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Structural inclination angle of near-field scalar fluctuations in a turbulent boundary layer
International Journal of Heat and Fluid Flow 81 (2020) 108521
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Structural inclination angle of near-field scalar fluctuations in a turbulent boundary layer
T
Krishna M. Talluru ,a, Kapil A. Chauhanb ⁎
a b
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia School of Civil Engineering, The University of Sydney, NSW 2006, Australia
ARTICLE INFO
ABSTRACT
Keywords: Two-point correlation Meandering plume Structure inclination angle
Two-point concentration measurements are obtained in a meandering passive-scalar plume released at five different heights within the fully-turbulent region of a high-Reynolds number turbulent boundary layer (TBL). Mean statistics of two-point concentration measurements are found to agree very well with the single-point measurements previously reported in Talluru et al. (2017a). The two-point correlation results of concentration indicate strong coherence in the scalar field similar to the large-scale coherence observed in the streamwise velocity fluctuations in a TBL (Marusic and Heuer, 2007). Particularly, the isocontours in the two-dimensional correlation map of concentration fluctuations illustrate that the scalar structures are inclined at 30∘ to the direction of the flow; such a trend is consistently observed for all the elevated plumes below z/δ ≤ 0.33. This observation of steeper inclination angle of scalar structures relative to the inclination angle of large-scale velocity fluctuations in a TBL is explained using the physical model put forth by Talluru et al. (2018). Most importantly, these results provide insights on the differences in the structural organisation of a passive scalar plume in the near- and the far-field regions.
1. Introduction The local concentration of a scalar (pollutant, heat, etc.) in a turbulent flow exhibits complex features over a broad range of length and time scales. As such, turbulent advection is crucial in most natural and engineering processes such as atmospheric dispersion, mixing and combustion. Although the underlying interaction between the turbulent velocity field and scalar fluctuations is not linear, there are many similarities between the statistics of scalar fluctuations and those of velocity, e.g. in a heated turbulent jet (Antonia et al., 1983). The possibility of exploiting any similarity between velocity and scalar fields in wall-bounded turbulent flows is also of practical significance, for example, prediction of ground level concentration of a pollutant emitted from an elevated source and vice-versa. Thus, many studies are aimed at characterising the role of coherent features of velocity in the dispersion and transport of a scalar (e.g. Talluru et al., 2018; Vanderwel and Tavoularis, 2016). When a contaminant plume is released into a turbulent boundary layer (TBL), the pollutants are mixed by the turbulent eddies in the flow. Further, when the size of the pollutant plume is relatively smaller than the turbulent eddies, the plume meanders and becomes highly intermittent (Fackrell and Robins, 1982; Hanna and Insley, 1989). This phenomenon leads to large fluctuations in the ⁎
pollutant concentration that are of the same order of magnitude as the mean concentration (Csanady, 1973). Such extreme fluctuations are of significant concern when the pollutant is a highly toxic or flammable material. Most dispersion models are capable of predicting the mean pollutant concentration and the amplitude of fluctuations about the mean. For example, Chatwin and Sullivan (1990) developed a simple two-parameter relationship between the mean and the standard deviation of concentration fluctuations in a statistically steady self-similar turbulent shear flows. Later, Sawford and Sullivan (1995); Mole and Clarke (1995) further developed the model to predict the first four moments of the concentration, but they concluded that such simple models cannot describe the near-field behaviour of a plume. Similar analyses were carried out by Lewis et al. (1997) on atmospheric field data, and more recently, by Schopflocher and Sullivan (2005) using wind tunnel data. Several other studies have developed models for the probability density functions (PDF) of the concentration. For instance, Yee and Chan (1997) proposed a three parameter clipped-Gamma distribution for a highly intermittent plume, which was tested against experimental data by Yee (2009) and Klein and Young, 2011. A more general approach for developing PDF models was based on meandering models
Corresponding author. E-mail addresses: [email protected] (K.M. Talluru), [email protected] (K.A. Chauhan).
https://doi.org/10.1016/j.ijheatfluidflow.2019.108521 Received 2 June 2019; Received in revised form 26 November 2019; Accepted 27 November 2019 0142-727X/ © 2019 Elsevier Inc. All rights reserved.
International Journal of Heat and Fluid Flow 81 (2020) 108521
K.M. Talluru and K.A. Chauhan
(Gifford, 1959; Yee et al., 1994; Luhar et al., 2000), pair-of-particles Lagrangian stochastic models ((Durbin, 1980; Sawford and Hunt, 1986), numerical models (Andronopoulos et al., 2001; Milliez and Carissimo, 2008), and large-eddy simulations (Xie et al., 2004; Philips et al., 2013), among others. Most importantly, none of the above models account for the spatial coherence of the scalar concentration field. This seriously limits the applicability of such models in predicting the scalar concentration at any point of space based on the information of scalar at a known location, in particular, in the near-field region, where the fluctuations about the mean are very strong due to plume meandering. To sum up, the existing models can estimate the statistical behaviour at a given point, but they fail to predict the behaviour of the plume simultaneously at several points in space. The present study aims to fill this gap by characterising the spatiotemporal correlation of concentration fluctuations in the vertical direction using experimental data. Specifically, this study focuses on the near-field behaviour of a passive plume, where the plume exhibits different characteristics compared to the far-field region. A long term aim of this study is to achieve real time predictions of concentration by developing linearised and stochastic models (machine learning techniques) based on the spatial correlations obtained here. Such models have been recently developed for predicting the behaviour of velocity in turbulent wall-bounded flows (Illingworth et al., 2018; Srinivasan et al., 2019; Sasaki et al., 2019). No such models are currently available for scalar concentration as there is no experimental data available to test them. It is hoped that the present paper will go some way towards fulfilling this need. The paper is organised as follows. At first, a brief discussion is presented on the two well-accepted results in turbulent boundary layers and plume dispersion that form the basis of the current study. Then we briefly outline our experiments of two-point measurements of instantaneous concentration in a point-source scalar plume. Results of time-averaged statistics, two-point correlation and spatial coherence follow thereafter.
1.2. Meandering of a slender passive scalar plume Observations of a smokestack plume in the atmosphere show that its instantaneous structure is puffy with narrow sections scattered between wide ones, and the plume meanders in a highly variable manner (Weil, 2012). This variability is caused by the turbulent wind field in which the plume is fully embedded, and the scale of atmospheric turbulence that influences the plume fluctuations will change as the plume size increases Sawford (1985). For example, in the near-field region, when the plume is smaller than the turbulent eddies, the plume is swept up and down about the plume axis by the turbulent eddies, but when the plume is larger than the eddies, i.e. in the far-field region, the eddies no longer advect the plume but merely cause minor fluctuations deep within the plume (Hanna and Insley, 1989; Yee et al., 1993). The meandering of plume results in the intermittent high and low concentrations in the time series signal obtained at a single point in space as observed in laboratory wind tunnel (Fackrell and Robins, 1982; Nironi et al., 2015) and full-scale atmospheric (Yee et al., 1993) measurements. A natural question that arises here is whether the structure of scalar fluctuations is similar to that of inclined velocity structures in a TBL, and secondly, how that structure varies as the plume size increases from the near-field region to the far-field region. The latter question is partly addressed in a recent numerical study of scalar dispersion in a turbulent channel flow (Srinivasan and Papavassiliou, 2011). They observed that the concentration fluctuations have a similar behaviour as that of velocity structures in the far-field region. However, little is known about the organisation of scalar structures in the near-field region. Recently, Talluru et al. (2018) conducted a study of point source plumes released at eight different elevations in a TBL and presented a physical model to explain the relative organisation of the large-scale coherent velocity structures and the concentration fluctuations in the plume. Through flow visualisation as well as joint-pdfs (probability density functions) and cross-spectra analysis, they comprehensively showed that the large-scale turbulent eddies cause meandering in the near-field region of the plume. Talluru et al. (2018) also observed that above the plume centreline, low-speed structures have a lead over the meandering plume, while high-momentum regions are seen to lag behind the plume below its centreline. The present work seeks to test the hypothesis proposed in Talluru et al. (2018) by performing two-point correlations of concentration measurements and determining the inclination angle of scalar structures. Confirming or denying our hypothesis will enable us to answer if the organisation of a passive scalar plume is same in the near- and the far-field regions, and how the organisation may be affected by the plume meandering.
1.1. Inclination angle of streamwise velocity fluctuations in a turbulent boundary layer Inclined structures, typically attached to the wall, are well-known features of streamwise velocity fluctuations in a turbulent boundary layer. One of the earliest studies that reported this behaviour is the flow visualisation study in a zero pressure gradient TBL (Head and Bandyopadhyay, 1981). They reported that at moderately high Reynolds number (Reθ > 2000), the TBL consisted of elongated hairpin vortices, inclined at a characteristic angle of approximately 45∘ to the wall. They suggested that the hairpin vortices could organise themselves at high Reynolds numbers to form larger structures that are typically oriented at an angle of 20∘ to the wall. The view of hairpin packets was further confirmed in the high-resolution PIV study of a turbulent boundary layer across a range of Reynolds number 930 ≤ Reθ ≤ 6845 (Adrian et al., 2000). They observed the inclined coherent features in the streamwise-wall-normal plane across the entire range of Reθ. Other studies (e.g., Ganapathisubramani et al., 2003; 2005; Hutchins et al., 2005) in laboratory turbulent boundary layers also indicated that the average coherent structure in the logarithmic region has a low inclination angle made up of organised packets of attached eddies. A more conclusive evidence was provided by Marusic and Heuer (2007), who showed that the two-point correlation map of streamwise velocity fluctuations obtained in laboratory and atmospheric boundary layers presented a characteristic inclination angle of 14∘. Very importantly, they observed that the inclination angle remained invariant over three orders of magnitude change in Reynolds number, 103 ≤ Reτ ≤ 106.
