## 2 Numerical Model
### Hydrodynamic Model of Corona Discharge
\[\
abla^{2}S_{ph}^{j}(r)-(\lambda_{j}P_{\text{O}_{2}})^{2}S_{ph}^{j}(r)=-A_{j}P _{\text{O}_{2}}^{2}I(r)\ \ \ j=1,2,3 \tag{5}\]  
\[S_{ph}(r)=\sum_{j=1}^{3}S_{ph}^{j}(r) \tag{6}\]  
where \(P_{O2}\) is the partial pressure of molecular oxygen; \(I(r)\) is the number of photons emitted from the radiation domain per unit time; \(\lambda_{j}\) and \(A_{j}\) are the fitting coefficients of the absorption function, and the subscript \(j\) refers to the fitting index. When positive ions strike the cathode, new electrons will be generated on the cathode surface, which is called the secondary electron emission. The related research shows that the secondary electron emission is the fundamental reason for generating continuous pulses, and the re-excited new electrons can be calculated as [21]  
\[N_{es}=\gamma N_{ps}\frac{\left|\mu_{p}E\right|}{\left|\mu_{e}E\right|} \tag{7}\]  
where \(\gamma\) is the secondary electron emission coefficient, taking as 0.04; \(N_{\text{s}}\) and \(N_{ps}\) are the densities of electrons and positive ions on the cathode surface, respectively. The ions in the discharge gap move directionally under the action of the electric field, which will form the current. The process of corona discharge follows the energy conservation, and the external input power is equal to the work performed by all ions in the whole discharge area per unit time. The corona current is mainly contributed by ions migration, and the displacement current generated by the capacitive behavior of the pin-plate geometry can be ignored, so the current is expressed as [22]  
\[I=\frac{e}{U}2\pi\iint_{s}r(\mu_{p}N_{p}-\mu_{e}N_{e}-\mu_{n}N_{n})\mathbf{E}\cdot E _{L}drdz \tag{8}\]  
where \(r\) and \(z\) are the radial and axial coordinates of the simulation domain, respectively; \(U\) is the applied voltage; \(E\) is the electric field; and \(E_{L}\) is the Laplace electric field, namely with no space charge. The corona model is simulated using a 2D axisymmetric geometry model, and the geometry size of this model is as follows: the length of the pin electrode is 3 mm, the radius of the pin tip curvature is 0.1 mm, the distance between the pin and plate electrode is 7 mm, and the simulation domain is 6 \(\times\) 10 mm\({}^{2}\). Meanwhile, when the corona field is discretized, the quality of mesh directly affects the stability of the model. In this paper, the Delaunay triangulation algorithm is used with triangular mesh elements. The electric field near the discharge channel varies violently, the fine mesh is used, and the maximum size is \(8\times 10^{-3}\) mm, with a maximum element growth rate of 1.02. There are small changes of the electric field in other places, so the coarse grid is used to reduce the computational cost. The average element quality is 0.94, and the mesh result of the computational domain is shown in Figure 2.  
The Poisson equation and Helmholtz equation of photoionization are solved discretely by the Galerkin FEM format. The continuity equations of three charged ions are solved by the position-state separation (POSS) method proposed by [PERSON] et al. [23,24]. The characteristic of POSS is that the continuity equation is decomposed into a position equation and state equation, and appropriate methods are used to solve them respectively. Compared with the traditional flux-corrected transport (FCT) method [25], POSS can improve computational efficiency at the premise of stability. Researchers have found that its time complexity increases almost linearly with the number of nodes when linear interpolation is used, and POSS owns better adaptability when dealing with irregular models. The corona model consists of five boundaries, including an inner boundary and four outer boundaries. The inner boundary is the axis of symmetry, and the outer boundaries are the pin electrode, plate electrode, and two artificial boundaries. Due to the fact that audible noise mainly comes from the positive corona, this paper mainly focuses on the research of the positive corona.