Cooling beyond the boundary value in supercritical fluids under vibration

Supercritical fluids when subjected to simultaneous quench and vibration have been known to cause various intriguing flow phenomena and instabilities depending on the relative direction of temperature gradient and vibration. Here we describe a surprising and interesting phenomenon wherein temperature in the fluid falls below the imposed boundary value when the walls are quenched and the direction of vibration is normal to the temperature gradient. We define these regions in the fluid as sink zones, because they act like sink for heat within the fluid domain. The formation of these zones is first explained using a one-dimensional (1D) analysis with acceleration in constant direction. Subsequently, the effect of various boundary conditions and the relative direction of the temperature gradient to acceleration are analyzed, highlighting the necessary conditions for the formation of sink zones. It is found that the effect of high compressibility and the action of self-weight (due to high acceleration) causes the temperature to change in the bulk besides the usual action of piston effect. This subsequently affects the overall temperature profile thereby leading to the formation of sink zones. Though the examined 1D cases differ from the current two-dimensional (2D) cases, owing to the direction of acceleration being normal as compared to parallel in case of former, the explanations pertaining to 1D cases are judiciously utilized to elucidate the formation of sink zones in 2D supercritical fluids subjected to thermal quench and vibrational acceleration. The appearance of sink zones is found to be dependent on several factors such as proximity to the critical point and acceleration. A surface three-dimensional plot illustrating the effect of these parameters on onset time of sink zones is presented to further substantiate these arguments.

Figure 1

Schematic of the test case.

Figure 2

$\left(T-T_w\right)$ contour plots for $T_i-T_c=100\mathrm{mK}$, $\delta T=10\mathrm{mK}$, $f=20\mathrm{Hz}$, and $A=5\mathrm{mm}$ at various time instances: (a) $t=4\mathrm{s}$, (b) $t=7\mathrm{s}$, (c) $t=10\mathrm{s}$. (d) Temperature profile at $y=0.3\mathrm{mm}$ at various time instances. Insets (a–c) and (d) highlight the growth of sink zones with time. The dashed lines in (a–c) represent the boundary where $T-T_w=0$.

Figure 3

Temperature profile for $T_i-T_c=100\mathrm{mK},\delta T=10\mathrm{mK},f=20\mathrm{Hz},A=5\mathrm{mm}$ at $10\mathrm{s}$. (a) Bottom wall adiabatic, sink zones are formed; (b) bottom wall isothermal ($\mathrm{i}.\mathrm{e}.$, at $T_i$), no sink zones are formed. The dashed lines in (a) represent the boundary where $T-T_w=0$.

Figure 4

Temperature contour plots for $T_i-T_c=100\mathrm{mK},\delta T=10\mathrm{mK},f=20\mathrm{Hz},A=5\mathrm{mm}$ at various instances of time along with streamlines depicting direction of motion of vibration. (a–b) For one-half period; (c–d) growth of sink zone with time.

Figure 5

Temperature contour plots illustrating sink zones (shaded regions) for various proximities to the critical point and other parameters as indicated. The temperature contours are represented by lines.

Figure 6

(a) Schematic of 1D case with both walls quenched; (b) evolution of the temperature field for $T_i-T_c=20\mathrm{mK},\delta T=10\mathrm{mK}$; solid lines: $100\vec{g}$, dotted lines: zero gravity.

Figure 7

(a) Schematic of 1D case with bottom wall quenched and top adiabatic; (b) evolution of the temperature field for $T_i-T_c=20\mathrm{mK},\delta T=10\mathrm{mK}$; solid lines: $100\vec{g}$, dotted lines: zero gravity.

Figure 8

(a) Schematic of 1D case with top wall quenched and bottom adiabatic; (b): evolution of the temperature field for $T_i-T_c=20\mathrm{mK},\delta T=10\mathrm{mK}$; solid lines) $100\vec{g}$, dotted lines: zero gravity.

Figure 9

Evolution of the temperature field ($T-T_w)$ (b, d) for $T_i-T_c=20\mathrm{mK},\delta T=10\mathrm{mK}$; solid lines: $100\vec{g},$ dotted lines: zero gravity for configurations shown in (a) and (c), respectively.

Figure 10

(a) Schematic of the phenomena leading to formation of sink zones in 2D flow; (b) streamlines near the corner for $T_i-T_c=100\mathrm{mK},\delta T=10\mathrm{mK},f=20\mathrm{Hz},A=5\mathrm{mm}$ at $t=10\mathrm{s}$.

Figure 11

Evolution of the temperature field ($T-T_w)$ for perfect gas (${\mathrm{H}}_2$) with $T_w=278\mathrm{K}$, quench $\delta T=10\mathrm{mK}$ under an acceleration of $100\vec{g}$ for schematic as in Fig. 6.

Figure 12

Time of appearance of sink zones for the schematic as shown in Fig. 6 for a quench of $\delta T=10\mathrm{mK}$ (a) for different proximities to the critical point for an acceleration of $100\vec{g}$. (b) Surface 3D plot for different accelerations and proximities to the critical point.