# Evolution of Turbulent Horseshoe Vortex System in Front of a Vertical Circular Cylinder in Open Channel

^{*}

## Abstract

**:**

## 1. Introduction

_{D}, of 1.0 × 10

^{5}and 2.2 × 10

^{5}. They observed that the flow field upstream of the cylinder is always dominated by a much more stable and larger primary HV (PHV). They also concluded the PHV wanders mostly in the streamwise direction under the action of large-scale structures in the incoming flow.

_{D}= 1.15 × 10

^{5}. They found the flow is dominated by a coherent PHV. However, for flows around a circular cylinder with lower Reynolds number, the DESs carried out by Escauriaza and Sotiropoulos [11] showed some differences on the evolution of the turbulent HV system. For the flow at Re

_{D}= 39,000, the system was found to be dominated by two primary horseshoe vortices (PHVs). These two HVs move upstream and downstream, respectively. On the other hand, the HV system at Re

_{D}= 20,000 is composed of four or five PHVs at all times, which move only in the downstream direction.

_{D}= 16,000 and 500,000, respectively. In both cases, the turbulent HV system was found to dominate by one PHV. The main mechanism for increasing the coherence of PHV is the merging and extraction of vorticity of the same sign from vortices originating in either the down-flow or the separation region. Meanwhile, the core of the PHV is larger when it is located away from the cylinder.

_{D}= 29,000, the flow upstream of the cylinder is not dominated by a PHV sometimes. However, the flows upstream of the cylinder with Re

_{D}= 47,000 and 123,000 are always dominated by a PHV. The PHV can maintain its coherence by merging vortices originating in the incoming flow, the down-flow along the cylinder face, or the upstream separation region. Schanderl et al. [18] obtained the flow fields in front of a circular cylinder on a flat bed with Re

_{D}= 39,000 by using PIV. The turbulent HV system was found to be dominated by a PHV.

_{D}= 9400 and 13,500 by using time-resolved PIV. The strength and size of the PHV were found to decrease gradually when it retreats back to the cylinder.

## 2. Datasets and Methodology

#### 2.1. Datasets

_{D}, defined with the diameter of the cylinder, D, and the bulk mean velocity, U

_{m}, was 8600, 10,200, and 13,900, respectively. The detailed flow characteristics for each experimental case are summarized in Table 1. For the sake of distinction, each case is named by its cylinder Reynolds number preceded by the letter “C”. According to the results of Wei et al. [7], the HV systems in the minimum and maximum Re

_{D}cases considered here are intermittently and fully turbulent, respectively. The 2D instantaneous velocity fields in the symmetry plane upstream of the cylinder were measured using a time-resolved PIV system. For each flow condition, two or three sequences of continuous velocity fields were measured with a frequency larger than 500 Hz. The total numbers of velocity fields for each case are 21,000, 21,000, and 24,000, respectively. More details about the experimental setup, PIV system, and the validation of the experiment data can be found in Chen et al. [13].

#### 2.2. Vortex Identification and Characterization Methods

_{ci}) as the vortex indicator [21]. When λ

_{ci}is non-zero at a point, the local streamline around the point is swirling, and the strength of the local swirling motion is represented by its value. For real vortices in turbulent flows, the strength of swirling motion shows maximum value at the center and decreases monotonously along the radius. Therefore, vortices are detected as non-zero clusters around the peaks of λ

_{ci}[22]. Since the imaginary part of the eigenvalues is always positive, we define Λ

_{ci}= λ

_{ci}∙sign(ω

_{z}) to distinguish between vortices rotating in the clockwise and counterclockwise sense, where ω

_{z}is the spanwise vorticity. The peak position of each cluster of Λ

_{ci}indicates the center of each vortex, and the peak value, Λ

_{v}, denotes the coherence of each vortex. Figure 2 presents the streamlines and contours of Λ

_{ci}in an instantaneous velocity field in front of the cylinder for case C8600. According to the streamline patterns, two HVs and one secondary vortex (SV) are observed in the junction region. The swirling strength criterion correctly identifies these vortices.

