Isentropic Gas Research Articles

Global discontinuous solution for $1D$ isentropic gas dynamics system

Global discontinuous solution for $1D$ isentropic gas dynamics system

Influence of Different Gases on the Design Point of an Industrial Axial Compressor and Deduced Aerodynamical Rematching Methodology

This paper presents a numerical investigation of the effects of compressing various gases, for example, carbon dioxide (CO2) and methane (CH4), on an eight-stage axial air compressor. Several adaptation methods are applied to achieve a similar operating point as for air. Theoretically, the operating point depends on Mach number, flow angles, Reynolds number and isentropic exponent. Numerical results show mismatch effects which arise in the parameters using a non-adapted geometry. A rematching procedure is described, including deduced speed adjustments, in order to achieve Mach number equality at compressor inlet. Only shroud modifications are performed to rematch the flow angles of the air simulation. Although Reynolds and Mach number are kept constant at compressor inlet, an inevitable deviation in downstream flow causes mismatches in efficiency and pressure ratio. Both analytical and numerical methodologies show that the scale of shroud adjustments, as well as the size of mismatch in Mach and Reynolds number, can be correlated to the isentropic gas exponent. In summary, the main impact on gas behavior in an axial air compressor is attributable to the change in isentropic exponent. Derivations of shroud adaptation and analyses of inevitable aerodynamic mismatch are therefore developed depending on the isentropic exponent.

Collisionless adiabatic afterglow

We study, by numerical and analytical means, the evolution of a collisionless plasma initiated between absorbing walls. The ensuing flow is described by rarefaction waves that travel inward from the boundaries, interact, and eventually vanish after crossing through, leading up to the asymptotic stage of the decay. Particle simulations indicate that the kinetic evolution strongly resembles one found in isentropic gas dynamics. Namely, a very gradual density profile forms in the expanding central region where the rarefaction waves interact, with an accompanying linear velocity profile. Asymptotically, the density falls off as 1/t. The density and the flux at the boundary show little variation over the period when rarefaction waves still exist. Plasma potential, on the other hand, drops quite rapidly (on the underlying ion-acoustic timescale) to less than initial electron temperature Te when over 70% of the particles still remain in the system. This is due to electron kinetics being governed by conservation of adiabatic invariant in a slowly varying potential well. Analytical model of the velocity distribution is presented to explain the simulations. The results have implications for afterglow plasmas used in material processing and also for ion-extraction devices. One property of potential interest is good uniformity of the decaying plasma that occurs after approximately one ion-acoustic time.

Large BV solution for isentropic gas dynamics system with initial data far from vacuum

We give the global existence results for the isentropic gas dynamics system with large BV initial data, assuming that the initial data are far from vacuum. The distance of the initial data from vacuum may be so large depending on the initial BV bounds. A new scaling framework is introduced, and we obtain the crucial BV bounds estimates of the vanishing viscosity solutions with the help of this framework coupled with the famous result for the corresponding small BV initial data given by Bianchini and Bressan.

Global stability to steady supersonic solutions of the 1-D compressible Euler equations with frictions

In this paper, using wave decomposition and establishing uniform a priori C1 estimates, we proved global stability of steady supersonic Fanno flows under small perturbations of initial-boundary values in a one-dimensional rectilinear finite duct with constant cross-sections. The flow is governed by the one-space dimensional compressible Euler equations with a nonlinear damping term representing frictions in the duct. Both isentropic and non-isentropic polytropic gases are considered.

Global smooth solutions to 3D irrotational Euler equations for Chaplygin gases

We study here the Cauchy problem associated with the isentropic and compressible Euler equations for Chaplygin gases. Based on the new formulation of the compressible Euler equations in J. Luk and J. Speck [The hidden null structure of the compressible Euler equations and a prelude to applications, J. Hyperbolic Differ. Equ. 17 (2020) 1–60] we show that the wave system satisfied by the modified density and the velocity for Chaplygin gases satisfies the weak null condition. We then prove the global existence of smooth solutions to the irrotational and isentropic Chaplygin gases without introducing a potential function, when the initial data are small perturbations to a constant state.

