Viscous lock-exchange in rectangular channels

TL;DR

This paper analytically calculates the viscous lock-exchange diffusion coefficient in rectangular channels of any aspect ratio, compares it with experiments, and discusses modeling approaches including the 2D Stokes-Darcy model with Brinkman correction.

Contribution

The paper provides the first analytical expression for the lock-exchange diffusion coefficient in rectangular channels of arbitrary aspect ratio and validates it with experimental data.

Findings

Analytical formula matches experimental results across aspect ratios.

2D Stokes-Darcy model with Brinkman correction approximates the flow well.

Diffusion coefficient depends on channel geometry and flow regime.

Abstract

In a viscous lock-exchange gravity current, which describes the reciprocal exchange of two fluids of different densities in a horizontal channel, the front between two Newtonian fluids spreads as the square root of time. The resulting diffusion coefficient reflects the competition between the buoyancy driving effect and the viscous damping, and depends on the geometry of the channel. This lock-exchange diffusion coefficient has already been computed for a porous medium, a 2D Stokes flow between two parallel horizontal boundaries separated by a vertical height, H, and, recently, for a cylindrical tube. In the present paper, we calculate it, analytically, for a rectangular channel (horizontal thickness b, vertical height, H) of any aspect ratio (H/b) and compare our results with experiments in horizontal rectangular channels for a wide range of aspect ratios (1/10-10). We also discuss the 2D Stokes-Darcy model for flows in Hele-Shaw cells and show that it leads to a rather good approximation, when an appropriate Brinkman correction is used.