# Self-Preservation of Turbulence Statistics in the Wall-Wake Flow of a Bed-Mounted Horizontal Pipe

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## Abstract

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## 1. Introduction

_{D}between 1000 and 7000. They dictated separation point from the cylinder surface and the variation of length of the primary and secondary downstream separation regions with Reynolds numbers. Akoz and Kirkgoz [5] investigated the turbulent flow field of a bed-mounted horizontal cylinder for Reynolds numbers between 1000 and 7000. Devi and Hanmaiahgari [6] experimentally analyzed a wall-mounted cylinder’s mean and turbulent characteristics at a downstream location. Devi et al. [7] conducted a comparative study for the flow field of a wall-mounted cylinder to determine the effect of gravel-bed roughness compared to sand bed roughness. Despite numerous research in this field, many issues related to the cylinder wake are still unresolved, especially in shallow flows.

## 2. Experimental Setup and Data Measurement Procedure

#### 2.1. Experimental Setup

_{50}equal to 2.54 mm, was coated over the bed to generate the rough surface. A user-configured tailgate was provided at the channel outlet to regulate the flow depth. The water surface levels were measured by employing two sets of Vernier point gauges with an accuracy of ±0.1 mm. The ADV system was carried on a movable trolley supported on two rails over the channel walls.

#### 2.2. Data Measurement Procedure

## 3. Results and Discussion

## 4. Conclusions

- The recirculation region is stretched up to $\widehat{x}=7.4$, which is more than the value found in the existing literature on the topic.
- The self-preserving characteristics such as turbulence intensities and Reynolds shear stresses start from $\widehat{x}=$ 3 and the flow field is found to be not yet recovered even at $\widehat{x}=12$.
- In the wall normal direction, self-preserving of velocity profiles, RSS profiles, and turbulence intensities is found for $z/D<2$ and self preservation is not found for $z/D>2$.
- The points of inflection (${d}^{2}\overline{u}/d{z}^{2}=0$) in the individual ${u}^{+}$ profiles are observed near the top level of the pipe, $z/D\approx 1$. Therefore, RSS and turbulence intensities are attaining their peaks at $z/D$ ≈ 1.
- The velocity deficit profile exhibits a single peak in its distribution at $z/{z}_{1}\approx 0.25$, the RSS deficit profiles display a single peak at $z/{z}_{1}\approx 0.6$, and the turbulence intensities deficit profiles display a single peak at $z/{z}_{1}\approx 0.8$.
- The streamwise decay of peak values of non-dimensional velocity defect, non-dimensional RSS defect, and non-dimensional turbulence intensity defect indicates recovery of flow.
- The half RSS defect profile width, half velocity defect profile width, and the half turbulence defect profile width grow in the streamwise direction, which indicates weaning of the wake strength.
- In the wake region, third-order moments ${M}_{03}$ and ${M}_{21}$ begin at the bed with small negative values, whereas ${M}_{30}$ and ${M}_{12}$ begin at the bed with small positive values. They change their signs at $z\approx \mathrm{D}$, which proves that sweep events are dominant below the level of the cylinder and ejection events are dominant above the level of the cylinder.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the experimental setup showing its plane view with the streamwise measurement locations for Run 1, Run 2, Run 3, and Run 4, and the coordinate system used for the experiments.

**Figure 2.**Vector plot of non-dimensional resultant velocity, ${U}^{+}$ along the central wall-normal plane ($xz$) of the channel for Run 3 in the wall-wake region of the pipe.

**Figure 3.**Wall-normal profiles of non-dimensional time-averaged streamwise velocity, ${u}^{+}$, in the wake region of the pipe at different streamwise locations, $\widehat{x}$, for Run 3 and Run 4.

**Figure 4.**Schematic representation of a typical wall-normal distribution of streamwise velocity deficit, $\Delta u$, in the wall-wake region of the pipe along with the definition of the half velocity profile width, ${z}_{1}$.

**Figure 5.**Self-preservation analysis of streamwise velocity distribution in the wall-wake region of the pipe for Run 1, Run 2, Run 3, and Run 4: (

**a**) variation of $\Delta u/{\left(\Delta u\right)}_{max}$ against $z/{z}_{1}$, depicting the self-preservation in velocity deficit ($\Delta u$) distribution; (

**b**) streamwise growth of non-dimensional half velocity profile width, ${z}_{1}/D$; and (

**c**) streamwise decay of non-dimensional maximum velocity deficit, ${\left(\Delta u\right)}_{max}/{u}_{*}$.

**Figure 6.**Wall-normal profiles of non-dimensional RSS, ${\tau}_{uw}{}^{+}$, in the wake region of the pipe at different streamwise locations, $\widehat{x}$, for Run 3 and Run 4.

