Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T01:26:44.925Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear development of subsonic modes on compressible mixing layers: a unified strongly nonlinear critical-layer theory

Published online by Cambridge University Press:  16 October 2008

CLIFFORD A. SPARKS  and
XUESONG WU
CLIFFORD A. SPARKS
Affiliation:
Department of Mathematics, Imperial College London, 180 Queens Gate, London SW7 2AZ, UK
XUESONG WU
Affiliation:
Department of Mathematics, Imperial College London, 180 Queens Gate, London SW7 2AZ, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with the nonlinear instability of compressible mixing layers in the regime of small to moderate values of Mach number M, in which subsonic modes play a dominant role. At high Reynolds numbers of practical interest, previous studies have shown that the dominant nonlinear effect controlling the evolution of an instability wave comes from the so-called critical layer. In the incompressible limit (M = 0), the critical-layer dynamics are strongly nonlinear, with the nonlinearity being associated with the logarithmic singularity of the velocity fluctuation (Goldstein & Leib, J. Fluid Mech. vol. 191, 1988, p. 481). In contrast, in the fully compressible regime (M = O(1)), nonlinearity is associated with a simple-pole singularity in the temperature fluctuation and enters in a weakly nonlinear fashion (Goldstein & Leib, J. Fluid Mech. vol. 207, 1989, p. 73). In this paper, we first consider a weakly compressible regime, corresponding to the distinguished scaling M = O(ε1/4), for which the strongly nonlinear structure persists but is affected by compressibility at leading order (where ε ≪ 1 measures the magnitude of the instability mode). A strongly nonlinear system governing the development of the vorticity and temperature perturbation is derived. It is further noted that the strength of the pole singularity is controlled by T′c, the mean temperature gradient at the critical level, and for typical base-flow profiles T′c is small even when M = O(1). By treating T′c as an independent parameter of O(ε1/2), we construct a composite strongly nonlinear theory, from which the weakly nonlinear result for M = O(1) can be derived as an appropriate limiting case. Thus the strongly nonlinear formulation is uniformly valid for O(1) Mach numbers. Numerical solutions show that this theory captures the vortex roll-up process, which remains the most prominent feature of compressible mixing-layer transition. The theory offers an effective tool for investigating the nonlinear instability of mixing layers at high Reynolds numbers.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 614 , 10 November 2008 , pp. 105 - 144
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Blumen, W. 1970 Shear layer instability of an inviscid compressible fluid. J. Fluid Mech. 40, 769–781.CrossRefGoogle Scholar
Blumen, W., Drazin, P. G. & Billings, D. F. 1975 Shear layer instability of an inviscid compressible fluid. Part 2. J. Fluid Mech. 71, 305–316.CrossRefGoogle Scholar
Broadwell, J. E. & Breidenthal, R. E. 1982 A simple model of mixing and chemical reaction in a turbulent shear layer. J. Fluid Mech. 125, 397–410.CrossRefGoogle Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.CrossRefGoogle Scholar
Clemens, N. T. & Mungal, M. G. 1995 Large-scale structure and entrainment in the supersonic mixing layer. J. Fluid Mech. 284, 171–216.CrossRefGoogle Scholar
Cowley, S. J. & Wu, X. 1994 Asymptotic approaches to transition modelling. In Progress in Transition Modelling, AGARD Report 793, Chapter 3, 1–38.Google Scholar
Curran, E. T., Heiser, W. H. & Pratt, D. T. 1996 Fluid phenomena in scramjet combustion systems. Annu. Rev. Fluid Mech. 28, 323–360.CrossRefGoogle Scholar
Day, M. J., Mansour, N. N. & Reynolds, W. C. 2001 Nonlinear stability and structure of compressible reacting mixing layers. J. Fluid Mech. 446, 375–408.CrossRefGoogle Scholar
Drazin, P. G. & Davey, A. 1977 Shear layer instability of an inviscid compressible fluid. Part 3. J. Fluid Mech. 82, 255–260.CrossRefGoogle Scholar
