Basics of Fluid Mechanics, 2014a

526 CHAPTER 12. COMPRESSIBLE FLOW 2–DIMENSIONAL
there is no nozzle (where is the nozzle?)? These questions are of interest in engineering
but are beyond the scope of this book (at least at this stage). Normally, the author
recommends that this function be used everywhere 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 to students show that many assume that the turning point can be sharp. At a
small Mach number, (1 + ɛ) the radial velocity is small ɛ. However, an increase in the
Mach number can result in a very significant radial velocity. The radial velocity is “fed”
through the reduction of the density. Aside from its close proximity to turning point,
mass balance is maintained by the reduction of the density. Thus, some researchers
recommend that, in many instances, the sharp point should be replaced by a smoother
transition.
12.4 The Maximum Turning Angle
The maximum turning angle is obtained when the
starting Mach number is 1 and the end Mach number
approaches infinity. In this case, Prandtl–Meyer
function becomes
slip line
Maximum
turning
Maximum Turning Angle
[√ ]
ν ∞ = π k +1
Fig. -12.25. Expansion of Prandtl-
2 k − 1 − 1 (12.98)
Meyer function when it exceeds the
maximum angle.
The maximum of the deflection point and the
maximum turning point are only a function of the specific heat ratios. However, the
maximum turning angle is much larger than the maximum deflection point because the
process is isentropic.
What happens when the deflection angel exceeds the maximum angle? The flow
in this case behaves as if there is almost a maximum angle and in that region beyond
the flow will became vortex street see Figure (12.25)
12.5 The Working Equations for the Prandtl-Meyer Function
The change in the deflection angle is calculated by
ν 2 − ν 1 = ν(M 2 ) − ν(M 1 ) (12.99)
12.6 d’Alembert’s Paradox