International Journal of Heat and Fluid Flowcoe Papers/JP_109.pdf · train ventilation system were carried out using well-resolved Hot-Wire Anemometry (HWA). Continuous wavelet transform

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International Journal of Heat and Fluid Flow

journal homepage: www.elsevier.com/locate/ijhff

Wavelet analysis on the turbulent flow structure of a T-junction

Bo Sua, Yantao Yina, Shicong Lia, Zhixiong Guob, Qiuwang Wanga, Mei Lina,⁎

a School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, ChinabDepartment of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, NJ 08854, USA

A R T I C L E I N F O

Keywords:Wavelet transformT-junctionTurbulent structureFluctuating velocity

A B S T R A C T

Experimental measurements in a T-junction with one inlet and two outlets mimicking the airflow in a high-speedtrain ventilation system were carried out using well-resolved Hot-Wire Anemometry (HWA). Continuous wavelettransform (CWT) is used to analyze the time-frequency contents of the instantaneous streamwise velocity; andthe discrete wavelet transform (DWT) is employed to determine the multi-resolution energy characteristics. Themeasurements and analysis are carried out at three representative streamwise locations, i.e., upstream, mid-center, and downstream of the T-junction. The results show that the normalized time-average velocity at themid-center of the T-junction is the largest near the wall region. Comparing the CWT data in the region near thewall, it is found that the dominant frequency of the periodic high energy coherent structures increases along thestreamwise direction, and the wavelet energy magnitude at mid-center of T-junction decreases with the increaseof velocity ratio. The DWT results show that apparent wavelet energy peak appears at the upstream anddownstream of the T-junction for different scales from 26 to 210, but not at the mid-center. However, the energyat scale 211 abruptly rises in all flow regions at all the three streamwise locations and this energy decreases withthe increase of the velocity ratio. Therefore, a higher velocity ratio is preferred for suppressing the generation oflarge-scale coherent structures to reduce drag forces and skin frictions for high-speed trains.

1. Introduction

Optimizing the ventilation system is critical for improving high-speed train's comfort, efficiency and safety. Fig. 1a shows the actualphysical unit of a high-speed train ventilation system, where the vanesare fixed to block the wild debris into the train's cabin. Some fresh airflowing through the ventilation port is sucked into the facility cabin bythe axial fan. Thus, turbulence and fluctuating forces will be generatedat the downstream of the ventilation port on train's surface. These in-crease train's surface frictions and flow drags. The domain marked bythe dotted rectangle in Fig. 1a is the test section in this study. Fig. 1bshows the plane graph of the ventilation port. Since it is difficult toperform a 1:1 scale experiment in laboratory, a small size model withthe scale of 1:4.8, will be used to conduct the experimental study. Thevelocity ratio, which is the ratio of the suction velocity to the train'svelocity, is a key parameter in ensuring the flow pattern similarity(Cambonie et al., 2014; Silva et al., 2003). On the contrary, Reynoldssimilarity is not strictly adopted due to high Reynolds number in thescaling analysis (Morii, 2008). In actual applications, the range of ve-locity ratio is from 0.08 to 0.15. The experimental model of the ven-tilation port is shown in Fig. 1c. It is necessary to understand the flowdynamics on the ventilation port in order to control the skin friction and

total drag and further improve the efficiency of high-speed trains.In experimental studies, the ventilation flow system in a wind

tunnel can be modeled as a T-junction channel (Wu et al., 2015), asshown in Fig. 2a. The airflow in the cross tube is regarded as the am-bient high-speed airflow passing through train's surface and part of theairflow in the cross tube is driven into the branch tube. The velocity inthe branch tube, ub, is much lower than the velocity in the cross tube,uc. The ratio of these two flow velocities, R= ub/uc, is the aforemen-tioned velocity ratio, controlling the flow rate ratio.

In the literature there are a number of researches on T-junctiontubes, spanning a wide range of industrial applications (Mitchell, 2001;Bodnar et al., 2008; Kumar et al., 2011; Alam et al., 2012). The T-junction diverging flows have been studied under laminar, turbulent,steady, or unsteady conditions. The early experiments of laminar-flowT-junction by Liepsch et al. (1982) using Laser Doppler Anemometer(LDA) discussed the effect of the Reynolds number and mass flow ratioon the velocity field, local shear stress and pressure drop. Meanwhile,they performed numerical simulations and a good agreement betweensimulation and experiment was found. Neary and Sotiropoulos (1996)numerically carried out calculations for various Reynolds numbers,discharge ratios and duct aspect ratios for steady laminar flow through90° diversion of rectangular cross-section. The result showed that even

https://doi.org/10.1016/j.ijheatfluidflow.2018.07.008Received 3 September 2017; Received in revised form 1 June 2018; Accepted 19 July 2018

⁎ Corresponding author.E-mail address: [email protected] (M. Lin).

International Journal of Heat and Fluid Flow 73 (2018) 124–142

0142-727X/ © 2018 Elsevier Inc. All rights reserved.

T

for large aspect ratio ducts, the flow at the symmetry plane is sig-nificantly affected by the distant top and bottom solid boundaries.Neofytou et al. (2014) simulated the shear-thinning and shear-thick-ening effects of laminar flow in a T-junction of rectangular ducts. Un-steady inlet condition is of interest for both the industrial and bioen-gineering applications. Anagnostopoulos and Mathioulakis (2004)numerically studied the pulsating flow field in a square T-junction ductwith equal branch flow rate. The flow can sustain much higher adversepressure gradients during acceleration before separating, compared tothe steady inlet condition. Afterward Miranda et al. (2008) carried outsteady and unsteady laminar flows in a two-dimensional (2D) T-junc-tion for Newtonian and non-Newtonian fluids. Matos andOliveira (2013) applied a generalized Newtonian fluid model in planar2D T-junction tubes having a dividing or bifurcating flow arrangement(one main channel with a side branch at 90°), and also analyzed thecompeting effects of inertia, shear thinning and extraction ratio for non-Newtonian inelastic flows. Chen et al. (2015) conducted a global linearsensitivity analysis of a complex flow through a pipe T-junction, fo-cusing on near the first Hopf bifurcation flow with Reynolds number at560.

Many other studies have also involved T-junction turbulent flows.Sierra-Espinosa et al. (2000a, 2000b) experimentally and numericallyinvestigated the turbulence structure of a water flow in the branch exitof a tee pipe junction, in which the Reynolds number was 1.26× 105.Three turbulent models of the standard k-ε, renormalization group(RNG k-ε), and Reynolds stress model (RSM), were used. A disagree-ment between predictions and measurements in the reattachment re-gion was observed. Also, the predictions did not reproduce the de-tachment region of the reverse flow as observed in the experiment.Costa et al. (2006) investigated the edge effects on the T-junction flow,and measured the pressure drop of a Newtonian fluid flow with sharpand round corners. It was found that the rounding corners reduce theenergy losses whereas the straight flow basically remains unaffected.Beneš et al. (2013) dealt with the numerical solution of laminar andturbulent flows of Newtonian and non-Newtonian fluids on branchchannels with two outlets. The channels considered were constantsquare or circular cross-sections. Some other mathematical models of

tube junction were also reported by Oka et al. (2005) andStogler (2006). Numerical results showed that the explicit algebraicReynolds stress (EARSM) turbulence model was capable of capturingthe secondary flows in channels of rectangular cross-section.

