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A spectral viscosity method for correcting the long-term behavior of POD models. (English) Zbl 1136.76412

Summary: Low-dimensional flow dynamical systems may converge to erroneous states after long-time integration, even if they are initialized with the correct state. In this paper, we investigate the accuracy of such two-dimensional models constructed from Karhunen–Loeve expansions for flows past a circular cylinder. We first demonstrate that although the short-term dynamics may be predicted accurately with only a handful of modes retained, drifting of the solution may arise after a few hundred vortex shedding cycles. We then propose a dissipative model based on a spectral viscosity (SV) diffusion convolution operator. The parameters of the SV model are selected rigorously based on bifurcation analysis. Our results show that this is an effective way of improving the accuracy of long-term predictions of low-dimensional Galerkin systems.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)

Software:

HomCont; AUTO
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[1] Bekooz, G.; Holmes, P.; Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Ann. Rev. Fluid Mech., 25, 539-575 (1993)
[2] Sirovich, L., Turbulence and the dynamics of coherent structures, Parts I, II and III, Quart. Appl. Math., XLV, 561-590 (1987) · Zbl 0676.76047
[3] Glezer, A.; Kadioglu, Z.; Pearlstein, A. J., Development of an extended proper orthogonal decomposition and its application to a time periodically forced plane mixing layer, Phys. Fluids, 1, 8, 1363 (1989)
[4] Citriniti, J. H.; George, W. K., Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition, J. Fluid Mech., 418, 137-166 (2000) · Zbl 1103.76333
[5] Arndt, R. E.; Long, D. F.; Glauser, M. N., The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet, J. Fluid Mech., 340, 1-33 (1997)
[6] Gordeyev, S. V.; Thomas, F. O., Coherent structure in the turbulent planar jet. Part 1. Extraction of proper orthogonal decomposistion eigenmodes and their self-similarity, J. Fluid Mech., 414, 145-194 (2000) · Zbl 0949.76511
[7] Delville, J.; Ukeiley, L.; Cordier, L.; Bonnet, J. P.; Glauser, M., Examination of large-scale structures in a turbulent plane mixing layer. Part 1. Proper orthogonal decomposition, J. Fluid Mech., 391, 91-122 (1999) · Zbl 0995.76030
[8] Deane, A. E.; Kevrekidis, I. G.; Karniadakis, G. E.; Orszag, S. A., Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders, Phys. Fluids A, 3, 10, 2337-2354 (1991) · Zbl 0746.76021
[9] Aubry, N.; Holmes, P.; Stone, J. L.; Lumley, J. L., The dynamics of coherent structures in the wall region of a turbulent boundary layer, J. Fluid Mech., 192, 115-173 (1988) · Zbl 0643.76066
[10] Rempfer, D.; Fasel, H. F., Evolution of three-dimensional coherent structures in a flat-plate boundary layer, J. Fluid Mech., 260, 351-375 (1994)
[11] Liakopoulos, A.; Blythe, P. A.; Gunes, H., A reduced dynamical model of convective flows in tall laterally heated cavities, Proc. R. Soc. Lond. A, 453, 663-672 (1997) · Zbl 0874.76022
[12] Cazemier, W.; Verstappen, R. W.; Veldman, A. E., Proper orthogonal decomposition and low-dimensional models for driven cavity flows, Phys. Fluids, 10, 7, 1685-1699 (1998)
[13] Singh, S. N.; James, H. M.; Gregory, A. A.; Siva, B.; James, K. H., Optimal feedback control of vortex shedding using proper orthogonal decomposition models, Trans. ACME, 123, 612-618 (2001)
[14] Foias, C.; Jolly, M. S.; Kevrekidis, I. G.; Titi, E. S., Dissipativity of the numerical schemes, Nonlinearity, 4, 591-613 (1991) · Zbl 0734.65080
