Results and Discussion

Figure 1 shows the pressure contours of the steady state cascade ft>w (Ap = 333Pa). This ft>w field corresponds to the case of the back pressure of 75,000 Pa. The inlet Mach number is 1.2 and the exit Mach number is 0.605. The inlet ft>w angle is 57.6 deg., and the exit ft>w angle is 52.35 deg. At the leading edge we have oblique shocks, and in the cascade passage a normal shock. Interaction between a sonic boom and cascaded blades is studied based on this steady condition. In order to save computational region as well as time we chose the incident N wave of a short period of 8 ms. In reality the sonic boom induced by SST may have a longer period of 200 ms, we thought the interaction between the shock and cascaded blades should be essential, but the period of two shocks may not be so important in the phenomena. This tendency has been suggested by the semi-actuator disk analysis.

The maximum static pressure amplitude of the N wave given at the inlet boundary is selected to be 325 Pa. This value corresponds to the increase of relative total pressure of 1013 Pa, 1% of the atmospheric pressure. In this case the maximum amplitude of the absolute total pressure is 1.19%, and the static pressure is 0.78% of the steady value, respectively.

Hereafter we investigate the pressure variation history along the grid points indicated in Fig.2. The points from 1 to 15 locate in the upstream field, those from 15 to 25 locate in the cascade passage, and points from 25 and greater locate in the downstream field.

Figure 1. Steady Pressure Contour (Ap:333Pa)

Figures 3 (a) to (e) show the results of pressure variation history for the inci­dent N-wave of period of 8ms. From the pressure history at the position 1, we can clearly recognize the incident N-wave followed by a little bit smaller N – wave refected back by the cascade at the leading edge plane. At positions 4, 6 and 8 the refected wave complicates the pressure signal significantly. Figures 3 (c) and 3 (d) show the pressure signals in front of and behind the passage shock, respectively. We see that high frequency components appear on the pressure signals behind the passage shock. Those components are probably due to var­ious types of resonance in the cascade passage. Further we go downstream the pressure signals seem to recover the N-shape as the transmitted wave.

Figure 4 shows the history of the non-steady aerodynamic force acting on blades during the passage of the N-wave. We see that at the passage of the front shock of the N-wave, the aerodynamic force once drops sharply then increases to have a peak. During the passage of the expansion portion of the N-wave, the

aerodynamic force increases gradually. At the passage of the end shock of the N-wave, the sharp variation of the aerodynamic force repeats again. The sharp variation can be explained as follows: First, the incident shock wave hits upon the suction side of blades reducing the aerodynamic force. Then, the shock wave refected at the suction side propagates downstream and hits upon the pressure side of blades increasing the aerodynamic force.

Figures 5 (a) to (d) show the blade surface pressure distribution along chord as measured by the perturbation from the steady state condition. We focus on the short period of the passage of the front shock, i. e., from 2.4ms to 3.0ms. The non-steady aerodynamic force experiences a sharp variation during this period. From 2.4ms to 2.6ms the suction side surface pressure increases be­cause of the incident shock wave. The pressure side surface pressure does not change during this time period. Then at 2.8ms the high pressure wave is diffracted around the trailing edge thus increasing the pressure side surface pressure near the trailing edge. In addition to this effect, at 3.0ms and after the pressure wave refected at the suction side of a blade arrives at the pressure

Figure 3. Pressure Variation History at Each Station

side of the adjacent blade increasing the whole level of the pressure side sur­face pressure. These behaviors of surface pressure clearly explain the variation of the non-steady aerodynamic force acting on blades.

Figure 5. Surface Pressure Distribution

The maximum amplitude of the non-steady aerodynamic force reaches 1.51% of the steady state aerodynamic force for the incident N-wave having the am­plitude of 1% of the steady state static pressure. Thus we can conclude that the amplitude of the non-steady aerodynamic force is of the same order of magnitude as the pressure amplitude of the incident N-wave.

2. Conclusions

Interaction between a sonic boom and a transonic cascade of blades is stud­ied numerically. The main results obtained can be summarized as follows.

For an incident N wave, the refbcted N wave and the transmitted N wave are clearly observed.

After passing through the passage shock the increase in the N wave am­plitude which has been suggested by the semi-actuator disk analysis is not clearly observed.

By the incidence of a sonic boom the aerodynamic force once drops sharply then rises to attain the peak pressure. The mechanism is the process such that first the incident shock wave hits upon the suction side of a blade then be refected to hit upon the pressure side of the adjacent blade.

The maximum amplitude of the unsteady aerodynamic force reaches 1.51% of the steady state aerodynamic force for the incident N wave having the amplitude of 1% of the steady state static pressure.

References

Kaji, S., “Interaction of Sonic Boom and Engine Fan Rotor”, Proc. 9th ISUAAAT, Sep. 2000, pp.362-374.

Paynter, G. C., Clark, L. T., and Cole, G. L., “Modeling the Response from a Cascade to an Upstream Acoustic Disturbance”, AIAA J., Vol.38, No.8, Aug. 2000.

Freund, D. and Sajben, M., “Experimental Investigation of Outfbw Boundary Conditions Used in Unsteady Inlet Flow Computations”, AIAA paper 97-0610.

Dorney, D. J., “Unsteady Acoustic Wave Propagation in a Transonic Compressor Cas­cade”, AIAA paper 96-0246.

“Study on Advanced Aircraft Technology Development” Report No. 1204, SJAC, 2001.