Enabling numerical accuracy of Navier-Stokes-$\alpha$ through deconvolution and enhanced stability
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 2, pp. 277-307.

We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatly improves accuracy in simulations. Standard finite element schemes for NS-α suffer from two major sources of error if their solutions are considered approximations to true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure error on the velocity error that arises from the (necessary) use of the rotational form nonlinearity. The proposed scheme “fixes” these two numerical issues through the combined use of a modified grad-div stabilization that acts in both the momentum and filter equations, and an adapted approximate deconvolution technique designed to work with the altered filter. We prove the scheme is stable, optimally convergent, and the effect of the pressure error on the velocity error is significantly reduced. Several numerical experiments are given that demonstrate the effectiveness of the method.

DOI : https://doi.org/10.1051/m2an/2010042
Classification : 65M12,  65M60,  76D05
Mots clés : ns-alpha, grad-div stabilization, turbulence, approximate deconvolution
@article{M2AN_2011__45_2_277_0,
author = {Manica, Carolina C. and Neda, Monika and Olshanskii, Maxim and Rebholz, Leo G.},
title = {Enabling numerical accuracy of Navier-Stokes-$\alpha$ through deconvolution and enhanced stability},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {277--307},
publisher = {EDP-Sciences},
volume = {45},
number = {2},
year = {2011},
doi = {10.1051/m2an/2010042},
zbl = {1267.76021},
language = {en},
url = {archive.numdam.org/item/M2AN_2011__45_2_277_0/}
}
Manica, Carolina C.; Neda, Monika; Olshanskii, Maxim; Rebholz, Leo G. Enabling numerical accuracy of Navier-Stokes-$\alpha$ through deconvolution and enhanced stability. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 2, pp. 277-307. doi : 10.1051/m2an/2010042. http://archive.numdam.org/item/M2AN_2011__45_2_277_0/

[1] N.A. Adams and S. Stolz, On the approximate deconvolution procedure for LES. Phys. Fluids 2 (1999) 1699-1701. | Zbl 1147.76506

[2] 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).

[3] G. Baker, Galerkin approximations for the Navier-Stokes equations. Harvard University (1976).

[4] J.J. Bardina, H. Ferziger and W.C. Reynolds, Improved subgrid scale models for large eddy simulation. AIAA Pap. (1983).

[5] L.C. Berselli, T. Iliescu and W.J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation. Springer (2006). | MR 2185509 | Zbl 1089.76002

[6] S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag (1994). | MR 1278258 | Zbl 1012.65115

[7] E. Burman, Pressure projection stabilizations for Galerkin approximations of Stokes' and Darcy's problem. Numer. Methods Partial Differ. Equ. 24 (2008) 127-143. | MR 2371351 | Zbl 1139.76029

[8] E. Burman and A. Linke, Stabilized finite element schemes for incompressible flow using Scott-Vogelius elements. Appl. Num. Math. 58 (2008) 1704-1719. | MR 2458477 | Zbl 1148.76031

[9] R. Camassa and D. Holm, An integrable shallow water equation with peaked solutions. Phys. Rev. Lett. 71 (1993) 1661-1664. | MR 1234453 | Zbl 0972.35521

[10] S. Chen, C. Foias, D. Holm, E. Olson, E. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett. 81 (1998) 5338-5341. | MR 1745983 | Zbl 1042.76525

[11] 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. | MR 1721139 | Zbl 1194.76069

[12] S. Chen, D. Holm, L. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model. Physica D 133 (1999) 66-83. | MR 1721140 | Zbl 1194.76080

[13] A.J. Chorin, Numerical solution for the Navier-Stokes equations. Math. Comp. 22 (1968) 745-762. | MR 242392 | Zbl 0198.50103

[14] B. Cockburn, G. Kanschat and D. Schotzau, A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comp. 74 (2005) 1067-1095. | MR 2136994 | Zbl 1069.76029

[15] R. Codina, Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4295-4321. | MR 1925888 | Zbl 1015.76045

