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Mathematical modeling is the most important tool for constructing theories of different kinds of two-phase flows. This review is devoted to the analysis of the introduction of mathematical modeling to two-phase flows, where solid particles mainly serve as the dispersed phase. The main problems and features of the study of gas-solid two-phase flows are included. The main characteristics of gas flows with solid particles are discussed, and the classification of two-phase flows is developed based on these characteristics. The Lagrangian and Euler approaches to modeling the motion of a dispersed phase (particles) are described. A great deal of attention is paid to the consideration of numerical simulation methods that provide descriptions of turbulent gas flow at different hierarchical levels (RANS, LES, and DNS), different levels of description of interphase interactions (one-way coupling (OWC), two-way coupling (TWC), and four-way coupling (FWC)), and different levels of interface resolution (partial-point (PP) and particle-resolved (PR)). Examples of studies carried out on the basis of the identified approaches are excluded, and they are also excluded for the mathematical modeling of various classes of gas-solid two-phase flows.

Mathematical modeling is the most important tool for constructing the theory of concentrated tornado-like vortices. A review and analysis of computational and theoretical works devoted to the study of the generation and dynamics of air tornado-like vortices has been conducted. Models with various levels of complexity are considered: a simple analytical model based on the Bernoulli equation, an analytical model based on the vorticity equation, a new class of analytical solutions of the Navier–Stokes equations for a wide class of vortex flows, and thermodynamic models. The approaches developed to date for the numerical simulation of tornado-like vortices are described and analyzed. Considerable attention is paid to developed approaches that take into account the two-phase nature of tornadoes. The final part is devoted to the analysis of modern ideas about the tornado, concerning its structure and dynamics (up to the breakup) and the conditions for its occurrence (tornadogenesis). Mathematical modeling data are necessary for interpreting the available field measurements while also serving as the basis for planning the physical modeling of tornado-like vortices in the laboratory.

This paper considers the problem associated with the numerical simulation of the interaction between the cocurrent stream occurring near a monoblock moving in the gas medium and solid fuel combustion products flowing from a solid fuel rocket engine (SFRE). The peculiarity of the approach used is the description of gas-dynamic processes inside the combustion chamber, in the nozzle block, and the down jet based on a single calculation methodology. In the formulated numerical methodology, the calculation of gas-dynamic parameters is based on the solution of unsteady Navier–Stokes equations and the application of a hybrid computational grid. A hybrid block-structured computational grid makes it possible to calculate gas flow near bodies of complex geometric shapes. The simulation of the main phase of interaction, corresponding to the stationary mode of rocket flight in the Earth’s atmosphere, has been carried out. A conjugated simulation of the internal ballistics of SFRE and interaction of combustion products jets is conducted.