# Asymmetry Propagation in a Pipe Flow Downstream of a 90° Sharp Elbow Bend

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## Abstract

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## Featured Application

**Sharp pipe elbows are commonly used in industrial applications where space is constrained. The flow downstream of a sharp elbow is highly asymmetric, and it is crucial in some cases to determine how far this asymmetry extends. For example, accurate flow meters are usually calibrated based on a fully developed symmetric flow profile. We showed that in turbulent pipe flows, the recovery length is systematically shorter at higher Reynolds numbers ($\mathit{R}\mathit{e}$) regardless of the criteria used. Specifically, a recovery length of 10–40 pipe diameters (D) is observed at $\mathit{R}\mathit{e}=5600$ and 10–30 D at $\mathit{R}\mathit{e}=\mathrm{10,000}$. However, for a laminar flow, even a length of 100 D might not be sufficient for the flow to fully recover to a symmetric profile.**

## Abstract

## 1. Introduction

## 2. Numerical Setup

#### 2.1. Geometry and Computational Domain

_{c}= 1.08R, as shown in Figure 2, where R = D/2 is the pipe radius. The length of the straight pipe sections upstream and downstream of the elbow were 10 D (250 mm) and 100 D (2500 mm), respectively.

#### 2.2. Governing Equations and Numerical Schemes

#### 2.3. Computational Mesh

#### 2.4. Boundary Conditions and Statistics Accumulation

#### 2.5. Validation

## 3. Results

#### 3.1. Asymmetry in Wall Shear Stress

#### 3.2. Asymmetry in Mean Velocity Profiles

#### 3.3. Asymmetry in Velocity Fluctuation Profiles

^{−7}is attributed mainly to the rounding error. For the turbulent flows, the recovery length is about 20–40 at $Re=5600$ and 10 D at $Re=\mathrm{10,000}$, which is qualitatively in agreement with our previous observations using WSS criterium. It can be concluded that the flow asymmetry in velocity fluctuations does not seem to be more persistent than the asymmetry in time-averaged velocity profiles.

