Aspects ThØoriques et NumØriques pour les Fluides .... Frey, intro.… · Geometry,Fluid...

Aspects Théoriques et Numériques pour les Fluides IncompressiblesTheoretical and Numerical Issues of Incompressible Fluid Flows

Instructors: Pascal Frey1, Yannick Privat2

1 Sorbonne Université, Paris2 Université de Strasbourg, CNRS

master ANEDP, 2019-2020

Outline of the lectures

Part I - Classical Methods

1. Fluid mechanics1.1 Notations, vectors, tensors1.2 Conservation laws,1.3 Flow models and simpli�cations.

2. The Stokes model2.1 Mathematical and numerical analysis, variational formulation,2.2 Finite element approximation, resolution,2.3 Unsteady Stokes problem2.4 The Finite Element Method

3. The Navier-Stokes model3.1 Steady state problem, analysis3.2 Discretization procedures

Aspects théoriques et numériques pour les �uides incompressibles M2 ANEDP, UPMC, 2019-2020 2/ 14

Outline of the lectures

Part II - Advanced Methods

4. Two fluid or two-phase flow problems4.1 Problem statement, modelling4.2 Evolution of the interface: level set formalism, scheme4.3 Numerical resolution

5. Shape optimization5.1 Framework of shape optimization, examples5.2 Shape sensitivity analysis using shape derivatives5.3 Céa’s method to compute derivatives5.4 Numerical issues

6. Error estimates and mesh adaptation facultative6.1 Residual and geometric estimates6.2 Mesh adaptation

A. Appendix◦ Variational approximation, convergence of approximation schemes


References

• Functional and numerical analysis

1. Allaire G., Numerical analysis and optimization, Oxford Science Publishing, (2007).

2. Brezis H., Analisis funcional, Teoria y aplicaciones, Alianza Editorial, (1983).

3. Ciarlet P.G., The Finite Element Method for Elliptic Problems, SIAM Classics, 40 (2002).

4. Ern A., Guermond J.L., Theory and Practice of Finite Elements, Applied Mathematical Series, 159,Springer, (2004).

5. Evans L.C., Partial di�erential equations, AMS, (2002).

6. Frey P., George P.L., Mesh generation, application to �nite element methods, Wiley, (2008).

7. Johnson C., Numerical Solution of Partial Di�erential Equations by the Finite Element Method,Cambridge University Press, (1987).

8. Lax P.D., Functional Analysis, Wiley Interscience, (2002).

9. Oden J.T., Applied Functional Analysis, Prentice-Hall, (1979).

10. Quarteroni A., Valli A., Numerical approximation of partial di�erential equations, 23, SpringerSeries in Computational Mathematics, Springer Verlag, (1997).

11. Quarteroni A., Sacco R., Saleri F., Numerical Mathematics, Springer, Texts in Applied Mathematics,37, (1991).

12. Rudin W., Functional Analysis, Mc-Graw Hill, (1973).

13. Solin, P., Partial Di�erential Equations and the Finite Element Method, Wiler Interscience, (2006).

14. Yosida K., Functional Analysis, Springer, (1980).


References (2)

• Fluid mechanics

1. Chorin A.J., Marsden J.E., A mathematical introduction to �uid mechanics, 3rd ed., Springer, (1992).

2. Durst F., Fluid Mechanics. An introduction to the Theory of Fluid Flows, Springer, (2008).

3. Landau L.D., Lifschitz E.M., Fluid mechanics, Course in Theoretical Physics, vol 6., 2nd ed., Perga-mon Press, (1987).

4. Pnueli D., Gut�nger C., Fluid Mechanics, Cambridge University Press, Cambridge, (1992).

5. Temam R., Miranville A., Mathematical modeling in continuum mechanics, Cambridge UniversityPress, (2005).

• Computational Fluid Dynamics

1. Acheson D.J., Elementary Fluid Dynamics, Oxford Applied Mathematics and Computing ScienceSeries, Oxford University Press, (2005).

2. Batchelor G.K, An introduction to �uid dynamics, Cambridge University Press, (2002).

3. Blazek J., Computational Fluid Dynamics Principles and Applications, Elsevier, (2005).

4. Donea J., Huerta A., Finite element methods for �ow problems, Wiley, (2003).

5. Feistauer M., Mathematical Methods in Fluid Dynamics, Longman Scienti�c & Technical, Harlow,(1993).

6. Ferziger J.H., Peric M., Computational Methods for Fluid Dynamics, Springer, (1999).


References (3)

• Computational Fluid Dynamics (cont’d)

7. Girault V., Raviart P.A., Finite element methods for Navier-Stokes equations. Theory and Algo-rithms, Springer, (1986).

