The deterministic chaos and random noise in turbulent jet Tian-Liang Yao, Hai-Feng Liu, Jian-Liang Xu, and Wei-Feng Li Citation: Chaos 24, 023132 (2014); doi: 10.1063/1.4883497 View online: http://dx.doi.org/10.1063/1.4883497 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The application of complex network time series analysis in turbulent heated jets Chaos 24, 024408 (2014); 10.1063/1.4875040 Efficient noncausal noise reduction for deterministic time series Chaos 11, 319 (2001); 10.1063/1.1357454 Detecting deterministic dynamics in stochastic systems AIP Conf. Proc. 502, 662 (2000); 10.1063/1.1302449 Pathological tremors: Deterministic chaos or nonlinear stochastic oscillators? AIP Conf. Proc. 502, 197 (2000); 10.1063/1.1302385 Estimating the amplitude of measurement noise present in chaotic time series Chaos 9, 436 (1999); 10.1063/1.166418

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CHAOS 24, 023132 (2014)

The deterministic chaos and random noise in turbulent jet Tian-Liang Yao,1,2,3 Hai-Feng Liu,1,a) Jian-Liang Xu,1 and Wei-Feng Li1 1

Key Laboratory of Coal Gasification and Energy Chemical Engineering of Ministry of Education, East China University of Science and Technology, P.O. Box 272, Shanghai 200237, China 2 Shanghai Institute of Space Propulsion, Shanghai 201112, China 3 Shanghai Engineering Research Center of Space Engine, Shanghai Institute of Space Propulsion, Shanghai 201112, China

(Received 12 December 2013; accepted 3 June 2014; published online 16 June 2014) A turbulent flow is usually treated as a superposition of coherent structure and incoherent turbulence. In this paper, the largest Lyapunov exponent and the random noise in the near field of round jet and plane jet are estimated with our previously proposed method of chaotic time series analysis [T. L. Yao, et al., Chaos 22, 033102 (2012)]. The results show that the largest Lyapunov exponents of the round jet and plane jet are in direct proportion to the reciprocal of the integral time scale of turbulence, which is in accordance with the results of the dimensional analysis, and the proportionality coefficients are equal. In addition, the random noise of the round jet and plane jet has the same linear relation with the Kolmogorov velocity scale of turbulence. As a result, the random noise may well be from the incoherent disturbance in turbulence, and the coherent C 2014 AIP Publishing LLC. structure in turbulence may well follow the rule of chaotic motion. V [http://dx.doi.org/10.1063/1.4883497]

“The problem of turbulence” has been seen as one of the great challenges of mathematics, physics, and engineering. Up to date, large numbers of evidences have shown that the chaotic phenomenon is existed in turbulence especially when the Reynolds numbers are not too large, and many researchers have studied this phenomenon and have made certain research progress. In this paper, we analyzed the velocity time series of air turbulent jet and found that turbulence can be decomposed into chaos and random noise. Finally, we studied the quantitative relation between the turbulence and the deterministic chaos as well as random noise in turbulent jet using the method of time series analysis.

I. INTRODUCTION

The problem of turbulence has been researched for more than a century.1 Since Lorenz proposed that the chaotic phenomenon could be existed in the determinate system, and Ruelle and Takens suggested that the onset of fluid turbulence could be described by chaotic attractors,2 the methods of chaotic dynamics and chaotic time series analysis have been widely used in the field of turbulence and fluid mechanics.3–6 A turbulent flow can be decomposed into coherent structure and incoherent turbulence, and the natures of coherent structure have been studied by many researchers.7–10 According to the double decomposition of turbulence,11 any instantaneous variable of turbulence consists of coherent structure and incoherent disturbance, i.e., f ¼ hf i þ fr ; a)

Electronic address: [email protected]

