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Application of the multigrid approach for solving 3D Navier-Stokes equations on hexahedral grids using the discontinuous Galerkin method

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Abstract

The Galerkin method with discontinuous basis functions is adapted for solving the Euler and Navier-Stokes equations on unstructured hexahedral grids. A hybrid multigrid algorithm involving the finite element and grid stages is used as an iterative solution method. Numerical results of calculating the sphere inviscid flow, viscous flow in a bent pipe, and turbulent flow past a wing are presented. The numerical results and the computational cost are compared with those obtained using the finite volume method.

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References

  1. J. Morrison and M. Hemsch, “Statistical Analysis of CFD Solutions,” in Proc. of the 3rd AIAA Drag Prediction Workshop, 2006; http://aaac.larc.nasa.gov/tsab/cfdlarc/aiaa-dpw/Workshop3/presentations/index.html.

  2. A. V. Wolkov and S. V. Lyapunov, “Investigation of the Efficiency of Using Numerical Schemes of a High Order of Accuracy for Solving Navier-Stokes and Reynolds Equations on Unstructured Adapted Grids,” Zh. Vychisl. Mat. Mat. Fiz. 46, 1894–1907 (2006) [Comp. Math. Math. Phys. 46, 1808–1820 (2006)].

    Google Scholar 

  3. B. Cockburn, G. Karniadakis, and C.-W. Shu, Discontinuous Galerkin Methods: Theory, Computation, and Applications. Lect. Notes Comput. Sci. (Springer, New York, 1999), pp. 69–224.

    Google Scholar 

  4. N. B. Petrovskaya and A. V. Wolkov, “The Issues of the Solution Approximation in Higher-Order Schemes on Distorted Grids,” Int. J. Comput. Meth. 4(2), (2007).

  5. F. Bassi, A. Crivellini, D. A. Di Pietro, and S. Rebay, “A High-Order Discontinuous Galerkin Solver for 3D Aerodynamic Turbulent Flows,” in Proc. ECCOMAS CFD Conf. 2006.

  6. H. Luo, J. D. Baum, and R. Löhner, A Fast, p-Multigrid Discontinuous Galerkin Method for Compressible Flows at all Speeds, AIAA Paper, 2006-110 (2006).

  7. Z. J. Wang, “High-Order Methods for the Euler and Navier-Stokes Equations on Unstructured Grids,” Progress in Aerospace Sci. 43, 1–41 (2007).

    Article  Google Scholar 

  8. R. P. Fedorenko, “A Relaxation Method for Solving Difference Elliptic Equations,” Zh. Vychisl. Mat. Mat. Fiz. 1, 922–927 (1961).

    MathSciNet  Google Scholar 

  9. E. V. Rönquist and A. T. Patera, “Spectral Element Multigrid. I. Formulation and Numerical Results,” J. Scient. Comput. 2(4), (1987).

  10. B. T. Helenbrook, D. Mavriplis, and H. L. Atkins, Analysis of p-Multigrid for Continuous and Discontinuous Finite Element Discretizations, AIAA Paper, 2003-3989 (2003).

  11. V. T. Zhukov, O. B. Feodoritova, and D. P. Yang, “Iterative Algorithms for High-Order Finite Element Schemes,” Mat. Modelir. 16(7), 117–128 (2004).

    MATH  Google Scholar 

  12. Li Wang and D. J. Mavriplis, “Implicit Solution of the Unsteady Euler Equations for High-Order Accurate Discontinuous Galerkin Discretizations,” AIAA Paper, 2006-109.

  13. P. L. Roe, “Approximate Riemann Problem Solvers, Parameter Vectors, and Difference Schemes,” J. Comput. Phys. 43, 357–372 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  14. H. L. Atkins and C. W. Shu, “Quadrature Free Implementation of Discontinuous Galerkin Method for Hyperbolic Equations,” AIAA J. 36(5), (1998).

  15. D. J. Mavriplis and V. Venkatakrishnan, “Agglomeration Multigrid for Two-Dimensional Viscous Flows,” Comput. Fluids 24, 553–570 (1995).

    Article  MATH  Google Scholar 

  16. www.numeca.be.

  17. A. Brandt, “Multilevel Adaptive Computations in Fluid Dynamics,” AIAA J. 18(10), (1980).

  18. T. J. Barth, “A Posteriori Estimation and Mesh Adaptivity for Finite Volume and Finite Element Method,” Lect. Notes in Comput. Sci. and Eng. (LNCSE) 41, (2004).

  19. M. Enayet, M. Gibson, A. Taylor, and M. Yianneskis, “Laser Doppler Measurements of Laminar and Turbulent Flow in a Pipe Bend,” NASA Contract Rep. CR-3551, 1982.

  20. U. R. Müller, B. Schulze B. and H. Henke, “Computation of Transonic Steady and Unsteady Flow about LANN Wing: Validation of CFD Codes and Assessment of Turbulence Models,” ECARP Rept. 58. pp. 479–500, Vieweg, 1996.

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  1. Zhukovski Central Institute of Aerohydrodynamics, ul. Zhukovskogo 1, Zhukovski, Moscow oblast, 140180, Russia

    A. V. Wolkov

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  1. A. V. Wolkov
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Correspondence to A. V. Wolkov.

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Original Russian Text © A.V. Wolkov, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 3, pp. 517–531.

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Wolkov, A.V. Application of the multigrid approach for solving 3D Navier-Stokes equations on hexahedral grids using the discontinuous Galerkin method. Comput. Math. and Math. Phys. 50, 495–508 (2010). https://doi.org/10.1134/S0965542510030103

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