# Influence of the Tip Clearance on the Aeroelastic Characteristics of a Last Stage Steam Turbine
## 4 Results
### Tip Clearance Flow Effects on Steam Turbine Aeroelastic Characteristics  
The tip clearance effects on the integrated aerodynamic damping coefficient at various IBPAs was firstly investigated, which is shown in Figure 7. The tip clearance flow showed a detectable impact on the aerodynamic damping, and the influence of tip clearance flow on aeroelastic stability was dependent on the analyzed IBPA. These characteristics are in agreement with previous research [3, 4, 8]. Since the aerodynamic damping coefficient at the least stable IBPA is most critical in the design of a turbine, the relationship between aerodynamic damping and tip gap height at the two unstable IBPAs is highlighted in Figure 8. The overall aerodynamic damping at the two unstable IBPAs monotonously increased with the tip gap height for the studied models. The increase of aerodynamic damping till the tip gap height equaled 5% of chord length is inconsistent with the previous analysis in low-speed turbines [2] or long last-stage steam turbines [7].  
Figure 6: Blade loading in the steady state on the suction side of the rotor near the tip.  
Figure 7: Aerodynamic damping versus IBPA with various tip gap heights.  
To investigate the difference caused by the tip clearance, the distribution of the integrated aerodynamic damping among spans at IBPA = \(-60^{\circ}\) is presented in Figure 9. The majority of aerodynamic work was done at the top 40% span, while the variation of aerodynamic damping among models mainly occurred above 85% span. This phenomenon is also verified in the distribution of wall work density at 80%, 90%, and 98% span in Figure 10. From Figure 10, the majority of the wall work coefficient difference occurred on the blade suction surface. Therefore, the distribution of local wall work density at the least stable IBPA (\(-60^{\circ}\)) near the blade tip is presented in Figure 11 to analyze the underlying physical mechanism.  
Figure 8: Aerodynamic damping versus tip gap height at the two unstable IBPAs.  
Figure 9: Aerodynamic damping versus span with various tip gap heights when IBPA = \(-60\) deg.  
The location of the wall work coefficient variation can also be divided into three regions, corresponding to the three regions of blade loading variation presented in Figure 6. Compared with the no tip gap model, the tip clearance flow increased the aeroelastic stability in Regions A and C, while the wall work coefficient in Region B transformed from negative to positive. The strength of the three phenomena increased with the tip gap height for the five models. The underlying mechanism behind the differences observed will be discussed in the next section.  
Figure 11: Distribution of the aerodynamic work coefficient on the suction side of rotor near the tip when IBPA \(=-60\deg\).  
Figure 10: Distribution of the wall work coefficient at **(a)** 80%, **(b)** 90%, and **(c)** 98% span; the abscissa is the ratio of chord coordinates to the length of chord, and \(-1\)–\(0\) indicate the trailing edge to the the leading edge on the suction surface, while \(0\)–\(1\) indicate the leading edge to the trailing edge on the pressure surface.