#### 3.4. Asymmetry in Vorticity Profiles

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Dean, W.R. LXXII. The Stream-Line Motion of Fluid in a Curved Pipe (Second Paper). Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1928**, 5, 673–695. [Google Scholar] [CrossRef] - Dean, W.R.; Hurst, J.M. Note on the Motion of Fluid in a Curved Pipe. Mathematika
**1959**, 6, 77–85. [Google Scholar] [CrossRef] - Prandtl, L. Essentials of Fluid Dynamics: With Applications to Hydraulics, Aeronautics, Meteorology and Other Subjects; Blackie & Son: Glasgow, UK, 1952. [Google Scholar]
- Kalpakli Vester, A.; Örlü, R.; Alfredsson, P.H. Turbulent Flows in Curved Pipes: Recent Advances in Experiments and Simulations. Appl. Mech. Rev.
**2016**, 68, 050802. [Google Scholar] [CrossRef] - Düz, H. Numerical and Experimental Study to Predict the Entrance Length in Pipe Flows. J. Appl. Fluid Mech.
**2019**, 12, 155–164. [Google Scholar] [CrossRef] - Doherty, J.; Ngan, P.; Monty, J.; Chong, M. The Development of Turbulent Pipe Flow. In Proceedings of the 16th Australasian Fluid Mechanics Conference, Gold Coast, Australia, 3–7 December 2007; pp. 266–270. [Google Scholar]
- Avila, M.; Barkley, D.; Hof, B. Transition to Turbulence in Pipe Flow. Annu. Rev. Fluid Mech.
**2023**, 55, 575–602. [Google Scholar] [CrossRef] - Mullin, T. Experimental Studies of Transition to Turbulence in a Pipe. Annu. Rev. Fluid Mech.
**2011**, 43, 1–24. [Google Scholar] [CrossRef] - Sudo, K.; Sumida, M.; Hibara, H. Experimental Investigation on Turbulent Flow in a Circular-Sectioned 90-Degree Bend. Exp. Fluids
**1998**, 25, 42–49. [Google Scholar] [CrossRef] - Sudo, K.; Sumida, M.; Hibarra, H. Experimental Investigation on Turbulent Flow through a Circular-Sectioned 180° Bend. Exp. Fluids
**2000**, 28, 51–57. [Google Scholar] [CrossRef] - Hellström, L.H.O.; Zlatinov, M.B.; Cao, G.; Smits, A.J. Turbulent Pipe Flow Downstream of a Bend. J. Fluid Mech.
**2013**, 735, R7. [Google Scholar] [CrossRef] - Tunstall, M.; Harvey, J. On the Effect of a Sharp Bend in a Fully Developed Turbulent Pipe-Flow. J. Fluid Mech.
**1968**, 34, 595–608. [Google Scholar] [CrossRef] - Sattarzadeh, S.S. Experimental Study of Complex Pipe Flow. Master’s Thesis, Royal Institute of Technology, Stockholm, Sweden, 2011. [Google Scholar]
- Kalpakli, A.; Örlü, R. Turbulent Pipe Flow Downstream a 90° Pipe Bend with and without Superimposed Swirl. Int. J. Heat Fluid Flow
**2013**, 41, 103–111. [Google Scholar] [CrossRef] - Kalpakli Vester, A.; Sattarzadeh, S.S.; Örlü, R. Combined Hot-Wire and PIV Measurements of a Swirling Turbulent Flow at the Exit of a 90° Pipe Bend. J. Vis.
**2016**, 19, 261–273. [Google Scholar] [CrossRef] - Canton, J.; Schlatter, P.; Örlü, R. Modal Instability of the Flow in a Toroidal Pipe. J. Fluid Mech.
**2016**, 792, 894–909. [Google Scholar] [CrossRef] - Lupi, V.; Canton, J.; Schlatter, P. Global Stability Analysis of a 90°-Bend Pipe Flow. Int. J. Heat Fluid Flow
**2020**, 86, 108742. [Google Scholar] [CrossRef] - Massaro, D.; Lupi, V.; Peplinski, A.; Schlatter, P. Global Stability of 180°-Bend Pipe Flow with Mesh Adaptivity. Phys. Rev. Fluids
**2023**, 8, 113903. [Google Scholar] [CrossRef] - Di Liberto, M.; Di Piazza, I.; Ciofalo, M. Turbulence Structure and Budgets in Curved Pipes. Comput. Fluids
**2013**, 88, 452–472. [Google Scholar] [CrossRef] - Noorani, A.; Schlatter, P. Swirl-Switching Phenomenon in Turbulent Flow through Toroidal Pipes. Int. J. Heat Fluid Flow
**2016**, 61, 108–116. [Google Scholar] [CrossRef] - Hufnagel, L.; Canton, J.; Örlü, R.; Marin, O.; Merzari, E.; Schlatter, P. The Three-Dimensional Structure of Swirl-Switching in Bent Pipe Flow. J. Fluid Mech.
**2018**, 835, 86–101. [Google Scholar] [CrossRef] - Röhrig, R.; Jakirlić, S.; Tropea, C. Comparative Computational Study of Turbulent Flow in a 90° Pipe Elbow. Int. J. Heat Fluid Flow
**2015**, 55, 120–131. [Google Scholar] [CrossRef] - He, X.; Apte, S.V.; Karra, S.K.; Doğan, Ö.N. An LES Study of Secondary Motion and Wall Shear Stresses in a Pipe Bend. Phys. Fluids
**2021**, 33, 115102. [Google Scholar] [CrossRef] - Al-Baghdadi, M.A.; Resan, K.K.; Al-Waily, M. CFD Investigation of the Erosion Severity in 3D Flow Elbow during Crude Oil Contaminated Sand Transportation. Eng. Technol. J.
**2017**, 35, 930–935. [Google Scholar] [CrossRef] - Ma, R.; Tang, R.; Gao, Z.; Yu, T. Optimized Design of Pipe Elbows for Erosion Wear. NATO Adv. Sci. Inst. Ser. E Appl. Sci.
