Incremental unknowns, multilevel methods and the numerical simulation of turbulence

Computer Methods in Applied Mechanics and Engineering

Volumn 159, Issue 1-2, 1998, Pages 123-189

  Dubois, T ^a   Jauberteau, F ^a   Temam, R ^b,c  

^a UNIVERSITÉ BLAISE PASCAL   (France)

^b UNIVERSITÉ PARIS SACLAY   (France)

^c INDIANA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; COMPUTER SIMULATION; NAVIER STOKES EQUATIONS; NUMERICAL ANALYSIS; STATISTICAL METHODS; TURBULENCE; TURBULENT FLOW;

APPROXIMATE INERTIAL MANIFOLDS (AIM); DYNAMIC MULTILEVEL (DML) METHOD; VELOCITY FIELD DECOMPOSITION;

COMPUTATIONAL FLUID DYNAMICS;

MATHEMATICAL MODELING; NAVIER-STOKES EQUATIONS; TURBULENCE;

FLOW MODELLING; NAVIER-STOKES EQUATIONS; TURBULENCE;

EID: 0032122346     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(98)80106-0     Document Type: Article

References (88)

1. A.N. Kolmogorov, The local structure of turbulence in incompressible viscous liquid, Dokl. Akad. Nauk. SSSR 30 (1941) 301-305.
2. A.N. Kolmogorov, On degeneration of isotropic turbulence in an incompressible viscous liquid, Dokl. Akad. Nauk. SSSR 31 (1941) 538-541.
3. R. Temam, Navier-Stokes equations (North-Holland Publ. Company, Amsterdam, 1984).
4. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 2nd revised edition (SIAM, Philadelphia, 1995).
5. G.K. Batchelor, The Theory of Homogeneous Turbulence, 2nd edition (Cambridge University Press, 1971).
6. S.A. Orszag, Lectures on the statistical theory of turbulence in: Proc. Summer School in Theoretical Physics, Les Houches (1973).
7. A.V. Babin and M.I. Vishik, Attractors of partial differential equations and estimate of their dimension. Uspekhi Mat. Nauk. 38 (1983) 133-187 (in Russian). Russian Math. Surveys 38 (1983) 151-213 (in English).
8. A.V. Babin and M.I. Vishik, Attractors of partial differential equations and estimate of their dimension. Uspekhi Mat. Nauk. 38 (1983) 133-187 (in Russian). Russian Math. Surveys 38 (1983) 151-213 (in English).
9. P. Constantin, C. Foias, O.P. Manley and R. Temam, Determining modes and fractal dimension of turbulent flows, J. Fluid Mech. 150 (1985) 427-440.
10. P. Constantin, C. Foias and R. Temam, Attractors representing turbulent flows, Memoirs of A.M.S. 53(114) (1988) 67 + vii pages.
11. V.X. Liu, A sharp lower bound for the Hausdorff dimension of the global attractor of the 2D Navier-Stokes equations, Comm. Math. Phys. 158 (1993) 327-339.
12. C. Foias, G.R. Sell and R. Temam, Variétés Inertielles des équations différentielles dissipatives, C.R. Acad. Sc. Paris, Série I 301 (1985) 139-142.
13. C. Foias, G.R. Sell and R. Temam, Inertial Manifolds for nonlinear evolutionary equations, J. Diff. Eqs. 73 (1988) 309-353.
14. C. Foias, M.S. Jolly, I.G. Kevrekidis, G.R. Sell and E.S. Titi, On the computation of inertial manifolds, Physics Lett. A 131 (1988) 433-436.
15. R. Temam, Induced trajectories and approximate inertial manifolds, Math. Mod. Numer. Anal. (M2AN) 23 (1989) 541-561.
16. R. Temam, Inertial manifolds, The Math. Intelligencer 12(4) (1990) 68-74.
17. E.S. Titi, On approximate inertial manifolds to the Navier-Stokes equations, J. Math. Anal. Appl. 149 (1990) 540-557.
18. A. Debussche and R. Temam, Inertial manifolds and the slow manifolds in meteorology, Diff. Integral Eqs. 4(5) (1991) 897-931.
