Abstract
In this chapter we collect results obtained within the IDIHOM project on the development of Discontinuous Galerkin (DG) methods and their application to aerodynamic flows. In particular, we present an application of multigrid algorithms to a higher order DG discretization of the Reynolds-averaged Navier-Stokes (RANS) equations in combination with the Spalart-Allmaras as well as the Wilcox-kω turbulence model. Based on either lower order discretizations or agglomerated coarse meshes the resulting solver algorithms are characterized as p- or h-multigrid, respectively. Linear and nonlinear multigrid algorithms are applied to IDIHOM test cases, namely theL1T2 high lift configuration and the deltawing of the second Vortex Flow Experiment (VFE-2) with rounded leading edge. All presented algorithms are compared to a strongly implicit single grid solver in terms of number of nonlinear iterations and computing time. Furthermore, higher order DG methods are combined with adaptive mesh refinement, in particular, with residual-based and adjoint-based mesh refinement. These adaptive methods are applied to a subsonic and transonic flow around the VFE-2 delta wing.
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References
Bassi, F., Botti, L., Colombo, A., Pietro, D.D., Tesini, P.: On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. Journal of Computational Physics 231(1), 45–65 (2012)
Bassi, F., Botti, L., Colombo, A., Rebay, S.: Agglomeration based discontinuous Galerkin discretization of the Euler and Navier-Stokes equations. Computers and Fluids 61(0), 77–85 (2012)
Bassi, F., Crivellini, A., Rebay, S., Savini, M.: Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-ω turbulence model equations. Computers & Fluids 34(4-5), 507–540 (2005)
Bassi, F., Ghidoni, A., Rebay, S., Tesini, P.: High-order accurate p-multigrid discontinuous Galerkin solution of the Euler equations. International Journal for Numerical Methods in Fluids 60, 847–865 (2009)
Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., Savini, M.: A high-order accurate discontinuous Finite Element method for inviscid and viscous turbomachinery flows. In: Decuypere, R., Dibelius, G. (eds.) 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Antwerpen, Belgium, March 5-7, pp. 99–108. Technologisch Instituut (1997)
Burgess, N., Nastase, C., Mavriplis, D., Martinelli, L.: Efficient solution techniques for discontinuous Galerkin discretizations of the Navier-Stokes equations on hybrid anisotropic meshes. 48th AIAA Aerospace Sciences Meeting. AIAA 2010-1448 (2010)
Crippa, S.: Advances in Vortical Flow Prediction Methods for Design of Delta-Winged Aircraft. PhD thesis, KTH Engineering Sciences, Stockholm, Sweden (2008)
Ilinca, D.P.F.: Positivity preservation and adaptive solution for the k-w model of turbulence. AIAA, J. 36, 44–50 (1998)
Fejtek, I.: Summary of code validation results for a multiple element airfoil test case. In: 28th AIAA Fluid Dynamics Conference. AIAA Paper 97-1932 (1997)
Fidkowski, K.J., Oliver, T.A., Lu, J., Darmofal, D.L.: p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. Journal of Computational Physics 207(1), 92–113 (2005)
Hartmann, R.: Higher-order and adaptive discontinuous Galerkin methods applied to turbulent delta wing flow. In: Dillmann, A., Heller, G., Kreplin, H.-P., Nitsche, W., Peltzer, I. (eds.) New Results in Numer. & Exp. Fluid Mech. NNFM, vol. 121, pp. 497–506. Springer, Heidelberg (2013)
Hartmann, R.: Higher-order and adaptive discontinuous Galerkin methods with shock-capturing applied to transonic turbulent delta wing flow. Int. J. Numer. Meth. Fluids 72(8), 883–894 (2013)
Hartmann, R., Held, J., Leicht, T.: Adjoint-based error estimation and adaptive mesh refinement for the RANS and k-ω turbulence model equations. Journal of Computational Physics 230(11), 4268–4284 (2011)
