1 Numerical Methods for Compressible Fluid Mechanics Numerical Methods for Compressible Fluid Mechanics Numerical Methods for Compressible Fluid Mechanics Numerical Methods for Compressible Fluid Mechanics 1. Introduction NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Elizabeth Mickaily-Huber, Dominique Charbonnier, Jan B. Vos [email protected][email protected][email protected][email protected]1 CFS Engineering CFS Engineering, PSE CFS Engineering, PSE CFS Engineering, PSE CFS Engineering, PSE-A, CH A, CH A, CH A, CH-1015 Lausanne 1015 Lausanne 1015 Lausanne 1015 Lausanne Numerical Methods for Compressible Fluid Mechanics Numerical Methods for Compressible Fluid Mechanics Numerical Methods for Compressible Fluid Mechanics Numerical Methods for Compressible Fluid Mechanics 1. Introduction The course lecturers 2 CFS Engineering Dr. Elizabeth Mickaily Huber Dr. Dominique Charbonnier Dr. Jan B. Vos
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
NUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWSNUMERICAL METHODS FOR COMPRESSIBLE FLOWS
Elizabeth Mickaily-Huber, Dominique Charbonnier, Jan B. Vos
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
The course lecturers
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Dr. Elizabeth Mickaily Huber Dr. Dominique Charbonnier Dr. Jan B. Vos
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Contents todays’ lecture
• Some words on who we are
• An example of studies we make• An example of studies we make
• Outline of the course
• Preparation of next week’s lecture
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Please switch off your mobile phones
Practical details
Course notes (polycopies) are the ones prepared by Dr. Alain Drotz + copies of the powerpoint presentations on the website of LIN
Please interrupt me to ask questions if something is not clear. Questions to me can be asked in English, French, German and Dutch
Exercises: two of the course days are reserved for exercises in a computer room
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Exercises: two of the course days are reserved for exercises in a computer room
Exam: orally, on the last day we will explain how the exam will be done
I have the habit to speak rapidly, please let me know if I go to fast !I have the habit to speak rapidly, please let me know if I go to fast !
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Who I am
Some words about myself
Studied Aerospace Engineering at Delft University in the Netherlands,specialization Theoretical Aerodynamics. Master degree in 1982 on thespecialization Theoretical Aerodynamics. Master degree in 1982 on thedevelopment of a 1D CFD (Computational Fluid Dynamics) code for MagnetoHydrodynamic Flow simulations
PhD in 1987 at the Delft University of Technology, topic Combustion in Solid FuelRamjets (development of a 2D CFD code with combustion)
Worked at EPFL from 1987 to 1999, mainly on the development of 3D CFDcodes for Aerospace Applications
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codes for Aerospace Applications
Founded CFS Engineering in 1999 and continued to work on the use anddevelopment of CFD codes
A large experience in developing and using CFD codes for a wide variety of A large experience in developing and using CFD codes for a wide variety of
applicationsapplications
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
• CFS Engineering (Computational Fluids & Structures) is a spin-off company created in 1999 and located at the Business park of EPFL (École Polytechnique Fédérale de Lausanne)
who we are: CFS Engineering
• The major shareholder of CFS Engineering is RUAG Aerospace
Mission of CFS Engineering
To offer services in the numerical simulation of To offer services in the numerical simulation of
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To offer services in the numerical simulation of To offer services in the numerical simulation of
Fluid Mechanics and Structural Mechanics problemsFluid Mechanics and Structural Mechanics problems
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFS Engineering
CFS Engineering – what do we do
Grid generation for fluid mechanics problems
Computational Fluid Dynamic simulations using the NSMB CFD code
Post-processing and analysis of the results
Coupled fluid dynamics-structural mechanics simulations using NSMB
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Adaptation of the NSMB code for specific applications
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Ansys ICEM CFD Tetra, Prism and Hexa for mesh generation
CFS Engineering
CFS Engineering – available tools
Ansys ICEM CFD Tetra, Prism and Hexa for mesh generation
Baspl++ and Paraview for visualization
B2000 for structural mechanics simulations
NSMB in house CFD code
Cluster of 10 Linux dual core PC’s for computing
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Several Linux PC workstations (organized in a cluster too)
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
The NSMB code was initially developed at EPFL in 1991. From 1994 to 2003 it was further developed in the so called NSMB project, composed of KTH
CFS Engineering: NSMB code
was further developed in the so called NSMB project, composed of KTH (Stockholm), SAAB Military Aircraft (Linkoping), CERFACS (Toulouse), Airbus France (Toulouse) and EPFL (Lausanne).