2. Experimental details Experiments are performed in the boundary layer wind tunnel at the University of Sydney. Full details of the facility are given in Talluru et al. (2017a) and a schematic of the test rig is shown in Fig. 1. In the present study, measurements are conducted in a turbulent boundary layer developing over a streamwise fetch of 13 m at a nominal free stream velocity (U∞) of 10 m/s. The boundary layer thickness (δ) and the friction velocity (Uτ) at the measurement location are approximately 0.31 m and 0.37 m/s, respectively, yielding a friction Reynolds number, Reτ ≡ δUτ/ν ≈ 7850 (ν is the kinematic viscosity of air). Here, upper case (e.g., C) and lower case letters (e.g., c) indicate mean and fluctuating quantities, respectively, and z represents the wallnormal coordinate. A point source tracer gas (a mixture of 1.5% isobutylene and 98.5% Nitrogen) is released at five different heights, sz/δ = 0.004, 0.044, 0.1, 0.25 and 0.33, at a streamwise distance sx / = 1 upstream of the measurement location, specifically to investigate the near-field behaviour of the plume. In our previous study (Talluru et al., 2017a), we found that δ does not change significantly (less than 5%) between the 2
International Journal of Heat and Fluid Flow 81 (2020) 108521
K.M. Talluru and K.A. Chauhan
Fig. 1. Schematic of the experimental set-up used for obtaining two-point measurements of concentration in the vertical direction across the plume.
source and the measurement locations. For this reason, sz is normalised by δ at the measurement location throughout this paper. The tracer gas is released isokinetically (i.e., the source velocity is matched with the local mean velocity of the flow) into the turbulent boundary layer. The sensor used for these experiments is the photoionisation detector (PID) manufactured by Aurora scientific, Inc. The sensor is calibrated by supplying the tracer gas at different concentrations and recording the sensor voltage output. A second-order polynomial is then fitted between the gas concentration and the sensor voltage during the calibration. Full description of the calibration procedure can be found in Talluru et al. (2019a). The frequency response of the PID sensor is measured using a custom built test rig that consists of a chopper wheel delivering a quasi-impulse of tracer gas to the PID sensor (see Talluru et al., 2017b, for details). Based on this test, the frequency response of PID is estimated to be 400 Hz. Hence, the fluctuating voltage signal from PID is low-pass filtered at a cut-off frequency of 400 Hz. The filtered signal is then sampled at 4096 Hz and is digitised using a 24-bit National Instruments A/D card, NI 4303. As shown in the schematic (cf. Fig. 1), a stationary PID is positioned at the source-height, z = sz , while a second PID is traversed along z. The spanwise gap between the two PIDs is approximately 3 mm, in accordance with the results of Metzger and Klewicki (2003), who showed that the PID has negligible influence on an adjacent probe when separated by a distance greater than 3d, where d = 0.76 mm is the inlet diameter of PID probes used in this study. It is important to note that the experimental test-rig and the measurement procedures used in this study are similar to that of Talluru et al. (2017a), however, the present study is unique and reports simultaneous measurements of concentration at two points in the plume to characterise the structure of scalar fluctuations in the flow. To our knowledge, such measurements using point-detection probes have not been reported before.
concentration in ground and elevated source plumes, we will briefly discuss the models that have been previously proposed for the mean concentration in a plume. Most continuous source plume models are based on the classical (Fickian) differential equation of diffusion, which have functions of Gaussian type as solutions. However, such models imply random and uncorrelated kind of turbulence, which is not characteristic of atmospheric turbulence (see Hutchins et al., 2012, for a full discussion on velocity length scales in atmospheric surface layer). Later, Gifford (1959) analysed a more generalised form of the diffusion equation for a meandering plume and showed that the Gaussian form is the asymptotic solution based on the central limit theorem in probability theory. The model was further developed into the reflectedGaussian model by Fackrell and Robins (1982), where the wall effect on the plume spread is accounted for.
C (z ) Cmax
exp
C (z ) Cmax
exp
z (1)
z
B
z
sz
2
+ exp
z
B
z + sz z
2
(2)
For a ground level source, C is found to have an exponential form as defined in Eq. 1, where Cmax represents the maximum concentration and α is the exponent that depends on surface roughness (Csanady, 1973). In our case, we found a value of = 1.5, which is close to the value reported by Fackrell and Robins (1982). In contrast, the mean concentration profiles in the elevated plumes show a reflected Gaussian behaviour, which is defined in Eq. 2, where B = ln(2) . The results shown in Fig. 2 are consistent with the experimental results previously reported by Fackrell and Robins (1982) and Nironi et al., 2015 in rough wall turbulent boundary layers. c (z )
3. Validation of two-point concentration measurements
c max
As a first step, we compare the statistics of concentration fluctuations obtained in the current study against single-point measurements reported in Talluru et al. (2017a) as a means of validating the two-point concentration measurements. Fig. 2(a) and (b) shows the normalised profiles of the mean (C/Cmax) and the root mean square (abbreviated as r.m.s) (σc/σcmax) of concentration along z when the plume is released at five different heights within the TBL. The open symbols in the figures represent the statistics obtained from the traversing PID, the solid symbols are from the stationary PID, and the grey solid lines represent the statistics taken from Talluru et al. (2017a). It is observed that the statistics from both the PID probes agree very well with those of Talluru et al. (2017a) to within ± 2%. Before commenting on the behaviour of mean statistics of
exp
B
z
sz z
2
(3)
On the other hand, the r.m.s of concentration fluctuations (σc) for all sources has a simple Gaussian behaviour along z as defined in Eq. 3. Note that δz and σz in eqns. 1 - 3 are the plume half-widths based on the mean and the r.m.s profiles of concentration, respectively. In general, the plume half-width is defined as the distance from the plume centreline, where the maximum concentration drops to half its value. A closer look at the profiles in Fig. 2 reveals that δz and σz vary with source height. The relationship between δz and σz and the turbulence intensities in the flow has been extensively discussed in our recent study Talluru et al. (2019b), and therefore will not be repeated here. It is known that no similarity exists between the mean statistics of concentration and velocity (or moments of their fluctuations). Nonetheless, it has been observed that a passive scalar plume meanders 3
International Journal of Heat and Fluid Flow 81 (2020) 108521
K.M. Talluru and K.A. Chauhan
because of the separation between the length-scales in the turbulent flow and the plume, i.e., when the integral length scale of velocity fluctuations is large compared to the plume width or the source diameter (Fackrell and Robins, 1982; Karnik and Tavoularis, 1989; Hanna and Insley, 1989). Since the integral length scale of a turbulent flow characterise the extent of coherent motions, it is anticipated that the coherent structures in the velocity will impose the instantaneous organisation of concentration field. In the remainder of this paper, we will identify, and then characterise the organisation of scalar fluctuations in the streamwise/wall-normal plane, particularly focussing on the inclination of scalar structures in the mean flow direction.
In contrast, the behaviour of autocorrelation functions Rcc(sz, z) in the wall-normal direction is not similar for different source heights, as evident in Fig. 3(b), which is anticipated since the spread of plume in the wall-normal direction (i.e., the plume half-width) varies with the source height. In this plot, the wall-normal shift (Δz) is presented with respect to the source height as (z sz )/ . It should be noted that the maximum in the vertical autocorrelation functions is less than 1, which is due to the gap (= 3 mm) between the two PID sensors in the transverse direction. It is seen that Rcc(sz, z) is not symmetric about the plume centreline especially when the plume spread is influenced by the wall as a result of the non-homogeneity of turbulence in the wallnormal direction. Interestingly, the autocorrelation functions for the plumes (sz/δ = 0.044, 0.1 and 0.25) exhibit both positive and negative correlations. This implies that there are instances at which the concentration fluctuations (about the mean value) at the plume centreline and at some distance away from the centreline are of opposite signs. In other words, this result presents a convincing evidence for the meandering behaviour of the plume in the vertical direction. Further, the magnitude of negative correlation drops with increasing source height from sz/δ = 0.044 to 0.25, which aligns well with the trend of wallnormal turbulence intensity in a zero pressure gradient TBL (see figure7 in Talluru et al., 2019b).
4. Single- and two-point correlation functions Using the time-series information of concentration fluctuations at two distinct points, the autocorrelation function along the streamwise direction is computed using Eq. 4. In the wall-normal direction, Eq. 5 represents the correlation coefficient at zero time-shift between the signals at the plume centreline (stationary PID) and at another z-location (traversing PID).
