Abstract
The influence of high-frequency vibrations on the convection of a liquid in an infinitely long horizontal cylinder of square cross-section, which undergoes vertical vibrations, is investigated. The problem was solved numerically on the basis of the averaged equations of thermal vibrational convection, written in terms of the vorticity of the average velocity and stream functions of the average and pulsating flows. The influence of vibrations on the system was determined by a dimensionless vibration parameter V proportional to the ratio of vibrational acceleration to gravitational acceleration and independent of the temperature difference. The values V ≥ 1 correspond to the case of low gravity conditions. The intensity of gravitational convection was characterized by the Grashof number Gr. All calculations were performed for the fixed value of the Prandtl number Pr = 100. For values 0 ≤ V ≤ 10 the evolution of average convective regimes was studied and a map of these regimes was plotted on the Gr—V parameter plane. The stability boundary of stationary average convection is determined. It is shown that after the loss of stability by a stationary average flow in a cavity, two oscillatory average convective regimes with different symmetries can be realized.
Similar content being viewed by others
References
Anisimov, I.A., Birikh R.V.: Hydrodynamic instability of vibrational advective flow under zero gravity. Vibration effects in hydrodynamics: Proceedings of the Perm State University, 17–24 (1998)
Batchelor, G.K.: Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quart. Appl. Math. 12(3), 209–233 (1954). https://doi.org/10.1090/qam/64563
Berkovskii, B.M., Polevikov, V.K.: Effect of the Prandtl number on the convection field and the heat transfer during natural convection. J. Eng. Phys. 24(5), 598–603 (1973). https://doi.org/10.1007/BF00838619
Birikh, R.V.: Vibrational convection in a plane layer with a longitudinal temperature gradient. Fluid Dyn. 25(4), 500–503 (1990). https://doi.org/10.1007/BF01049852
Boaro, A., Lappa, M.: Multicellular states of viscoelastic thermovibrational convection in a square cavity. Phys. Fluids 33(033105), 1–18 (2021). https://doi.org/10.1063/5.0041226
Bouarab, S., Mokhtari, F., Kaddeche, S., Henry, D., Botton, V., Medelfef1, A.: Theoretical and numerical study on high frequency vibrational convection: Influence of the vibration direction on the flow structure. Phys. Fluids. 31, 043605, 1–10 (2019). https://doi.org/10.1063/1.5090264
Crewdson, G., Lappa, M.: The zoo of modes of convection in liquids vibrated along the direction of the temperature gradient. Fluids. 6(30), 1–23 (2021). https://doi.org/10.3390/fluids6010030
Davis, G. de Vahl.: Laminar natural convection in an enclosed rectangular cavity. Int. J. Heat Mass Transf. 11(11), 1675–1693 (1968). https://doi.org/10.1016/0017-9310(68)90047-1
Davis, G. de Vahl, Thomas, R.W.: Natural convection between concentric vertical cylinders. Phys. Fluids. 12(12), II-198 – II-207 (1969). https://doi.org/10.1063/1.1692437
Demin, V.A., Gershuni, G.Z., Verkholantsev, I.V.: Mechanical quasiequilibrium and thermovibrational convective instability in an inclined fluid layer. Int. J. Heat Mass Transf. 39(9), 1979–1991 (1996). https://doi.org/10.1016/0017-9310(95)00239-1
Eckertf, E.R.G., Carlson, Walter O.: Natural convection in an air layer enclosed between two vertical plates with different temperatures. Int. J. Heat Mass Transf. 2(1–2), 106–110 (1961). https://doi.org/10.1016/0017-9310(61)90019-9
