Basics of Fluid Mechanics, 2014a

12.3. PRANDTL-MEYER FUNCTION 525
Now utilizing the expression that was obtained for U r and U θ equations (12.91) and
(12.90) results for the Mach number is
(
M 2 =1+ k +1
√ )
k − 1
k − 1 tan2 θ
(12.93)
k +1
or the reverse function for θ is
θ =
Reversed Angle
√ (√
k +1 k − 1 (
k − 1 tan−1 M 2 − 1 ))
k +1
(12.94)
What happens when the upstream Mach number is not 1? That is when the
initial condition for the turning angle doesn’t start with M =1but is already at a
different angle. The upstream Mach number is denoted in this segment as M starting .
For this upstream Mach number (see Figure (12.22))
√
tan ν = M 2 starting − 1 (12.95)
The deflection angle ν, has to match to the definition of the angle that is chosen here
(θ =0when M =1), so
ν(M) =θ(M) − θ(M starting ) (12.96)
ν(M) =
Deflection Angle
√ (√ )
k +1 k − 1 √
k − 1 tan−1 M 2
− 1
k +1
− tan −1 √ M 2 − 1
(12.97)
These relationships are plotted in Figure (12.26).
12.3.4 Comparison And Limitations between the Two Approaches
The two models produce exactly the same results, but the assumptions for the construction
of these models are different. In the geometrical model, the assumption is that the
velocity change in the radial direction is zero. In the rigorous model, it was assumed
that radial velocity is only a function of θ. The statement for the construction of the
geometrical model can be improved by assuming that the frame of reference is moving
radially in a constant velocity.
Regardless of the assumptions that were used in the construction of these models,
the fact remains that there is a radial velocity at U r (r =0)=constant. At this point
(r =0) these models fail to satisfy the boundary conditions and something else happens
there. On top of the complication of the turning point, the question of boundary layer
arises. For example, how did the gas accelerate to above the speed of sound when