Vol. 18, no. 3
Vol. 18, no. 3, 2022
The 200th Anniversary of the Navier – Stokes Equations
Rusyak I. G., Tenenev V. A., Korolev S. A.
Numerical Simulation of the Nonstationary Process of the Shot Based on the Navier – Stokes Equations
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 ke 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 selfsimilar 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.

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 highspeed
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 highspeed buffet
onset is investigated. The distribution surfaces of the velocity pulsation energy generation are
shown. The frequency of selfoscillations is determined on the basis of spectral analysis.

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 selfsimilar traveling waves is investigated.
It is shown that there are three independent families of such selfsimilar solutions. The characteristics
of the structure of nonpropagating exponentially growing waves are calculated. The
characteristic time of formation of such a structure is determined.

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 threedimensional 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 nearwall region of the channel. The results of numerical simulation are in
qualitative and quantitative agreement with the available data of physical and computational
experiments.

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.

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. 
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.

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 planeparallel 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.
