×

zbMATH — the first resource for mathematics

Incremental unknowns, multilevel methods and the numerical simulation of turbulence. (English) Zbl 0948.76070
Summary: The purpose of this work is to describe the dynamic multilevel (DML) methodology applied to the numerical simulation of turbulence. The general setting of the Navier-Stokes equations is recalled, and also a number of basic notions of the statistical theory of turbulence. The practical limitations of direct numerical simulation (DNS) and the needs for modeling are emphasized. We also discuss the modeling and numerical simulation of turbulent flows by multilevel methods related to the concept of approximate inertial manifolds (AIM). This mathematical concept stemming from the dynamical systems theory is briefly presented; AIM are based on a decomposition of velocity field into small and large scale components; they give a slaving law of small scales as a function of large scales. The novel aspect of the work presented here is the time-adaptative dynamical implementation of these multilevel methods. Indeed, the DML methodology is based on this decomposition of velocity field and on numerical arguments. We propose a numerical analysis by multilevel methods for a simple system. DML methods, with some numerical and physical justifications, are described for homogeneous turbulence. The numerical results obtained are discussed.

MSC:
76M99 Basic methods in fluid mechanics
76F65 Direct numerical and large eddy simulation of turbulence
76F20 Dynamical systems approach to turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kolmogorov, A.N., The local structure of turbulence in incompressible viscous liquid, Dokl. akad. nauk. SSSR, 30, 301-305, (1941) · JFM 67.0850.06
[2] Kolmogorov, A.N., On degeneration of isotropic turbulence in an incompressible viscous liquid, Dokl. akad. nauk. SSSR, 31, 538-541, (1941) · Zbl 0026.17001
[3] Temam, R., Navier—stokes equations, (1984), North-Holland Publ. Company Amsterdam · Zbl 0572.35083
[4] Temam, R., Navier—stokes equations and nonlinear functional analysis, CBMS-NSF regional conference series in applied mathematics, (1995), SIAM Philadelphia · Zbl 0833.35110
[5] Batchelor, G.K., The theory of homogeneous turbulence, (1971), Cambridge University Press · Zbl 0225.76003
[6] Orszag, S.A., Lectures on the statistical theory of turbulence, () · Zbl 0217.25803
[7] Babin, A.V.; Vishik, M.I.; Babin, A.V.; Vishik, M.I., Attractors of partial differential equations and estimate of their dimension, Uspekhi mat. nauk., Russian math. surveys, 38, 151-213, (1983), (in English) · Zbl 0541.35038
[8] Constantin, P.; Foias, C.; Manley, O.P.; Temam, R., Determining modes and fractal dimension of turbulent flows, J. fluid mech., 150, 427-440, (1985) · Zbl 0607.76054
[9] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Memoirs of A.M.S., 53, 114, 67 + vii pages, (1988) · Zbl 0567.35070
[10] Liu, V.X., A sharp lower bound for the Hausdorff dimension of the global attractor of the 2D navier—stokes equations, Comm. math. phys., 158, 327-339, (1993) · Zbl 0790.35085
[11] Foias, C.; Sell, G.R.; Temam, R., Variétés inertielles des équations différentielles dissipatives, C.R. acad. sc. Paris, Série I, 301, 139-142, (1985) · Zbl 0591.35062
[12] Foias, C.; Sell, G.R.; Temam, R., Inertial manifolds for nonlinear evolutionary equations, J. diff. eqs., 73, 309-353, (1988) · Zbl 0643.58004
[13] Foias, C.; Jolly, M.S.; Kevrekidis, I.G.; Sell, G.R.; Titi, E.S., On the computation of inertial manifolds, Physics lett. A, 131, 433-436, (1988)
[14] Temam, R., Induced trajectories and approximate inertial manifolds, Math. mod. numer. anal. (M2AN), 23, 541-561, (1989) · Zbl 0688.58036
