90939790-CFD.pdf

About CFD:- Computational Fluid Dynamics or CFD is the analysis of systems involving fluid flow, heat transfer and associated phenomena such as chemical reactions by means of computer-based simulation. The technique is very powerful and

spans a wide range of industrial and non-industrial application areas. Some examples are:

• Aerodynamics of aircraft and vehicles: lift and drag. • Hydrodynamics of ships.

• Power plant: combustion in IC engines and gas turbines. • Turbo machinery: flows inside rotating passages, diffusers etc. • Electrical and electronic engineering: cooling of equipment including

micro circuits. • Chemical process engineering: mixing and separation, polymer moulding. • External and internal environment of buildings: wind loading and heating

ventilation. • Marine engineering: loads on off-shore structures.

• Environmental engineering: distribution of pollutants and effluents. • Hydrology and oceanography: flows in rivers, estuaries, oceans. • Meteorology: weather prediction.

• Biomedical engineering: blood flows through arteries and veins.

The ultimate aim of developments in the CFD field is to provide a capability comparable to other CAE (Computer-Aided Engineering) tools such as stress

Introduction analysis codes. The main reason why CFD has lagged behind is the tremendous complexity of the underlying behaviour, which precludes a description of fluid flows that is at the same time economical and sufficiently complete.

The availability of affordable high performance computing hardware and the introduction of user- friendly interfaces have led to a recent upsurge of interest

and CFD is poised to make an entry into the wider industrial community in the 1990s. We estimate the minimum cost of suitable hardware to be between $5000 and

$10000 (plus annual maintenance costs). The perpetual licence fee for commercial software typically ranges from $10000 to $50000 depending on the

number of'added extras' required. CFD software houses can usually arrange annual licences as an alternative. Clearly the investment costs of a CFD capability are not small, but the total expense is not normally as great as that

of a high quality experimental facility.

CFD code work:- CFD codes are structured around the numerical algorithms that can tackle fluid flow problems. In order to provide easy access to their solving power all commercial CFD packages include sophisticated user interfaces to input

problem parameters and to examine the results.

Hence all codes contain three main elements: (i) a pre-processor,(ii) a solver and (iii) a post-processor. We briefly examine the function of each of these elements within the context of

a CFD code.

Pre-processor: Pre-processing consists of the input of a flow problem to a CFD program by

means of an operator-friendly interface and the subsequent transformation of this input into a form suitable for use by the solver.

The user activities at the pre-processing stage involve: • Definition of the geometry of the region of interest: the computational domain. • Grid generation-the sub-division of the domain into a number of smaller,

non- overlapping sub-domains: a grid (or mesh) of cells (or control volumes or elements).

• Selection of the physical and chemical phenomena that need to be modelled. • Definition of fluid properties. • Specification of appropriate boundary conditions at cells which coincide with

or touch the domain boundary.

The solution to a flow problem (velocity, pressure, temperature etc.) is defined at nodes inside each cell. The accuracy of a CFD solution is governed by the number of cells in the grid. In general, the larger the number of cells the better

the solution accuracy. Both the accuracy of a solution and its cost in terms of necessary computer hardware and calculation time are dependent on the fineness of the grid. Optimal meshes are often non-uniform finer in areas

where large variations occur from point to point and coarser in regions with relatively little change.

Efforts are under way to develop CFD codes with a self-adaptive meshing capability. Ultimately such programs will automatically refine the grid in areas

of rapid variations. A substantial amount of basic development work still needs to be done before these techniques are robust enough to be incorporated into commercial CFD

codes. At present it is still up to the skills of the CFD user to design a grid that is a suitable compromise between desired accuracy and solution cost.

Over 50% of the time spent in industry on a CFD project is devoted to the definition of the domain geometry and grid generation. In order to maximise

productivity of CFD personnel all the major codes now include their own CAD-style interface and or facilities to import data from proprietary surface

modellers and mesh generators such as PATRAN and I-DEAS.

Solver:

There are three distinct streams of numerical solution techniques: finite difference,finite element and spectral methods. In outline the numerical methods that form the basis of the solver perform the

following steps: • Approximation of the unknown flow variables by means of simple functions.

• Discretisation by substitution of the approximations into the governing flow equations and subsequent mathematical manipulations. • Solution of the algebraic equations.

