## 5 Numerical Findings
Figure 5d shows that the tape with a smaller attack angle, \(\beta\) = 15\({}^{\circ}\), has the most complex distribution and irregular vector distribution. The wall displays streams of swirling flow, while the continuous V-rib twisted tapes, positioned behind the tube core, induce longitudinal vortex flow. The CVRTs induced additional swirling flow near the ribs. Swirling and longitudinal vortex flows promote wall and core flow zone mixing. The vortex structure is dense and evenly distributed, while the boundary layer is periodically broken and reattached. The temperature distribution (Figure 6) is consistent with the TKE region, which verifies the effect of continuous V-ribs on turbulent disturbance with thermal boundary layer destruction and air mixing. Figure 4: Effect of V-rib attack angle (\(\beta\)) and tape arrangement on flow structure at \(Re\) = 10,000. (**a**) Typical twisted tape (TT). (**b**) Forward arrangement, CVRT, \(\beta\) = 45\({}^{\circ}\). (**c**) Backward arrangement, CVRT, \(\beta\) = 45\({}^{\circ}\). (**d**) Backward arrangement, CVRT, \(\beta\) = 15\({}^{\circ}\). Figure 5: Effect of V-rib angle (\(\beta\)) and tape arrangement on the flow field and TKE at \(Re\) = 10,000 and \(x/D\) = 2.0. (**a**) TT. (**b**) Forward arrangement, CVRT, \(\beta\) = 45\({}^{\circ}\). (**c**) Backward arrangement, CVRT, \(\beta\) = 45\({}^{\circ}\). (**d**) Backward arrangement, CVRT, \(\beta\) = 15\({}^{\circ}\). Figure 7a-d present the contour plots of streamlines and temperature fields for tubes fitted with a typical twisted tape (TT) and continuous V-rib twisted tapes (CVRTs) at various attack angles (\(\beta\)), evaluated at a Reynolds number of 10,000. The temperature distribution for the TT case reveals two distinct regions of extremely high and low temperatures, indicating poor thermal mixing and limited disruption of the thermal boundary layer. This non-uniformity suggests that the TT generates relatively weak secondary flow structures, which are insufficient to enhance heat transfer effectively across the tube's cross-section. In contrast, the CVRT configurations exhibit significantly more uniform temperature fields and thinner thermal boundary layers along the tube wall. The presence of continuous V-shaped ribs promotes stronger secondary flows and more vigorous fluid mixing. Figure 7: _Cont._  
Figure 8a-d illustrate the local Nusselt number along the tube wall with various continuous V-rib twisted tape inserts. Overall, the Nusselt number distributions in tubes with continuous V-rib twisted tapes exhibit more uniformity compared to those with a TT, as seen in Figure 8a. Swirl and longitudinal flows facilitate enhanced fluid mixing, resulting in more uniform Nusselt number distributions. In Figure 8a, elevated Nusselt numbers are observed at only one location adjacent to the borders of the TT. Nevertheless, the continuous V-rib twisted tape exhibits two or more locations with high Nusselt numbers, especially for the backward-arranged tape, which had two of these locations at the center of the continuous V-rib, as illustrated in Figure 8b-d. This is attributed to the greater protrusions of the CVRTs. Continuous V-rib twisted tapes generate enhanced turbulence, resulting in better heat transfer. These twisted tapes at all attack angles (\(\beta=15^{\circ},30^{\circ}\), and \(45^{\circ}\)) have superior heat transfer compared to a TT, especially at the lowest attack angle (\(\beta=15^{\circ}\)). Figure 7: Contour plots of streamlines and temperature fields for tubes equipped with typical twisted tape and V-rib twisted tapes at various attack angles (\(\beta\)) and arrangements under a Reynolds number of 10,000. (**a**) TT. (**b**) Forward arrangement, CVRT, \(\beta=45^{\circ}\). (**c**) Backward arrangement, CVRT, \(\beta=45^{\circ}\). (**d**) Backward arrangement, CVRT, \(\beta=15^{\circ}\).