Page 257 - 48Fundamentals of Compressible Fluid Mechanics

13.4. SOLUTION OF MACH ANGLE 219

Substituting the values of and equations (13.37)(13.38) into equation (13.28)
provides the equation to be solved for c .

"

< E <

"
.
.
(13.39)

"

) E
9

This author is not aware of any analytical demonstration in the literature which

16 . Nevertheless, this identity can

. Table

(13.6) is provided for the following demonstration. Substitution of all the above
showing that the solution is identity zero for c

.

Utilizing the symmetry and antisymmetry of the qualities of the and for

values into (13.28) results in
regardless to Mach number. Hence, the physical
be demonstrated by checking several points for example,
interpretation of this fact that either that no shock can exist and the flow is without

demonstrates that
17
any discontinuity or a normal shock exist . Note, in the previous case, positive
c

large deflection angle, there was transition from one kind of discontinuity to an-

other.
, the
question whether it is possible coefficients
for the oblique shock to exist?
In the range where c
The answer according to this

analysis and stability analysis is 1.0 , ,' , .
-3
-1
not. And according to this anal-
ysis no Mach wave can be gen- 2.0 3 0 -.
erated from the wall with zero
deflection. In other words, the -1 0
wall doesn’t emit any signal to -
the flow (assuming zero viscos-
ity) which contradicts the com-
Fig. 13.6: The various coefficients of three different
mon approach. Nevertheless,
is zero
in the literature, there are sev- Mach number to demonstrate that
18
eral papers suggesting zero strength Mach wave, other suggest singular point .
The question of singular point or zero Mach wave strength are only of mathematical
interest.
16
A mathematical challenge for those who like to work it out.
17 There are several papers that attempted to prove this point in the past. Once this analytical solution
was published, this proof became trivial. But for non ideal gas (real gas) this solution is only indication.
18 See for example, paper by Rosles, Tabak, “Caustics of weak shock waves,” 206 Phys. Fluids 10 (1)
, January 1998.