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Journal of Hydraulic Research Vol. 46, No. 2 (2008), pp. 247–264
© 2008 International Association of Hydraulic Engineering and Research
Kinematics of horseshoe vortex development in an evolving scour hole
at a square cylinder
Cinématique du développement de vortex en fer à cheval dans un processus
d’affouillement dû à une pile de section carrée
RAJKUMAR V. RAIKAR, IAHR Member, Doctoral Research Fellow, Department of Civil Engineering,
Indian Institute of Technology, Kharagpur 721302, West Bengal, India
SUBHASISH DEY, IAHR Member, Professor, Department of Civil Engineering, Indian Institute of Technology,
Kharagpur 721302, West Bengal, India. E-mail: sdey@iitkgp.ac.in
ABSTRACT
This paper presents an experimental investigation on the characteristics of the development of turbulent horseshoe vortex flow in an evolving (inter-
mediate stages and equilibrium) scour hole at square cylinder measured by an acoustic Doppler velocimeter (ADV). As the primary objective was
to study the turbulent flow characteristics of horseshoe vortex in an evolving scour hole, the flow zone downstream the cylinder was not considered.
Experiments were conducted with a square cylinder of width 12 cm embedded in the bed of uniform sand of median diameter 0.81 mm under the
approaching flow having undisturbed flow depth (=25cm) greater than twice the width of the cylinder and the depth-averaged approaching flow
velocity (=35.7 cm/s) equaling approximately 0.95 times the critical velocity for the uniform bed sand. The ADV flow measurements were taken
inside the intermediate scour holes (having depths of 0.25, 0.5 and 0.75 times the equilibrium scour depth) and the equilibrium scour hole (frozen by
spraying glue). The contours of the time-averaged velocities, turbulence intensities and Reynolds stresses at different azimuthal planes (0◦,45
◦and
90◦)are presented. The change of the characteristics of horseshoe vortex flow associated with a downflow from intermediate stages to equilibrium
condition of scour hole is revealed through the vector plots of the flow field at different azimuthal planes. Also, the flow characteristics of the horseshoe
vortex are analyzed from the point of view of similarity with the velocity and turbulence characteristic scales. The important observation is that the
flow and the turbulence intensities in horseshoe vortex flow in a developing scour hole are reasonably similar.
RÉSUMÉ
Cet article présente une recherche expérimentale sur les caractéristiques du développement de vortex turbulent en fer à cheval dans un affouillementdû
à une pile carrée; son évolution (étapes intermédiaires et équilibre) est mesurée par Vélocimétrie DopplerAcoustique (ADV). Le premier objectif étant
d’étudier les caractéristiques turbulentes du vortex en fer à cheval, la zone d’écoulement en aval de l’obstacle n’a pas été considérée. Les expériences
ont été entreprises avec une pile carrée de largeur 12cm, plantée dans le lit garni uniformément de sable de diamètre médian 0.81 mm, soumis à un
écoulement incident de tirant d’eau (=25 cm) supérieur à deux fois la largeur de l’obstacle, et de vitesse moyennée en profondeur (=35.7 cm/s)
égale approximativement à 0.95 fois la vitesse critique du lit de sable uniforme. Les mesures ADV de l’écoulement ont été faites à l’intérieur des
affouillements intermédiaires (de profondeurs 0.25, 0.5 et 0.75 fois la profondeur d’équilibre) et dans l’affouillement d’équilibre (fixé en pulvérisant
de la colle). Les profils des vitesses, des intensités de turbulence et des contraintes de Reynolds moyennées en temps, dans différents plans azimutaux
(0◦,45
◦et 90◦) sont présentés. L’évolution des caractéristiques du vortex en fer à cheval, liée à un écoulement descendant, depuis les étapes
intermédiaires jusqu’à l’état d’équilibre de l’affouillement, est indiquée par les tracés vectoriels du champ de courant dans différents plans azimutaux.
En outre, les caractéristiques du vortex en fer à cheval sont analysées du point de vue de la similitude avec les échelles de vitesse et de turbulence.
L’observation importante est que l’écoulement et les intensités de turbulence dans le vortex en fer à cheval d’un affouillement en formation sont
raisonnablement semblables.
Keywords: Bridge pier, hydraulics, open channel flow, scour, sediment transport, steady flow, three-dimensional flow,
turbulent flow
1 Introduction
Failure of bridges due to scour at bridge foundations is a common
phenomenon, and hence the prediction of scour depth at piers and
abutments is a topic of importance to the field engineers. Numer-
ous investigations on pier and abutment scour have been reported
by various researchers. Review of the important experimental
and field studies was given by Breusers et al. (1977), Breusers
Revision received November 20, 2006/Open for discussion until October 31, 2008.
