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Computational modelling of shock wave/boundary layer interaction with a cell-vertex scheme and transport models of turbulence

Published online by Cambridge University Press:  04 July 2016

M. A. Leschziner ,
K. P. Dimitriadis  and
G. Page
M. A. Leschziner
Affiliation:
University of Manchester Institute of Science and Technology Manchester, UK
K. P. Dimitriadis
Affiliation:
University of Manchester Institute of Science and Technology Manchester, UK
G. Page
Affiliation:
University of Manchester Institute of Science and Technology Manchester, UK
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Abstract

A calculation procedure for modelling the interaction between shock waves and attached or separated turbulent boundary layers is introduced. The numerical framework, applicable to general curved grids, combines cell-vertex storage, a Lax-Wendroff time-marching scheme and multigrid convergence acceleration. The main numerical ingredients of the procedure are documented in some detail, with particular emphasis placed on the inclusion of viscous and turbulence transport within the cell-vertex framework, which was originally formulated for inviscid flow. An investigation of the predictive performance of alternative transport models of turbulence has been the primary objective of the present work. Particular attention has been focused on a comparison between variants of low Reynolds number k-ε models and an algebraic variant of a Reynolds-stress transport closure in strong interaction situations, including shock-induced separation. The turbulence models are introduced, and important numerical issues affecting their stable implementation are discussed. The calculation procedure is then applied to two confined transonic flows over bumps — one incipiently and the other extensively separated (Delery Cases A and C) — and to the transonic flow around the RAE 2822 aerofoil at two angles of incidence (Cases 9 and 10). The investigation demonstrates that the eddy-viscosity models tend to seriously underestimate the strength of interaction, particularly when separation is extensive. The performance of the Reynolds-stress model is not entirely consistent across the range of conditions examined. In the case of bump flows, the model displays strong sensitivity to the shock, predicting excessive boundary layer displacement in Case A, a broadly correct separation pattern in Case C and insufficient rate of post shock recovery in both cases. The aerofoil flows are either attached or incipiently separated, and the benefits arising from Reynolds-stress modelling are modest. Neither the k-ε model nor the Reynolds-stress closure is able to return a satisfactory representation of the most challenging RAE 2822 Case 10; at least not with the recommended windtunnel corrections for freestream Mach number and angle of incidence.

Type
Research Article
Information
The Aeronautical Journal , Volume 97 , Issue 962 , February 1993 , pp. 43 - 61
Copyright
Copyright © Royal Aeronautical Society 1993 

