scholarly journals Analysis of the Stability of the Riemann Problem for a Simplified Model in Magnetogasdynamics

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Yujin Liu ◽  
Wenhua Sun
Keyword(s):  
Shock Wave ◽  
Riemann Problem ◽  
Contact Discontinuity ◽  
Ideal Gas ◽  
Simplified Model ◽  
Characteristic Method ◽  
Small Perturbations ◽  
Riemann Solutions ◽  
One Dimensional ◽  
The Stability

The generalized Riemann problem for a simplified model of one-dimensional ideal gas in magnetogasdynamics in a neighborhood of the origin(t>0)in the(x,t)plane is considered. According to the different cases of the corresponding Riemann solutions, we construct the perturbed solutions uniquely with the characteristic method. We find that, for some case, the contact discontinuity appears after perturbation while there is no contact discontinuity of the corresponding Riemann solution. For most cases, the Riemann solutions are stable and the perturbation can not affect the corresponding Riemann solutions. While, for some few cases, the forward (backward) rarefaction wave can be transformed into the forward (backward) shock wave which shows that the Riemann solutions are unstable under such local small perturbations of the Riemann initial data.

scholarly journals Riemann problem with delta initial data for the two-dimensional steady pressureless isentropic relativistic Euler equations

2021 ◽  
Author(s):  
Yu Zhang ◽  
Yanyan Zhang
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Riemann Problem ◽  
Contact Discontinuity ◽  
Global Solutions ◽  
Two Dimensional ◽  
Relativistic Euler Equations ◽  
Riemann Solutions ◽  
Isentropic Relativistic Euler Equations ◽  
The Stability

The Riemann problem for the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data is studied. First, the perturbed Riemann problem with three pieces constant initial data is solved. Then, via discussing the limits of solutions to the perturbed Riemann problem, the global solutions of Riemann problem with delta initial data are completely constructed under the stability theory of weak solutions. Interestingly, the delta contact discontinuity is found in the Riemann solutions of the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data.

scholarly journals Construction of the Global Solutions to the Perturbed Riemann Problem for the Leroux System

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Pengpeng Ji ◽  
Chun Shen
Keyword(s):  
Initial Data ◽  
Riemann Problem ◽  
Phase Plane ◽  
Wave Interaction ◽  
Global Solutions ◽  
Geometrical Structures ◽  
Small Perturbations ◽  
Riemann Solutions ◽  
Piecewise Constant ◽  
Rarefaction Curves

The global solutions of the perturbed Riemann problem for the Leroux system are constructed explicitly under the suitable assumptions when the initial data are taken to be three piecewise constant states. The wave interaction problems are widely investigated during the process of constructing global solutions with the help of the geometrical structures of the shock and rarefaction curves in the phase plane. In addition, it is shown that the Riemann solutions are stable with respect to the specific small perturbations of the Riemann initial data.

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

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Qixia Ding ◽  
Lihui Guo
Keyword(s):  
Shock Wave ◽  
Contact Discontinuity ◽  
Vacuum State ◽  
Chaplygin Gas ◽  
Delta Shock Wave ◽  
Generalized Chaplygin Gas ◽  
Riemann Solutions ◽  
Pressure Limit ◽  
Delta Shock ◽  
Riemann Solution

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 theory of a shock wave driven by a corrugated piston in a non-ideal fluid

2011 ◽  
Vol 691 ◽  
pp. 146-164 ◽  
Author(s):  
J. W. Bates
Keyword(s):  
Shock Wave ◽  
Ideal Gas ◽  
Mathematical Framework ◽  
Fluid Medium ◽  
Temporal Behaviour ◽  
Shock Stability ◽  
Linear Perturbations ◽  
Ripple Amplitude ◽  
The Stability ◽  
Appl Math