A number of literatures can be found on T-junction flows because ofthe mixing flow or separating flow for multi-phase systems in chemicaland power engineering field (Georgiou et al., 2017; Wang and Lu, 2015;Sakowitz et al., 2014; Solehati et al., 2014; Poole et al., 2013; Elazharyand Soliman, 2012; Silva et al., al.,2010).

It should be noticed that the aforementioned studies did not discussthe multi-resolution turbulent coherent structures on T-junction diver-ging flows, which is closely related to high wall-friction drag (Gad-el-Hak, 2000). Understanding the coherent structures is the key for con-trolling turbulence drag reduction (Kravchenko et al., 1993).

In the present study, the flow structure in the cross tube T-junctionchannel is similar to the train surface flow structure. The time series ofthe streamwise velocity components are collected at three re-presentative streamwise locations, i.e., the upstream, the mid-center,and the downstream of the T-junction with one inlet and two outlets.Various flow conditions with different velocities and velocity ratios areexamined. The experimental measurements are validated with pub-lished data in the literature. The wavelet transform is used to analyzethe time-frequency characteristics of a cross tube T-junction flow, andthe energy cascade process of the coherent structures is discussed fordifferent scales. These results will guide us to design an optimal ven-tilation system for high-speed trains.

2. Theory of wavelet analysis

Wavelet transform is a more accurate time-frequency domain ana-lytical method as compared to the Fourier transform. The Fourieranalysis transforms the signal into the frequency domain in the wholetime domain. Thus, the frequency characteristic of the signal could befully revealed whereas the time characteristic could not be found.Therefore, the Fourier transform is sufficient for analyzing stationarysignals. For non-stationary signals, however, it is difficult to extract thesignal feature through a Fourier transform. An improved Fourier

Fig. 1. Sketch of a ventilation system.

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analysis, named as short-time Fourier transform, was developed.However, its disadvantage is that once the window function is selected,the time width and frequency width for the window are also fixed. Theywill not change as the time and/or frequency change (Farge, 1992).

In contrast to the Fourier analysis, the wavelet analysis, which caneffectively analyze the local feature of a signal, is a very useful time-frequency analytical tool for non-stationary signals (Gupta et al., 2000;Gurley et al., 2003; Tomac et al., 2017; Wickersham et al., 2014; Zhenget al., 2017). It decomposes the signal into different scale components.With selection of an appropriate mother wavelet function which can betranslated and scaled, the signal can be analyzed at different frequencyranges and different time instants. Thus, a good time resolution butpoor frequency resolution can be obtained in the high-frequency signal;better frequency and temporal resolutions are obtained in the low-fre-quency signal. It overcomes the drawbacks of Fourier transform.Therefore, the wavelet transform has often been praised as a “mathe-matical microscope” (Hubbard, 1998), because it can implement multi-resolution analysis through the dilation and translation operations.

The continuous wavelet transform (CWT) turns an original signal f

(t) into a function with two variables: scale dilation and time, which arecalled the wavelet transform coefficients, WT(a, b). The one-dimen-sional CWT of f(t) is then defined as:

∫=WT a b f t ψ t dt( , ) ( ) * ( )R a b, (1)

= ⎛⎝

− ⎞⎠

ψ ta

ψ t ba

( ) 1a b, (2)

where Ψa,b(t) is the wavelet function at the scale dilation a (a∈R+) andtranslation b (b∈R) of the mother wavelet functionΨ(t). The asterisk (*)represents the complex conjugate and R denotes real number. f(t) andΨa,b(t) both belong to L2(R), which indicates the Hilbert space mea-surable, square-integrable one-dimensional function. Inserting Eq. (2)into Eq. (1), it follows that

∫= ⎛⎝

− ⎞⎠

WT a ba

f t ψ t ba

dt( , ) 1 ( ) *R (3)

The space-scale energy of a given signal in the wavelet energyspectrum (Farge, 1992; Rioul et al., 1992; Jiang et al., 2011), may be

Fig. 2. Schematic diagram of the experimental measurement.

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computed in Fourier space and is defined as

=E WT a b( , )c2 (4)

The wavelet spectrum is a 2D image or contour plot in which theabscissa represents the translation (time) and the ordinate representsthe scale or frequency.

The relationship between a given wavelet scale a and frequency fa isdescribed as (Daubechies, 1992):

=ff f

a.

ac s

(5)

where fa represents the corresponding frequency of scale a, fc is thecentral frequency of the wavelet, and fs is the signal sampling rate.According to the relationship in Eq. (5), the wavelet energy is expressedas a function of time and frequency. High frequency represents smallscale and vice versa.

According to Nyquist sampling principle, the actual frequency of thesignal should be limited to fs/2. Then the range of the scale frequency,fa, is from 0 to fs/2. The theoretical range of scale (a) should be from 2fcto infinite. However, the continuous wavelet transform is defined byusing the circular-convolution theorem, which is used the integrallimits (-∞, +∞). Since the length of the measured signals are finite,errors of the calculated CWTs occur due to the edge-effect (start andend-time). The cone of influence (COI) is defined as the width of thewavelet spectrum in which the edge-effect is significant. The effectiveregion of wavelet function is expressed at the edge-effect time wherethe wavelet power drops to a factor of e−2 (Torrence andCompo, 1998). For Morlet wavelet, the effective region is chosen at |b-2a, b+2a|. For 2 K length of signal series, the largest scale magnitudeshould be 2 K/4. Therefore, the ideal range of scale is a∈(2fc, 2 K/4).Since different mother wavelet functions correspond different centralfrequencies and different COI, the corresponding scale frequency andthe down and up limits of the scale should be changed for differentmother wavelet functions.

In practical applications, since the signal is often obtained by thediscrete sampling data, the discrete wavelet transform (DWT) can beexpressed as

∫=DT m n f t ψ t dt( , ) ( ) * ( )R m n, (6)

⎜ ⎟= ⎛⎝

− ⎞⎠

= −− − −ψ t a ψ t nb aa

a ψ a t nb( ) ( )m nm m

mm m

, 0 20 0

00 2 0 0

(7)

where m, n∈Z, a0>1, b0>0. a0 and b0 are the scale and time transformof wavelet function in first-level wavelet transform. If a0=2, b0=1,then Eq. (7) becomes

= −− −ψ t ψ t n( ) 2 (2 )m nm m

, 2 (8)

where a and b in Eq. (2) are replaced, respectively, by 2m and n2m, inwhich m and n are the dyadic dilation and translation parameters.

