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Hierarchical structures in a turbulent pipe flow
Fluid Dynamics Research 33 (2003) 493 – 508
Hierarchical structures in a turbulent pipe ow Zhengping Zoua; b , Yuanjie Zhua , Mingde Zhoua; c , Zhen-Su Shea; d;∗ a
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China b School of Jet Propulsion, Beijing University of Aeronautics and Astronautics, Beijing 100083, China c Department of Mechanical and Aerospace Engineering, University of Arizona, AZ 80621, USA d Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095, USA Received 27 January 2003; accepted 1 July 2003 Communicated by S. Kida
Abstract A hierarchical structure (HS) analysis (-test and -test) is applied to a fully developed turbulent pipe ow. Velocity signals are measured at two cross sections in the pipe and at a series of radial locations from the pipe wall. Particular attention is paid to the variation of turbulent statistics at wall units 10 ¡ y+ ¡ 3000. It is shown that at all locations the velocity uctuations satisfy the She–Leveque hierarchical symmetry (Phys. Rev. Lett. 72 (1994) 336). The measured HS parameters, and , are interpreted in terms of the variation of uid structures. Intense anisotropic uid structures generated near the wall appear to be more singular than the most intermittent structures in isotropic turbulence and appear to be more outstanding compared to the background uctuations; this yields a more intermittent velocity signal with smaller and . As turbulence migrates into the logarithmic region, small-scale motions are generated by an energy cascade and large-scale organized structures emerge which are also less singular than the most intermittent structures of isotropic turbulence. At the center, turbulence is nearly isotropic, and and are close to the 1994 She–Leveque predictions. A transition is observed from the logarithmic region to the center in which drops and the large-scale organized structures break down. We speculate that it is due to the growing eddy viscosity e?ects of widely spread turbulent uctuations in a similar way as in the breakdown of the Taylor vortices in a turbulent Couette–Taylor ow at high Reynolds numbers. c 2003 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. PACS: 47.27.Gs; 47.27.Jv Keywords: Turbulence; Intermittency; Hierarchical structure; Pipe ow
∗
Corresponding author. Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. Tel.: 310-8258576; fax: 310-2062679. E-mail address: [email protected] (Zhen-Su She). c 2003 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. 0169-5983/$30.00 All rights reserved. doi:10.1016/j. uiddyn.2003.07.002
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1. Introduction The wall-bounded ow has been the subject of intensive study for many years (see e.g. Coles (1962), Murlis et al. (1982), Kim et al. (1987), and Sreenivasan (1989) for some aspects of the work). The circular pipe ow is one of the typical wall-bounded ows, and is free of the corner e?ects compared to channel ows and of the side-wall e?ects compared to boundary ows over a at plate. It is hence a special ow for the study of turbulent structure evolution (den Toonder and Nieuwstadt, 1997; Antonia and Pearson, 2000). Recently, interests also arise because of the very high Reynolds number experiments in a superpipe at Princeton (Morrison et al., 2002). Considerable knowledge has been accumulated in understanding of the mean velocity proJle and the mean distribution of eddy energy. On the other hand, the study on intermittency e?ects (Frisch, 1995; Sreenivasan and Antonia, 1997) and on the evolution of the uctuation structures away from the wall has just begun (Toschi et al., 1999; Ruiz-Chavarria et al., 2000; Poggi et al., 2003). Most studies of intermittency e?ects near the wall were directed to the channel ow. In the present work, we undertake a detailed investigation of the multiple-scale structures in a turbulent pipe ow, especially the evolution of its multiple-scale correlations through the bu?er and logarithmic regions to the center. The correlations are characterized by the (relative) scaling property of the velocity structure functions which are studied in the framework of so-called hierarchical structure (HS) analysis (She and Liu, 2003; Liu and She, 2003). In particular, we attempt to relate the scaling analysis to the ow structural variation by interpreting the variation of the HS parameters of the streamwise velocity uctuations at various radial positions in the pipe. The goal is to obtain a quantitative signature for the evolution of turbulent structures across the strong shear near the pipe wall and in the logarithmic region. The HS study in turbulence began with the work of She and Leveque (She and Leveque, 1994) who suggested to deJne a hierarchy of uctuation intensities and assumed a relation among them, called hierarchical symmetry. The assumptions led to the She–Leveque scaling formula p = p=9 + 2(1 − ( 23 )p=3 ) for the longitudinal velocity structure functions, which are later shown to be mathematically realizable by a random log-Poisson multiplicative process (Dubrulle, 1994; She and Waymire, 1995). It was then developed as the HS model (She, 1998). She and Liu (2003) have further developed the HS analysis (Liu and She, 2003; She et al., 2001; Baroud et al., 2003) which consists in assessing the validity of the hierarchical symmetry (-test) and verifying the presence of the most intermittent structures (-test). The analysis has been successfully applied to analyze the Couette– Taylor ow (She et al., 2001), a liquid-helium swirling ow (Liu and She, 2003), three-dimensional isotropic Navier–Stokes turbulence by direct numerical simulations (She and Liu, 2003), temperature variations in climate changes (She et al., 2002) and in bio-molecular density distribution along E.coli DNA sequence (Wang et al., 2001). In these applications, the HS analysis constitutes a way to synthesize the correlations between structures of various scales and various intensities; such correlations are directly responsible for intermittency e?ects and are usually hidden in the quantitative measures of multi-scalings. The HS parameters unfold such measures, when they are interpreted in an appropriate way. There have been di?erent ways in which statistical physicists and uid physicists discuss about turbulent structures. Statistical physicists are familiar with a set of statistical measures like scaling exponents which are used to characterize the decay rate of multi-point correlations. In general, these scaling exponents could be sought of a way to quantify the macroscopic (statistical) state of a
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uctuation Jeld. With uid physicists, ow structures generally refer to as a certain kind of ow patterns which are obtained by a ow visualization technique or an analytical solution of the uid equations. There have been many attempts to bridge the two, and there are strong interests for doing that, because the physical intuitions come from ow patters but only the statistical correlations can be quantitatively obtained from experimental measurements in turbulent ows. The unsatisfactory state of our understanding of turbulence is partially re ected by the fact that there is no general consensus for linking the intuition and the measurements. The HS analysis attempts to link the multi-scaling measurements with the characterization of ow patterns. The HS model allows to summarize the whole set of scaling exponents in terms of two parameters, and , each of which is related to a speciJc organizational aspect of the ensemble of uctuations. In the present study with a concrete example of ow with strong shear, we explore the possible correspondence. The study of the (relative) scaling property of turbulent uctuations in inhomogeneous ows with a shear has been conducted by several groups (see e.g. Toschi et al., 1999; Ruiz-Chavarria et al., 2000; Poggi et al., 2003). It appears that even in the presence of strong shears, the streamwise longitudinal velocity uctuations satisfy the She–Leveque scaling (She and Leveque, 1994) formula, but needs to be generalized with appropriate HS parameters and . In the previous work mentioned above, there are several versions of the analysis of She–Leveque scaling parameters. We apply here a systematic procedure for verifying the HS property and measuring the HS parameters (She and Liu, 2003). We demonstrate in this paper that such a procedure is practical for analyzing inhomogeneous pipe ow and for obtaining interesting interpretation in ow structure evolution away from the wall. The validity of the interpretations will be subject to further study with more ow conJgurations. 2. Experimental setup and ow properties The experiments were carried out in a circular pipe with its inner diameter of 105 mm and total length of 22:5 m. The inner wall of pipe was covered with epoxy so that the wall can be regarded as a smooth surface. The pipe ow was run by a blower with a centrifugal fan. Measurements were made at two streamwise locations with distances from the pipe-inlet of 4.45 and 18:25 m which will be referred to as sections S1 and S2, respectively. The Reynolds number based on the mean velocity at the center of the pipe-outlet and on the inner diameter was 1:35 × 105 . Thus, the un-tripped pipe ow was a transitional rather than a fully developed turbulent ow because of the smooth surface and small Reynolds number. A tripping wire was then added on the inner wall near the pipe-inlet throughout the whole experiments. A hot wire anemometer (Model TSI-1050) was employed to measure the velocity. The sensor was a single-wire boundary-layer probe (Model 1261) with diameter of 5 , and length of 1 mm so that a satisfactory spatial resolution can be achieved. A 16-bit analog-digital converter (ADC) was applied for data acquisition and a cassette recorder (SONY PC204Ax) was used to record the data. For catching the high-frequency small-scale uctuations, the sampling frequency was set to 48k and the velocity signals were Jltered at 20k with an analog Jlter. Measurements were performed at various radial positions. The total sampling time at each test point was 20 min so that suMcient amount of samples were acquired for statistical analyses.
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Fig. 1. Mean velocity proJle and the rms velocity uctuation as a function of the distance to the wall in the wall unit in section S2. The points S2r2–S2r13 of long time series measurement are indicated on the mean velocity proJle.
