Page 289 - 48Fundamentals of Compressible Fluid Mechanics

14.3. THE MAXIMUM TURNING ANGLE 251

that the velocity in the radial direction is zero. While the rigorous model the as-
sumption was that radial velocity is only function of . Whence, the statement for
the construction of the geometrical can be improved by assuming that the frame of
reference moving in a constant velocity radially.
Regardless, to the assumption that were used in the construction of these
+*-, * .
) these models falls to satisfy the boundary conditions and

something else happen there. On top the complication of the turning point, the
question of boundary layer arises. For example, how the gas is accelerated to
% $'&)( (
above the speed of sound where there is no nozzle (where is the nozzle?)? These
models, the fact remains that that there is a radial velocity at "

questions have engineering interest but are beyond the scope of this book (at least
At this point (
at this stage). Normally, this author recommend to use this function every ever
beyond 2-4 the thickness of the boundary layer based on the upstream length.
In fact, analysis of design commonly used in the industry and even questions
posted for students shows that many assumed that the turning point can be sharp.
At small Mach number, the radial velocity is small . but increase of the
Mach number can result in a very significant radial velocity. The radial velocity is

“fed” through the reduction of the density. Aside to close proximity to turning point,
mass balance maintained by reduction of the density. Thus, some researchers rec-
ommend that in many instances, the sharp point should be replaced by a smother
transition.

14.3 The Maximum Turning Angle
The maximum turning angle is obtained when the starting Mach number is one and
end Mach number approach infinity. In this case, Prandtl–Meyer function became

(14.36)
*
*

The maximum of the deflection point and and maximum turning point are only

!
function of the specific heat ratios. However, the maximum turning angle is match

larger than the maximum deflection point because the process is isentropic.

What happen when the deflection angel exceeds the maximum angle? The
flow in this case behaves as if there almost maximum angle and in that region
beyond will became vortex street see Figure (14.5) i
14.4 The Working Equations For Prandtl-Meyer Func-
tion

The change in deflection angle is calculated by

(14.37)

* * *
*