Building your own shock tube

This report treats the development of a shock tube solver for the simulation of flows described by the one-dimensional Euler equations. A well-known one-dimensional flow problem is the initial Riemann problem, which treats the development of a flow due to two initially separated states. Removing the membrane separating these two states results in a characteristic wave pattern consisting in general of three waves, either a shock wave or an expansion fan, and a contact discontinuity. This Riemann problem can be solved exactly using gasdynamics theory. Implementing this theory in a computer program results in the exact Riemann solver. An elegant algorithm is used for the determination of the region containing the time axis. Several tests are performed to check this solver; all results complied with the literature. Solving more complex flow problems than the Riemann problem requires a numerical approach. A Finite-Volume Method is used to discretize the Euler equations. Hancock's predictor-corrector-type MUSCL scheme is used for marching in time, although also the first-order upwind scheme is implemented in the solver. The cell-interface fluxes required for this time stepping procedure are calculated using two different solvers; the exact Riemann solver, also called Godunov's method, and the approximate Riemann solver by Roe. Roe's method is implemented both with and without the entropy fix, which prevents the solution from becoming unphysical in the case of a transonic expansion fan. Further more, several averaging schemes used for the predictor step are implemented in the solver, being the Algebraic average, and the Double Minmod, Superbee and Koren's limiters. The shock tube solver is tested using the five test cases also used for testing the exact Riemann solver. The numerical results compare very well with the exact results. No significant differences between the Godunov and the Roe solver are present. Testing for convergence was done for both the first-order upwind scheme and the second-order Hancock scheme using the less-known fractional error norms such that the influence of the first order errors induced by the contact discontinuity and the shock is reduced. Both these schemes resulted in the expected order of convergence. The averaging schemes are also compared with each other. The limiters perform almost equally well, allowing practically no overshoot or wiggles. This behavior was visible in the case of the linear algebraic average. The entropy fix was tested using an adapted version of Sod's problem, with a transonic expansion fan. The solver of Roe without an entropy fix results in a discontinuity in the expansion fan, while the solver with the fix prevents this from happening. Further the Woodward-Colella double blast wave problem in a closed tube is treated. Two strong initial discontinuities in the pressure result in a complex wave pattern. The solver performs well on this test. Finally the interaction of two non-simple waves is treated. The special case in which the ratio of specific heats is taken equal to 3.0 and in which the characteristics become straight even in the non-simple region is treated too