Self-similar decay of the momentumless turbulent wake in a passive stratified medium

O. V. Kaptsov ; A. V. Fomina ; G. G. Chernykh ; A. V. Schmidt

Matematičeskoe modelirovanie, Tome 27 (2015) no. 1, pp. 84-98.

Résumé

Mathematical model of the far momentumless turbulent wake in a passive stratified medium based on the semi-empirical $k-\varepsilon$ model of turbulence is considered. The group-theoretical analysis of the model is performed. The model is reduced to a system of ordinary differential equations by the $B$-determining equations method. The system of ordinary differential equations is solved numerically. A comparison of the constructed solution with the results of numerical simulation of the model at a large distance from the body is performed. A close agreement is obtained.

Détail

Keywords: mathematical model of the far momentumless turbulent wake in a passive stratified medium, $k-\varepsilon$ model of turbulence, group-theoretical analysis, $B$-determining equations method, self-similar degeneration, numerical modeling.

Bibliographie

[1] O. F. Vasilev, B. G. Kuznetsov, Yu. M. Lytkin, G. G. Chernykh, “Development of the region of a turbulized liquid in a stratified medium”, Fluid Dynamics, 9:3 (1974), 368–373 | DOI

[2] Yu. M. Lytkin, G. G. Chernykh, “Podobie techeniia po plotnostnomu chislu Fruda i balans energii pri evoliutsii zony turbulentnogo cmesheniia v stratifitsirovannoi srede”, Matem. problemy mekhaniki sploshnykh sred, 47, In-t gidrodinamiki, Novosibirsk, 1988, 70–89

[3] S. Hassid, “Collapse of turbulent wakes in stable stratified media”, J. Hydronautics, 14:1 (1980), 25–32 | DOI

[4] A. Yu. Danilenko, B. I. Kostin, A. I. Tolstykh, O neiavnom algoritme rascheta techenii odnorodnoi i neodnorodnoi zhidkosti, preprint, VTS AN SSSR, M., 1985, 40 pp.

[5] G. S. Glushko, A. G. Gumilevskii, V. I. Polezhaev, “Evolution of the turbulent wakes of spherical bodies in a stably stratified media”, Fluid Dynamics, 29:11 (1994), 10–16 | DOI

[6] G. G. Chernykh, O. F. Voropayeva, “Numerical modeling of momentumless turbulent wake dynamics in a linearly stratified medium”, Computers and Fluids, 28:3 (1999), 281–306 | DOI

[7] O. F. Voropaeva, Yu. D. Chashechkin, G. G. Chernykh, “Diffusion of a passive admixture from a local source in a turbulent mixing zone”, Fluid Dynamics, 32:2 (1997), 212–218

[8] O. F. Voropayeva, “Dalnii bezympulsnyi turbulentnyi sled v passivno stratifitsirovannoi srede”, Vychisl. tekhnologii, 8:3 (2003), 32–46

[9] G. G. Chernykh, A. V. Fomina, N. P. Moshkin, “Numerical models of turbulent wake dynamics behind a towed body in a linearly stratified medium”, Russ. J. Numer. Anal. Math. Modelling, 21:5 (2006), 395–424 | DOI

[10] N. P. Moshkin, A. V. Fomina, G. G. Chernykh, “Chislennoe modelirovanie dinamiki turbulentnogo sleda za buksiruemym telom v lineino stratifitsirovannoi srede”, Matem. modelirovanie, 19:1 (2007), 29–56

[11] O. F. Voropaeva, “Anisotropic turbulence decay in a far momentumless wake in a stratified medium”, Math. Models and Comp. Simulations, 1:5 (2009), 605–619 | DOI

[12] G. G. Chernykh, O. A. Druzhinin, A. V. Fomina, N. P. Moshkin, “On numerical modeling of the dynamics of turbulent wake behind a towed body in linearly stratified medium”, J. Engineering Thermophysics, 21:3 (2012), 155–166 | DOI

[13] M. S. Bruzón, P. A. Clarkson, M. L. Gandarias, E. Medina, “The symmetry reductions of a turbulence model”, J. Phys. A: Math. Gen., 34 (2001), 3751–3760 | DOI

[14] S. Guenther, M. Oberlack, “Implication for Lie group symmetries for RANS modeling”, Proc. Appl. Math. Mech., 5 (2005), 563–564 | DOI

[15] O. V. Kaptsov, A. V. Schmidt, “Application of the B-Determining Equations Method to One Problem of Free Turbulence”, SIGMA, 8 (2012), 073, 10 pp.

[16] A. V. Schmidt, “Similarity reduction of a three-dimensional model of the far turbulent wake behind a towed body”, J. of Phys.: Conf. Ser., 318:4 (2011), 9 | DOI

[17] O. V. Kaptsov, A. V. Fomina, G. G. Chernykh, A. V. Schmidt, “Self-similar degeneration of the turbulent wake behind a body towed in a passively stratified medium”, J. Appl. Mech. and Tech. Phys., 53:5 (2012), 672–678 | DOI

[18] O. V. Kaptsov, I. A. Efremov, “Invariantnye svoistva modeli dalnego turbulentnogo sleda”, Vychisl. tekhnologii, 10:6 (2005), 45–51

[19] O. V. Kaptsov, I. A. Efremov, A. V. Shmidt, “Self-similar solutions of the second-order model of the far turbulent wake”, J. Appl. Mech. and Tech. Phys., 49:2 (2008), 217–221 | DOI

[20] I. A. Efremov, O. V. Kaptsov, G. G. Chernykh, “Avtomodelnye resheniia dvukh zadach svobodnoi turbulentnosti”, Matem. modelirovanie, 21:12 (2009), 137–144

[21] N. V. Aleksenko, V. A. Kostomakha, “Experimental study of an axisymmetric nonimpulsive turbulent jet”, J. Appl. Mech. and Tech. Phys., 28:1 (1987), 60–64 | DOI

[22] W. Rodi, “Examples of calculation methods for flow and mixing in stratified fluids”, J. Geophys. Res., 92:C5 (1987), 5305–5328 | DOI

[23] L. V. Ovsyannikov, Group analysis of differential equations, Academic press, N.Y., 1982, 416 pp.

[24] V. A. Gorodtsov, “Similarity and weak closing relations for symmetric free turbulence”, Fluid Dynamics, 14:1 (1979), 31–37 | DOI

[25] S. Hassid, “Similarity and decay laws of momentumless wakes”, Phys. Fluids, 23:2 (1980), 404–405 | DOI

[26] O. V. Kaptsov, “B-determining equations: applications to nonlinear partial differential equations”, Euro. J. Appl. Math., 6 (1995), 265–286 | DOI

[27] G. I. Barenblatt, Similarity, self-similarity, and intermediate asymptotics, Consultants Bureau, N.Y., 1979, 218 pp.

[28] O. F. Voropayeva, G. G. Chernykh, “O chislennom modelirovanii dinamiki oblastei turbulizovannoi zhidkosti v stratifitsirovannoi srede”, Vychislitelnye tekhnologii, 1, no. 1, IVT SO RAN, Novosibirsk, 1992, 93–104

[29] A. N. Tikhonov, A. A. Samarskii, Equations of mathematical physics, Pergamon Press, London, 1963, 765 pp.