[15] Temam, R., Inertial manifolds, The math. intelligencer, 12, 4, 68-74, (1990) · Zbl 0711.58025
[16] Titi, E.S., On approximate inertial manifolds to the navier—stokes equations, J. math. anal. appl., 149, 540-557, (1990) · Zbl 0723.35063
[17] Debussche, A.; Temam, R., Inertial manifolds and the slow manifolds in meteorology, Diff. integral eqs., 4, 5, 897-931, (1991) · Zbl 0753.35043
[18] Debussche, A.; Temam, R., Convergent families of approximate inertial manifolds, J. math. pures appliquées, 73, 485-522, (1994) · Zbl 0836.35063
[19] Debussche, A.; Dubois, T., Approximation of exponential order of the attractor of a turbulent flow, Physica D, 72, 372-389, (1994) · Zbl 0814.76030
[20] Jones, D.A.; Margolin, L.G.; Titi, E.S., On the effectiveness of the approximate inertial manifold: a computational study, Theoret. comput. fluid dyn., 7, 4, 243-260, (1995) · Zbl 0838.76066
[21] Jolly, M.S.; Xiong, C., On computing the long-time solution of the navier—stokes equations, Theoret. comput. fluid dyn., 7, 4, 261-278, (1995) · Zbl 0838.76065
[22] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer Verlag New York · Zbl 0658.76001
[23] Dubois, T.; Jauberteau, F.; Temam, R., Solution of the incompressible navier—stokes equations by the nonlinear Galerkin method, J. scientif. comput., 8, 2, 167-194, (1993) · Zbl 0783.76068
[24] Chen, S.; Doolen, G.D.; Kraichnan, R.H.; She, Z.S., On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence, Phys. fluids A, 5, 2, 458-463, (1993)
[25] Jimenez, J.; Wray, A.A.; Saffman, P.G.; Rogallo, R.S., The structure of intense vorticity in isotropic turbulence, J. fluid mech., 255, 65-90, (1993) · Zbl 0800.76156
[26] Lesieur, M., Turbulence in fluids, (1990), Kluwer Academic Publishers London · Zbl 0748.76004
[27] Monin, A.S.; Yaglom, A.M., ()
[28] Gikhman, I.I.; Skorokhod, A.V., Introduction to the theory of random processes, (1969), Saunders · Zbl 0429.60002
[29] Batchelor, G.K., Computation of the energy spectrum in homogeneous two-dimensional turbulence, high-speed computing in fluid dynamics, Phys. fluids suppl. II, 12, 233-239, (1969) · Zbl 0217.25801
[30] 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. · Zbl 0933.76042
[31] Reynolds, O., On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Phil. trans. roy. soc. lond. A, 186, 123-164, (1895) · JFM 26.0872.02
[32] Boussinesq, J., Théorie de l’écoulement tourbillonant, Mém. prés. acad. sci., Paris, 23-46, (1877)
[33] Speziale, C.G., Analytical methods for the development of Reynolds stress closures in turbulence, ICASE report 90-26, (1990)
[34] Smagorinsky, J., General circulation experiments with the primitive equations. I. the basic experiment, Mthly weather rev., 91, 99-164, (1963)
[35] Leonard, A., On the energy cascade in large-eddy simulations of turbulent fluid flows, Adv. geophys., 18 A, 237-248, (1974)
[36] Bardina, J.; Ferziger, J.H.; Reynolds, W.C., Improved turbulence models based on large-eddy simulation of homogeneous, incompressible, turbulent flows, (1983), Stanford University, Rep. TF-19
[37] Voke, P.R.; Collins, M.W., Large-eddy simulation: retrospect and prospect, J. physico chemical hydrodynam., 4, 2, 119-161, (1983)
[38] Speziale, C.G., Galilean invariance of subgrid-scale stress models in the large-eddy simulation of turbulence, J. fluid mech., 156, 52-62, (1985) · Zbl 0586.76099
[39] Erlebacher, G.; Hussaini, M.Y.; Speziale, C.G.; Zang, T.A., Toward the large-eddy simulation of compressible turbulent flows, J. fluid mech., 238, 155-185, (1992) · Zbl 0775.76059
