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On a non-linear theory of thin jets. Part 1

Published online by Cambridge University Press:  28 March 2006

Robert C. Ackerberg
Robert C. Ackerberg
Affiliation:
Polytechnic Institute of Brooklyn, Graduate Center, Farmingdale, New York
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Abstract

The injection of a two-dimensional jet into a uniform stream is considered, the fluids being assumed inviscid and incompressible. When the total head of the jet is much larger than that of the uniform flow, the motion is characterized by two disparate length scales, and uniformly valid asymptotic solutions can be found by the method of matched expansions. Inner and outer expansions are developed for the jet and the external flow. The first-order outer solution in the jet is the usual thin jet approximation which fails in the neighbourhood of the jet exit except for 90° injection, when it is uniformly valid. The basic non-linearity introduced by the pressure condition along the vortex sheet separating the jet from the external flow appears as a non-linear boundary condition for the first-order outer solution in the external flow. A novel feature of the analysis is the necessity of imposing a logarithmic singularity as an ‘inner’ boundary condition for the outer solution in the external flow. The first-order fluid speed and streamline deflection angle are shown to be given correctly to O(1) uniformly in the external flow (for all injection angles) by the first-order outer solution.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 31 , Issue 3 , 26 February 1968 , pp. 583 - 601
Copyright
© 1968 Cambridge University Press

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References

Ackerberg, R. C. & Pal, A. 1968 On the interaction of a two-dimensional jet with a parallel flow. J. Math. Phys. (To appear in volume 47.)Google Scholar
Clarke, N. S. 1965 On two-dimensional inviscid flow in a waterfall J. Fluid Mech. 22, 359–69.Google Scholar
Pal, A. 1965 Solution of a non-linear boundary value problem in fluid mechanics using a variational method. Polytechnic Institute of Brooklyn, PIBAL Rept. no. 890.Google Scholar
Preston, J. H. 1954 Note on the circulation in circuits which cut the streamlines in the wake of an aerofoil at right-angles. R. & M. no. 2957.Google Scholar
Spence, D. A. 1956 The lift coefficient of a thin, jet-flapped wing. Proc. Roy. Soc A 238, 46–68.Google Scholar
Stoker, J. J. 1957 Water Waves. New York: Interscience.
Taylor, G. I. 1954 The use of a vertical air jet as a windscreen. Jubile Scientifique de M. Dimitri P. Riabouchinsky, pp. 313–17. (Also in Collected Scientific Papers, III, p. 537.)Google Scholar
Van Dyke, M. 1965 Perturbation Methods in Fluid Mechanics. New York: Academic Press.