Skip to main content
SpringerLink
Log in

Diffusion velocity for a three-dimensional vorticity field

  • Original Article
  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

A discussion is presented on the existence of a diffusion velocity for the vorticity vector that satisfies extensions of the Helmholtz vortex laws in a three-dimensional, incompressible, viscous fluid flow. A general form for the diffusion velocity is derived for a complex-lamellar vorticity field that satisfies the property that circulation is invariant about a region that is advected with the sum of the fluid velocity and the diffusion velocity. A consequence of this property is that vortex lines will be material lines with respect to this combined velocity field. The question of existence of diffusion velocity for a general three-dimensional vorticity field is shown to be equivalent to the question of existence of solutions of a certain Fredholm equation of the first kind. An example is given for which it is shown that a diffusion velocity satisfying this property does not, in general, exist. Properties of the simple expression for diffusion velocity for a complex-lamellar vorticity field are examined when applied to the more general case of an arbitrary three-dimensional flow. It is found that this form of diffusion velocity, while not satisfying the condition of circulation invariance, nevertheless has certain desirable properties for computation of viscous flows using Lagrangian vortex methods. The significance and structure of the noncomplex-lamellar part of the viscous diffusion term is examined for the special case of decaying homogeneous turbulence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beaudoin, A., Huberson, S., Rivoalen, E.: Simulation of anisotropic diffusion by means of a diffusion velocity method. J. Comput. Phys. 186, 122–135 (2003)

    Google Scholar 

  2. Ogami, Y., Akamatsu, T.: Viscous flow simulation using the discrete vortex model—The diffusion velocity method. Comput. Fluids 19, 433–441 (1991)

    Article  MATH  Google Scholar 

  3. Strickland, J.H., Kempka, S.N., Wolfe, W.P.: Viscous diffusion of vorticity using the diffusion velocity concept. ESAIM: Proc. 1, 135–151 (1996)

    MATH  MathSciNet  Google Scholar 

  4. Chorin, A.J.: A numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–796 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  5. Ghoniem, A.F., Sherman, F.S.: Grid-free simulation of diffusion using random walk methods. J. Comput. Phys. 61, 1–37 (1985)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Rossi, L.: Resurrecting core spreading vortex methods: A new scheme that is both deterministic and convergent. SIAM J. Sci. Comput. 17, 370–397 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Shankar, S., van Dommelen, L.: A new diffusion procedure for vortex methods. J. Comput. Phys. 127, 88–109 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Marshall, J.S., Grant, J.R.: A Lagrangian vorticity collocation method for viscous, axisymmetric flows with and without swirl. J. Comput. Phys. 138, 302–330 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Degond, P., Mas-Gallic, S.: The weighted particle method for convection-diffusion equations. Part 1: The case of an isotropic viscosity. Math. Comput. 53, 485–507 (1989)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Gossler, A.A., Marshall, J.S.: Simulation of normal vortex–cylinder interaction in a viscous fluid. J. Fluid Mech. 431, 371–405 (2001)

    Article  MATH  ADS  Google Scholar 

  11. Marshall, J.S., Grant, J.R., Gossler, A.A., Huyer, S.A.: Vorticity transport on a Lagrangian tetrahedral mesh. J. Comput. Phys. 161, 85–113 (2000)

    Article  MATH  ADS  Google Scholar 

  12. Huberson, S., Rivoalen, E., Hauville, F.: Simulation numérique des équations de Navier-Stokes 3D par une méthode particulaire. C.R. Acad. Sci. Paris, Série IIb 324, 543–549 (1997)

    MATH  ADS  Google Scholar 

  13. Lacombe, G., Mas-Gallic, S.: Presentation and analysis of a diffusion-velocity method. ESAIM: Proc. 7, 225–233 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kida, S., Takaoka, M.: Vortex reconnection. Ann. Rev. Fluid Mech. 26, 169–189 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  15. Aris, R.: Vectors, Tensors and the Basic Equations of Fluid Dynamics. Dover, New York (1962)

  16. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1, p. 160. Wiley, New York (1953)

  17. Pogorzelski, G.: Integral Equations and Their Applications, pp. 155–163. Pergamon Press, Oxford (1966)

    MATH  Google Scholar 

  18. Vincent, A., Meneguzzi, M.: The spatial structure and statistical properties of homogeneous turbulence, J. Fluid Mech. 225, 1–20 (1991)

    Article  ADS  MATH  Google Scholar 

  19. Frisch, U.: Turbulence, pp. 112–115. Cambridge University Press, Cambridge (1995)

  20. Marshall, J.S., Beninati, M.L.: Analysis of subgrid-scale torque for large-eddy simulation of turbulence. AIAA J. 41, 1875–1881 (2003)

    ADS  Google Scholar 

  21. Marshall, J.S., Beninati, M.L.: External turbulence interaction with a columnar vortex. J. Fluid Mech. 540, 221–245 (2005)

    Article  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Anteon Inc., 1 Corporate Place, Middletown, RI, 02842-6277, USA

    J. R. Grant

  2. Department of Mechanical and Industrial Engineering and IIHR – Hydroscience and Engineering, The University of Iowa, Iowa City, Iowa, 52242, USA

    J. S. Marshall

Authors
  1. J. R. Grant
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. S. Marshall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. S. Marshall.

Additional information

Communicated by M. Y. Hussaini

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grant, J.R., Marshall, J.S. Diffusion velocity for a three-dimensional vorticity field. Theor. Comput. Fluid Dyn. 19, 377–390 (2005). https://doi.org/10.1007/s00162-005-0004-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-005-0004-8

Keywords

PACS

Search

Navigation