Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T02:00:45.208Z Has data issue: false hasContentIssue false
  • English
  • Français

A High-Order Unifying Discontinuous Formulation for theNavier-Stokes Equations on 3D Mixed Grids

Published online by Cambridge University Press:  16 May 2011

T. Haga ,
H. Gao  and
Z. J. Wang
Get access

Abstract

The newly developed unifying discontinuous formulation named the correction procedure viareconstruction (CPR) for conservation laws is extended to solve the Navier-Stokesequations for 3D mixed grids. In the current development, tetrahedrons and triangularprisms are considered. The CPR method can unify several popular high order methodsincluding the discontinuous Galerkin and the spectral volume methods into a more efficientdifferential form. By selecting the solution points to coincide with the flux points,solution reconstruction can be completely avoided. Accuracy studies confirmed that theoptimal order of accuracy can be achieved with the method. Several benchmark test casesare computed by solving the Euler and compressible Navier-Stokes equations to demonstrateits performance.

Keywords

high-ordermixed unstructured gridsNavier-Stokes equationsdiscontinuous Galerkinspectral collocationfinite difference
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 3: Computational aerodynamics , 2011 , pp. 28 - 56
Copyright
© EDP Sciences, 2011

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

Références

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 (1997), 267–279. CrossRefGoogle Scholar
F. Bassi, S. Rebay. GMRES discontinuous Galerkin solution of the compressible Navier- Stokes equations. In B. Cockburn, G.E. Karniadakis, and C. W. Shu, editors, Discontinuous Galerkin Methods: Theory, Computations and Applications, volume 11 of Lecture Note in Computational Science and Engineering. Springer, 2000.
Chen, Q. and Babuska, I.. Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle. Comput. Methods Appl. Mech. Eng., 128 (1995), 405-417. CrossRefGoogle Scholar
Cockburn, B., Lin, S. Y., Shu, C. W.. TVD Runge-Kutta local projection discontinuous Galerkin Finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys., 84 (1989), 90-113. CrossRefGoogle Scholar
Cockburn, B., Shu, C. W.. TVD Runge-Kutta local projection discontinuous Galerkin Finite element method for conservation laws II: general framework. Math. Comput., 52 (1989), 411-435. Google Scholar
Cockburn, B., Shu, C. W.. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal., 35 (1998), No. 6, 2440-2463. CrossRefGoogle Scholar
Cockburn, B., Shu, C. W.. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys., 141 (1998), 199-224. CrossRefGoogle Scholar
Fidkowski, K., Oliver, T. A., Lu, J., Darmofal, D.. p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys., 207 (2005), 92-113. CrossRefGoogle Scholar
H. Gao, Z. J. Wang. A high-order lifting collocation penalty formulation for the Navier- Stokes equations on 2D mixed grids. AIAA Paper 2009-3784, 2009.
Gassner, G. J., Lorcher, F., Munz, C-D., and Hesthaven, J. S.. Polymorphic nodal elements and their application in discontinuous Galerkin methods. J. Comput. Phys., 228 (2009), 1573-1590. CrossRefGoogle Scholar
Godunov, S. K.. A difference scheme for numerical computation of discontinuous solutions of equations of fluid dynamics. Math. Sbornik, 47 (1959), 271-306, In Russian. Google Scholar
T. Haga, M. Furudate, K. Sawada. RANS simulation using high-order spectral volume method on unstructured tetrahedral grids. AIAA Paper 2009–404, 2009.
Haga, T., Sawada, K., Wang, Z. J.. An implicit LU-SGS scheme for the spectral volume method on unstructured tetrahedral grids. Communications in Computational Physics, 6 (2009), No. 5, 978-996. CrossRefGoogle Scholar
Harris, R., Wang, Z. J., Liu, Y.. Efficient quadrature-free high-order spectral volume method on unstructured grids: Theory and 2D implementation. J. Comput. Phys., 227 (2008), 1620-1642. CrossRefGoogle Scholar
Hesthaven, J. S.. From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal., 35 (1998), No. 2, 655-676. CrossRefGoogle Scholar
H. T. Huynh. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Paper 2007–4079, 2007.
H. T. Huynh. A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion. AIAA Paper 2009–403, 2009.
Jameson, A.. Analysis and design of numerical schemes for gas dynamics. I. Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence. Int. J. Comput. Fluid Dyn., 4 (1994), 171–218. CrossRefGoogle Scholar
Johnson, T. A. and Patel, V. C.. Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech., 378 (1999), 19-70. CrossRefGoogle Scholar
Kopriva, D. A. and Kolias, J. H.. A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys., 125 (1996), 244–261. CrossRefGoogle Scholar
