# Chapter 20

# Flows with Shock Waves: Rotational Method of Characteristics

## 20.1. Shock wave and rotational flow

When calculating a supersonic flow by the method of characteristics, it may be necessary to take into account the presence of shock waves caused by discontinuities of direction (a wall with a sudden slope change), of pressure (flow at the exit of an overexpanded nozzle), or even focalization of compression waves. To address this problem, we must introduce *shock operators* which enable us to calculate the shock during the construction during the construction of the characteristic mesh.

As we know, crossing a shock wave (*C*) causes an increase in entropy *s*. If (*C*) is rectilinear, the downstream entropy *s* is constant, and thus we can apply the operators defined in section 18.2 as such, taking into account that the stagnation pressure has become *p*_{st} < *p*_{st0}. In fact, the shock waves encountered in practical problems are almost always curved. Locally, the shock equations remain the same and the relations of Chapter 13 apply. However, as the shock angle *σ* varies along (*C*), the entropy increase also varies from one streamline to another after the crossing of (*C*). According to the Crocco relation (see section 8.2.1), a non-zero vorticity value is associated with the entropy gradient: the flow downstream of a curved shock is rotational. Consequently, the quantities characterizing the flow will be three, namely: the pressure, the direction of the velocity, and the local entropy. Denoting ...