In the next step, the discrete wavelet basis may be convolved withthe time-series signal f(t), which has 2K data points. The wavelet detailcoefficients DT1m,n can be described as (Mallat, 1989):

∑==

−

DT f t ψ t1 ( ) * ( )m nn

m n,1

2

,

K m

(9)

The detail signal of f(t) can be expressed in the following form:

∑ ∑== =

−

f t DT ψ t( ) 1 ( )m

K

nm n m n

0 1

2

, ,

K m

(10)

The wavelet approximation coefficients (DT2c,n) have a scalingfunction, φc,n(t), that also produces an orthogonal basis by mean ofdilation and translation parameters:

= −− −φ t φ t n( ) 2 (2 )c nc c

, 2 (11)

∑==

−

DT f t φ t2 ( ) * ( )c nn

c n,1

2

( , )

K c

(12)

The approximation signal of f(t) at the mth resolution is:

∑==

−

f t DT φ t( ) 2 ( )mn

c n c nth1

2

, ,

K m

(13)

where all the sets ψm,n and φc,n are mutually orthogonal. So the waveletexpansion of signal f(t), can be describes as:

∑ ∑ ∑= += = =

− −

f t DT φ t DT ψ t( ) 2 ( ) 1 ( )n

c n c nm

n

nm n m n

1

2

, ,0 1

2

, ,

K m K m

(14)

where for 2K data points, the indexes should satisfy the inequality:1≤m≤c≤K and 1≤n≤2K-m. c is the index of the last decompositionlevel.

In the present paper, the detail energy E(m) in the sense of timetranslation n and for certain scale m can be defined as:

∑==

E m DT( ) 1in

m n1

2

,2

K

(15)

The total energy for all the scales can be summed up by:

∑==

E E m( )i

m

i1 (16)

The energy contribution ratio, E(m)/E, represents the percentage ofthe energy of each scale to the total energy. Note, the approximationcoefficient is ignored in the energy calculation process due to less en-ergy contribution.

The essence of signal decomposition based on the discrete wavelettransform is the cross-correlation between the signal and the filter bank,and the signal reconstruction is the convolution of the decompositionsignal and the image filter bank. In the discrete process, for orthogonalwavelet families, the DWT algorithm follows a sub-band coding schemewhich is based on the multiresolution analysis (Mallat, 1989). Theprocedure of multiresolution decomposition of an original signal (f(t))is shown in Fig. 3. At each step, two digital filters (a high-pass filter, H,and a low-pass filter, L) are implemented to decompose the originalsignal by down-sampling of a factor of 2 (↓2) (Sabouri et al., 2017; Zhanand Yao, 2017; Kumar et al., 2015). The symbol ↓ denotes the down-sampling process. The high and low frequency components for an ori-ginal signal pass through the high-pass and low-pass filters, respec-tively. Consequently, the high-pass filter generates the detailed coeffi-cients while the low-pass filter generates the approximationcoefficients. The implementation of m-level decomposition for an

Fig. 3. The procedure of m-level decomposition foran original signal.

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original signal is illustrated in Fig. 3. Therefore, for a DWT with mscales, an original signal consists of a set of m+1 coefficients whichinclude m detailed coefficients for different scales and one approx-imation coefficient for the last level. These coefficients give informationabout the different features presented in the original signal. The ap-proximation coefficient represents the rough feature of the originalsignal, while the detailed coefficient represents the detail feature. Thus,the original signal can be perfectly reconstructed by using these waveletcoefficients (Galliana-Merino et al., 2014; Mallat, 2009).

3. Experimental setup

The experiment was performed in a low-speed T-shape wind tunnelwith a maximum velocity of 57m/s, as shown in Fig. 2a. The cross tubewith a 143.3×161.7 mm2 rectangular cross-section is 36.5D long, 22Din front of the T-junction and 14.5D behind it. The length of the branchtube with a square cross-section is 14.2D and the side length is D =110mm. To ensure that a fully turbulent boundary layer is present inthe T-junction entrance, a 2.0mm rod is affixed to the tunnel side at thecross tube entrance. The test section is shown inside the dot-line rec-tangle, with details depicted in Fig. 2b, where the wall-normal height isw = 143.3 mm.

The origin of the coordinates is located at the center of the T-junction entrance. The velocity components are measured at threedifferent streamwise locations, i.e., x/D = −1, 0 and 1, of the sym-metric centerline (z = 0), where x is the streamwise coordinate, asshown in Fig. 3b. Eight test points are set at every location, and theyvary from y/w = 0.007 to 0.5. A special point is measured to monitorthe cross velocity at x/D=−2. The velocity components of u and w aremeasured simultaneously by using an X-type hot wire probe (TSI1240–20) in the (x, z) plane. The data sampling rate is 50 kHz with asample time of 40.6 s. So there are 2,097,152 sampled data for eachdynamic signal. It is well known that a complex three-dimensional flowfield exists in a branch tube where recirculation and secondary flowsare predominant. In our experiment, however, the concern is the tur-bulent coherent structures in the cross tube. According to the experi-mental data acquired by using the laser-doppler anemometry (seeCosta et al., 2006) and the numerical results by Beneš et al. (2013),there is no backflow in the cross tube. Therefore, the hot-wire anemo-metry is suitable to collect experimental data in the cross tube becauseof the high-response frequency and low velocity ratio (R<0.2), corre-sponding to the flow rate ratio less than 0.1.

The uncertainty of the hot-wire-measured mean cross velocityis± 1.0%, and the uncertainty of the branch bulk velocity using glassrotameter is± 4.3%.

The cross velocities considered in the present study are 30, 40 and50m/s, respectively. And the velocity ratios are 0.08, 0.13 and 0.18,respectively. The experimental conditions are illustrated in Table 1.

Before operating the experiment, the accuracy of our experimentalmeasurements has been validated via comparisons with the publishedresults measured by Gessner et al. (1979). Details have been previouslyreported in Yin et al. (2015). The comparison results at x/D=−5 isdisplayed in Fig. 4a. A reliable agreement between our results and thosein the referred literature is found.

4. Results and discussion

4.1. Time-average velocity distribution

To ensure a fully developed flow in the upstream of the T-junction, apre-measurement is performed at the location, x/D=−5. The frictionvelocities are found to be uτ=1.159 and 1.868m/s for uc= 30 and50m/s, respectively, which are obtained by fitting the mean velocityprofile with the empirical logarithmic law (u+= (1/k)ln(y+)+ B,k=0.41 and B=5.5). On the basis of the value, the correspondingfriction Reynolds numbers are estimated as Reτ= uτ(w/2)/ν=5427and 8747, respectively. The Reynolds number in the cross tube is de-fined as Rec= uaaw/ν, where, uaa= (w/2)∫ u y dy( )w

0/2

a , ua(y) is thetime-average velocity and ν is the kinematic viscosity. To compare thefriction velocity with the published value, another uτ value is obtainedaccording to the skin-friction coefficient (cf) extracted from Schultz andFlack (2013) given as:

⎜ ⎟= = ⎛⎝

⎞⎠

c τρu

uu0.5

2 τf

w

aa2

aa

2

(17)

where τw is the wall-shear stress and ρ is the fluid density.The comparison between the present result and the value of

Schultz and Flack (2013) is found to be within −0.7% for uc= 30m/s,in which the corresponding Reynolds number is Rec= 2.64×105. Thisis a good agreement. While for uc= 50m/s, the corresponding Rey-nolds number in the present study is Rec= 4.4×105; its skin-frictioncoefficient could not be extracted from Schultz and Flack (2013)

Table 1Experimental conditions (velocity unit: ms−1).

