TWO-DIMENSIONAL VORTEX SHEETS FOR THE NONISENTROPIC EULER EQUATIONS: LINEAR STABILITY

2008 ◽  
Vol 05 (03) ◽  
pp. 487-518 ◽  
Author(s):  
ALESSANDRO MORANDO ◽  
PAOLA TREBESCHI
Keyword(s):  
Linear Stability ◽  
Euler Equations ◽  
Compressible Flows ◽  
Tangential Velocity ◽  
Vortex Sheets ◽  
Isentropic Euler Equations ◽  
Contact Discontinuities ◽  
Two Space Dimensions ◽  
Loss Of Regularity ◽  
Additional Loss

We study the linear stability of contact discontinuities for the nonisentropic compressible Euler equations in two space dimensions. Assuming the jump of the tangential velocity across the discontinuity surface is sufficiently large, we derive a suitable energy estimate for the linearized boundary value problem. The found estimate extends to nonisentropic compressible flows the main result of Coulombel–Secchi for the isentropic Euler equations. Compared with this latter case, when the jump of the tangential velocity of the unperturbed flow attains a certain critical value in the region of weak stability, here an additional loss of regularity appears; this is related to the presence of a double root of the Lopatinskii determinant associated to the problem.

scholarly journals A New Flux Splitting Scheme Based on Toro-Vazquez and HLL Schemes for the Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Pascalin Tiam Kapen ◽  
Tchuen Ghislain
Keyword(s):  
Euler Equations ◽  
Compressible Flows ◽  
Arithmetic Mean ◽  
Shock Propagation ◽  
Test Cases ◽  
Splitting Scheme ◽  
Pressure System ◽  
Wave Speeds ◽  
Flux Splitting ◽  
Contact Discontinuities

This paper presents a new flux splitting scheme for the Euler equations. The proposed scheme termed TV-HLL is obtained by following the Toro-Vazquez splitting (Toro and Vázquez-Cendón, 2012) and using the HLL algorithm with modified wave speeds for the pressure system. Here, the intercell velocity for the advection system is taken as the arithmetic mean. The resulting scheme is more accurate when compared to the Toro-Vazquez schemes and also enjoys the property of recognition of contact discontinuities and shear waves. Accuracy, efficiency, and other essential features of the proposed scheme are evaluated by analyzing shock propagation behaviours for both the steady and unsteady compressible flows. The accuracy of the scheme is shown in 1D test cases designed by Toro.

scholarly journals A High-Order Method for Weakly Compressible Flows

2017 ◽  
Vol 22 (4) ◽  
pp. 1150-1174 ◽  
Author(s):  
Klaus Kaiser ◽  
Jochen Schütz
Keyword(s):  
Euler Equations ◽  
Discontinuous Galerkin ◽  
Compressible Flows ◽  
Temporal Integration ◽  
Reference Solution ◽  
Order Method ◽  
Asymptotic Preserving ◽  
High Order Method ◽  
Weakly Compressible ◽  
Isentropic Euler Equations

AbstractIn this work, we introduce an IMEX discontinuous Galerkin solver for the weakly compressible isentropic Euler equations. The splitting needed for the IMEX temporal integration is based on the recently introducedreference solutionsplitting [32, 52], which makes use of theincompressiblesolution. We show that the overall method isasymptotic preserving. Numerical results show the performance of the algorithm which is stable under a convective CFL condition and does not show any order degradation.

GLOBAL CENTERED WAVES AND CONTACT DISCONTINUITIES OF THE AXISYMMETRIC EULER EQUATIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 143-193 ◽  
Author(s):  
PAUL GODIN
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Contact Discontinuity ◽  
Existence Results ◽  
Smooth Solutions ◽  
Perfect Gas ◽  
Rotation Invariant ◽  
Contact Discontinuities ◽  
Positive Time ◽  
Two Space Dimensions

Global existence results have been obtained by Serre and Grassin–Serre for smooth solutions to the Euler equations of a perfect gas, provided the initial data belong to suitable spaces, the initial sound speed is small, and the initial velocity forces particles to spread out. We work in two space dimensions and start with initial data which are rotation invariant around 0 and of the type considered by Serre and Grassin–Serre. We then consider slightly perturbed initial data which are also rotation invariant around 0 and jump across a given circle centered at 0, in such a way that there is a solution with these perturbed initial data which presents two centered waves (in radial coordinates) and one contact discontinuity for small positive time. We show that this solution is global in positive time and keeps the same structure.

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

2019 ◽  
Vol 266 (9) ◽  
pp. 5397-5430 ◽  
Author(s):  
Alessandro Morando ◽  
Paola Trebeschi ◽  
Tao Wang
Keyword(s):  
Euler Equations ◽  
Nonlinear Stability ◽  
Two Dimensional ◽  
Vortex Sheets

scholarly journals Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations

2018 ◽  
Vol 52 (3) ◽  
pp. 893-944 ◽  
Author(s):  
Raphaèle Herbin ◽  
Jean-Claude Latché ◽  
Trung Tan Nguyen
Keyword(s):  
Energy Balance ◽  
Kinetic Energy ◽  
Internal Energy ◽  
Euler Equations ◽  
Left Hand ◽  
Isentropic Euler Equations ◽  
Space Discretization ◽  
Face Centered ◽  
The Stability ◽  
Stability And Consistency

In this paper, we build and analyze the stability and consistency of decoupled schemes, involving only explicit steps, for the isentropic Euler equations and for the full Euler equations. These schemes are based on staggered space discretizations, with an upwinding performed with respect to the material velocity only. The pressure gradient is defined as the transpose of the natural velocity divergence, and is thus centered. The velocity convection term is built in such a way that the solutions satisfy a discrete kinetic energy balance, with a remainder term at the left-hand side which is shown to be non-negative under a CFL condition. In the case of the full Euler equations, we solve the internal energy balance, to avoid the space discretization of the total energy, whose expression involves cell-centered and face-centered variables. However, since the residual terms in the kinetic energy balance (probably) do not tend to zero with the time and space steps when computing shock solutions, we compensate them by corrective terms in the internal energy equation, to make the scheme consistent with the conservative form of the continuous problem. We then show, in one space dimension, that, if the scheme converges, the limit is indeed an entropy weak solution of the system. In any case, the discretization preserves by construction the convex of admissible states (positivity of the density and, for Euler equations, of the internal energy), under a CFL condition. Finally, we present numerical results which confort this theory.

Sign in / Sign up

Export Citation Format

Share Document