The name comes from a method used to solve hyperbolic partial differential equations: Find "characteristic lines" (combinations of the independent variables) along which the partial differential equation reduces to a set of ordinary differential equations, or even, in some cases, to algebraic equations which are easier to solve.

In the context of aerodynamics, the method is very useful because it can be easily related to physical features of the flow. In this summary introduction to "MOC",  we present or link to several items:

1. A physical explanation
2. Detailed derivations for steady supersonic flow
3. Examples of calculating interior points and wall points
4. Examples of using MOC for steady supersonic flow
5. A calculator (a JAVA applet) to calculate nozzle contours using MOC in steady supersonic flow
6. Consideration of the usage of MOC in high-temperature gas dynamics
7. Application to unsteady subsonic flows.

1. Introduction using physical arguments.

We have seen that in supersonic flow, small disturbances cannot propagate upstream. (Why SMALL? Because very large disturbances such as detonations travel at the speed of sound appropriate to their internal conditions, which may be a lot faster than the ambient speed of sound. If a missile explodes behind a fighter flying at Mach 2, chances are that it is all over for the fighter. )

Using this speed-of-sound argument, we saw the idea of "Mach cones" or "Mach Waves", inclined at the "Mach angle" m to the flow direction. So, if there is a disturbance at a point along a streamline in a supersonic flow (disturbance = change in properties, be it velocity, pressure, temperature..), then this disturbance is first felt along a Mach wave originating from that point on the streamline. The flow upstream of the Mach wave is "undisturbed", that is, properties unchanged. Thus, changes in properties occur across Mach waves. Each Mach wave can be considered to be the dividing line between regions of slightly different properties.

In a 2-dimensional flow, we can think of Mach waves going out on both sides of the streamline. We might call these the "left-running Mach wave" and the "right-running Mach wave".

Lets jump up from our discussion and note (Perry Mason would call it "stipulate") that these Mach waves are "characteristic directions". Right ON the Mach wave, we have a sort-of-fence: we can't be sure if we are in the upstream region or the downstream region. Mathematicians go ga-ga at the prospect that these lines are thus lines along which something or other cannot be determined. A "determinant goes to zero" to keep the solution from blowing up: suddenly a new, simpler equation is available...

OK,  back to Mach waves. Lets say that I am a packet of gas ("hot air!!") just being blown out through the throat of a convergent-divergent nozzle, and have reached Mach 1.01. I know the pressure, temperature, Mach number and direction of my flight, and hence my velocity. I'd like to know where I am going to be, after I have gone a short distance downstream, and what the properties will be at all regions of the flow in the nozzle downstream.

Well, I know that if I move a short distance ds along my present direction at my present speed, I'll reach point C. I also know that my present condition is known and felt along the Mach waves emanating from point A, where I am at this instant. My friend is at point B, some distance to my left, at the same station along the nozzle, also moving at Mach 1.009. He knows the temperature, pressure etc. there.

We now have the elements of an MOC calculation. We step short, finite distances along streamlines, and assume that properties remain constant unless we cross a Mach wave emanating from somewhere. Each time there is such a change, we too send out Mach waves. We determine properties at a new point along each streamline by seeing what Mach waves intersect there (where did those Mach waves come from, and hence, what were the properties there?) We march downstream.

Lets see the detailed derivations now, where we will first show that the Mach waves are indeed characteristic directions of the partial differential equation describing the flow (in this case, the linearized potential equation for 2-D supersonic flow). We will then obtain relations between pressure and velocity along those lines, and use them to solve simultaneous algebraic equations to get conditions at each point.

Mathematics of the linearized potential equations

Derivation of the method of characteristics: getting the characteristic equations

Derivation of the method of characteristics: using the characteristic equations