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Early-period dynamics of an incompressible mixing layer

Published online by Cambridge University Press:  26 April 2006

Y. Bun  and
W. O. Criminale
Y. Bun
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
W. O. Criminale
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
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Abstract

The evolution of three-dimensional disturbances in an incompressible mixing layer in an inviscid fluid is investigated as an initial-value problem. A Green's function approach is used to obtain a general space–time solution to the problem using a piecewise linear model for the basic flow, thereby making it possible to determine complete and closed-form analytical expressions for the variables with arbitrary input. Structure, kinetic energy, vorticity, and the evolution of material particles can be ascertained in detail. Moreover, these solutions represent the full three-dimensional disturbances that can grow exponentially or algebraically in time. For large time, the behaviour of these disturbances is dominated by the exponentially increasing discrete modes. For the early time, the behaviour is controlled by the algebraic variation due to the continuous spectrum. Contrary to Squire's theorem for normal mode analysis, the early-time behaviour indicates growth at comparable rates for all values of the wavenumbers and the initial growth of these disturbances is shown to rapidly increase. In particular, the disturbance kinetic energy can rise to a level approximately ten times its initial value before the exponentially growing normal mode prevails. As a result, the transient behaviour can trigger the roll-up of the mixing layer and its development into the well-known pattern that has been observed experimentally.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 273 , 25 August 1994 , pp. 31 - 82
Copyright
© 1994 Cambridge University Press

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References

Ashurst, W. T. & Meiburg, E. 1988 Three-dimensional shear layers via vortex dynamics. J. Fluid Mech. 189, 87–116.Google Scholar
Balsa, T. F. 1987 On the spatial instability of piecewise linear free shear layers. J. Fluid Mech. 174, 553–563.Google Scholar
Balsa, T. F. 1988 On the receptivity of free shear layers to two-dimensional external excitation. J. Fluid Mech. 187, 155–177.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Benney, D. J. 1984 The evolution of disturbances in shear flows at high Reynolds numbers. Stud. Appl. Maths 70, 1–19.Google Scholar
Benney, D. J. & Gustavsson, C. H. 1981 A new mechanism for linear and nonlinear hydrodynamic stability. Stud. Appl. Maths 64, 185–209.Google Scholar
Bernal, L. P. & Roshko, A. 1986 Streamwise vortex structure in plane mixing layers. J. Fluid Mech. 170, 499–525.Google Scholar
Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. Academic.
Breidenthal, R. 1981 Structure in turbulent mixing layers and wakes using a chemical reaction. J. Fluid Mech. 109, 1–24.Google Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structures in turbulent mixing layers. J. Fluid Mech. 64, 775–816.Google Scholar
Case, K. M. 1960 Stability of inviscid plane Couette flow. Phys. Fluids 3, 143–148.Google Scholar
Case, K. M. 1961 Hydrodynamic stability and the inviscid limit. J. Fluid Mech. 10, 420–430.Google Scholar
Criminale, W. O. & Drazin, P. G. 1990 The evolution of linearized perturbations of parallel flows. Stud. Appl. Maths 83, 123–157.Google Scholar
Criminale, W. O., Long, B. & Zhu, M. 1991 General three-dimensional disturbances to inviscid Couette flow. Stud. Appl. Maths 85, 249–267.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.Google Scholar
Esch, R. E. 1957 The instability of a shear layer between two parallel streams. J. Fluid Mech. 3, 289–303.Google Scholar
Farrell, B. F. 1982 The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci. 39, 1663–1686.Google Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31, 2093–2102.Google Scholar
Ho, C. & Huang, L. 1982 Subharmonics and vortex merging in mixing layers. J. Fluid Mech. 119, 443–473.Google Scholar
Ho, C. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 119, 365–424.Google Scholar
Hussain, A. K. M. F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303–356.Google Scholar
Kelvin, W. 1871 Hydrokinetic solutions and observations. Phil. Mag. 42, 362–377.Google Scholar
Kelvin, W. 1887 Rectilinear motion of viscous fluid between two parallel plates. Phil. Mag. 24 (5), 188–196.Google Scholar
Konrad, J. H. 1976 An experimental investigation of mixing in two-dimensional turbulent shear flows with application to diffusion limited chemical reaction. PhD thesis, California Institute of Technology.
Lasheras, J. & Choi, H. 1988 Three-dimensional instabilities of a plane free shear layer. J. Fluid Mech. 189, 53–86.Google Scholar
Lin, C. C. 1955 The theory of Hydrodynamic Stability. Cambridge University Press.
Lin, C. C. 1961 Some mathematical problems in the theory of the stability of parallel flows. J. Fluid Mech. 10, 430–438.Google Scholar
Lin, S. J. & Corcos, G. M. 1984 The mixing layer: deterministic model of a turbulent flow. Part 3. J. Fluid Mech. 141, 139–178.Google Scholar
Lock, R. C. 1951 The velocity distribution in the laminar boundary layer between parallel streams. Q. J. Mech. Appl. Maths 4, 41–63.Google Scholar
Metcalfe, R. W., Orszag, S. A., Brachet, M. E., Menon, S. & Riley, J. J. 1987 Secondary instability of a temporally growing mixing layer. J. Fluid Mech. 184, 207–243.Google Scholar
Michalke, A. 1964 On the inviscid instability of the hyperbolic tangent velocity profile. J. Fluid Mech. 19, 543–556.Google Scholar
Michalke, A. 1965 Vortex formation in a free boundary layer according to stability theory. Part 2. J. Fluid Mech. 22, 371–383.Google Scholar
Orr, W. M'F. 1907a The stability or instability of the steady motions of a perfect liquid and a viscous liquid. Part I. Proc. R. Irish Acad. A 27, 9–68.Google Scholar
Orr, W. M'F. 1907b The stability or instability of the steady motions of a perfect liquid and a viscous liquid. Part II. Proc. R. Irish Acad. A 27, 69–138.Google Scholar
Oster, C. & Wygnanski, I. 1982 The forced mixing layer between parallel streams. J. Fluid Mech. 123, 91–130.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, 59–83.Google Scholar
Rayleigh, Lord 1894 The Theory of Sound, 2nd edn. Macmillan.
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1991 Pseudospectra of the Orr—Sommerfeld operator. SIAM J. Appl. Maths 53, 15–45.Google Scholar
Rogers, M. M. & Moser, R. D. 1989 The development of three-dimensional temporally evolving mixing layer. Seventh Symp. on Turbulent Shear Flows, Stanford, CA.
Tung, K. K. 1983 Initial-value problems for Rossby waves in a shear flow with critical level. J. Fluid Mech. 133, 443–469.Google Scholar
van Dyke, M. 1982 An Albumn of Fluid Motions. The Parabolic Press.