Theoretical and Numerical Issues of Incompressible Fluid Flows. Frey, intro.pdf · 1. Fluid mechanics 1.1Notations, vectors, tensors 1.2Conservation laws, 1.3Flow models and simpliﬁcations

Theoretical and Numerical Issues of Incompressible Fluid Flows

Instructor: Pascal FreyContributors: Yannick Privat, Charles Dapogny, Dena Kazerani,Thi Thu Cuc Bui, Thi Thanh Mai Ta

Sorbonne Université, CNRSLab. J.L. Lions & Inst. for computing and data sciences4, place Jussieu, Paris

University of Tehran, 2018

material can be downloaded at the page: https://www.ljll.math.upmc.fr/~frey/teheran2018.html

https://www.ljll.math.upmc.fr/~frey/teheran2018.html

Outline of the lectures

Part I - Classical Methods

1. Fluid mechanics1.1 Notations, vectors, tensors1.2 Conservation laws,1.3 Flow models and simplifications.

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

Theoretical and numerical issues for Incompressible Fluid Flows University of Tehran, April-May, 2018 2/ 15

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. Error estimates and mesh adaptation5.1 Residual and geometric estimates5.2 Mesh adaptation

6. Shape optimization6.1 Framework of shape optimization, examples6.2 Shape sensitivity analysis using shape derivatives6.3 Céa’s method to compute derivatives6.4 Numerical issues

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, AppliedMathematical Series, 159, Springer,(2004).

5. Evans L.C., Partial differential equations, AMS, (2002).

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

7. Johnson C., Numerical Solution of Partial Differential Equations by the Finite Element Method, Cam-bridge 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 differential equations, 23, Springer Seriesin 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 Differential 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 fluid 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., PergamonPress, (1987).

4. Pnueli D., Gutfinger C., Fluid Mechanics, Cambridge University Press, Cambridge, (1992).

5. Temam R., Miranville A., Mathematical modeling in continuummechanics, Cambridge University Press,(2005).

• Computational Fluid Dynamics

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

2. Batchelor G.K, An introduction to fluid 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 flow problems, Wiley, (2003).

5. Feistauer M., Mathematical Methods in Fluid Dynamics, Longman Scientific & 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 Algorithms,Springer, (1986).

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

9. Gresho P. M., Sani R.L., Incompressible flow and the finite element method, Wiley (1998).

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

11. Kwak D., Kiris C.C., Computation of viscous incompressible flows, Scientific Computation series, Springer,(2011).

12. Marion M., Temam R., Navier-Stokes equations: Theory and approximation, in Handbook of numericalanalysis, 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 fluids,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/ff++/ftp/freefem++doc.pdf.

2. Quarteroni A., Scientific Computing in Matlab and Octave, 2nd ed., Springer, Texts in ComputationalScience 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, 2nd EditionSpringer, (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 fluids, 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, andMaterials Science, Cambridge University Press, (1999).


Introduction to Fluid Dynamics

• Fluid flows are governed by partial differential equations (PDE) which represents con-servation laws (mass, momentum, and energy).

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

◦ provides a prediction of fluid flows 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 simulations

Investigation of the flow patterns to understand the flow phenomena by

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

experiments are expensive, slow, sequential• error sources: flow 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 classification

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,

• difficulty of tracking the interfaces between phase (or species).


CFD analysis

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

2. Mathematical modelling:• select flow model, identify forces,• write conservation laws (mass, momentum, energy): partial differential equations,• define computational domain and specify boundary + initial conditions

3. Space and time discretization:• mesh generation, references and domains identification,• discrete weak formulation (finite 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 certification 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 certain rangeof flow 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) simplifications or incorrect parameter values• approximation of PDEs (space and time discretizations)• convergence of iterative procedures• round-off (truncation) of computer arithmetic, computer programming


CFD codes

• Several CFD software are considered robust, reliable and efficient for performing accu-rate simulations. But they all require a knowledge of the underlying numerical methodsand 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) havenot 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

• Part I: Classical Methods◦ day 1: Preliminariesintroduction to fluid problems;

◦ day 2: Fluid modelsderivation of the incompressible fluid models (conservation laws, physical consid-erations,..);

◦ day 3: The Stokes equation Itheoretical and numerical analysis of steady Stokes problem, variational formulation,a saddle point approach, FE approximation of the problem, solution methods;

◦ day 4: The Stokes equation IIunsteady Stokes analysis, discretization in time, numerical issues;

◦ day 5: The Navier-Stokes equation INavier-Stokes model, variational formulation, theoretical results;

◦ day 6: The Navier-Stokes equation IINavier-Stokes problem, nonlinear iterative procedures, time discretization proce-dures, numerical issues;


Schedule

• Part II: Advanced Methods◦ day 7: Two-fluid / two-phase flowsproblem statement, level set formulation, numerical resolution;

◦ day 8: Error estimatesresidual based approaches, geometric approaches;

◦ day 9: Mesh generation Iterminology, triangulations methods;

◦ day 10: Mesh generation IIdifferential geometry, surface meshing;

◦ day 11: Mesh adaptation Idiscrete surfaces, anisotropic meshes;

◦ day 12: Mesh adaptation IIsizing function, mesh adaptation methods;

◦ day 13: Shape optimization IFramework, shape sensitivity - shape derivatives;

◦ day 14: Shape optimization IICéa’s method, numerical issues.


Documents

Theoretical and Numerical Issues of Incompressible Fluid Flows. Frey, intro.pdf · 1. Fluid mechanics 1.1Notations, vectors, tensors 1.2Conservation laws, 1.3Flow models and simpliﬁcations