesh cells containing faces approximating the radiative surfaces are joined in clusters by a special procedure that takes into account the face area and angle between normal and face in each partial cell. The cells intersected by boundaries between radiative surfaces of different emissivity are considered as belonging to one of these surfaces and cannot be combined in one cluster. This procedure is executed after constructing the computational mesh before the calculation and after each solution-adaptive mesh refinement, if any.

From each cluster, a number of rays are emitted, equally distributed over the enclosing unit hemisphere. Each ray is traced through the fluid and transparent solid bodies until it intercepts the computational domain’s boundary or a cluster belonging to another radiative surface, thus defining a ‘target’ cluster. Since the radiation heat is transferred along these rays only, their number and arrangement govern the accuracy of calculating the radiation heat coming from one radiative surface to another (naturally, the net heat radiated by a radiative surface does not depend on number of these rays). So, for each of the clusters, the hemisphere governed by the ray’s origin and the normal to the face at this origin is uniformly divided into several nearly equal solid angles generated by several zenith angles (at least 3 within the 0...90º range, including the

4L T Iε σ ρ= ⋅ ⋅ + ⋅

( )4N L I T Iε σ ρ − 1= − = ⋅ ⋅ + ⋅


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zero zenith angle of the normal to the face) and several azimuth angles (at least 12 within the 0...360º range).

The total number of emitted rays is

,

where m is the number of different latitude values for the rays (including the polar ray),

n is the number of different longitude values (n = 2 for 2D case),

Θ and Φ are the zenith (latitudinal) and azimuth (longitudinal) angles, respectively.

The value of m is defined directly by the View factor resolution level which can be changed by the user via the Calculation Control Options dialog box. The value of n

depends on m as follows: .

The higher the View factor resolution level, the better the accuracy of the radiation heat transfer calculation, but the calculation time and required computer resources increase significantly when high values of View factor resolution level are specified.

Periodically during the calculation, a radiation ray is emitted in each of the solid angles in a direction that is defined randomly within this solid angle. These radiation rays are traced until intersection with either another radiative surface or the boundary of the computational domain. To increase the accuracy of heat radiation calculation, the number of radiation rays emitted from each cluster can be increased automatically during the calculation, depending on the surface temperature and emissivity, to equalize the radiation heat emitting through the solid angles.

When a radiation ray intercepts a cluster of other radiative surfaces, the radiation heat

Fig.2.2Definition of rays emitted from cluster.

( ) 11 +⋅−= nmN

4⋅= mn

5-16

carried by this ray is uniformly distributed over the area of this cluster. The same procedure is performed if several radiation rays hit the same cluster. To smooth a possible non-uniformity of the incident radiation heat distribution over a radiative


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surface, a fraction of the radiation heat arriving with rays at a cluster can be transferred to the neighboring clusters also. In addition, small fluctuations are smoothed by the heat conduction in solid regions.

View Factor Calculation

The view factor between two clusters is the fraction of the total radiation energy emitted from one of the clusters that is intercepted by other clusters. The following relations are used in the code to define the view factor.

3D case

View factor for each ray (except for the polar ray) are defined as follows:

, , .

View factor for the Polar ray is:

.

2D case

, , .

.

Set of Equations

is the Incident radiation flux;

where:

ε is the emissivity coefficient, ρ is the reflection coefficient (ρ = 1-ε for walls, ρ = 0 for

openings), and is the Stefan-Boltzmann constant.

nF k

k

ε= ( ) ( )2

1112

+⋅−

⋅−=nm

nkkε 1,...,2,1 −= mk

( )( )

21

11 1polar

m nF

m n

− ⋅= − − ⋅ +

2k

kFε= ( ) ( ) ( )

−

−⋅⋅

−⋅

⋅= 12122

sin122

sin2 kmmk

ππε 1,...,2,1 −= mk

( )

−−⋅=12

1cos

m

mFpolar

π

∑=j

jjii LFI

4)( iij

jjiii TLFL σερ =− ∑

42810672.5 W−×=σ


Km


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Environment and Solar Radiation

Environmental and solar radiation can be applied to external and internal problems. In fact, an environment radiation is the energy flux generated by the walls of an imaginary huge «room» that surrounds the body. This flux has predefined radiation parameters. In contrast to the environment radiation, the solar radiation is modeled by the directional energy flux. Therefore, we define solar radiation via its power flow (intensity) and its directional vector. Directed energy fluxes can be emitted by the surfaces that have the «Solar opening» boundary condition.

