# Three-Dimensional Turbulent Flow Characteristics Near the Leading Edge of a Longitudinal Structure in the Presence of an Inclined Channel Bank

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup and Flow Diagnostics

#### 2.1. Physical Setup

^{2}openings and then it was securely attached to the channel bank and bed near the leading edge of the retaining wall.

#### 2.2. Flow Diagnostics, Measurement Domain, and Data Analysis

^{TM}4MP-180 cameras, a dual pulsed Nd:YAG laser with a wavelength of 532 nm, an articulating laser arm to assist delivering the laser at the desired locations, two cylindrical lenses with a focal length of −25 mm and −50 mm to expand the laser light in longitudinal and lateral directions, respectively, and a synchronizer. A mounting frame was used to fix the cameras near the channel side-wall. The cameras were calibrated after the flume was filled with water with a back-lit target with 0.2032 mm holes on its surface that were regularly spaced at 5 mm apart both in the horizontal and vertical directions. The VPIV laser light was projected to the area of interest from the free-surface. A thin Plexiglas board was placed on the free water surface, barely touching it, to minimize the laser light reflection and optical problems (Figure 2). It was supported with an aluminum arm attached to the channel side-wall.

## 3. Results

#### 3.1. Velocity Field

#### 3.2. JV System

#### 3.3. Bimodal Oscillations of the JV1

#### 3.4. POD Analysis of the Velocity Fluctuations

**X**form, such that each one of its columns consisted of measurements taken at a specific instant of time, and the number of rows in

**X**was equal to the number of data points within the measurement volume of interest times the number of velocity components (three in our case). Next, the singular value decomposition (SVD) of

**X**was calculated as $\mathbf{X}=\Psi \Sigma \mathbf{V}$, where $\Psi $ and $\mathbf{V}$ are unitary matrices characterizing an optimal (in an ${\ell}_{2}$ sense) rank-r truncation of the data matrix $\mathbf{X}$, and $\Sigma $ is a diagonal matrix characterizing the energy (variance) in each mode. The columns of $\Psi $ contain the spatial structure of each of the POD modes ${\varphi}_{j}\left(x\right)$ and the coefficients representing the time evolution of the modes are embedded in the matrix $\mathbf{V}$. The relative importance of the ${j}^{th}$ POD mode ${\varphi}_{j}\left(x\right)$ in the approximation of the matrix $\mathbf{X}$ is determined by its relative energy content:

**X**that are ordered in a descending order in the $\Sigma $.

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Sukhodolov, A.; Engelhardt, C.; Krüger, A.; Bungartz, H. Case study: Turbulent flow and sediment distributions in a groyne field. J. Hydraul. Eng.
**2004**, 130, 142–149. [Google Scholar] [CrossRef] - Radspinner, R.; Diplas, P.; Lightbody, A.; Sotiropoulos, F. River training and ecological enhancement potential using in-stream structures. J. Hydraul. Eng.
**2010**, 136, 967–980. [Google Scholar] [CrossRef] - Sumer, B.; Fredsøe, J. Scour around pile in combined waves and current. J. Hydraul. Eng.
**2001**, 127, 403–411. [Google Scholar] [CrossRef] - Coleman, S.; Lauchlan, C.; Melville, B. Clear-water scour development at bridge abutments. J. Hydraul. Res.
**2003**, 41, 521–531. [Google Scholar] [CrossRef] - Dey, S.; Barbhuiya, A. Flow Field at a Vertical-Wall Abutment. J. Hydraul. Eng.
**2005**, 131, 1126–1135. [Google Scholar] [CrossRef] - Duan, J.; He, L.; Fu, X.; Wang, Q. Mean flow and turbulence around experimental spur dike. Adv. Water Resour.
**2009**, 32, 1717–1725. [Google Scholar] [CrossRef] - Koken, M.; Constantinescu, G. An investigation of the flow and scour mechanisms around isolated spur dikes in a shallow open channel: 1. Conditions corresponding to the initiation of the erosion and deposition process. Water Resour. Res.
**2008**, 44, W08406. [Google Scholar] [CrossRef] - Paik, J.; Sotiropoulos, F. Coherent structure dynamics upstream of a long rectangular block at the side of a large aspect ratio channel. Phys. Fluids
**2005**, 17, 115104. [Google Scholar] [CrossRef] - Devenport, R. Time-dependent and time-averaged turbulence structure near the nose of a wing-body junction. J. Fluid Mech.
**1990**, 210, 23–55. [Google Scholar] [CrossRef] - Jeon, J.; Lee, J.; Kang, S. Experimental investigation of three-dimensional flow structure and turbulent flow mechanisms around a nonsubmerged spur dike with a low length-to-depth ratio. Water Resour. Res.
**2018**, 54, 3530–3556. [Google Scholar] [CrossRef] - Melville, B.; Raudkivi, A. Flow characteristics in local scour at bridge piers. J. Hydraul. Res.
**1977**, 15, 373–380. [Google Scholar] [CrossRef] - Paik, J.; Escauriaza, C.; Sotiropoulos, F. On the bimodal dynamics of the turbulent horseshoe vortex system in a wing-body junction. Phys. Fluids
**2007**, 19, 045107. [Google Scholar] [CrossRef] - Molinas, A.; Kheireldin, K.; Wu, B. Shear stress around vertical wall abutments. J. Hydraul. Eng.
**1998**, 124, 822–830. [Google Scholar] [CrossRef] - Kuhnle, R.; Jia, Y.; Alonso, C. Measured and simulated flow near a submerged spur dike. J. Hydraul. Eng.
**2008**, 134, 916–924. [Google Scholar] [CrossRef] - Kuhnle, R.; Alonso, C. Flow near a model spur dike with a fixed scoured bed. Int. J. Sediment Res.
**2013**, 28, 349–357. [Google Scholar] [CrossRef] - Rehman, K.; Hong, S.H. Influence of lateral flow contraction on bed shear stress estimation by using measured turbulent kinetic energy. Exp. Therm. Fluid Sci.
**2022**, 139, 110742. [Google Scholar] [CrossRef] - Launay, G.; Mignot, E.; Riviere, N. Laminar free-surface flow around emerging obstacles: Role of the obstacle elongation on the horseshoe vortex. Eur. J. -Mech.-B/Fluids
**2019**, 77, 71–78. [Google Scholar] [CrossRef][Green Version] - Zhang, H.; Nakagawa, H.; Kawaike, K.; Yasuyuki, B. Experiment and simulation of turbulent flow in local scour around a spur dyke. Int. J. Sediment Res.
**2009**, 24, 33–45. [Google Scholar] [CrossRef] - Heydari, N.; Diplas, P. Flow dynamics in the vicinity of a gravel embedded vertical retaining wall: Conditions corresponding to the initial stages of local erosion. Environ. Fluid Mech.
**2020**, 20, 203–225. [Google Scholar] [CrossRef] - Khosronejad, A.; Kozarek, J.; Diplas, P.; Hill, C.; Jha, R.; Chatanantavet, P.; Heydari, N.; Sotiropoulos, F. Simulation-based optimization of in-stream structures design: Rock vanes. Environ. Fluid Mech.
**2018**, 18, 695–738. [Google Scholar] [CrossRef] - Khosronejad, A.; Diplas, P.; Angelidis, D.; Zhang, Z.; Heydari, N.; Sotiropoulos, F. Scour depth prediction at the base of longitudinal walls: A combined experimental, numerical, and field study. Environ. Fluid Mech.
**2020**, 20, 459–478. [Google Scholar] [CrossRef] - Heydari, N.; Diplas, P.; Nathan Kutz, J.; Sadeghi Eshkevari, S. Modal Analysis of Turbulent Flow near an Inclined Bank–Longitudinal Structure Junction. J. Hydraul. Eng.
**2021**, 147, 04020100. [Google Scholar] [CrossRef] - Stellmacher, M.; Obermayer, K. A new particle tracking algorithm based on deterministic annealing and alternative distance measures. Exp. Fluids
**2000**, 28, 506–518. [Google Scholar] [CrossRef] - Koken, M.; Constantinescu, G. Investigation of flow around a bridge abutment in a flat bed channel using large eddy simulation. In Proceedings of the World Environmental And Water Resource Congress 2006: Examining The Confluence Of Environmental And Water Concerns, Omaha, Nebraska, 21–25 May 2006; pp. 1–11. [Google Scholar]
- Baker, C.; Taulbee, D.; George, W., Jr. Eddy Viscosity Calculations of Turbulent Buoyant Plumes. In Proceedings of the ASME/AIChE National Heat Transfer Conference, San Diego, CA, USA, 6–8 August 1979. [Google Scholar]
- Dargahi, B. The turbulent flow field around a circular cylinder. Exp. Fluids
**1989**, 8, 1–12. [Google Scholar] [CrossRef] - Koken, M.; Constantinescu, G. Flow and turbulence structure around abutments with sloped sidewalls. J. Hydraul. Eng.
**2014**, 140, 04014031. [Google Scholar] [CrossRef] - Agui, J.; Andreopoulos, J. Experimental investigation of a three-dimensional boundary layer flow in the vicinity of an upright wall mounted cylinder (data bank contribution). J. Fluids Eng.
**1992**, 114, 566–576. [Google Scholar] [CrossRef] - Dubief, Y.; Delcayre, F. On coherent-vortex identification in turbulence. J. Turbul.
**2000**, 1, 011. [Google Scholar] [CrossRef] - Rajaratnam, N.; Nwachukwu, B. Flow near groin-like structures. J. Hydraul. Eng.
**1983**, 109, 463–480. [Google Scholar] [CrossRef] - Apsilidis, N.; Diplas, P.; Dancey, C.L.; Bouratsis, P. Apsilidis Time-resolved flow dynamics and Reynolds number effects at a wall–cylinder junction. J. Fluid Mech.
**2015**, 776, 475–511. [Google Scholar] [CrossRef] - Paik, J.; Escauriaza, C.; Sotiropoulos, F. Coherent structure dynamics in turbulent flows past in-stream structures: Some insights gained via numerical simulation. J. Hydraul. Eng.
**2010**, 136, 981–993. [Google Scholar] [CrossRef] - Escauriaza, C.; Sotiropoulos, F. Lagrangian model of bed-load transport in turbulent junction flows. J. Fluid Mech.
**2011**, 666, 36–76. [Google Scholar] [CrossRef] - Escauriaza, C.; Sotiropoulos, F. Reynolds number effects on the coherent dynamics of the turbulent horseshoe vortex system. Flow, Turbul. Combust.
**2011**, 86, 231–262. [Google Scholar] [CrossRef] - Doligalski, T.; Smith, C.; Walker, J. Vortex interactions with walls. Annu. Rev. Fluid Mech.
**1994**, 26, 573–616. [Google Scholar] [CrossRef] - Kirkil, G.; Constantinescu, G. Flow and turbulence structure around an in-stream rectangular cylinder with scour hole. Water Resour. Res.
**2010**, 46, W11549. [Google Scholar] [CrossRef] - Lumley, J. Toward a turbulent constitutive relation. J. Fluid Mech.
**1970**, 41, 413–434. [Google Scholar] [CrossRef] - Berkooz, G.; Holmes, P.; Lumley, J. The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech.
**1993**, 25, 539–575. [Google Scholar] [CrossRef] - Gutierrez-Castillo, P.; Thomases, B. Proper Orthogonal Decomposition (POD) of the flow dynamics for a viscoelastic fluid in a four-roll mill geometry at the Stokes limit. J. -Non-Newton. Fluid Mech.
**2019**, 264, 48–61. [Google Scholar] [CrossRef] - Apsilidis, N.; Diplas, P.; Dancey, C.; Bouratsis, P. Effects of wall roughness on turbulent junction flow characteristics. Exp. Fluids
**2016**, 57, 1–16. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) A schematic of a section of the experimental flume along with the VPIV setup, (

