Frequency and energy distribution of seven level decompositions.
Abstract
To analyze the properties of the coherent structures in nearwall turbulence, an extraction method based on wavelet transform (WT) and a verification procedure based on correlation analysis are proposed in this work. The flow field of the turbulent boundary layer is measured using the hotfilm anemometer in a gravitational lowspeed water tunnel. The obtained velocity profile and turbulence intensity are validated with traditional boundary layer theory. The fluctuating velocities at three testing positions are analyzed. Using the power spectrum density (PSD) and WT, coherent and incoherent parts of the nearwall turbulence are extracted and analyzed. The probability density functions (PDFs) of the extracted signals indicate that the incoherent structures of turbulence obey the Gaussian distribution, while the coherent structures deviate from it. The PDFs of coherent structures and original turbulence signals are similar, which means that coherent structures make the most contributions to the turbulence entrainment. A correlation parameter is defined at last to prove the validity of our extraction procedure.
Keywords
 coherent structure
 wavelet transform
 correlation analysis
 turbulence
1. Introduction
Turbulence is a commonly seen but very complicated phenomenon in nature. Numerous tests have proven that turbulence is not a pure random process but contains different scales of fluctuations called coherent structures [1, 2, 3]. These structures significantly contribute to fluid entrainment and mass, momentum, and heat transfer [4, 5]. Therefore, investigating the coherent structures is of great significance to undercover the physics and to realize flow control.
Among the techniques of turbulence analysis, wavelet transform has been proven feasible and power to detect and extract the coherent structures in turbulence [6, 7, 8, 9]. Early works are based on continuous wavelet transform (CWT). Liandrant [10] and Jiang [11, 12] proposed the maximum energy principle, which considered the signal at the maximum energy scale as the burst events in turbulence. Kim [13] identified the coherent structure around a vibrating cantilever based on CWT. However, a drawback of CWT is that it is unable to reconstruct the signal if the mother wavelet is not orthogonal [14, 15, 16]. To solve this problem, Longo [17] used the multiresolution analysis technique based on the discrete wavelet transform (DWT) and extracted the structures in turbulence. DWT has evident advantages compared with CWT since it is invertible and multiscaled scales can be analyzed. Kadoch [18] combined DWT and direct numerical simulation (DNS), whose results proved that coherent structures preserve the vortical structures with only about 4
In this work, measurement of the turbulent boundary layer is carried out using hotfilm anemometer in a gravitational lowspeed water tunnel. A procedure based on the WT and correlation analysis is proposed to extract and verify the coherent and incoherent structure in turbulence.
2. Experimental tests and analysis
2.1 Experimental apparatus
A gravitational lowspeed water tunnel was constructed for the experiment. The gravity generated by the water level difference drives the water flow in the tunnel, and the flow can be tested in the experimental section (
Figure 1
). A maximum water speed of 2.0
2.2 Verification of turbulent boundary layer flow
By using the experimental setups in
Figure 2
, the flow velocity of the turbulence boundary layer was measured at a series of positions in the vertical direction. The mean velocity profile and the turbulence intensity distribution at the water speed 0.4
3. Theoretical background of wavelet transform
WT is a mapping of a time function, in a onedimensional case, to the two dimensional timescale joint representation. The temporal aspect of the signal can be preserved. The wavelet transform provides multiresolution analysis with dilated windows. The highfrequency part of the signal is analyzed using narrow windows, and the lowfrequency part is done using wide windows. WT decomposes the signal into different frequency components and then studies each component with a resolution matched to its scale. It has advantages over traditional Fourier methods in analyzing physics where the signal contains discontinuities and sharp spikes.
WT of a signal
where
Scale a and position b should be discretized for applications. Usually we choose
The corresponding DWT can be expressed as:
The orthogonality of
By choosing the scale
where
For turbulence, the fluctuating velocity of turbulence can be normally divided into two subparts:
where
By adopting the multiresolution analysis (
Figure 5
), the turbulence signal
where
4. Extraction and verification of turbulent structures
To extract the coherent structures in turbulence, the signals at the central area of turbulence should be selected. According to previous studies [20, 21, 22], the formation of the coherent structures in turbulence is formed in the area of
4.1 Preliminary evaluation of coherent structures
For preliminary evaluations of the coherent structures, CWT is first utilized for the analysis. CWT is a mathematical mapping similar to the Fourier transform [23, 24]. It is linear, invertible, and orthogonal. However, the Fourier transform uses basis functions, including the sines and cosines, which extend to infinity in time, while wavelet basis functions drop towards zero outside a finite domain (compact support). This allows for an effective localization in both time and frequency. CWT uses inner products to measure the similarity between the turbulence signal and the wavelet function, which defines a mapping between the two. CWT compares the turbulence signal to shifted and compressed/stretched versions of the wavelet function. Compressing/stretching is also referred to as dilation or scaling and corresponds to the physical notion of scale. By continuously varying the values of the scale parameter,
In the work, the 5th order of Daubechies wavelet was selected as the basis function, whose central frequency fc is 0.6667
4.2 Extraction of coherent structures
To obtain the frequency range of coherent structures in turbulence, power spectrum densities of the three selected signals were calculated in
Figure 8
. The centralized frequencies of the coherent structures are found in the range 0
WT of a signal is equivalent to local crosscorrelation analysis between the signal and wavelet function. OWT carries out the multiresolution analysis for both decomposition and reconstruction of the original turbulence signal. It is thought of the wavelet coefficients as digital filters as which the original signal is passed through lowpass filters to decompose into lowfrequency components and passed through highpass filters to analyze into highfrequency components.
Using the multiresolution analysis of OWT, the turbulence signal was split into seven scales as in
Table 1
, which eliminates most of the redundant signals. The frequency range of the approximate signal is mainly in the range 0
Signal  Frequency/

