## 2 Materials and Methods
### Numerical Simulation Method
#### 2.2.3 Computational Setting  
To optimize computational efficiency, we conducted simulations using a supercomputer. Our computational domain included both the interior and surrounding environment. The computational parameters are listed in Table 2, and key geometric dimensions of the ATH are shown in Figure 2. The inlet pressure was set to 3.06 atm and the rotational speed was fixed at 320,000 rpm, representing the actual operating conditions observed in the dental clinic. Using the BCM, we generated the computational grid for large-scale simulations within minutes. The simulation conditions were matched to the experimental setup to accurately reflect actual operational conditions, including inlet/outlet pressures and rotational speed. A fine mesh with a spacing of 0.025 mm was applied near the ATH to capture complex flow details. The total mesh cell count reached 100 million, offering precise information on critical regions, such as the gap between the head body and rotor \begin{table}
\begin{tabular}{l l l} \hline
**Equation Term** & **Algorithm** & **Numerical Method** \\ \hline Time advancement & Implicit & Adaptively switched time stepping \\ Convective terms & Roe scheme & Low Mach fix Roe [22] \\ Reconstruction & MUSCL & 5 th order (without limiter) \\ Viscous terms & Central difference & 2 nd order \\ \hline \end{tabular}
\end{table} 
Table 1: Numerical scheme. and the region near the airfoil of the impeller (Figure 1c). Although a full grid independence study was not performed due to the high computational cost associated with the mesh size (>100 million cells), the fine spatial and temporal resolutions were validated by the excellent agreement between simulated and experimental results (as discussed in Section 3.3). The time step size was set to 2 \(\times\) 10\({}^{-7}\) s to satisfy the CFL condition for numerical stability near the impeller region, allowing a Nyquist frequency of 2.5 MHz, which is significantly higher than the highest observed acoustic frequency (~21 kHz). This confirms the adequacy of the chosen computational settings for resolving relevant aeroacoustic phenomena. To prevent reflections from affecting result accuracy at the boundaries of the computational domain, we adopted an absorbing boundary condition for low-speed flow. Additionally, to accommodate the computational scale of hundreds of millions of cells, we utilized the Supercomputer Fugaku (RIKEN Center for Computational Science, Kobe, Hyogo, Japan). The simulations were performed using the powerful 2.2 GHz, 48-core A64 FX processors developed by Fujitsu. Message-passing interface libraries were used \begin{table}
\begin{tabular}{c c c} \hline
**Category** & **Parameter** & **Value** \\ \hline \multirow{8}{*}{Geometry} & Body diameter & 9 mm \\  & Length of the body & 13.0 mm \\  & Impeller diameter & 8.1 mm \\  & Length of the rotor & 10.7 mm \\ \cline{2-3}  & Stage 1 impeller blade count & 18 \\  & Stage 1 blade length (axial) & 2.0 mm \\  & Stage 1 blade height (radial) & 0.6 mm \\ \cline{2-3}  & Stage 2 impeller blade count & 18 \\  & Stage 2 blade length (axial) & 1.2 mm \\  & Stage 2 blade height (radial) & 0.7 mm \\ \hline \multirow{8}{*}{Numerical setup} & Atmospheric pressure \(P_{0}\) & 101,300.0 Pa \\  & Inlet pressure & 3.06 atm \\  & Outlet pressure & 1.15 atm \\ \cline{1-1}  & Rotational speed & 320,000 rpm \\ \cline{1-1}  & Velocity of the air turbine at impeller tip & 135 m/s \\ \cline{1-1}  & Computational time step (development phase) & 4 \(\times\) 10\({}^{-6}\) s flow under development \\ \cline{1-1}  & Computational time step (after steady state) & 2 \(\times\) 10\({}^{-7}\) s after quasi steady state \\ \hline \multirow{2}{*}{Mesh} & Finest cell size & 0.025 mm \\  & Cell number & 104,767,488 \\ \hline \end{tabular}
\end{table} 
Table 2: Computation parameter settings. Figure 2: Key geometric dimensions of the air turbine handpiece: (**a**) head body; (**b**) rotor. to establish communication between neighboring processors.