13.2.2 Wall Points

In Figure 13.6, point 4 is an internal flow point near a wall. Assume that we know all the flow properties at point 4. The C_ characteristic through point 4 intersects the wall at point 5. At point 5, the slope of the wall 9$ is known. The flow properties at the wall point, point 5, can be obtained from the known properties at point 4 as follows. Along the C_ characteristic, К is constant. Hence, (К )4 = (K-)5. Moreover, the value of К_ is known from Equation (13.17) evaluated at point 4:

(tf-)4 = (K-)s = 6*4 + v4 [13.25]

Evaluating Equation (13.17) at point 5, we have

(K_)5 = 9S + v5 [13.26]

In Equation (13.26), (K-)5 and 65 are known; thus V5 follows directly. In turn, all other flow variables at point 5 can be obtained from V5 as explained earlier. The characteristic line between points 4 and 5 is assumed to be a straight-line segment with average slope given by |(64 + 05) — ^(/x4 + /x5).

From the above discussion of both internal and wall points, we see that properties at the grid points are calculated from known properties at other grid points. Hence, in order to start a calculation using the method of characteristics, we have to know the flow properties along some initial data line. Then we piece together the characteristics mesh and associated flow properties by “marching downstream” from the initial data line. This is illustrated in the next section.

We emphasize again that the method of characteristics is an exact solution of inviscid, nonlinear supersonic flow. However, in practice, there are numerical er­rors associated with the finite grid; the approximation of the characteristics mesh by straight-line segments between grid points is one such example. In principle, the method of characteristics is truly exact only in the limit of an infinite number of characteristic lines.

We have discussed the method of characteristics for two-dimensional, irrota – tional, steady flow. The method of characteristics can also be used for rotational and three-dimensional flows, as well as unsteady flows. See Reference 21 for more details.