ECE 533 Final Project Decomposing non-stationary turbulent velocity in open channel...
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ECE 533 Final Project Decomposing non-stationary turbulent velocity in open channel flow Ying-Tien Lin 2005.12.20
Transcript of ECE 533 Final Project Decomposing non-stationary turbulent velocity in open channel...
ECE 533 Final Project
Decomposing non-stationary turbulent velocity in open
channel flow
Ying-Tien Lin
2005.12.20
2
Decomposing non-stationary turbulent velocity in open
channel flow
Ying-Tien Lin
1 Introduction
In natural environment, the flow of a fluid can be categorized as laminar or
turbulent flow. Observing that water run through a pipe, we can inject neutral dye to
investigate the flow characteristics (see the Figure 1 below). For “small enough
flowrates”, the dye streak will remain as a well-defined line as it flows along, with
only slight blurring due to molecular diffusion of the dye into the surrounding water.
For a somewhat “intermediate flowrates”, the dye streak fluctuates in time and space,
and intermittent bursts of irregular behaviors appear along the streak. On the other
hand, for ”large enough flowrates” the dye streak almost immediately becomes
blurred and spread across the entire pipe in a random fashion. These three
characteristics are denoted as laminar, transitional, and turbulent flow, respectively.
Figure 1 Experiment to illustrate type flow and dye streaks
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Suppose that we place a velocimetry probe at one point of the pipe, for laminar
flow, there is only one component for velocity; however, for turbulent flow the
predominant component of velocity is also along the pipe, but it is accompanied by
random components normal to the pipe axis. Slow motion pictures of the flow can
more clearly reveal the irregular, random, turbulent nature of the flow.
Figure 2 Time independent of fluid velocity at a point
Turbulent flows are beneficial to our daily life. Mixing processes and heat and
mass transfer processes are considerably enhanced in turbulent flow compared to
laminar flow. For example, to transfer the required heat between a solid and an
adjacent fluid (such as in the cooling coils of an air conditioner or a boiler of a power
plant) would require an enormously large heat exchanger if the flow were laminar.
Furthermore, it is considerably easier to mix cream into a cup of coffee (turbulent
flow) than to thoroughly mix two colors of a viscous paint (laminar flow). In order to
deal with turbulent flow, previous researcher represented the turbulent velocity as the
sum of time mean value, u and the fluctuating part of the velocity, 'u , that is:
'u u u= + (1)
Where 0
0
1 ( )t Ttu u t dt
T+= ∫ , the time interval, T, is considerable longer than the period
of the longest fluctuations, but considerably shorter than any unsteadiness of the
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average velocity. It is straightforward to prove that the time average of the fluctuation
velocity, 'u is zero. The products of the fluctuation velocity in x and y components
will account for the momentum transfer (hence, the shear stress). The total shear
stressτ can be shown as follows:
' ' lam turbdu u vdy
τ μ ρ τ τ= − = + (2)
Where μ is the viscosity of the fluid, ρ is the density of the fluid.
The customary random molecule-motion-induced laminar shear stress lamτ is
/du dyμ . For turbulent flow it is found that the turbulent shear stress, ' 'turb u vτ ρ= −
is positive. Hence, the shear stress is greater in turbulent than in laminar flow. Terms
of the form ' 'u vρ− are called Reynolds stresses in honor of Osborne Reynolds who
first discussed them in 1895. In a very narrow region near the wall, the laminar shear
stress is dominant. Away from the wall the turbulent portion of the shear stress is
dominant. In natural rivers, the shear stress is most relevant to the sediment transport,
which means that it affects the amount of sediment brought from upstream to
downstream (see Figure 3 below).
Figure 3 Sediment transport in natural river
As mentioned above, for extracting the time mean velocity, u , the unsteadiness
of the average velocity is supposed to be very small. Suppose the turbulent flow
follows a stationary process, the mean velocity u can be obtained easily by taking
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the average of the instantaneous velocity u , that is, u is not a function of time, .
Then, the fluctuation velocity 'u can be conventionally calculated by subtracting the
mean velocity u from the instantaneous velocities u in the realizations of the flow.