Today NSMB is further developed in a consortium composed of RUAG Aerospace (Emmen), IMFT (Toulouse), IMFS (Strassbourg), TU Munchen, Univ. of the Army (Munchen), ASTRIUM-ST (Les Mureaux), EPFL, ETHZ and CFS Engineering.
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NSMB offers all functionalities which can be expected from a modern CFD tool used in the Aerospace industry (turbulence modeling, numerical schemes, moving grids, flexibility for complex geometries).
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFS Engineering maintains the NSMB code, and is responsible for the parallelization of NSMB. CFS Engineering is working with PhD Students at
CFS Engineering: NSMB code
parallelization of NSMB. CFS Engineering is working with PhD Students at EPFL and ETHZ to extend the code with new turbulence and physical models.
CFS Engineering is working with SMR SA in Bienne to extend NSMB for the simulation of coupled engineering problems (fluid-heat transfer, fluid-structure).
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
NSMB has been used by Airbus-France in the design of aircraft (Airbus A330, A340, A380, A400M).
NSMB is used by ASTRIUM-ST for flows over missiles and re-entry vehicles (including CFD simulations over the Rafale Fighter Aircraft)
NSMB is used by IMFT in Toulouse to flows over oscillating airfoils and wings, etc.
NSMB is used by KTH in Stockholm for unsteady simulations over delta wings.
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NSMB is used by CFS Engineering and RUAG Aerospace to simulate the flow over the FA-18 fighter, the flow over UAVs, the flow in base bleed units, supersonic air intake flows, flows in nozzles, flows over re-entry vehicles, etc.
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Some figures (1)
Question: a numerical simulation that took 24 hours, 365 days in 1980 took how much time in 2005 ??
CFD: Introduction
much time in 2005 ??
1 second (a factor 32 million).
EPFL bought in 1989 a Cray 2 computer for more than 10 Million CHF. Today’s PC’s cost less than 1’000 CHF and deliver more computing power (a factor 10’000 in price).
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CFS Engineering: first PC cluster (1999) costed 20’000 CHF (6 PCs, 3GB total memory), last PC cluster (2008) costed 10’000 CHF (10 dual core PCs, 40 GB total memory) and is about 100 times faster.
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Some figures (2)
Over the last 20 years, large investments were made in developing CFD codes:
CFD: Introduction
Improvement of numerical methods including parallel computing
Improvement of physical modeling (turbulence, transition, ..)
The cost reduction of computing power, combined with more efficient numerical schemes has lead to an increase of use of CFD in industry, since it is cheaper and faster than experimental testing, and provides better understanding of the physics.
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physics.
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
An example of an application studied at CFS Engineering
CFD: Example of application
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Example of Application: FA-18 Fighter
Motivations to discuss this example:
CFD: Example of application
Motivations to discuss this example:
• Is concerned with CFD for compressible flows
• Shows an example of the use of CFD in industry
• Contains elements to future simulation environments
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
FA-18 studies
RUAG Aerospace and CFS Engineering performed CFD simulations from 2001 to 2007 on the FA-18 fighter.
CFD: Example of application
to 2007 on the FA-18 fighter.
2001: Compare CFD with Wind tunnel experiments.