In Fig. 1, we show the mean velocity proJle and the variation of the rms turbulent uctuation. In order to clearly exhibit the bu?er region and the logarithmic region, the x-axis is converted to the wall unit and plotted in a logarithmic scale. Our closest measurement to the wall reaches a distance from the wall of order y+ ∼ 10 (in normalized wall unit). A least-squares Jt to the logarithmic region shows a relation of 1 u+ = log y+ + B; (1) k where the von KNarmNan constant k = 0:41, and B = 5:20 in section S2 and k = 0:46, and B = 6:30 in section S1. These values are consistent with those in previously reported mean velocity proJles (Hinze, 1975). By comparison between several tests under the same condition, these values are proven to be repeatable within an accuracy of ±1%. For the analyses below, the Taylor frozen hypothesis is used to convert the time scale to the length scale in the streamwise direction. From Fig. 1, one can estimate that the turbulent intensity in the near wall region (10 ¡ y+ ¡ 100) is about 15 –20%, and is around 10% in the logarithmic region and much smaller in the center region. Although this conversion is problematic, it is believed to be acceptable to the kind of qualitative and quantitative information drawn in the present work. The validity of the hypothesis in the calculation of the velocity structure functions in a channel ow is discussed in Antonia et al. (1997). In Fig. 2, the streamwise uctuating velocity u is shown for three di?erent radial locations: S2r2 is at the center of the pipe; S2r5 is in the middle of the logarithmic region; S2R13 is at the closest position to the wall that we measured (see Fig. 1). A visual inspection of three signals indicates the presence of more ascending bursts at S2r13 (near the wall) and of more descending bursts at S2r2 (center). This is conJrmed by a plot of the probability density function (PDF) of the uctuating velocity u in Fig. 3, where it is shown that P(u ) for S2r2 is skewed to have a long left wing and
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Fig. 2. Streamwise uctuating velocity. From top to bottom, we show the streamwise velocity uctuation signals at three radial locations (a) S2r2, (b) S2r5, (c) S2r13 and the corresponding smoothed velocity signals (d) S2r2, (e) S2r5, (f) S2r13 obtained by a repeated three-point average which retain only large-scale structures.
Fig. 3. The PDF of the velocity at three locations in the pipe. S2r2 is at the center of the pipe. S2r5 is near the wall and close to the bottom of the log layer. S2r13 is the closest location to the wall with y+ = 5.
P(u ) for S2r13 is skewed to the right. At the middle of the logarithmic region S2r5, the uctuating velocity just happens to have a symmetric distribution. The balanced distribution is achieved at S2r5, from an inspection of Fig. 2b, by the production of more pronounced large-scale uctuation structures. In order to illustrate more explicitly the large-scale organized motion (OM) in the logarithmic regions in the HS discussion below, we present also in Fig. 2 three corresponding “smoothed” signals obtained from the original signals with a repeated three-point average (over 100,000 times). The smoothed signals retain only the large-scale structures. We can see that, at S2r5, there are remarkable large-scale patterns that are absent in both S2r2 (center) and S2r13 (near the wall). These patterns have larger uctuation amplitudes and are more extended in length scale when compared to S2r13. We refer to them as the signature of the OM in the logarithmic region. We believe that the presence of these motions will a?ect the way we model turbulent statistics in engineering. The wall-bounded shear layer is a rich source for the production of turbulent structures whose formation and dissipation process have been subjects of intensive studies (Robinson, 1991; Schoppa
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Fig. 4. Illustration of the hierarchy of intensities Fp (‘) as a function of moment order p at several length scales, obtained at S2r2.
and Hussain, 2000). Extensive e?orts have been invested to describe behavior of coherent structures in speciJc spatial regions such as viscous sub-layer, bu?er region, logarithmic region, etc. For example, the near-wall bu?er region is the most important zone of the boundary layer in terms of turbulence energy production and dissipation (Zhou and Liu, 1987); bu?er region activity is characterized by the bursting process (Lamballais et al., 1997); both outward ejection of low-speed uid and inward sweeps of high-speed uid occur intermittently throughout the layer, etc. Onorato et al. (2000) studied the statistical properties of streamwise velocity uctuations in fully developed turbulent channel ow and found that the maximum intermittency e?ects takes place between the bu?er layer and the inner part of the logarithmic region where the bursting phenomenon is the dominant dynamical feature. It is our goal to systematically describe the change of the multi-scale and multi-intensity characteristics of turbulent structures from near the wall to the center. We found that the HS framework is adequate to summarize such information. 3. Hierarchical structure analysis The HS analysis starts with the deJnition of a hierarchy of uctuation intensities. Let v‘ denote the velocity increment across a distance ‘, v‘ = v(x + ‘) − v(x), and Sp (‘) denote the velocity structure function, Sp (‘) = |v‘ |p . Introduce a hierarchy of function Sp+1 (‘) Fp (‘) = ; p = 0; 1; 2; : : : : (2) Sp (‘) Such a hierarchy covers the mean velocity uctuation intensity F0 = S1 , and a series of increasing intensities which, as p increases, approaches to the intensity of the so-called most intermittent structures, F∞ (‘)=limp→∞ Fp (‘) In Fig. 4, we illustrate such a hierarchy of intensities by plotting Fp (‘) vs. p at several length scales, which are computed with the uctuation signal at S2r2. Therefore, one can associate each intensity with an appropriate order p which varies continuously from 0 to inJnity.
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Note that the hierarchy of uctuation intensities, Fp (‘), are both function of ‘ and of p. In other words, the hierarchy spreads over both a range of scales and a range of intensities. This is an important feature of the HS analysis: we need not only to consider turbulence to be a collection of large and small eddies (a range of scales), but also a collection of strong and weak eddies (a range of intensities). Kolmogorov (1941) has established the concept of multiple scales; we propose here to include that of multiple intensities. Because of the existence of many intensities, it is natural to look for a relation among them. She and Leveque (1994) postulated that Fp+1 (‘) = Ap Fp (‘) F∞ (‘)1− ;
(3)
where 6 1 is a constant and Ap is independent of ‘. It is interesting that one can eliminate the term F∞ (‘)1− by considering the ratio, Fp+1 (‘) Ap Fp (‘) = : (4) F2 (‘) A1 F1 (‘) Both sides of Eq. (4) can be calculated from experimental data when the empirical PDFs (or histograms) are calculated from experimental uctuation signals. Furthermore, introduce a normalization for the ratio Fp =F1 by Fp (‘)=F1 (‘) : (5) p (‘) = Fp (‘0 )=F1 (‘0 ) Then, Eq. (4) is rewritten as p+1 (‘) = p (‘) :
(6)
A log–log plot of p+1 (‘) vs. p (‘) will be called a -test: if a linearity is observed, one says that the uctuation data passes the -test and the hierarchical symmetry is satisJed. When it happens, the value of can be obtained by measuring the slope. Furthermore, it is shown (She, 1998) that hypothesis (3) implies that a general formula of the scaling exponents of the velocity structure function deJned by Sp ∼ S3p is given by 1 − p p = p + (1 − 3) ; (7) 1 − 3 where 3 = 1 and the parameter is introduced for describing the most intermittent structure as F∞ (‘) ∼ ‘ . Then, a simple algebraic manipulation gives p − (p; ) = (p − 3(p; ));
(8)
where (p; ) = (1 − p )=(1 − 3 ). When we plot p − (p; ) vs. p − 3(p; ), the slope is . This constitutes the -test. Therefore, the -test veriJes whether turbulent uctuation intensities Fp (‘) obey a generalized similarity relation (3) over a range of scales (‘1 6 ‘ 6 ‘2 ) and over a range of intensities (p1 6 p 6 p2 ). Such a hierarchical symmetry is an indication of the self-organization of the ensemble of the uctuation events. When the hierarchical symmetry is present, the parameter measures the behavior of the self-organization. When = 1, all uctuation intensities have similar ‘-dependence. A notable example is Kolmogorov’s original 1941 picture of turbulence, a complete self-similar uctuation Jeld
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(Frisch, 1995). Recently, it is found that the coherent spiral state of the two-dimensional complex Ginzburg–Landau equation is such a self-similar state (Guo et al., 2003; Liu et al., 2003). The other limit is an extreme intermittency case of = 0. In this limit, only the most intermittent structures F∞ dominates the statistical properties such as moments. Randomly driven Burgers equation (She et al., 1992; Sinai, 1992) is an example where the only dissipative structures are shocks. A popular intermittency model of turbulence of Frisch, Nelkin and Sulem (Frisch et al., 1978) falls also into the = 0 case. Smaller is, more outstanding the most intermittent structures stand with respect to the background uctuations. For example, if some mechanism generates a few outstanding bursts into a nearly self-similar random uctuating ensemble, it may generally reduce the value of (when the -test still passes) and the resulting uctuation signal becomes more intermittent. Therefore, the value of is more intuitively related to the degree of intermittency. The parameter measures how singular the most intermittent structure is. By deJnition, F∞ (‘) ∼ ‘ ; in other words, is a property of the high-order moments p → ∞. From Fig. 2, we see that the intensities Fp have not attained nearly the values of F∞ for the typical range of p between 1 and 10 in our analysis (see below). So the value we obtained in our analysis is an extrapolated values based on the reasonably convergent moments below p = 10. It may be worthwhile to argue that obtained by such extrapolation is a meaningful measure of the very intense uctuation events in a turbulent ow. The results reported below support this argument. In summary, the and tests can be performed for any multi-scaling uctuation signal (or Jeld). They examine the properties of uctuation across a range of scales and intensities, and quantify the degree of self-organization (or intermittency) by the parameter (which measures how outstanding the most intermittent structure is compared to the rms uctuations) and (which measures how singular the most intermittent structure is). These are the essence of the HS description of multi-scaling uctuation Jeld. The resulting set of scaling exponents are given by Eq. (7). For isotropic turbulence, She and Leveque (1994) estimates that = ( 23 )1=3 ≈ 0:874 and = 19 . Several measurements in, e.g., turbulence behind a cylinder (Ruiz-Chavarria et al., 1994, 1995; Benzi et al., 1995, 1996) and in direct numerical simulations (Cao et al., 1996) support the prediction. Chen and Cao (1995) has an alternative suggestion that = ( 79 )1=3 ≈ 0:92 and = 0. It will be interesting to check if the isotropic incompressible ow has an asymptotic universal scaling. 4. Results 4.1. Energy dynamics First, let us examine the energy dynamics of the eddies from the wall to the center. Fig. 5 shows the time-averaged energy spectra calculated from the uctuating velocity signals from close to wall (S2rl3) to the center (S2r2). All spectra show a wide range of excitation in Fourier modes, consistent with the impression of developed turbulence from velocity signals (see Fig. 2). However, there are some qualitative and quantitative di?erences among the spectra. In the bu?er region (from S2rl3 to S2r7), there is a short k −1 power-law range in the spectra. As the probe moves further to the center, the power-law range extends further to smaller scales, and becomes steeper (E(k) ∼ k −1:5 ). At the two points S2r2 and S2r3, the large-scale motions suddenly get depleted, probably due to the
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Fig. 5. Energy spectra obtained from velocity signals at di?erent locations away from the wall. The spectra are shifted in the vertical direction for visibility. Note a nearly k −1 spectrum near the wall.