Hence, the total flow can be split into the mean part and the fluctuating component.
However, under non-stationary flow field, the time-varying mean velocity failed to be
obtained as easily as that in stationary flow. How to extract the time-varying mean
velocity from the velocity records has become the initial and critical step to
understand the turbulence characteristics of the non-stationary flow.
The most common non-stationary turbulent flow occurs in the flooding period,
where velocity profile changes with time rapidly. Due to the features in flooding
period, taking average velocities as the time mean average values is unsuitable any
more; hence, other methods are applied to decompose the instantaneous velocity
profile. Refer to previous investigations in fluid mechanics (Song and Graf (1996),
Nezu et al. (1997), Haung et al. (1998)) and some digital signal processing literatures
(Jansen (2001), and Rioul et al.(1991)) (the velocity in turbulent flow resembles the
contaminated signals in digital signal processing, my task is to separate the time mean
value and fluctuation velocity, somewhat similar to the de-noising processes.), three
methods will be applied in this project, which are Fourier decomposition method,
wavelet transformation, and Empirical mode decomposition, respectively. The
introductions of these methods are provided in the next section:
2 Decomposition methods
Three methods used to decompose the contaminated signals are briefly
introduced as follows:
(1) Fourier decomposition method:
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The Fourier component method is to transform the instantaneous velocity
profile{ }Niiu 1= into the frequency domain by using the discrete Fourier transform
(DFT). Only the frequency components lower than Tmf 2/)1( −= are used to
represent the mean velocity component; m is an odd integer, T is the time period of
the measurement of a hydrograph. The Fourier sum can be interpreted as the mean
velocity component.
( )∑ ++=−
=
2/)1(
10 sincos
21 m
iikkikki baaU ωω (3)
Where
∑−
=
=1
0
cos2 N
iikik u
Na ω , ∑
−
=
=1
0
sin2 N
iikik u
Nb ω
kNiik )/(2πω = ( )2/)1(,...,2,1,0 −= mk
This is like to pass the signal through a low pass filter with cut-off frequency of
Tmf 2/)1( −= . The frequency higher than the cut-off frequency will be eliminated,
only the lower frequency components will be retained. The Fourier transformation
establishes the relation between the time domain and the frequency domain, and the
sinusoidal function is its base. Thus, the spectrum shows the global strength with any
frequency included in this function. But, it fails to show how the frequencies vary
with time in the spectrum. In nature phenomena, turbulence flow has time-varying
frequencies features. Due to the characteristics of turbulence flow, the wavelet
transformation and empirical mode decompositions helps obtain the local frequencies
in time
(2) Wavelet transformations
In wavelet analysis, linear combinations of small wave functions (see figure
below) are used to represent signals.
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Figure 4 Small wave function
Wavelets separate a signal into multi-resolution components. The fine and coarse
resolution components capture the fine and coarse features in the signal, respectively.
The representation of the discrete wavelet transformation for a given scaling filter at
JN scales is:
( ) ( ) ( ) ( ) ( ) ( )1/ 2 / 22 2 2 2J J
JJ
JJ N J N j jJ N j
k j J N kf t c k t k d k t kφ ψ
−− −−
= −= − + −∑ ∑ ∑ (4)
Where there are 2JN = in ( )f t , and ( )tφ is the scaling function, and ( )tψ is the
wavelet. The scaling function is:
( ) 1tφ = for 0 1t< <
= 0 otherwise
There are two wavelets, one is Haar wavelet, which is a single square wave with
amplitude of one in ( )0,1t∈ ; the other one is Daubechise wavelet, which is more
sophisticated, see the figure below (length can be different, in this case, length=8 ).
