# Interactions between Tandem Cylinders in an Open Channel: Impact on Mean and Turbulent Flow Characteristics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{D}= 800–42,000) reported three flow regimes: the extended-body regime (L/D < 2, vortex street far behind the downstream cylinder), the reattachment regime (L/D = 2–5, shed vortices from upstream cylinder formed just behind the downstream cylinder), and the co-shedding regime (L/D > 5, where each cylinder sheds its own von Kármán vortices individually). The studies mentioned so far focused on the effects of cylinder spacing and Reynolds number on vortex development and movement in the wake.

## 2. Materials and Methods

#### 2.1. Experimental Setup

^{−3}. Two solid concrete circular cylinders of equal height (H

_{d}= 0.30 m) and diameter (D = 0.15 m) were mounted vertically along the channel centerline of a smooth flume bed at a distance of 11.0 m from the inlet to ensure that the turbulent flow was fully developed in the test section (the approach flow development is discussed in Section 3.1.). Then, 3 experimental scenarios (hereafter called S1, S2, and S3) were performed, where the cylinders were aligned in a tandem arrangement (Figure 1) with 3 different gap ratios of L/D = 3D in S1, L/D = 6D in S2, and L/D = 9D in S3 center to center, with a flow blockage ratio of 20%. We acknowledge that there may have been some effects of cylinder blockage on the flow characteristics in a narrow channel. Nevertheless, because the main objective in this study was related to flow characteristics along the centerline (behind the tandem cylinders in the plane of symmetry), this blockage ratio was not expected to affect the key findings of this study significantly. Further studies are recommended to examine the effect of blockage ratio on flow characteristics for tandem cylinders.

_{*}= 1.40 cm/s, estimated by the Reynolds shear stress method ($\tau =\rho {u}_{*}^{2}\left(1-z/\sigma \right)\cong -\rho \overline{{u}^{\prime}{w}^{\prime}}$) by Nezu and Nakagawa [17], using the measured Reynolds shear stress very close to the bed (10% depth from the bed), where τ is the total shear stress, σ is the distance from the bottom to the point of the maximum velocity, ρ is the water density, and ${u}^{\prime}$ and ${w}^{\prime}$ are the velocity fluctuations in the streamwise and vertical directions, respectively. All experiments were conducted at higher Reynolds numbers based on the diameter of the cylinder, $R{e}_{D}=\frac{{\overline{U}}_{0}D}{\vartheta}=52,900$, where $\vartheta $ is the coefficient of kinematic viscosity of water.

#### 2.2. ADV Measurements and Analysis

^{3}and assuming implicitly that such a duration was long enough to describe the turbulence. Some studies used relatively small number of samples for turbulence measurements, using ADVs as low as 3000 samples [20] and 1500 samples [19]. The raw data were first post-processed using WinADV [21] to remove spikes using the phase space threshold method of Goring and Nikora [22]. Velocity signal correlations (COR) and the signal-to-noise ratio (SNR) are commonly used to eliminate poor-quality data from measured time series. A filtering scheme with an average COR ≤ 70% and average SNR ≤ 15 dB was used to eliminate bad data from the velocity time series to yield highly reliable data [20]. Such criteria led to more than 90% of the data being retained in each experimental scenario (Table 1). Afterward, the velocity time series in 3 directions that had noisy signals (velocity spectra that generally displayed a flat slope instead of the Kolmogorov −5/3 slope in the inertial subrange) were removed by visual inspection, thus rejecting about 10% of data points. It should be noted that due to the limitations of the downward-facing ADV, flow information for the top 6.5 cm of the flow depth is not available.

_{p}) per unit mass density of fluid on a surface parallel to the bed:

_{p}was estimated leveraging the Reynolds shear stress in both vertical and transverse planes instead of using the measured Reynolds shear stress on the horizontal plane.

## 3. Results and Discussion

#### 3.1. Mean Flow Characteristics

_{*}) in the plane of symmetry for all three scenarios both midstream and downstream. The Figure confirms that the vertical profiles of the streamwise approach velocity satisfy the logarithmic law (Equation (1)), assuring a fully developed approach flow for a hydraulically rough flow regime.

