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Comparison of Flow and Morphological Characteristics in Uniform and Non-uniform Sand Bed Channel
Journal: Canadian Journal of Civil Engineering
Manuscript ID cjce-2017-0627.R3
Manuscript Type: Article
Date Submitted by the Author: 25-Jul-2019
Complete List of Authors: Sharma, Anurag; Tsinghua University, civil EngineeeringKumar, Bimlesh; IITGBalachandar, Ram; University of Windsor, Department of Civil and Environmental Engineering
Keyword: Hilbert-Huang spectrum, Semblance analysis, Turbulence
Is the invited manuscript for consideration in a Special
Issue? :Not applicable (regular submission)
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Comparison of Flow and Morphological Characteristics in Uniform and Non-
uniform Sand Bed Channel
Authors
1. Anurag Sharma
Post-Doctoral Research Fellow, Institute of River and Ocean Engineering, Tsinghua
University, China, [email protected]
2. Bimlesh Kumar [Corresponding Author]
Associate Professor, Department of Civil Engineering, Indian Institute of Technology
Guwahati, India-781039, 0091-361-2582420, [email protected]
3. Ram Balachandar
Professor, Department of Civil and Environmental Engineering, University of Windsor,
Windsor, Canada, [email protected]
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Abstract
An experimental investigation has been carried out to evaluate the turbulent flow and
morphological characteristics in an alluvial channel composed of uniform and non-uniform sand
beds. The measures of turbulent flow parameters show that the streamwise mean velocity and
Reynolds shear stress increase in the presence of the uniform sand bed, indicating a potential for
a greater movement of sediment particles. The turbulent integral scale increases significantly
with a uniform sand bed, which results in an increased energy and momentum transfer caused by
the larger eddy size in the near bed region. Frequency analysis of the velocity time series was
carried out with the help of the Hilbert-Huang transformation, which shows that the dominant
time scale of the velocity time series data in a uniform sand bed channel is smaller as compared
to the flow in a non-uniform sand bed channel.
Keywords: Hilbert-Huang spectrum; Semblance analysis; Turbulence; Velocity time series.
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1. Introduction
Aggradation and degradation of sediment owing to turbulent flow is important in fluvial
hydraulics. In recent years, sequences of research have been carried out to understand the
phenomena of fluvial hydraulics in alluvial rivers. Bennett and Best (1995) investigated the
dynamic relationship between suspended sediment particle size and flow velocity. They observed
that suspension of sediment is associated with high time-averaged vertical velocity, high
turbulence intensity and positive third-order velocity moments near the bed. Best et al. (1997)
studied the bed material movement in a coarse alluvial channel and found that the near bed flow
time averaged streamwise velocity and turbulence intensity were reduced due to bed material
movement. Bergeron and Carbonneau (1999) observed that the mean streamwise velocity of the
near bed flow is reduced due to the increase of friction factor linked with the apparent roughness
in the presence of bed particle movement. Nikora and Goring (2000) found a reduction in the
flow resistance in weakly mobile gravel beds as compared to static gravel beds because of
increased time mean streamwise velocity with weakly mobile gravel beds. Campbell et al.
(2005) studied the bed particle transport with both large (coarse grains, d50=3.96 mm) and small
particles (fine grains, d50=0.77 mm) and found a reduction in streamwise velocity for larger
material, while the velocity increased with smaller material. Researchers (Gust and Southward
1983; Bennett et al. 1998) investigated the universal nature of the von Kármán constant () for
smooth and transitional flow zones, and observed a reduction in the value from its universal
value of 0.41 due to sediment transport. Further, other researchers (Bennett and Bridge 1995;
Gallagher et al. 1999) also investigated the constancy of value for the full rough zone and
observed a fall in value because of sediment transport. Non-universal value of in sand bed
channels was reviewed by Gaudio et al. (2010) and behaves as a variable in flows with low
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submergence, or if there is bed and suspended-load transport. Field investigation of gravel
transport by Drake et al. (1988) and Robinson (1991) observed that bursting event sweep is the
principal parameter for gravel transport. Previously (Best 1992; Cao 1997; Dwivedi et al. 2010)
investigated the structure of flow turbulence over a flat sand bed channel in which bed is at an
incipient state of motion. The incipient motion of bed particle with the effect of turbulence
events was also examined in uniform sand bed channel (Dey et al. 2011; Sharma et al. 2015).
Also, others (Bennett and Bridge 1995; Sumer et al. 2003; Venditti et al. 2005) studied the flow
characteristics over a flat sand bed in which the bed material is in a mobile condition. In a recent
study, Saber et al. (2016) examined the effect of inertial particles on turbulent flow
characteristics in the inner and outer layer of the flow. They observed that flow mean velocity
and turbulence intensity are affected with the increase of volume fraction of inertial particles.