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References

1. Ni, R.H. A multiple grid scheme for solving the Euler equations, AIAA J, 1982, 20, pp 1565–1571.Google Scholar
2. Hall, M.G. Cell vertex multigrid schemes for solution of the Euler equations, In: Numerical Methods for Fluid Dynamics II, Clarendon Press, W., Morton and Baines, M.J. (Eds), 1986, pp 303–345.Google Scholar
3. Jameson, A. A vertex-based multigrid algorithm for three-dimensional compressible flow calculations, In: Numerical Methods for Compressible Flows — Finite Difference, Element and Volume Techniques, Tesduyar, T.E. and Hughes, T.J.R. (Eds), ASME/AMD, 1986, p 78.Google Scholar
4.Morton, K.W. and Paisley, M.F., A finite volume scheme with shock fitting for the steady Euler equations, J Comp Phys, 1989, 80, (1), p 168.Google Scholar
5. Jameson, A., Schmidt, W. and Turkel, E., Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes, AIAA paper 81-1259,1981.Google Scholar
6. A., Jameson Numerical solution of the Euler equations for compressible inviscid fluids, In: Numerical Methods for the Euler Equations of Fluid Dynamics, F., ANGRAND (Ed), SIAM, 1985.Google Scholar
7. Mackenzie, J.A. and Morton, K.W. Finite Volume Solutions of Convection Diffusion Test Problems, Oxford University Computing Laboratory, Numerical Analysis Group Report 90/7, 1990.Google Scholar
8. Rodi, W. 1980, Turbulence modelling and its application in hydraulics, IAHR Monograph, Delft, 1976.Google Scholar
9. Nallasamy, M. Turbulence models and their application to the prediction of internal flows: a review, Comput Fluid, 1987, 15, pp 151–194.Google Scholar
10. Lakschminarayana, B.J. Turbulence modelling for complex shear flows, AIAA J, 1986, 24, pp 1900–1917.Google Scholar
11. Sugavanam, A. Evaluation of low-Reynolds-number turbulence models for attached and separated flows, AIAA paper 85-0375, 1985.Google Scholar
12. Degrez, G. and Vandromme, D. Implicit Navier-Stokes calculations of transonic shock/turbulent boundary-layer interactions, In: Turbulent Shear-Layer/Shock-Wave Interactions, Delery, J. (Ed), Springer-Verlag, 1986, pp 53–64.Google Scholar
13. Viegas, J.R., Rubesin, M.W. and Horstman, C.C. On the use of wall functions as boundary conditions for two-dimensional separated compressible flows, AIAA paper 85-0180, 1985.Google Scholar
14. Escande, B. and Cambier, L. Turbulence modelling in transonic interactions, In: Turbulent Shear-Layer/Shock-Wave Interactions, Delery, J. (Ed), Springer-Verlag, 1986, pp 40–51.Google Scholar
15. Sahu, J. and Danberg, J.E. Navier-Stokes computations of transonic flows with a two-equation turbulence model, AIAA J, 1986, 24, (11), pp 1744–1751.Google Scholar
16. Wilcox, D.C. Reassessment of the scale-determining equation for advanced turbulence models, AIAA J, 1988, 26, pp 1299–1310.Google Scholar
17. Coakley, T.J., Viegas, J.R., Huang, P.G. and Rubesin, M.W. An assessment and application of turbulence models for hypersonic flows, 9th National Aerospace Plane Technology Symp, 106, Orlando, Florida, 1990.Google Scholar
18. Peace, A.J. Turbulent flow predictions for afterbody/nozzle geometries including base effects, AIAA paper 89-1865, 1989.Google Scholar
19. Stolcis, L. and Johnston, L.J. Compressible flow calculations using a two-equation turbulence model and unstructured grids, In: Numerical Methods in Laminar and Turbulent Flow, VII, TAYLOR, C., CHIN, J.H. and HOMSY, G.M. (Eds), 1991, pp 922–931.Google Scholar
20. Huang, P.G. and Coakley, T.J. An implicit Navier-Stokes code for turbulent flow modeling, AIAA paper 92-0547, 1992.Google Scholar
21. Launder, B.E. Second-moment closure: present and future?, Int J Heat Fluid Flow, 1989, 10, pp 282–299.Google Scholar
22. Leschziner, M.A. Modelling engineering flows with Reynolds-stress turbulence closure, J Wind Eng Indust Aerod, 1990, 35, pp 21–47.Google Scholar