[15] Jolly, M. S.; Kevrekidis, I. G.; Titi, E. S., Preserving dissipation in approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, J. Dynam. Diff. Eq., 3, 179-197 (1991) · Zbl 0738.35024
[16] Marion, M.; Temam, R., Nonlinear Galerkin methods, SIAM J. Numer. Anal., 26, 1139-1157 (1989) · Zbl 0683.65083
[17] Debussche, A.; Dubois, T.; Temam, R., The nonlinear Galerkin method: a multiscale method applied to the simulation of homogeneous turbulent flows, Theor. Comp. Fluid. Dyn., 7, 279-299 (1995) · Zbl 0838.76060
[18] Shen, J., Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38, 201-229 (1989) · Zbl 0684.65095
[19] Ma, X.; Karniadakis, G. E.; Park, H.; Gharib, M., DPIV-driven simulation: a new computational paradigm, Proc. R. Soc. Lond. A, 459, 547-565 (2003) · Zbl 1116.76413
[20] Tadmor, E., Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal., 26, 1, 30-44 (1989) · Zbl 0667.65079
[21] Karamanos, G. S.; Karniadakis, G. E., A spectral vanishing viscosity method for large-eddy simulations, J. Comp. Phys., 162, 22-50 (2000) · Zbl 0984.76036
[22] Kaber, S. M.O., A Legendre pseudospectral viscosity method, J Comput. Phys., 128, 165-180 (1996) · Zbl 0863.65065
[23] Andreassen, O.; Lie, I.; Wasberg, C. E., The spectral viscosity method applied to simulation of waves in a stratified atmosphere, J. Comput. Phys., 110, 257-273 (1994) · Zbl 0795.76056
[24] Tadmor, E., Super viscosity and spectral approximations of nonlinear conservation laws, (Baines, M. J.; Morton, K. W., Numerical Methods for Fluid Dynamics, IV (1993), Clarendon Press: Clarendon Press Oxford), 69 · Zbl 0805.76057
[25] Ma, H.-P., Chebyschev-Legendre spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 35, 3, 869-892 (1998) · Zbl 0912.35104
[26] Ma, H.-P., Chebyschev-Legendre super spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 35, 3, 893-908 (1998) · Zbl 0912.35105
[27] Ma, X.; Karniadakis, G. E., A low-dimensional model for simulating 3d cylinder flow, J. Fluid Mech., 458, 181-190 (2002) · Zbl 1001.76043
[28] Karniadakis, G. E.; Sherwin, S. J., Spectral/hp Element Methods for CFD (1999), Oxford University Press · Zbl 0954.76001
[29] Aubry, N.; Lian, W. Y.; Titi, E. S., Preserving symmetries in the proper orthogonal decomposition, SIAM J. Sci. Comput., 14, 2, 483-505 (1993) · Zbl 0774.65084
[30] S. Sirisup, Convergence and stability of low-dimensional flow models, Ph. D. Thesis, Division of Applied Mathematics, Brown University, in progress; S. Sirisup, Convergence and stability of low-dimensional flow models, Ph. D. Thesis, Division of Applied Mathematics, Brown University, in progress · Zbl 1070.35024
[31] Crandall, M. G.; Lions, P. L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc., 277, 1-42 (1983) · Zbl 0599.35024
[32] Maday, Y.; Ould Kaber, S. M.; Tadmor, E., Legendre pseudospectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 30, 321-342 (1993) · Zbl 0774.65072
[33] Tadmor, E., Total variation and error estimates for spectral viscosity approximations, Math. Comp., 60, 245 (1993) · Zbl 0795.65064
[34] E.J. Doedel, R.C. Paffenroth, A.R. Champneys, T.F. Fairgrieve, Yu.A. Kuznetsov, B. Sandstede, X. Wang, Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), Technical Report, Caltech, 2001; E.J. Doedel, R.C. Paffenroth, A.R. Champneys, T.F. Fairgrieve, Yu.A. Kuznetsov, B. Sandstede, X. Wang, Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), Technical Report, Caltech, 2001
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