Geodesic
An effective multigrid method for solving problems of high frequency vibrational convection
Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 2, pp. 171-187.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper describes an implemented algorithm for solving the problem of vibrational convection in a rectangular area filled with an unevenly heated incompressible fluid. The mathematical model is based on the solution of the Simonenko—Zenkovskaya equations obtained by averaging the Navier—Stokes equations under the assumption that the volume of liquid performs high-frequency translational vibrations. To solve the Poisson equations, an algebraic multigrid method is implemented in combination with a highly efficient dynamic programming method (based on the optimal control principle of R. Bellman) and fast Fourier transform. Mathematical software written in C/C++ has been developed. Examples of solving model problems with different directions of the heating flow of a square region relative to the vibration vector are given.
Keywords: multigrid method, high-frequency vibrational convection. Simonenko—Zenkovskaya equations, Bellman optimality principle
Mots-clés : discrete Fourier transform. . 
@article{SJIM_2023_26_2_a13,
     author = {A. I. Fedyushkin and K. A. Ivanov and A. A. Puntus},
     title = {An effective multigrid method for solving problems of high frequency vibrational convection},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {171--187},
     publisher = {mathdoc},
     volume = {26},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2023_26_2_a13/}
}
TY  - JOUR
AU  - A. I. Fedyushkin
AU  - K. A. Ivanov
AU  - A. A. Puntus
TI  - An effective multigrid method for solving problems of high frequency vibrational convection
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2023
SP  - 171
EP  - 187
VL  - 26
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2023_26_2_a13/
LA  - ru
ID  - SJIM_2023_26_2_a13
ER  - 
%0 Journal Article
%A A. I. Fedyushkin
%A K. A. Ivanov
%A A. A. Puntus
%T An effective multigrid method for solving problems of high frequency vibrational convection
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2023
%P 171-187
%V 26
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2023_26_2_a13/
%G ru
%F SJIM_2023_26_2_a13
A. I. Fedyushkin; K. A. Ivanov; A. A. Puntus. An effective multigrid method for solving problems of high frequency vibrational convection. Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 2, pp. 171-187. http://geodesic.mathdoc.fr/item/SJIM_2023_26_2_a13/

[1] M. Faraday, “On a peculiar class of acoustical figures”, Philos. Trans. Royal Soc., 1831, no. 121, 299–340

[2] Lord Rayleigh, “On the circulation of air observed in Kundt's tubes, and on some allied acoustical problems”, Trans. Royal Soc. London, 175:1 (1883), 1–21

[3] G. Z. Gershuni, D. V. Lubimov, Termal Vibrational Convection, John Willey Sons, 1998

[4] G. Z. Gershuni, Zhukhovitskii E. M., A. A. Nepomnyaschii, Ustoichivost konvektivnykh techenii, Nauka, M., 1989 | MR

[5] S. M. Zenkovskaya, I. B. Simoneneko, “O vliyanii vibratsii vysokoi chastoty na vozniknovenie konvektsii”, Izv. AN SSSR. Mekhanika zhidkosti i gaza, 1966, no. 5, 51–55

[6] A. Farooq, G. M. Homsy, “Streaming flows due to g-jitter-induced natural convection”, J. Fluid Mech., 271 (2006), 351–378 | DOI

[7] A. V. Zyuzgin, G. F. Putin, A. F. Kharisov, “Ground modeling of thermovibrational convection in real weightlessness”, Fluid Dynamics, 42:3 (2007), 354–361 | DOI | Zbl

[8] D. V. Lyubimov, T. P. Lyubimova, A. O. Ivantsov, “Vliyanie vibratsii na gidrodinamiku rasplava pri vyraschivanii kristallov beskontaktnym metodom Bridzhmena”, Vychisl. mekhanika sploshnykh sred, 4:4 (2011), 52–62 | DOI

[9] I. Kh. Avetisov, E. V. Zharikov, A. Yu. Zinovev, A. P. Sadovskii, “Novyi metod upravleniya teplomassoperenosom v rasplave pri vyraschivanii kristallov po Chokhralskomu”, Dokl. AN, 428:2 (2009), 177–179 | MR

[10] A. I. Fedyushkin, N. G. Bourago, V. I. Polezhaev, E. V. Zharikov, “The influence of vibration on hydrodynamics and heat-mass transfer during crystal growth”, J. Crystal Growth, 275 (2005), e1557–e1563 | DOI

[11] A. I. Fedyushkin, K. A. Ivanov, Gidrodinamika i teploobmen pri vibratsionnykh vozdeistviyakh na rasplav v protsessakh vyraschivaniya monokristallov, Preprint IPMekh RAN, No 1085, M., 2014

