Non-isentropic Euler Equations Research Articles

Global convergence of a two-fluid non-isentropic Euler–Poisson system in one space dimension

In this paper, we study the global in time quasi-neutral limit for a two-fluid non-isentropic Euler–Poisson system in one space dimension. We prove that the system converges to the non-isentropic Euler equations as the Debye length tends to zero. This problem is studied for smooth solutions near the constant equilibrium state. To prove this result, we establish uniform energy estimates and various dissipation estimates with respect to the Debye length and the time. These estimates allow to pass to the limit to obtain the limit system by compactness arguments. In addition, the global convergence rate is obtained by use of stream function technique.

Cartesian vector solutions for $ N $-dimensional non-isentropic Euler equations with Coriolis force and linear damping

<abstract><p>In this paper, we construct and prove the existence of theoretical solutions to non-isentropic Euler equations with a time-dependent linear damping and Coriolis force in Cartesian form. New exact solutions can be acquired based on this form with examples presented in this paper. By constructing appropriate matrices $ A(t) $, and vectors $ {\mathbf{b} }(t) $, special cases of exact solutions, where entropy $ s = \ln\rho $, are obtained. This is the first matrix form solution of non-isentropic Euler equations to the best of the authors' knowledge.</p></abstract>

Global existence and stability of temporal periodic solution to non-isentropic compressible Euler equations with a source term

<abstract><p>In this paper, the 1-D compressible non-isentropic Euler equations with the source term $ \beta\rho|u|^ \alpha u $ in a bounded domain are considered. First, we study the existence of steady flows which can keep the upstream supersonic or subsonic state. Then, by wave decomposition and uniform prior estimations, we prove the global existence and stability of smooth solutions under small perturbations around the steady supersonic flow. Moreover, we get that the smooth supersonic solution is a temporal periodic solution with the same period as the boundary, after a certain start-up time, once the boundary conditions are temporal periodic.</p></abstract>

Shock Formation and Vorticity Creation for 3d Euler

AbstractWe analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3, 4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self‐similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high‐order energy estimates in Sobolev spaces. © 2022 Wiley Periodicals LLC.

Formation of Singularities and Existence of Global Continuous Solutions for the Compressible Euler Equations

We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional non-isentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint a necessary and sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum -- there exists a compression in the initial data. For the non-isentropic Euler equations, we identify a sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum -- there exists a strong compression in the initial data. Furthermore, we identify two new phenomena -- decompression and de-rarefaction -- for the non-isentropic Euler flows, different from the isentropic flows, via constructing two respective solutions. For the decompression phenomenon, we construct a first global continuous non-isentropic solution, even though initial data contain a weak compression, by solving an inverse Goursat problem, so that the solution is smooth, except on several characteristic curves across which the solution has a weak discontinuity (i.e., only Lipschitz continuity). For the de-rarefaction phenomenon, we construct a continuous non-isentropic solution whose initial data contain isentropic rarefactions (i.e., without compression) and a locally stationary varying entropy profile, for which the solution still forms a shock wave in a finite time.

General limiting behavior of Riemann solutions to the non-isentropic Euler equations for modified Chaplygin gas

In this paper, the authors investigate the Riemann solutions of the non-isentropic Euler equations for the modified Chaplygin gas, which are the set of all rarefaction waves, shock waves, and contact discontinuity. For these Riemann solutions, the authors further consider their limiting behavior. In the process of analyzing the limiting behavior, the authors found that the delta shock wave is contained in the Riemann solutions of the non-isentropic Euler equations for the modified Chaplygin gas. Consequently, up to some appropriate conditions, there is unidirectional transformation from non-isentropic Euler equations for the modified Chaplygin gas to Chaplygin gas. On the basis of numerical simulations, the authors verified the reasonability of the theoretical analysis.

Well-posedness of non-isentropic Euler equations with physical vacuum

We consider the local well-posedness of the one-dimensional non-isentropic compressible Euler equations with moving physical vacuum boundary condition. The physical vacuum singularity requires the sound speed to be scaled as the square root of the distance to the vacuum boundary. The main difficulty lies in the fact that the system of hyperbolic conservation laws becomes characteristic and degenerate at the vacuum boundary. Our proof is based on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term. Then we construct the solutions to this degenerate parabolic problem and establish the estimates that are uniform with respect to the artificial viscosity parameter. Solutions to the compressible Euler equations are obtained as the limit of the vanishing artificial viscosity. Different from the isentropic case \[7, 12], our momentum equation of conservation laws has an extra term $p\_{S}S\_\eta$ that leads to some extra terms in the energy function and causes more difficulties even for the case of $\gamma=2$. Moreover, we deal with this free boundary problem starting from the general cases of $2\leq\gamma<3$ and $1<\gamma<2$ instead of only emphasizing the isentropic case of $\gamma=2$ in \[7, 12, 16].