presence of small-scale turbulent motions. This corresponds to the disappearance of the large eddies seen in Fig. 2. At the point S2r5, the spectrum seems to have the longest power-law range. In other words, in the middle of the logarithmic range, the ow has uctuations extending to a maximum range. This may have a consequence on the scaling behavior reported below. We speculate on the evolution of turbulent structures based on the change of the energy spectra. Near the wall (y+ ¡ 100), a strong shear introduces many short spikes of positive streamwise uctuations. They may be the sources for the k −1 energy spectrum. Here, the nonlinear interaction plays a role, but the cascade is not a free cascade as Kolmogorov assumes, because of anisotropic structures. As one moves further into the logarithmic region, more isotropic turbulent structures enter to the dynamics so that the cascade to small scales further proceeds, leading to steeper energy spectrum. Note that a k −1:5 energy spectrum has been observed in moderately high Reynolds number fully developed turbulence near the viscous cuto? with the presence of a spectral hump (She and Jackson, 1993). In the logarithmic region, there is an interesting mixture of large-amplitude anisotropic turbulent structures that are reminiscent of unstable wall-bounded shear ows and small-scale turbulent structures due to the cascade. In the center of the pipe, the anisotropic structures disappear due to turbulent di?usivity, and turbulence tends to be nearly homogeneous and isotropic.
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Fig. 6. PDFs of velocity increments across a range of distances ‘ (2 6 ‘ 6 384) obtained from the uctuating velocity signal (a) at the center S2r2; (b) at the logarithmic region S2r5; (c) near the wall S2r13.
4.2. - and -test It is instructive Jrst to visualize the evolution of PDFs of the velocity increments that carries the information about multi-scale and multi-intensity correlations. Figs. 6(a) – (c) show the PDFs of the velocity increments for a series of length scales (2 6 ‘ 6 380 in the unit of time resolution 1 St = 48;0000 s) at three radial locations. Although the velocity PDFs are skewed as mentioned earlier (see Fig. 3), the PDFs of velocity increments are nearly symmetric over this range of scales at all three locations. This indicates that upwards and downward uctuations are alike statistically. Near the wall at S2rl3 (Fig. 6c), the uctuation intensity is large through most of scales with a nearly Gaussian shape at large scales and a nearly exponential tail at small scales. At S2r5 (Fig. 6b), the uctuations have comparable magnitude with S2rl3, indicating that the same structures persist from S2rl3 to S2r5. This is supportive to our speculation above that anisotropic structures produced by the strong shear is dominant in the logarithmic region. At S2r2, the velocity increment has signiJcantly reduced the magnitude of the uctuations (narrower distributions in Fig. 6a). However, the general character that the PDFs change from a Gaussian shape to a nearly exponential tail is similar at the center than near the wall. They may only have some quantitative di?erence, which will be revealed by the HS analysis. Before starting the HS analysis, we verify the existence of the extended self-similarity (ESS) property (Benzi et al., 1993). This is reported in Fig. 7 which is obtained at S2r2. At other radial locations, similar ESS property is observed. The ESS analysis is a kind of pre-requisite for the HS analysis. It helps to identify the range of ‘’s and p’s over which the - and -test are performed. The range of scale ‘ and moment order p in the HS analysis below were adjusted to between
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Fig. 7. Extended self-similarity (ESS) test at two points (S2r2 and S2r13). Note that all other signals have similar ESS properties.
Fig. 8. -test for three representative data sets (S2r2, S2r5, S2r13) which are displayed vertically from top down. The slope is the measured values of .
1 2 6 ‘ 6 64 (in the units of the time resolution St = 48;000 s) and 1 6 p 6 10. Note that in our experiment, the Reynolds number is not very large to allow to see a clear inertial range from the ‘-dependence of the third-order moment. But our analysis do not require the presence of such an asymptotic inertial range, and the hierarchical symmetry scaling is a kind of generalized ESS scaling (Ching et al., 2002). In Figs. 8 and 9, the results of the - and -test are shown for three representative sets of turbulent signals: S2r2, S2r5, and S2rl3. They show a clear evidence that both - and -test pass with good straight lines, indicating that the turbulent pipe ow encompasses indeed a nicely self-organized hierarchical structures with well-deJned HS scaling properties. Furthermore, the - and -test reveal interesting transitions from the near wall (S2rl3) to the center (S2r2). From S2rl3 to S2r5 (the
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Fig. 9. -test for three representative data sets (S2r2, S2r5, S2r13) which are displayed vertically from top down. The slope is the measured values of .
Fig. 10. Scaling exponents p of the velocity structure functions at a set of points from S2r2 (center) to S2r13 (near the wall). Curves are the HS theoretical predictions based on the measured values of and . Note a rough separation into two groups for the scaling behavior before and after entering to the logarithmic region (y+ ∼ 100).
logarithmic region), the parameter changes from 0.80 for 0.88 with a signiJcant reduction of the variation range in p (‘), and the parameter has a more drastic change (from 0.095 to 0.142). Both the reduction of p (‘) and the increase of are consistent with the presence of more pronounced organized large-scale motions in the logarithmic region. From S2r5 to S2r2, the parameter changes little, whereas the parameter changes back around 0.11, close to the SL94 predicted value for isotropic turbulence. We interpret that as an evidence of the disappearance of the organized motions developed in the logarithmic region. We have observed similar transition in the measured relative (ESS) scaling exponents p shown in Fig. 10. Starting from S2rl3 (near the wall), the scaling deviates most strongly from K41, consistent
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Fig. 11. The variation of the values of and measured from the velocity uctuation signals. Note that lower values of and indicate stronger intermittency near the wall, due to a more outstanding role of wall shear structures and its singular property.
