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Water 2019, 11, 2079; doi:10.3390/w11102079 www.mdpi.com/journal/water
Article
Evolution of Turbulent Horseshoe Vortex System in
Front of a Vertical Circular Cylinder in Open Channel
Qigang Chen *, Zuolei Yang and Haojie Wu
School of Civil Engineering, Beijing Jiaotong University, Beijing, 100044, China; 18121147@bjtu.edu.cn (Z.Y.);
18125879@bjtu.edu.cn (H.W.)
* Correspondence: chenqg@bjtu.edu.cn; Tel.: +86-151-2000-1870
Received: 9 August 2019; Accepted: 1 October 2019; Published: 5 October 2019
Abstract: A turbulent horseshoe vortex (HV) system around a wall-mounted cylinder in open
channel is characterized by random variations in vortex features and an abundance of vortex
interactions. The turbulent HV system is responsible for initiating the local scour process in front of
the cylinder. The evolution of the turbulent HV system is investigated statistically and
quantitatively with time-resolved particle image velocimetry. The cylinder Reynolds numbers of
the flow are 8600, 10,200, and 13,600, respectively. A novel vortex tracking method was proposed to
obtain the variations in position, size, and strength of the primary HV (PHV) which dominates the
system most of the time. Relationships between the various features of the PHV during its
evolutionary process were obtained through correlation analyses. Results show that the
dimensionless mean lifespan of the PHV is about 5.0. Statistically, the downstream movement of
the PHV toward the cylinder is accompanied with its bed-approaching movement and decreasing
in size, and the opposite is true. The circulation strength of the PHV decreases and increases
dramatically in the region downstream of its time-averaged position when the PHV approaches
and departs from the cylinder, respectively. Meanwhile, mechanisms responsible for the
generation, movement, variation, and disappearance of the PHV are re-investigated and enriched
based on its interactions with vortices in the separation region and structures in the incoming flow.
The obtained change trends of the features of the PHV and the underlying mechanisms for its
evolution are valuable for predicting and controlling the initial stage of the local scour in front of
cylinders.
Keywords: turbulent horseshoe vortex; evolution of vortex; circular cylinder flow; open channel
turbulence
1. Introduction
Flows around a wall-mounted circular cylinder appear in a large number of technical and
environmental applications, such as river flows past bridge piers and pile foundations [1–3]. When
the natural wall-bounded flow encounters a vertical cylinder, horseshoe vortices (HVs) are formed
in the junction region between the bed and the cylinder [4–6]. The HVs are a system of necklace
structures wrapping around the front and flanks of the cylinder. Depending on the regime of the
approaching flow at the location of the cylinder, the horseshoe vortex (HV) system appears as a
laminar or turbulent system, respectively [7]. The turbulent HV system oscillates randomly in front
of the cylinder, inducing highly elevated means and fluctuations in bed shear stress [8]. Therefore, a
full understanding of the evolution of the turbulent HV system is necessary for understanding,
predicting, and controlling the initial stage of the local scour in front of cylinders.
Due to the highly dynamical nature of the turbulent HV system, many investigations have
focused on interpreting its underlying dynamics with models of low complexity [9–13].
Water 2019, 11, 2079 2 of 18
The evolution of the instantaneous turbulent HV system, however, has received less attention,
partly due to its complexity manifesting as aperiodic interactions and variations in number, position,
size, and strength. Dargahi [14] made one of the earliest descriptions on the evolutionary process of
the turbulent HV system based on experimental observations conducted with flow visualization
method. Results showed the HV closest to the cylinder oscillates upstream and downstream along
the bed. When the HV moves downstream toward the cylinder, it approaches the bed, and its
cross-section area and circulation strength decreases. Latter, Agui and Andreopoulos [15] visualized
the flow structures around a circular cylinder for cylinder Reynolds numbers, ReD, of 1.0 × 105 and
2.2 × 105. 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.
