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

^{1}

^{2}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

#### 1.1. Herschel–Bulkley Fluids

#### 1.2. A Wall Function for a Non-Newtonian Fluid

#### 1.3. Approach

## 2. Experiments

## 3. Methodology

#### 3.1. Solver and Numerics

**specified shear**boundary conditions as described in Mehta et al. [4]. Once a flow field is initialised, ${\tau}_{W}$ is calculated from the initial velocity field, following which the RANS equations are solved to obtain a new velocity field. This process is repeated until a converged solution is obtained. A solution is considered as converged once the iterative (absolute, not normalised) residuals for continuity, velocity, $\kappa $ and $\u03f5$ are below ${10}^{-6}$.

#### 3.1.1. An Appropriate Reynolds Number

#### 3.2. Mesh

## 4. Sensitivity Analysis

#### 4.1. Flow Velocity, Behaviour Index and Accuracy

**filled**(or ●, ■ and ◆), in which case, they correspond to ${\psi}_{2}$. A

**x**represents no solution with either wall function.

**without**${\psi}_{1}$ or ${\psi}_{2}$ i.e., these cases used the standard Newtonian wall function proposed by Launder and Spalding [5] described in Section 3.1. One notices that no converged solution is obtained for the considered flow velocities, for slurries apart from $S8$, $S10$ and $S14$. The region in which no solution is obtained is highlighted by dotted polygons in Figure 5, Figure 6, Figure 7 and Figure 8. Furthermore, the estimates for $S14$ with either RANS model can only be considered accurate once V is more than 1 m/s (the aim is to transport the turbulent slurry with a flow velocity ranging from 0.5 m/s to 1.5 m/s). Apart from the lowest velocities considered, the wall shear stresses generated by $S8$ and $S10$ are well-estimated ($e\le 5\%$) by either RANS model (regions outside the dotted polygons).

#### 4.2. Flow Velocity, Yield Stress and RANS Model

**x**means that none of the two RANS models led to a converged numerical solution or the error was beyond the value defined by the relevant plot. In Figure 10 and Figure 12, a filled symbol (●, ■ and ◆) represents ${\psi}_{2}$.

**x**) to a converged solution for test-cases with a ${\tau}_{W}/{\tau}_{y}$ ratio that is less than two orders of magnitude.

**x**(no solution) change to a solution obtained with either RANS models. In addition, solutions obtained with either $\kappa -\u03f5$ (○) or RSM (□) are obtainable with both models (◇) once the error margin is increased to $5\%<e\le 15\%$.