Case# uc ub R

1 30 3.9 0.132 40 3.2 0.083 40 5.2 0.134 40 7.2 0.185 50 6.5 0.13

Fig. 4. Comparison of our experimental data and the results in literature.

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because the restricted Reynolds number is up to Rec= 3.0× 105.The mean velocity profiles at x/D=−5 are plotted in the inner

scaling in Fig. 4b, where u+= ua/uτ, y+= yuτ/ν. It also shows thecomparison between our results and the experimental data obtained bySchultz and Flack (2013). The good agreement confirms that the flow inthe present study has become fully developed state over the region ofthe cross tube before the airflow enters the T-junction.

The effect of spatial resolution when measuring turbulence is a well-known source of experimental errors when using hot-wire anemometry(Ashol et al., 2012; Marusic et al., 2010; Hutchins et al., 2009). Thesensing element of the hot wire has a viscous scaled length of l+= l uτ/ν=76 and 122 for Reτ=5400 and 8700, respectively. To determinethe missing energy from the spatial filtering experienced on thestreamwise turbulence, Smits et al. (2011) proposed the attenuation ofthe streamwise Reynolds stress due to spatial filtering in wall-boundaryflows as described by:

=+

++ +u

uM l f yΔ ( ) ( )

2

2m (18)

where +uΔ 2 and +u m2 are the missing and measured streamwise Reynolds

stresses, respectively.The function M(l+) and f(y+) are taken the forms:

= −+ +M l l( ) 0.0091 0.069 (19)

= ++ +

++ − +f y

y e( ) 15 ln(2)

ln[ 1]y(15 ) (20)

Table 2 gives the attenuation of the streamwise Reynolds stress forthe wall-normal distance and probe/sensor geometries for two frictionReynolds numbers. It is seen that the experimental errors are almostidentical at the same outer units for different Reynolds numbers andspatial filtering. The error is maximized only 13% at y/w=0.007, thensharply reduces to 6% at y/w=0.014. Therefore, it could be thoughtthat the error is systematic due to use of same hot-wire probe, and thisdoes not adversely affect the results near the T-junction.

A series of the normalized streamwise components of velocities (ua/uc) at the three selected locations (upstream, mid-center, and down-stream of T-junction) are shown in Figs. 5–7. Fig. 5 depicts the nor-malized velocity distributions for three cross velocities (uc= 30, 40 and50m/s) along the y-direction at a given velocity ratio R=0.13. It isseen from Fig. 5a that, for uc= 30m/s, the velocity at the center of theT-junction, x/D=0, is the largest and the velocity at the upstream withx/D=−1, is the smallest near the wall of the cross tube, y/w<0.08.At 0.08< y/w<0.22, the velocity at x/D=0 is still the largest, whilethe velocity at the downstream x/D=1 is the smallest. At the flowregion far away from the wall, y/w>0.22, the velocity gradually de-creases along the streamwise direction. Similar velocity distributions

can be found in Fig. 5b and c for uc= 40 and 50m/s, respectively. Thisphenomenon can be explained as follows: at the near-wall flow regionof y/w<0.08 for x/D=0, the airflow in the cross tube turns 90° to thebranch tube so that the velocity becomes the largest. As for the up-stream location of x/D=−1, the flow field is still in an approximatelyfully developed turbulence state due to the weak suction so that thevelocity near the wall is the smallest. As for the downstream location ofx/D=1, although the airflow rate decreases, the fluid dives into thewall to generate a higher velocity compared to that at upstream x/D=−1. In addition, at upstream of x/D=−1, the velocity (ua/uc)shows a local maximum at y/w=0.28, other than y/w=0.5 (center ofthe cross duct). This phenomenon illustrates that the weak suction hasan important effect at the location of y/w=0.5. It is because the

Table 2Attenuation of the streamwise Reynolds stress at x/D=−5.

Reτ=5400, l+=76 Reτ=8700, l+=122

y/w y+ + +u uΔ / m2 2 y+ + +u uΔ / m2 2

0.007 76 0.127 122 0.1330.014 152 0.064 244 0.0660.021 228 0.042 367 0.0440.035 379 0.026 611 0.0270.056 607 0.016 978 0.0170.077 835 0.012 1345 0.0120.105 1138 0.008 1834 0.0090.14 1518 0.006 2446 0.0060.209 2276 0.004 3669 0.0040.279 3035 0.003 4892 0.0030.349 3794 0.002 6115 0.0020.419 4553 0.002 7338 0.0020.500 5437 0.002 8762 0.002

Fig. 5. Normalized streamwise velocity profiles at three streamwise locationswith R=0.13 for: (a) uc= 30m/s, (b) uc= 40m/s, and (c) uc= 50m/s.

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airflow at y/w=0.5 can be easily turned to the branch tube due to thecentripetal force resulted from the large curvature.

Fig. 6 compares the normalized velocity distributions for differentvelocities at different locations at a given velocity ratio, R=0.13. It isseen from Fig. 6a that the velocity distributions for the three crossvelocities are almost identical at the upstream of the T-junction, x/D=−1 due to the weak influence of suction and the fully developedcondition. The velocity near the wall region (y/w<0.15) slightly in-creases with the increase of the cross velocity for locations at the mid-center and downstream, x/D=0 and 1, respectively, as shown inFig. 6b and c. When the cross velocity increases, the entrance velocity in

the T-junction also increases under constant given velocity ratio.Therefore, the velocity near the wall region increases. In addition, theexternal flow velocity beyond the near wall (y/w>0.15) is almost thesame for different cross velocities. The reason is that, for a velocityratio, R=0.13, which corresponds to a flow rate ratio of 0.07, the ef-fect of suction is so weak that the external cross velocity could not beaffected.

Fig. 7 shows the normalized velocity distribution for different ve-locity ratios at a given velocity, uc= 40m/s. The considered velocityratio is R=0.08, 0.13 and 0.18, respectively. At x/D=−1, the velo-city profiles are almost the same for the three velocity ratios because ofweak suction, as shown in Fig. 7a. A remarkable change is observed thatthe velocity at x/D=0 in Fig. 7b increases with the increase of velocity

Fig. 6. Normalized streamwise velocity profiles for different cross velocitieswith R=0.13 at: (a) x/D=−1, (b) x/D=0, and (c) x/D=1.

Fig. 7. Normalized streamwise velocity profiles for different velocity ratioswith uc= 40m/s at: (a) x/D=−1, (b) x/D=0, and (c) x/D=1.

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ratio when y/w<0.25, whereas the velocity decreases with the in-crease of velocity ratio in the region y/w>0.25. Under the condition ofa given cross velocity, the air flow rate is constant at the upstream ofthe T-junction. The flow rate of the branch tube increases with the in-crease of velocity ratio. So the near-wall flow velocity of the cross tubeincreases, too. At x/D=1, the velocity profiles are separated into twodifferent regions. The velocity is almost identical for the three velocityratios in the region y/w<0.08, whereas it decreases with the increaseof velocity ratio in the region y/w>0.08. Since part of the airflow inthe cross tube bypasses to the branch tube, the rest of the airflow will beredistributed at the downstream of the T-junction. Therefore, the ve-locity profiles near the wall are almost the same due to both the weak

suction and the effect of wall friction at y/w<0.08. The decreases ofthe velocity in the external flow region can be attributed to the strongshunting of the T-junction at high velocity ratio at y/w>0.08.