1054-1500/2014/24(2)/023132/9/$30.00

(1)

where hf i and fr are the phase average of f and the incoherent disturbance of turbulence, respectively. Because the incoherent disturbance in the turbulent flow is random, we regard the incoherent disturbance as the random noise in current paper. Lyapunov exponents are the most useful dynamical diagnostic for chaotic systems and are the significant invariant for analyzing the dynamical behavior of a chaotic system.12 However, Lyapunov exponents calculated from time series are very sensitive to the random noise.13 In our previous paper, we proposed an algorithm to estimate simultaneously the largest Lyapunov exponent (LLE) and noise level from noisy chaotic time series.13 Although many researchers studied the chaotic characteristic in turbulence via the method of chaotic time series analysis, they rarely considered the influence of the random noise on the chaotic time series. In addition, the quantitative relation between the invariant of the chaotic characteristic and the scales of turbulence is also rarely studied. In this paper, the turbulent time series of the air jet for different exit Reynolds numbers and different flow positions are analyzed by the method of noisy chaotic time series analysis we proposed. What is more, the relationship between the LLE of the time series and the integral time scale of the turbulence as well as the relationship between the random noise of the time series and the Kolmogorov velocity scale of the turbulence are also studied. This paper is organized as follows. After this introduction, we describe the experiment apparatus and jet-signal collection in Sec. II. In Sec. III, we give a brief introduction of the algorithm we previously proposed.4 The calculated results and the analysis are listed in Sec. IV. In Sec. V, we give the conclusions based on the analysis. Finally, we summarize in Sec. VI.

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C 2014 AIP Publishing LLC V

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II. EXPERIMENT APPARATUS AND JET-SIGNAL COLLECTION

The sketch of experiment flowchart is drawn in Fig. 1. The work fluid was air and was supplied by a Roots blower. The bulk fluxes of the airflow from the buffer tank were controlled by a precision rotameter with accuracy of 61% of full scale deflection. A honeycomb which was 20 mm thick with a 1 mm cell size was also used to straighten out the flow and to dampen any large scale disturbances. The turbulent flow field of jet was obtained by an axisymmetric nozzle or a plane nozzle. The dimensions of the axisymmetric nozzle and the plane nozzle are plotted in Figs. 2(a) and 2(b). The exit diameter (D) of the axisymmetric nozzle was 10 mm and the exit height (H) of the plane nozzle was 5 mm. The jet Reynolds number at the nozzle exit was defined as Re ¼

UL ; v

(2)

where U was the bulk mean velocity at the nozzle exit, L ¼ D for the round jet and L ¼ H for the plane jet, and v was the kinematical viscosity of air, which was 15.53  106m2/s under experimental conditions. The velocity time series along the flow direction at the central line of the jet were acquired with a hot-wire anemometer (HWA). The sampling frequency was set at F ¼ 104 Hz and the sampling duration was T ¼ 100 s. The velocity time series of 0.05 s for the round jet at Re ¼ 2733 and x ¼ 3D and the plane jet at Re ¼ 926 and x ¼ 4H are plotted in Figs. 3(a) and 3(b), where x is the distance of the measurement position away from the nozzle exit, and u in Fig. 3 denotes the instantaneous velocity acquired by the HWA. As can be seen, these time series obey some determinate laws, but they are not exactly periodic. Meanwhile, the time series obviously contain a certain amount of random noise, which may well be the noisy chaotic time series.

III. A BRIEF INTRODUCTION OF THE ALGORITHM WE PREVIOUSLY PROPOSED

Let fxn g, where n ¼ 1,2,…,Ns, generated by a deterministic chaotic dynamical system, be a scalar time series. The number of observations is Ns and the sampling time interval

FIG. 2. Cross section of jet nozzles with all dimensions in millimeters. (a) Axisymmetric nozzle and (b) plane nozzle.

between measurements is Dt. The measured noisy signal yn is given by yn ¼ xn þ en ;

(3)

where en is the random noise. The attractor can be reconstructed using time-delay method14 in a d-dimensional phase space xdn ¼ ½xn ; xnþs ; :::; xnþðd1Þs ;

FIG. 1. Flow chart of experimental system. 1, roots fan; 2, buffer tank; 3, valve; 4, rotameter; 5, nozzle; 6, hot-wire probe; 7, hot-wire frame; 8, computer.

(4)

where s is the time lag, d is the embedding dimension, and fxdn g is the reconstructed attractor in d-dimensional embedding phase space. Analogously, we also have ydn ¼ ½yn ; ynþs ; :::; ynþðd1Þs  and edn ¼ ½en ; enþs ; :::; enþðd1Þs .

