Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki

For a nonlinear and linearized system of shallow water equations in a basin with an uneven bottom and gently sloping shores, the problem of short-wavelength asymptotic solutions describing waves excited by a time-harmonic spatially localized source is considered. In the linear approximation, such asymptotic solutions are essentially expressed via solutions of the Helmholtz equation, and the problem of constructing them is close to the problem of the asymptotics of the Green’s function. We use a recently developed approach based on the Maslov canonical operator and allowing one to find a global asymptotic solution of the linearized problem in any predetermined region, taking into account caustics and focal points, as well as Fermat’s variational principle, which, in combination with the canonical operator, makes it possible to construct such an asymptotic solution locally, i.e., in the neighborhood of a given observation point. The linearized problem is considered in a fixed domain, which is bounded by a shoreline corresponding to the fluid at rest. On this line, the equations degen- erate; accordingly, a correct statement of the problem does not require (and does not admit) classical boundary conditions; instead, the condition of finiteness of the energy integral is used. From the view- point of asymptotic theory, the shoreline is a nonstandard caustic, in the neighborhood of which the asymptotic solution of the linearized problem is expressed via a modified canonical operator. For the original nonlinear system, a free-boundary problem is considered: the position of the shoreline depends on the elevation of the free surface. According to a recently developed approach based on the modified Carrier–Greenspan transform, the asymptotic solution of the nonlinear system is expressed via the solution of the linearized system in the form of parametrically specified functions. The resulting formulas, in particular, describe the effects of wave inrush on the shore.

Burgers hierarchy equations are considered. It is shown that the well-known Cole-Hopf transformation for linearization of the classical Burgers equation generalizes to the case of equations of arbitrary order of the Burgers hierarchy. This fact allows us to find solitary and periodic waves described by the Burgers hierarchy equations, resembling the N-wave for the classical Burgers equation. A detailed consideration of the construction of solitary waves is presented for the third-order Sharma-Tasso-Olver equation and for the fourth-order hierarchy equation. It is found that for the third-order dissipative equation, N-wave type solitary waves have oscillations at the solution front. In the case of second and fourth order dissipative equations such oscillations are absent.

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.

Numerical solutions of the system of magnetoelasticity equations are considered. A numerical scheme based on central differences for spatial derivatives and the fourth-order Runge-Kutta method for temporal derivatives is applied. Smoothed step type data (discontinuity decay problem) are used as initial data. The bounds of existence of discontinuity structures of various types are determined type. A methodology for calculating the structure with radiation and internal weak gap is developed. It is found that, in the most general form, the statement of the problem on the gap decay assumes the presence of vibrational states, an appropriate calculation is made.

A method based on the application of artificial neural networks is proposed to determine in real time the additional forces and moments acting on an airplane in a vortex trail behind another airplane. The method is based on approximation by means of neural networks of additional forces and moments acting on the airplane caused by the influence of separate vortex sections. When forming a set of data for training neural networks, a program for calculating the transonic flow of the aircraft layout was used in the framework of solving the equations for the full potential in the external flow and solving the equations of the three-dimensional boundary layer on the surface. Accuracy and speed evaluations of the proposed method have been carried out.

A finite-depth fluid layer described by the Euler equations is considered. The ice cover is simulated by a geometrically nonlinear elastic Kirchhoff–Love plate. The trajectories of fluid particles under the ice cover are in the field of nonlinear surface periodic traveling waves of small, but finite amplitude. A solution describing such surface waves is allowed by the equations of the model. Periodic waves are described by Jacobi elliptic functions. The analysis uses explicit asymptotic expressions for solutions describing wave structures at the water–ice interface, such as a periodic wave against a zero-displacement surface, as well as asymptotic solutions for the velocity field in a fluid column generated by these waves.

The non-modal spatial development of stationary three-dimensional perturbations in a circular flooded jet at Re = 2850 is numerically investigated. The conditions of the laboratory experiment performed earlier at the Moscow State University Research Institute of Mechanical Engineering are reproduced. A method of calculation of optimal perturbations in the conditions of the main flow developing downstream is developed. The optimal perturbations corresponding to different azimuthal numbers are found. Their shape, character of development and degree of growth are determined. Nonlinear development of optimal perturbations at different values of their initial amplitude is studied. It is shown that nonlinear effects lead to a slowing down of the growth rate when they develop downstream. Currents deformed by stationary perturbations in the angular direction are investigated for stability to small unsteady perturbations. It is found that with increasing degree of deformation, the maximum growth rate of perturbations increases significantly due to the appearance of a specific short-wave mode of instability

Unsteady gas flow from a vertical well into a water-saturated reservoir is investigated within the axisymmetric formulation of the filtration problem. The influence of the shape of the gas-saturated zone on the injectivity coefficient of the well, i.e., on the maximum rate of gas injection, is estimated. It isshown that the well skin factor can be decomposed into two multipliers, the first of which — the shape parameter — characterizes the shape of the gas plume, and the second describes the growth of the skin factor with time. With the help of numerical modeling of filtration, diagrams describing the dependence of the noted multipliers on the similarity parameters have been constructed. It is shown that gas injection at any values of parameters and shapes of gas-saturated regions is accompanied by injectivity growth with time, and the maximum values of injectivity coefficients are reached at high rates of gas injection. For this limiting case, the relationship for the skin factor is explicitly obtained. The results of the study can be useful for determining effective ways of application of Carbon Capture, Utilization and Storage technology.