Hostname: page-component-5c6d5d7d68-wtssw Total loading time: 0 Render date: 2024-08-14T11:38:51.387Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-linear wave-number interaction in near-critical two-dimensional flows

Published online by Cambridge University Press:  29 March 2006

R. C. Diprima ,
W. Eckhaus  and
L. A. Segel
R. C. Diprima
Affiliation:
Department of Mathematics, Rensselaer Polytechnic Institute, Troy, N.Y. 12181
W. Eckhaus
Affiliation:
Technische Hogeschool, Delft, The Netherlands
L. A. Segel
Affiliation:
Department of Mathematics, Rensselaer Polytechnic Institute, Troy, N.Y. 12181
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Bénard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 49 , Issue 4 , 29 October 1971 , pp. 705 - 744
Copyright
© 1971 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

Arscott, F. M. 1964 Periodic Differential Equations. New York: Macmillan.
Chen, M. M. & Whitehead, J. A. 1968 Evaluation of two-dimensional Rayleigh convection cells of arbitrary wave-numbers. J. Fluid Mech., 31, 1–16.Google Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech., 21, 385–425.Google Scholar
Davey, A. 1962 The growth of Taylor vortices in flow betwen rotating cylinders. J. Fluid Mech., 14, 336–368.Google Scholar
Diprima, R. C. 1967 Vector eigenfunction expansions for the growth of Taylor vortices in the flow between rotating cylinders. Nonlinear Partial Differential Equations (ed. W. Ames). New York: Academic.
Diprima, R. C. & Grannick, R. N. 1970 A nonlinear investigation of the stability of flow between counter-rotating cylinders. A paper in Liepholz (1970), see below.
Diprima, R. C. & Habetler, G. J. 1969 A completeness theorom for nonself-adjoint eigen value problems in hydrodynamic stability. Arch. Rat. Mech. Anal., 34, 218–227.Google Scholar
Eckilaus, W. 1965 Studies in Non-linear Stability Theory. New York: Springer.
Eckhaus, W. 1970 On the stability of periodic solutions in fluid mechanics. Paper in Liepholz (1970), see below.
Görtler, H. & Velte, W. 1967 Recent mathematical treatments of laminar flow and transition problems. Phys. Fluid Suppl., 10, S3-10.Google Scholar
Keller, J. B. & Antman, S. (eds.)1968 Bifurcation Theory and Nonlinear Eigenvalue Problems. Courant Institute of Mathematical Sciences, New York University, New York, N.Y.
Kogelman, S. & DiPrima, R. C. 1970 Stability of spatially periodic supercritical flows in hydrodynamics. Phys. Fluids, 13, 1–11.Google Scholar
Krueger, E. R., Gross, A. & DiPrima, R. C. 1966 On the relative importance of Taylor-vortex and non-axisymmetric modes in flow between rotating cylinders. J. Fluid Mech., 24, 521–538.Google Scholar
Liepholz, H.(ed.)1971 Instability of Continuous Systems, I.U.T.A.M. Symposium, Herrenalb, Germany, 1969. Berlin: Springer.
Matkowsky, B. J. 1970 Nonlinear dynamic stability: a formal theory. SIAM J. Appl. Mech., 18, 872–883.Google Scholar
Newell, A. C., Lance, C. G. & Aucoin, P. J. 1970 Random convection. J. Fluid Mech., 40, 513–542.Google Scholar
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech., 38, 279–304.Google Scholar
Nguyen, H. P. F. & Davey, A. 1970 The expansion parameter for non-linear stability theory. Unpublished.
Odeh, F. 1968 A bifurcation problem in superconductivity. Chapter V of Keller & Antman (1968), see above.
Pekeris, C. L. & Shkoller, B. 1967 Stability of plane Poiseuille flow to periodic disturbances of finite amplitude in the vicinity of the neutral curve. J. Fluid Mech., 29, 31–38.Google Scholar
Ponomarenpo, Iu. B., 1968 Occurrence of space-periodic motions in hydrodynamics. Prikl. Mat. i Mekh., 32, 46–58.Google Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite amplitude instability of parallel shear flow. J. Fluid Mech., 27, 465–492.Google Scholar
Schade, H. 1964 Contribution to the nonlinear stability theory of inviscid shear layers. Phys. Fluids, 7, 623–628.Google Scholar
Segel, L. A. 1962 The non-linear interaction of two disturbances in the thermal convection problem. J. Fluid Mech., 14, 97–114.Google Scholar
Segel, L. A. 1969 Distance side-walls cause slow amplitude modulation of cellular convection. J. Fluid Mech., 38, 203–224.Google Scholar
Segel, L. A. 1970 On the effect of sidewalls in cellular convection. Paper in Liepholz (1970), see above.
Snyder, H. A. 1969 Wave-number selection a t finite amplitude in rotating Couette flow. J. Fluid Mech., 35, 273–298.Google Scholar
Stewartson, K. & Stuart, J. T. 1971 A non-linear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech., 48, 529–46.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanios of wave disturbances in stable and unstable parallel flows. Part I. J. Fluid Mech., 9, 353–370.Google Scholar
Watson, J. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 11. J. Fluid Mech., 9, 371–89.Google Scholar
Wollpind, D. J. & Segel, L. A. 1970 A non-linear stability analysis of the freezing of a dilute binary alloy. Proc. Roy. Soc. A 268, 351–380.Google Scholar