Enabling numerical accuracy of Navier-Stokes-$\alpha$ through deconvolution and enhanced stability

Manica, Carolina C.; Neda, Monika; Olshanskii, Maxim; Rebholz, Leo G.

ESAIM: Mathematical Modelling and Numerical Analysis , 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.

Classification : 65M12, 65M60, 76D05
Mots clés : ns-alpha, grad-div stabilization, turbulence, approximate deconvolution

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

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

[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 | Zbl

[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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

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

[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 | Zbl

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