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

Dhruv Mehta,Jules Van Lier,Adithya Thota Radhakrishnan,Francois Clemens

doi:10.3390/w11010019

Abstract

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 ψ 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. ψ 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 ψ 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 ψ 1 is a must for accuracy. One observes that, regardless of the fluid behaviour index, ψ 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 ψ 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.

Highlights

This introductory section summarises the content of Mehta et al [4] and provides details on the topics pertinent to this article
The term domestic slurry that is relevant to the authors is replaced with the term Herschel–Bulkley fluid in this article, in order to generalise it for other possible applications of the numerical methods discussed here
Based on the mentioned observations, one may conclude that the proposed wall functions ψ1 and ψ2 when combined with the standard κ − e or RSM could lead to accurate numerical quantification of the wall shear experienced by a circular pipe carrying a Herschel–Bulkley fluid in turbulent flow

Summary

Introduction

This introductory section summarises the content of Mehta et al [4] and provides details on the topics pertinent to this article. The term domestic slurry that is relevant to the authors is replaced with the term Herschel–Bulkley fluid in this article, in order to generalise it for other possible applications of the numerical methods discussed here. A non-Newtonian fluid experiences viscous stresses that depend on temperature and pressure and on the flow itself. Herschel and Bulkley [6] studied a certain class of non-Newtonian fluids that display a shear-thinning (pseudoplastic behaviour), which is the reduction in the apparent viscosity with increasing shear rate. It was observed that such fluids require a minimum shear stress before they flow like a fluid. This minimum stress is called the yield stress and, such fluids are called yield pseudoplastic fluids.

Objectives

Methods

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Conclusion