4.2. CWT analysis

The wavelet transform essentially identifies matching between thefunction f(t) and the wavelet ψ(t) (Walker, 1999). The selection of themother wavelet is important to accurately extract the signal char-acteristics of interest. It is difficult to choose the best-fit wavelet basisfor flow pattern analysis from many wavelet families. Two typicalwavelets are employed in the turbulence area of fluid mechanics, which

Fig. 8. Fluctuating velocity and wavelet energy spectrum for uc= 30m/s, R=0.13, y/w=0.007, and x/D=−1.

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are Mexican hat and the Morlet wavelet (Farge, 1992; Farge et al.,1996). The Mexican hat wavelet is a real-valued wavelet and is com-monly used to detect the peaks in the signal, which has a relatively hightemporal resolution with a lower frequency resolution; whereas theMorlet wavelet is a complex-valued wavelet that can provide not onlyunique phase information between signal components in two ortho-gonal spaces but also amplitude information similar to real wavelettransforms.

Compared with the Mexican hat wavelet, the Morlet wavelet has agood local balance in both the time and frequency domains. The Morletwavelet modulated by the Gaussian envelope of unit width in a planewave of wavevector, is mathematically defined as:

= − −φ t π e e( ) iω t t1/4 /20 2(21)

where ω0 is the non-dimensional frequency and here taken to beω0= 5, which can approximately satisfy the condition of admissibility(Farge, 1992).

In fact, the turbulent flow is not a complete random motion, but anexistence of the quasi-periodic coherent turbulent structure (Camussiet al., 1997; Kambea et al., 2000), which is recognizable and orderedlarge scale motions in shear turbulence (Kline et al., 1967; Kim et al.,1971; Hussain, 1986). The finding of the coherent structures is of im-portance in turbulence studies, which indicates that the large scaleorganized structures exist and be responsible for the transport of mass,momentum, heat and energy. Furthermore, the Morlet wavelet is also

Fig. 9. Fluctuating velocity and wavelet energy spectrum for uc= 50m/s, R=0.13, y/w=0.007, and x/D=−1.

Fig. 10. Fluctuating velocity and wavelet energy spectrum at x/D=0, R=0.13, and y/w=0.007.

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locally periodic, and it is widely applied to recognize the turbulentstructure in turbulence analysis (Baars et al., 2015; Bulusu et al., 2015;Farge et al., al.,1990; Kanani et al., 2015; Ruppert et al., 2005; Togeet al., 2017). Therefore, the Morlet wavelet is selected as the motherwavelet for the continuous wavelet transform in the current analysis.Actually, the wavelet transform of the fluctuating velocity signal isequivalent to the cross-correlation analysis of the signal and the waveletfunction at a certain scale. The wavelet coefficient represents the degreeof correlation between the signal and the wavelet function in this scale.And a high correlation indicates that wavelet function and signal havevery good similarities in this scale (Farge, 1992).

The wavelet-based results are generated by self-made code inMATLAB. The self-made code is validated by the results from theMATLAB toolbox. The deviation of results between our code and theMATLAB toolbox is less than 2.0% in 16,384 data points analyzed.

Figs. 8–12 show the time-frequency characteristics of the stream-wise fluctuating component (u´= u-ua). 32,768 data points were takenfor obtaining the wavelet spectral statistics in the time-series length.Such a time length of data could be sufficient for analyzing the energyspectra. Only a small fraction of data was adopted in some previousstudies (Balakumar et al., 2007; Baars et al., 2015; Thacker et al.,2010). In the plots the abscissa is time and the ordinate is the fluctu-ating velocity or the corresponding frequency of scale. The colorationrepresents the wavelet energy spectrum as defined in Eq. (4). The darkcolor in the spectrogram represents a small magnitude of the waveletenergy, while the red color indicates a larger wavelet energy, whichshows an intense turbulent event of the signal at the certain frequencyand at the given point in time. In addition, the dark region and the redregion alternately appear during a period of time. The wider is the

spectrogram band along the time axis (abscissa), the larger is the scaleof the coherent structure.

It is important to select an appropriate scale dilation a from thetheoretical range of scale a∈(2fc, 2 K/4) to perform continuous wavelettransform. Large bandwidths decrease the time resolution but increasethe frequency resolution and vice versa. On the other hand, the errors ofCWT will occur at the start and the end of the transformed signal usingthe circular-convolution theorem (Torrence et al., 1998; Hong et al.,2002; Simonovski et al., 2003). The width of the edge-effect is definedby the radius of trust:

= = =R kσ kaσ k aσ( /2)ta b t, (22)

where k represents the multiple of the time spread, σta,b is the timespread of the wavelet function. In this study, k = 4 is chosen for theMorlet wavelet (σ =1) (Boltezar and Slavic, 2004; Meyers et al., 1993).

Fig. 8 shows the wavelet energy spectrum at the first measuredpoint, y/w=0.007, under different values of dilation a for R=0.13,uc= 30m/s and x/D=−1, in which the frequency ranges of the di-lation a (ordinate) are 201–20,313 Hz, 12.7–203.1 Hz and 8.1–13.5 Hz,respectively. The smallest frequency bandwidth is 8.1 Hz and the lar-gest frequency bandwidth is 20,313 Hz. The COI is also indicated inFig. 8. False results are generated at the start time (green line) and atthe end time (red line) for each scale due to edge-effect. For the highfrequency range of (201–20,313 Hz) in Fig. 8a, it is found that no small-scale coherent structures present at high frequency bandwidths(fa≥ 495 Hz). In addition, the wavelet energy value is very small andthe largest value is only 780 m2.s−2. The value of COI is so small thatthe CWTs are reliable.

From the moderate frequency range in Fig. 8b, the universality

Fig. 11. Fluctuating velocity and wavelet energy spectrum at x/D=1, R=0.13, and y/w=0.007.

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coherent structures of turbulence were found. At the low frequencyregion (ordinate), about fa= 15.6 Hz, the black and bright streaks arewider and their number is fewer, which represents the existence of largescale structures. As the frequency increases (fa≥ 50.8 Hz), corre-sponding to the small scale structure, the black and bright streaks be-come narrow and their number increases, and the furcation of thesestreaks is obvious. This phenomenon reveals the energy cascade evo-lution of large-scale eddies splitting to small-scale eddies and inter-mittent structures. The largest wavelet energy value with the dominantfrequency of 15.6 Hz is more than 5000 m2.s−2 from 0.48 to 0.52 s,which demonstrates an intermittent event at the scale and the timemoment.

