canadian journal of civil engineering · phenomena of fluvial hydraulics in alluvial rivers....

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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)

https://mc06.manuscriptcentral.com/cjce-pubs

Canadian Journal of Civil Engineering

Draft

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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mailto:[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

Page 19 of 48



Draft

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

Page 20 of 48



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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

Page 21 of 48



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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.

References

Andrews, E. D. 1983. Entrainment of gravel from naturally sorted riverbed material. Geological

Society of America Bulletin, 94, 1225-1231.

Bennett, S. J., & Best, J. L. 1995. Particle size and velocity discrimination in a sediment-laden

turbulent flow using phase Doppler anemometry. Journal of Fluids Engineering, 117, 505-511.

Page 22 of 48



Draft

Bennett, S. J., & Bridge, J. S. 1995. An experimental study of flow, bedload transport and bed

topography under conditions of erosion and deposition and comparison with theoretical models.

Sedimentology, 42(1), 117-146.

Bennett, S. J., Bridge, J. S., & Best J. L. 1998. Fluid and sediment dynamics of upper stage plane

beds. Journal of Geophysical Research, 103(C1), 1239-1274.

Bergeron, N. E., & Carbonneau, P. 1999. The effect of sediment concentration on bedload

roughness. Hydrology Process, 13(16), 2583-2589.

Best, J. 1992. On the entrainment of sediment and initiation of bed defects: insights from recent

developments within turbulent boundary layer research. Sedimentology, 39(5), 797-811.

Best, J., Bennett, V., Bridge, J., & Leeder M. 1997. Turbulence modulation and particle

velocities over flat sand beds at low transport rates. Journal of Hydraulic Engineering, 123(12),

1118-1129.

Bridge, J. S., & Bennett, S. J. 1992. A model for the entrainment and transport of sediment

grains of mixed sizes, shapes and densities, Water Resources Research, 28(2), 337-363.

Campbell, L., McEwan, I., Nikora, V., Pokrajac, D., Gallagher, M., & Manes, C. 2005. Bedload

effects on hydrodynamics of rough bed open-channel flows. Journal of Hydraulic Engineering,

131(7), 576-585.

Cao, Z. 1997. Turbulent bursting-based sediment entrainment function. Journal of Hydraulic

Engineering, 123(3), 233-236.

Cao, Z., Li, Y., & Yue, Z. 2007. Multiple time scales of alluvial rivers carrying suspended

sediment and their implications for mathematical modeling. Advances in Water

Resources, 30(4), 715-729.

Page 23 of 48



Draft

Cooper, G. R. J., & Cowan, D. R. 2008. Comparing time series using wavelet based semblance

analysis. Computers & Geosciences, 34(2), 95-102.

Curran, J. C. 2007. The decrease in shear stress and increase in transport rates subsequent to an

increase in sand supply to a gravel-bed channel. Sedimentary Geology, 202, 572–580.

Curran, J. C., & Wilcock, P. R. 2005. Effect of sand supply on transport rates in a gravel-bed

channel. Journal of Hydraulic Engineering, 131(11), 961–967.

Deshpande, V., & Kumar, B. 2016. Turbulent flow structures in alluvial channels with curved

cross‐sections under conditions of downward seepage. Earth Surface Processes and

Landforms, 41(8), 1073-1087.

Dey, S., & Raikar, R. V. 2007. Characteristics of loose rough boundary streams at near-

threshold. Journal of Hydraulic Engineering, 133(3), 288-304.

Dey, S., Sarkar, S., & Solari, L. 2011. Near-bed turbulence characteristics at the entrainment

threshold of sediment beds. Journal of Hydraulic Engineering, 137(9), 945-958.

Dey, S., Ravi Kishore, G., Castro-Orgaz, O., & Ali, S. Z. 2017. Hydrodynamics of submerged

turbulent plane offset jets. Physics of Fluids, 29(6), 065112.

Donnelly, D. 2006. The fast Fourier and Hilbert-Huang transforms: a comparison.

In Computational Engineering in Systems Applications, IMACS Multiconference on IEEE, 1,

84-88.

Drake, T. G., Shreve, R. L., Dietrich, W. E., Whiting, P. J., & Leopold, L. B. 1988. Bedload

transport of fine gravel observed by motion-picture photography. Journal of Fluid Mechanics,

192, 193-217.

Page 24 of 48



Draft

Dwivedi, A., Melville, B., & Shamseldin, A. Y. 2010. Hydrodynamic forces generated on a

spherical sediment particle during entrainment. Journal of Hydraulic Engineering, 136(10), 756-

769.

Elhakeem, M., & Imran, J. 2016. Bedload Model for Non-uniform Sediment. Journal of

Hydraulic Engineering, 142(6), 06016004.

Fang, H. W., Han, D., & He, G. J., 2012. Flood management selections for the Yangtze River

midstream after the Three Gorges Project operation. Journal of Hydrology, 432, 1-11.

