Geodesic
Construction of complex shell models of turbulent systems by computer algebra methods
Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 41 (2022) no. 4, pp. 9-31 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

One popular class of small-scale turbulence models is the class of shell models. In these models, the fields of a turbulent system are represented by time-dependent collective variables (real or complex), which are understood as the field intensity meassure in a given range of spatial scales. The model itself is a certain system of quadratically nonlinear ordinary differential equations for collective variables. Construction a new shell model requires rather complex analytical transformations. This is due to the fact that the system of model equations in the absence of dissipation must have some quadratic invariants and phase-space volume is unchanged. In addition, there are limitations associated with the impossibility of non-linear interaction of some scales ranges. All this imposes limitations on the coefficients of the nonlinear terms of the model. Constraints form a system of equations with parameters. The complexity of this system increases sharply for non-local models, when the interaction is described not only close ranges of scales and when complex collective variables are used. The paper proposes a computational technology that allows automating the process of building shell models. It makes it easy to combine different invariants and the meassure of nonlocality. The technology is based on computer algebra methods. The process of constructing equations for unknown coefficients and their solution has been automated. As a result, parametric classes of cascade models are obtained that have the required analytical properties.
Mots-clés : turbulence
Keywords: shell models, computer algebra, automation of model development. 
@article{VKAM_2022_41_4_a0,
     author = {G. M. Vodinchar and L. K. Feshenko and N. V. Podlesnyi},
     title = {Construction of complex shell models of turbulent systems by computer algebra methods},
     journal = {Vestnik KRAUNC. Fiziko-matemati\v{c}eskie nauki},
     pages = {9--31},
     year = {2022},
     volume = {41},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VKAM_2022_41_4_a0/}
}
TY  - JOUR
AU  - G. M. Vodinchar
AU  - L. K. Feshenko
AU  - N. V. Podlesnyi
TI  - Construction of complex shell models of turbulent systems by computer algebra methods
JO  - Vestnik KRAUNC. Fiziko-matematičeskie nauki
PY  - 2022
SP  - 9
EP  - 31
VL  - 41
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VKAM_2022_41_4_a0/
LA  - ru
ID  - VKAM_2022_41_4_a0
ER  - 
%0 Journal Article
%A G. M. Vodinchar
%A L. K. Feshenko
%A N. V. Podlesnyi
%T Construction of complex shell models of turbulent systems by computer algebra methods
%J Vestnik KRAUNC. Fiziko-matematičeskie nauki
%D 2022
%P 9-31
%V 41
%N 4
%U http://geodesic.mathdoc.fr/item/VKAM_2022_41_4_a0/
%G ru
%F VKAM_2022_41_4_a0
G. M. Vodinchar; L. K. Feshenko; N. V. Podlesnyi. Construction of complex shell models of turbulent systems by computer algebra methods. Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 41 (2022) no. 4, pp. 9-31. http://geodesic.mathdoc.fr/item/VKAM_2022_41_4_a0/

[1] Mingshun J., Shida L., “Scaling behavior of velocity and temperature in a shell model for thermal convective turbulence”, Physical Review E, 56 (1997), 441 | DOI

[2] Hattori Y., Rubinstein R., Ishizawa A., “Shell model for rotating turbulence”, Physical Review E, 70 (2004), 046311 | DOI

[3] Ching E. S. C., Guo H., Cheng W. C., “Understanding the different scaling behavior in various shell models proposed for turbulent thermal convection”, Physica D: Nonlinear Phenomena, 237:4 (2008), 2009–2014 | DOI

[4] Frik P. G., Turbulentnost: podkhody i modeli, NITs «RKhD», Moskva–Izhevsk, 2010, 332 pp.

[5] Ditlevsen P., Turbulence and Shell Models, University Press, Cambridge, 2011, 152 pp.

[6] Devenport Dzh., Sire I., Turne E., Kompyuternaya algebra, Mir, M., 1991, 352 pp.

[7] Matrosov A. V., Maple 6. Reshenie zadach vysshei matematiki i mekhaniki, BKhV, SPb., 2001, 528 pp.

[8] Chichkarev E. A., Kompyuternaya matematika s Maxima, ALT Linux, M., 2012, 384 pp.

[9] Vodinchar G. M., Feschenko L. K., “Avtomatizirovannaya generatsiya kaskadnykh modelei turbulentnosti metodami kompyuternoi algebry”, Vychislitelnye tekhnologii, 26:5 (2021), 65–80

[10] Gledzer E. B., “Sistema gidrodinamicheskogo tipa, dopuskayuschaya kvadratichnykh integrala dvizheniya”, DAN SSSR, 209:5 (1973), 1046–1048

[11] Yamada M., Ohkitani K., “Lyapunov Spectrum of a Chaotic Model of Three-Dimensional Turbulence”, J. Phys. Soc. Jpn., 56 (1987), 4210 | DOI

[12] L'vov V., Podivilov E., Pomlyalov A., Procaccia I., Vandembroucq D., “Improved shell model of turbulence”, Phys. Rev. E, 58 (1998), 1811 | DOI

[13] Ditlevsen P., “Symmetries, invariants, and cascades in a shell model of turbulence”, Phys. Rev. E, 62 (2000), 484 | DOI

[14] Plunian F., Stepanov R., “A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence”, New Journal of Physics, 9 (2007), 294 | DOI

[15] Plunian F., Stepanov R. A., Frick P. G., “Shell models of magnetohydrodynamic turbulence”, Physics Reports, 523 (2013), 1–60 | DOI