## 4 Numerical Method
### Computational Domain  
Figure 3 illustrates the discretization of the computational domain using an unstructured mesh. An enhanced wall treatment model was selected for near-wall modeling, as it relies on a finely resolved near-wall mesh to accurately capture the viscous sublayer. To determine an optimal mesh size for the continuous V-rib twisted tapes (CVRTs), a grid independence study was conducted using Richardson extrapolation on grids with varying cell counts. Polyhedral cells were employed to mesh the surfaces of both the twisted tape and the tube wall, with localized grid refinement applied in the boundary layer regions.  
A grid independence study was performed using the periodic module of the typical twisted tape with a twist ratio (\(y/w\)) of 4.0 at a Reynolds number of 10,000 by using the RNG \(k\)-\(\varepsilon\) turbulence model. Five different grid densities: 52,055; 192,835; 371,245; 520,002; and 721,271 cells were evaluated. The results indicate that the Nusselt number and friction factor  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Element** & **Inlet** & **Outlet** & \
\begin{tabular}{l} **Continuous V-Rib** \\ **Twisted Tape** \\ \end{tabular} & **Tube Wall** \\ \hline \multirow{4}{*}{Boundary conditions} & \(T=298\) K & & & \\  & \(U=0.4748\) m/s & & & \\  & \(\rho=1.225\) kg/m\({}^{3}\) & & & \\  & \(\mu=1.7894\times 10^{-5}\) kg/m\(\cdot\)s & \(P=1.013\times 105\) Pa &
\begin{tabular}{l} Tape thickness is \\ considered zero \\ \end{tabular} & \(q^{\prime\prime}=1000\) W/m\({}^{2}\) \\  & \(k=0.0257\) W/m\(\cdot\)K & & & \\  & \(C_{p}=1006.43\) J/kg\(\cdot\)K & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: The boundary conditions of a tube fitted with continuous V-rib twisted tape.  
Figure 3: Computational domain of a periodic tube module fitted with continuous V-rib twisted tapes.  
exhibited negligible variation when the grid size increased from 520,002 to 721,271 cells. This suggests that the simulation with a grid number of 520,002 generates data that are sufficiently precise and requires less time to execute than the one with a grid number of 721,271. \(Nu\) and \(f\) are estimated to have a deviation of approximately \(\pm\)12.1% and \(\pm\)14.7%, respectively, in comparison to the current experimental data. For the continuous V-rib twisted tapes (CVRTs) with rib angles (\(\beta\)) of 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\), simulations were carried out using grids of 1,075,742; 1,083,549; and 1,057,525 cells, respectively. The predicted values of \(Nu\) and \(f\) were found to be sufficiently accurate. The deviations in \(Nu\) and \(f\) compared to experimental results were \(\pm\)12.13% and \(\pm\)14.71% for \(\beta\) = 15\({}^{\circ}\), \(\pm\)13.39% and \(\pm\)15.59% for \(\beta\) = 30\({}^{\circ}\), and \(\pm\)13.17% and \(\pm\)14.78% for \(\beta\) = 45\({}^{\circ}\), confirming the adequacy of the selected grid resolutions.  
The numerical study focused on tapes with specific geometric parameters. These include (1) a rib depth of 1.5 mm, (2) a twist ratio of 4.0, and (3) attack angles of 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\). Tapes arranged in forward and backward configurations were simultaneously considered. These configurations were selected to systematically investigate the continuous V-rib twisted tape effects on flow turbulence, thermal enhancement, and performance of the system.