The main differences between the three separate streams are associated with the way in which the flow variables are approximated and with the discretisation processes.

Finite difference methods: Finite difference methods describe the unknowns Ф of the flow problem by means of point samples at the node points of a grid of co-ordinate Introduction lines. Truncated Taylor series expansions are often

used to generate finite difference approximations of derivatives of Ф in terms of point samples of Ф at each grid point and its immediate neighbours.

Those derivatives appearing in the governing equations are replaced by finite differences yielding an algebraic equation for the values of Ф at each grid point. Smith (1985) gives a comprehensive account of

all aspects of the finite difference method.

Finite Element Method: Finite element methods use simple piecewise functions (e.g. linear or quadratic) valid on elements to describe the local variations of

unknown flow variables Ф. The governing equation is precisely satisfied by the exact solution Ф. If the piecewise approximating functions for Ф are substituted into the equation it will not hold exactly and a residual is defined to measure

the errors. Next the residuals (and hence the errors) are minimised in some sense by multiplying them by a set of weighting functions and integrating. As a

result we obtain a set of algebraic equations for the unknown coefficients of the approximating functions. The theory of finite elements has been developed initially for structural stress analysis. A

standard work for fluids applications is Zienkiewicz and Taylor (1991).

Spectral Methods: Spectral methods approximate the unknowns by means of truncated Fourier series or series of Chebyshev polynomials. Unlike the finite

difference or finite element approach the approximations are not local but valid throughout the entire computational domain. Again we replace the unknowns

in the governing equation by the truncated series. The constraint that leads to the algebraic equations for the coefficients of the Fourier or Chebyshev series is provided by a weighted residuals concept similar to the finite element method

or by making the approximate function coincide with the exact solution at a number of grid points. Further information on this specialised method can be found in Gottlieb and Orszag (1977).

The finite volume method: The finite volume method (FVM) is a common approach used in CFD codes.The governing equations are solved over discrete control volumes. Finite volume methods recast the governing partial differential

equations (typically the Navier-Stokes equations) in a conservative form, and then discretize the new equation. This guarantees the conservation of fluxes

through a particular control volume. Finite-volume methods have become popular

in CFD as a result, primarily, of two advantages. First, they ensure that the

discretization is conservative, i.e., mass, momentum, and energy are conserved

in a

discrete sense. While this property can usually be obtained using a finite-

difference

formulation, it is obtained naturally from a finite-volume formulation.

Second, finitevolume

methods do not require a coordinate transformation in order to be applied on

irregular meshes. As a result, they can be applied on unstructured meshes

consisting

of arbitrary polyhedra in three dimensions or arbitrary polygons in two

dimensions.

This increased flexibility can be used to great advantage in generating grids

about

arbitrary geometries.

Finite-volume methods are applied to the integral form of the governing

equations,

The finite volume method was originally developed as a

special finite difference formulation. This book shall be solely concerned

with this

most well-established and thoroughly validated general purpose CFD technique.

It is

central to four of the five main commercially available CFD codes: PHOENICS,

FLUENT, FL0W3D and STAR-CD. The numerical algorithm consists of the

following steps:

• Formal integration of the governing equations of fluid flow over all the

(finite)

control volumes of the solution domain.

• Discretisation involves the substitution of a variety of finite-difference-

type

approximations for the terms in the integrated equation representing flow

processes such as convection, diffusion and sources. This converts the

integral

equations into a system of algebraic equations.

• Solution of the algebraic equations by an iterative method. Post-processor

As in pre-processing a huge amount of development work has recently taken

place in

the post-processing field. Owing to the increased popularity of engineering

workstations, many of which have outstanding graphics capabilities, the

leading

CFD packages are now equipped with versatile data visualisation tools. These

include:

• Domain geometry and grid display

• Vector plots

• Line and shaded contour plots

• 2D and 3D surface plots

• Particle tracking

• View manipulation (translation, rotation, scaling etc.)

• Colour postscript output

More recently these facilities may also include animation for dynamic result

display

and in addition to graphics all codes produce trusty alphanumeric output and

have

data export facilities for further manipulation external to the code. As in

many other

branches of C AE the graphics output capabilities of CFD codes have

revolutionised

the communication of ideas to the non-specialist.

