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The Most Unstable Profiles of a Plane-Parallel Flow in a Channel

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Abstract

It is well known (after Rayleigh) that a plan-parallel flow in a channel can be unstable only if the basic velocity profile U(z) possesses inflection points. The profile determines (via the Rayleigh equation) the maximal increment ∣αc i∣ of small perturbations and 'eigenvalues' c (see Equation (1)). The increment and the imaginary parts c i of the eigenvalues c provide a quantitative characterization of the basic stability properties of the flow. Here we find some best possible bounds for these values. The bounds are determined by the following parameters: wave number α; enstrophy \(1/2\int_0^L {[U'(z)]^2 } {\text{d}}z\) of the basic flow; width of the channel L. A similar approach can be applied to models of atmosphere, ocean, plasma etc.

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  1. Hydrometeorological Center of Russia, Bolshoi Predtechenskii, per., 9-13, 123376, Russia, Moscow

    Vladimir A. Gordin

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  1. Vladimir A. Gordin
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Gordin, V.A. The Most Unstable Profiles of a Plane-Parallel Flow in a Channel. Meccanica 35, 39–53 (2000). https://doi.org/10.1023/A:1004842312649

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