Numerical Procedure

The harmonic wave equation (3) is solved using a second order finite dif­ference scheme. The algebraic system of equations is solved using an iterative BiConjugate Gradient Stabilized technique and incomplete LU factorization included in PETSc (Balay S. and S., 2000) software.

The model is computationally efficient and results for a single frequency can be obtained in minutes on a single processor SunBlade 1000 computer. Exten­sive code convergence tests and validation with the benchmark aeroacoustics rotor/stator interaction problem proposed by Hanson, 1999, has been presented in Elhadidi, 2002.

2. Results

We investigate the effect of vane lean or sweep in this section for two test geometries. In one case, we consider the geometry proposed by Hanson, 1999, for the benchmark aeroacoustic problem. In this case, B = 16, V = 24, rt = 1.0, and the hub radius, rh = 0.5. In the second case we modify the number of rotor blades, B = 23 to match the results calculated by Schulten, 1982. The chord, c = yf1, and the computational domain length, L = 3c.

The axial Mach number, Mx = 0.5, and reduced frequency ш = 3n, for all test cases. This corresponds to a rotor tip Mach number, Mt = 0.79.

We express the outgoing unsteady pressure in terms of the propagating and decaying pressure eigenfunctions which represent the unsteady pressure in the far-field (Golubev and Atassi, 1998),

p'(x)= ]T X+m0), (7)

m£Sm n =1

where m and n are the azimuthal and radial mode numbers for the acoustic modes. Sm is the set of all values of m, which are determined from the Tyler – Sofrin (Tyler and Sofrin, 1962) condition for tonal noise calculations. kmn and Pmn are the axial eigenvalue and corresponding eigenfunction of the mode mn. Pmn is normalized such that the maximum is equal to unity, and the coefficient cmn represents the magnitude of the unsteady pressure for mode mn.

To express the efficiency of noise reduction for lean or sweep, we calculate the relative acoustic power change (RAPC)to the case of zero lean and sweep,

p __ p в

Relative power change – RAPC = 100 x —-—Q~/g~°, (8)

Ра=в=0

where Pa=e=0 is the acoustic power at zero lean and sweep, a and в are the vane sweep and lean angles respectively. The acoustic sound power, P is cal­culated from, (Morfey, 1971),

where P is the sound power and Smn is the set of propagating modes. Efficient designs should have a negative RAPC.

Figure 2 shows the real component of the unsteady pressure difference across the stator vane span, for the first test case, with zero lean and sweep. The un­steady pressure difference exhibits a three-dimensional pressure distribution, where it is maximum at the leading edge and drops much faster at the hub as we move towards the trailing edge. This confirms the importance of three­dimensional calculations over the strip-theory approach.