Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-20T08:36:44.404Z Has data issue: false hasContentIssue false
  • English
  • Français

The physics of vortex-ring evolution in a stratified and shearing environment

Published online by Cambridge University Press:  12 April 2006

J. C. S. Meng
J. C. S. Meng
Affiliation:
Science Applications, Inc., P.O. Box 2351, La Jolla, California 92037
Get access Rights & Permissions [Opens in a new window]

Abstract

A semi-analytical numerical study was performed to simulate the development of a vortex ring in a stratified and/or shearing environment. Practical applications of results of this study can be found in turbulence modelling and in studies of plumes and wakes. The objective is to follow exactly the evolution of a vortex ring so that the three-dimensional vortex-stretching mechanisms due to stratification and the shear effects, respectively, can be understood.

The basic formulation consists of the solution of the vorticity equation in a stratified medium. The approach adopted is unique in that discrete vortex elements are used and arbitrary nonlinear interactions are allowed (therefore three-dimensional effects) among various vorticity generators. One of the two fundamental assumptions in this approach is that the vorticity is allowed to be generated only along the density discontinuity. The second assumption is that, while the vorticity carried by the vortex ring is modelled by vortex elements tangential to the vortex loop (which was a vortex ring initially), the vorticity generated by stratification effects is modelled by long vortex lines parallel to the axis of the vortex ring. This limits the validity of the present calculation to high Froude number flow.

Numerical stability is guaranteed by the finite core radius for each discrete vortex element and uniform spacing between them; the former is determined by consideration of the momentum integral over the vortex-ring plane. The latter is determined by a cubic spline interpolation method which conserves the circulation and centroids of the vorticity. The velocity of each vortex element is determined by the discretized Biot-Savart law, and motion of the vortex loop is calculated by a predicator-corrector time integration method.

Calculations were carried out for both momentum-carrying and momentumless vortex rings. A particular two-dimensional case gives good agreement with Kármán's theory. The evolution of the vortex loop reveals a process in which only the vorticity normal to the stratification is conserved; the remaining vorticity is dissipated through a simulated viscous dissipation. Evolution of a vortex loop on a shear layer reveals a vortex-loop rotation rate equal to the velocity shear, and a twisting motion due to the Magnus force which can lead to the turbulence energy cascade phenomenon. Numerical results demonstrate effects of each individual vorticity source and observed phenomena can be explained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 84 , Issue 3 , 13 February 1978 , pp. 455 - 469
Copyright
© 1978 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chorin, A. J. 1973 Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–796.Google Scholar
Fink, P. T. & Soh, W. K. 1974 Calculation of vortex sheets in unsteady flow and applications in ship hydrodynamics. Univ. New South Wales Rep. NAV/ARCH 74/1.Google Scholar
KÁrmÁn, T. Von 1940 The engineer grapples with nonlinear problems. Am. Math. Soc. Bull. 46, 615–683.Google Scholar
Kress, R. 1969 Treatment of the Prager problem of potential theory by the integral equation method. Phys. Fluids Suppl. 12, II 120–125.Google Scholar
Leonard, A. 1974 Numerical simulation of interacting, three-dimensional vortex filaments. In Lecture Notes in Physics, vol. 35, pp. 245–250. Springer.
Pao, Y. H. & Lin, J. T. 1973 Turbulent wake of a towed slender body in stratified and non-stratified fluids: analysis and flow visualizations. Presentation at 26th Ann. Meeting Fluid Dyn. Div. Am. Phys. Soc.
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. Roy. Soc. A 134, 170–192.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. M.I.T. Press.
Thomson, J. A. L. & Meng, J. C. S. 1974 Studies of free buoyant and shear flows by the vortex-in-cell method. In Lecture Notes in Physics, vol. 35, pp. 401–416. Springer.
Wu, J. 1969 Mixed region collapse with internal wave generation in a density-stratified medium. J. Fluid Mech. 35, 531–544.Google Scholar