Elliott, G. S., Samimy, M. & Arnette, S. A. 1995 The characteristics and evolution of large-scale structures in compressible mixing layers. Phys. Fluids 7, 864–876.CrossRefGoogle Scholar
Goldstein, M. E. 1994 Nonlinear interactions between oblique instability waves on nearly parallel shear flows. Phys. Fluids A 6, 724–735.CrossRefGoogle Scholar
Goldstein, M. E. & Hultgren, L. S. 1988 Nonlinear spatial evolution of an externally excited instability wave in a free shear layer. J. Fluid Mech. 197, 295–330.CrossRefGoogle Scholar
Goldstein, M. E. & Leib, S. J. 1988 Nonlinear roll-up of externally excited free shear layers. J. Fluid Mech. 191, 481–515.CrossRefGoogle Scholar
Goldstein, M. E. & Leib, S. J. 1989 Nonlinear evolution of oblique waves on compressible shear layers. J. Fluid Mech. 207, 73–96.CrossRefGoogle Scholar
Goldstein, M. E. & Wundrow, D. W. 1990 Spatial evolution of nonlinear acoustic mode instabilities on hypersonic boundary layers. J. Fluid Mech. 219, 585–607.CrossRefGoogle Scholar
Grosch, C. E. & Jackson, T. L. 1991 Inviscid spatial stability of a three-dimensional compressible mixing layer. J. Fluid Mech. 231, 35–50.CrossRefGoogle Scholar
Gutmark, E. J., Schadow, K. C. & Yu, K. H. 1995 Mixing enhancement in supersonic free shear flows. Annu. Rev. Fluid Mech. 27, 375–417.CrossRefGoogle Scholar
Ho, C. & Huerre, P. 1984 Perturbed free shear layers. Annu. Rev. Fluid Mech. 16, 365–422.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Hultgren, L. S. 1992 Nonlinear spatial equilibration of an externally excited instability wave in a free shear layer. J. Fluid Mech. 236, 635–664.CrossRefGoogle Scholar
Jackson, T. L. & Grosch, C. E. 1989 Inviscid spatial stability of a compressible mixing layer. J. Fluid Mech. 208, 609–637.CrossRefGoogle Scholar
Jackson, T. L. & Grosch, C. E. 1991 Inviscid spatial stability of a compressible mixing layer. Part 3. Effect of thermodynamics. J. Fluid Mech. 224, 159–175.CrossRefGoogle Scholar
Lees, L. & Lin, C. C. 1946 Investigation of the stability of the laminar boundary layer in a compressible fluid. NACA TN-1115.Google Scholar
Leib, S. J. 1991 Nonlinear evolution of subsonic and supersonic disturbances on a compressible free shear layer. J. Fluid Mech. 224, 551–578.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Liu, J. T. C. 1989 Coherent structures in transitional and turbulent free shear flows. Annu. Rev. Fluid Mech. 21, 285–315.CrossRefGoogle Scholar
Lock, R. C. 1951 The velocity distribution in the laminar boundary layer between parallel streams. Q. J. Mech. 4, 42–63.CrossRefGoogle Scholar
Mack, L. M. 1984 Boundary layer linear instability theory. In Special Course on Stability and Transition of Laminar Flow, AGARD Report R–709, pp. 3–1–3–81.Google Scholar
Ngan, K. & Shepherd, T. 1997 Chaotic mixing and transport in Rossby-wave critical layers. J. Fluid Mech. 334, 315–351.CrossRefGoogle Scholar
Olsen, M. G. & Dutton, J. C. 2003 Planar velocity measurements in a weakly compressible mixing layer. J. Fluid Mech. 486, 51–77.CrossRefGoogle Scholar
Ragab, S. A. & Sheen, S. 1992 The nonlinear development of supersonic instability waves in a mixing layer. Phys. Fluids A 4, 553–566.CrossRefGoogle Scholar
Rom-kedar, V., Leonard, A. & Wiggins, S. 1990 An analytic study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394.CrossRefGoogle Scholar
Sandham, N. & Reynolds, W. C. 1991 Three-dimensional simulations of a compressible mixing layer. J. Fluid Mech. 224, 133–158.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory. Springer.CrossRefGoogle Scholar
Sparks, C. A. 2006 Nonlinear instability of compressible mixing layers. PhD thesis, University of London, UK.Google Scholar
Stewartson, K. 1964 The Theory of Laminar Boundary Layers in Compressible Fluids. Oxford University Press.CrossRefGoogle Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. J. Fluid Mech. 9, 353–370.CrossRefGoogle Scholar
Urban, W. D. & Mungal, M. G. 2001 Planar velocity measurements in compressible mixing layers. J. Fluid Mech. 431, 189–222.CrossRefGoogle Scholar
Winant, C. D. & Browand, F. K. 1974 Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number. J. Fluid Mech. 63, 237–255.CrossRefGoogle Scholar