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The Navier-Stokes-alpha model of fluid turbulence. (English) Zbl 1037.76022
Summary: We review the properties of nonlinearly dispersive Navier-Stokes-alpha $(NS-\alpha)$ model of incompressible fluid turbulence -- also called viscous Camassa-Holm equations in the literature. We first re-derive the $NS-\alpha$ model by filtering the velocity of the fluid loop in Kelvin’s circulation theorem for Navier-Stokes equations. Then we show that this filtering causes the wavenumber spectrum of the translational kinetic energy for the $NS-\alpha$ model to roll off as $k^{-3}$ for $k\alpha>1$ in three dimensions, instead of continuing along the slower Kolmogorov scaling law, $k^{-5/3}$, that it follows for $k\alpha<1$. This roll off at higher wavenumbers shortens the inertial range for the $NS-\alpha$ model and thereby makes it more computable. We also explain how the $NS-\alpha$ model is related to large eddy simulation turbulence modeling and to the stress tensor for second-grade fluids. We close by surveying recent results in the literature for the $NS-\alpha$ model and its inviscid limit (the Euler-$\alpha$ model).

##### MSC:
 76F02 Fundamentals of turbulence 76D05 Navier-Stokes equations (fluid dynamics) 76F65 Direct numerical and large eddy simulation of turbulence
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##### References:
 [1] Germano, M.: Turbulence: the filtering approach. J. fluid mech. 238, 325-336 (1992) · Zbl 0756.76034 [2] Ghosal, S.: Mathematical and physical constraints on large-eddy simulation of turbulence. Aiaa j. 37, 425 (1999) [3] Chen, S.; Foias, C.; Holm, D. D.; Olson, E.; Titi, E. S.; Wynne, S.: The Camassa--Holm equations as a closure model for turbulent channel and pipe flow. Phys. rev. Lett. 81, 5338-5341 (1998) · Zbl 1042.76525 [4] Chen, S.; Foias, C.; Holm, D. D.; Olson, E.; Titi, E. S.; Wynne, S.: A connection between the Camassa--Holm equations and turbulence flows in pipes and channels. Phys. fluids 11, 2343-2353 (1999) · Zbl 1147.76357 [5] Chen, S.; Foias, C.; Holm, D. D.; Olson, E.; Titi, E. S.; Wynne, S.: The Camassa--Holm equations and turbulence in pipes and channels. Physica D 133, 49-65 (1999) · Zbl 1194.76069 [6] C. Foias, D.D. Holm, E.S. Titi, The three-dimensional viscous Camassa--Holm equations, and their relation to the Navier--Stokes equations and turbulence theory, J. Dyn. Diff. Eq., submitted for publication. · Zbl 0995.35051 [7] Chen, S. Y.; Holm, D. D.; Margolin, L. G.; Zhang, R.: Direct numerical simulations of the Navier--Stokes alpha model. Physica D 133, 66-83 (1999) · Zbl 1194.76080 [8] J.A. Domaradzki, D.D. Holm, Navier--Stokes-alpha model: LES equations with nonlinear dispersion, Special volume on LES, ERCOFTAC Bull., in press. [9] Holm, D. D.; Marsden, J. E.; Ratiu, T. S.: Euler--Poincaré models of ideal fluids with nonlinear dispersion. Phys. rev. Lett. 80, 4173-4176 (1998) [10] Holm, D. D.; Marsden, J. E.; Ratiu, T. S.: Euler--Poincaré equations and semidirect products with applications to continuum theories. Adv. math. 137, 1-81 (1998) · Zbl 0951.37020 [11] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934) 193--248. (Reviewed, e.g., in P. Constantin, C. Foias, B. Nicolaenko, R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, Appl. Math. Sci. 70 (1989).) [12] G. Gallavotti, Some rigorous results about 3D Navier--Stokes, in: R. Benzi, C. Basdevant, S. Ciliberto (Eds.), Les Houches 1992 NATO-ASI Meeting on Turbulence in Extended Systems, Nova Science, New York, 1993, pp. 45--81. [13] Kuz’min, G. A.: Ideal incompressible hydrodynamics in terms of the vortex momentum density. Phys. lett. A 96, 88-90 (1983) [14] V.I. Oseledets, New form of the Navier--Stokes equation. Hamiltonian formalism, Moskov. Matemat. Obshch. 