The Vanishing Pressure Limit of Riemann Solutions to the Non-Isentropic Euler Equations for Generalized Chaplygin Gas

We analyze the appearance of delta shock wave and vacuum state in the vanishing pressure limit of Riemann solutions to the non-isentropic generalized Chaplygin gas equations. As the pressure vanishes, the Riemann solution including two shock waves and possible one contact discontinuity converges to a delta shock wave solution. Both the densityρand the internal energyHsimultaneously present a Dirac delta singularity. And the Riemann solution involving two rarefaction waves and possible one contact discontinuity converges to a solution involving vacuum state of the transport equations.

On the initial-boundary value problem for the Euler equations in presence of a rarefaction wave

We study the initial-boundary value problem for the general non-isentropic 3D Euler equations with data which are incompatible in the classical sense, but are “rarefaction-compatible”. We show that such data are also rarefaction-compatible of infinite order and the initial-boundary value problem has a piece-wise smooth solution containing a rarefaction wave.

Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients

The linear stability with variable coefficients of the vortex sheets for the two-dimensional compressible elastic flows is studied. As in our earlier work on the linear stability with constant coefficients, the problem has a free boundary which is characteristic, and also the Kreiss-Lopatinskii condition is not uniformly satisfied. In addition, the roots of the Lopatinskii determinant of the para-linearized system may coincide with the poles of the system. Such a new collapsing phenomenon causes serious difficulties when applying the bicharacteristic extension method. Motivated by our method introduced in the constant-coefficient case, we perform an upper triangularization to the para-linearized system to separate the outgoing mode into a closed form where the outgoing mode only appears at the leading order. This procedure results in a gain of regularity for the outgoing mode which allows us to overcome the loss of regularity of the characteristic components at the poles, and hence to close all the energy estimates. We find that, analogous to the constant coefficient case, elasticity generates notable stabilization effects, and there are additional stable subsonic regions compared with the isentropic Euler flows. Moreover, since our method does not rely on the construction of the bicharacterisic curves, it can also be applied to other fluid models such as the non-isentropic Euler equations and the MHD equations.

Two-dimensional vortex sheets for the nonisentropic Euler equations: Nonlinear stability

We show the short-time existence and nonlinear stability of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on the weakly linear stability result of Morando and Trebeschi (2008) [20]. The missing normal derivatives are compensated through the equations of the linearized vorticity and entropy when deriving higher-order energy estimates. The proof of the resolution for this nonlinear problem follows from certain a priori tame estimates on the effective linear problem in the usual Sobolev spaces and a suitable Nash–Moser iteration scheme.

Global existence and nonlinear diffusion of classical solutions to non-isentropic Euler equations with damping in bounded domain

We consider classical solutions to the initial boundary value problem for the non-isentropic compressible Euler equations with damping in several space dimensions. We establish a global existence theory for classical solutions to this system when the initial data are sufficiently small. We prove that the pressure and the velocity converge exponentially toward some constants, while the entropy and the density in general do not approach constants. Finally, we prove that, as the time goes to infinity, the pressure and the velocity components of the non-isentropic Euler equations with damping converge exponentially toward those of an associated nonlinear diffusion system, and the global existence of classical solutions to these nonlinear diffusion equations is also established.

Long Time Existence of Entropy Solutions to the One-Dimensional Non-isentropic Euler Equations with Periodic Initial Data

The non-isentropic Euler system with periodic initial data in \({{\mathbb{R}}^1}\) is studied by analyzing wave interactions in a framework of specially chosen Riemann invariants, generalizing Glimm’s functionals and applying the method of approximate conservation laws and approximate characteristics. An \({{\mathcal O}(\varepsilon^{-2})}\) lower bound is established for the life span of the entropy solutions with initial data that possess \({\varepsilon}\) variation in each period.