Fang, H. W. & Rodi, W. 2003. Three dimensional calculations of flow and suspended sediment

transport in the neighborhood of the dam for the Three Gorges Project reservoir in the Yangtze

River. Journal of Hydraulic Research, 41(4), 379-394.

Fang, H. W., Chen, M. H., & Chen, Q. H. 2008. One-dimensional numerical simulation of non-

uniform sediment transport under unsteady flows. International Journal of Sediment Research.

23(4), 316-328.

Franceschini, S., & Tsai, C. W. 2010. Application of Hilbert-Huang transform method for

analyzing toxic concentrations in the Niagara river. Journal of Hydrologic Engineering, 15(2),

90-96.

Frostick L., Murphy B., & Middleton, R. 2008. Exploring the links between sediment character,

bed material erosion and landscape: Implications from a laboratory study of gravels and sand-

gravel mixtures. Geological Society, London, Special Publications, 296, 117–127.

Gallagher, M., McEwan I., & Nikora, V. 1999.The changing structure of turbulence over a self -

stabilising sediment bed. Internal Rep. No. 21, Department of Engineering, University of

Aberdeen, Aberdeen, U.K.

Page 25 of 48



http://apps.webofknowledge.com/full_record.do?product=WOS&search_mode=GeneralSearch&qid=1&SID=1EaWJv1IeZ9DSlqS5df&page=1&doc=8

http://apps.webofknowledge.com/full_record.do?product=WOS&search_mode=GeneralSearch&qid=1&SID=1EaWJv1IeZ9DSlqS5df&page=1&doc=8

http://apps.isiknowledge.com/full_record.do?product=WOS&search_mode=GeneralSearch&qid=1&SID=3E3m6fe@OBb4iKkk4EB&page=1&doc=2

http://apps.isiknowledge.com/full_record.do?product=WOS&search_mode=GeneralSearch&qid=1&SID=3E3m6fe@OBb4iKkk4EB&page=1&doc=2

Draft

Gaudio, R., Miglio, A., & Dey, S. 2010. Non- universality of von Karman’s k in fluvial streams.

Journal of Hydraulic Research, 48(5), 658-663.

Ghoshal, K., 2005. On velocity and suspension concentration in a sediment-laden flow:

Experimental and theoretical studies: Ph.D. Thesis, Jadavpur University, Calcutta.

Ghoshal, K., Mazumder, B. S., & Purkait, B., 2010. Grain size distributions of bedload:

inferences from flume experiments using heterogeneous sediment beds. Sedimentary Geology,

223 (1-2), 1-14.

Goring, D. G., & Nikora, V. I. 2002. Despiking acoustic Doppler velocimeter data. Journal of

Hydraulic Engineering, 128(1), 117-126.

Gust, G., & Southard, J. B 1983. Effect of weak bedload on the universal law of the wall. Journal

of Geophysical Research, 88(C10), 5939-5952.

Hayashi, T. S., Ozaki, I., & Ichibashi, T. 1980. Study on bedload transport of sediment mixture,

Proc. 24th Japanese Conference on Hydraulics.

Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., ... & Liu, H. H. 1998.

The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary

time series analysis. In Proceedings of the Royal Society of London A: mathematical, physical

and engineering sciences, The Royal Society, 454, 903-995.

Huang, Y., Schmitt, F. G., Lu, Z., & Liu, Y. 2009. Analysis of daily river flow fluctuations using

empirical mode decomposition and arbitrary order Hilbert spectral analysis. Journal of

Hydrology, 373(1), 103-111.

Juez, C., Ferrer-Boix, C., Murillo, J., Hassan, M. A., & García-Navarro, P. 2016. A model based

on Hirano-Exner equations for two-dimensional transient flows over heterogeneous erodible

beds. Advances in Water Resources, 87, 1-18.

Page 26 of 48



Draft

Karim, F. 1998. Bed material discharge prediction for nonuniform bed sediments. Journal of

Hydraulic Engineering, 124(6), 597-604.

Kuai, K. Z., & Tsai, C. W. 2012. Identification of varying time scales in sediment transport using

the Hilbert–Huang Transform method. Journal of hydrology, 420, 245-254.

Lacey, R. W., & Roy, A. G. 2008. Fine-scale characterization of the turbulent shear layer of an

instream pebble cluster. Journal of Hydraulic Engineering, 134(7), 925-936.

Mazumder, B. S., Ghoshal, K., & Dalal, D. C., 2005. Influence of bed roughness on sediment

suspension: Experimental and Theoretical Studies, Journal of Hydraulic Research, IAHR, 43(3),

245-257.

McLean, S. R. 1992. On the calculation of suspended load for noncohesive sediments. Journal of

Geophysical Research, 97 (C4), 5759–5770.

Nezu, I. 1977. Turbulent structure in open-channel flows. Kyoto University.