1.3 Problem solving with CFD

In solving fluid flow problems we need to be aware that the underlying physics

is

complex and the results generated by a CFD code are at best as good as the

physics

Introduction

(and chemistry) embedded in it and at worst as good as its operator.

A good understanding of the numerical solution algorithm is also crucial.

Three

mathematical concepts are useful in determining the success or otherwise of

such

algorithms: convergence, consistency and stability. Convergence is the

property of a

numerical method to produce a solution which approaches the exact solution as

the

grid spacing, control volume size or element size is reduced to zero.

Consistent

numerical schemes produce systems of algebraic equations which can be

demonstrated to be equivalent to the original governing equation as the grid

spacing tends to zero. Stability is associated with damping of errors as the

numerical

method proceeds. If a technique is not stable even roundoff errors in the

initial data

can cause wild oscillations or divergence.

. Engineers need CFD codes that

produce physically realistic results with good accuracy in simulations with

finite

(sometimes quite coarse) grids.

.

These are discussed further in Chapter 5;

here we highlight three crucial properties of robust methods:

conservativeness,

boundedness and transportiveness.

The finite volume approach guarantees local conservation of a fluid property

(p

for each control volume. Numerical schemes which possess the conservativeness

property also ensure global conservation of the fluid property for the entire

domain.

This is clearly important physically and is achieved by means of consistent

expressions for fluxes of <j> through the cell faces of adjacent control

volumes. The

boundedness property is akin to stability and requires that in a linear

problem

without sources the solution is bounded by the maximum and minimum boundary

values of the flow variable. Boundedness can be achieved by placing

restrictions on

Problem solving with CFD 7

the magnitude and signs of the coefficients of the algebraic equations.

Although flow

problems are non-linear it is important to study the boundedness of a finite

volume

scheme for closely related, but linear, problems.

Finally all flow processes contain effects due to convection and diffusion. In

diffusive phenomena, such as heat conduction, a change of temperature at one

location affects the temperature in more or less equal measure in all

directions

around it. Convective phenomena involve influencing exclusively in the flow

direction so that a point only experiences effects due to changes at upstream

locations. Finite volume schemes with the transportiveness property must

account

for the directionality of influencing in terms of the relative strength of

diffusion to

convection.

.

Good

CFD often involves a delicate balancing act between solution accuracy and

stability.

The user needs a thorough appraisal of the extent to which conservativeness,

boundedness and transportiveness requirements are satisfied by a code.

Performing the actual CFD computation itself requires operator skills of a

different kind. Specification of the domain geometry and grid design are the

main

tasks at the input stage and subsequently the user needs to obtain a

successful

simulation result. The two aspects that characterise such a result are

convergence of

the iterative process and grid independence.

Good initial grid design

relies largely on an insight into the expected properties of the flow. A

background in

the fluid dynamics of the particular problem certainly helps and experience

with

gridding of similar problems is also invaluable. The only way to eliminate

errors due

to the coarseness of a grid is to perform a grid dependence study, which is a

procedure of successive refinement of an initially coarse grid until certain

key results

do not change. Then the simulation is grid independent. A systematic search

for

grid-independent results forms an essential part of all high quality CFD

studies.

Every numerical algorithm has its own characteristic error patterns. Well-

known

CFD euphemisms for the word error are terms such as numerical diffusion, false

diffusion or even numerical flow. The likely error patterns can only be

guessed on

the basis of a thorough knowledge of the algorithms.

. It is the intention of this book to provide all the

necessary background material for a good understanding of the internal

workings of

a CFD code and its successful operation.

The field of computational fluid dynamics has a broad range of

applicability.Independentof the specific

applicationunderstudy,thefollowingsequence of steps generally

must be followed in order to obtain a satisfactory solution.

1.2.1 Problem Specification and Geometry Preparation The first step involves the specification of the problem, including

thegeometry, flowconditions, and the requirements of the simulation. The

geometry may result from1.2. BACKGROUND 3 measurements of an existing configuration or may be associated with a design

study.

