# American Institute of Mathematical Sciences

June  2016, 21(4): 1027-1050. doi: 10.3934/dcdsb.2016.21.1027

## Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization

 1 Dipartimento di Matematica, Università di Pisa, I-56127 Pisa, Italy 2 Department of Civil Infrastructure and Environmental Engineering, Khalifa University of Science, Technology & Research (KUSTAR), Abu Dhabi, United Arab Emirates 3 Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, United States

Received  May 2015 Revised  January 2016 Published  March 2016

We study mathematical and physical properties of a family of recently introduced, reduced-order approximate deconvolution models. We first show a connection between these models and the Navier-Stokes-Voigt model, and also that Navier-Stokes-Voigt can be re-derived in the approximate deconvolution framework. We then study the energy balance and spectra of the model, and provide results of some turbulent-flow computations that backs up the theory. Analysis of global attractors for the model is also provided, as is a detailed analysis of the Voigt model's treatment of pulsatile flow.
Citation: 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
##### References:
 [1] N. A. Adams and S. Stolz, Deconvolution Methods for Subgrid-Scale Approximation in Large Eddy Simulation, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, 2001. Google Scholar [2] N. A. Adams and S. Stolz, A subgrid-scale deconvolution approach for shock capturing, J. Comput. Phys., 178 (2002), 391-426. doi: 10.1006/jcph.2002.7034.  Google Scholar [3] H. Beirão da Veiga, Time periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains-Leray's problem for periodic flows, Arch. Ration. Mech. Anal., 178 (2005), 301-325. ibidem 198 (2010), 1095 doi: 10.1007/s00205-005-0376-3.  Google Scholar [4] L. C. Berselli, Analysis of a Large Eddy Simulation model based on anisotropic filtering, J. Math. Anal. Appl., 386 (2012), 149-170. doi: 10.1016/j.jmaa.2011.07.044.  Google Scholar [5] L. C. Berselli, Towards fluid equations by approximate deconvolution models, in Mathematical Aspects of Fluid Mechanics, vol. 402 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2012, 1-22.  Google Scholar [6] L. C. Berselli and L. Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal., 75 (2012), 117-130. doi: 10.1016/j.na.2011.08.011.  Google Scholar [7] L. C. Berselli, F. Guerra, B. Mazzolai and E. Sinibaldi, Pulsatile viscous flows in elliptical vessels and annuli: solution to the inverse problem, with application to blood and cerebrospinal fluid flow, SIAM J. Appl. Math., 74 (2014), 40-59. doi: 10.1137/120903385.  Google Scholar [8] L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation, Springer-Verlag, Berlin, 2006. doi: 10.1007/b137408.  Google Scholar [9] L. C. Berselli and R. Lewandowski, Convergence of approximate deconvolution models to the mean Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 171-198. doi: 10.1016/j.anihpc.2011.10.001.  Google Scholar [10] L. C. Berselli and S. Spirito, On the Construction of Suitable Weak Solutions to the 3D Navier-Stokes Equations in a Bounded Domain by an Artificial Compressibility Method, Technical Report 1504.07800, arXiv, 2015, URL http://arxiv.org/abs/1504.07800. To appear in Commun. Contemp. Math. Google Scholar [11] A. L. Bowers, T.-Y. Kim, M. Neda, L. G. Rebholz and E. Fried, The Leray-$\alpha\beta$-deconvolution model: Energy analysis and numerical algorithms, Appl. Math. Model., 37 (2013), 1225-1241. doi: 10.1016/j.apm.2012.03.040.  Google Scholar [12] Y. Cao, E. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848, \urlprefixhttp://projecteuclid.org/euclid.cms/1175797613. doi: 10.4310/CMS.2006.v4.n4.a8.  Google Scholar [13] R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976, Mathematics in Science and Engineering, Vol. 127.  Google Scholar [14] T. Chacón Rebollo and R. Lewandowski, Mathematical and Numerical Foundations of Turbulence Models and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, New York, 2014. doi: 10.1007/978-1-4939-0455-6.  Google Scholar [15] S. Chen, G. Doolen, R. Kraichnan and Z. She, On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence, Phys. Fluids A, 5 (1993), 458-463. doi: 10.1063/1.858897.  Google Scholar [16] S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Phys. D, 133 (1999), 66-83, Predictability: quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998). doi: 10.1016/S0167-2789(99)00099-8.  Google Scholar [17] 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 [18] A. Dunca and Y. Epshteyn, On the Stolz-Adams deconvolution model for the large-eddy simulation of turbulent flows, SIAM J. Math. Anal., 37 (2006), 1890-1902 (electronic). doi: 10.1137/S0036141003436302.  Google Scholar [19] A. A. Dunca, A two-level multiscale deconvolution method for the large eddy simulation of turbulent flows, Math. Models Methods Appl. Sci., 22 (2012), 1250001, 30pp. doi: 10.1142/S0218202512500017.  