Three-dimensional navier–stokes formulation, finite element discretization, stability analysis, and validation for crude oil pipeline flow

Sarah Khaindi Wandabwa; David Angwenyi; Frankline Tireito

doi:10.51867/asarev.maths.3.1.21

Auteurs-es

Sarah Khaindi Wandabwa Department of Mathematics, Masinde Muliro University of Science and Technology, Kakamega, Kenya Auteur-e
David Angwenyi Department of Mathematics, Masinde Muliro University of Science and Technology, Kakamega, Kenya Auteur-e https://orcid.org/0000-0002-6958-2817
Frankline Tireito Department of Mathematics, Masinde Muliro University of Science and Technology, Kakamega, Kenya Auteur-e https://orcid.org/0000-0002-4106-4022

DOI :

https://doi.org/10.51867/asarev.maths.3.1.21

Mots-clés :

Navier–Stokes equations, finite element method, crude oil pipeline, Taylor–Hood elements, energy stability, $k$-$\epsilon$ turbulence model

Résumé

This paper presents a comprehensive formulation of the three-dimensional Navier–Stokes equations for viscous, incompressible crude oil flow in pipelines, including finite element discretization, energy stability analysis, and thorough validation. The governing equations are derived from first principles using the Reynolds transport theorem. A Newtonian constitutive relation appropriate for light to medium crude oils is adopted. The weak formulation is obtained using the Galerkin method, while the finite element discretization employs Taylor–Hood P2–P1 elements that satisfy the discrete inf–sup condition. The Crank–Nicolson scheme is used for temporal integration. For turbulent flows, the standard k–ε model is incorporated. Energy stability is established for both the continuous and semidiscrete formulations. The model is validated against the analytical Hagen–Poiseuille solution for laminar flow, achieving maximum relative errors below 0.05% and a root mean square error of 8.2 × 10⁻⁵ m/s. For turbulent flow, validation against Laufer’s experimental data at Re = 50,000 yields a coefficient of determination of R² = 0.996, with errors below 4% for wall shear stress and the skin friction coefficient. A detailed implementation section covering mesh generation, solver specifications, and computational costs is also provided. These results confirm the accuracy and reliability of the proposed model for predicting velocity profiles, pressure drops, and turbulence characteristics in pipeline flows.

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