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Received: 7 October 2016 Revised: 6 March 2017 Accepted: 7 March 2017
DOI 10.1002/env.2442
RESEARCH ARTICLE
Prediction of plane-wise turbulent events to the Reynolds stress in
a flow over scour-bed
Haradhan Maity1Bijoy Singha Mazumder2
1Post-Doctoral Fellow, Tata Institute of
Fundamental Research (TIFR), Centre for
Applicable Mathematics, Bangalore, 560065, India
2Professor (Ret.), Emeritus Fellow, Fluvial
Mechanics Laboratory, Physics and Applied
Mathematics Unit, Indian Statistical Institute,
Kolkata, 700108, India
Present address: Bijoy Singha Mazumder, Fluid
Mechanics and Hydraulic Laboratory (FMHL),
Department of Aerospace Engineering and
Applied Mechanics, Indian Institute of
Engineering Science and Technology (IIEST),
Shibpur, Howrah 711103, India.
Correspondence
Haradhan Maity, Post-Doctoral Fellow, Tata
Institute of Fundamental Research (TIFR), Centre
for Applicable Mathematics, Bangalore, 560065,
India.
Email: hmaity@tifrbng.res.in
In this paper, we intend to quantify the contribution of turbulent events to the total
Reynolds shear stresses u′v′,u′w′,andv′w′from the four different quadrants of three
different planes (xy,xz,yz); and to make a comparative study among the planes
in the scour geometry developed by short circular cylinder of fixed length with a
fixed diameter placed over the sand bed transverse to the flow. We also intend to
predict the magnitude of covariance terms u′v′,u′w′,andv′w′and their contribu-
tions in the four quadrants by making use of the conditional probability distribution
of the Reynolds shear stresses −u′v′,−u′w′,and−v′w′, which can be derived by
applying the cumulant-discard method to the Gram-Charlier probability distribution
of the two variables. This consideration motivates the work on the flow over the
obstacle marks generated on sand bed using different short cylinders. The contri-
butions of burst-sweep cycles to the Reynolds shear stresses from the planes over
and within the scour around the obstacle are computed using the quadrant analysis
to identify the leading shear stress plane, which are responsible to form the scour
geometry. It is discovered that the yz and xy- planes are much more important in
the scouring regions, whereas xz- plane is important for the smooth surface. Using
cumulant-discard method (taking into account the cumulants of less than fourth
order), it is shown that the qualitative behaviours of turbulent events agree well
with experimental data. Thus, it is confirmed that even the third-order probability
distribution of the Reynolds stresses can describe the experimental results very well.
KEYWORDS
conditional probability, covariance terms, Gaussian distribution, open channel flow,
scour-bed, turbulence
1INTRODUCTION
Turbulent flow has always been a challenge for scientists,
that is common in nature and has an important role in sev-
eral geophysical processes related to a variety of phenomena
such as river morphology, landscape modelling, atmospheric
dynamics, and ocean currents. As the turbulent flows are
Nomenclature: ar=Dc∕L, cylinder aspect ratio; d50, mean sediment size; Dc, diameter of cylinder; Fs, sediment Froude number; Hw, water depth; h′,
thickness of sand bed; h=Hw−h′, water depth over the sand bed; L, length of cylinder; Q, flow discharge; Re, flow Reynolds number; u,v,w,flow
velocities along stream-wise, transverse and vertical to the flow; um, maximum flow velocity; ̄u,̄v,̄w, time-averaged flow velocities; u′,v′,w′, fluctuations in
u,vand w;ws, width of the scour hole; x,y,z, Cartesian coordinates; 𝜈, kinematic viscosity of the fluid; 𝜌, fluid density; 𝜎g, geometric standard deviation
of the grain size distribution; Fr, froude number; 𝜏xy ,𝜏
xz,𝜏
yz, shear stresses; Fku, stream-wise flux of turbulent kinetic energy; Fkw , vertical flux of turbulent
kinetic energy; Su, coefficients of skewness in the direction of u;Sw, coefficients of skewness in the direction of w;σu=u′2, turbulence intensity (r.m.s
value) in xdirection; σv=v′2, turbulence intensity (r.m.s value) in ydirection; σw=w′2, turbulence intensity (r.m.s value) in zdirection.
irregular, seemingly random (chaotic) and complex, till today
no analytical solutions exist for turbulent flows. We believe
that even after 516 years (Leonardo da Vinci around 1500,
see Gad-El-Hak, 2000), turbulence studies are still in their
infancy. We are still discovering how turbulence behaves,
in many respects. We do have a crude, practical, working
Environmetrics. 2017;28:e2442. wileyonlinelibrary.com/journal/env Copyright © 2017 John Wiley & Sons, Ltd. 1of14
https://doi.org/10.1002/env.2442
2of14 MAITY AND MAZUMDER
understanding of many turbulent phenomena but certainly not
a comprehensive theory, and nothing that provide predictions
of an accuracy demanded by designers.
While coherent phenomena have long been familiar in the
context of turbulent flow, intensive research on the dynamics
of bursting process in the water and air flows over smooth and
rough surfaces has been performed to investigate the nature
of coherent structures in the (x, z)-plane (vertical plane along
the mean flow) by many investigators in the last 40 years.