Riverbeds are generally composed of non-uniform sand and the finer material tends to get
transported. Previous literature (McLean 1992; Wilcock 1998) related to sediment aggradation
and degradation is limited to the homogeneous sediment mixture. For most of the studies based
on the sediment mixture, the correction factor for bedload based on hiding and exposure factor is
used (Parker et al. 1982; Hayashi et al. 1980; Andrews 1983). Considering the correction factor
of the bed material, fractional bedload transport was estimated (Bridge and Bennett 1992; Proffit
and Sutherland 1983; Karim 1998). Based on experimental studies (Wilcock and Crowe 2003;
Curran and Wilcock 2005; Fang and Wang 2000; Fang et al. 2012), it was observed that the
sand-gravel mixture in the heterogeneous sediment affects the overall transport rate under
laboratory conditions. The transport of gravel fractions in channel beds is larger due to the
increase in sediment supply to the bed channel, which can cause bed degradation and sediment
migration from the channel bed (Curran 2007). Frostick et al. (2008) carried out flume
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experiments to observe the influence of fine material on the mobility of coarse sand, and
suggested that the fine materials increase the movement of gravel, provoking the development of
different bed forms. Patel and Range Raju (1996) assumed that the lift force and drag force cause
the movement of fine and coarse material respectively, proposing a hiding-exposure correction
factor. The transport rates of individual fractions of the bed mixture were estimated based on
laboratory data (Samaga et al. 1986; Fang and Rodi 2003). Ghoshal (2005) and Mazumder et al.
(2005) carried out a series of experiments to study the effect of wall roughness on the transport
of suspended material in heterogeneous mixtures. Roos et al. (2007) investigated the influence of
sediment mixture on the hydrodynamics of offshore tidal sandbanks. Ghoshal et al. (2010)
conducted the experiments to investigate the transport rate of individual fraction of sand-gravel
beds and observed that the movement trend of individual fractions depends upon the discharge
and wall roughness. Based on laboratory and field data, Wu et al. (2010) proposed an expression
for fractional transport rates for bed and suspended loads in sediment mixtures. They tested this
formula with the data obtained in the laboratory and field. Juez et al. (2015) developed a two-
dimensional numerical finite volume model for water flowing over erodible beds by considering
non-uniform grain size. Elhakeem and Imran (2016) incorporated a model based on density
function of the bed material movement to develop a bedload transport model for non-uniform
sediment. In a recent study, Sharma and Kumar (2017a) investigated the turbulent structure in a
sand bed channel composed of non-uniform sand bed channel. However, the study did not
compare the characteristics that in a uniform sand bed channel.
Multiple time scales occur in the process of flow and sediment transport. Kuai and Tsai (2012)
introduced the HHT method to the field of sediment transport for time series data analysis for
identifying multiple time scale embedded in the flow and sediment data. HHT can be used to
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analyse the non-stationary and nonlinear time series data by filtering out the spikes of non-
stationary data sets and to obtain the various time scale underlying in the observed data (Huang
et al. 1998; Donnelly 2006; Franceschini and Tsai, 2010). In the field of hydrology, Huang et al.
(2009) applied HHT to analyse daily river flow fluctuations. Franceschini and Tsai (2010) used
HHT for analysing toxic concentrations in a natural river. Considering the importance of various
time scales in the field of sediment transport, HHT is used in the present study. HHT uses two
steps: first, the Empirical Mode Decomposition (EMD) which decomposes the datasets into
several Intrinsic Mode Functions (IMF) and second, Hilbert Spectral Analysis (HSA) which
extracts instantaneous frequency data from the entire signal. Also, the EMD may use to find
times scales in the velocity signal. In this process, HHT decomposes the signal into different
modes, which are intrinsic to the function. These modes are known as IMF, can have varying
amplitude and frequency. First, one IMF is extracted, and then it is subtracted from the original
time history. Then, the next IMF is again extracted, and it is subtracted from the previous
resulting signal. This process is carried on until the standard deviation between the two
consecutive IMFs lies between 0.25-0.3. Each of the extracted IMFs is used to form an analytical
signal, which is represented by a real part and an imaginary part. The real part is the IMF itself
and the imaginary part is the IMF convoluted with (1/ (πt)). The original signal is the summation
of the real parts of these analytical functions and a residue term. Let, zi(t) be the analytical
function obtained from the ith IMF. Then, zi(t) can be expressed as:
(1)( )( )( ) ( ) ( ) ii
j t dtj ti i iz t a t e a t e
where, is the local magnitude of the original signal and is the instantaneous or local ( )ia t i t
phase of the original signal. The instantaneous frequency ω(t) is represented as the derivative of
the local phase (Franceschini and Tsai 2010).
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The importance of similarity measures is increasing continuously in order to compare the
multiple signals from different fields. The semblance analysis provides similarity between two
datasets based on correlations between their phase angles in terms of frequency. Cooper and
Cowan (2008) used wavelet built semblance analysis for comparing series of synthetic data. The
semblance analysis using continuous wavelet transform (CWT) permits the local phase
interaction between the two signals in terms of both scale and time. The comparison between two
time series datasets using CWT can be expressed as (Torrence and Compo 1998):
(2)1,2 1 2f f fW W W
The quantity Wf1, 2 is a complex nature with an amplitude X and local phase θ is given by
(3)1,2fX W
(4) 11,2 1,2tan Im (W ) Re (W )f faginary al
When the local phase θ of two datasets are calculated, the semblance at each frequency is
measured as (Cooper and Cowan 2008),
(5) cosnS
where, n is an odd number. The advantage of using S compared with θ is that the values of S will
vary from -1 to 1. Values of S=-1 and S=1 indicates inverse and perfect correlation between two
datasets, respectively, while S=0 represent no correlation between signals. Since, S and θ gives
information of phase angles rather than amplitude, the advantage of comparison of two signals
using S and θ is that signals are not required to have the same units. The absence of amplitude
information leads to noise sensitivity, which can be furthered express as:
(6) 1 2cosnf fD W W
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when n=1, the value of D is approximately equal to the vector dot product of two wavelet
coefficients for each scale and position. D indicates the information of amplitude X with
information of phase S that can be beneficial if the phase correlations of the higher amplitude
components of the signal are best noticeable.
