Euler Variables Research Articles

Numerical modelling of the IMF influence on the large-scale structure of the ionosphere, including the poles displacement

The interplanetary magnetic field (IMF), frozen into the solar wind plasma, determines the level of geomagnetic activity and significantly affects the large-scale structure of the high-latitude ionosphere. Variations in the IMF components lead to non-stationarity of the magnetospheric convection electric field, causing changes in the spatio-temporal distribution of charged particles in the ionosphere. The situation is complicated by the fact that the geographic and geomagnetic poles do not coincide, which leads to universal time control (UT control) of the parameters of the high-latitude ionosphere. The effect of the poles displacement is most manifest in winter. Therefore, in this work, within the framework of a nonstationary mathematical model of the F-region of the ionosphere in Euler variables, taking into account the displacement between the geographic and geomagnetic poles, we study the influence of the interplanetary magnetic field components on the large-scale structure of the high-latitude ionosphere in winter. For this, the Weimer model of high-latitude electric potentials, which depends on the IMF, was used. As a result, of numerical modelling, it is shown that the IMF components affect the shape, size and spatio-temporal location of the main structural features of the high-latitude ionosphere depending on universal time. The obtained effect of ionisation tongue separation requires experimental confirmation. The obtained results demonstrate that the developed non-stationary three-dimensional model of the ionosphere in Euler variables can be applied to the study of ionospheric disturbances, including ionospheric disturbances associated with changes in the IMF components.

Conformally Invariant Gravity and Gravitating Mirages

The action of an ideal fluid in Euler variables with a variable number of particles is used for the phenomenological description of the processes of particle creation in strong external fields. It has been demonstrated that the conformal invariance of the creation law imposes quite strict restrictions on the possible types of sources. It is shown that combinations with the particle number density in the creation law can be interpreted as dark matter within the framework of this model.

Kinematic Characteristics of Deformed Porous Structures

Experimental and computational methods of studying the stress state in the plastic region are characterized by various methods and accuracy of measurements, methods of mathematical processing of experimental information, and interpretation of results. The experimentally determined kinematics as a starting point is the most widely used method to study the stress-strain state in the plastic region. When studying the process of plastic deformation of porous blanks, the model of a rigid-plastic isotropic-strengthening porous body with a loading surface that has the shape of an ellipsoid with semi-axes. It depends on the amount of porosity and the ratio of the associated flow law as a mechanical model of the material. In the axisymmetric extrusion of porous blanks, the viscoplasticity method was used to determine the field of flow velocities based on the results of experimental studies. R-functions were applied to approximate experimentally obtained values. The problem of finding approximations was formulated in a variational statement. Cubic splines of one argument were used to interpolate functions. As a result, an approach was proposed, which consists of a particular sequence of calculating the derivatives of the coordinates of the nodes in time in combination with the R-functions approach. All the calculations were performed in Euler variables, eliminating the need to switch from Lagrangian variables and simplifying the solution. Additionally, this method allowed for working with an irregular and non-rectangular grid in areas with any shape of boundaries. This approach is more effective from the point of view of the approximation’s accuracy and the speed of calculations. Finally, the equation for calculating the porosity in the volume of the deformable workpiece based on the information about the distortion of the dividing grid elements was obtained. For stationary axisymmetric processes, a technique was developed that allowed for replacing the calculation of the accumulated deformation of the base material along the deformation trajectory by integration over the region. A technique was developed for determining the stress-strain state at unstable and stable stages of axisymmetric plastic deformation of porous blanks. The calculation results were compared based on the proposed experimental and calculation techniques and the finite element method.

