Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-22T10:56:42.330Z Has data issue: false hasContentIssue false

Solution of the Reynolds-averaged Navier-Stokes equations for transonic aerofoil flows

Published online by Cambridge University Press:  04 July 2016

L. J. Johnston*
Affiliation:
Department of Mechanical Engineering, UMIST, Manchester

Summary

A computational method to predict the viscous transonic flow development around two-dimensional aerofoil sections is described. The Reynolds-averaged Navier-Stokes equations applicable to turbulent flow are discretised in space using a cell-centred finite-volume formulation. A multi-stage, explicit, time-marching scheme is used to advance the unsteady flow equations in time to a steady-state solution. Turbulence closure is achieved using either the Baldwin-Lomax algebraic model, or a one-equation model based on the turbulent kinetic energy equation. This latter equation is solved using essentially identical procedures to those for the mean-flow equations. Results are presented for the RAE 2822, RAE 5225, CAST 7 and MBB-A3 transonic aerofoil sections. The relative performance and limitations of the two turbulence models are discussed.

Type
Research Article
Copyright
Copyright © Royal Aeronautical Society 1991 

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. Whitcomb, R. T. Review of NASA supercritical airfoils, ICAS Paper No 74-10, 1974.Google Scholar
2. Collyer, M. R. and Lock, R. C. Prediction of viscous effects on steady transonic flow past an aerofoil, Aeronaut Q, August 1979, 30, pp 485505.Google Scholar
3. Melnik, R., Brook, J. and Mead, H. GRUMFOIL a computer code for the computation of viscous transonic flow over airfoils, AIAA Paper 87-0414, 1987.Google Scholar
4. Lock, R. C. A modification to the method of Garabedian and Korn, Notes on Numerical Fluid Mechanics, Vol 3, Numerical Methods for the Computation of Inviscid Transonic Flows with Shock Waves, Vieweg, 1980.Google Scholar
5. Whitfield, D. L., Thomas, J. L., Jameson, A. and Schmidt, W. Computation of Transonic Viscous-Inviscid Interacting Flow, NASA TM 85203, 1983.Google Scholar
6. Drela, M. and Giles, M. B. Viscous-inviscid analysis of transonic and low Reynolds number airfoils, AIAA Paper 87- 0424, 1987.Google Scholar
7. Chen, L. T., Li, S. and Chen, H. Calculation of transonic airfoil flows by interaction of Euler and boundary-layer equations, AIAA Paper 87-0521, 1987.Google Scholar
8. Lock, R. C. and Williams, B. R. Viscous-inviscid interactions in external aerodynamics, Prog Aerospace Sci, 1987, 24, pp 55171.Google Scholar
9. East, L. F. A Representation of Second-Order Boundary Layer Effects in the Momentum Integral Equation and in Viscous-Inviscid Interactions, RAE TR 81002, January 1981.Google Scholar
10. Ashill, P. R., Wood, R. F. and Weeks, D. J. A Semi-Inverse Version of the Viscous Garabedian and Korn Method, RAE TR 87002, 1987.Google Scholar
11. Holst, T. L. Viscous Transonic Airfoil Workshop compendium of results, AIAA Paper 87-1460, 1987.Google Scholar
12. Johnson, D. A. and Kino, L. S. A mathematically simple turbulence closure model for attached and separated turbulent boundary layers, AIAA J, Nov 1985, 23, (11), pp 16841692.Google Scholar
13. Coakley, T. J. Numerical simulation of viscous transonic airfoil flows, AIAA Paper 87-0416, 1987.Google Scholar
14. Jameson, A., Schmidt, W. and Turkel, E. Numerical solutions of the Euler equations by finite volume methods using Runge- Kutta time-stepping schemes, AIAA Paper 81-1259, 1981.Google Scholar
15. Baldwin, B. S. and Lomax, H. Thin-layer approximation and algebraic model for separated turbulent flows, AIAA Paper 78-257, 1978.Google Scholar
16. Jameson, A. and Baker, T. J. Solution of the Euler equations for complex configurations, AIAA Paper 83-1929, 1983.Google Scholar
17. Jameson, A. Solution of the Euler equations for two dimensional transonic flow by a multigrid method, Appl Math Comput, 1983, 13, pp 327355.Google Scholar
18. Coakley, T. J. Numerical method for gas dynamics combining characteristic and conservation concepts, AIAA Paper 81-1257, 1981.Google Scholar
19. Martinelli, L., Jameson, A. and Grasso, F. A multigrid method for the Navier-Stokes equations, AIAA Paper 86-0208, 1986.Google Scholar
20. Martinelli, L. and Jameson, A. Validation of a multigrid method for the Reynolds averaged equations, AIAA Paper 88- 0414, 1988.Google Scholar
21. Cebeci, T. and Smith, A. M. O. Analysis of turbulent boundary layers, Academic Press, 1974.Google Scholar
22. Stock, H. W. and Haase, W. The determination of turbulent length scales in algebraic turbulence models for attached and slightly separated flows using Navier-Stokes methods, AIAA Paper 87-1302, 1987.Google Scholar
23. Granville, P. S. Baldwin-Lomax factors for turbulent boundary layers in pressure gradients, AIAA J, Dec 1987, 25, (12), pp 16241627.Google Scholar
24. Mitcheltree, R., Salas, M. and Hassan, H. A one equation turbulence model for transonic airfoil flows, AIAA Paper 89- 0557, 1989.Google Scholar
25. Wolfshtein, M. The velocity and temperature distribution in one-dimensional flow with turbulence augmentation and pressure gradient, Int J Heat Mass Transfer, 1969, 12, pp 301318.Google Scholar
26. Chen, H. C. and Patel, V. C. Practical near-wall turbulence models for complex flows including separation, AIAA Paper 87-1300, 1987.Google Scholar
27. Rizzi, A. Computational mesh for transonic airfoils, Notes on Numerical Fluid Mechanics, Vol 3, Numerical Methods for the Computation of Inviscid Transonic Flows with Shock Waves, Vieweg, 1980, pp 222253.Google Scholar
28. 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, May 1979, A6-1 to A6-77.Google Scholar
29. Lock, R. C. Aerodynamic design methods for transonic wings, Aeronaut J, Jan 1990, 94, (931), pp 116.Google Scholar
30. Stanewsky, E., Puffert, W., Muller, R. and Bateman, T. E. B. Supercritical Airfoil CAST 7 — Surface Pressure, Wake and Boundary Layer Measurements, AGARD AR 138, May 1979, A3-1 to A3-35.Google Scholar
31. Stanewsky, E., Demurie, F., Ray, E. J. and Johnson, C. B. High Reynolds number tests of the CAST-10-2/DOA 2 transonic airfoil at ambient and cryogenic temperature conditions, AGARD CP 348, 1983, pp 10–1 to 10–13.Google Scholar
32. Swanson, R. C., Radespiel, R. and McCormick, V. E. Comparison of two- and three-dimensional Navier—Stokes solutions with NASA experimental data for CAST-10 airfoil, NASA CP-3052, CAST-10-2/DOA-2 Airfoil Studies Workshop Results, 1989, pp 233–258.Google Scholar
33. Bucciantini, G., Oggiano, M. S. and Onorato, M. Supercritical airfoil MBB-A3 surface pressure distributions, wake and boundary condition measurements, AGARD AR 138, May 1979, A8-1 to A8-25.Google Scholar