19. A. Debussche and R. Temam, Convergent families of approximate inertial manifolds, J. Math. Pures Appliquées 73 (1994) 485-522.
20. A. Debussche and T. Dubois, Approximation of exponential order of the attractor of a turbulent flow, Physica D 72 (1994) 372-389.
21. D.A. Jones, L.G. Margolin and E.S. Titi, On the effectiveness of the approximate inertial manifold: a computational study, Theoret. Comput. Fluid Dyn. 7(4) (1995) 243-260.
22. M.S. Jolly and C. Xiong, On computing the long-time solution of the Navier-Stokes equations, Theoret. Comput. Fluid Dyn. 7(4) (1995) 261-278.
23. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics (Springer Verlag, New York, 1988).
24. T. Dubois, F. Jauberteau and R. Temam, Solution of the incompressible Navier-Stokes equations by the nonlinear Galerkin method, J. Scientif. Comput. 8(2) (1993) 167-194.
25. S. Chen, G.D. Doolen, R.H. Kraichnan and Z.S. She, On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence, Phys. Fluids A 5(2) (1993) 458-463.
26. J. Jimenez, A.A. Wray, P.G. Saffman and R.S. Rogallo, The structure of intense vorticity in isotropic turbulence, J. Fluid Mech. 255 (1993) 65-90.
27. M. Lesieur, Turbulence in Fluids, 2nd edition (Kluwer Academic Publishers, London, 1990).
28. A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 2 (MIT Press, Cambridge, 1975).
29. I.I. Gikhman and AV. Skorokhod, Introduction to the Theory of Random Processes (Saunders, 1969).
30. G.K. Batchelor, Computation of the energy spectrum in homogeneous two-dimensional turbulence, High-speed computing in fluid dynamics, Phys. Fluids Suppl. II 12 (1969) 233-239.
31. T. Dubois and F. Jauberteau, A dynamical model for the simulation of the small structures in three-dimensional homogeneous isotropic turbulence, J. Sci. Comput., to be published.
32. O. Reynolds, On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Phil. Trans. Roy. Soc. Lond. A 186 (1895) 123-164.
33. J. Boussinesq, Théorie de l'écoulement tourbillonant, Mém. Prés. Acad. Sci., Paris (1877) 23-46.
34. C.G. Speziale, Analytical methods for the development of Reynolds stress closures in turbulence, ICASE Report 90-26, 1990.
35. J. Smagorinsky, General circulation experiments with the primitive equations. I. The basic experiment, Mthly Weather Rev. 91 (1963) 99-164.
36. A. Leonard, On the energy cascade in large-eddy simulations of turbulent fluid flows, Adv. Geophys. 18 A (1974) 237-248.
37. J. Bardina, J.H. Ferziger and W.C. Reynolds, Improved turbulence models based on large-eddy simulation of homogeneous, incompressible, turbulent flows, Stanford University, Rep. TF-19 (1983).
38. P.R. Voke and M.W. Collins, Large-eddy simulation: retrospect and prospect, J. Physico Chemical Hydrodynam. 4(2) (1983) 119-161.
39. C.G. Speziale, Galilean invariance of subgrid-scale stress models in the large-eddy simulation of turbulence, J. Fluid Mech. 156 (1985) 52-62.
40. G. Erlebacher, M.Y. Hussaini, C.G. Speziale and T.A. Zang, Toward the large-eddy simulation of compressible turbulent flows, J. Fluid Mech. 238 (1992) 155-185.
41. M. Germano, U. Piomelli, P. Moin and W.H. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A 3(7) (1991) 1760-1765.
42. D.K. Lilly, A proposed modification of the Germano subgrid-scale closure method, Phys. Fluids A 4(3) (1992) 633-635.
43. S. Ghosal, T.S. Lund, P. Moin and K. Akselvoll, A dynamic localization model for large-eddy simulation of turbulent flows, J. Fluid Mech. 286 (1995) 229-255.
