Flow structures and vertical transport in tilting salt fingers with a background shear

In this work we study the fingering double diffusive convection, namely, the buoyancy-driven convection flow within a fluid layer experiencing an unstable salinity gradient and a stable thermal gradient. In particular, we investigate the influences from a background shear with uniform strength. Linear stability analysis indicates that the unstable modes shift from a circular shape to a sheetlike shape as the shear becomes stronger. Three-dimensional direct numerical simulations are conducted for five groups of cases, each of which has the same combination of thermal and salinity gradients (measured by corresponding Rayleigh numbers) and gradually increasing shear strength. The same properties of seawater are used for all simulations, i.e., the Prandtl number $\text{Pr}=7$ (the ratio of kinematic viscosity to thermal diffusivity) and the Schmidt number $\text{Sc}=700$ (the ratio of kinematic viscosity to salinity diffusivity). Numerical results reveal that a very weak shear organizes the salt fingers into a very regular pattern, which enhances the salinity flux. This enhancement effect, however, reduces as the Rayleigh number increases. For stronger shear the dominant structures shift from salt fingers to salt sheets, and the coherence length scale increases in the streamwise direction. Meanwhile, salinity and heat fluxes decrease. These findings suggest that even a weak shear can notably alter the morphology and transport properties of fingering double diffusive convection.

Figure 1

Sketch of the domain geometry and flow setup. In the horizontal directions the periodic boundary condition is used for all flow quantities.

Figure 2

The real part of the temporal growth factor ${\omega }_r$ in $(k_y,k_x)$ space for three cases with $\text{Ra}={10}^7$, $\mathrm{\Lambda }=1$: (a) unsheared case, (b) $\text{Ri}={10}^4$, (c) $\text{Ri}={10}^2$, and (d) $\text{Ri}=1$. The black solid line denotes the neutral line (${\omega }_r=0$).

Figure 3

The real part of the temporal growth factor ${\omega }_r$ as a function of $(k_x,\text{Ra})$ and $(k_x,\mathrm{\Lambda })$ with (a) $\mathrm{\Lambda }=0.5$, (b) $\mathrm{\Lambda }=1$, (c) $\mathrm{\Lambda }=2$, and (d) $\text{Ra}={10}^7$. For all panels $k_y=0$ and $\text{Ri}=10$. The black dashed lines denote the fastest-growing modes, and dots mark the spanwise length scales of the dominant structures in DNS results, respectively. The scaling laws in (a) and (b) are given by linear fitting with slopes 0.13 and 0.24, respectively. The relationships in (c) and (d) are directly obtained from Eqs. (8) and shown by the black solid lines, respectively.

Figure 4

The initial time evolution of the mean salinity vertical profile, the mean temperature vertical profile, the Turner angle profile, and the Reynolds number (from top to bottom), with (a) $\mathrm{\Lambda }=0.5$, (b) $\mathrm{\Lambda }=1$, and (c) $\mathrm{\Lambda }=2$ (from left to right). The other control parameters read $\text{Ra}={10}^7$ and $\text{Ri}={10}^4$.

Figure 5

Three-dimensional volume rendering of the instantaneous salinity field with (a) no background shear, (b) $\text{Ri}={10}^4$, (c) $\text{Ri}={10}^3$, (d) $\text{Ri}={10}^2$, and (e) $\text{Ri}=10$. The color is determined by the salinity in two end values (left) and a single end value (right). The other control parameters all read $\text{Ra}={10}^7$, $\mathrm{\Lambda }=1$. The spanwise half wavelength $d$ and the streamwise length scale ${\lambda }_y$ calculated by the autocorrelation function of salinity (Fig. 6) are shown in the right column. In (e) the width $d^l$ predicted by the linear stability analysis is also shown.

Figure 6

The autocorrelation coefficient $C_s$ of salinity for the streamwise separation ${\delta }_y$ and the spanwise separation ${\delta }_x$ in the vertical midplane. Different shear strengths are set as unsheared for (a) and $\text{Ri}={10}^5$, $4\times {10}^4$, ${10}^4$, $4\times {10}^3$, ${10}^3$, ${10}^2$, and 10 for (b–h), respectively. The other control parameters all read $\text{Ra}={10}^7$, $\mathrm{\Lambda }=1$. The spanwise half wavelength $d$ and the streamwise length scale ${\lambda }_y$ are denoted by black solid lines in each panel.

Figure 7

The streamwise correlation length scale determined from the autocorrelation function versus the shear strength. For all cases Ra is fixed at ${10}^7$.

Figure 8

The mean profiles of (a) salinity, (b) temperature, (c) standard deviation of salinity, and (d) standard deviation of temperature. The bar and the bracket stand for the temporal and horizontal average value, respectively. In each panel three cases with $\text{Ra}={10}^7,\mathrm{\Lambda }=1$ and different shear strengths are shown.

Figure 9

(a) The central salinity gradient ${\langle {\partial }_z\overline{s}\rangle }_b$ vs ${\text{Ri}}^{-1}$. (b) The salinity boundary layer thickness ${\lambda }_s$ vs ${\text{Ri}}^{-1}$. For all cases Ra is fixed at ${10}^7$.

Figure 10

The time-averaged salinity contour in the $z=0.9$ plane for (a) the unsheared case, (b) $\text{Ri}={10}^4$, and (c) $\text{Ri}=10$. For all the cases $\text{Ra}={10}^7,\mathrm{\Lambda }=1$.

Figure 11

The global fluxes for different shear strengths normalized by the values of no-shear cases. (a) Salinity Nusselt number, (b) heat Nusselt number, and (c) Reynolds number. For all cases Ra is fixed at ${10}^7$. For $\mathrm{\Lambda }=(0.5,1,2)$, ${\text{Nu}}_s^0=(33.8,32.3,31.0)$, ${\text{Nu}}_{\theta }^0=(1.32,1.17,1.09)$, and ${\text{Re}}_z^0=(0.93,0.78,0.65).$

Figure 12

Three-dimensional volume rendering of the instantaneous salinity field with (a) $\text{Ra}={10}^8$ and (b) $\text{Ra}={10}^9$. The other control parameters read $\text{Ri}={10}^4$, $\mathrm{\Lambda }=1$.

Figure 13

The streamwise correlation length scale determined from the autocorrelation function versus the shear strength. $\mathrm{\Lambda }$ is fixed at 1.

Figure 14

The global fluxes for different shear strengths normalized by the values of no-shear cases. (a) Salinity Nusselt number, (b) heat Nusselt number, and (c) Reynolds number. $\mathrm{\Lambda }$ is fixed at 1. For $\text{Ra}=({10}^7,{10}^8,{10}^9)$, ${\text{Nu}}_s^0=(32.3,63.8,135)$, ${\text{Nu}}_{\theta }^0=(1.17,1.35,1.77)$, ${\text{Re}}_z^0=(0.78,2.02,5.50).$