Large Scale Scientific Computing pp 209-231 | Cite as

# Vortex Dynamics Studied by Large-Scale Solutions to the Euler Equations

## Abstract

Numerical solutions to the Euler equations have been found to exhibit the realistic behaviour of some vortex flows, but the fundamental question surrounding these is the way by which the vorticity initially is created in the flow. One explanation lies in the nonlinear occurrence of vortex sheets admitted as a solution to the Euler equations. The sheets may arise from the build up and breaking of simple shear waves, and seems associated with singular points of the surface velocity field where the flow separates from the body. In order to gain insight into this situation a numerical method to solve either the compressible or incompressible Euler equation in three dimensions is described. The computational procedure is of finite-volume type and highly vectorized, achieving a rate of 125 M flops on the CYBER 205, and allows the use of very dense meshes necessary to resolve the local features like the singular points. Several flow problems designed to test the vortex-sheet hypothesis are solved upon a mesh with over 600,000 grid cells. These solutions, even when the body is smooth, present numerical observations of such vortex sheets and singular points which support the hypothesis. One such computation runs in about 2 hours of CPU time on the CYBER 205 supercomputer.

## Keywords

Singular Point Euler Equation Truncation Error Vortex Sheet Delta Wing## Preview

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## References

- 1.Rizzi, A., Eriksson, L.E., Schmidt, W. and Hitzel, S.: “Numerical Solutions of the Euler Equations Simulating Vortex Flows Around Wings”, AGARD-CP-342, 1983.Google Scholar
- 2.Koeck, C.: “Computation of Three Dimensional Flow Using the Euler Equations and a Multiple-Grid Schemes”, Int. J. Num. Meth. Fluids, Vol. 5, 1985, pp. 483–500.MATHCrossRefGoogle Scholar
- 3.Raj, P. and Sikora, J.S.: “Free-Vortex Flows Recent Encounters with an Euler Code”, AIAA Paper 84–0135, Jan 1984.Google Scholar
- 4.Hoeijmakers, H.W.M. and Rizzi, A.: “Vortex-Fitted Potential Solution Compared with Vortex-Captured Euler Solution for Delta Wing with Leading-Edge Vortex Separation”, AIAA Paper 84–2144, 1984.Google Scholar
- 5.Krause, E., Shi, X.G. and Hartwich, P.M.: “Computation of Leading-Edge Vortices”, AIAA Paper 83–1907, 1983.Google Scholar
- 6.Salas, M.D.: “Foundations for the Numerical Solution of the Euler Equations”, in Advances in Computational Transonics, ed. W.G. Habashi, Pmeridge Press, 1985.Google Scholar
- 7.Schmidt, W. and Jameson, A.: “Euler Solvers as an Analysis Tool for Aircraft Aerodynamis”, in Advances in Computational Transonic, ed. W.G. Habashi, Pineridge Press, 1985.Google Scholar
- 8.Meyer, R.E.: “Introduction to Mathematical Fluid Dynamics, Wiley-Interscience, New York, 1971, p7 75ff.MATHGoogle Scholar
- 9.Rizzi, A. and Eriksson L.E.: “Computation of Inviscid Incompressible Flow with Rotation”, J. Fluid Mech., Vol. 153, April 1985, pp. 275–312.MATHCrossRefGoogle Scholar
- 10.Powell, K., Murman, E., Perez, E, and Baron, J.: “Total Pressure Loss in Vortical Solutions of the Concial Euler Equations”, AIAA Paper 85–1701, 1985.Google Scholar
- 11.Smith, J.H.B.: “Remarks on the Structure of Conical Flow”, in Progress in Aerospace Sciences, ed. D. Kuchemann, Pergamon Press, Oxford, 1972.Google Scholar
- 12.Eriksson, L.E.: “Generation of Boundary-Conforming Grids around Wing-Body Configurations using Transfinite Interpolation”, AIAA J., Vol. 20, Oct 1982, pp. 1313–1320.MATHCrossRefGoogle Scholar
- 13.Rizzi, A. and Eriksson, L.E.: “Computation of Flow Around Wings Based on the Euler Equations”, J. Fluid Mech., Vol. 148, Nov 1984, pp. 45–71.CrossRefGoogle Scholar