## 5 Numerical Findings  
Before the simulation of tubes equipped with continuous V-rib twisted tapes, the numerical results were verified by comparing the current findings (Nusselt number and friction factor) for the plain tube with established correlations found in the literature by [PERSON] et al. [30]. The validation was conducted for Reynolds numbers (\(Re\)) of 10,000 in turbulent flow conditions. In the current simulation, the Nusselt numbers diverge from the Dittus-Boelter correlation by \(\pm\)6.8%, while the numerical friction factor deviates from the Blasius correlation by \(\pm\)8.9%, respectively. This study also evaluates the reliability of the renormalized group (RNG) \(k\)-\(\epsilon\) turbulence model by comparing numerical findings from a tube with a classical twisted tape to previous experimental data [31]. It can be found that the current numerical Nusselt number and friction factor deviate from previous data by \(\pm\)12.1% and \(\pm\)14.7%, respectively. The flow structures within the tubes, including the continuous V-rib twisted tapes, are demonstrated to raise understanding of the heat transfer mechanisms. Figure 4a-d present the flow structures. All twisted tape inserts, on both the left and right sides of the tape, clearly display axial swirl flows. The boundary layer in the typical twisted tape (TT) configuration is considerably thicker than that in the continuous V-rib twisted tape (CVRT), which contributes to a larger flow separation zone. The continuous V-rib induces longitudinal vortex flows by deflecting the mainstream fluid along its inclined surfaces to both sides, generating a symmetrical secondary flow that enhances fluid mixing and heat transfer. Compared to a conventional twisted tape (TT), the continuous V-rib twisted tape exhibits distinct thermal boundary layer behavior. On the protrusion side of the V-rib, the thermal boundary layer is notably thinner, whereas it becomes thicker on the depression side. These alternating protrusion and depression features enhance flow disturbances, promoting more vigorous fluid motion. The continuous V-ribs generate multiple vortices that form a dense reattachment zone, which significantly enhances fluid mixing and improves near-wall heat transfer performance. Backward V-ribs create a secondary flow along the flow direction and better enhance fluid mixing. Forward V-ribs create a forward vortex flow, which produces a different mixing effect. For a large 45\({}^{\circ}\) attack angle, the flow separation is gentle, and the vortex scale is large, but the intensity is weak. As the attack angle decreases, flow separation and reattachment are balanced. The boundary layer becomes increasingly thinner. The vortex structure is denser, the turbulence intensity is significantly enhanced, and the boundary layer is periodically shed, forming a region of high turbulent kinetic energy. The continuous V-rib twisted tapes create both swirling and longitudinal vortex flows, which improve fluid mixing, especially at a smaller attack angle (\(\beta\) = 15\({}^{\circ}\)) arranged backward. This is because they have the largest blockage area, which creates the strongest longitudinal vortex flows. Figures 5 and 6 demonstrate the contours of velocity vectors, TKE, and fluid temperature within a round tube containing twisted tape inserts. As displayed in Figure 5a, a typical twisted tape insert provides relatively low and symmetrical TKE distribution. The flow is smooth, no significant vortex structure can be found, and the boundary layer is thick. The continuous V-rib twisted tape inserts exhibit higher TKE with a distribution mode, which is accompanied by significantly alternating and disordered vector directions. The phenomenon indicates that the V-ribs increase flow complexity. Figure 5b,c illustrate the differences between the forward and backward arrangement. The backward-arranged tape creates more disturbance.