Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 073, 10 pages      arXiv:1210.4254      http://dx.doi.org/10.3842/SIGMA.2012.073
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

Application of the B-Determining Equations Method to One Problem of Free Turbulence

Oleg V. Kaptsov and Alexey V. Schmidt
Institute of Computational Modeling SB RAS, Akademgorodok, Krasnoyarsk, 660036, Russia

Received May 17, 2012, in final form October 04, 2012; Published online October 16, 2012

Abstract
A three-dimensional model of the far turbulent wake behind a self-propelled body in a passively stratified medium is considered. The model is reduced to a system of ordinary differential equations by a similarity reduction and the B-determining equations method. The system of ordinary differential equations satisfying natural boundary conditions is solved numerically. The solutions obtained here are in close agreement with experimental data.

Key words: turbulence; far turbulent wake; B-determining equations method.

pdf (485 kb)   tex (159 kb)

References

  1. Andreev V.K., Kaptsov O.V., Pukhnachov V.V., Rodionov A.A., Applications of group theoretical methods in hydrodynamics, Mathematics and its Applications, Vol. 450, Kluwer Academic Publishers, Dordrecht, 1998.
  2. Barenblatt G.I., Galerkina N.L., Luneva M.V., Evolution of a turbulent burst, J. Eng. Phys. Thermophys. 53 (1987), 1246-1252.
  3. Cazalbou J.B., Spalart P.R., Bradshaw P., On the behavior of two-equation models at the edge of a turbulent region, Phys. Fluids 6 (1994), 1797-1804.
  4. Chashechkin Yu.D., Chernykh G.G., Voropaeva O.F., The propagation of a passive admixture from a local instantaneous source in a turbulent mixing zone, Int. J. Comp. Fluid Dyn. 19 (2005), 517-529.
  5. Chernykh G.G., Fedorova N.N., Moshkin N.P., Numerical simulation of turbulent wakes, Russian J. Theor. Appl. Mech. 2 (1992), 295-304.
  6. Chernykh G.G., Fomina A.V., Moshkin N.P., Numerical models for turbulent wake dynamics behind a towed body in a linearly stratified medium, Russian J. Numer. Anal. Math. Modelling 21 (2006), 395-424.
  7. Efremov I.A., Kaptsov O.V., Chernykh G.G., Self-similar solutions of two problems of free turbulence, Mat. Model. 21 (2009), 137-144 (in Russian).
  8. Gibson M.M., Launder B.E., On the calculation of horizontal, turbulent, free shear flows under gravitational influence, J. Heat Transfer 98 (1976), 81-87.
  9. Hassid S., Collapse of turbulent wakes in stable stratified media, J. Hydronautics 14 (1980), 25-32.
  10. Hinze J.O., Turbulence: an introduction to its mechanism and theory, McGraw-Hill Series in Mechanical Engineering, McGraw-Hill Book Co., Inc., New York, 1959.
  11. Hulshof J., Self-similar solutions of Barenblatt's model for turbulence, SIAM J. Math. Anal. 28 (1997), 33-48.
  12. Kaptsov O.V., B-determining equations: applications to nonlinear partial differential equations, European J. Appl. Math. 6 (1995), 265-286.
  13. Kaptsov O.V., Efremov I.A., Invariant properties of the far turbulent wake model, Comput. Technol. 10 (2005), no. 6, 45-51 (in Russian).
  14. Kaptsov O.V., Efremov I.A., Schmidt A.V., Self-similar solutions of the second-order model of the far turbulent wake, J. Appl. Mech. Tech. Phys. 49 (2008), 217-221.
  15. Kaptsov O.V., Shan'ko Yu.V., Family of self-similar solutions of one model of the far turbulent wake, in Proceedinds of International Conference "Computational and Information Technologies in Sciences, Engineering, and Education" (September 20-22, 2006, Pavlodar, Kazakhstan), Vol. 1, TOO NPF "EKO", Pavlodar, 2004, 576-579 (in Russian).
  16. Launder B.E., Spalding D.B., Mathematical models of turbulence, Academic Press, London, 1972.
  17. Lin J.T., Pao Y.H., Wakes in stratified fluids, Ann. Rev. Fluid Mech. 11 (1979), 317-338.
  18. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
  19. Ovsiannikov L.V., Group analysis of differential equations, Academic Press Inc., New York, 1982.
  20. Pope S.B., Turbulent flows, Cambridge University Press, Cambridge, 2000.
  21. Rodi W., Examples of calculation methods for flow and mixing in stratified fluids, J. Geophys. Res. 92 (1987), 5305-5328.
  22. Schlichting H., Boundary layer theory, McGraw-Hill, New York, 1955.
  23. Vasiliev O.F., Kuznetsov B.G., Lytkin Yu.M., Cherhykh G.G., Development of the turbulized fluid region in a stratified medium, Fluid Dyn. (1974), no. 3, 45-52 (in Russian).
  24. Voropaeva O.F., Far momentumless turbulent wake in a passively stratified medium, Comput. Technol. 8 (2003), no. 3, 32-46 (in Russian).
  25. Voropaeva O.F., Chernykh G.G., On numerical simulation of the dynamics of the turbulized fluid regions in stratified medium, Comput. Technol. 1 (1992), no. 1, 93-104 (in Russian).
  26. Voropaeva O.F., Moshkin N.P., Chernykh G.G., Internal waves generated by turbulent wakes in a stably stratified medium, Dokl. Phys. 48 (2003), 517-521.
  27. Wilcox D.C., Turbulence modeling for CFD, DCW Industries, Canada, 1994.

Previous article  Next article   Contents of Volume 8 (2012)