with the results of several other recent studies (see e.g. Ruiz-Chavarria et al., 2000). In the HS analysis, this is associated with the presence of very singular intense structure with smaller ∼ 0:095, and with its more dominant role in the hierarchy with smaller . Moving towards the logarithmic region, the ESS scaling approaches K41 when both and signiJcantly increase. From a HS point of view, a larger is associated with the development of large-scale less-singular pattern as the “leader”, and a large implies that the “leader” structures are less outstanding from the background uctuation structures. This is a state of uctuation which is less intermittent than the SL94 prediction. Finally, at the center (S2r2), the ESS scaling approaches again the SL94 prediction, which is consistent with the disappearance of organized motions. In Fig. 10, we present also the HS prediction of the scaling exponents with the measured and (see lines in Fig. 10). The good agreement between the experimental data (points) and HS theoretical predictions (curves) gives a strong evidence that the HS statistical structure persists in the pipe shear ow turbulence. We have thus a sound foundation to discuss the physical signiJcance of the parameters and . The quantitative results of the - and -test are reported in Fig. 11. Here, the variations of and at all radial locations show a pattern: near the wall (y+ ¡ 100), and are both smaller, indicating a more distinct role of the most intense/intermittent structures in energy dissipation (small ) and its more singular nature (small ). Our interpretation is that strong-shear driven anisotropic turbulent structures are the most intermittent structures near the wall. In the hierarchy of uctuations in this region (S2rl3–S2r7), these structures have a distinct singular character ( ¡ 0:1) and they play the most dominant role for energy dissipation. But in the logarithmic region (S2r6 –S2r3), these anisotropic structures change their character and become less distinct ( ∼ 0:88) and less singular ( ∼ 0:13); the rise in may be due to the emergence of other uctuation structures of lower intensities. There appears to have a new transition from S2r3 to S2r2 (center), where remains essentially unchanged, but drops to 0.11. This transition is somewhat similar to the breaking of the Taylor vortices in a turbulent Couette–Taylor ow (She et al., 2001) as the Reynolds number increases. It is likely that there is a breaking of a kind of large-scale smooth structures as turbulence
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moves to the center with a big reduction of the rms velocity uctuations. We speculate that an interaction between wall-shear related anisotropic structures may be the cause for this transition and this process takes place most eMciently at the center. It is seen from Fig. 11 that the scaling exponents at the center S2r2 tend to the She–Leveque (1994) prediction. This is a limited Reynolds number result. It may suggest that the She–Leveque scaling formula predict the asymptotic scaling of longitudinal velocity structure functions in an incompressible truly isotropic turbulence. The present HS analysis then leads to the speculation that the increase of intermittency towards the wall regions is due to the presence of intensive anisotropic coherent shear structures, which are destroyed during the transition from the logarithmic region to the center. Finally, we note that the actual values of and obtained from the HS analysis may vary slightly with the range of the scale ‘ and of the moment order p. One needs to be cautious to compare and obtained with di?erent procedures. However, when the procedure is determined with the deJned ranges by means of the ESS analysis, the variations in and show consistent physical meaning. Only then, the comparison of the HS parameters between di?erent ows becomes meaningful and interesting. 5. Conclusions We have carried out an experimental study of the evolution of turbulent structures from the wall to the center in a turbulent pipe ow in the framework of the hierarchical structure model. First, we show that small-scale turbulent uctuations (over a range of scales more than a decade) at all radial positions (10 6 y+ 6 3000) satisfy the She–Leveque hierarchical symmetry relation. In other words, turbulent uctuations at all positions are hierarchically organized across scales and across intensities through (3). Then, the HS analysis (- and -test) allows one to quantify the degree of multi-intensity correlations () and the degree of the singularity of the most intermittent structures (). We Jnd that the scaling exponents of the streamwise longitudinal velocity structure function are roughly separated into two groups: near-wall regions y+ ¡ 100 and far region y+ ¿ 100, the variations of and as functions of the distance from the wall give rise to a more detailed picture. Near the wall (S2rl3–S2r7), both and are smaller, indicating a more singular nature of the most intense structure (small ) that is more outstanding from the background uctuations (small ). In the logarithmic regions, both and increase signiJcantly. The velocity signal at S2r5 exhibits the rise of some large-scale patterns, which are interpreted to be associated with the small intense structure. In a turbulent Couette–Taylor ow, one observes a decrease of corresponding to the breaking of the Taylor vortices as the Reynolds number increases (She et al., 2001). We speculate that an opposite process takes place with the formation of the smooth structures. As the probe moves to the center S2r2, a further decrease of signiJes the disappearance of the large-scale smooth structures, and the almost constancy of reveals that the overall organization of the hierarchical uctuations is kept the same. This last transition may be similar to the Taylor-vortex-breaking transition in the turbulent Couette–Taylor ow. The organized large-scale motions in the intermediate logarithmic regions have not been emphasized before. Our HS analysis is able to detect such a transition, which suggests an interest of the HS analysis. Like in the turbulent Couette–Taylor ow, the - and -test can help to monitor the
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change of the properties of the ow structures. This link has long been the goal of statistical analysis of turbulent ows. Note that what the HS analysis can detect is a feature of an organized motion recorded in a streamwise velocity component; the lack of three-dimensional ow information does not allow for a detailed characterization of the organized ow structure. Yet, detecting the presence or disappearance of organized structures based on a scaling analysis (the HS analysis) may be signiJcant. Note an important limitation of the present work, that is, uctuation structures are only characterized through the relative scalings. Other quantity such as the absolute scaling of the third-order structure function related to the mean energy transfer may be important to consider in strong shear ows. Acknowledgements We have beneJted from useful discussions with many people at the LTCS of Peking University, specially Prof. Wei Zhonglei, Prof. Wei Qingding, Prof. Zhang Boyin, Prof. Gu Zhifu, Prof. Wu Jiezhi and Prof. Su Weidong and Dr. Fu Qiang. This work was supported by the projects 10032020 and 10225210 from the NNSF of China. References Antonia, R.A., Pearson, B.R., 2000. Reynolds number dependence of velocity structure functions in a turbulent pipe ow. Flow, Turbulence Combust. 64, 95–117. Antonia, R.A., Zhou, T., Romano, G.P., 1997. Second- and third-order longitudinal velocity structure functions in a fully developed turbulent channel ow. Phys. Fluids 9 (11), 3465–3471. Baroud, C.N., Plapp, B.B., Swinney, H., She, Z.-S., 2003. Scaling in three-dimensional and quasi-two-dimensional rotating turbulent ows. Phys. Fluids 15 (8), 2091–2104. Benzi, R., Ciliberto, S., Baudet, C., Massaioli, F., Tripicione, R., Succi, S., 1993. Extended self-similarity in turbulent ows. Phys. Rev. E 48, 29–32. Benzi, R., Ciliberto, S., Baudet, C., Ruiz Chavarria, G., 1995. On the scaling of three dimensional homogeneous and isotropic turbulence. Physica D 80, 385–398. Benzi, R., Biferale, L., Ciliberto, S., Struglia, M.V., Tripiccione, R., 1996. Generalized scaling in fully developed turbulence. Physica D 96, 162–181. Cao, N., Chen, S., She, Z.-S., 1996. Scalings and relative scalings in the Navier–Stokes turbulence. Phys. Rev. Lett. 76, 3711–3714. Chen, S.Y., Cao, N.Z., 1995. Inertial range scaling in turbulence. Phys. Rev. E 52, R5757. Ching, E.S.C., She, Z.-S., Su, W.D., Zou, Z.P., 2002. Phys. Rev. E 65, 066303. Coles, D.E., 1962. The turbulent boundary layer in a compressible uid. Rand Corp. Report No. R-403-PR. den Toonder, J.M.J., Nieuwstadt, F.T.M., 1997. Reynolds number e?ects in a turbulent pipe ow for low to moderate Reynolds number. Phys. Fluids 9, 3398–3409. Dubrulle, B., 1994. Intermittency in fully developed turbulence: log-Poisson statistics and scale invariance. Phys. Rev. Lett. 73, 959–962. Frisch, U., 1995. Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge. Frisch, U., Nelkin, M., Sulem, P.-L., 1978. J. Fluid Mech. 87, 719. Guo, H., Li, L., Ouyang, Q., Liu, J., She, Z.-S., 2003. A systematic study of spirals and spiral turbulence in a reactiondi?usion system. J. Phys. Chem. 118 (11), 5038–5044. Hinze, J.O., 1975. Turbulence, 2nd Edition. McGraw-Hill, New York. Kim, J., Moin, P., Moser, R.D., 1987. Turbulence statistics in fully developed channel ow at low Reynolds number. J. Fluid Mech. 177, 133–166.
