Inviscid Flow through Cascades in Oscillatory and Distorted Flow

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The present paper deals with the flow through a staggered cascade of airfoils in which unsteady disturbances from the upstream are swept down with the flow, as in the case of unsteady or distorted inlet flow conditions in an axial flow compressor. In the oscillatory case, unsteady normal velocity fluctuations which are carried down with the steady-state flow, like traveling waves, cause varying angles of attack along the chord lengths of the airfoils. Disturbances due to steady-state circumferential inlet distortion can be decomposed into Fourier components of oscillatory flows with some phage lag between the blades. The problem has been formulated under the assumptions of incompressibl e potential flow. An approximate method of solution has been developed for the integral equations involved and numerical results have been obtained for oscillatory flows. The unsteady lift coefficient of the airfoils has been obtained as a function of the frequency of the oscillations. The amplitude of the fluctuating lift for a cascade decreases from the steady state value with increasing frequency more slowly than it does for a single airfoil. The quantitative behavior of this has been studied for different values of stagger angle and solidity of the cascade. Nomenclature J a = distance between the trailing edges of two adjacent airfoils of a cascade c = semichord length of an airfoil CL = lift coefficient denned in Eq. (28) H = transfer function denned in Eq. (29) i = (-1)1/2 ix = unit vector in x direction iy = unit vector in y direction I = I(x) defined in Eq. (19) 3 = (~D1/2 k = number of cycles of distortion around the circumference in the distorted flow K = K(z) defined in Eq. (A13) L = L(x) defined in Eq. (18) N = number of blades of the rotor Ap* = pressure difference—Eq. (26) RI = inner radius of the compressor Rz = outer radius of the compressor -Ravg = average radius of the compressor t* = time £o* = time for one revolution of the rotor T = cascade spacing parameter—a/c vs* = see Eq. (Al) vp* = see Eq. (A2) vax = axial velocity of flow in the annulus Vrot = rotational velocity of the airfoils vrei = flow velocity relative to the blades vx* = x*component of the velocity field ^j/* = ?/*component of the velocity field F* - F*(2*)—Eq. (9) Vx* = real part of V* Vy* = real part of iV* w = unsteady normal velocity distribution of the flow relative to the blades