Moving shock

Though the form of the Rankine-Hugoniot equation, Eqs. (1.42)-(1.44), is obtained when a stationary shock wave is created in a moving coordinate system, the same relationship is obtained for a moving shock wave in a stationary coordinate system. In a stationary coordinate system, the velocity of the moving shock wave is Ml and the particle velocity is given by u = u M2. The ratios of temperature, pressure, and density are the same for both moving and stationary coordinates. 

Type II supernovae are massive stars, ones that progress in their nuclear fuels well past the fusion of carbon and the fusion of oxygen at their centers. When their cores run out of nuclear fuel, those central regions collapse to form a neutron star, or in some cases a black hole. The incredibly intense emission of neutrinos from the newly born neutron star so heats the overlying layers, aided by an outward moving shock wave of pressure, that those layers pardy explode and are ejected. The last of these thatwas visible to the naked eye occurred in 1987, and demonstrated for the first time the correctness of the intense neutrino burst that is their main energy output. 

The discontinuities diagrammed in Fig. 5.4.1 are termed kinematic shocks in that they represent discontinuities in density. Let us calculate the speed at which the top discontinuity moves down and the bottom one up. For specificity consider a downward-moving shock. With respect to a coordinate system moving down with the speed of the discontinuity u (Fig. 5.4.2A), the flow is steady and conservation of mass for the one-dimensional picture considered gives... 

The matching of asymptotic expansions at the moving shock front (see section 2.5) shows that both conditions (27,28) must be satisfied. 

Derivation of boundary condition at the moving shock front. 

Jump Conditions Across a Moving Shock Knowing T2, the speed of sound 2 (Eq. 1) and then flow velocity U2 can be calculated. Assuming that the distance between stations 2 and 3 is negligible, and knowing the pressure ratio across the shock Ils = P3IP2, conditions at station 3 can be obtained by the normal shock relations (Eqs. 9 and 10) ... 

Figure 21 shows the water saturation as obtained with the simulator ECLIPSE. It is obvious, that the moving shocks are much more smeared than in the case of the ADER schemes. On the other hand, the general behaviour of the solutions is very similar as well as the jump height of the discontinuity. The results obtained by the simulator FrontSim are displayed in Figure 22. Here, the numerical diffusion is very small and the interface at the oil-water contact is resolved very sharply.

Derivation of Boundary Condition at the Moving Shock Front... 

---