More detailed descriptions on the evolution of the turbulent HV system were not available until
eddy-resolving numerical simulations were applied. Kirkil et al. [16] performed one of the earliest
large eddy simulations (LES) on flow around a circular cylinder at a Reynolds number of 18.000.
They found the junction region is not always dominated by a large relatively stable PHV. The PHV
can generate through growing of a small vortex or merging of several vortices with the same rotation
sense. Moreover, some of the vortices are separated from the attached boundary layer on the
upstream cylinder face instead of the upstream separation region. During its lifetime, the PHV gains
or loses vorticity due to interaction with co-rotating or counter-rotating vortices, respectively. In
some cases, the PHV was split or practically destroyed. Later, Paik et al. [10] carried out detached
eddy simulations (DES) for flow around a wing mounted on a flat bed at ReD = 1.15 × 105. 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 ReD = 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 ReD = 20,000 is composed of
four or five PHVs at all times, which move only in the downstream direction.
Recently, Kirkil and Constantinescu [17] adopted the LES and DES approaches to study flows
around a circular cylinder with ReD = 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.
Quantitative experimental studies on the evolution of the turbulent HV system upstream of a
circular cylinder were reported recently by several groups. Apsilidis et al. [12] reported the influence
of cylinder Reynolds number on the dynamics of the turbulent HV system by using time-resolved
PIV. For the case with ReD = 29,000, the flow upstream of the cylinder is not dominated by a PHV
sometimes. However, the flows upstream of the cylinder with ReD = 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 ReD = 39,000 by
using PIV. The turbulent HV system was found to be dominated by a PHV.
The PHV undergoes relatively large streamwise displacements, and the closer to the cylinder it
locates the more coherent it would be. Recently, Li et al. [19] observed part of the evolutionary
process of turbulent HV in front of a circular cylinder with ReD = 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.
To summarize, despite many valuable contributions by using flow visualization methods,
eddy-resolving simulations, and quantitative measurements, the evolutionary process of the
turbulent HV system is far to be well understood. On one hand, existing observations based on
different methods, especially numerical simulations, have shown some discordances. On the other
hand, many existing investigations on the evolution of turbulent HV system have been based merely
Water 2019, 11, 2079 3 of 18
on the observations of instantaneous velocity fields, which is insufficient for building a statistically
reliable description on the evolutionary process of the turbulent HV system. Therefore, the present
study uses quantitative and statistical methods to investigate the evolution of the turbulent HV
system. It expands upon the previously published work by Chen et al [13]. Especially, relationships
between the various features of the PHV dominating the turbulent HV system during its evolution
were investigated. Meanwhile, the underlying mechanisms responsible for the generation,
disappearance, large-scale streamwise movements, and strength variations of the PHV were
re-investigated.
2. Datasets and Methodology
2.1. Datasets
Three experimental datasets were used for the present investigation. The experiments were
carried out in a recirculating tilting water flume that was 20 m long and 0.3 m wide. Figure 1 shows
two diagrams and a photograph of the experimental setup. A circular cylinder with a diameter of 3.2
cm was mounted vertically on the flat bed of the flume. The length of the cylinder is larger than the
water depth to avoid submergence. The distance between the axis of the cylinder and the entrance of
the flume is 12 m to ensure a fully developed turbulent open channel flow. The cylinder Reynolds
number ReD, defined with the diameter of the cylinder, D, and the bulk mean velocity, Um, 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 ReD 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].
As shown in Figure 1, the origin of the coordinate system used to describe the results is defined
to lie at the center of the cylinder on its bottom surface. The x axis is oriented along the main flow
and parallel to the flume bed. The y axis is normal to the bed and coincides with the central axis of
the cylinder. The transverse z axis is normal to the side walls of the flume. The instantaneous
velocity components in the x, y, and z directions are denoted by U, V, and W, respectively.
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Figure 1. (a) Cross section, (b) side view, and (c) a photograph of the experimental setup.
