|
Article information
2021 , Volume 26, ¹ 5, p.65-80
Vodinchar G.M., Feshchenko L.K.
Automated generation of turbulence shell models by the methods of computer algebra
There is a great variety of shell turbulence models. Such models reproduce certain characteristics of turbulence. A model that could reproduce all turbulence regimes does not exist at the moment. Information about a particular model is contained in a set of persistent quantities, which are some quadratic forms of turbulent fields. These quadratic forms should be formal analogs of the exact conserved quantities. It is important to note that the main idea of Shell models presupposes a refusal to describe the geometric structure of movements. At the same time, it is well known that turbulent processes in spaces of two and three dimensions behave differently. Therefore, the provision of certain combinations of conserved quantities allows indirect introducing into the shell model the information about the dimension of the physical space in which the turbulent process develops. Purpose. The aim of this work was to create software tools that would quickly generate classes of models that satisfy one or another set of conservation laws. The choice of a specific model within these classes can then be specified using additional physical considerations, for example, the existence of a given probability distribution for the interaction of certain shells. Methods. The developed technique for generating equations of shell turbulence models is carried out using symbolic computation systems (computer algebra systems — CAS). Note that symbolic packages are used not for studying ready-made shell models, but for the automated generation of the equations of these models themselves. The technique allows varying the value of the scale nonlocality of the model, the form of the quadratic conservation laws and spectral laws, the denominator of the geometric progression of scales. It allows quickly and accurately generating the entire set of classes of the models. It can be used to develop shell models of hydrodynamic, magnetohydrodynamic and convective turbulent systems. Findings. It seems that the proposed technique will be useful for studying the properties of turbulence in the framework of cascade models.
[full text]
Keywords: turbulence, cascade models, computer algebra, automation of model development
doi: 10.25743/ICT.2021.26.5.006
Author(s):
Vodinchar Gleb Mikhailovich
PhD. , Associate Professor
Position: Leading research officer
Office: Institute of Cosmophysical Researches and Radio Wave Propagation FEB RAS
Address: 684034, Russia, Paratunka, Mirnaya str., 7
Phone Office: (4152) 426824
E-mail: gvodinchar@ikir.ru
SPIN-code: 2079-6494Feshchenko Lyubov Konstantinovna
PhD.
Position: Research Scientist
Office: Institute of Cosmophysical Researches and Radio Wave Propagation FEB RAS
Address: 684034, Russia, Paratunka, Mirnaya str., 7
Phone Office: (4152) 426824
E-mail: feshenko.lk@yandex.ru
References:
1. Ditlevsen P.D. Turbulence and shell models. N.Y.: Cambridge University Press; 2011: 152.
2. Frick P.G. Turbulentnost’: Podkhody i modeli [Turbulence: Approaches and models]. M.; Izhevsk: RKhD; 2010: 332. (In Russ.)
3. Plunian F., Stepanov R.A., Frick P.G. Shell models of magnetohydrodynamic turbulence. Physics Reports. 2013; (523):1–60.
4. Frick P., Dubrulle B., Babiano A. Scaling properties of a class of shell models. Physical Review E. 1995; (51):5582–5593.
5. Hattori Y., Rubinstein R., Ishizawa A. Shell model for rotating turbulence. Physical Review E. 2004; 70(4):046311.
6. Plunian F., Stepanov R. A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence. New Journal of Physics. 2007; (9):294. DOI:10.1088/1367-2630/9/8/294.
7. Davenport J., Siret Y., Tournier E. Calcul formel: Syst’emes et algorithmes de manipulations alg’ebriques. Paris: Masson; 1987: 263.
8. Chichkarev E.A. Komp’yuternaya matematika s Maxima [Computer mathematics with Maxima]. Moscow: ALT Linux; 2012: 384. (In Russ.)
9. Matrosov A.V. Maple 6. Reshenie zadach vysshey matematiki i mekhaniki [Maple 6. Solution of problems of advanced mathematics and mechanics]. SPb.: BKhV; 2001: 528. (In Russ.)
10. Vodinchar G., Feshchenko L. Symbolic computations in simulations of hydromagnetic dynamo. E3S Web of Conferences. 2017; (20):02017.
11. Vodinchar G. Using symbolic calculations to calculate the eigenmodes of the free damping of a geomagnetic field. E3S Web of Conferences. 2018; (62):02018
Bibliography link:
Vodinchar G.M., Feshchenko L.K. Automated generation of turbulence shell models by the methods of computer algebra // Computational technologies. 2021. V. 26. ¹ 5. P. 65-80
|