[16] J. Connors, Convergence analysis and computational testing of the finite element discretization of the Navier-Stokes-alpha model. Numer. Methods Partial Differ. Equ. (to appear). | MR 2732382 | Zbl pre05814514

[17] C. Ethier and D. Steinman, Exact fully 3d Navier-Stokes solutions for benchmarking. Int. J. Numer. Methods Fluids 19 (1994) 369-375. | Zbl 0814.76031

[18] L.P. Franca and S.L. Frey, Stabilized finite element methods. II. The incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 99 (1992) 209-233. | MR 1186727 | Zbl 0765.76048

[19] C. Foias, D. Holm and E. Titi, The Navier-Stokes-alpha model of fluid turbulence. Physica D 152-153 (2001) 505-519. | MR 1837927 | Zbl 1037.76022

[20] C. Foias, D. Holm and E. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory. J. Dyn. Diff. Equ. 14 (2002) 1-35. | MR 1878243 | Zbl 0995.35051

[21] T. Gelhard, G. Lube, M.A. Olshanskii and J.-H. Starcke, Stabilized finite element schemes with LBB-stable elements for incompressible flows. J. Comput. Appl. Math. 177 (2005) 243-267. | MR 2125317 | Zbl 1063.76054

[22] V. Gravemeier, W.A. Wall and E. Ramm, Large eddy simulation of turbulent incompressible flows by a three-level finite element method. Int. J. Numer. Methods Fluids 48 (2005) 1067-1099. | MR 2152771 | Zbl 1070.76034

[23] A.E. Green and G.I. Taylor, Mechanism of the production of small eddies from larger ones. Proc. Royal Soc. A 158 (1937) 499-521. | JFM 63.1358.03

[24] J.L. Guermond, J.T. Oden and S. Prudhomme, An interpretation of the Navier-Stokes-alpha model as a frame-indifferent Leray regularization. Physica D 177 (2003) 23-30. | MR 1965324 | Zbl 1082.35120

[25] M. Gunzburger, Finite Element Methods for Viscous Incompressible Flow: A Guide to Theory, Practice, and Algorithms. Academic Press, Boston (1989). | MR 1017032 | Zbl 0697.76031

[26] P. Hansbo and A. Szepessy, A velocity-pressure streamline diffusion method for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 84 (1990) 175-192. | MR 1087615 | Zbl 0716.76048

[27] V. John and A. Kindl, Numerical studies of finite element variational multiscale methods for turbulent flow simulations. Comput. Methods Appl. Mech. Engrg. 199 (2010) 841-852. | MR 2581347 | Zbl pre05685178

[28] V. John and W.J. Layton, Analysis of numerical errors in Large Eddy Simulation. SIAM J. Numer. Anal. 40 (2002) 995-1020. | MR 1949402 | Zbl 1026.76028

[29] V. John and A. Liakos, Time dependent flow across a step: the slip with friction boundary condition. Int. J. Numer. Methods Fluids 50 (2006) 713-731. | MR 2199095 | Zbl 1086.76040

[30] W. Layton, A remark on regularity of an elliptic-elliptic singular perturbation problem. Technical report, University of Pittsburgh (2007).

[31] W. Layton, Introduction to the numerical analysis of incompressible viscous flows. SIAM (2008). | MR 2442411 | Zbl 1153.76002

[32] W. Layton, C. Manica, M. Neda and L. Rebholz, Numerical analysis and computational testing of a high-accuracy Leray-deconvolution model of turbulence. Numer. Methods Partial Differ. Equ. 24 (2008) 555-582. | MR 2382797 | Zbl 1191.76061

[33] W. Layton, C. Manica, M. Neda, M.A. Olshanskii and L. Rebholz, On the accuracy of the rotation form in simulations of the Navier-Stokes equations. J. Comput. Phys. 228 (2009) 3433-3447. | MR 2513841 | Zbl 1161.76030