The external radiation view factor can be calculated as , where Fi are the

view factors for the rays that have reached the boundaries of the computational domain, and S is the cluster area. Each «Solar opening» boundary condition produces one ray that follows the directional vector. After it reaches the outer boundary or the surface having appropriate radiation boundary condition, the view factor can be estimated as

.

Radiative Surface Types

For the «Wall» and «Wall to ambient» boundary condition, the program gets Tr from the current results set.

For the «Opening/Outer Boundary» boundary condition, Tr is taken from the Engineering Database.

The rays are emitted only from surfaces and boundaries on which the «Wall» or «Opening» boundary conditions are applied.

Radiative surface type Prescribed values Dependent values

Wall ε, Tr ρ = 1-ε, α = εOpening/Outer boundary ε, Tr ρ = 0, α = 1

Solar opening n, W ε = 1

Symmetry No parameters

Absorbent wall No parameters

Wall to ambient ε, Tr ρ = 1-ε, α = εNon-radiative No parameters

SFFi

i ∗=∑

( ) SnnF clustsolar ∗= ,

5-18

Surfaces with the specified «Absorbent wall» boundary condition are taken into account during the calculation but they can act as absorbents only. This wall type takes all heat from the radiation that reaches it and does not emit any heat.


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The «Symmetry» boundary condition forces the walls to which it is applied to reflect rays as an ideal mirror.

The «Solar opening» boundary condition requires the wall to emit radiation like the outer solar radiation. It is specified by the direction vector and intensity. The solar radiation at the computational domain boundaries can be specified not only by the direction vector and intensity, but also by the location (on the surface of the Earth) and time.

«Wall to ambient» reproduces the most elementary phenomenon among the radiation effects. The walls with this condition does not interact with any other surfaces. They can only exhaust energy into the space that surrounds the computational domain. Heat flux from the surface could be calculated as:

,

where is the temperature of the environmental radiation.

«Non-radiative» boundary condition removes specific surfaces from the radiation heat transfer analysis, so they do not affect the results.

After rays reach the surfaces for which «Opening», «Solar opening» or «Wall to ambient» radiative surface types are specified, they disappear. All energy that is carried out by these rays also dies away.

Viewing Results

The main result of the heat radiation calculation is the solids’ surface or internal temperatures. But these temperatures are influenced by calculations of heat transfer in solids and solid/fluid heat transfer also. To see the radiation calculation’s results only, the User may view the Leaving radiant flux and the distributions of the Net radiant flux over the selected radiative surfaces as Surface Plots, and their maximum, minimum, and average values over these surfaces in the Surface Parameters dialog boxes, as well as the Leaving radiation rate and Net radiation rate as an integral over these surfaces in the Surface Parameters dialog boxes.

2.5 Global Rotating Reference Frame

The rotation of the coordinate system is taken into account via the following mass-distributed force:

where eijk is the Levy-Civita symbols (function), Ω is the angular velocity of the rotation,

r is the vector coming to the point under consideration from the nearest point lying on the

( )44outout TrTrO ⋅−⋅= εσ

outTr

ikjijkrotationi rueS 22 Ω+Ω−= ρρ ,


rotation axis.


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2.6 Local rotating regions

This option is employed for calculating time-dependent (transient) or steady-state flows in regions surrounding rotating non-axisymmetrical solids (e.g. impellers, mixers, propellers, etc), when a single global rotating reference cannot be employed. For example, local rotating regions can be used in analysis of the fluid flow in the model including several components rotating over different axes and/or at different speeds or if the computational domain has a non-axisymmetrical (with respect to a rotating component) outer solid/fluid interface. In accordance with the employed approach, each rotating solid component is surrounded by an axisymmetrical (with respect to the component's rotation axis) Rotating region, which has its own coordinate system rotating together with the component. If the model includes several rotating solid components having different rotation axes, the rotating regions surrounding these components must not intersect with each other. The fluid flow equations in the stationary (non-rotating) regions of the computational domain are solved in the inertial (non-rotating) Cartesian Global Coordinate System. The influence of the rotation's effect on the flow is taken into account in the equations written in each of the rotating coordinate systems.

To connect solutions obtained within the rotating regions and in non-rotating part of the computational domain, special internal boundary conditions are set automatically at the fluid boundaries of the rotating regions. Since the coordinate system of the rotating region rotates, the rotating region’s boundaries are sliced into rings of equal width as shown on the Fig.2.3. Then the values of flow parameters transferred as boundary conditions from the adjacent fluid regions are averaged circumferentially over each of these rings.