2002: Extract Aerodynamic loads from CFD and compare with the Boeing Loads data base
2003: Sensitivity analysis different aircraft configurations
2004: Develop tool for static wing deformation
2005: New grid, study influence LEX fence, unsteady CFD for loads
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2005: New grid, study influence LEX fence, unsteady CFD for loads
2006: Dynamic Fluid Structure Interaction, influence SIWA fins on loads
2007: Study of Vertical Tail Buffeting
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Components FAComponents FA--18 Fighter 18 Fighter –– CFD model 2005CFD model 2005
VTAIL
CFD: Example of application
forward fuselage
center fuselage
aft fuselage
TEF
HSTAB
VTAIL
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forward fuselage
LEXILEF
OLEF
wing rootwing fold
SIWA
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Grid Generation FAGrid Generation FA--1818
Different aircraft configurations
various control surfaces deflections
with and without weapons or fuel tanks!
CFD: Example of application
Grid generated by ICEM CFD Hexa
contains ~ 14.0 Million grid points
contains around 3000 blocks
Replay files for control surface deflections and components addition or removal
various control surfaces deflections!
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To permit loads calculation on each aircraft component,
faces of a block single CAD surface familyMesh topology
each aircraft component
single CAD surface familyCAD surface families =
=
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Cut on wingCut on wing
CFD: Example of application
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Comparison CFD and Experiments, Mach=0.5Comparison CFD and Experiments, Mach=0.5
CFD: Example of application
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Flow FeaturesFlow Features
CFD: Example of application
StreamlinesMach = 0.95
Mach number contourssymmetry plan,
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symmetry plan, Mach = 0.95
Shock wavecanopy and fuselage near vertical fin
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Aerodynamic Load Studies for the FAAerodynamic Load Studies for the FA--18 fighter18 fighter
The US Navy executed in the 1980s a flight test program with an
CFD: Example of application
instrumented FA-18 fighter yielding the so called F4 Flight Test Data Base. Boeing used this data base to define different load cases for the Swiss FA-18 fleet, this is the Boeing Loads data base.
These data bases are incomplete for the conditions and usage of the Swiss FA-18 fighter (Swiss usage of the FA-18 fighter is three times more severe).
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A Full Scale Fatigue Test (FSFT) facility was build at RUAG.
CFD is used to provide the aerodynamic loads for different configurations (flap deflections, with/without fuel tanks), and flight conditions to complement the available load data bases.
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Full Scale Fatigue Test rig at RUAG AerospaceFull Scale Fatigue Test rig at RUAG Aerospace
CFD: Example of application
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFD CFD –– Boeing Boeing -- and Flight data base correlationand Flight data base correlation
CFD: Example of application
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Ingredients for dynamic Fluid Structure Interaction:
CFD: Example of application
Ingredients for dynamic Fluid Structure Interaction:
- Unsteady CFD solver with ALE formulation
- FSI transfer tool
- CSM solver (modal integration)
- Volume mesh deformation tool
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
AGARD 445.6 WingAGARD 445.6 Wing
The AGARD 445.6 wing was has a 45o quarter chord sweep, and a
CFD: Example of application
constant NACA64A004 symmetric profile
Measurements were made in the NASA Langley Transonic wind tunnel in 1963 to determine stability characteristics
Most published results are available for the so-called weakened wing in air
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For FSI calculations, four modes are considered, 2 bending modes and 4 torsional modes
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
AGARD 445.6 WingAGARD 445.6 Wing
Mode 1
CFD: Example of application
Mode 2
Mode 3 Mode 4
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Mode 3 Mode 4
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
AGARD 445.6 WingAGARD 445.6 Wing
CFD Parameters:
CFD: Example of application
Mach = 0.95
Rho_inf = 0.061 kg/m3
P_inf = 3500/4600/7000 Pa
Flutter index = 0.27/0.31/0.37
For this case the flutter boundary has a flutter index of 0.32.
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For this case the flutter boundary has a flutter index of 0.32.
Tests were made using different grid densities, different outer time steps, different time integration scheme, different values of the structural damping, different values of inner loop convergence criterium.
