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Stochastic low-dimensional modelling of a random laminar wake past a circular cylinder

Published online by Cambridge University Press:  10 July 2008

DANIELE VENTURI ,
XIAOLIANG WAN  and
GEORGE EM KARNIADAKIS
DANIELE VENTURI
Affiliation:
Department of Energy, Nuclear and Environmental Engineering, University of Bologna, Italy
XIAOLIANG WAN
Affiliation:
Division of Applied Mathematics, Brown University, RI 02912, USA
GEORGE EM KARNIADAKIS*
Affiliation:
Division of Applied Mathematics, Brown University, RI 02912, USA
*
†Author to whom correspondence should be addressed: gk@dam.brown.edu.
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Abstract

We present a new compact expansion of a random flow field into stochastic spatial modes, hence extending the proper orthogonal decomposition (POD) to noisy (non-coherent) flows. As a prototype problem, we consider unsteady laminar flow past a circular cylinder subject to random inflow characterized as a stationary Gaussian process. We first obtain random snapshots from full stochastic simulations (based on polynomial chaos representations), and subsequently extract a small number of deterministic modes and corresponding stochastic modes by solving a temporal eigenvalue problem. Finally, we determine optimal sets of random projections for the stochastic Navier–Stokes equations, and construct reduced-order stochastic Galerkin models. We show that the number of stochastic modes required in the reconstruction does not directly depend on the dimensionality of the flow system. The framework we propose is general and it may also be useful in analysing turbulent flows, e.g. in quantifying the statistics of energy exchange between coherent modes.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 606 , 10 July 2008 , pp. 339 - 367
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Acharjee, S. & Zabaras, N. 2006 A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs. Intl J. Numer. Anal. Model. 66 (12), 1934–1954.Google Scholar
Amit, D. J. & Martín-Mayor, V. 2005 Field Theory, The Renormalization Group and Critical Phenomena, 3rd Edn.World Scientific.CrossRefGoogle Scholar
Aubry, N. 1991 On the hidden beauty of the proper orthogonal decomposition. Theor. Comput. Fluid Dyn. 2, 339–352.CrossRefGoogle Scholar
Aubry, N., Guyonnet, R. & Lima, R. 1995 Spatio-temporal symmetries and bifurcations via bi-orthogonal decomposition. J. Nonlinear Sci. 2, 183–215.CrossRefGoogle Scholar
Belkin, M. & Niyogi, P. 2003 Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15, 1373–1396.CrossRefGoogle Scholar
Blanchard, G., Bousquet, O. & Zwald, L. 2007 Statistical properties of kernel principal component analysis. Mach. Learn. 66, 259–294.CrossRefGoogle Scholar
Bodner, S. E. 1969 Turbulence theory with a time-varying Wiener–Hermite basis. Phys. Fluids 12 (1), 33–38.CrossRefGoogle Scholar
Burkardt, J. & Webster, C. 2007 Reduced order modelling of some nonlinear stochastic partial differential equations. Intl J. Numer. Anal. Model. 4 (3–4), 368–391.Google Scholar
Deane, A. E., Kevrekidis, I. G., Karniadakis, G. E. & Orszag, S. A. 1991 Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids 3 (10), 2337–2354.CrossRefGoogle Scholar
Delville, J., Ukeiley, L., Cordier, L., Bonnet, J. P. & Glauser, M. 2003 Examination of large-scale structures in a turbulent plane mixing layer. Part 1. Proper orthogonal decomposition. J. Fluid Mech. 497, 335–363.Google Scholar
