Gas Dynamics & Supersonic Flow


 
Compressible 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 Expansion Shocks are not possible for steady flow, then expansions will be produced by a series on Mach waves. The expansion will be an isentropic process.


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 Prandtl-Meyer Expansion fan.


Figure 34 : Prandtl-Meyer Expansion

A Prandtl-Meyer fan is shown in Fig. 34, through which a flow expands from a Mach Number, M1 to M2 . 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 θ and the change in Mach Number can be found.

Considering a differential element within the fan (Fig. 35) then the previous rules for 1-D flow with weak waves shown in the previous sections canbe applied. The Mach Number changes from M to M + dM as the flow turns through an angle .


Figure 35 : Prandtl-Meyer Expansion, continued

For a weak disturbance (Mach Wave),

$$-dθ=√{M^2-1}{dw}/w$$

where w is the initial flow velocity before the wave.

On integrating it gives,

$$-θ+\text"constant"= ∫√{M^2-1}{dw}/w = ν(M)$$

where a new function ν has been introduced. This is called the Prandtl-Meyer function.

Starting from the results of the previous 1-D flow section,

$${dw}/w={dM}/M+{da}/a= {dM}/M(1/{1+{γ-1}/2M^2})$$

then, after manipulation and substitution,

$$ν(M)=√{{γ+1}/{γ-1}}\tan^{-1}(√{{γ-1}/{γ+1}(M^2-1)})-\tan^{-1}(√{M^2-1})$$

The Prandtl-Meyer function is a significant tool for calculating supersonic flows. Note that for M=1, ν=0. For every Mach Number greater than one there is a unique Prandtl-Meyer function value. Tables for supersonic flow at the end of this section list ν as a function of Mach Number. The practice, ν is the angle through which a sonic flow should be turned in order to reach a Mach Number of M. In addition, a flow turning through an angle θ will experience a change in Prandtl-Meyer function value of,

$$ν_2=ν_1 + θ$$

With the knowledge of ν, M2 can be calculated for any value of θ. This function can also calculate the Mach Number following an isentropic compression,

$$ν_2=ν_1 - θ$$

Figure 36 : Using Prandtl-Meyer Function

It should be noted that ν decreases in compression and increases in expansion. See Fig. 36