247
and Raudkivi (1991), Dey (1997), Hoffmans and Verheij (1997),
Melville and Coleman (2000), Richardson and Davis (2001),
Sumer and Fredsøe (2002) and Barbhuiya and Dey (2004). How-
ever, these studies primarily focus on the estimation of maximum
scour depth at piers and abutments. The comprehensive under-
standing of the scour mechanism from the viewpoint of the flow
and turbulence characteristics of the horseshoe vortex aids the
precise prediction of the scour depth. The characteristics of
248 R. V. Raikar and S. Dey Journal of Hydraulic Research Vol. 46, No. 2 (2008)
horseshoe vortex at the junction of the cylinder and the base
plate were well explored (see Ballio et al., 1998; Simpson, 2001),
while the turbulent flow fields inside a scour hole at circular cylin-
ders were studied by Melville (1975), Melville and Raudkivi
(1977), Dey et al. (1995), Dey (1995), Ahmed and Rajaratnam
(1998), Istiarto and Graf (2001), Graf and Istiarto (2002) and Dey
and Raikar (2007). Importantly, Lin et al. (2003) classified the
horseshoe vortex system at the junction of the square cylinder and
the base plate as steady, periodical oscillatory and turbulence-like
chaotic vortex systems based on the cylinder Reynolds number
and the ratio of the cylinder width to the displacement boundary
layer thickness.
The present study reports on how the flow and turbulence char-
acteristics of the horseshoe vortex change with the development
of the scour hole at a square cylinder. Also, the flow and turbu-
lence characteristics scales are proposed from the point of view
of similarity. However, the flow zone downstream the cylinder
is beyond the scope of the investigation, as downstream (beyond
90◦)the cylinder, the horseshoe vortex attenuates considerably
and is insignificant.
2 Experimental setup and procedure
The laboratory experiments were conducted in a glass-walled
flume of 15 m long, 0.9 m wide and 0.7 m deep at the Hydraulic
and Water Resources Engineering Laboratory, Indian Institute of
Technology, Kharagpur. Perspex made square cylinder of width
b=12 cm was embedded vertically in the middle of a sand
recess (2.4 m long, 0.9 m wide and 0.3 m deep), with its side
facing the approaching flow. The sand recess was located at 10 m
from the flume inlet. The uniform sand of median diameter d50 =
0.81 mm, having angle of repose 30◦, critical bed shear stress =
0.408 Pa and geometric standard deviation of the particle size
distribution σg=1.34, was used. For uniformly graded sand,
σg[= (d84/d16 )0.5]is less than 1.4 (Dey et al., 1995). A false floor
was constructed at an elevation of 0.3m from the flume bottom
through out the length of the flume to maintain the equal bed
level of the sand in the recess. The same uniform sand that was
used for the scour test was glued at the top surface of the false
floor to simulate the turbulent flow over a rough planar sand-bed
during the experiment. The flow discharge was measured using
a calibrated V-notch weir fitted at the inlet of the flume. The
flow depth in the flume was adjusted by a downstream tailgate.
During the experiments, the approaching flow depth hwas set
as 25 cm and the depth-averaged approaching flow velocity U
was maintained as 35.7 cm/s, which was about 0.95 times the
critical velocity for the uniform sand satisfying the clear-water
scour condition. The depth-averaged approaching flow velocity
was determined form the measured profile of the approaching
flow velocity at 2 m upstream the cylinder where the effect of
the cylinder did not exist. The instantaneous scour depth at a
cylinder was measured observing the position of the base of the
scour hole by sliding a periscope up and down in the cylinder. An
intense light enabled to read the scour depth from the graduation
made on the transparent body of the cylinder with an accuracy
of 1 mm. When negligible (1 mm or less) difference of scour
depth was observed at an interval of two hours after 72h, it was
considered that an equilibrium stage of scour hole was attained.
After the run was stopped, the equilibrium scour depth, observed
at the upstream base of the cylinder, was then carefully measured
by a Vernier point gauge. The equilibrium scour depth dse was
measured as 21 cm. In addition to the equilibrium scour hole, a
number of intermediate scour holes (having intermediate scour
depths dsof 0.25, 0.5 and 0.75 times the equilibrium scour depth
dse)were stabilized for the acoustic Doppler velocimeter (ADV)
measurements. Once the equilibrium scour hole was obtained, the
same run was repeated to achieve the intermediate scour holes
for the desired fraction of the equilibrium scour depth. At the
end of the experimental run, the water was carefully drained
out from the scoured bed. When the bed was reasonably dry,
the scoured sand-bed was frozen by spraying a synthetic resin
mixed with water (1 : 3 by volume). The bed was sufficiently
impregnated becoming rock-hard with the resin when it was left
to set for a period of 72 h, facilitating measurements by the ADV.
The contour lines of the scour hole are shown in Fig. 1(a)–1(d).
