## 2 Establishment of the Fluid-Solid Coupling Structure for DD-PMSG
### Mathematical Modeling  
The flow of the cooling medium and heat transfer within the generator must satisfy the physical conservation laws, namely the conservation of mass, the conservation of momentum, and the conservation of energy [21]. 1. Conservation of mass: \[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{ \partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0\] (1) where \(\rho\) represents the density (kg/m\({}^{3}\)); \(t\) represents time (s); and \(u\), \(v\), and \(w\) are the fluid velocity components in the \(x\), \(y\), and \(z\) directions, respectively. 2. Conservation of energy: \[\frac{\partial(\rho T)}{\partial t}+\frac{\partial(\rho uT)}{\partial x}+\frac {\partial(\rho vT)}{\partial y}+\frac{\partial(\rho wT)}{\partial z}=\frac{ \partial}{\partial x}\bigg{(}\frac{\lambda}{C_{\rm p}}\frac{\partial T}{ \partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(}\frac{\lambda}{C_{\rm p }}\frac{\partial T}{\partial y}\bigg{)}+\frac{\partial}{\partial z}\bigg{(} \frac{\lambda}{C_{\rm p}}\frac{\partial T}{\partial z}\bigg{)}+S_{\rm T}\] (2) where \(T\) represents the temperature (\({}^{\circ}\)C); \(\lambda\) represents the thermal conductivity of the fluid (W/(m\(\cdot\)K)); \(C_{\rm p}\) represents the specific heat capacity (J/(kg\({}^{\circ}\)C)); and \(S_{\rm T}\) represents the heat generated by internal heat sources within the fluid and the conversion of fluid mechanical energy into thermal energy (W). 3. Conservation of momentum: \[\left\{ \begin{array}{l}\frac{\partial(\rho u)}{\partial t}+\frac{\partial( \rho uu)}{\partial x}+\frac{\partial(\rho uv)}{\partial y}+\frac{\partial( \rho uv)}{\partial z}=-\frac{\partial p}{\partial x}+\frac{\partial x_{\rm xx }}{\partial x}+\frac{\partial y_{\rm xx}}{\partial y}+\frac{\partial x_{\rm xx }}{\partial z}+F_{\rm x}\\ \frac{\partial(\rho v)}{\partial t}+\frac{\partial(\rho vu)}{\partial x}+\frac {\partial(\rho vv)}{\partial y}+\frac{\partial(\rho vw)}{\partial z}=-\frac{ \partial p}{\partial y}+\frac{\partial x_{\rm yy}}{\partial x}+\frac{ \partial y_{\rm yy}}{\partial y}+\frac{\partial x_{\rm yy}}{\partial z}+F_{ \rm y}\\ \frac{\partial(\rho vt)}{\partial t}+\frac{\partial(\rho wt)}{\partial x}+\frac {\partial(\rho wt)}{\partial y}+\frac{\partial(\rho wt)}{\partial z}=-\frac {\partial p}{\partial z}+\frac{\partial x_{\rm yy}}{\partial x}+\frac{ \partial y_{\rm zz}}{\partial y}+\frac{\partial x_{\rm zz}}{\partial z}+F_{ \rm z}\end{array} \right.\] (3) where \(p\) represents the pressure on the fluid element; \(\tau_{\rm xx}\), \(\tau_{\rm yy}\), and \(\tau_{\rm xx}\) represent the viscous stress components on the surface of the fluid element in the \(x\), y, and z directions, respectively; and \(F_{\rm x}\), \(F_{\rm y}\), and \(F_{\rm z}\) represent the body force components acting on the fluid element in the \(x\), \(y\), and \(z\) directions (\(n\)), respectively. \[\left\{ \begin{array}{l}\frac{\partial}{\partial x}\bigg{(}\lambda_{x}\frac{ \partial T}{\partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(}\lambda_{y} \frac{\partial T}{\partial y}\bigg{)}+\frac{\partial}{\partial z}\bigg{(} \lambda_{z}\frac{\partial T}{\partial z}\bigg{)}=-q\\ \frac{\partial T}{\partial t}=0\\ \lambda\frac{\partial T}{\partial t}=-\alpha(T-T_{\rm F})\end{array} \right. \tag{4}\]  
Figure 2: Schematic diagram of internal ventilation system flow of proposed generator. where \(T\) represents the temperatures of the generator's respective components (\({}^{\circ}\)C); \(\lambda_{\rm{x}}\), \(\lambda_{\rm{y}}\), and \(\lambda_{\rm{z}}\) represent the thermal conductivities of the materials of the solid components in the x, y, and z directions (W/(m\(\cdot\)K)); \(q\) represents the heat source density within the solid components of the generator (W/m\({}^{3}\)); \(\alpha\) represents the heat transfer coefficient of the cooling surfaces of the solid components (W/(m\({}^{2}\)-K)); and \(T_{\rm{F}}\) represents the temperature of the fluid surrounding the cooling surfaces of the solid components of the generator (\({}^{\circ}\)C).