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r ¼ re . In practice, we usually use the linear least squares method to compute the LLE rather than the difference quotient of Eq. (8). So, we compute the slope of an intermediate range via the linear least squares method to get the estimate of f ðT; d; r2 Þ and use fa ðd; r2 Þ to denote the approximate value of f ðT; d; r2 Þ. In this case, we can easily get the following conclusions: (1) when r2 < re 2 , fa ðd; r2 Þ decreases with d; (2) when r2 ¼ re 2 , fa ðd; r2 Þ is invariable for different d, and fa ðd; r2 Þ ¼ k1 ; 2 (3) when re 2 < r2 < re 2 þ hn ðT;dÞ 2d , fa ðd; r Þ increases with d.

FIG. 3. Velocity time series of air jet. (a) Round jet at Re ¼ 2733 and x ¼ 3D and (b) plane jet at Re ¼ 926 and x ¼ 4H.

Now let us take an arbitrary point, xdn , in d-dimension embedding phase space. All points falling into the e-neighborhood of xdn are indicated as fxdn;m g, where m ¼ 1,2,…,N, and N is the number of these points, and e is the radius of the neighborhood. After the evolution of a time interval t ¼ TDt, xdn;m will proceed to xdn;m ðTÞ. The N points are divided into two subgroups, fxdn;i g and fxdn;iþNP g, where i ¼ 1,2,…,NP, and NP is the integer part of N=2. We have jxdn;i ðTÞ  xdn;iþNP ðTÞj ¼ jxdn;i  xdn;iþNP j  ek1 TDt ;

(5)

where j•j denotes the Euclidean distance in this paper and k1 is the LLE. Then we obtain hlnhn ðT; dÞi ¼ 2k1 TDt þ hlnhn ð0; dÞi;

(6)

P

P jxdn;i ðTÞ  xdn;iþNP ðTÞj2 and hn ðT; 0Þ where hn ðT; dÞ ¼ N1P Ni¼1 P P jxdn;i  xdn;iþNP j2 and h•i denotes the average over ¼ N1P Ni¼1

all values of n. For the measured noisy signal, when Ns tends to infinity, we get Hn ðT; dÞ ¼ hn ðT; dÞ þ 2dre 2 ;

(7)

P P d where Hn ðT; dÞ ¼ N1P Ni¼1 jyn;i ðTÞ  ydn;iþNP ðTÞj2 and re denotes the standard deviation of the noise. Substituting Eq. (7) in to Eq. (6), we can get the rescaled formula for calculating the k1 when the time series is contaminated with noise. However, re is usually unknown a priori, so we suppose that the standard deviation of the noise is r, and define a ternary function f ðT; d; r2 Þ ¼

SðT þ DT; d; r2 Þ  SðT; d; r2 Þ ; 2DT  Dt

(8)

where SðT; d; r2 Þ ¼ hln½Hn ðT; dÞ  2dr2 i. Obviously, 2 f ðT; d; r Þ ¼ k1 and do not depend on the dimension d at

As a result, we can get the re from the variation trend of fa ðd; r2 Þ in different dimensional embedding phase spaces, and obtain the LLE at the same time. However, fa ðd; r2 Þ still fluctuates slightly at r ¼ re in different dimensional embedding phase spaces. So, we compute Sf ðrÞ, the slopes which fa ðd; r2 Þ varies with the embedding dimension d, with the linear least squares method, and then taking rc , the estimate of the random noise intensity, to be the value of r which makes Sf ðrÞ ¼ 0 with the linear interpolation method. In addition, we compute fa ðd; rc 2 Þ for different d, and take the average of fa ðd; rc 2 Þ to be the estimate of the LLE, i.e., k1c ¼

dM X 1 fa ðd; rc 2 Þ; dM  dm þ 1 d¼d

(9)

m

where dm and dM are the minimum embedding dimension and the maximal embedding dimension, respectively. The detailed derivation process and computational details of this algorithm can refer to the authors’ former paper.13 IV. RESULTS AND ANALYSIS A. Example

