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Numerical solutions of the steady-state vorticity equation in rectangular basins

Published online by Cambridge University Press:  28 March 2006

John A. T. Bye
John A. T. Bye
Affiliation:
University of California, San Diego, Scripps Institution of Oceanography La Jolla, California
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Abstract

The Navier-Stokes vorticity equation is solved numerically for the circulation induced in a vertical plane, by a constant stress acting on a liquid, enclosed in a basin of uniform depth and vertical sides.

Solutions of the linearized vorticity equation are obtained for all Reynolds numbers (τsD2/4ρν2 where νs is the surface stress, ρ is the density, ν is the kinematic viscosity, and D is the depth of the liquid) and solutions of the complete vorticity equation for Reynolds numbers 0–400.

The notable feature of the solutions is the totally different end circulations. At the upwind end the flow becomes very slack, and the vorticity equation has a boundary-layer limit, while at the downwind end a damped wave occurs and the equation has an inviscid limit.

At Reynolds numbers between 400 and 600, the streamlines at the downwind end lead to a condition of hydrodynamic instability, in approximate agreement with some experimental observations by G. H. Keulegan.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 26 , Issue 3 , November 1966 , pp. 577 - 598
Copyright
© 1966 Cambridge University Press

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References

Bye, J. A. T. 1962 Ph.D. Thesis, London University.
Bye, J. A. T. 1965 Computer J. 8, 53–56.
Bye, J. A. T. 1966 Computer J. (submitted for publication).
Hidaka, K. 1939 Mem. Imp. Mar. Obs. 7, no. 1.
Keulegan, G. H. 1951 J. Res. Nat. Bur. Standards 46, 358–381.
Schlichting, H. 1962 Boundary Layer Theory. New York: McGraw-Hill.