# Numerical Investigations on the Blade Tip Clearance Excitation Forces in an Unshrouded Turbine
## 1 Introduction  
In an unshrouded turbine with an eccentric rotor, tip clearance excitation forces are generated owing to the variation of blade efficiency with the tip clearance in the circumferential direction. These forces, also known as the \"Thomas-Alford forces,\" could result in self-excited vibration and severe rotordynamic instabilities since they increase with the turbine load. Therefore, to ensure the stability of a rotor-bearing system, it is of great significance to investigate the characteristics of the excitation forces in the unshrouded turbine. As mentioned above, these forces were first founded by [PERSON] [1] and [PERSON] [2]. They both theoretically analyzed the mechanism of these forces and proposed respective models. [PERSON] indicated that these forces are related to the stage torque \(T\) and mean tip clearance \(t_{m}\) as described in Equation (1):  
\[F=\beta\frac{T}{R_{m}}\frac{t_{m}}{h}, \tag{1}\]  
where \(R_{m}\) is the mean radius, \(h\) is the blade height, and \(\beta\) is a correction factor, which is related to the blade efficiency. Based on their theory, several experimental and theoretical analyses were performed to investigate these forces. [PERSON] [3] and [PERSON] [4] measured the excitation forces ina low-power test turbine and found a direct force contribution that was not mentioned by [PERSON]. Also, it was found that the excitation forces increased significantly as the mean tip clearance decreased. [PERSON] et al. (2018) conducted an experiment in a relatively higher-power turbine with five different unshrouded configurations and one shrouded configuration. They found that the forces increase as the hub gap between the stator and rotor decreases, and the force coefficient decreases with the operation flow coefficient. [PERSON] and [PERSON] (2018) verified [PERSON]'s mathematical prediction model through a small high-speed axial flow test fan and found that the force is speed-dependent and easily affected by inlet flow conditions. [PERSON] et al. (2018) experimentally and theoretically investigated the effects of rotor eccentricity, which causes circumferentially nonuniform tip clearance, unsteady flow, and destabilizing aerodynamic forces, and proposed a simplified method for predicting the stability of aerodynamically excited turbomachinery. [PERSON] and [PERSON] (2018, 2019) studied the force generation mechanism and proposed an actuator disk model that describes the flow response to a finite clearance at the rotor tip. [PERSON] and [PERSON] (2018) presented experimental results from a scale model of the steam turbine and analyzed the destabilizing forces by considering a partial admission condition. In addition, [PERSON] et al. (2018, 2019) recently proposed a semi-empirical model for the predictions of steam turbine excitation forces and discussed the effect of an additional tip leakage injection on the resulting Thomas-Alford forces. In recent years, extensive studies on the destabilizing fluid forces in seals have been performed ([PERSON] and [PERSON], 2018; [PERSON], 2018; [PERSON], 2018). However, there are few studies on the blade tip clearance excitation fluid forces, except for the studies mentioned above. Hence, there is still a need to study the characteristics of these forces. Meanwhile, quite few of these existing studies were based on computational fluid dynamic (CFD) analysis, which is a reliable way to reveal more details of the complex flow and also a popular method for studying the tip leakage flow and the forces in seals. Moreover, many recent studies have experimentally and numerically investigated the effects of unshrouded blade tip leakage flow on the blade efficiency and heat transfer characteristics ([PERSON] and [PERSON], 2018; [PERSON], 2018; [PERSON], 2018). However, few of them focused on the blade tip clearance excitation forces. Therefore, this paper utilized the three-dimensional CFD numerical method to investigate the tip clearance excitation forces in an unshrouded turbine.