_{v}is the swirling strength at the center of each vortex. The threshold is selected based on the following relation holding for a standard Oseen vortex [24]

#### 2.3. Vortex Tracking Method

_{ci}in a given velocity field as the PHV.

_{ci}within its vortex core as the gray-level intensity of a particle image. Following some basic ideas of the particle tracking velocimetry technique, the method for tracking vortex contains the following steps, as illustrated in Figure 3.

_{c}is the convective velocity of the vortex V0 defined as the instantaneous velocity at its center; Δt is the time interval between two successive velocity fields; R

_{v}is the radius of the vortex V0. The introduction of R

_{v}is based on the observation that the center of HV can vary dramatically within its core.

_{ci}in each instantaneous velocity field is converted into a frame of a video animation. By replaying the videos, incorrect tracking results are identified and revised. Meanwhile, the videos are used to assist analyzing the evolution behaviors of the PHV.

_{v}, vertical position, y

_{v}, coherence, Λ

_{v}, core area, A

_{v}, and circulation strength, Γ

_{v}, of several PHVs that appeared one after another in case C13900. A color change on the time histories indicates the disappearance of an old PHV and the generation of a new PHV. The horizontal dashed line indicates the time-averaged value of the corresponding feature. Figure 4a,b reveal the aperiodic oscillation, both streamwise and vertical, of the PHV. The streamwise position of the PHV changes from −0.91D to −0.53D with a mean value of about −0.7D. The vertical position lies in the range of 0.02D to 0.13D, with a mean value of about 0.06D. The instantaneous and mean positions of the PHV obtained by vortex tracking are in accordance with previous observations [12,18,25]. These agreements validate the reasonableness of the current definition of PHV and the reliability of the tracking results. In Figure 4c–e, the size, coherence, and circulation strength of the PHV also show chaotic variations with time. These features indicate that the investigated HV system is fully turbulent, at least in the case of C13900. The mean diameter and circulation strength are 0.033D and −0.08DU

_{m}, respectively.

#### 2.4. Correlation Analysis Between the Features of PHV

## 3. Results and Discussion

#### 3.1. Generation of PHV

_{D}= 18,000 and by Apsilidis et al. [12] at Re

_{D}= 29,000. For cases with higher Reynolds number, both experimental and numerical results showed the persistent existence of the PHV [11,12,18].

_{D}= 16,000. However, it is unclear whether such a mechanism exists for junction flows with larger cylinder Reynolds number.

#### 3.2. Lifespan of PHV

_{m}, 4.81D/U

_{m}, and 4.64D/U

_{m}for case C8600, C10900, and C13900, respectively. Even though the mean lifespan of PHVs was not reported directly, Kirkil and Constantinescu [17] showed the mean period of a full transition among flow modes of the turbulent HV system is about 5.32D/U

_{m}at Re

_{D}= 16.000. Since the PHV is usually found to lose coherence strongly when the HV system transits from the backflow mode to the zero-flow mode [10], it is reasonable to conjecture that the disappearance of PHVs happens mainly after the mode transition. Therefore, the mean period of the transition could be a reasonable estimation of the mean lifespan of PHVs. To this end, the calculated mean lifespans are not inconsistent with existing findings. Meanwhile, as a preliminary estimation, the value of 5.0D/U

_{m}seems a reasonable approximation of the mean lifespan of PHVs.

#### 3.3. Oscillation of PHV

_{xy}, and of the streamwise position and the core area, R

_{xA}. In Figure 8a, the correlation function, R

_{xy}, exhibits a distinct negative peak on the right-hand side of the origin. It means that the PHV tends to approach the bed while moving toward the cylinder and the opposite is true. This movement trend agrees with the evolution model proposed by Dargahi [14]. In Figure 8b, the correlation function, R

_{xA}, also exhibits a distinct negative peak on the right-hand side of the origin. It means that the PHV decreases in size while approaching the cylinder and the opposite is also true. The statistical result supports the observations by Dargahi [14], Kirkil and Constantinescu [17], and Li et al. [19] in instantaneous velocity fields. The positions of the peaks indicate that the variations in vertical position and core size occur slightly later than the corresponding variation in streamwise position. Therefore, it is conjectured that the streamwise movement induces the variations in the vertical position and size of the PHV.