On the other hand, Hurther and Lemmin (2000) examined
the dynamics of bursting process in open-channel flow over
smooth and rough beds to describe the statistical properties
of covariance terms in (x, z)-plane and (y, z)-plane (across
the mean flow). But, as far as we know, none of them exam-
ined the dynamics of bursting process in (x, y)-plane (parallel
to the mean flow). Recently, Maity and Mazumder (2014)
investigated experimentally the impact of coherent flow
structures over equilibrium scour hole generated by an obsta-
cle using quadrant threshold technique (Brodkey, Wallace,
& Eckelmann, 1974; Wallace, Eckelmann, & Brodkey, 1972)
and described the structure of turbulence in (x, z)-plane
along the mean flow, and also predicted the magnitudes of
the contribution to the Reynolds shear stress from each event
(Nakagawa & Nezu, 1977). They observed that the magnitude
of the contribution to the Reynolds stress from each bursting
event depends on vertical height and horizontal location over
the scour hole. In turbulent boundary layers, coherent struc-
tures with large flux events have been proposed to explain
the “bursting” phenomena responsible for two types of eddy
motions name as “ejections” and “sweeps” (Cantwell, 1981;
Robinson, 1991). These events are traditionally detected
by conditional sampling through quadrant analysis in the
(x, z)-plane (Willmarth & Lu, 1972) and their statistics have
been investigated for a variety of flows and wall-roughness
conditions, for example, experiments in open-channel
(Hurther & Lemmin, 2000; Hurther, Lemmin, & Terray,
2007; Mazumder, 2007; Mazumder, Pal, Ghoshal, & Ojha,
2009; Nakagawa & Nezu, 1977; Nelson, Shreve, McLean,
& Drake, 1995; Ojha & Mazumder, 2008; Venditti & Bauer,
2005); in wind-tunnel (Raupach, 1981); under an-ice bound-
ary layer (Fer, McPhee, & Sirevaag, 2004); in atmospheric
boundary layers (Hurther & Lemmin, 2003; Katul, Kuhn,
Schieldge, & Hsieh, 1997; Katul, Poggi, Cava, & Finnigan,
2006; Sterk, Jacobs, & van Boxel, 1998), and in scour around
vertical circular cylinders (Debnath, Manik, & Mazumder,
2012; Kirkil, Constantinscu, & Ettema, 2008; Sarkar,
Chakraborty, & Mazumder, 2015; Sarkar, Chakraborty, &
Mazumder, 2016). Besides the dominant role of the sweeps
close to a rough wall, an “equilibrium region” is often
observed in fully developed turbulent flows (Krogstad,
Antonia, & Browne, 1992). It is known from experiments
for flow over smooth surface that ejections and sweeps are
the dominant events near bed region, whereas outward and
inward interactions are dominant near the water surface, but
till now in the scouring region the nature of busting events is
very much unpredictable.
In the light of the above, this paper examines the nature
of bursting processes using quadrant threshold technique to
describe the statistical properties of all three covariance terms
u′v′,u′w′,andv′w′in (x, y), (x, z), and (y, z)-planes, respec-
tively, over nearly equilibrium scour hole structure generated
by a short cylinder on sandy bed; and to verify the exist-
ing theory formulated by Nakagawa and Nezu (1977) in
(x, z)-plane. It hitherto remains unclear whether the statisti-
cal properties of all three covariance terms, u′v′,u′w′,and
v′w′are equally important over the scour hole structure or
there is one leading shear stress plane out of the three involv-
ing the bursting events over and within the scour geometry to
be investigated. The development of scour mark around the
obstacle on the sandy bed under the action of turbulent flow
has the potential to be useful to researchers or scientists who
study the dynamics of pipelines and short cylinders placed
on riverbeds, seabeds, and coastal areas (Sumer & Fredsoe,
2002a, 2002b; Voropayev, Cense, McEachern, Boyer, &
Fernando, 1999; Voropayev, Testik, Fernando, & Boyer,
2003); the flow field around bridge piers (Melville & Raud-
kivi, 1977); and the development of crescentic scour marks
in a recent stream (Bhattacharya, 1993; Maity, Dasgupta, &
Mazumder, 2013; Sarkar, et al., 2015, 2016; Sengupta, 1966,
2007) for prediction of palaeocurrent directions.
2EXPERIMENTAL SETUP
Experiments were conducted in a recirculating “closed cir-
cuit” laboratory flume (Mazumder, Ray, & Dalal, 2005) spe-
cially designed at the Fluvial Mechanics Laboratory of the
Physics and Earth Science Division, Indian Statistical Insti-
tute, Kolkata. Both the experimental and the recirculating
channels of the flume have identical dimensions of 10 m
length, 0.50 m width, and 0.50 m height. For details of the
test channel, experimental conditions and the results associ-
ated with the mean velocities and turbulence characteristics
around the scour geometry generated by obstacles, the paper
by Mazumder, Maity, and Chadda (2011) may be referred.
For the test run, a sand bed of thickness h′(=4cm)
and 5 m long covering the entire width (50 cm) of the
flume was laid down at the bottom of the flume. The
median particle diameter d50 of the sand was 0.25 mm
and the standard geometric deviation 𝜎g=0.685. The specific
gravity of sediments was 2.65. Flow depth was kept constant
at Hw=0.30 m. The flow discharge Q=0.0147 m3/sec was
chosen in such a way that the local flow velocity was less than
the critical velocity to initiate the sediment particle move-
ment. Hydraulic slope of the flume was at the order of 0.0001.
The flow Reynolds number was Re(=umh∕𝜈)=6.81 ×104,
where umis the maximum fluid velocity, h=Hw−h′is the
depth of water above the sand bed, and 𝜈is the kinematic
viscosity of fluid. The velocity data were collected using
acoustic Doppler velocimeter (ADV) over the plane bed sur-
MAITY AND MAZUMDER 3of14
FIGURE 1 Schematic diagram of the experimental set-up. ADV =acoustic Doppler velocimeter
TABLE 1 Experimental values of flow parameters
Dc(cm)arQ×10−2(m3∕sec)um(cm∕sec)Fr(= um
gh )FsRe=umh
νws(cm)
4.2 0.42 1.472 26.20 0.164 4.12 68120 4.5
where umis the maximum fluid velocity, ar(= Dc∕L)is the cylinder aspect ratio, Fr is the Froude number, wsis
the scour width and Fs=um∕(𝛾s−1)gd50)is the sediment Froude number.
face. The data were processed to remove the spikes using a
phase space threshold despiking method (Goring & Nikora,
2002). From the ADV data analysis, it was noted that the flow
was free from the secondary currents at the central line of the
flume. The quantitative evidence of lateral and vertical mean
velocity profiles throughout the depth was shown in Figure 3
of Maity and Mazumder (2014).