Figure 5 Daubechise wavelet
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The ( )JJ Nc k− is the scaling coefficients, and the ( )jd k is the wavelet
coefficients. The energy of ( )f t can be defined as:
( ) ( ) ( )12
JJ
J
J N jk j J N k
f t dt c k d k−
−= −
= +∑ ∑ ∑∫ (5)
For de-noising procedure, we can set up some threshold value γ and, then the
de-noised estimate of f say f has wavelet and scaling coefficients ( )jd k and
( )JJ Nc k− where
( ) ( ) ( )( )
0 <
j j j
j
d k d k d k
d k
γ
γ
= ≥
=
( ) ( ) ( )( )
0 < J J
J
J N J J N J N
J N
c k c k c k
c k
γ
γ
− − −
−
= ≥
=
In this study, the universal threshold 2 lnuniv Nγ σ= (where N is sampling
points in the signal, 2σ is the noise variance) well-known is applied in this project.
(3) Empirical decomposition method
There are several drawbacks in Fourier and Wavelet analysis. Firstly, both of
them assume that the system is linear, and then the signal in the system is supposed to
be the superpositions of sinusoidal and small wave functions with different
amplitudes and frequencies. However, although many phenomena in the natures can
be approximated by linear systems, the nonlinear feature makes the imperfection of
data analysis.
Huang et al. (1998) recently developed a new data processing method termed
Hilbert-Huang Transformation (HHT) to analyze the non-stationary time-series data.
In performing HHT, two steps are required. First, is uses a so-called empirical mode
decomposition (EMD) to deintegrate the complicated time series into a finite number
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of local characteristic oscillations called intrinsic mode functions (IMF). Given
)(tu is a velocity profile obtained in unsteady flow, all the local maxima and minima
are connected by a cubic spline line as the upper and lower envelopes. The mean of
the upper and lower envelopes are designated as 1m . The difference between the
original velocity profile )(tu and the mean value 1m are the first IMF, )(1 tc , if it
meets two criteria:(1) the number of extrema and the number of zero crossings are
equal or differs at most by one for the whole data set; and (2) at any point, the mean
value of the envelope defined by the local maxima and the enveloped defined by the
local minima is zero. This is the so-called sifting process. Through a series of sifting
processes, the riding waves and asymmetry of the profile can be eliminated. Finally,
the original flow velocity )(tu in unsteady flow can be decomposed into different
timescales of distinct IMF components.
)()()(1
trtctU N
N
jj +∑=
= (6)
Where N is the number of the IMF components; and )(trN is the final
residue. The final residue )(trN is usually a monotonic function. The mean
time-varying velocity can be regarded as the sum of the last few IMFs and the final
residue.
In this project, initially, some given functions with the additive noises following
a Gaussian distribution with zero mean and unit covariance will be used to test these
three de-noising method. Then, some non-stationary turbulent flow data collected in
open channel flow will be decomposed and the corresponding Reynolds shear stress
will be evaluated by the best-estimated method. Finally, the results will be compared
with the laboratory observations.
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3 Selection of decomposition method
In order to test the feasibility of the non-stationary turbulence velocity, four
given function, including Heavi-Sine function, Exponential-Sine function, Bessel
function and Hydrograph in flooding periods, with additive noises (Gaussian
distribution with zero mean and unit variance) will be decomposed by the
above-mentioned methods. The sampling points are 4096 with a sampling frequency
of 50Hz (duration is 81.92 seconds), corresponding to the frequency of the equipment
used to measure velocity profile in the laboratory flume. The reasons for choosing
these four functions lie in their variety, nonlinearity and similarity to the situations in
flooding periods. In Fourier decomposition, the cut-off frequency should be
predetermined. From Nezu’s study (1997), he mentioned that the cut-off frequency
should be much smaller than the bursting frequency of turbulence. In the selection
procedures, due to the artificial generation of the noise of the signal, the cut-off
frequency will be chosen until the mean squared errors (MSE) between the fitting and
original signals approach to steady state, namely, the difference percentages of MSE
between adjacent frequencies are less than 0.1%. Then, in wavelet analysis, the
threshold values will be determined by the universal threshold mentioned previously.
Finally, by using EMD methods, the fitting function will be the sum of the residual
and last “several” terms. Here, the number of the “several” terms will be based on the
minimum MSE values. The details procedures will be provided in the “Heavi-Sine”
case. In Fourier decomposition method, the steady MSE value, which is 0.032, occurs
at cut-off frequency of 0.195Hz (m=33) (See Figure 6).
