The non-Boussinesq lock-exchange problem. Part 2. High-resolution simulations

Abstract

The present investigation explores the unsteady dynamics of large density contrast non-Boussinesq lock-exchange flows by means of high-resolution two-dimensional simulations of the incompressible variable-density Navier–Stokes equations, employing a combination of spectral and compact finite-difference methods. For small density contrasts, the simulations closely reproduce earlier Boussinesq results for corresponding flows. Across the entire range of density contrasts, good agreement is obtained between the computed front propagation velocities and corresponding experimental observations reported in Part 1 of this investigation and by other authors. The simulations yield the required quantitative information with respect to the light and dense front heights, their propagation velocities, and the spatial structure of the dissipation fields in order to determine conclusively which of the scenarios developed in Part 1 is observed in reality. Simulations are conducted for fluids with the same kinematic viscosity, as well as for fluids with the same dynamic viscosity. For both slip and no-slip boundary conditions, and for all $\hbox{\it Re}$ values, we find that for larger density contrasts, the dense front dissipates an increasing amount of energy. In contrast, the energy dissipated by the light front remains near its Boussinesq level for all values of the density ratio. In addition, for all density ratios, the height of the light front is very close to half the channel height, and it propagates with a non-dimensional velocity close to a half. This provides strong evidence that the dynamics of the light front is indeed approximated by the energy-conserving solution described in an earlier theoretical analysis. In contrast, the height of the dense front is substantially less than half the channel height. In addition, its velocity is close to the value derived in Part 1 for a dissipative gravity current. Together with the above results for the dissipation field, this confirms that the dense front behaves as a dissipative gravity current.