Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Boundary conditions and stability of inviscid plane-parallel flows


Author: Joel C. W. Rogers
Journal: Quart. Appl. Math. 31 (1973), 199-216
MSC: Primary 76.65
DOI: https://doi.org/10.1090/qam/418649
MathSciNet review: 418649
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Abstract: The problem of describing the manner in which disturbances in an incompressible flow propagate downstream, when the infinite speed of signal transmission would make it theoretically possible for the disturbances to travel upstream also, and indeed would make upstream values dependent on boundary conditions imposed at arbitrarily great distances downstream, arises in numerous cases of physical interest. One such case is the growth of perturbations on plane-parallel flows in long channels, where the nature of the perturbation at any particular location is ``quasi-steady'' in time, although the perturbation grows with distance downstream. Another such case arises with the inevitable numerical disturbances in numerical ``experiments'' with computers, where an infelicitous choice of downstream boundary conditions may lead to results wholly at variance with observations. The purpose of this paper is to indicate how the choice of boundary conditions can drastically influence the phenomena observed, and to select those boundary conditions most consistent with reality. Our ideas are applied in some detail to the description of the linear theory of growth of perturbations on a two-dimensional inviscid plane-parallel basic flow. In particular, we show that the stability or instability of the flow depends on the nature of the boundary conditions imposed, and we find boundary conditions whose application leads to results in conformity with actual experimental situations.


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DOI: https://doi.org/10.1090/qam/418649
Article copyright: © Copyright 1973 American Mathematical Society


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