Alternatively, in a design context, no geometry need be supplied. Instead, a

set

of objectives and constraints must be specified. Flow conditions might

include, for

example, the Reynolds number and Mach number for the flow over an airfoil. The

requirements of the simulation include issues such as the level of accuracy

needed, the

turnaround time required, and the solution parameters of interest. The first

two ofSelection of Governing Equations and Boundary Conditions Once the problem has been specified, an appropriate set of governing equations

and

boundary conditions must be selected. It is generally accepted that the

phenomena of

importance to the field of continuum fluid dynamics are governed by the

conservation

of mass, momentum, and energy. The partial differential equations resulting

from

these conservation laws are referred to as the Navier-Stokes equations.

However, in

the interest of efficiency, it is always prudent to consider solving

simplified forms

of the Navier-Stokes equations when the simplifications retain the physics

which are

essential to the goals of the simulation. Possible simplified governing

equations include

the potential-flow equations, the Euler equations, and the thin-layer Navier-

Stokes

equations. These may be steady or unsteady and compressible or incompressible.

Boundary types which may be encountered include solid walls, inflow and

outflow

boundaries, periodic boundaries, symmetry boundaries, etc. The boundary

conditions

which must be specified depend upon the governing equations. For example, at a

solid

wall, the Euler equations require flow tangency to be enforced, while the

Navier-Stokes

equations require the no-slip condition. If necessary, physical models must be

chosen

for processes which cannot be simulated within the specified constraints.

Turbulence

is an example of a physical process which is rarely simulated in a practical

Selection of Gridding Strategy and Numerical Method Next a numerical method and a strategy for dividing the flow domain into

cells, or

elements, must be selected. We concern ourselves here only with numerical

methods

requiring such a tessellation of the domain, which is known as a grid, or

mesh.context (at

the time of writing) and thus is often modelled. The success of a simulation

depends

greatly on the engineering insight involved in selecting the governing

equations and

physical models based on the problem specification.

these requirements are often in conflict and compromise is necessary. As an

example

of solution parameters of interest in computing the flowfield about an

airfoil, one may

be interested in i) the lift and pitching moment only, ii) the drag as well as

the lift

and pitching moment, or iii) the details of the flow at some specific

location. Many different gridding strategies exist, including structured,

unstructured, hybrid,

composite, and overlapping grids. Furthermore, the grid can be altered based

on

the solution in an approach known as solution-adaptive gridding. The numerical

methods generally used in CFD can be classified as finite-difference, finite-

volume,

finite-element, or spectral methods. The choices of a numerical method and a

gridding

strategy are strongly interdependent. For example, the use of finite-

difference

methods is typically restricted to structured grids. Here again, the success

of a simulation

can depend on appropriate choices for the problem or class of problems of

interest. Assessment and Interpretation of Results Finally, the results of the simulation must be assessed and interpreted. This

step can

require post-processing of the data, for example calculation of forces and

moments,

and can be aided by sophisticated flow visualization tools and error

estimation techniques.

It is critical that the magnitude of both numerical and physical-model errors

be well understood.

Solution algorithms Discretization in space produces a system of ordinary differential equations for unsteady problems and algebraic equations for steady problems. Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of

(usually) nonlinear algebraic equations. Applying a Newton or Picard iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefinite in the presence of incompressibility. Such systems, particularly in 3D, are frequently too large for direct solvers, so iterative methods are used, either stationary methods such as successive overrelaxation or Krylov subspace methods. Krylov methods such as GMRES, typically used with preconditioning, operate by minimizing the residual over successive subspaces generated by the preconditioned operator. Multigrid has the advantage of asymptotically optimal performance on many problems. Traditional[according to whom?] solvers and

preconditioners are effective at reducing high-frequency components of the residual, but low-frequency components typically require many iterations to reduce. By operating on multiple scales, multigrid reduces all components of the residual by similar factors, leading to a mesh-independent number of iterations.[citation needed] For indefinite systems, preconditioners such as incomplete LU factorization, additive Schwarz, and multigrid perform poorly or fail entirely, so the problem structure must be used for effective preconditioning.[16] Methods commonly used in CFD are the SIMPLE and Uzawa algorithms which exhibit mesh-dependent convergence rates, but recent advances based on block LU factorization combined with multigrid for the resulting definite

