3.2.1 NumericalExperimental comparison on unsteady pressure
distribution. The amplitude (normalized by outlet value) and phase angle of the unsteady pressure distribution is plotted in figure 5 for a perturbation frequency of 100Hz. A pressure amplification of factor three can be observed downstream of the shock location for both experimental and RANS calcula
tion results. It is interesting to note that this amplification is not observed in the Euler simulation and might thus originate viscous or turbulent effects, or even possibly the Shock Boundary Layer Interaction (SBLI).
The analysis of the phase angle distribution is facilitated by considering the behaviour of travelling pressure waves in duct fbws. Similarly to potential interaction in turbomachines, outlet static pressure fhctuations propagate upstream at a relative velocity of cU. As long as the propagating speed is unchanged, the slope of the phase angle also remains constant, which is the case in the outflow region. However, in the vicinity downstream of the shock, the phase angle stops decreasing and even increases, which would actually correspond to a downstream propagating pressure wave.
Figure 5. DFSD on unsteady pressure distribution over 2D bump for Fp=100Hz

At higher perturbation frequency (Fp=500Hz on figure 6), the pressure amplification for both experimental and 2D RANS simulation exhibits an attenuation downstream of the shock whereas the phase angle distribution presents an important phase shift (about 160o) at the same location (x=95mm). Furthermore the same "increasing phase angle” behaviour is still observed downstream of the shock. It is noteworthy that the Euler simulation differs both in the amplitude and phase distribution and does not present the same characteristics.
Figure 6. DFSD on unsteady pressure distribution over 2D bump for Fp =500Hz

Generally, a fairly good agreement was achieved between experiments and turbulent computations. However, although the phase angle distribution for experimental measurements and 2D RANS simulation are very similar, there seems to be an "offset" between the two distributions. This may be explained by the location of the experimental reference on the side wall. As the pressure perturbations are generated by the rotating ellipse and propagate upstream in the test section, there might be a possibility of a phase shift between the lower and side wall pressure measurements as the ft>w is non uniform in the diffusor downstream of the shock.
Considering the above observation on phase shift (increasing phase angle in the vicinity of the shock) it is believed that the unsteady pressure distribution at this location results from the superposition of an upstream and downstream travelling wave with respective varying amplitudes and phase angles. Assuming that only plane waves can propagate in the nozzle at 100Hz, a one dimensional acoustic decomposition (see equation 1) was performed on the 2D RANS numerical results using the isentropic flow velocity and sound speed on the bump surface.
respective amplitude 1.7 and 0.8. The relative phase angle distribution is more delicate to understand. For upstream propagating wave, the phase distribution tends to increase in the vicinity of the shock. This might be explained by the fact that the relative propagating velocity cU tends toward zero in the vicinity of the shock. The wave length thus also tends toward zero locally, which is interpreted as an increase of the phase in the decomposition algorithm. Concerning the downstream propagating waves, it is not yet clear why the phase distribution is decreasing between x=80mm and x=210mm. It is assumed that the decomposition model is not fully accurate in the region and that other phenomena are not yet taken into account.
1.5.2 Unsteady pressure distribution at different channel heights.
Considering the fairly good agreement between experiments and 2D RANS simulations, a complementary analysis of the unsteady pressure distribution was conducted within the numerical domain at different positions in the channel’s height. In addition to the upper and lower walls, three other locations were selected as illustrated in figure 2(c): the SBLI region (probe C), the middle of the shock (probe B), and the region where the sonic line meets the shock (probe A).
Figure 8. DFSD on unsteady pressure distribution over 2D bump for Fp = 100Hz

At low perturbation frequency (100Hz), the highest pressure amplification occurs in the BL on the upper and lower walls. Although not located in the BL, a slight amplification occurs downstream of the shock at the location of probe A and C whereas the pressure amplification in the middle of the channel (probe B) is very similar to the one observed in Euler computation. Beside, the phase angle distribution exhibits the behaviour described previously concerning the increasing phase angle downstream of the shock exclusively for the location where a shock occurs. It is noteworthy that this phenomenon is most pronounced in the middle of the channel although no pressure amplification was noticed. It thus seems as the observed phenomenon on the phase angle is
rather due to the presence of the shock wave than to viscous or turbulent effects.
Figure 9. DFSD on unsteady pressure distribution over 2D bump for Fp = 500Hz