R cc (sz , x ) = R cc (sz , z ) =
c (sz , x ) c (sz , x + x ) , [ c (sz ) c (sz )]
c ( s z ) c (z ) , [ c (sz ) c (z )]
(4)
5. Two-point correlation and inclination angle
(5)
Before discussing the two-point correlation map of concentration fluctuations, we first examine the time-series signals of concentration for the plume released at sz/δ = 0.004, i.e., the ground level source in our measurements. Fig. 4 shows a sample of 1.5 second record of concentration measurements at the wall (z = 0 ) and at another location in the log-region of TBL (i.e., z/ = 0.06). One can readily appreciate the qualitative similarity of these two signals, indicating that there is a strong correlation between the signals, however with a finite shift in time. This similarity is quantified using the correlation function which also provides an estimate of the time-shift or equivalently the spatialshift ( x = Uc t ). The correlation coefficient, Rcc between two signals is defined as,
It is important to understand that the temporal information of concentration fluctuations is converted to the spatial domain using the relationship x = Uc t, where Δt is the time-lag between two concentration signals and Uc is the convection velocity taken to be the mean velocity at the corresponding wall-normal height. Note that since the plume evolves rapidly in the streamwise direction, particularly close to the source, Taylor’s hypothesis of frozen turbulence cannot be applied to the scalar field precisely. However in this study, we aim to compare the inclination angle of scalar coherence with the inclination angle of streamwise velocity fluctuations, where the Taylor’s hypothesis is extensively used to convert temporal domain to a spatial one. Thus to make an equivalent comparison the relation, x = Uc t is also used for the scalar field. This aspect is further discussed in section 5. The results of spatial autocorrelation functions computed with reference to the plume source height along x and z are plotted in Fig. 3(a, b), respectively. Looking at Fig. 3(a), it is evident that the correlation of concentration fluctuations sustains for longer distance in the case of a ground level source. The integral length scale ( c ) (which is obtained as c = 0 Rcc dx ) for the ground level plume is found to be five times larger than that for the elevated plumes. This can be explained as follows. In a ground level plume, the concentration fluctuations tend to persist for longer time since the streamwise mean velocity is relatively low in the near-wall region. Further, since the concentration flux is zero at the wall, longer correlations and higher integral time-scales of concentration fluctuations are observed in the ground level plume in comparison to the elevated plumes. This is further corroborated by the fact that, for a ground plume, the plume width in the spanwise direction is larger than in the vertical direction (see Talluru et al., 2019b, for further discussion). On the other hand, the turbulence in the flow forces the elevated plume to sway above and below the centreline causing intermittency, which in turn, causes the correlation to drop quickly. Interestingly, the curves of Rcc(sz, Δx) are almost identical for all the elevated sources. This behaviour can be understood by considering the distribution of integral length scales of velocity fluctuations ( u ) in the TBL. In our previous work (Talluru et al., 2018), we observed that u is constant in the region 0.1 ≤ z/δ ≤ 0.5, which implies the velocity structures will impose identical meandering of the elevated plumes. Hence, Rcc(sz, Δx) are expected to be similar in the plumes released in that region of the TBL.
R cc (z, x ) =
c (sz , x ) c (z , x + x ) , [ c (sz ) c (z )]
(6)
where overbar indicates time average. The correlation coefficient between the sample signals shown in Fig. 4 is plotted in Fig. 5, along with the correlation coefficient between c(z ≈ 0) and signals at other z-locations. High levels of correlation are observed across the plume, however, the peak correlation magnitude decreases with increasing distance from the wall. Further, it is seen that with increasing distance from the wall, the location of peak correlation occurs at a non-zero shift on the abscissa. Thus, one can define the inclination angle, = arctan[(z sz )/ x *], where z is the wall-normal position in the plume and Δx* is the separation distance corresponding to the peak value in Rcc. Similarly, the inclination angles inferred from the correlation functions at different heights in the plume are calculated, and the average angle for the ground-level source is found to be 29∘, as indicated in Fig. 6(e) for sz / = 0.004 . The above procedure is repeated for the plume released at four other source locations that span the near wall, log-, and outer-regions of the TBL. The average inclination angle is calculated using correlation in the region where σc(z)/ σc(sz) > 0.2. Contour maps of correlation coefficient Rcc are plotted as a function of Δx/δ and z/δ in Figs. 6(a-e) for better visualization of the coherence in the scalar field and its inherent inclination. It is expected that the cross-correlation at the source height along the dot-dashed lines in each of the Figs. 6(a-e) should be symmetrical about x/ = 0 as it also represents the autocorrelation function since the two PID sensors are at the same vertical location. However, there is slight asymmetry in the 4
International Journal of Heat and Fluid Flow 81 (2020) 108521
K.M. Talluru and K.A. Chauhan
Fig. 2. Normalised profiles of (a) mean (C/Cmax) and (b) r.m.s (σc/σcmax) of concentration in the vertical direction. Open symbols represent the data from the traversing PID in a point source plume released at different source heights; sz/δ = 0.004 ( ); 0.044 ( ); 0.1 ( ) ; 0.25 ( ); 0.33 ( ). Solid symbols represent the measurement from the stationary PID and the grey solid lines represent the data taken from Talluru et al. (2017a).
Fig. 3. (a) Autocorrelation functions in the streamwise direction using Eq. (4). (b) Zero time-shift correlation coefficient in the wall-normal direction computed as per Eq. (5). Symbols and colours are same as in Fig. 2.
5
International Journal of Heat and Fluid Flow 81 (2020) 108521
K.M. Talluru and K.A. Chauhan
Fig. 4. Sample records of simultaneous signals of concentration at the wall and a point in the log-region of turbulent boundary layer. Horizontal dashed lines indicate mean concentration values. The dashed rectangles highlight the similarity in the two signals with a finite time shift.
Fig. 5. Cross correlation of c (z = 0) at the wall and c(z > 0) at elevated positions in the plume released at ground-level. Red lines indicate the magnitude of Δx*.
cross-correlation for low levels of correlations at the source height, which is due to the gap between two PID sensors in the spanwise direction. The nominal inclination angle of scalar structure is found to be approximately 30∘ for plumes released at four other source locations, sz/ δ = 0.045, 0.1, 0.25 and 0.33. Note that the height of ordinate axis in each sub-plot is proportional to the plume spread. Some inferences can be made by comparing the average structure of concentration fluctuations against that of velocity structures previously reported in Hutchins et al., 2011 and Talluru et al. (2014). Firstly, these results show that the size of correlated regions scale with δ, suggesting that scalar structure is governed by length scales in the outer-region of TBL. Recently, Talluru et al. (2018) provided the evidence for the role of large-scale velocity fluctuations in the scalar transport across the TBL through their role in streamwise scalar flux. Secondly, it is noticed that the positively correlated regions of concentration fluctuations are inclined to the horizontal axis at θ ≈ 30∘ within the region, 0.5 (z sz )/ z 0.5, for all sources in the fully turbulent region, i.e. sz/δ ≤ 0.33. This angle is considerably steeper than the corresponding inclination angle of ≈ 14∘ noticed in the
Fig. 6. Two dimensional two-point correlation map of concentration fluctuations in the streamwise/wall-normal plane at five source heights, sz / = (a) 0.33; (b) 0.25; (c) 0.1; (d) 0.045 and (e) 0.004. Contour levels are in increments of 0.15 from 0.1 to 1. The dashed lines represent the inclination angles indicated on the top-right of each sub-plot.
