Receptivity of a Super-Sonic Acoustic Mode Through a Normal Shock

Acoustic fluctuations traveling through a normal shock at supersonic velocities in atmospheric flight are of prime importance for capturing and predicting the laminar to turbulent transition process. The derivation of the one-dimensional fluctuation traveling through a normal shock-wave is compared to linearized numerical simulations and full high-fidelity simulations. The receptivity process can be confidently captured using either the direct solve or the algebraic relations. The Rankine-Hugoniot equations are combined with what is known about linearized inviscid modes to result in algebraic relations between the inflow conditions and the post-shock conditions. Both the amplitude of the resulting acoustic and entropy modes are characterized along with the specified phase. The results are compared to direct numerical simulations for each of the primitive variables and the results are in excellent agreement for both the fast and slow acoustic mode traveling through a normal shock wave at various conditions. The Rayleigh-Pitot tube formula was also used to compare to the wall pressure fluctuation along two sphere cones at two conditions near Mach 10. The resulting stagnation line and surface stagnation point pressure fluctuations can be predicted accurately using the analytical inviscid relations as well as the linearized direct solution. Additionally, the steady-state wall pressure ratio from CFD can be used in conjunction with the analytical inviscid stagnation relation to approximate the pressure fluctuation further down the body prior to any modal growth due to Mack's second mode. The resulting analytical relations here allow one to approximate the acoustic fluctuation over the surface of the vehicle provided one knows the freestream conditions and surface pressure.