systems have led to preconditioners that deliver mesh-independent convergence rates. Methodology In all of these approaches the same basic procedure is followed. During preprocessing The geometry (physical bounds) of the problem is defined. The volume occupied by the fluid is divided into discrete cells (the mesh). The mesh may be uniform or non uniform. The physical modeling is defined – for example, the equations of motions + enthalpy + radiation + species conservation Boundary conditions are defined. This involves specifying the fluid behaviour

and properties at the boundaries of the problem. For transient problems, the initial conditions are also defined. The simulation is started and the equations are solved iteratively as a steady-state or transient. Finally a postprocessor is used for the analysis and visualization of the resulting solution. [edit]Discretization methods The stability of the chosen discretization is generally established numerically rather than analytically as with simple linear problems. Special care must also be taken to ensure that the discretization handles discontinuous solutions gracefully. The Euler equations and Navier–Stokes equations both admit shocks, and contact surfaces. Some of the discretization methods being used are:

[edit]Finite volume method Main article: Finite volume method The finite volume method (FVM) is a common approach used in CFD codes.[citation needed] The governing equations are solved over discrete control volumes. Finite volume methods recast the governing partial differential equations (typically the Navier-Stokes equations) in a conservative form, and then discretize the new equation. This guarantees the conservation of fluxes through a particular control volume. The finite volume equation yields governing equations in the form, where Q is the vector of conserved variables, F is the vector of fluxes (see Euler equations or Navier–Stokes equations), V is the volume of the control volume element, and is the

surface area of the control volume element. [edit]Finite element method Main article: Finite element method The finite element method (FEM) is used in structural analysis of solids, but is also applicable to fluids. However, the FEM formulation requires special care to ensure a conservative solution. The FEM formulation has been adapted for use with fluid dynamics governing equations.[citation needed] Although FEM must be carefully formulated to be conservative, it is much more stable than the finite volume approach[4] However, FEM can require more memory than FVM.[5] In this method, a weighted residual equation is formed:

where Ri is the equation residual at an element vertex i, Q is the conservation equation expressed on an element basis, Wi is the weight factor, and Ve is the volume of the element. [edit]Finite difference method Main article: Finite difference method The finite difference method (FDM) has historical importance[citation needed] and is simple to program. It is currently only used in few specialized codes.[citation needed] Modern finite difference codes make use of an embedded boundary for handling complex geometries, making these codes highly efficient and accurate.[citation needed] Other ways to handle geometries include use of overlapping grids, where the solution is interpolated across each grid.

where Q is the vector of conserved variables, and F, G, and H are the fluxes in the x, y, and z directions respectively. [edit]Spectral element method Main article: Spectral element method Spectral element method is a finite element type method. It requires the mathematical problem (the partial differential equation) to be casted in a weak formulation. This is typically done by multiplying the differential equation by an arbitrary test function and integrating over the whole domain. Purely mathematically, the test functions are completely arbitrary - they belong to an infinitely dimensional function space. Clearly an infinitely dimensional function space cannot be represented on a discrete spectral element mesh. And this is where the spectral element

discretization begins. The most crucial thing is the choice of interpolating and testing functions. In a standard, low order FEM in 2D, for quadrilateral elements the most typical choice is the bilinear test or interpolating function of the form v(x,y) = ax + by + cxy + d. In a spectral element method however, the interpolating and test functions are chosen to be polynomials of a very high order (typically e.g. of the 10th order in CFD applications). This guarantees the rapid convergence of the method. Furthermore, very efficient integration procedures must be used, since the number of integrations to be performed in a numerical codes is big. Thus, high order Gauss integration quadratures are employed, since they achieve the highest accuracy with the smallest number of computations to be carried out. At the time there are some

academic CFD codes based on the spectral element method and some more are currently under development, since the new time-stepping schemes arrise in the scientific world. You can refer to the C-CFD website to see movies of incompressible flows in channels simulated with a spectral element solver or to the Numerical Mechanics (see bottom of the page) website to see a movie of the lid-driven cavity flow obtained with a compeletely novel unconditionally stable time-stepping scheme combined with a spectral element solver.

Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is performed using a wind tunnel with the final validation coming in full-scale testing, e.g. flight tests.