Energy/








0

100  100  100 

0

85.6498  77.0677  81.3847 

260

0.0147  0.0200  0.0259 

520

0.0062  0.0086  0.0129 

1042

0.0019  0.0036  0.0042 

2083

0.0464  0.0703  0.0868 

4167

0.3957  0.8099  0.8505 

83,334

3.5993  5.5974  5.2322 

16,668

10.2411  16.4225  12.4028 
The extracted signals of each level are shown in Figure 9 , where “A7” is the approximate signal, i.e., the coherent structures; where “sD7” is the incoherent structures, which is calculated by:
and where “s” is the original signal. “D1
4.3 Verification of extracted signals
To characterize the properties of the extracted signals, the probability density functions (PDFs) were analyzed in Figure 10 . It can be observed that the incoherent structures are approximately Gaussian, demonstrating isotropic characteristics. The PDFs of coherent structures deviate from the Gaussian distribution, presenting strong anisotropic characteristics. And the PDFs of the coherent structures resemble that of the original turbulence signals. This means that coherent structures contribute the most to turbulence entrainment.
For further validation of the extracted coherent and incoherent structures, correlation analysis was carried out here. A correlation parameter
where
5. Conclusion
The flow field of the turbulence boundary layer was measured using hotfilm anemometer in a gravitational lowspeed water tunnel. The coherent and incoherent structures in turbulence were separated successfully with an extraction method based on WT. With CWT, the turbulent structures can be observed in various scales. With DWT, multiresolution analysis can be carried out for the decomposition and reconstruction of vortical structures in different scales. The PDF of the incoherent structures was found to obey the Gaussian distribution, while that of the coherent structures deviate from it. The similarity of the PDFs of the coherent structures and the original turbulence signal demonstrate that the coherent structures make most contributions to turbulence. A correlation parameter between coherent and incoherent structures was defined, which proves the successful separation of coherent structure from turbulence.
Acknowledgments
The authors acknowledge the support from the National Natural Science Foundation of China (Grant No. 51879218, 51679203) and Fundamental Research Funds for the Central Universities (Grant No. 3102018gxc007, 3102020HHZY030004).
Appendix I: complex wavelet transform
The continuous wavelet transform (CWT) has the drawback of redundancy. As the dilation parameter a and the shift parameter b take continuous values, the resulting CWT is a very redundant representation. Therefore, the discrete wavelet transform was proposed to overcome this problem by setting the scale and shift parameters on a discrete set of basis functions. Their discretization is performed by:
where
and the discrete wavelet decomposition of a signal
where
The basis function set
The advantage of the DWT is the multiresolution analysis ability. Although the standard DWT is powerful, it has three major disadvantages that undermine its applications: shift sensitivity, poor directionality, and absence of phase information.
Complex wavelet transform can be used to overcome these drawbacks. It uses complexvalued filtering and decomposes the signal into real and imaginary parts, which can be used to calculate the amplitude and phase information.
For turbulence analysis, the complex wavelet transform should be used since the modulus of the wavelet coefficients allows characterizing the evolution of the turbulent energy in both the time and frequency domains. The realvalued wavelets will make it difficult to sort out the features of the signal or the wavelet. On the contrary, the complexvalued wavelets can eliminate these spurious oscillations. The complex extension of a real signal
where
The complex wavelet transform is able to remove the redundancy for turbulence analysis where the directionality and phase information play important roles.
Notes
Reprinted (adapted) with permission from Chinese Physics B, 2013, 22(7): 074703.
Abbreviations
WT  wavelet transform 
PSD  power spectrum density 
OWT  orthogonal wavelet transform 
probability density function  
CWT  continuous wavelet transform 
DWT  discrete wavelet transform 
DNS  direct numerical simulation 
s t  original turbulence signal 
φ m , n t  scaling function 
a  scale parameter 
b  position parameter 
m 0  critical scale 
ψ m , n t  wavelet function 
s ˜  coherent part of the signal 
s ′  incoherent part of the signal 
f  frequency 
f s  sampling frequency 
f c  central frequency of particular wavelet basis 
y +  dimensionless wall distance 
