# Numerical Analysis of Fluid Forces for Flow Past a Square Rod with Detached Dual Control Rods at Various Gap Spacing

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

_{1}= 1–5 and g

_{2}= 0.5–5 (where g

_{1}is the gap between the upstream control rod and the main rod, and g

_{2}is the space between the main rod and the downstream control rod) at Re = 160. The simulation results were obtained in the form of vorticity contour, drag and lift coefficients, Strouhal number, and force statistics. Under the effect of gap spacing, three different flow modes were found and named according to their behavior. It was found that the mean drag coefficient showed decreasing behavior by increasing the value of g

_{2}continually at a fixed value of g

_{1}. The largest value of $Cdmean$ was found at (g

_{1}, g

_{2}) = (1, 1) and the greatest percentage reduction in $Cdmean$ was obtained at (g

_{1}, g

_{2}) = (1, 3), which is 139.72%. The effect of thrust was also noticed for all selected values of g

_{1}and g

_{2}. Furthermore, it was noticed that the Strouhal number and the root mean square values of the drag and lift coefficients smaller values than the single rod values, except for the Clrms value of (g

_{1}, g

_{2}) = (1, 3) and (1, 4).

## 1. Introduction

## 2. The Lattice Boltzmann Method

## 3. Problem Statements and Boundary Conditions

## 4. Computational Domain, Grid Independence and Code Validation Study

## 5. Results and Discussion

#### 5.1. Vorticity Contours Visualization, Time-History Analysis of Drag and Lift Coefficients, and Energy Spectra Analysis of the Lift Coefficient

#### 5.2. Physical Parameters

## 6. Conclusions

- (i)
- Three different types of flow modes were found and were named (a) shear layer reattachment (SLR), (b) steady flow mode (SF), and (c) semi-developed vortex shedding (SDVS).
- (ii)
- The $\mathit{C}\mathit{d}\mathit{m}\mathit{e}\mathit{a}\mathit{n}$ values were negative for all selected combinations of ${\mathit{g}}_{\mathbf{1}}$ and ${\mathit{g}}_{\mathbf{2}}$ due to the effect of trust.
- (iii)
- The values of $\mathit{C}\mathit{d}\mathit{m}\mathit{e}\mathit{a}\mathit{n}$ decreased by increasing the gap spacing. The maximum value of $\mathit{C}\mathit{d}\mathit{m}\mathit{e}\mathit{a}\mathit{n}$ was obtained at $\left({\mathit{g}}_{\mathbf{1}},{\mathit{g}}_{\mathbf{2}}\right)=\left(\mathbf{1},\mathbf{1}\right){i}.{e}.,-\mathbf{0.3956}.$
- (iv)
- The values of $\mathit{C}\mathit{d}\mathit{r}\mathit{m}\mathit{s}$ and $\mathit{C}\mathit{l}\mathit{r}\mathit{m}\mathit{s}$ increased by increasing the value of ${\mathit{g}}_{\mathbf{2}}$ at fixed values of ${\mathit{g}}_{\mathbf{1}}.$ The maximum values of $\mathit{C}\mathit{d}\mathit{r}\mathit{m}\mathit{s}$ and $\mathit{C}\mathit{l}\mathit{r}\mathit{m}\mathit{s}$ were obtained at 0.0084 and 0.2910, respectively.
- (v)
- The greatest reduction in $\mathit{C}\mathit{d}\mathit{m}\mathit{e}\mathit{a}\mathit{n}$ was obtained at $\left({\mathit{g}}_{\mathbf{1}},\text{}{\mathit{g}}_{\mathbf{2}}\right)=\text{}\left(\mathbf{1},\text{}\mathbf{3}\right),$ and this value was 139.72%.
- (vi)
- The minimum reduction was acquired at $\left({\mathit{g}}_{\mathbf{1}},\text{}{\mathit{g}}_{\mathbf{2}}\right)=\left(\mathbf{1},\text{}\mathbf{1}\right)$, and this value was 132.1%.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Cd | Drag |

Cl | Lift |

$Cdmean$ | Mean drag force |

$Cdrms$ | Root-mean-square value of drag force |

$Clrms$ | Root-mean-square value of lift force |

${C}_{s}$ | Speed of sound |

$h$ | Height of the control rods |

$d$ | Size of the main rod |

$l$ | Length of the control rods |

${e}_{i}$ | Velocities direction |

${F}_{d}$ | Horizontal component of force |

${F}_{1}$ | Transverse component of force |

f_{s} | Vortex shedding |

${h}_{i}$ | Density distribution function |

${h}_{i}{}^{\left(eq\right)}$ | Equilibrium distribution function |

$Lu$ | Upstream position |

$Ld$ | Downstream position |

$Q$ | Number of particles |

$Re$ | Reynolds number |

$St$ | Strouhal number |

${U}_{\infty}$ | Uniform inflow velocity |

SF | Steady flow |

SLR | Shear layer reattachment |

SDVS | Semi developed vortex shedding |

SR | Single rod |

Greek Symbols | |

$\nu \text{}$ | Kinematic viscosity |

$\rho \text{}$ | Fluid density |

$\xi $ | Weighting coefficients |

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**Figure 2.**Schematic diagram for flow over a square rod with detached upstream/downstream control rods at fixed Re = 160.