Nezu, I., & Nakagawa, H. 1993. Turbulence in open channels. IAHR/AIRH Monograph.

Balkema, Rotterdam, The Netherlands.

Nikora, V. I., & Goring, D. G. 1998. ADV measurements of turbulence: Can we improve their

interpretation?. Journal of Hydraulic Engineering, 124(6), 630-634.

Nikora, V., & Goring, D. G. 2000. Flow turbulence over fixed and weakly mobile gravel beds.

Journal of Hydraulic Engineering, 126(9), 679-690.

Parker, G., Kilingeman, P. C., & Mclean, D. G. 1982. Bedload and size distribution in paved

gravel-bed streams. Journal of Hydraulic Division, ASCE, 108(4), 544-571.

Patel, M., Deshpande, V., & Kumar, B. 2015. Turbulent characteristics and evolution of sheet

flow in an alluvial channel with downward seepage. Geomorphology, 248, 161-171.

Page 27 of 48



Draft

Patel, P. L., & Ranga Raju, K. G. 1996. Fraction wise calculation of bedload transport. Journal of

Hydraulic Research, 34(3), 363-379.

Proffit, G. T. & Sutharland., A. J. 1983. Transport of nonuniform sediment, Journal of Hydraulic

Research, 21(1), 33-43.

Rilling, G., Flandrin, P., Gonçalves, P., & Lilly, J. M. 2007. Bivariate empirical mode

decomposition. IEEE signal processing letters, 14(12), 936-939.

Robinson, S. K. 1991. The kinematics of turbulent boundary layer structure. NASA TM-103859,

Ames Research Centre, Moffett Field, USA.

Roos, P. C., Wemmenhove, R., Hulscher, S. J., Hoeijmakers, H. W., & Kruyt, N. P. 2007.

Modeling the effect of nonuniform sediment on the dynamics of offshore tidal

sandbanks. Journal of Geophysical Research: Earth Surface, 112(F2).

Saber, A., Lundström, T. S., & Hellström, J. G. I. 2016. Influence of inertial particles on

turbulence characteristics in outer and near wall flow as revealed with high resolution particle

image velocimetry. Journal of Fluids Engineering, 138(9), 091303.

Samaga, B. R., Raju, K. G. R., & Garde, R. J. 1986. Bedload transport of sediment mixtures.

Journal of Hydraulic Engineering, 112 (11), 1003-1017.

Sharma, A., Patel, M., & Kumar, B. 2015. Turbulent parameters and corresponding sediment

transport in curved cross section channel. ISH Journal of Hydraulic Engineering, 21(3), 333-342.

Sharma, A., & Kumar, B. 2016. Probability distribution of turbulence in curvilinear cross section

mobile bed channel. Water Science and Technology, 73(6), 1472-1482.

Sharma, A., & Kumar, B. 2017a. Structure of turbulence over non uniform sand bed channel

with downward seepage. European Journal of Mechanics - B/Fluids, 65, 530-551

Page 28 of 48



http://www.sciencedirect.com/science/journal/09977546

Draft

Sharma, A., & Kumar, B. 2017b. Boundary layer development over non-uniform sand rough bed

channel. ISH Journal of Hydraulic Engineering, 25(2), 162-169

Sumer, B. M., Chua, L. H. C., Cheng, N. S., & Fredsoe, J. 2003. Influence of turbulence on bed

load sediment transport. Journal of Hydraulic Engineering, 129(8), 585-596.

Taylor, G. I. 1935. Statistical theory of turbulence. In Proceedings of the Royal Society of

London A: Mathematical, Physical and Engineering Sciences, 151, 421-444.

Torrence, C., & Compo, G. P. 1998. A practical guide to wavelet analysis. Bulletin of the

American Meteorological society, 79(1), 61-78.

Venditti, J. G., Church, M., & Bennett, S. J. 2005. Morphodynamics of small‐scale superimposed

sand waves over migrating dune bed forms. Water Resources Research, 41(10).

Wilcock, P. R. 1993. Critical shear stress of natural sediments. Journal of Hydraulic Engineering,

119(4), 491-505.

Wilcock, P. R., & Crowe, J. C. 2003. Surface-based transport model for mixed-size sediment.

Journal of Hydraulic Engineering, 129(2), 120-128.

Wu, F. C., & Yang, K. H. 2004. Entrainment probabilities of mixed-size sediment incorporating

near bed coherent flow structures. Journal of Hydraulic Engineering, 130(12), 1187-1197.

Wu, W., Wang, S. S., & Jia, Y. 2010. Nonuniform sediment transport in alluvial rivers. Journal

of Hydraulic Research, 38(6), 427-434.

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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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Draft

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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Draft

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*


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


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


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

Page 41 of 48



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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Draft

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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Draft

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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Draft

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


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


Scale

20 40 60 80 100 120 140 160 180050100

20 40 60 80 100 120 140 160 1802030 Data 2


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

Page 48 of 48



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