AbstractIn the context of an Eulerian fluid description, we investigate the dynamics of a shock wave that is driven by the steady impulsively initiated motion of a two-dimensional planar piston with small corrugations superimposed on its surface. This problem was originally solved by Freeman (Proc. Royal Soc. A, vol. 228, 1955, pp. 341–362), who showed that piston-driven shocks are unconditionally stable when the fluid medium through which they propagate is an ideal gas. Here, we generalize Freeman’s mathematical framework to account for a fluid characterized by an arbitrary equation of state. We find that a sufficient condition for shock stability is $\ensuremath{-} 1\lt h\lt {h}_{c} $, where $h$ is the D’yakov parameter and ${h}_{c} $ is a critical value less than unity. For values of $h$ within this range, linear perturbations imparted to the front by the piston at time $t= 0$ attenuate asymptotically as ${t}^{\ensuremath{-} 3/ 2} $. Outside of this range, the temporal behaviour of perturbations is more difficult to determine and further analysis is required to assess the stability of a shock front under such circumstances. As a benchmark of the main conclusions of this paper, we compare our generalized expression for the linearized shock-ripple amplitude with an independent Bessel-series solution derived by Zaidel’ (J. Appl. Math. Mech., vol. 24, 1960, pp. 316–327) and find excellent agreement.

On the stability of plane shock waves

1957 ◽  
Vol 2 (4) ◽  
pp. 397-411 ◽  
Author(s):  
N. C. Freeman
Keyword(s):  
Shock Wave ◽  
Mach Number ◽  
Shock Waves ◽  
Plane Shock ◽  
Two Dimensional ◽  
Small Perturbations ◽  
Diverging Channel ◽  
Maximum Stability ◽  
The Stability ◽  
Fourier Type

The decay of small perturbations on a plane shock wave propagating along a two-dimensional channel into a fluid at rest is investigated mathematically. The perturbations arise from small departures of the walls from uniform parallel shape or, physically, by placing small obstacles on the otherwise plane parallel walls. An expression for the pressure on a shock wave entering a uniformly, but slowly, diverging channel already exists (given by Chester 1953) as a deduction from the Lighthill (1949) linearized small disturbance theory of flow behind nearly plane shock waves. Using this result, an expression for the pressure distribution produced by the obstacles upon the shock wave is built up as an integral of Fourier type. From this, the shock shape, ξ, is deduced and the decay of the perturbations obtained from an expansion (valid after the disturbances have been reflected many times between the walls) for ξ in descending power of the distance, ζ, travelled by the shock wave. It is shown that the stability properties of the shock wave are qualitatively similar to those discussed in a previous paper (Freeman 1955); the perturbations dying out in an oscillatory manner like ζ−3/2. As before, a Mach number of maximum stability (1·15) exists, the disturbances to the shock wave decaying most rapidly at this Mach number. A modified, but more complicated, expansion for the perturbations, for use when the shock wave Mach number is large, is given in §4.In particular, the results are derived for the case of symmetrical ‘roof top’ obstacles. These predictions are compared with data obtained from experiments with similar obstacles on the walls of a shock tube.

scholarly journals Interaction of waves in one-dimensional dusty gas flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Pooja Gupta ◽  
Rahul Kumar Chaturvedi ◽  
L. P. Singh
Keyword(s):  
Shock Wave ◽  
High Frequency ◽  
Gas Flow ◽  
Ideal Gas ◽  
Transport Equations ◽  
Dust Particles ◽  
Multiple Time Scales ◽  
Dusty Gas ◽  
One Dimensional ◽  
Geometrical Acoustics

AbstractThe present study uses the theory of weakly nonlinear geometrical acoustics to derive the high-frequency small amplitude asymptotic solution of the one-dimensional quasilinear hyperbolic system of partial differential equations characterizing compressible, unsteady flow with generalized geometry in ideal gas flow with dust particles. The method of multiple time scales is applied to derive the transport equations for the amplitude of resonantly interacting high-frequency waves in a dusty gas. These transport equations are used for the qualitative analysis of nonlinear wave interaction process and self-interaction of nonlinear waves which exist in the system under study. Further, the evolutionary behavior of weak shock waves propagating in ideal gas flow with dust particles is examined here. The progressive wave nature of nonresonant waves terminating into the shock wave and its location is also studied. Further, we analyze the effect of the small solid particles on the propagation of shock wave.