Fig.2.3

Computational domain or fluid subdomainFlow parameters are calculated in the inertial Global Coordinate System

Rotation axisFlow parameters areaveraged over these rings

Local rotating regionFlow parameters are calculatedin the local rotating coordinatesystem

5-20

To solve the problem, an iterative procedure of adjusting the flow solutions in the rotating regions and in the adjacent non-rotating regions, therefore in the entire computational domain, is performed with relaxations.


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Please note that even in case of time-dependent (transient) analysis the flow parameters within the rotating regions are calculated using a steady-state approach and averaged on the rotating regions' boundaries as described above.

2.7 Mass Transfer in Fluid Mixtures

The mass transfer in fluid mixtures is governed by species conservation equations. The equations that describe concentrations of mixture components can be written as

Here Dmn, are the molecular and turbulent matrices of diffusion, Sm is the rate of

production or consumption of the m-th component.

In case of Fick's diffusion law:

The following obvious algebraic relation between species concentrations takes place:

2.8 Flows in Porous Media

Porous media are treated in COSMOSFloWorks as distributed resistances to fluid flow, so they can not occupy the whole fluid region or fill the dead-end holes. In addition, if the Heat conduction in solids option is switched on, the heat transfer between the porous solid matrix and the fluid flowing through it is also considered. Therefore, the porous matrix act on the fluid flowing through it via the Si, Siui, and (if heat conduction in solids is considered) QH terms in Eqs. (5.2) and (5.3), whose components related to porosity are

defined as:

( ) ( ) M 1,2,..., , =+

∂∂+

∂∂=

∂∂+

∂∂

mSx

yDD

xyu

xt

ym

i

ntmnmn

imi

i

m ρρ (5.26)

Dmnt

σµδδ t

mntmnmnmn D DD ⋅=⋅= , (5.27)

∑ =m

my 1 . (5.28)

jijporousi ukS ρδ−= , (5.29)

, (5.30)QHporosity γ Tp T–( )⋅=


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where k is the resistance vector of the porous medium (see below), γ is the user-defined volumetric porous matrix/fluid heat transfer coefficient, Tp is the temperature of the porous matrix, T is temperature of the fluid flowing through the matrix, and the other designations are given in Section 1. In addition, the fluid density in Eqs. (5.1)-(5.3) is multiplied by the porosity n of the porous medium, which is the volume fraction of the interconnected pores with respect to the total medium volume.

In the employed porous medium model turbulence disappears within a porous medium and the flow becomes laminar.

If the heat conduction in porous matrix is considered, then, in addition to solving Eqs. (5.1)-(5.3) describing fluid flow in porous medium, an Eq. (5.25) describing the heat conduction in solids is also considered within the porous medium. In this equation the source QH due to heat transfer between the porous matrix and the fluid is defined in the

same manner as in Eq. (5.30), but with the opposite sign. The values of γ and c for the porous matrix may differ from those of the corresponding bulk solid material and hence must be specified independently. Density of the solid material is multiplied by the solid volume fraction in the porous matrix, i.e. by (1-n).

Thermal conductivity of the porous matrix can be specified as anisotropic in the same manner as for the solid material.

The conjugate heat transfer problem in a porous medium is solved under the following restrictions:

• heat conduction in a porous medium not filled with a fluid is not considered,

• porous media are considered transparent for radiation heat transfer,

• heat sources in the porous matrix can be specified in the forms of heat generation rate or volumetric heat generation rate only; heat sources in a form of constant or time-dependent temperature can not be specified.

To perform a calculation in COSMOSFloWorks, you have to specify the following porous medium properties: the effective porosity of the porous medium, defined as the volume fraction of the interconnected pores with respect to the total medium volume. Later on, the permeability type of the porous medium must be chosen among the following:

• isotropic (i.e., the medium permeability is independent of direction),

• unidirectional (i.e., the medium is permeable in one direction only),

• axisymmetrical (i.e., the medium permeability is fully governed by its axial and transversal components with respect to a specified direction),

• orthotropic (i.e., the general case, when the medium permeability varies with direction and is fully governed by its three components determined along three principal directions).

Then you have to specify some constants needed to determine the porous medium

5-22

resistance to fluid flow, i.e., vector k defined as k = - grad(P)/(ρ⋅V), where P, ρ, and V are fluid pressure, density, and velocity, respectively. It is calculated according to one of the following formulae:


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• k = ∆P⋅S/(m⋅L), where ∆P is the pressure difference between the opposite sides of a sample parallelepiped porous body, m is the mass flow rate through the body, S and L are the body cross-sectional area and length in the selected direction, respectively. You can specify ∆P as a function of m, whereas S and L are constants. Instead of mass flow rate you can specify volume flow rate, v. In this case COSMOSFloWorks calculates m = v⋅ρ. All these values do not specify the porous body for the calculation, but its resistance k only.