A SonTek made 5 cm downlooking ADV (16 MHz MicroADV
Lab Model) having a sampling rate and volume of 50 Hz and
0.09 cm3, respectively, was used to measure the instantaneous
three-dimensional velocity components. The output data from
the ADV was filtered using a spike removal algorithm. The sam-
pling durations were ranged from 3 to 10 min, depending on
the turbulence intensity, to have a statistically time independent
average-velocity. Near the scoured bed, the sampling durations
were relatively long. TheADV readings were taken along several
vertical lines at different azimuthal planes. The lowest vertical
resolution of the ADV measurements was 0.2 cm. On the other
hand, the lowest horizontal resolution of the ADV measurements
was 1 cm. However, the vertical resolution of the measurements
was higher above the scour hole. In order to avoid overlapping
or congested plots, the experimental data at the lowest resolution
are not shown in the plots. The measurement by the ADV probe
was not possible in the zone 4.5 mm above the bed.
3 Analyses of flow and turbulence fields
Figure 1(e) shows the schematic of a scour hole at a square cylin-
der. A cylindrical polar coordinate system is used to represent
the flow and turbulence fields [see Fig. 1(e)]. It is convenient
to use a cylindrical polar coordinate system, as the upstream
scour contours are almost concentric. The time-averaged veloc-
ity components in (θ,r,z) are represented by (u,v,w), whose
corresponding fluctuations are (u,v
,w
). The velocity and tur-
bulence fields are plotted in an rz-plane at different azimuthal
angles θ(=0◦,45
◦and 90◦). In order to avoid the solid portion
of the cylinder in the diagrammatic representations, the abscissa
scale is chosen as r0(=r−0.5b). Different resolutions are
used along abscissa and ordinate to have a clear representation
of the flow field inside the intermediate scour holes, which are
smaller in dimension in actual scale. Though it stretches the fig-
ures along the horizontal, it makes possible to show the flow
Journal of Hydraulic Research Vol. 46, No. 2 (2008) Kinematics of horseshoe vortex development in an evolving scour hole 249
Figure 1 Scour contours for (a) ds=0.25dse, (b) ds=0.5dse , (c) ds=0.75dse , (d) ds=dse and (e) coordinate system
and turbulence contours. It is important to mention that the outer
radius of the ADV sensor was 2.5 cm having three receiving trans-
ducers mounted on short arms around the transmitting transducer
at 120◦azimuth intervals, which made possible to measure the
flow as close as 2 cm (approximately) to the cylinder surface.
3.1 Flow fields
The time-averaged velocity vectors, whose magnitude and direc-
tion are (v2+w2)0.5and arc tan(w/v), respectively, at different
azimuthal planes (0◦,45
◦and 90◦) for intermediate scour holes
and the equilibrium scour hole are depicted in Fig. 2(a)–2(c).
The vector plots at 0◦and 45◦exhibit the characteristics of the
horseshoe vortex along with the downflow along the upstream
face of the cylinder. The vortices are apparently stretched due to
the different length scales of the axes (ordinate and abscissa)
for intermediate scour holes. In the initial stage of the scour
(ds=0.25dse), the vortical flow is not clear. With an increase in
the dimension of the scour hole, the size of the horseshoe vortex
core, which is confined to inside the scour hole, becomes bigger
and well defined. The horseshoe vortex is a forced vortex type of
flow, because the swirl velocity increases in the outward direction
from the center of the vortex. The shape of the vortex is elliptical
in cross section with its major axis approximately bisecting the
angle made by the slope of the scour hole with the horizontal. It
is noticeable that the height (length of the minor axis) of the ellip-
tical vortex at 0◦is larger that that at 45◦. In general, the length
of the major axis is about 2–4 times the length of the minor axis.
The vortical flow is strongest at 0◦, while it decreases with an
increase in θ. Above the scour hole (z>0), the flow is horizontal
and towards the cylinder, but it is downward close to the cylin-
der. At 90◦, the vortical flow is not distinct due to the separated
flow. However, a close examination of the vector field at 90◦
reveals that a feeble vortical flow prevails near the scoured bed.
Figure 3(a)–3(c) represent the time-averaged absolute velocity
V[= (u2+v2+w2)0.5]contours at different azimuthal planes
(0◦,45
◦and 90◦)for intermediate scour holes and the equilib-
rium scour hole. It exhibits the exclusive vortical flow at 0◦due
to the absence of tangential velocity u. On the other hand, at 90◦,
tangential velocity uis a predominant flow feature. At 45◦, the
250 R. V. Raikar and S. Dey Journal of Hydraulic Research Vol. 46, No. 2 (2008)
Figure 2 Velocity vectors at azimuthal planes: (a) θ=0◦, (b) θ=45◦and (c) θ=90◦
Journal of Hydraulic Research Vol. 46, No. 2 (2008) Kinematics of horseshoe vortex development in an evolving scour hole 251
Figure 3 Contours of V(in cm/s) at azimuthal planes: (a) θ=0◦, (b) θ=45◦and (c) θ=90◦
252 R. V. Raikar and S. Dey Journal of Hydraulic Research Vol. 46, No. 2 (2008)
diminishing nature of the horseshoe vortex with θis displayed.