Aforementioned studies have focused either on the flow hydrodynamics or turbulent
characteristics in uniform and non-uniform sand bed channel. However, to the best of our
knowledge, none of the earlier studies have compared the flow and morphological characteristics
between uniform and non-uniform sand bed channels. In order to investigate the effect of the
uniform sand and non-uniform sand on the flow characteristics, a four-beam downlooking
acoustic Doppler velocimeter (ADV) probe (Nortek® Vectrino) was used to measure the
instantaneous velocity components at a point. The measured data will deliver significant
evidence linked to the flow turbulence, such as velocity, Reynolds shear stress, turbulence
intensity, turbulent anisotropy, turbulent scale and von Kármán constant. In order to investigate
the effect of the uniform sand and non-uniform sand on the bed morphology, HHT technique is
introduced in the zone of bed particle movement for velocity time series data. Also, the
semblance analysis is performed on velocity time series datasets to present their correlations in
terms of both scale and time.
2. Methodology
The tests were carried out in a 17.20 m length, 1.00 m width and 0.72 m depth rectangular flume
as depicted in Figure 1. The flow was conditioned with flow straighteners prior to entering the
channel with a 2.80 m length, 1.50 m width, and 1.50 m depth inlet tank placed at the upstream
end of the flume. The flow discharge applied in the channel is 0.0429 m3/s. The flow regulation
in the channel was done with a valve installed at the inlet tank. The flow depth in the channel
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was controlled with a tail gate provided at the downstream end of the flume. The flow was fully
developed in the test section region of the flume, which was was located 5 m upstream of the tail
gate (Sharma and Kumar 2017b). The digital point gauge fixed to a moving trolley was used to
measure the channel flow depth. The channel flow discharge was measured by taking the flow
depth over a rectangular weir with a discharge coefficient (Cd) of 0.82. A Pitot-static tube
attached with digital manometer, assembled on a moving trolley, was used to measure the water
surface slope in experiments. A Total Station was used to measure the bed slope in the channel.
In the present experiment, the channel bed was eroded at the upstream because of the increased
bed shear stress and the eroded material from the bed at a section was deposited on the channel
bed at the adjacent section towards downstream and carried forward by the water towards the
longitudinal length of main channel. The amount of sediment eroded from the bed is transported
in such a way that the particles roll over one another. Since the channel bed is in mobile
condition, a bedload sampler 0.50 m long, 1.00 m wide and 0.21 m depth was placed near the tail
gate to accumulate the transported bed material. Additional information about the
experimentation can be found in Sharma and Kumar (2016).
Insert Figure 1
Both uniform and non-uniform sand were used in the study. Figure 2 depicts the distributions of
grain size used for the experiments. The standard deviation ( ) for uniform sand was 84
16g
dd
1.3 and for the non-uniform sand was 1.79. The flow parameters and the details of sediment
mixture are summarized in Table 1. In Table 1, d50 and 50 100
i i i i0 50
M d p d p represents
respectively the median class and Kramer’s coefficient of the sediment mixture, where di
correspond to the symmetrical proportions of two successive classes and pi is the percentage of
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bed material. Kramer’s uniformity parameter M = 1 for uniform sediment and M<1 for non-
uniform sediment.
Insert Figure 2
Insert Table 1
In the experimental run as shown in table 1, water was supplied to the flume by slowly regulating
the inlet tank valve so that discharge, Q and corresponding flow depth, h is recorded.
Instantaneous velocity measurements were taken at the centre line of flume by the Vectrino+
Acoustic Doppler Velocimeter (ADV). At each vertical location, a sampling rate of 100 Hz was
used for the data acquisition. The data was acquired with an acoustic frequency of 10 MHz
having an adjustable cylindrical sampling volume of 6 mm diameter and 1 to 4 mm in height.
The height of the sampling volume was set at 4 mm when the measurement location was away
from the bed and 1 mm when very near to the bed so that the sampling volume did not touch the
particles on the bed surface. In recent study (Dey et al. 2017), a sampling length of 1 mm was
found to be adequate to capture the correct descriptions in the shear layer and near-boundary
zones. The ADV collects data in the sampling volume located 50 mm below the central
transmitter. As such, the data could not be collected in the top 50 mm zone from the water
surface. Velocity measurements were carried out in a vertical profile and at each location,
instantaneous velocity samples were collected for a duration of 300 s. This sampling time was
found to be sufficient to achieve statistically time-independent time-averaged velocity. The
acceleration threshold method is used to remove the spikes from the ADV data (Goring and
Nikora 2002). Lacey and Roy (2008) suggested that power spectra of streamwise velocity in
inertial sub range should satisfy Kolmogorov -5/3 scaling law. Velocity power spectra Fuu (f) at
z=4 mm are displayed in the Figure 3, where, f is frequency and z is the distance from bed
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surface. Figure 3 shows that the velocity power spectra in inertial subrange for the filtered data
satisfies the Kolmogorov’s −5/3 law. Further, close investigation of Figure 3 indicates that
profiles of velocity power spectra of filtered data are similar for both sands.
Ten recordings of instantaneous velocity were recorded at 10 mm above the bed to obtain the
uncertainty associated with the measured data. In Table 2, u, v and w are the mean velocity in the
streamwise, spanwise, and vertical directions respectively, while , and are the fluctuations u v w
components of velocity in the streamwise, spanwise, and vertical directions respectively.