Table 1. Details of the bed and flow characteristics: S, bed slope; H, water depth; Um, bulk mean
velocity; Re, Reynolds number;
Case S (‰) H (cm)
U
m (cm/s) Re ReD 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
ReD, cylinder Reynolds number; Fr, Froude number.
2.2. Vortex Identification and Characterization Methods
The turbulent HVs are usually identified as closed or spiral streamlines. However, this method
fails when the convection velocity of a vortex becomes non-ignorable [20]. Meanwhile, quantitative
features of vortices can hardly be read out from the streamline patterns. Therefore, a number of
recent investigations used kinematics implied by the velocity gradient tensor as the indicators of
HVs. For instance, Paik et al. [10], Kirkil and Constantinescu [17], and Apsilidis et al. [12] adopted
the Q criterion to visualize vortices within the HV system.
The present investigation uses the swirling strength criterion to identify HVs. The criterion
adopts the imaginary part of the eigenvalues of the velocity gradient tensor (termed swirling
strength and denoted by λci) as the vortex indicator [21]. When λci 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 λci [22]. Since the imaginary part of the eigenvalues is always
positive, we define Λci = λci∙sign(ωz) to distinguish between vortices rotating in the clockwise and
counterclockwise sense, where ωz is the spanwise vorticity. The peak position of each cluster of Λci
indicates the center of each vortex, and the peak value, Λv, denotes the coherence of each vortex.
Figure 2 presents the streamlines and contours of Λci 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.
Figure 2. Streamlines and contours of swirling strength in an instantaneous flow field in front of the
cylinder for case 8600.
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The swirling strength criterion can determine the center and coherence of each vortex, but
methods are still required to determine their core size and circulation strength. Traditionally, two
categories of methods were widely used. The first kind of methods define the boundary of a vortex
as an isoline of vortex indicators, of which the value equals to a given threshold [22]. The circulation
strength of the vortex is then calculated by integrating the vorticity within the vortex core. These
methods are simple and intuitive, but the results depend on the selected threshold. The second kind
of methods extract the features of a vortex by fitting the local velocity field with a standard vortex
model [23]. By properly selecting the model, results of this kind of method would be more objective.
However, it is computationally intense and may lose its superiority in treating instantaneous
vortices.
We adopted a revised version of the threshold method to determine the size and circulation
strength of instantaneous HVs. Instead of selecting a universal and constant threshold for all
vortices, we defined the threshold for each vortex based on its own coherence as
0.25=v
εΛ
(1)
where Λv 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]
()
0.25=≈
ci v v
rR
ΛΛ
. (2)
2.3. Vortex Tracking Method
In order to quantitatively investigate the evolution of turbulent HV system, vortices comprising
the system should be tracked. However, as shown in Figure 2, the instantaneous turbulent HV
system is usually composed of more than one vortex with different size and strength, making it
impractical to track all these vortices. According to existing investigations, most of the time the
turbulent HV system is dominated by a primary vortex which relates tightly with the dynamics of
the system. Therefore, only the PHV is selected as the representative vortex in the turbulent HV
system and tracked. Nevertheless, one difficulty still exists, as no clear and universal definition for
the PHV is available even though it is commonly used. In a laminar HV system, the PHV is typically
the standing HV closest to the cylinder base [7]. For the turbulent HV system, however, the most
coherent HV seems a more intuitively accepted definition for the PHV. Therefore, we chose the HV
with the highest peak of Λci in a given velocity field as the PHV.
To track a PHV in successive velocity fields, we treated the spatial distribution of Λci 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.
Step 1. Vortices in all instantaneous velocity fields of a sequence are identified with the swirling
strength criterion. The core of each vortex is represented by the contours of swirling strength within
its boundary, defined by Equation (1).
Step 2. The PHV in the first snapshot (V0) is selected as the tracking object and candidates of
this vortex in the next snapshot (V1–V4) are determined by a displacement threshold (denoted by a
circle). The threshold is defined as
=+
th c v
DVtR
Δ
(3)
where Vc 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; Rv is the radius of the vortex V0.