Experiments were conducted using circular cylinder of
diameter Dc=4.2 cm and length L=10 cm placed on the
sediment bed of known grain-size distribution. The cylin-
der was placed at the center line over the smooth sand bed
transverse to the flow at the measuring station 6 m down-
stream of the channel inlet (Figure 1). For each experiment,
a cylinder was placed over the sand bed at a distance 6 m
downstream of the channel inlet. The length of the cylinder
was shorter than the width of the channel and hence flow was
essentially three dimensional. Flow discharge Qwas fixed to
form scour-shaped structure around the cylinder, and conse-
quently it attained nearly equilibrium condition after a longer
time. Then the vertical velocity profiles from upstream to
downstream along the centerline and the offline (away from
the centerline) were collected using ADV for 180 seconds
at a sampling rate of 40 Hz to ensure full turbulence phe-
nomena under the same flow Reynolds number Re.Forthe
present case, only centerline velocity data were used for anal-
ysis. Using the similar filtering method of Goring and Nikora
(2002), the velocity data were analyzed. A schematic dia-
gram of the experimental setup is shown in Figure 1. The
ranges of flow parameters used for the experiments are pro-
vided in Table 1. The photographic picture of the equilibrium
scour mark developed around the cylinder and the schematic
diagram of the locations of velocity data collection along
FIGURE 2 The equilibrium scour-hole developed at the upstream of
cylinder diameter (Dc=4.0 cm) with flow discharge (Q)=0.015 m3∕sec
(after Maity and Mazumder, 2014)
and within the scour mark are shown in Figures 2 and 3,
respectively.
3THEORETICAL CONSIDERATIONS
For the instantaneous velocity components (u,v,w)inthe
(x,y,z)-directions, the following three relations can be
4of14 MAITY AND MAZUMDER
FIGURE 3 Schematic diagram of equilibrium scour-hole developed at the upstream of the cylinder of diameter (Dc=4.2cm), where A to H are the
sampling locations
written as
u=̄u+u′,v=̄v+v′,w=̄w+w′,(1)
where over bar denotes time-averaged velocity and the prime
denotes its fluctuations. Here u,v,ware the instantaneous
flow velocities in longitudinal, transverse, and vertical direc-
tions, respectively.
The time-averaged local Reynolds shear stresses at a point
in the (u′,v′), (u′,w′), and (v′,w′)-planes are determined as
𝜏xy =−𝜌u′v′,𝜏
xz =−𝜌u′w′,𝜏
yz =−𝜌v′w′(2)
with
u′v′=1
n
n
i=1
(ui−̄u)(vi−̄v),u′w′=1
n
n
i=1
(ui−̄u)(wi−̄w),
v′w′=1
n
n
i=1
(vi−̄u)(wi−̄w),
(3)
where 𝜌is the fluid density and nis the total number of veloc-
ity observations at each point. Figure 4 shows all the three
planes: parallel (u′,v′), longitudinal (u′,w′), and transverse
(v′,w′) in the Cartesian coordinate system (x,y,z), and the
four quadrants of each plane.
3.1 Mean velocities and turbulence characteristics
Examination of velocity data at A on the plane surface away
from the cylinder revealed that the stream-wise mean veloc-
ity profile followed the log-law up to flow depths about 18 cm
from the sand bed, and the lateral and vertical mean veloc-
ities were almost zero throughout the depth. The turbulence
intensities along stream-wise and vertical directions matched
well (Clifford, 1998; Mazumder et al., 2009; Nezu & Rodi,
1986; Ojha & Mazumder, 2008, 2010). Therefore, a reason-
able uniformity in the flow was observed under the given flow
conditions (Maity & Mazumder, 2014).
Subsequently, velocity data at different locations B to H
across the nearly equilibrium scour hole identified the spa-
tial changes of mean flows, turbulence intensities, and the
Reynolds shear stress associated with the scour hole struc-
tures; and facilitated a comparison with different cylinder
FIGURE 4 Sketch of plane-wise quadrant decomposition
diameters (Mazumder et al., 2011). They observed that the
flow reversal occurred at the scour holes due to the formation
of horseshoe vortices (Figure 5) causing frontal scouring at
locations C and D (close to the cylinder) and resulting in flow
instability in the near-bed region. The turbulent kinetic energy
(TKE), turbulent diffusions in both stream-wise and verti-
cal directions at different locations across and within scour
marks were studied by Maity and Mazumder (2012). It was
observed that near the scour hole, the magnitude of TKE was
maximum due to the influence of the pressure gradient and
eddy motions. The stream-wise Fku =0.5(u′u′u′+u′v′v′+u′w′w′)
u3
m
and vertical Fkw =0.5(w′u′u′+w′v′v′+w′w′w′)
u3
m
fluxes of TKE were
discussed along and within the scour holes. Within the scour
region, negative values of Fku indicated the energy transport
in the upward direction from the near-bed region, and the
negative values of Fkw indicated the transport of energy down-
ward. It was also observed that the vertical distributions of
Fku were qualitatively similar to those of Fkw with higher
magnitude (Maity & Mazumder, 2014).