Observing the low frequency range in Fig. 8c, it is found that thewavelet energy spectrum widely distributes at the time frequencybandwidth of (8.1–13.5 Hz). As expected, the large CWTs is fault be-cause of locating in the region of COI. For the middle region in time-scale plane, no high energy coherent structures present. A better wa-velet analysis result is obtained at the frequency bandwidth12.7–203.1 Hz as shown in Fig. 8b, where the high and low frequencycoherent structures can be clearly seen. The size of the COI is appro-priate and, the difference between the noise and the harmonic com-ponent can be distinguished. For the next wavelet analysis, the band-width 12.7–203.1 Hz is selected as a reference bandwidth to illuminatethe time-frequency characteristics of the flow in T-junction.

Fig. 12. Fluctuating velocity and wavelet energy spectrum for uc= 40m/s, y/w=0.007, and x/D=0.

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Fig. 9 shows the wavelet spectrogram at the upstream of the T-junction (x/D=−1) with a given velocity ratio R=0.13 foruc= 50m/s and y/w=0.007. Compared with Fig. 8b, with an increaseof the cross velocity, the large-scale structures extend and the corre-sponding dominant frequency changes gradually from 15.6 Hz to12.7 Hz. It is seen that the dominant frequency near 12.7–15.6 Hz from0.32 to 0.44 s can be dedicated the large wavelet energy value of10,000 m2.s−2. This peak is twice that of uc= 30m/s.

Figs. 10 and 11 plot the fluctuating velocity signal and the waveletenergy spectrogram at the mid-center or downstream of the T-junction,respectively. Other parameters are R=0.13, y/w=0.007, uc= 30 and50m/s. It is seen that the turbulence large-scale structures at the mid-center evidently extend to higher frequency compared with that at theupstream, which shows the large energy structure contains moreabundant frequency components. For the low cross velocity (uc= 30m/s), shown in Fig. 10a, the dominant frequency with large energystructure ranges from 15.6 to 29.0 Hz at the time of 0.24–0.40 s, and thelargest wavelet energy value is 8600 m2.s−2. While it is also found thatthe frequency of the large energy structure extends to low frequencycomponents beside of high frequency components for high velocity(uc= 50m/s), as shown in Fig. 10b. The dominant frequency range ofthe large structure is from 12.7 to 20.3 Hz at time of 0.24–0.36 s, whereit has a larger wavelet energy of 10,000 m2.s−2. In addition, a largenumber of multi-scale coherent structures with high frequency of50.8 Hz are widely spread from 0.20 to 0.44 s in the time-frequencyplane. The other coherent structures with 20.3–29 Hz are also peri-odically presented at time of 0.14–0.20 s. The event at the time of 0.15 sis the sole occurrence of such an event for the entire duration of thevelocity signal, which contains high energy at lower frequency of about10 Hz, whose figure is not shown here due to identical ordinate.

From Fig. 11 in the downstream it can be seen that the waveletenergy value obviously decreases compared with that at the mid-centerbecause of existence of turbulence flow near the wall. The waveletenergy increases with the increase of velocity. The largest wavelet en-ergy for uc= 50m/s is 2.5 times greater than that for uc= 30m/s. Thedominant frequency of large energy structure for uc= 50m/s isfa= 50.8 Hz, which is much higher than 20.3–29.0 Hz for uc= 30m/s.As observed carefully from Fig. 11a, two clearly periodic coherentstructure series exist at the frequency of 24 Hz at time of 0.08–0.32 sand the frequency of 44 Hz at time of 0.52–0.58 s, respectively. It in-dicates the turbulence contains many medium-scale structures. Incontrast to Fig. 11b for high cross velocity (uc= 50m/s), the number ofhigh energy coherent structure with low frequency (12.7 Hz) re-markably reduces for short duration time (0.16–0.20 s). Many highenergy coherent structures with higher frequency (50.8–115 Hz) peri-odically occur in the time-frequency plane for long duration time(0.04–0.44 s). Another dominant frequency with 15.6–20.3 Hz is spreadat the time of 0.34–0.54 s. This indicates that the turbulence hasevolved into higher frequency structure and contains much more en-ergy of small-scale structures.

The influence of velocity ratio on the wavelet energy and frequencyis displayed in Fig. 12 for uc= 40m/s and x/D=0. From Fig. 12a atthe low velocity ratio R=0.08, it is seen that the high energy coherentstructure with a single frequency of 14.3 Hz exists quasi-periodically inthe entire time series. Here, the high energy structure is distinguishedby more than 0.33 of the maximum energy value. The high energycoherent structure (bright color) in the spectrogram is present att=0.12, 0.15, 0.19, 0.22, 0.25, 0.29, 0.42, 0.46 and 0.49 s, respec-tively. Moreover, the maximum magnitude of the wavelet energy isEc= 45,000 m2/s2. From 0.24 to 0.32 s and from 0.48 to 0.52 s, re-spectively, there is a clear frequency shift from 14.5 to 29.0 Hz, whichindicates splitting of the large-scale structures into small-scale struc-tures. With the velocity ratio increasing to R=0.13, as shown inFig. 12b, the high energy coherent structures with 13.3 Hz are presentat t=0.16, 0.20, 0.24, 0.28, 0.32 and 0.36 s, respectively. The periodicfeature of the high energy coherent structures is quite evident. Theenergy peak decreases to 20,000 m2/s2. Meanwhile, another dominantfrequency of 65 Hz appears vaguely at time of t=0.16–0.18 s and0.40–0.42 s, respectively. Compared with the low velocity ratio(R=0.08), it is found that the turbulence field contains larger scale(lower frequency) and smaller scale (higher frequency) structures forR=0.13, which indicates a more complicated energy cascade struc-ture. As the velocity ratio further increases to 0.18, as shown inFig. 12c, the high energy coherent structures with different frequencieswidely spread in the time domain. And the energy peak further de-creases to 10,400 m2/s2. Inspecting carefully, these coherent structuresat 12.7 Hz are present at t=0.16, 0.20, 0.24, 0.28, 0.36, 0.40, 0.44 and0.48 s, respectively. Their energy ranges from 3500 to 7000 m2/s2. Inaddition, the energy peak is located at fa= 20.3 Hz and t=0.20 s,where the energy magnitude is about 10,400 m2/s2. The other highenergy coherent structure for (20.3–29 Hz) occurs at t=0.08–0.24 sand t=0.48–0.56 s, respectively. At the time of 0.44–0.48 s, the highenergy structures with about 5000 m2/s2 has a frequency of 36.9 Hz. Asa result, the dominant frequency of the high energy structure shiftsfrom low frequency to high frequency, that is, energy transfer fromlarge-scale structures to small-scale structures. Thus, with the increaseof the velocity ratio, the number and the magnitude of large-scale high-energy coherent structures decrease, but the number of small-scalecoherent structures increase. This suggests that the high velocity ratio ishelpful in suppressing the generation of coherent structures and ac-celerate the energy cascade. It is beneficial to reduce the drag and skinfriction for high-speed train vents. The mechanism is that the crossvelocity in the T-junction tube decreases with the increase of the ve-locity ratio, and the low cross velocity and recirculation will create ahigh turbulent motion in the T-junction entrance, which could cause theincrease of large-scale (low-frequency) fluctuating energy. The bursting

Fig. 13. Root mean square for uc= 40m/s, y/w=0.007, and x/D=0.

