Skip to main content
SpringerLink
Log in

Application of mosaic-skeleton approximations in the simulation of three-dimensional vortex flows by vortex segments

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

The computations in a vortex method for three-dimensional fluid dynamics simulation are accelerated by applying mosaic-skeleton approximations of matrices in the velocity computations. A modified vortex segment method is proposed in which mosaic-skeleton matrix approximations are effectively used to solve the problem of a vorticity field developing in an unbounded three-dimensional domain and in separated flow problems. Examples of the numerical solution of model fluid dynamic problems, such as the motion of a pair of vortex rings, the flow past a hemisphere, and the flow past an octahedral cylinder, are given that illustrate the capability of accelerating the computations while preserving the qualitative and quantitative description of the flow.

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. E. E. Tyrtyshnikov, “Fast Multiplication Methods and the Solution of Equations,” in Matrix Methods and Computations (Inst. Vychisl. Mat. Ross. Akad. Nauk, Moscow, 1999) [in Russian].

    Google Scholar 

  2. E. E. Tyrtyshnykov, “Mosaic-Skeleton Approximations,” Calcolo 33(1/2), 47–57 (1996).

    Article  MathSciNet  Google Scholar 

  3. J. Barnes and P. Hut, “A Hierarchical O(N log N) Force Calculation Algorithm,” Nature 324, 446–449 (1986).

    Article  Google Scholar 

  4. N. Kornev, A. Leder, and K. Mazaev, “Comparison of Two Fast Algorithms for the Calculation of Flow Velocities Induced by a Three Dimensional Vortex Field,” Schiffbauforschung 40(1), 47–55 (2001).

    Google Scholar 

  5. C. Anderson, “A Method of Local Corrections for Computing the Velocity Field due to a Distribution of Vortex Blobs,” J. Comput. Phys. 62, 111–123 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  6. G. Ya. Dynnikova, “Calculation of Flow Around a Circular Cylinder on the Basis of Two-Dimensional Navier-Stokes Equations at Large Reynolds Numbers with High Resolution in a Boundary Layer,” Dokl. Akad. Nauk 422, 755–757 (2008) [Dokl. Phys. 53, 544–547 (2008)].

    Google Scholar 

  7. S. A. Goreinov, “Mosaic-Skeleton Approximations of Matrices Generated by Asymptotically Smooth Oscillatory Kernels,” in Matrix Methods and Computations (Inst. Vychisl. Mat. Ross. Akad. Nauk, Moscow, 1999) [in Russian].

    Google Scholar 

  8. V. Yu. Kiryakin and A. V. Setukha, “On the Convergence of a Vortical Numerical Method for Three-Dimensional Euler Equations in Lagrangian Coordinates,” Differ. Uravn. 43, 1263–1276 (2007) [Differ. Equations 43, 1295–1310 (2007)].

    MathSciNet  Google Scholar 

  9. V. A. Gutnikov, I. K. Lifanov, and A. V. Setukha, “Simulation of the Aerodynamics of Buildings and Structures by Means of the Closed Vortex Loop Method,” Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza, No. 4, 78–92 (2006).

  10. I. K. Lifanov, Method of Singular Integral Equations and Numerical Experiments (Yanus, Moscow, 1995) [in Russian].

    Google Scholar 

  11. V. Yu. Kiryakin, “Flow Simulation Based on the Discrete Vortex Method with a Vortex Sheet Represented by Vortex Particles,” Nauchn. Vestn. Mosk. Gos. Tekh. Univ. Grazhdan. Aviatsii, Ser. Aeromekh. Prochnost’, No. 125 (1), 78–82 (2008).

  12. A. A. Gurzhii, M. Yu. Konstantinov, and V. V. Meleshko, “Interaction of Coaxial Vortex Rings in an Ideal Fluid,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 78–84 (1988).

  13. D. V. Bogomolov, I. K. Marchevskii, A. V. Setukha, and G. A. Shcheglov, “Numerical Simulation of the Motion of Two Vortex Rings in an Ideal Fluid by Discrete Vortex Methods,” Inzh. Fiz., No. 4, 8–14 (2008).

  14. Handbook of the Aircraft Designer, Vol. 1: Aerodynamics of Airplane (TsAGI, Moscow, 1937) [in Russian].

  15. G. A. Savitskii, Wind Effects on Buildings and Structures (Izd. Literatury po Stroitel’stvu, Moscow, 1972) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Numerical Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119333, Russia

    A. A. Aparinov

  2. Department of Mathematics, Zhukovsky Air Force Engineering Academy, ul. Planetnaya 3, Moscow, 125190, Russia

    A. V. Setukha

Authors
  1. A. A. Aparinov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Setukha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Aparinov.

Additional information

Original Russian Text © A.A. Aparinov, A.V. Setukha, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 5, pp. 937–948.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aparinov, A.A., Setukha, A.V. Application of mosaic-skeleton approximations in the simulation of three-dimensional vortex flows by vortex segments. Comput. Math. and Math. Phys. 50, 890–899 (2010). https://doi.org/10.1134/S096554251005012X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S096554251005012X

Key words

Search

Navigation