Mixing and nonlinear internal waves in a shallow flow of a three-layer stratified fluid

Abstract

We propose a mathematical model describing the mixing and propagation of nonlinear long waves in a shear three-layer flow of a stratified fluid under a lid. The shallow water equations describe the fluid flow in the outer almost potential homogeneous layers. In the intermediate mixing layer, the fluid is inhomogeneous and its flow is turbulent. Kinematic boundary conditions at the interfaces ensure the interaction between the layers. In the Boussinesq approximation, we reduced the governing equations to an evolutionary system of balance laws that is hyperbolic for a small difference in velocities in the outer layers. Using the characteristic velocities of the proposed model, we define the concept of a supercritical (subcritical) three-layer flow. We study classes of stationary flows and construct examples of continuous and discontinuous oscillating solutions that describe the spatial evolution of the mixing layer in a stratified flow with upstream shear. The problem of transcritical flow over an obstacle is considered. Depending on the relative velocity, the obtained solutions describe qualitatively different flow regimes on the leeward side of the obstacle. The proposed one-dimensional model was compared against experimental data and numerical results based on the Navier–Stokes equations. The model quite accurately describes the region of intense mixing and characteristic features of the flow.