At higher perturbation frequency (500Hz), a pressure attenuation can be observed close to the bump surface whereas a slight amplification still occurs at other location in the channel. Surprisingly, the increasing phase angle behaviour only occurs on the bump surface whereas the phase angle distribution refects a single upstream propagating wave behaviour at all other locations.
The particular amplitude and phase angle distribution is not yet clearly understood and might be related to the lower amplitude of the motion of the shock, its inertia to oscillate, or the smaller wave length of the perturbation at higher frequencies.
Shock motion. The amplitude and phase angle of the shock motion throughout the channel’s height are presented in figure 10 and 11 respectively for the perturbation frequencies 100Hz and 500Hz. A fairly good agreement is achieved between experiments and numerics regarding both the amplitude and phase distribution of the unsteady shock motion. For both frequencies, the amplitude of shock motion increases with the height of the channel. It is noteworthy that the same trend is observed also in the Euler simulation. The amplitude of motion of the shock is thus related to the mean fbw gradients rather than to the SBLI. Beside, the amplitude clearly decreases with the frequency. Assuming that the shock has a certain inertia to oscillate or to respond to a back pressure variation, the decreasing amplitude of motion of the shock at higher frequencies is then probably due to shorter perturbation wave length although the amplitude of perturbation remains the same.
On the other hand, the shock motion phase angle slightly differs both between the two frequencies and the two numerical calculations. At low frequency, the phase angle seems to be constant throughout the channel height, which corresponds to a "rigid" motion between the foot and the top of the shock. At higher frequency however, there is an important phase lag (up to
Figure 10. DFSD on shock motion over 2D bump for Fp = 100Hz

90o for experiments) between the foot and the top of the shock. The "head" of the shock seems to oscillate with a certain time lag (delay) compared to the foot. This tendency can be observed both on experimental visualization and 2D RANS simulation, but not on Euler calculation where the phase angle is still constant throughout the channel height. This effect might be related to the higher propagation speed within the BL than in the free stream. As a result, the pressure perturbations reach the foot of the shock slightly in advance in the SBLI region. The same behaviour should theoretically also be seen at lower frequencies, but since the wave propagation speed is the same, the perturbations wave length is longer and thus the phase lag between perturbations in the BL and in the free stream is lower.
Figure 11. DFSD on shock motion over 2D bump for Fp =500Hz
1.5.3 Relation between unsteady pressure and shock motion. The
phase lag between unsteady pressure distribution over the bump and the shock motion (closest from the bump) has been calculated and plotted in figure 12 as a function of the perturbation frequency. Experimental results show an increasing phase lag with the perturbation frequency, meaning that the shock response with a certain increasing time delay with the perturbation frequency. For numerical simulation however, the trend is not as clear and further calculations
should be performed in order to clearly define a tendency.
Figure 12. Phase Lag between Ps & Shock Figure 13. Phase shift through shock

Additionally, the unsteady pressure phase angle jump through underneath the shock location was plotted as a function of the perturbation frequency in figure 13. A fairly good agreement was found between experimental pressure measurements and 2D RANS numerical simulation. This unsteady pressure phase shift, which seems to increase linearly with the perturbation frequency, is extremely important considering aeroelastic stability prediction. Indeed, the unsteady aerodynamic load on an airfoil is directly infhenced by the value of the phase shift and the overall stability of the airfoil might change from stable to unstable (and vice versa) for a certain value of this phase shift.
3.2.5 Unsteady separated region motion. The unsteady motion of the separated zone was evaluated as a function of the shock motion for the 2D RANS numerical simulation, and is presented in figure 14 for two perturbation frequencies (100Hz and 500Hz). The advantage to display the separation versus the shock motion is to be able to see both the amplitude of motion and the phase lag between the separation/reattachment and the shock oscillations. Clearly, the separation oscillates with the same amplitude and nearly in phase with the shock wave. This behaviour does not seem to change with the frequency and confirms the idea of a shock induced separation. On the other hand, the reattachment seems to oscillate with a much larger amplitude and a certain phase lag with the shock. Both the amplitude and phase lag seem to be related to the perturbation frequency. It is however not possible to state upon a clear tendency and further calculation as well as experimental measurement should be performed.
Figure 14. Separated region motion for 2D RANS calculations

2. Summary
Unsteady pressure measurements and high speed Schlieren visualizations were conducted together with 2D RANS and Euler numerical simulations over a convergent divergent nozzle geometry in order to investigate the Shock Boundary Layer interaction. A fairly good agreement between experiments and numerical simulations was obtained both regarding the unsteady pressure distribution and shock motion.
Results showed that the unsteady pressure distributions, both on the bump and within the channel, result from the superposition of upstream and downstream propagating waves. It is believed that outlet pressure perturbations propagate upstream within the the nozzle, interact in the high subsonic ft>w region according to the acoustic blockage theory, and are partly refected or absorbed by the oscillating shock, depending on the frequency of the perturbations. The amplitude of motion of the shock was found to be related to the mean flow gradients and the local wave length of the perturbations rather than to the shock boundary layer interaction. The phase angle between unsteady pressure distribution on the bump and the shock motion for experimental results was found to increase with the perturbation frequency. However, no clear tendency could be defined for numerical results.
At last, but not least, the phase angle "jump" underneath the shock location was found to linearly increase with the perturbation frequency. The phase shift is critical regarding aeroelastic stability since it might have a significant impact on the phase angle of the overall aerodynamic force acting on the blade and shift the aerodynamic damping from stable to exciting.