11
0
2
4
6
8
10
12
14
16
0 20 40 60 80 100
Time(sec)
Velo
city
(cm
/s)
Noise-free signal
Noise signal
Figure 6 The Heavi-Sine function with and without additive noises by using Fourier
decomposition method
For wavelet analysis, the universal threshold 2 lnuniv Nγ σ= is 4.08. The
energy distribution with the universal threshold is shown below.
Figure 7 Energy distributions and the universal threshold value in wavelet analysis
(This is a enlarged plot)
Figure 7 shows that most of the energy is less than the universal threshold value.
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The energy larger than the threshold means the time-varying mean velocity. The
smaller energy components are responsible for noise motions. Applying inverse
discrete wavelet transformation (IDWT), the velocities with and without additive
noise are provided in Figure 8. The MSE value for this case is 0.047.
Figure 8 The Heavi-Sine function with and without additive noises by using wavelet
transformation method
Lastly, the EMD method will be employed to deintegrate the contaminated signal.
Figure 9 shows some selective intrinsic mode functions (IMF).
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Imf=1
Imf=5
Imf=8 Imf=14
Imf=19 Imf=23 (residue)
Figure 9 Selective intrinsic mode functions in Heavi-Sine function
Each IMF has similar frequency component. In this case, the fitting function is the
sum of residue plus IMFs from 11 to 23 (see Figure 10). The MSE value is 0.024
better than the other two methods..
14
0
2
4
6
8
10
12
14
16
0 10 20 30 40 50 60 70 80 90
Time(sec)
Velo
city
(cm
/s)
Origin
Fourier
Figure 10 The original Heavi-Sine function and its evaluation by using Empirical
Mode Decomposition method
In other three testing functions, the same procedures are followed. The plots for
these four functions and their fitting curves from different decomposition methods are
provided followings. Table 1 shows the MSE values for each case.
15
0
2
4
6
8
10
12
14
16
0 10 20 30 40 50 60 70 80 90
Time(sec)
Velo
city
(cm
/s)
OriginFourierWaveletEMD
(a) Heavi-Sin function simulations
0
5
10
15
20
25
30
35
0 10 20 30 40 50 60 70 80 90
Time(sec)
Velo
city
(cm
/s)
OriginFourierWaveletEMD
(b) Hydrograph function simulations
0
5
10
15
20
25
30
0 10 20 30 40 50 60 70 80 90
Time(sec)
Velo
city
(cm
/s)
OriginFourierWaveletEMD
(c) Exponential-Sine function simulations
-4
-2
0
2
4
6
8
10
12
14
16
0 10 20 30 40 50 60 70 80 90
Time(sec)
Velo
city
(cm
/s)
OriginFourierWaveletEMD
(d) Bessel function simulations
Figure 11 Simulation results for three decomposition methods
Table 1 MSE values for these four cases
Heavi-Sine Hydrograph Exponential-Sine Bessel
Fourier 0.026 0.0026 0.0018 0.0021
Wavelet 0.047 0.0095 0.0280 0.0085
EMD 0.024 0.0037 0.0015 0.0027
According to Figure 11 and Table 1, we can find that in each case, estimated
functions are almost fitting to the original function. The differences between them are
hard to discern. Compared with their MSE values, Fourier and EMD methods always
obtain better results than the wavelet transformations. Due to the ability of the EMD
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to know how frequency changes with time, the EMD will be applied to the data
measured in the laboratory flume later.
4 Experimental measurements and Analysis
The experiments were conducted in a 12.19m (40ft) long and 1.22m (4ft) wide
slope-adjustable recirculating flume. Water flow rate was adjusted through a butterfly
valve located immediately upstream of a tee. A flowmeter, upstream of the butterfly
valve, continuously measured the discharge coming through the system (see Figure
12).
Figure 12 Experimental flume located in Engineering Hall Basement
The instrument used for measuring instantaneous velocities in this study was a
Sontek, 10-Mhz, Acoustic Doppler Velocimeter (ADV) (see Figure 13).