**Figure 3.**Comparison of different uniform inflow velocities at (

**a**) U

_{∞}= 0.03456217, (

**b**) 0.0438596, and (

**c**) 0.05383556.

**Figure 4.**(

**a**–

**e**) Vorticity contour visualization of the shear layer reattachment (SLR) flow mode. (

**f**–

**k**) Time trace analysis of drag and lift coefficients for the SLR flow mode. (

**l**–

**n**) Energy spectrum analyses of lift coefficients for the SLR flow mode.

**Figure 6.**(

**a**–

**c**) Vorticity contour visualization of the SDVS flow mode. (

**d**–

**i**) Time trace analysis of drag and lift coefficients for the SDVS flow mode. (

**j**–

**l**) Energy spectrum analyses of lift coefficients for the SDVS flow mode.

**Figure 7.**(

**a**–

**e**) The effect of the spacing ratio (g

_{1}= 1–5, g

_{2}= 0–5) of the upstream and downstream control rods to the main rod at a fixed Re = 160 in terms of statistics force and percentage reduction in Cdmean.

**Table 1.**Physical parameters for different computational domains at g

_{1}= 1, g

_{2}= 2, and Re = 160.

Cases | Cdmean | Clrms | St | |
---|---|---|---|---|

(I) | Lu = 7.0 d; Ld = 33.0 d; H = 8.0 d | −0.425 | 0.1099 | 0.102 |

(II) | Lu = 8.0 d; Ld = 33.0 d; H = 8.0 d | −0.426 | 0.1095 | 0.096 |

(III) | Lu = 9.0 d; Ld = 33.0 d; H = 11.0 d | −0.427 | 0.1091 | 0.096 |

(IV) | Lu = 8.0 d; Ld = 30.0 d; H = 8.0 d | −0.426 | 0.1097 | 0.099 |

(V) | Lu = 8.0 d; Ld = 35.0 d; H = 8.0 d | −0.426 | 0.1096 | 0.102 |

(VI) | Lu = 8.0 d; Ld = 33.0 d; H = 7.0 d | −0.422 | 0.0609 | 0.099 |

(VII) | Lu = 8.0 d; Ld = 33.0 d; H = 9.0 d | −0.492 | 0.3623 | 0.108 |

Cases | Cdmean | Cdrms | Clrms | St |
---|---|---|---|---|

d = 10.0 | 1.5932 | 0.0285 | 0.3688 | 0.3197 |

d = 20.0 | 1.5272 | 0.0229 | 0.3152 | 0.1712 |

d = 30.0 | 1.5458 | 0.0428 | 0.3033 | 0.2106 |

d = 40.0 | 1.5528 | 0.5921 | 0.3104 | 0.2193 |

**Table 3.**Comparison of Cdmean, St, Cdrms, and Clrms for flow past a single square cylinder at Re = 200.

Re = 200 | Cdmean | St |
---|---|---|

Saha et al. [33] | 1.670 | 0.163 |

Sohankar et al. [34] | 1.424 | 0.165 |

Okajima [35] | 1.480 | 0.138 |

Norberg [36] | 1.450 | 0.152 |

Abograis and Alshayji [37] | 1.488 | 0.153 |

Present | 1.519 | 0.155 |

Re = 200 | Cdrms | Clrms |

Sohankar et al. [34] | 0.012 | 0.012 |

Abograis and Alshayji [37] | 0.027 | 0.027 |

Present | 0.038 | 0.038 |

Flow Modes | (g_{1}, g_{2}) |
---|---|

Shear Layer Reattachment | (1, 0.5), (1, 1), (1, 1.5), (1, 2), (2, 0), (2, 0.5), (2, 1), (2, 1.5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 0), (3, 1.5), (3, 2), (3, 3), (3, 4), (3, 5), (4, 1.5), (4, 2), (4, 3), (4, 4), (4, 5), (5, 1.5), (5, 2), (5, 3), (5, 4), (5, 5) |

Steady | (3, 0.5), (3, 1), (4, 0), (4, 0.5), (4, 1), (5, 0), (5, 0.5), (5, 1) |

Semi-Developed Vortex Shedding | (1, 3), (1, 4), (1, 5) |

% Reduction Cdmean | g_{2} = 0.5 | g_{2} = 1 | g_{2} = 1.5 | g_{2} = 2 | g_{2} = 3 | g_{2} = 4 | g_{2} = 5 |
---|---|---|---|---|---|---|---|

g_{1} = 1 | 135.21 | 132.11 | 135.06 | 134.58 | 139.72 | 139.22 | 139.22 |

g_{1} = 2 | 137.11 | 136.94 | 137.58 | 139.09 | 139.37 | 139.03 | 138.85 |

g_{1} = 3 | 137.31 | 137.45 | 137.64 | 137.82 | 138.08 | 137.88 | 137.65 |

g_{1} = 4 | 136.52 | 136.57 | 136.58 | 135.91 | 136.39 | 136.62 | 136.63 |

g_{1} = 5 | 134.30 | 134.36 | 133.87 | 132.79 | 134.21 | 134.60 | 134.53 |

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**MDPI and ACS Style**

Manzoor, R.; Ghaffar, A.; Baleanu, D.; Nisar, K.S.
Numerical Analysis of Fluid Forces for Flow Past a Square Rod with Detached Dual Control Rods at Various Gap Spacing. *Symmetry* **2020**, *12*, 159.
https://doi.org/10.3390/sym12010159

**AMA Style**

Manzoor R, Ghaffar A, Baleanu D, Nisar KS.
Numerical Analysis of Fluid Forces for Flow Past a Square Rod with Detached Dual Control Rods at Various Gap Spacing. *Symmetry*. 2020; 12(1):159.
https://doi.org/10.3390/sym12010159

**Chicago/Turabian Style**

Manzoor, Raheela, Abdul Ghaffar, Dumitru Baleanu, and Kottakkaran Sooppy Nisar.
2020. "Numerical Analysis of Fluid Forces for Flow Past a Square Rod with Detached Dual Control Rods at Various Gap Spacing" *Symmetry* 12, no. 1: 159.
https://doi.org/10.3390/sym12010159