Developing horizontal convection against stable temperature stratification in a rectangular container

The effect of background stable temperature stratification in the developing stage of horizontal convection is studied by conducting laboratory experiments. By imposing horizontally differential heating at the top of a layer of low-temperature water, both vertical and horizontal temperature differences are explicitly defined. In developing horizontal convection of the present study, the flow structures are driven only by the baroclinic torque produced by the horizontal temperature difference, and braked by the restoring force of stable temperature stratification. We thus defined a nondimensional stratification parameter, which represents the balance of the braking force and the baroclinic torque, in addition to the Rayleigh number. Various features of the flow structures, maximum velocity, stream function, roll thickness, circulation of the roll, total kinetic energy, and Reynolds number, which are quantified via particle-tracking velocimetry, are summarized in spaces of the two nondimensional parameters. In the developing horizontal convection, the quantified flow features are well organized by power laws of the nondimensional parameters. The finite domain of the fluid container augments the effect of the apparent braking force, and the bulk quantities of the roll structures are suppressed by the stable temperature stratification. These results are the evidences for the significance of the nondimensional stratification parameter in the developing horizontal convection, unlike thermally equilibrated horizontal convection conventionally considering destabilizing thermal buoyancy as the primary driving force.

Figure 1

Conceptual diagrams of two different problem settings for HC: (a) thermally equilibrated HC and (b) developing HC. $T_{\mathrm{i}}$ denotes the interior temperature, and $T_{\mathrm{h}}$ and $T_{\mathrm{l}}$ are the temperatures set at the same horizontal plane, $T_{\mathrm{h}}>T_{\mathrm{l}}$.

Figure 2

Schematic of the rectangular fluid container: (a) front view and (b) side view.

Figure 3

Parameters explored in the present study: (a) plots of experimentally controllable parameters, horizontal temperature difference $\mathrm{\Delta }\theta$, and vertical temperature difference $\mathrm{\Delta }T$ and (b) plots of nondimensional parameters in the $\mathrm{Ra}$–${\gamma }_{\mathrm{lin}}^{\prime }$ domain. Color contours in (a) represent ${\gamma }_{\mathrm{lin}}^{\prime }$ values.

Figure 4

Example of flow visualization using the TLC particles of two different temperature ranges at $t=30\min$ with the condition of $\mathrm{\Delta }T=2\mathrm{K}$ and $\mathrm{\Delta }\theta =5\mathrm{K}$. Red and black panels above the fluid layer indicate the locations of the heating copper blocks and insulating rubber sheets. A magnified view of the region enclosed by the solid square of $0.2L\times 0.1L$ is presented in the inset, and qualitative isotherms are indicated by dotted lines. Colors of the TLC particles change as red, green, and blue according to the increase in fluid temperature. The color contrast is adjusted to enhance visibility.

Figure 5

Example of temperature measurements made beneath the copper blocks at $z\sim 0.98L$ under the conditions $\mathrm{\Delta }T=5\mathrm{K}$ and $\mathrm{\Delta }\theta =5\mathrm{K}$: (a) temperatures measured by the thermistor sensors (squares) and corresponding fitting curves and (b) horizontal temperature gradients derived from the fitting curves. Each plot and curve is colored according to the time elapsed from the initiation of heating. Horizontal dotted lines in (a) show the set temperatures of $T=T_0, T_0+\mathrm{\Delta }T$, and $T_0+\mathrm{\Delta }T+\mathrm{\Delta }\theta$ from bottom to top. Horizontal dashed lines indicate typical temperature gradients obtained using the set constants. Regions shaded gray show horizontal positions of the insulating rubber sheet, at which the horizontal temperature difference $\mathrm{\Delta }\theta$ is imposed.

Figure 6

Example of velocity field measurements at $t=90\min$ under the conditions of $\mathrm{\Delta }T=2\mathrm{K}$ and $\mathrm{\Delta }\theta =5\mathrm{K}$: (a) particles tracked through PTV analysis indicated by dots colored according to speed; (b) magnified view of the particle vectors in the region enclosed by the dashed square in (a); (c) overview of the interpolated velocity fields; and (d) magnified view. Only 1% and 4% of the tracked particles are displayed in (a) and (b), respectively. Interpolated velocity vectors are superposed on the color contours of speeds as white arrows in (c) and (d), and only one out of 10 and one out of two vectors are plotted in the $x$ and $z$ directions, respectively, for the sake of clarity in (c).

Figure 7

Example of vorticity and stream function fields at $t=150\min$ under the conditions $\mathrm{\Delta }T=2\mathrm{K}$ and $\mathrm{\Delta }\theta =5\mathrm{K}$: (a) vorticity $\eta$ field and (b) stream function $\psi$ field. Positive and negative contours are drawn using solid and dashed lines.

