Vertically Bounded Double Diffusive Convection in the Finger Regime: Comparing No-Slip versus Free-Slip Boundary Conditions

Vertically bounded fingering double diffusive convection is numerically investigated, focusing on the influences of different velocity boundary conditions, i.e., the no-slip condition, which is inevitable in the lab-scale experimental researches, and the free-slip condition, which is an approximation for the interfaces in many natural environments, such as the oceans. For both boundary conditions the flow is dominated by fingers and the global responses follow the same scaling laws, with enhanced prefactors for the free-slip cases. Therefore, the laboratory experiments with the no-slip boundaries serve as a good model for the finger layers in the ocean. Moreover, in the free-slip case, although the tangential shear stress is eliminated at the boundaries, the local dissipation rate in the near-wall region may exceed the value found in the no-slip cases, which is caused by the stronger vertical motions of horizontally focused fingers and sheet structures near the free-slip boundaries. This counterintuitive result might be relevant for properly estimating and modeling the mixing and entrainment phenomena at free surfaces and interfaces widespread in oceans and geophysical flows.

Figure 1

The explored parameters shown in the ${\mathrm{Ra}}_S–\mathrm{\Lambda }$ plane and colored by ${\mathrm{Ra}}_T$. The black circle marks the case shown in Fig. 2.

Figure 2

The volume rendering of salt fingers for $({\mathrm{Ra}}_S,\mathrm{\Lambda })=(5\times 10^7,2.0)$ (marked by a black circle in Fig. 1) with (a) the no-slip boundary condition and aspect ratio 2.0 and (b) the free-slip boundary condition and aspect ratio 2.4, respectively. The two plots share the same color map and opacity settings.

Figure 3

System responses versus ${\mathrm{Ra}}_S$. (a) ${\mathrm{Nu}}_S$ compensated by ${\mathrm{Ra}}_S^{1/3}$, (b) ${\mathrm{Nu}}_T-1$, (c) Re compensated by ${\mathrm{Ra}}_S^{1/2}$, and (d) $({\lambda }_s/L{)}^{-1}$ compensated by ${\mathrm{Ra}}_S^{1/3}$. The free-slip cases are marked by open symbols and the no-slip cases by solid symbols, respectively. Symbols are colored according to the density ratio $\mathrm{\Lambda }$. The dashed line in (a) represents the Grossmann-Lohse prediction.

Figure 4

(a) The mean profiles of $D_a=⟨S_{ij}{S}_{ij}{⟩}_h$ (black dashed lines) and the contributions from the diagonal components of the strain-rate tensor $D_d=⟨{\mathrm{\Sigma }}_{i=j}(S_{ij}{S}_{ij}){⟩}_h$ (red lines) and the off-diagonal components $D_o=⟨{\mathrm{\Sigma }}_{i\ne j}(S_{ij}{S}_{ij}){⟩}_h$ (blue lines) for the free-slip case and no-slip cases. The control parameters are ${\mathrm{Ra}}_S=5\times 10^7$ and $\mathrm{\Lambda }=2.0$. Because of the symmetry about $z=0.5$, only the lower half of the domain is shown here. (b) The averaged dissipation rate ${ϵ}_b$ at boundary. The symbols and color map are the same as in Fig. 3.

Figure 5

The contours on a horizontal plane near the bottom plate at $z/L=0.005$ for the flow field shown in Fig. 2 with the free-slip boundary condition. (a) The vertical velocity $u_3$, (b) the salinity $s$, and (c) $D_d\equiv {\mathrm{\Sigma }}_{i=j}(S_{ij}{S}_{ij})$, which dominates the local dissipation rate.