Figure 13 Acoustic Doppler Velocimeter
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The semi-intrusive instrument is superior to other, more intrusive methods in that
the probe is situated approximately 5 cm above the location of the actual sample
volume (where measurements are taken). The ADV determines water velocities based
on the Doppler shift principle. This principle states that the change in frequency of a
sound source relative to the receiver is proportional to the velocity with which the
source is moving. The probe consists of one central transmitter surrounded by three
receiving transducers. The transmitter emits a short pulse of sound at a specific known
frequency to travel downward through the water column in a narrow cone. As the
acoustic pulse travels, it reflects in all directions from the particulate matter and air
bubbles in the flow, some of which is reflected back towards the receivers. The
receivers then record the reflected sound from a very narrow range and each receiver
uses the computed Doppler shift to measure the instantaneous velocity along its
bistatic axis. Since the three receivers are all narrowly focused on a small sample
volume and located about 5 cm below the transmitter; the velocity is computed for the
three bistatic axes all corresponding to said sample volume. Given the known probe
geometry and sample volume location, the instrument is then able to convert these
velocities into Cartesian velocities for that parcel of water. In this study, the
x-component and y-component indicate the mean stream and transverse directions
respectively, and the z-component means the vertical direction (see Figure 14). The
details of the principle of ADV can be referred to Sontek (1997).
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Figure 14 Coordinate system in the flume
The ADV is installed in 6m downstream of flume inlet to ensure the achievement
of fully-developed flow conditions. For investigating turbulence nature
comprehensively, a rather high sampling frequency of 50Hz of ADV was chosen for
this experiment, i.e. 50 samples were taken each second.
In the experiments, the flow was initially maintained at uniform flow conditions,
then increased gradually until it reaches the prescribed peak discharge, and then
decreased close to the initial discharge. The procedures were finished by manual
controls due to lack of computer-controlled system.
There three-component velocity profiles collected with duration of 300 seconds
and sampling rates of 50 Hz in the flume is provided as follows.
0
5
10
15
20
25
30
0 50 100 150 200 250 300
Time(sec)
Veol
city
(cm
/s)
(a) x-component velocity profiles
x: Mean stream direction
y: Transverse direction
z: Vertical direction
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-8
-6
-4
-2
0
2
4
6
8
10
0 50 100 150 200 250 300
Time(sec)
Veol
city
(cm
/s)
(b) y-component velocity profiles
-6
-4
-2
0
2
4
6
0 50 100 150 200 250 300
Time(sec)
Veol
city
(cm
/s)
(c) z-component velocity profiles
Figure 15 Velocity profiles collected in the laboratory flume
Then, we apply the EMD method to decompose the signal as many IMF terms,
which start from the shortest period (IMF1) and end at the longest period (Residual).
The last IMF is a monotonic function. By comparing the fitting curve with the
instantaneous velocity profiles, the time-varying mean velocity can be decided by the
following combinations:
Case 1: Sums from 15th IMF to 27th IMF (residue).
Case 2 : Sums from 14th IMF to 27th IMF (residue).
Case 3: Sums from 13th IMF to 27th IMF (residue).
There are too many oscillations from the sums of 12th IMF to the residue.
Moreover, the combinations from 16th IMF to the residue do not fit the instantaneous
velocities very well. Figure 16 shows the three cases and original velocity profiles
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with time.