Global Weak Solutions for a Nonlinear Hyperbolic System

In this paper, we study the global existence of weak solutions for the Cauchy problem of the nonlinear hyperbolic system of three equations (1.1) with bounded initial data (1.2). When we fix the third variable s, the system about the variables ρ and u is the classical isentropic gas dynamics in Eulerian coordinates with the pressure function $$P(\rho ,s) = {{\rm{e}}^s}{{\rm{e}}^{ - {1 \over \rho }}}$$ , which, in general, does not form a bounded invariant region. We introduce a variant of the viscosity argument, and construct the approximate solutions of (1.1) and (1.2) by adding the artificial viscosity to the Riemann invariants system (2.1). When the amplitude of the first two Riemann invariants (w1(x, 0), w2(x, 0)) of system (1.1) is small, (w1(x, 0), w2(x, 0)) are nondecreasing and the third Riemann invariant s(x, 0) is of the bounded total variation, we obtained the necessary estimates and the pointwise convergence of the viscosity solutions by the compensated compactness theory. This is an extension of the results in [1].

Solution of the mixed formulation for generalized Forchheimer flows of isentropic gases

This paper focuses on the generalized Forchheimer flows of isentropic gas in a porous medium, described by a system of two nonlinear degenerate partial differential equations of first order. We prove the existence and uniqueness of the Dirichlet problem for the stationary problem. The technique of semi-discretization in time is used to prove the existence for the time-dependent problem.

The Riemann Problem for Chaplygin Gas Flow in a Duct with Discontinuous Cross-Section

The fluid flows in a variable cross-section duct are nonconservative because of the source term. Recently, the Riemann problem and the interactions of the elementary waves for the compressible isentropic gas in a variable cross-section duct were studied. In this paper, the Riemann problem for Chaplygin gas flow in a duct with discontinuous cross-section is studied. The elementary waves include rarefaction waves, shock waves, delta waves and stationary waves.

Radon measures as solutions of the Cauchy problem for evolution equations

We consider a standard Navier–Stokes system on the n-D torus $${\mathbb {T}}^n={\mathbb {R}}^n/{\mathbb {Z}}^n$$ , valid when the flow is not very compressible and the temperature does not vary too much. We construct a sequence of approximate solutions that tend to satisfy the equations in a weak sense for arbitrary physical initial conditions. By weak compactness, we obtain Radon measure solutions in density and momentum when the velocity of the flow is finite, as numerically observed in all tests, and the absence of void regions in the flow. We notice that the method also applies without viscosity and complements results on other systems of evolution equations such as the systems of isothermal and isentropic gas flows considered in Colombeau (Z Angew Math Phys 66(5):2575–2599, 2015).

Existence of global solutions for isentropic gas flow with friction

In this paper, we study the existence of global entropy solutions for the Cauchy problem of the extended river flow system with a friction α(x) depending on the space variable x. First, under the boundedness of , we apply the viscosity-flux approximation method coupled with the maximum principle to obtain the a priori L∞ estimates for the approximation solutions. Second, we prove the pointwise convergence of the approximation solutions by using the compactness framework from the compensated compactness theory. Finally, as a by-product, we obtain the existence of global solutions for isentropic gas flow in the Laval nozzle with friction.

A Typical Dropping Cusp Singularity of Solutions to Equations of a One-Dimensional Isentropic Gas Flow

Solutions are considered for the system of nonlinear equations of a one-dimensional isentropic gas flow with low densities. A typical singularity of these solutions corresponding to a cusp catastrophe is described. A model solution with such a singularity was obtained for the special case of shallow-water equations. Within the terms of this model solution, an asymptotic solution is obtained for the system of nonlinear equations of a one-dimensional isentropic gas flow.

The Riemann problem for an isentropic ideal dusty gas flow with a magnetic field

The Riemann problem for an isentropic ideal dusty gas flow with a magnetic field

Kinetic relaxation to entropy based coupling conditions for isentropic flow on networks

We consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions. First, we construct approximate solutions by a vector-valued BGK model with a kinetic coupling function. Introducing so-called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations. We will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution. These inequalities define the macroscopic coupling condition. Our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles. We give several examples for coupling functions and prove corresponding entropy flux inequalities. We prove also new existence results for solid wall boundary conditions and pipelines with discontinuous cross-sectional area.