_{m}. Both streamlines and contours of the instantaneous streamwise velocity are plotted for better interpretation. At the beginning of this process, the PHV locates at about x/D = −0.78 in Figure 9a. Then it moves unidirectionally downstream toward the cylinder in Figure 9b–e. In Figure 9f, the PHV arrives at about x/D = −0.61 and dissipates quickly.

_{m}. In Figure 10a, the HV system contains a relatively weak PHV at about x/D = −0.62 and a stronger HV (HV1) at about x/D = −0.77. Then the PHV moves gradually upstream and arrives at x/D = −0.83 in Figure 10f. During the movement process, another HV (HV2) is shedding from the separation region in Figure 10c,d. However, due to the obstacle of the secondary vortex upstream of the PHV, vortices HV1 and HV2 fail to merge with the PHV before their disappearances. The engine of the upstream movement is hard to be recognized. However, two exclusive features are observed during the movement. Firstly, no inrush of high-momentum fluid from the incoming flow down to the bed upstream of the PHV is observed. Secondly, the PHV is fed vertically by a patch of high-momentum fluid from the incoming flow above it. However, such a feed is unnecessary for PHVs undergoing downstream movements.

_{D}= 123,000) that the rapid upstream motion of the PHV is originated from an influx of high-momentum fluid from the incoming boundary layer feeding it directly. Our present investigation shows that a similar mechanism exists in flows with relatively low Reynolds numbers as well. In general, it can be confidently concluded that the large-amplitude streamwise movements of the PHV are related with large-scale and high-momentum structures in the incoming flow.

#### 3.4. Strength Variation of PHV

_{xΛ}, and of the streamwise position and circulation strength, R

_{xΓ}, respectively. Both functions contain peaks on both sides of the origin, and the correlation peaks on the right-hand side are stronger than those on the left-hand side. The bimodal correlations indicate that a PHV might undergo two different variation trends in strength with the variation of streamwise position. Therefore, conditional cross-correlation analyses between the streamwise position and swirling strength and between the streamwise position and circulation strength were conducted based on the streamwise positions of HVs. Specifically, the correlation function defined in Equation (4) was calculated separately for the instances when PHVs locate at the upstream and downstream of their time-averaged streamwise position, x

_{vavg}. The region downstream of the time-averaged streamwise position is termed the near-cylinder region hereafter.

#### 3.5. Disappearance of PHV

## 4. Conclusions

_{m}. During its lifetime, the PHV undergoes large-amplitude streamwise oscillation due to the action of large-scale and high-momentum structures from the incoming flow. Meanwhile, the size and strength of the PHV varies dramatically due to frequent interactions with vortices and the bed. The interaction of the PHV with secondary vortices lifted off the wall is responsible for triggering the zero-flow mode. However, such interactions neither have preferential locations nor always result in the downstream movement of the PHV. These mechanisms imply that the evolution of the turbulent HV can be controlled indirectly through manipulating coherent structures in the incoming flow and the down-flow.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Streamlines and contours of swirling strength in an instantaneous flow field in front of the cylinder for case 8600.

**Figure 4.**Time histories of (

**a**) streamwise position, (

**b**) vertical position, (

**c**) coherence, (

**d**) core area, and (

**e**) circulation strength of several primary horseshoe vortices for case C13900.

**Figure 5.**Sequence of instantaneous velocity fields illustrating the generation of a PHV for case C13900. (

**a**) t = 0; (

**b**) t = 0.29D/U

_{m}; (

**c**) t = 0.58D/U

_{m}; (

**d**) t = 0.87D/U

_{m}; (

**e**) t = 1.13D/U

_{m}; (

**f**) t = 1.40D/U

_{m.}

**Figure 6.**Sequence of instantaneous velocity fields illustrating the generation of a PHV for case C13900. (

**a**) t = 0; (

**b**) t = 0.40D/U

_{m}; (

**c**) t = 0.60D/U

_{m}; (

**d**) t = 0.81D/U

_{m}; (

**e**) t = 1.01D/U

_{m}; (

**f**) t = 1.22D/U

_{m.}

**Figure 8.**Correlation functions of (

**a**) streamwise and vertical positions, (

**b**) streamwise position and core area.

**Figure 9.**Sequence of instantaneous velocity fields illustrating the large-amplitude downstream motion of a PHV for case C13900. (

