Solution of the problems of nonhomogeneous incompressible fluid dynamics by the CABARET method

Abstract

Method for solving a nonhomogeneous viscous incompressible fluid dynamics is proposed using CABARET scheme. We study the problem of spot dynamics in a fluid that is stably stratified by density. The comparison with the results of other works and with analytic solution is considered. The statement of the problem was obtained from the consideration of a spot with mixed salt water placed in a solution with the steady-state distributions of the salinity and density fields. We will assume that the spot is placed in a rectangular tank with sufficiently remote liquid-tight walls. The spot has the form of a cylinder. Density stratification is given by a linear function of height, and buoyancy forces are modeled as a deviation of density from a stable stratification. The problem is solved in a two-dimensional formulation at different flow regimes determined by Reynolds and Froude numbers. There are no restrictions on the smallness of the density deviations, therefore theoretically this technique can be applied to problems with a complex dependence of density on height. For example, in the problem of the occurrence of internal waves in a stratified atmospheric layer, where, as a result of solar radiation, the heated at the surface air forms ascending currents.