Amplitude Blowup in Radial Isentropic Euler Flow

We show that the compressible Euler system for isentropic gas flow admits unbounded solutions. The examples are radial flows of similarity type and describe a spherically symmetric and continuous w...

Structural stability of shock waves in 2D compressible elastodynamics

We study the two-dimensional structural stability of shock waves in a compressible isentropic inviscid elastic material in the sense of the local-in-time existence and uniqueness of discontinuous shock front solutions of the equations of compressible elastodynamics in two space dimensions. By the energy method based on a symmetrization of the wave equation and giving an a priori estimate without loss of derivatives for solutions of the constant coefficients linearized problem we find a condition sufficient for the uniform stability of rectilinear shock waves. Comparing this condition with that for the uniform stability of shock waves in isentropic gas dynamics, we make the conclusion that the elastic force plays stabilizing role. In particular, we show that, as in isentropic gas dynamics, all compressive shock waves are uniformly stable for convex equations of state. Moreover, for some particular deformations (and general equations of state), by the direct test of the uniform Kreiss–Lopatinski condition we show that the stability condition found by the energy method is not only sufficient but also necessary for uniform stability. As is known, uniform stability implies structural stability of corresponding curved shock waves.

Global entropy solutions to multi-dimensional isentropic gas dynamics with spherical symmetry

We are concerned with spherically symmetric solutions to Euler equations for multi-dimensional compressible fluids which have many applications in diverse real physical situations. The system can be reduced to one-dimensional isentropic gas dynamics with geometric source terms. Due to the presence of the singularity at the origin, there are few papers devoted to this problem. The present paper proves two existence theorems of global entropy solutions. The first one focuses on a case excluding the origin in which negative velocity is allowed, and the second one corresponds to a case which includes the origin with non-negative velocity. The compensated compactness framework and vanishing viscosity method are applied to prove the convergence of approximate solutions. In the second case, we show that if the blast wave initially moves outwards and the initial densities and velocities decay to zero with certain rates near the origin, then the densities and velocities tend to zero with the same rates near the origin for any positive time. In particular, the entropy solutions in two existence theorems are uniformly bounded with respect to time.

CFD simulations of violent breaking wave impacts on a vertical wall using a two-phase compressible solver

Four different types of breaking wave impacts on a vertical wall are simulated using a two-dimensional two-phase Computational Fluid Dynamic (CFD) model. Air is considered as an isentropic ideal gas without solving an additional energy equation and water is treated as an incompressible liquid. The discontinuity in fluid properties across the free surface is treated using the Ghost Fluid Method, which accounts for the jump in density and compressibility. The numerical results are compared with large-scale experimental data for five cases in terms of surface elevations, total forces and pressure distributions along the wall, and a reasonable agreement is obtained overall. The characteristics of impact pressures under different breaking wave conditions are discussed and compared to each other. The two largest total forces on the wall occur in the ‘flip-through’ and ‘large air pocket’ cases. The peak pressure of the flip-through impact is localized in both time and space. The pressure within the trapped air pocket is nearly uniform with a smaller peak value and a much longer duration than that of the ‘flip-through’ case. The compression and expansion of the air pocket results in pressure oscillations, which are overestimated in frequency and amplitude due to the inaccuracy in capturing the air escape. The broken wave case has the smallest total force, but the largest local pressure among the present numerical cases, which demonstrates the necessity to study all different impact types.

On a Difference Scheme on Triangular Meshes for Gas Dynamics Equations

The paper proposes a difference scheme for the equations of an isentropic ideal gas on triangular meshes. The scheme satisfies the mass conservation law and guarantees the positiveness of the density function. The solution of the scheme satisfies the energy inequality. The solution of the scheme is proven to exist.

Riemann problem for the isentropic Chaplygin gas Cargo-LeRoux model

The so-called Cargo-LeRoux model is a hyperbolic system of conservation laws derived from the Euler equations with a constant gravity. This paper is concerned with such a model for an isentropic Chaplygin gas. The Riemann problem is solved with four kinds of structures. The first is composed of the contact discontinuities, the second contains an overcompressible δ-wave (δ-shock wave), and the third and fourth involve a contact discontinuity and a kind of non-overcompressible δ-wave. One of the highlights of this paper lies in the introduction of the non-overcompressible δ-waves.