23. Vandromme, P. and Ha Minh, H. Physical analysis for turbulent boundary-layer/shock-wave interactions using second-order closure predictions, In: Turbulent Shear-Layer/Shock-Wave Interactions, DELERY, J. (Ed), Springer-Verlag, 1985, pp 127–136.Google Scholar
24. Benay, R., Coet, M.C. and Delery, J.A. A study of turbulence modelling in transonic shock-wave/boundary-layer interactions, Proc 6th Symp on Turbulence Shear Flows, Toulouse, 1987, pp 8.2.1-8.2.6.Google Scholar
25. Huang, P.G. Modeling hypersonic boundary-layer flows with sec ond-moment closure, Centre for Turbulence Research, University of Stanford, Annual Research Briefs, 1990, pp 1-13.Google Scholar
26. Dimitriadis, K.P. and Leschziner, M.A. Computation of turbulent transonic and supersonic flow with a cell-vertex algorithm and second-moment closure, 12th Int Conf on Numerical methods in Fluid Mechanics, In: Lecture Notes in Physics, 1990, 371, Morton, K.W. (Ed), Springer-Verlag, pp 111–115.Google Scholar
27. Davidson, L. Calculation of the flow around a high-lift airfoil using explicit code and an algebraic Reynolds-stress model, In: Numerical Methods in Laminar and Turbulent Flow VII, Taylor, C., Chin, J.H., Homsey, G.M. (Eds), Pineridge Press, 1991, pp 852–862.Google Scholar
28. Dimitriadis, K.P. and Leschziner, M.A. Approximation of viscous and turbulent transport in transonic-flow cell-vertex algorithm, In: Numerical Methods in Laminar and Turbulent Flows, VI, Taylor, C., Gresho, P.M. and Saw, R.L. (Eds), Pineridge Press, 1989, pp 861–881.Google Scholar
29. Dimitriadis, K.P. and Leschziner, M.A. Multilevel convergence acceleration for viscous and turbulent transonic flows computed with a cell vertex method, Proc 4th Copper Mountain Conference on Multigrid Methods, SIAM, 1989, pp 130-148.Google Scholar
30. Dimitriadis, K.P. and Leschziner, M.A. A cell-vertex TVD scheme for transonic viscous flow, In: Numerical Methods in Laminar and Turbulent Flow VII, Taylor, C., Chin, J.H. and Homsby, G.M. (Eds.), Pineridge Press, 1991, pp 874–886.Google Scholar
31. Wolfshtein, M. The velocity and temperature distribution in one-dimensional flow with turbulence augmentation and pressure gradient, Int J Heat Mass Trans, 1969, 12, pp 301–312.Google Scholar
32. Gibson, M.M. and Launder, B.E. Ground effects on pressure fluctuations in the atmospheric boundary layer, J Fluid Mech, 1978, 86, pp 4911-511.Google Scholar
33. Rodi, W. A new algebraic relation for calculating the Reynolds-stresses, Zamm, 1976, 56, p 219–222.Google Scholar
34. Chien, K.Y. Predictions of channel and boundary-layer flows with a low-Reynolds number turbulence model, AIAA J, 1982, 20, pp 33–38.Google Scholar
35. Launder, B.E. and Sharma, B.I. Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc, Lett Heat Mass Transfer, 1974, 1, pp 131–138.Google Scholar
36. Jones, W.P. and Launder, B.E. The prediction of laminarisation with a two-equation model of turbulence, Int J Heat Mass Transfer, 1972, 15, pp 301–314.Google Scholar
37. Launder, B.E. and Spalding, D.B. The numerical computation of turbulent-flow, Comp Meths Appl Mech Engng, 1974, 3, pp 269–289.Google Scholar
38. Delery, J., Copy, C. and Reisz, J. Analyse au velocimetre laser bi-directionnel d'une interaction choc-couche limite turbulente avec decollement etendu, ONERA Rapport Technique No. 37/7078 AY 014, Chatillon, August 1980.Google Scholar
39. Delery, J. and Reisz, J. Analyse experimentale d'une interaction choc-couche limite turbulente a Mach 1·3 (decollment naissant), ONERA, Rapport Technique No. 42/7078 AY 014, Chatillon, December 1980.Google Scholar
40. Cook, P.H., Mcdonald, M.A. and Firmin, M.C.P. Aerofoil RAE 2822 — Pressure Distributions and Boundary Layer and Wake Measurements, AGARD AR 138, 1979, pp A6-1-A6-77.Google Scholar
41. Lien, F.S. and Leschziner, M.A. Modelling shock/turbulent-bound ary-layer interaction with second-moment closure within a pressure-velocity strategy, Proc 13fh Int Conf on Numerical Methods in Fluid Dynamics, Rome 6-10 July 1992 (to appear).Google Scholar