• k = (A⋅V+B)/ρ, where V is the fluid velocity, A and B are constants, ρ is the fluid density. Here, only A and B are specified, since V and ρ are calculated.

• k= µ/(ρ⋅D2), where µ and ρ are the fluid dynamic viscosity and density, D is the reference pore size determined experimentally. Here, only D is specified, since µ and ρ are calculated.

• k= µ/(ρ⋅D2)⋅f(Re), differing from the previous formula by the f(Re) factor, yielding a more general formula. Here, in addition to D, f(Re) as a formula dependency is specified.

To define a certain porous body, you specify both the body position in the model and, if the porous medium has a unidirectional or axisymmetrical permeability, the reference directions in the porous body.

2.9 Two-phase (fluid + particles) Flows

COSMOSFloWorks calculates two-phase flows as a motion of spherical liquid particles (droplets) or spherical solid particles in a steady-state flow field. COSMOSFloWorks can simulate dilute two-phase flows only, where the particle’s influence on the fluid flow (including its temperature) is negligible (e.g. flows of gases or liquids contaminated with particles). Generally, in this case the particles mass flow rate should be lower than about 30% of the fluid mass flow rate.


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The particles of a specified (liquid or solid) material and constant mass are assumed to be spherical. Their drag coefficient is calculated with Henderson’s formula (Ref. 1), derived for continuum and rarefied, subsonic and supersonic, laminar, transient, and turbulent flows over the particles, and taking into account the temperature difference between the fluid and the particle. The particle/fluid heat transfer coefficient is calculated with the formula proposed in Ref. 2. If necessary, the gravity is taken into account. Since the particle mass is assumed constant, the particles cooled or heated by the surrounding fluid change their size. The interaction of particles with the model surfaces is taken into account by specifying either full absorption of the particles (that is typical for liquid droplets impinging on surfaces at low or moderate velocities) or ideal or non-ideal reflection (that is typical for solid particles). The ideal reflection denotes that in the impinging plane defined by the particle velocity vector and the surface normal at the impingement point, the particle velocity component tangent to surface is conserved, whereas the particle velocity component normal to surface changes its sign. A non-ideal reflection is specified by the two particle velocity restitution (reflection) coefficients, en and eτ, determining values of these particle velocity components after reflection, V2,n and V2,τ, as their ratio to

the ones before the impingement, V1,n and V1,τ:

As a result of particles impingement on a solid surface, the total erosion mass rate, RΣerosion, and the total accretion mass rate, RΣaccretion, are determined as follows:

,

,

where:

N is the number of fractions of particles specified by user as injections in COSMOSFloWorks (the user may specify several fractions of particles, also called injections, so that the particle properties at inlet, i.e. temperature, velocity, diameter, mass flow rate, and material, are constant within one fraction),i is the fraction number,Mp i is the mass impinging on the model walls in unit time for the i-th particle fraction,

Ki is the impingement erosion coefficient specified by user for the i-th particle fraction,Vp i is the impingement velocity for the i-th particle fraction,

b is the user-specified velocity exponent (b = 2 is recommended), f1 i (αp i) is the user-specified dimensionless function of particle impingement angle αp i,

n,1Vn,2V

ne =ττ

τ,1V,2V

e =

ipipi2ipi1b

ip

N

1i Mierosion md)d(f)(fVKR

ip

&⋅⋅⋅=∑ ∫=

∑ α

∑=

∑ =N

1iipaccretion MR

5-24

f2 i (dp i) is the user-specified dimensionless function of particle diameter dp i.


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2.10 Cavitation

A liquid subjected to a low pressure above a threshold ruptures and forms vaporous cavities. More specifically, when the local pressure at a point in the liquid falls below the liquid's vapour pressure at the local temperature, the liquid undergoes phase transition and form cavities filled with the liquid's vapor with an addition of gas that has been dissolved in the liquid. This phenomenon is called cavitation.

A homogeneous equilibrium model of cavitation in water is employed. The computational simplicity is the main advantage of the homogeneous model. The stationary homogeneous flow approach provides a simple technique for analyzing two-phase flows. It is accurate enough to handle a variety of practically important processes, including localized boiling of water due to intense heating.

The fluid is assumed to be a homogenous gas-liquid mixture with the volume-averaged parameters and the gaseous phase comprising the liquid vapour and non-condensable (dissolved) gas. The liquid vapour to gas ratio is defined at the local equilibrium thermodynamic conditions. By default, the mass fraction of non-condensable air is set to

10-5. This is a typical value under normal conditions and appropriate in most cases but it

can be modified by the user in the range of 10-4…10-8.

