Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T15:19:21.630Z Has data issue: false hasContentIssue false
  • English
  • Français

Enabling numerical accuracy of Navier-Stokes-α through deconvolution and enhanced stability*

Published online by Cambridge University Press:  02 August 2010

Carolina C. Manica ,
Monika Neda ,
Maxim Olshanskii  and
Leo G. Rebholz
Carolina C. Manica
Affiliation:
Departmento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Brazil. carolina.manica@ufrgs.br; http://chasqueweb.ufrgs.br/~carolina.manica
Monika Neda
Affiliation:
Department of Mathematics, University of Nevada, Las Vegas, USA. Monika.Neda@unlv.edu; http://www.pitt.edu/~mon5
Maxim Olshanskii
Affiliation:
Department of Mechanics and Mathematics, Moscow State M. V. Lomonosov University, Moscow 119899, Russia. Maxim.Olshanskii@mtu-net.ru; http://www.mathcs.emory.edu/~molshan
Leo G. Rebholz
Affiliation:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA. rebholz@clemson.edu; http://www.math.clemson.edu/~rebholz
Get access

Abstract

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.

Keywords

NS-alphagrad-div stabilizationturbulenceapproximate deconvolution
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 2 , March 2011 , pp. 277 - 307
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, N.A. and Stolz, S., On the approximate deconvolution procedure for LES. Phys. Fluids 2 (1999) 1699–1701.
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).
G. Baker, Galerkin approximations for the Navier-Stokes equations. Harvard University (1976).
J.J. Bardina, H. Ferziger and W.C. Reynolds, Improved subgrid scale models for large eddy simulation. AIAA Pap. (1983).
L.C. Berselli, T. Iliescu and W.J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation. Springer (2006).
S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag (1994).
Burman, E., Pressure projection stabilizations for Galerkin approximations of Stokes' and Darcy's problem. Numer. Methods Partial Differ. Equ. 24 (2008) 127–143. CrossRef
Burman, E. and Linke, A., Stabilized finite element schemes for incompressible flow using Scott-Vogelius elements. Appl. Num. Math. 58 (2008) 1704–1719. CrossRef
Camassa, R. and Holm, D., An integrable shallow water equation with peaked solutions. Phys. Rev. Lett. 71 (1993) 1661–1664. CrossRef
Chen, S., Foias, C., Holm, D., Olson, E., Titi, E. and Wynne, S., The Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett. 81 (1998) 5338–5341. CrossRef
Chen, S., Foias, C., Holm, D., Olson, E., Titi, E. and Wynne, S., The Camassa-Holm equations and turbulence. Physica D 133 (1999) 49–65. CrossRef
Chen, S., Holm, D., Margolin, L. and Zhang, R., Direct numerical simulations of the Navier-Stokes alpha model. Physica D 133 (1999) 66–83. CrossRef
Chorin, A.J., Numerical solution for the Navier-Stokes equations. Math. Comp. 22 (1968) 745–762. CrossRef
Cockburn, B., Kanschat, G. and Schotzau, D., A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comp. 74 (2005) 1067–1095. CrossRef
Codina, R., Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4295–4321. CrossRef
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).
Ethier, C. and Steinman, D., Exact fully 3d Navier-Stokes solutions for benchmarking. Int. J. Numer. Methods Fluids 19 (1994) 369–375. CrossRef
Franca, L.P. and Frey, S.L., Stabilized finite element methods. II. The incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 99 (1992) 209–233. CrossRef
Foias, C., Holm, D. and Titi, E., The Navier-Stokes-alpha model of fluid turbulence. Physica D 152-153 (2001) 505–519. CrossRef
Foias, C., Holm, D. and Titi, E., 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. CrossRef
Gelhard, T., Lube, G., Olshanskii, M.A. and Starcke, J.-H., Stabilized finite element schemes with LBB-stable elements for incompressible flows. J. Comput. Appl. Math. 177 (2005) 243–267. CrossRef
Gravemeier, V., Wall, W.A. and Ramm, E., Large eddy simulation of turbulent incompressible flows by a three-level finite element method. Int. J. Numer. Methods Fluids 48 (2005) 1067–1099. CrossRef
