Previous |  Up |  Next

Article

Keywords:
incompressible turbulent flows; Large Eddy Simulation (LES); commutation errors; Variational Multiscale (VMS) methods
Summary:
Numerical simulation of turbulent flows is one of the great challenges in Computational Fluid Dynamics (CFD). In general, Direct Numerical Simulation (DNS) is not feasible due to limited computer resources (performance and memory), and the use of a turbulence model becomes necessary. The paper will discuss several aspects of two approaches of turbulent modeling—Large Eddy Simulation (LES) and Variational Multiscale (VMS) models. Topics which will be addressed are the detailed derivation of these models, the analysis of commutation errors in LES models as well as other results from mathematical analysis.
References:
[1] R. A.  Adams, J. J. F.  Fournier: Sobolev Spaces. Academic Press, New York, 2003, 2nd  edition. MR 2424078
[2] A. A.  Aldama: Filtering Techniques for Turbulent Flow Simulation. Lecture Notes in Engineering, Vol.  56. Springer-Verlag, Berlin, 1990. DOI 10.1007/978-3-642-84091-3 | MR 1097828
[3] L. C.  Berselli, C. R.  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 “U.  Dini”, 2004. MR 2761078
[4] L. C.  Berselli, T. Iliescu, and W. J.  Layton: Mathematics of Large Eddy Simulation of Turbulent Flows. Springer-Verlag, Berlin, 2006. MR 2185509
[5] L. C.  Berselli, V. John: Asymptotic behavior of commutation errors and the divergence of the Reynolds stress tensor near the wall in the turbulent channel flow. Math. Methods Appl. Sci. (2006), In press. DOI 10.1002/mma.750 | MR 2248564
[6] R. A.  Clark, J. H.  Ferziger, and W. C.  Reynolds: Evaluation of subgrid-scale models using an accurately simulated turbulent flow. J.  Fluid Mech. 91 (1979), 1–16. DOI 10.1017/S002211207900001X
[7] S. S.  Collis: Monitoring unresolved scales in multiscale turbulence modeling. Physics of Fluids 13 (2001), 1800–1806. DOI 10.1063/1.1367872 | Zbl 1184.76110
[8] P. A.  Davidson: Turbulence. An Introduction for Scientists and Engineers. Oxford University Press, Oxford, 2004. MR 2077129 | Zbl 1061.76001
[9] A. Dunca, V. John, and W. J.  Layton: The commutation error of the space averaged Navier-Stokes equations on a bounded domain. In: Contributions to Current Challenges in Mathematical Fluid Mechanics, J. G. Heywood G. P. Galdi, and R. Rannacher (eds.), Birkhäuser-Verlag, Basel, 2004, pp. 53–78. MR 2085847
[10] C. L.  Fefferman: Existence & smoothness of the Navier-Stokes equations. http://www.claymath.org/millennium/Navier-Stokes_Equations/ (2000).
[11] P. Fischer, T. Iliescu: Large eddy simulation of turbulent channel flows by the rational LES  model. Phys. Fluids 15 (2003), 3036–3047. DOI 10.1063/1.1604781
[12] P. Fischer, T. Iliescu: Backscatter in the rational LES  model. Comput. Fluids 33 (2004), 783–790. DOI 10.1016/j.compfluid.2003.06.011
[13] G. B.  Folland: Introduction to Partial Differential Equations. Mathematical Notes, Vol. 17. Princeton University Press, Princeton, 1995, 2nd edition. MR 1357411
[14] G. P.  Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I: Linearized Steady Problems. Springer Tracts in Natural Philosophy, Vol 38. Springer-Verlag, New York, 1994. MR 1284205
[15] G. P.  Galdi: An introduction to the Navier-Stokes initial-boundary value problem. In: Fundamental Directions in Mathematical Fluid Dynamics, G. P.  Galdi, J. G.  Heywood, and R. Rannacher (eds.), Birkhäuser-Verlag, Basel, 2000, pp. 1–70. MR 1798753 | Zbl 1108.35133
[16] G. P.  Galdi, W. J.  Layton: Approximation of the larger eddies in fluid motion. II:  A model for space filtered flow. Math. Models Methods Appl. Sci. 10 (2000), 343–350. DOI 10.1142/S0218202500000203 | MR 1753115
[17] M. Germano, U. Piomelli, P. Moin, and W. Cabot: A dynamic subgrid-scale eddy viscosity model. Phys. Fluids  A 3 (1991), 1760–1765. DOI 10.1063/1.857955
[18] V. Gravemeier: The variational multiscale method for laminar and turbulent incompressible flow. PhD.  Thesis, Institute of Structural Mechanics, University of Stuttgart, 2003.