44, No. 3 (267) (1989) 169--170 (in Russian). · Zbl 0850.76130 [15] S. Gama, U. Frisch, Local helicity, a material invariant for the odd-dimensional incompressible Euler equations, in: M.R.E. Proctor, P.C. Mathews, A.M. Rucklidge (Eds.), Proceedings of NATO-ASI on Theory of Solar and Planetary Dynamos, Cambridge University Press, Cambridge, 1993, pp. 115--119. [16] Holm, D. D.; Kupershmidt, B. A.: Poisson brackets and clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity. Physica D 6, 347-363 (1983) · Zbl 1194.76285 [17] Foias, C.; Temam, R.: Gevrey class regularity for the solutions of the Navier--Stokes equations. J. funct. Anal. 87, 359-369 (1989) · Zbl 0702.35203 [18] Doering, C. R.; Titi, E. S.: Exponential decay rate of the power spectrum for solutions of the Navier--Stokes equations. Phys. fluids 7, 1384-1390 (1995) · Zbl 1023.76513 [19] Holm, D. D.: Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion. Physica D 133, 215-269 (1999) · Zbl 1194.76062 [20] S. Shkoller, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of fluid, J. Diff. Geom., to appear. · Zbl 1044.35061 [21] Kraichnan, R. H.: Inertial ranges in two-dimensional turbulence. Phys. fluids 10, 1417-1423 (1967) [22] C. Foias, What do the Navier--Stokes equations tell us about turbulence? in: Harmonic Analysis and Nonlinear Differential Equations, Riverside, CA, 1995; Contemp. Math. 208 (1997) 151--180, Amer. Math. Soc., Providence, RI. · Zbl 0890.76030 [23] S.Y. Chen, D.D. Holm, L.G. Margolin, R. Zhang, Direct numerical simulations of the Navier--Stokes alpha model in the limit {$\alpha$}\to\infty, in preparation. · Zbl 1194.76080 [24] Oboukov, A. M.: On the distribution of energy in the spectrum of a turbulent flow. Dokl. akad. Sci. nauk SSSR 32A, 22-24 (1941) [25] M. Lesieur, Turbulence in fluids, Fluid Mechanics and its Applications, Vol. 40, 3rd Edition, Kluwer Academic Publishers, London, 1997, p. 179. · Zbl 0876.76002 [26] Leith, C. E.; Kraichnan, R. H.: Predictability of turbulent flows. J. atmos. Sci. 29, 1041-1058 (1972) [27] Dunn, J. E.; Fosdick, R. L.: Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade. Arch. rat. Mech. anal. 56, 191-252 (1974) · Zbl 0324.76001 [28] Dunn, J. E.; Rajagopal, K. R.: Fluids of differential type: critical reviews and thermodynamic analysis. Int. J. Eng. sci. 33, 689-729 (1995) · Zbl 0899.76062 [29] Cioranescu, D.; Girault, V.: Solutions variationelles et classiques d’une famille de fluides de grade deux. CR acad. Sci. Paris série 322, No. 1, 1163-1168 (1996) · Zbl 0853.76003 [30] Cioranescu, D.; Girault, V.: Weak and classical solutions of a family of second grade fluids. Int. J. Non-lin. Mech. 32, 317-335 (1997) · Zbl 0891.76005 [31] Busuioc, V.: On second grade fluids with vanishing viscosity. CR acad. Sci. serie I 328, 1241-1246 (1999) · Zbl 0935.76004 [32] Arnold, V. I.: Sur la géometrie differentielle des groupes de Lie de dimenson infinie et ses applications à l’hydrodynamique des fluids parfaits. Ann. inst. Fourier Grenoble 16, 319-361 (1966) [33] Ebin, D.; Marsden, J. E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. math. 92, 102-163 (1970) · Zbl 0211.57401 [34] Rivlin, R. S.: The relation between the flow of non-Newtonian fluids and turbulent Newtonian fluids. Q. appl. Math. 15, 212-215 (1957) · Zbl 0079.17905 [35] Chorin, A. J.: Spectrum, dimension, and polymer analogies in fluid turbulence. Phys. rev. Lett. 60, 1947-1949 (1988) [36] Shih, T. H.; Zhu, J.; Lumley, J. L.: A new Reynolds stress algebraic equation model. Comput. meth. Appl. mech. Eng. 125, 287-302 (1995) [37] Yoshizawa, A.: Statistical analysis of the derivation of the Reynolds stress from its eddy-viscosity representation. Phys. fluids 27, 1377-1387 (1984) · Zbl 0572.76048 [38] Rubinstein, R.; Barton, J. M.: Nonlinear Reynolds stress models and the renormalization group. Phys. fluids A 2, 1472-1476 (1990) · Zbl 0709.76068 [39] D.D. Holm, J.E. Marsden, T.S. Ratiu, The Euler--Poincaré equations in geophysical fluid dynamics, in: Proceedings of the Isaac Newton Institute Programme on the Mathematics of Atmospheric and Ocean Dynamics, Cambridge University Press, Cambridge, in press (see Section 3). [40] A. Leonard, Personal communication, June 1999. [41] M. Oliver, S. Shkoller, The vortex blob method as a second-grade non-Newtonian fluid, Preprint, 1999. · Zbl 0983.35106 [42] Camassa, R.; Holm, D. D.: An integrable shallow water equation with peaked solitons. Phys. rev. Lett. 71, 1661-1664 (1993) · Zbl 0972.35521 [43] D.D. Holm, S. Kouranbaeva, J.E. Marsden, T. Ratiu, S. Shkoller, A nonlinear analysis of the averaged Euler equations, unpublished. [44] Shkoller, S.: Geometry and curvature of diffeomorphism groups with H1 metric and mean hydrodynamics. J. funct. Anal. 160, 337-355 (1998) · Zbl 0933.58010 [45] J.E. Marsden, T. Ratiu, S. Shkoller, A nonlinear analysis of the averaged Euler equations and a new diffeomorphism group, Geom. Funct. Anal. 10 (2000) 582--599. · Zbl 0979.58004