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Evaluation of a high-order discontinuous Galerkin method for the DNS of turbulent flows. (English) Zbl 1391.76209
Summary: The purpose of this paper is to assess the accuracy of a discontinuous Galerkin (DG) method for the direct numerical simulation (DNS) of freely decaying and wall-bounded turbulent flows. The 2D dipole-wall collision problem, the 3D Taylor-Green vortex flow, and the turbulent channel flow are considered. The results of the DG-based DNS are compared with reference data from DNS based on classical pseudo-spectral methods. For all three configurations the high-order DG method is found to predict accurately second-order statistics. The tests show that the level of accuracy provided by high-order DG discretizations is comparable to that of spectral methods for an equivalent number of degrees of freedom. Emphasis is placed on the benefits of increasing the polynomial order, \(p\), over increasing the mesh resolution, \(h\). A detailed \(hp\)-convergence study for the dipole-wall configuration reveals that an order of approximation higher than \(2 (p>1)\) is necessary to obtain accurate results at minimum computational cost.

MSC:
76F65 Direct numerical and large eddy simulation of turbulence
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics
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[1] Wang, Z.; Fidkowski, K.; Abgrall, R.; Bassi, F.; Caraeni, D.; Cary, A., High-order CFD methods: current status and perspective, Int J Numer Meth Fluids, 1-42, (2012)
[2] Kroll, N.; Bieler, H.; Deconinck, H.; Couaillier, V.; vanderVen, H.; Sorensen, K., ADIGMA-a European initiative on the development of adaptive higher-order variational methods for aerospace applications: results of a collaborative research project funded by the European Union, 2006-2009, vol. 113, (2010), Springer
[3] B. Cockburn, G. Karniadakis, C. Shu, Discontinuous Galerkin methods: theory, computation, and applications, Lecture notes in computational science and engineering, vol. 11. 2000.
[4] S. Collis, Discontinuous Galerkin methods for turbulence simulation, in: Center for Turbulence Research, Proceedings of the Summer Program, 2002, pp. 155-167.
[5] Ainsworth, M., Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods, J Comput Phys, 198, 1, 106-130, (2004) · Zbl 1058.65103
[6] Ghosal, S., An analysis of numerical errors in large-eddy simulations of turbulence, J Comput Phys, 125, 1, 187-206, (1996) · Zbl 0848.76043
[7] Moin, P.; Mahesh, K., Direct numerical simulation: a tool in turbulence research, Annu Rev Fluid Mech, 30, 1, 539-578, (1998) · Zbl 1398.76073
[8] W. Reed, T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479.
[9] Cockburn, B.; Shu, C., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math Comp, 52, 186, 411-435, (1989) · Zbl 0662.65083
[10] Cockburn, B.; Lin, S.; Shu, C., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J Comput Phys, 84, 1, 90-113, (1989) · Zbl 0677.65093
[11] Cockburn, B.; Hou, S.; Shu, C., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math Comp, 54, 190, 545-581, (1990) · Zbl 0695.65066
[12] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J Comput Phys, 131, 2, 267-279, (1997) · Zbl 0871.76040