Elder, J.W.: Laminar free convection in a vertical slot. J. Fluid Mech. 23(1), 77–98 (1965). https://doi.org/10.1017/S0022112065001246
Elder, J.W.: Numerical experiments with free convection in a vertical slot. J. Fluid Mech. 24(4), 823 – 843 (1966). https://doi.org/10.1017/S0022112066001022
Fernandez, J., Salgado Sanchez, P., Tinao, I., Porter, J., Ezquerro, J.M.: The CFVib experiment: control of fluids in microgravity with vibrations. Microgravity Sci. Technol. 29, 351–364 (2017)
Gandikota, G., Chatain, D., Lyubimova, T., Beysens, D.: Dynamic equilibrium under vibrations of H2 liquid-vapor interface at various gravity levels. Physic. Rev. E. 89(6), 063003 (2014a)
Gandikota, G., Chatain, D., Amiroudine, S., Lyubimova, T., Beysens, D.: Faraday instability in a near-critical fluid under weightlessness. Phys. Rev. E. 89(1), 013022 (2014b)
Gandikota, G., Chatain, D., Amiroudine, S., Lyubimova, T., Beysens, D.: Frozen-wave instability in near-critical hydrogen subjected to horizontal vibration under various gravity fields. Physic. Rev. E. 89(1), 012309 (2014c)
Gershuni, G.Z., Zhukhovitskii, E.M., Tarunin, E.L.: Numerical investigation of convective motion in a closed cavity. Fluid Dyn. 1(5), 38–42 (1966). https://doi.org/10.1007/BF01022148
Gershuni, G.Z., Zhukhovitskii, E.M.: On free thermal convection in a vibrational field under conditions of weightlessness. DAN USSR. 249(3), 580–584 (1979)
Gershuni, G.Z., Zhukhovitskii, E.M.: Convective instability of a fluid in a vibration field under conditions of weightlessness. Fluid Dyn. 16(4), 498–504 (1981). https://doi.org/10.1007/BF01094590
Gershuni, G.Z., Zhukhovitskii, E.M.: Plane-parallel advective flows in a vibration field. J. Eng. Phys. 56(2), 160–163 (1989). https://doi.org/10.1007/BF00870570
Gershuni, G.Z., Lyubimov, D.V.: Thermal Vibrational Convection. Wiley, N.Y. et al (1998)
Landau, L.D. and Lifshits, E.M.: Theoretical physics: Mechanics. Nauka, FizMatLit, Moscow, 1, (2001)
Lappa, M.: Control of convection patterning and intensity in shallow cavities by harmonic vibrations. Microgravity Sci. Technol. 28, 29–39 (2016). https://doi.org/10.1007/s12217-015-9467-4
Lizée, A., Alexander, J.I.D.: Chaotic thermovibrational flow in a laterally heated cavity. Phys. Rev. E 56(4), 4152–4156 (1997). https://doi.org/10.1103/PhysRevE.56.4152
Lyubimov, D.V.: Thermovibrational flows in a fluid with a free surface. Microgravity q. 4(2), 107–112 (1994)
Lyubimov, D.V.: Convective flows under the influence of high frequency vibrations. Eur. J. Mech. B/Fluids. 14(4), 439–458 (1995)
Lyubimov, D.V., Cherepanov, A.A., Lyubimova, T.P., Roux, B.: The flows induced by a heated oscillating sphere. Int. J. Heat and Mass Transf. 38(11), 2089–2100 (1995). https://doi.org/10.1016/0017-9310(94)00327-R
Lyubimov, D.V., Lyubimova, T.P., Roux, B.: Mechanisms of vibrational control of heat transfer in a liquid bridge. Int. J. Heat Mass Transf. 40(17), 4031–4042 (1997). https://doi.org/10.1016/S0017-9310(97)00053-7
Lyubimov, D., Kolchanova, E., Lyubimova, T.: Vibration effect on the nonlinear regimes of thermal convection in a two-layer system of fluid and saturated porous medium. Transp. Porous Med. 106(2), 237–257 (2015). https://doi.org/10.1007/s11242-014-0398-0
Lyubimov, D.V., Lyubimova, T.P., Tcherepanov, A.A.: Dynamics of Fluid Interfaces in Vibrational Fields. Moscow, Phys. Math Lit., 216 (2003)
Lyubimov, D.V., Khilko, G.L., Ivantsov, A.O., Lyubimova, T.P.: Viscosity effect on the longwave instability of a fluid interface subjected to horizontal vibrations. J. Fluid Mech. 814, 24–41 (2017)
