# Computational Modeling of Flow and Scour around Two Cylinders in Staggered Array

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## Abstract

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## 1. Introduction

## 2. Numerical Model

#### 2.1. Hydrodynamic

_{i}, x

_{j}are the coordinates, t is the time, $\overline{P}$ is the modified pressure, ρ is the density of the fluid, u

_{i}and u

_{j}are the filtered velocity components, υ is the kinematic viscosity, and S

_{ij}is the resolved strain rate tensor:

_{t}is the turbulent viscosity, C

_{s}is the Smagorinsky constant (=0.16), and Δ is the filter size given as follows:

_{p}is the flow velocity parallel to the bed at the distance of y

_{p}from the bed and κ is the von Kármán constant, u

_{*}is the local shear velocity, y

_{0}= k

_{s}/30. Here, k

_{s}is the roughness height and is evaluated by k

_{s}= 2.5d where d is the diameter of sand particle.

#### 2.2. Sediment Transport

_{D}is the drag coefficient evaluated by the relations proposed by Morsi and Alexander [29], $\overrightarrow{g}$ is the gravity vector, $\overrightarrow{n}$ is the unit normal vector, V

_{p}is a particle volume, and D/Dt is a material derivative (=$\partial /\partial \mathrm{t}+\overrightarrow{u}\xb7\nabla $). The lift force term on the right-hand side of Equation (9) is replaced by the theoretical and experimental relationships proposed by McLaughlin [30] and Mei [31]. It is also possible to estimate sediment transport rate within a scour hole through an empirical equation. For example, Dodaro et al. [32] and Dodaro et al. [33] modified the Einstein sediment transport model to simulate local scour evolution downstream of a rigid bed.

_{n}and v

_{t}are the normal and tangential velocities, respectively. The coefficients of β

_{n}and β

_{t}are set to 0.65 and 1.0 [34].

_{c}is the critical shear stress based on the criterion of initial rotation of the particle.

#### 2.3. Morphodynamic

_{2}and A

_{3}are the shape coefficient for spherical particles, which are π/4 and π/6, respectively. n

_{depo}and n

_{pickup}are the number of particles of deposition and pickup, respectively. The second term of right-hand side represents the effect of sliding. The bed elevation was defined in the cell center and was interpolated across other points by bilinear interpolation with second order accuracy.

## 3. Numerical Conditions

_{bulk}) was 0.25 m/s. In order to achieve the desired bulk velocity containing turbulent fluctuations at the upstream, a fringe region with periodic boundary conditions was placed at the upstream. The fully developed turbulence yielding from this region flowed into the main domain. The corresponding Reynolds and Froude numbers were 41,500 and 0.23, respectively. The bed consisted of non-cohesive fine sand materials with a uniform size of d = 0.85 mm. Figure 2 shows a sketch of two cylinders and present simulation cases. To investigate the effect of spacing ratio and alignment angle of the cylinders, a total of four spacing ratios (s/D = 1.25, 2.5, 3.75 and 5.0) and five alignment angles (α = 0°, 30°, 45°, 60°, 90°) were considered.

## 4. Numerical Results

#### 4.1. Flow Field

#### 4.2. Scour Process

_{s}) at each cylinder was normalized by the maximum scour depth of a single cylinder (d

_{s1}) [23]. For scour depth at the front cylinder, the bed rapidly evolved during tU

_{bulk}/D < 25, and the growth rate of the scour slowed down with time. Over time, the bed reached the equilibrium. The trend of maximum scour depth around the front cylinder was similar for each other, regardless of the alignment angle and spacing ratio. However, the scour depth and the growth rate of scour depended on them. For example, the growth rate of scour increased with an increase in the alignment angle for s/D = 1.25 at the initial stage (tU

_{bulk}/D < 25). In this time, the intense shear layers shed from sides of the cylinder generated an acceleration of flow at approximately 45° from the plane of the cylinder, which contributed to a rapid increase in local scour depth. In addition, the interference of flow was highly dominated due to an increase in the projected area, and thus it could result in high dynamic interactions between the turbulent flow and the bed. Such mechanisms were responsible for the bed scouring at the initial stage. At the later stage, the growth rate of scour was similar to each other, even though the scour depths at the equilibrium were estimated to be d

_{s}/d

_{s1}= 1.08, 1.08, 1.13, 1.15, 1.16 in α = 0°, 30°, 45°, 60°, 90°. For s/D = 2.5, the time evolution of scour for α = 0°, 90° was relatively faster than that of α = 30°, 45°, 60° which indicated that the turbulent stress near the bed was higher due to interference effect for α = 0°, 90°. During later stage, the deepening of the bed occurred gradually, and the growth rate of scour becomes slow. The temporal evolution of maximum scour was almost identical for s/D = 5.0, due to significant decrease in the interference effect. Unlike the local scouring around the front cylinder, the rate of scour growth around the rear cylinder was significantly affected by both the alignment angle and spacing ratio. In particular, apparent difference depended on the alignment angle appears in s/D = 1.25. For an alignment angle of α = 60° and 90°, the rate of scour growth increased rapidly in the initial time (tU