correlations of streamwise velocity structures (Marusic and Heuer, 2007). Here, we revisit the question about the validity of Taylor’s hypothesis, and evaluate its effect on the structure inclination angle. The two main factors that can affect the calculation of the inclination angle are (i) plume meandering and (ii) estimate of the convection velocity of scalar plume. The effect of these two factors on the inclination angle is discussed in the following. First, in these experiments, the plume width in the vertical direction is typically about 0.1δ ( ~ 0.03 m) and the wall-normal velocity fluctuations are about σw ≈ 0.4 m/s at the source height. Therefore, 6
International Journal of Heat and Fluid Flow 81 (2020) 108521
K.M. Talluru and K.A. Chauhan
the time for the plume to meander from the centreline to the edge of the plume is O(0.1) seconds. It is then reasonable to consider that this is the time over which the instantaneous structure of the plume changes significantly and Taylor’s hypothesis cannot be applied to scalar field over time lags that are O(0.1) seconds. During this time the mean flow within the boundary layer travels a distance of approximately 0.6 m or 2δ in the streamwise direction. However, looking at the results in Fig. 6, we see that the correlation length scales for scalar are much shorter ~ 0.2δ (corresponding to time lead/lag of O(0.01) seconds). Thus the time scale of meandering is much larger compared to the time over which significant correlation persists between two distinct positions in the plume. Hence the correlation of scalar fluctuations over shorter distances is not influenced by the meandering, justifying the use of Taylor’s hypothesis in estimating the inclination angle of scalar structures. Secondly, regarding the uncertainty about the inclination angle due to error in estimating the convection velocity, it is recalled that the inclination angle is equivalent to tan 1 [ z /(Uc· t )], where Δt is the time-lag corresponding to the peak-correlation between two signals separated by distance Δz. In our analysis we have considered Uc = U (z ) resulting in an inclination angle of ~ 30∘. Now if the actual convection velocity is within ± 10% of the local mean velocity, the resultant inclination angle would be between 28 and 33 degrees (note that lower convection velocity leads to larger inclination angle). Even a 50% variation of the convection velocity from U considered here results in estimates of inclination angle between 21∘ and 40∘. Thus the observed deviation of structure inclination angle of scalar from the well-known 14∘ of u-fluctuations is robust. In a recent study, Renard and Deck (2015) established that the convection velocity of streamwise fluctuations closely matches with the local streamwise velocity across most part of turbulent boundary layer except in the very near wall region (y/δ ≤ 0.004). Since the source height of the scalar plume sz/ δ ≥ 0.004 for all the cases in our study, and for the passive scalar, advection (U˜ · C˜ / x ) is significant for its streamwise transport; the findings of Renard and Deck (2015) supports the use of mean velocity as convection velocity of the scalar as a best estimate. The phenomenological model of Talluru et al. (2018) that outlines the preferential organisation of the scalar motions with respect to the large-scale low- and high-speed structures in the flow provides an explanation of this observation. A schematic of the physical model is presented in Fig. 7, where a meandering plume along with a cluster of low- and high-speed regions are indicated. In this schematic, the lowspeed and high-speed regions of velocity fluctuations are indicated at 14∘ to the horizontal, as the correlation between similarly signed (lowor high-speed) fluctuations will contribute to the 14∘ angle observed in
numerous past studies (Adrian, 2007; Hutchins et al., 2011; Marusic and Heuer, 2007; Talluru et al., 2014). Recently, Talluru et al. (2018) have shown that above the plume centreline, low-speed regions in velocity are correlated with high-concentration scalar regions, resulting in uc < 0, whereas below the centreline, high-speed regions are correlated with the high-concentration region resulting in uc > 0 . Thus, instantaneously, even though the velocity fluctuations might have the same sign across the whole plume width (at a nominal angle of 14∘), the concentration field itself is not consistently correlated with velocity fluctuations. Regions of coherent scalar fluctuations in the meandering plume thereby attains a much steeper inclination angle as shown in Fig. 7, that is approximately 30∘, as observed in this study. Thus, we find that although the mean and r.m.s of concentration fluctuations has a symmetric distribution (Gaussian type), instantaneously it has a characteristic inclination angle that manifests as the inclination observed in the correlation map of Rcc. Thus, the degree of meandering is not symmetric about the plume centreline and has a spatial preference. In a recent numerical study of a turbulent channel flow, Srinivasan and Papavassiliou (2011) showed that the forward correlations of scalar structures are oriented at an angle of 76∘ with the vertical (equivalent to 14∘ to the horizontal), which is different than present results. This difference could be attributed to two reasons - (i) how the scalar is released in the flow or the computational domain and (ii) whether the plume is in the near- or the far-field region. In our wind tunnel experiments, the scalar is released via a point source and the measurements are performed in the near-field region, where the plume is still meandering. On the other hand, Srinivasan and Papavassiliou (2011) studied a flow, where the scalar is released uniformly from a line source in the wall-normal plane of the computational domain. This implies that there is no concentration gradient in the vertical direction (or a Gaussian profile) in the numerical simulations performed by Srinivasan and Papavassiliou (2011). Further, their simulations are in the far-field region and hence their results are not directly comparable to the present study. We would like to emphasise that the observations made in this study are closely linked to the meandering nature of the plume. It is very likely that the correlation map of concentration fluctuations further downstream, i.e., in the farfield region, where the scalar plume is thoroughly dispersed, will show an inclination angle of 14∘, consistent with the inclination of velocity structures in a turbulent boundary layer. It would be insightful to further investigate the inclination angle of scalar structures (released via a point source) at large source distances in the future studies.
Mean flow
Meandering plume
14o 30o
negative
positive
Fig. 7. A schematic of the inclination angles of velocity (14∘, dashed lines) and scalar (30∘, dot-dashed line) structures in a meandering plume based on the physical model given by Talluru et al. (2018). 7
International Journal of Heat and Fluid Flow 81 (2020) 108521
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6. Conclusions
dimensional conditional structure of a high Reynolds-number turbulent boundary layer. J. Fluid Mech. 673, 255–285. Illingworth, S.J., Monty, J.P., Marusic, I., 2018. Estimating large-scale structures in wall turbulence using linear models. J. Fluid Mech. 842, 146–162. Karnik, U., Tavoularis, S., 1989. Measurements of heat diffusion from a continuous line source in a uniformly sheared turbulent flow. J. Fluid Mech. 202, 233–261. Klein, P.M., Young, D.T., 2011. Concentration fluctuations in a downtown urban area. Part I: Analysis of joint urban 2003 full-scale fast-response measurements. Environ. Fluid Mech. 11 (1), 23–42. Lewis, D.M., Chatwin, P.C., Mole, N., 1997. Investigation of the collapse of the skewness and kurtosis exhibited in atmospheric dispersion data. Il Nuovo Cimento 20 (3), 385–398. Luhar, A.K., Hibberd, M.F., Borgas, M.S., 2000. A skewed meandering plume model for concentration statistics in the convective boundary layer. Atmos. Environ. 34 (21), 3599–3616. Marusic, I., Heuer, W.D.C., 2007. Reynolds number invariance of the structure inclination angle in wall turbulence. Phys. Rev. Lett. 99 (11), 114504. Metzger, M.M., Klewicki, J.C., 2003. Development and characterization of a probe to measure scalar transport. Meas. Sci. Tech. 14 (8), 1437. Milliez, M., Carissimo, B., 2008. Computational fluid dynamical modelling of concentration fluctuations in an idealized urban area. Bound. Layer Meteorol. 127 (2), 241–259. Mole, N., Clarke, E.D., 1995. Relationships between higher moments of concentration and of dose in turbulent dispersion. Bound. Layer Meteorol. 73 (1–2), 35–52. Nironi, C., Salizzoni, P., Marro, M., Mejean, P., Grosjean, N., Soulhac, L., 2015. Dispersion of a passive scalar fluctuating plume in a turbulent boundary layer. Part I: Velocity and concentration measurements. Bound. Layer Meteorol. 156 (3), 415–446. Philips, D.A., Rossi, R., Iaccarino, G., 2013. Large-eddy simulation of passive scalar dispersion in an urban-like canopy. J. Fluid Mech. 723, 404–428. Renard, N., Deck, S., 2015. On the scale-dependent turbulent convection velocity in a spatially developing flat plate turbulent boundary layer at Reynolds number Reθ = 13000. J. Fluid Mech. 775, 105–148. Sasaki, K., Vinuesa, R., Cavalieri, A.V.G., Schlatter, P., Henningson, D.S., 2019. Transfer functions for flow predictions in wall-bounded turbulence. J. Fluid Mech. 864, 708–745. Sawford, B.L., 1985. Lagrangian statistical simulation of concentration mean and fluctuation fields. J. Climate App. Meteorol. 24 (11), 1152–1166. Sawford, B.L., Hunt, J.C.R., 1986. Effects of turbulence structure, molecular diffusion and source size on scalar fluctuations in homogeneous turbulence. J. Fluid Mech. 165, 373–400. Sawford, B.L., Sullivan, P.J., 1995. A simple representation of a developing contaminant concentration field. J. Fluid Mech. 289, 143–157. Schopflocher, T.P., Sullivan, P.J., 2005. The relationship between skewness and kurtosis of a diffusing scalar. Bound. Layer Meteorol. 115 (3), 341–358. Srinivasan, C., Papavassiliou, D.V., 2011. Direction of scalar transport in turbulent channel flow. Phys. Fluids 23 (11), 115105. Srinivasan, P.A., Guastoni, L., Azizpour, H., Schlatter, P., Vinuesa, R., 2019. Predictions of turbulent shear flows using deep neural networks. Phys. Rev. Fluids 4 (5), 054603. Talluru, K.M., Baidya, R., Hutchins, N., Marusic, I., 2014. Amplitude modulation of all three velocity components in turbulent boundary layers. J. Fluid Mech. 746, R1. Talluru, K.M., Hernandez-Silva, C., Chauhan, K.A., 2019. A robust calibration technique for concentration measurements using ionisation detectors. Meas. Sci. Tech. 30 (7), 075103. Talluru, K.M., Hernandez-Silva, C., Philip, J., Chauhan, K.A., 2017. Measurements of scalar released from point sources in a turbulent boundary layer. Meas. Sci. Tech. 28 (5), 055801. Talluru, K.M., Hernandez-Silva, C., Philip, J., Chauhan, K.A., 2017. Measurements of velocity and concentration in a high Reynolds number turbulent boundary layer. 10th International Symposium on Turbulence and Shear Flow Phenomena, Chicago, USA. Talluru, K.M., Philip, J., Chauhan, K.A., 2018. Local transport of passive scalar released from a point source in a turbulent boundary layer. J. Fluid Mech. 846, 292–317. Talluru, K.M., Philip, J., Chauhan, K.A., 2019. Self-similar spectra of point source scalar plumes in a turbulent boundary layer. J. Fluid Mech. 870, 698–717. Vanderwel, C., Tavoularis, S., 2016. Scalar dispersion by coherent structures in uniformly sheared flow generated in a water tunnel. J. Turb. 17 (7), 633–650. Weil, J.C., 2012. Atmospheric dispersion. In: Fernando, H.J. (Ed.), Handbook of Environmental Fluid Dynamics, Volume Two: Systems, Pollution Modeling and Measurements. CRC press, pp. 163–174. Xie, Z., Hayden, P., Voke, P.R., Robins, A.G., 2004. Large-eddy simulation of dispersion: comparison between elevated and ground-level sources. J. Turb. 5 (31), 1–3. Yee, E., 2009. Probability law of concentration in plumes dispersing in an urban area. Environ. Fluid Mech. 9 (4), 389–407. Yee, E., Chan, R., 1997. A simple model for the probability density function of concentration fluctuations in atmospheric plumes. Atmos. Environ. 31 (7), 991–1002. Yee, E., Chan, R., Kosteniuk, P.R., Chandler, G.M., Biltoft, C.A., Bowers, J.F., 1994. Incorporation of internal fluctuations in a meandering plume model of concentration fluctuations. Bound. Layer Meteorol. 67 (1–2), 11–39. Yee, E., Kosteniuk, P.R., Chandler, G.M., Biltoft, C.A., Bowers, J.F., 1993. Statistical characteristics of concentration fluctuations in dispersing plumes in the atmospheric surface layer. Bound. Layer Meteorol. 65 (1–2), 69–109.
