A MULTI-SCALE APPROACH FOR THE ANALYSIS OF PROPER SAMPLED DATA SCALE IN HOT-WIRE EXPERIMENT OF SQUARE DUCT FLOW*

ZHANG Bin, WANG Tong, GU Chuan-gang

Key Laboratory for Power Machinery and Engineering, Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: sjtu2009@sjtu.edu.cn

DAI Zheng-yuan

Key Laboratory for Power Machinery and Engineering, Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China

Trane’s Asia-Pacific Research Center, Shanghai 200001, China

A MULTI-SCALE APPROACH FOR THE ANALYSIS OF PROPER SAMPLED DATA SCALE IN HOT-WIRE EXPERIMENT OF SQUARE DUCT FLOW*

ZHANG Bin, WANG Tong, GU Chuan-gang

Key Laboratory for Power Machinery and Engineering, Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: sjtu2009@sjtu.edu.cn

DAI Zheng-yuan

Key Laboratory for Power Machinery and Engineering, Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China

Trane’s Asia-Pacific Research Center, Shanghai 200001, China

Without rational criteria to determine the Proper Sampled Data Scale (PSDS), it would result in the expense of the too much unnecessary processing time and storage space in turbulent experiments. A novel approach for PSDS was established herein on the basis of turbulence theory and statistics. The specific procedure was given by using wavelet tools. A case study to prove the reliability and rationality of this approach was reported, where the sampled hot-wire data were from the experiment of square duct flow and turbulence kinetic energy was selected as the concerned turbulence parameter. It is shown that 220quantities of the sampled data are enough to analyze turbulence kinetic energy in the present experiment. The PSDSs of three turbulence parameters at different Reynolds numbers (Re=4.60×104, 7.68×104and 1.23×105) were studied. The results illustrate that the PSDSs increase with the increment of the Reynolds number and the order of concerned turbulence parameter.

PSDS, turbulence, statistics, wavelet, hot-wire, square duct flow The data size is often much greater than the minimal needed data size for the analysis of the concerned turbulence parameter in turbulence experiments, and it would result in the expense of the too much unnecessary processing time and storage space. If the data size can be set as the Proper Sampled Data Scale (PSDS) according to the experimental purpose, the efficiency of data storage and analysis could be greatly improved. Zhang et al.[3]pointed out that the

1. Introduction

To study turbulent flows,the measuring instruments are adopted to sample the turbulent data such as hot-wire, PIV, dynamic pressure transducer, PDA, etc.[1,2]. A great amount of turbulent data is sampled to collect sufficient turbulence information. PSDS is related to the sizes of the largest energy-containing eddies and smallest dissipation eddies. They qualitatively provided the range of PSDS, but it is not enough to determine it. Many researchers[4-7]focused on the effect of sampled data size on the mean velocity and turbulence intensity inPIV measurements. They found that post-processing results varied with the data size and a convergent distribution would be obtained with sufficient sampled data. Li et al.[8]found that the turbulence auto-variance and co-variance are more and more sensitive to the hot-wire data size with the increase of the mean velocity. But they did not discuss how to determine PSDS. Dai et al.[9]put forward a new reduction method of sampling size in turbulence experiments. However, it was a little lack of theoretical support to use the maximal turbulent energy density as the basis of data reduction algorithm. And it is still disputable that the scale of the maximal turbulence energy density could represent turbulent coherent structures[10,11].

In this work, the experiment of square duct flow was conducted to collect velocity data with the hot-wire sampling system. With the aid of the turbulence theory and statistical principles, a rational approach for PSDS is proposed and the specific procedure was given for wavelet analysis. And a case study is described to discuss the feasibility and rationality of the approach.

2. Experimental set-up

The experiment was conducted in an open loop wind tunnel which consists of an inflow section, a filter box, a converging nozzle, a test section, a controlling valve, a centrifugal air fan used to draw the ambient air into the facility. The data sampling system included a PC, data collecting instruments, velocity transducers and some sensors. And the schematic of the experimental loop is shown in Fig.1.

Fig.1 Diagram of the experimental loop

A 300 mm×300 mm×100 mm filter box enclosed the flow inlet. The speed of the centrifugal fan driven by an electric motor was controlled by an adjustable frequency AC controller. The test section was an 80 mm×80 mm×1000 mm straight square duct. The converging nozzle was placed in 600 mm upstream from the entrance of the test section in which the flow properties were not fully developed but developing. The duct continued up to 800 mm downstream before flowing into the bend duct. The turbulence sampling point was located at the central point of the geometric volume of the test section. A 1-D hot-wire probe (kulite) was adopted to sample instantaneous velocity signals. The sampling frequency was chosen as 100 kHz and the Reynolds number was 4.60×104, 7.68×104and 1.23×105.