The introduction of Rv is based on the observation that the center of HV can vary dramatically within
its core.
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Figure 3. Major steps of the method for tracking a vortex in successive velocity fields.
Step 3. Counterpart of the object vortex in the second snapshot is detected by a
cross-correlation analysis. The cross-correlation analysis is performed between the swirling strength
distribution of snapshot one and that of snapshot two. In the resulted correlation map, the
maximum peak represents the correlation coefficient between the object vortex and its counterpart,
and its coordinate represents the displacement of the object vortex during Δt.
After obtaining the displacement of the object vortex, the position of its counterpart in the
second snapshot (V1) is determined. The detected counterpart is defined as the new object vortex in
the next step.
Step 4. Steps 2 and 3 are conducted repeatedly until the correlation coefficient between the
object vortex and its counterpart is smaller than 0.5. The threshold of 0.5 has been selected based on
pre-conducted sensitive analysis. Such a circumstance means that the old PHV disappears. Then a
new PHV is defined in the next snapshot and the tracking process restarts.
Step 5. The tracking results for each velocity field sequence are double checked manually to
avoid possible mismatch. The image of streamlines and contours of Λci 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.
Figure 4 presents the time histories of the streamwise position, xv, vertical position, yv,
coherence, Λv, core area, Av, and circulation strength, Γv, of several PHVs that appeared one after
another in case C13900. A color change on the time histories indicates the disappearance of an old
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PHV and the generation of a new PHV. The horizontal dashed line indicates the time-averaged
value of the corresponding feature. Figures 4(a) and 4(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
Figures 4(c)–4(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 -0.08DUm,
respectively.
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.
2.4. Correlation Analysis Between the Features of PHV
Despite their chaotic variations, a scrutiny of Figure 4 reveals that the variations of different
features are quasi-synchronous with each other. For example, the moment indicated by the arrows
corresponds to local minima in the streamwise position, coherence, and circulations strength, and
local maxima in the vertical position and core area, respectively. To explore the inter-relationships
among these features, temporal cross-correlation analyses were conducted between the streamwise
position and the other features of the PHV. As an example, the temporal cross-correlation function
between the streamwise and vertical positions is calculated as follows:
Water 2019, 11, 2079 8 of 18
()
() ()
() ()
,,
1
22
,,
11
=
==
−+−
=
−+−
N
vn v vn v
n
xy NN
vn v vn v
nn
xtx yt ty
Rt
xtx yt ty
Δ
Δ
Δ
(4)
Here, N is the sample number of velocity fields and the overbar represents time-averaging. For
the sake of interpretation, absolute value of the coherence (swirling strength) and the circulation
strength, which were negative by definition, were used to calculate the correlation functions.
3. Results and Discussion
3.1. Generation of PHV
Instantaneous velocity fields show that although the junction region between the bed and the
cylinder is dominated by a PHV most of the time, there are instants where no typical HV is observed
for all cases in the present study. Therefore, each PHV undergoes a complete evolutionary process
consisted of generation, life, and disappearance stages. The occasional absence of a PHV in the
junction region was also observed by Kirkil et al. [16] at a Reynolds number of ReD = 18,000 and by
Apsilidis et al. [12] at ReD = 29,000. For cases with higher Reynolds number, both experimental and
numerical results showed the persistent existence of the PHV [11,12,18].
The sequence of instantaneous velocity fields in Figure 5 shows the most frequently observed
mechanisms for the generation of the PHV. In Figure 5a,b, no coherent HV is observed in the
junction region. Then a weak HV, as indicated by the black arrow, is generated in the separation
region in Figure 5c. This vortex increases in size and strength while moving downstream slowly.
Finally, it grows up into a PHV dominating the junction region in Figure 5d. The growth of a HV
shedding from the separation region into a PHV is natural and well-accepted since all PHVs are
generated in the separation region for a laminar HV system [26].
