Euler System Of Equations Research Articles

Interaction of a plane shock wave with an area of ionization instability of discharge plasma in air

The conducted study is devoted to the research area of control of high speed flows and shock-wave configurations. Experimental and numerical results on the interaction of a plane shock wave (M = 5–6) with the area formed by ionization instability in gas discharge plasma at a pressure of 3–7 Torr and gas-discharge current of 100–400 mA are presented. The region of ionization instability in plasma of high degree of ionization and non-equilibrium (gas temperature T ∼ 300–600 K, electron temperature Te∼5⋅103 K) has been obtained experimentally in air. This instability is characterized by spherical strata of changing shape. Two types of a shape of the ionization instability have been obtained: large-scale type and small-scale type. Also, the transitional type from large-scaled strata to low-scaled ones has been obtained. The influence of the ionization instability was established to cause the distortion of an initially plane shock wave up the complete disappearance of its front. Numerical modeling the interaction of the initially plane shock wave with ionization-unstable plasma using the Euler system of equations has been conducted. The area formed by ionization unstable plasma was modeled via a set of thermal layers with varying densities (and temperatures). Changing the physical–chemical properties of the gas medium was taken into account varying the ratio of specific heats. Layered structures with wavy shock-wave and contact discontinuity fronts have been observed. The multiple generations of shear layer instabilities and the Richtmyer–Meshkov instabilities have been obtained as the result of the interaction of a shock wave with thermal strata. It has confirmed that the shapes of the both discontinuities change acquiring an unstable character. Examples of the flow modes with the disappearance of the shock wave front causing by the origination of a structure with cellular order of shear layer instabilities have been presented. A good agreement at the qualitative level with the experimental Schlieren images has been demonstrated.

A cure for numerical shock instability in HLLC Riemann solver using antidiffusion control

Various forms of numerical shock instabilities are known to plague many contact and shear preserving approximate Riemann solvers, including the popular Harten-Lax-van Leer with Contact (HLLC) scheme, during high speed flow simulations governed by the Euler system of equations. In this paper we propose a simple and inexpensive novel strategy to prevent the HLLC scheme from developing such spurious solutions without compromising on its linear wave resolution ability. The cure is primarily based on a reinterpretation of the HLLC scheme as a combination of its well-known diffusive counterpart, the HLL scheme, and an antidiffusive term responsible for its accuracy on linear wavefields. In our study, a linear analysis of this alternate form indicates that shock instability in the HLLC scheme could be triggered due to the unwanted activation of the antidiffusive term appearing in its mass and interface-normal momentum flux component discretizations on interfaces that are not aligned with the shock front. This inadvertent activation results in weakening of the favourable dissipation provided by its inherent HLL scheme and causes unphysical mass flux variations along the shock front. To mitigate this, we propose a modified HLLC scheme that employs a simple differentiable pressure-ratio based multidimensional shock sensor to achieve smooth control of these critical antidiffusive terms near shocks. Using a linear perturbation analysis and a matrix based stability analysis, we establish that the resulting scheme, called HLLC-ADC (Anti-Diffusion Control), is shock stable over a wide range of freestream Mach numbers. Results from standard numerical test cases demonstrate that the HLLC-ADC scheme is indeed free from the most common manifestations of shock instability including the Carbuncle phenomenon without significant loss of accuracy on shear dominated viscous flows.

