3-4. Characteristics

Basic supersonic equations and integration method

Near sonic and supersonic flow

Method of characteristics
for solving sonic initial value problems (2D plane flow), compilation of the analytical basis:

The coefficients K13, K23
The coefficient K is a function of the velocity variable (nue), via the local Mach number M. Averaging of K between gridpoints 1 and 3, and between 2 and 3 to K13 and K23, (which for the present applications are the same because nue(1) and nue(2) are equal), allows 2nd order accurate integration of the whole triangular domain to obtain a solution Phi, Psi(nue, theta). Synchroneous integration of the physical plane (x, y) result in a flow pattern with Mach waves ( = characteristics), and streamlines (Psi = const).

Characteristic equations in the near sonic domain

(In Lecture 3): Basic equations (2) in the domain M ~ 1 for the variables U, V, X, Y in a parameter plane (s, t),

for j = 0 (plane 2D flow), k= 1/3, and l = 1 (flow domain part where M > 1).

These equations are transformed to 'characteristic variables" in the half plane s > 0:

Lit.: Sobieczky, H., Qian, Y. J.: Extended Mapping and Characteristics Techniques. Proc. Third Int. Conf. on Inverse Design Concepts and Optimization in Engineering Sciences. Washington, D.C. pp 1 - 7 (1991)