Measure-Valued Solutions to a Non-Strictly Hyperbolic System with Delta-Type Riemann Initial Data

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Shuangrong Li ◽  
Chun Shen
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Riemann Problem ◽  
Contact Discontinuity ◽  
Initial Condition ◽  
Global Measure ◽  
Delta Shock Wave ◽  
Dirac Delta ◽  
Delta Shock ◽  
Delta Type

AbstractThis paper is concerned with the construction of global measure-valued solutions to the extended Riemann problem for a non-strictly hyperbolic system of two conservation laws with delta-type initial data. The wave interaction problems have been extensively studied for all kinds of situations by using the initial condition consisting of constant states in three pieces instead of delta-type initial data under the perturbation method. The measure-valued solutions of the extended Riemann problem are achieved constructively when the perturbed parameter tends to zero. During the process of constructing solutions, a new and interesting nonlinear phenomenon is discovered, in which the initial Dirac delta function travels along the trajectory of either delta shock wave or contact discontinuity (or delta contact discontinuity). Moreover, a delta shock wave is separated into a delta contact discontinuity and a shock wave during the process of delta shock wave penetrating a composite wave composed of a rarefaction wave and a contact discontinuity. In addition, we further consider the constructions of global measure-valued solutions when the initial condition contains Dirac delta functions at two different initial points.

Kinetic Vlasov simulations of contact discontinuities

2020 ◽  
Author(s):  
Takayuki Umeda ◽  
Naru Tsujine ◽  
Yasuhiro Nariyuki
Keyword(s):  
Contact Discontinuity ◽  
High Density ◽  
Stable Structure ◽  
Low Density ◽  
Sharp Transition ◽  
Thermal Pressure ◽  
Transition Layers ◽  
One Dimensional ◽  
Contact Discontinuities ◽  
The Stability

<p>The stability of contact discontinuities formed by the relaxation of two Maxwellian plasmas with different number densities but the same plasma thermal pressure is studied by means of a one-dimensional electrostatic full-Vlasov simulation. Our simulation runs with various combinations of ion-to-electron ratios of the high-density and low-density regions showed that transition layers of density and temperature without jump in the plasma thermal pressure are obtained when the electron temperatures in the high-density and low-density regions are almost equal to each other. However, the stable structure of the contact discontinuity with a sharp transition layer on the Debye scale is not maintained. It is suggested that non-Maxwellian velocity distributions are necessary for the stable structure of contact discontinuities. A direct comparison between full- and hybrid-Vlasov simulations is also made. </p>

The exact Riemann solutions to an isentropic non-ideal dusty gas flow under a magnetic field

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yicheng Pang ◽  
Jianjun Ge ◽  
Zuozhi Liu ◽  
Min Hu
Keyword(s):  
Magnetic Field ◽  
Riemann Problem ◽  
Analytical Solutions ◽  
Gas Flow ◽  
Numerical Solutions ◽  
Transverse Magnetic ◽  
Dusty Gas ◽  
Riemann Solutions ◽  
One Dimensional ◽  
A New Technique

Abstract We analyse exact solutions to the Riemann problem for a one-dimensional isentropic and perfectly conducting non-ideal dusty gas flow in the presence of a transverse magnetic field. We give the expression of wave curves as well as the behaviors of these wave curves. A new technique is provided to get a complete list of analytical solutions with the corresponding criteria. Moreover, the numerical solutions to the Riemann problem are also given. It is shown that the analytical solutions match well with the corresponding numerical solutions.

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