MAITY AND MAZUMDER 5of14
FIGURE 5 Definition sketch for flow and scour around an immovable horizontal circular cylinder. (a) initial stage; (b) scouring stage; and (c) equilibrium
stage
The third-order moments provide a measure of degree of
symmetry of the distribution of velocity fluctuations with
respect to mean velocity components. The positive or negative
values of skewness represent the useful statistical information
of the distributions. The third-order moment was sensitive
to the occurrence of burst-sweep cycles and turbulent flux
contributions; and hence it was directly related to coherent
structures through turbulence characteristics (Gad-El-Hak &
Bandyopadhyay, 1994). The coefficients of skewness in the
directions of uand ware defined as Su=u′3
𝜎3
u
and Sw=w′3
𝜎3
w
;
and these are computed across the scour marks from the loca-
tion A to H. Within the scour hole region, the positive and
negative values of Suin the direction of uindicated respec-
tively the dominance of sweep (high-speed fluid streaks) and
ejection (low-speed fluid streaks) events; and opposite results
were obtained for positive and negative values of skewness Sw
in the wdirection (Maity & Mazumder, 2014).
3.2 Quadrant analysis
The stream-wise turbulent velocity field in the near-wall
region of a boundary layer is known to be organized
and coherent in alternating streaks of high and low speed
velocity, which are persistent in time. Related to this
structure are intermittent, quasi-periodic events, consisting
of ejections of low-speed fluid moves far from the
bed and sweeps of high-speed fluid moving towards the
bed, called coherent structures (i.e., the bursting phenom-
ena). Coherent structures are composed of many eddies,
and their sizes vary continuously from the largest scale to the
smallest. Coherent eddies occur periodically in laminar and
transitional flows, whereas they occur randomly in space and
time in turbulent shear flows, in flows at higher Reynolds
numbers. Thus, one must resort to a statistical analysis or
technique that includes a detecting condition of bursting phe-
nomena that the sequence of events, called turbulent events.
This statistical technique is called “quadrant threshold tech-
nique” or quadrant analysis (Brodkey et al., 1974; Lu &
Willmarth, 1973; Wallace et al., 1972).
From the observations, it revealed that the turbulent bound-
ary layer was directly associated with large-scale coherent
structures occurring irregularly. Coherent structures had been
proposed to explain the bursting phenomena responsible for
the transport processes, turbulence production, and mixing.
These structures were quasi-periodic and they occupied the
total boundary layer depth. The turbulence along the flow
was examined through the quadrant analysis to estimate the
major turbulent events characterizing the coherent flow struc-
tures. The measured instantaneous Reynolds shear stress was
analyzed by employing a framework of quadrant threshold
6of14 MAITY AND MAZUMDER
FIGURE 6 Quadrant decomposition of turbulent events in (u′,v′)plane
technique. To develop this framework in three planes, we first
consider (x,y)-plane and then consider other two planes, (x,z)
and (y,z).
The total Reynolds shear stress u′v′at a point in the
(u′,v′)-plane parallel to the flow is the sum of contributions in
four quadrants. The (u′,v′)plane was divided into five regions
(four quadrants beyond the hyperbolic curves and a hole
region) as shown in Figure 6. In this figure, the cross-hatched
region is called a “hole,” and is bounded by the curves u′v′
=H. We label the turbulent events defined by the four quad-
rants i as: (Qxy
i:i=1, u′>0, v′>0), (Qxy
i:i=2, u′<0,
v′>0), (Qxy
i:i=3, u′<0, v′<0), and (Qxy
i:i=4,
u′>0, v′<0), where lower iand upper xy indices represent
the quadrant and plane, respectively. The turbulent events are
designated according to the sign of the fluctuating velocity
components. Here, quadrants Qxy
1and Qxy
3give positive con-
tributions and quadrants Qxy
2and Qxy
4negative contributions
to the turbulent flux of momentum. Additionally, a hyperbolic
hole His defined, excluding from the analysis a region of
instantaneous values of u′v′that are greater than H.u′v′.
Systematic variation of the hole size Hallows the investiga-
tion of the contributions of events to the total Reynolds shear
stress, whether they are large, small, or frequent. The quad-
rant threshold technique for direct estimation of observed data
of conditional statistics of Reynolds shear stress is presented
briefly.
If <> denotes a conditional average, we have
<u′v′>i,H= lim
T→∞
1
TT
0
u′(t)v′(t)Ixy
i,H(u′(t),v′(t))dt,(4)
where Ixy
i,His the indicator function, defined as
Ixy
i,H(u′,v′)=1,if (u′,v′)is in quadrant i and if u′v′⩾Hu′v′
=0,otherwise.(5)
The stress fraction for quadrant i(= 1,2,3&4),Sxy
i,H,isthen
Sxy
i,H=<u′v′>i,H
u′v′(6)
and it satisfies
Sxy
1,0+Sxy
2,0+Sxy
3,0+Sxy
4,0=1(7)
and
4
i=1
Sxy
i,H+Sxy
5,H=1(H0),(8)
where Sxy
5,His the stress fraction for hole region.
Furthermore, sign of Sxy
i,Hdepends on sign of u′v′.
If u′v′<0, then Sxy
1,H,Sxy
3,H<0andSxy
2,H,Sxy
4,H>0.
If u′v′>0, then Sxy
1,H,Sxy
3,H>0andSxy
2,H,Sxy
4,H<0.
In addition, the time fraction (mean time) Txy
i,Hspent in
making a contribution Sxy
i,His defined as
Txy
i,H= lim
T→∞
1
TT
0
ΔTiIxy
i,H(u′(t),v′(t))dt,(9)
where ΔTiis the time during which the function Ixy
i,Hof
Equation 5 is unity.
The corresponding probability Pxy
i,Hof turbulent event (Sxy
i,H)
is defined by
Pxy
i,H= lim
T→∞
1
TT
0
Ixy
i,H(u′(t),v′(t))dt (10)
such that
4
i=1
Pxy
i,0=1&0⩽Pxy
i,H⩽1 (11)
and
4
i=1
Pxy
i,H+Pxy
5,H=1(H0),(12)
where Pxy
5,His the probability for hole region.