_{*}) in the symmetry plane both midstream and downstream for all three scenarios. Midstream, based on velocity vector and u/u

_{*}≤ 0, the flow recirculation length on the symmetry plane was x/D~1.80 in S1, and ~1.50 in S2 and S3 almost over the entire depth, and was stronger near the free surface than the bed level. On the other hand, downstream, flow recirculation was observed immediately behind the cylinder and was strong near the bed level. The size of the recirculation was x/D~0.90 in S1, ~1.10 in S2, and ~1.30 in S3, with values much smaller and weaker than that midstream. For an emergent cylinder and high Reynolds number, Sadeque et al. [6] determined x/D~1.50 for the entire water depth and for the rough bed; Kirkil and Constantinescu [16] found x/D~1.60 at the near-free surface; and Kirkil and Constantinescu [16] reported x/D~1.20 at the near-bed. By comparing the length of the recirculation zone (x/D ~ 1.20–1.6) for a single cylinder, it is clear that the downstream cylinder influences the longer recirculation zone midstream in S1. The smaller recirculation zone downstream in S1 and S2 are affected by the upstream cylinder. Similarly, Lin et al. [11] found that the upstream cylinder substantially altered the near-wake flow behind the downstream cylinder for L/D = 1.15 to 5.10 at a cylinder Reynolds number (Re

_{D}) = 1 × 10

^{4}. As the flow approached the downstream (away from the recirculation zone), a rapid increase in velocity and the profiles approaching developed from x/D = 3.80 in S3 midstream, x/D = 3.20 in S1, and 3.80 in S2 and S3 downstream. In midstream, the flow deceleration was evident as it approached the downstream cylinder.

_{d}/u

_{*}) along the plane of symmetry both midstream and downstream. Midstream, u

_{d}/u

_{*}= 0 at x/D = 2.00 in S1 and at x/D = 1.70 in S2 and S3 (Figure 4a). In S1, the flow recirculation zone occupied almost the full space midstream, where u

_{d}/u

_{*}varied from −5.20 to 0.40. In S2, a rapid increase was found in u

_{d}/u

_{*}= −4.40 at x/D = 1.10 to u

_{d}/u

_{*}= 13.20 at x/D = 3.20; then the velocity slowed, and the maximum u

_{d}/u

_{*}= 14.10 occurred at x/D = 4.50. Finally, the velocity dipped as the flow approached the downstream cylinder. In S3, similar to S2, u

_{d}/u

_{*}increased sharply from −5.10 at x/D = 0.90 to 14.30 at x/D = 3.20. The rate of increase slowed and the maximum u

_{d}/u

_{*}= 16.60 occurred at x/D = 7.20; thereafter it decreased as the flow approached the downstream cylinder. In contrast to midstream, u

_{d}/u

_{*}= 0 at x/D 0.80, 1.10, and 1.20 in S1, S2, and S3, respectively (Figure 4b). The variation patterns of u

_{d}/u

_{*}in all three scenarios were similar, with a rapid increase in u

_{d}/u

_{*}(varying from −3.80 to 17.40) between x/D = 0.70 and x/D = 3.80 and values then remaining almost constant. Interestingly, downstream the value of u

_{d}/u

_{*}slightly decreased from S1 to S3 due to the effect of their corresponding recirculation lengths, which increased from S1 (x/D~0.90) to S3 (x/D~1.30) (as discussed above). Therefore, Figure 3 and Figure 4 show that the downstream cylinder controls the flow recirculation length midstream in S1 and has zero interruption in S2 and S3.

#### 3.2. Turbulent Flow Characteristics

_{e}, which characterizes the intensity of the turbulence, and the resultant Reynolds shear stress, ${\tau}_{p}$, which may promote sediment movement, are analyzed here to understand the turbulence flow characters better. Figure 5 shows the vertical profiles of dimensionless k

_{e}/u

_{*}

^{2}in the symmetry plane for scenarios S1, S2, and S3 both midstream and downstream. Both midstream and downstream, the elevated k

_{e}/u

_{*}

^{2}grew with the streamwise distance until the end of the recirculation zone. With a further streamwise distance, despite the obvious reduction, the k

_{e}/u

_{*}

^{2}remained at a relatively high level. The pronounced k

_{e}/u

_{*}

^{2}downstream of the cylinders is attributed to large-scale vortices generated in the recirculation region [25].

