Hypersonic Flows for Reentry Problems pp 451-471 | Cite as

# The Application of an Adaptive Unstructured Grid Method to the Solution of Hypersonic Flows Past Double Ellipse and Double Ellipsoid Configurations

## Abstract

In this contribution, we use an adaptive finite element algorithm for the solution of inviscid and laminar compressible viscous flows past double ellipse and double ellipsoid configurations. The spatial discretisation is achieved with linear triangular elements in two dimensions and linear tetrahedral elements in three dimensions, while the time discretisation is accomplished in either a fully explicit or in an implicit/explicit fashion. In the analysis of a given problem, the elements in the computational grid may be partitioned into an explicit group and an implicit group and the appropriate form of the algorithm used directly within each group. The implicit formulation uses one of a family of finite difference methods devised originally by Lerat and co-workers [1,2], while the complete algorithm has the desirable feature that, in its explicit form, it reduces to a solution scheme that we have previously employed [3,4]. The explicit form of the algorithm is applied in the solution of inviscid flows while viscous flows are solved using the explicit/implicit version. In two dimensional simulations, several authors [5,6], while nominally employing an unstructured grid method to simulate viscous flows, have used a structured grid in the vicinity of solid surfaces. In the present context, such an approach leads to a natural partitioning in which the elements which are treated implicitly lie in the vicinity of solid walls so that the grid structure, in both the normal and tangential directions, can be utilised in a line relaxation procedure for the solution of the resulting equation system. However, in the simulation of three dimensional flows, the physical boundaries will be represented by an unstructured assembly of triangular elements. Now a grid which is structured only in the normal direction is employed near the solid surfaces and the implicit equation system is solved by appealing to a newly developed line relaxation process for unstructured grids [7].

### Keywords

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### References

- [1]A. Lerat, ‘Implicit methods of second order accuracy’, AIAA J. 23, 33–40, 1985.CrossRefMATHADSMathSciNetGoogle Scholar
- [2]H. Hollanders, A. Lerat and R. Peyret, ‘Three dimensional calculation of transonic viscous flows by an implicit method’, AIAA J. 23, 1670–1678, 1985CrossRefMATHADSMathSciNetGoogle Scholar
- [3]K. Morgan and J. Peraire, ‘Finite element methods for compressible flows’, von Karman Institute for Fluid Dynamics Lecture Series 1987–04, Brussels, 1987.Google Scholar
- [4]J. Peraire, L. Formaggia, J. Peiro, K. Morgan and O. C. Zienkiewicz, ‘Finite element Euler computations in 3D’, AIAA Paper 88 - 0032, 1988.Google Scholar
- [5]K. Nakahashi, ‘FDM-FEM zonal approach for computations of compressible viscous flows’, Lecture Notes in Physics 264, 494–498, Springer, 1986.CrossRefADSGoogle Scholar
- [6]R. R. Thareja, J. R. Stewart, O. Hassan, K. Morgan and J. Peraire, ‘A point implicit unstructured grid solver for the Euler and Navier-Stokes equations’, AIAA Paper 88 - 0036, 1988.Google Scholar
- [7]O. Hassan, K. Morgan and J. Peraire, ‘An implicit finite element method for high speed flows’, AIAA Paper 90 - 0402, 1990.Google Scholar
- [8]J. Peraire, M. Vahdati, K. Morgan and O. C. Zienkiewicz, ‘Adaptive remeshing for compressible flow computations’, J.Comp. Phys. 72, 449–466, 1987.CrossRefMATHADSGoogle Scholar
- [9]R. M. Beam and R. F. Warming, ‘An implicit factored scheme for the compressible Navier-Stokes equations’, AIAA J. 16, 393–401, 1978.CrossRefMATHADSGoogle Scholar
- [10]O. C. Zienkiewicz and K. Morgan, Finite Elements and Approximation, Wiley, New York, 1985.Google Scholar
- [11]L. Formaggia, J. Peraire, K. Morgan and J. Peiro, ‘Implementation of a 3D explicit Euler solver on a CRAY computer’, Proc. 4th Int. Symp. on Science and Engineering on CRAY Supercomputers, 45–65, Minneapolis, 1988.Google Scholar