The contours lines of Vare concentrated near the scoured bed,
indicating a region of rapid change of the magnitude of velocity.
With the development of the scour hole, Vdecreases near the
scoured bed. At 90◦, the magnitudes of vand ware negligible
close to the cylinder as the side of the cylinder is parallel to the
flow and the flow separation takes place at 45◦from the sharp
edge of the cylinder, resulting in the lower magnitude of Vnear
the cylinder.
The characteristics of the individual velocity components (u,v
and w) were analyzed, though they are not furnished in this paper.
Nevertheless, a close observation of Figs 2(a)–2(c) and 3(a)–3(c)
provides with adequate information on the characteristics of u,
vand w. The tangential velocity ucharacterizes the passage of
the approaching flow by the side of the cylinder and it steers the
horseshoe vortex towards the cylinder downstream. The mag-
nitude of the tangential velocity ualong the upstream axis of
symmetry (that is at 0◦azimuthal plane) is essentially zero, and
it becomes finite increasing with an increase in azimuthal angle
θ. For instance, the magnitude of uat 90◦is 1.3–1.5 times greater
than that of uat 45◦at the corresponding locations. At 45◦and
90◦, the magnitude of uis larger when the dimensions of scour
holes are small, while it decreases gradually with an increase
in scour depth because of an increase in flow area. The magni-
tude of udecreases with an increase in radial distance r0from
the cylinder and remains almost constant over the flat bed. On
the other hand, uincreases with an increase in z. The vertical
gradient of u(i.e. ∂u/∂z) inside the scour hole (i.e. z≤0) is
more than that above the scour hole (i.e. z>0). The magnitude
of ∂u/∂z is greater inside the smaller scour holes and decreases
progressively with the development of scour hole. This indicates
the rapid change of uin an evolving scour hole. At 0◦, the radial
velocityvinside the scour hole (z≤0) changes direction, which
occurs approximately at a depth of 0.55–0.65 times the local scour
depth below the original bed level. This indicates the separation
of the approaching flow below the edge of the scour hole forming
a reversal flow inside the scour hole in the cylinder upstream con-
firming the existence of a strong horseshoe vortex inside the scour
hole. Inside the scour hole (z≤0), the magnitude of vis reduced
suddenly along z(from z=0 towards the bed) and becomes
zero at the depths that are 0.55 – 0.65 times the local depth of the
scour hole. Near the bed, vbecomes positive (outwards the cylin-
der) as the flow returns from the base of the cylinder resulting
in a reversed flow along the sloping bed of the scour hole. The
magnitude of the maximum reversed velocity that occurs near the
base of the cylinder is approximately 0.2U. Above the scour hole
(z>0), the radial velocity v, where the variation of valong z
is comparatively less, is negative (towards the cylinder) and uni-
directional. Further, the magnitude of vattains maximum value
near the free surface. At 45◦, the magnitude of vis smaller than
that at the corresponding locations of the flow zone at 0◦. The
reducing nature of vwith an increase in θfrom 0◦to 90◦is due
to the attenuation of horseshoe vortex, while at 0◦the distribu-
tion of vis strongest. As the flow separates at 45◦from the sharp
edge of the square cylinder, the horseshoe vortex detaches at 45◦
azimuthal plane. Hence, the flow at 90◦is out of phase from those
at 0◦and 45◦.At90
◦,vacts towards the cylinder inside the scour
hole (z≤0); while above the scour hole (z>0) the flow deflects
outwards by the side of the cylinder. The magnitude of vertical
velocity wincreases in the downward direction from the free sur-
face indicating the existence of a downward negative pressure
gradient. The maximum-woccurs near the cylinder just below
the original bed level, and then wdecreases towards the base of
the scour hole. The maximum-w, that exists near to the original
bed level for the scour hole having ds=0.25dse, moves deeper as
the scour depth increases. After ds=0.75dse, the position of the
maximum-wremains approximately at a depth of 0.45 times the
scour depth. The maximum magnitude of wregistered was 0.62U
at θ=0◦and z=−0.1 m for ds=0.75dse; while at equilibrium,
it becomes 0.6Uthat is equal to that of Istiarto and Graf’s (2001)
0.6Ufor circular cylinders. However, it is slightly lower than that
of Melville’s (1975) 0.8Ufor circular cylinders, as theADV mea-
surement very close to the cylinder-wall, where wmight possibly
be greatest, was not possible. The magnitude of wdecreases with
an increase in θindicating the attenuation of the horseshoe vortex
towards the downstream. For instance, the maximum magnitudes
of downflow at 45◦and 90◦are 0.75 and 0.5 times that at 0◦.
