The subject of transonic flow is far too large a topic to cover in any kind of depth in a document such as this, but for the absolute novice, some rudimentary explanation is required in order to understand what the solutions represent. The term transonic implies that the flow is mixed, with some regions subsonic (slower than the speed of sound) and some regions supersonic (faster than the speed of sound). This is actually a common occurrence. When an aircraft is flying at a high subsonic speed, the air accelerates over the top of the wing to create lift, and it is possible for the accelerated flow to exceed the speed of sound over part of the upper surface of the wing, culminating in a shockwave that slows the flow back down to subsonic speeds. This occurs regularly on commercial airliners, and it can often be seen by the naked eye. The density of the flow changes abruptly from one side of a shockwave to the other, and this density gradient refracts the light traveling through it. The result is that, with the proper lighting, a shadow appears on the wing surface below the shockwave, usually seen as a faint dark line that is almost always curved and discontinuous and jumps around as the air conditions outside the aircraft change. Thus, while the aircraft is flying subsonically, part of the flowfield is supersonic, resulting in transonic flow. This is shown in the sequence of figures below, for a 10 percent thick biconvex airfoil at Mach 0.80, 0.85, 0.90 and 0.95. In the images, the green line is the sonic line, where the flow smoothly accelerates from subsonic to supersonic, and the cyan line is the shock, where the flow abruptly decelerates from supersonic to subsonic flow.

M=0.80	M=0.85	M=0.90	M=0.95

If the incoming flow is supersonic, but not too high, than a bow-shock is formed in front of the wing, creating a small subsonic region downstream. The bow shock is what creates the sonic boom that we hear when an aircraft breaks the sound barrier. In this case, while the aircraft is flying faster than the speed of sound, a small portion of the flow is decelerated by a shock to subsonic speeds, again creating transonic flow. This is shown in the sequence of figures below, for a 10 percent thick biconvex airfoil at Mach 1.25, 1.30, 1.35 and 1.40, respectively.

M=1.25	M=1.30	M=1.35	M=1.40

Understanding the true difficulty in predicting transonic flow requires a pretty healthy understanding of partial differential equations. As it turns out the subsonic problem is elliptic in nature (a boundary-value problem), and the supersonic problem is hyperbolic in nature (an initial value problem). Boundary value problems are a sort of averaging where the solution at any point is affected by the values on the boundaries on all sides of the point. This can be seen in the subsonic regions, where the Mach number (indicated by the colored shading) varies smoothly, without indicating any specific directions in the flow. Initial value problems, on the other hand, have a defined direction (or directions) along which disturbances propagate. This is apparent in the supersonic regions where the Mach number is constant along essentially straight lines angling back from the airfoil. These lines are called characteristics, and they define domains of influence and dependence for the flow. Without going into detail, the supersonic figures above provide a quick picture of characteristics and the domain of influence. The orange region in front of the airfoil is the incoming flow, and you can see that the flow is undisturbed above and below the wing outside of a wedge-shaped area emanating from the leading edge of the wing. The outer boundaries of the wedge-shaped region defines the domain of influence of the wing. At the specified Mach number, the wing cannot influence the flow outside of this boundary. A numerical method that attempts to model this flow must reflect this concept.

If the flow is assumed to be two-dimensional, steady, inviscid and irrotational, then it can be described by the potential equation

where the Greek letter Phi is the velocity potential, the subscripts x and y indicate partial differentiation in the respective directions, and a is the speed of sound. The velocity potential is defined such that the partial derivative with respect to x is the velocity in the x-direction, and the partial derivative with respect to y is the velocity in the y-direction. The full potential equation is a second order, non-linear partial differential equation, and is quite difficult to solve in this form, thus, simplifications are generally made, as detailed below.

To simplify the full potential equation, the flow is assumed to be essentially freestream, with only small perturbations. It is assumed that the the potential takes on the form

where U-infinity is the freestream velocity, and little phi is the perturbation potential. Thus, the partial derivatives with respect to x and y become

and

.

The idea behind perturbation theory is that if we assume that the perturbation quantities are small, then the product of two or more of these small quantities is so small that it can be neglected.

Substituting these perturbation equations into the full-potential equation and simplifying by dropping the higher order terms yields the transonic small disturbance equation

where M-infinity is the freestream Mach number and the Greek letter gamma is the ratio of specific heats, as described in an earlier section. This is the equation that is solved numerically in this program. Note that this is a second order non-linear partial differential equation, which is still quite difficult to solve.

It turns out that if the flow is entirely subsonic or supersonic, then the right-hand-side of the transonic small disturbance equation can also be neglected, and the governing equation we must solve reduces to the linear form