8. Glowinski R., Finite Element Methods fo Incompressible Viscous Flows, in Handbook of numericalanalysis, vol. 9 (part 3), North-Holland, (2003).

9. Gresho P. M., Sani R.L., Incompressible �ow and the �nite element method, Wiley (1998).

10. Gunzburger M., Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Prac-tice, and Algorithms, Academic Press, (1989).

11. Kwak D., Kiris C.C., Computation of viscous incompressible �ows, Scienti�c Computation series,Springer, (2011).

12. Marion M., Temam R., Navier-Stokes equations: Theory and approximation, in Handbook of nu-merical analysis, vol. 6, 503-689, North-Holland, (1998).

13. Peyret R., Taylor T.D., Computational Methods for Fluid Flow, Springer, (1983).

14. Pironneau O., Finite element methods for �uids,Wiley & Sons, (1989).

15. Wesseling P., Principles of Computational Fluid Dynamics, Springer, (2000).

• Numerical programming

1. Hecht F. et al., FreeFem++, UPMC, http://www.freefem.org/�++/ftp/freefem++doc.pdf.

2. Quarteroni A., Scienti�c Computing in Matlab and Octave, 2nd ed., Springer, Texts in Computa-tional Science and Engineering, (2006).


References (4)

• Shape optimization

1. Allaire G., Conception optimale de structures, Mathématiques et Applications 58, Springer, (2006).

2. Bendsøe M.P. and Sigmund O., Topology Optimization, Theory, Methods and Applications, 2ndEdition Springer, (2003).

3. Henrot A., and Pierre M., Variation et optimisation de formes, une analyse géométrique, Springer,(2005).

4. Mohammadi B. and Pironneau O., Applied shape optimization for �uids, Oxford University Press,28, (2001).

5. Pironneau O., Optimal Shape Design for Elliptic Systems, Springer, (1984).

6. Sethian J.A., Level Set Methods and Fast Marching Methods : Evolving Interfaces in ComputationalGeometry,Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press,(1999).


Introduction to Fluid Dynamics

• Fluid �ows are governed by partial di�erential equations (PDE) which representsconservation laws (mass, momentum, and energy).

• Computational Fluid Dynamics (CFD): virtual experimental laboratory◦ consists in replacing suitably the PDE problem by algebraic equations that arethen solved numerically using computers,

◦ provides a prediction of �uid �ows based on◦ mathematical models (continuous)◦ numerical schemes (discrete)◦ algorithmic techniques (meshers, solvers, visualization).

What is CFD?

Computational Fluid Dynamics (CFD) provides a qualitative (and

sometimes even quantitative) prediction of fluid flows by means of

• mathematical modeling (partial di!erential equations)

• numerical methods (discretization and solution techniques)

• software tools (solvers, pre- and postprocessing utilities)

CFD enables scientists and engineers to perform ‘numerical experiments’

(i.e. computer simulations) in a ‘virtual flow laboratory’

real experiment CFD simulation

Why use CFD?

Numerical simulations of fluid flow (will) enable

• architects to design comfortable and safe living environments

• designers of vehicles to improve the aerodynamic characteristics

• chemical engineers to maximize the yield from their equipment

• petroleum engineers to devise optimal oil recovery strategies

• surgeons to cure arterial diseases (computational hemodynamics)

• meteorologists to forecast the weather and warn of natural disasters

• safety experts to reduce health risks from radiation and other hazards

• military organizations to develop weapons and estimate the damage

• CFD practitioners to make big bucks by selling colorful pictures :-)

What is CFD?

Computational Fluid Dynamics (CFD) provides a qualitative (and

sometimes even quantitative) prediction of fluid flows by means of

• mathematical modeling (partial di!erential equations)

• numerical methods (discretization and solution techniques)

• software tools (solvers, pre- and postprocessing utilities)

CFD enables scientists and engineers to perform ‘numerical experiments’

(i.e. computer simulations) in a ‘virtual flow laboratory’

real experiment CFD simulation

Why use CFD?

Numerical simulations of fluid flow (will) enable

• architects to design comfortable and safe living environments

• designers of vehicles to improve the aerodynamic characteristics

• chemical engineers to maximize the yield from their equipment

• petroleum engineers to devise optimal oil recovery strategies

• surgeons to cure arterial diseases (computational hemodynamics)

• meteorologists to forecast the weather and warn of natural disasters

• safety experts to reduce health risks from radiation and other hazards

• military organizations to develop weapons and estimate the damage

• CFD practitioners to make big bucks by selling colorful pictures :-)

real simulation numerical simulation


The two pilars: experiments and simulationsInvestigation of the �ow patterns to understand the �ow phenomena by

1. Experiments: description of the phenomena using measurements• main features: laboratory scale, one variable at a time, at a few locations, oper-ating conditions

experiments are expensive, slow, sequential• error sources: �ow measurements, probes, instruments, interferences,

calibration, reproducibility

2. Simulations: prediction of the phenomena using computers• main features: scale 1:1, high resolution, virtually any problem, run scenarios

software codes are versatile, portable, easy to modify, cheap, fast, parallel,• error sources: hypothesis, modelling, discretization, implementation,

robustness, stability and convergence issues.