A 7 , D 1 ∼ D 7  detailed signal of each level 
v  fluctuating velocity signal 
β  correlation parameter 
References
 1.
Kowal G, Lazarian A. Velocity field of compressible magnetohydrodynamic turbulence: Wavelet decomposition and mode scalings. The Astrophysical Journal. 2010; 720 (1):742  2.
XiaoBing L, ZhengQing C, ChaoQun L. Latestage vertical structures and eddy motions in a transitional boundary layer. Chinese Physics Letters. 2010; 27 (2):024706  3.
Rinoshika A, Omori H. Orthogonal wavelet analysis of turbulent wakes behind various bluff bodies. Experimental Thermal and Fluid Science. 2011; 35 (7):12311238  4.
Rinoshika A, Watanabe S. Orthogonal wavelet decomposition of turbulent structures behind a vehicle external mirror. Experimental Thermal and Fluid Science. 2010; 34 (8):13891397  5.
Okamoto N, Yoshimatsu K, Schneider K, et al. Coherent vortices in high resolution direct numerical simulation of homogeneous isotropic turbulence: A wavelet viewpoint. Physics of Fluids. 2007; 19 (11):115109  6.
De Stefano G, Vasilyev OV. A fully adaptive waveletbased approach to homogeneous turbulence simulation. Journal of Fluid Mechanics. 2012; 695 :149172  7.
Futatani S, Bos WJT, delCastilloNegrete D, et al. Coherent vorticity extraction in resistive driftwave turbulence: Comparison of orthogonal wavelets versus proper orthogonal decomposition. Comptes Rendus Physique. 2011; 12 (2):123131  8.
de la Llave PM, Cant S, Prosser R. On the use of biorthogonal interpolating wavelets for largeeddy simulation of turbulence. Journal of Computational Physics. 2012; 231 (20):67546769  9.
Khujadze G, Schneider K, Oberlack M, et al. Coherent vorticity extraction in turbulent boundary layers using orthogonal wavelets. Journal of Physics: Conference Series. 2011; 318 (2):022011  10.
Liandrat J, MoretBailly F. The wavelet transformsome applications to fluid dynamics and turbulence. European Journal of Mechanics  B/Fluids. 1990; 9 :119  11.
Nan J, Jin Z. Detecting multiscale coherent eddy structures and intermittency in turbulent boundary layer by wavelet analysis. Chinese Physics Letters. 2005; 22 (8):1968  12.
Jianhua L, Nan J, Zhendong W, et al. Multiscale coherent structures in turbulent boundary layer detected by locally averaged velocity structure functions. Applied Mathematics and Mechanics. 2005; 26 (4):495504  13.
Kim YH, Cierpka C, Wereley ST. Flow field around a vibrating cantilever: Coherent structure education by continuous wavelet transform and proper orthogonal decomposition. Journal of Fluid Mechanics. 2011; 669 :584606  14.
Yoshimatsu K, Schneider K, Okamoto N, et al. Intermittency and geometrical statistics of threedimensional homogeneous magnetohydrodynamic turbulence: A wavelet viewpoint. Physics of Plasmas. 2011; 18 (9):092304  15.
Baars WJ, Talluru KM, Hutchins N, et al. Wavelet analysis of wall turbulence to study largescale modulation of small scales. Experiments in Fluids. 2015; 56 (10):188  16.
Camussi R. Coherent structure identification from wavelet analysis of particle image velocimetry data. Experiments in Fluids. 2002; 32 (1):7686  17.
Longo S. Turbulence under spilling breakers using discrete wavelets. Experiments in Fluids. 2003; 34 (2):181191  18.
Arimitsu T, Arimitsu N. Analysis of PDFs for energy transfer rates from 40963 DNSverification of the scaling relation within MPDFT. Journal of Turbulence. 2011; 12 :N1  19.
Gang D, ShiSheng Z, Yang L. Time series prediction using wavelet process neural network. Chinese Physics B. 2008; 17 (6):1998  20.
Asai M, Minagawa M, Nishioka M. The instability and breakdown of a nearwall lowspeed streak. Journal of Fluid Mechanics. 2002; 455 :289314  21.
Chen L, Tang DB. Study on turbulent spots in plane Couette flow. Transactions of Nanjing University of Aeronautics & Astronautics. 2007; 24 (3):211217  22.
Hu HB, Du P, Huang SH, Wang Y. Extraction and verification of coherent structures in nearwall turbulence. Chinese Physics B. 2013; 22 (7):074703  23.
Huang L, Kemao Q, Pan B, et al. Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry. Optics and Lasers in Engineering. 2010; 48 (2):141148  24.
Canal MR. Comparison of wavelet and short time Fourier transform methods in the analysis of EMG signals. Journal of Medical Systems. 2010; 34 (1):9194  25.
Du P, Wen J, Zhang Z, et al. Maintenance of air layer and drag reduction on superhydrophobic surface. Ocean Engineering. 2017; 130 :328335  26.
Haibao H, Peng D, Feng Z, et al. Effect of hydrophobicity on turbulent boundary layer under water. Experimental Thermal and Fluid Science. 2015; 60 :148156  27.
Sang YF. A review on the applications of wavelet transform in hydrology time series analysis. Atmospheric Research. 2013; 122 :815