Global Existence and Asymptotic Behavior for the 3D Compressible Non–isentropic Euler Equations with Damping

We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

A new model of roll waves: comparison with Brock’s experiments

AbstractWe derive a mathematical model of shear flows of shallow water down an inclined plane. The non-dissipative part of the model is obtained by averaging the incompressible Euler equations over the fluid depth. The averaged equations are simplified in the case of weakly sheared flows. They are reminiscent of the compressible non-isentropic Euler equations where the flow enstrophy plays the role of entropy. Two types of enstrophies are distinguished: a small-scale enstrophy generated near the wall, and a large-scale enstrophy corresponding to the flow in the roller region near the free surface. The dissipation is then added in accordance with basic physical principles. The model is hyperbolic, the corresponding ‘sound velocity’ depends on the flow enstrophies. Periodic stationary solutions to this model describing roll waves were obtained. The solutions are in good agreement with the experimental profiles of roll waves measured in Brock’s experiments. In particular, the height of the vertical front of the waves, the shock thickness and the wave amplitude are well captured by the model.

Quasi-neutral limit of the full bipolar Euler-Poisson system

The quasi-neutral limit of the multi-dimensional non-isentropic bipolar Euler-Poisson system is considered in the present paper. It is shown that for well-prepared initial data the smooth solution of the nonisentropic bipolar Euler-Poisson system converges strongly to the compressible non-isentropic Euler equations as the Debye length goes to zero.

Anelastic Limits for Euler-Type Systems

In this paper, we present rigorous derivations of anelastic limits for compressible Euler type systems when the Mach (or Froude) number tends to zero. The first and main part is to prove local existence and uniqueness of strong solution together with uniform estimates on a time interval independent of the small parameter. The key new remark is that the systems under consideration can be written in a form where ideas from \cite{MS} can be adapted. The second part of the analysis is to pass to the limit as the parameter tends to zero. In this context, the main problem is to study the averaged effect of fast acoustic waves on the slow incompressible motion. In some cases, the averaged system is completely decoupled from acoustic waves. The first example studied in this paper enters this category: it is a shallow-water type system with topography and the limiting system is the inviscid lake equation (rigid lid approximation). This is similar to the low Mach limit analysis for prepared data, following the usual terminology, where the acoustic wave disappears in a pure pressure term for the limit equation. The decoupling also occurs in infinite domains where the fast acoustic waves are rapidly dispersed at infinity and therefore have no time to interact with the slow motion (see \cite{Sc,MS, Al}). In other cases, and this should be expected in general for bounded domains or periodic solutions, this phenomenon does not occur and the acoustic waves leave a nontrivial averaged term in the limit fluid equation, which cannot be incorporated in the pressure term. In this case, the limit system involves a fluid equation, coupled to a nontrivial infinite dimensional system of differential equations which models the energy exchange between the fluid and some remanent acoustic energy. This was suspected for the periodic low Mach limit problem for nonisentropic Euler equations in \cite{MeSc} and proved for finite dimensional models. The second example treated in this paper, namely Euler type system with heterogeneous barotropic pressure law, is an example where this scenario is rigorously carried out. To the authors' knowledge, this is the first example in the literature where such a coupling is mathematically justified.

Computation of Indirect Combustion Noise by a CAA Method

Indirect combustion noise, which is generated by the acceleration of hot gases from a turbulent swirling flame, has become an important issue to understand turbo-machinery core noise. This indirect combustion noise it is dominated by a strong interaction of hydrodynamic and acoustic perturbations due to the transonic base flow in the turbine stages and combustion chamber exit nozzle. Current numerical simulation methods for the phenomenon based on compressible Large Eddy Simulation (LES) are very expansive. Thus LES applications are still limited to one design or few variations with immense computational costs. The most simple model fully covering indirect combustion noise, are the linearized non-isentropic Euler equations over an arbitrary mean flow. Therefore the gap between theory and LES is closed by a method adopted from Computational Aeroacoustics (CAA). It combines a high cost efficiency in terms of required points per wavelength with a high adaptability to realistic mean flow conditions. The CAA method is first validated with a simplified experiment and then applied to predict the acoustic properties of two model combustion chambers, one of them with transonic termination nozzle. By means of the acoustic intensity the sources of the noise are located.

Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions

We study the zero Mach number limit of classical solutions to the compressible Euler equations for nonisentropic fluids in a domain $\Omega \subset \mathbb R^d$ ($d=2$ or $3$). We consider the case of general initial data. For a domain $\Omega$, bounded or unbounded, we first prove the existence of classical solutions for a time independent of the small parameter. Then, in the exterior case, we prove that the solutions converge to the solution of the incompressible Euler equations.

Stability of contact discontinuities for the nonisentropic Euler equations

We study the stability of contact discontinuities for the nonisentropic Euler equations in two or three space dimensions. A simple criterion predicting neutral stability or violent instability is given.