# Energy Analysis of a Quasi-Two-Dimensional Friction Model for Simulation of Transient Flows in Viscoelastic Pipes

^{1}

^{2}

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## Abstract

**:**

_{P}) and frictional work (D

_{f}) increase with an increase in Re. However, when the initial Re values are high, the D

_{f}values are much larger than the W

_{P}values. In addition, for Re < 3 × 10

^{5}, the 1D model underestimated the viscoelastic terms. There was no significant difference between the two models for Re > 3 × 10

^{5}. In the case of different initial Re values, the two models produced almost the same values for W

_{P}. This study provides a theoretical basis for investigating transient flow from the perspective of energy analysis.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Governing Equations

#### 2.2. Kelvin-Voigt Model

_{0}= 1/E

_{0}, E

_{0}is the elastic modulus, ${J}_{k}=1/{E}_{k}$ is the creep compliance of the k-th element, ${\tau}_{k}$ is the retarded time of the k-th element, and ${E}_{k}$ is the elasticity modulus of the k-th element.

_{e}) and retarded component (ε

_{r}):

#### 2.3. Numerical Scheme of 1D and Quasi-2D Models

_{j}is the radial distance between the outer surface of cylinder j and the pipe centre, and r

_{cj}is the radial distance between the centre of cylinder j and the pipe centre.

_{pi,j}, and K

_{ni,j}are known values, the elements of which depend on H, u, and q at the previous time level.

#### 2.4. ITE Method Based on Quasi-2D Friction Model

_{f}is the total rate of frictional dissipation, W

_{E}is the total rate of work from the ends of the pipe, and W

_{P}is the total rate of work from the pipe wall, expressed as follows:

## 3. Results: Experimental Setup and Validation

^{−4}. The K-V parameters were calibrated according to literature [19], and their values were as follows: 1D model (E

_{0}= 1556 N/mm

^{2}, E

_{1}= 7820 N/mm

^{2}, ${\tau}_{1}$ = 582.6 ms, E

_{2}= 18,370 N/mm

^{2}, ${\tau}_{2}$ = 59.76 ms, E

_{3}= 6842 N/mm

^{2}, and ${\tau}_{3}$ = 21,570 ms); 2D model (E

_{0}= 1563 N/mm

^{2}, E

_{1}= 9596 N/mm

^{2}, ${\tau}_{1}$ = 562.3 ms, E

_{2}= 19,490 N/mm

^{2}, ${\tau}_{2}$ = 52.18 ms, E

_{3}= 5834 N/mm

^{2}, and ${\tau}_{3}$ = 19,680 ms).

## 4. Discussion: Energy Analysis at Different Reynolds Numbers

#### 4.1. D_{f} in 1D and 2D Models

_{f}are shown in Figure 7. The results show that larger D

_{f}values correspond to greater values of the cumulative sum of the wall-shear stress of the entire pipeline. This indicates that the work done by the friction term becomes larger as Re values increase.

_{f}values calculated using the 2D model are larger than those calculated using the 1D model. It can be seen that the 1D model underestimated the instantaneous wall shear stress under flow conditions with initial Re values in the range of 1.0 × 10

^{5}–3.0 × 10

^{5}.

_{f}differed only slightly between the 1D and 2D models. This means that the instantaneous wall-shear stress calculated by the 1D model was close to that calculated by the 2D model, with an initial Re value of 4.0 × 10

^{5}–7.0 × 10

^{5}.

_{f}was maximal near the initial time and gradually decayed with time in both the 2D and 1D models.

_{f}values were positive for both models in the critical region of turbulence. Thus, the D

_{f}values show the dissipation of the friction energy in the transient flow. This is consistent with the description of transient flow energy in pipelines [10]. The D

_{f}values also increased with an increasing initial Re.

#### 4.2. W_{P} in 1D and 2D Models

_{P}(the work done by the viscoelastic term of the pipe wall per unit time) are shown in Figure 8. The W

_{P}values gradually increased with the initial Re values.

_{P}values from the 1D model were similar at all Re values to those from the 2D model because the governing equations of both models were consistent in calculating the retarded strain. The work due to the viscoelastic term of the pipe wall in the two friction models was essentially the same under different initial Re values. In particular, the maximum value of W

_{P}calculated by both friction models appeared at approximately 2 L/a for all Re values, and the W

_{P}values gradually decayed over time. In the critical region of turbulence, W

_{P}had both positive and negative values. This indicates that the interaction between the fluid and the pipe wall during transient flow involves both energy transfer and energy dissipation, which agrees with the description of the energy variation of transient flows in pipelines in the literature [10].

_{f}became significantly greater than W

_{P}. This shows that the work done by the friction term had a significant influence on the energy dissipation when the initial Re values were relatively high [13].