**2024**, 14, 984. [Google Scholar] [CrossRef] - Ma, G.; Ma, H.; Sun, Z. Simulation of Two-Phase Flow of Shotcrete in a Bent Pipe Based on a CFD–DEM Coupling Model. NATO Adv. Sci. Inst. Ser. E Appl. Sci.
**2022**, 12, 3530. [Google Scholar] [CrossRef] - Henríquez Lira, S.; Torres, M.J.; Guerra Silva, R.; Zahr Viñuela, J. Numerical Characterization of the Solid Particle Accumulation in a Turbulent Flow through Curved Pipes by Means of Stokes Numbers. NATO Adv. Sci. Inst. Ser. E Appl. Sci.
**2021**, 11, 7381. [Google Scholar] [CrossRef] - Khalifa, A.; Gollwitzer, J.; Breuer, M. LES of Particle-Laden Flow in Sharp Pipe Bends with Data-Driven Predictions of Agglomerate Breakage by Wall Impacts. Fluids
**2021**, 6, 424. [Google Scholar] [CrossRef] - Gotfredsen, E.; Kunoy, J.D.; Mayer, S.; Meyer, K.E. Experimental Validation of RANS and DES Modelling of Pipe Flow Mixing. Heat Mass Transf.
**2020**, 56, 2211–2224. [Google Scholar] [CrossRef] - Wojewodka, M.M.; White, C.; Shahpar, S.; Kontis, K. Numerical Study of Complex Flow Physics and Coherent Structures of the Flow through a Convoluted Duct. Aerosp. Sci. Technol.
**2022**, 121, 107191. [Google Scholar] [CrossRef] - Rütten, F.; Schröder, W.; Meinke, M. Large-Eddy Simulation of Low Frequency Oscillations of the Dean Vortices in Turbulent Pipe Bend Flows. Phys. Fluids
**2005**, 17, 035107. [Google Scholar] [CrossRef] - Alabdalah, A.; Wnos, L. Numerical Simulation and Flow Analysis of a 90-Degree Elbow. Int. J. Civ. Mech. Energy Sci.
**2020**, 6, 10–13. [Google Scholar] [CrossRef] - Bilde, K.G.; Sørensen, K.; Hærvig, J. Decay of Secondary Motion Downstream Bends in Turbulent Pipe Flows. Phys. Fluids
**2023**, 35, 015102. [Google Scholar] [CrossRef] - Tunstall, R.; Laurence, D.; Prosser, R.; Skillen, A. Large Eddy Simulation of a T-Junction with Upstream Elbow: The Role of Dean Vortices in Thermal Fatigue. Appl. Therm. Eng.
**2016**, 107, 672–680. [Google Scholar] [CrossRef] - Zhou, M.; Costa Garrido, O.; Ma, S.; Zhang, N. Numerical Investigation of Turbulent Thermal Stratification at a Horizontally Oriented 90° Pipe-Elbow with Varied Elbow Radiuses. Int. J. Therm. Sci.
**2023**, 185, 108092. [Google Scholar] [CrossRef] - Kren, J.; Frederix, E.M.A.; Tiselj, I.; Mikuž, B. Numerical Study of Taylor Bubble Breakup in Counter-Current Flow Using Large Eddy Simulation. Phys. Fluids
**2024**, 36, 023311. [Google Scholar] [CrossRef] - Kren, J.; Zajec, B.; Tiselj, I.; Shawish, S.E.; Perne, Ž.; Tekavčič, M.; Mikuž, B. Dynamics of Taylor Bubble Interface in Vertical Turbulent Counter-Current Flow. Int. J. Multiph. Flow
**2023**, 165, 104482. [Google Scholar] [CrossRef] - OpenFOAM V10. Available online: https://openfoam.org/version/10/ (accessed on 17 February 2024).
- HPC Vega. Available online: https://en-vegadocs.vega.izum.si/ (accessed on 17 February 2024).
- Cerkovnik, K. Simulacija Turbuletnega Toka z Metodo Velikih Vrtincev v Cevi s Kolenom. Master’s Thesis, Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia, 2023. [Google Scholar]
- Nicoud, F.; Ducros, F. Subgrid-Scale Stress Modelling Based on the Square of the Velocity Gradient Tensor. Flow Turbul. Combust.
**1999**, 62, 183–200. [Google Scholar] [CrossRef] - Komen, E.; Shams, A.; Camilo, L.; Koren, B. Quasi-DNS Capabilities of OpenFOAM for Different Mesh Types. Comput. Fluids
**2014**, 96, 87–104. [Google Scholar] [CrossRef] - Mikuž, B.; Tiselj, I. Wall-Resolved Large Eddy Simulation in Grid-Free 5×5 Rod Bundle of MATiS-H Experiment. Nucl. Eng. Des.
**2016**, 298, 64–77. [Google Scholar] [CrossRef] - Frederix, E.; Tajfirooz, S.; Hopman, J.; Fang, J.; Merzari, E.; Komen, E. Two-Phase Turbulent Kinetic Energy Budget Computation in Co-Current Taylor Bubble Flow. Nucl. Sci. Eng.
**2023**, 197, 2585–2601. [Google Scholar] [CrossRef] - Fukagata, K.; Kasagi, N. Highly Energy-Conservative Finite Difference Method for the Cylindrical Coordinate System. J. Comput. Phys.
**2002**, 181, 478–498. [Google Scholar] [CrossRef] - El Khoury, G.K.; Schlatter, P.; Noorani, A.; Fischer, P.F.; Brethouwer, G.; Johansson, A.V. Direct Numerical Simulation of Turbulent Pipe Flow at Moderately High Reynolds Numbers. Flow Turbul. Combust.
**2013**, 91, 475–495. [Google Scholar] [CrossRef] - Komen, E.M.J.; Camilo, L.H.; Shams, A.; Geurts, B.J.; Koren, B. A Quantification Method for Numerical Dissipation in Quasi-DNS and under-Resolved DNS, and Effects of Numerical Dissipation in Quasi-DNS and under-Resolved DNS of Turbulent Channel Flows. J. Comput. Phys.
**2017**, 345, 565–595. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Dean vortices downstream of the pipe bend; (

