# Unconfined Unsteady Laminar Flow of a Power-Law Fluid across a Square Cylinder

## Abstract

**:**

## 1. Introduction

## 2. The Mathematical Model and Numerical Code

^{−10}was used for the x- and y-velocity components, as well as for the residuals of continuity. In order to treat an unbounded domain the four computational boundaries were placed far away from the cylinder (at distance 25,000D in all directions) with AB = BC = CE = EA = 50,000D. The grid near the surface of the cylinder was sufficiently fine in order to adequately resolve the flow characteristics near the cylinder. For the refinement of the grid at the cylinder surface the grid-adaptation function of ANSYS FLUENT was used. This very large calculation domain has been used by the present author in three recent papers [10,11,12].

## 3. Results and Discussion

## 4. Conclusions

- The drag coefficient and the Strouhal number are higher in a confined flow compared to those of an unconfined flow. This is valid both for Newtonian and non-Newtonian fluids.
- In a confined flow some minor frequencies are suppressed and disappear. This means that some flow characteristics are lost in a confined flow.
- Complete results for the drag coefficient in the entire shear-thinning and shear-thickening region have been produced. Each drag curve is high at low and high power-law index values and shows a minimum at the shear-thinning region.
- The influence of the Re number on the drag coefficient is opposite in shear-thinning and shear-thickening regions. In the first the ${c}_{D}$ increases with increasing Re and in the second the ${c}_{D}$ decreases with increasing Re.
- Complete results for the Strouhal number in the entire shear-thinning and shear-thickening region have been produced. Each St curve is low at low and high power-law index valuesand shows a maximum at the shear-thinning region.
- The influence of the Re number on the St number is opposite in shear-thinning and shear-thickening regions. In the first, the St decreases with increasing Re and, in the second, the St increases with increasing Re.
- A remarkable correlation exists between the drag coefficient and the Strouhal number. The drag coefficient becomes minimum when the Strouhal number reaches its maximum.
- In shear-thinning fluid chaotic structures exist which diminish at higher values of the power-law index.

## Conflicts of Interest

## References

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**Figure 3.**Variation of drag coefficient across a square cylinder for a Newtonian fluid as a function of Reynolds number at different non-dimensional heights of computational domain.

**Figure 4.**Variation of Strouhal number in a square cylinder for a Newtonian fluid as a function of the Reynolds number at different non-dimensional heights of the computational domain.

**Figure 5.**Unsteady flow for a Newtonian fluid with Re = 100: (

**A**) time series for ${c}_{D}$; (

**B**) time series for ${c}_{L}$; (

**C**) power spectrum for ${c}_{L}$; and (

**D**) power spectrum for ${c}_{D}$.

**Figure 6.**Values of drag coefficients for different Reynolds numbers for shear-thinning and shear-thickening fluids: (

**A**) Re = 80; (

**B**) Re = 100; (

**C**) Re = 120; (

**D**) Re = 160. The solid line represents the present work with H/D = 50,000; the broken line represents the work of Sahu et al. [8] with H/D = 20.

**Figure 7.**Values of Strouhal number for different Reynolds numbers for shear-thinning and shear-thickening fluids: (

**A**) Re = 80; (

**B**) Re = 100; (

**C**) Re= 120; (

**D**) Re = 160. The solid line represents the present work with H/D = 50,000; the broken line represents the work of Sahu et al. [8] with H/D = 20.

**Figure 8.**Values of the drag coefficient for different Reynolds numbers for shear-thinning and shear-thickening fluids in an unconfined field.

**Figure 9.**Values of the Strouhal number for different Reynolds numbers for shear-thinning and shear-thickening fluids in an unconfined field.

**Figure 10.**Power spectra of the lift coefficient for Re = 100 and different values of the power-law index: (

**A**) n =0.1; (

**B**) n = 0.3; (

**C**) n = 1; (

**D**) n = 2.

Number of Grid Pints | n = 0.5 | n = 1.6 |
---|---|---|

21,500 | 1.2968 | 1.6165 |

38,000 | 1.3023 | 1.6234 |

150,000 | 1.3168 | 1.6261 |

340,000 | 1.3168 | 1.6261 |

**Table 2.**Comparison of drag coefficients and Strouhal numbers for n = 1.4 and different Reynolds numbers.

Re | Source | ${\mathit{c}}_{\mathit{D}}$ | S_{t} |
---|---|---|---|

60 | Nikfarjam and Sohankar [18] | 1.78 | 0.1120 |

60 | present | 1.67 | 0.1110 |

100 | Nikfarjam and Sohankar [18] | 1.66 | 0.1300 |

100 | present | 1.56 | 0.1295 |

160 | Nikfarjam and Sohankar [18] | 1.58 | 0.1450 |

160 | present | 1.47 | 0.1422 |

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**MDPI and ACS Style**

Pantokratoras, A.
Unconfined Unsteady Laminar Flow of a Power-Law Fluid across a Square Cylinder. *Fluids* **2016**, *1*, 37.
https://doi.org/10.3390/fluids1040037

**AMA Style**

Pantokratoras A.
Unconfined Unsteady Laminar Flow of a Power-Law Fluid across a Square Cylinder. *Fluids*. 2016; 1(4):37.
https://doi.org/10.3390/fluids1040037

**Chicago/Turabian Style**

Pantokratoras, Asterios.
2016. "Unconfined Unsteady Laminar Flow of a Power-Law Fluid across a Square Cylinder" *Fluids* 1, no. 4: 37.
https://doi.org/10.3390/fluids1040037