Table 3Frequency range for different scales (m=16, fs= 50 kHz).

Scale level (m) Frequency (Hz) Central frequency (Hz)

1 50,000–25,000 37,5002 25,000–12,500 18,7503 12,500–6250 9375… … …7 781–390 5868 390–195 2929 195–97.5 14610 97.5–48 7311 48–24 36… … …16 1.5–0.75 1.1

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events will increase for low velocity ratio (see Fig. 12a). The maximumfluctuating velocity value is −15m/s at t=0.16 s, −10m/s att=0.06 s, and −7m/s at t=0.18 s for R=0.08, 0.13 and 0.18, re-spectively. Furthermore, the above mechanism can be explained as theRMS (root mean square) at the same measurement point decreases withthe increase of velocity ratio, as shown in Fig. 13.

4.3. DWT analysis

CWT analysis mainly reveals the energy cascade stage of large-scalecoherent structures changing to small-scale coherent structures at thescale-time plane, in which the turbulent kinetic energy of differentscales could not be rapidly calculated for velocity fluctuation because ofhighly redundant coefficients (Farge, 1992; Burrus et al., 1998). In-stead, DWT based on multiresolution analysis (Mallat, 1989) can be fastdecomposed the original velocity signals to get the DWT coefficient of

different scales due to less computation. In this study “Daubechies 4”wavelet is selected as the mother wavelet (Kulkarni et al., 2001; Liet al., 2007) and a 16-level DWT is employed to generate the detailcoefficients (DT11, DT12…DT116) and the approximation coefficient(DT216). The frequency range of each scale for a case in which m=16and fs= 50 kHz is illustrated in Table 3 by the Nyquist theorem. Thewhole time series signal of 40.6 s duration is discrete to obtain the DWTcoefficient using the MATLAB toolbox. Figs. 14–16 show the discretewavelet turbulent kinetic energy of different levels from velocity fluc-tuation decomposition and the energy contribution for the three loca-tions at x/D=−1, 0 and 1 under a given velocity ratio R=0.13. Theleft-hand figures illustrate the energy distributions (E(m)), and theright-hand figures show the corresponding energy contribution ratio (E(m)/E). The abscissa and the ordinate use a base 2 logarithmic valueand 10 logarithmic value, respectively. From the left-hand side figure ofFig. 14, it is evident that the energy decreases as y/w becomes larger at

Fig. 14. DWT energy and energy contribution ratio for x/D=−1, R=0.13: (a) uc= 30m/s, (b) uc= 40m/s, and (c) uc= 50m/s.

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x/D=−1. It is separated into two regions by the wall-normal distancefrom the wall, internal flow region, y/w<0.28 and external flow re-gion, y/w≥ 0.28. Near the wall, y/w≤ 0.077, the energy for all thescales is approximately equal and symmetric along the scale axis.Moreover, one maximum appears at the level m=9 for uc= 30m/s(the left-hand side of Fig. 14a) and m=8 for uc= 50m/s (the left-handside of Fig. 14c), respectively. In the external region, y/w≥ 0.28, theenergy of each scale slightly decreases with the increase of the wall-normal distance. The energy curve is not smooth at the large scalebands (2m>210) especially for the low cross velocities. It indicates thatthe number and energy for the large-sale structure relatively increasesand that of the small-scale structures decreases in the process of energycascade. The maximum of the energy is located at the level m=7 in theexternal region. The corresponding energy contribution ratio shows the

profile of each scale energy to the total energy. It is observed that theenergy ratio of each scale is almost equal in the region near the wall (y/w≤ 0.14) for the three cross velocities. The energy ratio of each scaleremarkably shifts to the high frequency at the external region (y/w≥ 0.28), as shown in the right-hand side of Fig. 14. The second en-ergy peak at the large scale level (m=11) for the low cross velocitydisappears with the increase of the cross velocity. This could be ex-plained by the backflow of interface of T-junction to the external regionat x/D=−1. On the other hand, the effect of suction has an importantrole on the external region for the low cross velocity. Therefore, the twofactors interact together to result an increase of the large-scale energy.As a result, the energy contribution for the small scale increases andthat for the large scale decreases in wall-normal region. That is to say,with the increase of wall-normal distance, the energy contribution

Fig. 15. DWT energy and energy contribution ratio for x/D=0, R=0.13: (a) uc= 30m/s, (b) uc= 40m/s, (c) uc= 50m/s.

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transforms from large scale to small scale, and with the increase of thecross velocity, the frequency shift has been enhanced, e.g., the value offrequency interval distance (d) is increased from 0.7 at uc= 30m/s to1.2 at 50m/s.

Fig. 15 shows the DWT energy and the energy ratio distributions atx/D=0. At the mid-center of the T-junction, the large energy of dif-ferent scales concentrates mainly in the scale ranging from m=6 to 11.Furthermore, the energy of m=11 near the wall such as at y/w=0.007 and 0.021, has a sudden jump for the middle and high ve-locities of uc= 40 and 50m/s. In general, each scale energy close to thewall slightly decreases as the wall-normal distance from the wall in-creases, whereas in the external region it is nearly identical. The cor-responding energy contribution in the external region slightly shifts tothe high frequency compared with that at the upstream of the T-

junction. However, the energy contribution ratio close to the wall hastwo large values which located at the scale level m=9 and 11 for themiddle velocity, uc= 40m/s. As for uc= 50m/s, it rapidly reduces atm=11 in the internal region while suddenly rises in the external re-gion at the same scale (see the right-hand side of Fig. 15c). In conclu-sion, for free-boundary flow, the energy close to the wall has a complexdistribution which can achieve another peak at the large scale. Thesudden peak will decrease as the cross velocity increases.

The DWT energy and its contribution ratio of each scale at thedownstream of T-junction (x/D=1) are depicted in Fig. 16. The energydistributions of each scale gradually decrease as the wall-normal dis-tance increases. But the energy of each scale in the external region isstill the same for a given cross velocity. The maximum energy frequencyis relatively steady and its scale level is m=8 close to the wall. These

Fig. 16. DWT energy and energy contribution ratio for x/D=1, R=0.13: (a) uc= 30m/s, (b) uc= 40m/s, and (c) uc= 50m/s.

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trends continue for the external region (y/w≥ 0.28) in which the lowfrequency energy is unsteady, especially with the middle to high ve-locity, uc= 40 and 50m/s, as shown in the left-hand side in Fig. 16band c, respectively. The corresponding energy ratio also shows similartrends. The energy ratio of each scale close to the wall is almost iden-tical at a given velocity. The energy ratio of high frequency in the ex-ternal region is also identical at the scale level m≤ 7. However, it hasanother maximum when the scale level is m=11 for the middle andhigh velocities, uc= 40 and 50m/s (see the right-hand side in Fig. 16band c). This indicates that the large-scale coherent structure will beenhanced in the external region with the increase of cross velocity.Frequency shift does not occur at the downstream. The reason is thatthe airflow at the downstream will be redistributed in the channel sothat the energy of different scale will be redistributed. Observing the

wavelet spectrogram in Fig. 11, the different scale energy uniformlyspreads across the time-frequency plane.

