# American Institute of Mathematical Sciences

• Previous Article
Even solutions of the Toda system with prescribed asymptotic behavior
• CPAA Home
• This Issue
• Next Article
A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities
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 & 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.  Google Scholar [2] G. Baker, Galerkin approximations for the Navier-Stokes equations, Tech. report, Harvard University, 1976. Google Scholar [3] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976.  Google Scholar [4] V. Barbu, "Analysis and Control of Nonlinear Infinite Dimensional Systems," Academic Press, Boston, 1993.  Google Scholar [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.  Google Scholar [6] H. Brezis, "Opérateurs Maximaux et Semigroupes de Contractions dans les Espaces de Hilbert," North-Holland, New York, 1973.  Google Scholar [7] A. Caglar, A finite element approximation of the Navier-Stokes-alpha model, PIMS, preprint series (2003), PIMS 03-14. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] A. Chorin and J. Marsden, "A Mathematical Introduction to Fluid Mechanics," Springer, 2000.  Google Scholar [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.  Google Scholar [13] P. Ditlevsen and P. Giuliani, Cascades in helical turbulence, Phys. Rev. E, 63 (2001), 036304/1-4. Google Scholar [14] U. Frisch, "Turbulence," Cambridge, 1995. Google Scholar [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.  Google Scholar [16] M. Germano, Differential filters for the large eddy numerical simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757.  Google Scholar [17] M. Germano, Differential filters of elliptic type, Phys. Fluids, 29 (1986), 1757-1758. doi: 10.1063/1.865649.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [22] R. Kraichnan, Inertial-range transfer in two- and three-dimensional turbulence, J. Fluid Mech., 47 (1971), 525. doi: 10.1017/S0022112071001216.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [31] J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.  Google Scholar [32] J. L. Lions, "Quelques méthodes de résolution des problémes aux limites non linéaires," Études mathématiques, Dunod Gauthiers-Villars, 1969.  Google Scholar [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.  Google Scholar [34] J. J. Moreau, Constantes d'unilot tourbilloinnaire en fluide parfait barotrope, C. R. Acad. Sci. Paris, 252 (1961), 2810-2812. Google Scholar [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.  Google Scholar [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.  Google Scholar [37] E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, 1970.  Google Scholar [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.  Google Scholar [39] L. Tartar, Topics in nonlinear analysis, Publications Mathématiques d'Orsay, vol. 13, Université de Paris-Sud, Départment de Mathématique, Orsay, 1978.  Google Scholar [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.  Google Scholar [41] M. I. Vishik, E. S. Titi and V. V. Chepyzhov, Dokl. Akad. Nauk, Russian Math Dokladi, 400 (2005), 583-586.  Google Scholar