Lyubimova, T., Lyubimov, D., Parshakova, Y.: Vertical vibration effect on stability of conductive state of two-layer system with deformable interface. Int. J. Heat Mass Transf. 92, 1158–1165 (2016)
Lyubimova, T., Ivantsov, A., Garrabos, Y., Lecoutre, C., Gandikota, G., Beysens, D.: Band instability in near-critical fluids subjected to vibration under weightlessness. Phys. Rev. E. 95(1), art. no. 013105 (2017)
Lyubimova, T., Ivantsov, A., Garrabos, Y., et al.: Faraday waves on band pattern under zero gravity conditions. Physic Rev. Fluids. 4(6), 064001 (2019)
Lyubimova, T.P., Perminov, A.V.: Vibration effect on a stability of stationary flow of pseudoplastic fluid in vertical slot. Int. J. Heat Mass Transf. 126, 545–556 (2018). https://doi.org/10.1016/j.ijheatmasstransfer.2018.05.044
Mialdun, A., Ryzhkov, I.I., Melnikov, D., Shevtsova, V.: Experimental Evidence of Thermal Vibrational Convection in Nonuniformly Heated Fuid in a Reduced Gravity Environment. Phys. Rev. Lett. 101(084501), 1–4 (2008). https://doi.org/10.1103/PhysRevLett.101.084501
Perminov, A.V.: Stability of the rigid state of a generalized newtonian fluid. Fluid Dyn. 49(2), 140–148 (2014). https://doi.org/10.1134/S0015462814020033
Poots, G.: Heat transfer by laminar free convection in enclosed plane gas layers. The Quarterly Journal of Mechanics and Applied Mathematics. 11(3), 257–273 (1958). https://doi.org/10.1093/qjmam/11.3.257
Porter, J., Salgado Sanchez, P., Shevtsova, V., Yasnou, V.: A review of fluid instabilities and control strategies with applications in microgravity. Mathematical Modelling of Natural Phenomena. 16, 24 (2021)
Quon, Ch.: High Rayleigh Number Convection in an Enclosure — A Numerical Study. Physic. Fluids. 15(12), 12 – 19 (1972). https://doi.org/10.1063/1.1693729
Rubel, A., Landis, F.: Numerical Study of Natural Convection in a Vertical Rectangular Enclosure. Physic. Fluids. 12(12), II-208 – II-213 (1969). https://doi.org/10.1063/1.1692438
Shevtsova, V., Ryzhkov, I.I., Melnikov, D.E., Gaponenko, Y.A., Mialdun, A.: Experimental and theoretical study of vibration-induced thermal convection in low gravity. J. Fluid Mech. 648, 53–82 (2010). https://doi.org/10.1017/S0022112009993442
Shevtsova, V., Gaponenko, Y.A., Sechenyh, V., Lyubimova, T., Melnikov, D.E., Mialdun, A.: Dynamics of a binary mixture subjected to a temperature gradient and oscillatory forcing. J. Fluid Mech. 767, 290–322 (2015)
Salgado Sanchez, P., Yasnou, V., Gaponenko, Y., Mialdun, A., Porter, J., Shevtsova, V.: Interfacial phenomena in immiscible liquids subjected to vibrations in microgravity. J. Fluids Mech. 865, 850–883 (2019a)
Salgado Sanchez, P., Fernandez, J., Tinao, I. and Porter, J.: Vibroequilibria in microgravity: comparison of experiments and theory. Physic. Rev. E. 99, 042803 (2019b)
Tarunin, E.L.: Numerical study of free convection. Proceedings of the Perm State University: Hydrodynamics Series. 184(1), 135–168 (1968)
Thom, A., Apelt, C.J.: Field Computations in Engineering and Physics. Van Nostrand, London (1961)
Vorobev, A., Lyubimova, T.: Vibrational convection in a heterogeneous binary mixture. Part 1. Time-Averaged Equations. J. Fluid Mech. 870, 543–562 (2019a)
Vorobev, A., Lyubimova, T.: Vibrational convection in a heterogeneous binary mixture. Part 2. Frozen waves. J. Fluid Mech. 870, 563–594 (2019b)