**b**) Pipe bends with different bend angles and bend radii.

**Figure 2.**The entire computational domain with the 10 D long inlet section and a 100 D long straight pipe downstream of the sharp elbow. The origin of the coordinate system is set at the centre of the inlet, and the sharp pipe bending detail is shown in the bottom right side.

**Figure 3.**Two details of the computational meshes: (

**a**) Insight into the elbow section for the middle mesh. (

**b**) Pipe cross-section for the fine mesh.

**Figure 4.**Time-averaged streamwise velocity profiles for fully developed flow: (

**a**) $Re=5600$; (

**b**) $Re=\mathrm{10,000}$.

**Figure 5.**Time-averaged streamwise velocity fluctuations for fully developed flow: (

**a**) $Re=5600$; (

**b**) $Re=\mathrm{10,000}$.

**Figure 6.**Instantaneous velocity magnitude field for the three simulated cases: (

**a**) $Re=1400$; (

**b**) $Re=5600$; (

**c**) $Re=\mathrm{10,000}$. Blue and red color correspond to small and large velocity magnitude, respectively.

**Figure 7.**Evolution of Dean vortices downstream of the pipe elbow shows that these structures persist at a distance as far as 25 D. The coloured fields illustrate the direction of vorticity component x: blue is negative (counterclockwise) and red is positive (clockwise). Clearly, there is a shift (reversal) in vorticity direction at a distance between 8 D and 10 D. These results have been calculated on a time-averaged velocity field for flow at $Re=5600$.

**Figure 8.**The original profile (red) and its mirrored counterpart (black) of the time-averaged velocity across the pipe diameter at the distance of 5 D (

**a**) and 10 D (

**b**) downstream of the elbow. The profile’s asymmetry is proportional to the purple area between the two profiles.

**Figure 9.**Calculated time-averaged wall shear stress downstream of the pipe elbow is a function of axial and azimuthal coordinates, as well as $Re$: (

**a**) $Re=1400$; (

**b**) $Re=5600$; (

**c**) $Re=\mathrm{10,000}$. In order to visualize small variations in the calculated WSS, we plotted $\tau -{\tau}_{0}$ in logarithmic scale with ${\tau}_{0}$ being a positive value of ${\tau}_{0}=1.18\times {10}^{-5}{\mathrm{m}}^{2}/{\mathrm{s}}^{2}$ at $Re=1400$, ${\tau}_{0}=1.30\times {10}^{-4}{\mathrm{m}}^{2}/{\mathrm{s}}^{2}$ at $Re=5600$ and ${\tau}_{0}=3.25\times {10}^{-4}{\mathrm{m}}^{2}/{\mathrm{s}}^{2}$ at $Re=\mathrm{10,000}$.

