## 4 Numerical Method
### Procedure for Resolution  
In this study, a finite volume technique was used to discretize the steady-state incompressible Navier-Stokes equations, together with a turbulent model, which would achieve stable and reliable solutions. The convective factors were addressed with the QUICK technique to accurately characterize heat and momentum transfer, whilst the diffusive terms were estimated using central differencing to depict viscous and thermal diffusion effects. To enhance numerical stability, the resultant nonlinear equations were solved implicitly. Furthermore, the SIMPLE algorithm was performed to integrate pressure/velocity fields, guaranteeing precise pressure resolution and mass conservation. The tube surface was reinforced with impermeability and no-slip conditions, while the lower wall of the test area was exposed to an isothermal flux to simulate regulated heating. To restrict heat transfer solely within the test section, both the external tube surface and the transverse twisted-tape surfaces were thermally insulated. Fluid flow periodicity was simulated by applying a periodic boundary condition to emulate cyclical flow characteristics. Under-relaxation coefficients of 0.4 and 0.6 were applied to the pressure and momentum equations, respectively, to maintain a stable iterative solution procedure. Convergence thresholds required the residuals of the mass, momentum, and energy conservation equations to be below 10\({}^{-6}\), guaranteeing precise adherence to fundamental physical conservation principles. The numerical simulations were performed at a Reynolds number of 10,000, defined based on the inlet flow conditions. The following assumptions were applied:  
* A constant mass flow rate was used to maintain periodic flow conditions at each Reynolds number. * The thermophysical properties of air were assumed constant and evaluated at the bulk mean temperature. * A constant wall heat flux of 1000 W/m\({}^{2}\) was applied to the tube surface. * The continuous V-rib twisted tapes were assumed to be adiabatic, modeled by imposing a high thermal resistance at their surfaces. In this study, the Reynolds number is used to characterize the flow regime, which is determined by the mass flow rate at the inlet and outlet of the test section. The average inlet velocity is 0.4748 m/s, corresponding to a Reynolds number (\(Re\)) of 10,000, indicating fully developed, single-phase turbulent flow within the tube. In the absence of inserts, the flow inside the tube is purely axial. When a continuous V-rib twisted tape insert is introduced, the flow becomes more complex, transitioning from axial to swirling due to the tape-induced flow modification. For simplicity, the tape thickness is assumed to be negligible. The inlet temperature, working fluid, and thermal boundary conditions are maintained consistently for both with and without the tape insert. A constant wall heat flux of 1000 W/m\({}^{2}\) is applied as the boundary condition on the tube surface. The interaction between the fluid (air) and the continuous V-rib twisted tape is modeled using a fluid-structure interaction boundary condition. The presence of the twisted tape significantly alters the velocity and thermal boundary layers, leading to enhanced heat transfer performance. These changes arise from the generation of swirl flow, which results in a distinct velocity profile compared to the plain tube under identical conditions. To evaluate the effects of this modified flow on heat transfer and friction factor, the plain tube (without inserts) is used as a reference case. In this study, fluid flow behavior is governed by pressure differences, as the flow is pressure-driven. Since the precise pressure values are not critical in this context, the inlet boundary condition is set to zero gauge pressure. The outlet pressure is defined as atmospheric pressure (\(P=1.013\times 10^{5}\) Pa). The pressure drop can be estimated from the difference between the inlet and outlet pressures.