## 2 Numerical Simulation
### Governing Equations and Boundary Conditions  
In the present work, we aim to investigate the turbulent heat transfer in a circular tube, prescribing a range in the Reynolds number of 5000-35,000. To simplify the numerical model, the following assumptions are made:  
1. The physical properties of the fluid are constant; the computational region is \(L_{0}\) = 880 mm, with an upstream section of \(L_{1}\) = 210 mm and a downstream section of \(L_{2}\) = 210 mm. A series of important geometric parameters employed in the numerical simulation are listed in Table 2, where \(D_{8}\) and \(D_{\rm b}\) respectively represent the diameter of the small and big balls, \(BDR\) is the diameter ratio of the small ball to the big ball, and \(S\) is the spacer length between them. Water is used as the working fluid. The main focus of this work was concerned with the effects of two geometric parameters, i.e., different ball diameter ratios (\(BDR\) = 0.5, 0.75 and 1) and spacer lengths (\(S\) = 40, 51.77 and 62.5 mm) under a constant condition of wall heat flux. Furthermore, the influence of grid generation on the prediction results is also reported. 2. The fluid is turbulent, steady, and incompressible. 3. The effect of gravity, natural convection, and thermal radiation are negligible. 4. The thin rod, keeping the entire insert in the axial position is also neglected. Equations of continuity, momentum, and energy are given below in a tensor form: Continuity equation: \[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{i})}{\partial x_{i}}= 0.\] (1) Momentum equation: \[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial\big{(}\rho u_{i}u_{j} \big{)}}{\partial x_{j}}=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\bigg{[}\mu\bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\bigg{)}-\rho\overline{u^{\prime}_{i}u^{\prime }_{j}}\bigg{]}\] (2) \[-\rho\overline{u^{\prime}_{i}u^{\prime}_{j}}=\mu_{t}\bigg{(}\frac{\partial u _{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\bigg{)}-\frac{2}{ 3}k\delta_{ij}.\] (3) Energy equation: \[\frac{\partial}{\partial t}(\rho E)+\frac{\partial}{\partial x_{i}}[\mu_{i}( \rho E+\rho)]=\frac{\partial}{\partial x_{i}}\bigg{[}k_{e\!t\!t}\frac{\partial T }{\partial x_{i}}\bigg{]}.\] (4)  
Here, the shear stress transport (SST) \(k\)-\(\omega\) turbulence model is used, which is suited for the thermal-hydraulic calculation, particularly for the cases where there is a stream separation. The main feature of the SST model is its ability to take into account a viscous sublayer by applying the \(k-\omega\) model near the wall and to use the standard \(k-\varepsilon\) model in the turbulent core. This flow pattern exists in the tested inserts. [PERSON] and [PERSON] made a detailed analysis of this model in References [21; 23]. The SST model includes two transport equations:  
Turbulence kinetic energy:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho u_{i}k)}{\partial x_{ i}}=\widetilde{P}_{k}-\beta^{*}\rho\omega k+\frac{\partial}{\partial x_{i}} \bigg{[}(\mu+\sigma_{k}\mu_{t})\frac{\partial k}{\partial x_{i}}\bigg{]}. \tag{5}\]  
Specific dissipation rate:  
\[ \begin{array}{c}\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho u _{i}\omega)}{\partial x_{i}}=\frac{\alpha}{v_{i}}\widetilde{P}_{k}-\beta\rho \omega^{2}+\frac{\partial}{\partial x_{i}}\Big{[}(\mu+\sigma_{\omega}\mu_{t}) \frac{\partial\omega}{\partial x_{i}}\Big{]}\\ +2\rho(1-1)\sigma_{\omega}\frac{1}{\partial x}\frac{\partial}{\partial x_{i}} \frac{\partial\omega}{\partial x_{i}}\end{array}