Delusions in Theoretical Hydrodynamics

Theoretical hydrodynamics may lead one into serious delusions. This article is focused on three of them. First, using flowing around a sphere as an example it is shown that the known potential solutions of the flow-around problems are not unique and there exist nonpotential solutions. A nonpotential solution has been obtained for flowing around a sphere. A general solution of the problem of flowing around an arbitrary surface has been obtained in the quadrature form. To single out a physically realisable solution among a great number of others, it is necessary to add supplementary conditions to the known boundary ones, in particular, to find a solution with the minimum total energy. The hypothesis explaining the reason for stalled flows by viscosity is erroneous. When considering a flow-around problem one should use stalled and broken solutions of the continuity equation along with the continuous ones. If the minimum total energy is achieved by the continuous solution, it is a continuous flow that will be implemented. If it is achieved by the broken solution, a stalled flow will be realised. Second, the hydrodynamics of a flow is considered exclusively at each point of it. Differential equations are used to describe the flows that are written for a randomly small volume of a flow, i.e., for a point. The integral characteristics of a flow and its inertial properties are neglected in the consideration, which results in the misunderstanding of the mechanism of the formation of a vortex. The reason for the formation of vortices is related to viscosity, which is a mistake. The formation of vortices is the result of the inhomogeneity of the acceleration field and the inertial properties of a flow. Third, the fictitious values of viscous stresses are used in hydrodynamics. As a matter of fact, viscosity is the momentum diffusion and it should be described by the diffusion equation included into the Euler system of equations for a viscous fluid. The momentum diffusion leads to the necessity of including the volume momentum sources produced by diffusion into the continuity equation and excluding the viscosity forces from the equation of motion. The problem of a viscous fluid flowing around a thin plate has been solved analytically, the velocity profiles satisfying the experiment have been obtained. The superfluidity of helium is not its property. It is the interaction of helium with a streamlined surface that is responsible for the mechanism of superfluidity. At low temperatures when the quantum properties are most pronounced the momentum transfer from the helium atoms to the streamlined wall becomes impossible, since the value of the energy transferred in the collision of a helium atom with that of the wall is smaller than the permitted quantum of energy. This mechanism takes place in the case of a flow in capillaries. Under a hydrodynamic flow-around superfluidity does not manifest due to the occurrence of stalled flows. The hypothesis of the disappearance of the viscous stresses at low temperatures is erroneous. The viscous stresses cannot disappear since they do not exist in nature. The theory of representing superfluidity as a phase transition accompanied by the formation of the combined viscous and nonviscous phases is a mistake.

Conical shock wave for non-isentropic compressible Euler system of equations

Conical shock wave is generated when a sharp conical projectile flies supersonically in the air. We study the linear stability and existence of steady conical shock waves in supersonic flow for the equations of complete Euler system in 3D non-isentropic gas-dynamics.

Detailed simulation of the pulsating detonation wave in the shock-attached frame

The paper is devoted to the numerical investigation of the stability of propagation of pulsating gas detonation waves. For various values of the mixture activation energy, detailed propagation patterns of the stable, weakly unstable, irregular, and strongly unstable detonation are obtained. The mathematical model is based on the Euler system of equations and the one-stage model of chemical reaction kinetics. The distinctive feature of the paper is the use of a specially developed computational algorithm of the second approximation order for simulating detonation wave in the shock-attached frame. In distinction from shock capturing schemes, the statement used in the paper is free of computational artifacts caused by the numerical smearing of the leading wave front. The key point of the computational algorithm is the solution of the equation for the evolution of the leading wave velocity using the second-order grid-characteristic method. The regimes of the pulsating detonation wave propagation thus obtained qualitatively match the computational data obtained in other studies and their numerical quality is superior when compared with known analytical solutions due to the use of a highly accurate computational algorithm.

On spatial discretization of the one-dimensional quasi-gasdynamic system of equations with general equations of state and entropy balance

The one-dimensional quasi-gasdynamic system of equations in the form of mass, momentum, and total energy conservation laws with general gas equations of state is considered. A family of three-point symmetric spatial discretizations of this system is studied for which the internal energy equation has a suitable form (without imbalance terms). An entropy balance equation is derived, and the influence exerted by the choice of discretizations of various terms on the form of difference imbalance terms in this equation is determined. Special discretizations are presented for which the corresponding nondivergence imbalance terms are zero. The Euler system of equations is solved numerically in the cases of a perfect polytropic gas, stiffened gas, and the van der Waals equations of state.