[13] Lomtev, I.; Karniadakis, G., A discontinuous Galerkin method for the Navier-Stokes equations, Int J Numer Meth Fluids, 29, 5, 587-603, (1999) · Zbl 0951.76041
[14] Baumann, C.; Oden, J., A discontinuous hp finite element method for the Euler and Navier-Stokes equations, Int J Numer Met Fluids, 31, 1, 79-95, (1999) · Zbl 0985.76048
[15] Diosady, L.; Darmofal, D., Preconditioning methods for discontinuous Galerkin solutions of the Navier-Stokes equations, J Comput Phys, 228, 11, 3917-3935, (2009) · Zbl 1185.76812
[16] Dolejší, V., On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations, Int J Numer Methods Fluids, 45, 10, 1083-1106, (2004) · Zbl 1060.76570
[17] Hartmann, R.; Houston, P., Symmetric interior penalty DG methods for the compressible Navier-Stokes equations I: method formulation, Int J Numer Anal Model, 3, 1-20, (2006) · Zbl 1129.76030
[18] Borrel, M.; Ryan, J., The elastoplast discontinuous Galerkin (EDG) method for the Navier-Stokes equations, J Comput Phys, 231, 1, 1-22, (2012) · Zbl 1452.76199
[19] Hindenlang, F.; Gassner, G. J.; Altmann, C.; Beck, A.; Staudenmaier, M.; Munz, C.-D., Explicit discontinuous Galerkin methods for unsteady problems, Comput Fluids, 61, 86-93, (2012) · Zbl 1365.76117
[20] Birken P, Gassner G, Haas M, Munz C-D. Preconditioning for modal discontinuous Galerkin methods for unsteady 3D Navier-Stokes equations, J Comput Phys, 2013. · Zbl 1426.76520
[21] Warburton, T.; Karniadakis, G., A discontinuous Galerkin method for the viscous MHD equations, J Comput Phys, 152, 2, 608-641, (1999) · Zbl 0954.76051
[22] Dubiner, M., Spectral methods on triangles and other domains, J Sci Comput, 6, 4, 345-390, (1991) · Zbl 0742.76059
[23] Sherwin, S.; Karniadakis, G., A new triangular and tetrahedral basis for high-order (hp) finite element methods, Int J Numer Meth Eng, 38, 22, 3775-3802, (1995) · Zbl 0837.73075
[24] Karniadakis, G.; Sherwin, S., Spectral/hp element methods for computational fluid dynamics, (2005), Oxford University Press · Zbl 1116.76002
[25] Gassner, G. J.; Lörcher, F.; Munz, C.-D.; Hesthaven, J. S., Polymorphic nodal elements and their application in discontinuous Galerkin methods, J Comput Phys, 228, 5, 1573-1590, (2009) · Zbl 1267.76062
[26] Bassi, F.; Botti, L.; Colombo, A.; Di Pietro, D. A.; Tesini, P., On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations, J Comput Phys, 231, 1, 45-65, (2012) · Zbl 1457.65178
[27] Wei, L.; Pollard, A., Direct numerical simulation of compressible turbulent channel flows using the discontinuous Galerkin method, Comput Fluids, 47, 1, 85-100, (2011) · Zbl 1271.76142
[28] Wei, L.; Pollard, A., Interactions among pressure, density, vorticity and their gradients in compressible turbulent channel flows, J Fluid Mech, 673, 1-18, (2011) · Zbl 1225.76168
[29] Coleman, G.; Kim, J.; Moser, R., A numerical study of turbulent supersonic isothermal-wall channel flow, J Fluid Mech, 305, 159-184, (1995) · Zbl 0960.76517
[30] Gassner, G.; Beck, A., On the accuracy of high-order discretizations for underresolved turbulence simulations, Theor Comput Fluid Dyn, 1-17, (2012)
[31] Clercx, H.; Bruneau, C., The normal and oblique collision of a dipole with a no-slip boundary, Comput Fluids, 35, 3, 245-279, (2006) · Zbl 1160.76328
[32] Cant R. Fergus, a user guide, technical report. Cambridge University Engineering Department.