Google Scholar [20] C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519, Advances in nonlinear mathematics and science. doi: 10.1016/S0167-2789(01)00191-9.  Google Scholar [21] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, vol. 83 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.  Google Scholar [22] U. Frisch, Turbulence, The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.  Google Scholar [23] Y. C. Fung, Biomechanics. Circulation, Springer-Verlag, 1997. Google Scholar [24] K. J. Galvin, L. G. Rebholz and C. Trenchea, Efficient, unconditionally stable, and optimally accurate FE algorithms for approximate deconvolution models, SIAM J. Numer. Anal., 52 (2014), 678-707. doi: 10.1137/120887412.  Google Scholar [25] D. F. Hinz, T.-Y. Kim and E. Fried, Statistics of the Navier-Stokes-alpha-beta regularization model for fluid turbulence, J. Phys. A, 47 (2014), 055501, 21pp. doi: 10.1088/1751-8113/47/5/055501.  Google Scholar [26] K. J. K. Galvin, Advancements in Finite Element Methods for Newtonian and Non-Newtonian Flows, PhD thesis, Clemson University, 2013.  Google Scholar [27] V. K. Kalantarov, B. Levant and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152. doi: 10.1007/s00332-008-9029-7.  Google Scholar [28] V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714. doi: 10.1007/s11401-009-0205-3.  Google Scholar [29] T.-Y. Kim, M. Cassiani, J. D. Albertson, J. E. Dolbow, E. Fried and M. E. Gurtin, Impact of the inherent separation of scales in the Navier-Stokes-$\alpha\beta$ equations, Phys. Rev. E (3), 79 (2009), 045307, 4pp. doi: 10.1103/PhysRevE.79.045307.  Google Scholar [30] T.-Y. Kim, L. G. Rebholz and E. Fried, A deconvolution enhancement of the Navier-Stokes-$\alpha\beta$ model, J. Comput. Phys., 231 (2012), 4015-4027. doi: 10.1016/j.jcp.2011.12.003.  Google Scholar [31] P. Kuberry, A. Larios, L. G. Rebholz and N. E. Wilson, Numerical approximation of the Voigt regularization for incompressible Navier-Stokes and magnetohydrodynamic flows, Comput. Math. Appl., 64 (2012), 2647-2662. doi: 10.1016/j.camwa.2012.07.010.  Google Scholar [32] A. Larios, The Inviscid Voigt-Regularization for Hydrodynamic Models: Global Regularity, Boundary Conditions, and Blow-Up Phenomena, PhD thesis, Univ. California, Irvine, 2011.  Google Scholar [33] A. Larios and E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 603-627. doi: 10.3934/dcdsb.2010.14.603.  Google Scholar [34] W. J. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128 (electronic). doi: 10.3934/dcdsb.2006.6.111.  Google Scholar [35] W. J. Layton and L. Rebholz, Approximate Deconvolution Models of Turbulence Approximate Deconvolution Models of Turbulence, vol. 2042 of Lecture Notes in Mathematics, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-24409-4.  Google Scholar [36] B. Levant, F. Ramos and E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Commun. Math. Sci., 8 (2010), 277-293. doi: 10.4310/CMS.2010.v8.n1.a14.  Google Scholar [37] A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 98-136, Boundary value problems of mathematical physics and related questions in the theory of functions, 7.  Google Scholar [38] A. P. Oskolkov, On the theory of unsteady flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 115 (1982), 191-202, 310, Boundary value problems of mathematical physics and related questions in the theory of functions, 14.  Google Scholar [39] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, in Handbook of numerical analysis. Vol. XII, Handb. Numer. Anal., XII, North-Holland, Amsterdam, 2004, 3-127. doi: 10.1016/S1570-8659(03)12001-7.  Google Scholar [40] L. G. Rebholz, Well-posedness of a reduced order approximate deconvolution turbulence model, J. Math. Anal. Appl., 405 (2013), 738-741. doi: 10.1016/j.jmaa.2013.04.036.  Google Scholar [41] P. Sagaut, Large Eddy Simulation for Incompressible Flows, Scientific Computation, Springer-Verlag, Berlin, 2001, An introduction, With an introduction by Marcel Lesieur, Translated from the 1998 French original by the author. doi: 10.1007/978-3-662-04416-2.  Google Scholar [42] Z. S. She, E. Jackson and S. A. Orszag, Statistical aspects of vortex dynamics in turbulence, in New perspectives in turbulence (Newport, RI, 1989), Springer, New York, 1991, 315-328. doi: 10.1007/978-1-4612-3156-1_12.  Google Scholar [43] I. Stanculescu and C. C. Manica, Numerical analysis of Leray-Tikhonov deconvolution models of fluid motion, Comput. Math. Appl., 60 (2010), 1440-1456. doi: 10.1016/j.camwa.2010.06.026.  Google Scholar [44] S. Stolz and N. A. Adams, An approximate deconvolution procedure for large-eddy simulation, Phys. Fluids, 11 (1999), 1699-1701. doi: 10.1063/1.869867.  Google Scholar [45] S. Stolz, N. A. Adams and L. Kleiser, An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows, Phys. Fluids, 13 (2001), 997-1015. doi: 10.1063/1.1350896.  Google Scholar [46] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [47] J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol., 127 (1955), 553-563. doi: 10.1113/jphysiol.1955.sp005276.  Google Scholar

