Page 252 - 48Fundamentals of Compressible Fluid Mechanics

214 CHAPTER 13. OBLIQUE-SHOCK

and where the definitions of and are

(13.29)
"
, E ,
and "

(13.30)

)
" ", , E , E ,
Only three roots can exist for Mach angle, . From mathematical point of view, if
9
all the roots
.
all the roots

one root is real and two roots are complex. For the case
are real and unequal.
.
are real and at least two are identical. In the last case where
The physical meaning of the above analysis demonstrates that in the range

10
occurs when no shock angle can be found so that the shock normal component is
no solution exist because no imaginary solution can exist .
where
reduced to be subsonic and yet be parallel to inclination angle.
the solution has physical

has to be examined in the light
Furthermore, only in some cases when
of other issues to determine the validity of the solution.
meaning. Hence, the solution in the case of
the three unique roots are reduced to two roots at least for

11
steady state because the thermodynamics dictation . Physically, it can be shown

When
that the first solution(13.23), referred sometimes as thermodynamically unstable
root which also related to decrease in entropy, is “unrealistic.” Therefore, the first

solution dose not occur in reality, at least, in steady state situations. This root has
12
only a mathematical meaning for steady state analysis .
These two roots represents two different situations. One, for the second root,
the shock wave keeps the flow almost all the time as supersonic flow and it refereed
to as the weak solution (there is a small branch (section) that the flow is subsonic).
Two, the third root always turns the flow into subsonic and it refereed to a strong
solution. It should be noted that this case is where entropy increases in the largest
amount.
In summary, if there was hand which moves the shock angle starting with
the deflection angle, reach the first angle that satisfies the boundary condition,
however, this situation is unstable and shock angle will jump to the second angle
(root). If additional “push” is given, for example by additional boundary conditions,
10 A call for suggestions, should explanation about complex numbers and imaginary numbers should
be included. Maybe insert example where imaginary solution results in no physical solution.
11 This situation is somewhat similar to a cubical body rotation. The cubical body has three symmetri-
cal axises which the body can rotate around. However, the body will freely rotate only around two axis
with small and large moments of inertia. The body rotation is unstable around the middle axes. The
reader simply can try it.
12 There is no experimental evidence, that this author found, showing that it is totally impossible.
Though, those who are dealing with rapid transient situations should be aware that this angle of the
oblique shock can exist. The shock initially for a very brief time will transient in it and will jump from this
angle to the thermodynamically stable angles.