, and are the turbulence intensity of , and respectively. Here, the 0.5u u uuur 0.5
v v 0.5w w u v w
term standard uncertainty describes the standard deviation of the mean for a set of several
repeated pulses of instantaneous velocity. Standard uncertainty is calculated as:
du
SS
n (7)
where, Sd is the standard deviation and n is the number of measurements in the set. The data
shown in Table 2 are within ±5% for the time-averaged velocity and rms quantities,
corroborating the capability of 100 Hz frequency of measurements by the vectrino (Dey et al.
2012).
Insert Figure 3
Insert Table 2
3. Results
In order to understand the changes in hydrodynamics of the flow with uniform and non-uniform
sand beds, turbulent flow parameters such as mean velocity, Reynolds shear stresses (RSS),
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turbulence intensity, flow anisotropy and turbulent scale are evaluated in the middle of the test
section.
3.1 Flow characteristics
Reynolds shear stress (RSS) defines the momentum flux of the streamwise velocity in the
vertical direction caused by the fluctuating velocity field. Dimensional RSS is obtained for both
the tests and the vertical distributions are displayed in Figure 4 (a). The shear velocity u* [=
], where τ0=boundary shear stress, which is evaluated by the linear projection of the 0.50
RSS profile to the channel bed, as given by Nezu (1977). The calculated 0 0uw zu w
shear velocity for uniform and non-uniform sand bed channel is obtained as 12.64 mm/s and
12.27 mm/s, respectively. There is a 2.90 % increase in the shear velocity in the case of uniform
sand bed channel as compared with the non-uniform sand bed channel.
Figure 4 (b) displays the profiles of non-dimensional RSS normalised by . Figure 4(b) shows 2*u
that RSS increases towards the channel boundary correspond to higher momentum transfer
towards the boundary to sustain bed material movement by overcoming the bed resistance. The
magnitude of RSS reaches a peak value in between 0.05<z/h<0.2 and consecutively reduces
towards the boundary due to existence of a roughness sub layer in the vicinity of the bed. The
data trends of RSS distribution are observed to be similar for both the sands but with the
increased magnitude of RSS in the uniform sand bed channel. Figure 4 shows that RSS is
increased by ~8-15% with uniform sand bed channel as compared to non-uniform sand bed
channel, confirming increased momentum transfer toward the channel boundary in the presence
of uniform sand. This increased momentum transferred towards the bed increases the sediment
transport.
Insert Figure 4
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Figure 5 (a) displays the profiles of time-averaged velocity against dimensionless depth (z/h,
where z is distance from channel boundary and h is the flow depth) for both the tests. It is
observed that mean velocity increases with the application of a uniform sand bed channel. These
increased velocity cause increase in sediment movement along the flow in the channel. Figure
5(a) shows that velocity near the bed are increased by ~3-5% with uniform sand as compared to
non-uniform sand while away from the boundary, the magnitude and profiles of time averaged
velocity is almost same for both sands, which suggests that the effect of the sand on the flow
velocity profile is not apparent in the flow outer layer. Experimental data is fitted to the log law
(Sharma and Kumar, 2017a)
(8) *
1 1ln lnu z zu
where, z+=z/d50, Δz+= Δz/d50, ε+= z0/d50, Δz and z0, respectively, represents the virtual bed level
and zero-velocity level from the channel boundary surface, u* and , respectively, represents the
shear velocity and von Karman constant. Figure 5(b) shows the velocity logarithmic law for flow
on both the sand beds. The experimental data sets for both the sands are well fitted with the
logarithmic law given by equation (8). The values of , Δz and z0 are tabulated in Table 3. Value
of for non-uniform sand is observed to be slightly higher than the universal value (0.41), while
for uniform sand; is slightly lower than the universal value (0.41). The flow subjected to
uniform sand bed influences the bed particles starts to move rapidly and the value of decreases
when compared to the non-uniform sand bed channel. Further, from the regression equation
shown in Figure 5(b), it is observed that with uniform sand bed channel, the value of Δz and z0
increase suggesting an exposure of increased streamwise velocity component on the bed surface
sand particles.
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Insert Figure 5
Insert Table 3
The normalised profiles of streamwise (σu) and vertical (σw) root mean square of velocity
fluctuations are displayed in Figures 6. The normalised streamwise turbulence intensity increases
with the non-uniform sand, while the vertical turbulence intensity decreases in comparison with
the flow on the uniform sand. The vertical profiles of turbulence intensity are important within
the near bed flow zone because in the case of non-uniform sand bed channel, the degree of
damping in the streamwise (σu) and vertical (σw) turbulence intensity, respectively, increases and
decreases by an average value of 3.73 % and 0.3 % as compared to the uniform sand bed
channel.
Insert Figure 6
The several laws have been suggested to describe the profile of turbulence intensity in
streamwise direction (Nezu and Nakagawa 1993; Nikora and Goring 1998). Figure 7 shows that
the streamwise turbulence intensity ( ) of the experimental datasets does not fit into *ˆu u u
the universal law provided in existing literature (Nezu and Nakagawa 1993; Nikora and Goring
1998). Regression analysis is used to derive a new empirical constant for flows in uniform and
non-uniform sand bed channels. The value of empirical constants is tabulated in table 4.