The homogeneous equilibrium cavitation model does not describe the detailed structure of the cavitation area, and the migration of individual vapour bubbles in the counter-gradient direction is not considered. The velocities and temperatures of the gaseous (including vapour and non-condensable gas) and liquid phases are assumed to be the same.

The density of the gas-liquid mixture is calculated as:

, ,

where v is the specific volume of the gas-liquid mixture, vl is the specific volume of

liquid, zv (T,P) is the vapour compressibility ratio, P is the local static pressure, T is the

local temperature, yv is the mass fraction of vapour, yg is the mass fraction of the

non-condensable gas; µg is the molar mass of the non-condensable gas, µv is the molar

mass of vapour, Runiv is the universal gas constant.

The mass fraction of vapour yv is computed numerically from the following non-linear

equation for the full enthalpy gas-liquid mixture:

,

v

1=ρv

vunivvlvg

g

univg P

PTTzRyPTvyy

P

TRyv

µµ),(

),()1( +−−+=

2

2),(),()1(),(

vIPThyPThyyPThyH C

vvlvggg ++−−+=


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where temperature of the mixture T is a function of pressure P and yv. Here hg, hl, hv are

the enthalpies of non-condensable gas, liquid and vapour, respectively,

is the squared impulse.

The model has the following limitations and assumptions:

• Cavitation is currently available only for incompressible water (when defining the project fluids you should select Water SP from the list of Pre-Defined liquids); cavitation in mixtures of different liquids cannot be calculated.

• The properties of the non-condensable gas are set to be equal to those of air.

• The temperature and pressure in the phase transition areas should be within the following ranges:

T = 277.15 - 583.15 K, P = 800 - 107 Pa.

• If the calculation has finished or has been stopped and the Cavitation option has been enabled or disabled, the calculation cannot be resumed or continued and must be restarted from the beginning.

• The Cavitation option should not be selected if you calculate a water flow in the model without flow openings (inlet and outlet).

• The model does not describe the detailed structure of the cavitation area, i.e parameters of individual vapour bubbles.

• The fluid region where cavitation occurs should be well resolved by the computational mesh.

• The parameters of the flow at the inlet boundary conditions must be such that the volume fraction of liquid water in the inlet flow would be at least 0.1.

2.11 Boundary Conditions

Internal Flow Boundary Conditions

For internal flows, i.e., flows inside models, COSMOSFloWorks offers the following two options of specifying the flow boundary conditions: manually at the model inlets and outlets (i.e. model openings), or to specify them by transferring the results obtained in another COSMOSFloWorks calculation in the same coordinate system (if necessary, the calculation can be performed with another model, the only requirement is the flow regions at the boundaries must coincide).

With the first option, all the model openings are classified into "pressure" openings, "flow" openings, and "fans", depending on the flow boundary conditions which you intend to specify on them.

222 )()()( zyxC uuuI ρρρ ++=


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A "pressure" opening boundary condition, which can be static pressure, or total pressure, or environment pressure is imposed in the general case when the flow direction and/or magnitude at the model opening are not known a priori, so they are to be calculated as part of the solution. Which of these parameters is specified depends on which one of them is known. In most cases the static pressure is not known, whereas if the opening connects the computational domain to an external space with known pressure, the total pressure at the opening is known. The Environment pressure condition is interpreted by COSMOSFloWorks as a total pressure for incoming flows and as a static pressure for outgoing flows. If, during calculation, a vortex crosses an opening with the Environment pressure condition specified at it, this pressure considered as the total pressure at the part of opening through which the flow enters the model and as the static pressure at the part of opening through which the flow leaves the model.

Note that when inlet flow occurs at the "pressure" opening, the temperature, fluid mixture composition and turbulence parameters have to be specified also.

A "flow" opening boundary condition is imposed when dynamic flow properties (i.e., the flow direction and mass/volume flow rate or velocity/ Mach number) are known at the opening. If the flow enters the model, then the inlet temperature, fluid mixture composition and turbulence parameters must be specified also. The pressure at the opening will be determined as part of the solution. For supersonic flows the inlet pressure must be specified also.

A "fan" condition simulates a fan installed at a model opening. In this case the dependency of volume flow rate on pressure drop over the fan is prescribed at the opening. These dependencies are commonly provided in the technical documentation for the fans being simulated.

