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.
Lord Rayleigh (18S0) Proc. London Math. Soc. XI, 57 (Scientific Payers, vol. 1, 474, Dover Publications, 1934)
Joel C. W. Rogers, thesis, Dept. Math., Massachusetts Institute of Technology, 1967
- J. Watson, On spatially-growing finite disturbances in plane Poiseuille flow, J. Fluid Mech. 14 (1962), 211–221. MR 145787, DOI https://doi.org/10.1017/S0022112062001172
F. K. Browand, Dept. Aero. Astro., Massachusetts Institute of Technology, Report ASRL TR 92-4, 1965
G. I. Taylor, Phil. Trans. Roy. Soc. London A 223, 289 (1923)
- M. Gaster, A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability, J. Fluid Mech. 14 (1962), 222–224. MR 149776, DOI https://doi.org/10.1017/S0022112062001184
- Louis N. Howard, Note on a paper of John W. Miles, J. Fluid Mech. 10 (1961), 509–512. MR 128233, DOI https://doi.org/10.1017/S0022112061000317
Lord Rayleigh (18S0) Proc. London Math. Soc. XI, 57 (Scientific Payers, vol. 1, 474, Dover Publications, 1934)
Joel C. W. Rogers, thesis, Dept. Math., Massachusetts Institute of Technology, 1967
J. Watson, J. Fluid Mech. 14, 211 (1962)
F. K. Browand, Dept. Aero. Astro., Massachusetts Institute of Technology, Report ASRL TR 92-4, 1965
G. I. Taylor, Phil. Trans. Roy. Soc. London A 223, 289 (1923)
M. Gaster, J. Fluid Mech. 14, 222 (1962)
Louis N. Howard, J. Fluid Mech. 10, 509 (1961)
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Article copyright:
© Copyright 1973
American Mathematical Society