Methodology of investigation the dynamics of longitudinally moving systems under the action of impulse perturbations

A methodology for researching oscillations of one-dimensional elastic systems under the influence of impulse disturbances has been developed. The mathematical model of the considered process is boundary value problems for hyperbolic equations of the second order with an irregular right-hand side and a mixed derivative with respect to linear and time variables. As for the irregularity of the right part, it takes into account the impulse effect on the research object and has the most important theoretical and practical periodic character; and with the help of the mixed derivative in the Euler variables, the longitudinal movement of the studied systems is partially taken into account. Using the basic ideas of asymptotic integration of equations with partial derivatives, the basic propositions of the wave theory of motion adapted to the considered class of problems, the principle of single-frequency oscillations in nonlinear systems with concentrated masses and distributed parameters, the method of regularization of impulse disturbances, equations in the standard form were obtained that describe the determining parameters of nonlinear oscillations for resonant and non-resonant cases. It is shown: that the longitudinal movement of the systems has a significant effect on the natural frequency of oscillations, and therefore on the period of impulse disturbances during which the resonance process takes place; the amplitude of the passage of resonance depends significantly on the speed of movement. The results obtained in the work can serve as a basis for choosing the operating modes of some classes of systems characterized by longitudinal movement in order to avoid resonant processes in them.

Theoretical study of stability of nodal completely conservative difference schemes with viscous filling for gas dynamics equations in Euler variables

For the equations of gas dynamics in Eulerian variables, a family of twolayer time-fully conservative difference schemes (FCDS) with space-profiled time weights is investigated. Nodal schemes and a class of divergent adaptive viscosities for FCDS with spacetime profiled weights connected with variable masses of moving nodal particles of the medium are developed. Considerable attention is paid to the methods of constructing regularized flows of mass, momentum and internal energy that preserve the properties of fully conservative difference schemes of this class, to the analysis of their stability and to the possibility of their use on uneven grids. The effective preservation of the internal energy balance in this class of divergent difference schemes is ensured by the absence of constantly operating sources of difference origin that produce “computational” entropy (including entropy production on the singular features of the solution). Developed schemes may be used in modelling of hightemperature flows in temperature-disequilibrium media, for example, if it is necessary to take into account the electron-ion relaxation of temperature in a short-living plasma under conditions of intense energy input.

Двухслойные полностью консервативные схемы газовой динамики с узловой аппроксимацией и адаптивной регуляризацией решения в переменных Эйлера

The paper investigates the stability of a family of two-layer in time completely conservative difference schemes with space-profiled time weights for the system of equations of gas dynamics in Euler variables using adaptive artificial viscosity. Regularization of divergent flows of mass, momentum and internal energy of the equations of gas dynamics using adaptive artificial viscosity that does not violate the properties of complete conservatism of schemes of this class is proposed. Regularized flows make the scheme quasi-monotonic. The results are numerically tested based on Einfeldt problems and shock wave calculations.

Asymptotic method and wave theory of motion in studying the effect of periodic impulse forces on systems characterized by longitudinal motion velocity

A methodology for researching dynamic processes of one-dimensional systems with distributed parameters that are characterized by longitudinal component of motion velocity and are under the effect of periodic impulse forces has been developed. The boundary problem for the generalized non-linear differential Klein–Gordon equation is the mathematical model of dynamics of the systems under study in Euler variables. Its specific feature is that the unexcited analogue does not allow applying the known classical Fourier and D'Alembert methods for building a solution. Non-regularity of the right part for the excited non-linear analogue is another problem. This study shows that the dynamic process of the respective unexcited motion can be treated as overlapping of the direct and reflected waves of different lengths but equal frequencies. Analytical dependencies have been obtained for describing the aforesaid parameters of the waves. They show that the dynamic process in such mechanical systems depends not only on their main physical and mechanical parameters and boundary conditions, but also on the longitudinal motion velocity (relative momentum). As relative momentum increases, the frequency of the process decreases. To describe excited motion, we use the principle of single frequency of oscillations in non-linear systems with concentrated masses and distributed parameters as well as regularization of periodic impulse excitations. The main idea of asymptotic integration of systems with small non-linearity into the class of dynamic systems under study has been generalized. A standard equation for the resonance and non-resonance cases has been obtained. It has been established that for the first approximation, in the non-resonance case, impulse excitation affects only the partial change of the form of oscillations. Resonance processes are possible at a specific relation between the impulse excitation period, the motion velocity of the medium, and physical-mechanical features of the body. The amplitude of transition through resonance becomes higher when impulse actions are applied closer to the middle of the body. As the longitudinal motion velocity increases, it initially increases and then decreases.

