*2.3. Governing Equations*

No. 25# transformer oil is used as the working fluid, due to its good viscosity temperature property. Considering that the temperature change in the whole fluid domain is very small, the viscosity of oil is assumed to be constant. As pointed out by Chai et al. [13], the standard *k* − *ε* turbulence model is the most widely used and validated model, so the standard *k* − *ε* turbulence model is used as the mathematical model for numerical simulation, the near-wall treatment is enhanced wall treatment, and the governing equations include continuity equation, momentum equation, and energy equation, which are expressed as follows:

Continuity equation:

$$\frac{\partial}{\partial \mathbf{x}\_i} \left( \rho\_o \mathbf{u}\_i \right) = 0 \tag{1}$$

Momentum equation:

$$\frac{\partial}{\partial t}(\rho\_o u\_i) + \frac{\partial}{\partial \mathbf{x}\_w}(\rho\_o u\_i u\_w) = \frac{\partial}{\partial \mathbf{x}\_w}(\mu \frac{\lambda\_o \partial u\_i}{\partial \mathbf{x}\_w} - \rho\_o \overline{u\_i' u\_w'}) - \frac{\partial p}{\partial \mathbf{x}\_i} \tag{2}$$

Energy equation:

$$\frac{\partial \mathbf{x}}{\partial t}(\rho\_o T) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho\_o u\_i T) = \frac{\partial}{\partial \mathbf{x}\_i}(\frac{\lambda\_o}{\mathbb{C}\_p} \frac{\partial T}{\partial \mathbf{x}\_i} - \rho\_o \overline{u\_i' T'}) \tag{3}$$

*k* equation:

$$\frac{\partial}{\partial t}(\rho\_o k) + \frac{\partial}{\partial \mathbf{x}\_l}(\rho\_o k u\_i) = \frac{\partial}{\partial \mathbf{x}\_w} \left[ \left( \mu + \frac{\mu\_l}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_w} \right] + G - \rho\_o \varepsilon \tag{4}$$

*ε* equation:

$$\frac{\partial}{\partial t}(\rho\_o \varepsilon) + \frac{\partial}{\partial \mathbf{x}\_l}(\rho\_o \varepsilon u\_l) = \frac{\partial}{\partial \mathbf{x}\_{\mathcal{U}}} \left[ \left( \mu + \frac{\mu\_l}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_{\mathcal{w}}} \right] + \frac{\mathbf{C}\_1 \varepsilon}{k} G - \mathbf{C}\_2 \rho\_o \frac{\varepsilon^2}{k} \tag{5}$$
