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Model reduction for compressible flows using POD and Galerkin projection. (English) Zbl 1098.76602
Summary: We present a framework for applying the method of proper orthogonal decomposition (POD) and Galerkin projection to compressible fluids. For incompressible flows, only the kinematic variables are important, and the techniques are well known. In a compressible flow, both the kinematic and thermodynamic variables are dynamically important, and must be included in the configuration space. We introduce an energy-based inner product which may be used to obtain POD modes for this configuration space. We then obtain an approximate version of the Navier–Stokes equations, valid for cold flows at moderate Mach number, and project these equations onto a POD basis. The resulting equations of motion are quadratic, and are much simpler than projections of the full compressible Navier–Stokes equations.

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76F99 Turbulence
76M25 Other numerical methods (fluid mechanics) (MSC2010)
Full Text: DOI
[1] Sirovich, L., Turbulence and the dynamics of coherent structures, parts I-III, Quart. appl. math., XLV, 3, 561-590, (1987) · Zbl 0676.76047
[2] Aubry, N.; Holmes, P.; Lumley, J.L.; Stone, E., The dynamics of coherent structures in the wall region of a turbulent boundary layer, J. fluid mech., 192, 115-173, (1988) · Zbl 0643.76066
[3] P. Holmes, J.L. Lumley, G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, 1996. · Zbl 0890.76001
[4] R. Abraham, J.E. Marsden, T.S. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd ed., Applied Mathematical Sciences, No. 75, Springer-Verlag, 1988.
[5] Rempfer, D., On low-dimensional Galerkin models for fluid flow, Theor. comput. fluid dyn., 14, 2, 75-88, (2000) · Zbl 0984.76068
[6] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. · Zbl 0152.44402
[7] C.W. Rowley, Modeling, simulation, and control of cavity flow oscillations, Ph.D. Thesis, California Institute of Technology, 2002.
[8] C.W. Rowley, T. Colonius, R.M. Murray, Dynamical models for control of cavity oscillations, AIAA Paper 2001-2126, May 2001.
[9] Zank, G.P.; Matthaeus, W.H., The equations of nearly incompressible fluids. I. hydrodynamics, turbulence, and waves, Phys. fluids, 3, 1, 69-82, (1991) · Zbl 0718.76049
[10] J.L. Lumley, A. Poje, Low-dimensional models for flows with density fluctuations, Phys. Fluids 9 (7).
[11] T. Colonius, J.B. Freund, Reconstruction of large-scale structures and acoustic radiation from a turbulent M=0.9 jet using proper orthogonal decomposition, in: I.P. Castro, P.E. Hancock, T.G. Thomas (Eds.), Advances in Turbulence IX, CIMNE, Barcelona, 2002.
[12] J.B. Freund, T. Colonius, POD analysis of sound generation by a turbulent jet, AIAA Paper 2002-0072, 2002.
[13] Rowley, C.W.; Colonius, T.; Basu, A.J., On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities, J. fluid mech., 455, 315-346, (2002) · Zbl 1147.76607
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