Определение типов разрывов в вычислительной газовой динамике

The paper proposes a technique for determining the types of discontinuities in gas dynamics problems in Euler variables. This technique makes it possible to determine the discontinuities of the solution without involving the Riemann problem, which makes it possible to solve gas dynamics problems with an arbitrary equation of state. This approach works efficiently for multidimensional problems and can be effectively used on parallel computing machines, since the type of the discontinuity is determined locally in each computational cell. This method allows to adapt the computational algorithm to regions occupied by different types of discontinuities, which provides an optimal artificial viscosity for obtaining monotonic solutions and ensuring the condition of non-diminishing entropy. This paper gives an example of a gas dynamics problem, in which different types of discontinuities are present.

Energy Mechanisms of Free Vibrations and Resonance in Elastic Bodies

The scientific novelty of this work is determined by the rationale for the participation in transformations, along with the kinetic energy of particles, of four types of elastic energy, identified by the peculiarities of their phase changes in the oscillation process. Two types are converted into kinetic energy, while the other two types change the deformed state of particles in accordance with the equations of motion due to internal sources. The result is obtained based on the use of the superposition principle in the space of Lagrange variables with the imposition of forced and free oscillations, as well as a new model of mechanics based on the concepts of space, time, and energy with a new scale of average stresses that takes into account the energy of particles in the initial state. In such a model of mechanics, a generalized measure of the elastic energy of particles is a quadratic invariant of asymmetric tensor whose components are partial derivatives of Euler variables with respect to Lagrange variables. The concept of kinematic energy parameters is introduced, which differ from the corresponding volumetric energy densities by a multiplier equal to the modulus of elasticity, which is directly proportional to the density and heat capacity of the material, and inversely proportional to the volumetric compression coefficient. Comparison of the values of kinematic parameters shows that most of the energy required for oscillations is associated with the deformation of particles and comes from internal sources. The mechanisms of transformation of forced vibrations into their own for transverse, torsional, and longitudinal vibrations are considered, as well as the occurrence of resonance when free and forced vibrations are superimposed with the same or a similar frequency. The formation of a new free wave after each cycle of external influences with an increase in amplitude, which occurs mainly due to internal, and not external, energy sources is justified.

Investigation of Twist Waves Distribution along Structurally Nonuniform Yarn

Abstract This paper presents the features of yarn structure formation on spinning machine, i.e. yarn twist change when winding. It was considered that the twist distribution was one of the reasons for its decrease along the formed yarn. In this paper, based on analysis of changes in thickness and twist due to axial deformation, we consider a yarn moving at constant speed. Moving dynamics of yarn are studied here by using Euler variables. The correspondences of forward and reverse twist waves’ distribution speeds on presented frequency at various vibration forms are obtained. The parameters of Doppler effect for the waves distributed along the yarn are determined.

A numerical method for the solid mechanics with Euler variables

Based on the requirement of field engineering, we proposed a new numerical method for analyzing the solid mechanics with Euler variables. By replacing the Lagrange variables with the Euler variables, there will be three new items appearing in the equation. We use the fixed Euler grid to calculate solid mechanics, and use the boundary of material regiment to describe the movement of the boundary of material, after the boundary of material regiment moves beyond the Euler element grid size, new Euler element grids will appear and the corresponding old Euler element grids which have been out of the region of material will disappear. By this way, the movement of solid could be calculated in the Euler fixed grids. Then we did some test cases, we get the result that the new method is right and could be used to calculate the solid mechanic problems.

Theoretical and Numerical Analysis of One Euler Two-Layer Completely Conservative Difference Scheme of Gas Dynamics with Adaptive Viscosity

For the equations of gas dynamics in Euler variables, using the operator approach, a family of two-layer in time completely conservative difference schemes (CCDS) with space-profiled weight factors used to approximate the momentum and energy fluxes in time is constructed, theoretically and numerically studied. The schemes have second-order accuracy and are implemented using simple iterative processes. A class of divergent adaptive viscosities for these CCDS is developed and a theoretical analysis of their stability is carried out. The regularization of flux terms of the gas dynamics equations with the help of adaptive artificial viscosity is proposed and, using the example of well-known Einfeld problem, is numerically investigated. This regularization effectively eliminates nonphysical oscillations of the solution, entropy peaks and preserves the property of complete conservatism of schemes of this class.