Similarly, we define stress fractions, time fractions, and
probabilities in the (u′,w′)and (v′,w′)-planes: let Sxz
i,Hand Syz
i,H
be the stress fractions in the longitudinal (u′,w′)and trans-
verse (v′,w′)planes, respectively; Txz
i,Hand Tyz
i,Hbe the time
fractioninthe(u′,w′)and (v′,w′)-planes; and Pxz
i,Hand Pyz
i,H
be the probabilities in the (u′,w′)and (v′,w′)-planes. Here
i(= 1,2,3&4)refers to the ith quadrant. Qxz
1,Q
xz
2,Q
xz
3,andQ
xz
4
are the four quadrants in (u′,w′)-plane and Qyz
1,Q
yz
2,Q
yz
3,and
Qyz
4are the four quadrants in (v′,w′)-plane.
MAITY AND MAZUMDER 7of14
3.2.1 Stress fractions
The coherent structures are the vortices generated by the inter-
action of the incoming flow with the scour mark and the
cylinder, which are essentially three-dimensional. Therefore,
the perturbation in the flow field affects the flow in all three
directions and alters the nature of bursting processes. Thus,
it is necessary to study the statistical properties of all three
covariance terms u′v′,u′w′,andv′w′corresponding to (x, y),
(x, z), and (y, z)-planes, respectively, over the equilibrium
scour hole structure generated by a short cylinder on sandy
bed. For comparison, contributions of stress fractions Si,Hto
the individual Reynolds shear stress along (u′,v′),(u′,w′),and
(v′,w′)-planes are estimated from the velocity data collected
in the presence of Dc=4.2 cm.
The stress fractions Sxy,xz,yz
i,Halong the planes at the bottom
level are plotted against the threshold parameter H(H=0to
20) for each of the four quadrants at the location A on the
plane surface in Figure 7a; at location D within the deepest
part of the equilibrium scour hole in Figure 7b; and at location
H above the deposited materials (on the ridge) in Figure 7c.
From the quadrant analysis, the following relationships are
developed among the events and planes:
a. Location A: (1) relationship among the events in (u′,v′)
plane: Sxy
2,0>Sxy
4,0>Sxy
1,0>Sxy
3,0; (2) relation-
ship among the events in (u′,w′)plane: Sxz
4,0>Sxz
1,0>
Sxz
3,0>Sxz
2,0; (3) relationship among the events in
(v′,w′)plane: Syz
1,0>Syz
3,0>Syz
4,0>Syz
2,0;(4)
relationships among all three planes: (i) in the quadrant
i=1: Sxz
1,0>Sxy
1,0>Syz
1,0, (ii) in the quadrant i=2:
Sxz
2,0>Sxy
2,0>Syz
2,0, (iii) in the quadrant i=3:
Sxz
3,0>Sxy
3,0>Syz
3,0, and (iv) in the quadrant i=4:
Sxz
4,0>Sxy
4,0>Syz
4,0.
b. Location D: (1) relationship among the events in (u′,v′)
plane: Sxy
1,0>Sxy
4,0>Sxy
3,0>Sxy
2,0; (2) relation-
ship among the events in (u′,w′)plane: Sxz
4,0>Sxz
2,0>
Sxz
3,0>Sxz
1,0; (3) relationship among the events in
(v′,w′)plane: Syz
4,0>Syz
3,0>Syz
2,0>Syz
1,0;(4)
relationships among all three planes: (i) in the quadrant
FIGURE 7 Plots of stress fractions Si,Hagainst hole size Hat bottom label for Dc=4.2 cm. (a) At the location A: ◦,xy-plane; +,xz-plane; ⊳,yz-plane, (b) at
D close to the cylinder: ◦,xy-plane; +,xz-plane; ⊳,yz-plane, and (c) at H: ◦,xy-plane; +,xz-plane; ⊳,yz-plane
8of14 MAITY AND MAZUMDER
i=1: Sxy
1,0>Syz
1,0>Sxz
1,0, (ii) in the quadrant i=2:
Syz
2,0>Sxy
2,0>Sxz
2,0, (iii) in the quadrant i=3:
Syz
3,0>Sxy
3,0>Sxz
3,0, and (iv) in the quadrant i=4:
Syz
4,0>Sxy
4,0>Sxz
4,0.
c. Location H: (1) relationship among the events in (u′,v′)
plane: Sxy
4,0>Sxy
2,0>Sxy
1,0>Sxy
3,0; (2) relation-
ship among the events in (u′,w′)plane: Sxz
1,0>Sxz
3,0>
Sxz
2,0>Sxz
4,0; (3) relationship among the events in
(v′,w′)plane: Syz
2,0>Syz
4,0>Syz
1,0>Syz
3,0;(4)
relationships among all three planes: (i) in the quadrant
i=1: Sxz
1,0>Sxy
1,0>Syz
1,0, (ii) in the quadrant i=2:
Syz
2,0>Sxy
2,0>Sxz
2,0, (iii) in the quadrant i=3:
Sxz
3,0>Sxy
3,0>Syz
3,0, and (iv) in the quadrant i=4:
Sxy
4,0>Syz
4,0>Sxz
4,0.
It is observed that the quadrant-wise estimates of stress
fractions Si,Halong the transverse (v′,w′)and parallel
(u′,v′)-planes from the velocity measurements are more
important over the scour hole structure than that along the
longitudinal (u′,w′)-plane (Figure 7b), whereas at the loca-
tion A on the plane surface, the quadrant-wise estimates of
the Reynolds shear stress fraction Si,Halong the (u′,w′)-plane
are more prominent than other planes (Figure 7a). It is also
observed that contribution of the 4th quadrant over (v′,w′)-
plane is more important than other quadrants.