Figure 8

Extraction of the roll region according to the intensity of the stream function field at $t=180\min$ under the conditions $\mathrm{\Delta }T=2\mathrm{K}$ and $\mathrm{\Delta }\theta =2\mathrm{K}$: (a) stream function field normalized by the maximum value ${\psi }_{\max }$; (b) binarized stream function field with a threshold at 5% intensity of the maximum value; and (c) bottom line of the roll structure fitted by the Gompertz function, showing thickness of the roll structure ${\delta }_{\mathrm{roll}}\left(x\right)$. The position of $\psi ={\psi }_{\max }$ is indicated by a white star in (a). Contour lines of $\psi /{\psi }_{\max }$ are superposed on (b) and (c) as gray solid (positive) and dashed (negative) lines.

Figure 9

Temporal evolution of roll structures at $t\ge 10\min$ under the conditions $\mathrm{\Delta }T=2\mathrm{K}$ and $\mathrm{\Delta }\theta =2\mathrm{K}$ observed in (a)–(h) stream function fields and (i) characteristic values, ${\psi }_{\max }$ and $\overline{{\delta }_{\mathrm{roll}}}$. Color contours shown in (a) to (h) represent the overall view of stream function fields and red dashed lines indicate the roll thickness ${\delta }_{\mathrm{roll}}$. Circles and squares in (i) show changes in ${\psi }_{\max }$ and $\overline{{\delta }_{\mathrm{roll}}}$ with time. Left- and right-pointing triangles are changes in ${\delta }_{\mathrm{roll}}$ at $x=0$ and $x=4L$, respectively. Blue and red solid lines are least-squares fits for ${\psi }_{\max }$ and ${\delta }_{\mathrm{roll}}$. Vertical dashed lines in (i) correspond to the flow fields shown in (a) to (h).

Figure 10

Plots of characteristic values of horizontal convection for all the explored parameters measured at $t\ge 30\min$ in $\mathrm{Ra}$ or ${\gamma }_{\mathrm{lin}}^{\prime }$ domains: (a) and (b) the maximum horizontal velocity $u_{\max }$; (c) and (d) maximum stream function values; and (e) and (f) roll thickness $\overline{{\delta }_{\mathrm{roll}}}$ averaged over the $x$ direction. Colors of plots in (a), (c), and (e) represent the $\mathrm{Ra}$ values while those in (b), (d), and (f) are the stratification parameter ${\gamma }_{\mathrm{lin}}^{\prime }$. Blue arrows in (a), (c), and (e) guide the temporal evolution.

Figure 11

Plots of characteristic values of horizontal convection for all the explored parameters after $t\ge 30\min$ in $\mathrm{Ra}$ or ${\gamma }^{\prime }$ domains: (a) and (b) the maximum horizontal velocity $u_{\max }$; (c) and (d) maximum stream function values; (e) and (f) roll thickness $\overline{{\delta }_{\mathrm{roll}}}$ averaged over the $x$ direction. Colors of plots in (a), (c), and (e) represent the $\mathrm{Ra}$ values while those in (b), (d), and (f) are the stratification parameter ${\gamma }^{\prime }$. Black dashed lines in (a), (c), and (e) are power laws that seemingly fit the present data, and red solid lines in (b), (d), and (f) are power laws predicted in previous studies.

Figure 12

Plots of bulk representative values of horizontal convection after $t\ge 30\min$ as functions of $\mathrm{Ra}$ and ${\gamma }^{\prime }$: (a) and (b) the circulation of the roll region ${\mathrm{\Gamma }}_{\mathrm{c}}$; (c) and (d) total kinetic energy ${\text{KE}}_{\mathrm{t}}$; and (e) and (f) root-mean-squared velocity $U_{\mathrm{rms}}$ (left axis) and Reynolds number $\mathrm{Re}$ (right axis). Colors of plots in (a), (c), and (e) represent the $\mathrm{Ra}$ values, while those in (b), (d), and (f) are the stratification parameter ${\gamma }^{\prime }$. Black dashed lines in (a), (c), and (e) are power laws of ${\gamma }^{\prime }$, while those in (b), (d), and (f) are power laws of $\mathrm{Ra}$, seemingly fitting the present data.

Figure 13

Various quantities plotted in the ${\gamma }^{\prime }$–$\mathrm{Ra}$ domain; (a) maximum horizontal velocity $u_{\max }$, (b) maximum stream function ${\psi }_{\max }$, (c) average roll thickness $\overline{{\delta }_{\mathrm{roll}}}$, (d) circulation of the roll region ${\mathrm{\Gamma }}_{\mathrm{c}}$, (e) total kinetic energy ${\text{KE}}_{\mathrm{t}}$, and (f) the Reynolds number $\mathrm{Re}$. Colors of the plots represent the quantities shown above each panel. Colored contour lines shown behind the plots are the planes fitted in the SVD method, and the scaling factors computed in the method are noted at the upper left of each panel.