Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T06:35:34.875Z Has data issue: false hasContentIssue false
  • English
  • Français

A cell-vertex upwind scheme for two-dimensional supersonic Euler flows

Published online by Cambridge University Press:  04 July 2016

C. B. Allen  and
S. P. Fiddes
C. B. Allen
Affiliation:
Aerospace Engineering Department, University of Bristol, Bristol, UK
S. P. Fiddes
Affiliation:
Aerospace Engineering Department, University of Bristol, Bristol, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

An implicit cell-vertex upwind method is presented for the solution of the two-dimensional Euler equations for supersonic flows. It is based on the two-point difference scheme introduced by S.F. Wornom, but contains many significant developments. Particular attention is given to the implementation of boundary conditions, to ensure that no “numerical” boundary conditions are required, and “image” cells are avoided, making the scheme ideal for use in conjunction with a multiblock approach. Examples are given for quasi-one-dimensional nozzle flows, and in two dimensions, oblique shock reflection, a shock interaction problem, and a supersonic intake problem. The accuracy and efficiency of the scheme is demonstrated, along with its ease of use.

Type
Research Article
Information
The Aeronautical Journal , Volume 101 , Issue 1005 , May 1997 , pp. 199 - 208
Copyright
Copyright © Royal Aeronautical Society 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Shaw, J.A., Forsey, C.R., Weatherill, N.P. and Rose, K.E. A block-structured mesh-generation technique for aerodynamic geometries, in First International Conference on Numerical Grid Generation in Computational Fluid Dynamics, Landshut Pineridge Press, 1986.Google Scholar
2. Wornom, S.F. Implicit conservative characteristic modelling schemes for the Euler equations — a new approach, AIAA paper 83-1939 Danvers, Massachusetts, 1983.Google Scholar
3. Wornom, S.F. A two-point difference scheme for computing steady-state solutions to the conservative one-dimensional Euler equations, Comp Fluids, 1984, 12, (1), pp 11–30.Google Scholar
4. Wornom, S.F. and Hafez, M.M. Calculation of quasi-one-dimensional flows with shocks, Comp Fluids, 1986, 14, (2), pp 131–140.Google Scholar
5. Wornom, S.F. and Hafez, M.M. Implicit conservative schemes for the Euler equations, AIAA J, 1986. 24, (2). pp 215–223.Google Scholar
6. Allen, C.B. An Efficient Euler Solver For Predominantly Supersonic Flows With Embedded Subsonic Pockets, University of Bristol, Department of Aerospace Engineering, PhD Dissertation, 1992.Google Scholar
7. Denton, J.E. A Time-Marching Method for Two- and Three-Dimen-sional Blade-to-Blade Flows, Aeronautical Researsh Council R and M No 3775, 1974.Google Scholar
8. Marsh, H. and Merryweather, H. The calculation of subsonic and supersonic flows in nozzles, University of Cambridge Engineering Department CUED/A-Turbo/TR3, 1969.Google Scholar
9. Shubin, G.R., Stephen, A.B. and Glaz, H.M. Steady shock tracking and Newton's method applied to one-dimensional duct flow, J Comp Phys, 1981, 39, pp 364–374.Google Scholar
10. Ni, R-H. A Multiple-grid scheme for solving the Euler equations, AIAA paper 81-1025, Palo Alto, California, 1981, see also AIAA J, November 1982, 20, (11), pp 1565–1571.Google Scholar
11. Hall, M.G. Fast multigrid solution of the Euler equations using a finite volume scheme of the Lax-Wendroff type, RAE Technical Report 84013, 1984.Google Scholar
12. Hall, M.G. Cell-Vertex Multigrid Schemes for Solution of the Euler Equations, in Numerical Methods for Fluid Dynamics II, Morton, K.W. and Baines, M.J. (Eds), Oxford University Press, 1985.Google Scholar
13. Yee, H.C., Warming, R.F. and Harten, A. Implicit total variation diminishing (TVD) schemes for steady state calculations, AIAA paper 83-1902, Danvers, Massachusetts, 1983.Google Scholar
14. Glaz, H.M. and Wardlaw, A.B. A high-order Godunov scheme for steady supersonic gas dynamics, J Comp Phys, 1985, 58, pp 157–187.Google Scholar
15. Toro, E.F. and Chakraborty, A. Development of an approximate Riemann solver for the steady Euler equations, Aeronaut J, November 1994, 98, (979), pp 325–339.Google Scholar
16. Rubin, S.G. and Tannehill, J.C. Parabolized/reduced Navier-Stokes computational techniques, Annual Revue Fluid Mech, 1992, 24, pp 117–144.Google Scholar
17. Srinivas, K. An explicit spatial marching algorithm for Navier-Stokes equations, Comps and Fluids, 1992, 21, pp 291–299.Google Scholar
18. Srinivas, K. Computation of Hypersonic Flow Past a Compression Corner by a Spatial-Marching Scheme, Hypersonic Flows for Reentry Problems, Vol 3, Aborall, R., Desideri, J.A., Glowinsk, R., Mallet, M. and Periaux, J. (Eds), Springer Verlag, 1992, pp 338–341.Google Scholar
19. Srinivas, K. An explicit finite-volume spatial marching method for reduced Navier-Stokes equations 2, Proceedings, Ninth International Conference on Numerical Methods in Laminar and Turbulent Flow, Atlanta, Georgia, July 1995, pp 516–527.Google Scholar
20. Roe, P.L. Approximate Riemann solvers, parameter vectors, and difference schemes, J Comp Phys, 1981, 43, pp 357–372.Google Scholar
21. Van-leer, B. Flux-vector splitting for the Euler equations, Lecture Notes in Physics, 1982, 170, pp 507–512.Google Scholar
22. LIou, M-S. and Van-leer, B. Choice of implicit and explicit operators for the upwind differencing method, AIAA paper 88-0624, Reno, Nevada, 1988.Google Scholar
23. Kumar, A.J. Numerical Analysis of the Scramjet-Inlet Flow by using Two-Dimensional Navier-Stokes Equations, NASA Technical Paper 1940, 1981.Google Scholar