532.5.032 Influence of Rayleigh, Prandtl numbers and boundary conditions on convective fluid flows in horizontal layers

Fedyushkin A. I. (Ishlinsky Institute for Problems in Mechanics)

NUMERICAL SIMULATION, CONVECTION, COUNTER FLOW, BOUNDARY CONDITIONS, THE STRUCTURE OF THE FLOW


doi: 10.18698/2309-3684-2020-1-2844


The paper considers two problems of thermal convection of an incompressible liquid in a horizontally extended layer: with lateral heat supply and with heating of the horizontal layer from below-the Rayleigh-Benard problem. An influence of boundary conditions and Prandtl numbers on the convective flow structure and temperature distribution is considered. The solutions of these problems are obtained using numerical modeling. The simulation is based on the numerical solution of a system of non-stationary 2D Navier-Stokes equations for an incompressible fluid, as well as for the Rayleigh-Benard problem for the case of a two-layers gas-liquid system. Navier-Stokes equations are solved using two numerical methods: the finite difference method and the control volume method. To verify the models, the results of calculations obtained by different numerical methods were compared with each other and compared with experimental data. The paper presents the results of numerical simulation of convective flows and heat and mass transfer in horizontal liquid layers under different defining dimensionless parameters and boundary conditions. The non-linear features of convective flows in horizontal liquid layers are shown, in particular, the occurrence of a counter-flow inside the layer-a liquid flow with a direction opposite to the main convective flow. The influence of boundary conditions and Rayleigh and Prandtl numbers on the existence of a countercurrent is considered. A simulation of the convective flow of a liquid in a horizontal layer when heated from the side at small Prandtl numbers, as well as at a Prandtl number equal to zero, is performed. The simulation results showed that for laminar convection (the Rayleigh number is greater than 10^5), the flow with Prandtl numbers equal to or less than 10^-2 is qualitatively different from the flow of a liquid with a zero Prandtl number. Therefore, the approximation of the zero value of the Prandtl number may not always be correct. Nonlinear peculiarities of convective flows in horizontal liquid layers are presented. It is shown that in long horizontal layers laterally heated only by thermal laminar convection (without the presence of impurities and concentration convection), it is possible to create a stable vertical density stratification of the fluid and, as a result, lead to the appearance of layered structures.


[1] Gershuni G.Z., Zhukhovsitskii E.M. Konvektivnaya ustoychivost' neszhimayemoy zhidkosti [Convective Stability of an Incompressible Liquid]. Moscow, Nauka Publ., 1972, 392 p.
[2] Polezhaev V.I., Bello M.S., Verezub N.A., Dubovik K.G., Lebedev A.P., Nikitin S.A., Pavlovskii D.S., Fedyushkin A.I. Konvektivnyye protsessy v nevesomosti [Convective Processes in Weightlessness]. Moscow, Nauka Publ., 1991, 240 p.
[3] Gershuni G.Z., Zhukhovitsky E.M. Sbornik "Gidrodinamika" — collection "Hydrodynamics",1970, iss. 2, pp. 207–217.
[4] Birikh R.V. Prikladnaya mekhanika i tekhnicheskaya fizika — Applied Mechanics and Technical Physics,1966, no. 3, pp. 43–47.
[5] Cormack D. E., Leal L.G., Imberger J. Natural convection in a shallow cavity with differentially heated end walls. Pt. 1, Asymptotic theory. Journal of Fluid Mechanics, 1974, vol. 65, pp. 209–229.
[6] Zimin V. D., Lyakhov Yu. N., Shaidurov G. F. Sbornik "Gidrodinamika" — Collection "Hydrodynamics", 1971, iss. 3, pp. 126–138.
[7] Bejan A., Al-Homoud A.A., Imberger J. Experimental study of high Rayleigh number convection in a horizontal cavity with different end temperatures. Journal of Fluid Mechanics, 1981, vol. 109, pp. 283–299.
[8] Kirdyashkin A.G. Teplovyye gravitatsionnyye techeniya i teploobmen v astenosfere [Thermal gravitational flows and heat transfer in the asthenosphere]. Novosibirsk, «Nauka» Siberian branch Publ., 1989, 81 p.
[9] Kirdyashkin A.G., Polezhaev V.I., Fedyushkin A.I. Prikladnaya mekhanika i tekhnicheskaya fizika — Applied Mechanics and Technical Physics, 1983, no. 6., pp. 122–128.
[10] Drummond J.E., Korpella S.A. Natural convection in a shallow cavity. Journal of Fluid Mechanics, 1987, vol. 182, pp. 543–564.
[11] Cormack D.E., Leal L.G., Seinfield J.H. Natural convection in a shallow cavity with differentially heated end walls. Pt. 2. Numerical solutions. Journal of Fluid Mechanics, 1974, vol. 65, pp. 231–246.
[12] Fedyushkin A. I., Puntus A. A. Trudy MAI — Trudy MAI, 2018, vol.102, pp. 1–20.
[13] Polezhaev V.I., Bune A.V., Verezub N.A., and others. Matematicheskoye modelirovaniye konvektivnogo teplomassoobmena na osnove uravneniy Nav'ye-Stoksa [Mathematical modeling of convective heat and mass transfer based on Navier-Stokes equations]. Moscow, Nauka Publ., 1987, 272 p.
[14] Polezhaev V. I., Gryaznov V. L. Doklady Akademii nauk SSSR — Proceedings of the USSR Academy of Sciences, 1974, vol. 219, no. 2, pp. 301–304.
[15] Fedyushkin A. I. Issledovaniye matrichnogo metoda resheniya uravneniy konvektsii. Kompleks programm «MARENA» [Research of a matrix method for solving convection equations. Complex of programs "MARENA"]. Moscow, IPM of the USSR Academy of Sciences Publ., 1990, 32 p.
[16] Mazhorova O.S., Popov Yu.P. Doklady Akademii nauk SSSR — Proceedings of the USSR Academy of Sciences,1981, vol. 259, no 3, pp. 64–81.
[17] Fedyushkin A.I., Rozhkov A.N. Spreading of drops on impact on solid surfaces. Actual problems of applied mathematics, computer science and mechanics. Proceedings of the International scientific conference. 2018, pp. 966–977.
[18] Fedyushkin A.I., Rozhkov A.N. Numerical modeling of droplet coalescence. Actual problems of applied mathematics, computer science and mechanics. Proceedings of the International scientific conference. 2018, pp. 978-986.
[19] Patankar S.V. Chislennyye metody resheniya zadach teploobmena i dinamiki zhidkosti [Numerical Heat Transfer and Fluid Flow]. Moscow, Energoatomizdat Publ., 1984. 152 p.
[20] Brackbill J. U., Kothe D. B., Zemach C. A. Continuum Method for Modeling Surface Tension. Journal of Computational Physics, 1992, vol. 100, pp. 335–354.
[21] Muzaferija S., Peric M., Sames P., Schellin. T. A Two-Fluid Navier-Stokes Solver to Simulate Water Entry. In Proc. 22nd Symposium on Naval Hydrodynamics, 1998, pp. 277–289.


Федюшкин А.И. Влияние чисел Рэлея, Прандтля и граничных условий на конвективные течения жидкости в горизонтальных слоях. Математическое моделирование и численные методы. 2020. № 1. с. 28–44.


Работа выполнена при финансовой поддержке программы AAAA-
A20-120011690131-7.


Download article

Количество скачиваний: 217