Using the foregoing findings, the physics of the flow inside
the scour hole at a square cylinder can be detailed. The flow char-
acteristic at a square cylinder is primarily featured by a relatively
large vortex flow and the skewed flow velocity distributions by
the side of the cylinder. To be more explicit, the approaching
flow deflects under the influence of the transverse pressure gra-
dient created by the cylinder. The slower-moving fluid near the
cylinder-wall and the bed with less momentum turns more than
the faster-moving fluid away from the cylinder and the bed. Thus,
the direction as well as the magnitude of the approaching velocity
vector changes with the distance from the cylinder-wall and the
bed causing the skewed velocity distributions. On the other hand,
a horseshoe vortex, which is almost analogous to a ground roller
downstream of a dune crest, is formed inside the scour hole due to
the flow separation at the upstream edge of the scour hole, termed
the separation line. Thus, the scour hole acts as if it were a zone
of separation. The downflow is developed due to the stagnation
pressure gradient of the nonuniform approaching flow velocity
(maximum at the free surface and zero at the bed) adjacent to the
cylinder-wall upstream. The downflow is pushed further down by
the horseshoe vortex. Importantly, the process of flow separation
can be explained using the concept of limiting streamline (see
Dey, 1995). The two limiting streamlines along the original bed
upstream (due to the approaching flow) and the sloping bed (due
to the reversed flow) unite at the edge of the scour hole, forming
a separated streamline. Thus, a surface of separation is formed in
the form of an envelope by the separated streamlines. In this pro-
cess, the approaching flow bends down and rolls into the scour
hole to form a horseshoe vortex (like a helicoidal flow), which
migrates downstream by the side of the cylinder.
3.2 Vorticity and circulation
Figure 4(a)–4(c) shows the vorticity ω(=∂v/∂z−∂w/∂r) contours
at different azimuthal planes (0◦,45
◦and 90◦)for intermediate
Journal of Hydraulic Research Vol. 46, No. 2 (2008) Kinematics of horseshoe vortex development in an evolving scour hole 253
Figure 4 Contours of ω(in s−1)at azimuthal planes: (a) θ=0◦, (b) θ=45◦and (c) θ=90◦
254 R. V. Raikar and S. Dey Journal of Hydraulic Research Vol. 46, No. 2 (2008)
Table 1 Circulations for different azimuthal angles and scour depths
ds(m2/s)
θ=0◦θ=45◦θ=90◦
0.25dse 1.17 ×10−29.55 ×10−32.65 ×10−3
0.5dse 4.27 ×10−23.425 ×10−28.17 ×10−3
0.75dse 8.252 ×10−26.54 ×10−21.565 ×10−2
dse 8.923 ×10−27.077 ×10−21.689 ×10−2
scour holes and the equilibrium scour hole. The vorticity contours
are computed from the contours of vand w(not shown in this
paper). The left hand convention that is positive in anti-clockwise
direction is adopted to define the vorticity. The concentration of
the vorticity inside the scour hole is revealed in Fig. 4(a)–4(c).
From the observation of the vector plots and the vorticity con-
tours, it corroborates that it is a forced vortex type flow. The
higher magnitude of vorticity near the center of the horseshoe
vortex indicates the vortex core. In the initial stage of scour, the
vortex core is smaller in size and it grows with the development
of scour hole. Also, the size of the vortex core decreases with an
increase in θ, becoming smallest at 90◦. The vorticity is consid-
erably weak at 90◦. The circulation of the horseshoe vortex is
calculated from the vorticity contours as shown in Fig. 4(a)–4(c)
by using the Stokes theorem. It is
=c
V·ds=A
ω·dA(1)
where
Vis the velocity vector; dsis the differential displacement
vector over a closed curve; and Ais an area enclosed. Table 1 fur-
nishes the circulations computed for different azimuthal planes
and scour depths. The magnitudes of decrease and increase with
an increase in θand scour hole size, respectively. The magnitudes
of at 45◦and 90◦are 0.79–0.82 and 0.18–0.23 times at 0◦,
respectively, for all intermediate scour holes and the equilibrium
scour hole. However, for all azimuthal planes (0◦,45
◦and 90◦)
for intermediate scour depths having ds=0.25dse,ds=0.5dse
and ds=0.75dse are 0.13–0.16, 0.46–0.49 and 0.9–0.93 times
of equilibrium scour condition, respectively.