3. Wavelet analysis

The Continuous Wavelet Transform[12](CWT) of a signal x( t) could be expressed as

where ψ(t) is the mother wavelet, and the other wavelets are its dilated and translated versions, aand bare the dilation parameter and translation parameter, respectively. In practice, the Discrete WT (DWT) coefficients are given as

4. A novel approach for PSDS

In this article, the novel approach can be described as follows.M

(1) Assume that the signal u( t) of 2 data size is sampled firstly.

(2) The initial value of the dividing parameter Kis set to be 0.

(3) As K=K+1, the original signalu( t) could be divided into 2Kdata segments (or named as “sub-signal”) of 2M−Ksize.

(4) Select one or more concerned turbulence parameters in the present experiment.

(5) Collect their turbulent characteristic information from the original signals and all of sub-signals respectively.

(6) By using the statistical methods, determine if the turbulent characteristic information of the original signal and all of sub-signals is in an accepted statistical scope (or “same” statistically).

(7) If the condition (6) is matched, it is shown that the sampled data of 2M−Ksize can offer theconcerned turbulent characteristic information. Repeat Steps (3)-(6) until the condition (6) is not met. In the end, PSDS could be chosen as 2M−K.

(8) If the condition (6) is not matched, it means that the concerned turbulent characteristic information should be given by more data size. After the turbulence data of more size (such as: 2M+1or more) are sampled, repeat Steps (1)-(8).

Based on statistical theory[15], the statistical laws of a random variable can be fully obtained by the probability density function. Main relations are described as follows:

whereu(t) is a random signal,P(u) the probabil ity density function,K(z)the characteristic function,E(un)then-th order of statistical moments. Equati on (3) leads to:

thare the significance level parameters and logical return values (“1” if the result is true, otherwise “0”) in theJ-andt- testing methods respectively, andhis the result of logical AND betweenhjbthand.

Fig.2 Original signalu(t) and its wavelet spectrum of the turbulent kinetic energy

5. Results and discussion

5.1Case study

Fig.3 Wavelet spectrum of the turbulence kinetic energy at different division levels

Table 1 Statistics for testing results at different division levels

Fig.4 Wavelet spectrum of the turbulent kinetic energy at different sampled data size

Fig.5 The statistical value of total turbulent kinetic energy vs. sampled data size

5.2The effect of pre-set parameters on PSDS

In view that the pre-set parameters (including wavelet function and two significance level parameters) stand for the accuracy requirement of the approach to some extent, the effect of these parameters on PSDS should be further discussed. The analysis data are given in Table 2. The results indicate that the PSDS iss not sensitive to the change in wavelet function. It is also demonstrated that the novel approach for the determination of PSDS has a goodstability and reliability in the choice of wavelet function. The change in the significance level parameterJphas slight effect on PSDS, but PSDS increases slowly with the decrease of the parameterJpin thet- testing method. It is noted that the testing result of normal distribution is still true when the testing result of confidence limit was false, and this is the case frequently. It meant that the influence ofTpon PSDS is greater than that ofJp.

Table 2 Effect of pre-set parameters on PSDS

Fig.6 Time series of sampled hot-wire data withRe=1.23× 105

5.3Further research

To investigate PSDS of different turbulence parameters at different Reynolds numbers, three 7.68×104and 1.23×105are sampled, where a length of groups of hot-wire signals withRe=4.60×104, the signal withRe=1.23× 105is shown in Fig.6. Three turbulence parameters[19], including turbulence kinetic energyE, skewness coefficientkSkand flatness coefficientFl, were selected as the concerned parameters here. With the same statistical testing and wavelet basis as the test case, the analysis results of PSDS are listed in Table 3. The results illustrate that PSDS increases with the increment of Re for the same concerned turbulence parameter. It is because more data should be sampled for largerRe. PSDS ofSkandFlare larger than that ofEkfor the same Reynolds number. It could be concluded that PSDS should be larger with the higher order of concerned turbulence parameter. Furthermore, the variation tendency of the statistical values of three turbulent parameters with different sampled data sizes should be further discussed at different Reynolds numbers. All parameters are normalized as follows:

Table 3 PSDS of different at different

Fig.7 Three frequently-used turbulent parameters vs. sampled data size

6. Conclusions

(1) Based on turbulence theory and statistical principles, a novel multi-scale approach for PSDS has been put forward here. And the specific procedure is given by usingwavelet tools.

(2) A case study is provided to prove the feasibility and rationality of this approach, where the sampled hot-wire data are from the experiment of square duct flow and turbulence kinetic energy is selected as the concerned turbulence parameter. The results show that the 220quantities of sampled data size are sufficient for the analysis of turbulence kinetic energy asRe=7.68× 104in the present experiment.

