A Study of Solutions to Euler Equations for a One Dimensional Unsteady Flow

In this paper we deal with the Euler equations for Isothermal gas. In analyzing the equations we obtain two real and distinct eigenvalues which enables us to determine the wave structure of the possible solutions to the Riemann problem set up. By considering the Rankine-Hugoniot condition we obtain the shock wave solution analytically. The rarefaction wave solution is determined analytically by considering the fact that rarefaction wave lies along integral curves. To obtain the numerical solution to the Riemann p roblem that we set up, we use a relaxation scheme to d iscretize the Euler equations for isothermal gas. Finally we present the simulation results of the numerical solutions, that is, the approximate shock and rarefaction wave solutions are shown, graphically, and explained. Consider the Eu ler equations for isothermal gas. The system consists of Euler equations and is strictly hyperbolic with t wo real and distinct eigenvalues, whereby one is greater than the other. Depending on the init ial data the eigenvalues may represent shock and rarefaction waves. The resolution of the discontinuities of a self-similar solution of the compressible Euler equation is sharper than the corresponding initial value solution, Ravi(4). A shock tube problem is the study of the propagation of shock waves in a one dimensional tube. The energy of a shock wave dissipates relatively quickly within d istance. Moving shocks are usually generated by the interaction of two bodies of gas at different pressure, with a shock wave propagating into the lower pressure gas and an expansion wave propagating into the higher pressure gas. The numerical co mputation of the shock tube problem by means on F wave digital principle showed that the MD Kichoffs network can successfully be extended by taking viscosity into account to represent the Navier Stokes equation Mengel(2). A Riemann problem consists of equations together with the discontinuous initial data. The nu merical dissipation of eight different schemes and five delimiters to nu merical