Basics of Fluid Mechanics, 2014a

518 CHAPTER 12. COMPRESSIBLE FLOW 2–DIMENSIONAL
M x M ys M yw θ s θ w δ
P 0y
P 0x
5.0000 0.43016 3.8006 86.9122 21.2845 12.0000 0.80600
And the additional conditions are
M x M yw θ w δ
P y
P x
T y
T x
P 0y
P 0x
5.0000 3.8006 21.2845 12.0000 1.6963 1.6625 0.80600
The conditions in zone 4 and zone 3 have two things that are equal. They are the
pressure and the velocity direction. It has to be noticed that the velocity magnitudes in
zone 3 and 4 do not have to be equal. This non–continuous velocity profile can occur
in our model because it is assumed that fluid is non–viscous.
If the two sides were equal because of symmetry the slip angle is also zero. It
is to say, for the analysis, that only one deflection angle exist. For the two different
deflection angles, the slip angle has two extreme cases. The first case is where match
lower deflection angle and second is to match the higher deflection angle. In this case,
it is assumed that the slip angle moves half of the angle to satisfy both of the deflection
angles (first approximation). Under this assumption the conditions in zone 3 are solved
by looking at the deflection angle of 12 ◦ +1.5 ◦ =13.5 ◦ which results in
M x M ys M yw θ s θ w δ
P 0y
P 0x
3.5040 0.47413 2.6986 85.6819 27.6668 13.5000 0.88496
with the additional information
M x M yw θ w δ
P y
P x
T y
T x
P 0y
P 0x
3.5040 2.6986 27.6668 13.5000 1.6247 1.5656 0.88496
And in zone 4 the conditions are due to deflection angle of 13.5 ◦ and Mach 3.8006
M x M ys M yw θ s θ w δ
P 0y
P 0x
3.8006 0.46259 2.9035 85.9316 26.3226 13.5000 0.86179
with the additional information
M x M yw θ w δ
P y
P x
T y
T x
P 0y
P 0x
3.8006 2.9035 26.3226 13.5000 1.6577 1.6038 0.86179