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The stability of thermally stratified plane Poiseuille flow

Published online by Cambridge University Press:  28 March 2006

K. S. Gage  and
W. H. Reid
K. S. Gage
Affiliation:
The University of Chicago
W. H. Reid
Affiliation:
The University of Chicago
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Abstract

In studying the stability of a thermally stratified fluid in the presence of a viscous shear flow, we have a situation in which there is an important interaction between the mechanism of instability due to the stratification and the Tollmien-Schlichting mechanism due to the shear. A complete analysis has been carried out for the Bénard problem in the presence of a plane Poiseuille flow and it is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, a theorem of Squire's type does not follow unless the Richardson number exceeds a certain small negative value. This conclusion follows from the fact that, when the stratification is unstable and the Prandtl number is unity, the equivalent two-dimensional problem becomes identical mathematically to the stability problem for spiral flow between rotating cylinders and, from the known results for the spiral flow problem, Squire's transformation can then be used to obtain the complete three-dimensional stability boundary. For the case of stable stratification, however, Squire's theorem is valid and the instability is of the usual Tollmien—Schlichting type. Additional calculations have been made for this case which show that the flow is completely stabilized when the Richardson number exceeds a certain positive value.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 33 , Issue 1 , 12 July 1968 , pp. 21 - 32
Copyright
© 1968 Cambridge University Press

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References

Deardorff, J. W. 1965 Phys. Fluids, 8, 1027.
Drazin, P. G. & Howard, L. N. 1966 In Advances in Applied Mechanics, vol. 9, pp. 1–89 (ed. G. Kuerti). New York: Academic Press.
Gallagher, A. P. & Mercer, A. MCD. 1965 Proc. Roy. Soc. A, 286, 117.
Hughes, T. H. & Reid, W. H. 1968 Phil. Trans. A, 263, 57.
Isaacson, E. & Keller, H. B. 1961 Analysis of Numerical Methods. New York: John Wiley and Sons.
Koppel, D. 1964 J. Mathematical Phys. 5, 963.
Kuo, H.-L. 1963 Phys. Fluids, 6, 195.
Mack, L. M. 1965 In Methods in Computational Physics, vol. 4, pp. 247–300 (ed. B. Alder, S. Fernbach & M. Rotenberg). New York: Academic Press.
Miles, J. W. 1961 J. Fluid Mech. 10, 496.
Priestley, C. H. B. 1959 Turbulent Transfer in the Lower Atmosphere. University of Chicago Press.
Schlichting, H. 1935 Z. angew. Math. Mech. 15, 313. (Also as NACA Tech. Memo. no. 1262, 1950.)
Squire, H. B. 1933 Proc. Roy. Soc. A, 142, 621.
Townsend, A. A. 1962 Quart. J. Roy. Met. Soc. 88, 51.