3.3 Turbulence fields
The contours of the turbulence intensities u+[= (uu)0.5, where
uis the fluctuation of u],v+[= (vv)0.5, where vis the fluc-
tuation of v] and w+[= (ww)0.5, where wis the fluctuation
of w] at different azimuthal planes (0◦,45
◦and 90◦)for inter-
mediate scour holes and the equilibrium scour hole are shown
in Figs 5(a)–5(c), 6(a)–6(c) and 7(a)–7(c), respectively. The tur-
bulence intensities are the root-mean-square (RMS) values of
the velocity fluctuations. From Figs 5(a)–5(c), 6(a)–6(c) and
7(a)–7(c), it is revealed that the distributions of u+,v+and w+are
almost similar. A core of high turbulence intensity occurs inside
the scour hole as a result of flow separation and then the turbu-
lence intensities decrease with an increase in r0and z. There is
insignificant change in the distribution patterns of u+,v+and w+
with θ. However, the sizes of the high turbulence intensity cores
increase with the development of the scour hole. Except for θr-
plane (that is the horizontal plane), the turbulence characteristics
are non-isotropic, because the mean values (and standard devia-
tion values) of the ratios of u+/v+,v+/w+and w+/u+are 0.996
(0.234), 2.612 (0.758) and 0.433 (0.151), respectively. The con-
tours of the turbulent kinetic energy k[= 0.5(u+2+v+2+w+2)]
are given in Fig. 8(a)–8(c) to illustrate the distributions of kthat
are similar to those of turbulence intensities.
Figures 9(a)–9(c), 10(a)–10(c) and 11(a)–11(c) represent the
contours of the Reynolds stresses τuv[= −ρ(uv), where ρis the
mass density of water], τvw[= −ρ(vw)] and τwu [= −ρ(wu)]
at different azimuthal planes (0◦,45
◦and 90◦)for intermediate
scour holes and the equilibrium scour hole, respectively. In the
plots, the Reynolds stresses are shown relative to the mass density
ρof water. The contours of τuv changes sign from negative above
the scour hole to positive inside the scour hole, while the contours
of τvw and τwu change from positive above the scour hole to
negative inside the scour hole. There is a slight variation of the
Reynolds stresses τvw and τwu above the scour hole, while they
increase significantly inside the scour hole owing to the turbulent
mixing of fluid as a result of the vortical flow. Therefore, a core of
higher magnitudes of τvw and τwu occurs at the central portion of
the scour hole. But, close to the scoured bed, τvw and τwu reduce
due to the lower magnitude (smooth flow) of upward velocity
along the inclined bed of the scour hole. In general, inside the
scour hole, the Reynolds stresses at 45◦are higher than those at
0◦and 90◦. With the development of the scour hole, the Reynolds
stresses increase inside the scour hole.
4 Comparison of the characteristics of horseshoe
vortex at square and circular cylinders
It is, however, important to compare the findings of the charac-
teristics of horseshoe vortex at a squire cylinder with those at a
circular cylinder, which was studied by Dey and Raikar (2007).
The comparisons of the present findings with those of a circular
cylinder in Dey and Raikar (2007) are as follows:
•The magnitudes of uat a square cylinder are smaller than that
at the corresponding locations of the flow zone at a circular
cylinder, as the size of the scour hole at a square cylinder is
larger than that at a circular cylinder.
•The magnitudes of vat a square cylinder are slightly greater
than that at the corresponding locations of the flow zone at a
circular cylinder by approximately 10%.
•The magnitude of downflow at a square cylinder is greater than
that at the corresponding locations of the flow zone at a circular
cylinder by approximately 40%.
•In case of a square cylinder, the size of the horseshoe vortex
core is slightly larger than that of circular cylinder.
•In general, the vorticity ωat a square cylinder is marginally
stronger than that at a circular cylinder.
•The magnitude of circulation for square cylinder is 1.33–1.35
times for circular cylinder.
•In case of a square cylinder, the turbulence intensities and the
Reynolds stresses are greater than those for a circular cylinder.
