Alexander A. Chesnokov

This paper extends the two-layer high-level Green–Naghdi (HLGN) internal-wave model to study boundary time-varying problems, involving moving bottom or surface disturbances. The equations for the two-layer HLGN model with time-varying boundaries are presented, accompanied by a time-domain algorithm for solving these equations. The wave profiles predicted by the HLGN model for internal waves generated by boundary disturbances, whether occurring at the bottom or at the surface, show excellent agreement with results obtained by the fully nonlinear potential-flow (FNPF) solution. For internal waves generated by a surface moving disturbance, the results obtained by the HLGN model show good agreement with the experimental observations and the FNPF solution, including the relationships between the disturbance speed and the resulting wave amplitude and phase speed. Furthermore, the HLGN model is applied to analyse the evolution of the wave profiles and speed generated by the surface disturbance with different moving speeds. In addition, the extended HLGN model incorporates background linear shear currents to examine internal waves generated by a moving bottom disturbance with a linear shear current. The results reveal that background vorticity exerts a pronounced modulation effect on the wave profile and velocity field. Counter-flow narrows the waves and increases their phase speed, whereas co-flow broadens the waves and enhances their amplitude.

A one-dimensional evolutionary system of equations is proposed that describes the motion of a thin bottom gravity current in a submerged environment of a lighter fluid in the Boussinesq approximation, taking into account the development of shear instability and the formation of an intermediate mixing layer. For hydrostatic flows, the propagation velocities of disturbances are determined and the concept of subcritical (supercritical) flow is formulated. An unsteady problem of the mixing layer is considered. Evidently, a monotonic or an oscillating mixing layer as a function of the Froude number of the incoming flow is formed. In the first case of a monotonic mixing layer, maximum entrainment is attained and a stationary solution is determined over a finite interval. For flows with a nonhydrostatic pressure distribution in the lower layer, steady solutions are constructed in the form of second-mode solitary waves adjacent to a given constant flow. Unsteady computations of the formation and propagation of large-amplitude bottom waves are performed.

A three-layer shallow water model in the Boussinesq approximation with allowance for nonlinearity, dispersion, and mixing is used to describe the propagation and breaking of large-amplitude internal waves interacting with uneven bottom topography. The proposed equations of motion are solved numerically by applying the Godunov method with additional inversion of an elliptic operator at each time step. Stationary solutions in the form of mode-1 solitary waves are constructed. Mixing processes induced by breaking internal solitary waves due to their interaction with a single or combined obstacle are modeled. It is shown that the numerical results are in good agreement with experimental data and direct numerical simulation.

We consider a nonlinear long-wave Boussinesq-type model describing the propagation of breaking internal solitary waves in a three-layer flow between two rigid boundaries. The Green–Naghdi-type equations govern the fluid flow in the top and bottom homogeneous layers. In the intermediate hydrostatic layer, the fluid is non-homogeneous, and its flow is described by the depth-averaged shallow water equations for shear flows. The velocity shear in the outer layers can lead to the development of the Kelvin–Helmholtz instability and turbulent mixing. To take this into account, we propose a simple law of vertical mixing, which governs the interaction of these layers. Stationary solutions and non-stationary calculations show the effect of mixing (or breaking) for waves of sufficiently large amplitude. We construct steady-state soliton-like solutions of the three-layer model adjacent to a given constant flow. The obtained theoretical profiles of breaking solitary waves are consistent with laboratory experiments.

A mathematical model is proposed that describes the formation of internal hydraulic jumps and the mixing of codirectional flows of an ideal stratified fluid in the Boussinesq approximation. The model is based on a three-layer representation of the flow taking into account the entrainment of fluid from the outer layers into the intermediate vortex layer and is represented as a system of inhomogeneous conservation laws. The rate of entrainment is given by the equilibrium condition within the framework of a more general model for the evolution of the mixing layer. The velocity of propagation of disturbances and the concepts of subcritical and supercritical flows are formulated. It is shown that the model is suitable for describing flow mixing and splitting in deep-sea currents. Solutions were constructed that correspond to the flow around an obstacle with the formation of an internal hydraulic jump and a region of intense mixing. The results of numerical simulation were shown to be in good agreement with experimental data.

We consider the propagation of plane waves in an ideal gas in the presence of external sources of energy inflow and dissipation. Using the Whitham criterion, we obtain conditions under which small perturbations of a constant solution are transformed into nonlinear quasiperiodic wave packets of finite amplitude that move in opposite directions. The structure of these wave packets is shown to be similar to roll waves in inclined open channels. We perform numerical calculations of the development of self-oscillations and the nonlinear interaction of waves. The calculations show that under a small harmonic perturbation of the initial equilibrium state, two types of wave structures can develop: roll waves and periodic two-peak standing waves.