## 2 Numerical and Control Methods  
### Numerical Method for Flow-Induced Acoustic Resonance Frequency Prediction  
The frequency of flow-induced acoustic resonance is predicted numerically in this paper, where the LES model and the FW-H equation are used to calculate the flow and the acoustic field, respectively. The LES model is employed to simulate the transient flow field in ducts with cavities. The governing equations for LES are established by filtering the Navier-Stokes (N-S) equations in the physical space. The continuity equation and the N-S equation for incompressible fluids can be expressed as [29]  
\[\frac{\partial\overline{u}_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial\overline{u}_{i}}{\partial t}+\frac{\partial\overline{u}_{i} \overline{u}_{j}}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\left[v( \frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline{u}_{ j}}{\partial x_{i}})\right]-\frac{1}{\rho}\frac{\partial\overline{p}}{\partial x_{i}}- \frac{\partial\overline{v}_{ij}}{\partial x_{j}} \tag{2}\]  
where \(x_{i}\) and \(x_{j}\) represent the physical space, \(\overline{u}_{i}\) and \(\overline{u}_{j}\) are the fluid velocity components in the physical space, \(t\) is the time, \(\overline{p}\) is the static pressure, \(\rho\) is the density, \(v\) is the kinematic viscosity and \(\tau_{ij}\) is the subgrid-scale (SGS) stress. Here \(\tau_{ij}\) is calculated as follows  
\[\tau_{ij}-\frac{1}{3}\tau_{kk}\Delta_{ij}=-2\mu_{i}\overline{S}_{ij} \tag{3}\]  
In (3), \(\tau_{kk}\) is the isotropic part of the subgrid scale stress, \(\Delta_{ij}\) is the Kronecker Delta symbol, \(\mu_{t}\) is the subgrid scale turbulent viscosity and \(\overline{S}_{ij}\) is the rate-of-strain tensor for the resolved scale  
\[\overline{S}_{ij}=\frac{1}{2}\left(\frac{\partial\overline{u}_{i}}{\partial x _{j}}+\frac{\partial\overline{u}_{j}}{\partial x_{i}}\right) \tag{4}\]  
In the Smagorinsky-Lilly (SL) model, the subgrid-scale turbulent viscosity \(\mu_{t}\) is determined by (5) to (7)  
\[\mu_{t}=\rho L_{s}^{2}|\overline{S}| \tag{5}\]  
\[L_{s}=\min\Bigl{(}kd,C_{s}V^{1/3}\Bigr{)}, \tag{6}\]  
\[|\overline{S}|=\sqrt{2\overline{S}_{ij}\overline{S}_{ij}}, \tag{7}\]  
where \(L_{s}\) is the mixing length for subgrid scales, \(k\) is the Von Karman constant, \(d\) is the distance to the closest wall, \(V\) is the volume of the computational cell and \(C_{s}\) is the Smagorinsky constant, which is taken as the default value of 0.1 in this paper.  
The FW-H equation based on [PERSON]'s acoustic analogy theory is used to simulate the characteristics of the flow-induced resonance noise. The FW-H equation can be expressed as [30]  
\[\begin{array}{c}\frac{1}{c_{0}^{2}}\frac{\partial^{2}p^{\prime}}{\partial t ^{2}}-\
abla^{2}p^{\prime}=\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}[ T_{ij}H(f)]-\frac{\partial}{\partial x_{i}}\{[P_{ij}n_{j}+\rho u_{i}(u_{n}-v_{n})] \Delta(f)\}\\ \hskip 14.226378 pt+\frac{\partial}{\partial t}\{[\rho_{0}v_{n}+\rho(u_{n}-v_{ n})]\Delta(f)\}\end{array} \tag{8}\]  
where \(u_{i}\) is the fluid velocity component in direction \(x_{i}\), \(u_{n}\) is the fluid velocity component perpendicular to the object surface, \(v_{i}\) is the object surface velocity component in direction \(x_{i}\), \(v_{n}\) is the object surface normal velocity component, \(\Delta(f)\) is the Dirac delta function, \(H(f)\) is the Heaviside function and \(T_{ij}\) is the Lighthill stress tensor.  
In this work, FLUENT is used for geometrical modelling and structured meshing. The flow field calculation is the most computationally intensive step in the whole calculation process. It is stabilized by the condition that the main flow trends are fully developed and the statistical values of the time-varying flow variables are constant. In the calculation of the acoustic signal, several source surfaces need to be selected simultaneously to investigate their contribution to the sound field.