Thin Airfoils in Supersonic Flow

For M > 1 the differential equation governing <p may be written

-B„ + *>гг = 0, (5-86)

where В = /M2 — 1. This equation is hyperbolic, which greatly simplifies the problem. A completely general solution of (5-86) is easily shown to be

<p = F(x — Be) + G(x + Be). (5-87)

Notice the great similarity of (5-87) to the complex representation (5-41) of <p in the incompressible case; in fact (5-87) may be obtained in a formal way from (5-41) simply by replacing z by ±iBz. The lines

x — Bz = const

, R. (5-88)

x + Bz = const

are the characteristics of the equation, in the present context known as Mach lines. Disturbances in the flow propagate along the Mach lines. (This can actually be seen in schlieren pictures of supersonic flow.) In the first-order solution the actual Mach lines are approximated by those of the undisturbed stream.

Since the disturbances must originate at the airfoil, it is evident that in the solution (5-87) G must be zero for z > 0, whereas F = 0 for z <0. The solution satisfying (5-30) is thus

<P = — – gzu{x — Be) for z > 0 <P = ^zi(x + Bz) for z < 0.

From (5-31) it therefore follows that

c~ = lfr (5-90)

This formula was first given by Ackeret (1925). Comparisons of this simple result with experiments are shown in Fig. 5-8. It is seen that the first-order theory tends to underestimate the pressure and in general is

Fig. 5-8. Comparison of theoretical and experimental supersonic pressure distribution at M = 1.85 on a 10% thick biconvex airfoil at 0° angle of attack.

less accurate than that for incompressible flow. The deviation near the trailing edge is due to shock wave-boundary layer interaction which tends to make the higher pressure behind the oblique trailing-edge shocks leak upstream through the boundary layer.

The much greater mathematical simplicity of supersonic-flow problems than subsonic ones so strikingly demonstrated in the last two sections is primarily due to the absence of upstream influence for M > 1. Hence the flows on the upper and lower sides of the airfoil are independent and there is no need to separate the flow into its thickness and lifting parts. In later chapters cases of three-dimensional wings will be considered for which there is interaction between the two wing sides over limited regions.