Том 63, № 4 (2023)

A simple criterion for evaluating the grid resolution needed to obtain a grid-independent solution of turbulent flow problems in the framework of statistical approach to turbulence simulation is proposed. The criterion is derived on the basis of an a posteriori estimate of the local interpolation error of the field of turbulent kinetic energy. A good resolution should ensure a small local interpolation error of the discrete turbulent kinetic energy. The equation for the transport of turbulent kinetic energy and realizability conditions for the turbulent stress tensor made it possible to reduce the estimation of relative interpolation error to an explicit formula for the estimate of the maximal grid step required for obtaining a grid-independent solution. The proposed criterion is applied to a steady problem for a flow past a backward facing step and to the problem of unsteady flow around a half-circular profile arranged at a zero angle of attack for the Reynolds number Re = 45 000. A numerical study showed that the proposed criterion gives a good estimate of the grid resolution required for obtaining a grid-independent solution away from a wall. This criterion can be used both for estimating the grid independence of the solution and for adapting the computation grid.

The modeling of physical and technological processes often involves solving stiff initial value problems. In most cases, their exact solution is difficult to find, while numerical schemes sometimes fail to produce a sufficiently accurate solution in acceptable computation time. Moreover, for some classes of problems, numerical solution schemes are unsuitable because of their insufficient stability. This paper deals with numerical methods based on solution continuation with respect to arguments of various types that make it possible to enhance the stability of explicit numerical schemes. Most frequently, the used best argument is hardly applicable to problems in which the integral curves grow at a superpower or nearly exponential rate. Previously, the authors proposed a modification of the best argument that alleviates these disadvantages. In the present paper, we estimate the domain of absolute stability of the explicit Euler scheme as applied to problems transformed to a modified best argument of special form and refine the proof of a similar estimate for initial value problems transformed to the best argument. A test initial value problem is used to verify the resulting theoretical estimates and to analyze the application of the modified best argument of solution continuation.

An abstract Cauchy problem for a second-order differential equation with nonlinear operator coefficients is considered. The local solvability of the problem in suitable spaces of abstract continuous and differentiable functions is proved. Sufficient conditions for finite-time blow-up of its solutions are obtained.

The paper presents the results of the symmetry classification of nonlinear integrable 2-field evolutionary systems of the third order with a constant separant.

The aggregation kinetics of settling particles is studied theoretically and numerically using the advection–diffusion equation. Agglomeration caused by these mechanisms (diffusion and advection) is important for both small particles (e.g., primary ash or soot particles in the atmosphere) and large particles of identical or close size, where the spatial inhomogeneity is less pronounced. Analytical results can be obtained for small and large Péclet numbers, which determine the relative importance of diffusion and advection. For small numbers (spatial inhomogeneity is mainly due to diffusion), an expression for the aggregation rate is obtained using an expansion in terms of Péclet numbers. For large Péclet numbers, when advection is the main source of spatial inhomogeneity, the aggregation rate is derived from ballistic coefficients. Combining these results yields a rational approximation for the whole range of Péclet numbers. The aggregation rates are also estimated by numerically solving the advection–diffusion equation. The numerical results agree well with the analytical theory for a wide range of Péclet numbers (extending over four orders of magnitude).

For the mathematical model of the sea thermodynamics developed at the Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, the problem of variational assimilation of observational data in order to recover heat fluxes on the sea surface is considered. The sensitivity of functionals of the solution to the input data on the heat flux in this problem is studied, and the results of numerical experiments for the model of Black Sea dynamics are presented.

The study is limited to two-dimensional disturbances of the interface between media. A computational technology based on the explicit interface separation in the form of one of the lines of a regular grid is used. A method for visualizing spontaneous disturbances at an early stage when they cannot yet be seen on the interface profile is proposed. It is shown that the computer rounding error plays an insignificant role in their formation. For the late stage of the disturbance development, a method for obtaining the profile of the local oscillation amplitude along the interface is proposed. The features of spontaneous disturbance at different stages of its development are studied. It is shown that the spontaneous disturbance tends to grid convergence, at least until the beginning of the process of formation of a quasi-stationary shockless combustion wave. It is shown that during the formation of a quasi-stationary wave and its subsequent motion, an additional spontaneous disturbance arises. The interaction of a specified sinusoidal disturbance having an initial amplitude of up to 0.1 of the wavelength with a quasi-stationary combustion wave is studied. It is shown that the Kelvin–Helmholtz instability is the main mechanism for the development of instability at the nonlinear stage. The combustion wave is not destroyed. The profiles of the oscillation amplitude of the given disturbance are obtained, from which it is possible to extract the universal time-independent part.