44. S.A. Orszag, Analytical theories of turbulence, J. Fluid Mech. 41(2) (1970) 363-386.
45. D. Foster, D. Nelson and M. Stephen, Large-distance and long-time properties of a randomly stirred fluid, Phys. Rev. A 16 (1977) 732.
46. H.A. Rose, Eddy diffusivity, eddy noise and subgrid-scale modelling, J. Fluid Mech. 81 (1977) 719-734.
47. V. Yakhot and S.A. Orszag, Renormalization group analysis of turbulence. I. Basic theory, J. Scientific Comput. 1(1) (1986) 3-55.
48. Y. Zhou, G. Vahala and M. Hossain, Renormalization-group theory for the eddy-viscosity in subgrid modeling, Physical Rev. A 37(7) (1988) 2590-2598.
49. S.B. Pope, Pdf methods for turbulent reactive flows, Prog. Energy Combust. Sci. 11 (1985) 119-192.
50. M. Marion and R. Temam, Nonlinear Galerkin methods, SIAM J. Numer. Anal. 26 (1989) 1139-1157.
51. M. Marion and R. Temam, Nonlinear Galerkin methods: the finite elements case, Numer. Math. 57 (1990) 205-226.
52. R. Temam, Inertial manifolds and multigrid methods, SIAM J. Math. Anal. 21(1) (1990) 154-178.
53. R. Temam, Stability analysis of the nonlinear Galerkin method, Math. Comput. 57(196) (1991) 477-505.
54. R. Temam, Méthodes multirésolutions en analyse numérique, in: C. Baiocchi and J.L. Lions, eds., Boundary Value Problems for Partial Differential Equations and Applications (Masson and Wiley Publishers, 1993).
55. R. Temam, Applications of inertial manifolds to scientific computing: a new insight in multilevel methods, Trends and Perspectives in Applied Mathematics, Volume in honor of Fritz John, J. Marsden and L. Sirovich, eds., Springer-Verlag, Applied Mathematics Series 100 (1994) 315-358.
56. J.B. Burie and M. Marion, Multi-level methods in space and time for the Navier-Stokes equations, SIAM J. Numer. Anal. 34(4) (1997) 1514-1599.
57. J.L. Lions, R. Temam and S. Wang, Splitting up methods and numerical analysis of some multiscale problems, Comput. Fluid Dyn. J., special issue dedicated to A. Jameson, 5(2) (1996) 157-202.
58. C. Foias, O.P. Manley and R. Temam, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows, Math. Mod. Numer. Anal. (M2AN) 22(1) (1988) 93-114.
59. Y. Zhou, M. Hossain and G. Vahala, A critical look at the use of filters in large eddy simulation, Phys. Lett. A 139(7) (1989) 330-332.
60. D.C. Leslie and G.L. Quarini, The application of turbulence theory to the formulation of subgrid modeling procedures, J. Fluid Mech. 91(1) (1979) 65-91.
61. J.W. Deardorff, A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, J. Fluid Mech. 41(2) (1970) 453-480.
62. D. Gottlieb and S.A. Orszag, Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, 1977).
63. P. Moin and J. Kim, Numerical investigation of turbulent channel flow, J. Fluid Mech. 118 (1982) 341-377.
64. S. Ghosal and P. Moin, The basic equations for the large eddy simulation of turbulent flows in complex geometry, J. Comput. Phys. 118 (1995) 24-37.
65. F. Jauberteau, Résolution numérique des équations de Navier-Stokes instationnaires par méthodes spectrales. Méthode de Galerkin non linéaire, Thèse Université de Paris-Sud (1990).
66. T. Dubois, Simulation numérique d'écoulements homogènes et non homogènes par des méthodes multi-résolution, Thèse Université de Paris-Sud (1993).
67. A. Debussche, T. Dubois and R. Temam, The nonlinear Galerkin method: A multiscale method applied to the simulation of homogeneous turbulent flows, Theoret. Comput. Fluid Dynamics 7(4) (1995) 279-315.
