# Vibroacoustic Optimization Study for the Volute Casing of a Centrifugal Fan
## 3 Vibroacoustic Aerodynamics  
### Numerical Methods  
CFD (Computational fluid dynamics) technology has been proven to be a very useful tool in the analysis of the internal flow of turbomachinery both in design and performance prediction. It has been widely applied to simulate the fully three-dimensional (3D) unsteady flow in centrifugal turbomachinery. In this part, the unsteady aerodynamic forces that excite vibrational noise are calculated by CFD. The whole unsteady flow for the entire impeller-volute configuration was conducted using the computational fluid dynamics (CFD) code ANSYS CFX. The numerical simulation is based on a finite-volume numerical method that employs an incompressible flow model to solve the Unsteady Reynolds Averaged Navier-Stokes equations (URANS). The characteristic Mach number of the simulated fan described by the blade tip circumferential velocity was \(u_{2}/c\) = 0.18 (<0.3); therefore, the flow was guaranteed to be incompressible. The continuity equation and momentum equations were solved independently of the energy equation because of the isothermal flow. The standard \(k\)-\(\varepsilon\) turbulence model, which [PERSON] [30] and [PERSON] [31] applied to capture wall pressure fluctuations, was used in the present simulation of the unsteady flow field. A coupled solver, which uses a fully implicit discretization scheme to solve all of the equations (corresponding to the velocity and pressure), was used. However, a second-order high-resolution discretization scheme was used for the convection terms, and a second-order backward Euler scheme was used for the transient terms. For the three-dimensional calculations, a couple of high-quality hexahedral structural grids were employed to define the flow domains. Details of the grid features and meridian grid cross-section, including the radical gap and cavity around the volute, are shown in Figure 2. More details about the grids of this fan have been reported in the previous work ([PERSON] et al.) [13]. The numerical deviations result from the grid number needed to be removed, and a grid-independent validation of the fan total pressure coefficient and efficiency was performed. Figure 3 shows the influence of the grid size on the fan total pressure coefficient and efficiency. From Figure 3, we can see that the grid was independent when the total number of grid points exceeded 2.8 million. In addition, by increasing the grid size to 5.7 million, the total pressure coefficient with respect to the flow rate was nearly unchanged, compared with the smaller total pressure coefficient over a small flow rate range that was close to the best efficiency point (BEP). \begin{table}
\begin{tabular}{c c} \hline
**Parameters Description** & **Parameter Values** \\ \hline Impeller blade outlet diameter \(D_{2}\) (mm) & 520 \\ Impeller blade inlet diameter \(D_{1}\) (mm) & 328 \\ Impeller outlet width \(b_{2}\) (mm) & 102 \\ Impeller inlet width \(b_{1}\) (mm) & 138 \\ Blade number \(Z_{n}\) & 12 \\ Volute width \(B\) (mm) & 286 \\ Impeller-tongue distance (mm) & 38 \\ Impeller-tongue distance (\% of \(D_{2}\)) & 13.2\% \\ \hline \end{tabular}
\end{table} 
Table 1: Fan dimensions. The modeled boundary conditions were considered to have greater physical meaning for turbomachinery flow simulations. In this case, the CFD simulation process began with a steady flow calculation using the frozen-rotor approach, and nonslip conditions were specified at the solid walls. In addition, for the near-wall flow region, a scalable wall function treatment based on the logarithmic law [32] was applied to cause the mean value of \(y^{+}\) (\(y+\) is the dimensionless distance from the wall; it is used to check the location of the first node away from a wall) to vary between 30-300 (the recommended values by ANSYS CFD code to ensure a high calculation accuracy), as shown in Figure 4. Therefore, by applying this method, the number of the grid points was greatly reduced without reducing the calculation accuracy.