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Kolmogorov, A.N., 1941. The local structure of turbulence in incompressible viscous uid for very large Reynolds number. Dokl. Akad. Nauk. SSSR. V. 30, 9–13. Lamballais, E., Lesieur, M., Metais, O., 1997. Probability distribution functions and coherent structures in a turbulent channel. Phys. Rev. E 56 (6), 6761–6766. Liu, L., She, Z.-S., 2003. Hierarchical structure description of intermittent structures of turbulence. Fluid Dyn. Res. 33 (3), 261–286. Liu, J., She, Z.-S., Ouyang, Q., He, X.T., 2003. Int. J. Mod. Phys., to appear. Morrison, J.F., Jiang, W., McKeon, B.J., Smits, A.J., 2002. Reynolds number dependence of streamwise velocity spectra in turbulent pipe ow. Phys. Rev. Lett. 88, 214501–1. Murlis, J., Tsai, H.M., Bradshaw, P., 1982. Structure of turbulent boundary layers at low Reynolds numbers. J. Fluid Mech. 122, 13–56. Onorato, M., Camussi, R., Iuso, G., 2000. Small scale intermittency and bursting in a turbulent channel ow. Phys. Rev. E 61 (2), 1447–1454. Poggi, D., Porporato, A., RidolJ, L., 2003. Analysis of the small-scale structure of turbulence on smooth and rough walls. Phys. Fluids 15 (1), 35. Robinson, S.K., 1991. Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639. Ruiz-Chavarria, G., Baudet, C., Ciliberto, S., 1994. Hierarchy of the energy dissipation moments in fully developed turbulence. Phys. Rev. Lett. 74, 1986–1989. Ruiz-Chavarria, G., Baudet, C., Benzi, R., Ciliberto, S., 1995. Hierarchy of the velocity structure functions in fully developed turbulence. J. Phys. II 5, 485–490. Ruiz-Chavarria, G., Ciliberto, S., Baudet, C., Leveque, E., 2000. Scaling properties of the streamwise component of velocity in a turbulent boundary layer. Physica D 141, 183–198. Schoppa, W., Hussain, F., 2000. Coherent structure dynamics in near-wall turbulence. Fluid Dyn. Res. 26, 119–139. She, Z.-S., 1998. Prog. Theor. Phys. 130, 87–102. She, Z.-S., Jackson, E., 1993. On the universal form of energy-spectra in fully developed turbulence. Phys. Fluids 5, 1526. She, Z.-S., Leveque, E., 1994. Phys. Rev. Lett. 72, 336–339. She, Z.-S., Liu, L., 2003. Acta Mech. Sinica, to appear. She, Z.-S., Waymire, E.C., 1995. Quantized energy cascade and Log-Poisson statistics in fully developed turbulence. Phys. Rev. Lett. 74, 262–265. She, Z.-S., Aurell, E., Frisch, U., 1992. The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623–637. She, Z.-S., Ren, K., Lewis, G.S., Swinney, H.L., 2001. Phys. Rev. E. 64, 016308. She, Z.-S., Fu, Z.-T., Chen, J., Liang, S., Liu, S.-D., 2002. Hierarchical structures in climate and atmospheric turbulence. Prog. Natl. Sci. 12 (10), 747. Sinai, Ya.G., 1992. Statistics of the inviscid Burgers equation with initial Brownian data. Commun. Math. Phys. 148, 601–622. Sreenivasan, K.R., 1989. The turbulent boundary layer. In: Gad-el-Hak, M. (Ed.), Frontiers in Experimental Fluid Mechanics. Springer, Berlin, pp. 159–209. Sreenivasan, K.R., Antonia, R.A., 1997. The phenomenology of small-scale turbulence. Ann. Rev. Fluid Mech. 29, 435–472. Toschi, F., Amati, G., Succi, S., Benzi, R., Piva, R., 1999. Intermittency and structure functions in channel ow turbulence. Phys. Rev. Lett. 82 (25), 5044–5047. Wang, J., Zhang, Q., Ren, K., She, Z.-S., 2001. Multi-scaling hierarchical structure analysis of the sequence of E. coli complete genome. Chin. Sci. Bull. 46 (23), 1988–1992. Zhou, M.D., Liu, T.S., 1987. Stability investigation in normally 2-D laminar boundary layers by means of heat plusing. In: Meier, H.U. (Ed.), Perspectives in Turbulence Studies. Springer, Berlin, pp. 47–70.
Hierarchical structures in a turbulent pipe ow Zhengping Zoua; b , Yuanjie Zhua , Mingde Zhoua; c , Zhen-Su Shea; d;∗ a
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China b School of Jet Propulsion, Beijing University of Aeronautics and Astronautics, Beijing 100083, China c Department of Mechanical and Aerospace Engineering, University of Arizona, AZ 80621, USA d Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095, USA Received 27 January 2003; accepted 1 July 2003 Communicated by S. Kida
Abstract A hierarchical structure (HS) analysis (-test and -test) is applied to a fully developed turbulent pipe ow. Velocity signals are measured at two cross sections in the pipe and at a series of radial locations from the pipe wall. Particular attention is paid to the variation of turbulent statistics at wall units 10 ¡ y+ ¡ 3000. It is shown that at all locations the velocity uctuations satisfy the She–Leveque hierarchical symmetry (Phys. Rev. Lett. 72 (1994) 336). The measured HS parameters, and , are interpreted in terms of the variation of uid structures. Intense anisotropic uid structures generated near the wall appear to be more singular than the most intermittent structures in isotropic turbulence and appear to be more outstanding compared to the background uctuations; this yields a more intermittent velocity signal with smaller and . As turbulence migrates into the logarithmic region, small-scale motions are generated by an energy cascade and large-scale organized structures emerge which are also less singular than the most intermittent structures of isotropic turbulence. At the center, turbulence is nearly isotropic, and and are close to the 1994 She–Leveque predictions. A transition is observed from the logarithmic region to the center in which drops and the large-scale organized structures break down. We speculate that it is due to the growing eddy viscosity e?ects of widely spread turbulent uctuations in a similar way as in the breakdown of the Taylor vortices in a turbulent Couette–Taylor ow at high Reynolds numbers. c 2003 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. PACS: 47.27.Gs; 47.27.Jv Keywords: Turbulence; Intermittency; Hierarchical structure; Pipe ow
∗
Corresponding author. Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. Tel.: 310-8258576; fax: 310-2062679. E-mail address: [email protected] (Zhen-Su She). c 2003 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. 0169-5983/$30.00 All rights reserved. doi:10.1016/j. uiddyn.2003.07.002
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1. Introduction The wall-bounded ow has been the subject of intensive study for many years (see e.g. Coles (1962), Murlis et al. (1982), Kim et al. (1987), and Sreenivasan (1989) for some aspects of the work). The circular pipe ow is one of the typical wall-bounded ows, and is free of the corner e?ects compared to channel ows and of the side-wall e?ects compared to boundary ows over a at plate. It is hence a special ow for the study of turbulent structure evolution (den Toonder and Nieuwstadt, 1997; Antonia and Pearson, 2000). Recently, interests also arise because of the very high Reynolds number experiments in a superpipe at Princeton (Morrison et al., 2002). Considerable knowledge has been accumulated in understanding of the mean velocity proJle and the mean distribution of eddy energy. On the other hand, the study on intermittency e?ects (Frisch, 1995; Sreenivasan and Antonia, 1997) and on the evolution of the uctuation structures away from the wall has just begun (Toschi et al., 1999; Ruiz-Chavarria et al., 2000; Poggi et al., 2003). Most studies of intermittency e?ects near the wall were directed to the channel ow. In the present work, we undertake a detailed investigation of the multiple-scale structures in a turbulent pipe ow, especially the evolution of its multiple-scale correlations through the bu?er and logarithmic regions to the center. The correlations are characterized by the (relative) scaling property of the velocity structure functions which are studied in the framework of so-called hierarchical structure (HS) analysis (She and Liu, 2003; Liu and She, 2003). In particular, we attempt to relate the scaling analysis to the ow structural variation by interpreting the variation of the HS parameters of the streamwise velocity uctuations at various radial positions in the pipe. The goal is to obtain a quantitative signature for the evolution of turbulent structures across the strong shear near the pipe wall and in the logarithmic region. The HS study in turbulence began with the work of She and Leveque (She and Leveque, 1994) who suggested to deJne a hierarchy of uctuation intensities and assumed a relation among them, called hierarchical symmetry. The assumptions led to the She–Leveque scaling formula p = p=9 + 2(1 − ( 23 )p=3 ) for the longitudinal velocity structure functions, which are later shown to be mathematically realizable by a random log-Poisson multiplicative process (Dubrulle, 1994; She and Waymire, 1995). It was then developed as the HS model (She, 1998). She and Liu (2003) have further developed the HS analysis (Liu and She, 2003; She et al., 2001; Baroud et al., 2003) which consists in assessing the validity of the hierarchical symmetry (-test) and verifying the presence of the most intermittent structures (-test). The analysis has been successfully applied to analyze the Couette– Taylor ow (She et al., 2001), a liquid-helium swirling ow (Liu and She, 2003), three-dimensional isotropic Navier–Stokes turbulence by direct numerical simulations (She and Liu, 2003), temperature variations in climate changes (She et al., 2002) and in bio-molecular density distribution along E.coli DNA sequence (Wang et al., 2001). In these applications, the HS analysis constitutes a way to synthesize the correlations between structures of various scales and various intensities; such correlations are directly responsible for intermittency e?ects and are usually hidden in the quantitative measures of multi-scalings. The HS parameters unfold such measures, when they are interpreted in an appropriate way. There have been di?erent ways in which statistical physicists and uid physicists discuss about turbulent structures. Statistical physicists are familiar with a set of statistical measures like scaling exponents which are used to characterize the decay rate of multi-point correlations. In general, these scaling exponents could be sought of a way to quantify the macroscopic (statistical) state of a
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uctuation Jeld. With uid physicists, ow structures generally refer to as a certain kind of ow patterns which are obtained by a ow visualization technique or an analytical solution of the uid equations. There have been many attempts to bridge the two, and there are strong interests for doing that, because the physical intuitions come from ow patters but only the statistical correlations can be quantitatively obtained from experimental measurements in turbulent ows. The unsatisfactory state of our understanding of turbulence is partially re ected by the fact that there is no general consensus for linking the intuition and the measurements. The HS analysis attempts to link the multi-scaling measurements with the characterization of ow patterns. The HS model allows to summarize the whole set of scaling exponents in terms of two parameters, and , each of which is related to a speciJc organizational aspect of the ensemble of uctuations. In the present study with a concrete example of ow with strong shear, we explore the possible correspondence. The study of the (relative) scaling property of turbulent uctuations in inhomogeneous ows with a shear has been conducted by several groups (see e.g. Toschi et al., 1999; Ruiz-Chavarria et al., 2000; Poggi et al., 2003). It appears that even in the presence of strong shears, the streamwise longitudinal velocity uctuations satisfy the She–Leveque scaling (She and Leveque, 1994) formula, but needs to be generalized with appropriate HS parameters and . In the previous work mentioned above, there are several versions of the analysis of She–Leveque scaling parameters. We apply here a systematic procedure for verifying the HS property and measuring the HS parameters (She and Liu, 2003). We demonstrate in this paper that such a procedure is practical for analyzing inhomogeneous pipe ow and for obtaining interesting interpretation in ow structure evolution away from the wall. The validity of the interpretations will be subject to further study with more ow conJgurations. 2. Experimental setup and ow properties The experiments were carried out in a circular pipe with its inner diameter of 105 mm and total length of 22:5 m. The inner wall of pipe was covered with epoxy so that the wall can be regarded as a smooth surface. The pipe ow was run by a blower with a centrifugal fan. Measurements were made at two streamwise locations with distances from the pipe-inlet of 4.45 and 18:25 m which will be referred to as sections S1 and S2, respectively. The Reynolds number based on the mean velocity at the center of the pipe-outlet and on the inner diameter was 1:35 × 105 . Thus, the un-tripped pipe ow was a transitional rather than a fully developed turbulent ow because of the smooth surface and small Reynolds number. A tripping wire was then added on the inner wall near the pipe-inlet throughout the whole experiments. A hot wire anemometer (Model TSI-1050) was employed to measure the velocity. The sensor was a single-wire boundary-layer probe (Model 1261) with diameter of 5 , and length of 1 mm so that a satisfactory spatial resolution can be achieved. A 16-bit analog-digital converter (ADC) was applied for data acquisition and a cassette recorder (SONY PC204Ax) was used to record the data. For catching the high-frequency small-scale uctuations, the sampling frequency was set to 48k and the velocity signals were Jltered at 20k with an analog Jlter. Measurements were performed at various radial positions. The total sampling time at each test point was 20 min so that suMcient amount of samples were acquired for statistical analyses.
