# A Wall Boundary Condition for the Simulation of a Turbulent Non-Newtonian Domestic Slurry in Pipes

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## Abstract

**:**

## 1. Introduction

**bold**, whereas vectors are underlined. Above, $\underline{u}$ is the velocity vector and $\nabla \underline{u}$ is the velocity gradient tensor or $\frac{\partial {u}_{j}}{\partial {x}_{i}}$. In the fields of rheology and non-Newtonian fluid mechanics, it is common practice to replace $2\mathit{\u03f5}$ as follows:

#### 1.1. Herschel–Bulkley Fluids

#### 1.2. Computational Fluid Dynamics (CFD) and Non-Newtonian Fluids

**perhaps**comparable with studies used to validate the proposed closure.

#### 1.3. Wall-Bounded Flows

**numerical model**. We would like to mention again that our purpose is to explore suitable modelling techniques and not to formulate a theoretical basis for the turbulence of non-Newtonian fluids.

#### 1.4. Non-Newtonian Wall Functions

#### 1.5. Approach

## 2. Methodology

#### 2.1. Solver and Numerics

- The Reynolds stresses in the wall-adjacent cells are calculated explicitly in terms of the wall shear stress. This setting shall be referred to as
**RSM1**. - A transport equation for $\kappa $ is solved to obtain the Reynolds stresses in the cells adjacent to the wall, which shall be quoted as
**RSM2**.

#### 2.2. Wall Modelling: Specified Shear Approach

#### 2.2.1. An Appropriate Reynolds Number

**wall effective viscosity**defined below. This viscosity can only be determined once the wall shear stress is known, and hence it serves only as a post-experimental (numerical) parameter.

#### 2.2.2. Specified Wall Shear Based on Prandtl’s Mixing Length

**specified shear**boundary conditions.

#### 2.3. Mesh

## 3. Experiments

## 4. Observations

**two y-axes**. The left y-axis represents ${\tau}_{W}$, while the right y-axis represents the Reynolds number $R{e}_{R}$. Therefore, each figure simultaneously shows the relationship of the velocity V on the x-axis with ${\tau}_{W}$ and $R{e}_{R}$. For simplicity, $R{e}_{R}$ is shown with a dashed line with crosses at the relevant data points.

**does not**imply that a given experimental ${\tau}_{W}$ value holds for all velocities V on the x-axis. Rather, the velocity at which a given experimental ${\tau}_{W}$ is known is made clear with a green error bar on the relevant grey line, representing a $5\%$ deviation from the experimental value, and with an additional magenta bar that indicates a $15\%$ deviation.

## 5. Conclusions and Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Rheograms of time-independent flow behaviour (adapted from [1]).

**Figure 3.**The law of the wall (adapted from ANSYS [40]).

**Figure 4.**The effective mixing region in a pipe carrying a Herschel–Bulkley fluid with yield stress ${\tau}_{y}$. The unyielding plug represents the region wherein $\tau <{\tau}_{y}$.

**Figure 6.**A schematic of the experimental set-up showing the location of the pressure sensors and the dimensions of the various sections [49] (courtesy: Stichting Deltares).

**Figure 7.**Normalised residuals for one of the test cases. Applying ${\psi}_{1}$ improved the numerical prediction of the wall shear stress.

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

KERS2408 | 1061 | 1.04 | 0.0136 | 0.8031 | 0.079 | 380 | Slatter [13] |

KERS0608 | 1071 | 1.88 | 0.0102 | 0.8428 | 0.079 | 380 | Slatter [13] |

PARK1 | 1012 | 9.30 | 0.0894 | 0.7254 | 0.051 | 590 | Park et al. [48] |

S17 | 1113 | 0.16 | 0.0328 | 0.6043 | 0.100 | 450 | Thota Radhakrishnan et al. [49] |

S21 | 1146 | 0.43 | 0.0831 | 0.5207 | 0.100 | 450 | Thota Radhakrishnan et al. [49] |

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/w10020124

**AMA Style**

Mehta D, Thota Radhakrishnan AK, Van Lier J, Clemens F.
A Wall Boundary Condition for the Simulation of a Turbulent Non-Newtonian Domestic Slurry in Pipes. *Water*. 2018; 10(2):124.
https://doi.org/10.3390/w10020124

**Chicago/Turabian Style**

Mehta, Dhruv, Adithya Krishnan Thota Radhakrishnan, Jules Van Lier, and Francois Clemens.
2018. "A Wall Boundary Condition for the Simulation of a Turbulent Non-Newtonian Domestic Slurry in Pipes" *Water* 10, no. 2: 124.
https://doi.org/10.3390/w10020124