Supersonic Flow

In a compressible flow if the local speeds of air particles exceed the local speed of sound, the supersonic flow condition emerges. In such cases different types of waves appear in the flow depending on geometry or back pressure change in the flow. The most pronounced among these waves are the shock waves. The simplest shock wave is the normal shock wave which occurs normal to the flow direction of one dimensional flows. The physically possible flow is the one being the supersonic flow before and the subsonic flow after the shock (Shapiro 1953). This way the entropy increases after the shock in accordance with the second law of thermo­dynamics. The normal shocks are strong shocks therefore the entropy increase is large and this large entropy causes the flow to change its regime from supersonic to subsonic. A simple model of a normal shock is shown in Fig. 5.9. After the normal shock, the pressure temperature and the density of the flow increase whereas the Mach number and the stagnation pressures decrease.

The oblique shock waves, on the other hand, appear to be two dimensional but they can be treated with one dimensional flow analysis. Oblique waves are inclined with the free stream but they are straight like normal shocks. The free stream of the flow deflects and slows down after passing an oblique shock. The amount of deflection in the flow is inversely proportional with the inclination of the shock. The oblique shocks are known as weak shocks because they cause smaller entropy change than that of a normal shock. The reason for the occurrence of the oblique shock is in general the narrowing of the flow area as shown in Fig. 5.10. The flow after the oblique shock goes parallel to the wall which is narrowed down with angle в. As indicated in Fig. 5.10, after the shock the component of the freestream parallel to the shock remains the same, however, the normal component slows down. Since the shock is weak the entropy change after the shock is small.

Fig. 5.10 Oblique shock, velocity relations

fact and the shock being a straight line enable us to assume potential flow before and after the shock.

Another reason for waves to appear in the supersonic flow is the expansion of the flow field. In this case the expansion waves are created in the flow so that the flow passing through this waves is made to turn until it remains parallel to the wall. The expansion process as shown in Fig. 5.11 is an isentropic process.

After seeing all the wave types likely to occur in supersonic flow, we can analyze the flow about a profile immersed in a supersonic stream. Shown in Fig. 5.12 are the expansion and the shock waves over the upper and lower surfaces of a diamond shaped profile immersed in a supersonic stream with an angle of attack.

According to Fig. 5.12, on the upper surface of the profile the flow expands after the leading edge and continues to expand and speed up at the mid chord until the trailing edge where the oblique shock slows down and changes the direction of the flow. At the lower surface, however, because of compression at the leading edge an oblique shock is generated to slow down the flow and the flow expands after mid chord and at the trailing edge to become parallel with the flow of the upper surface. After the trailing edge, we may have upper and lower surface speeds different from each other so that a slip line is created without any pressure dis­continuity across. In supersonic flow the pressure difference at the wake is zero and that is how we satisfy the Kutta condition.

So far we have seen a summary of types of waves which can occur because of presence of a profile in a steady supersonic flow. Let us study furthermore, the time dependent creation of these waves from the lifting surfaces for which we are interested to find unsteady aerodynamic coefficients.