[40] Germano, M.; Piomelli, U.; Moin, P.; Cabot, W.H., A dynamic subgrid-scale eddy viscosity model, Phys. fluids A, 3, 7, 1760-1765, (1991) · Zbl 0825.76334
[41] Lilly, D.K., A proposed modification of the Germano subgrid-scale closure method, Phys. fluids A, 4, 3, 633-635, (1992)
[42] Ghosal, S.; Lund, T.S.; Moin, P.; Akselvoll, K., A dynamic localization model for large-eddy simulation of turbulent flows, J. fluid mech., 286, 229-255, (1995) · Zbl 0837.76032
[43] Orszag, S.A., Analytical theories of turbulence, J. fluid mech., 41, 2, 363-386, (1970) · Zbl 0191.25601
[44] Foster, D.; Nelson, D.; Stephen, M., Large-distance and long-time properties of a randomly stirred fluid, Phys. rev. A, 16, 732, (1977)
[45] Rose, H.A., Eddy diffusivity, eddy noise and subgrid-scale modelling, J. fluid mech., 81, 719-734, (1977) · Zbl 0373.76054
[46] Yakhot, V.; Orszag, S.A., Renormalization group analysis of turbulence. I. basic theory, J. scientific comput., 1, 1, 3-55, (1986) · Zbl 0648.76040
[47] Zhou, Y.; Vahala, G.; Hossain, M., Renormalization-group theory for the eddy-viscosity in subgrid modeling, Physical rev. A, 37, 7, 2590-2598, (1988)
[48] Pope, S.B., Pdf methods for turbulent reactive flows, Prog. energy combust. sci., 11, 119-192, (1985)
[49] Marion, M.; Temam, R., Nonlinear Galerkin methods, SIAM J. numer. anal., 26, 1139-1157, (1989) · Zbl 0683.65083
[50] Marion, M.; Temam, R., Nonlinear Galerkin methods: the finite elements case, Numer. math., 57, 205-226, (1990) · Zbl 0702.65081
[51] Temam, R., Inertial manifolds and multigrid methods, SIAM J. math. anal., 21, 1, 154-178, (1990) · Zbl 0715.35039
[52] Temam, R., Stability analysis of the nonlinear Galerkin method, Math. comput., 57, 196, 477-505, (1991) · Zbl 0734.65079
[53] Temam, R., Méthodes multirésolutions en analyse numérique, () · Zbl 0801.65118
[54] Temam, R., 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, (), 315-358
[55] Burie, J.B.; Marion, M., Multi-level methods in space and time for the navier—stokes equations, SIAM J. numer. anal., 34, 4, 1514-1599, (1997) · Zbl 0897.76070
[56] Lions, J.L.; Temam, R.; Wang, S., Splitting up methods and numerical analysis of some multiscale problems, Comput. fluid dyn. J., 5, 2, 157-202, (1996), special issue dedicated to A. Jameson
[57] Foias, C.; Manley, O.P.; Temam, R., Modelling of the interaction of small and large eddies in two-dimensional turbulent flows, Math. mod. numer. anal. (M2AN), 22, 1, 93-114, (1988) · Zbl 0663.76054
[58] Zhou, Y.; Hossain, M.; Vahala, G., A critical look at the use of filters in large eddy simulation, Phys. lett. A, 139, 7, 330-332, (1989)
[59] Leslie, D.C.; Quarini, G.L., The application of turbulence theory to the formulation of subgrid modeling procedures, J. fluid mech., 91, 1, 65-91, (1979) · Zbl 0411.76045
[60] Deardorff, J.W., A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, J. fluid mech., 41, 2, 453-480, (1970) · Zbl 0191.25503
[61] Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods: theory and applications, CBMS—NSF regional conference series in applied mathematics, (1977), Society for Industrial and Applied Mathematics Philadelphia · Zbl 0412.65058
[62] Moin, P.; Kim, J., Numerical investigation of turbulent channel flow, J. fluid mech., 118, 341-377, (1982) · Zbl 0491.76058
[63] Ghosal, S.; Moin, P., The basic equations for the large eddy simulation of turbulent flows in complex geometry, J. comput. phys., 118, 24-37, (1995) · Zbl 0822.76069
[64] Jauberteau, F., Résolution numérique des équations de navier—stokes instationnaires par méthodes spectrales. Méthode de Galerkin non linéaire, (1990), Thèse Université de Paris-Sud