Figure 5. Sequence of instantaneous velocity fields illustrating the generation of a PHV for case
C13900. (a) t = 0; (b) t = 0.29D/Um; (c) t = 0.58D/Um; (d) t = 0.87D/Um; (e) t = 1.13D/Um; (f) t = 1.40D/Um
Figure 6 presents another less frequently observed mechanism for the generation of the PHV.
In Figure 6a, no PHV is observed in the junction region. Instead, a newly generated HV (HV1) is
shedding away from the separation region, and a relatively weak prograde vortex (PV1, vortex
with negative vorticity) originating in the incoming flow is convected toward the junction region by
the down-flow in front of the cylinder. The vortex PV1 becomes stronger while approaching the
bed. At the same time, a second HV (HV2) is generated in the separation region in Figure 6b.
Instead of growing up into PHVs, the vortex HV1 dissipates gradually, and the vortex HV2 is partly
engulfed by the vortex PV1 in Figure 6c,d. Meanwhile, another prograde vortex (PV2) convected by
Water 2019, 11, 2079 9 of 18
the down-flow is engulfed by the vortex PV1 in Figure 6e. After merging with vortices HV1 and
PV2, the vortex PV1 becomes a relatively large and coherent PHV in Figure 6f. The generation of a
PHV through the merging of several prograde vortices originating in the incoming flow and the
down-flow is observed in all experimental cases of the present study. Very occasionally, even a
single prograde vortex is observed to grow up into a PHV. A similar mechanism for the generation of a
PHV was also observed by Kirkil et al. [16] in their LES investigation at ReD = 16,000. However, it is
unclear whether such a mechanism exists for junction flows with larger cylinder Reynolds number.
Figure 6. Sequence of instantaneous velocity fields illustrating the generation of a PHV for case
C13900. (a) t = 0; (b) t = 0.40D/Um; (c) t = 0.60D/Um; (d) t = 0.81D/Um; (e) t = 1.01D/Um; (f) t = 1.22D/Um
3.2. Lifespan of PHV
The lifespan of each PHV can be obtained directly from the vortex tracking method. However,
due to the finite number of PHVs measured in each experimental case, only the probability density
function (PDF) of the lifespan of PHVs is presented in Figure 7. The result shows the lifespan of
PHVs varies in quite a large range. No obvious Reynolds number dependence of the lifespan is
observed. Although not converged yet, the calculated mean lifespans of PHVs are 6.04D/Um,
4.81D/Um, and 4.64D/Um 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/Um at ReD = 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/Um seems a reasonable approximation of the mean lifespan of PHVs.
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Figure 7. Probability density function (PDF) of the lifespan of PHVs.
3.3. Oscillation of PHV
During their lifetimes, the PHVs undergo oscillations in both the streamwise and vertical
directions. Meanwhile, the core size of PHVs was observed to change with their streamwise
position. Figure 8 presents the temporal cross-correlation functions of the streamwise position and
the vertical position, Rxy, and of the streamwise position and the core area, RxA. In Figure 8a, the
correlation function, Rxy, 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
8(b), the correlation function, RxA, 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.
Figure 8. Correlation functions of (a) streamwise and vertical positions, (b) streamwise position and
core area.
The mechanism for the streamwise oscillation is complicated. However, observations of
instantaneous velocity fields indicate that the large-amplitude streamwise movements relate tightly
with the incoming flow. Figure 9 presents the downstream movement of a PHV during a time interval
of about 1.6D/Um. 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.
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One notable observation is that the forward movement of the PHV is accompanied by an
inrush of high-momentum fluid from the incoming flow down to the bed upstream of it. The
downwards inclined streamlines and the relatively large streamwise velocity upstream of the PHV
in Figure 9 indicate clearly such an inrush structure. The inrush fluid penetrates the near-wall
separation region and pushes the PHV forward to the cylinder.