With the second option, you specify the boundary conditions by transferring the results obtained in another COSMOSFloWorks calculation in the same coordinate system. If necessary, the calculation can be performed with another model, the only requirement is the flow regions at the boundaries must coincide. At that, you select the created boundary conditions’ type: either as for external flows (so-called "ambient" conditions, see the next Section), or as for "pressure" or "flow" openings, see above. If a conjugate heat transfer problem is solved, the temperature at the part of the boundary lying in a solid body is transferred from the other calculation.

Naturally, the flow boundary conditions specified for an internal flow problem with the first and/or second options must be physically consistent with each other, so it is expedient to specify at least one "pressure"-type boundary condition and at least one "flow"-type boundary condition, if not only "ambient" boundary conditions are specified.

External Flow Boundary Conditions

For external problems such as flow over an aircraft or building, the parameters of the external incoming flow (so-called "ambient" conditions) must be defined. Namely the


velocity, pressure, temperature, fluid mixture composition and turbulence parameters must be specified. Evidently, during the calculation they can be partly violated at the flow boundary lying downstream of the model.


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Wall Boundary Conditions

In COSMOSFloWorks the default velocity boundary condition at solid walls corresponds to the well-known no-slip condition. The solid walls are also considered to be impermeable. In addition, the wall surface's translation and/or rotation (without changing the model's geometry) can be specified. If a calculation is performed in a rotating coordinate system, then some of the wall surfaces can be specified as stationary, i.e. a backward rotation in this coordinate system (without changing the model geometry). COSMOSFloWorks also provides the "Ideal Wall" condition that corresponds to the well-known slip condition. For example, Ideal Walls can be used to model planes of flow symmetry.

If the flow of non-Newtonian liquids is considered, then the following slip condition at solid walls can be specified: if the shear stress τ exceeds the yield stress value τ0,slip, then

a slip velocity vslip determined from , where C1 and C2, as well

as τ0,slip, can be specified by user, if they are not specified in the model of non-Newtonian

liquid. If conjugate heat transfer in fluid and solid media is not considered, one of the following boundary conditions can be imposed at solid walls: either the wall temperature

or the heat flux,

being positive for heat flows from fluid to solid, equal to zero for adiabatic (heat-insulated) walls, and negative for heat flows from solid to fluid.

When considering conjugate heat transfer in fluid and solid media, the heat exchange between fluid and solid is calculated by COSMOSFloWorks, so heat wall boundary conditions are not specified at the walls.

Internal Fluid Boundary Conditions

If one or several non-intersecting axisymmetric rotating regions (local rotating reference frames) are specified, the flow parameters are transferred from the adjacent fluid regions and circumferentially averaged over rotating regions’ boundaries as boundary conditions.

2Cslip0,1slip )τ(τCv −=

wTT = (5.31),

wqq = (5.32)


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Periodic Boundary Conditions

The "periodicity" condition may be applied if the model consist of identical geometrical features arranged in periodic linear order. Periodic boundary conditions are specified at the pair of computational domain boundaries for the selected direction in which a geometrical feature or a group of features repeats regularly over distance. Periodic boundary conditions allows to reduce the analysis time by calculating the fluid flow only for a small group of identical geometrical features or even just for one feature, but taking into account influence of other identical features in the pattern. Please note that the number of basic mesh cells along the direction in which the "periodicity" condition is applied must be no less than five.

3 Numerical Solution Technique

The numerical solution technique employed in COSMOSFloWorks is robust and reliable, so it does not require any user knowledge about the computational mesh and the numerical methods employed. But sometimes, if the model and/or the problem being solved are too complicated, so that the COSMOSFloWorks standard numerical solution technique requires extremely high computer resources (memory and/or CPU time) which are not available, it is expedient to employ COSMOSFloWorks options which allow the adjustment of the automatically specified values of parameters governing the numerical solution technique. To employ these options properly and successfully, take into account the information presented below about COSMOSFloWorks’ numerical solution technique.

Briefly, COSMOSFloWorks solves the governing equations with the finite volume (FV) method on a spatially rectangular computational mesh designed in the Cartesian coordinate system with the planes orthogonal to its axes and refined locally at the solid/fluid interface and, if necessary, additionally in specified fluid regions, at the solid/solid surfaces, and in the fluid region during calculation. Values of all the physical variables are stored at the mesh cell centers. Due to the FV method, the governing equations are discretized in a conservative form. The spatial derivatives are approximated with implicit difference operators of second-order accuracy. The time derivatives are approximated with an implicit first-order Euler scheme. The viscosity of the numerical scheme is negligible with respect to the fluid viscosity.