Journal of Hydraulic Research Vol. 46, No. 2 (2008) Kinematics of horseshoe vortex development in an evolving scour hole 255
Figure 5 Contours of u+(in cm/s) at azimuthal planes: (a) θ=0◦, (b) θ=45◦and (c) θ=90◦
256 R. V. Raikar and S. Dey Journal of Hydraulic Research Vol. 46, No. 2 (2008)
Figure 6 Contours of v+(in cm/s) at azimuthal planes: (a) θ=0◦, (b) θ=45◦and (c) θ=90◦
Journal of Hydraulic Research Vol. 46, No. 2 (2008) Kinematics of horseshoe vortex development in an evolving scour hole 257
Figure 7 Contours of w+(in cm/s) at azimuthal planes: (a) θ=0◦, (b) θ=45◦and (c) θ=90◦
258 R. V. Raikar and S. Dey Journal of Hydraulic Research Vol. 46, No. 2 (2008)
Figure 8 Contours of k(in cm2/s2)at azimuthal planes: (a) θ=0◦, (b) θ=45◦and (c) θ=90◦
Journal of Hydraulic Research Vol. 46, No. 2 (2008) Kinematics of horseshoe vortex development in an evolving scour hole 259
Figure 9 Contours of −uv(=τuv/ρ,incm
2/s2)at azimuthal planes: (a) θ=0◦, (b) θ=45◦and (c) θ=90◦
260 R. V. Raikar and S. Dey Journal of Hydraulic Research Vol. 46, No. 2 (2008)
Figure 10 Contours of −vw(=τvw/ρ,incm
2/s2)at azimuthal planes: (a) θ=0◦, (b) θ=45◦and (c) θ=90◦
Journal of Hydraulic Research Vol. 46, No. 2 (2008) Kinematics of horseshoe vortex development in an evolving scour hole 261
Figure 11 Contours of −wu(=τwu/ρ,incm
2/s2)at azimuthal planes: (a) θ=0◦, (b) θ=45◦and (c) θ=90◦
262 R. V. Raikar and S. Dey Journal of Hydraulic Research Vol. 46, No. 2 (2008)
5 Similarity of velocity and turbulence profiles
Two length scales dand hare introduced for the flow zone inside
(z≤0) and above (z>0) the scour hole, respectively, in order
to group all the flow data together in horseshoe vortex flow at a
square cylinder. Here, dis the local depth of the scour hole mea-
sured from the original bed level. The scales of the velocities u,
vand ware considered as [u]max,[v]max and [w]max for the indi-
vidual profiles, respectively. The flow data for scour hole with
ds=0.25dse are not considered, as the scour hole dimension was
small. Also, for v- and w-profiles, the data at θ=90◦are not
included, as the vortex flow is not prominent at θ=90◦. On the
other hand, for u-profile, the data at θ=0◦are not accounted
because the magnitude of uis negligible. Figure 12(a)–12(c)
illustrate the collapse of all the flow data on a single band by
using the proposed length and velocity scales. For w-profiles,
an emphasis is given on the downflow profiles at the cylinder
upstream. In case of turbulence intensities, consideration of the
corresponding maximum turbulence intensity for the individual
profiles facilitates to bring down all the data approximately on a
single band. Figure 13(a)–13(c) exhibit some degree of collapse
Figure 12 Similarities of (a) u-profiles, (b) v-profiles and (c) w-profiles
Figure 13 Similarities of (a) u+-profiles, (b) v+-profiles and (c)
w+-profiles
of the data plots of turbulence intensities, though the plots have
considerable scatter. However, it is not possible to collapse the
Reynolds stresses (tangential stress) data on a single band. The
possible reasons are due to a rigorous mixing of fluid as a result
of the vortex flow, and hence the Reynolds stresses that are very
sensitive to the turbulent fluctuations subject to uncertain attenu-
ation. Nevertheless, the flow and the turbulence intensities in the
horseshoe vortex flow in an evolving scour hole are reasonably
similar.
6 Applications
The characteristics of developing horseshoe vortex that is
believed to be the primary agent of scour at a cylinder are revealed
from Figs 2–11. In addition, Table 1 provides with information on
the circulation that is considered to be the strength of the horse-
shoe vortex. Furthermore, the turbulence characteristics have
significant role towards the mobility of sediment particles inside
the scour hole. Therefore, the present findings have immense
Journal of Hydraulic Research Vol. 46, No. 2 (2008) Kinematics of horseshoe vortex development in an evolving scour hole 263
importance in developing a mathematical model of local scour
process at a square cylinder including the influence of turbulence.
7 Conclusions
Experiments were conducted to measure the turbulent horseshoe
vortex flow in an evolving (intermediate stages and equilib-
rium) scour hole at a square cylinder by an acoustic Doppler
velocimeter. The contours of time-averaged velocities, turbu-
lence intensities and Reynolds stresses at different azimuthal
planes for intermediate and the equilibrium scour holes have been
presented. The velocity is reversal inside the scour hole forming a
horseshoe vortex. The maximum downflow at cylinder upstream
registered was 0.62 times the average approaching flow veloc-
ity at θ=0◦and z=−0.1 m for ds=0.75dse. The vector
plots of the flow field provide with a good understanding of the
change of the horseshoe vortex in an evolving scour hole. The
horseshoe vortex is a forced vortex type of flow. The size of the
horseshoe vortex core (having the shape of an ellipse) being con-
fined to inside the scour hole increases with the development of
the scour hole. The circulations of horseshoe vortex have been
computed from the vorticity contours by using the Stokes the-
orem. The circulations decrease and increase with an increase
in azimuthal angle and scour hole size, respectively. A core of
higher magnitude of turbulence intensities and Reynolds stresses
exists inside the scour hole and increases with the development of
the scour hole. The characteristics of horseshoe vortex flow have
been analyzed from the viewpoint of similarity proposing the pos-
sible velocity and turbulence characteristic scales. Interestingly,
both the flow and turbulence intensities in horseshoe vortex flow
in an evolving scour hole are found to be plausibly similar.