[19] V. Gravemeier, W. A.  Wall, and E. Ramm: A three-level finite element method for the instationary incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 193 (2004), 1323–1366. DOI 10.1016/j.cma.2003.12.027 | MR 2068898
[20] J.-L.  Guermond: Stabilization of Galerkin approximations of transport equations by subgrid modeling. M2AN, Math. Model. Numer. Anal. 33 (1999), 1293–1316. DOI 10.1051/m2an:1999145 | MR 1736900 | Zbl 0946.65112
[21] J.-L.  Guermond, J. T.  Oden, and S. Prudhomme: Mathematical perspectives on large eddy simulation models for turbulent flows. J.  Math. Fluid Mech. 6 (2004), 194–248. MR 2053583
[22] L. Hörmander: The Analysis of Linear Partial Differential Operators  I. Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin-Heidelberg-New York, 1990, 2nd edition.
[23] T. J. R.  Hughes, L. Mazzei, and K. E.  Jansen: Large eddy simulation and the variational multiscale method. Comput. Vis. Sci. 3 (2000), 47–59. DOI 10.1007/s007910050051
[24] T. J. R.  Hughes: Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng. 127 (1995), 387–401. DOI 10.1016/0045-7825(95)00844-9 | MR 1365381 | Zbl 0866.76044
[25] T. J. R.  Hughes, L. Mazzei, A. A.  Oberai, and A. A.  Wray: The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence. Phys. Fluids 13 (2001), 505–512. DOI 10.1063/1.1332391
[26] T. J. R.  Hughes, A. A.  Oberai, and L. Mazzei: Large eddy simulation of turbulent channel flows by the variational multiscale method. Phys. Fluids 13 (2001), 1784–1799. DOI 10.1063/1.1367868
[27] T. Iliescu, V. John, W. J.  Layton, G. Matthies, and L. Tobiska: A numerical study of a class of LES  models. Int. J.  Comput. Fluid Dyn. 17 (2003), 75–85. DOI 10.1080/1061856021000009209 | MR 1961907
[28] T. Iliescu, W. J.  Layton: Approximating the larger eddies in fluid motion. III: The Boussinesq model for turbulent fluctuations. An. Stiin. Univ. Al. I. Cuza Iasi, ser. Noua, Mat. 44 (1998), 245–261. MR 1783204
[29] V. John: Large Eddy Simulation of Turbulent Incompressible Flows. Analytical and Numerical Results for a Class of LES  Models. Lecture Notes in Computational Science and Engineering, Vol. 34. Springer-Verlag, Berlin, 2004. MR 2018955
[30] V. John: An assessment of two models for the subgrid scale tensor in the rational LES model. J.  Comput. Appl. Math. 173 (2005), 57–80. DOI 10.1016/j.cam.2004.02.022 | MR 2098628 | Zbl 1107.76040
[31] V. John, S. Kaya: A finite element variational multiscale method for the Navier-Stokes equations. SIAM J.  Sci. Comput. 26 (2005), 1485–1503. DOI 10.1137/030601533 | MR 2142582
[32] V. John, S. Kaya: Finite element error analysis of a variational multiscale method for the Navier-Stokes equations. Adv. Comp. Math. (2006), In press. MR 2358041
[33] V. John, S. Kaya, and W. Layton: A two-level variational multiscale method for convection-dominated convection-diffusion equations. Comput. Methods Appl. Mech. Eng. 195 (2006), 4594–4603. DOI 10.1016/j.cma.2005.10.006 | MR 2229851
[34] V. John, W. J.  Layton: Analysis of numerical errors in large eddy simulation. SIAM J.  Numer. Anal. 40 (2002), 995–1020. DOI 10.1137/S0036142900375554 | MR 1949402
[35] S. Kaya, W. J.  Layton: Subgrid-scale eddy viscosity models are variational multiscale methods. Technical Report TR-MATH 03-05, University of Pittsburgh, 2003.
