Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T06:54:01.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Gravity waves over a non-uniform flow

Published online by Cambridge University Press:  29 March 2006

H. G. M. Velthuizen  and
L. Van Wijngaarden
H. G. M. Velthuizen
Affiliation:
Twente Institute of Technology, Enschede, The Netherlands
L. Van Wijngaarden
Affiliation:
Twente Institute of Technology, Enschede, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with the propagation of small amplitude gravity waves over a flow with non-uniform velocity distribution. For such a flow Burns derived a relation for the velocity of propagation in terms of the velocity distribution of the mean flow. This result is derived here in another way and some of its implications are discussed. It is shown that one of these is hardly acceptable physically. Burns's result holds only when a real value of the propagation velocity is assumed; the mentioned difficulties vanish if complex values are allowed for, implying damping or growth of the waves. Viscous effects which are the cause of damping or growth are important in the wall layer near the bottom and also at the critical depth, which is present when the wave speed is between zero and the fluid velocity at the free surface.

In § 2 the basic equations for the present problem are given. In § 3 exchange of momentum and energy between wave and primary flow is discussed. This is analogous to what happens at the critical height in a wind flow over wind-driven gravity waves. In § 4 the viscous effects at the bottom are included in the analysis and the complex equation for the propagation velocity is derived. Finally in § 5 illustrations of the theory are given for long waves over running flow and for the flow along a ship advancing in a wavy sea. In these examples a negative curvature of the mean velocity profile is shown to have a stabilizing effect.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 39 , Issue 4 , 15 December 1969 , pp. 817 - 829
Copyright
© 1969 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

Benjamin, T. B. 1959 J. Fluid Mech. 6, 161.
Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. New York: Academic.
Burns, J. C. 1953 Proc. Camb. Phil. Soc. 49, 695.
Joosen, W. P. A. 1966 Proc. 6th Symposium on Naval Hydrodynamics. Ed. R. D. Cooper and S. W. Doroff, Washington D.C.
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Miles, J. W. 1957 J. Fluid Mech. 3, 185.
Miles, J. W. 1960 J. Fluid Mech. 8, 593.