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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 |
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] | |
[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] | |
[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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