#### 4.3. Reynolds Number, Yield Stress and RANS Model

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NS | Navier-Stokes |

CFD | Computational fluid dynamics |

RANS | Reynolds-averaged Navier-Stokes |

RSM | Reynolds stress model |

SIMPLE | Semi-implicit method for pressure-linked equations |

NWO | Nederlandse Organisatie voor Wetenschappelijk Onderzoek |

TTW | Toegepaste en Technische Wetenschappen |

## References

- Thota Radhakrishnan, A.; van Lier, J.; Clemens, F. Rheological characterisation of concentrated domestic slurry. Water Res.
**2018**, 141, 235–250. [Google Scholar] [CrossRef] - Thota Radhakrishnan, A.K.; van Lier, J.; Clemens, F. Rheology of Un-Sieved Concentrated Domestic Slurry: A Wide Gap Approach. Water
**2018**, 10, 1287. [Google Scholar] [CrossRef] - Chabbra, R.P.; Richardson, J.F. Non-Newtonian Flow in the Process Industries, 1st ed.; Butterworth-Heinemann: Oxford, UK, 1999. [Google Scholar]
- Mehta, D.; Thota-Radhakrishnan, A.K.; van Lier, J.; Clemens, F. A wall boundary condition for the simulation of a turbulent non-Newtonian domestic slurry in pipes. Water
**2018**, 10, 124. [Google Scholar] [CrossRef] - Launder, B.E.; Spalding, D.B. The numerical computation of turbulent flows. Comput. Methods Appl. Mech. Eng.
**1974**, 3, 269–289. [Google Scholar] [CrossRef] - Herschel, W.H.; Bulkley, R. Konsistenzmessungen von Gummi-Benzollösungen. Kolloid-Zeitschrift
**1926**, 39, 291–300. [Google Scholar] [CrossRef] - Oldroyd, J.G. A rational formulation of the equations of plastic flow for a Bingham solid. Math. Proc. Camb. Philos. Soc.
**1947**, 43, 100–105. [Google Scholar] [CrossRef] - Skelland, A.H.P. Non-Newtonian Flow and Heat Transfer; John Wiley & Sons: Hoboken, NJ, USA, 1967. [Google Scholar]
- Govier, G.W.; Aziz, K. The Flow of Complex Mixtures in Pipes; R.E. Krieger Pub. Co.: New York, NY, USA, 1972. [Google Scholar]
- Bird, R.B.; Dai, G.C.; Yarusso, B.J. The rheology and flow of viscoplastic materials. Rev. Chem. Eng.
**1983**, 1, 1–70. [Google Scholar] [CrossRef] - Bird, R.B.; Armstrong, R.C.; Hassager, O. Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics, 2nd ed.; John Wiley & Sons: New York, NY, USA, 1987. [Google Scholar]
- Heywood, N.I.; Cheng, D.C.H. Comparison of methods for predicting head loss in turbulent pipe flow of non-Newtonian fluids. Trans. Inst. Meas. Control
**1984**, 6, 33–45. [Google Scholar] [CrossRef] - Torrance, B.M. Friction factors for turbulent non-Newtonian fluid flow in circular pipes. S. Afr. Mech. Eng.
**1963**, 13, 89–91. [Google Scholar] - Hanks, R.W. Low Reynolds number turbulent pipeline flow of pseudohomogeneous slurries. In Proceedings of the Hydrotransport 5 Conference, Hanover, Germany, 8–11 May 1978; pp. 8–11. [Google Scholar]
- Ferziger, J.; Perić, M. Computational Methods for Fluid Dynamics, 3rd ed.; Springer Science and Business Media: New York, NY, USA, 2012. [Google Scholar]
- Davidson, P.A. Turbulence—An Introduction for Scientists and Engineers; Oxford University Press: Oxford, UK, 2004. [Google Scholar]
- Schlichting, H. Boundary-Layer Theory, 8th ed.; Springer: New York, NY, USA, 2017. [Google Scholar]
- Prandtl, L. Zur turbulenten Strömung in glatten Röhren. Z. Angew. Math. Mech.
**1925**, 5, 136–139. [Google Scholar] - Prandtl, L. Neure Ergebnisse der Turbulenzforschung. Z. Ver. Deutsch. Ingenieure
**1933**, 77, 105–114. [Google Scholar] - von Kármán, T. Mechanische Ähnlichkeit und Turbulenz; Sonderdrucke aus den Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen: Mathematisch-physische Klasse; Weidmannsche Buchh: Berlin, Germany, 1930. [Google Scholar]
- Rudman, M.; Blackburn, H.M. Direct numerical simulation of turbulent non-Newtonian flow using a spectral element method. App. Math. Model.
**2006**, 30, 1229–1248. [Google Scholar] [CrossRef] - Slatter, P.T. Transitional and Turbulent Flow of Non-Newtonian Slurries in Pipes. Ph.D. Thesis, University of Cape Town, Cape Town, South Africa, 1995. [Google Scholar]
- Park, J.T.; Mannheimer, R.J.; Grimley, T.A.; Morrow, T.B. Pipe flow measurements of a transparent non-Newtonian slurry. J. Fluids Eng.
**1989**, 111, 331–336. [Google Scholar] [CrossRef] - ANSYS. ANSYS FLUENT User’s Guide; Ansys Inc.: Canonsburg, PA, USA, 2011. [Google Scholar]
- Rudman, M.; Blackburn, H.M.; Graham, L.J.W.; Pullum, L. Turbulent pipe flow of shear-thinning fluids. J. Non-Newton. Fluid Mech.
**2004**, 118, 33–48. [Google Scholar] [CrossRef] - Malin, M.R. The turbulent flow of Bingham plastic fluids in smooth circular tubes. Int. Commun. Heat Mass Transf.
**1997**, 24, 793–804. [Google Scholar] [CrossRef] - Malin, M.R. Turbulent pipe flow of Herschel-Bulkley fluids. Int. Commun. Heat Mass Transf.
**1998**, 25, 321–330. [Google Scholar] [CrossRef] - Bartosik, A.S. Modification of κ-ϵ model for slurry flow with yield stress. In Proceedings of the 10th International Conference on Numerical Methods in Laminar and Turbulent Flows, Swansea, UK, 21–25 July 1997; Volume 10, pp. 265–274. [Google Scholar]
- Bartosik, A.S. Modelling of a turbulent flow using the Herschel-Bulkley rheological model. Chem. Process Eng. Inzynieria Chem. Proces.
**2006**, 27, 623–632. [Google Scholar] - Tanner, R.I.; Milthorpe, J.H. Numerical simulation of the flow of fluids with yield stress. In Numerical Methods in Laminar and Turbulent Flow, Proceedings of the Third International Conference, Seattle, WA, 8–11 August 1983; Pineridge Press: Swansea, UK, 1983. [Google Scholar]
- Mitsoulis, E. Flows of Viscoplastic Materials: Models and Computations; Rheology Reviews, British Society of Rheology: London, UK, 2007; pp. 135–178. [Google Scholar]
- Metzner, A.B.; Reed, J.C. Flow of non-Newtonian fluids—Correlation of the laminar, transition and turbulent-flow regions. AIChE J.
**1955**, 1, 434–440. [Google Scholar] [CrossRef] - Dean, W. XVI. Note on the motion of fluid in a curved pipe. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1927**, 4, 208–223. [Google Scholar] [CrossRef] - Tunstall, M.J.; Harvey, J.K. On the effect of a sharp bend in a fully developed turbulent pipe-flow. J. Fluid Mech.
**1968**, 34, 595–608. [Google Scholar] [CrossRef] - Rütten, F.; Schröder, W.; Meinke, M. Large-eddy simulation of low frequency oscillations of the Dean vortices in turbulent pipe bend flows. Phys. Fluids
**2005**, 17, 035107. [Google Scholar] [CrossRef] - Kalpakli, A.; Örlü, R. Turbulent pipe flow downstream a 90° pipe bend with and without superimposed swirl. Int. J. Heat Fluid Flow
**2013**, 41, 103–111. [Google Scholar] [CrossRef] - Wilson, K.C.; Thomas, A.D. A new analysis of the turbulent flow of non-Newtonian fluids. Can. J. Chem. Eng.
**1985**, 63, 539–546. [Google Scholar] [CrossRef] - Wilcox, D.C. Turbulence Modeling in CFD, 3rd ed.; DCW Industries: La Canada Flintridge, CA, USA, 2006. [Google Scholar]

