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A high order Discontinuous Galerkin Finite Element solver for the incompressible Navier-Stokes equations. (English) Zbl 1431.76011
Summary: The paper presents an unsteady high order Discontinuous Galerkin (DG) solver that has been developed, verified and validated for the solution of the two-dimensional incompressible Navier-Stokes equations. A second order stiffly stable method is used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Symmetric Interior Penalty Galerkin formulation. The non-linear terms in the Navier-Stokes equations are expressed in the convective form and approximated through the Lesaint-Raviart fluxes modified for DG methods.
Verification of the solver is performed for a series of test problems; purely elliptic, unsteady Stokes and full Navier-Stokes. The resulting method leads to a stable scheme for the unsteady Stokes and Navier-Stokes equations when equal order approximation is used for velocity and pressure. For the validation of the full Navier-Stokes solver, we consider unsteady laminar flow past a square cylinder at a Reynolds number of 100 (unsteady wake). The DG solver shows favourably comparisons to experimental data and a continuous Spectral code.

MSC:
76-06 Proceedings, conferences, collections, etc. pertaining to fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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[1] Arnold DN. An interior penalty finite element method with discontinuous elements. PhD thesis, The University of Chicago, Chicago, IL; 1979.
[2] Arnold, D.N.; Brezzi, F.; Cockburn, B.; Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J numer anal, 39, 1749-1779, (2001) · Zbl 1008.65080
[3] Babuška, I., The finite element method with penalty, Math comp, 27, 221-228, (1973) · Zbl 0299.65057
[4] Baumann CE. An hp-adaptive discontinuous finite element method for computational fluid dynamics. PhD thesis, University of Texas at Austin, Austin, TX; 1997.
[5] Cockburn, B.; Shu, C., Runge – kutta discontinuous Galerkin methods for convection-dominated flows, J sci comput, 16, 173-263, (2001) · Zbl 1065.76135
[6] Darekar, R.M.; Sherwin, S.J., Flow past a square-section cylinder with a wavy stagnation face, J fluid mech, 426, 263-295, (2001) · Zbl 1016.76015
[7] Deville, M.O.; Fischer, P.F.; Mund, E.H., High-order methods for incompressible fluid flows. Cambridge monographs on applied and computational mathematics, (2002), Cambridge University Press · Zbl 1007.76001
[8] Epshteyn, Y.; Riviere, B., Estimation of penalty parameters for symmetric interior penalty Galerkin methods, J comp appl math, 206, 843-872, (2007) · Zbl 1141.65078
[9] Ferrer E, Willden RHJ. Development of a high order incompressible Discontinuous Galerkin Finite Element solver. In: Proceedings of the European conference on computational fluid dynamics, ECCOMAS-CFD 2010, Lisbone, Portugal; 2010.
[10] Girault, V.; Wheeler, F., Discontinuous Galerkin methods, Comp meth appl sci, 16, 3-26, (2008) · Zbl 1149.65323
[11] Guermond, J.L.; Minev, P.; Shen, J., An overview of projection methods for incompressible flows, Comp meth appl mech eng, 195, 6011-6045, (2006) · Zbl 1122.76072
[12] Guermond, J.L.; Shen, J., Velocity-correction projection methods for incompressible flows, SIAM J numer anal, 41, 112-134, (2003) · Zbl 1130.76395
[13] Hesthaven, J.S.; Warburton, T., Nodal discontinuous Galerkin methods – algorithms, analysis, and applications, (2008), Springer · Zbl 1134.65068
[14] Karniadakis, G.E.; Israeli, M.; Orszag, S.A., High-order splitting methods for the incompressible navier – stokes equations, J comp phys, 97, 414-443, (1991) · Zbl 0738.76050
[15] Karniadakis, G.E.; Sherwin, S.J., Spectral/hp element methods for computational fluid dynamics, (2005), Oxford Science Publications · Zbl 1116.76002
[16] Lions, J.L., Problèmes aux limites non homogènes à donées irrégulières: une méthode d’approximation, (), 283-292
[17] Nitsche, J.A., Über ein variationsprinzip zur Lösung Dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unteworfen sind, abh, Math sem univ Hamburg, 36, 9-15, (1971) · Zbl 0229.65079
[18] Okajima, A., Strouhal numbers of rectangular cylinders, J fluid mech, 123, 379-398, (1982)
[19] Riviere, B., Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation, SIAM front appl math, (2008) · Zbl 1153.65112
[20] Shahbazi, K.; Fischer, P.F.; Ethier, C.R., A high-order discontinuous Galerkin method for the unsteady incompressible navier – stokes equations, J comp phys, 222, 391-407, (2007) · Zbl 1216.76034
[21] Vos, P.; Sherwin, S.J.; Kirby, R.M., From h to p efficiently: implementing finite and spectral h/p element discretisations to achieve optimal performance at low and high order approximations, J comp phys, 229, 5161-5181, (2010) · Zbl 1194.65138
[22] Wheeler, M.F., An elliptic collocation-finite element method with interior penalties, SIAM J numer anal, 15, 152-161, (1978) · Zbl 0384.65058
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