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November  2011, 10(6): 1763-1777. doi: 10.3934/cpaa.2011.10.1763

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

1. 

University of Pittsburgh, Department of Mathematics, Pittsburgh, PA 15260

2. 

Farquhar College of Arts and Sciences, Nova Southeastern University, Ft. Lauderdale, FL 33314, United States

3. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, United States

Received  June 2010 Revised  March 2011 Published  May 2011

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.
Citation: W. Layton, Iuliana Stanculescu, Catalin Trenchea. Theory of the NS-$\overline{\omega}$ model: A complement to the NS-$\alpha$ model. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1763-1777. doi: 10.3934/cpaa.2011.10.1763
References:
[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.

show all references

References:
[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.

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