Incompressible flow A flow is said to be incompressible if the density of a fluid element does not change during its motion. It is a property of the flow and not of the fluid. The rate of change of density of a material fluid element is given by the material derivative From the continuity equation we have

Hence the flow is incompressible if the divergence of the velocity field is identically zero. Note that the density field need not be uniform in an incompressible flow. All that is required is that the density of a fluid element should not change in time as it moves through space. For example, flow in the ocean can be considered to be incompressible even though the density of water is not uniform due to stratification. Compressible flow can with good accuracy be approximated as incompressible for steady flow if the Mach number is below 0.3.

2.1 Lid-driven cavity flow This tutorial will describe how to pre-process, run and post-process a case involving isothermal, incompressible flow in a two-dimensional square domain. The geometry is shown in Figure 2.1 in which all the boundaries of the square are walls. The top wall moves in the -direction at a speed of 1 m/s while the other 3 are stationary. Initially, the flow will be assumed laminar and will be solved on a uniform mesh using the icoFoam solver for laminar, isothermal, incompressible flow. During the course of the tutorial, the effect of increased mesh resolution and mesh

grading towards the walls will be investigated. Finally, the flow Reynolds number will be increased and the pisoFoam solver will be used for turbulent, isothermal, incompressible flow. Figure 2.1: Geometry of the lid driven cavity. Features of OpenFOAM

The OpenFOAM® (Open Field Operation and Manipulation) CFD Toolbox is a free, open source CFD software package produced by OpenCFD Ltd. It has a large user base across most areas of engineering and science, from both commercial and academic organisations. OpenFOAM has an extensive range of features to solve anything from complex fluid flows involving chemical reactions, turbulence and heat transfer, to solid dynamics and electromagnetics. It includes tools for meshing, notably snappyHexMesh, a parallelised mesher for complex CAD geometries, and for pre- and post-processing. Almost everything (including meshing, and pre- and post-processing) runs in parallel as standard, enabling users to take full advantage of computer hardware at their disposal.

By being open, OpenFOAM offers users complete freedom to customise and extend its existing functionality, either by themselves or through support from OpenCFD. It follows a highly modular code design in which collections of functionality (e.g. numerical methods, meshing, physical models, …) are each compiled into their own shared library. Executable applications are then created that are simply linked to the library functionality. OpenFOAM includes over 80 solver applications that simulate specific problems in engineering mechanics and over 170 utility applications that perform pre- and post-processing tasks, e.g. meshing, data visualisation, etc.

Standard Solvers An extensive set of OpenFOAM solvers has evolved (and is forever growing) that are available to users. OpenFOAM is used mainly for CFD but has found use in other areas such as stress analysis, electromagnetics and finance because it is fundamentally a tool for solving partial differential equations rather than a CFD package in the traditional sense. Below is a list of solvers available to our users. „Basic‟ CFD codes

laplacianFoam Solves a simple Laplace equation, e.g. for thermal diffusion in a solid potentialFoam Simple potential flow solver which can be used to generate starting fields for full Navier-Stokes codes scalarTransportFoam Solves a transport equation for a passive scalar Incompressible flow adjointShapeOptimizationFoam Steady-state solver for incompressible, turbulent flow of non-Newtonian fluids with optimisation of duct shape by applying ”blockage” in regions causing pressure loss as estimated using an adjoint formulation boundaryFoam Steady-state solver for incompressible, 1D turbulent flow, typically to generate boundary layer

conditions at an inlet, for use in a simulation channelFoam Incompressible LES solver for flow in a channel icoFoam Transient solver for incompressible, laminar flow of Newtonian fluids MRFSimpleFoam Steady-state solver for incompressible, turbulent flow of non-Newtonian fluids with MRF regions nonNewtonianIcoFoam Transient solver for incompressible, laminar flow of non-Newtonian fluids pimpleDyMFoam Transient solver for incompressible, flow of Newtonian fluids on a moving mesh using the PIMPLE (merged PISO-SIMPLE) algorithm pimpleFoam Large time-step transient solver for incompressible,

flow using the PIMPLE (merged PISO-SIMPLE) algorithm pisoFoam Transient solver for incompressible flow porousSimpleFoam Steady-state solver for incompressible, turbulent flow with implicit or explicit porosity treatment shallowWaterFoam Transient solver for inviscid shallow-water equations with rotation simpleFoam Steady-state solver for incompressible, turbulent flow SRFSimpleFoam Steady-state solver for incompressible, turbulent flow of non-Newtonian fluids in a single rotating frame windSimpleFoam Steady-state solver for incompressible, turbulent flow with external source in the momentum equation

