# Orbit Decomposition Method for Rotordynamic Coefficients Identification of Annular Seals
## 1 Introduction
The complete Navier-Stokes equations are solved. The prediction capability and accuracy are improved. The CFD method can be classified either as steady [14] or transient [15]. The steady CFD method assumes that the rotor performs a circular orbit whirl around the stator center. The unsteady problem can be transformed into a steady one by solving the three-dimensional flow field in the frame of  
Figure 1: Fluid-induced forces acting on the whirl rotor. reference attached to the whirling rotor [13]. Curve-fitting technologies are subsequently employed to identify the rotordynamic coefficients [13]. With this method, the obtained rotordynamic coefficients are frequency independent [13; 14]. The latest research results show that many annular seals possess frequency-dependent rotordynamic coefficients, including the labyrinth seal [16; 17], pocket damper seal [17], hole-pattern seal [18; 19], etc. Interaction between the acoustic waves in a circumferential annulus and the rotor excitation frequency is the cause of the frequency dependency [20]. To identify the frequency-dependent rotordynamic coefficients of annular seals, the transient CFD method is applied. The transient CFD method directly solves the unsteady flow field with a mesh deformation technique. The rotor performs the periodic motion. The time-dependent fluid-induced forces are obtained to identify the frequency-dependent rotordynamic coefficients. According to the difference in whirl orbit, the transient CFD method can be divided into a linear orbit model [21], circular orbit model [22], and elliptical orbit model [23]. In 2007, [PERSON] et al. [21] employed the linear orbit model to determine the frequency-dependent rotordynamic coefficients of a hole-pattern seal. The rotor was given a periodic motion in one direction. The frequency-dependent rotordynamic coefficients can be obtained with one transient computation for each frequency. Subsequently, [PERSON] et al. [22] advanced [PERSON] et al. 's model by applying the circular orbit model in 2011. Motion in two orthogonal directions was imposed on the rotor. The amplitudes in two orthogonal directions were equal. The results showed that the circular orbit model has a superior accuracy to the linear orbit model. However, the circular orbit model is more time-consuming because two transient computations of the forward and backward whirls are required for each frequency. Considering the computational time, the elliptical orbit model was applied by [PERSON] et al. [23] in 2012. Compared with the circular orbit model, the consumed time was reduced by 50% because one transient computation was required for each frequency. The accuracy of the elliptical orbit model was equivalently high. In addition, the elliptical orbit model is more representative of the real rotor excitation. The elliptical orbit model is widely used to predict the frequency-dependent rotordynamic coefficients of annular seals [24; 25; 26]. Although a significant reduction of computational time has been achieved by the elliptical orbit model, the computational cost is still high because of transient computations [23]. In this paper, the orbit decomposition method is proposed to reduce the computational time. The elliptical whirl orbit is decomposed into the forward and backward circular whirl orbits. Under small perturbation circumstances, the rotordynamic problem can be linearized. The fluid-induced forces of the elliptical orbit model can be obtained by the linear superposition of the fluid-induced forces arising from the two decomposed circular orbit models. As the fluid-induced forces of the circular orbit model can be calculated with the steady CFD method, the transient computations can be replaced with the steady ones in the solution process of the elliptical orbit whirl model. The computational time is largely reduced. To validate the orbit decomposition method, its rotordynamic results are compared with those of the transient CFD method and experimental data.