Page 258 - 48Fundamentals of Compressible Fluid Mechanics

220 CHAPTER 13. OBLIQUE-SHOCK

Suppose that there is a Mach wave at the wall at zero inclination (see Figure
13.7). Obviously, another Mach wave occurs after a small distance. But because
the velocity after a Mach wave (even for a extremely weak shock wave) is reduced,
). If the situation is keeping on occurring

over a finite distance there will be a point where the Mach number will be one and
a normal shock will occur according the common explanation. However, the reality
is that no continues Mach wave can occur because the viscosity (boundary layer).
thus, the Mach angle will be larger (
In reality, there are imperfections
in the wall and in the flow and there is
the question of boundary layer. It is well
known, in engineering world, that there
no such thing as a perfect wall. The im- µ 1
µ 2
µ 3
µ ∞
perfections of the wall can be, for sim-
plicity sake, assumed to be as a sinu-
Fig. 13.7: The Mach waves that supposed to
soidal shape. For such wall the zero
be generated at zero inclination
inclination changes from small positive
value to a negative value. If the Mach
number is large enough and wall is
rough enough there will be points where a weak 19 weak will be created. On the
other hand, the boundary layer covers or smooths the bumps. With these con-
flicting mechanisms, and yet both not allowing situation of zero inclination with
emission of Mach wave. At the very extreme case, only in several points (depends
on the bumps) at the leading edge a very weak shock occurs. Therefor, for the pur-
pose of introductory class, no Mach wave at zero inclination should be assumed.
Furthermore, if it was assumed that no boundary layer exist and wall is per-
fect, any deviations from the zero inclination angle creates a jump between a pos-
itive angle (Mach wave) to a negative angle (expansion wave). This theoretical
jump occurs because in Mach wave the velocity decreases while in expansion
waves the velocity increases. Further, the increase and the decrease depend on
the upstream Mach number but in different direction. This jump has to be in real-
ity either smoothed or has physical meaning of jump (for example, detach normal
shock). The analysis started by looking at normal shock which occurs when there
is a zero inclination. After analysis of the oblique shock, the same conclusion must
be found, i.e. that normal shock can occur at zero inclination. The analysis of the
oblique shock impose that the inclination angle is not the source (boundary con-
dition) that creates the shock. There must be another boundary condition(s) that
forces a shock. In the light of this discussion, at least for a simple engineering
analysis, the zone in the proximity of zero inclination (small positive and negative
inclination angle) should be viewed as zone without any change unless the bound-
ary conditions force a normal shock.
Nevertheless, emission of Mach wave can occur in other situations. The

19 It is not a mistake, there two “weaks.” These words mean two different things. The first “weak”
means more of compression “line” while the other means the weak shock.