Handbook of Compressible Aerodynamics

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 ...