_{P}values of viscoelastic term were compared with the 1D model at different Reynolds number (Figure 9). From these figures, the difference of W

_{P}values between 1D model considering both friction and viscoelastic effects, and only considering the viscoelastic effect, increase with Re values, but compared with the increase of D

_{f}values at different Reynolds number, there was a slight increase in W

_{P}values in orders of magnitude. This is because the viscoelastic effect does not dominate under different initial Reynolds numbers [10,20].

#### 4.3. Energy integral in 2D Models

## 5. Conclusions

^{5}, the 1D (but not the 2D) model underestimated the work due to the friction term D

_{f}. For Re > 3.0 × 10

^{5}, this error was reduced.

_{f}> W

_{P}for a large initial Re.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | cross-sectional area of the pipeline |

a | wave speed |

D | pipe diameter |

D_{f} | total rate of frictional dissipation |

${E}_{k}$ | elasticity modulus of the k-th element |

e | wall thickness |

g | gravitational acceleration |

H | pressure head |

J | creep compliance of the k-th element |

j | subscript representing the radial grid number |

N_{r} | number of segments along the radius |

N_{r0} | number of cylinders along the radius |

Q | discharge |

q | radial flux |

r | radial distance from the pipe centre |

r_{cj} | radial distance between the centre of the cylinder j cross-section and the pipe centre |

r_{j} | radial distance between the outer surface of the cylinder j cross-section and the pipe centre |

T | total kinetic energy of the system |

t | time |

U | total internal energy of the system |

u | longitudinal velocity |

v | radial velocity |

W_{E} | total rate of work from the ends of the pipe |

W_{P} | total rate of work from the pipe wall |

x | axial coordinate along the pipe |

Greek Symbols | |

$\alpha $ | constraint coefficient |

ε, θ | weighting coefficients |

${\epsilon}_{r}$ | retarded strain |

$\rho $ | density |

$\tau $ | shear stress |

${\tau}_{k}$ | retarded time of the k-th element |

${\tau}_{w}$ | pipe-wall shear stress |

$\gamma $ | bulk weight |

Abbreviations | |

quasi-2D | quasi-two-dimensional |

1D | one-dimensional |

Re | Reynolds number |

K-V | Kelvin-Voight |

HDPE | high-density polyethylene pipe |

MOC | method of characteristics |

## Appendix A. Derivation of Equation (29)

## References

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**Figure 2.**Variation in pressure head with time: (

**a**) results of experiment and two simulations at Re = 55,940; (

**b**) simulation results at different Re values.

**Figure 5.**The time difference between the extreme value of retarded strain and instantaneous strain.

**Figure 7.**Frictional work D

_{f}at different Re values in the critical region: (

**a**) 1D model; (

**b**) 2D model.

**Figure 8.**Viscoelastic work, W

_{P}, at different Re values in the critical region: (

**a**) 1D model; (

**b**) 2D model.

**Figure 9.**Work done of W

_{P}by 1D model including only viscoelastic effect: (

**a**) Re = 13,880; (

**b**) Re = 69,680.

**Figure 10.**Energy variation of 2D model in critical region at different Re values: (

**a**) Re = 13,880; (

**b**) Re = 28,320; (

**c**) Re = 40,950; (

**d**) Re = 55,940; (

**e**) Re = 69,680.

**Table 1.**Experimental settings [19].

Case | Q (L/s) | Re | H_{0} (m) | T_{c} (s) |
---|---|---|---|---|

1 | 1 | 13,380 | 21.63 | 0.0875 |

2 | 2.04 | 28,320 | 21.13 | 0.0752 |

3 | 2.95 | 40,950 | 20.74 | 0.1188 |

4 | 4.03 | 55,940 | 20.34 | 0.1575 |

5 | 5.02 | 69,680 | 19.82 | 0.1533 |

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**MDPI and ACS Style**

Wu, K.; Feng, Y.; Xu, Y.; Liang, H.; Liu, G.
Energy Analysis of a Quasi-Two-Dimensional Friction Model for Simulation of Transient Flows in Viscoelastic Pipes. *Water* **2022**, *14*, 3258.
https://doi.org/10.3390/w14203258

**AMA Style**

Wu K, Feng Y, Xu Y, Liang H, Liu G.
Energy Analysis of a Quasi-Two-Dimensional Friction Model for Simulation of Transient Flows in Viscoelastic Pipes. *Water*. 2022; 14(20):3258.
https://doi.org/10.3390/w14203258

**Chicago/Turabian Style**

Wu, Kai, Yujie Feng, Ying Xu, Huan Liang, and Guohong Liu.
2022. "Energy Analysis of a Quasi-Two-Dimensional Friction Model for Simulation of Transient Flows in Viscoelastic Pipes" *Water* 14, no. 20: 3258.
https://doi.org/10.3390/w14203258