[12] A. I. Fedyushkin, N. G. Burago, A. A. Puntus, “Convective heat and mass transfer modeling under crystal growth by vertical Bridgman method”, J. Phys. Conf. Ser., 1479 (2020), 012029 | DOI

[13] A. I. Fedyushkin, “Vliyanie upravlyaemykh vibratsii na gidrodinamiku i teploperenos pri roste kristallov metodom Chokhralskogo”, Aktualnye problemy prikladnoi matematiki, informatiki i mekhaniki, Tr. Mezhd. nauchn. konf. (Voronezh, 17-19 dekabrya 2018 g.), Voronezh, 2018, 955–962

[14] A. I. Fedyushkin, “Vliyanie vibratsii na pogranichnye sloi v zadachakh svobodnoi i vynuzhdennoi konvektsii”, Materialy XIII Mezhd. konf. po prikladnoi matematike i mekhanike v aerokosmicheskoi otrasli, AMMAI'2020 (6-13 sentyabrya 2020 g.), Alushta, 2020, 594–596 http://www.npnj.ru/files/npnj2020_web.pdf

[15] A. I. Fedyushkin, “Vliyaniya upravlyaemykh vibratsii na gidrodinamiku i teploperenos pri roste kristallov”, Sovremennye problemy mekhaniki sploshnoi sredy, Tr. XX Mezhd. konf. (Rostov-na-Donu, 18-21 iyunya 2020 g.), v. 2, Izd-vo Yuzhnogo fed. un-ta, Rostov-na-Donu, 2020, 236–240

[16] A. I. Fedyushkin, “Heat and mass transfer during crystal growing by the Czochralski method with a submerged vibrator”, J. Phys. Conf. Ser., 1359 (2019), 012054 | DOI

[17] A. I. Fedyushkin, “The effect of controlled vibrations on Rayleigh-Benard convection”, J. Phys. Conf. Ser., 2057:1 (2021), 012012 | DOI

[18] A. I. Fedyushkin, “Numerical simulation of gas-liquid flows and boiling under effect of vibrations and gravity”, J. Phys. Conf. Ser., 2020, 1479 | DOI

[19] K. A. Ivanov, A. A. Zolotarev, A. N. Yurasov, Chislennye metody resheniya statsionarnykh zadach teploprovodnosti mnogosetochnymi metodami, izd. VNIIgeosistem, M., 2021

[20] A. I. Fedyushkin, Issledovanie matrichnogo metoda resheniya uravnenii konvektsii. Kompleks programm «MARENA», Preprint In-ta prikl. matematiki No 471, M., 1990

[21] R. P. Fedorenko, “Relaksatsionnyi metod resheniya raznostnykh ellipticheskikh uravnenii”, Zhurn. vychisl. matematiki i mat. fiziki, 1:5 (1961), 922–927 | Zbl

[22] P. N. Vabischevich, A. N. Pavlov, A. G. Churbanov, “Metody rascheta nestatsionarnykh neszhimaemykh techenii v estestvennykh peremennykh na neraznesennykh setkakh”, Mat. modelirovanie, 8:7 (1996), 1–29

[23] K. N. Volkov, Yu. N. Deryugin, V. N. Emelyanov i dr, Metody uskoreniya gazodinamicheskikh raschetov na nestrukturirovannykh setkakh, Fizmatlit, M., 2014

[24] M. K. Ermakov, “Ispolzovanie mnogosetochnogo metoda v zadachakh fizicheskoi mekhaniki”, Fiz. khim. kinetika v gazovoi dinamike, 15:2 (2014) http://chemphys.edu.ru/issues/2014-15-2/articles/116/

[25] R. Bellman, E. Endzhel, Dinamicheskoe programmirovanie i uravneniya v chastnykh proizvodnykh, Mir, M., 1974

[26] V. G. Davis de, I. P. Jones, “Natural convection in square cavity: A comparison execise”, Internat. J. Numer. Meth. Fluids, 3 (1983), 227 | DOI | Zbl

[27] S. Patankar, Chislennye mody resheniya zadach teploobmena i dinamiki zhidkosti, Energoatomizdat, M., 1984

[28] Rouch P., Vychislitelnaya gidrodinamika, 1980, Mir, M. | Zbl

[29] V. M. Paskonov, V. I. Polezhaev, L. A. Chudov, Chislennoe modelirovanie protsessov teplo- i massoobmena, Nauka, M., 1984

[30] K. A. Ivanov i dr., Svidetelstvo No 2021668343 o gosudarstvennoi registratsii programmy AliceFlow2D v rossiiskom Reestre programm dlya EVM ot 15.11.2021 g.