Abstract
In the spirit of Ha Minh's semi-deterministic model, we propose a new method for computing fully-developed turbulent flows, called Coherent Vortex Simulation (CVS). It is based on the observation that turbulent flows contain both an organized part, the coherent vortices, and a random part, the incoherent background flow. The separation into coherent and incoherent contributions is done using the wavelet coefficients of the vorticity field and the Biot–Savart kernel to reconstruct the coherent and incoherent velocity fields. The evolution of the coherent part is computed using a wavelet basis, adapted at each time step to resolve the regions of strong gradients, while the incoherent part is discarded during the flow evolution, which models turbulent dissipation. The CVS method is similar to LES, but it uses nonlinear multiscale band-pass filters, which depend on the instantaneous flow realization, while LES uses linear low-pass filters, which do not adapt to the flow evolution. As example, we apply the CVS method to compute a time developing two-dimensional mixing layer and a wavelet forced two-dimensional homogeneous isotropic flow. We also demonstrate how walls or obstacles can be taken into account using penalization and compute a two-dimensional flow past an array of cylinders. Finally, we perform the same segmentation into coherent and incoherent components in a three-dimensional homogeneous isotropic turbulent flow. We show that the coherent components correspond to vortex tubes, which exhibit non-Gaussian statistics and long-range correlation, with the same k −5/3power-law energy spectrum as the total flow. In contrast, the incoherent components correspond to an homogeneous random background flow which does not contain organized structures and presents an energy equipartition together with a Gaussian PDF of velocity. This justifies their elimination during the CVS computation to model turbulent dissipation.
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Angot, P., Bruneau, C.H. and Fabrie, P., A penalization method to take into account obstacles in viscous flows. Numer. Math. 81 (1999) 497–500.
Arquis, E. and Caltagirone, J.P., Sur les conditions hydrodynamique au voisinage d'une interface millieux fluide-millieu poreux: Application à la convection naturelle. C.R. Acad. Sci. Paris II 299 (1984) 1–4.
Burke Hubbard B., The World According to Wavelets: The Story of a Mathematical Technique in the Making. AK Peters, Natick, MA (1998).
Casella, G. and Berger, R.L., Statistical Inference. Wadsworth and Brooks/Cole, Pacific Grove (1990).
Daubechies I., Ten Lectures on Wavelets. SIAM, Philadelphia, PA (1992).
Domaradzki, J.A., Nonlocal triad interactions and the dissipation range of isotropic turbulence. Phys. Fluids A 4 (1992) 2037.
Donoho, D., Unconditional bases are optimal bases for data compression and statistical estimation. Appl. Comput. Harmon. Anal. 1 (1993) 100.
Donoho, D. and Johnstone, I., Ideal spatial adaption via wavelet shrinkage, Biometrica 81 (1994) 425–455.
Farge, M. and Rabreau, G., Transformée en ondelettes pour détecter et analyser les structures cohérentes dans les écoulements turbulents bidimensionnels. C.R. Acad. Sci. Paris, Série II 307 (1988) 433.
Farge, M., Holschneider, M. and Colonna, J.F., Wavelet analysis of coherent structures in two-dimensional turbulent flows. In: Moffatt, H.K. and Tsinober, A. (eds), Topological Fluid Mechanics. Cambridge University Press, Cambridge (1989) pp. 765–776.
Farge, M., Wavelet transforms and their applications to turbulence. Ann. Rev. Fluid Mech. 24 (1992) 395–457.
Farge, M., Goirand, E., Meyer, Y., Pascal, F. and Wickerhauser, M.V., Improved predictability of two-dimensional turbulent flows using wavelet packet compression. Fluid Dynam. Res. 10 (1992) 229.
Farge, M. and Philipovitch, T., Coherent structure analysis and extraction using wavelets. In: Meyer, Y., Murenzi, R. and Roques, S. (eds) Progress in Wavelet Analysis and Applications. Editions Frontières, Paris (1993) p. 477.
Farge, M., Kevlahan, N.K., Perrier, V. and Goirand, E., Wavelets and turbulence. Proc. IEEE 84(4) (1996) 639–669.
Farge, M., Kevlahan, N.K., Perrier, V. and Schneider, K., Turbulence analysis, modelling and computing using wavelets. In: van den Berg, J.C. (ed.), Wavelets in Physics Cambridge University Press, Cambridge (1999) pp. 117–200.
Farge, M., Schneider, K. and Kevlahan, N.K., Non-Gaussianity and Coherent Vortex Simulation for two-dimensional turbulence using an adaptive orthonormal wavelet basis. Phys. Fluids 11(8) (1999) 2187–2201.
Farge, M., Schneider, K., Pellegrino, G., Wray, A.A. and Rogallo, R. S., CVS decomposition of 3D homogeneous turbulence using orthogonal wavelets. Center for Turbulence Research: Summer Program (2000) pp. 305–317.
Farge, M., Pellegrino, G. and Schneider, K., Coherent vortex extraction in 3D turbulent flows using orthogonal wavelets. Phys. Rev. Lett. 87(5) (2001).
Ferziger, J.H., Large Eddy Simulation. In: Gatski, T.B., Hussaini, M.Y. and Lumley, J.L. (eds), Simulation and Modeling of Turbulent Flows, ICASE Series in Computational Science and Engineering. Springer-Verlag, New York (1992) p. 109.