Doostan, A., Ghanem, R. & Red-Horse, J. 2007 Stochastic model reduction for chaos representation. Comput. Meth. Appl. Mech. Engng 196, 3951–3966.CrossRefGoogle Scholar
Dozier, R. B. & Silverstein, J. W. 2007 On the empirical distribution of eigenvalues of large dimensional information plus-noise type matrices. J. Multivariate Anal. 98 (4), 678–694.CrossRefGoogle Scholar
Everson, R. M. & Roberts, S. J. 2000 Inferring the eigenvalues of covariance matrices from limited, noisy data. IEEE Trans. Sig. Process. 48 (7), 2083–2091.CrossRefGoogle Scholar
Gerstner, T. & Griebel, M. 1998 Numerical integration using sparse grids. Numer. Algorithms 18 (3–4), 209–232.CrossRefGoogle Scholar
Ghanem, R. G. & Spanos, P. D. 1998 Stochastic Finite Elements: A Spectral Approach. Springer.Google Scholar
Gordeyev, S. V. & Thomas, F. O. 2000 Coherent structure in the turbulent planar jet. Part 1. Extraction of proper orthogonal decomposition eigenmodes and their self-similarity. J. Fluid Mech. 414, 145–194.CrossRefGoogle Scholar
Hachem, W., Loubaton, P. & Najim, J. 2006 On the empirical distribution of eigenvalues of a Gram matrix with a given variance profile. Ann. l'Inst. Henri Poincaré (B), Probability Statist. 42 (6), 649–670.CrossRefGoogle Scholar
Ham, J., Lee, D. D., Mika, S. & Scholköpf, B. 2003 A kernel view of the dimensionality reduction of manifolds. Tech. Rep. TR-110. Max Plank Institute for Biological Cybernetics.CrossRefGoogle Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.CrossRefGoogle Scholar
Hoyle, D. C. & Rattray, M. 2004 A statistical mechanics analysis of Gram matrix eigenvalue spectra. In Learning Theory, 17th Annual Conf. on Learning Theory, COLT 2004, Banff, Canada, July 1–4, 2004, Proc. (ed. Shawe-Taylor, J. & Singer, Y.), pp. 579–593. Springer.CrossRefGoogle Scholar
Jenssen, R., Eltoft, T., Girolami, M. & Erdogmus, D. 2007 Kernel maximum entropy data transformation and an enhanced spectral clustering algorithm. In Advances in Neural Information Processing Systems (NIPS) 19, pp. 633–640. MIT Press.Google Scholar
Kamiński, M. & Carey, G. F. 2005 Stochastic perturbation-based finite element approach to fluid flow problems. Int. J. Numer. Meth. Heat Fluid FLow 15 (7), 671–697.CrossRefGoogle Scholar
Karniadakis, G. E. & Sherwin, S. 2005 Spectral/hp Element Methods for CFD, 2nd Edn.Oxford University Press.Google Scholar
Kato, T. 1995 Perturbation Ttheory for Linear Operators, 4th Edn.Springe.CrossRefGoogle Scholar
Lee, C. P., Meecham, W. C. & Hogge, H. D. 1982 Application of the Wiener–Hermite expansion to turbulence of moderate Reynolds number. Phys. Fluids 25 (8), 1322–1327.CrossRefGoogle Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic.Google Scholar
Ma, X., Karamanos, G. S. & Karniadakis, G. E. 2000 Dynamics and low-dimensionality of turbulent near-wake. J. Fluid Mech. 410, 29–65.CrossRefGoogle Scholar
Ma, X., Karniadakis, G. E., Park, H. & Gharib, M. 2003 DPIV-driven simulation: a new computational paradigm. Proc. R. Soc. Lond. A 459, 547–565.CrossRefGoogle Scholar
Meecham, W. C. & Jeng, D. T. 1968 Use of the Wiener–Hermite expansion for nearly normal turbulence. J. Fluid Mech. 32 (2), 225–249.CrossRefGoogle Scholar
Meecham, W. C. & Siegel, A. 1964 Wiener–Hermite expansion in model turbulence at large Reynolds numbers. Phys. Fluids 7 (8), 1178–1190.CrossRefGoogle Scholar
Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.CrossRefGoogle Scholar
Noack, B. R., Papas, P. & Monkiewitz, P. A. 2005 The need of a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365.CrossRefGoogle Scholar
