# Numerical Investigations on the Blade Tip Clearance Excitation Forces in an Unshrouded Turbine
## 4 Conclusions  
A three-dimensional CFD numerical method, along with an improved mesh deformation technique based on the transient analysis and multi-frequency elliptical orbit whirling model, was proposed to investigate the tip clearance excitation forces in an unshrouded turbine. The tip clearance excitation forces were represented as the rotordynamic coefficients (stiffness and damping coefficients) in this paper. The effects of several variables, including the whirling frequency, rotational speed, inlet flow angle, eccentricity, and the mean tip clearance, on the rotordynamic coefficients and aerodynamic performance were analyzed in detail. The conclusions are as follows:  
(1) Positive direct stiffness coefficients confirmed the existence of a direct force contribution in the tip clearance excitation forces. The damping effects were found to be quite small and could be ignored. Both the stiffness and damping showed a dependency on the rotor whirling frequency. Specifically, the direct stiffness increased with the frequency, while the cross-coupling stiffness decreased with the whirling frequency.  
(2) The increase of the rotational speed caused a considerable decrease of the direct stiffness and cross-coupling stiffness due that the torque on the rotor decreasing with the rotational speed, which indicates a decreasing tip clearance excitation effect. The increase of the mean tip clearance led to a decreasing direct stiffness and a complicated change of the cross-coupling stiffness. For turbines with a small mean tip clearance (\(t_{m}/h=1\%\)), increasing the mean tip clearance caused the increase of cross-coupling stiffness, which led to rotor instability. For a turbine with a higher mean tip clearance (\(t_{m}/h=3\%\)), the cross-coupling stiffness decreased with the increase of the mean tip clearance, which enhanced the rotor stability. The effects of the inlet flow angle and eccentricity on the rotordynamic coefficients were relatively smaller.  
(3) The rotor eccentricity caused a pressure redistribution of both upstream and downstream flow. A higher eccentricity and higher mean tip clearance resulted in a greater nonuniformity of the circumferential static pressure distribution. The characteristics of leakage flow changed with the local  
Figure 14: Distributions of selected variables: (**a**) numerical result under \(ER=0.7\), \(t=0.05\)s, \(n=10\),000 r/min, \(\alpha=48.36^{\circ}\), \(t_{m}/h=2\%\), and x-direction excitation, and (**b**) experimental results in Song [9].  
tip clearance. The phase difference between the local tip clearance and the local torque on the blade was about 120\({}^{\circ}\), which led to the direct force contribution.  
Conceptualization, Y.P. and Q.Y.; methodology, Y.P.; software, G.H.; validation, J.G.; formal analysis, Y.P.; investigation, Y.P. and G.H.; resources, Y.P.; data curation, J.G.; writing--original draft preparation, Y.P. and Q.Y.; writing--review and editing, Y.P.; visualization, G.H.; supervision, P.L. and G.Z.; project administration, Q.Y.; funding acquisition, Q.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (grant number 11872289) and the Key Project of Natural Science Foundation of Xi'an Jiao Tong University (grant number ZKZD2017025).  
The authors declare no conflict of interest.