A finite element variational multiscale method for the Navier-Stokes equations

SIAM Journal on Scientific Computing

Volumn 26, Issue 5, 2005, Pages 1485-1503

  John, Volker ^a   Kaya, Songul ^b  

^a OTTO VON GUERICKE UNIVERSITY   (Germany)

^b Illinois Institute of Technology   (United States)

Author keywords

Finite element method; Navier Stokes equations; Turbulent flows; Variational multiscale method

Indexed keywords

COMPUTER SIMULATION; FINITE ELEMENT METHOD; MATHEMATICAL MODELS; TENSORS; TURBULENT FLOW; VARIATIONAL TECHNIQUES; VELOCITY MEASUREMENT; VISCOSITY;

TURBULENT VISCOSITY; VARIATIONAL MULTISCALE METHOD (VMS); VELOCITY DEFORMATION;

NAVIER STOKES EQUATIONS;

EID: 25444505626     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/030601533     Document Type: Article

References (33)

1. R. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. L. BERSELLI, C. GRISANTI, AND V. JOHN, On Commutation Errors in the Derivation of the Space Averaged Navier-Stokes Equations, Preprint 12, Università di Pisa, Dipartimento di Matematica Applicata, 2004.
3. L. BERSELLI AND V. JOHN, On the Comparison of a Commutation Error and the Reynolds Stress Tensor for Flows Obeying a Wall Law, Preprint 18, Università di Pisa, Dipartimento di Matematica Applicata, 2004.
4. F. BREZZI, T. J. R. HUGHES, L. D. MARINI, A. RUSSO, AND E. SÜLI, A priori error analysis of residual-free bubbles for advection-diffusion problems, SIAM J. Numer. Anal., 36 (1999), pp. 1933-1948.
5. M. BRISTEAU, R. GLOWINSKI, AND J. PERIAUX, Numerical methods for the Navier-Stokes equations: Applications to the simulation of compressible and incompressible viscous flows, Comput. Phys. Rep., 6 (1987), pp. 73-187.
6. S. COLLIS, Monitoring unresolved scales in multiscale turbulence modeling, Phys. Fluids, 13 (2001), pp. 1800-1806.
7. P. COMTE, M. LESIEUR, AND E. LAMBALLAIS, Large- and small-scale stirring of vorticity and a passive scalar in a 3-d temporal mixing layer, Phys. Fluids A, 4 (1992), pp. 2761-2778.
8. T. DUBOIS, F. JAUBERTEAU, AND R. TEMAM, Dynamic Multilevel Methods and the Numerical Simulation of Turbulence, Cambridge University Press, Cambridge, UK, 1998.
9. A. DUNCA, V. JOHN, AND W. LAYTON, The commutation error of the space averaged Navier-Stokes equations on a bounded domain, in Contributions to Current Challenges in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech. 3, G. P. Galdi, J. G. Heywood, and R. Rannacher, eds., Birkhäuser, Basel, 2004, pp. 53-78.
10. P. GRESHO AND R. SANI, Incompressible Flow and the Finite Element Method, John Wiley, Chichester, UK, 2000.
11. J.-L. GUERMOND, Stabilization of Galerkin approximations of transport equations by subgrid modeling, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 1293-1316.
12. T. HUGHES, L. MAZZEI, AND K. JANSEN, Large eddy simulation and the variational multiscale method, Comput. Vis. Sci., 3 (2000), pp. 47-59.
13. T. HUGHES, L. MAZZEI, A. OBERAI, AND A. WRAY, The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence, Phys. Fluids, 13 (2001), pp. 505-512.
14. T. HUGHES, A. OBERAI, AND L. MAZZEI, Large eddy simulation of turbulent channel flows by the variational multiscale method, Phys. Fluids, 13 (2001), pp. 1784-1799.
15. E. HYLIN AND J. MCDONOUGH, Chaotic small-scale velocity fields as prospective models for unresolved turbulence in an additive decomposition of the Navier-Stokes equations, Int. J. Fluid Mech. Res., 26 (1999), pp. 539-567.
16. V. JOHN, Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 40 (2002), pp. 775-798.
17. V. JOHN, Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equations - numerical tests and aspects of the implementation, J. Comput. Appl. Math., 147 (2002), pp. 287-300.
18. V. JOHN, Large Eddy Simulation of Turbulent Incompressible Flows. Analytical and Numerical Results for a Class of LES Models, Lect. Notes Comput. Sci. Eng. 34, Springer-Verlag, Berlin, Heidelberg, New York, 2004.
19. V. JOHN, An assessment of two models for the subgrid scale tensor in the rational LES model, J. Comput. Appl. Math., 173 (2005), pp. 57-80.
20. V. JOHN, Reference values for drag and lift of a two-dimensional time dependent flow around a cylinder, Internat. J. Numer. Methods Fluids, 44 (2004), pp. 777-788.
21. V. JOHN AND S. KAYA, Finite Element Error Analysis of a Variational Multiscale Method for the Navier-Stokes Equations, preprint, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik, 2004.
22. V. JOHN AND G. MATTHIES, Higher order finite element discretizations in a benchmark problem for incompressible flows, Internat. J. Numer. Methods Fluids, 37 (2001), pp. 885-903.
23. V. JOHN AND G. MATTHIES, MooNMD - a program package based on mapped finite element methods, Comput. Vis. Sci., 6 (2004), pp. 163-170.
24. S. KAYA AND W. LAYTON, Subgrid-Scale Eddy Viscosity Models are Variational Multiscale Methods, Tech. report TR-MATH 03-05, University of Pittsburgh, 2003.
25. W. LAYTON, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput., 133 (2002), pp. 147-157.
26. Y. MADAY AND E. TADMOR, Analysis of the spectral vanishing viscosity method for periodic conservation laws, SIAM J. Numer. Anal., 26 (1989), pp. 854-870.
27. W. RODI, J. FERZIGER, M. BREUER, AND M. POURQUIÉ, Status of large eddy simulation: Results of a workshop, J. Fluids Eng., 119 (1997), pp. 248-262.
28. M. ROGERS AND R. MOSER, Direct simulation of a self-similar turbulent flow, Phys. Fluids, 6 (1994), pp. 903-923.
29. Y. SAAD, A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14 (1993), pp. 461-469.
30. P. SAGAUT, Large Eddy Simulation for Incompressible Flows, 2nd ed., Springer-Verlag, Berlin, Heidelberg, New York, 2003.
31. J. SMAGORINSKY, General circulation experiments with the primitive equations, Mon. Wea. Rev., 91 (1963), pp. 99-164.
32. S. TUREK, Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach, Lect. Notes Comput. Sci. Eng. 6, Springer-Verlag, Berlin, Heidelberg, New York, 1999.
33. B. VREMAN, B. GUERTS, AND H. KUERTEN, Large-eddy simulation of turbulent mixing layer, J. Fluid Mech., 339 (1997), pp. 357-390.