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Divergence-free \(H(\operatorname{div})\)-FEM for time-dependent incompressible flows with applications to high Reynolds number vortex dynamics. (English) Zbl 1392.35210

Summary: In this article, we consider exactly divergence-free \(H(\operatorname{div})\)-conforming finite element methods for time-dependent incompressible viscous flow problems. This is an extension of previous research concerning divergence-free \(H^1\)-conforming methods. For the linearised Oseen case, the first semi-discrete numerical analysis for time-dependent flows is presented whereby special emphasis is put on pressure- and Reynolds-semi-robustness. For convection-dominated problems, the proposed method relies on a velocity jump upwind stabilisation which is not gradient-based. Complementing the theoretical results, \(H(\operatorname{div})\)-FEM are applied to the simulation of full nonlinear Navier-Stokes problems. Focussing on dynamic high Reynolds number examples with vortical structures, the proposed method proves to be capable of reliably handling the planar lattice flow problem, Kelvin-Helmholtz instabilities and freely decaying two-dimensional turbulence.

MSC:

35Q30 Navier-Stokes equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D17 Viscous vortex flows
76M10 Finite element methods applied to problems in fluid mechanics
76E06 Convection in hydrodynamic stability
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Ahmed, N, On the Grad-div stabilization for the steady Oseen and Navier-Stokes equations, Calcolo, 54, 471-501, (2017) · Zbl 1375.35305
[2] Ahmed, N; Chacón Rebollo, T; John, V; Rubino, S, Analysis of a full space-time discretization of the Navier-Stokes equations by a local projection stabilization method, IMA J. Numer. Anal., 37, 1437-1467, (2017) · Zbl 1407.76059
[3] Ahmed, N., Linke, A., Merdon, C.: On really locking-free mixed finite element methods for the transient incompressible Stokes equations (2017). doi:10.20347/WIAS.PREPRINT.2368 · Zbl 1422.65218
[4] Bazilevs, Y; Michler, C; Calo, VM; Hughes, TJR, Weak Dirichlet boundary conditions for wall-bounded turbulent flows, Comput. Methods Appl. Mech. Engrg., 196, 4853-4862, (2007) · Zbl 1173.76397
[5] Bertozzi, AL, Heteroclinic orbits and chaotic dynamics in planar fluid flows, SIAM J. Math. Anal., 19, 1271-1294, (1988) · Zbl 0656.76025
[6] Boffetta, G; Ecke, RE, Two-dimensional turbulence, Annu. Rev. Fluid Mech., 44, 427-451, (2012) · Zbl 1350.76022
[7] Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer-Verlag, Berlin (2013) · Zbl 1277.65092
[8] Botti, L; Pietro, DA, A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure, J. Comput. Phys., 230, 572-585, (2011) · Zbl 1283.76030
[9] Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Springer Science & Business Media, New York (2013) · Zbl 1286.76005
[10] Bracco, A; McWilliams, JC; Murante, G; Provenzale, A; Weiss, JB, Revisiting freely decaying two-dimensional turbulence at millennial resolution, Phys. Fluids, 12, 2931-2941, (2000) · Zbl 1184.76069
[11] Burman, E, Interior penalty variational multiscale method for the incompressible Navier-Stokes equation: monitoring artificial dissipation, Comput. Methods Appl. Mech. Engrg., 196, 4045-4058, (2007) · Zbl 1173.76332
[12] Burman, E; Fernández, MA, Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence, Numer. Math., 107, 39-77, (2007) · Zbl 1117.76032
[13] Burman, E; Fernández, MA; Hansbo, P, Continuous interior penalty finite element method for oseen’s equations, SIAM J. Numer. Anal., 44, 1248-1274, (2006) · Zbl 1344.76049
[14] Charnyi, S; Heister, T; Olshanskii, MA; Rebholz, LG, On conservation laws of Navier-Stokes Galerkin discretizations, J. Comput. Phys., 337, 289-308, (2017)
[15] Cockburn, B; Kanschat, G; Schötzau, D, A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comp., 74, 1067-1095, (2005) · Zbl 1069.76029
[16] Cockburn, B; Kanschat, G; Schötzau, D, A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J. Sci. Comput., 31, 61-73, (2007) · Zbl 1151.76527
[17] Dallmann, H; Arndt, D; Lube, G, Local projection stabilization for the Oseen problem, IMA J. Numer. Anal., 36, 796-823, (2016) · Zbl 1433.76079
[18] Davidson, P.A.: Turbulence: An Introduction for Scientist and Engineers. Oxford University Press, (2004) · Zbl 1061.76001
[19] Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer-Verlag, Berlin (2012) · Zbl 1231.65209
[20] Durst, F.: Fluid Mechanics: An Introduction to the Theory of Fluid Flows. Springer-Verlag, Berlin (2008) · Zbl 1153.76001