The contributions of all three stress fractions Si,Halong
the (u′,v′),(u′,w′),and (v′,w′)-planes to the Reynolds shear
stresses are plotted against the threshold parameter H(H=
0 to 20) at the level z∕h=0.50 for each of the four quad-
rants at the location A at the height z∕h=0.50 in Figure 8a;
at location D at height z∕h=0.50 in Figure 8b; and at
location H above the deposited materials (on the ridge) in
Figure 8c. From the quadrant analysis, we determined the
following relationships among the events and planes:
a. Location A: (1) relationship among the events in (u′,v′)
plane: Sxy
2,0>Sxy
4,0>Sxy
3,0>Sxy
1,0; (2) relation-
ship among the events in (u′,w′)plane: Sxz
1,0>Sxz
3,0>
Sxz
2,0>Sxz
4,0; (3) relationship among the events in
(v′,w′)plane: Syz
4,0>Syz
2,0>Syz
1,0>Syz
3,0;(4)
relationships among all three planes: (i) in the quadrant
i=1: Syz
1,0>Sxz
1,0>Sxy
1,0, (ii) in the quadrant i=2:
FIGURE 8 Plots of stress fractions Si,Hagainst hole size Hat z∕h=0.50 for Dc=4.2 cm. (a) At the location A: ◦,xy-plane; +,xz-plane; ⊳,yz-plane, (b) at
D close to the cylinder: ◦,xy-plane; +,xz-plane; ⊳,yz-plane and (c) at H: ◦,xy-plane; +,xz-plane; ⊳,yz-plane
MAITY AND MAZUMDER 9of14
FIGURE 9 Depth averaged stress fractions Si,0(i=1,2,3,4)againstx∕hfor different planes at the locations A, B, C, D, E, F, G, and H with the location A
starting from the first point of the figures (a), (b), (c). ◦,i=1;□,i=2;+,i=3;⊳,i=4
Syz
2,0>Sxy
2,0>Sxz
2,0, (iii) in the quadrant i=3:
Syz
3,0>Sxz
3,0>Sxy
3,0, and (iv) in the quadrant i=4:
Syz
4,0>Sxy
4,0>Sxz
4,0.
b. Location D: (1) relationship among the events in (u′,v′)
plane: Sxy
2,0>Sxy
4,0>Sxy
3,0>Sxy
1,0; (2) relation-
ship among the events in (u′,w′)plane: Sxz
1,0>Sxz
3,0>
Sxz
4,0>Sxz
2,0; (3) relationship among the events in
(v′,w′)plane: Syz
3,0>Syz
1,0>Syz
4,0>Syz
2,0;(4)
relationships among all three planes: (i) in the quadrant
i=1: Syz
1,0>Sxz
1,0>Sxy
1,0, (ii) in the quadrant i=2:
Syz
2,0>Sxz
2,0>Sxy
2,0, (iii) in the quadrant i=3:
Syz
3,0>Sxz
3,0>Sxy
3,0, and (iv) in the quadrant i=4:
Syz
4,0>Sxz
4,0>Sxy
4,0.
c. Location H: (1) relationship among the events in (u′,v′)
plane: Sxy
2,0>Sxy
4,0>Sxy
1,0>Sxy
3,0; (2) relation-
ship among the events in (u′,w′)plane: Sxz
1,0>Sxz
3,0>
Sxz
2,0>Sxz
4,0; (3) relationship among the events in
(v′,w′)plane: Syz
2,0>Syz
4,0>Syz
3,0>Syz
1,0;(4)
relationships among all three planes: (i) in the quadrant
i=1: Syz
1,0>Sxz
1,0>Sxy
1,0, (ii) in the quadrant i=2:
Syz
2,0>Sxz
2,0>Sxy
2,0, (iii) in the quadrant i=3:
Syz
3,0>Sxz
3,0>Sxy
3,0, and (iv) in the quadrant i=4:
Syz
4,0>Sxz
4,0>Sxy
4,0.
From the above analysis it is observed that the
quadrant-wise estimates of stress fractions Si,Halong the
transverse (v′,w′)plane from the velocity measurements
are more important over the scour hole structure than that
along the longitudinal (u′,w′)and parallel (u′,v′)-planes
(Figure 8a,b,c). It is also observed that contributions of 2nd
and 4th quadrants over transverse (v′,w′)and parallel (u′,v′)-
planes are greater than the contributions of 1st and 3rd
quadrants. Similar results (Nakagawa & Nezu, 1977) have
been found in the longitudinal (u′,w′)-plane that the con-
tributions of 1st and 3rd quadrants are greater than the
contributions of 2nd and 4th quadrants.
Depth averaged stress fractions Si,0(i=1,2,3,4) against
x∕hfor different planes (x, y), (x, z), and (y, z) at the locations
A, . …,H are plotted in Figure (9a,b,c), respectively. Stress
fraction behavior along the longitudinal direction over the
scour geometry shows oscillatory in nature for all the planes.
It is observed that the maximum stress fraction occurs at the
(y, z)-plane. Figure 10 shows the contributions of stress frac-
tions Si,0(i=1,2,3,4) averaged over both vertical and
horizontal directions for all the different planes. The stress
fraction Si,0yz is the largest value for each of the quadrant.