For the first time, a unique set of two-point concentration measurements in the vertical direction are analysed with the aim of characterising the structure of passive scalar fluctuations in a TBL. For this, experiments are conducted by varying the location of point source at five positions that span the near-wall, log-, and outer-regions of TBL. The separation distance between the source and the measurement location is chosen to be 1δ in order to investigate the near-field meandering behaviour of the plume. Autocorrelation functions computed using single-point measurements at the plume centreline revealed that the correlation is sustained for longer distances in the case of a ground level plume due to zero vertical scalar flux at the wall. Further, the concentration fluctuations along z has both positively and negatively correlated regions indicating that the plume meanders in the vertical direction (and likewise in the spanwise plane). Based on the two-point correlation analysis, a relatively steeper structural angle of approximately 30∘ is observed in the scalar structures in the region ( 0.5 (z sz )/ z 0.5) close to the plume axis. These results are found to be entirely consistent with the physical model put forth by Talluru et al. (2018) for the large-scale organisation of scalar concentration relative to the large-scale velocity structures in a TBL, i.e., the preference of scalar to correlate with low- or high-speed regions in a turbulent boundary layer results in the coherent organisation of the scalar structures. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References Adrian, R.J., 2007. Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301. Adrian, R.J., Meinhart, C.D., Tomkins, C.D., 2000. Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54. Andronopoulos, S., Grigoriadis, D., Robins, A., Venetsanos, A., Rafailidis, S., Bartzis, J.G., 2001. Three-dimensional modelling of concentration fluctuations in complicated geometry. Environ. Fluid Mech. 1 (4), 415–440. Antonia, R.A., Chambers, A.J., Elena, M., 1983. Points of symmetry in turbulent shear flows. Intl. Commun. Heat Mass Transf. 10, 395–402. Chatwin, P.C., Sullivan, P.J., 1990. A simple and unifying physical interpretation of scalar fluctuation measurements from many turbulent shear flows. J. Fluid Mech. 212, 533–556. Csanady, G., 1973. Turbulent Diffusion in the Environment. Reidel, Dordrecht, The Netherlands. Durbin, P.A., 1980. A stochastic model of two-particle dispersion and concentration fluctuations in homogeneous turbulence. J. Fluid Mech. 100 (2), 279–302. Fackrell, J.E., Robins, A.G., 1982. Concentration fluctuations and fluxes in plumes from point sources in a turbulent boundary layer. J. Fluid Mech. 117 (1), 26. Ganapathisubramani, B., Hutchins, N., Hambleton, W.T., Longmire, E.K., Marusic, I., 2005. Investigation of large-scale coherence in a turbulent boundary layer using twopoint correlations. J. Fluid Mech. 524, 57–80. Ganapathisubramani, B., Longmire, E.K., Marusic, I., 2003. Characteristics of vortex packets in turbulent boundary layers. J. Fluid Mech. 478, 35–46. Gifford, F., 1959. Statistical properties of a fluctuating plume dispersion model. Adv. Geophys. 6, 117–137. Hanna, S.R., Insley, E.M., 1989. Time series analyses of concentration and wind fluctuations. Boundary Layer Studies and Applications. Springer, pp. 131–147. Head, M.R., Bandyopadhyay, P., 1981. New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297–338. Hutchins, N., Chauhan, K., Marusic, I., Monty, J., Klewicki, J., 2012. Towards reconciling the large-scale structure of turbulent boundary layers in the atmosphere and laboratory. Bound. Layer Meteorol. 145 (2), 273–306. Hutchins, N., Hambleton, W.T., Marusic, I., 2005. Inclined cross-stream stereo particle image velocimetry measurements in turbulent boundary layers. J. Fluid Mech. 541, 21–54. Hutchins, N., Monty, J.P., Ganapathisubramani, B., Ng, H.C.H., Marusic, I., 2011. Three-
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Structural inclination angle of near-field scalar fluctuations in a turbulent boundary layer
T
Krishna M. Talluru ,a, Kapil A. Chauhanb ⁎
a b
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia School of Civil Engineering, The University of Sydney, NSW 2006, Australia
ARTICLE INFO
ABSTRACT
Keywords: Two-point correlation Meandering plume Structure inclination angle
Two-point concentration measurements are obtained in a meandering passive-scalar plume released at five different heights within the fully-turbulent region of a high-Reynolds number turbulent boundary layer (TBL). Mean statistics of two-point concentration measurements are found to agree very well with the single-point measurements previously reported in Talluru et al. (2017a). The two-point correlation results of concentration indicate strong coherence in the scalar field similar to the large-scale coherence observed in the streamwise velocity fluctuations in a TBL (Marusic and Heuer, 2007). Particularly, the isocontours in the two-dimensional correlation map of concentration fluctuations illustrate that the scalar structures are inclined at 30∘ to the direction of the flow; such a trend is consistently observed for all the elevated plumes below z/δ ≤ 0.33. This observation of steeper inclination angle of scalar structures relative to the inclination angle of large-scale velocity fluctuations in a TBL is explained using the physical model put forth by Talluru et al. (2018). Most importantly, these results provide insights on the differences in the structural organisation of a passive scalar plume in the near- and the far-field regions.
1. Introduction The local concentration of a scalar (pollutant, heat, etc.) in a turbulent flow exhibits complex features over a broad range of length and time scales. As such, turbulent advection is crucial in most natural and engineering processes such as atmospheric dispersion, mixing and combustion. Although the underlying interaction between the turbulent velocity field and scalar fluctuations is not linear, there are many similarities between the statistics of scalar fluctuations and those of velocity, e.g. in a heated turbulent jet (Antonia et al., 1983). The possibility of exploiting any similarity between velocity and scalar fields in wall-bounded turbulent flows is also of practical significance, for example, prediction of ground level concentration of a pollutant emitted from an elevated source and vice-versa. Thus, many studies are aimed at characterising the role of coherent features of velocity in the dispersion and transport of a scalar (e.g. Talluru et al., 2018; Vanderwel and Tavoularis, 2016). When a contaminant plume is released into a turbulent boundary layer (TBL), the pollutants are mixed by the turbulent eddies in the flow. Further, when the size of the pollutant plume is relatively smaller than the turbulent eddies, the plume meanders and becomes highly intermittent (Fackrell and Robins, 1982; Hanna and Insley, 1989). This phenomenon leads to large fluctuations in the ⁎
pollutant concentration that are of the same order of magnitude as the mean concentration (Csanady, 1973). Such extreme fluctuations are of significant concern when the pollutant is a highly toxic or flammable material. Most dispersion models are capable of predicting the mean pollutant concentration and the amplitude of fluctuations about the mean. For example, Chatwin and Sullivan (1990) developed a simple two-parameter relationship between the mean and the standard deviation of concentration fluctuations in a statistically steady self-similar turbulent shear flows. Later, Sawford and Sullivan (1995); Mole and Clarke (1995) further developed the model to predict the first four moments of the concentration, but they concluded that such simple models cannot describe the near-field behaviour of a plume. Similar analyses were carried out by Lewis et al. (1997) on atmospheric field data, and more recently, by Schopflocher and Sullivan (2005) using wind tunnel data. Several other studies have developed models for the probability density functions (PDF) of the concentration. For instance, Yee and Chan (1997) proposed a three parameter clipped-Gamma distribution for a highly intermittent plume, which was tested against experimental data by Yee (2009) and Klein and Young, 2011. A more general approach for developing PDF models was based on meandering models
Corresponding author. E-mail addresses: [email protected] (K.M. Talluru), [email protected] (K.A. Chauhan).
https://doi.org/10.1016/j.ijheatfluidflow.2019.108521 Received 2 June 2019; Received in revised form 26 November 2019; Accepted 27 November 2019 0142-727X/ © 2019 Elsevier Inc. All rights reserved.
International Journal of Heat and Fluid Flow 81 (2020) 108521
K.M. Talluru and K.A. Chauhan
(Gifford, 1959; Yee et al., 1994; Luhar et al., 2000), pair-of-particles Lagrangian stochastic models ((Durbin, 1980; Sawford and Hunt, 1986), numerical models (Andronopoulos et al., 2001; Milliez and Carissimo, 2008), and large-eddy simulations (Xie et al., 2004; Philips et al., 2013), among others. Most importantly, none of the above models account for the spatial coherence of the scalar concentration field. This seriously limits the applicability of such models in predicting the scalar concentration at any point of space based on the information of scalar at a known location, in particular, in the near-field region, where the fluctuations about the mean are very strong due to plume meandering. To sum up, the existing models can estimate the statistical behaviour at a given point, but they fail to predict the behaviour of the plume simultaneously at several points in space. The present study aims to fill this gap by characterising the spatiotemporal correlation of concentration fluctuations in the vertical direction using experimental data. Specifically, this study focuses on the near-field behaviour of a passive plume, where the plume exhibits different characteristics compared to the far-field region. A long term aim of this study is to achieve real time predictions of concentration by developing linearised and stochastic models (machine learning techniques) based on the spatial correlations obtained here. Such models have been recently developed for predicting the behaviour of velocity in turbulent wall-bounded flows (Illingworth et al., 2018; Srinivasan et al., 2019; Sasaki et al., 2019). No such models are currently available for scalar concentration as there is no experimental data available to test them. It is hoped that the present paper will go some way towards fulfilling this need. The paper is organised as follows. At first, a brief discussion is presented on the two well-accepted results in turbulent boundary layers and plume dispersion that form the basis of the current study. Then we briefly outline our experiments of two-point measurements of instantaneous concentration in a point-source scalar plume. Results of time-averaged statistics, two-point correlation and spatial coherence follow thereafter.