**Figure 10.**Obtained asymmetry parameter for time-averaged velocity field $\overline{{U}_{x}}$ calculated in y- and z-profiles at different $Re$: (

**a**) $Re=1400$; (

**b**) $Re=5600$; (

**c**) $Re=\mathrm{10,000}$. Red curve indicates same quantity calculated in the recycling inlet pipe section, where no asymmetry is expected.

**Figure 11.**Asymmetry parameter for streamwise velocity fluctuations $\overline{{{u}_{x}}^{2}}$ calculated in y- and z-profiles at different $Re$: (

**a**) $Re=1400$; (

**b**) $Re=5600$; (

**c**) $Re=\mathrm{10,000}$. Red curve indicates same quantity calculated in the recycling inlet pipe section, where no asymmetry is expected.

**Figure 12.**Asymmetry parameter for vorticity field $\overline{\Vert \omega \Vert}$ calculated in y- and z-profiles at different $Re$: (

**a**) $Re=1400$; (

**b**) $Re=5600$; (

**c**) $Re=\mathrm{10,000}$. Red curve indicates same quantity calculated in the recycling inlet pipe section, where no asymmetry is expected.

**Table 1.**Meshing parameters for the three applied meshes. Cell sizes in wall units are calculated for the flow conditions at $Re=5600$.

Mesh | Coarse | Middle | Fine |
---|---|---|---|

Number of cells [in millions] | 0.305 | 5.95 | 13.7 |

Number of near-wall prism layers | 31 | 31 | 37 |

Stretching ratio (SR) | 1.0 | 1.1 | 1.1 |

Max. aspect ratio AR | 52.6 | 108 | 127 |

Non-orthogonality (mean/max) [°] | 5.6/35 | 4.1/45 | 3.9/46 |

Maximum skewness | 1.22 | 0.86 | 0.88 |

First cell height at walls [mm] | 0.3–0.6 | 0.04–0.56 | 0.02–0.56 |

Streamwise cell size [mm] | 11.6 | 1.2 | 0.6 |

First cell height at walls [wall units] | 2.7–5.4 | 0.36–4.8 | 0.21–4.8 |

Streamwise cell size [wall units] | 100 | 10 | 5 |

Reynolds Number (Re) | 1400 | 5600 | 10,000 |
---|---|---|---|

Mean (bulk) velocity ${U}_{b}$ [m/s] | 0.04469 | 0.1792 | 0.32 |

Time-averaging duration [s] | 800 | 400 | 55 |

Convective time units $(D/{U}_{b})$ | 1880 | 3750 | 950 |

Number of time steps | 10 × 10^{6} | 26 × 10^{6} | 15 × 10^{6} |

Asymmetry Parameter $\mathit{\alpha}$ | ${\mathit{\alpha}}_{\mathit{U}}$ | ${\mathit{\alpha}}_{\mathit{\sigma}}$ | ${\mathit{\alpha}}_{\mathit{\omega}}$ |
---|---|---|---|

Variable $\mathit{f}(\mathit{x},\mathit{r},\mathit{\vartheta})$ | $\overline{{U}_{x}}$ | $\overline{{{u}_{x}}^{2}}$ | $\overline{\Vert \omega \Vert}$ |

Normalization N | ${U}_{b}\text{}R$ | ${{U}_{b}}^{2}\text{}R$ | ${U}_{b}$ |

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**MDPI and ACS Style**

Mikuž, B.; Cerkovnik, K.; Tiselj, I.
Asymmetry Propagation in a Pipe Flow Downstream of a 90° Sharp Elbow Bend. *Appl. Sci.* **2024**, *14*, 7895.
https://doi.org/10.3390/app14177895

**AMA Style**

Mikuž B, Cerkovnik K, Tiselj I.
Asymmetry Propagation in a Pipe Flow Downstream of a 90° Sharp Elbow Bend. *Applied Sciences*. 2024; 14(17):7895.
https://doi.org/10.3390/app14177895

**Chicago/Turabian Style**

Mikuž, Blaž, Klemen Cerkovnik, and Iztok Tiselj.
2024. "Asymmetry Propagation in a Pipe Flow Downstream of a 90° Sharp Elbow Bend" *Applied Sciences* 14, no. 17: 7895.
https://doi.org/10.3390/app14177895