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Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods

Published online by Cambridge University Press:  12 April 2006

J. C. S. Meng  and
J. A. L. Thomson
J. C. S. Meng
Affiliation:
Physical Dynamics, Inc., P.O. Box 1069, Berkeley, California 94701 Present address: Science Applications, Inc., P.O. Box 2351, La Jolla, California 92037.
J. A. L. Thomson
Affiliation:
Physical Dynamics, Inc., P.O. Box 1069, Berkeley, California 94701
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Abstract

A class of nonlinear hydrodynamic problems is studied. Physical problems such as shear flow, flow with a sharp interface separating two fluids of different density and flow in a porous medium all belong to this class. Owing to the density difference across the interface, vorticity is generated along it by the interaction between the gravitational pressure gradient and the density gradient, and the motion consists of essentially two processes: the creation of a vortex sheet and the subsequent mutual induction of different portions of this sheet.

Two numerical methods are investigated. One is based upon the well-known Green's function method, which is a Lagrangian method using the Biot-Savart law, while the other is the vortex-in-cell (VIC) method, which is a Lagrangian-Eulerian method. Both methods treat the interface as sharp and represent it by a distribution of point vortices. The VIC method applies the FFT (fast Fourier transform) to solve the stream-function/vorticity equation on an Eulerian grid, and computational efficiency is further improved by using the reality properties of the physical variables.

Four specific problems are investigated numerically in this paper. They are: the Rayleigh-Taylor instability, the Saffman-Taylor instability, transport of aircraft trailing vortices in a wind shear, and the gravity current. All four problems are solved using the VIC method and the results agree well with results obtained by previous investigators. The first two problems, the Rayleigh-Taylor instability and the Saffman-Taylor instability, are also solved by the Green's function method. Comparisons of results obtained by the two methods show good agreement, but, owing to its computational economy, the VIC method is concluded to be the better method for treating the class of hydrodynamic problems considered here.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 84 , Issue 3 , 13 February 1978 , pp. 433 - 453
Copyright
© 1978 Cambridge University Press

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References

Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209.Google Scholar
Blackdar, A. K. & Tennekes, H. 1968 Asymptotic similarity in neutral barotropic planetary boundary layers. J. Atmos. Sci. 25, 1015.Google Scholar
Brashears, M. R. & Hallock, J. N. 1973 Aircraft wake vortex transport model. A.I.A.A. Paper no. 73–679.Google Scholar
Chorin, A. J. & Bernard, P. S. 1973 Discretization of a vortex street with an example of roll-up. J. Comp. Phys. 13, 423.Google Scholar
Cooley, J. W. & Tukey, J. W. 1965 An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19, 297.Google Scholar
Donaldson, C. Dup., Snedeker, R. S. & Sullivan, R. D. 1973 Calculation of the wakes of three transport aircraft in holding, takeoff, and landing configurations, and comparison with experimental measurements. Air Force Office Sci. Res. Wash. Rep. AFOSR-TR-73-1594 (also FAA-RD-73-42).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
Fohl, T. 1967 Optimization of flow for forcing stack wastes to high altitudes. J. Air Pollution Control Ass. 17, 730.Google Scholar
Hinze, J. O. 1959 Turbulence. McGraw-Hill.
KÁrmÁn, T. Von 1940 The engineer grapples with nonlinear problems. Am. Math. Soc. Bull. 46, 615.Google Scholar
Meng, J. C. S. 1977 The physics of vortex ring evolution in a stratified and shearing environment. A.I.A.A. Paper no. 77–12.Google Scholar
Owen, P. R. 1970 The decay of a turbulent trailing vortex. Aero. Quart. 21, 69.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. Roy. Soc. A 245, 312.Google Scholar
Scorer, R. S. 1958 Natural Aerodynamics. Pergamon.
Thomson, J. A. & Meng, J. C. S. 1976 Scanning laser doppler velocimeter system simulation for sensing aircraft wake vortices. J. Aircraft 13, 605.Google Scholar
Turner, J. S. 1959 A comparison between buoyant vortex rings and vortex pairs. J. Fluid Mech. 7, 419.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Woods, J. D. 1969 On Richardson's numbers as criteria for laminar–turbulent–laminar transition in the ocean and atmosphere. Radio Sci. 4, 1289.Google Scholar
Wu, J. 1969 Mixed region collapse with internal wave generation in a density-stratified medium. J. Fluid Mech. 35, 531.Google Scholar