On the existence of global solution of the system of equations of liquid movement in porous medium

The initial-boundary value problem for the system of one-dimensional isothermal motion of viscous liquid in deformable viscous porous medium is considered. Local theorem of existence and uniqueness of problem is proved in case of compressible liquid. In case of incompressible liquid the theorem of global solvability in time is proved in Holder classes. A feature of the model of fluid filtration in a porous medium considered in this paper is the inclusion of the mobility of the solid skeleton and its poroelastiс properties. The transition from Euler variables to Lagrangian variables is used in the proof of the theorems.

Some Reasons for Nonmonotonic Variation of Descrete-Phase Concentration in a Turbulent Two-Phase Jet

Abstract—A mathematical model of a two-phase turbulent jet flow is presented, the averaged equations of which are written for each phase in Euler variables. Analysis of the results of calculations performed using this mathematical model made it possible to identify the origins of a nonmonotonic variation in the volume concentration of the dispersed phase along the axis of the two-phase jet. The influence of particle size, transverse velocity, particle concentration, and phase velocity slip in the initial section of the jet on the variation of the particle concentration in the jet is considered. It is shown that the listed factors cause a nonmonotonic variation in the particle volume concentration in a two-phase jet only when the value of the Stokes number based on the phase parameters in the initial section of the jet is greater than 0.14–0.15. At lower values of the Stokes number, when the particles can be regarded as a passive admixture, their volume fraction in the jet decreases monotonically along the jet axis, same as the impurity concentration in a gas jet of variable composition.

Exact solution of the compressible Euler–Helmholtz equation and the millennium prize problem generalization

SummaryFor the first time the exact strong solution of the three-dimensional (3D) compressible Euler–Helmholtz (EH) vortex equation in unbounded space is obtained. It corresponds to the general solution obtained in Euler variables for the 3D Riemann–Hopf (RH) equation and has a singularity in finite time. The exact closed description for the evolution of all one and multi point moments of the hydrodynamic fields and their spectra is now possible and thus, the mysterious turbulence problem has the exact solution for this case. As an example, for the first time the two-dimensional (2D) energy spectrum in the inertial scale interval is obtained directly from the exact solution of compressible 2D Navier–Stokes (NS) equation. For this purpose, a smooth continuation at all times for these exact solutions to the EH and RH equations is obtained by introducing a super-threshold homogeneous friction or any even extremely low effective viscosity. A new, smooth for all time, analytic solutions to the 1D, 2D and 3D NS equations is obtained. This solution coincides with the above-mentioned modified by viscosity solutions to the EH and RH equations, when also taking into account a linear relation between the pressure and the velocity field divergence, which is well known for the systems far from the equilibrium with relatively large second viscosity. This gives the positive answer to the generalization of the millennium prize problem (www.claymath.org) for the case of the compressible NS equation. Indeed, earlier only a negative answer to the problem of the existence of smooth solutions for any finite time has been a priori considered for the compressible case with the finite initial energy of the smooth velocity field in the unbounded space.

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ТРЕХМЕРНЫХ ПРОЦЕССОВ РАЗГОНА УПРУГОПЛАСТИЧЕСКИХ ТЕЛ ВЗРЫВОМ

A methodology for numerically modeling 3D processes of acceleration of deformable bodies by detonation products of solid explosives in the air is described. Simultaneous motion of the explosion products, the air and the elastoplastic medium is numerically analyzed in Euler variables, using a modified Godunov scheme, which is applicable for both fluid dynamics and elastoplastic flows, and an exact solution of Riemann problem of discontinuity breakage in the media and along the gas-elastic body interface. The methodology uses three types of spatial grids. The grids of the first type are sets of triangles (STL files) defining the surfaces of the objects and updating those surfaces in the process of motion. The grids of the second type are base fixed Cartesian grids nested into each medium. The grids of the third type are local orthogonal movable grids coupled with each of the triangles of the grids of the first type. The integration of dynamic equations of continua and mutual interpolation of parameters among the different kinds of grids is done on base and local grids. To simulate the detonation propagation process in a solid explosive, an algorithm based on Huygens principle and the account of energy release upon the arrival of the detonation wave into the integrated cell are used. Examples of numerically modeling propagation of elastic and elastoplastic disc-shaped, tetrahedral and cubic bodies of various materials adjacent to the charge are given. The results of the analyses testify to the applicability of the above methodology for modeling coupled processes of acceleration of deform-able bodies by detonation products up to deceleration in the ambient medium. Certain laws of the acceleration process have been found for the case where detonation is initiated in the center of a spherical charge. In particular, the duration of the acceleration of a body is comparable with the time of the arrival of the detonation wave at the surface of the contact; the major residual strains altering the geometry of the bodies occur at the initial stage of the acceleration and affect both the interaction with the detonation products and the parameters of the motion of the bodies; the final acceleration velocity is observed to depend considerably on the initial geometry of the accelerated body, its orientation relative to the detonation front and the deformation properties for the same mass.Keywords: modeling, detonation, 3D processes, multi-grid approach, interpolation, Godunov scheme, nonlinear coupled problems, aeroelastoplasticity, acceleration of bodies.