3.3 Mathematical modeling
In order to apply the cumulant discard method, we define
the following variables: u′,v′,andw′are respectively the
zero mean fluctuating stream-wise, lateral and vertical veloc-
ity components; and ̂u,̂v,and̂ware respectively equal to
u′∕u′2,v′∕v′2,andw′∕w′2. We shall quantify the
contribution from the different planes of the relative shear
stress for the covariance terms 𝜏1=u′v′
u′v′,𝜏2=u′w′
u′w′,and
𝜏3=v′w′
v′w′. Three characteristic functions 𝜒1(𝛼, 𝛽),𝜒2(𝛼, 𝛾)
and 𝜒3(𝛽,𝛾), expressed as the Fourier transform of the joint
probability density functions p1(̂u,̂v),p2(̂u,̂w),andp3(̂v,̂w),
respectively, can be expressed as function of the moment
and cumulant generating functions in which m1,st =̂uŝvt,
m2,st =̂uŝwt,andm3,st =̂vŝwtdenote the moments of (s+t)th
order and q1,st,q2,st and q3,st correspond to the cumulants
10 of 14 MAITY AND MAZUMDER
FIGURE 10 Longitudinal averaged stress fractions Si,0(i=1,2,3,4) for different planes at the all locations (A, B, C, D, E, F, G, and H)
of (s+t)th order. Nakagawa and Nezu (1977) expressed the
conditionally sampled probability density over the four quad-
rant of covariance events 𝜏2. The mathematical calculations
are completely described in Nakagawa and Nezu (1977). The
following equations are given:
pj,2(𝜏j)=pjG(𝜏j)+𝜓−
j(𝜏j)(𝜏j>0)
pj,4(𝜏j)=pjG(𝜏j)−𝜓−
j(𝜏j)(𝜏j>0)
pj,1(𝜏j)=pjG(𝜏j)+𝜓+
j(𝜏j)(𝜏j<0)
pj,3(𝜏j)=pjG(𝜏j)−𝜓+
j(𝜏j)(𝜏j<0)
,(13)
where the index iin pj,idenotes the quadrant index in the jth
plane with planes j=1, j=2, and j=3 corresponding to
the xy,xz,andyz-planes, respectively. The probability density
pjG(𝜏j)is directly developed from the corresponding bivariate
normal distribution where
pjG(𝜏j)= Rj
2𝜋eRjξjK0(ξj)
(1−R2
j)
ψ+
j(𝜏j)= Rj
2𝜋eRjξjK1
2
(ξj)ξj
(1−Rj)2(1+Rj)S+
j
3+D+
jξj−(2−Rj)
3S+
j+D+
j
ψ−
j(𝜏j)= Rj
2𝜋eRjξjK1
2
(ξj)ξj
(1+Rj)2(1−Rj)S−
j
3+D−
jξj−(2+Rj)
3S−
j+D−
j
ξj=Rj𝜏j
(1−R2
j);S±
j=qj,03±qj,30
2;D±
j=qj,21±qj,12
2(j=1,2&3)
R1=−u′v′
σuσv
;R2=−u′w′
σuσw
;R3=−v′w′
σvσw
,(14)
where K𝜆(𝜉j)(𝜆=0&0.5)is the 𝜆th order modified Bessel
function of the second kind. The non-conditionally sampled
probability function of shear stress is
pj(𝜏j)=pj,1(𝜏j)+pj,2(𝜏j)+pj,3(𝜏j)+pj,4(𝜏j)=2pjG(𝜏j).(15)
The fractional contribution Sj
i,Hto the Reynolds stress,
corresponds to each quadrant and each plane represented by
Sj
i,H=∞
H𝜏jpj,i(𝜏j)d𝜏j(i=2,4)
Sj
i,H=−H
−∞ 𝜏jpj,i(𝜏j)d𝜏j(i=1,3); j=1,2and 3
(16)
and corresponding probability Pxy
i,Hcan represent in the turbu-
lent boundary layer is as follows:
Pj
i,H=∞
Hpj,i(𝜏j)d𝜏j(i=2,4)
Pj
i,H=−H
−∞ pj,i(𝜏j)d𝜏j(i=1,3); j=1,2and 3.
(17)
Note that : j=1, j=2andj=3 correspond to xy,xz,and
yz-planes.
3.4 Verifications with experimental data
and discussions
From the quadrant analysis it is discovered that in the scouring
regions, the contribution of turbulent events to the Reynolds
shear stress of yz-plane was much more important than other
MAITY AND MAZUMDER 11 of 14
FIGURE 11 Plot of p3(𝜏3)against 𝜏3=v′w′
v′w′at the location D: ◦(·−),
z∕h=−0.081;+(- -), z∕h=0;⊳(-), z∕h=0.2. Here dash-dot, dash and
solid lines represent the theoretical value
two planes xy and xz. Therefore, in the present discussion the
above theoretical results have been verified with the exper-
imental data in yz-plane only. We have also verified above
theoretical results in other two planes, but not presented here.
3.4.1 Probability density distribution of normalized
Reynolds shear stress
The measured and computed values of p3(𝜏3)of normal-
ized Reynolds shear stress 𝜏3are plotted in Figure 11 for
the location D(within the scour region) for three vertical
heights z∕h=−0.081 (bottom surface), z∕h=0.0(level
surface), and z∕h=0.2 (above the level surface). It is noted
from the figure that the probability density of (𝜏3)calculated
from Equation 15 is fairly well with experimental data at all
three heights. It is observed from Figure 11 that peak of the
FIGURE 12 Plot of stress fractions Syz
i,Hagainst hole size Hat the location D: ◦(·−), z∕h=−0.081;+(- -), z∕h=0;⊳(-), z∕h=0.2. Here dash-dot, dash
and solid lines represent the theoretical value
distribution profile decreases with the increase of vertical
height. In fact, the maximum peak occurs at the near-bed
(z∕h=−0.081). In a theoretical equation p3(𝜏3)have the sin-
gularity at (𝜏3)=0, that is, origin because K0(𝜏3=0)=
∞. Consequently, the unconditional probability distribution
p3(𝜏3)of Reynolds shear stress is represented by one directly
derived from a Gaussian distribution with high accuracy, as
verified by Antonia and Atkinson (1973), Lu and Willmarth
(1973), and Nakagawa and Nezu (1977) in the xz-plane.
