This review explores recent advancements in modeling the flow behavior of Herschel-Bulkley (HB) fluids in pipes, discussing theoretical, semi-empirical, computational, and experimental methods. While the laminar flow of non-Newtonian HB fluids can be effectively modeled using first-principle physics, significant challenges remain in turbulent and transitional flow regimes. Existing turbulence models, though widely used, may not always fully align with experimental data, often requiring further validation or complex mathematical tuning, leading to higher computational costs. Further, the transition to turbulence in HB fluids is influenced by shear-thinning and yield stress, yet current models often fail to account for this delayed transition. Consequently, stability and Reynolds number-based transition models can exhibit inconsistencies, limiting their broader applicability. Progress is further hindered by limited experimental studies, constrained by resolution, attenuation, cost, and material combinations. Inaccuracies in rheological modeling—due to inappropriate shear rate ranges, curve-fitting techniques, or simplifying assumptions such as homogeneity and non-elasticity—further complicate flow predictions. Through this review, we delve deeper into the state-of-the-art modeling of HB fluids, highlighting progress and these challenges. Addressing these limitations requires advanced experimental and numerical studies, particularly for near-wall measurements, to better capture flow complexities and improve model predictions. This could also facilitate the development of data-driven approaches and operational envelopes that define their validity thresholds. Future research should also prioritize the independent effects of yield stress and shear-thinning properties while considering material attributes and settling phenomena in non-Newtonian suspensions. Ultimately, these advancements will enable more accurate flow predictions and practical solutions for industrial applications.

Modeling fully developed turbulent flow for Herschel–Bulkley (HB) fluids in pipes is a long-standing challenge. Existing semi-empirical, theoretical, and numerical methods are either inconsistent with experimental data or are validated for low Reynolds numbers. This study focuses on validating a novel approach using rheology-based wall functions within Reynolds-averaged Navier–Stokes solvers. Simulations of wall shear stress and velocity profiles were conducted across a wide range of Reynolds numbers using a single-phase HB fluid, with measurements taken both upstream and downstream of a 90 pipe bend. Two turbulence closure models, the k–e model and the Reynolds stress model, were employed with the wall function implemented as a specified shear boundary condition. Results demonstrate significant improvements over the Newtonian-based models, such as standard wall function by Launder–Spalding or with available semi-empirical models, achieving strong statistical correlations and minimal deviation (from the experimental findings) at high Reynolds numbers. The study also examines the utility of the wall viscosity Reynolds number and assesses the reliability of semi-empirical models for HB fluids. These findings offer valuable insights for enhancing modeling accuracy in complex fluid flow scenarios, with potential applications spanning industries like mining, chemical processing, petroleum transportation, and sanitation systems, providing practical alternatives to costly experimental procedures in pipe systems.