Decomposing nonstationary turbulent velocity in open channel
flow
Ying-Tien Lin
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 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 flowrate”, 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.
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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.
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'
Where u 
(1)
1 t0 T
 u (t )dt , the time interval, T, is considerable longer than the period
T t0
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
stress  can be shown as follows:
 
du
  u ' v '   lam   turb
dy
(2)
Where  is the viscosity of the fluid,  is the density of the fluid.
The customary random molecule-motion-induced laminar shear stress  lam is
 d u / dy . For turbulent flow it is found that the turbulent shear stress,  turb    u ' v '
is positive. Hence, the shear stress is greater in turbulent than in laminar flow. Terms
of the form   u ' v ' 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 the amount of sediment brought from upstream to downstream in nature
rivers. 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 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
realization of the flow. That is, the total flow can be split into the mean part and the
fluctuating component. However, under nonstationary flow field, the time-varying
mean velocity failed to be obtained as easily as that in nonstationary 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
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nonstationary flow.
The most common nonstaionary turbulent flow occurs in the flooding period,
where velocity profile changes with time rapidly. Therefore, other methods are used to
decompose the instantaneous velocity profile. Refer to previous investigations and
some digital signal processing literature (the velocity in turbulent flow resembles the
contaminated signals in digital signal processing, my task is also to separate the time
mean value and fluctuation velocity, somewhat similar to the denoising processes.),
three methods will be applied in this studies, which are Fourier decomposition method,
Empirical mode decomposition, Wavelet analysis, respectively. The introductions of
these methods are provided as follows:
(1) Fourier decomposition method:
The Fourier component method is to transform the instantaneous velocity
profile ui i 1 into the frequency domain by using the discrete Fourier transform
N
(DFT). Only the frequency components lower than f  (m  1) / 2T 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.
Ui 
( m 1) / 2
1
a 0   a k c o 
s ik  bk s i n ik 
i 1
2
(3)
Where
ak 
2
N
N 1
 ui c o s ik , bk 
i 0
ik  2 (i / N )k
2
N
N 1
u
i 0
i
sin  ik
k  0,1,2, . . (.m,  1) / 2
(2) Empirical decomposition method
Huang et al. (1998) recently developed a new data processing method termed
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Hilbert-Huang Transformation (HHT) to analyze the nonstationary time-series data. In
performing HHT, two steps are required. First, is uses a so-called
empirical mode
decomposition (EMD) to deinterrate the complicated time series into a finite number
of local characteristic oscillations called intrinsic mode functions (IMF).
Given
u (t ) 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 m1 . The difference between the
original velocity profile u (t ) and the mean value m1 are the first IMF, c1 (t ) , 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 u (t ) in unsteady flow can be decomposed into different
timescales of distinct IMF components.
N
U (t )   c j (t )  rN (t )
(4)
j 1
Where N is the number of the IMF components; and rN (t ) is the final
residue.
The final residue rN (t ) 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.
Secondly, Huang et al. (1998) also introduced the concepts of instantaneous
frequency and instantaneous amplitude in their model. After sifting process, the
Hilbert transformation is then employed to these IMF c j t  , the instantaneous
frequency and instantaneous amplitude can be determined. The Hilbert transform of
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each IMF c j t  is:
cˆ j t  
P




  dt
cj t'
t t
'
(5)
'
where P is the Cauchy principal value. Hence, an analytic signal can be define as
z j t   c j t   cˆ j t   a j t e
i j t 
(6)
The amplitude a j t  and phase  j t  for each IMF are:
a j t   c j t   cˆ j t 
2
2
(7)
 cˆ j t 




c
t
j


 j t   tan 1 
(8)
The instantaneous frequency f j t  can be defined as:
f j t  
1 d j t 
2 dt
(9)
From eq.(7), (8), and (9), the original data can be expressed by
i 2f j t dt
U t   Re  a j t e 
n
(10)
j 1
Compared with the constant amplitude and frequency in Fourier transform, the
amplitude and instantaneous frequency of HHT are both function of time. Thus, HHT
can characterize the features of nonstationary time series data. The time-frequency
distribution of the amplitude or squared amplitude was designated as the Hilbert
amplitude spectrum and Hilbert energy spectrum (Huang et al. (1998)). In order to
compare with the power spectrum in turbulence, the Hilbert energy spectrum is
employed and denoted as Hilbert spectrum H  f , t  in this study. With the Hilbert
spectrum, the marginal spectrum h f  can be defined as:
h f   0 H  f , t dt
T
The marginal Hilbert spectrum h f  represents the cumulated energy over the entire
data and provides the total energy contributions from each frequency.
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(2) Wavelet analysis
In addition to the Fourier decomposition, empirical mode decomposition, the
wavelet analysis is applied as well. The Fourier transformation establishes the relation
between the time domain and the frequency domain, and the sinusoidal function is its
base. However, Fourier transformation is unable to show how the signal in frequency
domain varies with time. In nature phenomena, turbulence flow has time-varying
frequencies features. Due to the characteristics of turbulence flow, the wavelet
transformation helps us obtain the local frequencies in time, that is, the wavelet
transformation has similar functions as empirical mode decomposition. In wavelet
analysis, the linear combinations of wavelet functions used to represent signals.
Wavelets separate a signal into multiresolution 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
N J scales is:
f  t    c J  N J  k  2
k
J  NJ  / 2
  2 J  N t  k     d j  k  2 j / 2  2 j t  k 
J 1
J
jJ NJ k
(11)
Where there are N  2 J in f  t  , and   t  is the scaling function, and  t  is the
wavelet. The scaling function is:
 t   1 for 0  t  1
= 0 otherwise
There are two wavelets, one is Haar wavelet, which is a single square wave with
amplitude of one in t   0,1 ; the other one is Daubechise wave, which is more
sophisticated, see the figure below (length can be different, in this case, length=8 ).
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The cJ  NJ  k  is the scaling coefficients, and the d j  k  is the wavelet
coefficients. The energy of f  t  can be defined as:
J 1
 f  t  dt   cJ  N J  k     d j  k 
2
k
(12)
jJ NJ k
For denoising procedure, we can set up some threshold value  and them the
denoised estimate of f say f has wavelet and scaling coefficients d j  k  and
c J  NJ  k 
where
d j k   d j k  d j k   
d j  k  <
0
c J  N J  k   cJ  N J  k  cJ  N J  k   
cJ  N J  k  <
0
The denosing results will be compared with the results from Fourier
decomposition and empirical method decomposition methods.
In this study, firstly, some given functions with the added noises following a
normal distribution with zero mean and unit covariance will be used to test these three
denoising method. Then, some nonstationary turbulent flow data collected in open
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channel flow will be decomposed and the corresponding Reynolds shear stress will be
evaluated. Finally, the results will be compared with the field observations.
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