It is needed to reconstruct the dynamics from acquired velocity time series with HWA firstly when we use the algorithm of Yao et al.13 We take the lag where the autocorrelation function drops to 1-1/e of its initial value, as in the work of Rosenstein et al.,15 for the reconstruction lag, and use the singular-value decomposition method to determine the embedding dimensions.16 Take the velocity time series of the round jet at Re ¼ 2733 and x ¼ 3D, for example, its autocorrelation function CðsÞ is shown in Fig. 4. As can be seen, the autocorrelation function is closest to 1-1/e of its initial value at s ¼ 6, so we take l ¼ 6 as the time-delay of phase space reconstruction. Figure 5 plots the singular-value spectrum Sm of the time series. We can obtain that Sm is almost invariant at m  4 from Fig. 5. Strictly speaking, S5 is still smaller slightly than S4 . So, the embedding dimension should at least choose to be d ¼ 5 to unfold the trajectory of the time series. For the sake of safety, we take the minimum embedding dimension as dm ¼ 6 in the practical computation. Because it is very difficult to depict the high-dimensional phase space, we project it into two-dimensional phase space to look over the approximate shape of the reconstitute

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FIG. 4. The CðsÞ  s curve of the velocity time series in the round jet at Re ¼ 2733 and x ¼ 3D.

attractor. The two-dimensional reconstructed phase spaces from the time series of round jet at Re ¼ 2733 and x ¼ 3D and plane jet at Re ¼ 926 and x ¼ 4H are plotted in Fig. 6. We can see that the reconstructed attractor has a self-similar fractal property, but the trajectory is not smooth, which may be contaminated by the random noise. According to the basic idea of the algorithm of Yao et al.,13 we take a series of different values of r, and then we can get the values of fa ðd; r2 Þ (the meaning of fa ðd; r2 Þ is entirely the same as the one in Yao et al.13) for different embedding dimensions d. Figure 7 gives the relationship between fa ðd; r2 Þ and d for the two time series in Fig. 3. As can be seen, fa ðd; r2 Þ decreases with the increase in d when r is small, and increases with the increase in d when r is large, and must exist a r which makes fa ðd; r2 Þ almost invariant for different d. Obviously, this property is totally consonant with the one of the chaotic time series which is contaminated with noise. Namely, it indeed exist lowdimensional chaos in the near field of the air jet, but it also inevitably contains a certain amount of random noise in this deterministic chaos. Based on the algorithm of Yao et al.,13 we have k1 ¼ 403:6 s1 and re ¼ 0:0619 m/s for this time series of round jet, and k1 ¼ 297:6 s1 and re ¼ 0:0667 m/s for this time series of plane jet, where k1 is the largest Lyapunov exponent and re is the standard deviation of the random noise.

FIG. 6. Two-dimensional reconstructed attractor of velocity time series. (a) Round jet at Re ¼ 2733 and x ¼ 3D and (b) plane jet at Re ¼ 926 and x ¼ 4H.

FIG. 5. The Sm  m curve of the velocity time series in the round jet at Re ¼ 2733 and x ¼ 3D.

FIG. 7. The curve of fa ðd; r2 Þ varied with d for (a) round jet at Re ¼ 2733 and x ¼ 3D and (b) plane jet at Re ¼ 926 and x ¼ 4H.

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B. The variation of the LLE for different Re and different flow positions

Using the similar method, we obtain the k1 and re of the acquired velocity time series of different exit Reynolds numbers and different measuring positions. The calculated k1 of different exit Reynolds numbers are graphed as a function of flow positions in Fig. 8. Obviously, for the same flow position, the exponent shows an increase with the exit Reynolds number, indicating that the “degree of chaos” as well as the unpredictability of the flow increases with Re and the dynamics of the system are becoming more complex as the Re increases. For the same Re, the calculated k1 increases first and then decreases gradually with downstream distance in the potential core region of the jets. As the jet structure developed downstream, due to the nonlinear interaction and the hydrodynamic instability, the flow will transit to fully developed turbulence and the flow will become more complex. So, k1 increases first in the near field of the air jet. However, after a certain distance away from the nozzle exit, the large-scale coherent structures with a remarkable periodicity begin to generate in the mixing layer. Under the influence of the coherent structures, the periodicity of the flow becomes stronger in the potential core region. In this case, the largest exponent k1 will gradually decrease. Overall, it is reasonable that the calculated k1 increases first and then decreases gradually with downstream distance at the central line of the jet, and a maximum value of k1 , k1;max , exists for

