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An experimental study of the instability of the laminar Ekman boundary layer

Published online by Cambridge University Press:  28 March 2006

Alan J. Faller
Alan J. Faller
Affiliation:
Woods Hole Oceanographic Institution, Woods Hole, Mass.
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Abstract

This study concerns the stability of the steady laminar boundary-layer flow of a homogeneous fluid which occurs in a rotating system when the relative flow is slow compared to the basic speed of rotation. Such a flow is called an Ekman boundary-layer flow after V. W. Ekman who considered the theory of such flows with application to the wind-induced drift of the surface waters of the ocean.

Ekman flow was produced in a large cylindrical rotating tank by withdrawing water from the centre and introducing it at the rim. This created a steady-state symmetrical vortex in which the flow from the rim to the centre took place entirely in the shallow viscous boundary layer at the bottom. This boundary-layer flow became unstable above the critical Reynolds number $Re_c = vD|v = 125 \pm 5$ where v is the tangential speed of flow, $D = (v| \Omega)^{\frac {1}{2}}$ is the characteristic depth of the boundary layer, v is the kinematic viscosity, and Ω is the basic speed of rotation. The initial instability was similar to that which occurs in the boundary layer on a rotating disk, having a banded form with a characteristic angle to the basic flow and with the band spacing proportional to the depth of the boundary layer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 15 , Issue 4 , April 1963 , pp. 560 - 576
Copyright
© 1963 Cambridge University Press

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References

Arons, A. B., Ingersoll, A. P. & Green, T. 1961 Experimentally observed instability of a laminar Ekman flow in a rotating basin. Tellus, 13, 31–9.Google Scholar
Arx, W. S. von 1955 An experimental study of the dependence of the primary ocean circulation on the mean zonal wind field. Doctoral dissertation, M.I.T.
Bridgman, P. W. 1931 Dimensional Analysis, 2nd ed. New Haven: Yale University Press.
Cochran, W. G. 1934 The flow due to a rotating disc. Proc. Camb. Phil. Soc. 30, 365.Google Scholar
Dorsey, N. E. 1940 Properties of ordinary water substance. Amer. Chem. Soc. Mono. no. 81. New York: Rheinhold Publ. Co.
Ekman, V. W. 1905 On the influence of the earth's rotation on ocean currents. Ark. Mat. Astr. Fys. 2, no. 11Google Scholar
Faller, A. J. 1960 Further examples of stationary planetary flow patterns in bounded basins. Tellus, 12, 159–71.Google Scholar
Faller, A. J. 1962 The development of fluid model analogues of atmospheric circulations. W.H.O.I., Final Rep. Contract AF 19 (604)-4982, GRD, AFCRL.Google Scholar
Fettis, H. E. 1955 On the integration of a class of differential equations occurring in boundary layer and other hydrodynamic problems. Proc. 4th Midwest Conf. Fluid Mech.
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans. A, 248, 155–99.Google Scholar
Gregory, N. & Walker, W. S. 1960 Experiments on the effect of suction on the flow due to a rotating disc. J. Fluid Mech. 9, 225–34.Google Scholar
Haurwitz, B. 1935 On the change of wind with elevation under the influence of viscosity in curved air currents. Gerland's Beitrage zur Geophysik, 45, 243–67.Google Scholar
Hoel, P. G. 1947 Introduction to Mathematical Statistics. London: Chapman and Hall.
Howard, L. N. 1961 A note on the existence of certain viscous flows. J. Math. Phys., 40, 172–6.Google Scholar
Kármán, Th. von 1921 Laminare und turbulente Reibung. Z. angew. Math. Mech. 1, 233–52.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Rogers, M. H. & Lance, G. N. 1960 The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disc. J. Fluid Mech. 7, 617–31.Google Scholar
Schlichting, H. 1960 Boundary Layer Theory, 4th ed. New York: McGraw-Hill.
Stern, M. E. 1960 Instability of Ekman flow at large Taylor number. Tellus, 12, 399–417.Google Scholar
Stewartson, K. 1953 On the flow between two rotating co-axial dises. Proc. Camb. Phil. Soc. 49, 333.Google Scholar
Stuart, J. T. 1954 On the effects of uniform suction on the steady flow due to a rotating disc. Quart. J. Mech. Appl. Math., 7, 446–57.Google Scholar