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# Vol. 18, no. 3, 2022 The 200th Anniversary of the Navier – Stokes Equations

 Abstract Citation: Editorial Board, On the 200th Anniversary of the Derivation of the Navier – Stokes Equations, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp. 331
 Rusyak I. G.,  Tenenev V. A.,  Korolev S. A. Abstract This paper gives a spatial mathematical formulation of the problem of internal ballistics based on the Navier – Stokes equations, taking into account the swirl of the flow due to the rotation of the projectile. The k-e model of turbulent viscosity is used. The control volume method is used for the numerical solution of systems of equations. The gas parameters at the boundaries of the control volumes are determined by the method of S. K. Godunov using a self-similar solution to the problem of the decay of an arbitrary discontinuity. The MUSCL scheme is used to increase the order of approximation of the difference method. For equations written in a cylindrical coordinate system, an orthogonal difference grid is constructed using the complex boundary element method. A comparative analysis of the results obtained with different approaches to modeling the process of an artillery shot is given. Quantitative data are presented on the influence of factors not previously taken into account on the characteristics of the process. Keywords: internal ballistics, mathematical model of a shot, mechanics of heterogeneous media, Navier – Stokes equations, axisymmetric swirling flow, computational algorithms Citation: Rusyak I. G.,  Tenenev V. A.,  Korolev S. A., Numerical Simulation of the Nonstationary Process of the Shot Based on the Navier – Stokes Equations, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp. 333-348 DOI:10.20537/nd220301
 Lipanov A. M.,  Karskanov S. A. Abstract The results of the theoretical solution of aerodynamic problems based on direct numerical simulation by integrating the Navier – Stokes equations without involving additional models and empirical constants are shown. Modern approaches to the theoretical study of high-speed flows are determined. The advantages, problems, development trends and scientific directions of research on various approaches are revealed. The advantages and disadvantages of the direct numerical simulation are analyzed. The velocity vectors of laminar and transient flows in a rectangular channel with a sudden expansion at the inlet are presented in different planes. The convergence of the method is studied when the computational domain is quantized in space. It is discovered that fast relaminarization is characteristic of transitional flows. A mathematical model for calculating bottom drag is presented. The numerical results are compared with the data of physical experiments and the results of other methods. It is shown that the results of simulation based on DNS are not inferior in accuracy to RANS and LES results. The results of a parametric study of a transonic flow around a profile are presented. The high-speed buffet onset is investigated. The distribution surfaces of the velocity pulsation energy generation are shown. The frequency of self-oscillations is determined on the basis of spectral analysis. Keywords: direct numerical simulation, Navier – Stokes equations, transient flows, base drag, baffet onset Citation: Lipanov A. M.,  Karskanov S. A., Direct Numerical Simulation of Aerodynamic Flows Based on Integration of the Navier – Stokes Equations, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp. 349-365 DOI:10.20537/nd220302
 Safronov A. A. Abstract The wave capillary flow of the surface of an inviscid capillary jet, initiated by a single $\delta$-perturbation of its surface, is studied. It is shown that the wave pattern has a complex structure. The perturbation generates both fast traveling damped waves and a structure of nonpropagating exponentially growing waves. The structure of self-similar traveling waves is investigated. It is shown that there are three independent families of such self-similar solutions. The characteristics of the structure of nonpropagating exponentially growing waves are calculated. The characteristic time of formation of such a structure is determined. Keywords: instability, capillary flow, nonviscous jet Citation: Safronov A. A., Investigation of the Structure of Waves Generated by a $\delta$-perturbation of the Surface of a Capillary Jet, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp. 367-378 DOI:10.20537/nd220303
 Emelyanov V. N.,  Volkov K. N. Abstract Direct numerical simulation of a fully developed turbulent flow of a viscous compressible fluid containing spherical solid particles in a channel is carried out. The formation of regions with an increased concentration of solid particles in a fully developed turbulent flow in a channel with solid walls is considered. The fluid flow is simulated with unsteady three-dimensional Navier – Stokes equations. The discrete trajectory approach is applied to simulate the motion of particles. The distributions of the mean and fluctuating characteristics of the fluid flow and distribution of the concentration of the dispersed phase in the channel are discussed. The formation of regions with an increased concentration of particles is associated with the instantaneous distribution of vorticity in the near-wall region of the channel. The results of numerical simulation are in qualitative and quantitative agreement with the available data of physical and computational experiments. Keywords: Navier – Stokes equation, direct numerical simulation, turbulence, duct, particle, concentration Citation: Emelyanov V. N.,  Volkov K. N., Direct Numerical Simulation of Fully Developed Turbulent Gas–Particle Flow in a Duct, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp. 379-395 DOI:10.20537/nd220304
