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Stochastic modelling and diffusion modes for proper orthogonal decomposition models and small-scale flow analysis

  • Valentin Resseguier (a1) (a2), Etienne Mémin (a1), Dominique Heitz (a1) (a3) and Bertrand Chapron (a2)


We present here a new stochastic modelling approach in the constitution of fluid flow reduced-order models. This framework introduces a spatially inhomogeneous random field to represent the unresolved small-scale velocity component. Such a decomposition of the velocity in terms of a smooth large-scale velocity component and a rough, highly oscillating component gives rise, without any supplementary assumption, to a large-scale flow dynamics that includes a modified advection term together with an inhomogeneous diffusion term. Both of those terms, related respectively to turbophoresis and mixing effects, depend on the variance of the unresolved small-scale velocity component. They bring an explicit subgrid term to the reduced system which enables us to take into account the action of the truncated modes. Besides, a decomposition of the variance tensor in terms of diffusion modes provides a meaningful statistical representation of the stationary or non-stationary structuration of the small-scale velocity and of its action on the resolved modes. This supplies a useful tool for turbulent fluid flow data analysis. We apply this methodology to circular cylinder wake flow at Reynolds numbers $Re=100$ and $Re=3900$ . The finite-dimensional models of the wake flows reveal the energy and the anisotropy distributions of the small-scale diffusion modes. These distributions identify critical regions where corrective advection effects, as well as structured energy dissipation effects, take place. In providing rigorously derived subgrid terms, the proposed approach yields accurate and robust temporal reconstruction of the low-dimensional models.


