Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-18T18:15:00.710Z Has data issue: false hasContentIssue false
  • English
  • Français

On finite amplitude oscillations in laminar mixing layers

Published online by Cambridge University Press:  28 March 2006

J. T. Stuart
J. T. Stuart
Affiliation:
Department of Mathematics, Imperial College, London
Get access Rights & Permissions [Opens in a new window]

Abstract

In the first part of the paper, a mixing layer of tanh y form is considered, and twodimensional solutions of the non-linear inviscid equations are found representing periodic perturbations from the neutral wave of linearized stability theory. To second order in amplitude the solutions are equivalent to the equilibrium state calculated by Schade (1964), who studied the development of perturbations in time and found an evolution towards that equilibrium state. The present calculation, however, is taken to fourth-order in amplitude. It is noted that the solutions presented in this paper are regular, even though viscosity is ignored; and the relationships to the singular (if inviscid) time-dependent solutions of Schade are explained. Such regular, inviscid solutions have been found only for odd velocity profiles, such as tanh y.

Although the details of the velocity distributions are not of the form observed experimentally, it is shown that the amplitude ratios of fundamental and first harmonic, for a given absolute amplitude, are comparable with those observed.

In part 2 some exact non-linear solutions are presented of the inviscid, incompressible equations of fluid flow in two or three spatial dimensions. They illustrate the flows of part 1, since they are periodic in one co-ordinate (x), have a shear in another (y) and are independent of the third. Included, as two-dimensional cases, are (i) the tanh y velocity distribution for a flow wholly in the x-direction, (ii) the well-known solution for the flow due to a set of point vortices equi-spaced on the axis, and (iii) an example of linearized hydrodynamic (Orr-Sommerfeld) stability theory. The flows may involve concentrations of vorticity. In three-dimensional cases the z component of velocity is even in y, whereas the x component is odd. Consequently, the class of flows represents, in general, small or large periodic perturbations from a skewed shear layer. Time-dependent solutions, representing waves travelling in the x direction may be obtained by translation of axes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 29 , Issue 3 , 6 September 1967 , pp. 417 - 440
Copyright
© 1967 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

Bateman, H. 1952 Partial Differential Equations of Mathematical Physics, pp. 166–169. Cambridge University Press.
Benney, D. J. 1961 A non-linear theory for oscillations in a parallel flow J. Fluid Mech. 10, 209–236.Google Scholar
Betchov, R. & Szewczyk, A. 1963 Stability of a shear layer between parallel streams Phys. Fluids 6, 1391–1396.Google Scholar
Bieberbach, L. 1916 &\Delta U = eu und die automorphen Funktionen. Math. Ann. 77, 173–212.Google Scholar
Bradshaw, P. 1966 The effect of initial conditions on the development of a free shear layer J. Fluid Mech. 26, 225–236.Google Scholar
Brodetsky, S. 1924 Vortex Motion. Proc. 1st Int. Cong. Appl. Math., Delft, pp. 374–379.
Browand, F. K. 1966 An experimental investigation of the instability of an incompressible, separated shear layer J. Fluid Mech. 26, 281–307.Google Scholar
Curle, N. 1956 Hydrodynamic stability of unlimited antisymmetrical velocity profiles. British A.R.C. Rept. no. 18564.Google Scholar
Davies, H. T. 1962 Introduction to Nonlinear Differential and Integral Equations, pp. 20–21, 377. Dover.
Drazin, P. G. & Howard, L. N. 1962 The instability to long waves of unbounded parallel inviscid flow J. Fluid Mech. 14, 257–283.Google Scholar
Freymuth, P. 1965 See appendix to Michalke. (1965). Also 1966 J. Fluid Mech. 25, 683–704.
Garcia, R. V. 1956 Barotropic waves in straight parallel flow with curved velocity profiles Tellus 8, 82–93.Google Scholar
Jones, C. W. & Watson, E. J. 1963 Two-dimensional boundary layers. In Laminar Boundary Layers (ed. Rosenhead, L.), pp. 198–257. Oxford: Clarendon Press.
Kelly, R. E. 1967 On the stability of an inviscid shear layer which is periodic in space and time J. Fluid Mech. 27, 657–689.Google Scholar
Küchemann, D., Crabtree, L. F. & Sowerby, L. 1963 Three-dimensional boundary layers, in Laminar Boundary Layers (ed. Rosenhead, L.). Oxford: Clarendon Press.
Lamb, H. 1932 Hydrodynamics, Section 156, pp. 224–5, 6th edition. Cambridge University Press (also Dover Reprint).
Lichtenstein, L. 1913 Intégration de l’équation Δ2u = keu sur une surface fermée. Comptes Rendus 157, 1508–1511.Google Scholar
Lichtenstein, L. 1915 Intégration der Differentialgleichungen Δ2u = keu aub geschlossenen Flächen: Methode der unendlichvielen Variabeln. Acta Math. 140, 1–34.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Liouville, J. 1853 Sur l'eacute;quation aux différences partielles ∂2 log λ/∂u ∂v±λ/2a2 = 0. J. Math. 18 (1), 71–72.Google Scholar
Michalke, A. 1964 On the inviscid instability of the hyperbolic-tangent profile J. Fluid Mech. 19, 543–556.Google Scholar
Michalke, A. 1965 On spatially-growing disturbances in an inviscid shear layer J. Fluid Mech. 23, 521–544.Google Scholar
Picard, E. 1893 De l'eacute;quation Δ u = keu sur une surface de Riemann fermée. J. Math. 9 (4), 273–291.Google Scholar
Picard, E. 1998 Sur l'eacute;quation Δ u = eu. J. Math. 4 (5), 313–316.Google Scholar
Picard, E. 1905 De l'ntegration de l'eacute;quation Δu = eu sur une surface de Riemann fermée. J. Math. 130, 243–258.Google Scholar
Poincaré, H. 1898 Les functions fuchsiennes et l'quation Δu = eu. J. Math. 4 (5), 137–230. Also Oeuvres II, 1916, 512–591.Google Scholar
Sato, H. 1956 Experimental investigation on the transition of a laminar separated layer. J. Phys. Soc. Japan 11, 702–709, 1128.Google Scholar
Schade, H. 1964 Contribution to the nonlinear stability theory of inviscid shear layers Phys. Fluids, 7, 623–628.Google Scholar
Schmid-Burgk, J. 1965 Zweidimensionale selbstkonsistente Lösungen der stationären Wlassowgleichung für Zweikomponentenplasmen. Thesis, Munich University.
Stuart, J. T. 1960 Nonlinear effects in hydrodynamic stability. Proc. Xth Int. Cong. Appl. Mech., Stresa (Elsevier, Amsterdam, 1962).Google Scholar
Stuart, J. T. 1965 The production of intense shear layers by vortex stretching and convection. N.P.L. Aero Rept. 1147, Agard Rept. 514.Google Scholar
Walker, G. W. 1915 Some problems illustrating the forms of nebulae. Proc. Roy. Soc A 91, 410–420.Google Scholar