Abstract
The effect of an injection of specific momentum at the bottom boundary of an axisymmetric, quasisteady maintained vortex is studied by extending the earlier investigation of Kuo (1971). Non-zero vertical velocities at the top of the surface layer representing the lower extremity of a vortex are prescribed. Positive values of the superimposed vertical velocity signified pumping; negative, sucking. The two second-order ordinary differential equations governing the tangential and radial velocities of the vortex are solved by employing Newton's iterative method.
The result, viz., that pumping produces a deeper inflow layer and destabilizes the motion while suction depresses the inflow layer and produces stability confirmed an earlier finding of certain fluid dynamicists. Modifications of the boundary-layer structure produced by spatially varying the angular momentum distribution of a vortex are analogous to those caused by the imposition of the Taylor boundary condition at the lower extremity of the vortex. They are also similar to those rendered by varying pumping or suction. The latter result is believed to be new while the former simply agrees with an earlier theoretical deduction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bode, L., Leslie, L. M., and Smith, R. K.: 1975, ‘A Numerical Study of Boundary Effects on Concentrated Vortices with Application to Tornadoes and Waterspouts’, Quart. J. Roy. Meteorol. Soc. 101, 313–324.
Boedewadt, von, U. T.: 1940, ‘Die Drehströmung über festem Grunde’, Z. Angew Math. Mech. 20, 241–253.
Brandes, E. E.: 1977, ‘Flow in Severe Thunderstorms Observed by Dual-Doppler Radar’, Mon. Wea. Rev. 105, 113–120.
Burggraf, O., Stewartson, K., and Belcher, R.: 1971; ‘Boundary Layer Induced by a Potential Vortex’, Phys. Fluids, 14, 1821–1833.
Burggraf, O. R. and Stewartson, K.: 1975, ‘The Ladder Structure of the Generalized Vortex’, Z. Angew. Math. Phys. 26, 549–559.
Chi, S. W.: 1974, ‘Numerical Modeling of the Three-Dimensional Flows in the Ground Boundary Layer of a Maintained Axisymmetrical Vortex’, Tellus 4, 444–455.
Collatz, L.: 1966, Functional Analysis and Numerical Mathematics, Academic Press, 473 pp.
Conte, S. D. and deBoor, C.: 1972, Elementary Numerical Analysis: An Algorithmic Approach, McGraw-Hill, 396 pp.
Davies-Jones, R. P. and Vickers, G. T.: 1971, ‘Numerical Simulation of Convective Vortices’, NOAA Tech. Memo, ERL-NSSL 1957, Norman, Okla., 27 pp.
Debnath, L. and Mukherjee, S.: 1973, ‘Unsteady Multiple Boundary Layers on a Porous Plate in a Rotating System’, Phys. Fluids 16, 1418–1421.
Eliassen, A.: 1971, ‘On the Ekman Layer in a Circular Vortex’, J. Meteorol. Soc. Jap. 49, 209–214.
Evans, D. J.: 1969, ‘The Rotationally Symmetric Flow of a Viscous Fluid in the Presence of an Infinite Rotating Disc with Uniform Suction’, Quart. J. Mech. Appl. Math. 22, 467–485.
Froese, C.: 1962, ‘On solving y″ = fy + g with a Boundary Condition at Infinity’, Math. Comp. 16, 492–494.
Fujita, T. T., and Carcena, F.: 1977, ‘An Analysis of.Three Weather-Related Aircraft Accidents’, Bull. Amer. Meteorol. Society 58, 1164–1181.
Keller, H. B.: 1968, Numerical Methods for Two Point Boundary Value Problems, Blaisdell Publishing Co., 184 pp.
King, W. S. and Lewellen, W. S.: 1964, ‘Boundary-Layer Similarity Solutions for Rotating Flows with and without Magnetic Interaction’, Phys. Fluids 7, 1674–1680.
Kuiken, H. K.: 1971, ‘The Effect of Normal Blowing on the Flow Near a Rotating Disk of Infinite Extent’, J. Fluid Mech. 47, 789–798.
Kuo, H. L.: 1971, ‘Axisymmetric Flow in the Boundary Layer of a Maintained Vortex’, J. Atmos. Sci. 28, 20–41.
Mack, L. M.: 1962, ‘The Laminar Boundary Layer on a Disk of Finite Radius in a Rotating Flow’, Jet Propulsion Lab., Tech. Rept. 32-224, California Institute of Technology, 46 pp.
Nguyen, N. D., Ribault, J. P., and Florent, P.: 1975, ‘Multiple Solutions for Flow between Coaxial Disks’, J. Fluid Mech, 68, 369–388.
Nilson, F. N.: 1970, ‘Cubic Splines on Uniform Meshes’, Com. of the ACM, 13, 255–258.
Ray, P. S.: 1976, ‘Vorticity and Divergence Fields within Tornadic Storms from Dual Doppler Observations’, J. Appl. Meteorol., 15, 879–890.
Rogers, M. H. and Lance, G. N.: 1960, ‘The Rotationally Symmetric Flow of a Viscous Fluid in the Presence of an Infinite Disk’, J. Fluid Mech. 7, 617–631.
Rogers, M. H. and Lance, G. N.: 1964, ‘The Boundary Layer on a Disc of Finite Radius in a Rotating Fluid’, Quart. J. Mech. Appl. Math. 17, 319–330.
Rott, N., and Lewellen, W. S.: 1964, ‘Boundary Layers in Rotating Flows’, Proc. Eleventh Intern. Congress Applied Mechanics, Berlin, Springer-Verlag, 1030–1036.
Schlichting, H.: 1968, Boundary Layer Theory, McGraw-Hill, 748 pp.
Serrin, J.: 1972, ‘The Swirling Vortex’, Phil. Trans. 271, 325–360.
Sparrow, F. M., and Gregg, J. L.: 1960, ‘Mass Transfer, Flow and Heat Transfer about a Rotating Disk’, J. Heat Transf. 82, 294–302.
Stuart, J. T.: 1954, ‘On the Effect of Uniform Suction on the Steady Flow Due to a Rotating Disk’, Quart. J. Mech. Appl. Math. 7, 446–457.
Taylor, G. I.: 1915, ‘Eddy Motion in the Atmosphere’, Phil. Trans. Roy. Soc. London A215, 1–26.
Varga, R. S.: 1962, Matrix Iterative Analysis, Prentice-Hall, 322 pp.
Watson, E. J.: 1966, ‘The Equation of Similar Profiles in Boundary-Layer Theory with Strong Blowing’, Proc. Roy. Soc. A294, 208–234.
Wylie, C. R., Jr.: 1966, Advanced Engineering Mathematics, McGraw-Hill, 813 pp.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rao, G.V., Raymond, W.H. The effect of a lower boundary condition on the boundary-layer structure of an axisymmetric vortex. Boundary-Layer Meteorol 14, 525–541 (1978). https://doi.org/10.1007/BF00121892
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00121892