68. A. Vincent and M. Ménéguzzi, The spatial structure and statistical properties of homogeneous turbulence, J. Fluid Mech. 225 (1991) 1-20.
69. L.M. Smith and W.C. Reynolds, The dissipation-range spectrum and the velocity-derivative skewness in turbulent flows, Phys. Fluids A 3 (1991) 992.
70. O.P. Manley, The dissipation range spectrum, Phys. Fluids A 4(6) (1992) 1320-1321.
71. R.H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids 10 (1967) 1417-1423.
72. R. Temam, Infinite dimensional dynamical systems in mechanics and physics, Springer-Verlag, New York, Applied Mathematical Sciences Series 68 (1988), 2nd revised edition (1997).
73. F. Jauberteau, C. Rosier and R. Temam, The nonlinear Galerkin method in computational fluid dynamics, Appl. Numer. Math. 6 (1989) 361-370.
74. F. Jauberteau, C. Rosier and R. Temam, A nonlinear Galerkin method for the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 80 (1990) 245-260.
75. T. Dubois and R. Temam, Separation of scales in turbulence using the nonlinear Galerkin method, in: W.G. Habashi and M. Hafez, eds., Advances in Computational Fluid Dynamics (Gordon and Breach Publishers, 1993).
76. R.M. Kerr, Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence, J. Fluid Mech. 153 (1985) 31-58.
77. Z.S. She, E. Jackson and S.A. Orszag, Structure and dynamics of homogeneous turbulence: models and simulations, Proc. R. Soc. Lond. A 434 (1991) 101-124.
78. J. Jimenez, Resolution requirements for velocity gradients in turbulence, Annual Research Briefs, Center for Turbulence Research, Stanford University (1994).
79. T. Dubois and R. Temam, The nonlinear Galerkin method applied to the simulation of turbulence in a channel flow, in: N. Napolitano and F. Sabetta, eds., Proc. ICNMFD 13 (Rome), Springer Verlag, Lecture Notes in Physics (1992).
80. J. Shen, Efficient spectral-Galerkin method I: direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Scientific Comput. 15(6), (1995) 1489-1505.
81. J. Shen, Efficient spectral-Galerkin method II: direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Scientific Comput. 16(1) (1995) 74-87.
82. J. Laminie, F. Pascal and R. Temam, Implementation of finite element nonlinear Galerkin methods using hierarchical bases, J. Comput. Mech. 11 (1993) 384-407.
83. J. Laminie, F. Pascal and R. Temam, Implementation and numerical analysis of the nonlinear Galerkin methods with finite elements discretization, Appl. Numer. Math. 15 (1994) 219-246.
84. C. Calgaro, J. Laminie and R. Temam, Dynamical multilevel schemes for the solution of evolution equations by hierarchical finite element discretization, Appl. Numer. Math. 21 (1997) 1-40.
85. M. Chen and R. Temam, Incremental unknowns for solving partial differential equations, Numer. Math. 59 (1991) 255-271.
86. M. Chen and R. Temam, Incremental unknowns in finite differences: condition number of the matrix, SIAM J. Matrix Anal. Applic. (SIMAX) 14(2) (1993) 432-455.
87. J.P. Chehab and R. Temam, Incremental unknowns for solving nonlinear eigenvalue problems: new multiresolution methods, Numerical Methods for Partial Differential Eqs. 11(3) (1995) 199-218.
88. J.P. Chehab, A nonlinear adaptative multi-resolution method in finite differences with incremental unknowns, Math. Model. Numer. Anal. (M2AN) 29(11) (1995) 451-475.