Liou, M. S.. A sequel to AUSM, Part II: AUSM+-up for all speeds. J. Comput. Phys., 214 (2006), 137-170. CrossRefGoogle Scholar
Y. Liu, M. Vinokur, and Z. J. Wang. Discontinuous spectral difference method for conservation laws on unstructured grids. In Proceedings of the Third International Conference on Computational Fluid Dynamics, Toronto, Canada, July 2004.
Liu, Y., Vinokur, M., and Wang, Z. J.. Spectral difference method for unstructured grids I: Basic formulation. J. Comput. Phys., 216 (2006), 780-801. CrossRefGoogle Scholar
Liu, Y., Vinokur, M., Wang, Z. J.. Spectral (finite) volume method for conservation laws on unstructured grids V: Extension to three-dimensional systems. J. Comput. Phys., 212 (2006), 454-472. CrossRefGoogle Scholar
Luo, H., Baum, J. D., and Lohner, R.. A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids. J. Comput. Phys., 227 (2008), 8875-8893. CrossRefGoogle Scholar
Mavriplis, D. J.. Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes. J. Comput. Phys., 145 (1998), 141-165. CrossRefGoogle Scholar
G. May, A. Jameson. A spectral difference method for the Euler and Navier-Stokes equations. AIAA Paper 2006–304, 2006.
Nastase, C. R., Mavriplis, D. J.. High-order discontinuous Galerkin methods using an hp-multigrid approach. J. Comput. Phys., 213 (2006), 330-357. CrossRefGoogle Scholar
Osher, S.. Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal., 21 (1984), 217-235. CrossRefGoogle Scholar
W. H. Reed, T. R. Hill. Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory Report LA-UR-73-479, 1973.
Roe, P. L.. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43 (1981), 357-372. CrossRefGoogle Scholar
Rusanov, V. V.. Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys., 1 (1961), 267-279. Google Scholar
Sherwin, S. J., Karniadaks, G. E.. A new triangular and tetrahedral basis for high-order (hp) finite element methods. Int. J. Num. Meth. Eng., 38 (1995), 3775–3802. CrossRefGoogle Scholar
Shu, C. W.. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing, 9 (1988), 1073-1084. CrossRefGoogle Scholar
C. W. Shu. Essentially non-oscillatory and weighted and non-oscillatory schemes for hyperbolic conservation laws. In B. Cockburn, C. Johnson, C.-W. Shu, and E. Tadmor, editors, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, volume 1697 of Lecture Note in Mathematics. Springer, 1998.
Sun, Y., Wang, Z. J., and Liu, Y.. High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids. Communications in Computational Physics, 2 (2007), 310-333. Google Scholar
Taneda, S.. Experimental investigations of the wake behind a sphere at low reynolds nombers. J. Phys. Soc. Japan, 11 (1956), 1104-1108. CrossRefGoogle Scholar
Tomboulides, A. G., Orzag, S. A.. Numerical investigation of transitional and weak turbulent flow past a sphere. J. Fluid Mech., 416 (2000), 45-73. CrossRefGoogle Scholar
Van den Abeele, K. and Lacor, C.. An accuracy and stability study of the 2D spectral volume method. J. Comput. Phys., 226 (2007), 1007-1026. CrossRefGoogle Scholar
Van den Abeele, K., Lacor, C., Wang, Z. J.. On the stability and accuracy of the spectral difference method. J. Sci. Comput., 37 (2008), 162-188. CrossRefGoogle Scholar
Van Leer, B.. Towards the ultimate conservative difference scheme V. A second order sequel to GodunovŠs method. J. Comput. Phys., 32 (1979), 110-136. CrossRefGoogle Scholar
B. Van Leer, S. Nomura. Discontinuous Galerkin for diffusion. AIAA Paper 2005–5108, 2005.
Wang, Z. J.. Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation. J. Comput. Phys., 178 (2002), 210-251. CrossRefGoogle Scholar
Wang, Z. J.. High-order methods for the Euler and Navier-Stokes equations on unstructured grids. Progress in Aerospace Sciences, 43 (2007), 1-41. CrossRefGoogle Scholar
Wang, Z. J., Gao, H.. A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. J. Comput. Phys., 228 (2009), 8161-8186. CrossRefGoogle Scholar
Wang, Z. J., Liu, Y.. Spectral (finite) volume method for conservation laws on unstructured grids II: Extension to two-dimensional scalar equation. J. Comput. Phys., 179 (2002), 665-697. CrossRefGoogle Scholar
Wang, Z. J., Liu, Y.. Spectral (finite) volume method for conservation laws on unstructured grids III: One-dimensional systems and partition optimization. Journal of Scientific Computing, 20 (2004), No. 1, 137-157. CrossRefGoogle Scholar
Wang, Z. J., Zhang, L., Liu, Y.. Spectral (finite) volume method for conservation laws on unstructured grids IV: Extension to two-dimensional systems. J. Comput. Phys., 194 (2004), 716-741. CrossRefGoogle Scholar
Warburton, T.. An explicit construction of interpolation nodes on the simplex. J. Eng. Math., 56 (2006), 247-262. CrossRefGoogle Scholar
O. C. Zienkiewicz, R. L. Taylor. The Finite Element Method The Basics, vol. 1. Butterworth-Heinemann, Oxford, England, 2000.