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Similarity transformations for the two-dimensional, unsteady, stream-function equation

Cantwell, Brian J. (1978) Similarity transformations for the two-dimensional, unsteady, stream-function equation. Journal of Fluid Mechanics, 85 (2). pp. 257-271. ISSN 0022-1120. doi:10.1017/S0022112078000634.

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The methods described by Bluman & Cole (1974) are used to derive the infinitesimals of the general invariance group of the unsteady, two-dimensional, stream-function equation for the case where the kinematic viscosity v is equal to a constant and the case where v = 0. The infinitesimals in each case involve ten independent parameters, seven of which appear explicitly and three of which are contained implicitly in three arbitrary functions of time. The various finite groups and similarity transformations which may be derived from the infinitesimals are discussed through examples. Two of the arbitrary functions of time are non-trivial and represent invariance of the stream-function equation under a transformation to a co-ordinate system which moves in a non-uniform irrotational fashion. A general similarity form is derived for which the equations dx/dt = u(x, y, t) and dy/dt = v(x, y, t) for the particle paths may be reduced to an autonomous system. This form is general enough to suggest the hypothesis that, under certain restrictions, the entrainment processes of unsteady flows dominated by two-dimensional large-scale motions may be displayed diagrammatically on a phase-plane plot of particle trajectories.

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Additional Information:© 1978 Cambridge University Press. Received 2 March 1977 and in revised form 12 July 1977. Published Online April 12 2006. I should like to express thanks to Prof. A. E. Perry, who first convinced me of the value of phase-plane diagrams for interpreting fluid flows. This work was supported, in part, by the National Science Foundation.
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Official Citation:Similarity transformations for the two-dimensional, unsteady, stream-function equation. Journal of Fluid Mechanics, 85 , pp 257-271 doi:10.1017/S0022112078000634
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:32773
Deposited By: Ruth Sustaita
Deposited On:30 Jul 2012 15:15
Last Modified:09 Nov 2021 21:29

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