Green, A.E. and Taylor, G.I., Mechanism of the production of small eddies from larger ones. Proc. Royal Soc. A 158 (1937) 499–521.
Guermond, J.L., Oden, J.T. and Prudhomme, S., An interpretation of the Navier-Stokes-alpha model as a frame-indifferent Leray regularization. Physica D 177 (2003) 23–30. CrossRef
M. Gunzburger, Finite Element Methods for Viscous Incompressible Flow: A Guide to Theory, Practice, and Algorithms. Academic Press, Boston (1989).
Hansbo, P. and Szepessy, A., A velocity-pressure streamline diffusion method for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 84 (1990) 175–192. CrossRef
John, V. and Kindl, A., Numerical studies of finite element variational multiscale methods for turbulent flow simulations. Comput. Methods Appl. Mech. Engrg. 199 (2010) 841–852. CrossRef
John, V. and Layton, W.J., Analysis of numerical errors in Large Eddy Simulation. SIAM J. Numer. Anal. 40 (2002) 995–1020. CrossRef
John, V. and Liakos, A., Time dependent flow across a step: the slip with friction boundary condition. Int. J. Numer. Methods Fluids 50 (2006) 713–731. CrossRef
W. Layton, A remark on regularity of an elliptic-elliptic singular perturbation problem. Technical report, University of Pittsburgh (2007).
W. Layton, Introduction to the numerical analysis of incompressible viscous flows. SIAM (2008).
Layton, W., Manica, C., Neda, M. and Rebholz, L., Numerical analysis and computational testing of a high-accuracy Leray-deconvolution model of turbulence. Numer. Methods Partial Differ. Equ. 24 (2008) 555–582. CrossRef
Layton, W., Manica, C., Neda, M., Olshanskii, M.A. and Rebholz, L., On the accuracy of the rotation form in simulations of the Navier-Stokes equations. J. Comput. Phys. 228 (2009) 3433–3447. CrossRef
Layton, W., Manica, C., Neda, M. and Rebholz, L., Numerical analysis and computational comparisons of the NS-omega and NS-alpha regularizations. Comput. Methods Appl. Mech. Engrg. 199 (2010) 916–931. CrossRef
Layton, W., Rebholz, L. and Sussman, M., Energy and helicity dissipation rates of the NS-alpha and NS-alpha-deconvolution models. IMA J. Appl. Math. 75 (2010) 56–74. CrossRef
Lunasin, E., Kurien, S., Taylor, M. and Titi, E.S., A study of the Navier-Stokes-alpha model for two-dimensional turbulence. J. Turbulence 8 (2007) 751–778.
Marsden, J.E. and Shkoller, S., 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. CrossRef
Matthies, G., Lube, G. and Roehe, L., Some remarks on residual-based stabilisation of inf-sup stable discretisations of the generalised Oseen problem. Comput. Meth. Appl. Math. 198 (2009) 368–390.
W. Miles and L. Rebholz, Computing NS-alpha with greater physical accuracy and higher convergence rates. Numer. Methods Partial Differ. Equ. (to appear).
Moffatt, H. and Tsoniber, A., Helicity in laminar and turbulent flow. Ann. Rev. Fluid Mech. 24 (1992) 281–312. CrossRef
Muschinsky, A., A similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type LES. J. Fluid Mech. 325 (1996) 239–260. CrossRef
Olshanskii, M.A., 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. CrossRef
Olshanskii, M.A. and Reusken, A., Grad-Div stabilization for the Stokes equations. Math. Comput. 73 (2004) 1699–1718. CrossRef
Olshanskii, M.A., Lube, G., Heiste, T. and Löwe, J., Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3975–3988. CrossRef
Rebholz, L., Conservation laws of turbulence models. J. Math. Anal. Appl. 326 (2007) 33–45. CrossRef
Rebholz, L., 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. CrossRef
Rebholz, L. and Sussman, M., On the high accuracy NS- $\alpha$ -deconvolution model of turbulence. Math. Models Methods Appl. Sci. 20 (2010) 611–633. CrossRef
Scott, L.R. and Vogelius, M., 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. CrossRef
Stolz, S., Adams, N. and Kleiser, L., An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows. Phys. Fluids 13 (2001) 997. CrossRef
Svaček, P., Application of finite element method in aeroelasticity. J. Comput. Appl. Math. 215 (2008) 586–594. CrossRef
Tafti, D., 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. CrossRef
Taylor, G.I., On decay of vortices in a viscous fluid. Phil. Mag. 46 (1923) 671–674. CrossRef