_{bulk}/D < 10). However, the scouring rate of α = 90° evolved much slower than that of α = 60°, and its value of the final scour depth at the equilibrium became lower. The maximum scouring regions were placed near the side, and the gap for α = 60° and 90°, respectively. For α = 90°, the maximum scoured region was located at the gap between two cylinders, resulting from local acceleration of flow at the contraction area. We speculate that the shear layer from the front cylinder for α = 60° drove much higher momentum to the side of the rear cylinder than that of α = 90°, which could contribute to an increase in turbulent stress near the bed. The growth rate and scour depth decreased with a decrease in the alignment angle. For α = 0°, the two cylinders were regarded as a single structure elongated to the downstream direction, and the growth rate of scour declined considerably, due to the shielding effect which interrupted the horseshoe vortices. Thus, the scouring at the nose of the rear cylinder was almost not observed, and scour appeared only at the sides of the rear cylinder. The local scour depth was estimated to be d

_{s}/d

_{s1}= 0.77. For s/D = 2.5, the rate of growth scour for α = 45°, 60°, 90° showed a similar trend while it slowed down, and scour depth decreased for α = 0° and 30°. This demonstrates that the shielding effect due to the front cylinder was still valid for s/D ≤ 2.5, with α ≤ 30° corresponding to the reattachment regime. As the spacing ratio increased, the shielding effect decreased and the growth rate of scour around the rear cylinder in s/D = 5.0 was similar except for α = 0°.

#### 4.3. Equilibrium Bed Topography

#### 4.4. Scour Depth

_{s}/d

_{s1}= 1.06 to 1.14. The computed scour depth was in good agreement with experimental data [36]. For s/D = 1.25, the scour depth increased as the alignment angle increased, and it reached a maximum value of d

_{s}/d

_{s1}= 1.37 at α = 90°. This trend demonstrates that when the alignment angle becomes larger, the contraction effect and the size of turbulent horseshoe vortices increase due to an increase in the projected area of the cylinders. For s/D = 2.5, the scour depth decreased first with increasing alignment angle, and subsequently it increased slightly. When the spacing ratio was larger than s/D = 2.5, the trend of scour depth versus the alignment angle became similar. Even though the scour depth around the front cylinder was affected by the alignment angle, the scour depth varied with a range of d

_{s}/d

_{s1}= 1.06 to 1.37.

_{s}/d

_{s1}> 1.0. From the computed result, the behavior of bed scouring around the rear cylinder was significantly linked to the shielding effect.

_{s}/d

_{s1}= 1.0.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Schematic representation of flow around two cylinders and simulation cases with staggered array: The approaching velocity U

_{bulk}is from left to right. The α is the alignment angle between two cylinders and the dot represent the center location of another cylinder.

**Figure 3.**Computational grids around two cylinders with an alignment angle of 45°: (

**a**) s/D = 1.25; (

**b**) s/D = 5.0.

**Figure 4.**Instantaneous isosurface of Q-criterion on equilibrium bed at different alignment angles of two cylinders with three spacing ratio (s/D = 1.25 (

**Left**); s/D = 2.5 (

**Middle**); s/D = 1.25 (

**Right**)): (

**a**) α = 0°; (

**b**) α = 30°; (

**c**) α = 60°.

**Figure 5.**Time series of instantaneous shear velocity (α = 0° (

**Left**); α = 30° (

**Middle**); α = 60° (

**Right**)): (

**a**) s/D=1.25 and (

**b**) s/D=5.0.

**Figure 6.**Instantaneous views of the bed elevation for an alignment angle of 45° with s/D = 1.25 at 10 s, 240 s, 1800 s and equilibrium: (

**a**) Time = 10 s; (

**b**) Time = 240 s; (

**c**) Time = 1800 s; (

**d**) Time = Equilibrium.

**Figure 7.**Instantaneous views of the bed elevation for an alignment angle of 45° with s/D = 5.0 at 10 s, 240 s, 1800 s and final time: (

**a**) Time = 10 s; (

**b**) Time = 240 s; (

**c**) Time = 1800 s; (

**d**) Time = Equilibrium.

**Figure 8.**Maximum scour profiles around each cylinder for s/D = 1.25 (

**Upper**), s/D = 2.5 (

**Middle**), s/D = 5.0 (

**Lower**): (

**a**) Front cylinder; (

**b**) Rear cylinder.

**Figure 9.**Bed topography at the equilibrium for alignment angles of α = 0°, 30°, 60°: (

**a**) s/D = 1.25; (

**b**) s/D = 2.5; (

**c**) s/D = 3.75; (

**d**) s/D = 5.0.

**Figure 10.**Local scour depth: (

**a**) Schematic view of maximum scour areas around two cylinders; (

**b**) Maximum scour depth at the front cylinder; (

**c**) Maximum scour depth at the rear cylinder; (

**d**) Local scour depth at the center location between two cylinders.

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**MDPI and ACS Style**

Kim, H.S.; Roh, M.; Nabi, M.
Computational Modeling of Flow and Scour around Two Cylinders in Staggered Array. *Water* **2017**, *9*, 654.
https://doi.org/10.3390/w9090654

**AMA Style**

Kim HS, Roh M, Nabi M.
Computational Modeling of Flow and Scour around Two Cylinders in Staggered Array. *Water*. 2017; 9(9):654.
https://doi.org/10.3390/w9090654

**Chicago/Turabian Style**

Kim, Hyung Suk, Min Roh, and Mohamed Nabi.
2017. "Computational Modeling of Flow and Scour around Two Cylinders in Staggered Array" *Water* 9, no. 9: 654.
https://doi.org/10.3390/w9090654