Vibration-induced thermal instabilities in supercritical fluids in the absence of gravity

Supercritical fluids (SCFs) are known to exhibit anomalous behavior in their thermophysical properties such as diverging compressibility and vanishing thermal diffusivity on approaching the critical point. This behavior leads to a strong thermomechanical coupling when SCFs are subjected to simultaneous thermal perturbation and mechanical vibration. The behavior of the thermal boundary layer leads to various interesting dynamics such as thermovibrational instabilities, which become particularly ostensive in the absence of gravity. In the present paper, two types of instabilities, Rayleigh-vibrational and parametric instabilities, have been numerically investigated under zero gravity in a two-dimensional configuration using a mathematical model wherein density is calculated directly from the continuity equation. A comparison of experimental observations with numerical simulations is also presented. The peculiarity of the model warrants the investigation of instabilities in a more stringent manner (in terms of higher quench percentage and closer proximity to the critical point), unlike the previous studies wherein the equation of state was linearized around the considered state for the calculation of density, resulting in a less precise analysis. In addition to providing an explanation of the physical causes of these instabilities, we analyze the effect of various parameters on the critical amplitude for the onset of these instabilities. Furthermore, various attributes such as wavelength of the instabilities, their behavior under various factors (quench percentage and acceleration), and the effect of cell size on the critical amplitude are also investigated. Finally, a three-dimensional stability plot is shown describing the type of instability (Rayleigh-vibrational or parametric or both) to be expected for the operating condition in terms of amplitude, frequency, and quench percentage for a given proximity to the critical point.

Figure 1

Schematic of the test case (square cavity of length $h=7\mathrm{mm}$).

Figure 2

Comparison between (a) numerical results and (b) experimental results [21] and for thermovibrational instabilities ($f=20\mathrm{Hz}$ and $A=0.875\mathrm{mm}$). There exists a difference in experimental and numerical conditions (refer to text for explanation).

Figure 3

Schematic illustration for the mechanism leading to Rayleigh-vibrational instabilities.

Figure 4

Critical amplitude (${\mathit{A}}_{\mathit{cr}}$) for the onset of Rayleigh-vibrational instabilities as a function of frequency and distance from the critical point for two different quench temperatures, $\delta T=\text{–}5$ and $\text{–}2.5\mathrm{mK}$. Distance in terms of proximity varies from $T_i-T_c=5$ to $2000\mathrm{mK}$.

Figure 5

Comparison of wavelengths (average), experimental [21] and numerical, for different proximities to the critical point for the same values of $\delta T=\text{–}400\mathrm{mK}$, $A=0.6125\mathrm{mm}$, and $f=30\mathrm{Hz}$. Proximities vary from $T_i-T_c=0.6$ to $2.6\mathrm{K}$.

Figure 6

(a)–(d) Instant velocity-vector plot colored by temperature contours illustrating the merging of fingers [marked in Figs. 6] with time ($T_i-T_c=100\mathrm{mK},\delta T=-10\mathrm{mK},f=10\mathrm{Hz},A=4\mathrm{mm}$).

Figure 7

Wavelength in Rayleigh-vibrational instabilities for different quench percentage as a function of acceleration reduced by earth acceleration for $T_i-T_c=500\mathrm{mK}$.

Figure 8

Wavelength in Rayleigh-vibrational instabilities vs nondimensional acceleration ($\mathit{A}{\mathbf{\omega }}^2/\mathit{g})$ for $T_i-T_c=500\mathrm{mK}$, $q=10\%$ for two different cell sizes ($h=7$ and 14 mm).

Figure 9

Critical Rayleigh-vibrational number with proximity to the critical point for $f=10$ and 30 Hz on a log-log plot.

Figure 10

Parametric instabilities for one time period of vibration for $T_i-T_c=500\mathrm{mK},\delta T=-50\mathrm{mK},f=10\mathrm{Hz},A=10\mathrm{mm}$. Arrows represent the direction of cell movement at that time instant.

Figure 11

Schematic illustration of the mechanism leading to parametric instabilities: (a) formation of a thin TBL and (b) equivalent configuration representing parametric (Faraday) instability such as configuration under an acceleration field.

Figure 12

Parametric instabilities near left wall $AD$ for $T_i-T_c=500\mathrm{mK},\delta T=-50\mathrm{mK},f=20\mathrm{Hz},A=6\mathrm{mm}$ with symmetric condition on the horizontal walls ($AB$ and $CD$).

Figure 13

Velocity vector near the corner ($A$) with a wall condition on the top plate for $T_i-T_c=500\mathrm{mK},\delta T=50\mathrm{mK},f=20\mathrm{Hz},A=6\mathrm{mm}$ illustrating the (a) boundary layer velocity profile and (b) formation of vortices (scale 1:$8\times {10}^5\mathrm{cm}/\mathrm{m}{\mathrm{s}}^{-1}$).

Figure 14

Critical amplitude (${\mathit{A}}_{\mathit{cr}}$) for the onset of parametric instabilities as a function of frequency distance from the critical point for two different quench temperatures, $\delta T=\text{–}100\mathrm{mK}$ (for $T_i-T_c=2000,1500,1000$, and $500\mathrm{mK}$) and $\delta T=\text{–}2.5\mathrm{mK}$ (for $T_i-T_c=100,50$, and 20 mK).

Figure 15

Critical amplitude for parametric instabilities for cells with different dimensions for two proximities to the critical point, $T_i-T_c=500$ and $20\mathrm{mK}$ for $q=10\%$ and $f=20\mathrm{Hz}$.

Figure 16

Wavelength vs reduced acceleration for $T_i-T_c=500\mathrm{mK}$ with different quench percentages.

Figure 17

Stability plot describing critical amplitude for onset of Rayleigh-vibrational and parametric instabilities as a function of quench percentage and frequency.

Figure 18

Stability plot illustrating different instabilities based on their operating conditions for $T_i-T_c=500\mathrm{mK}$.