**Figure 1.**A schematic of a circular horizontal pipe [4].

**Figure 2.**The effective mixing region in a pipe carrying an Herschel–Bulkley fluid with yield stress ${\tau}_{y}$. The unyielding plug represents the region wherein $\tau <{\tau}_{y}$.

**Figure 4.**The computational grid for the experimental set-up. The cross section is an O-grid, even across the bend.

**Figure 7.**Error with the $\kappa -\u03f5$ model with ${\psi}_{1}$ (or ${\psi}_{2}$, filled symbols).

**Figure 10.**Performance of the standard RANS models combined with ${\psi}_{1}$ to achieve an error $e\le 5\%$.

**Figure 12.**Performance of the standard RANS models combined with ${\psi}_{1}$ to achieve an error $5\%<e\le 15\%$.

**Figure 14.**Performance of the standard RANS models combined with ${\psi}_{1}$ to achieve an error $e\le 5\%$.

**Figure 16.**Performance of the standard RANS models combined with ${\psi}_{1}$ to achieve an error $5\%<e\le 15\%$.

Case | $\mathit{\rho}$ (kg/m${}^{3}$) | ${\mathit{\tau}}_{\mathit{y}}$ (Pa) | m (Pas${}^{\mathit{n}}$) | n | D (m) | $\frac{\mathit{L}}{\mathit{D}}$ | Reference |
---|---|---|---|---|---|---|---|

KERS2408 | 1061 | 1.04 | 0.0136 | 0.8031 | 0.079 | 380 | Slatter [22] |

KERS0608 | 1071 | 1.88 | 0.0102 | 0.8428 | 0.079 | 380 | Slatter [22] |

PARK1 | 1012 | 9.30 | 0.0894 | 0.7254 | 0.051 | 590 | Park et al. [23] |

S8 | 1052 | 0.0014 | 0.0041 | 0.7900 | 0.100 | 450 | Thota Radhakrishnan et al. [1] |

S10 | 1068 | 0.0052 | 0.0071 | 0.7000 | 0.100 | 450 | Thota Radhakrishnan et al. [1] |

S14 | 1091 | 0.0490 | 0.0124 | 0.6500 | 0.100 | 450 | Thota Radhakrishnan et al. [1] |

S17 | 1113 | 0.1585 | 0.0328 | 0.6043 | 0.100 | 450 | Thota Radhakrishnan et al. [1] |

S21 | 1146 | 0.4316 | 0.0831 | 0.5207 | 0.100 | 450 | Thota Radhakrishnan et al. [1] |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mehta, D.; Thota Radhakrishnan, A.K.; Van Lier, J.; Clemens, F.
Sensitivity Analysis of a Wall Boundary Condition for the Turbulent Pipe Flow of Herschel–Bulkley Fluids. *Water* **2019**, *11*, 19.
https://doi.org/10.3390/w11010019

**AMA Style**

Mehta D, Thota Radhakrishnan AK, Van Lier J, Clemens F.
Sensitivity Analysis of a Wall Boundary Condition for the Turbulent Pipe Flow of Herschel–Bulkley Fluids. *Water*. 2019; 11(1):19.
https://doi.org/10.3390/w11010019

**Chicago/Turabian Style**

Mehta, Dhruv, Adithya Krishnan Thota Radhakrishnan, Jules Van Lier, and Francois Clemens.
2019. "Sensitivity Analysis of a Wall Boundary Condition for the Turbulent Pipe Flow of Herschel–Bulkley Fluids" *Water* 11, no. 1: 19.
https://doi.org/10.3390/w11010019