# Large Eddy Simulation of Pulsatile Flow through a Channel with Double Constriction

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

#### 2.1. Boundary Conditions and Computational Parameters

#### 2.2. Overview of Numerical Procedures

## 3. Data Processing

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Streamwise inlet velocity $\overline{v}/{\overline{V}}_{max}$ (

**a**) near the wall $x/L=0.0057$, (

**b**) at the centre of the channel $x/L=0.5$ and (

**c**) at the different phases of the pulse while $Re=2000$.

**Figure 4.**Grid independence test for the mean streamwise velocity, $\langle \overline{v}\rangle /{\overline{V}}_{max}$, at (

**a**) $y/L=$ inlet, (

**b**) $y/L=0.0$, (

**c**) $y/L=1.0$, (

**d**) $y/L=2.0$, (

**e**) $y/L=3.0$, (

**f**) $y/L=4.0$, (

**g**) $y/L=5.0$, (

**h**) $y/L=6.0$, (

**i**) $y/L=8.0$, (

**j**) $y/L=10.0$, (

**k**) $y/L=12.0$ and (

**l**) $y/L=$ outlet, while $Re=2000$, Case 1: $50\times 300\times 50$, Case 2: $50\times 350\times 50$ and Case 3: $70\times 350\times 50$ control volumes.

**Figure 5.**Grid independence test for the turbulent kinetic energy (TKE), $\frac{1}{2}}\langle {u}_{j}^{\prime \prime}{u}_{j}^{\prime \prime}\rangle /{\overline{V}}_{max}^{2$, (

**a**) $y/L=$ inlet, (

**b**) $y/L=0.0$, (

**c**) $y/L=1.0$, (

**d**) $y/L=2.0$, (

**e**) $y/L=3.0$, (

**f**) $y/L=4.0$, (

**g**) $y/L=5.0$, (

**h**) $y/L=6.0$, (

**i**) $y/L=8.0$, (

**j**) $y/L=10.0$, (

**k**) $y/L=12.0$ and (

**l**) $y/L=$ outlet, while $Re=2000$; Case 1: $50\times 300\times 50$, Case 2: $50\times 350\times 50$ and Case 3: $70\times 350\times 50$ control volumes.

**Figure 6.**Dynamic Smagorinsky constant, ${C}_{s}$, for (

**a**) $Re=1000$, (

**b**) $Re=1400$, (

**c**) $Re=1700$ and (

**d**) $Re=2000$ at $t/T=10.25$.

**Figure 7.**Normalised sub-grid scale (SGS) eddy viscosity, ${\mathsf{\mu}}_{sgs}/\mathsf{\mu}$, for (

**a**) $Re=1000$, (

**b**) $Re=1400$, (

**c**) $Re=1700$ and (

**d**) $Re=2000$ at $t/T=10.25$.

**Figure 8.**Instantaneous cross-stream velocity vectors based on the velocity components $\overline{u}-\overline{v}$, (

**a**) $y/L=$ inlet, (

**b**) $y/L=0.0$, (

**c**) $y/L=1.0$, (

**d**) $y/L=2.0$, (

**e**) $y/L=3.0$, (

**f**) $y/L=4.0$, (

**g**) $y/L=6.0$ and (

**h**) $y/L=8.0$ while $Re=2000$.

**Figure 9.**Mean streamwise streamlines appended on the mean streamwise velocity, $<\overline{v}>/{\overline{V}}_{max}$, for (

**a**) $Re=1000$, (

**b**) $Re=1400$, (

**c**) $Re=1700$ and (

**d**) $Re=2000$.

**Figure 10.**Contour plot of the instantaneous streamwise velocity, $\overline{v}/{\overline{V}}_{max}$, for (

**a**) $Re=1000$, (

**b**) $Re=1400$, (

**c**) $Re=1700$ and (

**d**) $Re=2000$ at $t/T=10.25$.

**Figure 11.**Instantaneous wall shear stress, ${\mathsf{\tau}}_{w}/\mathsf{\rho}{\overline{V}}_{max}^{2}$, at the (

**a**) upper wall and (

**b**) lower wall for the different Reynolds numbers at $t/T=10.25$.

**Figure 12.**Time-mean pressure, $\langle \overline{p}\rangle /\mathsf{\rho}{\overline{V}}_{max}^{2}$, at the (

**a**) upper wall and (

**b**) lower wall for the different Reynolds numbers.

**Figure 13.**rms of the cross-stream velocity fluctuations, ${\langle {u}^{\prime \prime}\rangle}_{rms}/{\overline{V}}_{max}$, at the different axial locations: (

**a**) $y/L=$ inlet, (

**b**) $y/L=0.0$, (

**c**) $y/L=1.0$, (

**d**) $y/L=1.5$, (

**e**) $y/L=2.0$, (

**f**) $y/L=2.5$, (

**g**) $y/L=3.0$, (

**h**) $y/L=3.5$, (

**i**) $y/L=4.0$, (

**j**) $y/L=4.5$, (

**k**) $y/L=5.0$, (

**l**) $y/L=5.5$, (

**m**) $y/L=6.0$, (

**n**) $y/L=7.0$, (

**o**) $y/L=8.0$, (

**p**) $y/L=10.0$, (

**q**) $y/L=12.0$ and (

**r**) $y/L=$ outlet, for the different Reynolds numbers.

**Figure 14.**rms of the streamwise velocity fluctuations, ${\langle {v}^{\prime \prime}\rangle}_{rms}/{\overline{V}}_{max}$, at the different axial locations: (

