# Influence of the Tip Clearance on the Aeroelastic Characteristics of a Last Stage Steam Turbine
## 3 Steam Turbine Flutter Test Case  
The KTH Steam Turbine Flutter Test Case was applied as the research object. The geometry of the test case was initially designed by Durham University [15]. A schematic figure of the KTH Steam Turbine Flutter Test Case and the definition of span surfaces are shown in Figure 1. The diffuser in the original design was included in the rotor computation domain to extend the fluid domain and reduce the influence of acoustic reflection at the outlet boundary [16]. The geometry and boundary conditions of the KTH Steam Turbine Flutter Test Case are available online [17]. The rotor rotated at 3000 rpm, and the average length of the rotor blade was 920 mm. A high stagger angle of 67 degrees presented near the tip of the rotor blade. The designed working point [15] was applied in this study. The average isentropic Mach number at the rotor exit was 1.12, and the flow at tip region was transonic. In conclusion, the test case had representative geometrical parameters and flow-field characteristics of modern steam turbines.  
Figure 1: Schematic figure of the steam turbine model with span lines and the location of the rotor blade tip.  
The blade material was assumed to be 17 PH4 high strength steel in the modal analysis, the properties of which are shown in Table 1. The first blade-dominated bending mode of the KTH Steam Turbine Flutter Test Case was applied in the flutter analysis, as shown in Figure 2, which has been verified to be aeroelastically unstable at this working point [8].  
The modal frequency of the first flap mode was 92.953 Hz, and therefore, the corresponded reduced frequency was about 0.2. The reduced frequency is a non-dimensional parameter that describes the level of unsteadiness, which is computed by Equation (3):  
\[\omega^{*}=\frac{\omega c}{V_{ref}}=\frac{2\pi fc}{V_{ref}} \tag{3}\]  
where \(f\) is the modal frequency, \(c\) is the chord length, and \(V_{ref}\) is the average relative velocity at the turbine exit. The Durham steam turbine test case was designed based on aerodynamic considerations, but not structural dynamics, and as a result, the strength of the rotor blade was insufficient. The reduced frequency of the first bending mode was lower than that of a typical industry steam turbine. To build a representative real-scale steam turbine flutter test case, the reduced frequency was modified as 0.3 in the KTH Steam Turbine Flutter Test Case. The modified modal frequency calculated by Equation (3) equaled 132.08 Hz. The maximum blade vibration amplitude was set as 2 mm, which was 1.225% of chord length at the blade tip.  
Five rotor models based on the original geometry are studied in this paper: one of them was set as no tip clearance, and the other four had different tip gap heights, as shown in Table 2. The blade tip surface was designed to be parallel to the shroud surface, as revealed in Figure 1. The computational meshes for the fluid domain of the five models were generated by TurboGrid. TurboGrid uses a non 1:1 mesh interface to connect the meshes from the pressure and suction side in the tip clearance. The number of mesh nodes and layers used in the spanwise direction in the tip clearance after mesh independence verification [18] is also presented in Table 2. The average y-plus of the cell height on the walls was around 20 to meet the requirement of the automatic turbulence wall function. Representative slices of the mesh in the fluid domain are revealed in Figure 3.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Material Properties** & **Value** \\ \hline Density & 7750 kg/m\({}^{3}\) \\ Modulus of elasticity & 210 GPa \\ Poisson’s ratio & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of the rotor blade.  
Figure 2: The first flap mode shape of the KTH Steam Turbine Flutter Test Case magnified by a factor of fifty. LE: the leading edge; TE: the trailing edge.