Prandtl-Meyer Expansion Waves When a supersonic flow is turned around a corner, an expansion fan occurs producing a higher speed, lower pressure, etc. in an isentropic process. fan Forward Mach line Rearward Mach line µ1 M1 > 1 µ2 θ waves ∗ Just as we saw with shock waves, if we apply conservation of mass and momentum across a single wave, the tangential velocity is unchanged. ∗ Unlike a shock wave, an expansion wave is isentropic. So let’s pick out a single wave: Mach wave w V u µ V V + dv From law of sines: ⎞ ⎛π sin ⎜ + µ ⎟ V + dV ⎠ ⎝2 = V ⎞ ⎛π sin ⎜ − µ − dθ ⎟ ⎠ ⎝2 1 , we can find : M dV dθ = M 2 − 1 V Using dθ → 0 & sin µ = dθ du Prandtl-Meyer Expansion Waves dv dM da = + v M a Using adiabatic relationships, we can re-write: Next, v = M a ⇒ γRTo γRT = ao Tc ∂ −1 2 M = = 1+ a T ∂ −1 da ⎛ γ −1⎞ ⎛ γ −1 2 ⎞ ⇒ = −⎜ M ⎟⎟ dM ⎟ M ⎜⎜1 + a γ ⎝ 2 ⎠ ⎝ ⎠ dv dM 1 M 2 − 1 dM ⇒ = ⇒ dθ = γ −1 2 M v γ −1 2 M M 1+ M 1+ 2 2 Finally, integrating dθ we find: θ = v( M 2 ) − v( M 1 ) Where γ + 1 −1 γ − 1 2 v( M ) = tan ( M − 1) − tan −1 M 2 − 1 γ −1 γ −1 Prandtl-Meyer function Problem: Estimate the rates of the pressure inside the pitot probe to the freestream static pressure, pa p∞ . M∞ = 3 16.100 2002 15° Pressure in pitot probe = pa 2