Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-06T06:16:33.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional simulations of large eddies in the compressible mixing layer

Published online by Cambridge University Press:  26 April 2006

N. D. Sandham  and
W. C. Reynolds
N. D. Sandham
Affiliation:
DLR, Institute for Theoretical Fluid Mechanics, Gottingen, Germany
W. C. Reynolds
Affiliation:
Stanford University, Stanford, CA 94305, USA and NASA Ames Research Center, Moffett Field, CA 94035, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The effect of Mach number on the evolution of instabilities in the compressible mixing layer is investigated. The full time-dependent compressible Navier–Stokes equations are solved numerically for a temporally evolving mixing layer using a mixed spectral and high-order finite difference method. The convective Mach number Mc (the ratio of the velocity difference to the sum of the free-stream sound speeds) is used as the compressibility parameter. Simulations with random initial conditions confirm the prediction of linear stability theory that at high Mach numbers (Mc > 0.6) oblique waves grow more rapidly than two-dimensional waves. Simulations are then presented of the nonlinear temporal evolution of the most rapidly amplified linear instability waves. A change in the developed large-scale structure is observed as the Mach number is increased, with vortical regions oriented in a more oblique manner at the higher Mach numbers. At convective Mach numbers above unity the two-dimensional instability is found to have little effect on the flow development, which is dominated by the oblique instability waves. The nonlinear structure which develops from a pair of equal and opposite oblique instability waves is found to resemble a pair of inclined A-vortices which are staggered in the streamwise direction. A fully nonlinear computation with a random initial condition shows the development of large-scale structure similar to the simulations with forcing. It is concluded that there are strong compressibility effects on the structure of the mixing layer and that highly three-dimensional structures develop from the primary inflexional instability of the flow at high Mach numbers.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 224 , March 1991 , pp. 133 - 158
Copyright
© 1991 Cambridge University Press

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

Anderson, D. A., Tannehill, J. C. & Pletcher, R. H., 1984 Computational Fluid Mechanics and Heat Transfer. McGraw-Hill.
Bebnal, L. P. & Roshko, A., 1986 Streamwise vortex structure in plane mixing layers. J. Fluid Mech. 170, 499–525.Google Scholar
Birch, S. F. & Eggers, J. M., 1973 A critical review of the experimental data for developed free turbulent shear layers. NASA –SP-321, pp. 11–40.Google Scholar
Bogdanoff, D. W.: 1983 Compressibility effects in turbulent shear layers. AIAA J. 21, 926–927.Google Scholar
Brown, G. L. & Roshko, A., 1974 On density effects and large scale structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.Google Scholar
Clemens, N. T. & Mungal, M. G., 1990 Two– and three-dimensional effects in the supersonic mixing layer. AIAA/ASME/SAE/ASEE 26th Joint Propulsion Conf. July 1990, Orlando, Florida, AIAA Paper 90-1978Google Scholar
Clemens, N. T., Mungal, M. G., Berger, T. E. & Vandsburger, U., 1990 Visualizations of the structure of the turbulent mixing layer under compressible conditions. AIAA Paper 90–0500.Google Scholar
Eberhardt, S., Riley, J. J., Soetrisno, M. & Greenotjgh, J. A., 1988 A numerical study of inviscid, supersonic mixing layers. Intl Workshop on the Physics of Compressible Turbulent Mixing, Princeton, Oct. 1988.
Greenough, J., Riley, J., Soetrisno, M. & Ebebhardt, D., 1989 The effects of walls on a compressible mixing layer. AIAA Paper 89–0372.Google Scholar
Gropbngiesser, H.: 1969 Beitrag zur Stabilität freier Grenzschichten in kompressiblen Medien. DLR FB 69–25. (English transl. NASA TT F-12,786, 1970.)Google Scholar
Herbert, T.: 1988 Secondary instability of boundary layers. Ann. Rev. Fluid Mech. 20, 487–526.Google Scholar
Lele, S.: 1989 Direct numerical simulation of compressible free shear flows. AIAA Paper 89–0374.Google Scholar
Lele, S.: 1990 Compact finite difference schemes with spectral-like resolution. CTR Manuscript 107. Center for Turbulence Research, Stanford University, CA. (Submitted to J. Comput. PhysGoogle Scholar
Lessen, M., Fox, J. A. & Zien, H. M. 1965 On the inviscid stability of the laminar mixing of two parallel streams of a compressible fluid. J. Fluid Mech. 23, 355–367.Google Scholar
Lessen, M., Fox, J. A. & Zien, H. M. 1966 Stability of the laminar mixing of two parallel streams with respect to supersonic disturbances. J. Fluid Mech. 25, 737–742.Google Scholar
Lin, S. J. & Cohcos, G. M., 1984 The mixing layer: deterministic models of a turbulent flow. part 3. The effect of plane strain on the dynamics of streamwise vortices. J. Fluid Mech 141, 139–178.Google Scholar
Mack, L. M.: 1984 Boundary-layer stability theory. Special Course on Stability and Transition of Laminar Flow, AOARD Rep. 709, 3–1 to 3–81.Google Scholar
Mungal, M. G. & Dimotakis, P. E. Mixing and combustion with low heat release in a turbulent shear layer. J. Fluid Mech. 148, 349–382.
Mungal, M. G. & Hollingsworth, D. K., 1989 Organized motion in a very high Reynolds number jet. Phys. Fluids A 1, 1615–1623.Google Scholar
Papamoschou, D.: 1989 Structure of the compressible turbulent shear layer. AIAA Paper 89–01 26.Google Scholar
Papamoschou, D. & Roshko, A., 1986 Observations of supersonic free shear layers. AIAA Paper 86–0162.Google Scholar
Papamoschou, D. & Roshko, A., 1988 The compressible turbulent mixing layer: an experimental study. J. Fluid Mech. 197, 453–477.Google Scholar
Pierrehumbert, R. T. & Widnall, S. E., 1982 The two– and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–62.Google Scholar
Ragab, S. A. & Wu, J. L., 1988 Instabilities in the free shear layer formed by two supersonic streams. AIAA Paper 8–0038Google Scholar
Rogers, M. M. & Moseh, R. D., 1989 The development of three-dimensional temporally-evolving mixing layers. Seventh Symp. on Turbulent Shear Flows, Stanford, CA, August 21–23, 1989Google Scholar
Sandham, N. D. & Reynolds, W. C., 1989 A numerical investigation of the compressible mixing layer. Rep. TF-45. Thermosciences Division, Mechanical Engineering Department, Stanford University.
Sandham, N. D. & Reynolds, W. C., 1990 Compressible mixing layer: linear theory and direct simulation. AIAA J. 8, 618–624.Google Scholar
Sandham, N. D. & Yee, H. C., 1989 A numerical study of a class of TVD schemes for compressible mixing layers. NASA TM -102194.Google Scholar
Thompson, K. W.: 1987 Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys. 68, 1–24Google Scholar
White, F. M.: 1974 Viscous Fluid Flow McGraw-Hill.
Wray, A. A.: 1986 Very low storage time-advancement schemes. Internal Rep. NASA-Ames Research Center, Moffett Field, California.Google Scholar
Zhuanfg, M., Dimotakis, P. & Kubota, T., 1990 The effect of walls on a spatially growing supersonic shear layer. Phys. Fluids A 2, 599–604.Google Scholar