Physics – Fluid Dynamics
Scientific paper
1998-09-17
Physics
Fluid Dynamics
Latex, 28 pages, 9 figures. Accepted by Studies in Applied Mathematics
Scientific paper
The linear stability of inviscid, incompressible, two-dimensional, plane parallel, shear flow was considered over a century ago by Rayleigh, Kelvin, and others. A principal result on the subject is Rayleigh's celebrated inflection point theorem {R80}, which states that for an equilibrium flow to be unstable, the equilibrium velocity profile must contain an inflection point. That is, if the velocity profile is given by $U(y)$, where $y$ is the cross-stream coordinate, then there must be a point, $y=y_I$, for which $U''(y_I)=0$. Much later, in 1950, Fj{\o}rtoft {F50} generalized the theorem by showing that, moreover, if there is one inflection point, then $U'''(y_I)/U'(y_I)<0$ is required for instability (see {Bar} for further extensions). Both Rayleigh's Theorem and Fj{\o}rtoft's subsequent generalization are necessary conditions for instability, but they are not sufficient. That is, even though an equilibrium profile may contain a vorticity minimum, it is not necessarily unstable. The point of this paper is to derive, for a large class of equilibrium velocity profiles, a condition that is necessary and sufficient for instability.
BALMFORTH Neil J.
Morrison Philip J.
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