Chaos 24, 023132 (2014)

every exit Re. k1;max of different exit Reynolds numbers are graphed as a function of Re in Fig. 9. We find that k1;max at the central line of the jet is just directly proportional to the exit Reynolds numbers, i.e., it has the form k1;max ¼ kRe:

(10)

Using the least-squares fit method, we get k ¼ 0.15 and the correlation coefficient is R2 ¼ 0.99 for the round jet, and k ¼ 0.34 and the correlation coefficient is R2 ¼ 0.99 for the plane jet. From Fig. 8, we also observe that the position at which the maximum k1 appears gradually close to the nozzle exit with the increase in the Re. This is mainly because the generated position of the coherent structure in the mixing layer is gradually close to the nozzle exit with increasing the exit Re. Figure 10 gives the xm , the distance away from the nozzle exit of the position which the maximum k1 appears, for different Re. We can see that it should have the form xm ¼ AReB ; L

(11)

where L ¼ D for the round jet and L ¼ H for the plane jet. Using the least-squares fit method, we have A ¼ 95, B ¼ 0.46, and the correlation coefficient is R2 ¼ 0.94 for the round jet, and A ¼ 72, B ¼ 0.46, and R2 ¼ 0.90 for the plane jet. Comparatively speaking, the position which xm appears of plane jet is closer to the nozzle exit than that of the round jet for the same exit Reynolds number, but the positions are directly proportional to Re0.46 for both kinds of jets. C. The relationship between the LLE and the integral time scale in turbulent jet

The integral time scale in turbulence is obtained from the autocorrelation function R(t) of the streamwise velocity fluctuation17 ð1 I¼ RðtÞdt; (12) 0 0

0

where RðtÞ ¼ u ðt0 Þu0 2ðt0 þtÞ , u0 ¼ u  u, and • denotes the mean u

value. In Fig. 11, we plot the relationship between the LLE

FIG. 8. k1 for different Re and different flow positions in (a) round jet and (b) plane jet.

FIG. 9. The relationship between k1;max and Re.

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conjecture that the random noise mainly arises from the random motion of small eddies. In Fig. 12, we plot the re for different Re and different flow positions for the round jet and plane jet. As can be seen, for the two kinds of jets, the random noise increases with the increase in the exit Re or the distance away from nozzle exit. This can be interpreted as follows: (1) with the increase in Re, the movement velocities of small eddies increase and more hierarchy small eddies are excited by the turbulence, so the total value of random motion will increase and then the random noise will increase with Re. (2) as the jet develops downstream, the small eddies are generally unstable to smaller eddies, so the total values of random motion will increase and then the random noise will increase with the distance away from nozzle exit. FIG. 10. The relationship between xm and Re.

E. The relationship between the random noise and the Kolmogorov scale in turbulent jet

and the integral time scale for the two kinds of jets. It is amazing that they almost show an identical form of power function for both jets. According to the dimensional analysis, the LLE k1 and the integral time scale I should have the form

Kolmogorov scale is indicative of the smallest eddies present in the flow, and the scale at which the energy is dissipated. Kolmogorov velocity scale is uniquely determined by the viscosity  and the energy dissipation rate per unit mass e:17,18

k1 ¼ aI1 :

ug ¼ ðe Þ1=4 ;

(13)

Using the least-squares fit method for the whole results of the two kinds of jets, we get a ¼ 0.24 and the correlation coefficient is R2 ¼ 0.97. The integral time scale represents the characteristic of larger eddies in turbulence. The LLE, if positive, is the most important evidence for chaos and is an important invariant of characterizing chaos produced from a dynamical system. Because the LLE and the integral time scale are strong correlation, the motion of large eddies in turbulence may well follow the rule of chaotic motion.