Notation
A=Area enclosed between successive vorticity contours
b=Width of square cylinder
d=Local depth of scour hole
d16 =16% finer sand diameter
d50 =Median diameter of sand
d84 =84% finer sand diameter
ds=Differential displacement vector
ds=Intermediate scour depth
dse =Equilibrium scour depth
h=Approaching flow depth
k=Turbulent kinetic energy
r=Radial distance
r0=r−0.5b
U=Depth-averaged approaching flow velocity
u=Time-averaged tangential velocity
u=Fluctuation of u
u+=(uu)0.5
V=Time-averaged absolute velocity
V=Time-averaged velocity vector
v=Time-averaged radial velocity
v=Fluctuation of v
v+=(vv)0.5
w=Time-averaged vertical velocity
w=Fluctuation of w
w+=(ww)0.5
z=Vertical distance
θ=Azimuthal angle
=Circulation
σg=Geometric standard deviation
ρ=Mass density of water
τuv =−ρ(uv)
τvw =−ρ(vw)
τwu =−ρ(wu)
ω=Vorticity
References
Ahmed, F., Rajaratnam, N. (1998). “Flow Around Bridge Piers”.
J. Hydraul. Eng. 124(3), 288–300.
Ballio, F., Bettoni, C., Franzetti, S. (1998). “A Survey of Time-
Averaged Characteristics of Laminar and Turbulent Horseshoe
Vortices”. J. Fluids Eng. 120(4), 233–242.
Barbhuiya, A.K., Dey, S. (2004). “Local Scour at Abutments: A
Review”. Sadhana,Proc. Indian Acad. Sci. 29, 449–476.
Breusers, H.N.C., Nicollet, G., Shen, H. W. (1977). “Local Scour
Around Cylindrical Piers”. J. Hydraul. Res. 15(3), 211–252.
Breusers, H.N.C., Raudkivi, A.J. (1991). Scouring. IAHR
Hydraulic Design Manual 2, A.A. Balkema, Rotterdam, The
Netherlands.
Dey, S. (1995). “Three-Dimensional Vortex Flow Field Around
a Circular Cylinder in a Quasi-equilibrium Scour Hole”.
Sadhana,Proc. Indian Acad. Sci. 20, 871–885.
Dey, S. (1997). “Local Scour at Piers, Part 1: A Review of
Development of Research”. Int. J. Sediment Res. 12(2), 23–44.
Dey, S., Bose, S.K., Sastry, G.L.N. (1995). “ClearWaterScour at
Circular Piers: A Model”. J. Hydraul. Eng. 121(12), 869–876.
Dey, S., Raikar, R.V. (2007). “Characteristics of Horseshoe Vor-
tex in Developing Scour Holes at Piers”. J. Hydraul. Eng.
133(4), 399–413.
Graf, W.H., Istiarto, I. (2002). “Flow Pattern in the Scour Hole
Around a Cylinder”. J. Hydraul. Res. 40(1), 13–20.
Hoffmans, G.J.C.M., Verheij, H.C. (1997). Scour Manual. A.A.
Balkema, Rotterdam, The Netherlands.
Istiarto, I., Graf, W.H. (2001). “Experiments on Flow Around
a Cylinder in a Scoured Channel Bed”. Int. J. Sediment Res.
16(4), 431–444.
Lin, C., Lai, W.J., Chang, K.A. (2003). “Simultaneous Particle
Image Velocimetry and Laser Doppler Velocimetry Measure-
ments of Periodical Horseshoe Vortex System Near Square
Cylinder-Base Plate Juncture”. J. Eng. Mech. 129(10), 1173–
1188.
Melville, B.W. (1975). “Local Scour at Bridge Sites”. Report
No. 117, School of Engineering, University of Auckland,
Auckland, New Zealand.
264 R. V. Raikar and S. Dey Journal of Hydraulic Research Vol. 46, No. 2 (2008)
Melville, B.W., Coleman, S.E. (2000). Bridge Scour. Water
Resources Publications, Fort Collins, Colorado.
Melville, B.W., Raudkivi, R.J. (1977). “Flow Characteristics in
Local Scour at Bridge Piers”. J. Hydraul. Res. 15(4), 373–380.
Richardson, E.V., Davis, S.R. (2001). “Evaluating Scour
at Bridges”. HEC18 FHWA NHI-001, Federal Highway
Administration, US Department of Transportation, Washing-
ton, DC, USA.
Simpson, R.L. (2001). “Junction Flows”. Annu. Rev. Fluid Mech.
33(1), 415–443.
Sumer, B.M., Fredsøe, J. (2002). The Mechanics of Scour in the
Marine Environment. World Scientific, Singapore.
