## Gas Dynamics & Supersonic FlowCompressible Flow Equations of Motion 1-D Isentropic Relations Wave Propagation Flow through Nozzles and Ducts 2-D Compressible Flow Prandtl-Meyer Expansion Shock Interactions Shock-Expansion Techniques for Aerofoils Method of Characteristics Unsteady Supersonic Flow Flow Tables/Software |
## Prandtl-Meyer Expansion
In this section the expansion of a flow is considered. A supersonic flow
negotiating a convex corner undergoes expansion. As Figure 33: Flow expansion through a shock?
The waves may be centered at a convex corner or spread out as in the case of a convex surface.
The Mach Waves are divergent in both the cases. A centered wave
is called a Figure 34 : Prandtl-Meyer Expansion
A Prandtl-Meyer fan is shown in M . The leading wave
is inclined to the flow at an angle $μ_1=\sin^{-1}(1\/M_1)$
and the expansion terminates in a wave inclined at an angle, $μ_2=\sin^{-1}(1\/M_2)$
. An expression
connecting the flow turning angle _{2}θ and the change in Mach
Number can be found.
Considering a differential element within the fan ( Figure 35 : Prandtl-Meyer Expansion, continued
For a weak disturbance (Mach Wave), where On integrating it gives,
where a new function Starting from the results of the previous 1-D flow section, then, after manipulation and substitution,
The Prandtl-Meyer function is a significant tool for calculating
supersonic flows. Note that for
With the knowledge of θ.
This function can also calculate the Mach Number following
an isentropic compression,
Figure 36 : Using Prandtl-Meyer Function
It should be noted that |