Page 262 - 48Fundamentals of Compressible Fluid Mechanics

224 CHAPTER 13. OBLIQUE-SHOCK
13.4.5 Flow in a Semi–2D Shape

The discussion so far was about the straight infinite long wedge 20 which is
a “pure” 2–D configuration. Clearly, for any finite length of the wedge, the
analysis needs to account for edge effects. The end of the wedge must
have a different configuration (see Figure 13.10). Yet, the analysis for mid-
dle section produces close results to reality (because symmetry). The sec-
tion where the current analysis is close to reality can be estimated from a
dimensional analysis for the required accuracy or by a numerical method.
The dimensional analysis shows that
only doted area to be area where cur- flow direction no shock
rent solution can be assumed as cor- intermidiate analysis
range
21
rect . In spite of the small area normal analysis {
range
{
were the current solution can be as- no shock { {
sumed, this solution is also act as “re- {
ality check” to any numerical analysis.
The analysis also provides additional
value of the expected range.
Another geometry that can be
considered as two dimensional is the
cone (some referred to as Taylor– 2-D oblique shock
on both sides
Maccoll flow). Eventhough, the cone edge analysis
is a three dimensional problem, the range
symmetrical nature of the cone cre-
ates a semi–2D problem. In this case Fig. 13.10: Schematic of finite wedge with zero
there are no edge effects and the ge- angle of attack
ometry dictates slightly different re-
sults. The mathematics is much more complicated but there are three solutions.
As before, the first solution is thermodynamical unstable. Experimental and ana-
lytical work shows that the weak solution is the stable solution and a discussion is
provided in the appendix of this chapter. As oppose to the weak shock, the strong
shock is unstable, at least, for steady state and no know experiments showing that
it exist can be found the literature. All the literature, known to this author, reports
that only a weak shock is possible.

13.4.6 Small “Weak Oblique shock”

This topic has interest mostly from academic point of view. It is recommend to
skip this issue and devote the time to other issues. This author, is not aware of a
20 Even finite wedge with limiting wall can be considered as example for this discussion if the B.L. is
neglected.
21 At this stage dimensional analysis is not competed. This author is not aware of any such analysis
in literature. The common approach is to carry numerical analysis. In spite recent trends, for most
engineering application, simple tool are sufficient for limit accuracy. In additionally, the numerical works
require many times a “reality check.”