[34] W. Layton, C. Manica, M. Neda and L. Rebholz, Numerical analysis and computational comparisons of the NS-omega and NS-alpha regularizations. Comput. Methods Appl. Mech. Engrg. 199 (2010) 916-931. | MR 2581353 | Zbl pre05685184

[35] W. Layton, L. Rebholz and M. Sussman, Energy and helicity dissipation rates of the NS-alpha and NS-alpha-deconvolution models. IMA J. Appl. Math. 75 (2010) 56-74. | MR 2587566 | Zbl pre05681027

[36] E. Lunasin, S. Kurien, M. Taylor and E.S. Titi, A study of the Navier-Stokes-alpha model for two-dimensional turbulence. J. Turbulence 8 (2007) 751-778. | MR 2392989 | Zbl 1039.35078

[37] J.E. Marsden and S. Shkoller, Global well-posedness for the lagrangian averaged Navier-Stokes (lans-alpha) equations on bounded domains. Philos. Trans. Roy. Soc. London A 359 (2001) 14-49. | MR 1853633 | Zbl 1006.35074

[38] G. Matthies, G. Lube and L. Roehe, Some remarks on residual-based stabilisation of inf-sup stable discretisations of the generalised Oseen problem. Comput. Meth. Appl. Math. 198 (2009) 368-390. | MR 2641303 | Zbl 1245.76051

[39] W. Miles and L. Rebholz, Computing NS-alpha with greater physical accuracy and higher convergence rates. Numer. Methods Partial Differ. Equ. (to appear).

[40] H. Moffatt and A. Tsoniber, Helicity in laminar and turbulent flow. Ann. Rev. Fluid Mech. 24 (1992) 281-312. | MR 1145012 | Zbl 0751.76018

[41] A. Muschinsky, A similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type LES. J. Fluid Mech. 325 (1996) 239-260. | Zbl 0891.76045

[42] M.A. Olshanskii, A low order Galerkin finite element method for the Navier-Stokes equations of steady incompressible flow: a stabilization issue and iterative methods. Comp. Meth. Appl. Mech. Eng. 191 (2002) 5515-5536. | MR 1941488 | Zbl 1083.76553

[43] M.A. Olshanskii and A. Reusken, Grad-Div stabilization for the Stokes equations. Math. Comput. 73 (2004) 1699-1718. | MR 2059732 | Zbl 1051.65103

[44] M.A. Olshanskii, G. Lube, T. Heiste and J. Löwe, Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3975-3988. | MR 2557485 | Zbl 1231.76161

[45] L. Rebholz, Conservation laws of turbulence models. J. Math. Anal. Appl. 326 (2007) 33-45. | MR 2277764 | Zbl 1110.76028

[46] L. Rebholz, A family of new high order NS-alpha models arising from helicity correction in Leray turbulence models. J. Math. Anal. Appl. 342 (2008) 246-254. | MR 2440794 | Zbl 1138.35078

[47] L. Rebholz and M. Sussman, On the high accuracy NS-$\alpha$-deconvolution model of turbulence. Math. Models Methods Appl. Sci. 20 (2010) 611-633. | MR 2647034 | Zbl 1187.76720

[48] L.R. Scott and M. Vogelius, Norm estimates for a maximum right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér. 19 (1985) 111-143. | Numdam | MR 813691 | Zbl 0608.65013

[49] S. Stolz, N. Adams and L. Kleiser, An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows. Phys. Fluids 13 (2001) 997. | Zbl 1184.76530

[50] P. Svaček, Application of finite element method in aeroelasticity. J. Comput. Appl. Math. 215 (2008) 586-594. | MR 2406660 | Zbl 1134.76031

[51] D. Tafti, Comparison of some upwind-biased high-order formulations with a second order central-difference scheme for time integration of the incompressible Navier-Stokes equations. Comput. Fluids 25 (1996) 647-665. | MR 1416431 | Zbl 0888.76061

[52] G.I. Taylor, On decay of vortices in a viscous fluid. Phil. Mag. 46 (1923) 671-674. | JFM 49.0607.02