Hartmann, R., Held, J., Leicht, T., Prill, F.: Discontinuous Galerkin methods for computational aerodynamics – 3D adaptive flow simulation with the DLR PADGE code. Aerosp. Sci. Technol. 14, 512–519 (2010)
Hartmann, R., Leicht, T.: Higher order and adaptive DG methods for compressible flows. In: Deconinck, H. (ed.) VKI LS 2014-03: 37th Advanced VKI CFD Lecture Series: Recent Developments in Higher Order Methods and Industrial Application in Aeronautics, Belgium, December 9-12. Von Karman Institute for Fluid Dynamics, Rhode Saint Genèse (2014)
Hummel, D., Redeker, G.: A new vortex flow experiment for computer code validation. In: RTO-AVT Symposium on Vortex Fow and High Angle of Attack, Loen, Norway, 7.-11.05 (2001)
Konrath, R., Klein, C., Engler, R., Otter, D.: Analysis of PSP results obtained for the VFE-2 65° delta wing configuration at sub- and transonic speeds. In: 44th AIAA Aerospace Sciences Meeting and Exhibit. AIAA 2006-59-624 (2006)
Landmann, B., Kessler, M., Wagner, S., Krämer, E.: A parallel, high-order discontinuous Galerkin code for laminar and turbulent flows. Computers & Fluids 37(4), 427–438 (2008)
Langer, S., Schwöppe, A., Kroll, N.: The DLR Flow Solver TAU - Status and Recent Algorithmic Developments. In: 52nd AIAA Aerospace Sciences Meeting. AIAA Paper 2014-0080 (2014)
Luo, H., Baum, J.D., Löhner, R.: A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids. Journal of Computational Physics 227(20), 8875–8893 (2008)
Moulitsas, I., Karypis, G.: Multilevel algorithms for generating coarse grids for multigrid methods. In: Proceedings of the 2001 ACM/IEEE Conference on Supercomputing, p. 45. ACM (2001)
Mulder, W.A., Leer, B.V.: Experiments with implicit upwind methods for the Euler equations. Journal of Computational Physics 59(2), 232–246 (1985)
Allmaras, S.R., Johnson, F.T., Spalart, P.R.: Modifications and clarifications for the implementation of the spalart-allmaras turbulence model. In: ICCFD7, p. 1902 (2012)
Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics 43(2), 357–372 (1981)
Spalart, P., Allmaras, S.: One-equation turbulence model for aerodynamic flows. Recherche Aerospatiale (1), 5–21 (1994)
Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press (2001)
Wallraff, M., Leicht, T.: 3D application of higher order multigrid algorithms for a RANS-kω DG-solver. In: Abgrall, R., Beaugendre, H., Congedo, P.M., Dobrzynski, C., Perrier, V., Ricchiuto, M. (eds.) High Order Nonlinear Numerical Methods for Evolutionary PDEs. LNCSE, vol. 99, pp. 77–88. Springer, Heidelberg (2013)
Wallraff, M., Leicht, T.: Higher order multigrid algorithms for a discontinuous galerkin rans solver. In: 52nd AIAA Aerospace Sciences Meeting. AIAA Paper 2014-0936 (2014)
Wallraff, M., Leicht, T., Lange-Hegermann, M.: Numerical flux functions for Reynolds-averaged Navier-Stokes and kω turbulence model computations with a line-preconditioned p-multigrid discontinuous Galerkin solver. International Journal for Numerical Methods in Fluids 71(8), 1055–1072 (2013)
Wilcox, D.C.: Reassessment of the scale-determining equation for advanced turbulence models. AIAA Journal 26(11), 1299–1310 (1988)
Wilcox, D.C.: Turbulence Modeling for CFD. DCW Industries, Inc., La Canada CA (1993)
**e, Z.Q., Sevilla, R., Hassan, O., Morgan, K.: The generation of arbitrary order curved meshes for 3D finite element analysis. Computational Mechanics 51(3), 361–374 (2013)
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Wallraff, M., Hartmann, R., Leicht, T. (2015). Multigrid Solver Algorithms for DG Methods and Applications to Aerodynamic Flows. In: Kroll, N., Hirsch, C., Bassi, F., Johnston, C., Hillewaert, K. (eds) IDIHOM: Industrialization of High-Order Methods - A Top-Down Approach. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-319-12886-3_9
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DOI: https://doi.org/10.1007/978-3-319-12886-3_9
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