(3) Through the analysis of pre-set parameters in this approach, the results illustrate that PSDS is not sensitive to the change in wavelet function and the significance level parameter in the normal distribution testing. It demonstrates that the novel approach has a good stability for the determination of PSDS.Re=4.60×104, 7.68×104and 1.23×105have been

(4) The PSDSs of three turbulence parameters at further discussed. It is found that the PSDS increases with the increment of the Reynolds number and the order of concerned turbulence parameter.

[1] YU Jun, SHI Liu-liu and WANG Wei-zhe et al. Conditional averaging of TR-PIV measurements of wake behind square cylinder using an improved cross-correlation approach[J].Journal of Hydrodynamics,2010, 22(1): 29-34.

[2] PAN Dong-yuan, WANG Tong and ZHANG Bin et al. PIV measurement on rotating disks flow in cylinder[J].ChineseJournal of Hydrodynamics,2009, 24(2): 200-206(in Chinese).

[3] ZHANG Zhao-shun, CUI Gui-xiang and XU Chun-xiao.Theory and modeling of turbulence[M]. Beijing: Tsinghua University Press, 2006(in Chinese).

[4] SINHA M., KATZ J. Quantitative visualization of the flow in a centrifugal pump with diffuser Vanes-I: On flow structures and turbulence[J].Journal of Fluids Engineering,2000, 122: 97-107.

[5] UZOL O., CAMCI C. Aerodynamic loss characteristics of a turbine blade with trailing edge coolant ejection Part 2: External aerodynamics, total pressure losses and Predictions[J].Journal of Turbo machinery,2001, 123: 249-257.

[6] UZOL O., CAMCI C. The effect of sample size, turbulence intensity and the velocity field on the experimental accuracy of ensemble averaged PIV measurements[C].4th International Symposium on PIV.Gottingen, Germany, 2001, PIV 01 Paper 1096.

[7] KUKUKOVA A., NOEL B. and KRESTA S. et al. Impact of sampling method and scale on themeasurement of mixing and the coefficient of variance[J].AIChE Journal,2008, 54(12): 3068-3083.

[8] LI Fu-yu, WANG kai and ZHANG Hong-sheng et al. The influences of the sampling length and frequency on the processing of atmospheric turbulent data[J].Meteorological, Hydrological and Marine Instruments,2003, 3: 25-31(in Chinese).

[9] DAI Zheng-yuan, GU Chuan-gang and WANG Tong et al. Decision of sample sizes in experiments of turbulent flow based on wavelet analysis[J].Journal of Hydrodynamics, Ser. B,2006, 18(3): 324-328.

[10] LIU Jian-hua, JIANG Nan and WANG Zhen-dong et al. Multi-scale coherent structures in turbulent boundary layer detected by locally averaged velocity structure functions[J].Applied Mathematics and Mechanics (English Edition),2005, 26(4): 495-504.

[11] JIANG Nan, ZHANG Jin. Detecting multi-scale coherent eddy structures and intermittency layer in turbulent boundary layer by wavelet analysis[J].Chinese Physics Letters,2005, 22(8): 1968-1971.

[12] BORDEIANUL C., LANDAU R. and PAEZ M. Wavelet analyses and applications[J].European Journal of Physics,2009, 30: 1049-1062.

[13] GU Wei-guo, WANG De-zhong and KAWAGUCHI Y. et al. Study on the characteristics of 2D CTAC solution flow by wavelets transform[J].Chinese Journal of Hydrodynamics,2009, 24(4): 389-397(in Chinese).

[14] PERCIVAL D., WALDEN A.Wavelet methods for time series analysis[M]. Cambridge, UK: Cambridge University Press, 2000.

[15] HAYES M.Statistical digital signal processing and modeling[M]. New York: John Wiley and Sons, Inc., 1996.

[16] RUPPERT-FELSOT J., FARGE M. and PETITJEANS P. Wavelet tools to study intermittency: Application to vortex bursting[J].Journal of Fluid Mechanics,2009, 636: 427-453.

[17] FOIAS C., MANEY O. and ROSA R. et al.Navier-Stokes Equations and Turbulence[M]. Cambridge, UK: Cambridge University Press, 2001.

[18] WANG Yan, SUI Si-lian and WANG Ai-qing.Mathematical statistics and MATLAB analysis of engineering data[M]. Tsinghua University Press, 2006(in Chinese).

[19] CHERNYSHOV A., KARELSKY K. and PETROSYAN A. Validation of large eddy simulation method for study of flatness and skewness of decaying compressible magneto hydrodynamic turbulence[J].Theoretical and Computational Fluid Dynamics,2009, 23: 154-470.

February 1, 2009, Revised May 13, 2010)

2010,22(3):438-444

10.1016/S1001-6058(09)60075-5

* Project supported by the National Natural Science Foundation of China (Grant No. 50776056) the National High Technology Research and Development of China (863 Program, Grant No. 2009AA05Z201).

Biography:ZHANG Bin (1983- ), Male, Ph. D. Candidate

WANG Tong,

E-mail: twang@sjtu.edu.cn