Analysis of the Motion of Frenkel-Kontorova Dislocations in Single Crystals of Aluminum with Allowance for the Peierls Barrier

The regularities of the motion of a one-dimensional Frenkel-Kontorova dislocation in pure aluminum at helium temperatures are studied. Computer simulation was carried out using the sine Gordon equation, written in dimensionless variables. It is proven that when the transition to dimensionless variables the discreteness of the model is preserved. The dependence of the true values of stresses on deformation in the Euler variables, as well as the velocity distribution of the dislocation fragments along the coordinate for successive instants of time, are obtained. It is shown that under these conditions dislocation motion is realized by quantum tunneling of the dislocation bends. The quantum-mechanical estimate confirms the possibility of quantum tunneling of the kink of dislocations in aluminum at low temperatures.

Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction

The $\mathcal{KS}$ map is revisited in terms of an $S^1$-action in $T^*\mathbb{H}_0$ with the bilinear function as the associated momentum map. Indeed, the $\mathcal{KS}$ transformation maps the $S^1$-fibers related to the mentioned action to single points. By means of this perspective a second twin-bilinear function is obtained with an analogous $S^1$-action. We also show that the connection between the 4-D isotropic harmonic oscillator and the spatial Kepler systems can be done in a straightforward way after regularization and through the extension to 4 degrees of freedom of the Euler angles, when the bilinear relation is imposed. This connection incorporates both bilinear functions among the variables. We will show that an alternative regularization separates the oscillator expressed in Projective Euler variables. This setting takes advantage of the two bilinear functions and another integral of the system including them among a new set of variables that allows to connect the 4-D isotropic harmonic oscillator and the planar Kepler system. In addition, our approach makes transparent that only when we refer to rectilinear solutions, both bilinear relations defining the $\mathcal{KS}$ transformations are needed.

Gas dynamics and mixture formation in swirled flows with precession of air flow

The effect of precessing air flow on the processes of mixture formation in the wake of the front winding devices of the combustion chambers is considered. Visual observations have shown that at different times the shape of the atomized jet is highly variable and has signs of precessing motion. The experimental data on the distribution of the velocity and concentration fields of the droplet fuel in the working volume of the flame tube of a typical combustion chamber are obtained. The method of calculating flows consisted in integrating the complete system of Reynolds equations written in Euler variables and closed with the two-parameter model of turbulence k-ε. Calculation of the concentration fields of droplet and vapor fuel is based on the use of models for disintegration into droplets of fuel jets, fragmentation of droplets and analysis of motion and evaporation of individual droplets in the air flow. Comparison of the calculation results with experimental data showed their good agreement.

Ill posedness of force-free electrodynamics in Euler potentials

We prove that the initial value problem for Force-free Electrodynamics in Euler variables is not well posed. We establish this result by showing that a well-posedness criterion provided by Kreiss fails to hold for this theory, and using a theorem provided by Strang. To show the nature of the problem we display a particular bounded (in Sobolev norms) sequence of initial data for the Force-free equations such that at any given time as close to zero as one wishes, the corresponding evolution sequence is not bounded. Thus, the Force-free evolution is non continuous in that norm with respect to the initial data. We furthermore prove that this problem is also ill-posed in the Leray-Ohya sense.