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/Um; (c) t = 0.65D/Um; (d) t = 0.96D/Um; (e) t =
1.29D/Um; (f) t = 1.60D/Um
Compared to the downstream movements, the large-amplitude upstream movements of PHVs
are less frequently observed. Figure 10 presents the upstream movement of a PHV during a time
interval of about 1.75D/Um. 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.
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/Um; (c) t = 0.70D/Um; (d) t = 1.04D/Um; (e) t =
1.75D/Um; (f) t = 2.08D/Um
The influence of the incoming flow on the streamwise movements of the PHV was firstly
reported by Agui and Andreopoulos [15]. They found the streamwise oscillation of the PHV is
forced by large scale structures originating in the incoming flow.
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Later, Praisner and Smith [27] observed in their flow visualization and PIV measurement
results that the erupting of counter-rotating secondary vortex upstream of the PHV is followed by
an inrush of fluid from the incoming flow. During this eruptive process, the PHV is characterized
by obvious streamwise motions. On the other hand, Apsilidis et al. [12] observed in their
high-Reynolds-number experimental case (ReD = 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.
The streamlines in Figures 9 and 10 also show that the vertical positions of PHVs are lower
when they locate closer to the cylinder. The mechanism for this trend should be the strengthening of
the down-flow which pushes the PHV toward the wall while approaching the cylinder. Meanwhile,
the core sizes of PHVs decrease gradually while approaching the cylinder. The decreasing in size of
the PHV should be attributed to the enhanced stretching along its axis.
3.4. Strength Variation of PHV
Figure 11a,b present the temporal correlation functions of the streamwise position and swirling
strength, R
xΛ, and of the streamwise position and circulation strength, RxΓ, 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, xvavg. The region downstream of the time-averaged streamwise
position is termed the near-cylinder region hereafter.
Figure 11. Correlation functions of (a) streamwise position and swirling strength, and (b)
streamwise position and circulation.
Figure 12 shows the conditional cross-correlation functions between the streamwise position
and swirling strength of PHVs. In Figure 12a, the correlation functions exhibit a weak positive peak
on the left-hand side of the origin. It implies that for PHVs locating upstream of the time-averaged
position, increasing in coherence is necessary, at least partly, for their downstream movements. On
the other hand, typical negative peaks are observed on the right-hand side of the origin in Figure
12b. It suggests that for PHVs locating in the near cylinder region, the downstream movements
cause PHVs to lose coherence. The magnitude difference of the peaks indicates that the change of
coherence with the streamwise position is more pronounced when PHVs appear in the near-cylinder
region.
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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 presents the conditional cross-correlation functions between the streamwise position
and circulation strength of PHVs. In Figure 13a, the peaks on both sides of the origin are comparable,
indicating no dominant variation trend of the circulation strength with the streamwise position. This
is reasonable, as the coherence and size of PHVs vary differently with the changing of the
streamwise position in the region upstream of the time-averaged stream position. On the other hand,
typical negative peaks are observed on the right-hand side of the origin in Figure 13b, indicating the
weakening of PHVs while they move downstream in the near-cylinder region. Such a tendency
agrees with the results that both the size and coherence of PHVs decrease when they are moving
downstream in the near-cylinder region. Meanwhile, the statistical results support the evolution
model proposed by Dargahi [14] and the observations of Li et al. [19].
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.
The mutual interactions between the PHV and other vortices are found to be one of the major
mechanisms responsible for its variation in strength. The sequence of instantaneous velocity fields
in Figure 14 shows typical examples of such interactions. In Figure 14a, the turbulent HV system
consists of a PHV, an upstream HV (HV1), and a wall-attached secondary vortex (SC) with opposite
rotation sense between the two HVs. Meanwhile, several prograde vortices (PV1, PV2, and PV3) are
convected toward the junction region by the incoming flow and the down-flow in front of the
cylinder. These prograde vortices are engulfed partly or entirely by the PHV in Figure 14b–d,
causing the PHV to become larger and stronger. At the same time, the secondary vortex (SC) grows
and strengthens gradually. Finally, it is lifted off the wall in Figure 14f and interacts with the PHV,
causing the PHV becoming smaller and weaker.
