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Influence of High Frequency Vertical Vibrations on Convective Regimes in a Closed Cavity at Normal and Low Gravity Conditions

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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.

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Authors and Affiliations

  1. Perm National Research Polytechnic University, Perm Komsomolsky prospect, 29, 614990, Perm, Russia

    A. V. Perminov &  S.A.Nikulina

  2. Institute of Continuous Media Mechanics, Ural Branch of the Russian Academy of Sciences, Perm, Acad. Koroleva Str., 1, 614013, Perm, Russia

    T. P. Lyubimova

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  1. A. V. Perminov
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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.

Fig. 18
figure 18

Dependence of Nu a and ψm b on the number of grid nodes N along the axis x for V = 0 and Gr = 3500

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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.

Fig. 19
figure 19

Dependences of Nu a and ψm b on the number of the grid nodes N along the axis x for V = 0.1 and Gr = 300

Full size image

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.

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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

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