dorsal/arxiv
View SchemaA necessary and sufficient instability condition for inviscid shear flow
| Authors | N. J. Balmforth, P. J. Morrison |
|---|---|
| Categories | |
| ArXiv ID | physics/9809024 |
| URL | https://arxiv.org/abs/physics/9809024 |
Abstract
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.
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"abstract": "The linear stability of inviscid, incompressible, two-dimensional, plane\nparallel, shear flow was considered over a century ago by Rayleigh, Kelvin, and\nothers. A principal result on the subject is Rayleigh\u0027s celebrated inflection\npoint theorem {R80}, which states that for an equilibrium flow to be unstable,\nthe equilibrium velocity profile must contain an inflection point. That is, if\nthe velocity profile is given by $U(y)$, where $y$ is the cross-stream\ncoordinate, then there must be a point, $y=y_I$, for which $U\u0027\u0027(y_I)=0$. Much\nlater, in 1950, Fj{\\o}rtoft {F50} generalized the theorem by showing that,\nmoreover, if there is one inflection point, then $U\u0027\u0027\u0027(y_I)/U\u0027(y_I)\u003c0$ is\nrequired for instability (see {Bar} for further extensions). Both Rayleigh\u0027s\nTheorem and Fj{\\o}rtoft\u0027s subsequent generalization are necessary conditions\nfor instability, but they are not sufficient. That is, even though an\nequilibrium profile may contain a vorticity minimum, it is not necessarily\nunstable. The point of this paper is to derive, for a large class of\nequilibrium velocity profiles, a condition that is necessary and sufficient for\ninstability.",
"arxiv_id": "physics/9809024",
"authors": [
"N. J. Balmforth",
"P. J. Morrison"
],
"categories": [
"physics.flu-dyn"
],
"title": "A necessary and sufficient instability condition for inviscid shear flow",
"url": "https://arxiv.org/abs/physics/9809024"
},
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