## 2 Numerical Calculation Methodology
### Dissipation Loss Theory  
Tip leakage flow leads to dissipation losses within the impeller. To clarify the relationship between the tip leakage flow field and dissipation loss, the entropy production theory is introduced to evaluate the energy loss caused by fluid dissipation. [PERSON] and [PERSON] presented the entropy production formula for turbulent flow [25; 26; 27]. This study neglects the losses caused by heat transfer and considers only the losses generated by fluid dissipation. Therefore, the entropy production can be calculated as follows:  
\[\rho\bigg{(}\frac{\partial s}{\partial t}+u_{1}\frac{\partial s}{\partial x_{1 }}+u_{2}\frac{\partial s}{\partial x_{2}}+u_{3}\frac{\partial s}{\partial x_{ 3}}\bigg{)}=-\text{div}\left(\frac{\overset{\rightarrow}{q}}{T}\right)+\frac{ \Phi}{T} \tag{6}\]  
where \(\rho\) is the air density, kg/m\({}^{3}\); \(u_{1}\), \(u_{2}\), and \(u_{3}\) are the velocity components, m/s; \(T\) is the thermodynamic temperature, K; \(\overset{\rightarrow}{q}\) is the heat flux density vector, W/m\({}^{2}\); \(s\) is the entropy, J/(kg\(\cdot\)K); \(x_{1}\), \(x_{2}\), and \(x_{3}\) are the coordinate components, m; and \(\Phi\) is the dissipation function.  
For incompressible fluids, the dissipation loss can be simplified as follows:  
\[\dot{N}_{\text{EP,D}}=\Phi=2\mu\bigg{[}\bigg{(}\frac{\partial u_{1}}{\partial x _{1}}\bigg{)}^{2}+\bigg{(}\frac{\partial u_{2}}{\partial x_{2}}\bigg{)}^{2}+ \bigg{(}\frac{\partial u_{3}}{\partial x_{3}}\bigg{)}^{2}\bigg{]}+\mu\bigg{[} \bigg{(}\frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial u_{2}}{\partial x _{1}}\bigg{)}^{2}+\bigg{(}\frac{\partial u_{1}}{\partial x_{3}}+\frac{ \partial u_{3}}{\partial x_{1}}\bigg{)}^{2}+\bigg{(}\frac{\partial u_{3}}{ \partial x_{2}}+\frac{\partial u_{2}}{\partial x_{3}}\bigg{)}^{2}\bigg{]} \tag{7}\]  
where \(\dot{N}_{\text{EP,D}}\) is the unit volume energy loss, W/m\({}^{3}\).  
Figure 3: Comparison of fan efficiency at different grid nodes.  
Figure 2: Computational domain mesh of contra-rotating axial fan.  
In this study, the Reynolds-averaged Navier-Stokes equations are employed to solve the turbulent flow. The energy loss is divided into direct dissipation \(\dot{N}_{\text{EP,D}}\) and turbulent dissipation \(\dot{N}_{\text{EP,D}^{\prime}}\):  
\[\dot{N}_{\text{EP,D}}=\dot{N}_{\text{EP,D}}+\dot{N}_{\text{EP,D}^{\prime}} \tag{8}\]  
\[\dot{N}_{\text{EP,D}}=\mu\left[\left(\frac{\partial\overline{u}_{1}}{\partial x _{2}}+\frac{\partial\overline{u}_{2}}{\partial x_{1}}\right)^{2}+\left(\frac{ \partial\overline{u}_{1}}{\partial x_{3}}+\frac{\partial\overline{u}_{3}}{ \partial x_{1}}\right)^{2}+\left(\frac{\partial\overline{u}_{3}}{\partial x_ {2}}+\frac{\partial\overline{u}_{2}}{\partial x_{3}}\right)^{2}\right]+2\mu \left[\left(\frac{\partial\overline{u}_{1}}{\partial x_{1}}\right)^{2}+\left( \frac{\partial\overline{u}_{2}}{\partial x_{2}}\right)^{2}+\left(\frac{ \partial\overline{u}_{3}}{\partial x_{3}}\right)^{2}\right] \tag{9}\]  
\[\dot{N}_{\text{EP,D}^{\prime}}=\beta\rho\omega k \tag{10}\]  
where \(\dot{N}_{\text{EP,D}}\) is the unit volume energy loss induced by direct dissipation, \(\text{W}/\text{m}^{3}\); \(\dot{N}_{\text{EP,D}^{\prime}}\) is the energy loss unit volume induced by turbulent dissipation, \(\text{W}/\text{m}^{3}\); \(\overline{u}_{1}\), \(\overline{u}_{2}\), and \(\overline{u}_{3}\) are the time-average velocity component, \(\text{m}/\text{s}\); and \(\beta=0.09\).