**b**) an upstream looking cross sectional view of the water recirculating flume.

**Figure 2.**A schematic of the measurement volumes located in the vicinity of the upstream face of the longitudinal structure.

**Figure 3.**Dimensionless time-averaged streamwise (<u>*), lateral (<v>*), and vertical (<w>*) flow velocity fields within the representative planes in the immediate vicinity of the retaining wall. The flow direction is shown by a blue arrow.

**Figure 4.**Dimensionless time-averaged velocity fields at (

**a**–

**c**) $\frac{Z}{H}=0.18$, (

**d**–

**f**) $\frac{Z}{H}=0.32$, and (

**g**–

**i**) $\frac{Z}{H}=0.49$. The first, second, and third columns show the lateral, streamwise, and vertical velocity components, respectively.

**Figure 5.**Dimensionless time-averaged vertical vorticity fields at (

**a**) $\frac{Z}{H}=0.19$, (

**b**) $\frac{Z}{H}=0.28$, and (

**c**) $\frac{Z}{H}=0.37$ from the channel bed.

**Figure 6.**Isosurfaces of the dimensionless time-averaged horizontal vorticity field corresponding to the isovalue of −1.25.

**Figure 7.**(

**a**–

**c**) show the $\frac{<-{u}^{{}^{\prime}}{v}^{{}^{\prime}}>}{{U}_{0}^{2}}$ at $\frac{Z}{H}=0.17$, 0.22, and 0.26 horizontal planes, respectively.

**Figure 8.**(

**a**–

**c**) show the $\frac{<-{u}^{{}^{\prime}}{w}^{{}^{\prime}}>}{{U}_{0}^{2}}$ component at $\frac{Z}{H}=0.17$, 0.22, and 0.26 horizontal planes, respectively.

**Figure 9.**The dimensionless instantaneous velocity vectors over the dimensionless (

**a**) spanwise, and (

**b**) streamwise vorticity fields within the representative planes identified in the image inserted in the second row. The flow moves in the positive x-direction.

**Figure 10.**(

**a**–

**f**) Instantaneous dimensionless velocity vectors laied over the corresponding dimensionless vorticity fields (in the spanwise direction), emphasizing the position and size of JV1, from over the channel bank to within the main channel.