**a**) t = 0; (

**b**) t = 0.32D/U

_{m}; (

**c**) t = 0.65D/U

_{m}; (

**d**) t = 0.96D/U

_{m}; (

**e**) t = 1.29D/U

_{m}; (

**f**) t = 1.60D/U

_{m.}

**Figure 10.**Sequence of instantaneous velocity fields illustrating the large-amplitude upstream motion of a PHV for case C13900. (

**a**) t = 0; (

**b**) t = 0.35D/U

_{m}; (

**c**) t = 0.70D/U

_{m}; (

**d**) t = 1.04D/U

_{m}; (

**e**) t = 1.75D/U

_{m}; (

**f**) t = 2.08D/U

_{m}.

**Figure 11.**Correlation functions of (

**a**) streamwise position and swirling strength, and (

**b**) streamwise position and circulation.

**Figure 12.**Conditional correlation functions between the streamwise position and swirling strength of PHVs locating at (

**a**) upstream and (

**b**) downstream of their time-averaged streamwise position.

**Figure 13.**Conditional correlation functions between streamwise position and circulation strength of PHVs locating at (

**a**) upstream and (

**b**) downstream of their time-averaged streamwise position.

**Figure 14.**Sequence of instantaneous velocity fields illustrating the interactions between a PHV and other vortices for case C13900. (

**a**) t = 0; (

**b**) t = 0.17D/U

_{m}; (

**c**) t = 0.34D/U

_{m}; (

**d**) t = 0.51D/U

_{m}; (

**e**) t = 0.68D/U

_{m}; (

**f**) t = 0.84D/U

_{m}; (

**g**) t = 1.02D/U

_{m}; (

**h**) t = 1.18D/U

_{m}; (

**i**) t = 1.35D/U

_{m.}

**Figure 15.**(

**a**–

**f**) Sequence of instantaneous velocity fields illustrating a PHV dissipating in the junction region, and (

**g**) variations of the core area and circulation strength during the dissipation process for case C13900. Each dashed line in (

**g**) indicates the time of the flow field labelled by the letter.

**Table 1.**Details of the bed and flow characteristics: S, bed slope; H, water depth; U

_{m}, bulk mean velocity; Re, Reynolds number.

Case | S (‰) | H (cm) | U_{m} (cm/s) | Re | Re_{D} | Fr |
---|---|---|---|---|---|---|

C8600 | 0.5 | 3.0 | 22.4 | 8100 | 8600 | 0.41 |

C10200 | 0.5 | 4.0 | 26.4 | 12,700 | 10,200 | 0.42 |

C13900 | 1.5 | 3.5 | 36.9 | 15,200 | 13,900 | 0.63 |

_{D}, cylinder Reynolds number; Fr, Froude number.

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**MDPI and ACS Style**

Chen, Q.; Yang, Z.; Wu, H.
Evolution of Turbulent Horseshoe Vortex System in Front of a Vertical Circular Cylinder in Open Channel. *Water* **2019**, *11*, 2079.
https://doi.org/10.3390/w11102079

**AMA Style**

Chen Q, Yang Z, Wu H.
Evolution of Turbulent Horseshoe Vortex System in Front of a Vertical Circular Cylinder in Open Channel. *Water*. 2019; 11(10):2079.
https://doi.org/10.3390/w11102079

**Chicago/Turabian Style**

Chen, Qigang, Zuolei Yang, and Haojie Wu.
2019. "Evolution of Turbulent Horseshoe Vortex System in Front of a Vertical Circular Cylinder in Open Channel" *Water* 11, no. 10: 2079.
https://doi.org/10.3390/w11102079