# Numerical Investigations on the Blade Tip Clearance Excitation Forces in an Unshrouded Turbine
## 2 Numerical Setup
### Rotor Motion and Improved Mesh Deformation Method  
The multi-frequency elliptical orbit whirling model was used in this paper to obtain the entire set of the frequency-dependent rotordynamic coefficients. This model has been validated in [PERSON] et al. [23]. The rotor motion Equations (3) and (4) are defined as harmonic functions with specific vibration amplitudes. Table 3 presents the details of the rotor motion parameters. The vibration amplitudes were obtained based on the small motion theory [24].  
For x-direction excitation:  
\[x=a\sum_{i=1}^{N}(\Omega_{i}t),\;y=b\sum_{i=1}^{N}(\Omega_{i}t), \tag{3}\]  
and for the y-direction excitation:  
\[x=b\sum_{i=1}^{N}(\Omega_{i}t),\;y=a\sum_{i=1}^{N}(\Omega_{i}t). \tag{4}\]  
Figure 4: Comparison of the isentropic Mach number between the numerical results and reference data.  
Figure 3: Variations of the mass flow rate and stage torque with element number.  
As mentioned above, the model we used was one of a concentric unshrouded rotor. Therefore, to realize the specific rotor motion, a mesh deformation method [25], which can dynamically control the movement of the mesh nodes, was needed. It should be noted that the mesh quality, especially for grids of boundary layers, is likely to become worse during the mesh deformation procedure. Moreover, compared to the seal model in [PERSON] et al. [23], the unshrouded turbine model was more complicated. Hence, to guarantee the mesh quality, an improved mesh deformation method that could precisely set specific locations of every node at every moment was proposed. Figure 5 presents the two partitions of the cross-section and the illustration of the node movement.  
For the nodes in part A (green area, \(R_{h}<R_{0}<R_{t}\)), the specific locations of the nodes at the time \(t\) (\(X_{t}\), \(Y_{t}\)) could be described using:  
\[\left\{\begin{array}{l}X_{t}=X_{0}+x\\ Y_{t}=Y_{0}+y\end{array}\right., \tag{5}\]  
and for the nodes in part B (yellow area, \(R_{t}<R_{0}<R_{c}\)), the specific locations could be described using:  
\[\left\{\begin{array}{l}X_{t}=X_{0}+\frac{R_{c}-R_{0}}{R_{c}-R_{0}}x\\ Y_{t}=Y_{0}+\frac{R_{c}-R_{0}}{R_{c}-R_{0}}y\end{array}\right., \tag{6}\]  
where (\(X_{0}\), \(Y_{0}\)) depict the locations of nodes at the initial time, \(R_{0}\) can be calculated using \(R_{0}=\sqrt{{X_{0}}^{2}+{Y_{0}}^{2}}\), and (\(x\), \(y\)) can be obtained via Equations (3) and (4). Additionally, the mesh deformation method could also be used to alter the mean tip clearance without building a new model, which dramatically improved the computing efficiency. For an unshrouded turbine with a new mean tip clearance \(t^{\prime}_{m}\), the specific locations of the nodes in part A (green area, \(R_{h}<R_{0}<R_{t}\)) could be described using:  
\[\left\{\begin{array}{l}X_{t}=X_{0}+\frac{t^{\prime}_{m}}{t_{m}}x\\ Y_{t}=Y_{0}+\frac{t^{\prime}_{m}}{t_{m}}y\end{array}\right., \tag{7}\]  
Figure 5: Illustration of the two partitions and node movement.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Units** & **Values** \\ \hline Vibration amplitudes, \(a\) & mm & 0.005 \\ Vibration amplitudes, \(b\) & mm & 0.0025 \\ Number of frequency, \(N\) & - & 13 \\ Frequency, \(\Omega_{i}\) & Hz & 20, 40,..., 240, 260 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Details of rotor motion parameters.  
where \(t_{m}^{\prime}/t_{m}\) is the correction factor of the vibration amplitude. The specific locations of the nodes in part B (yellow area, \(R_{t}<R_{0}<R_{c}\)) could be described using:  
\[\left\{\begin{array}{l}X_{t}=\bigg{(}R_{t}+\frac{t_{m}^{\prime}}{t_{m}}(R_{0}- R_{t})\bigg{)}\frac{X_{0}}{R_{0}}+\frac{R_{c}-R_{0}}{R_{c}-R_{t}}x\\ Y_{t}=\bigg{(}R_{t}+\frac{t_{m}^{\prime}}{t_{m}}(R_{0}-R_{t})\bigg{)}\frac{Y_{0}} {R_{0}}+\frac{R_{c}-R_{0}}{R_{c}-R_{t}}y\end{array}\right., \tag{8}\]  
where the first items to the right of the equal sign are the new coordinates of nodes in this part caused by the mean tip clearance change.