Insert Figure 7
Insert Table 4
Figure 8 displays the profiles of turbulence anisotropy ( ) against z/h, and it can be ˆ ˆw u
observed that the flow is highly anisotropic as <1. The ratio is approximately 0.27 ˆ ˆw u ˆ ˆw u
and 0.23 near the boundary for flow subjected to uniform and non-uniform sand beds,
respectively, varying almost linearly with the flow depth. Nezu and Nakagawa (1993) observed
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the flow anisotropy has a universal value of 0.55 for a smooth boundary throughout the flow
depth, while Dey and Raikar (2007) observed a universal value of 0.6 for mobile gravel
boundaries. But, the value of in the present result remains lower than the previous ˆ ˆw u
literature (Nezu and Nakagawa 1993; Dey and Raikar 2007). However, the turbulent flow is
more anisotropic in a uniform sand bed channel as compared to the flow in a non-uniform sand
bed channel.
Insert Figure 8
3.5 Integral Scale
An integral length scale represents the large eddy size in the flow and the corresponding time
scale indicates the eddy turnover time at a given point. The transfer of momentum and turbulent
kinetic energy in the flow are linked to these eddies. Venditti et al. (2005) reported that near bed
flow integral scale is the reason behind the formation of bed features. Therefore, velocity time
series data near the bed (10.0 mm above the bed surface) is used to measure the integral length
scale. The Eulerian integral time scale is expressed as:
(9)0
( )T
TE R t dt
where, R(t) and dt, respectively, represents the autocorrelation function and the lag distance, and
T represent time where, R(t) nearly reaches zero. The integral length scale is expressed as
(Taylor 1935):
(10)L TE E u
Table 5 shows the values of and , where, the eddy length and the eddy turnover time close TE LE
to boundary surface increases considerably with a uniform sand bed channel. Higher eddy size in
the near boundary surface tends to higher energy and momentum transfer from the flow to the
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bed particles. Therefore, the flow turbulence increases with the higher eddy size, which results in
higher rate of sediment transport with a uniform sand bed channel.
Insert Table 5
3.6 Bedload Transport
As indicated earlier, the transported bed particles were collected in a bedload sampler. The
collected bed materials were in wet condition. The particles are dried in an oven to obtain the
weight of dry bed material and the bedload transport rate per unit width was calculated. Table 6
shows the value of the bedload transport for uniform and non-uniform sand bed flows. The
bedload transport is higher in a uniform sand bed channel. The change in morphodynamical
characteristics in terms of erosion and deposition of bed materials can be directly linked to the
turbulent flow characteristics. The turbulent parameters such as Reynolds shear stress (Figure 4),
time average streamwise velocity (Figure 5), turbulence intensity (Figure 6), turbulent anisotropy
(Figure 8) and turbulent integral scale (Table 5) is increased with flow over a uniform sand bed,
which justifies the higher rate of sediment transport.
Insert Table 6
Figure 9 shows the particle size distribution of the bedload transport. The median grain size and
the standard deviation of bedload transport in a uniform sand bed channel is approximately the
same as the original bed material distribution, which suggests that the uniformity of the bedload
transport remains same as the original bed material distribution. The result is slightly different
with a non-uniform sand bed channel, in which the median grain size and the standard deviation
of bedload transport is lower than the original bed material distribution. The bed material
transport of heterogeneous sediment is complex. The criterion for entrainment of a bed particle
size and its transport rate is influenced by the presence of other particle sizes in the mixture. The
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finer material transported as bedload are sheltered by the coarse materials, and the coarser
material are exposed more to the action of flow past them. In this manner, the shear stress
obtained by a particular fraction of heterogeneous mixture is different from what it would be at
the same shear stress if the material were uniform.
Insert Figure 9
3.7 Hilbert-Huang Transform (HHT)
An Empirical Mode Decomposition (EMD) package (Rilling et al. 2007) is used to perform the
Hilbert spectral analysis (HAS). The velocity time series data of flow over the uniform and the
non-uniform sand bed channel are shown in Figure 10. In the first step, EMD decomposes the
velocity time series into various IMFs. The IMFs reveal the presence of multi scale signals in the
velocity time series (Figure 11). Figure 11 shows that the first IMF (IMF1) is descriptive of
extremely fluctuating components and the frequency of fluctuations is reduced with the increase
of IMF. Thus, IMF12 represents the signals of lowest fluctuating values. The last diagram
represents the residual, which displays the major part in the time series datasets after the noise
removal. It can be seen from the residual function that the predominant trend decreases with the
increase of time scale. It should be noted that the lowest measurable time scale is at least equal to
the sampling time interval (0.01 sec in this study), that may be suitable for data collection and
acquisition.
Insert Figure 10
Insert Figure 11
In the second step, HSA was used to get more insight into time-frequency-energy of the signals
by using the Hilbert spectrum. Figure 12 shows the Hilbert spectrum with different colors and
each color denotes a specific energy level over time (x-axis) and over the frequency (y-axis). The
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darker points represent higher energy levels. The collection of points with the most energy
represents the dominant frequency (dominant normalized frequency), or time scale (Franceschini
and Tsai 2010). The frequency corresponding to the maximum number of high energy level
points signifies the dominant time scale (= (1/dominant normalized frequency) x measuring time
interval). The scattered points near the high frequency correspond to different individual time
scales. It should be noticed that most energy points are located in the dominant relative
frequency (y-axis) and the dominant time scale (x-axis) is not constant but fluctuates with time.
