Kinetic energy correction factor model for elliptical engineering pipelines under laminar–turbulent transition flow

In the fields of oil and gas, aerospace, and marine engineering, elliptical pipes are utilized for optimizing space usage, reducing aerodynamic drag, and enhancing structural resilience against bending and torsional forces. Predicting and optimizing fluid dynamics within such configurations remains challenging due to complex velocity distributions impacting kinetic energy calculations. This study explores the interaction between the kinetic energy correction factor and velocity distribution in elliptical pipes by uniquely solving the Navier–Stokes equations in an elliptical coordinate system. We begin by deriving transformation relationships between elliptical and Cartesian coordinates and defining gradient and Laplacian operators to express fundamental fluid dynamics quantities such as velocity vectors and pressure distributions. After analyzing the steady-state Navier–Stokes equations, we establish a set of partial differential equations that describe fluid velocity and pressure. Employing the method of separation of variables and linearizing into an eigenvalue problem, we analytically solve these equations. To confirm the validity of our theoretical models, we utilize computational fluid dynamics with the Wray–Agarwal turbulence model, alongside physical experiments with Preston tubes, which corroborate the predicted velocity distributions in elliptical pipes. Comparative analysis with experimental data using the root mean square error (RMSE) and Bland–Altman methods demonstrates that RMSE values are less than 0.05 and all differences fall within the 95% confidence interval. This level of precision significantly improves the reliability of kinetic energy correction factor predictions and supports the design and study of fluid dynamics in elliptical pipes.