Article Contents
Article Contents

# Theory of the NS-$\overline{\omega}$ model: A complement to the NS-$\alpha$ model

• We study a recent regularization of the Navier-Stokes equations, the NS-$\overline{\omega}$ model. This model has similarities to the NS-$\alpha$ model, but its structure is more amenable to be used as a basis for numerical simulations of turbulent flows. In this report we present the model and prove existence and uniqueness of strong solutions as well as convergence (modulo a subsequence) to a weak solution of the Navier-Stokes equations as the averaging radius decreases to zero. We then apply turbulence phenomenology to the model to obtain insight into its predictions.
Mathematics Subject Classification: 65M12, 65M60, 76D05, 76F65.

 Citation:

•  [1] C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140. [2] G. Baker, Galerkin approximations for the Navier-Stokes equations, Tech. report, Harvard University, 1976. [3] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976. [4] V. Barbu, "Analysis and Control of Nonlinear Infinite Dimensional Systems," Academic Press, Boston, 1993. [5] V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, J. Math. Anal. Appl., 255 (2001), 281-307.doi: 10.1006/jmaa.2000.7256. [6] H. Brezis, "Opérateurs Maximaux et Semigroupes de Contractions dans les Espaces de Hilbert," North-Holland, New York, 1973. [7] A. Caglar, A finite element approximation of the Navier-Stokes-alpha model, PIMS, preprint series (2003), PIMS 03-14. [8] Q. Chen, S. Chen and G. Eyink, The joint cascade of energy and helicity in three dimensional turbulence, Physics of Fluids, 15 (2003), 361-374.doi: 10.1063/1.1533070. [9] S. Chen, C. Foias, D. Holm, E. Olson, E. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65.doi: 10.1016/S0167-2789(99)00098-6. [10] A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 629-649.doi: 10.1098/rspa.2004.1373. [11] A. Chorin and J. Marsden, "A Mathematical Introduction to Fluid Mechanics," Springer, 2000. [12] J. Connors, Convergence analysis and computational testing of the finite element discretization of the Navier-Stokes alpha model, Numer. Methods for Partial Differential Equations, 26 (2010), 1328-1350. [13] P. Ditlevsen and P. Giuliani, Cascades in helical turbulence, Phys. Rev. E, 63 (2001), 036304/1-4. [14] U. Frisch, "Turbulence," Cambridge, 1995. [15] G. P. Galdi and W. J. Layton, Approximation of the larger eddies in fluid motions. II. A model for space-filtered flow, Math. Models Methods Appl. Sci., 10 (2000), 343-350. [16] M. Germano, Differential filters for the large eddy numerical simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757. [17] M. Germano, Differential filters of elliptic type, Phys. Fluids, 29 (1986), 1757-1758.doi: 10.1063/1.865649. [18] B. J. Geurts and D. D. Holm, Regularization modeling for large-eddy simulation, Phys. Fluids, 15 (2003), L13-L16.doi: 10.1063/1.1529180. [19] B. J. Geurts and D. D. Holm, Leray and LANS-alpha modeling of turbulent mixing, J. of Turbulence, 7 (2006), 1-33.doi: 10.1080/14685240500501601. [20] J. P. Graham, D. Holm, P. Mininni and A. Pouquet, Comparison on three regularized model of the NSE when viewed as large eddy simulations, Tech. report, 2007. [21] A. A. Ilyin, E. M. Lunasin and E. S. Titi, A modified Leray-$\alpha$ subgrid model of turbulence, Nonlinearity, 19 (2006), 879-897.doi: 10.1088/0951-7715/19/4/006. [22] R. Kraichnan, Inertial-range transfer in two- and three-dimensional turbulence, J. Fluid Mech., 47 (1971), 525.doi: 10.1017/S0022112071001216. [23] A. Labovschii, W. Layton, C. Manica, M. Neda, L. Rebholz, I. Stanculescu and C. Trenchea, Mathematical architecture of approximate deconvolution models of turbulence, Quality and Reliability of Large-Eddy Simulations, ERCOFTAC, Springer, 2007. [24] W. Layton and R. Lewandowski, A high accuracy Leray-deconvolution model of turbulence and its limiting behavior, Analysis and Applications, 6 (2008), 23-49.doi: 10.1142/S0219530508001043. [25] W. Layton, C. Manica, M. Neda and L. Rebholz, Helicity-Energy cascade for homogeneous, isotropic turbulence generated by approximate deconvolution models, Adv. Appl. Fluid Mech., 4 (2008), 1-46. [26] W. Layton, C. Manica, M. Neda and L. Rebholz, Numerical analysis and computational testing of a high order Leray-deconvolution turbulence model, Numer. Methods Partial Differential Equations, 24 (2008), 555-582.doi: 10.1002/num.20281. [27] W. Layton, C. Manica, M. Neda and L. Rebholz, Numerical analysis and computational comparisons of the NS-alpha and NS-omega regularizations, Comput. Methods Appl. Mech. Engrg., 199 (2010), 916-931.doi: 10.1016/j.cma.2009.01.011. [28] W. Layton and M. Neda, A similarity theory of approximate deconvolution models of turbulence, J. Math. Anal. Appl., 333 (2007), 416-429.doi: 10.1016/j.jmaa.2007.01.063. [29] W. Layton and M. Neda, Truncation of scales by time relaxation, J. Math. Anal. Appl., 325 (2007), 788-807.doi: 10.1016/j.jmaa.2006.02.014. [30] J. Leray, Essay sur les mouvements plans d'une liquide visqueux que limitent des parois, J. Math. Pur. Appl., Paris Ser. IX, 13 (1934), 331-418. [31] J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.doi: 10.1007/BF02547354. [32] J. L. Lions, "Quelques méthodes de résolution des problémes aux limites non linéaires," Études mathématiques, Dunod Gauthiers-Villars, 1969. [33] W. W. Miles and L. G. Rebholz, An enhanced-physics-based scheme for the NS-$\alpha$ turbulence model, Numer. Methods Partial Differential Equations, 26 (2010), 1530-1555. [34] J. J. Moreau, Constantes d'unilot tourbilloinnaire en fluide parfait barotrope, C. R. Acad. Sci. Paris, 252 (1961), 2810-2812. [35] A. Muschinski, A similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type LES, J. Fluid Mech., 325 (1996), 239-260.doi: 10.1017/S0022112096008105. [36] L. G. Rebholz, Conservation laws of turbulence models, J. Math. Anal. Appl., 326 (2007), 33-45.doi: 10.1016/j.jmaa.2006.02.026. [37] E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, 1970. [38] S. Stolz, N. A. Adams and L. Kleiser, An approximate deconvolution model for large eddy simulation with application to wall-bounded flows, Phys. Fluids, 13 (2001), 997-1015.doi: 10.1063/1.1350896. [39] L. Tartar, Topics in nonlinear analysis, Publications Mathématiques d'Orsay, vol. 13, Université de Paris-Sud, Départment de Mathématique, Orsay, 1978. [40] M. van Reeuwijk, H. J. J. Jonker and K. Hanjalić, Incompressibility of the Leray-$\alpha$ model for wall-bounded flows, Phys. Fluids, 18 (2006), 018103, 4. [41] M. I. Vishik, E. S. Titi and V. V. Chepyzhov, Dokl. Akad. Nauk, Russian Math Dokladi, 400 (2005), 583-586.