Numerical Investigation of Two-Phase Flow-Induced Forces in a U-Bend Pipe: A Third-Order Response Surface of Flow Regime Dependence

Two-phase flow induces a spectrum of forces in pipe structures significantly impacting the structural integrity and operational performance of piping systems. Accurate prediction of these forces remains challenging due to varying gas–liquid-phase distributions across existing flow patterns. This study aims to predict the forces induced by two-phase flow across stratified, wavy, slug, elongated bubble, annular, and dispersed bubble flow regimes. An experimentally validated numerical model was utilized to achieve a total of 135 induced force signals for varying gas and liquid superficial velocities. The Root Mean Square (RMS) force value of each signal was calculated using Simpson's 1/3 rule (third-order accurate) for precise evaluation of both hydrodynamic force magnitude and their fluctuating energy content. The article proposes 7 third-order deterministic models as explicit functions of phasal superficial velocities, predicting flow regime-specific induced forces. The polynomial models demonstrated high predictive accuracy by achieving maximum Root Mean Square Error (RMSE) of 4.7% and minimum R² of 0.95 across all six flow regimes. This work offers a data-driven approach for predicting two-phase force spectra essential for structural integrity assessment.