Summary
The aim of this project was to develop a higher order upwind finite volume scheme on unstructured grids of tetrahedra to solve the nonstationary compressible Navier-Stokes equations with high Reynolds numbers. The new aspects in this numerical scheme are a new definition of a limiter function for a special class of linear reconstruction function to get an upwind finite volume scheme of higher order on an unstructured grid of simplices, a new criterion to adapt the unstructured grid locally and a new kind of discretization of second derivatives. With several numerical tests we will show that we can obtain better results with these new approaches than with existing ones, which we have also used in numerical tests. So it turns out that the new upwind scheme of higher order for conservation laws is really of higher order in regions where the solution is smooth and has no oscillations at discontinuities. The quality of the new discretization of second derivatives will be shown also on grids with large aspect ratios. We will apply the solver for the compressible Euler equations to the flow in a simplified two-stroke engine with a moving piston in 3D. For the modelling of the moving boundary we developed a new technique which guarantees the conservation of mass during the calculations.
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References
Bänsch, E.: An Adaptive Finite-Element Strategy for the Three-dimensional Time-dependent Navier-Stokes equations. Journal of Computational and Applied Mathematics 36 (1991), 3–28.
Barth, T. J.; Jesperson, D. C.: The Design and Application of Upwind Schemes on Unstructured Meshes. AIAA-89-0366.
Bayyuk, S. A.; Powell, K. G.; v. Leer, B.: A Simulation Technique for 2-D Unsteady Inviscid Flows Around Arbitrarily Moving and Deforming Bodies of Arbitrary Geometry. AIAA 93-3391
Chorin, A. J.: Random Choice Solution of Hyperbolic Systems. Journal of Computational Physics 22 (1976), 517–533.
E. Dick, Multigrid Methods for Steady Euler-and Navier-Stokes equations based on polynomial Flux-Difference Splitting, in: International Series of Numerical Mathematics, Vol. 98, Birkhäuser, 1991.
Durlofsky, L. J.; Engquist, B.; Osher, S.: Triangle Based Adaptive Stencils for the Solution of Hyperbolic Conservation Laws. Journal of Computational Physics 98 (1992), 64–73.
Epstein, P. H.; Reitz, R. D.; Foster, D. E.: Computations of a Two-Stroke Engine Cylinder and Port Scavenging Flows. SAE 910672.
Frink, N. T.; Parikh, P.; Pirzadeh, S.: A Fast Upwind Solver for the Euler Equations on Three-Dimensional Unstructured Meshes. AIAA-91-0102.
Herbin, R.: An Error Estimate for a Finite Volume Scheme for a Diffusion-Convection Problem on a Triangular Mesh. Numerical Methods for Partial Differential Equations 11, 165–173, 1995.
Kröner, D.; Noelle, S.; Rokyta, M.: Convergence of Higher Order Finite Volume Schemes on Unstructured Grids for Scalar Conservation Laws in Two Space Dimensions. Numerische Mathematik 71, 1995.
Kröner, D.; Rokyta, M.; Wierse, M.: Apriori Error Estimates for Upwind Finite Volume Schemes in Several Space Dimensions.(in preparation)
Lai, Y. G.; Przekwas, A. J.; Sun, R. L. T.: CFD Simulation of Automative I. C. Engines with Advanced Moving Grid and Multi-Domain Methods. AIAA-93-2953.
Pan, D.; Cheng, J.-C.; Upwind Finite-Volume Navier-Stokes Computations on Unstructured Triangular Meshes. AIAA Journal Vol. 31, No. 9, 1993.
Schlichting, H.: Grenzschicht-Theorie. Verlag G. Braun, Karlsruhe (1965)
Shu, C. W.; Osher, S.: Efficient Implementation of Essentially Non-Oscillatory Shockcap-turing Schemes. Journal of Computational Physics 77 (1988), 439–471.
Steger, J.: On the Use of Composite Grid Schemes in Computational Aerodynamics. Computer Methods in Applied Mechanics and Engineering 64 (1987), 301–320.
Steger, J. L.; Warming R. F.: Flux Vector Splitting of the Inviscid Gasdynamic Equations with Application to Finite-Difference Methods. Journal of Computational Physics 40 (1981), 263–293. Pitman Press, London (1979).
Nessyahu, H.; Tassa, T.; Tadmor, E.: The Convergence Rate of Godunov Type Schemes. SIAM J. Numer. Anal., Vol. 31, No. 1, February 1994.
Vilsmeier, R., Hänel D.: Computational Aspects of Flow Simulation on 3-D, Unstructured, Adaptive Grids. In this publication.
van Leer, B: Flux Vector Splitting for the Euler Equations, Proc. 8 th International Conference on Numerical Methods in Fluid Dynamics, Berlin. Springer Verlag, 1982.
Wierse, M.: Higher Order Upwind Schemes on Unstructured Grids for the Compressible Euler Equations in Timedependent Geometries in 3D. Dissertation, Freiburg, 1994. Preprint 393, SFB 256, Bonn.
Wierse, M.: Cell-Centered Upwind Finite Volume Scheme on Triangles for Scalar Convection Diffusion Equations, in preparation.
Woodward, P. R.; Colella, P.: The Numerical Simulation of Two-Dimensional Flow with Strong Shocks. Journal of Computational Physics 54 (1984), 115–173.
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© 1996 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden
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Wierse, M., Kröner, D. (1996). Higher Order Upwind Schemes on Unstructured Grids for the Nonstationary Compressible Navier-Stokes Equations in Complex Timedependent Geometries in 3D. In: Hirschel, E.H. (eds) Flow Simulation with High-Performance Computers II. Notes on Numerical Fluid Mechanics (NNFM), vol 48. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-89849-4_27
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DOI: https://doi.org/10.1007/978-3-322-89849-4_27
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