**Figure 11.**(

**a**–

**c**) Instantaneous non-dimensional velocity vectors plotted over the ${\omega}_{X}^{*}$ in representative planes perpendicular to the main flow direction to show that OV is pronounced in the immediate vicinity of the retaining wall.

**Figure 12.**Visualization of the mean flow using Q-criterion. (

**a**) shows a 3D view, and (

**b**) illustrates a plan view for the same levels of iso value (2).

**Figure 13.**Distribution of the dimensionless bed mean bed shear stress values, (

**a**) shows a 3D view, and (

**b**) shows a top view along with the isosurfaces of the JV system at isovalue of 1.5.

**Figure 14.**Time series of the dimensionless instantaneous bed shear stress at ($\frac{X}{H}$,$\frac{Y}{H}$, $\frac{Z}{H}$) = (0.03, 0.08, 0.07).

**Figure 15.**Distribution of TKE* within the representative vertical sections, parallel to the main flow direction, (

**a**–

**d**) over the channel bank and (

**e**,

**f**) within the main channel. The location of the sections parallel to the main flow direction shown here was defined in Figure 10.

**Figure 16.**Instantaneous iso-surfaces of the dimensionless Q-criterion in a representative section over the channel bank at the upstream face of the retaining wall showing the (

**a**) backflow, and (

**b**) zeroflow modes of the JV1 vortex. The color coding corresponds to the vertical velocity component.

**Figure 17.**(

**a**–

**f**) Instantaneous spanwise vorticity fields showing the transition over−time from backflow mode to zeroflow mode within a representative plane, that is vertical and is in the streamflow direction, located at $\frac{Y}{H}=-0.28$. The time interval between consecutive images is 0.02 s.

**Figure 18.**(

**a**–

**e**) The pdfs of the vertical velocity fluctuations just upstream of the core of JV1 (in the mean flow) within the planes identified in Figure 10.

**Figure 19.**The relative energy content of the first twenty POD modes within (

**a**) the individual measurement volumes over the channel bank, (

**b**) vertical planes aligned with the flow and located beyond the retaining wall in the spanwise direction, and (

**c**) vertical planes perpendicular to the main flow direction within the main channel.

**Figure 20.**Reconstruction of the dimensionless streamwise velocity fluctuations (red line) using the first ten POD modes, compared to the original data (black solid line).

**Figure 21.**The first energetic POD mode. The color coding represents the Euclidean norm of the modes calculated as ${\left|\varphi \right|}_{2}=\sqrt{{{\varphi}_{u}}^{2}+{{\varphi}_{v}}^{2}+{{\varphi}_{w}}^{2}}$. Note that ${\varphi}_{u}$, ${\varphi}_{v}$ and ${\varphi}_{w}$ are the POD sub-modes associated with the streamwise, lateral, and vertical velocities, respectively.

H | ${\mathit{U}}_{0}$ | S | ${\mathit{u}}_{\mathit{s}}$ | ${\mathit{F}}_{\mathit{r}}$ | Re | ${\mathit{u}}_{\mathit{s}}/{\mathit{u}}_{\mathbf{cr}}$ |
---|---|---|---|---|---|---|

(m) | (m/s) | (%) | (m/s) | - | - | - |

0.18 | 0.57 | 0.14 | 0.05 | 0.43 | 282,430 | 0.90 |

_{0}= approach depth-averaged flow velocity; S = channel bed slope; u

_{s}= shear velocity = $\sqrt{\frac{{\tau}_{0}}{\rho}}$; ${\tau}_{0}=\gamma HS$ = mean bed shear stress; γ = specific weight of water; ${F}_{r}=\frac{{U}_{0}}{\sqrt{{g}^{H}}}$ = Froude number; g = gravitational acceleration; $Re=\frac{{U}_{0}4R}{v}$ = Reynolds number; $R=\frac{A}{P}$ = hydraulic radius; A = cross-sectional area; P = wetted perimeter; v = kinematic viscosity of water; u

_{cr}= critical shear velocity.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Heydari, N.; Diplas, P. Three-Dimensional Turbulent Flow Characteristics Near the Leading Edge of a Longitudinal Structure in the Presence of an Inclined Channel Bank. *Water* **2022**, *14*, 3524.
https://doi.org/10.3390/w14213524

**AMA Style**

Heydari N, Diplas P. Three-Dimensional Turbulent Flow Characteristics Near the Leading Edge of a Longitudinal Structure in the Presence of an Inclined Channel Bank. *Water*. 2022; 14(21):3524.
https://doi.org/10.3390/w14213524

**Chicago/Turabian Style**

Heydari, Nasser, and Panayiotis Diplas. 2022. "Three-Dimensional Turbulent Flow Characteristics Near the Leading Edge of a Longitudinal Structure in the Presence of an Inclined Channel Bank" *Water* 14, no. 21: 3524.
https://doi.org/10.3390/w14213524