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Analytical and numerical studies of the structure of steady separated flows

Published online by Cambridge University Press:  28 March 2006

Odus R. Burggraf
Odus R. Burggraf
Affiliation:
Department of Aeronautical and Astronautical Engineering, The Ohio State University, Columbus, Ohio
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Abstract

The viscous structure of a separated eddy is investigated for two cases of simplified geometry. In § 1, an analytical solution, based on a linearized model, is obtained for an eddy bounded by a circular streamline. This solution reveals the flow development from a completely viscous eddy at low Reynolds number to an inviscid rotational core at high Reynolds number, in the manner envisaged by Batchelor. Quantitatively, the solution shows that a significant inviscid core exists for a Reynolds number greater than 100. At low Reynolds number the vortex centre shifts in the direction of the boundary velocity until the inviscid core develops; at large Reynolds number, the inviscid vortex core is symmetric about the centre of the circle, except for the effect of the boundary-layer displacement-thickness. Special results are obtained for velocity profiles, skin-friction distribution, and total power dissipation in the eddy. In addition, results of the method of inner and outer expansions are compared with the complete solution, indicating that expansions of this type give valid results for separated eddies at Reynolds numbers greater than about 25 to 50. The validity of the linear analysis as a description of separated eddies is confirmed to a surprising degree by numerical solutions of the full Navier–Stokes equations for an eddy in a square cavity driven by a moving boundary at the top. These solutions were carried out by a relaxation procedure on a high-speed digital computer, and are described in § 2. Results are presented for Reynolds numbers from 0 to 400 in the form of contour plots of stream function, vorticity, and total pressure. At the higher values of Reynolds number, an inviscid core develops, but secondary eddies are present in the bottom corners of the square at all Reynolds numbers. Solutions of the energy equation were obtained also, and isotherms and wall heat-flux distributions are presented graphically.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 24 , Issue 1 , January 1966 , pp. 113 - 151
Copyright
© 1966 Cambridge University Press

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References

Batchelor, G. 1954 Quart. J. Appl. Math. 12, 20.
Batchelor, G. 1956a J. Fluid Mech. 1, 38.
Batchelor, G. 1956b J. Fluid Mech. 1, 17.
Burggraf, O. R. 1965 Proc. 1965 Heat Transfer and Fluid Mech. Institute. Stanford University Press.
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids (2nd ed.). Oxford University Press.
Chapman, D. R., Kuehn, D. M. & Larson, H. K. 1956 NACA RM A55L14.
Collatz, L. 1960 The Numerical Treatment of Differential Equations (3rd ed.) Berlin: Springer.
Dean, W. R. & Montagnon, P. E. 1949 Proc. Camb. Phil. Soc. 45, 38.
Dwight, H. B. 1961 Tables of Integrals and Other Mathematical Data (4th ed.). New York: MacMillan.
Feynman, R. P. & Lagerstrom, P. A. 1956 IX International Congress of Applied Mechanics, Brussels.
Glacomelli, R. 1934 Aerodynamic Theory, Div. D. (ed. Wm. F. Durand). Berlin: Springer.
Greenspan, D., Jain, P. C., Manohar, R., Noble, B. & Sakurai, A. 1964 Numerical studies of the Navier — Stokes equations. MRC TR no. 482. Univ. of Wisconsin, May 1964.Google Scholar
Grove, A. S. 1963 Ph.D. Thesis, University of California, Berkeley.
Grove, A. S., Shair, F. H., Petersen, E. E. & Acrivos, A. 1964 J. Fluid Mech. 19, 6.
Hoerner, S. 1958 Fluid Dynamic Drag (publ. by author). Midland Park, New Jersey, U.S.A.
Howarth, L. 1953 Modern Developments in Fluid Dynamics: High Speed Flow. Oxford: Clarendon Press.
Imai, I. 1960 Proc. X Int. Congr. of Appl. Mech., Stresa, Italy.
Kaplun, S. 1957 J. Math. & Mech. 6, 59.
Kaplun, S. & Lagerstrom, P. 1957 J. Math. & Mech. 6, 58.
Kawaguti, M. 1961 J. Phys. Soc. Japan, 16, 2307.
Lamb, H. 1932 Hydrodynamics, 6th ed. New York: Dover.
Lewis, J. A. & Carrier, G. F. 1949 Quart. Appl. Math. 7, 22.
Lighthill, M. J. 1962 Fourier Analysis and Generalized Functions. Cambridge University Press.
Mills, R. D. 1965 J. Roy. Aero. Soc. 69, 11.
Moffatt, H. K. 1964 J. Fluid Mech. 18, 1.
Muskhelishvili, N. I. 1963 Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, the Netherlands.
Nash, J. F. 1962 NPL Aero. Rep. no. 1006.
Prandtl, L. 1934 Applied Hydro- and Aeromechanics. New York: McGraw-Hill.
Proudman, I. & Pearson, J. R. A. 1957 J. Fluid Mech. 2, 23.
Simuni, L. M. 1964 Inzhenernii Zhournal, (USSR), 4, 446.
Squire, H. 1956 J. Roy. Aero. Soc. 60, 20.
Taylor, G. I. 1960 Aeronautics and Astronautics (eds. N. J. Hoff & W. G. Vincenti). Oxford: Pergamon Press.
Taylor, G. I. 1962 Miszaelaneen der Angewandten Mechanik (ed. M. Schafer). Berlin: Akademie.
Van Dyke, M. D. 1964 Perturbation Methods in Fluid Mechanics. New York: Academic Press.
Whittaker, E. T. & Watson, G. N. 1952 A Course of Modern Analysis, 4th ed. Cambridge University Press.