[65] Dubois, T., Simulation numérique d’écoulements homogènes et non homogènes par des méthodes multi-résolution, (1993), Thèse Université de Paris-Sud
[66] Debussche, A.; Dubois, T.; Temam, R., The nonlinear Galerkin method: A multiscale method applied to the simulation of homogeneous turbulent flows, Theoret. comput. fluid dynamics, 7, 4, 279-315, (1995) · Zbl 0838.76060
[67] Vincent, A.; Ménéguzzi, M., The spatial structure and statistical properties of homogeneous turbulence, J. fluid mech., 225, 1-20, (1991) · Zbl 0721.76036
[68] Smith, L.M.; Reynolds, W.C., The dissipation-range spectrum and the velocity-derivative skewness in turbulent flows, Phys. fluids A, 3, 992, (1991)
[69] Manley, O.P., The dissipation range spectrum, Phys. fluids A, 4, 6, 1320-1321, (1992)
[70] Kraichnan, R.H., Inertial ranges in two-dimensional turbulence, Phys. fluids, 10, 1417-1423, (1967)
[71] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, (), (1997) · Zbl 0871.35001
[72] Jauberteau, F.; Rosier, C.; Temam, R., The nonlinear Galerkin method in computational fluid dynamics, Appl. numer. math., 6, 361-370, (1989) · Zbl 0702.76077
[73] Jauberteau, F.; Rosier, C.; Temam, R., A nonlinear Galerkin method for the navier—stokes equations, Comput. methods appl. mech. engrg., 80, 245-260, (1990) · Zbl 0722.76039
[74] Dubois, T.; Temam, R., Separation of scales in turbulence using the nonlinear Galerkin method, ()
[75] Kerr, R.M., Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence, J. fluid mech., 153, 31-58, (1985) · Zbl 0587.76080
[76] She, Z.S.; Jackson, E.; Orszag, S.A., Structure and dynamics of homogeneous turbulence: models and simulations, Proc. R. soc. lond. A, 434, 101-124, (1991) · Zbl 0726.76047
[77] Jimenez, J., Resolution requirements for velocity gradients in turbulence, annual research briefs, center for turbulence research, (1994), Stanford University
[78] Dubois, T.; Temam, R., The nonlinear Galerkin method applied to the simulation of turbulence in a channel flow, ()
[79] Shen, J., Efficient spectral-Galerkin method I: direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. scientific comput., 15, 6, 1489-1505, (1995) · Zbl 0811.65097
[80] Shen, J., Efficient spectral-Galerkin method II: direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. scientific comput., 16, 1, 74-87, (1995) · Zbl 0840.65113
[81] Laminie, J.; Pascal, F.; Temam, R., Implementation of finite element nonlinear Galerkin methods using hierarchical bases, J. comput. mech., 11, 384-407, (1993) · Zbl 0771.76039
[82] Laminie, J.; Pascal, F.; Temam, R., Implementation and numerical analysis of the nonlinear Galerkin methods with finite elements discretization, Appl. numer. math., 15, 219-246, (1994) · Zbl 0816.65064
[83] Calgaro, C.; Laminie, J.; Temam, R., Dynamical multilevel schemes for the solution of evolution equations by hierarchical finite element discretization, Appl. numer. math., 21, 1-40, (1997) · Zbl 0879.76046
[84] Chen, M.; Temam, R., Incremental unknowns for solving partial differential equations, Numer. math., 59, 255-271, (1991) · Zbl 0712.65103
[85] Chen, M.; Temam, R., Incremental unknowns in finite differences: condition number of the matrix, SIAM J. matrix anal. applic. (SIMAX), 14, 2, 432-455, (1993) · Zbl 0773.65080
[86] Chehab, J.P.; Temam, R., Incremental unknowns for solving nonlinear eigenvalue problems: new multiresolution methods, Numerical methods for partial differential eqs., 11, 3, 199-218, (1995) · Zbl 0828.65124
[87] Chehab, J.P., A nonlinear adaptative multi-resolution method in finite differences with incremental unknowns, Math. model. numer. anal. (M2AN), 29, 11, 451-475, (1995) · Zbl 0836.65114
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.