3.1 Computational Mesh

COSMOSFloWorks computational mesh is rectangular everywhere in the computational domain, so the mesh cells’ sides are orthogonal to the specified axes of the Cartesian coordinate system and are not fitted to the solid/fluid interface. As a result, the solid/fluid interface cuts the near-wall mesh cells. Nevertheless, due to special measures, the mass and heat fluxes are treated properly in these cells named partial.

The rectangular computational domain is automatically constructed (may be changed


manually), so it encloses the solid body and has the boundary planes orthogonal to the specified axes of the Cartesian coordinate system. Then, the computational mesh is constructed in the following several stages.


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Chapter Numerical Solution Technique

First of all, a basic mesh is constructed. For that, the computational domain is divided into slices by the basic mesh planes, which are evidently orthogonal to the axes of the Cartesian coordinate system. The user can specify the number and spacing of these planes along each of the axes. The so-called control planes whose position is specified by user can be among these planes also. The basic mesh is determined solely by the computational domain and does not depend on the solid/fluid interfaces.

Then, the basic mesh cells intersecting with the solid/fluid interface are split uniformly into smaller cells in order to capture the solid/fluid interface with mesh cells of the specified size (with respect to the basic mesh cells). The following procedure is employed: each of the basic mesh cells intersecting with the solid/fluid interface is split uniformly into 8 child cells; each of the child cells intersecting with the interface is in turn split into 8 cells of next level, and so on, until the specified cell size is attained.

At the next stage of meshing, the mesh obtained at the solid/fluid interface with the previous procedure is refined (i.e. the cells are split further or probably merged) in accordance with the solid/fluid interface curvature. The criterion to be satisfied is established as follows: the maximum angle between the normals to the surface inside one cell should not exceeds certain threshold, otherwise the cell is split into 8 cells.

Finally, the mesh obtained with these procedures is refined in the computational domain to satisfy the so-called narrow channel criterion: for each cell lying at the solid/fluid interface, the number of the mesh cells (including the partial cells) lying in the fluid region along the line normal to the solid/fluid interface and starting from the center of this cell must not be less than the criterion value. Otherwise each of the mesh cells on this line is split into 8 child cells.

As a result of all these meshing procedures, a locally refined rectangular computational mesh is obtained and used then for solving the governing equations on it.

Since all the above-mentioned meshing procedures are performed before the calculation, the obtained mesh is unable to resolve all the solution features well. To overcome this disadvantage, the computational mesh can be refined further at the specified moments during the calculation in accordance with the solution spatial gradients (both in fluid and in solid, see User’s Guide for details). As a result, in the low-gradient regions the cells are merged, whereas in the high-gradient regions the cells are split. The moments of the computational mesh refinement during the calculation are prescribed either automatically or manually.


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3.2 Spatial Approximations

The cell-centered finite volume (FV) method is used to obtain conservative approximations of the governing equations on the locally refined rectangular mesh. The governing equations are integrated over a control volume which is a grid cell, and then approximated with the cell-centered values of the basic variables. The integral conservation laws may be represented in the form of the cell volume and surface integral equation:

are replaced by the discrete form

The second-order upwind approximations of fluxes F are based on the implicitly treated modified Leonard's QUICK approximations (Ref. 3) and the Total Variation Diminishing (TVD) method (Ref. 4).

In COSMOSFloWorks, especially consistent approximations for the convective terms, div and grad operators are employed in order to derive a discrete problem that maintains the fundamental properties of the parent differential problem in addition to the usual properties of mass, momentum and energy conservation.

Spatial Approximations at the Solid/fluid Interface

Considering equation (5.34) for partial mesh cells (i.e., for the mesh cells cut by the solid/fluid interface), we introduce the additional boundary faces and the corresponding boundary fluxes taking the boundary conditions and geometry into account (see Fig.3.1), as well as use a special calculation procedure for them. As a result, the solid/fluid interface influence on the problem solution both in the fluid and in the solid is calculated very accurately.

∫∫∫ =⋅+∂∂

dvdsdv QFUt

(5.33)

( ) QvSFUv =⋅+∂∂ ∑

facescellt

(5.34)


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3.3 Temporal Approximations

Time-implicit approximations of the continuity and convection/diffusion equations (for momentum, temperature, etc.) are used together with an operator-splitting technique (Ref. 5, Ref. 6, and Ref. 7). This technique is used to efficiently resolve the problem of pressure-velocity decoupling. Following the SIMPLE-like approach (Ref. 8), an elliptic type discrete pressure equation is derived by algebraic transformations of the originally derived discrete equations for mass and momentum, and taking into account the boundary conditions for velocity.

Fig.3.1 Computational mesh cells at the solid/fluid interface.