Novak, E. & Ritter, K. 1996 High-dimensional integration of smooth functions over cubes. Numer. Math. 75, 79–97.CrossRefGoogle Scholar
Novak, E. & Ritter, K. 1999 Simple cubature formulas with high polynomial exactness. Constructive Approximation 15, 499–522.CrossRefGoogle Scholar
Paiva, A. R. C., Xu, J. & Principe, J. C. 2006 Kernel principal components are maximum entropy projections. In Proc. 6th Intl Conf. on Independent Component Analysis and Blind Signal Separation (ICA), Charleston, SC, USA, pp. 846–853.Google Scholar
Rempfer, D. 2003 Low-dimensional modelling and numerical simulation of transition in simple shear flows. Annu. Rev. Fluid. Mech. 35, 229–265.CrossRefGoogle Scholar
Rényi, A. 1961 On measures of information and entropy. In Proc. 4th Berkeley Symposium on Mathematics, Statistics and Probability.Google Scholar
Saul, K. L. & Roweis, S. T. 2003 Think globally, fit locally: unsupervised learning of low dimensional manifolds. J. Machine Learning Res. 4, 119–155.Google Scholar
Scholköpf, B. & Smola, A. J. 2002 Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond. MIT Press.Google Scholar
Scholköpf, B., Smola, A. J. & Müller, K. R. 1998 Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation 10, 1299–1319.CrossRefGoogle Scholar
Segall, A. & Kailath, T. 1976 Orthogonal functionals of independent-increment processes. IEEE Trans. Inf. Theory 22 (3), 287–298.CrossRefGoogle Scholar
Sengupta, M. & Mitra, P. P. 1999 Distribution of singular values of some random matrices. Phys. Rev. E 60, 3389–3392.Google ScholarPubMed
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures: 1, 2, 3. Q. Appl. Maths. 45 (3), 561–590.CrossRefGoogle Scholar
Tenenbaum, J. B., de Silva, V. & Langford, J. C. 2000 A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323.CrossRefGoogle ScholarPubMed
Venturi, D. 2006 On proper orthogonal decomposition of randomly perturbed fields with applications to flow past a cylinder and natural convection over a horizontal plate. J. Fluid Mech. 559, 215–254.CrossRefGoogle Scholar
Wan, X. & Karniadakis, G. E. 2006 a Long term behavior of polynomial chaos in stochastic flow simulations. Comput. Meth. Appl. Mech. Engng 195, 5582–5596.CrossRefGoogle Scholar
Wan, X. & Karniadakis, G. E. 2006 b Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 (3), 901–928.CrossRefGoogle Scholar
Webster, C. 2007 Sparse collocation techniques for the numerical solution of stochastic partial differential equations. PhD thesis, The Florida State University.Google Scholar
Weinberger, K. Q. & Saul, L. K. 2006 Unsupervised learning of image manifolds by semidefinite programming. Intl J. Computer Vision 70 (1), 77–90.CrossRefGoogle Scholar
Weinberger, K. Q., Sha, F. & Saul, L. K. 2004 Learning a kernel matrix for nonlinear dimensionality reduction. In Proc. 21st Int Conf. on Machine Learning, Banff, Alberta, Canada, p. 106.Google Scholar
Wiener, N. 1966 Nonliner Problems in Random Theory. MIT Press.Google Scholar
Xiu, D. & Karniadakis, G. E. 2002 The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2), 619–644.CrossRefGoogle Scholar
Xiu, D. & Karniadakis, G. E. 2003 Modelling uncertainty in flow simulations via generalized polynomial chaos. J. Comuput. Phys. 187, 137–167.CrossRefGoogle Scholar
Zdravkovich, M. M. 1997 Flow around Circular Cylinders, vol. 1,2. Oxford Univ. Press.CrossRefGoogle Scholar
Zinn-Justin, J. 2002 Quantum Field Theory and Critical Phenomena, 4th Edn.Oxford University Press.CrossRefGoogle Scholar