Forestier, M., Etude par méthode spectrale de sillages tridimensionnels en fluide stratifié. Thèse de doctorat, Université de Nice-Sophia Antipolis (2000).
Fröhlich, J. and Schneider, K., An adaptive wavelet-vaguelette algorithm for the solution of PDEs. J. Comput. Phys. 130 (1997) 174–190.
Germano, M., Piomelli, U., Moin, P. and Cabot, W.H., A dynamic subgrid scale model eddy viscosity model. Center for Turbulence Research: Summer Report (1990) pp. 5–17.
Glowinski, R., Pan, T.W. and Periaux, J., A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 112 (1994) 3–17.
Grossmann, A. and Morlet, J., Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15 (1984) 723.
Ha Minh H., La modélisation statistique de la turbulence: ses capacités et ses limitations. C.R. Acad. Sci. Paris, Série IIb 327 (1999) 343–358.
Ha Minh, H. and Kourta, A., Semi-deterministic turbulence modelling for flows dominated by strong organized structures. In: Proceedings 9th Symposium on Turbulent Shear Flows, Kyoto, August 16–18 (1993) pp. 10.5.1–10.5.6.
Ha Minh, H., Order and disorder in turbulent flows: their impact on turbulence modelling. In: Osborne Reynolds Centenary Symposium. UMIST, Manchester (1994) pp. 1–30.
Kevlahan, N.K. and Farge, M., Vorticity filaments in two-dimensional turbulence: Creation, stability and effect. J. Fluid Mech. 346 (1997) 49–76.
Kevlahan, N.K. and Ghidaglia, J.M., Computation of turbulent flow past an array of cylinders using a spectralmethod with Brinkman penalization. Eur. J.Mech. B/Fluids 20 (2001) 333–350.
Kraichnan, R.H., Inertial ranges of two-dimensional turbulence. Phys. Fluids 10 (1967) 1417–673.
Leonard, A., Energy cascade in Large Eddy Simulations of turbulent fluid flows. Adv. Geophys. 18A (1974) 237.
Lemarié, P.G. and Meyer, Y., Ondelettes et bases Hilbertiennes, Rev. Math. Ibero Am. 2 (1986) 1.
Mallat, S., A Wavelet Tour of Signal Processing. Academic Press, New York (1998).
Méneveau, C., Analysis of turbulence in the orthonormal wavelet representation. J. FluidMech. 232 (1991), 469.
Moffatt, H.K., Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. J. Fluid Mech. 150 (1985), 359.
Peskin, C.S., Numerical analysis of blood flow around heart valves: A numerical study. J. Comput. Phys. 10 (1972) 252.
Reynolds, O., On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philos. Trans. Roy. Soc. London 186 (1894) 123.
Reynolds, W.C. and Hussain, A.K.M.F., The mechanism of an organized wave in turbulent shear flow. J. Fluid Mech. 54 (1971) 263–288.
Schneider, K., Kevlahan, N. and Farge, M., Comparison of an adaptive wavelet method and nonlinearly filtered pseudo-spectral methods for two-dimensional turbulence. Theoret. Comput. Fluid Dynam. 9 (1997) 191–206.
Schneider, K. and Farge, M., Wavelet approach for modelling and computing turbulence. In: Lecture Series 1998–05: Advances in Turbulence Modelling, von Karman Institute for Fluid Dynamics, Bruxelles (1998) pp. 1–132.
Schneider, K., Farge, M. and Kevlahan, N., Spatial intermittency in two-dimensional turbulence: A wavelet approach. J. Fluid Mech. (2001) (submitted).
Schneider, K. and Farge, M., Wavelet forcing for numerical simulation of two-dimensional turbulence. C.R. Acad. Sci. Paris, Série IIb 325 (1997) 263–270.
Schneider, K. and Farge, M., Numerical simulation of a temporally growing mixing layer in an adaptive wavelet basis. C.R. Acad. Sci. Paris, Série II b 328 (2000) 263–269.
Schneider, K., Farge, M., Pellegrino, G. and Rogers, M., CVS filtering of 3D turbulent mixing layers using orthogonal wavelets. Center for Turbulence Research: Summer Program (2000) pp. 319–330.
Schneider, K., Modélisation et simulation numérique en base d'ondelettes adaptatives, applications en mécanique des fluides et en combustion. Habilitation à diriger des recherches, Université Louis Pasteur, Strasbourg (2001) pp. 16–19.
Schneider, K. and Farge, M., Coherent Vortex Simulation (CVS) of two-dimensional flows in complex geometries. Preprint CMI, Université de Provence, Marseille (2001).
Vincent, A. and Meneguzzi, M., The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225 (1991) 1–20.
Winckelmans, G.S., Some progress in large-eddy simulation using the 3D vortex particle method. Center for Turbulence Research: Annual Research Briefs (1995) pp. 391–415.
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Farge, M., Schneider, K. Coherent Vortex Simulation (CVS), A Semi-Deterministic Turbulence Model Using Wavelets. Flow, Turbulence and Combustion 66, 393–426 (2001). https://doi.org/10.1023/A:1013512726409
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DOI: https://doi.org/10.1023/A:1013512726409