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Fig. 1. Mean velocity proJle and the rms velocity uctuation as a function of the distance to the wall in the wall unit in section S2. The points S2r2–S2r13 of long time series measurement are indicated on the mean velocity proJle.
In Fig. 1, we show the mean velocity proJle and the variation of the rms turbulent uctuation. In order to clearly exhibit the bu?er region and the logarithmic region, the x-axis is converted to the wall unit and plotted in a logarithmic scale. Our closest measurement to the wall reaches a distance from the wall of order y+ ∼ 10 (in normalized wall unit). A least-squares Jt to the logarithmic region shows a relation of 1 u+ = log y+ + B; (1) k where the von KNarmNan constant k = 0:41, and B = 5:20 in section S2 and k = 0:46, and B = 6:30 in section S1. These values are consistent with those in previously reported mean velocity proJles (Hinze, 1975). By comparison between several tests under the same condition, these values are proven to be repeatable within an accuracy of ±1%. For the analyses below, the Taylor frozen hypothesis is used to convert the time scale to the length scale in the streamwise direction. From Fig. 1, one can estimate that the turbulent intensity in the near wall region (10 ¡ y+ ¡ 100) is about 15 –20%, and is around 10% in the logarithmic region and much smaller in the center region. Although this conversion is problematic, it is believed to be acceptable to the kind of qualitative and quantitative information drawn in the present work. The validity of the hypothesis in the calculation of the velocity structure functions in a channel ow is discussed in Antonia et al. (1997). In Fig. 2, the streamwise uctuating velocity u is shown for three di?erent radial locations: S2r2 is at the center of the pipe; S2r5 is in the middle of the logarithmic region; S2R13 is at the closest position to the wall that we measured (see Fig. 1). A visual inspection of three signals indicates the presence of more ascending bursts at S2r13 (near the wall) and of more descending bursts at S2r2 (center). This is conJrmed by a plot of the probability density function (PDF) of the uctuating velocity u in Fig. 3, where it is shown that P(u ) for S2r2 is skewed to have a long left wing and
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Fig. 2. Streamwise uctuating velocity. From top to bottom, we show the streamwise velocity uctuation signals at three radial locations (a) S2r2, (b) S2r5, (c) S2r13 and the corresponding smoothed velocity signals (d) S2r2, (e) S2r5, (f) S2r13 obtained by a repeated three-point average which retain only large-scale structures.
Fig. 3. The PDF of the velocity at three locations in the pipe. S2r2 is at the center of the pipe. S2r5 is near the wall and close to the bottom of the log layer. S2r13 is the closest location to the wall with y+ = 5.
P(u ) for S2r13 is skewed to the right. At the middle of the logarithmic region S2r5, the uctuating velocity just happens to have a symmetric distribution. The balanced distribution is achieved at S2r5, from an inspection of Fig. 2b, by the production of more pronounced large-scale uctuation structures. In order to illustrate more explicitly the large-scale organized motion (OM) in the logarithmic regions in the HS discussion below, we present also in Fig. 2 three corresponding “smoothed” signals obtained from the original signals with a repeated three-point average (over 100,000 times). The smoothed signals retain only the large-scale structures. We can see that, at S2r5, there are remarkable large-scale patterns that are absent in both S2r2 (center) and S2r13 (near the wall). These patterns have larger uctuation amplitudes and are more extended in length scale when compared to S2r13. We refer to them as the signature of the OM in the logarithmic region. We believe that the presence of these motions will a?ect the way we model turbulent statistics in engineering. The wall-bounded shear layer is a rich source for the production of turbulent structures whose formation and dissipation process have been subjects of intensive studies (Robinson, 1991; Schoppa
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Fig. 4. Illustration of the hierarchy of intensities Fp (‘) as a function of moment order p at several length scales, obtained at S2r2.
and Hussain, 2000). Extensive e?orts have been invested to describe behavior of coherent structures in speciJc spatial regions such as viscous sub-layer, bu?er region, logarithmic region, etc. For example, the near-wall bu?er region is the most important zone of the boundary layer in terms of turbulence energy production and dissipation (Zhou and Liu, 1987); bu?er region activity is characterized by the bursting process (Lamballais et al., 1997); both outward ejection of low-speed uid and inward sweeps of high-speed uid occur intermittently throughout the layer, etc. Onorato et al. (2000) studied the statistical properties of streamwise velocity uctuations in fully developed turbulent channel ow and found that the maximum intermittency e?ects takes place between the bu?er layer and the inner part of the logarithmic region where the bursting phenomenon is the dominant dynamical feature. It is our goal to systematically describe the change of the multi-scale and multi-intensity characteristics of turbulent structures from near the wall to the center. We found that the HS framework is adequate to summarize such information. 3. Hierarchical structure analysis The HS analysis starts with the deJnition of a hierarchy of uctuation intensities. Let v‘ denote the velocity increment across a distance ‘, v‘ = v(x + ‘) − v(x), and Sp (‘) denote the velocity structure function, Sp (‘) = |v‘ |p . Introduce a hierarchy of function Sp+1 (‘) Fp (‘) = ; p = 0; 1; 2; : : : : (2) Sp (‘) Such a hierarchy covers the mean velocity uctuation intensity F0 = S1 , and a series of increasing intensities which, as p increases, approaches to the intensity of the so-called most intermittent structures, F∞ (‘)=limp→∞ Fp (‘) In Fig. 4, we illustrate such a hierarchy of intensities by plotting Fp (‘) vs. p at several length scales, which are computed with the uctuation signal at S2r2. Therefore, one can associate each intensity with an appropriate order p which varies continuously from 0 to inJnity.