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
AGARD 445.6 WingAGARD 445.6 Wing
CFD: Example of application
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
AGARD 445.6 WingAGARD 445.6 Wing
CFD: Example of application
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
FAFA--18 C2S825 Load case18 C2S825 Load case
Unsteady calculation with and without dynamic FSI
CFD: Example of application
2000 time steps made to simulate 0.5 seconds of real time
Calculation time: about 10 days on a cluster of 10 PCs
Generated more than 1.4 TeraByte of data
Post processing took 2 days
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Post processing took 2 days
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFD: Example of application
Flight results
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Source: NASA
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Final Remarks FAFinal Remarks FA--18 studies18 studies
An example of the use of CFD in industry has been discussed
CFD: Example of application
In 2001 we used CFD to predict steady aerodynamic forces and loads
In 2007 we used CFD to study coupled CFD-CSM unsteady phenomena
Today people are planning the simulation of the so called digital aircraft (CFD + Structures + Flight Mechanics)
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The way industry is using CFD is changing rapidly !
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFD: Course Outline
Outline of the CourseOutline of the Course
The course contains 3 modulesThe course contains 3 modules
I Introduction to unsteady flows
II Construction of higher order schemes
III Monotonic schemes of higher order
Which corresponds to the modules of the course given by Dr. Alain Drotz in
previous years.
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We modified the course, less theory, more directed to the use of numerical
methods in practice.
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFD: Course Outline
Outline of the CourseOutline of the Course
Module I: Introduction to unsteady flowsModule I: Introduction to unsteady flows
1. Introduction to the course (today)
2. 3D Euler equations
3. 1D Euler equations
4. Unsteady 1 dimensional flows
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4. Unsteady 1 dimensional flows
5. Introduction to the Riemann problem
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFD: Course Outline
Outline of the CourseOutline of the Course
Module II Construction of higher order schemesModule II Construction of higher order schemes
6. Conservative discretization schemes
7. Exercises
8. Classical finite difference schemes
9. Riemann problem and Roe scheme
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9. Riemann problem and Roe scheme
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFD: Course Outline
Outline of the CourseOutline of the Course
Module III Construction of higher order schemesModule III Construction of higher order schemes
10. Roe and AUSM schemes
11. Higher order monotonic schemes
12. Boundary conditions and preparation exercise 2
13. Exercises 2
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14. Preparation of Exam and Assessing the quality of numerical
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFD: Course Outline
What is a compressible flow?What is a compressible flow?
A compressible flow is a flow for which the density can not be consideredA compressible flow is a flow for which the density can not be considered
constant.
In general this occurs for air flows with a free stream Mach number larger than
0.3
What is a hypersonic flow?What is a hypersonic flow?
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Flows with a free stream Mach number larger than 5. At very high free stream
Mach numbers chemistry effects become important.
Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFD: Course Outline
Flow over an Airfoil Flow over an Airfoil –– Mach number contoursMach number contours
Free stream Mach 0.745
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
CFD: Course Outline
Flow over an Airfoil Flow over an Airfoil -- CpCp
Free stream Mach 0.745
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Navier Stokes equations
The The NavierNavier Stokes equations in vector & differential formStokes equations in vector & differential form
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Navier Stokes equations
Closure relations for the Closure relations for the NavierNavier Stokes equationsStokes equations
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Navier Stokes equations
How to solve these equations?How to solve these equations?
1. The difficulty of solving the Navier Stokes equations are the inviscid or1. The difficulty of solving the Navier Stokes equations are the inviscid or
convective terms => ignore for the moment the viscous terms (they are in
general approximated using 2nd order differences)
2. The compressible Euler and Navier Stokes equations permit
discontinuities in the solution (shock waves, expansion waves). The
numerical formulation needs to resolve these discontinuities.
3. Often different strategies for incompressible or compressible flows due to
the nature of the equations
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the nature of the equations
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Navier Stokes equations
The Euler equations The Euler equations –– conservative formatconservative format
Incompressible flows: ρ is constant, energy equation is constant, energy equation is often not neededUnknowns: p, u, v, w
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Numerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid MechanicsNumerical Methods for Compressible Fluid Mechanics1. Introduction
Navier Stokes equations
To summarize (1)To summarize (1)
1. Compressible Euler equations: hyperbolic in time, unknowns are ρ, ρu,1. Compressible Euler equations: hyperbolic in time, unknowns are ρ, ρu,