1.2. Meandering of a slender passive scalar plume Observations of a smokestack plume in the atmosphere show that its instantaneous structure is puffy with narrow sections scattered between wide ones, and the plume meanders in a highly variable manner (Weil, 2012). This variability is caused by the turbulent wind field in which the plume is fully embedded, and the scale of atmospheric turbulence that influences the plume fluctuations will change as the plume size increases Sawford (1985). For example, in the near-field region, when the plume is smaller than the turbulent eddies, the plume is swept up and down about the plume axis by the turbulent eddies, but when the plume is larger than the eddies, i.e. in the far-field region, the eddies no longer advect the plume but merely cause minor fluctuations deep within the plume (Hanna and Insley, 1989; Yee et al., 1993). The meandering of plume results in the intermittent high and low concentrations in the time series signal obtained at a single point in space as observed in laboratory wind tunnel (Fackrell and Robins, 1982; Nironi et al., 2015) and full-scale atmospheric (Yee et al., 1993) measurements. A natural question that arises here is whether the structure of scalar fluctuations is similar to that of inclined velocity structures in a TBL, and secondly, how that structure varies as the plume size increases from the near-field region to the far-field region. The latter question is partly addressed in a recent numerical study of scalar dispersion in a turbulent channel flow (Srinivasan and Papavassiliou, 2011). They observed that the concentration fluctuations have a similar behaviour as that of velocity structures in the far-field region. However, little is known about the organisation of scalar structures in the near-field region. Recently, Talluru et al. (2018) conducted a study of point source plumes released at eight different elevations in a TBL and presented a physical model to explain the relative organisation of the large-scale coherent velocity structures and the concentration fluctuations in the plume. Through flow visualisation as well as joint-pdfs (probability density functions) and cross-spectra analysis, they comprehensively showed that the large-scale turbulent eddies cause meandering in the near-field region of the plume. Talluru et al. (2018) also observed that above the plume centreline, low-speed structures have a lead over the meandering plume, while high-momentum regions are seen to lag behind the plume below its centreline. The present work seeks to test the hypothesis proposed in Talluru et al. (2018) by performing two-point correlations of concentration measurements and determining the inclination angle of scalar structures. Confirming or denying our hypothesis will enable us to answer if the organisation of a passive scalar plume is same in the near- and the far-field regions, and how the organisation may be affected by the plume meandering.
1.1. Inclination angle of streamwise velocity fluctuations in a turbulent boundary layer Inclined structures, typically attached to the wall, are well-known features of streamwise velocity fluctuations in a turbulent boundary layer. One of the earliest studies that reported this behaviour is the flow visualisation study in a zero pressure gradient TBL (Head and Bandyopadhyay, 1981). They reported that at moderately high Reynolds number (Reθ > 2000), the TBL consisted of elongated hairpin vortices, inclined at a characteristic angle of approximately 45∘ to the wall. They suggested that the hairpin vortices could organise themselves at high Reynolds numbers to form larger structures that are typically oriented at an angle of 20∘ to the wall. The view of hairpin packets was further confirmed in the high-resolution PIV study of a turbulent boundary layer across a range of Reynolds number 930 ≤ Reθ ≤ 6845 (Adrian et al., 2000). They observed the inclined coherent features in the streamwise-wall-normal plane across the entire range of Reθ. Other studies (e.g., Ganapathisubramani et al., 2003; 2005; Hutchins et al., 2005) in laboratory turbulent boundary layers also indicated that the average coherent structure in the logarithmic region has a low inclination angle made up of organised packets of attached eddies. A more conclusive evidence was provided by Marusic and Heuer (2007), who showed that the two-point correlation map of streamwise velocity fluctuations obtained in laboratory and atmospheric boundary layers presented a characteristic inclination angle of 14∘. Very importantly, they observed that the inclination angle remained invariant over three orders of magnitude change in Reynolds number, 103 ≤ Reτ ≤ 106.
2. Experimental details Experiments are performed in the boundary layer wind tunnel at the University of Sydney. Full details of the facility are given in Talluru et al. (2017a) and a schematic of the test rig is shown in Fig. 1. In the present study, measurements are conducted in a turbulent boundary layer developing over a streamwise fetch of 13 m at a nominal free stream velocity (U∞) of 10 m/s. The boundary layer thickness (δ) and the friction velocity (Uτ) at the measurement location are approximately 0.31 m and 0.37 m/s, respectively, yielding a friction Reynolds number, Reτ ≡ δUτ/ν ≈ 7850 (ν is the kinematic viscosity of air). Here, upper case (e.g., C) and lower case letters (e.g., c) indicate mean and fluctuating quantities, respectively, and z represents the wallnormal coordinate. A point source tracer gas (a mixture of 1.5% isobutylene and 98.5% Nitrogen) is released at five different heights, sz/δ = 0.004, 0.044, 0.1, 0.25 and 0.33, at a streamwise distance sx / = 1 upstream of the measurement location, specifically to investigate the near-field behaviour of the plume. In our previous study (Talluru et al., 2017a), we found that δ does not change significantly (less than 5%) between the 2
International Journal of Heat and Fluid Flow 81 (2020) 108521
K.M. Talluru and K.A. Chauhan
Fig. 1. Schematic of the experimental set-up used for obtaining two-point measurements of concentration in the vertical direction across the plume.
source and the measurement locations. For this reason, sz is normalised by δ at the measurement location throughout this paper. The tracer gas is released isokinetically (i.e., the source velocity is matched with the local mean velocity of the flow) into the turbulent boundary layer. The sensor used for these experiments is the photoionisation detector (PID) manufactured by Aurora scientific, Inc. The sensor is calibrated by supplying the tracer gas at different concentrations and recording the sensor voltage output. A second-order polynomial is then fitted between the gas concentration and the sensor voltage during the calibration. Full description of the calibration procedure can be found in Talluru et al. (2019a). The frequency response of the PID sensor is measured using a custom built test rig that consists of a chopper wheel delivering a quasi-impulse of tracer gas to the PID sensor (see Talluru et al., 2017b, for details). Based on this test, the frequency response of PID is estimated to be 400 Hz. Hence, the fluctuating voltage signal from PID is low-pass filtered at a cut-off frequency of 400 Hz. The filtered signal is then sampled at 4096 Hz and is digitised using a 24-bit National Instruments A/D card, NI 4303. As shown in the schematic (cf. Fig. 1), a stationary PID is positioned at the source-height, z = sz , while a second PID is traversed along z. The spanwise gap between the two PIDs is approximately 3 mm, in accordance with the results of Metzger and Klewicki (2003), who showed that the PID has negligible influence on an adjacent probe when separated by a distance greater than 3d, where d = 0.76 mm is the inlet diameter of PID probes used in this study. It is important to note that the experimental test-rig and the measurement procedures used in this study are similar to that of Talluru et al. (2017a), however, the present study is unique and reports simultaneous measurements of concentration at two points in the plume to characterise the structure of scalar fluctuations in the flow. To our knowledge, such measurements using point-detection probes have not been reported before.
concentration in ground and elevated source plumes, we will briefly discuss the models that have been previously proposed for the mean concentration in a plume. Most continuous source plume models are based on the classical (Fickian) differential equation of diffusion, which have functions of Gaussian type as solutions. However, such models imply random and uncorrelated kind of turbulence, which is not characteristic of atmospheric turbulence (see Hutchins et al., 2012, for a full discussion on velocity length scales in atmospheric surface layer). Later, Gifford (1959) analysed a more generalised form of the diffusion equation for a meandering plume and showed that the Gaussian form is the asymptotic solution based on the central limit theorem in probability theory. The model was further developed into the reflectedGaussian model by Fackrell and Robins (1982), where the wall effect on the plume spread is accounted for.
