Two-dimensional variable-density Navier–Stokes simulations have been conducted in order to investigate the effects of a slope on the classical lock-exchange flow. Simulations of full lock releases show that the flow goes through an initial quasi-steady phase that is characterized by a constant front velocity. This quasi-steady front velocity has a maximum for slope angles around 40°, and it persists up to a dimensionless time of the order of 10. The flow subsequently undergoes a transition to a second phase with a larger, unsteady, front velocity. These computational findings were confirmed by experimental observations of lock-exchange flows in a tube of circular cross-section.
The reason for the observed transition from a quasi-steady front velocity to a larger, unsteady, value is found in the continuous acceleration of the stratified fluid layers connecting the two fronts by the streamwise component of the gravity vector. This acceleration leads to a situation where the fluid layers behind the current front move faster than the front itself. Initially the resulting addition of fluid to the current front from behind affects only the size of the front, while its velocity remains unchanged. Eventually, the current front is unable to absorb more fluid from behind and its velocity has to increase, thereby triggering the transition to the second, unsteady, phase. The transition time is determined as a function of the slope and the density ratio of the two fluids. For increasing density contrast, the transition is seen to occur earlier for the denser current.
Conceptually simple models based on the analysis by Thorpe (1968) are compared with simulation results for the flow in the region connecting the fronts. For the early stages of the flow a two-layer stratification model is found to be appropriate, while the later stages require a three-layer stratification model, owing to the intense mixing in the central part of the channel cross-section. These models are employed to estimate the time after which the accelerating stratified fluid layers will affect the velocities of the current fronts. They provide upper and lower estimates for the transition time which are in good agreement with the simulation results.