0
5
10
15
20
25
30
0 50 100 150 200 250 300 350
Time(sec)
Velo
city
(cm
/s)
Original dataCase 1(IMF15-IMF27)Case2(IMF14-IMF27)Case3(IMF13-IMF27)
Figure 16 (a) The mean velocity in x-component done by the empirical decomposition mode method
-8
-6
-4
-2
0
2
4
6
8
10
0 50 100 150 200 250 300 350
Time(sec)
Velo
city
(cm
/s)
Original dataCase 1(IMF15-IMF27)Case2(IMF14-IMF27)Case3(IMF13-IMF27)
Figure 16 (b) The mean velocity in y-component done by the empirical decomposition mode method
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-6
-4
-2
0
2
4
6
0 50 100 150 200 250 300 350
Time(sec)
Velo
city
(cm
/s)
Original dataCase 1(IMF15-IMF27)Case2(IMF14-IMF27)Case3(IMF13-IMF27)
Figure 16 (c) The mean velocity in z-component done by the empirical decomposition mode method
Because the fitting mean velocity in y-component for Case 2 and Case 3 perform
abrupt oscillations in the end points, these portions will be cut in order to avoid the
errors. For case 1, these oscillations occur in z-component and last for a long time,
due to that, we will neglect the Case 1 for later discussions. The distributions of the
fluctuating velocity in x,y,and z components show similar Gaussian distribution(see
Figure 17).
22
0
0.05
0.1
0.15
0.2
0.25
0.3
-15 -10 -5 0 5 10 15
Fluctuating velocity(cm/s)
Prob
abilit
y de
nsity
Case 2Case 3
Figure 17 (a) Distributions of fluctuating velocity in x-component obtained from EMD
0
0.05
0.1
0.15
0.2
0.25
0.3
-10 -5 0 5 10
Fluctuating velocity(cm/s)
Prob
abilit
y de
nsity
Case 2Case 3
Figure 17 (b) Distributions of fluctuating velocity in y-component obtained from EMD
23
0
0.05
0.1
0.15
0.2
0.25
0.3
-10 -5 0 5 10
Fluctuating velocity(cm/s)
Prob
abilit
y de
nsity
Case 2Case 3
Figure 17 (c) Distributions of fluctuating velocity in z-component obtained from EMD
Figure 17 shows the features of Gaussian distribution, corresponding to the
features of noise in nature environment. In terms of turbulence characteristics,
turbulence intensity and Reynolds shear stress are the most important factors. The
turbulence intensity is defined as ( )2'uu
, where u is time mean velocity, and 'u
is fluctuating velocity. In this project, turbulence-intensity values and Reynolds shear
stress during the hydrograph will be provided. Before computing these values, an
interval TΔ used to average the data should be determined first. Then, the statistics
of turbulence at time T was time averaged from 2/TT Δ− to 2/TT Δ+ .
Traditionally, the TΔ is chosen as 0.5 seconds. Table 2 provides the maximum
turbulence intensity and the corresponding occurrence time in x,y,z directions.
24
Table 2 Maximum turbulent intensity and the corresponding occurring time
Case 2'u
(cm/s) t(sec)
2'v
(cm/s) t(sec)
2'w
(cm/s) t(sec)
2 5.45 140.75 4.32 103.25 2.94 106.25 3 5.56 129.25 4.26 103.25 2..88 106.25
Table 2 shows that the turbulence intensity values in x direction are larger than
these in y and z direction. The results are similar to what Nezu et al. (1997) concluded.
Figure 18 provides the comparisons of normalized turbulent intensities with water
depth (all quantities are normalizing by their maximum values).
Figure 18 (a) Non-dimensional turbulent intensities and water depth against time for
Case2
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
x-directionh/hmax
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
y-directionh/hmax
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
z-directionh/hmax
25
Figure 18 (b) Non-dimensional turbulent intensities and water depth against time for
Case3
The results also correspond to the conclusions obtained by Nezu et al. (1997) and
Song et al. (1996)’s conclusions that in y and z directions, turbulence intensity is
stronger in the rising stage than that in the falling stage. The maximum Reynolds
shear stresses for ''vuρ− , ''wvρ− and ''wuρ− are listed in Table 3.
Table 3 Maximum Reynolds shear stress and the corresponding occurring time
Case ''vuρ−
(N) t(sec)
''wvρ−
(N) t(sec)
''wuρ−
(N) t(sec)
2 0.757 93.75 0.324 218.75 1.0572 129.25 3 0.698 91.75 0.328 100.25 1.223 129.25
Figure 19 shows the non-dimensional Reynold shear stress and water depth
against time (all quantities are normalized by their maximum values).