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Note that the hierarchy of uctuation intensities, Fp (‘), are both function of ‘ and of p. In other words, the hierarchy spreads over both a range of scales and a range of intensities. This is an important feature of the HS analysis: we need not only to consider turbulence to be a collection of large and small eddies (a range of scales), but also a collection of strong and weak eddies (a range of intensities). Kolmogorov (1941) has established the concept of multiple scales; we propose here to include that of multiple intensities. Because of the existence of many intensities, it is natural to look for a relation among them. She and Leveque (1994) postulated that Fp+1 (‘) = Ap Fp (‘) F∞ (‘)1− ;
(3)
where 6 1 is a constant and Ap is independent of ‘. It is interesting that one can eliminate the term F∞ (‘)1− by considering the ratio, Fp+1 (‘) Ap Fp (‘) = : (4) F2 (‘) A1 F1 (‘) Both sides of Eq. (4) can be calculated from experimental data when the empirical PDFs (or histograms) are calculated from experimental uctuation signals. Furthermore, introduce a normalization for the ratio Fp =F1 by Fp (‘)=F1 (‘) : (5) p (‘) = Fp (‘0 )=F1 (‘0 ) Then, Eq. (4) is rewritten as p+1 (‘) = p (‘) :
(6)
A log–log plot of p+1 (‘) vs. p (‘) will be called a -test: if a linearity is observed, one says that the uctuation data passes the -test and the hierarchical symmetry is satisJed. When it happens, the value of can be obtained by measuring the slope. Furthermore, it is shown (She, 1998) that hypothesis (3) implies that a general formula of the scaling exponents of the velocity structure function deJned by Sp ∼ S3p is given by 1 − p p = p + (1 − 3) ; (7) 1 − 3 where 3 = 1 and the parameter is introduced for describing the most intermittent structure as F∞ (‘) ∼ ‘ . Then, a simple algebraic manipulation gives p − (p; ) = (p − 3(p; ));
(8)
where (p; ) = (1 − p )=(1 − 3 ). When we plot p − (p; ) vs. p − 3(p; ), the slope is . This constitutes the -test. Therefore, the -test veriJes whether turbulent uctuation intensities Fp (‘) obey a generalized similarity relation (3) over a range of scales (‘1 6 ‘ 6 ‘2 ) and over a range of intensities (p1 6 p 6 p2 ). Such a hierarchical symmetry is an indication of the self-organization of the ensemble of the uctuation events. When the hierarchical symmetry is present, the parameter measures the behavior of the self-organization. When = 1, all uctuation intensities have similar ‘-dependence. A notable example is Kolmogorov’s original 1941 picture of turbulence, a complete self-similar uctuation Jeld
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(Frisch, 1995). Recently, it is found that the coherent spiral state of the two-dimensional complex Ginzburg–Landau equation is such a self-similar state (Guo et al., 2003; Liu et al., 2003). The other limit is an extreme intermittency case of = 0. In this limit, only the most intermittent structures F∞ dominates the statistical properties such as moments. Randomly driven Burgers equation (She et al., 1992; Sinai, 1992) is an example where the only dissipative structures are shocks. A popular intermittency model of turbulence of Frisch, Nelkin and Sulem (Frisch et al., 1978) falls also into the = 0 case. Smaller is, more outstanding the most intermittent structures stand with respect to the background uctuations. For example, if some mechanism generates a few outstanding bursts into a nearly self-similar random uctuating ensemble, it may generally reduce the value of (when the -test still passes) and the resulting uctuation signal becomes more intermittent. Therefore, the value of is more intuitively related to the degree of intermittency. The parameter measures how singular the most intermittent structure is. By deJnition, F∞ (‘) ∼ ‘ ; in other words, is a property of the high-order moments p → ∞. From Fig. 2, we see that the intensities Fp have not attained nearly the values of F∞ for the typical range of p between 1 and 10 in our analysis (see below). So the value we obtained in our analysis is an extrapolated values based on the reasonably convergent moments below p = 10. It may be worthwhile to argue that obtained by such extrapolation is a meaningful measure of the very intense uctuation events in a turbulent ow. The results reported below support this argument. In summary, the and tests can be performed for any multi-scaling uctuation signal (or Jeld). They examine the properties of uctuation across a range of scales and intensities, and quantify the degree of self-organization (or intermittency) by the parameter (which measures how outstanding the most intermittent structure is compared to the rms uctuations) and (which measures how singular the most intermittent structure is). These are the essence of the HS description of multi-scaling uctuation Jeld. The resulting set of scaling exponents are given by Eq. (7). For isotropic turbulence, She and Leveque (1994) estimates that = ( 23 )1=3 ≈ 0:874 and = 19 . Several measurements in, e.g., turbulence behind a cylinder (Ruiz-Chavarria et al., 1994, 1995; Benzi et al., 1995, 1996) and in direct numerical simulations (Cao et al., 1996) support the prediction. Chen and Cao (1995) has an alternative suggestion that = ( 79 )1=3 ≈ 0:92 and = 0. It will be interesting to check if the isotropic incompressible ow has an asymptotic universal scaling. 4. Results 4.1. Energy dynamics First, let us examine the energy dynamics of the eddies from the wall to the center. Fig. 5 shows the time-averaged energy spectra calculated from the uctuating velocity signals from close to wall (S2rl3) to the center (S2r2). All spectra show a wide range of excitation in Fourier modes, consistent with the impression of developed turbulence from velocity signals (see Fig. 2). However, there are some qualitative and quantitative di?erences among the spectra. In the bu?er region (from S2rl3 to S2r7), there is a short k −1 power-law range in the spectra. As the probe moves further to the center, the power-law range extends further to smaller scales, and becomes steeper (E(k) ∼ k −1:5 ). At the two points S2r2 and S2r3, the large-scale motions suddenly get depleted, probably due to the
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Fig. 5. Energy spectra obtained from velocity signals at di?erent locations away from the wall. The spectra are shifted in the vertical direction for visibility. Note a nearly k −1 spectrum near the wall.
presence of small-scale turbulent motions. This corresponds to the disappearance of the large eddies seen in Fig. 2. At the point S2r5, the spectrum seems to have the longest power-law range. In other words, in the middle of the logarithmic range, the ow has uctuations extending to a maximum range. This may have a consequence on the scaling behavior reported below. We speculate on the evolution of turbulent structures based on the change of the energy spectra. Near the wall (y+ ¡ 100), a strong shear introduces many short spikes of positive streamwise uctuations. They may be the sources for the k −1 energy spectrum. Here, the nonlinear interaction plays a role, but the cascade is not a free cascade as Kolmogorov assumes, because of anisotropic structures. As one moves further into the logarithmic region, more isotropic turbulent structures enter to the dynamics so that the cascade to small scales further proceeds, leading to steeper energy spectrum. Note that a k −1:5 energy spectrum has been observed in moderately high Reynolds number fully developed turbulence near the viscous cuto? with the presence of a spectral hump (She and Jackson, 1993). In the logarithmic region, there is an interesting mixture of large-amplitude anisotropic turbulent structures that are reminiscent of unstable wall-bounded shear ows and small-scale turbulent structures due to the cascade. In the center of the pipe, the anisotropic structures disappear due to turbulent di?usivity, and turbulence tends to be nearly homogeneous and isotropic.
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Fig. 6. PDFs of velocity increments across a range of distances ‘ (2 6 ‘ 6 384) obtained from the uctuating velocity signal (a) at the center S2r2; (b) at the logarithmic region S2r5; (c) near the wall S2r13.
4.2. - and -test It is instructive Jrst to visualize the evolution of PDFs of the velocity increments that carries the information about multi-scale and multi-intensity correlations. Figs. 6(a) – (c) show the PDFs of the velocity increments for a series of length scales (2 6 ‘ 6 380 in the unit of time resolution 1 St = 48;0000 s) at three radial locations. Although the velocity PDFs are skewed as mentioned earlier (see Fig. 3), the PDFs of velocity increments are nearly symmetric over this range of scales at all three locations. This indicates that upwards and downward uctuations are alike statistically. Near the wall at S2rl3 (Fig. 6c), the uctuation intensity is large through most of scales with a nearly Gaussian shape at large scales and a nearly exponential tail at small scales. At S2r5 (Fig. 6b), the uctuations have comparable magnitude with S2rl3, indicating that the same structures persist from S2rl3 to S2r5. This is supportive to our speculation above that anisotropic structures produced by the strong shear is dominant in the logarithmic region. At S2r2, the velocity increment has signiJcantly reduced the magnitude of the uctuations (narrower distributions in Fig. 6a). However, the general character that the PDFs change from a Gaussian shape to a nearly exponential tail is similar at the center than near the wall. They may only have some quantitative di?erence, which will be revealed by the HS analysis. Before starting the HS analysis, we verify the existence of the extended self-similarity (ESS) property (Benzi et al., 1993). This is reported in Fig. 7 which is obtained at S2r2. At other radial locations, similar ESS property is observed. The ESS analysis is a kind of pre-requisite for the HS analysis. It helps to identify the range of ‘’s and p’s over which the - and -test are performed. The range of scale ‘ and moment order p in the HS analysis below were adjusted to between
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Fig. 7. Extended self-similarity (ESS) test at two points (S2r2 and S2r13). Note that all other signals have similar ESS properties.
Fig. 8. -test for three representative data sets (S2r2, S2r5, S2r13) which are displayed vertically from top down. The slope is the measured values of .
1 2 6 ‘ 6 64 (in the units of the time resolution St = 48;000 s) and 1 6 p 6 10. Note that in our experiment, the Reynolds number is not very large to allow to see a clear inertial range from the ‘-dependence of the third-order moment. But our analysis do not require the presence of such an asymptotic inertial range, and the hierarchical symmetry scaling is a kind of generalized ESS scaling (Ching et al., 2002). In Figs. 8 and 9, the results of the - and -test are shown for three representative sets of turbulent signals: S2r2, S2r5, and S2rl3. They show a clear evidence that both - and -test pass with good straight lines, indicating that the turbulent pipe ow encompasses indeed a nicely self-organized hierarchical structures with well-deJned HS scaling properties. Furthermore, the - and -test reveal interesting transitions from the near wall (S2rl3) to the center (S2r2). From S2rl3 to S2r5 (the
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Fig. 9. -test for three representative data sets (S2r2, S2r5, S2r13) which are displayed vertically from top down. The slope is the measured values of .