 Burmasheva N. V.,  Prosviryakov E. Y. Abstract In this paper, we report on several classes of exact solutions for describing the convective flows of multilayer fluids. We show that the class of exact Lin – Sidorov – Aristov solutions is an exact solution to the Oberbeck – Boussinesq system for a fluid discretely stratified in density and viscosity. This class of exact solutions is characterized by the linear dependence of the velocity field on part of coordinates. In this case, the pressure field and the temperature field are quadratic forms. The application of the velocity field with nonlinear dependence on two coordinates has stimulated further development of the Lin – Sidorov – Aristov class. The values of the degrees of the forms of hydrodynamical fields satisfying the Oberbeck – Boussinesq equation are determined. Special attention is given to convective shear flows since the reduced Oberbeck – Boussinesq system will be overdetermined. Conditions for solvability within the framework of these classes are formulated. Keywords: exact solution, multilayer fluids, Oberbeck – Boussinesq equations, shear flows, self-similar flows with spatial acceleration Citation: Burmasheva N. V.,  Prosviryakov E. Y., Exact Solutions to the Navier – Stokes Equations for Describing the Convective Flows of Multilayer Fluids, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp. 397-410 DOI:10.20537/nd220305
 Korepanov M. A.,  Koroleva M. R.,  Mitrukova E. A.,  Nechay A. N. Abstract This paper considers krypton flow in a micronozzle with a cylindrical tube. A standardized conical nozzle elongated with cylindrical portion performs gas discharge into a vacuum chamber at a pressure of $10^{−2}$ Pa. Under such conditions, a low temperature area is formed in the central part of the jet with gas condensation. The particles are entrained by the gas flow. The portion with a constant section behind the nozzle should focus the supersonic flow part and the condensed particle flow and also decrease particle dispersion behind the nozzle throat. The paper expresses a mathematical model of homogeneous gas motion with respect to formation processes and the growth of condensation nuclei. Since the condensed particles are small, the research is carried out with a single velocity motion model. The results obtained have shown that the application of the cylindrical tube leads to nonlinear flow effects. The flow responds to: the geometrical exposure related to flow transition from the conical diverging nozzle into the cylindrical tube, heat exposure and mass outflow due to particle formation and growth, and considerable friction force exposure due to the small sizes of the channel. The sum total ofthese factors leads to an insignificant deceleration of the supersonic flow part and highly impacts condensation. Keywords: micronozzle, krypton, Navier – Stokes equations, condensed phase, numerical modeling Citation: Korepanov M. A.,  Koroleva M. R.,  Mitrukova E. A.,  Nechay A. N., Nonlinear Effects of Krypton Flow in a Micronozzle with a Cylindrical Tube, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp. 411-422 DOI:10.20537/nd220306
 Kotsur O. S.,  Shcheglov G. A.,  Marchevsky I. K. Abstract This paper is concerned with the equation for the evolution of vorticity in a viscous incompressible fluid, for which approximate weak solutions are sought in the class of vortex filaments. In accordance with the Helmholtz theorem, a system of vortex filaments that is transferred by the flow of an ideal barotropic fluid is an exact solution to the Euler equation. At the same time, for viscous incompressible flows described by the system of Navier – Stokes equations, the search for such generalized solutions in the finite time interval is generally difficult. In this paper, we propose a method for transforming the diffusion term in the vorticity evolution equation that makes it possible to construct its approximate solution in the class of vortex filaments under the assumption that there is no helicity of vorticity. Such an approach is useful in constructing vortex methods of computational hydrodynamics to model viscous incompressible flows. Keywords: weak solution, vortex filament, helicity of vorticity, diffusion velocity, viscosity Citation: Kotsur O. S.,  Shcheglov G. A.,  Marchevsky I. K., Approximate Weak Solutions to the Vorticity Evolution Equation for a Viscous Incompressible Fluid in the Class of Vortex Filaments, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp. 423-439 DOI:10.20537/nd220307
 Vetchanin E. V.,  Portnov E. A. Abstract In this paper we present a method for constructing inhomogeneous velocity fields of an incompressible fluid using expansions in terms of eigenfunctions of the Laplace operator whose weight coefficients are determined from the problem of minimizing the integral of the squared divergence. A number of examples of constructing the velocity fields of plane-parallel and axisymmetric flows are considered. It is shown that the problem of minimizing the integral value of divergence is incorrect and requires regularization. In particular, we apply Tikhonov’s regularization method. The method proposed in this paper can be used to generate different initial conditions in investigating the nonuniqueness of the solution to the Navier – Stokes equations. Keywords: inhomogeneous velocity field, expansion in terms of eigenfunctions, ill-conditioned system of linear algebraic equations Citation: Vetchanin E. V.,  Portnov E. A., Construction of Inhomogeneous Velocity Fields Using Expansions in Terms of Eigenfunctions of the Laplace Operator, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp. 441-464 DOI:10.20537/nd220308

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