_{e}/u

_{*}

^{2}for the plane of symmetry are also plotted in Figure 6a for midstream and in Figure 6b for downstream. Midstream, it is apparent that the k

_{e}keeps on increasing as the flow progresses. The highest k

_{e}/u

_{*}

^{2}= 154 was observed at x/D = 1.83 in S1, while k

_{e}/u

_{*}

^{2}= 152 was observed at x/D = 1.83 in S2, and k

_{e}/u

_{*}

^{2}= 157 was observed at x/D = 2.17 in S3. In all scenarios, a rapid increase in k

_{e}/u

_{*}

^{2}(by about 3.9 times of the lowest k

_{e}/u

_{*}

^{2}) was found over a short longitudinal distance (x/D = 0.70) to 2.17. After that, the peak k

_{e}/u

_{*}

^{2}gradually decreased by about 4.5 times over a distance x/D = 2.17 to 7.83 in S3. Downstream, the highest k

_{e}/u

_{*}

^{2}= 156 was observed at x/D = 0.90 in S1, k

_{e}/u

_{*}

^{2}= 159 was observed at x/D = 1.30 in S2, and k

_{e}/u

_{*}

^{2}= 108 was observed at x/D = 1.10 in S3. In all scenarios, a rapid increase in k

_{e}/u

_{*}

^{2}(by about 1.6, 2.1, and 2.8 times the lowest k

_{e}/u

_{*}

^{2}) over a short longitudinal distance x/D = 0.70 to 0.90, 0.70 to 1.30, and 0.70 to 1.10 in S1, S2, and S3, respectively, was observed. After that, the k

_{e}/u

_{*}

^{2}peaks gradually decreased by about 4.3, 4.4, and 3.9 times over a distance x/D = 0.90 to 5.83, 1.30 to 5.83, and 1.10 to 5.83 in S1, S2, and S3, respectively. Therefore, peaks of k

_{e}/u

_{*}

^{2}in both midstream and downstream for all three scenarios occurred near the end of their corresponding recirculation zones, except for S3 midstream where the peak occurred outside the recirculation zone.

_{e}towards the downstream cylinder had a significant influence on the k

_{e}just downstream of this cylinder. For S1, S2, and S3, the k

_{e}/u

_{*}

^{2}just upstream of downstream cylinder ended with 150, 56, and 35, respectively, and their corresponding k

_{e}/u

_{*}

^{2}downstream started with 95, 74, and 38, indicating that the k

_{e}downstream increased with the increased k

_{e}midstream.

_{p}/u

_{*}

^{2}) in the plane of symmetry for S1, S2, and S3 both midstream and downstream. It is apparent that τ

_{p}/u

_{*}

^{2}dominated near the recirculation zone for all scenarios. The magnitude of vertical profiles outside of the recirculation zone and in farther downstream locations was generally low and slightly elevated the near-free surface. Figure 8 shows the variations of maximum τ

_{p}/u

_{*}

^{2}for the plane of symmetry midstream and downstream, which are relatively chaotic as compared to the variation of k

_{e}/u

_{*}

^{2}with x/D.

_{p}/u

_{*}

^{2}= 15 was observed at x/D = 1.83 in S1 and τ

_{p}/u

_{*}

^{2}= 14 was observed at x/D = 2.50 in S2 and S3 (Figure 8a). Similarly, downstream the highest τ

_{p}/u

_{*}

^{2}= 13 was observed at x/D = 1.10 in S1, where a value of 19 was observed at x/D = 1.30 in S2, and a value of 17 was obtained at x/D = 1.50 in S3 (Figure 8b). Though the plots are scattered, in general the lowest values of τ

_{p}/u

_{*}

^{2}just downstream of the cylinders increase rapidly in the streamwise distance, and the values reach a peak in or outside the recirculation region; thereafter the values gradually decrease with the downstream distance. In all three scenarios, the peaks of τ

_{p}/u

_{*}

^{2}in downstream are higher than those midstream.

#### 3.3. Near-Bed Shear Stress

_{e}/u

_{*}

^{2}and τ

_{p}/u

_{*}

^{2}with the relative streamwise distance (x/D) in Figure 6 and Figure 8.

## 4. Conclusions

- The analysis of vertical profiles of streamwise normalized velocity revealed that the rate of flow development downstream was faster than that midstream. The midstream profiles were influenced by the downstream cylinder in all scenarios.
- The flow recirculation zones midstream (x/D~1.50–1.80) were larger and stronger than those downstream (x/D~0.90–1.30). The longest recirculation zone in S1 midstream was affected by the downstream cylinder, and there was zero interruption from the downstream cylinder in S2 and S3.
- The length of the recirculation zone increased with the increase in c/c spacing of cylinders. The rate of flow development to achieve cross-sectional mean velocity became slower with increasing c/c spacing.
- The maximum turbulent kinetic energy for all three scenarios occurred approximately near the end of their respective recirculation zones. The midstream k
_{e}had a significant influence on the k_{e}in downstream. - The variation of resultant Reynolds shear stress was relatively chaotic as compared to the turbulent kinetic energy. The maximum shear stresses occurred within the recirculation zone for all scenarios.
- The peaks of dimensionless maximum near-bed shear stress (${\tau}_{b}^{*}$) were higher (5–20%) and occurred over an extended length of about 0.5D in midstream as compared to the peaks of ${\tau}_{b}^{*}$ downstream, which occurred over a short length of 0.2D. The highest value of ${\tau}_{b}^{*}$ in the TKE method was about 50% and 70% higher than that in the Reynolds and modified TKE methods, respectively.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Plan view of the position of the measurement stations in the symmetry plane midstream and downstream for 3 scenarios: S1, S2, and S3. The + symbol shows the position of the measurement stations. U/S and D/S indicate upstream and downstream, respectively.