**a**) $y/L=$ inlet, (

**b**) $y/L=0.0$, (

**c**) $y/L=1.0$, (

**d**) $y/L=1.5$, (

**e**) $y/L=2.0$, (

**f**) $y/L=2.5$, (

**g**) $y/L=3.0$, (

**h**) $y/L=3.5$, (

**i**) $y/L=4.0$, (

**j**) $y/L=4.5$, (

**k**) $y/L=5.0$, (

**l**) $y/L=5.5$, (

**m**) $y/L=6.0$, (

**n**) $y/L=7.0$, (

**o**) $y/L=8.0$, (

**p**) $y/L=10.0$, (

**q**) $y/L=12.0$ and (

**r**) $y/L=$ outlet, for the different Reynolds numbers.

**Figure 15.**rms of the spanwise velocity fluctuations, ${\langle {w}^{\prime \prime}\rangle}_{rms}/{\overline{V}}_{max}$, at the different axial locations: (

**a**) $y/L=$ inlet, (

**b**) $y/L=0.0$, (

**c**) $y/L=1.0$, (

**d**) $y/L=1.5$, (

**e**) $y/L=2.0$, (

**f**) $y/L=2.5$, (

**g**) $y/L=3.0$, (

**h**) $y/L=3.5$, (

**i**) $y/L=4.0$, (

**j**) $y/L=4.5$, (

**k**) $y/L=5.0$, (

**l**) $y/L=5.5$, (

**m**) $y/L=6.0$, (

**n**) $y/L=7.0$, (

**o**) $y/L=8.0$, (

**p**) $y/L=10.0$, (

**q**) $y/L=12.0$ and (

**r**) $y/L=$ outlet, for the different Reynolds numbers.

**Figure 16.**rms of the pressure fluctuations, ${\langle {p}^{\prime \prime}\rangle}_{rms}/\mathsf{\rho}{\overline{V}}_{max}^{2}$, at the different axial locations: (

**a**) $y/L=$ inlet, (

**b**) $y/L=0.0$, (

**c**) $y/L=1.0$, (

**d**) $y/L=1.5$, (

**e**) $y/L=2.0$, (

**f**) $y/L=2.5$, (

**g**) $y/L=3.0$, (

**h**) $y/L=3.5$, (

**i**) $y/L=4.0$, (

**j**) $y/L=4.5$, (

**k**) $y/L=5.0$, (

**l**) $y/L=5.5$, (

**m**) $y/L=6.0$, (

**n**) $y/L=7.0$, (

**o**) $y/L=8.0$, (

**p**) $y/L=10.0$, (

**q**) $y/L=12.0$ and (

**r**) $y/L=$ outlet, for the different Reynolds numbers.

**Figure 17.**Streamwise velocity fluctuations ${v}^{\prime \prime}/{v}_{max}^{\prime \prime}$ against the time cycle $t/T$ of pulsation at the different axial locations while $Re=2000$ and $x/L=z/L=0.5$.

**Figure 18.**Streamwise velocity $v/{\overline{V}}_{max}$ against the time cycle $t/T$ of pulsation at (

**a**) $y/L$ = inlet, (

**b**) $y/L$ = 0 (centre of the first stenosis), (

**c**) $y/L$ = 1.0, (

**d**) $y/L$ = 2.0, (

**e**) $y/L$ = 3.0 (centre of the second stenosis), (

**f**) $y/L$ = 4.0 and (

**g**) $y/L$ = 5.0, while $Re=2000$ and $x/L=z/L=0.5$.

**Table 1.**Values of ${M}_{n}$ and ${\mathsf{\varphi}}_{n}$ for different harmonics according to Womersley [30].

${\mathit{N}}_{\mathit{h}}$ | ${\mathit{M}}_{\mathit{n}}$ | ${\mathsf{\varphi}}_{\mathit{n}}$ |
---|---|---|

1 | $0.78$ | $0.0113446$ |

2 | $1.32$ | $-1.4442599$ |

3 | $-0.74$ | $0.4625122$ |

4 | $-0.41$ | $-0.2879793$ |

Case | $\mathit{Re}$ | ${\mathit{N}}_{\mathit{x}}$ | ${\mathit{N}}_{\mathit{y}}$ | ${\mathit{N}}_{\mathit{z}}$ | $\mathbf{\Delta}\mathit{t}$ |
---|---|---|---|---|---|

0 | <2000 | 50 | 300 | 50 | ${10}^{-3}$ |

1 | 2000 | 50 | 300 | 50 | ${10}^{-3}$ |

2 | 2000 | 50 | 350 | 50 | ${10}^{-3}$ |

3 | 2000 | 70 | 350 | 50 | ${10}^{-3}$ |

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**MDPI and ACS Style**

Molla, M.M.; Paul, M.C.
Large Eddy Simulation of Pulsatile Flow through a Channel with Double Constriction. *Fluids* **2017**, *2*, 1.
https://doi.org/10.3390/fluids2010001

**AMA Style**

Molla MM, Paul MC.
Large Eddy Simulation of Pulsatile Flow through a Channel with Double Constriction. *Fluids*. 2017; 2(1):1.
https://doi.org/10.3390/fluids2010001

**Chicago/Turabian Style**

Molla, Md. Mamun, and Manosh C. Paul.
2017. "Large Eddy Simulation of Pulsatile Flow through a Channel with Double Constriction" *Fluids* 2, no. 1: 1.
https://doi.org/10.3390/fluids2010001