Mesh Generation OpenFOAM supports unstructured meshes of cells of any shape; cells in OpenFOAM can have any number of faces and faces can have any number of edges. Such complete freedom on cell shape gives greater flexibility for the generation of meshes around complex shapes, embedded refinement, etc. There is a need in CFD for multiple meshing tools that cover a range of complexity of meshing task. At one

extreme, there is meshing software that allows the user to define simple geometries and mesh to those geometries. At the other extreme, there is software that meshes to highly complex CAD surfaces. In between, there is room for one or two tools that generate optimal meshes for moderately complex surfaces. blockMesh At present, OpenFOAM has meshing software that operate at the extremes of geometry complexity. For simple geometries, there is blockMesh, a multi-block mesh generator that generates meshes of hexahedra from a text configuration file. The OpenFOAM distribution contains numerous example configuration files for blockMesh to generate meshes for

flows around simple geometries, e.g. a cylinder, a wedge, etc. snappyHexMesh For complex geometries, there is snappyHexMesh that meshes to surfaces from CAD, but also allows the user to define simple geometric entities such as boxes, spheres, planes, etc. The snappyHexMesh utility can run in parallel, so can generate meshes of 100s of millions of cells, given a sufficient number of CPUs and memory. It performs automatic load balancing when generating a mesh in parallel, i.e. redistributes cells across the CPUs, so the final mesh has approximately the same number cells per processor, for optimal performance during simulation. Other mesh generation tools

There are some other tools that can be used to generate meshes in OpenFOAM: extrudeMesh: Generates a mesh by extruding cells from a patch of an existing mesh, or from a surface mesh; can do a range of extrusions to produce linear, wedge or spherical meshes. polyDualMesh: Creates the dual of a polyMesh, e.g. can be used to generate a “honeycomb” polyhedral mesh from a tetrahedral mesh

The Finite Volume Method for

Diffusion Problems

4.1 Introduction

The nature of the transport equations governing fluid flow and heat transfer

and the

formal control volume integration were described in Chapter 2. Here we develop

the

numerical method based on this integration, the finite volume (or control

volume)

method, by considering the simplest transport process of all: pure diffusion

in the

steady state. The governing equation of steady diffusion can easily be derived

from

the general transport equation B.39) for property (f> by deleting the

transient and

convective terms. This gives

div(T grad <j>) + 5^ = 0 D.1)

The control volume integration, which forms the key step of the finite volume

method that distinguishes it from all other CFD techniques, yields the

following

form:

f div{T grad 4>)dV + f S^dV = f n . (r grad cj>)dA + I S^dV = 0

CV CV A CV

D.2)

By working with the one-dimensional steady state diffusion equation the

approximation techniques that are needed to obtain the so-called discretised

equations are introduced. Later the method is extended to two- and three-

dimensional diffusion problems. Application of the method to simple one-

dimensional steady state heat transfer problems is illustrated through a

series of

worked examples and the accuracy of the method is gauged by comparing

numerical

results with analytical solutions.

The Finite Volume Method for

Convection-Diffusion Problems

5.1 Introduction

In problems where fluid flow plays a significant role we must account for the

effects

of convection. Diffusion always occurs alongside convection in nature so here

we

examine methods to predict combined convection and diffusion. The steady

convection-diffusion equation can be derived from the transport equation B.39)

for

a general property </> by deleting the transient term

div(pu(f)) = div(T grad </>) + S^ E.1)

Formal integration over a control volume gives

n. {P(f)u)dA = f n. (F grad <j>)dA + f S+dV E.2)

a a cr

This equation represents the flux balance in a control volume. The left hand

side

gives the net convective flux and the right hand side contains the net

diffusive flux

and the generation or destruction of the property <f> within the control

volume.

The principal problem in the discretisation of the convective terms is the

calculation of the value of transported property <$> at control volume faces

and its

convective flux across these boundaries. In Chapter 4 we introduced the

central

differencing method of obtaining discretised equations for the diffusion and

source

terms on the right hand side of equation E.2). It would seem obvious to try

out this

practice, which worked so well for diffusion problems, on the convective

terms.