[33] Moser, R.; Kim, J.; Mansour, N., Direct numerical simulation of turbulent channel flow up to re=590, Phys Fluids, 11, 943, (1999) · Zbl 1147.76463
[34] Arnold, D.; Brezzi, F.; Cockburn, B.; Marini, L., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal, 39, 5, 1749-1779, (2002) · Zbl 1008.65080
[35] Bassi, F.; Crivellini, A.; Rebay, S.; Savini, M., Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-\(\omega\) turbulence model equations, Comput Fluids, 34, 4-5, 507-540, (2005) · Zbl 1138.76043
[36] Bassi, F.; Crivellini, A.; Di Pietro, D.; Rebay, S., An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations, J Comput Phys, 218, 2, 794-815, (2006) · Zbl 1158.76313
[37] Gottlieb, S.; Shu, C.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev, 43, 1, 89-112, (2001) · Zbl 0967.65098
[38] Keetels, G.; D’Ortona, U.; Kramer, W.; Clercx, H.; Schneider, K.; Van Heijst, G., Fourier spectral and wavelet solvers for the incompressible Navier-Stokes equations with volume-penalization: convergence of a dipole-wall collision, J Comput Phys, 227, 2, 919-945, (2007) · Zbl 1301.76062
[39] Laizet, S.; Lamballais, E., High-order compact schemes for incompressible flows: a simple and efficient method with quasi-spectral accuracy, J Comput Phys, 228, 16, 5989-6015, (2009) · Zbl 1185.76823
[40] Kramer, W.; Clercx, H.; van Heijst, G., Vorticity dynamics of a dipole colliding with a no-slip wall, Phys Fluids, 19, 12, 126603, (2007) · Zbl 1182.76399
[41] Mavriplis, C., Adaptive mesh strategies for the spectral element method, Comput Methods Appl Mech Eng, 116, 1, 77-86, (1994) · Zbl 0826.76070
[42] Shu, C.; Don, W.; Gottlieb, D.; Schilling, O.; Jameson, L., Numerical convergence study of nearly incompressible, inviscid Taylor-Green vortex flow, J Sci Comput, 24, 1, 1-27, (2005) · Zbl 1161.76535
[43] Larsson, J.; Lele, S.; Moin, P., Effect of numerical dissipation on the predicted spectra for compressible turbulence, Annu Res Briefs, Center Turbul Res, 47, (2007)
[44] Van Rees, W.; Leonard, A.; Pullin, D.; Koumoutsakos, P., A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high Reynolds numbers, J Comput Phys, 230, 8, 2794-2805, (2011) · Zbl 1316.76066
[45] Taylor, G.; Green, A., Mechanism of the production of small eddies from large ones, Proc R Soc A, 158, 895, 499, (1937) · JFM 63.1358.03
[46] Brachet, M.; Meiron, D.; Orszag, S.; Nickel, B.; Morf, R.; Frisch, U., Small-scale structure of the Taylor-Green vortex, J Fluid Mech, 130, 41, 1452, (1983)
[47] Brachet, M.; Meneguzzi, M.; Vincent, A.; Politano, H.; Sulem, P., Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows, Phys Fluids, 4, 2845, (1992) · Zbl 0775.76026
[48] Mbengoue D, Genet D, Lachat C, Martin E, Mogé M, Perrier V, et al. Comparison of algorithm in aerosol and aghora for compressible flows. · Zbl 1329.76182
[49] Brun, C.; Petrovan Boiarciuc, M.; Haberkorn, M.; Comte, P., Large eddy simulation of compressible channel flow, Theor Comput Fluid Dyn, 22, 3, 189-212, (2008) · Zbl 1161.76493
[50] Huang, P.; Coleman, G.; Bradshaw, P., Compressible turbulent channel flows: DNS results and modelling, J Fluid Mech, 305, 1, 185-218, (1995) · Zbl 0857.76036
[51] Lechner, R.; Sesterhenn, J.; Friedrich, R., Turbulent supersonic channel flow, J Turbulence, 2, 1, 1-25, (2001) · Zbl 1001.76510
[52] Morinishi, Y.; Tamano, S.; Nakabayashi, K., A DNS algorithm using B-spline collocation method for compressible turbulent channel flow, Comput Fluids, 32, 5, 751-776, (2003) · Zbl 1083.76542
[53] Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J Fluid Mech, 177, 1, 133-166, (1987) · Zbl 0616.76071
[54] Dubief, Y.; Delcayre, F., On coherent-vortex identification in turbulence, J Turbulence, 1, 11, (2000) · Zbl 1082.76554
[55] Kravchenko, A.; Moin, P., On the effect of numerical errors in large eddy simulations of turbulent flows, J Comput Phys, 131, 2, 310-322, (1997) · Zbl 0872.76074
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