Internal Points

Consider the internal grid points 1, 2, and 3 as shown in Figure 13.5. Assume that we know the location of points 1 and 2, as well as the flow properties at these points. Define point 3 as the intersection of the C__ characteristic through point 1 and the C+ characteristic through point 2. From our previous discussion, (K…) = (K_)^

because К.. is constant along a given C characteristic. The value of (K_) = (К _) у

is obtained from Equation (13.17) evaluated at point 1:

(K-) з = (A_)i = в + и і

Figure 13.5 Characteristic mesh used for the location of point 3 and the calculation of flow conditions at point 3, knowing the locations and flow properties at points 1 and 2.

Similarly, (К+)г = (К+)з because K+ is constant along a given C+ characteristic. The value of (K+)2 = (К+)з is obtained from Equation (13.18) evaluated at point 2:

(K+h = (K+h = 6>2 – v2 [13.30]

Now evaluate Equations (13.17) and (13.18) at point 3:

e3 + v3 = (K-h [13.31]

and 03 – v3 = (K+)3 [13.33]

In Equations (13.21) and (13.22), (К_)з and (K+ )3 are known values, obtained from Equations (13.19) and (13.20). Hence, Equations (13.21) and (13.22) are two alge­braic equations for the two unknowns в3 and v3. Solving these equations, we obtain

въ = [{К-)1 + {К+)2] [13.33]

v3 = ±[(K-)i ~(K+)2] [13.34]

Knowing 03 and v3, all other flow properties at point 3 can be obtained as follows:

1. From Уз, obtain the associated M3 from Appendix C.

2. From M3 and the known po and 7b for the flow (recall that for inviscid, adiabatic flow, the total pressure and total temperature are constants throughout the flow), find p3 and r3 from Appendix A.

3. Knowing 7з, compute a3 = RT3. In turn, V3 = M3a3.

As stated earlier, point 3 is located by the intersection of the C_ and C+ characteristics through points 1 and 2, respectively. These characteristics are curved lines; however, for purposes of calculation, we assume that the characteristics are straight-line seg­ments between points 1 and 3 and between points 2 and 3. For example, the slope of the C – characteristic between points 1 and 3 is assumed to be the average value between these two points, that is, j($i +03) — ^(j± + д3). Similarly, the slope of the C+ characteristic between points 2 and 3 is approximated by ^ (02 + #з) + (дг +Дз)-