C (z ) Cmax
exp
C (z ) Cmax
exp
z (1)
z
B
z
sz
2
+ exp
z
B
z + sz z
2
(2)
For a ground level source, C is found to have an exponential form as defined in Eq. 1, where Cmax represents the maximum concentration and α is the exponent that depends on surface roughness (Csanady, 1973). In our case, we found a value of = 1.5, which is close to the value reported by Fackrell and Robins (1982). In contrast, the mean concentration profiles in the elevated plumes show a reflected Gaussian behaviour, which is defined in Eq. 2, where B = ln(2) . The results shown in Fig. 2 are consistent with the experimental results previously reported by Fackrell and Robins (1982) and Nironi et al., 2015 in rough wall turbulent boundary layers. c (z )
3. Validation of two-point concentration measurements
c max
As a first step, we compare the statistics of concentration fluctuations obtained in the current study against single-point measurements reported in Talluru et al. (2017a) as a means of validating the two-point concentration measurements. Fig. 2(a) and (b) shows the normalised profiles of the mean (C/Cmax) and the root mean square (abbreviated as r.m.s) (σc/σcmax) of concentration along z when the plume is released at five different heights within the TBL. The open symbols in the figures represent the statistics obtained from the traversing PID, the solid symbols are from the stationary PID, and the grey solid lines represent the statistics taken from Talluru et al. (2017a). It is observed that the statistics from both the PID probes agree very well with those of Talluru et al. (2017a) to within ± 2%. Before commenting on the behaviour of mean statistics of
exp
B
z
sz z
2
(3)
On the other hand, the r.m.s of concentration fluctuations (σc) for all sources has a simple Gaussian behaviour along z as defined in Eq. 3. Note that δz and σz in eqns. 1 - 3 are the plume half-widths based on the mean and the r.m.s profiles of concentration, respectively. In general, the plume half-width is defined as the distance from the plume centreline, where the maximum concentration drops to half its value. A closer look at the profiles in Fig. 2 reveals that δz and σz vary with source height. The relationship between δz and σz and the turbulence intensities in the flow has been extensively discussed in our recent study Talluru et al. (2019b), and therefore will not be repeated here. It is known that no similarity exists between the mean statistics of concentration and velocity (or moments of their fluctuations). Nonetheless, it has been observed that a passive scalar plume meanders 3
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because of the separation between the length-scales in the turbulent flow and the plume, i.e., when the integral length scale of velocity fluctuations is large compared to the plume width or the source diameter (Fackrell and Robins, 1982; Karnik and Tavoularis, 1989; Hanna and Insley, 1989). Since the integral length scale of a turbulent flow characterise the extent of coherent motions, it is anticipated that the coherent structures in the velocity will impose the instantaneous organisation of concentration field. In the remainder of this paper, we will identify, and then characterise the organisation of scalar fluctuations in the streamwise/wall-normal plane, particularly focussing on the inclination of scalar structures in the mean flow direction.
In contrast, the behaviour of autocorrelation functions Rcc(sz, z) in the wall-normal direction is not similar for different source heights, as evident in Fig. 3(b), which is anticipated since the spread of plume in the wall-normal direction (i.e., the plume half-width) varies with the source height. In this plot, the wall-normal shift (Δz) is presented with respect to the source height as (z sz )/ . It should be noted that the maximum in the vertical autocorrelation functions is less than 1, which is due to the gap (= 3 mm) between the two PID sensors in the transverse direction. It is seen that Rcc(sz, z) is not symmetric about the plume centreline especially when the plume spread is influenced by the wall as a result of the non-homogeneity of turbulence in the wallnormal direction. Interestingly, the autocorrelation functions for the plumes (sz/δ = 0.044, 0.1 and 0.25) exhibit both positive and negative correlations. This implies that there are instances at which the concentration fluctuations (about the mean value) at the plume centreline and at some distance away from the centreline are of opposite signs. In other words, this result presents a convincing evidence for the meandering behaviour of the plume in the vertical direction. Further, the magnitude of negative correlation drops with increasing source height from sz/δ = 0.044 to 0.25, which aligns well with the trend of wallnormal turbulence intensity in a zero pressure gradient TBL (see figure7 in Talluru et al., 2019b).
4. Single- and two-point correlation functions Using the time-series information of concentration fluctuations at two distinct points, the autocorrelation function along the streamwise direction is computed using Eq. 4. In the wall-normal direction, Eq. 5 represents the correlation coefficient at zero time-shift between the signals at the plume centreline (stationary PID) and at another z-location (traversing PID).
R cc (sz , x ) = R cc (sz , z ) =
c (sz , x ) c (sz , x + x ) , [ c (sz ) c (sz )]
c ( s z ) c (z ) , [ c (sz ) c (z )]
(4)
5. Two-point correlation and inclination angle
(5)
Before discussing the two-point correlation map of concentration fluctuations, we first examine the time-series signals of concentration for the plume released at sz/δ = 0.004, i.e., the ground level source in our measurements. Fig. 4 shows a sample of 1.5 second record of concentration measurements at the wall (z = 0 ) and at another location in the log-region of TBL (i.e., z/ = 0.06). One can readily appreciate the qualitative similarity of these two signals, indicating that there is a strong correlation between the signals, however with a finite shift in time. This similarity is quantified using the correlation function which also provides an estimate of the time-shift or equivalently the spatialshift ( x = Uc t ). The correlation coefficient, Rcc between two signals is defined as,
It is important to understand that the temporal information of concentration fluctuations is converted to the spatial domain using the relationship x = Uc t, where Δt is the time-lag between two concentration signals and Uc is the convection velocity taken to be the mean velocity at the corresponding wall-normal height. Note that since the plume evolves rapidly in the streamwise direction, particularly close to the source, Taylor’s hypothesis of frozen turbulence cannot be applied to the scalar field precisely. However in this study, we aim to compare the inclination angle of scalar coherence with the inclination angle of streamwise velocity fluctuations, where the Taylor’s hypothesis is extensively used to convert temporal domain to a spatial one. Thus to make an equivalent comparison the relation, x = Uc t is also used for the scalar field. This aspect is further discussed in section 5. The results of spatial autocorrelation functions computed with reference to the plume source height along x and z are plotted in Fig. 3(a, b), respectively. Looking at Fig. 3(a), it is evident that the correlation of concentration fluctuations sustains for longer distance in the case of a ground level source. The integral length scale ( c ) (which is obtained as c = 0 Rcc dx ) for the ground level plume is found to be five times larger than that for the elevated plumes. This can be explained as follows. In a ground level plume, the concentration fluctuations tend to persist for longer time since the streamwise mean velocity is relatively low in the near-wall region. Further, since the concentration flux is zero at the wall, longer correlations and higher integral time-scales of concentration fluctuations are observed in the ground level plume in comparison to the elevated plumes. This is further corroborated by the fact that, for a ground plume, the plume width in the spanwise direction is larger than in the vertical direction (see Talluru et al., 2019b, for further discussion). On the other hand, the turbulence in the flow forces the elevated plume to sway above and below the centreline causing intermittency, which in turn, causes the correlation to drop quickly. Interestingly, the curves of Rcc(sz, Δx) are almost identical for all the elevated sources. This behaviour can be understood by considering the distribution of integral length scales of velocity fluctuations ( u ) in the TBL. In our previous work (Talluru et al., 2018), we observed that u is constant in the region 0.1 ≤ z/δ ≤ 0.5, which implies the velocity structures will impose identical meandering of the elevated plumes. Hence, Rcc(sz, Δx) are expected to be similar in the plumes released in that region of the TBL.
R cc (z, x ) =
c (sz , x ) c (z , x + x ) , [ c (sz ) c (z )]
(6)
where overbar indicates time average. The correlation coefficient between the sample signals shown in Fig. 4 is plotted in Fig. 5, along with the correlation coefficient between c(z ≈ 0) and signals at other z-locations. High levels of correlation are observed across the plume, however, the peak correlation magnitude decreases with increasing distance from the wall. Further, it is seen that with increasing distance from the wall, the location of peak correlation occurs at a non-zero shift on the abscissa. Thus, one can define the inclination angle, = arctan[(z sz )/ x *], where z is the wall-normal position in the plume and Δx* is the separation distance corresponding to the peak value in Rcc. Similarly, the inclination angles inferred from the correlation functions at different heights in the plume are calculated, and the average angle for the ground-level source is found to be 29∘, as indicated in Fig. 6(e) for sz / = 0.004 . The above procedure is repeated for the plume released at four other source locations that span the near wall, log-, and outer-regions of the TBL. The average inclination angle is calculated using correlation in the region where σc(z)/ σc(sz) > 0.2. Contour maps of correlation coefficient Rcc are plotted as a function of Δx/δ and z/δ in Figs. 6(a-e) for better visualization of the coherence in the scalar field and its inherent inclination. It is expected that the cross-correlation at the source height along the dot-dashed lines in each of the Figs. 6(a-e) should be symmetrical about x/ = 0 as it also represents the autocorrelation function since the two PID sensors are at the same vertical location. However, there is slight asymmetry in the 4
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Fig. 2. Normalised profiles of (a) mean (C/Cmax) and (b) r.m.s (σc/σcmax) of concentration in the vertical direction. Open symbols represent the data from the traversing PID in a point source plume released at different source heights; sz/δ = 0.004 ( ); 0.044 ( ); 0.1 ( ) ; 0.25 ( ); 0.33 ( ). Solid symbols represent the measurement from the stationary PID and the grey solid lines represent the data taken from Talluru et al. (2017a).
Fig. 3. (a) Autocorrelation functions in the streamwise direction using Eq. (4). (b) Zero time-shift correlation coefficient in the wall-normal direction computed as per Eq. (5). Symbols and colours are same as in Fig. 2.
5
International Journal of Heat and Fluid Flow 81 (2020) 108521
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Fig. 4. Sample records of simultaneous signals of concentration at the wall and a point in the log-region of turbulent boundary layer. Horizontal dashed lines indicate mean concentration values. The dashed rectangles highlight the similarity in the two signals with a finite time shift.