Effect of the Earth’s compression on the physical libration of the Moon

The effect of the Earth’s compression on the physical libration of the Moon is studied using a new vector method. The moment of gravitational forces exerted on the Moon by the oblate Earth is derived considering second order harmonics. The terms in the expression for this moment are arranged according to their order of magnitude. The contribution due to a spherically symmetric Earth proves to be greater by a factor of 1.34 × 106 than a typical term allowing for the oblateness. A linearized Euler system of equations to describe the Moon’s rotation with allowance for external gravitational forces is given. A full solution of the differential equation describing the Moon’s libration in longitude is derived. This solution includes both arbitrary and forced oscillation harmonics that we studied earlier (perturbations due to a spherically symmetric Earth and the Sun) and new harmonics due to the Earth’s compression. We posed and solved the problem of spinorbital motion considering the orientation of the Earth’s rotation axis with regard to the axes of inertia of the Moon when it is at a random point in its orbit. The rotation axes of the Earth and the Moon are shown to become coplanar with each other when the orbiting Moon has an ecliptic longitude of L☾ = 90° or L☾ = 270°. The famous Cassini’s laws describing the motion of the Moon are supplemented by the rule for coplanarity when proper rotations in the Earth-Moon system are taken into account. When we consider the effect of the Earth’s compression on the Moon’s libration in longitude, a harmonic with an amplitude of 0.03″ and period of T8 = 9.300 Julian years appears. This amplitude exceeds the most noticeable harmonic due to the Sun by a factor of nearly 2.7. The effect of the Earth’s compression on the variation in spin angular velocity of the Moon proves to be negligible.

Classification of some modifications of the euler system of equations

The classical Euler system of equations, which is mathematically a system of first-order quasilinear hyperbolic equations, describes inviscid gas flows. There are numerous modifications of the system, primarily including the viscous Navier‐Stokes equations, which are of the compound hyperbolic‐parabolic type. This paper addresses different modifications. First, we analyze a typical quasi-gasdynamic system of equations from [1]. It is shown that the system is parabolic in the sense of Petrovskii (in the physically most interesting range of parameters). Second, we consider the generalized Euler system of equations from [2]. It is established that the system is Petrovskii elliptic (which, in our view, makes its application problematic from the point of view of mechanics and computational mathematics). Third, two related modifications are examined, which are shown to be systems of secondorder quasilinear hyperbolic equations. Sketches of the proofs are presented. 1. The system of quasi-gasdynamic equations from

Existence of solutions to the transonic pressure gradient equations of the compressible euler equations in elliptic Regions

We establish the existence of a smooth solution in tis elliptic region in the self—similar plane to the pressure—gradient system arisen from the wave—particle splitting of the two—dimensional compressible Euler system of equations. The pressure—gradient system takes the form Here (u,v) is the velocity, ρ is the density which is independent of time resulted from the splitting procedurep is the pressure, and is the energy. The natural (parabolically degenerate) boundary value is used.

Composite, Navier-Stokes and Euler unsteady-flow computations in boundary layers

Computational approaches based on previous nonlinear theoretical findings are developed for a Composite system, the Navier-Stokes system, and the Euler system of equations, in turn, with applications to incompressible boundary-layer transition and dynamic stall. The emphasis is on schemes appropriate for medium-to-large Reynolds numbers, and so as a start the computations are kept to two spatial dimensions. The Composite scheme is developed first, including significant normal-pressure-gradient effects as suggested by the theory. The same scheme is then modified to accommodate, iteratively, the Navier-Stokes form and then the Euler form, for comparison. The agreement between the three sets of results tends to be very close in the parameter ranges studied. The necessary extensions of the work are also discussed.

A FINITE VOLUME SOLVER FOR THE SHALLOW WATER EQUATIONS IN VARYING BED ENVIRONMENTS

Abstract A numerical scheme for solving the shallow-water equations is presented. An analogy is made between flows governed by shallow-water equations and the Euler system of equations used in gas dynamics. An emphasis is placed on the difference presented by the bathymetry in hydraulic systems. The discretization of the governing equations is based on Roe's flux difference-splitting solver, initially developed for solving inviscid compressible flows. The spatial discretization is handled within a finite-volume context by using triangles or quadrilaterals as the basic control-volume cells. This approach enables an easy and flexible treatment of general geometries. A development of the boundary conditions tailored for the current scheme is given. Fundamental validation tests are presented.