3.4.2 Stress fractions
Figure 12 presents the theoretical and experimental quad-
rant fractional contributions of the relative covariance
term 𝜏3=v′w′
v′w′calculated from Equation 16 using the
v′w′-quadrant technique at location D (close to the cylin-
der) in the scour region, contribution of Qyz
4event is
greater than the Qyz
2event for all the vertical heights
z∕h=−0.081 (bottom surface), z∕h=0.0 (level surface),
and z∕h=0.2 (above the level surface). At z∕h=−0.081
for H=0, the contributions from the quadrants are
Syz
1,0=−0.816,Syz
2,0=1.108,Syz
3,0=−1.714,Syz
4,0=2.422,
the sum of which is 1. Corresponding values at H=8are,
Syz
1,8=−0.353,Syz
2,8=0.493,Syz
3,8=−1.355,Syz
4,8=2.068,
the sum of which is 0.853, that is, 85%of −v′w′occurs in
events more than 2 times the −v′w′, pointing the intermit-
tent nature of the turbulence. The intensities of all events
increase with increasing distance (z∕h⩽0.2) from the
scour-bed. A comparison of the results at three different
depths confirms the Qyz
4-Qyz
2dominance in contributions.
Good agreement is seen between the theoretical and the
experimental distributions for a threshold level H.Itis
12 of 14 MAITY AND MAZUMDER
FIGURE 13 Plot of probabilities Pyz
i,Hagainst hole size Hat the location D: ◦(·−), z∕h=−0.081;+(- -), z∕h=0;⊳(-), z∕h=0.2. Here dash-dot, dash and
solid lines represent the theoretical value
verified that even the third-order conditional probability dis-
tribution of the Reynolds shear stress shows fairly good
agreement with the experimental results and that the sequence
of events in the bursting process, that is, Qyz
1,Qyz
2,Qyz
3,and
Qyz
4, is directly related to the turbulent energy budget in the
form of turbulent diffusion. Also, we found that the scour
effect is marked in the area from the wall to the level bed
surface region, and that Qyz
4appear to be more important
than Qyz
2.
3.4.3 Probabilities of turbulent event
The probabilities of occurrence of turbulent events in the
ith quadrant at each plane and at each vertical height are
to be determined by setting a threshold parameter Hfrom
Equation 10. Figure 13 shows the plots of probabilities
Pyz
i,H(i=1,2,3,4),occurrence of all events for vertical
heights z∕h=−0.081,0.0,&0.2 against the threshold level
Hat the location D(within the scour region). In the figure
◦,+,&⊳are the measured values calculated from
Equation 10; dash-dot, dash, and solid lines represent the the-
oretical value calculated from Equation 17, the agreement
between the experimental data and the predicted values is
fairly good over a wide range of hole size H,sothatitmay
be expected that the third-order probability distribution repre-
sents the correct picture for a sequence of bursting processes.
It is observed from the figure that the occurrence to Pi,Hfor a
fixed threshold parameter Hfrom all four quadrants increases
with increase of vertical distance. At z∕h=−0.081 for H=0,
the occurrence of turbulent events from the quadrants are
Pyz
1,0=0.265,Pyz
2,0=0.327,Pyz
3,0=0.200,Pyz
4,0=0.208,
the sum of which is 1. Corresponding values at H=4are,
Pyz
1,4=0.049,Pyz
2,4=0.073,Pyz
3,4=0.092,Pyz
4,4=0.098, the
sum of which is 0.312, that is, 31%occurrence of all turbulent
events.
4CONCLUSIONS
The purpose of this study was to ascertain the quadrant thresh-
old technique for direct estimation of conditional statistics of
Reynolds shear stresses (u′v′,u′w′,andv′w′) in the turbulent
boundary layer over the crescentic scour hole, and subse-
quently, a theoretical model of probability density function of
momentum flux variable (Nakagawa & Nezu, 1977; Raupach,
1981) was employed to verify the threshold technique for all
three components of shear stresses. The following conclu-
sions may be drawn as:
The contributions of burst-sweep cycles to the Reynolds
shear stresses for all three planes (xy,xz,andyz)overand
within the scour mark around an obstacle are computed using
the quadrant analysis to identify the leading shear stress
plane involving the bursting events, which are responsible to
form the scour geometry. Partitioning the shear stress into
quadrant-wise turbulent events (i =1; i =2; i =3; and i
=4), it is observed that contribution of turbulent events to
the yz-plane is much more important in the scouring region
than that in xy and xz-planes; whereas at the plane sur-
face, contribution to the xz-plane is more prominent than the
other two.
The contributions of turbulent events to the Reynolds shear
stresses −u′v′,−u′w′,and−v′w′are derived by applying the
cumulant-discard method to the Gram-Charlier probability
distribution of the two variables. The conditional statistics of
the shear stresses show a good agreement with the experimen-
tal data.
MAITY AND MAZUMDER 13 of 14
It was verified that even the third-order conditional proba-
bility distribution of the Reynolds shear stresses shows fairly
good agreement with the experimental data and that the
sequence of events in the bursting process is directly related to
the turbulent energy budget in the form of turbulent diffusion.
The study of scour marks around the objects on the sandy
bed under the action of turbulent flows in natural environ-
ments has the potential to be useful to researchers who con-
sider the dynamics of pipelines and short cylinders placed on
riverbeds, seabeds, and shallow water regions of coastal areas;
and who examine the formations of crescentic scour marks
in a recent stream to establish palaeocurrent directions. This
study would be helpful for engineers to modify their structural
designs or using some protective measures for meeting the
problems faced due to scouring in the presence of turbulence.
ACKNOWLEDGMENTS
One of the authors (Haradhan Maity) would like to express his
sincere thanks to Professor A. S. Vasudeva Murthy of TIFR
for his fruitful discussions. Haradhan Maity is also thank-
ful to TIFR Centre for Applicable Mathematics, Bangalore
for providing him the post-doctoral fellowship. The authors
acknowledge Professor Supriya Sengupta and Professor J. K.
Ghosh for introducing them the research topic and suggesting
the problem.
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How to cite this article: Maity H, Mazumder BS.
Prediction of plane-wise turbulent events
to the Reynolds stress in a flow over
scour-bed. Environmetrics. 2017;28:e2442.
https://doi.org/10.1002/env.2442
