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.

This article concerns the turbulent flow of Herschel–Bulkley slurries through circular horizontal pipes; in particular, that of concentrated domestic slurry obtained upon separation of domestic waste water and reduction in the use of water for domestic purposes. Experiments with a rheologically equivalent clay (kaolin) slurry indicated a non-Newtonian behaviour of the Herschel–Bulkley type. A modified wall function was developed to enable the Reynolds-averaged Navier–Stokes simulation of Herschel–Bulkley slurries to estimate the wall shear stress. Despite the accuracy achieved, the use of Reynolds-averaged Navier–Stokes models for an entire waste water system is impractical. Therefore, this article assesses the accuracy of semi-empirical models in estimating frictional losses. It also discusses possible modifications of existing models to encompass Herschel–Bulkley behaviour. An evaluation suggests that most existing models deliver estimates of comparable accuracy; however, the probability of these estimates being reliable, while accounting for experimental errors in quantifying the actual frictional losses, is rather low.

The concentration of domestic slurry has two advantages, it promotes resource recovery (nutrients and biomass) and saves water. But the design of a relevant sewerage requires a clear understanding of the frictional losses incurred during the transport of the slurry. This abstracts describes numerical & CFD-based methods to estimate losses while the concentrated slurry flows through circular pipes in a fully-turbulent flow. To model turbulent flows through circular pipes, one can rely on either the Newtonian Moody Charts appropriate for engineering applications or a computational fluid dynamics (CFD)-based analysis, made possible through the Newtonian universal law of the wall. However, our studies reveal that concentrated domestic slurry behaves like a non-Newtonian fluid, of the Herschel-Bulkley type. Therefore, the analysis of such a slurry would require modifications to both, existing engineering models and CFD methods. This abstract summarises a modified law of the wall suitable for Herschel-Bulkley fluids, which has been validated against experiments on concentrated domestic slurry. It further details possible non-Newtonian numerical engineering models that could be modified to assess frictional losses incurred by Herschel-Bulkley fluids. The latter will be a quicker and perhaps reliable alternative to computationally expensive CFD-analyses.

This article follows from a previous study by the authors on the computational fluid dynamics-based analysis of Herschel-Bulkley fluids in a pipe-bounded turbulent flow. The study aims to propose a numerical method that could support engineering processes involving the design and implementation of a waste water transport system, for concentrated domestic slurry. Concentrated domestic slurry results from the reduction in the amount of water used in domestic activities (and also the separation of black and grey water). This primarily saves water and also increases the concentration of nutrients and biomass in the slurry, facilitating efficient recovery. Experiments revealed that upon concentration, domestic slurry flows as a non-Newtonian fluid of the Herschel-Bulkley type. An analytical solution for the laminar transport of such a fluid is available in literature. However, a similar solution for the turbulent transport of a Herschel-Bulkley fluid is unavailable, which prompted the development of an appropriate wall function to aid the analysis of such flows. The wall function (called ψ₁ hereafter) was developed using Launder and Spalding's standard wall function as a guide and was validated against a range of experimental test-cases, with positive results.ψ₁ is assessed for its sensitivity to rheological parameters, namely the yield stress, the fluid consistency index and the behaviour index and their impact on the accuracy with which ψ₁ can correctly quantify the pressure loss through a pipe. This is done while simulating the flow of concentrated domestic slurry using the Reynolds-Averaged Navier-Stokes (RANS) approach for turbulent flows. This serves to establish an operational envelope in terms of the rheological parameters and the average flow velocity within which ψ₁ is a must for accuracy. One observes that, regardless of the fluid behaviour index, ψ₁ is necessary to ensure accuracy with RANS models only in flow regimes where the wall shear stress is comparable to the yield stress within an order of magnitude. This is also the regime within which the concentrated slurry analysed as part of this research flows, making ψ₁ a requirement. In addition, when the wall shear stress exceeds the yield stress by more than one order (either due to an inherent lower yield stress or a high flow velocity), the regular Newtonian wall function proposed by Launder and Spalding is sufficient for an accurate estimate of the pressure loss, owing to the relative reduction in non-Newtonian viscosity as compared to the turbulent viscosity.

The concentration (using a lesser amount of water) of domestic slurry promotes resource recovery (nutrients and biomass) while saving water. This article is aimed at developing numerical methods to support engineering processes such as the design and implementation of sewerage for concentrated domestic slurry. The current industrial standard for computational fluid dynamics-based analyses of turbulent flows is Reynolds-averaged Navier-Stokes (RANS) modelling. This is assisted by the wall function approach proposed by Launder and Spalding, which permits the use of under-refined grids near wall boundaries while simulating a wall-bounded flow. Most RANS models combined with wall functions have been successfully validated for turbulent flows of Newtonian fluids. However, our experiments suggest that concentrated domestic slurry shows a Herschel-Bulkley-type non-Newtonian behaviour. Attempts have been made to derive wall functions and turbulence closures for non-Newtonian fluids; however, the resulting laws or equations are either inconsistent across experiments or lack relevant experimental support. Pertinent to this study, laws or equations reported in literature are restricted to a class of non-Newtonian fluids called power law fluids, which, as compared to Herschel-Bulkley fluids, yield at any amount of applied stress. An equivalent law for Herschel-Bulkley fluids that require a minimum-yield stress to flow is yet to be reported in literature. This article presents a theoretically derived (with necessary approximations) law of the wall for Herschel-Bulkley fluids and implements it in a RANS solver using a specified shear approach. This results in a more accurate prediction of the wall shear stress experienced by a circular pipe with a turbulent Herschel-Bulkley fluid flowing through it. The numerical results are compared against data from our experiments and those reported in literature for a range of Reynolds numbers and rheological parameters that are relevant to the prediction of pressure losses in a sewerage transporting non-Newtonian domestic slurry. Nonetheless, the application of this boundary condition could be extended to areas such as chemical and food engineering, wherein turbulent non-Newtonian flows can be found.