Supersonic Expansion

Expansion Fan

We have already seen that compression of supersonic flow takes while passing through the shock. In other words, when the supersonic flow turns into itself then it undergoes the compression through a shock. Exactly opposite situation can be encountered when the supersonic flow turns out of itself where, expansion of the supersonic flow takes place. This expansion unlike compression takes place smoothly through infinite expansion waves hence called as expansion fan. This expansion fan is comprised of infinite number of expansion waves or Mach waves where every wave is responsible for infinitesimal amount of deflection. Typical expansion fan in the supersonic flow is shown in Fig. 1.where supersonic flow turns outward by an angle θ.
 


For better understanding of expansion of supersonic flow, consider that p₁, T₁ and M₁ be the properties of flow before expansion or upstream of the expansion fan and p₂, T₂ and M₂ be the properties of the flow after expansion or downstream of the expansion fan due to outward deflection by an angle θ. For the known upstream flow properties and deflection angle it should be possible for us to calculate the downstream flow properties. Since the expansion is the continuous and smooth process carried out via infinite Mach waves, lets consider one such wave across upstream of which velocity is V and Mach number is M. Angle made by this Mach wave with the upstream velocity vector is μ. Lets consider dV be the change is velocity brought by the Mach wave by turning through an angle dθ. Hence V+dV is downstream velcoity and M+dM is the downstream Mach number. If we construct the velocity triangle as shown in Fig. 2 then we can use the sin law for triangle as,
 




But we know that

 
And,

 
Hence we can re-write Eq. (1) as,



We can approximate as

sin dθ ≈ dθ and cos dθ ≈ 1, Therefore Eq. (2) can be simplified as,
 


Since dθ tan μ < 1, lets recall the expansion for x<1,

Neglecting higher order term, we can express Eq. (3) as,

But we know that



Hence above equation becomes,

 
Prandtl Mayer Function

We can see that since for positive value of dθ, we get positive dV which leads to expansion. This formula is also valid for small angles for compression where we get negative dV. If we integrate this formula for the total expansion angle then we can get the downstream Mach number.



Before integrating we can express the integrant in Mach number,

V = Ma

ln V = ln M + ln a



We can express here the second term on right hand side in terms of Mach number using the isentropic relations as,
     


Using Eq. (5) and (6) we can re-write Eq. (4) as,

 
Intergration of right hand side is as,

 
Here, v is called as the Prandtl-Meyer function.

θ = v(M₂) - v(M₁)                  (7)

Therefore upstream Mach number (M₁) we can calculate the upstream Prandtl-Meyer functions. Hence for known flow deflection angle and upstream Mach number we can get the downstream Prandtl-Meyer function and hence the downstream Mach number.

Process of expansion of supersonic flow is an isentropic process. However, while passing through the expansion fan, pressure, temperature and density of the flow decreases while Mach number and velocity increases for the supersonic flow. Moreover, all the total properties remain constant. We can calculate the total pressure, temperature and density upstream of the expansion using isentropic relations for the known flow Mach number. From the calculated downstream Mach number, we can calculate all the static flow properties from known stagnation or total properties.

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