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Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations. (English) Zbl 1134.35387
Summary: The proper orthogonal decomposition (POD) and the singular value decomposition (SVD) are used to study the finite difference scheme (FDS) for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations. The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD. Combining the above procedures with a Galerkin projection approach yields a new optimized FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations. The errors between POD approximate solutions and FDS solutions are analyzed. It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results. Moreover, it is also shown that this validates the feasibility and efficiency of the POD method.
35Q30Stokes and Navier-Stokes equations
76M20Finite difference methods (fluid mechanics)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
[1]Temam R. Navier-Stokes Equations. Amsterdam: North-Holland, 1984
[2]Fu D X. Numerical Simulations of Fluid Mechanics (in Chinese). Beijing: National Defence Industry Press, 1994
[3]Xin X K, Liu R X, Jiang B C. Computational Fluid Dynamics (in Chinese). Changsha: National Defence Science Technology Press, 1989
[4]Holmes P, Lumley J L, Berkooz G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge: Cambridge University Press, 1996
[5]Fukunaga K. Introduction to Statistical Recognition. Academic Press, 1990
[6]Jolliffe I T. Principal Component Analysis. Springer-Verlag, 2002
[7]Crommelin D T, Majda A J. Strategies for model reduction: comparing different optimal bases. J Atmospheric Sci, 61: 2306–2317 (2004)
[8]Majda A J, Timofeyev I, Vanden-Eijnden E. Systematic strategies for stochastic mode reduction in climate. J Atmospheric Sci, 60: 1705–1723 (2003) · doi:10.1175/1520-0469(2003)060<1705:SSFSMR>2.0.CO;2
[9]Selten F. Barophilic empirical orthogonal functions as basis functions in an atmospheric model. J Atmospheric Sci, 54: 2100–2114 (1997)
[10]Lumley J L. Coherent structures in turbulence. In: Meyer R E, ed. Transition and Turbulence. Academic Press, 1981, 215–242
[11]Aubry N, Holmes P, Lumley J L, et al. The dynamics of coherent structures in the wall region of a turbulent boundary layer. J Fluid Mech, 192: 115–173 (1988) · Zbl 0643.76066 · doi:10.1017/S0022112088001818
[12]Sirovich L. Turbulence and the dynamics of coherent structures: Part I-III. Q Appl Math, 45(3): 561–590 (1987)
[13]Roslin R D, Gunzburger M D, Nicolaides R, et al. A self-contained automated methodology for optimal flow control validated for transition delay. AIAA Journal, 35: 816–824 (1997) · Zbl 0901.76067 · doi:10.2514/2.7452
[14]Ly H V, Tran H T. Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Q Appl Math, 60: 631–656 (2002)
[15]Moin P, Moser R D. Characteristic-eddy decomposition of turbulence in channel. J Fluid Mech, 200: 417–509 (1989) · Zbl 0659.76062 · doi:10.1017/S0022112089000741
[16]Rajaee M, Karlsson S K F, Sirovich L. Low dimensional description of free shear flow coherent structures and their dynamical behavior. J Fluid Mech, 258: 1401–1402 (1994) · Zbl 0800.76190 · doi:10.1017/S0022112094003228