[36] S. Kaya, B. Riviere: A discontinuous subgrid eddy viscosity method for the time dependent Navier-Stokes equations. SIAM J.  Numer. Anal. 43 (2005), 1572–1595. DOI 10.1137/S0036142903434862 | MR 2182140
[37] A. N.  Kolmogoroff: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. (Dokl). Akad. Nauk URSS 30 (1941), 301–305. (Russian) MR 0004146
[38] R. H.  Kraichnan: Inertial ranges in two dimensional turbulence. Phys. Fluids 10 (1967), 1417–1423. DOI 10.1063/1.1762301
[39] O. A.  Ladyzhenskaya: New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them. Tr. Mat. Inst. Steklova 102 (1967), 85–104. MR 0226907 | Zbl 0202.37802
[40] L. D.  Landau, E. M.  Lifshitz: Fluid Mechanics. Vol.  6 of Course of Theoretical Physics. Pergamon Press, Oxford, 1987, 2nd edition. MR 0961259
[41] W. J.  Layton: A connection between subgrid scale eddy viscosity and mixed methods. Appl. Math. Comput. 133 (2002), 147–157. DOI 10.1016/S0096-3003(01)00228-4 | MR 1923189 | Zbl 1024.76026
[42] A. Leonard: Energy cascade in large eddy simulation of turbulent fluid flows. Adv. Geophys. 18A (1974), 237–248.
[43] M. Lesieur: Turbulence in Fluids. Fluid Mechanics and Its Applications, Vol.  40. Kluwer Academic Publishers, , 1997, 3rd  edition. MR 1447438
[44] D. K.  Lilly: A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids  A 4 (1992), 633–635. DOI 10.1063/1.858280
[45] B. Mohammadi, O. Pironneau: Analysis of the K-Epsilon Turbulence Model. John Wiley & Sons, New York, 1994. MR 1296252
[46] N. V.  Nikitin, F. Nicoud, B. Wasistho, K. D.  Squires, and P. R.  Spalart: An approach to wall modeling in large-eddy simulation. Phys. Fluids 12 (2000), 1629–1632. DOI 10.1063/1.870414
[47] U. Piomelli, E. Balaras: Wall-layer models for large eddy simulation. Annu. Rev. Fluid Mech. 34 (2002), 349–374. DOI 10.1146/annurev.fluid.34.082901.144919 | MR 1893771
[48] S. B.  Pope: Turbulent Flows. Cambridge University Press, Cambridge, 2000. MR 1881598 | Zbl 0966.76002
[49] L. F.  Richardson: Weather Prediction by Numerical Process. Cambridge University Press, Cambridge, 1922. MR 2358797
[50] W. Rudin: Functional Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, New York, 1991, 2nd edition. MR 1157815
[51] P. Sagaut: Large Eddy Simulation for Incompressible Flows. An Introduction. Springer-Verlag, Berlin, 2002. MR 1941970 | Zbl 1020.76001
[52] H. Schlichting: Boundary-Layer Theory. McGraw-Hill, New York, 1979. MR 0076530 | Zbl 0434.76027
[53] H. Sohr: The Navier-Stokes Equations. An Elementary Functional Analytic Approach. Birhäuser Advanced Texts. Birkhäuser-Verlag, Basel, 2001. MR 1928881
[54] A. Świerczewska: Mathematical analysis of large eddy simulation of turbulent flows. PhD.  Thesis, TU Darmstadt, 2004. Zbl 1081.76002
[55] F. van der Bos, B. J.  Geurts: Commutator errors in the filtering approach to large-eddy simulation. Physics of Fluids 17 (2005), . MR 2136436
[56] E. R.  van Driest: On turbulent flow near a wall. J.  Aeronaut. Sci. 23 (1956), 1007–1011. DOI 10.2514/8.3713 | Zbl 0073.20802
[57] Y. Zhang, R. L.  Street, and J. R.  Koseff: A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Phys. Fluids  A 5 (1993), 3186–3196. DOI 10.1063/1.858675
Partner of
EuDML logo