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Except for the wall-attached secondary vortex, video animations show that the PHV also
interacts with retrograde vortices originating in the incoming flow and the down-flow.
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/Um; (c) t = 0.34D/Um; (d) t = 0.51D/Um; (e) t =
0.68D/Um; (f) t = 0.84D/Um; (g) t = 1.02D/Um; (h) t = 1.18D/Um; (i) t = 1.35D/Um
The merging of the PHV with other co-rotating vortices has been widely reported in previous
research [12,14–19]. However, one notable observation is that with the increasing of Reynolds
number, more vortices being engulfed by the PHV are originating in the incoming flow and the
down-flow instead of in the separation region. One observed mechanism is that the HVs shedding
from the separation region become relatively weaker than the counter-rotating secondary vortex
appearing upstream of the PHV with the increasing of Reynolds number. Accordingly, it becomes
more difficult for the HVs to overcome the interactions with the secondary vortex in flows with
higher Reynolds number. Therefore, the secondary vortex works like a barrier and prevents the
merging of these HVs with the PHV. For example, the HV that appears in Figure 14a fails to merge
with the PHV until it is dissipated in Figure 14e. The experimental results of Apsilidis et al. [12] also
showed a similar mechanism. Meanwhile, previous research [28] has shown that the population of
prograde vortices in the incoming turbulent open channel flow changes with the Reynolds number.
Therefore, the evolution and dynamics of the turbulent HV system are anticipated to be influenced
by the Reynolds number, as indicated by previous investigations [12,17]. Another noteworthy
feature presented in Figure 14e–h is that the turbulent HV system is at the zero-flow mode when
the secondary vortex is lifted off the wall. This feature agrees well with previous research on the
multi-mode dynamics of the turbulent HV system [10–13]. However, two findings different from
previous suggestions in the literature on the back-flow mode can be read out in these figures.
Firstly, the zero-flow mode does not have preferential locations. It can take place at locations far
away from the cylinder. Similar conclusions were also given recently by Apsilidis et al. [12] and
Chen et al. [13]. Secondly, the PHV does not always retreat downstream toward the cylinder during
the interaction with the lifted secondary vortex, as is observed by Paik et al. [10] and Escauriaza and
Sotiropoulos [11]. As an example, the streamwise position of the PHV in Figure 14 is nearly
unchanged during the whole interaction process.
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In addition to the interactions with counter-rotating vortices, interaction with the bed plays an
important role in the weakening processes of PHVs. As indicated in Figure 8a, PHVs get closer to the
bed while moving downstream.
The shortening in distance intensifies the interactions between the PHV and the bed, causing
the weakening of the PHV. Therefore, for PHVs locating in the near-cylinder region, losing strength
is due to the dominance of interactions with counter-rotating vortices and the bed when they
approach closer to the cylinder.
3.5. Disappearance of PHV
Due to the enhanced stretching and bed interaction, PHVs become smaller and weaker while
approaching the cylinder. Therefore, the disappearances of PHVs are observed to take place mainly
in the near-cylinder region. Figure 15a–f presents a process by which a PHV disappears, and Figure
15g shows the variations of the strength and size of the PHV during this process. In Figure 15a, the
PHV locates at about x/D = -0.68. Then it moves gradually toward the cylinder through Figure
15b–d. The time histories of the circulation strength and core area of the PHV indicate that its
strength and size decrease significantly during the forward moving process. In Figure 15d, a weak
HV generates in the separation region upstream of the PHV. However, the PHV fails to merge with
this vortex and dissipates continually until its disappearance in Figure 15e.
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.
Several other mechanisms for the disappearance of the PHV were also reported in previous
research. Paik et al. [10] found that after being ground up by hairpin vortices originating underneath
it, the remnant of the PHV retreats downstream closer to the cylinder and is dissipated here.
