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Lee waves in a stratified flow. Part 4. Perturbation approximations

Published online by Cambridge University Press:  28 March 2006

John W. Miles  and
Herbert E. Huppert
John W. Miles
Affiliation:
Also Department of Aerospace and Mechanical Engineering Sciences.
Herbert E. Huppert
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, La Jolla Also Department of Aerospace and Mechanical Engineering Sciences.
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Abstract

A two-dimensional stratified flow over an obstacle in a half space is considered on the assumptions that the upstream dynamic pressure and density gradient are constant (Long's model). A general solution of the resulting boundary-value problem is established in terms of an assumed distribution of dipole sources. Asymptotic solutions for prescribed bodies are established for limiting values of the slenderness ratio ε (height/breadth) of the obstacle and the reduced frequency k (inverse Froude number based on the obstacle breadth) as follows: (i) ε → 0 with k fixed; (ii) k → 0 with ε fixed; (iii) k → ∞ with kε fixed. The approximation (i) is deveoped to both first (linearized theory) and second order in ε in terms of Fourier integrals. The approximation (ii), which constitutes a modification of Rayleigh-scattering theory, is obtained by the method of matched asymptotic expansions and depends essentially on the dipole form (which is proportional to the sum of the displaced and virtual masses) of the obstacle with respect to a uniform flow. A simple approximation to this dipole form is proposed and validated by a series of examples in an appendix. The approximation (iii) is obtained through the reduction of the original integral equation to a singular integral equation of Hilbert's type that is solved by the techniques of function theory. A composite approximation to the lee-wave field that is valid in each of the limits (i)-(iii) also is obtained. The approximation (iii) yields an estimate of the maximum value of kε for which completely stable lee-wave formation behind a slender obstacle is possible. The differential and total scattering cross-sections and the wave drag on the obstacle are related to the power spectrum of the dipole density. It is shown that the drag is invariant under a reversal of the flow in the limits (i) and (ii), but only for a symmetric obstacle in the limit (iii). The results are applied to a semi-ellipse, an asymmetric generalization thereof, the Witch of Agnesi (Queney's mountain), and a rectangle. The approximate results for the semi-ellipse are compared with the more accurate results obtain by Huppert & Miles (1969). It appears from this comparison that the approximate solutions should be adequate for any slender obstacle within the range of kε for which completely stable lee-wave formation is possible. The extension to obstacles in a channel of finite height is considered in an appendix.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 35 , Issue 3 , 20 February 1969 , pp. 497 - 525
Copyright
© 1969 Cambridge University Press

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