Viscous lock-exchange in rectangular channels

J Martin,N Rakotomalala,L Talon,D Salin

doi:10.1017/s0022112010006208

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 two-dimensional (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 and 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 to 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.

Highlights

The lock-exchange configuration refers to the release, under gravity, of the interface between two fluids of different densities, confined in the section of a horizontal channel
The viscous lock-exchange diffusion coefficient reflects the competition between the buoyancy-driving effect and the viscous damping, and depends on the geometry of the channel
We recall its computation for a porous medium as already found by Huppert & Woods (1995), and compute it for a 2D Stokes flow between two parallel horizontal boundaries separated by a vertical height, H

Summary

Introduction

The lock-exchange configuration refers to the release, under gravity, of the interface between two fluids of different densities, confined in the section of a horizontal channel. The slumping phase refers to the initial regime where inertia dominates over viscous forces, which typically applies for the case of salted and fresh water in a tank. In this regime, Benjamin (1968) and, more recently, Shin, Dalziel & Linden (2004) have shown that in the presence of a small density contrast (i.e. in the Boussinesq approximation ρ ρ), the two opposite currents travelled at the same constant velocity.

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