Standard Configuration 10

Standard configuration 10 [14] is a two-dimensional compressor cascade of a modified NACA0006 profile at a stagger angle of 45 degrees with a gap to chord ratio of 1.0. The unsteady inviscid pressure response due to the blades harmonically pitching about their mid-chord position is sought.

Solutions for Case 1 (inlet Mach number M1 = 0.7, inlet fbw angle в і = 55.0o, reduced frequency = 0.5 based on chord and interblade phase angle

a = 0) are shown in Fig. 1. The solutions were calculated using Roe’s scheme. The comparison with the previous numerical work is very good and the current method appears to give the correct answer for this case.

0.25

Figure 1. Imaginary unsteady pressure coefficients for Standard Configuration 10 Case 1

Mi = 0.7, h = 55.0 , u> = 0.5 and a = 0

pressure surface

suction surface

0.25

Figure 2. Imaginary unsteady pressure coefficients for Standard Configuration 10 Case 6: Mi = 0.7, ві = 55.0o, = 1.5 and a = 90o

Solutions for Case 6 (M1 = 0.7, ві = 55.0o, w* = 0.5, a = 90o) are shown in Fig. 2. These solutions were also calculated using Roe’s scheme. This case is significantly more challenging than Case 1 because it is at a resonant condition at the downstream far-field as the sum of the acoustic waves here form planar waves which are travelling in a direction perpendicular to the axis. Conversely, for Case 1 where a = 0, planar waves are formed in both far-fields, which are travelling parallel to the axis, and a one-dimensional treatment is sufficient to ensure non-refection of waves at these boundaries. The difficulty of Case 6 is evident in the fact that there is no consensus between the previous solutions. The solutions are similar but one would expect a better agreement (as achieved for Case 1) for such a well defined problem. Solutions from the current method are shown at two different grid resolutions and also for an extended mesh which has the same grid resolution as the mesh with 12116 cells but with the far-field boundaries located two chord length from the profile as opposed to one chord length for the other meshes. Note that the solutions from the current method are independent of grid resolution and the location of the far-field boundaries. Also note that a higher grid resolution than Case 1 was required to achieve grid convergence for this case. The fact that the current solutions are independent of far-field boundary location suggests that the non-refecting boundary condition has been correctly implemented. Further verification of this can be seen in the good comparison of the aerodynamic

damping versus interblade phase angle with previous work shown in Fig. 3, particularly the prediction of the peaks at resonant conditions.

Solutions for Case 17 (M1 = 0.8, ві = 58.0o, u* = 0.5 and a = 0) are shown in Fig. 4. The inlet Mach number is higher for this case than the previous two cases and causes a shock to form on the suction surface at approximately x/c = 0.25. Solutions calculated with Roe’s scheme, AUSMDV and EFM at a high grid resolution (27261 cells) are shown. For each solution shown, the same flix scheme was used for the the steady-state and linear solution. The Roe solution exhibits unphysical peaks in the solution near the shock. This is probably due to lack of numerical dissipation at the high grid resolution. These peaks are still present in the AUSMDV solution but are significantly smaller. The peaks are not present in the EFM solution due to the highly dissipative nature of the scheme. The conclusion is made that for high resolution Euler calculations, EFM is the better choice because its solutions at high grid resolu­tion are less likely to exhibit unphysical noise and the extra dissipation of the scheme only has a small affect on the accuracy of the solution [10].

The shape of the shock impulse (unsteady pressure response due to motion of shock) predicted by Huff is wider with a lower peak than that predicted by the current method. This is because Huff used a pitching amplitude of a0 = 2.0o, and the current method uses a very small amplitude (a0 = 1.0 x 10-6 o) in order to calculate the linear response. An EFM solution at a lower grid res­olution (12216 cells) is also shown and it can be seen that grid convergence has not been achieved. However, the work done on the blade by the shock im­pulse does converge. The aerodynamic dampings calculated by EFM solutions on grids with 3029, 12216 and 27261 cells were 0.2122, 0.2344 and 0.2340 respectively. The validity of using linearised unsteady analysis for ft>ws with shocks for flutter investigations was recognized by Lindquist and Giles [16].