Unsteady disturbance theory and modeling for centrifugal compressors

Abstract

We present a model for the instability dynamics of a centrifugal compressor. The working fluid is approximated as incompressible and inviscid. The compressor is modeled as a cylindrical inlet duct of infinitesimal thickness followed by a perpendicular planar outlet duct also of infinitesimal thickness, with the blade rows located at the junction of the two ducts. The flow in the inlet duct only is assumed to be irrotational. For our analysis we use a Fourier series expansion of the periodic dynamical variables along the circumference and solve explicitly for the coefficients. The resulting model is based on the first mode approximation only and incorporates explicitly the nonlinear compressor map. Although based on identical assumptions, our model is significantly more complex than the well-known Moore-Greitzer model describing similar instabilities in axial compressors. The reason is the underlying geometric complexity of a centrifugal compressor. A first implication of our model is that, unlike the case of axial compressors, the vorticity field at the outlet duct plays a significant role in the dynamics of the instabilities developed at the compressor blades. Partial Nomenclature PT = stagnation pressure upstream of compressor Ps = plenum static pressure r = radial coordinate 6 = circumferential coordinate z = axial coordinate t = time u = radial velocity component v = circumferential velocity component w = axial velocity component p = static pressure U = transformed radial velocity component, Eq. (11) V = transformed circumferential velocity component, Eq. (11) P = transformed static pressure, Eq. (11) R; = radius of compressor inlet, also blade inlet radius Re = blade exit radius R* = exit radius from compressor outlet to the plenum pi = fluid density at the inlet (assumed constant) pc = fluid density at the outlet (assumed constant) Q = blade angular velocity bj = thickness of inlet duct be = thickness of outlet duct li = inlet duct length ^L = l°' compressor map function (for blade section only) f = global compressor map function (for entire compressor) y = generic representation of u, U, v, V, p, or P n = subscript for n-th Fourier mode H = n/R (), = denotes a circumferentially averaged (CA) value ()i = denotes a perturbation, measured from a CA value ()s = denotes the Fourier sin n0 amplitude of a variable ()c = denotes the Fourier cos n6 amplitude of a variable All other symbols are auxiliary and defined locally in the text.