Apsilidis et al. [12] observed that the disappearance of the PHV is preceded by the collision of it with
a fluid patch originating from the down-flow. Even though the proposed mechanisms for it show
Water 2019, 11, 2079 16 of 18
some differences, all observations agree the disappearance of the PHV mainly occurs in the region
very close to the cylinder surface.
The turbulent HV system triggers the local scour process in front of vertical piers on a flat loose
bed by significantly amplifying the bed shear stress in this region [6,8,14]. The amplification of the
bed shear stress is tightly related to the vertical position, size, and strength of the HVs as its
magnitude is determined directly by the wall-normal gradient of the streamwise velocity, ∂U/∂y,
induced by the HVs at the bed surface. The present investigation shows that the vertical position,
size, and strength of the PHV vary regularly with its streamwise position. These change trends are
valuable for understanding and predicting the spatial distribution of the bed shear stress in front of
the cylinder. For instance, when other features of a PHV remain unchanged, the downward
movement and the weakening in circulation strength result in an increase and decrease in the bed
shear stress, respectively. Therefore, the peak value of the time-averaged bed shear stress is
anticipated to occur around the time-averaged streamwise position of PHVs, as has been reported in
previous investigations [5]. Meanwhile, both the present and many existing investigations indicate
that the evolution of turbulent HV system is strongly influenced by structures in the incoming flow
and the down-flow. The underlying mechanisms provide possible ways of controlling the local
scour around piers indirectly through manipulating the coherent structures in the incoming flow
and the down-flow. Nevertheless, due to the relatively low cylinder Reynolds number of flows
investigated here, experiments with much higher cylinder Reynolds number are needed for
completing our understanding on the evolution of turbulent HV system in practical flows.
4. Conclusions
The evolution of the turbulent HV system around a circular cylinder mounted on the flat bed of
an open channel was investigated quantitatively and statistically with time-resolve PIV
measurements. The cylinder Reynolds numbers of the flows were 8600, 10.600, and 13.600,
respectively. Among vortices comprising the turbulent HV system, the PHV was selected as the
representative vortex, and was tracked continuously in successive velocity fields with a novel
method to obtain the time histories of its position, size, coherence, and circulation strength. These
histories were cross-correlated to investigate the evolution trends of the PHV. Meanwhile,
mechanisms underlying the evolutionary process of the turbulent HV system were re-investigated
and enriched by using the time-resolved instantaneous velocity fields.
The PHV undergoes a complete evolutionary process composed of generation, life, and
disappearance stages. The generation of the PHV originates either from the growth of a HV
shedding from the separation region or the merging of several vortices from the incoming flow or
the down-flow. The dissipation of the PHV takes place mainly in the region very close to the cylinder
due to the enhanced interactions with counter-rotating vortices and the bed. The mean lifespan of
the PHV is estimated to be about 5.0D/Um. 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.
The statistical relationships between the features of the PHV during its evolutionary process
were obtained. When a PHV moves downstream from the separation region toward the cylinder, it
approaches the bed and decreases in size. Its coherence first increases slightly and then decreases
quickly. Consequently, the circulation strength of the PHV shows no tendency variation and decreases
dramatically in the region upstream and downstream of its time-averaged position, respectively. The
opposite is true when the PHV moves upstream from the region very close to the cylinder. The change
trends of the vertical position, size, and circulation strength can correctly interpret and predict the spatial
distribution of the time-averaged bed shear stress in front of a vertical cylinder.
Water 2019, 11, 2079 17 of 18
Author Contributions: Q.C. conducted the experiment, proposed the methods, and wrote the initial paper.
Q.C. and Z.Y. completed the data analysis; H.W. helped with review and editing of the manuscript.
Funding: This research was funded by the Fundamental Research Funds for the Central Universities
(2016RC033) and the National Natural Science Foundation of China (51609002).
Acknowledgments: The authors thank Danxun Li and Qiang Zhong for their constructive suggestions for
preparing the paper.
Conflicts of Interest: The authors declare no conflict of interest.
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