# Sensitivity Analysis of a Wall Boundary Condition for the Turbulent Pipe Flow of Herschel–Bulkley Fluids

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Sanitary Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology, 2628 CN Delft, The Netherlands

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Stichting Deltares, 2629 HV Delft, The Netherlands

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Author to whom correspondence should be addressed.

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Current address: Sanitary Engineering, Stevinweg 1, 2628 CN Delft, The Netherlands.

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This document is a follow-up study to Mehta et al. and is a collaborative effort.

Received: 17 October 2018 / Revised: 13 December 2018 / Accepted: 19 December 2018 / Published: 22 December 2018

(This article belongs to the Special Issue Smart Hydraulics in Wastewater Transport)

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 ${\psi}_{1}$ 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. ${\psi}_{1}$ 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 ${\psi}_{1}$ 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 ${\psi}_{1}$ is a must for accuracy. One observes that, regardless of the fluid behaviour index, ${\psi}_{1}$ 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 ${\psi}_{1}$ 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.