Fig. 10. Scaling exponents p of the velocity structure functions at a set of points from S2r2 (center) to S2r13 (near the wall). Curves are the HS theoretical predictions based on the measured values of and . Note a rough separation into two groups for the scaling behavior before and after entering to the logarithmic region (y+ ∼ 100).
logarithmic region), the parameter changes from 0.80 for 0.88 with a signiJcant reduction of the variation range in p (‘), and the parameter has a more drastic change (from 0.095 to 0.142). Both the reduction of p (‘) and the increase of are consistent with the presence of more pronounced organized large-scale motions in the logarithmic region. From S2r5 to S2r2, the parameter changes little, whereas the parameter changes back around 0.11, close to the SL94 predicted value for isotropic turbulence. We interpret that as an evidence of the disappearance of the organized motions developed in the logarithmic region. We have observed similar transition in the measured relative (ESS) scaling exponents p shown in Fig. 10. Starting from S2rl3 (near the wall), the scaling deviates most strongly from K41, consistent
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Fig. 11. The variation of the values of and measured from the velocity uctuation signals. Note that lower values of and indicate stronger intermittency near the wall, due to a more outstanding role of wall shear structures and its singular property.
with the results of several other recent studies (see e.g. Ruiz-Chavarria et al., 2000). In the HS analysis, this is associated with the presence of very singular intense structure with smaller ∼ 0:095, and with its more dominant role in the hierarchy with smaller . Moving towards the logarithmic region, the ESS scaling approaches K41 when both and signiJcantly increase. From a HS point of view, a larger is associated with the development of large-scale less-singular pattern as the “leader”, and a large implies that the “leader” structures are less outstanding from the background uctuation structures. This is a state of uctuation which is less intermittent than the SL94 prediction. Finally, at the center (S2r2), the ESS scaling approaches again the SL94 prediction, which is consistent with the disappearance of organized motions. In Fig. 10, we present also the HS prediction of the scaling exponents with the measured and (see lines in Fig. 10). The good agreement between the experimental data (points) and HS theoretical predictions (curves) gives a strong evidence that the HS statistical structure persists in the pipe shear ow turbulence. We have thus a sound foundation to discuss the physical signiJcance of the parameters and . The quantitative results of the - and -test are reported in Fig. 11. Here, the variations of and at all radial locations show a pattern: near the wall (y+ ¡ 100), and are both smaller, indicating a more distinct role of the most intense/intermittent structures in energy dissipation (small ) and its more singular nature (small ). Our interpretation is that strong-shear driven anisotropic turbulent structures are the most intermittent structures near the wall. In the hierarchy of uctuations in this region (S2rl3–S2r7), these structures have a distinct singular character ( ¡ 0:1) and they play the most dominant role for energy dissipation. But in the logarithmic region (S2r6 –S2r3), these anisotropic structures change their character and become less distinct ( ∼ 0:88) and less singular ( ∼ 0:13); the rise in may be due to the emergence of other uctuation structures of lower intensities. There appears to have a new transition from S2r3 to S2r2 (center), where remains essentially unchanged, but drops to 0.11. This transition is somewhat similar to the breaking of the Taylor vortices in a turbulent Couette–Taylor ow (She et al., 2001) as the Reynolds number increases. It is likely that there is a breaking of a kind of large-scale smooth structures as turbulence
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moves to the center with a big reduction of the rms velocity uctuations. We speculate that an interaction between wall-shear related anisotropic structures may be the cause for this transition and this process takes place most eMciently at the center. It is seen from Fig. 11 that the scaling exponents at the center S2r2 tend to the She–Leveque (1994) prediction. This is a limited Reynolds number result. It may suggest that the She–Leveque scaling formula predict the asymptotic scaling of longitudinal velocity structure functions in an incompressible truly isotropic turbulence. The present HS analysis then leads to the speculation that the increase of intermittency towards the wall regions is due to the presence of intensive anisotropic coherent shear structures, which are destroyed during the transition from the logarithmic region to the center. Finally, we note that the actual values of and obtained from the HS analysis may vary slightly with the range of the scale ‘ and of the moment order p. One needs to be cautious to compare and obtained with di?erent procedures. However, when the procedure is determined with the deJned ranges by means of the ESS analysis, the variations in and show consistent physical meaning. Only then, the comparison of the HS parameters between di?erent ows becomes meaningful and interesting. 5. Conclusions We have carried out an experimental study of the evolution of turbulent structures from the wall to the center in a turbulent pipe ow in the framework of the hierarchical structure model. First, we show that small-scale turbulent uctuations (over a range of scales more than a decade) at all radial positions (10 6 y+ 6 3000) satisfy the She–Leveque hierarchical symmetry relation. In other words, turbulent uctuations at all positions are hierarchically organized across scales and across intensities through (3). Then, the HS analysis (- and -test) allows one to quantify the degree of multi-intensity correlations () and the degree of the singularity of the most intermittent structures (). We Jnd that the scaling exponents of the streamwise longitudinal velocity structure function are roughly separated into two groups: near-wall regions y+ ¡ 100 and far region y+ ¿ 100, the variations of and as functions of the distance from the wall give rise to a more detailed picture. Near the wall (S2rl3–S2r7), both and are smaller, indicating a more singular nature of the most intense structure (small ) that is more outstanding from the background uctuations (small ). In the logarithmic regions, both and increase signiJcantly. The velocity signal at S2r5 exhibits the rise of some large-scale patterns, which are interpreted to be associated with the small intense structure. In a turbulent Couette–Taylor ow, one observes a decrease of corresponding to the breaking of the Taylor vortices as the Reynolds number increases (She et al., 2001). We speculate that an opposite process takes place with the formation of the smooth structures. As the probe moves to the center S2r2, a further decrease of signiJes the disappearance of the large-scale smooth structures, and the almost constancy of reveals that the overall organization of the hierarchical uctuations is kept the same. This last transition may be similar to the Taylor-vortex-breaking transition in the turbulent Couette–Taylor ow. The organized large-scale motions in the intermediate logarithmic regions have not been emphasized before. Our HS analysis is able to detect such a transition, which suggests an interest of the HS analysis. Like in the turbulent Couette–Taylor ow, the - and -test can help to monitor the
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change of the properties of the ow structures. This link has long been the goal of statistical analysis of turbulent ows. Note that what the HS analysis can detect is a feature of an organized motion recorded in a streamwise velocity component; the lack of three-dimensional ow information does not allow for a detailed characterization of the organized ow structure. Yet, detecting the presence or disappearance of organized structures based on a scaling analysis (the HS analysis) may be signiJcant. Note an important limitation of the present work, that is, uctuation structures are only characterized through the relative scalings. Other quantity such as the absolute scaling of the third-order structure function related to the mean energy transfer may be important to consider in strong shear ows. Acknowledgements We have beneJted from useful discussions with many people at the LTCS of Peking University, specially Prof. Wei Zhonglei, Prof. Wei Qingding, Prof. Zhang Boyin, Prof. Gu Zhifu, Prof. Wu Jiezhi and Prof. Su Weidong and Dr. Fu Qiang. This work was supported by the projects 10032020 and 10225210 from the NNSF of China. References Antonia, R.A., Pearson, B.R., 2000. Reynolds number dependence of velocity structure functions in a turbulent pipe ow. Flow, Turbulence Combust. 64, 95–117. Antonia, R.A., Zhou, T., Romano, G.P., 1997. Second- and third-order longitudinal velocity structure functions in a fully developed turbulent channel ow. Phys. Fluids 9 (11), 3465–3471. Baroud, C.N., Plapp, B.B., Swinney, H., She, Z.-S., 2003. Scaling in three-dimensional and quasi-two-dimensional rotating turbulent ows. Phys. Fluids 15 (8), 2091–2104. Benzi, R., Ciliberto, S., Baudet, C., Massaioli, F., Tripicione, R., Succi, S., 1993. Extended self-similarity in turbulent ows. Phys. Rev. E 48, 29–32. Benzi, R., Ciliberto, S., Baudet, C., Ruiz Chavarria, G., 1995. On the scaling of three dimensional homogeneous and isotropic turbulence. Physica D 80, 385–398. Benzi, R., Biferale, L., Ciliberto, S., Struglia, M.V., Tripiccione, R., 1996. Generalized scaling in fully developed turbulence. Physica D 96, 162–181. Cao, N., Chen, S., She, Z.-S., 1996. Scalings and relative scalings in the Navier–Stokes turbulence. Phys. Rev. Lett. 76, 3711–3714. Chen, S.Y., Cao, N.Z., 1995. Inertial range scaling in turbulence. Phys. Rev. E 52, R5757. Ching, E.S.C., She, Z.-S., Su, W.D., Zou, Z.P., 2002. Phys. Rev. E 65, 066303. Coles, D.E., 1962. The turbulent boundary layer in a compressible uid. Rand Corp. Report No. R-403-PR. den Toonder, J.M.J., Nieuwstadt, F.T.M., 1997. Reynolds number e?ects in a turbulent pipe ow for low to moderate Reynolds number. Phys. Fluids 9, 3398–3409. Dubrulle, B., 1994. Intermittency in fully developed turbulence: log-Poisson statistics and scale invariance. Phys. Rev. Lett. 73, 959–962. Frisch, U., 1995. Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge. Frisch, U., Nelkin, M., Sulem, P.-L., 1978. J. Fluid Mech. 87, 719. Guo, H., Li, L., Ouyang, Q., Liu, J., She, Z.-S., 2003. A systematic study of spirals and spiral turbulence in a reactiondi?usion system. J. Phys. Chem. 118 (11), 5038–5044. Hinze, J.O., 1975. Turbulence, 2nd Edition. McGraw-Hill, New York. Kim, J., Moin, P., Moser, R.D., 1987. Turbulence statistics in fully developed channel ow at low Reynolds number. J. Fluid Mech. 177, 133–166.
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