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3.4 Form of the Numerical Algorithm

Let index 'n' denotes the time-level, and '*' denotes intermediate values of the flow parameters. The following numerical algorithm is employed to calculate flow parameters on time-level (n+1) using known values on time-level (n):

ρ* = ρ(pn+δp,T*,y*),

Here U = (ρu, ρT, ρκ, ρε, ρy)Tis the full set of basic variables excluding pressure p,

u=(u1,u2,u3)T is the velocity vector, y = (y1, y2, ..., yM)T is the vector of component

concentrations in fluid mixtures, and δp = pn+1 - pn is an auxiliary variable that is called a pressure correction. These parameters are discrete functions stored at cell centers. They are to be calculated using the discrete equations (5.35)-(5.40) that approximate the governing differential equations. In equations (5.35)-(5.40) Ah, divh, gradh and Lh =

divhgradh are discrete operators that approximate the corresponding differential operators

with second order accuracy.

Equation (5.35) corresponds to the first step of the algorithm when fully implicit discrete convection/diffusion equations are solved to obtain the intermediate values of momentum and the final values of turbulent parameters, temperature, and species concentrations.

The elliptic type equation (5.36) is used to calculate the pressure correction δp. This

equation is defined in such a way that the final momentum field ρun+1 calculated from (5.35) satisfies the discrete fully implicit continuity equation. Final values of the flow parameters are defined by equations (5.37)-(5.40).

( ) nnh SpA

t=+

∆*

*UU

U-U,n

n, (5.35)

( ) ,

tttpL

nh

h ∆−

∆+

∆= ρρρδ

** 1udiv (5.36)

(5.37), 1 pt h

n δρρ grad⋅∆−=+ *uu

,ppp nn δ+=+1 (5.38)

, , , ,TT 1nn1nn **1**1 yy ρρρερερκρκρρ ==== ++++ (5.39)

( ) y 1111 ,, ++++ = nnnn Tpρρ (5.40).


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3.5 Methods to Resolve Linear Algebraic Systems

Iterative Methods for Nonsymmetrical Problems

To solve the asymmetric systems of linear equations that arise from approximations of momentum, temperature and species equations (5.35), a preconditioned generalized conjugate gradient method (Ref. 9) is used. Incomplete LU factorization is used for preconditioning.

Iterative Methods for Symmetric Problems

To solve symmetric algebraic problem for pressure-correction (5.36), an original double-preconditioned iterative procedure is used. It is based on a specially developed multigrid method (Ref. 10).

Multigrid Method

The multigrid method is a convenient acceleration technique which can greatly decrease the solution time. Basic features of the multigrid algorithm are as follows. Based on the given mesh, a sequence of grids (grid levels) are constructed, with a decreasing number of nodes. On every such grid, the residual of the associated system of algebraic equations is restricted onto the coarser grid level, forming the right hand side of the system on that grid. When the solution on the coarse grid is computed, it is interpolated to the finer grid and used there as a correction to the result of the previous iteration. After that, several smoothing iterations are performed. This procedure is applied repeatedly on every grid level until the corresponding iteration meets the stopping criteria.

The coefficients of the linear algebraic systems associated with the grid are computed once and stored.


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References

1 Henderson, C.B. Drag Coefficients of Spheres in Continuum and Rarefied Flows. AIAA Journal, v.14, No.6, 1976.

2 Carlson, D.J. and Hoglund, R.F. Particle Drag and Heat Transfer in Rocket Nozzles. AIAA Journal, v.2, No.11, 1964.

3 Roache, P.J., (1998) Fundamentals of Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, New Mexico, USA.

4 Hirsch, C., (1988). Numerical computation of internal and external flows. John Wiley and Sons, Chichester.

5 Glowinski, R. and P. Le Tallec, (1989). Augmented Lagrangian Methods and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia.

6 Marchuk, G.I., (1982). Methods of Numerical Mathematics, Springer-Verlag, Berlin.

7 Samarskii, A.A., (1989). Theory of Difference Schemes, Nauka, Moscow (in Russian).

8 Patankar, S.V., (1980). Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, D.C.

9 Saad, Y. (1996). Iterative methods for sparse linear systems, PWS Publishing Company, Boston.

10 Hackbusch, W. (1985). Multi-grid Methods and Applications, Springer-Verlag, NY, USA.

11 Reid R.C., Prausnitz J.M., Poling B.E. (1987). The properties of gases and liquids, 4th edition, McGraw-Hill Inc., NY, USA.

12 Idelchik, I.E. (1986). Handbook of Hydraulic Resistance, 2nd edition, Hemisphere, New York, USA.


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