show all references

##### References:
 [1] N. A. Adams and S. Stolz, Deconvolution Methods for Subgrid-Scale Approximation in Large Eddy Simulation, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, 2001. Google Scholar [2] N. A. Adams and S. Stolz, A subgrid-scale deconvolution approach for shock capturing, J. Comput. Phys., 178 (2002), 391-426. doi: 10.1006/jcph.2002.7034.  Google Scholar [3] H. Beirão da Veiga, Time periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains-Leray's problem for periodic flows, Arch. Ration. Mech. Anal., 178 (2005), 301-325. ibidem 198 (2010), 1095 doi: 10.1007/s00205-005-0376-3.  Google Scholar [4] L. C. Berselli, Analysis of a Large Eddy Simulation model based on anisotropic filtering, J. Math. Anal. Appl., 386 (2012), 149-170. doi: 10.1016/j.jmaa.2011.07.044.  Google Scholar [5] L. C. Berselli, Towards fluid equations by approximate deconvolution models, in Mathematical Aspects of Fluid Mechanics, vol. 402 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2012, 1-22.  Google Scholar [6] L. C. Berselli and L. Bisconti, On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models, Nonlinear Anal., 75 (2012), 117-130. doi: 10.1016/j.na.2011.08.011.  Google Scholar [7] L. C. Berselli, F. Guerra, B. Mazzolai and E. Sinibaldi, Pulsatile viscous flows in elliptical vessels and annuli: solution to the inverse problem, with application to blood and cerebrospinal fluid flow, SIAM J. Appl. Math., 74 (2014), 40-59. doi: 10.1137/120903385.  Google Scholar [8] L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation, Springer-Verlag, Berlin, 2006. doi: 10.1007/b137408.  Google Scholar [9] L. C. Berselli and R. Lewandowski, Convergence of approximate deconvolution models to the mean Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 171-198. doi: 10.1016/j.anihpc.2011.10.001.  Google Scholar [10] L. C. Berselli and S. Spirito, On the Construction of Suitable Weak Solutions to the 3D Navier-Stokes Equations in a Bounded Domain by an Artificial Compressibility Method, Technical Report 1504.07800, arXiv, 2015, URL http://arxiv.org/abs/1504.07800. To appear in Commun. Contemp. Math. Google Scholar [11] A. L. Bowers, T.-Y. Kim, M. Neda, L. G. Rebholz and E. Fried, The Leray-$\alpha\beta$-deconvolution model: Energy analysis and numerical algorithms, Appl. Math. Model., 37 (2013), 1225-1241. doi: 10.1016/j.apm.2012.03.040.  Google Scholar [12] Y. Cao, E. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848, \urlprefixhttp://projecteuclid.org/euclid.cms/1175797613. doi: 10.4310/CMS.2006.v4.n4.a8.  Google Scholar [13] R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976, Mathematics in Science and Engineering, Vol. 127.  Google Scholar [14] T. Chacón Rebollo and R. Lewandowski, Mathematical and Numerical Foundations of Turbulence Models and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, New York, 2014. doi: 10.1007/978-1-4939-0455-6.  Google Scholar [15] S. Chen, G. Doolen, R. Kraichnan and Z. She, On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence, Phys. Fluids A, 5 (1993), 458-463. doi: 10.1063/1.858897.  Google Scholar [16] S. Chen, D. D. Holm, L. G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Phys. D, 133 (1999), 66-83, Predictability: quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998). doi: 10.1016/S0167-2789(99)00099-8.  Google Scholar [17] 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 [18] A. Dunca and Y. Epshteyn, On the Stolz-Adams deconvolution model for the large-eddy simulation of turbulent flows, SIAM J. Math. Anal., 37 (2006), 1890-1902 (electronic). doi: 10.1137/S0036141003436302.  Google Scholar [19] A. A. Dunca, A two-level multiscale deconvolution method for the large eddy simulation of turbulent flows, Math. Models Methods Appl. Sci., 22 (2012), 1250001, 30pp. doi: 10.1142/S0218202512500017.  Google Scholar [20] C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519, Advances in nonlinear mathematics and science. doi: 10.1016/S0167-2789(01)00191-9.  