(14)

where ug is the Kolmogorov velocity scale. Using the Taylor’s frozen hypothesis and some manipulation, the Kolmogorov velocity scale is expressed as (see Appendix)

D. The variation of the random noise for different Re and different flow positions

We also find the velocity time series include inevitable part of random noise from the calculating results. Because the motion of small eddies in turbulence is random, we

FIG. 11. The relationship between k1 of round jet and plane jet and the integral time scale.

FIG. 12. re for different Re and different flow positions in (a) round jet and (b) plane jet.

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"

15 2 ug ¼ u2



2 #1=4 @u : @t

(15)

Kolmogorov velocity scale and the calculated noise have the same dimension. If the random noise arises from the random motion of small eddies in turbulence, the random noise and the Kolmogorov velocity scale should have positive correlation. In Fig. 13, we plot the relationship between the random noise re and the Kolmogorov velocity scale ug . Obviously, re and ug have the same order of magnitude, i.e., re  ug . In addition, re almost has an identical linear relation with ug for the two kinds of jets. So, re and ug should have the form re ¼ Cug þ D:

(16)

Using the least-squares fit method for the whole results of the two kinds of jets, we get C ¼ 2.6, D ¼ 0.19, and the correlation coefficient is R2 ¼ 0.95. Because the random noise and the Kolmogorov velocity scale are strong correlation, we believe more confidently that the random noise mainly arises from the random motion of small eddies in turbulence. V. DISCUSSIONS A. Comparison with other studies

Other researchers also studied the chaotic characteristic using the method of chaotic time series analysis in turbulence. For instance, McMackin et al.4 presented the results of an experiment to analyze and characterize the flow structure of an air jet using chaotic measures. The fluid flow in their study is a heated free jet air flow, and the average temperature of the air in the jet is approximately 10  C above ambient. The exit diameter (D) of the jet nozzle was 25 mm, and the jet Reynolds number at the nozzle exit was about 7244, which can be compared with the round jet of the Re ¼ 7288 in our paper. The LLE of their time series for several downstream positions ranging from 0.5D to 8.0D was calculated using the Wolf method.19 The calculated LLEs are graphed as a function of flow positions in Fig. 14. As can be seen, the exponent shows a variation between 100 and

FIG. 14. Lyapunov exponents as a function of downstream position.4

800, which is different from our results whose range is between 700 and 1100, but their orders of magnitudes are equal. The difference may have two reasons: (1) The periodicity of the coherent structure of the heated free air jet may be stronger than our free jet, so their positive Lyapunov exponents are smaller than ours. (2) As McMackin stated, their calculation of the Lyapunov exponents is often unstable and the results can be precariously dependent on the input parameters, so McMackin’s results may have some deviations. In practice, Lyapunov exponent is very sensitive to the random noise.13 When the noise is existed in the time series, the estimated results of the Lyapunov exponents using the traditional method (e.g., Wolf method19) of chaotic time series analyze are not reliable. In our opinion, they did not consider the influence of the random noise on the time series. So, McMackin’s results may have several biases. We need more robust method to calculate the Lyapunov exponents of the turbulent time series. The algorithm proposed by Yao et al.13 considers the influence of the random noise on the LLE, and can simultaneously obtain the LLE and noise level from a noisy chaotic time series reliably. In this paper, the results of the LLE and random noise obviously correspond to the rule of turbulent motion. Objectively, the method of Yao et al.13 has several disadvantages. First, a large database is needed, e.g., 1 000 000 data points are used to calculate the LLE and random noise in our paper. Second, it is difficult to calculate if the random noise is too large. So, we did not calculate the result of the time series in the far field of the air jets in this paper. B. Further discussion

Based on the foregoing computation and analysis, we spontaneously have the following conjectures:

FIG. 13. The relationship between re and ug in round jet and plane jet.

