Wall effects in non-Boussinesq density currents

Abstract

We report on the results of a numerical study of nearly immiscible contrasted density currents aimed at shedding light on the influence of wall effects on current dynamics in the lock-exchange configuration. The numerical approach is an interface-capturing method which does not involve any explicit reconstruction of the interface. Navier–Stokes equations are solved on a fixed grid and a hyperbolic equation is used for the transport of the local volume fraction of one of the fluids. This allows us to describe the density currents for the complete range of density contrast 10−3≤ρL/ρH≤0.99 (ρL and ρH being the density of the light and heavy fluids) and a wide range of Reynolds number 70≤Re≤5×104 (based on the channel height and the viscosity of the heavy fluid). The use of free-slip vs. no-slip boundary conditions enables us to separate the dissipation at the interface from the dissipation at the boundaries. Present results reveal that wall effects play a significant role on the propagation of contrasted density currents, unlike dissipation at the interface. It is first shown that when wall friction can be neglected, theoretical models based on the inviscid shallow-water approximations and Benjamin's steady-state result describe fairly well the light and heavy front velocities of density currents for the complete range of density ratio. However, when wall friction cannot be neglected, the results depart significantly from the prediction of inviscid theories. It is observed that most of the dissipation in highly contrasted currents takes place at the bottom wall and is a maximum at the head of the heavy current. This dissipation is shown to be responsible for the decrease of the front velocity. We propose a simple model based on Benjamin's analysis that includes wall friction. Keeping in mind the simplicity and limitations of the present model, the prediction of the front velocity of both the heavy and light currents is observed to be in good agreement with the numerical results for the complete range of density contrast. This gives further support to the idea that wall effects are the crucial ingredient for accurately predicting the front velocity of highly contrasted density currents.