Riemann problem shock

The theory of hyperbolic systems makes it possible to solve the problem known as the Riemann Problem (R.P.) where one seeks to connect two constant states (see also Glueckauf, 1949). It corresponds to the transfirrmation of an homogeneous rock which is the downstream medium (first constant state) by a fluid of constant composition (upstream medium, second constant state). Because the fluid circulates from the left to the right, the downstream medium is said on the right ("a droite", d) and the upstream medium is said on the left ("a gauche", g)). The theory shows that the solution of the R.P. is made of 3 constant states in addition to the two states (g) and (d), another constant state (intermediate, i) appears, which is connected to (g) and (d) by shocks, "detentes" or contact... 

Figure 3. Representation in both the composition space (left) and the (composition, space) coordinates (rig)it) of the evolution of solid solution composition for different types of initial and boundary conditions, giving rise to different types of Riemann problems. The two independent composition variables arc termed u and v. The craistant states of the Riemann problems are called u , v and u v" respectively for each case. Shock, ddtente waves appear, so as intermediate plateaus. The 1-waves and 2-waves are labelled by I and 2 respectively.

Here we apply the finite-volume scheme to simulate two different examples of inhomogeneous kinetic equations. The first example is a non-equilibrium Riemann shock problem with different values of the collision time t. The second example is two ID crossing jets with different collision times. In reality, the collision time is controlled by the number density Moo, which we normalize with respect to unity in these examples. Thus, the reader can interpret the different values of t as different values of the unnormalized number density. As noted above, for the multi-Gaussian quadrature we compute the spatial fiuxes using Ml = 14 and Mo = 4 with a CFL number of unity. 

The initial conditions for the nonzero moments in the Riemann shock problem are shown in Figure 8.9. On the left half of the domain the normalized number density is Moo = 1, and on the right half it is 0.1. Initially the RMS velocity is 1 on both sides of the domain and the mean velocity is null. The moments are initialized as Maxwellian, so that the initial conditions are at local equilibrium. As time increases, a shock wave starting at x = 0 moves to the right and a deflation wave moves to the left. In the limit of t = 0, the solution is the same as for the Euler equation of gas dynamics. Sample results for the moments with T = 100 at time t = 0.5 are shown in Figure 8.10. For this case, the collisions are very weak, and thus there is little transfer of kinetic energy from the M-component to the... 

The above referenced experiments (adiabatic) occur in flow geometries much more complicated than the simple (isothermal) planar Riemann (shock structure) problems discussed above. Furthermore, the interesting results on shock splitting given in [27], [41-44] all involved piston and not Riemann initial value problems. Hence a direct comparison with the local shock structure theory given here may be difficult. Nonetheless for completeness we record some obvious consistencies and discrepancies between the experimental results and traveling wave theory. 

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