Hence, CFD is a now a complement and tends to be a substitute to experiments.


Fluid characteristics

1. Flow properties (macroscopic)

ρ densityµ viscosityp pressurev velocityT temperature

2. Flow classi�cation

viscous inviscidcompressible incompressiblelaminar turbulentsteady unsteadysingle-phase multi-phase

3. Reliability issues in simulations

• related to mathematical models: Stokes, Navier-Stokes, Euler, Saint-Venant, . . .

• input data may be inaccurate,

• accuracy related to computer features (memory, architecture),

• sensitivity of turbulence models,

• di�culty of tracking the interfaces between phase (or species).


CFD analysis

1. Problem analysis:• physical phenomena, type of �ow (laminar, steady, . . . )• domain geometry, interfaces, free surfaces,• objectives: computation of integral values (lift, drag), shape optimization

2. Mathematical modelling:• select �ow model, identify forces,• write conservation laws (mass, momentum, energy): partial di�erential equa-tions,

• de�ne computational domain and specify boundary + initial conditions

3. Space and time discretization:• mesh generation, references and domains identi�cation,• discrete weak formulation (�nite elements),• approximation of temporal derivatives: explicit vs implicit scheme

4. Resolution and visualization:• solve sparse algebraic systems• compute derived quantities (streamlines, vorticity), visualization.


CFD analysis: problems at stake

1. Validation of models and certi�cation of CFD codes:• check if model is adequate for solving problem• compare numerical solutions with experimental results• introduce sensitivity analysis• switch between models, domain geometry and boundary conditions

i.e. the goal is to ensure that the codes produce reasonable results for a certainrange of �ow problems

2. Uncertainty: usually related to the lack of knowledge (e.g. turbulence models)

3. Errors: may have various causes:• physical modelling due to (over) simpli�cations or incorrect parameter values• approximation of PDEs (space and time discretizations)• convergence of iterative procedures• round-o� (truncation) of computer arithmetic, computer programming


CFD codes

• Several CFD software are considered robust, reliable and e�cient for performingaccurate simulations. But they all require a knowledge of the underlying numericalmethods and physics.

• Below is a (non exhaustive) list of general-purpose free CFD softwares:

OpenFOAM open-source http://www.openfoam.comCode Saturn open-source http://code-saturne.org/cmsGeris open-source http://gfs.sourceforge.netOpenFVM open-source http://openfvm.sourceforge.netFeatFlow open-source http://www.featflow.deOpenFlower open-source http://openflower.sourceforge.net

A list of commercial codes can be found at:http://www.cfd-online.com/Wiki/Codes#Commercial_codes

• Recent research approaches (stochastic models, error estimates, complex models)have not been yet fully integrated and require further developments.


http://www.openfoam.com

http://code-saturne.org/cms

http://gfs.sourceforge.net

http://openfvm.sourceforge.net

http://www.featflow.de

http://openflower.sourceforge.net

http://www.cfd-online.com/Wiki/Codes#Commercial_codes

Schedule

Parti I: lectures 21 hours• class 1: Preliminaries, fluid models 3 hoursintroduction to �uid problems and derivation of the incompressible �uid models(conservation laws, physical considerations,..);

• class 2: Stokes I 3 hourstheoretical and numerical analysis of steady Stokes problem, variational formula-tion, a saddle point approach, FE approximation of the problem, solution methods;

• class 3: Stokes II + Navier-Stokes I 1.5 + 1.5 hoursunsteady Stokes analysis, discretization in time, numerical issues. Navier-Stokesmodel, variational formulation, theoretical results;

• class 4: Navier-Stokes II 3 hoursNavier-Stokes problem, nonlinear iterative procedures, time discretization proce-dures, numerical issues;

• class 5: Two fluids or two-phase flows 3 hoursproblem statement, level set formulation, numerical resolution.

• classes 6+7: Introduction to shape optimization 3+3 hoursFramework, shape sensitivity - shape derivatives, Céa’s method, numerical issues.

Part II: numerical experiments using FreeFem++ 21 hours.Aspects théoriques et numériques pour les �uides incompressibles M2 ANEDP, UPMC, 2019-2020 14/ 14

Aspects ThØoriques et NumØriques pour les Fluides .... Frey, intro.… · Geometry,Fluid...

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