**Figure 7.**Schematic representation of a typical wall-normal distribution of RSS deficit, $\Delta {\tau}_{uw}$, distribution in the wall-wake region of the pipe along with the definition of the half RSS profile width, ${z}_{2}$.

**Figure 8.**Self- preservation analysis of RSS distribution in the wall-wake region of the pipe for Run 1, Run 2, Run 3, and Run 4: (

**a**) variation of $\Delta {\tau}_{uw}/{\left(\Delta {\tau}_{uw}\right)}_{max}$ against $z/{z}_{2}$, depicting the self-preservation in RSS deficit ($\Delta {\tau}_{uw}$) distribution; (

**b**) streamwise growth of non-dimensional half RSS profile width, ${z}_{2}/D$; and (

**c**) streamwise decay of non-dimensional maximum RSS deficit,$-{\left(\Delta {\tau}_{uw}\right)}_{max}/{u}_{*}{}^{2}$.

**Figure 9.**Wall-normal profiles of non-dimensional streamwise turbulence intensity, ${\sigma}_{u}{}^{+}$ in the wake region of the pipe at different streamwise locations, $\widehat{x}$, for Run 3 and Run 4.

**Figure 10.**Wall-normal profiles of non-dimensional wall-normal turbulence intensity, ${\sigma}_{w}{}^{+}$, in the wake region of the pipe at different streamwise locations, $\widehat{x}$, for Run 3 and Run 4.

**Figure 11.**Self- preservation analysis of turbulence intensities in the wall-wake region of the circular pipe for Run 1, Run 2, Run 3, and Run 4: (

**a**) variation of $\Delta {\sigma}_{u}/{\left(\Delta {\sigma}_{u}\right)}_{max}$ against $z/{z}_{2}$, depicting the self-preservation in streamwise turbulence intensity deficit ($\Delta {\sigma}_{u}$) distribution; (

**b**) variation of $\Delta {\sigma}_{w}/{\left(\Delta {\sigma}_{w}\right)}_{max}$ against $z/{z}_{2}$, depicting the self-preservation in wall-normal turbulence intensity deficit ($\Delta {\sigma}_{w}$) distribution; (

**c**) streamwise decay of non-dimensional maximum streamwise turbulence intensity deficit,$-{\left(\Delta {\sigma}_{u}\right)}_{max}/{u}_{*}$; (

**d**) streamwise decay of non-dimensional maximum wall-normal turbulence intensity deficit,$-{\left(\Delta {\sigma}_{w}\right)}_{max}/{u}_{*}$.

**Figure 12.**The wall-normal profiles of the third-order correlations, ${M}_{ij}$ in the wake region of the pipe at different $\widehat{x}$ for Run 3: (

**a**) ${M}_{30}$ and ${M}_{03}$, (

**b**)${M}_{21}$ and ${M}_{12}$; and for Run 4: (

**c**)${M}_{30}$ and ${M}_{03}$, and (

**d**) ${M}_{21}$ and ${M}_{12}$.

Exp. Run | $\mathbf{D}$ (m) | $\mathbf{h}$ (m) | $\mathbf{U}$ (m/s) | ${\mathbf{u}}_{*}$ (m/s) | ${\mathbf{F}}_{\mathbf{r}}$ | $\mathbf{R}{\mathbf{e}}_{\mathbf{D}}$ |
---|---|---|---|---|---|---|

Run 1 | 0.05 | 0.3 | 0.15 | 0.0071 | 0.111 | 7500 |

Run 2 | 0.05 | 0.3 | 0.19 | 0.0092 | 0.111 | 9500 |

Run 3 | 0.08 | 0.3 | 0.15 | 0.0071 | 0.087 | 12,000 |

Run 4 | 0.08 | 0.3 | 0.19 | 0.0092 | 0.111 | 15,200 |

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**MDPI and ACS Style**

Devi, K.; Hanmaiahgari, P.R.; Balachandar, R.; Pu, J.H.
Self-Preservation of Turbulence Statistics in the Wall-Wake Flow of a Bed-Mounted Horizontal Pipe. *Fluids* **2021**, *6*, 453.
https://doi.org/10.3390/fluids6120453

**AMA Style**

Devi K, Hanmaiahgari PR, Balachandar R, Pu JH.
Self-Preservation of Turbulence Statistics in the Wall-Wake Flow of a Bed-Mounted Horizontal Pipe. *Fluids*. 2021; 6(12):453.
https://doi.org/10.3390/fluids6120453

**Chicago/Turabian Style**

Devi, Kalpana, Prashanth Reddy Hanmaiahgari, Ram Balachandar, and Jaan H. Pu.
2021. "Self-Preservation of Turbulence Statistics in the Wall-Wake Flow of a Bed-Mounted Horizontal Pipe" *Fluids* 6, no. 12: 453.
https://doi.org/10.3390/fluids6120453