Scaling laws and flow structures of double diffusive convection in the finger regime

Abstract

Direct numerical simulations are conducted for double diffusive convection (DDC) bounded by two parallel plates. The Prandtl numbers, i.e. the ratios between the viscosity and the molecular diffusivities of scalars, are similar to the values of seawater. The DDC flow is driven by an unstable salinity difference (here across the two plates) and stabilized at the same time by a temperature difference. For these conditions the flow can be in the finger regime. We develop scaling laws for three key response parameters of the system: The non-dimensional salinity flux Nu S mainly depends on the salinity Rayleigh number Ra S , which measures the strength of the salinity difference, and exhibits a very weak dependence on the density ratio Λ, which is the ratio of the buoyancy forces induced by two scalar differences. The non-dimensional flow velocity Re and the non-dimensional heat flux Nu T are dependent on both Ra S and Λ. However, the rescaled Reynolds number ReΛ α eff u and the rescaled convective heat flux (Nu T − 1)Λ α eff T depend only on Ra S . The two exponents are dependent on the fluid properties and are determined from the numerical results as α eff u = 0.25 ± 0.02 and α eff T = 0.75 ± 0.03. Moreover, the behaviors of Nu S and ReΛ α eff u agree with the predictions of the Grossmann-Lohse theory which was originally developed for the Rayleigh-Bénard flow. The non-dimensional salt-finger width and the thickness of the velocity boundary layers, after being rescaled by Λ α eff u /2 , collapse and obey a similar power-law scaling relation with Ra S . When Ra S is large enough, salt fingers do not extend from one plate to the other and horizontal zonal flows emerge in the bulk region. We then show that the current scaling strategy can be successfully applied to the experimental results of a heat-copper-ion system (Hage and Tilgner, Phys. Fluids, 22, 076603, 2010). The fluid has different properties and the exponent α eff u takes a different value 0.54 ± 0.10.

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

“…The agreement is very good, which has also been found in our previous studies [22][23][24]. Our previous studies with no-slip boundary conditions revealed that for fixed Ra S , as Λ increases within the finger regime, Nu S only changes slightly and Re decreases significantly [22,24].…”

“…Those experiments provide a good platform to investigate the vertical scalar transport and flow structures of finger layers. Our previous DNS successfully reproduced most key observations of the experiments, and good agreement was obtained between our numerical results and the experimental results [22][23][24].…”

“…Our previous studies with no-slip boundary conditions revealed that for fixed Ra S , as Λ increases within the finger regime, Nu S only changes slightly and Re decreases significantly [22,24]. The present results suggest that it is also true for the free-slip cases; see…”

A Reduced Model for Salt-Finger Convection in the Small Diffusivity Ratio Limit

“…The dynamics of the IFSC model (20) are driven by the linear salt-finger instability, which has the distinctive feature that the optimal (horizontal) wavenumber is finite because of the coexistence of large-and small-scale damping stemming from the stabilizing temperature and viscosity, respectively. In the following, we therefore assume that the characteristic horizontal scale of the system is determined by the optimal wavenumber k opt and confirm this assumption in Figure 9, which shows the correspondence between the peak in the energy spectrum obtained from numerical simulations ( Figure 6) and expression (24) for the optimal wavenumber. Here, the constant ratio e 0.15 ≈ 1.16 between k finger and k opt confirms the validity of our assumption.…”

“…In this paper, we therefore study the properties of this model at some length, albeit in two dimensions. To do so, we employ a doubly-periodic setting as appropriate for oceanic and astrophysical applications with distant boundaries, in contrast to earlier studies of salt-finger convection in vertically-bounded domains, e.g., [21][22][23][24][25]. In this formulation, elevator modes consisting of vertically-invariant upward and downward moving fingers are exact nonlinear solutions, and we believe that analogous modes are crucial even in vertically-bounded domains because of their efficiency in extracting energy from the salinity field.…”