Google Scholar [21] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, vol. 83 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.  Google Scholar [22] U. Frisch, Turbulence, The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.  Google Scholar [23] Y. C. Fung, Biomechanics. Circulation, Springer-Verlag, 1997. Google Scholar [24] K. J. Galvin, L. G. Rebholz and C. Trenchea, Efficient, unconditionally stable, and optimally accurate FE algorithms for approximate deconvolution models, SIAM J. Numer. Anal., 52 (2014), 678-707. doi: 10.1137/120887412.  Google Scholar [25] D. F. Hinz, T.-Y. Kim and E. Fried, Statistics of the Navier-Stokes-alpha-beta regularization model for fluid turbulence, J. Phys. A, 47 (2014), 055501, 21pp. doi: 10.1088/1751-8113/47/5/055501.  Google Scholar [26] K. J. K. Galvin, Advancements in Finite Element Methods for Newtonian and Non-Newtonian Flows, PhD thesis, Clemson University, 2013.  Google Scholar [27] V. K. Kalantarov, B. Levant and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152. doi: 10.1007/s00332-008-9029-7.  Google Scholar [28] V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714. doi: 10.1007/s11401-009-0205-3.  Google Scholar [29] T.-Y. Kim, M. Cassiani, J. D. Albertson, J. E. Dolbow, E. Fried and M. E. Gurtin, Impact of the inherent separation of scales in the Navier-Stokes-$\alpha\beta$ equations, Phys. Rev. E (3), 79 (2009), 045307, 4pp. doi: 10.1103/PhysRevE.79.045307.  Google Scholar [30] T.-Y. Kim, L. G. Rebholz and E. Fried, A deconvolution enhancement of the Navier-Stokes-$\alpha\beta$ model, J. Comput. Phys., 231 (2012), 4015-4027. doi: 10.1016/j.jcp.2011.12.003.  Google Scholar [31] P. Kuberry, A. Larios, L. G. Rebholz and N. E. Wilson, Numerical approximation of the Voigt regularization for incompressible Navier-Stokes and magnetohydrodynamic flows, Comput. Math. Appl., 64 (2012), 2647-2662. doi: 10.1016/j.camwa.2012.07.010.  Google Scholar [32] A. Larios, The Inviscid Voigt-Regularization for Hydrodynamic Models: Global Regularity, Boundary Conditions, and Blow-Up Phenomena, PhD thesis, Univ. California, Irvine, 2011.  Google Scholar [33] A. Larios and E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 603-627. doi: 10.3934/dcdsb.2010.14.603.  Google Scholar [34] W. J. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128 (electronic). doi: 10.3934/dcdsb.2006.6.111.  Google Scholar [35] W. J. Layton and L. Rebholz, Approximate Deconvolution Models of Turbulence Approximate Deconvolution Models of Turbulence, vol. 2042 of Lecture Notes in Mathematics, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-24409-4.  Google Scholar [36] B. Levant, F. Ramos and E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Commun. Math. Sci., 8 (2010), 277-293. doi: 10.4310/CMS.2010.v8.n1.a14.  Google Scholar [37] A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 98-136, Boundary value problems of mathematical physics and related questions in the theory of functions, 7.  Google Scholar [38] A. P. Oskolkov, On the theory of unsteady flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 115 (1982), 191-202, 310, Boundary value problems of mathematical physics and related questions in the theory of functions, 14.  Google Scholar [39] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, in Handbook of numerical analysis. Vol. XII, Handb. Numer. Anal., XII, North-Holland, Amsterdam, 2004, 3-127. doi: 10.1016/S1570-8659(03)12001-7.  Google Scholar [40] L. G. Rebholz, Well-posedness of a reduced order approximate deconvolution turbulence model, J. Math. Anal. Appl., 405 (2013), 738-741. doi: 10.1016/j.jmaa.2013.04.036.  Google Scholar [41] P. Sagaut, Large Eddy Simulation for Incompressible Flows, Scientific Computation, Springer-Verlag, Berlin, 2001, An introduction, With an introduction by Marcel Lesieur, Translated from the 1998 French original by the author. doi: 10.1007/978-3-662-04416-2.  Google Scholar [42] Z. S. She, E. Jackson and S. A. 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