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Andrews, D. & McIntyre, M. 1976 Planetary waves in horizontal and vertical shear: the generalized Eliassen- Palm relation and the zonal mean acceleration. J. Atmos. Sci. 33, 20312048.
Artana, G., Cammilleri, A., Carlier, J. & Mémin, E. 2012 Strong and weak constraint variational assimilations for reduced order fluid flow modeling. J. Comput. Phys. 231 (8), 32643288.
Aspden, A., Nikiforakis, N., Dalziel, S. & Bell, J. B. 2008 Analysis of implicit LES methods. Appl. Maths Comput. Sci. 3 (1), 103126.
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115173.
Bardina, J., Ferziger, J. H. & Reynolds, W. C. 1980 Improved subgrid scale models for large eddy simulation. In 13th Fluid Mechanics & Plasma Dynamics Conference. AIAA.
Boris, J. P., Grinstein, F. F., Oran, E. S. & Kolbe, R. L. 1992 New insights into large-eddy simulation. Fluid Dyn. Res. 10, 199228.
Boussinesq, J.1877 Essai sur la théorie des eaux courantes. Mémoires présentés par divers savants à l’Académie des Sciences, 23 (1) 1–680.
Brooke, M., Kontomaris, K., Hanratty, T. & McLaughlin., J. 1992 Turbulent deposition and trapping of aerosols at a wall. Phys. Fluids 825834.
Buffoni, M., Camarri, S., Iollo, A. & Salvetti, M. V. 2006 Low-dimensional modelling of a confined three-dimensional wake flow. J. Fluid Mech. 569, 141150.
Cammilleri, A., Gueniat, F., Carlier, J., Pastur, L., Mémin, E., Lusseyran, F. & Artana, G. 2013 POD-spectral decomposition for fluid flow analysis and model reduction. Theor. Comput. Fluid Dyn. 27, 787815.
Caporaloni, M., Tampieri, F., Trombetti, F. & Vittori, O. 1975 Transfer of particles in non- isotropic air turbulence. J. Atmos. Sci. 32, 565568.
Cazemier, W., Verstappen, R. & Veldman, A. 1998 Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids 10 (7), 16851699.
Chandramouli, P., Heitz, D., Laizet, S. & Mémin, E. 2016 Coarse large-eddy simulations in a transitional wake flow with flow models under location uncertainty. J. Fluid Mech. (submitted).
Cordier, L., Noack, B. R., Tissot, G., Lehnasch, G., Delville, J., Balajewicz, M., Daviller, G. & Niven, R. K. 2013 Identification strategies for model-based control. Exp. Fluids 54 (8), 121.
Couplet, M., Basdevant, C. & Sagaut, P. 2005 Calibrated reduced-order POD-Galerkin system for fluid flow modelling. J. Comput. Phys. 207 (1), 192220.
D’Adamo, J., Papadakis, N., Mémin, E. & Artana, G. 2007 Variational assimilation of POD low-order dynamical systems. J. Turbul. 8 (9), 122.
Deane, A., Kevrekidis, I., Karniadakis, G. & Orszag, S. 1991 Low-dimensional models for complex geometry fows: application to grooved channels and circular cylinders. Phys. Fluids 3 (10), 23372354.
Galetti, B., Botaro, A., Bruneau, C.-H. & Iollo, A. 2007 Accurate model reduction of transient and forced wakes. Eur. J. Mech. (B/Fluids) 26 (3), 354366.
Gautier, R., Laizet, S. & Lamballais, E. 2014 A DNS study of jet control with microjets using an immersed boundary method. Intl J. Comput. Fluid Dyn. 28 (6–10), 393410.
Genon-Catalot, V., Laredo, C. & Picard, D. 1992 Non-parametric estimation of the diffusion coefficient by wavelets methods. Scand. J. Stat. 317335.
Gent, P., Willebrand, J., Mcdougall, T. & Mcwilliams, J. 1995 Parameterising eddy-induced tracer transports in ocean circulation models. J. Phys. Oceanogr. 25, 463474.
Haworth, D. & Pope, S. 1986 A generalized langevin model for turbulent flows. Phys. Fluids 29, 387405.
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherence Structures, Dynamical Systems and Symetry. Cambridge University Press.
Kalb, V. & Deane, A. 2007 An intrinsic stabilization scheme for proper orthogonal decomposition based low-dimensional models. Phys. Fluids 19 (5), 054106.
Karamanos, G. & Karniadakis, G. 2000 A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys. 163 (1), 2250.
Koopman, B. 1931 Hamiltonian systems and transformations in Hilbert space. Proc. Natl Acad. Sci. USA 17 (5), 315318.
Kraichnan, R. 1987 Eddy viscosity and diffusivity: exact formulas and approximations. Complex Syst. 1 (4–6), 805820.
Laizet, S. & Lamballais, E. 2009 High-order compact schemes for incompressible flows: a simple and efficient method with the quasi-spectral accuracy. J. Comput. Phys. 228 (15), 59896015.
Lamballais, E., Fortuné, V. & Laizet, S. 2011 Straightforward high-order numerical dissipation via the viscous term for direct and large eddy simulation. J. Comput. Phys. 230, 32703275.
Lilly, D. 1992 A proposed modification of the Germano subgrid-scale closure. Phys. Fluids 3, 27462757.
Ma, X., Karamanos, G.-S. & Karniadakis, G. E. 2000 Dynamics and low-dimensionality of a turbulent near wake. J. Fluid Mech. 410, 2965.
Ma, X., Karniadakis, G. E., Park, H. & Gharib, M. 2002 DPIV-driven simulation: a new computational paradigm. Proc. R. Soc. Lond. A 459, 547565.
Macinnes, J. & Bracco, F. 1992 Stochastic particles dispersion modelling and the tracer-particle limi. Phys. Fluids 4 (12), 28092824.
Mémin, E. 2014 Fluid flow dynamics under location uncertainty. Geophys. Astrophys. Fluid Dyn. 108 (2), 119146.
Mezic, I. 2005 Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41 (1–3), 309325.
Noack, B., Morzynski, M. & Tadmor, G. 2010 Reduced-Order Modelling for Flow Control, CISM Courses and Lectures, vol. 528. Springer.
Noack, B., Papas, P. & Monkevitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339365.
Östh, J., Noack, B., Krajnović, S., Barros, D. & Borée, J. 2014 On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an ahmed body. J. Fluid Mech. 747, 518544.
Ouvrard, H., Koobus, B., Dervieux, A. & Salvetti, M. V. 2010 Classical and variational multiscale LES of the flow around a circular cylinder on unstructured grids. Comput. Fluids 39 (7), 10831094.
Parnaudeau, P., Carlier, J., Heitz, D. & Lamballais, E. 2008 Experimental and numerical studies of the flow over a circular cylinder at reynolds number 3900. Phys. Fluids 20 (8), 085101.
Pasquetti, R. 2006 Spectral vanishing viscosity method for large-eddy simulation of turbulent flows. J. Sci. Comput. 27 (1–3), 365375.
Perret, L., Collin, E. & Delville, J. 2006 Polynomial identification of POD based low-order dynamical system. J. Turbul. 7, N17.
Piomelli, U., Cabot, W. H., Moin, P. & Lee, S. 1991 Subgrid scale backscatter in turbulent and transitional flows. Phys. Fluids 3 (7), 17661771.
Pope, S. 1994 Lagrangian pdf methods for turbulent flows. Annu. Rev. Fluid Mech. 26 (1), 2363.
Pope, S. 2000 Turbulent Flows. Cambridge University Press.
Protas, B., Noack, B. R & Östh, J. 2015 Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows. J. Fluid Mech. 766, 337367.
Reeks, M. 1983 The transport of discrete particles in inhomogeneous turbulence. J. Aero. Sci. 14 (6), 729739.
Rempfer, D. & Fasel, H. F. 1994 Evolution of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech. 260, 351375.
Resseguier, V., Mémin, E. & Chapron, B. 2017a Geophysical flows under location uncertainty, part I: random transport and general models. Geophys. Astrophys. Fluid Dyn. 111 (3), 149176.
Resseguier, V., Mémin, E. & Chapron, B. 2017b Geophysical flows under location uncertainty, part II: quasi-geostrophic models and efficient ensemble spreading. Geophys. Astrophys. Fluid Dyn. 111 (3), 177208.
Resseguier, V., Mémin, E. & Chapron, B. 2017c Geophysical flows under location uncertainty, part III. SQG and frontal dynamics under strong turbulence. Geophys. Astrophys. Fluid Dyn. 111 (3), 209227.
Rowley, C., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115127.
Sawford, B. 1986 Generalized random forcing in random-walk models of turbulent dispersion model. Phys. Fluids 29, 35823585.
Schmid, P. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 528.
Sehmel, G. 1970 Particle deposition from turbulent air flow. J. Geophys. Res. 75, 17661781.
Semaan, R., Kumar, P., Burnazzi, M., Tissot, G., Cordier, L. & Noack, B. R. 2016 Reduced-order modeling of the flow around a high-lift configuration with unsteady coanda blowing. J. Fluid Mech. 800, 72110.
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Q. J. Appl. Maths 45, 561590.
Smagorinsky, J. 1963 General circulation experiments with the primitive equation: I. The basic experiment. Mon. Weath. Rev. 91, 99165.
Tadmor, E. 1989 Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal. 26 (1), 3044.
Takens, F. 1981 Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898, pp. 366381. Springer.
Yang, Y. & Mémin, E. 2017 High-resolution data assimilation through stochastic subgrid tensor and parameter estimation from 4DEnVar. Tellus A 69 (1), 1308772.
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Stochastic modelling and diffusion modes for proper orthogonal decomposition models and small-scale flow analysis

  • Valentin Resseguier (a1) (a2), Etienne Mémin (a1), Dominique Heitz (a1) (a3) and Bertrand Chapron (a2)


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