Insert Figure 12
The small sampling time interval of 1/100 s is used in order to understand the high intermittence
(high frequency part) to the frequency–time spectrum (Figure 12). It is observed that the greater
energy points are accumulated near the low frequency zones. In addition, high energy level
signals are more scattered or show a wider range of frequencies. Huang et al. (1998) reported
that the turbulent flows are dominated by common properties like intermittency and frequency
modulations. These provide evidence that the effect of turbulent flow on the instantaneous
sediment transport is at a low frequency. Most of the turbulent fluctuations of the velocity signal
are filtered out by the high acquisition rate. Even in steady flow nature, the velocity time series
rate can be highly unsteady until the data sampling duration is small enough. Hence, the velocity
signal fluctuates with time scales. It can be seen from the spectrum that the dominant time scale
of uniform and non-uniform sand bed flows is detected in the zone with normalized frequency
between to 0.006 and 0.007. The time interval between two consecutive data sets was 1/100 s in
the present experimental study, therefore, the dominant time scale is (1/dominant relative
frequency) (measuring time interval) = (1/0.007) (1/100) = 1.43 s for uniform sand bed channel
and 1.66 s for non-uniform sand bed channel. A large time scale corresponds to a slower
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variation of the physical quantity in time (Cao et al. 2007). Since the dominant time scale for
uniform sand bed channel is lower, the rate of sediment transport is higher in uniform sand bed
channel. In the field, extraction of such type of information can be very useful to monitor
changes in alluvial channels.
3.8 Wavelet Based Semblance Analysis
In the present study, semblance analysis is also evaluated for the velocity time series. Figure 13
shows the semblance analyses of velocity time series for n=1(Fig. 13a) and n=3 (Fig. 13b). The
first and second row of the figure indicates the velocity time series and CWT for uniform beds.
Similarly, the third and fourth row of the figure indicates the velocity time series and CWT for
non-uniform beds. Lastly, the fifth and sixth row of the figure indicates the semblance and dot
product of the velocity time series of uniform and non-uniform beds. The semblance analysis
gives similarity between two datasets based on correlations between their phase angles in terms
of frequency. In the first row of Figure 13, the real part of the complex continuous wavelet
transform (CWT) indicates the occurrence of velocity time series for the duration of the dataset
in a uniform sand bed channel. In the third row of Figure 13, the velocity time series dataset of
flow in a non-uniform sand bed channel exists but this dataset differs in phase angle from those
in the first dataset in terms of position. The last row shows the semblance S and dot product D
calculated with n = 1. In the time portion (0–60) s of the data series, the longer and shorter
wavelength components, respectively, represents the perfectly correlated (S = 1) and inversely
correlated (S = -1). In the fifth row of Figure 13 (a) shows a broad blue patch in the portion (0-
60) s at a wavelength of around 25 units, representing a negative correlation between the two
data series. In the portion (60–120) s of the plot, the phase relationship between the velocity time
series components of the two data series was changed, so that they are correlated, uncorrelated or
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inversely correlated as a function of position. In the time portion (120–180) s of the data series,
the larger wavelength component is inversely correlated (S = -1), while the shorter wavelength
component shows a broad red patch at a wavelength of approximately 15 units, representing a
positive correlation between the two data series. There is improvement in the simplicity of plot
by dot product when information of amplitude is incorporated concurrently. However, if there is
an interest to compute the phase relationship of signals corresponding to small values of
amplitude, then displaying this plot might not be appropriate. The longitudinal defining
restrictions of this methodology are scale dependant i.e., visible longitudinal deviations in
semblance is sharper at lower wavelengths as compared to higher wavelengths. These happen
because of the wavelet transform nature. Figure 13 (b) shows the effect of increasing the
parameter n on the plots of S and D. As n is increased to 3, the spreading in the areas of positive
and negative correlation is gradually decreased. The higher values of n will have a tendency to
grind down the degree of these areas mainly at higher wavelengths creating the selection of an
optimum value dependant on the signals.
Insert Figure 13
Discussion and Conclusions
In the present study, the basic results corresponding to the flow characteristics and
morphological features are compared between a uniform and a non-uniform sand bed channel.
One of the motivating factors of the present study compared to previous studies is the use of non-
uniform sediment mixture, which is practically observed in riverbeds. Though in non-uniform
sand, the particle size distribution of sediment in transport is finer than the distribution of the
original bed material because of selective transport, the coarser fraction of sand present in the
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sediment mixture induces more resistance to the flow. As a result, the profiles of turbulent flow
structure in non-uniform sediment beds are more or less in general agreement with flow
characteristics in case of uniform sand bed. The transport of sediment mixtures in open channels
flow is complicated because the condition for the initiation of the motion of a given sediment
size as well as its transport rate is affected by the presence of the other sizes in the mixture. It is
observed that bedload transport increases (shown in Table 6) with the application of a uniform
sand bed channel.
For the uniform sand bed channel, the uniformity of the bedload transport remains the same as
compared to original bed material distribution. The result is slightly different with non-uniform
sand bed channel in which the non-uniformity of the bedload transport is decreased as compared
to original bed material distribution. Analysis of such behaviour requires a close and careful
observation of the internal flow structure. Therefore, instantaneous velocity data were taken by
ADV to observe the turbulent flow characteristics in uniform and non-uniform sand bed channel,
which leads to a number of insights. Analysis of mean velocity profile (Figure 5(a)) reveals that
the velocity near the bed increases on average by (3-5) % for the uniform bed.The increases in
velocity near the bed causes the bed material to move faster. The increase in the depth of the
virtual bed level and the zero velocity level (Figure 5 (b)) reflects that the particles on the
uniform sand bed channel were exposed to a higher component of streamwise velocity. The
decrease in the value of von Karman’s constant (Table 3) suggests the prevalence of sediment
transport in a uniform sand bed channel. The increase in maximum Reynolds shear stress (Figure
4) by (8-15) % with uniform sediment corresponds to increases in shear velocity by 2.9%,
thereby, suggesting a greater momentum transfer towards the boundary in the uniform sand bed
channel. The turbulence is strongly anisotropic in a uniform sand bed channel and varies
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approximately linear with flow depth. The analysis of turbulent integral scale (Table 5) suggests
that the higher levels of turbulence is observed near the bed with an increased eddy size, which
results in higher rate of sediment transport in a uniform sand bed channel.
