# Orbit Decomposition Method for Rotordynamic Coefficients Identification of Annular Seals
## 1 Introduction  
Annular seals are widely used in turbomachinery to control the leakage flow through rotor-stator clearances from high-pressure regions to low-pressure regions [1, 2]. Besides leakage flow reduction, annular seals are intended to serve as vibration damping devices. However, the flow in seals often generates fluid-induced forces that can destabilize the rotor [34]. As modern turbomachinery is moving towards higher operating parameters, the fluid-induced forces and their influence on the rotordynamic stability are becoming more pronounced [5, 6]. Accurate and fast identification of the rotordynamic coefficients is essential in the seal design and evaluation process. For small orbital motion of the rotor about the stator center, four forces are generated, \(F_{\mu\tau}\), \(F_{\mu\theta}\), \(F_{tr}\), and \(F_{\mu\theta}\), as shown in Figure 1. For the four forces, the first subscript represents the force direction, where \"\(\tau\)\" means that the force is in the displacement direction, and \"\(\tau\)\" means that the force is perpendicular to the displacement direction. The second subscript represents the disturbance parameter, where \"\(\tau\)\" represents the displacement disturbance, and \"\(\theta\)\" represents the whirl velocity disturbance. A rotor displacement disturbance produces two forces. One force is \(F_{\pi\eta}\), which is in the displacement direction. The other force is \(F_{tr}\), which is perpendicular to the displacement direction. The two forces are proportional to the rotor displacement (whirl orbit radius \(r\)). The force \(F_{rr}\) is divided by the rotor displacement \(r\) to yield the direct stiffness \(K\). The force \(F_{tr}\) is divided by the rotor displacement \(r\) to yield the cross-coupled stiffness \(k\). The rotor whirl velocity disturbance also produces two forces. One force is \(F_{t\theta}\), which is perpendicular to the displacement direction. The other force is \(F_{r\theta}\), which is in the displacement direction. The two forces are proportional to the rotor whirl velocity. The force \(F_{t\theta}\) is divided by the rotor whirl velocity \(\Omega r\) to yield the direct damping \(C\). The force \(F_{r\theta}\) is divided by the rotor whirl velocity \(\Omega r\) to yield the cross-coupled damping \(c\). The methods for rotordynamic coefficient identification of annular seals can be grouped into the analytical method based on bulk-flow theory [7] and the numerical method based on computational fluid dynamics (CFD) simulation [6]. The bulk-flow method was developed in the early 1980s. This method permits quick calculations with some simplifying assumptions. The one-control volume model is the simplest one [8]. Each seal cavity is represented with one control volume. Variations in pressure, velocity, and density in the control volume are neglected. The average values of variables are used. Considering that the jet flow region and vortex flow region have different flow patterns, the two-control volume model was developed by [PERSON] et al. [9] in 1984. Each seal cavity is divided into two parts. In 1991, the three-control volume model was developed to take into account the jet flow beneath the seal tooth [10]. All these control volume models use friction factor corrections to link the wall stress with average flow velocity [8; 9; 10]. The numerical accuracy of the bulk-flow model largely relies on these corrections [11; 12]. In practice, it is hard to accurately model/prescribe these corrections, because they change with seal geometries and operating conditions. From the one-control volume model to three-control volume model, the description of the seal flow field becomes more detailed. When a large number of control volumes are employed to fill the seal flow domain, the analysis results tend to reach the real flow field. This led to the development of the CFD method [13]. With the progress of computer technologies, the CFD method has received increased attention since the 1990s. The seal flow domain is discretized by numerous grid cells.