Flow through pipes is a central problem in fluid mechanics and has major industrial relevance, particularly for oil transportation systems. Understanding velocity distributions, pressure loss, and vorticity dynamics is important for improving flow efficiency and anticipating transition to turbulence.

This study investigates axisymmetric incompressible flow in two Sudanese pipeline systems: Greater Nile (GN) and Petrodar (PD). Two complementary numerical approaches are used:

1)Simplified simulated flow profiles, where velocity, pressure, and vorticity are generated using FDM, FVM, and FEM to compare numerical stability and convergence through SOR residual analysis.

2)A full Navier-Stokes solver in axisymmetric coordinates, which provides a validated physical solution of steady axial velocity and vorticity fields.

The numerical results are validated using the analytical Poiseuille solution for fully developed laminar pipe flow.

Incompressible flow in pipes has been a key subject in fluid mechanics because of its theoretical importance and real-world uses, such as water supply and oil pipelines in Sudan [1][2]. Reynolds’ classical experiments determined the critical Reynolds number that separates laminar from turbulent flow, offering essential insights into internal flows [3]. In laminar flows, analytical solutions like the Poiseuille profile illustrate a parabolic velocity distribution, acting as a standard for verifying computational techniques [4][5]. The numerical simulation of incompressible flows depends significantly on discretization techniques like Finite Difference (FDM), Finite Volume (FVM), and Finite Element Methods (FEM). FDM, among the first techniques, estimates derivatives through Taylor series expansions and works well on structured grids, but has difficulties with intricate geometries [6]. FVM maintains conservation principles at the control volume level, ensuring its reliability and extensive usage in industrial CFD applications [7]. FEM offers adaptability for intricate geometries and boundary conditions; however, it frequently necessitates stabilization techniques like Streamline Upwind Petrov-Galerkin (SUPG) or Variational Multiscale (VMS) approaches to ensure numerical stability in convection-dominated flows [8][9].

The governing equations for incompressible flows are the Navier–Stokes equations, which, when averaged over time for turbulent conditions, yield the Reynolds-Averaged Navier-Stokes (RANS) equations [10]. The closure of RANS equations necessitates turbulence modeling, and the k-ε and k-ω models are commonly used for simulating turbulent flow in pipes [11][12]. High-fidelity methods, like Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS), capture more detailed turbulent structures but entail much greater computational expense [13][14].

Pressure-velocity coupling methods are essential for solvers dealing with incompressible flow. The SIMPLE (Semi-Implicit Method for Pressure Linked Equations) and PISO (Pressure Implicit with Splitting of Operators) methods guarantee mass conservation during the iterative solution of momentum equations [15][16]. Iterative methods like Successive Over-Relaxation (SOR), Conjugate Gradient (CG), and Multigrid techniques are commonly employed for effectively solving the resulting linear systems [17][18].

Multiple studies have confirmed numerical methods by comparing them to analytical solutions for laminar flow. The parabolic Poiseuille profile, for instance, has been widely employed to check the precision of FDM, FVM, and FEM discretization [4][19]. Residual convergence analysis shows that FVM and FEM typically reach faster and more stable convergence than FDM because of their fundamental conservation characteristics [20][21].

Modeling turbulent flow in pipes has been the subject of many studies. Studies, both experimental and computational fluid dynamics (CFD), indicate that velocity profiles become flatter at elevated Reynolds numbers, while vorticity accumulates near the boundary because of sharp velocity gradients [22][23]. Numerical analyses indicate that the choice of solver, grid resolution, and relaxation settings greatly affect convergence and precision [24][25]. The integration of turbulence modeling with high-resolution grids is crucial for accurately capturing near-wall effects and pressure loss [26][27]. Recent uses of CFD in pipeline systems highlight practical design and operational factors. Simulations of multiphase and slurry flows in pipelines illustrate the necessity for strong numerical methods to manage intricate interactions among phases [28][29]. Additionally, research on Sudanese pipelines emphasizes the necessity of dependable computational models for forecasting pressure drops and flow allocation in oil and water transportation systems [30].

In summary, the literature emphasizes the importance of integrating analytical solutions, turbulence modeling, and numerical method evaluation to obtain accurate simulations of pipe flow. This research expands on these principles by contrasting simulated turbulent flow with a proven Navier–Stokes solver and assessing the effectiveness of FDM, FVM, and FEM related to Sudanese pipeline flows.

The simplified simulations provide fast qualitative understanding of velocity, pressure, and vorticity patterns and allow numerical comparison of FDM, FVM, and FEM. The full Navier–Stokes solver provides a physically accurate steady-state solution and reproduces the analytical Poiseuille flow profile, validating numerical stability and correctness.

The observed differences between GN and PD pipelines are explained by differences in viscosity and pressure gradient, which influence peak velocity, velocity profile shape, and vorticity magnitude.

The main limitation is that axial diffusion and convective acceleration were neglected, which is appropriate for fully developed laminar/transitional assumptions but not for fully turbulent three-dimensional flow.

Both the simplified turbulent-like simulations and the full Navier-Stokes solver successfully represent pipeline flow in GN and PD systems.GN shows higher velocities and reduced viscous effects compared with PD.FVM achieves faster residual convergence in SOR iterations compared with FDM and FEM.The Navier-Stokes solver reproduces the analytical Poiseuille solution, confirming validation.The study provides a practical computational framework for pipeline flow modelling in Sudanese systems.

1) Extend the work to full 3D turbulent Navier–Stokes simulations to capture real turbulence effects.

2) Include temperature-dependent viscosity and density for more realistic pipeline conditions.

3) Use adaptive mesh refinement near the wall to better resolve shear and vorticity.

4) Apply the solver to optimization studies targeting pressure-loss reduction and energy efficiency.

5) Compare with experimental pipeline measurements for further validation.

The authors acknowledge the staff of Greater Nile Petroleum pipelines and Petrodar for providing operational data used in this study. Their collaboration supported validation of the numerical models and improved the practical relevance of the results.