The energy contribution ratios at two special points (y/w=0.007and 0.5) are shown in Figs. 17 and 18, respectively, for different ve-locity ratios and cross velocities. The left-hand figures depict threevelocity ratios at a given velocity uc= 40m/s. The right-hand figuresshow the three cross velocities at a given velocity ratio R=0.13. FromFig. 17a, at x/D=−1, it is readily apparent that the energy ratio ofeach scale is almost identical for the three velocity ratios, R=0.08,0.13 and 0.18. This indicates that the scale energy is proportional tovelocity ratio, whereas the energy ratio of each scale shifts to highfrequency as the cross velocity becomes larger. The maximum energyratio turns from scale level of 9 to 8, as shown in the right-hand side inFig. 17a. Meanwhile, the large scale energy contribution increases with

Fig. 17. DWT energy contribution ratio at y/w=0.007: (a) x/D=−1, (b) x/D=0, and (c) x/D=1.

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the increase of the cross velocities at 4≤m≤ 7, while at m>8, thelarger scale energy contribution decreases. This phenomenon for m≤ 8is consistent with the results observed by Marusic et al. (2010). For thesmallest scale (m<3), the energy ratio is almost identical for differentcross velocities. The same trend can be seen at very large scale m>13.This reveals that the energy ratio is proportional to the cross velocity inthe two special regions.

Fig. 17b shows the complex energy distributions at x/D=0, wherethe effect of velocity ratio on the energy ratio is obvious at the scalelevel range of 7≤m≤ 12. Two maximum energy ratios exist at m=7and 11, respectively. As the velocity ratio increases, the energy ratio atm=11 decreases whereas that at 7≤m≤ 10 increases. In accordancewith the previous discussion, at the low velocity ratio, the velocity inthe branch tube is too low to cause a large backflow region in the entry

interface of the T-junction. The backflow region area reduces as thevelocity ratio increases. The fluctuating flow of backflow region willremarkably affect the cross flow. In addition, for the effect of crossvelocities, the energy ratio of each scale still noticeably shifts to highfrequency as the cross velocity increases. Besides the scale level ofm=7, another maximum energy ratio at m=11 appears for the crossvelocity of 40m/s, then it decreases as the cross velocity increases to50m/s. The generation mechanism of the largest scale level of m=11may be related to the local Reynolds shear stress (Flores et al., 2006;Kim et al., 2016).

Fig. 17c exhibits the energy ratio distributions at downstream of theT-junction (x/D=1). The energy ratio is not completely coincident ascompared with that at x/D=−1 for different velocity ratios, and thetrends slightly shifts to high frequency as the velocity ratio increases.

Fig. 18. DWT energy contribution ratio at y/w=0.5: (a) x/D=−1, (b) x/D=0, and (c) x/D=1.

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The same phenomenon can be observed for the effect of cross velocitieswith respect to that at x/D=−1.

Fig. 18 shows the energy ratio distributions in the external region(y/w=0.5). Note, the energy ratio distributions on the upstreamcontrol point at (x/D=−2, y/w=0.5) is almost identical to that of x/D=−1 for different velocities and velocity ratios. First, as shown inthe left-hand side of Fig. 18, the energy ratio is almost equal as thevelocity ratio increases at the locations of x/D=−1 and 0, whereas atthe position of x/D=1, another maximum energy ratio appears atm=11 besides m=7 at the low velocity ratio of 0.08 and it decreasesas the velocity ratio increases. Then it tends to be equal when the ve-locity ratio increases from 0.13 to 0.18. In the right-hand side of Fig. 18,a common feature is found that the energy ratio for the scale level ofm<7 increases with the increase of cross velocities for all the threelocations. The energy ratio of each scale at the external region shifts tohigh frequency with the increase of the cross velocity. The difference isthat the energy ratio of large scale level of m=11 decreases with theincrease of cross velocity at the location of x/D=−1 in the fully de-veloped flow zone. But at x/D=0 and 1, the energy ratio of m=11increases with the increase of the cross velocity, where are the free-shear flow and developing flow zones, respectively.

The DWT-results show that, with an increasing in the cross velocity,the energy of the large-scale structures decreases and that of the small-scale structures increase. This reveals the energy transfer from low-frequency (large-scale) coherent structures to high-frequency (small-scale) coherent structures. As the velocity ratio increases, however, thisphenomenon disappears. On the other hand, a second peak of energythat appears at the large-scale level of m=11 contributes to the in-crease of the flow drag and skin friction at high-speed train surfaces.The second energy peak is undermined with increase of velocity ratio.Therefore, increasing velocity ratio can suppress the generation of thelarge-scale structures and enhance the energy cascade.

5. Conclusions

The time-frequency contents of fluctuating velocity measurementsin a T-junction are investigated using the CWT and the DWT.Experimental measurements are taken at the range of Reτ=5400–8700using the hot wire anemometer. The experimental method and data arevalidated with the published results in the literature. Three streamwiselocations, i.e., upstream, mid-center, and downstream of the T-junction,are examined, which represent the fully developed, free-shear, anddeveloping turbulence flow, respectively. Characteristics of multi-re-solution turbulent structures in the cross tube of the T-junction throughthe wavelet analysis are drafted in followings:

(1) The normalized time-average velocity in the region close to the wallis the largest at the mid-center of the T-junction and the smallest atthe upstream. It decreases along the streamwise direction in theexternal flow region. The normalized velocity at the upstream of theT-junction is approximately identical as the velocity ratio and crossvelocity increase due to the weak suction effect. For the other twolocations at the mid-center and downstream, however, the nor-malized velocity slightly changes with the cross velocity.

(2) Continuous wavelet spectrum provides abundant energy distribu-tion information about the time-frequency characteristics of theflow in the T-junction. From the CWT-results near the wall, it isfound that the high energy coherent structures are periodicallypresent at the time-frequency plane. The dominant frequency of thehigh energy structures shifts from the low frequency (large-scalestructures) to the high frequency (small-scale structures). The wa-velet energy magnitude of the high energy structures decreases withthe increase of velocity ratio at mid-center of T-junction, but in-creases with the increase of the cross velocity for three streamwiselocations. The findings near the wall (y/w=0.007) from the CWTanalysis are similar to those from the DWT analysis.

(3) Discrete wavelet turbulent kinetic energy is nearly equal for thenear wall region at the upstream of the T-junction. However, at themid-center of the T-junction, the energy of coherent structures doesnot exhibit any apparent peak between the scale of 26 and 210 andnon-uniformly reduces as the wall-normal distance increases. At thedownstream of the T-junction, the energy of coherent structuresdecreases more uniformly with the increase of wall-normal dis-tance.

The energy at the scale of 211 abruptly rises either in the externalregion or near the wall region for all the cross velocity conditionsstudied and this energy decreases with the increase of the velocity ratio,i.e., increasing the velocity ratio can enhance the energy cascade andsuppress the generation of large-scale coherent structures. Therefore, ahigher velocity ratio will reduce the drag force and skin friction of highspeed trains.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant No.51376145 and No.51236003).

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