Yu, W.C., Chen, Z.B., Hsu, W.T., Roux, B., Lyubimova, T.P., Lan, C.W.: Effects of angular vibration on the flow, segregation, and interface morphology in vertical Bridgman crystal growth. Int. J. Heat Mass Transf. 50(1–2), 58–66 (2007). https://doi.org/10.1016/j.ijheatmasstransfer.2006.06.040
Zen’kovskaya, S.M., Simonenko, I.B.: Effect of high frequency vibration on convection initiation. Fluid Dynamics. 1(5), 35–37 (1966). https://doi.org/10.1007/BF01022147
Zharikov, E.V., Prihod’ko, L.V., Storozhev, N.R.: J. Crystal Growth. 99(1–4), 910–914 (1990). https://doi.org/10.1016/S0022-0248(08)80051-6
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A
Appendix A
All calculations were performed on a uniform spatial grid with square cells of the size Δh = 1/40, which corresponded to N = 80 grid points along the coordinate axes. The step of the spatial grid was chosen by studying the convergence of the solution with decreasing Δh, which was carried out over the entire range of values of the vibration parameter V. The typical behavior of the solution to the problem with decreasing the spatial step (with an increase in the number of grid nodes N along the coordinate axis) is shown in Figs. 18 and 19.
Figure 18 shows the dependence of the Nusselt number Nu (a), determined by the formula (4), and the maximal value of the stream function ψm (b) on the number of grid nodes N along the x axis for V = 0 and Gr = 3500. Calculations have shown that for N < 80 (Δh > 1/40) in the absence of vibrations, oscillatory solutions are observed. The solid lines in Fig. 18 correspond to the average, and the dashed lines to the maximal and minimal values of Nu and ψm. With increasing N (decreasing Δh), the oscillation amplitude decreases. When N ≥ 80 (Δh ≤ 1/40), stationary convection is observed (see Fig. 2), and the values of Nu and ψm stop changing with increasing N.
In figure 2 пthe dependence of the Nusselt number Nu (a) and the maximal value of the stream function ψm (b) on the number of the grid nodes N along the axis x for V = 0.1 and Gr = 300, which corresponds to the domain IV in Fig. 2, i.e. to the oscillatory regime of convection.
As before, in Fig. 19, the solid lines correspond to the average, and the dashed lines to the amplitude values of Nu and ψm. It is seen that with increasing N (decreasing Δh), the Nusselt number and the maximal value of the stream function asymptotically tend to certain values. The relative change of Nu and ψm with an increase in the number of nodes from N = 80 (Δh = 1/40) to 100 (Δh = 1/50) does not exceed 1%.
As it was mentioned before, at high Grashof numbers in region IV in Fig. 2, near the solid walls of the cavity the boundary layers are formed in which large gradients of velocity and / or temperature are observed. The minimal thickness of the boundary layer is observed near the vertical wall and, depending on the parameters of the problem, can be 0.1 dimensionless units or more. For a spatial grid step Δh = 1/40, the boundary layer contains no less than 4—5 nodes. Based on the study of the convergence of the solution with decreasing Δh, it can be argued that this number of nodes is sufficient to resolve the boundary layers.
Rights and permissions
About this article
Cite this article
Perminov, A.V., Lyubimova, T.P. & S.A.Nikulina Influence of High Frequency Vertical Vibrations on Convective Regimes in a Closed Cavity at Normal and Low Gravity Conditions. Microgravity Sci. Technol. 33, 55 (2021). https://doi.org/10.1007/s12217-021-09898-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12217-021-09898-0