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
x-directionh/hmax
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
y-directionh/hmax
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
z-directionh/hmax
26
Figure 19 (a) Non-dimensional Reynold shear stress and water depth against time
for Case2
Figure 19 (b) Non-dimensional Reynold shear stress and water depth against time
for Case3
-1.5
-1
-0.5
0
0.5
1
1.5
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
u'v'h/hmax
-1.5
-1
-0.5
0
0.5
1
1.5
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
v'w'h/hmax
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350
Time(sec)
Rat
iou'w'h/hmax
-1
-0.5
0
0.5
1
1.5
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
u'v'h/hmax
-1.5
-1
-0.5
0
0.5
1
1.5
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
v'w'h/hmax
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350
Time(sec)
Rat
io
v'w'h/hmax
27
The ''vuρ− term distributes dispersedly and randomly as the ''wvρ− term
does. But, the ''wuρ− term performs different patterns, whose peak value is apparent
and always located right after the peak of the hydrograph. Further, the direction of
the ''wuρ− term is toward the x-diretcion (mean stream direction), i.e. it is most
related to the sediment transport. The larger the ''wuρ− term is, the more the
sediment yields will be brought from upstream to downstream. Thus, the ''wuρ−
term can be used an index to observe how the unsteadiness effects the sediment
transport in open channel flow. Based on this viewpoint, the result shows that the
strongest shear stress occurs right after the hydrograph peak, namely, the sediment
transport rate will possibly reach its peak at this moment. Field and lab measurements
(Kuhnle, 1992; Lee et al., 2004) showed the sediment transport processes during the
passage of the flooding are quite different with those in steady flow. Owing to the
temporal lag, the total sediment yield in rising period was less than that in the falling
period. Furthermore, measured total sediment yield in the unsteady flow experiments
was larger than the predicted value, which was estimated by using the results obtained
from the equivalent steady flow experiment. In this project, we focused on the
characteristics of turbulence flow. From turbulent intensities and Reynolds stress
obtained from the decomposition velocities, some of these values indeed are larger in
the falling stage, i.e. their maximum values occur after the peak of the hydrograph.
These turbulence characteristics prove that the phenomena occur in open channel flow.
Compared our results with previous investigators’ measurements, we found that the
turbulence characteristics show the same trend as well. These comparisons show the
feasibility of our decomposition methods.
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5 Conclusions
In the project, we attempted to compare the signal de-noising results by using
Fourier decomposition method, wavelet transformations, and empirical mode
decomposition. There are four given functions with additive noises used to test these
methods. The simulation results showed that the mean square errors obtained from
empirical mode decomposition will be less than others. Therefore, EMD method was
employed to decompose the turbulence velocity profile collected by acoustic Doppler
velocimeter (ADV) in the laboratory flume. With the decomposition velocities,
turbulence intensities and Reynolds shear stress, most associated to the sediment
transport yields, can be evaluated. According to the theoretical analysis, the larger the
turbulence intensities and Reynolds shear stress, the more sediment yields will be.
Compared with the field and laboratory results, these turbulence characteristics
provided in our decomposition results can successfully explain the phenomena
happened in the rising and falling stages during the flooding period. This is also to
prove the feasibility of the EMD method, although more evidence and strict
procedures are needed in the future. On the other hand, there are still many drawbacks
in EMD method. The most significant one is that how many IMFs should be kept to
represent the time-varying mean velocity. Many researchers are working on this issue.
Recently, some researchers applied 2-D EMD to solve the problems in image
compressions and image restorations. This is a new way which we can work on.
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6 References
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Lee, K.T., Liu, Y.L. and Cheng K.H. (2004). “Experimental investigation of bedload
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Maarten Jansen. Noise Reduction and by Wavelet Thresholding, volume 161.
Springer Verlag, United States of America, 1st edition, 2001
Nezu, I., Kadota, A., and Nakagawa, H. (1997). “Turbulent structure in unsteady
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Sontek (1997). Sontek ADV Acoustic Doppler Velocimeter Technical Documentation.
