Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-22T13:14:19.263Z Has data issue: false hasContentIssue false
  • English
  • Français

An Investigation of the Diffusion Errors in Diffusion Vortex Methods

Published online by Cambridge University Press:  05 May 2011

M.-J. Huang  and
Y.-Y. Chen
M.-J. Huang*
Affiliation:
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
Y.-Y. Chen*
Affiliation:
Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
*
*Associate Professor
**Graduate student
Get access

Abstract

To model the viscous effect, the core area of particles in Leonard's core spreading vortex method must grow linearly in time, which however results in growing convective errors and consequently causes the failure of a correct convergence of the method to the Navier-Stokes equations. The so-called diffusion vortex method was proposed and claimed to have smaller core-area growth rates because part of the viscous effect is modeled into the movement of particles. The growth rates however are non-uniform and an additional diffusion error arises due to the nonzero divergence of the diffusion velocity. The goal of this work is to analyze the associated errors of several existing versions of the diffusion vortex method and compare them with that of Leonard's. Simulations of two axisymmetric flows are performed to measure the involved diffusion errors and consequently distinguish these diffusion vortex methods. The results show that the circulation conservation is important, besides a small core-area growth rate, in obtaining a good accuracy. Under the consideration of both efficiency and accuracy, the diffusion vortex method in which each vortex particle conserves its own circulation on its core area alone is recommended.

Keywords

Vortex methodDiffusion velocityCirculation
Type
Articles
Information
Journal of Mechanics , Volume 21 , Issue 2 , June 2005 , pp. 63 - 70
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2005

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

REFERENCES

1.Cottet, G.-H. and Koumoutaskos, P., Vortex Methods: Theory and Practice, Cambridge Univ. Press, Cambridge, UK (1999).Google Scholar
2.Leonard, A., “Vortex Methods for Flow Simulations”, J. Comput. Phys., 37, pp. 289–335 (1980).CrossRefGoogle Scholar
3.Greengard, C., “The Core Spreading Vortex Method Approximates the Wrong Equation,” J Comput. Phys., 61, pp. 345–348 (1985).CrossRefGoogle Scholar
4.Chang, C. and Chern, R., “A Numerical Study of Flow Around an Impulsively Started Circular Cylinder by a Deterministic Vortex Method,” J. Fluid Mech., 233, p. 243 (1991).CrossRefGoogle Scholar
5.Chorin, A. J., “Numerical Study of Slighly Viscous Flow,” J. Fluid Mech., 57, pp. 785–796 (1973).Google Scholar
6.Degond, P. and Mas-Gallic, S., “The Weighted Particle Method for Convection-Diffusion Equations, Part 1: The Case of an Isotropic Viscosity,” Math. Comput., 53, pp. 485–507 (1989).Google Scholar
7.Fishelov, D., “A New Vortex Scheme for Viscous Flow,” J. Comput. Phys., 86, pp. 211–224 (1990).CrossRefGoogle Scholar
8.Ogami, Y. and Akamatsu, T., “Viscous Flow Simulation using the Discrete Vortex Model— The Diffusion Velocity Method,” Computers and Fluids, 19, pp. 433–441 (1991).Google Scholar
9.Kempkal, S. N. and Strickland, J. H., “A Method to Simulate Viscous Diffusion of Vorticity by Convective Transport of Vortices at a Non-Solenoidal Velocity,” Sandia Report, Sandia National Lab., SAND93-1763.UC-700 (1993).Google Scholar
10.Shintani, M. and Akamatsu, T., “Investigation of Two Dimensional Discrete Vortex Method with Viscous Diffusion Model,” Computational Fluid Dynamics, 3, pp. 237–254 (1994).Google Scholar
11.Huang, M. J., “Circulation Conserved Diffusion Vortex Method,” Transactions of the Aeronautical and Astronautical Society of the Republic of China, 35, pp. 65–72 (2003).Google Scholar
12.Haberman, R., Elementary Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems, Prentice-Hall Inc. (1983).Google Scholar