The flow over bodies at supersonic speeds is very different from that
at subsonic speeds. As discussed earlier, the differential equations for
inviscid flow become hyperbolic, rather than elliptic as the Mach number
As disturbances propagate at a speed slower than the freestream speed, waves form.
The angle of the waves generated when the disturbances are small is called the Mach angle:
Sin β = a dt / V dt = 1/M
The Mach wave is a boundary between areas that are affected by the presence of very small disturbances and those that are unaffected. A shock wave, however, is produced by larger disturbances. As the flow is compressed, the temperature changes, the speed of sound changes, and the shock angle becomes greater than the Mach angle.
When the flow is forced to turn a corner, a shock wave is created. The relationship between the initial Mach number, the Mach number after the shock, the turning angle δ, and the shock angle θ, can be computed from the shock jump conditions: relationships derived from continuity, momentum, and energy.
The relationship between the turning angle and the shock angle is given in many textbooks on compressible flow. The important point here is that there are two solutions, one corresponding to weak shock waves and one corresponding to strong shocks. The weak shock solution is the one that actually occurs in most external aerodynamics problems. This solution has a maximum value of δ and θ for any incident Mach number. This means that there is a maximum value of turning angle that is possible.
Attempting to turn the flow by an angle greater than this results in a detached, bow shock and much higher drag. This is why the leading edges of supersonic airfoils are sharp. These sharp leading edges lead to great simplifications in the analysis of supersonic airfoils because the entire flow may be quite well approximated with small perturbations.
Further details are left to courses in compressible flow theory. You should be familiar with oblique shocks, shock jump conditions, Prandtl-Meyer expansions, etc. Additional information on sonic booms and linear interference effects will be included in the next edition of these notes.
Solutions to the Prandtl-Glauert equation in supersonic flow are particularly
simple. Since the differential equation is still linear, we can still superimpose
known solutions, such as vortices, sources and doublets. For example, a
3-D supersonic source can be described by:
But because of the change in the character of the equation, we can write down a simple, general solution to the equation (in 2-D for now):
One can verify that the solution:
does satisfy the differential equation. Where F and G are arbitrary functions!
This solution is very different in character from the subsonic solution because of the fact that disturbances cannot propagate ahead of the wave front. This is known as the law of forbidden signals. The two functions F and G in our solution represent characteristic lines moving away from the upper and lower surfaces of the body respectively. Consider the part that describes the upper surface flow field:
The expressions for the perturbation velocity components are then:
If the boundary condition is approximated by:
Thus, the total velocity at the airfoil surface is given approximately by:
The pressure may then be computed using the compressible Bernoulli equation. Since we have assumed small perturbations in the freestream, the linearized form of the Bernoulli equation is appropriate:
The pressure on the surface of the airfoil is thus:
Note that θ is positive when the flow is being turned away from the freestream direction, and negative when it is being turned back.
Since the same solution applies all along the characteristic line, this is the pressure in the flow field as well, with changes due to dispersion and dissipation.
Unlike the incompressible case, we require supersonic airfoils to produce small perturbations if they are to satisfy the equations. This means small surface slopes everywhere: no blunt leading edges. This means that we can express the surface slope as:
Note that the Cp equations are linear in θ, thus the net Cp is the sum of that due to thickness, camber, and angle of attack.
The net lift and moment is then also the sum of those associated with each of the 3 components.
Note that the flat plate at angle of attack is the only component that contributes to the integral.
The result for any supersonic airfoil is then:
(The zero lift angle of attack is always 0)
The camber line contributes to the moment however with the result that the moment
coefficient about the leading edge is:
All three components contribute to the drag:
There are cross terms such as angle of attack * thickness slope, but a useful canceling of cross terms occurs so that the total drag is the sum of the drags of each component (the drag due to a flat plate at angle of attack, the drag of the symmetrical thickness form at zero lift, and the drag of the camber line at zero lift.
The drag of the flat plate is:
This implies that the net force on the plate acts normal to the plate. Well, this seems obvious: we are integrating surface pressures which act normal to the surface. How could it be any other way? In fact how can we explain that in subsonic flow Cd = 0, even for a flat plate at angle of attack?
The reason that we can get zero drag in subsonic flow is that for thick airfoils, there is indeed some surface area facing in the freestream direction.
As the airfoil is made thinner, this forward facing area gets smaller, but the Cp gets more negative (a pressure peak develops). In the limit as t/c goes to 0, a singularity develops. The combined effects always cancel, so Cd = 0. For supersonic sections no singularity can form; in fact, the pressure peak has a finite limit, so Cd is greater than zero.
The forward facing pressure component is called leading edge suction and the amount of leading edge suction that can be achieved depends on the angle of attack, Mach number, and sharpness of the leading edge.
To appreciate the effect of airfoil shape and angle of attack on performance at supersonic speeds, try changing the airfoil below and examine the effect on lift, moment, and L/D. Click on the upper half of the plot to increase angle of attack and on the lower half to decrease it. The pitching moment is measured about the 50% chord point. You may also visualize complete flow field pressures by clicking the second button, but note that this may take a while on slower computers.