Computation of the generalized aerodynamic forces

The unsteady aerodynamic forces are computed from a basis of mn real mode shapes Ф of the reference sector, for an oscillation frequency и and an inter-blade phase angle an. By expressing the displacements of the reference sector as a linear combination of the modes Ф and by assuming that the struc­tural motion is harmonic and that all the sectors have the same motion with a constant phase angle an between two adjacent sectors, the generalized aerody­namic forces generated by the displacement un (t) are written, by linearity:

fagn (un (t), un (t)) Fagn (^, i t) fin •

Fagn (Ф, i u, t) is the time-dependant aerodynamic coefficient matrix whose (i, j)-term is obtained by projecting the unsteady aerodynamic force generated by the harmonic motion of the j-th mode on the i-th mode:

Fagn, ij (Ф, iu, t) = – / [Pn (M, , iu, t) – Ps (M)]tt(M). (M) dS

JM ЄЕ

(11)

where Pn is the unsteady pressure generated by the j-th mode ^j, Ps the steady pressure, Фі is the displacement vector of the i-th mode, it is the unit external normal vector to the surface S and dS is an elementary surface of S. Let’s introduce the aerodynamic coefficient matrix An (Ф, iu, t) obtained from the integral in Eq. (10) with Pn and Ps replaced by the associated pres­sure coefficient Cp = (P — P00)/(^ Poo V^>), where P^, Рос and are the pressure, the density and the velocity of the upstream unperturbed fhid.

By performing a Fourier analysis of Fagn (Ф, i u, t) and by keeping only the first harmonic term, we have:

Fagn ^, iu, t) – Fagn (ф iи)Є

The generalized aerodynamic forces generated by un (t) become:

fagn (un (t)> un (t)) – Fagn ( Ф, i u) fin еІ

Fagn (Ф, i u) and An (Ф, i u) are complex, asymmetric square matrices of di­mension mn. They are computed for иш oscillation frequencies.

In the flitter equation (8), the aerodynamic coefficient matrix should be de­termined from the mn complex modes Фп. Using the linearity hypothesis, the aerodynamic coefficient matrix Fagn (Фп, i u) generated by Фп can be ex­tracted from the aerodynamic coefficient matrix computed from the basis of the 2 mn real vectors formed by the real and imaginary parts of Фп.

The unsteady aerodynamic forces are obtained solving the Euler equations for an ideal gas using an aerodynamic code called CANARI and developed for years at ONERA Dugeai et al., 2000. Because of the cyclic symmetry of the flow, a chorochronic boundary condition is applied to the simulated channel :

F(r,9 + к (3,z, t) = F(r,9,z, t + for n

F is any fbwfield variable, r is the rotation radius, and 9 the azimuthal an­gle. In a first step, a steady state is computed depending on the rotation speed, on the pressure ratio and on the far-field total temperature, total pressure, and velocity. In a second step, unsteady simulations are performed by forcing an oscillating blade motion at different frequencies. These simulations depend on the steady flowfield previously computed and used as initial conditions, on the inter-blade phase angle, and on the forced motion shape and frequency. A blowing condition is then used to simulate the blade motion. Once a pseudo­steady oscillating state has been reached (no transient effect), a Fourier trans­form is performed over the pressure to get the unsteady aerodynamic forces.