**Figure 2.**Vertical profiles of dimensionless streamwise velocity (u/u

_{*}) in the symmetry plane: (

**a**) midstream and (

**b**) downstream. A solid black line shows the classical logarithmic law.

**Figure 3.**Velocity vector and isocontours of dimensionless streamwise velocity (u/u

_{*}) on the symmetry plane for the 3 experimental scenarios S1, S2, and S3, combining both midstream and downstream.

**Figure 4.**Variation of dimensionless depth-averaged streamwise velocity (u

_{d}/u

_{*}) in the symmetry plane: (

**a**) midstream and (

**b**) downstream.

**Figure 5.**Vertical profiles of dimensionless turbulent kinetic energy (k

_{e}/u

_{*}

^{2}) in the symmetry plane for the 3 experimental scenarios S1, S2, and S3, combining both midstream and downstream.

**Figure 6.**Variation of dimensionless maximum turbulent kinetic energy (k

_{e}/u

_{*}

^{2}) in the symmetry plane: (

**a**) midstream and (

**b**) downstream.

**Figure 7.**Vertical profiles of dimensionless resultant Reynolds shear stress (τ

_{p}/u

_{*}

^{2}) in the symmetry plane for the 3 experimental scenarios S1, S2, and S3, combining both midstream and downstream.

**Figure 8.**Variation of dimensionless maximum projected Reynolds shear stress (τ

_{p}/u

_{*}

^{2}) in the symmetry plane: (

**a**) midstream and (

**b**) downstream.

**Figure 9.**Variation of dimensionless maximum near-bed shear stress (${\tau}_{b}^{*}$) using the Reynolds (Equation (3)), TKE (Equation (4)), and modified TKE (Equation (5)) methods in the symmetry plane: (

**a**) midstream and (

**b**) downstream.

Experimental Scenarios | Cylinder Diameter (D) | Cylinder Spacing * | Flow Rate | Approach Water Depth (H) | Approach Flow Velocity ^{θ} | Reynolds Number ^{Ϯ} (R_{d}) | Shear Velocity ^{±} (u *) | Average | Average | Data Retained |
---|---|---|---|---|---|---|---|---|---|---|

COR | SNR | |||||||||

- | (cm) | (times) | (m^{3}/s) | (cm) | (cm/s) | - | (cm/s) | (%) | (dB) | (%) |

S1 | 15 | 3D | 0.069 | 26.5 | 35.3 | 52,900 | 1.4 | 80.6 | 16.7 | 90.1 |

S2 | 15 | 6D | 0.069 | 26.5 | 35.3 | 52,900 | 1.4 | 80.6 | 16.9 | 90.4 |

S3 | 15 | 9D | 0.069 | 26.5 | 35.3 | 52,900 | 1.4 | 78.3 | 15.6 | 91.2 |

^{±}The Reynolds shear stress method was applied to estimate u

_{*}using measured Reynolds shear stress very close to the bed (lower 10% depth).

^{Ϯ}Reynolds number is based on the diameter of the cylinder. * Spacing is in center to center.

^{θ}Depth-averaged approach flow velocity magnitude.

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**MDPI and ACS Style**

Zobeyer, H.; Baki, A.B.M.; Nowrin, S.N. Interactions between Tandem Cylinders in an Open Channel: Impact on Mean and Turbulent Flow Characteristics. *Water* **2021**, *13*, 1718.
https://doi.org/10.3390/w13131718

**AMA Style**

Zobeyer H, Baki ABM, Nowrin SN. Interactions between Tandem Cylinders in an Open Channel: Impact on Mean and Turbulent Flow Characteristics. *Water*. 2021; 13(13):1718.
https://doi.org/10.3390/w13131718

**Chicago/Turabian Style**

Zobeyer, Hasan, Abul B. M. Baki, and Saika Nowshin Nowrin. 2021. "Interactions between Tandem Cylinders in an Open Channel: Impact on Mean and Turbulent Flow Characteristics" *Water* 13, no. 13: 1718.
https://doi.org/10.3390/w13131718