(1) The round jets are different from the plane jets, but the relationship between the LLE and the integral time scale as well as the relationship between the random noise and the Kolmogorov velocity scale are the same for the two kinds of jets. Because the two kinds of jets are both the free shear flow, we conjecture that these relationships are the common characteristic for free shear flow. (2) The velocity time series of round jet and plane jet are both totally consonant with the properties of noisy chaotic time series. In addition, for the two kinds of jets, the

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LLE and the integral time scale in turbulence are strong correlation, and the random noise and the Kolmogorov velocity scale are also strong correlation. So, we conjecture that turbulence neither is totally random, nor can be thoroughly described by the deterministic chaos. In this view, the turbulence is composed of the deterministic chaos and the random noise, where the motion of large eddies in turbulence follows the rule of chaotic motion and the random noise arises from the random motion of small eddies in turbulence. (3) There are large numbers of coherent structures in turbulent flow. Generally, the coherent structures derive from the large scale structure in turbulence, and the incoherent disturbance is random which is very similar to the random noise. So, we conjecture that the random noise is mainly from the incoherent disturbance in turbulence, meanwhile the coherent structure in turbulence exhibits the nature of deterministic chaotic.

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follows the rule of chaotic motion and the random noise arises from the random motion of small eddies in turbulence. In addition, the random noise is mainly from the incoherent disturbance in turbulence and the coherent structure in turbulence exhibits the nature of deterministic chaotic. ACKNOWLEDGMENTS

This work was partly supported by the National Basic Research Program of China (Grant No. 2010CB227005), and the National Natural Science Foundation of China (Grant No. 21176079) and the Shanghai Engineering Research Center of Space Engine (Grant No. 13DZ2250600). APPENDIX: THE DERIVATION OF THE KOLMOGOROV VELOCITY SCALE

Kolmogorov velocity scale is uniquely determined by the viscosity  and the energy dissipation rate per unit mass e:17,18

VI. CONCLUSIONS

Many researchers studied the chaotic characteristic in turbulence using the method of chaotic time series analysis, but they rarely considered the influence of the random noise on the time series and the quantitative relation between the turbulence and the deterministic chaos. In our paper, we analyzed the turbulent time series via the method we proposed in our previous paper and studied the chaotic nature as well as the random noise of the air jet. Based on the computational results, the conclusions are mainly as follows: (1) The velocity time series of round jet and plane jet are both totally consonant with the properties of chaotic time series which are contaminated with noise. (2) For both jets, we find the calculated LLE increases with the exit Reynolds number for the same flow position, and increases first and then decreases gradually with downstream distance in the potential core region for the same Re. (3) For both jets, the maximum value of the LLE, k1;max , at the central line of the jet is just directly proportional to the exit Reynolds numbers. Using the least-squares fit method, we get k1;max ¼ 0:15Re for the round jet and k1;max ¼ 0:34Re for the plane jet. (4) For both jets, the distance away from the nozzle exit of the position which the maximum k1 appears has a form of power function. We have xDm ¼ 95Re0:46 for the round jet and xLm ¼ 72Re0:46 for the plane jet. (5) For both jets, the LLE and the integral time scale almost show an identical form of power function and we have k1 ¼ 0:24I1 . (6) For both jets, the random noise re and the Kolmogorov velocity scale ug have the same order of magnitude, and they almost have an identical linear relation. We obtain re ¼ 2:6ug  0:19 for both jets. According to the analyzed results, we consider that the turbulence is composed of the deterministic chaos and the random noise, where the motion of large eddies in turbulence

ug ¼ ðe Þ1=4 ;

(A1)

where ug is the Kolmogorov velocity scale. The energy dissipation rate e can be given by the following equation, which comes from the analytically derived conservation equation for turbulent kinetic energy:18 e ¼ 2sij sij ; where sij ¼

1 2



@ui @xj

(A2)

 @u þ @xij . Working out Eq. (A2) further for

isotropic turbulence (mainly bookkeeping for all the terms) results in18 e ¼ 15 ð@u=@xÞ2 :

(A3)

Using the Taylor’s frozen hypothesis, we have x ¼ ut;

(A4)

where u is the mean value of u. Substituting Eq. (A4) in Eq. (A3), we obtain 15 e¼ 2 u



2 @u : @t

(A5)

Substituting Eq. (A5) in Eq. (A1), we get "

 2 15 @u ug ¼ @t u2 2

#1=4 :

(A6)

1

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The deterministic chaos and random noise in turbulent jet.

A turbulent flow is usually treated as a superposition of coherent structure and incoherent turbulence. In this paper, the largest Lyapunov exponent a...
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