TAFSM

Team for Advanced Flow Simulation and Modeling



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Supersonic Flow Past a Delta-Wing

In this problem, we consider the flow of air past a delta-wing model of an aerospace vehicle at Mach 3. The Reynolds number, based on free-stream values and the maximum chord length (along the plane of symmetry), is 1.1 million. Due to the assumed symmetry of the solution, only half of the flow over the delta-wing is computed.

The delta-wing has a wedge type cross section as an underbody, and the corners merge smoothly to the flat surface on top. The delta- wing has a unit length in the chord-wise direction, and tapers from 0.0 to 0.69 units in the span-wise direction. The geometry of the delta-wing was provided to us by Dr. Chien Li from NASA-JSC.

Our preliminary steady-state solution of this problem was obtained using a relatively coarse mesh, with 152,397 nodes and 143,920 hexahedral elements. For each time step there were 725,000+ nonlinear equations solved. This solution is presented in the figure below, the top image shows the side view of the delta-wing together with the Mach number distribution around it, and the bottom image shows the front view. This image appeared on the cover page of the Slide Book of the ARPA High Performance Computing Software PI Meeting, Norfolk, Virginia, March 17-18, 1993.

Later, we solved this problem with a mesh consisting of 1,032,328 nodes and 1,002,684 hexahedral elements. In this case, 5,001,031 nonlinear equations were solved iteratively. This computation was carried out on a CM-5 using 512 processing nodes at a sustained speed of 9.8 GigaFlops. The cost of a single nonlinear iteration with 1 GMRES outer iteration and a Krylov subspace dimension of 10 is 14.9 seconds. Below, the figure on the left shows the pressure distribution on the wing surface and at a cross section, and the figure on the right shows the top view of the delta- wing together with the pressure field around it. The mesh generator, flow solver, and flow visualization sofware (based on Wavefront) were developed by the T*AFSM.

References:

1. T.E. Tezduyar and T.J.R. Hughes, "Finite Element Formulations for Convection Dominated Flows with Particular Emphasis on the Compressible Euler Equations", AIAA Paper 83-0125, Proceedings of AIAA 21st Aerospace Sciences Meeting, Reno, Nevada (1983).

2. T.J.R. Hughes and T.E. Tezduyar, "Finite Element Methods for First-order Hyperbolic Systems with Particular Emphasis on the Compressible Euler Equations", Computer Methods in Applied Mechanics and Engineering, 45 (1984) 217-284.

3. G.J. Le Beau and T.E. Tezduyar, "Finite Element Computation of Compressible Flows with the SUPG Formulation", Advances in Finite Element Analysis in Fluid Dynamics (eds. M.N. Dhaubhadel, M.S. Engelman and J.N. Reddy), FED-Vol. 123, ASME, New York (1991) 21-27.

4. G.J. Le Beau and T.E. Tezduyar, "Finite Element Solution of Flow Problems with Mixed-Time Integration", Journal of Engineering Mechanics, 117 (1991) 1311-1330.

5. T.E. Tezduyar, S.K. Aliabadi, M. Behr and S. Mittal, "Massively Parallel Finite Element Simulation of Compressible and Incompressible Flows", Computer Methods in Applied Mechanics and Engineering, 119 (1994) 157-177.