Mathematical Formulation

We consider a non-viscous, non-heat-conducting fhid and use the Euler Equations as the governing equations. The ft>w quantities are then expanded as follows

U(x, t) = Uo(x) + u(x, t) p(x, t) = Po(x) + p'(x, t), p(x, t) = po(x) + p'(x, t),

where x stands for the position vector, t for time, and Uo, po, po are the steady mean velocity, pressure, and density, respectively. The corresponding unsteady perturbation quantities, u, p, p are such that |u(x, t)| ^ |Uo(x)|, |p'(x, t)| <Po(x) and |p'(x, t)| <po(x).

The mean fbw is assumed axisymmetric and of the form,

U (x) = Ux (r)ex + Ue (r)ee, (4)

where Ux and Ue are the mean velocity components in the axial and circum­ferential directions, respectively. ex and ee represent unit vectors in the axial and circumferential directions, respectively.

We further assume the flow to be isentropic and nondimensionalize lengths with respect to the mean radius rm, and define the reduced frequency as C =

<-orm

com

We assume time-harmonic disturbances of the form e-iwt and we use the linearized Euler equations. These equations are solved using the second order accurate Lax Wendroff scheme in the passage between two unloaded blades for an incoming upstream disturbance. The blades are placed in the middle third of the computational domain along the circumferential boundaries. The impermeability condition is imposed at the hub and tip radii and on the blades’ surfaces.