Problem Formulation Governing Equations

We apply the linearized Euler equations in this study. We also assume that the mean ft>w is isentropic. The ft>w variables, velocity U, pressure p and density p, are considered to be the sum of their mean values Uo, po and po, and their disturbances u, p, p. For this investigation, we assume that the mean ft>w is uniform and of the form, U = U x ex, where Ux is the constant axial mean ft>w component, and e x is the unit vector along the machine axis.

It is convenient to represent the unsteady velocity disturbance as the sum of a vortical component uR and an irrotational component Уф,

u(x, t) = uR + Уф.

For uniform mean flow the vortical component is divergence free, and is purely convected, and the governing equations reduce to (Goldstein, 1976, Atassi,

1 D2 __ 0

c2 Dt2

where c0 is the mean fbw speed of sound, jff = + Ux is the material

derivative associated with the mean fbw. The pressure is defined as,

Without loss of generality an incoming vortical gust can be represented as,

(5)

where mg is the azimuthal mode number for the vortical disturbance, and amg represents the gust Fourier modes. и is the gust angular frequency and kx = jj- is the axial wave number. For rotor/stator interaction problems, и = mgfl, where ^ is the shaft rotational speed. The gust azimuthal mode m g = pB, B is the number of upstream rotor blades, and p is an integer that represents the blade passing frequency (BPF). For this study we only consider the first BPF, p = 1. At incidence, we neglect the disturbance radial component, thus the incident vortical disturbance is of the form,

where ae is the blade up-wash. The mean ft>w velocity and density are nor­malized with respect to co, and po, respectively. The unsteady pressure, p’ is normalized with respect to po co ae. All length scales are normalized with respect to the mean radius, rm, and the reduced frequency Cj = ^f2-. The reduced frequency can be related to the rotor tip Mach number, Mt, by the relation, Cj = pBMt where г/ is the tip radius.

We consider a single Fourier component of (5) and take ф = фв-ші. Equa­tion (3) is an elliptic equation and will be solved numerically.