The presence of numerous time scales in the velocity time series becomes a significant part in the
modeling of flow velocity prediction. The technique Hilbert Huang Transform (Figure 12)
provides the distribution of physical parameters of sediment transport at different time scales. It
is observed that the dominant time scale is 1.43 s for uniform sand bed channel and 1.66 s for
non-uniform sand bed channel. The dominant time scale for a uniform sand bed channel is lower
by 16%, which suggests that the rate of sediment transport is increased in a uniform sand bed
channel. In order to quantify the wide disparity in physical and dynamic characteristics of
sediment transport among a range of scales, HSA helps to provide a clear and meaningful
insight.The semblance analysis (Figure 13) gives similarity between two datasets based on
correlations between their phase angles in terms of frequency. The phase relationship between
the velocity time series components of the uniform and non-uniform beds was observed in such a
way that the phase relationship may be correlated, un-correlated or inversely correlated as a
function of position. The methodology of semblance analysis is applicable to an extensive range
of problems in fluvial hydraulics.
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Figure Caption
Figure 1 Schematic diagram of tilting flume
Figure 2 Particle size distribution curve
Figure 3 Velocity power spectra after spike removal with Kolmogorov’s -5/3 law in the inertial sub range at z = 4 mm for streamwise velocities.
Figure 4 (a) Dimensional and (b) Non-dimensional Reynolds shear stress profiles of flow
Figure 5 (a) Velocity profiles of flow and (b) Velocity log law
Figure 6 Turbulent intensity profiles of flow in streamwise and vertical direction
Figure 7 Comparison of streamwise turbulent intensity of flow with existing literatures
Figure 8 Turbulent anisotropy
Figure 9 Particle size distribution of resultant bed material
Figure 10 Instantaneous velocity (U) time series datasets
Figure 11 IMFs and residue extracted from the velocity time series of flow subjected to uniform
and non-uniform sand bed channel
Figure 12 Identification of varying time scales from the Hilbert-Huang spectrum of velocity time series
Figure 13 Semblance analyses of velocity time series for order (a) n=1and (b) n=3
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. Table 1 Details of sediment mixture and flow parameters used in experiments
Type of Sand
Sand size d50 (mm)
Standarddeviation
σg
KramersCoefficient
M
Bed slope,
S0
Flow depth h
(m)
Discharge, Q (m3/s)
Flow Reynolds number
Non-uniform
0.425 1.79 0.14 0.001 0.118 0.0429 42900
Uniform 0.425 1.30 1 0.001 0.118 0.0429 42900
Table 2 Uncertainty associated with ADV data
u (m/sec) v
(m/sec)
w
(m/sec) 0.5u u uuur
(m/sec) 0.5v v
(m/sec) 0.5w w
(m/sec)
Standard
deviation
4.32x10-3 9.6x10-4 4.3x10-4 1.09x10-3 9.39x10-4 3.46x10-4
Uncertainty
%
0.32 0.065 0.80 0.092 0.078 0.04
Table 3 Coefficient value observed from log law equation
Type of sand k Δz (mm) z0 (mm)
Non-uniform 0.421 0.382 0.0011
Uniform 0.405 0.595 0.0040
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Table 4 Values of empirical constants
Type of sand Du Cu
Non-uniform 3.0 -0.34
Uniform 3.3 -0.44
Table 5 Time averaged integral length scale and time scale at 10mm above the bed surface
Type of Sand Eulerian time scale
(sec)TE
Time Average Velocity,
u (m/sec)
Eulerian length scale
(m)LE
Non-uniform 0.42 0.2397 0.1006
Uniform 0.55 0.2475 0.1361
Table 6 Bedload values
Non-uniform sand bed channel
Uniform sand bed channel
Bedload (kg/hr) 0.042 0.057
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Inlet tank
Tail tank
Tail gate
SupportingFramework
Point gauge Pitot tube
Seepage length 15.2 m
Side View
Pumpin
g unit
10 HP
each
1.5 m
2 m 5 mTest reach 5 m
0.22 mPressure chamber0.72 m
Over headstorage tank
Dischargecontrolling valve
Flow
Baffle
Flow
Channel Length, 17.20 m
1.5 m
2.8 m
1m
1.7m
1.9m
Plan View
Baffle
Sand bed
ADV Bed loadsampler
Bedloadsampler
Figure 1 Schematic diagram of tilting flume
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0.01 0.1 1 100102030405060708090
100
Percen
tage fi
ner
Sieve Size (mm)
Non-uniform sand (d50=0.425, g=1.79) Uniform sand (d50=0.425, g=1.30)
Figure 2 Particle size distribution curve
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Draft0.1 1 10 100 10001E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
F uu(m2 /s)
f (Hz)
Non-uniform Sand Uniform Sand Kolmogorov -5/3 law
Figure 3 Velocity power spectra after spike removal with Kolmogorov’s -5/3 law in the inertial sub range at z = 4 mm for streamwise velocities.
