Three-Dimensional Results (Bending)

To further validate the method, we compare computed three-dimensional results to the unsteady semi-analytical analysis of Namba et al., 2000. The configuration considered here is composed of two rows (rotor/stator) of blades. The upstream rotor has helical surfaces with zero steady pressure loading. The downstream rotor blades are fkt untwisted plates aligned with fbw that in the nonrotating frame of reference is uniform and axial. The details of this configuration are given in Table 2. The axial gap between the neighboring rows is 0.058973. In this case the rotor blades plunge with a reduced frequency и о of 1.0 based on the local chord and the local axial velocity. The displacement of the blade is normal to the local blade chord. The amplitude of the displacement as a function of the radius is given by

where h (r) is the radial distribution of the amplitude of a canti-lever beam for the first bending mode.

As an example, the unsteady solution was obtained for an interblade phase angle of 94.7° using nine spinning modes. Figure 4 compares the real and the imaginary parts of the first harmonic of the unsteady pressure difference across the blade computed using the present method to the results obtained using the semi-analytical analysis of Namba et al., 2000. It can be seen that

the agreement between the two theories is excellent. These results indicate that the present method is ‘mode-converged” using just nine spinning modes.

Next, the interblade phase angle was varied between — 180o and 180o and the unsteady lift coefficients was computed. Figure 5 shows the computed un­steady sectional lift coefficients at the mid-span station. The overall agreement over the full range of the possible interblade phase angles is very good. Also, one can easily see the effect of multistage coupling in the same figure.