[21] Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Springer, New York (2004) · Zbl 1059.65103
[22] Evans, J.A.: Divergence-free B-spline Discretizations for Viscous Incompressible Flows. Ph.D. thesis, The University of Texas at Austin (2011) · Zbl 1344.76049
[23] Evans, JA; Hughes, TJR, Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations, J. Comput. Phys., 241, 141-167, (2013) · Zbl 1349.76054
[24] Frutos, J; García-Archilla, B; John, V; Novo, J, Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements, J. Sci. Comput., 66, 991-1024, (2016) · Zbl 1462.65138
[25] Gravemeier, V; Wall, WA; Ramm, E, Large eddy simulation of turbulent incompressible flows by a three-level finite element method, Int. J. Numer. Meth. Fluids, 48, 1067-1099, (2005) · Zbl 1070.76034
[26] Groesen, E, Time-asymptotics and the self-organization hypothesis for 2D Navier-Stokes equations, Physica A, 148, 312-330, (1988) · Zbl 0678.76020
[27] Guzmán, J., Shu, C.W., Sequeira, F.A.: H(div) conforming and DG methods for incompressible Euler’s equations. IMA J. Numer. Anal. (2016). doi:10.1093/imanum/drw054 · Zbl 1433.76079
[28] Hillewaert, K.: Development of the Discontinuous Galerkin Method for High-Resolution, Large Scale CFD and Acoustics in Industrial Geometries. Ph.D. thesis, Université catholique de Louvain (2013)
[29] Jenkins, EW; John, V; Linke, A; Rebholz, LG, On the parameter choice in Grad-div stabilization for the Stokes equations, Adv. Comput. Math., 40, 491-516, (2014) · Zbl 1426.76272
[30] John, V, An assessment of two models for the subgrid scale tensor in the rational LES model, J. Comput. Appl. Math., 173, 57-80, (2005) · Zbl 1107.76040
[31] John, V.: Finite Element Methods for Incompressible Flow Problems. Springer International Publishing, (2016) · Zbl 1358.76003
[32] John, V; Linke, A; Merdon, C; Neilan, M; Rebholz, LG, On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev., 59, 492-544, (2017) · Zbl 1426.76275
[33] Lesieur, M.: Turbulence in Fluids, 4th edn. Springer, Netherlands (2008) · Zbl 1138.76004
[34] Lesieur, M; Staquet, C; Roy, P; Comte, P, The mixing layer and its coherence examined from the point of view of two-dimensional turbulence, J. Fluid Mech., 192, 511-534, (1988)
[35] Linke, A; Merdon, C, Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 311, 304-326, (2016)
[36] Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, (2002) · Zbl 0983.76001
[37] Melander, M.V., Zabusky, N.J., McWilliams, J.C.: Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303-340 (1988) · Zbl 0653.76020
[38] Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. SIAM (2008) · Zbl 1426.76275
[39] Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems, 2nd edn. Springer-Verlag, Berlin (2008) · Zbl 1155.65087
[40] San, O; Staples, AE, High-order methods for decaying two-dimensional homogeneous isotropic turbulence, Comput. & Fluids, 63, 105-127, (2012) · Zbl 1365.76064
[41] Schlichting, H., Gersten, K.: Boundary-Layer Theory, 8th edn. Springer-Verlag, Berlin (2000) · Zbl 0940.76003
[42] Schroeder, P.W., Lube, G.: Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier-Stokes flows. J. Numer. Math. (2017). doi:10.1515/jnma-2016-1101 · Zbl 1388.76151
[43] Scott, LR; Vogelius, M, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, ESAIM: M2AN, 19, 111-143, (1985) · Zbl 0608.65013
[44] Tabeling, P, Two-dimensional turbulence: a physicist approach, Phys. Rep., 362, 1-62, (2002) · Zbl 1001.76041
[45] Thuburn, J; Kent, J; Wood, N, Cascades, backscatter and conservation in numerical models of two-dimensional turbulence, Quart. J. Roy. Meteor. Soc., 140, 626-638, (2014)
[46] Tritton, D.J.: Physical Fluid Dynamics, 2nd edn. Oxford University Press, New York (1988) · Zbl 0383.76001
[47] Vemaganti, K, Discontinuous Galerkin methods for periodic boundary value problems, Numer. Methods Partial Differential Equations, 23, 587-596, (2007) · Zbl 1114.65144
[48] Wang, J; Wang, Y; Ye, X, A robust numerical method for Stokes equations based on divergence-free H(div) finite element methods, SIAM J. Sci. Comput., 31, 2784-2802, (2009) · Zbl 1407.76074
[49] Wang, J; Ye, X, New finite element methods in computational fluid dynamics by H(div) elements, SIAM J. Numer. Anal., 45, 1269-1286, (2007) · Zbl 1138.76049
[50] Zistl, C; Hilbert, R; Janiga, G; Thévenin, D, Increasing the efficiency of postprocessing for turbulent reacting flows, Comput. Vis. Sci, 12, 383-395, (2009)
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