However, the diffusion process affects the distribution of a transported

quantity

along its gradients in all directions, whereas convection spreads influence

only in the

flow direction. This crucial difference manifests itself in a stringent upper

limit to the

grid size, that is dependent on the relative strength of convection and

diffusion, for

stable convection-diffusion calculations with central differencing.

Naturally, we also present the case for a number of alternative discretisation

practices for the convective effects which enable stable computations under

less

104 the finite volume method for convection-diffusion problems

restrictive conditions. In the current analysis no reference will be made to

the

evaluation of face velocities. It is assumed that they are 'somehow' known.

The

method of computing velocities will be discussed in Chapter 6.

The field of computational fluid dynamics has a broad range of

applicability.Independentof the specific

applicationunderstudy,thefollowingsequence of steps generally

must be followed in order to obtain a satisfactory solution.

1.2.1 Problem Specification and Geometry Preparation The first step involves the specification of the problem, including

thegeometry, flowconditions, and the requirements of the simulation. The

geometry may result from1.2. BACKGROUND 3 measurements of an existing configuration or may be associated with a design

study.

Alternatively, in a design context, no geometry need be supplied. Instead, a

set

of objectives and constraints must be specified. Flow conditions might

include, for

example, the Reynolds number and Mach number for the flow over an airfoil. The

requirements of the simulation include issues such as the level of accuracy

needed, the

turnaround time required, and the solution parameters of interest. The first

two ofSelection of Governing Equations and Boundary Conditions Once the problem has been specified, an appropriate set of governing equations

and

boundary conditions must be selected. It is generally accepted that the

phenomena of

importance to the field of continuum fluid dynamics are governed by the

conservation

of mass, momentum, and energy. The partial differential equations resulting

from

these conservation laws are referred to as the Navier-Stokes equations.

However, in

the interest of efficiency, it is always prudent to consider solving

simplified forms

of the Navier-Stokes equations when the simplifications retain the physics

which are

essential to the goals of the simulation. Possible simplified governing

equations include

the potential-flow equations, the Euler equations, and the thin-layer Navier-

Stokes

equations. These may be steady or unsteady and compressible or incompressible.

Boundary types which may be encountered include solid walls, inflow and

outflow

boundaries, periodic boundaries, symmetry boundaries, etc. The boundary

conditions

which must be specified depend upon the governing equations. For example, at a

solid

wall, the Euler equations require flow tangency to be enforced, while the

Navier-Stokes

equations require the no-slip condition. If necessary, physical models must be

chosen

for processes which cannot be simulated within the specified constraints.

Turbulence

is an example of a physical process which is rarely simulated in a practical

Selection of Gridding Strategy and Numerical Method Next a numerical method and a strategy for dividing the flow domain into

cells, or

elements, must be selected. We concern ourselves here only with numerical

methods

requiring such a tessellation of the domain, which is known as a grid, or

mesh.context (at

the time of writing) and thus is often modelled. The success of a simulation

depends

greatly on the engineering insight involved in selecting the governing

equations and

physical models based on the problem specification.

these requirements are often in conflict and compromise is necessary. As an

example

of solution parameters of interest in computing the flowfield about an

airfoil, one may

be interested in i) the lift and pitching moment only, ii) the drag as well as

the lift

and pitching moment, or iii) the details of the flow at some specific

location. Many different gridding strategies exist, including structured,

unstructured, hybrid,

composite, and overlapping grids. Furthermore, the grid can be altered based

on

the solution in an approach known as solution-adaptive gridding. The numerical

methods generally used in CFD can be classified as finite-difference, finite-

volume,

finite-element, or spectral methods. The choices of a numerical method and a

gridding

strategy are strongly interdependent. For example, the use of finite-

difference

methods is typically restricted to structured grids. Here again, the success

of a simulation

can depend on appropriate choices for the problem or class of problems of

interest. Assessment and Interpretation of Results Finally, the results of the simulation must be assessed and interpreted. This

step can

require post-processing of the data, for example calculation of forces and

moments,

and can be aided by sophisticated flow visualization tools and error

estimation techniques.

It is critical that the magnitude of both numerical and physical-model errors

be well understood.