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Numerical Solutions of the Viscous Flow Equations for a Class of Closed Flows

Published online by Cambridge University Press:  04 July 2016

Ronald D. Mills
Ronald D. Mills*
Affiliation:
Department of Mechanical Engineering, University of Strathclyde
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Extract

The Navier-Stokes equations are solved iteratively on a small digital computer for the class of flows generated within a rectangular “cavity” by a surface passing over its open end. Solutions are presented for depth/breadth ratios ƛ=0.5 (shallow), 10 (square), 20 (deep) and Reynolds number 100. Flow photographs ore obtained which largely confirm the predicted flows. The theoretical velocity profiles and pressure distributions through the centre of the vortex in the square cavity are calculated.

In an appendix an improved finite difference formula is given for the vorticity generated at a moving boundary.

Since Thorn began his pioneering work some thirty-five years ago the number of numerical solutions which have been obtained for the equations of incompressible viscous fluid motion remains small (see bibliographies of Thom and Apelt, Fromm). The known solutions are principally for steady streaming flows, although two methods have now been used with success for non-steady flows (Payne jets and Fromm flow past obstacles). By contrast this paper is concerned with the class of closed flows generated in a rectangular region of varying depth/breadth ratio by a surface passing over an open end. This problem has been considered for a number of reasons.

Type
Research Article
Information
The Aeronautical Journal , Volume 69 , Issue 658 , October 1965 , pp. 714 - 718
Copyright
Copyright © Royal Aeronautical Society 1965

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References

1.Thom, A. and Apelt, C. J.Field Computations in Engineering and Physics. D. Van Nostrand & Co, London, 1961.Google Scholar
2.Fromm, J. E. A Method of Computing Nonsteady, Incompressible, Viscous Fluid Flows. Los Alamos Scientific Laboratory LA 2910, September 1963.Google Scholar
3.Payne, R. B. A Numerical Method for Calculating the Starting and Perturbation of a Two-dimensional Jet at Low Reynolds Number. ARC R & M 3047, June 1956 (HMSO 1958).Google Scholar
4.Mills, R. D.On the Closed Motion of a Fluid in a Square Cavity. Journal of the Royal Aeronautical Society, Vol. 69, pp 116–120, February 1965.CrossRefGoogle Scholar
5.Thom, A. and Apelt, C. The Pressure in a Two-dimensional Static Hole at Low Reynolds Number. ARC R & M 3090, February 1957 (HMSO 1958).Google Scholar
6.Squire, H. B.Note on the Motion Inside a Region of Recirculation (Cavity Flow). Journal of the Royal Aeronautical Society, Vol. 60, pp 203–205, March 1956.Google Scholar
7.Batchelor, G. K.On Steady Laminar Flow with Closed Streamlines at Large Reynolds Numbers. Journal Fluid Mechanics, Vol. 1(2), pp 177–190, July 1956.Google Scholar
8.Roshko, A. Some Measurements of Flow in a Rectangular Cut-out. NACA TN 3488, 1955.Google Scholar
9.Allen, D. N. DE G. and Southwell, R. V. Relaxation Methods Applied to Determine the Motion, in Two Dimensions, of a Viscous Fluid Past a Fixed Cylinder. Quarterly Journal Mech and Applied Math, Vol. 8(2), pp 129–145, 1955.Google Scholar
10.Mabey, D. G. The Formation and Decay of Vortices, MSc Thesis, London University, 1955.Google Scholar
11.Thom, A. and Apelt, C. J. Note on the Convergence of Numerical Solutions of the Navier-Stokes Equations. ARC R & M 3061, June 1956 (HMSO 1958).Google Scholar
12.Wieghardt, K. Erhohung des Turbulenten Reibungswider-standes durch Oberflachenstorungen, Schiffstechnik, 2 Heft, Hansa Verlag, Hamburg, 1953.Google Scholar
13.Baturin, W. W.Luftungsanlagen für Industriebauten, 2nd Ed., Veb Verlag Technik, Berlin, p 187; 1959.Google Scholar
14.Mills, R. D. Flow in Rectangular Cavities. PhD Thesis, London University, 1961.Google Scholar
15.Woods, L. C.The Numerical Solution of 4th Order Differential Equations. Aeronautical Quarterly, Vol. V, pp 176–184, 1954.CrossRefGoogle Scholar