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Shock Shock Interaction

Interaction of two shocks is of great interest for various reasons. The major reason is the prediction of the flow field after interaction. This flow field is greatly dependant on the family of the shock (right running or left running) and strength of the shocks in turn free stream properties. Aerodynamics of the body is highly dependent on the shock-shock (S/S) interaction. Edney* has defined six types of shock-shock interactions based on family and strength of the interacting shocks and post shock flow field. Herewith, basics of the shock-shock interaction are mentioned.

Interaction of shocks of different family

Let’s consider the situation given in Fig. 1 to understand the S/S interaction for shocks from two different families.

Here the right running shock originated from the top wedge interacts with the left running shock originated from the bottom wedge, hence this is the S/S interaction for shocks of different families. For simplicity we have considered flow deflection angles to be same (θ) for both the wedges. The top wedge, in this situation, is responsible for right running shock wave while bottom wedge is responsible for left running shock wave. Now consider this streamline ABC passing through the right running shock and PQR passing through the left running shock. The right running shock wave induces negative deflection to the streamline ABC at B, while left running shock wave induces positive deflection to the streamline PQR at Q. The corresponding increase in pressure is associated with the flow turning are shown in Fig. 2 using the P- θ diagram. Here we can clearly see that points A & P are identical in this plot, since both the point are in the region upstream of shocks. Since streamline ABC gets negative deflection and streamline PQR gets positive deflection of same magnitude, points B and Q represent same pressure but different deflections of same magnitude. Suppose, both the shocks are weak enough to keep the flow supersonic in the region given by points B and Q of the streamlines. The flow properties will be identical downstream of the first shocks (at B and Q), except the y-direction velocity. After interaction we can see that two different shocks originate from the interaction point. The main reason for this shock formation is to avoid the collision and intersection of streamlines ABC and PQR.

Therefore, streamline ABC passes through a left running shock and streamline PQR passes through the right running shock wave. Hence the further pressure rise is evident in Fig. 2 in the presence of post interaction shocks.

Moreover, these shocks cancel out the deflection so as to make the both the streamlines parallel to each other without avoiding their crossing. For flow downstream of first shock, we have to draw the pressure-deflection diagram from points B & Q respectively corresponding to their Mach numbers (which are identical in the present situation). Again it can be seen that, points C and R represent same pressure and deflection.

Interaction of shocks of different family

Now consider the same S/S interaction problem for shocks of different strengths. As shown in Fig. 3 wedge angles and hence the flow deflection angles are shown different to imitate the two different shock strength for S/S interaction understanding.

Here the wedge angles are θ₁ and θ₂ for top wedge and bottom wedge respectively, where θ₁ > θ₂ in magnitude. Similar to the earlier situation, the top wedge is responsible for right running shock wave while bottom wedge is responsible for left running shock wave. Consider the streamlines ABC passing through the right running shock originated from top wedge and streamline PQR passing through the left running shock originated from bottom wedge. Initial deflection acquired by both the streamlines will be different in this case since the flow deflection angles are different. Flow deflection angles and corresponding pressures at point B and Q are shown using the P-θ diagram in Fig. 4. Here we can clearly see that points A & P are identical due to the fact that both the points belong to the region upstream of shocks.

In this situation, streamline ABC gets negative deflection of higher magnitude at points B in comparison with the deflection of streamline PQR at point Q. Due to higher strength of the right running shock, pressure at point B will be more than that of point Q. Therefore unlike earlier situation, all the flow properties will be different downstream of the first shocks at B and Q. Suppose, both the shocks are weak enough to keep the flow supsonic in the region given by points B and Q of the streamlines. However, when both (right and left running) shocks intersect, we can see similar interaction from where two new shocks originate. The reason for presence of these two shocks is to make the flows (passing over top wedge and passing over bottom wedge) parallel to each other. Streamline ABC which has higher deflection, now passes through the left running shock but cannot cancel the earlier deflection completely. Hence, the part of the flow passed from the top wedge retains certain negative deflection in the region of point C. At the same time, the part of the flow passing from the bottom wedge, which has positive deflection with respective passes though the right running shock which intern cancels its positive deflection and induces negative deflection in the region of point R so that streamlines ABC and PQR become parallel to each other at the end of the interaction region. Hence the strength of the shock encountered by the streamline PQR while passing from Q to R is higher than the strength of the shock encountered by the streamline ABC while passing from B to C. The new entity which we can observe is the slip line (shown by dotted line) which originates from the point of intersection of the shocks. The main purpose of this line is to avoid mixing of two flows passing from two different shock patterns which have different entropies. However pressure and direction of velocity remain conserved across this line. Therefore the final points C and R are shown in Fig. 4 portry finite amount of same deflection and pressure.