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.  Google Scholar [2] G. Baker, Galerkin approximations for the Navier-Stokes equations, Tech. report, Harvard University, 1976. Google Scholar [3] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976.  Google Scholar [4] V. Barbu, "Analysis and Control of Nonlinear Infinite Dimensional Systems," Academic Press, Boston, 1993.  Google Scholar [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.  Google Scholar [6] H. Brezis, "Opérateurs Maximaux et Semigroupes de Contractions dans les Espaces de Hilbert," North-Holland, New York, 1973.  Google Scholar [7] A. Caglar, A finite element approximation of the Navier-Stokes-alpha model, PIMS, preprint series (2003), PIMS 03-14. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] A. Chorin and J. Marsden, "A Mathematical Introduction to Fluid Mechanics," Springer, 2000.  Google Scholar [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.  Google Scholar [13] P. Ditlevsen and P. Giuliani, Cascades in helical turbulence, Phys. Rev. E, 63 (2001), 036304/1-4. Google Scholar [14] U. Frisch, "Turbulence," Cambridge, 1995. Google Scholar [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.  Google Scholar [16] M. Germano, Differential filters for the large eddy numerical simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757.  Google Scholar [17] M. Germano, Differential filters of elliptic type, Phys. Fluids, 29 (1986), 1757-1758. doi: 10.1063/1.865649.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [22] R. Kraichnan, Inertial-range transfer in two- and three-dimensional turbulence, J. Fluid Mech., 47 (1971), 525. doi: 10.1017/S0022112071001216.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [31] J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.  Google Scholar [32] J. L. Lions, "Quelques méthodes de résolution des problémes aux limites non linéaires," Études mathématiques, Dunod Gauthiers-Villars, 1969.  Google Scholar [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.  Google Scholar [34] J. J. Moreau, Constantes d'unilot tourbilloinnaire en fluide parfait barotrope, C. R. Acad. Sci. Paris, 252 (1961), 2810-2812. Google Scholar [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.  Google Scholar [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.  Google Scholar [37] E. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, 1970.  Google Scholar [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.  Google Scholar [39] L. Tartar, Topics in nonlinear analysis, Publications Mathématiques d'Orsay, vol. 13, Université de Paris-Sud, Départment de Mathématique, Orsay, 1978.  Google Scholar [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.  Google Scholar [41] M. I. Vishik, E. S. Titi and V. V. Chepyzhov, Dokl. Akad. Nauk, Russian Math Dokladi, 400 (2005), 583-586.  Google Scholar
 [1] Junyoung Jang, Kihoon Jang, Hee-Dae Kwon, Jeehyun Lee. Feedback control of an HBV model based on ensemble kalman filter and differential evolution. Mathematical Biosciences & Engineering, 2018, 15 (3) : 667-691. doi: 10.3934/mbe.2018030 [2] T. Tachim Medjo. A Cahn-Hilliard-Navier-Stokes model with delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2663-2685. doi: 10.3934/dcdsb.2016067 [3] Yejuan Wang, Tongtong Liang. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3713-3740. doi: 10.3934/dcdsb.2018312 [4] W. Layton, R. Lewandowski. On a well-posed turbulence model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 111-128. doi: 10.3934/dcdsb.2006.6.111 [5] T. Tachim Medjo. Robust control of a Cahn-Hilliard-Navier-Stokes model. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2075-2101. doi: 10.3934/cpaa.2016028 [6] Yann Brenier. Approximation of a simple Navier-Stokes model by monotonic rearrangement. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1285-1300. doi: 10.3934/dcds.2014.34.1285 [7] Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697 [8] Alexey Cheskidov, Susan Friedlander, Nataša Pavlović. An inviscid dyadic model of turbulence: The global attractor. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 781-794. doi: 10.3934/dcds.2010.26.781 [9] Peter R. Kramer, Joseph A. Biello, Yuri Lvov. Application of weak turbulence theory to FPU model. Conference Publications, 2003, 2003 (Special) : 482-491. doi: 10.3934/proc.2003.2003.482 [10] Changjing Zhuge, Xiaojuan Sun, Jinzhi Lei. On positive solutions and the Omega limit set for a class of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2487-2503. doi: 10.3934/dcdsb.2013.18.2487 [11] Luigi C. Berselli, Tae-Yeon Kim, Leo G. Rebholz. Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1027-1050. doi: 10.3934/dcdsb.2016.21.1027 [12] T. Tachim Medjo. Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1281-1305. doi: 10.3934/cpaa.2011.10.1281 [13] Julia García-Luengo, Pedro Marín-Rubio, Gabriela Planas. Attractors for a double time-delayed 2D-Navier-Stokes model. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4085-4105. doi: 10.3934/dcds.2014.34.4085 [14] T. Tachim Medjo. The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1117-1138. doi: 10.3934/cpaa.2019054 [15] Julia García-Luengo, Pedro Marín-Rubio, Gabriela Planas. Some regularity results for a double time-delayed 2D-Navier-Stokes model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3929-3946. doi: 10.3934/dcdsb.2018337 [16] Shuguang Shao, Shu Wang, Wen-Qing Xu. Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation. Kinetic & Related Models, 2018, 11 (1) : 179-190. doi: 10.3934/krm.2018009 [17] Anne Bronzi, Ricardo Rosa. On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 19-49. doi: 10.3934/dcds.2014.34.19 [18] Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115 [19] Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397 [20] Tae-Yeon Kim, Xuemei Chen, John E. Dolbow, Eliot Fried. Going to new lengths: Studying the Navier--Stokes-$\alpha\beta$ equations using the strained spiral vortex model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2207-2225. doi: 10.3934/dcdsb.2014.19.2207

2020 Impact Factor: 1.916