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The development of concentrated vortices: a numerical study

Published online by Cambridge University Press:  29 March 2006

L. M. Leslie
L. M. Leslie
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria, Australia Present address: Commonwealth Meteorology Research Centre, P.O. Box 5089 AA, Melbourne, Victoria, Australia, 3001
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Abstract

Amongst the more important laboratory experiments which have produced concentrated vortices in rotating tanks are the sink experiments of Long and the bubble convection experiments of Turner & Lilly. This paper describes a numerical experiment which draws from the laboratory experiments those features which are believed to be most relevant to atmospheric vortices such as tornadoes and waterspouts.

In the numerical model the mechanism driving the vortices is represented by an externally specified vertical body force field defined in a narrow neighbourhood of the axis of rotation. The body force field is applied to a tank of fluid initially in a state of rigid rotation and the subsequent flow development is obtained by solving the Navier–Stokes equations as an initial-value problem.

Earlier investigations have revealed that concentrated vortices will form only for a restricted range of flow parameters, and for the numerical experiment this range was selected using an order-of-magnitude analysis of the steady Navier–Stokes equations for sink vortices performed by Morton. With values of the flow parameters obtained in this way, concentrated vortices with angular velocities up to 30 times that of the tank are generated, whereas only much weaker vortices are formed at other parametric states. The numerical solutions are also used to investigate the comparative effect of a free upper surface and a no-slip lid.

The concentrated vortices produced in the numerical experiment grow downwards from near the top of the tank until they reach the bottom plate whereupon they strengthen rapidly before reaching a quasi-steady state. In the quasi-steady state the flow in the tank typically consists of the vortex at the axis of rotation, strong inflow and outflow boundary layers at the bottom and top plates respectively, and a region of slowly-rotating descending flow over the remainder of the tank. The flow is cyclonic (i.e. in the same sense as the tank) in the vortex core and over most of the bottom half of the tank and is anticyclonic over the upper half of the tank away from the axis of rotation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 48 , Issue 1 , 13 July 1971 , pp. 1 - 21
Copyright
© 1971 Cambridge University Press

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References

Arakawa, A. 1966 Computational design for long term numerical integration of the equations of fluid motion. Two-dimensional incompressible flow. Part 1 J. Comput. Phys. 1, 119.Google Scholar
Barcilon, A. I. 1967 Vortex decay above a stationary boundary J. Fluid Mech. 27, 155.Google Scholar
Burgers, J. M. 1948 A mathematical model illustrating the theory of turbulence Adv. Appl. Mech. 1, 197.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Lewellen, W. S. 1962 A solution for three-dimensional vortex flows with strong circulation J. Fluid Mech. 14, 420.Google Scholar
Lilly, D. K. 1964 Numerical solutions for the shape-preserving two dimensional thermal convection element J. Atmos. Sci. 21, 83–98.Google Scholar
Long, R. R. 1956 Sources and sinks at the axis of a rotating liquid Quart. J. Mech. Appl. Math. 9, 385.Google Scholar
Long, R. R. 1958 Vortex motion in a viscous fluid J. Meteor. 15, 108.Google Scholar
Long, R. R. 1961 A vortex in an infinite viscous fluid J. Fluid Mech. 11, 611.Google Scholar
Morton, B. R. 1966 Geophysical vortices. In Progress in Aeronautical Sciences, volume 7, p. 145 (ed. D. Küchemann). Pergamon.
Morton, B. R. 1969 The strength of vortex and swirling core flows J. Fluid Mech. 38, 315.Google Scholar
Pearson, C. E. 1965 A computational method for viscous flow problems J. Fluid Mech. 21, 611.Google Scholar
Phillips, N. A. 1959 An example of non-linear computational instability. In The Atmosphere and the Sea in Motion (Rossby Memorial Volume), pp. 501–504. Rockefeller Institute Press.
Platzman, G. W. 1963 The dynamic prediction of wind tides on Lake Erie Meteor. Monog. 4, no.Google Scholar
Rogers, M. H. & Lance, G. N. 1960 The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk J. Fluid Mech. 7, 617.Google Scholar
Sullivan, R. D. 1959 A two-cell vortex solution of the Navier—Stokes equations J. Aero. Space Sci. 26, 767.Google Scholar
Turner, J. S. 1966 The constraints imposed on tornado-like vortices by the top and bottom boundary conditions J. Fluid Mech. 25, 377.Google Scholar
Turner, J. S. & Lilly, D. K. 1963 The carbonated-water tornado vortex J. Atmos. Sci. 20, 468.Google Scholar
Williams, G. P. 1967 Thermal convection in a rotating fluid annulus, Part I J. Atmos. Sci. 24, 144.Google Scholar