Periodicity, Superposition, and 3D Effects in Supersonic Compressor Flutter Aerodynamics

Abstract

In this work, a family of methods, both blade-to-blade surface and 3D, based on the numerical integration of the unsteady Euler equations, are used in studying various aspects of the unsteady aerodynamics of vibrating compressor cascades, in the supersonic flutter region. Most aerodynamic methods assume a traveling wave assembly mode of structural vibration, and suppose that the associated chorochronical periodicity is also encountered in the flowfield. This hypothesis has been tested by simulating the flow in a full annular cascade and has been verified in all the cases we have studied. When analyzing the aeroelastic stability of cascades, some assembly modal basis must be used. This, especially in presence of mistuning, relies on modal superposition hypotheses and linearity assumptions. The superposition assumption seems to be justified and the linear range of the amplituderesponse is fairly large, although it varies greatly with frequency. Finally, an assessment of the importance of unsteady 3D effects is attempted using the 3D method. Nomenclature 'Cp = first pressure coefficient harmonic (Pt -P)|nlet 5 vib f = Vibration frequency F = aeromechanical variable (Eq. 1) k = blade number Μ = Mach number Ν = number of blades Ρ = pressure τ, Φ = traveling-wave order R = radius Sr x = chord based Strouhal number; Sr^ : t = time χ -= coordinate along the engine axis W = relative flow velocity Ζ = set of all integer numbers <*vib = torsional vibration amplitude (deg.) ßr = interblade phase-angle 5 vib = vibratory amplitude 5 vib = T( m s a u X rfaL ade V(6: + 5 2 R+ 6j)) θ = angular position ω = angular vibration frequency Subscripts (. ) t = total thermodynamic conditions '(.) = first harmonic (•) x,R ,0 = vector coordinates

Filtered chorochronic interface as a capability for 3-D unsteady throughflow analysis of multistage turbomachinery

“…The phase-shifts (1f, 1g) are valid for the general travelling-wave (1d, 1e), and its 1-harmonic, and define the phase-shift of the time-harmonics of the waves by (Gerolymos et al 2002a, (12), 1143). There is a distinct analogy between these travelling-wave descriptions (1) of the flow and of the travelling-wave description of flutter (Gerolymos 1988(Gerolymos , 1990a(Gerolymos , 1993aGerolymos, Blin, and Quiniou 1991;Gerolymos and Vallet 1996), and of aeroelastic modes (Gerolymos 1990b(Gerolymos , 1993b of an isolated rotor, implying that the same numerical algorithms can be used to apply pitchwise periodicity conditions in both cases (He 2010).…”