Comparative study of buoyancy-driven instability for thermohaline stratification in a Hele-Shaw cell and extended geometry

Abstract

Studies on buoyancy-driven instabilities are of considerable interest for its wide application in oceanic, environmental and industrial fields. For miscible two-layer stratifications in a Hele-Shaw cell, the base state density profile in the diffusion regime can be predicted by linear stability analysis such that the buoyancy-driven instabilities can be classified in (R, δ) parameter space, where R is the initial density stability ratio and δ is the diffusion coefficient ratio. However, flow dynamics in extended geometry do not satisfy Darcy's law. Hence, linear stability analysis cannot be applied to unbounded flow with nonlinear momentum convection. In this study, we carry out a series of numerical simulations to investigate buoyancy-driven instability for thermohaline stratification in both a Hele-Shaw cell and extended geometry at a high thermal Rayleigh number, Ra T = 7 · 10 6 , with R = 0.05, 0.25, 0.8, 1, 4, and 20. The time evolution of the density stability ratio, temperature, salinity and base state density are calculated and compared. Fingers of unbounded thermohaline stratification have a more distorted shape due to the nonlinear momentum convection. However, the base state density profile of thermohaline stratification in extended geometry is found to be similar to that of a Hele-Shaw cell; thus, the classification of buoyancy-driven instabilities in (R, δ) space for miscible two-layer stratifications in Hele-Shaw cells can be extended to the same flow in extended geometry.

Three-dimensional buoyancy-driven convection structures in thermohaline stratification

“…8 Recently, we have found that this classification of the convection modes can also be applied to the unbounded thermohaline flow, which does not obey Darcy's law. 9 In this study, we provide a report on the three-dimensional convection structures in the thermohaline stratification with hot salted water located below the cold fresh water, also referred to simply as the salt-heat system. With the rapid development of supercomputer and parallel computing technology, three-dimensional studies on the miscible two-layers flow have attracted more and more interests.…”

“…(4) are the reciprocals of the respective values in Ref. 9, because the hot salted water is located below the cold fresh water in this study. The unit vector in the vertical direction is denoted as k. The dimensionless position x, velocity u, pressure p, time τ, temperature T, salinity S, Prandtl number Pr and Rayleigh number Ra T are given by:…”

“…Then, according to Ref. 9, this system follows the RT mode for R < R C ≈ 0.508; the MM mode for R c ≤ R < 1; and the DLC mode for R ≥ 1. Then, three values of the density stability ratio, R=0.5, 0.8, and 1.1, are carefully chosen for this comparative study of the salt-heat system under the three convection modes.…”

“…The nondimensional governing equations of the thermohaline stratification, in the Boussinesq approximation, 9 are:…”

“…The governing equations can be written in the stream function-vorticity form and resolved in two dimensions using the implemented self-code based on the HODIE (High-order Differences with Identity Expansion) finite difference scheme. 2,9,13 The corresponding two-dimensional and three-dimensional numerical simulations are also performed using the commercially available CFD code, ANSYS Fluent.…”