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Draft0.0000 0.0001 0.0002 0.00030.0
0.10.20.30.40.50.60.7
-u'w'(m2/s2)
Non-uniform Sand Uniform Sand
z/h
'
(a)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.60.7
-u'w'/u2*
Non-uniform Sand Uniform Sand
z/h
'
(b)
Figure 4 (a) Dimensional and (b) Non-dimensional Reynolds shear stress profiles of flow
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Draft5 10 15 20 25 30 350.0
0.10.20.30.40.50.60.7 Non-uniform Sand
Uniform Sand
z/h
u/u*
(a)
1 2 3 4 512
15
18
21
24
27
ln(z++z+)
Non-uniform sand 2.374*ln(z++z+)+14.117 Uniform Sand 2.467*ln(z++z+)+11.475
(b)
u/u*
Figure 5 (a) Velocity profiles of flow and (b) Velocity log law
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Draft1.5 2.0 2.5 3.0 3.5 4.0 4.50.0
0.10.20.30.40.50.60.7
Non-uniform Sand Uniform Sand
z/h
u/u*
0.4 0.6 0.8 1.0 1.2 1.4 1.60.00.10.20.30.40.50.60.7
z/h
w/u* Figure 6 Turbulent intensity profiles of flow in streamwise and vertical direction
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Draft1E-3 0.01 0.1 1 101
2
3456
ˆu
z/h
Non-uniform Sand Uniform Sand Nezu and Nakagawa (1993) Nikora and Goring (1998)
Figure 7 Comparison of streamwise turbulent intensity of flow with existing literatures
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Draft0.20 0.25 0.30 0.35 0.40 0.450.0
0.10.20.30.40.50.60.7
wu
Non-uniform Sand Uniform Sand
z/h
Figure 8 Turbulent anisotropy
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Draft0.01 0.1 1 100102030405060708090
100
Percen
tage fi
nerSieve size (mm)
Non-uniform sand (d50=0.40, g=1.54) Uniform sand (d50=0.426, g=1.32)
Figure 9 Particle size distribution of resultant bed material
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Draft0 20 40 60 80 100 120 140 160 1800
10
20
30
40
50
U (cm
/s)
t (s)
Uniform Sand
0 20 40 60 80 100 120 140 160 1800
10
20
30
40
50
U (cm
/s)
t (s)
Non-uniform Sand
Figure 10 Instantaneous velocity (U) time series datasets
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Uniform Sediment Non-uniform Sediment
0 20 40 60 80 100 120 140 160 180-10
-5
0
5
10Am
plitude
(cm)
Tim e scale (s)
IM F 1
0 2 0 4 0 6 0 8 0 1 00 1 2 0 1 4 0 1 6 0 1 8 0-1 0
-5
0
5
1 0
Amplit
ude (cm
)
T im e sc a le (s)
IM F 1
0 2 0 4 0 6 0 8 0 1 0 0 1 20 1 4 0 1 6 0 1 8 0-1 0
-5
0
5
1 0
T im e sc a le (s)
IM F 4
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0-1 0
-5
0
5
1 0
T im e s c a le ( s )
I M F 4
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Draft0 20 4 0 6 0 80 1 0 0 12 0 1 40 1 60 1 8 0-10
-5
0
5
10
T im e sca le (s )
IM F 7
0 20 40 60 80 100 120 140 160 180-10
-5
0
5
10
Tim e scale (s)
IM F 7
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0-1 0
-5
0
5
1 0
T im e sc a le (s )
IM F 1 0
0 20 40 60 80 100 120 140 160 180-10
-5
0
5
10
T im e scale (s)
IM F 10
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0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0-1 0
-5
0
5
1 0
T im e sc a le (s )
IM F 1 2
0 2 0 4 0 60 8 0 1 0 0 12 0 1 4 0 1 60 1 8 0-1 0
-5
0
5
1 0
T im e sc a le (s)
IM F 1 2
0 20 4 0 60 80 1 00 12 0 140 160 18023 .5
24 .0
24 .5
T im e sca le (s)
R esidue
0 20 40 60 80 100 120 140 160 18022232425262728
T im e sca le (s)
R esidue
Figure 11 IMFs and residue extracted from the velocity time series of flow subjected to uniform and non-uniform sand bed channel
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Figure 12 Identification of varying time scales from the Hilbert-Huang spectrum of velocity time series
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20 40 60 80 100 120 140 160 1802030
Data 1
Data 1 CWT Real Part
Scale
20 40 60 80 100 120 140 160 180050
100
20 40 60 80 100 120 140 160 1802030
Data 2
Data 2 CWT Real Part
Scale
20 40 60 80 100 120 140 160 180050
100
Semblance
Scale
20 40 60 80 100 120 140 160 180050
100
Dot Product
Scale
20 40 60 80 100 120 140 160 180050
100-101
(a)
20 40 60 80 100 120 140 160 1802030
Data 1
Data 1 CWT Real Part
Scale
20 40 60 80 100 120 140 160 180050100
20 40 60 80 100 120 140 160 1802030 Data 2
Data 2 CWT Real Part
Scale
20 40 60 80 100 120 140 160 180050100
Semblance
Scale
20 40 60 80 100 120 140 160 180050
100
Dot Product
Scale
20 40 60 80 100 120 140 160 180050
100-101
(b)
Figure 13 Semblance analyses of velocity time series for order (a) n=1and (b) n=3
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