Question

For a given Prandtl-Meyer expansion, the upstream Mach number is 3 and the pressure ratio across the wave is P2/P1 = 0.4. Calculate teh angles of the forward and rearward mach lines of the expansion fan relative to the free-stream direction

Answer 1

The angle for the forward Mach line is 19.47°

The angle for the rearward Mach line is 5.21°

Step-by-step explanation:

From table A-1 (Modern Compressible Flow: with historical perspective):

`(P_(o) )/(P_(1) ) =36.73` (M₁ = 3)

If Po₁ = Po₂

`(P_(o2) )/(P_(2) ) =(P_(o1) )/(P_(1) ) *(P_(1) )/(P_(2) ) =36.73*(1)/(4) =91.825`

Table A-1:

`(P_(o2) )/(P_(2) ) =91.825,M_(2) =3.63`

Table A-5:

v₁ = 49.76°

μ₁ = 19.47°

v₂ = 60.55°

μ₂ = 16°

θ = 60.55 - 49.76 = 10.79°

The angle for the forward Mach line is:

μ₁ = 19.47°

The angle for the rearward Mach line is:

θr = μ₂ - θ = 16 - 10.79 = 5.21°