Fig. 5. Cross correlation of c (z = 0) at the wall and c(z > 0) at elevated positions in the plume released at ground-level. Red lines indicate the magnitude of Δx*.
cross-correlation for low levels of correlations at the source height, which is due to the gap between two PID sensors in the spanwise direction. The nominal inclination angle of scalar structure is found to be approximately 30∘ for plumes released at four other source locations, sz/ δ = 0.045, 0.1, 0.25 and 0.33. Note that the height of ordinate axis in each sub-plot is proportional to the plume spread. Some inferences can be made by comparing the average structure of concentration fluctuations against that of velocity structures previously reported in Hutchins et al., 2011 and Talluru et al. (2014). Firstly, these results show that the size of correlated regions scale with δ, suggesting that scalar structure is governed by length scales in the outer-region of TBL. Recently, Talluru et al. (2018) provided the evidence for the role of large-scale velocity fluctuations in the scalar transport across the TBL through their role in streamwise scalar flux. Secondly, it is noticed that the positively correlated regions of concentration fluctuations are inclined to the horizontal axis at θ ≈ 30∘ within the region, 0.5 (z sz )/ z 0.5, for all sources in the fully turbulent region, i.e. sz/δ ≤ 0.33. This angle is considerably steeper than the corresponding inclination angle of ≈ 14∘ noticed in the
Fig. 6. Two dimensional two-point correlation map of concentration fluctuations in the streamwise/wall-normal plane at five source heights, sz / = (a) 0.33; (b) 0.25; (c) 0.1; (d) 0.045 and (e) 0.004. Contour levels are in increments of 0.15 from 0.1 to 1. The dashed lines represent the inclination angles indicated on the top-right of each sub-plot.
correlations of streamwise velocity structures (Marusic and Heuer, 2007). Here, we revisit the question about the validity of Taylor’s hypothesis, and evaluate its effect on the structure inclination angle. The two main factors that can affect the calculation of the inclination angle are (i) plume meandering and (ii) estimate of the convection velocity of scalar plume. The effect of these two factors on the inclination angle is discussed in the following. First, in these experiments, the plume width in the vertical direction is typically about 0.1δ ( ~ 0.03 m) and the wall-normal velocity fluctuations are about σw ≈ 0.4 m/s at the source height. Therefore, 6
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the time for the plume to meander from the centreline to the edge of the plume is O(0.1) seconds. It is then reasonable to consider that this is the time over which the instantaneous structure of the plume changes significantly and Taylor’s hypothesis cannot be applied to scalar field over time lags that are O(0.1) seconds. During this time the mean flow within the boundary layer travels a distance of approximately 0.6 m or 2δ in the streamwise direction. However, looking at the results in Fig. 6, we see that the correlation length scales for scalar are much shorter ~ 0.2δ (corresponding to time lead/lag of O(0.01) seconds). Thus the time scale of meandering is much larger compared to the time over which significant correlation persists between two distinct positions in the plume. Hence the correlation of scalar fluctuations over shorter distances is not influenced by the meandering, justifying the use of Taylor’s hypothesis in estimating the inclination angle of scalar structures. Secondly, regarding the uncertainty about the inclination angle due to error in estimating the convection velocity, it is recalled that the inclination angle is equivalent to tan 1 [ z /(Uc· t )], where Δt is the time-lag corresponding to the peak-correlation between two signals separated by distance Δz. In our analysis we have considered Uc = U (z ) resulting in an inclination angle of ~ 30∘. Now if the actual convection velocity is within ± 10% of the local mean velocity, the resultant inclination angle would be between 28 and 33 degrees (note that lower convection velocity leads to larger inclination angle). Even a 50% variation of the convection velocity from U considered here results in estimates of inclination angle between 21∘ and 40∘. Thus the observed deviation of structure inclination angle of scalar from the well-known 14∘ of u-fluctuations is robust. In a recent study, Renard and Deck (2015) established that the convection velocity of streamwise fluctuations closely matches with the local streamwise velocity across most part of turbulent boundary layer except in the very near wall region (y/δ ≤ 0.004). Since the source height of the scalar plume sz/ δ ≥ 0.004 for all the cases in our study, and for the passive scalar, advection (U˜ · C˜ / x ) is significant for its streamwise transport; the findings of Renard and Deck (2015) supports the use of mean velocity as convection velocity of the scalar as a best estimate. The phenomenological model of Talluru et al. (2018) that outlines the preferential organisation of the scalar motions with respect to the large-scale low- and high-speed structures in the flow provides an explanation of this observation. A schematic of the physical model is presented in Fig. 7, where a meandering plume along with a cluster of low- and high-speed regions are indicated. In this schematic, the lowspeed and high-speed regions of velocity fluctuations are indicated at 14∘ to the horizontal, as the correlation between similarly signed (lowor high-speed) fluctuations will contribute to the 14∘ angle observed in
numerous past studies (Adrian, 2007; Hutchins et al., 2011; Marusic and Heuer, 2007; Talluru et al., 2014). Recently, Talluru et al. (2018) have shown that above the plume centreline, low-speed regions in velocity are correlated with high-concentration scalar regions, resulting in uc < 0, whereas below the centreline, high-speed regions are correlated with the high-concentration region resulting in uc > 0 . Thus, instantaneously, even though the velocity fluctuations might have the same sign across the whole plume width (at a nominal angle of 14∘), the concentration field itself is not consistently correlated with velocity fluctuations. Regions of coherent scalar fluctuations in the meandering plume thereby attains a much steeper inclination angle as shown in Fig. 7, that is approximately 30∘, as observed in this study. Thus, we find that although the mean and r.m.s of concentration fluctuations has a symmetric distribution (Gaussian type), instantaneously it has a characteristic inclination angle that manifests as the inclination observed in the correlation map of Rcc. Thus, the degree of meandering is not symmetric about the plume centreline and has a spatial preference. In a recent numerical study of a turbulent channel flow, Srinivasan and Papavassiliou (2011) showed that the forward correlations of scalar structures are oriented at an angle of 76∘ with the vertical (equivalent to 14∘ to the horizontal), which is different than present results. This difference could be attributed to two reasons - (i) how the scalar is released in the flow or the computational domain and (ii) whether the plume is in the near- or the far-field region. In our wind tunnel experiments, the scalar is released via a point source and the measurements are performed in the near-field region, where the plume is still meandering. On the other hand, Srinivasan and Papavassiliou (2011) studied a flow, where the scalar is released uniformly from a line source in the wall-normal plane of the computational domain. This implies that there is no concentration gradient in the vertical direction (or a Gaussian profile) in the numerical simulations performed by Srinivasan and Papavassiliou (2011). Further, their simulations are in the far-field region and hence their results are not directly comparable to the present study. We would like to emphasise that the observations made in this study are closely linked to the meandering nature of the plume. It is very likely that the correlation map of concentration fluctuations further downstream, i.e., in the farfield region, where the scalar plume is thoroughly dispersed, will show an inclination angle of 14∘, consistent with the inclination of velocity structures in a turbulent boundary layer. It would be insightful to further investigate the inclination angle of scalar structures (released via a point source) at large source distances in the future studies.
Mean flow
Meandering plume
14o 30o
negative
positive
Fig. 7. A schematic of the inclination angles of velocity (14∘, dashed lines) and scalar (30∘, dot-dashed line) structures in a meandering plume based on the physical model given by Talluru et al. (2018). 7
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6. Conclusions
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For the first time, a unique set of two-point concentration measurements in the vertical direction are analysed with the aim of characterising the structure of passive scalar fluctuations in a TBL. For this, experiments are conducted by varying the location of point source at five positions that span the near-wall, log-, and outer-regions of TBL. The separation distance between the source and the measurement location is chosen to be 1δ in order to investigate the near-field meandering behaviour of the plume. Autocorrelation functions computed using single-point measurements at the plume centreline revealed that the correlation is sustained for longer distances in the case of a ground level plume due to zero vertical scalar flux at the wall. Further, the concentration fluctuations along z has both positively and negatively correlated regions indicating that the plume meanders in the vertical direction (and likewise in the spanwise plane). Based on the two-point correlation analysis, a relatively steeper structural angle of approximately 30∘ is observed in the scalar structures in the region ( 0.5 (z sz )/ z 0.5) close to the plume axis. These results are found to be entirely consistent with the physical model put forth by Talluru et al. (2018) for the large-scale organisation of scalar concentration relative to the large-scale velocity structures in a TBL, i.e., the preference of scalar to correlate with low- or high-speed regions in a turbulent boundary layer results in the coherent organisation of the scalar structures. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References Adrian, R.J., 2007. Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301. Adrian, R.J., Meinhart, C.D., Tomkins, C.D., 2000. Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54. Andronopoulos, S., Grigoriadis, D., Robins, A., Venetsanos, A., Rafailidis, S., Bartzis, J.G., 2001. Three-dimensional modelling of concentration fluctuations in complicated geometry. Environ. Fluid Mech. 1 (4), 415–440. Antonia, R.A., Chambers, A.J., Elena, M., 1983. Points of symmetry in turbulent shear flows. Intl. Commun. Heat Mass Transf. 10, 395–402. Chatwin, P.C., Sullivan, P.J., 1990. A simple and unifying physical interpretation of scalar fluctuation measurements from many turbulent shear flows. J. Fluid Mech. 212, 533–556. Csanady, G., 1973. Turbulent Diffusion in the Environment. Reidel, Dordrecht, The Netherlands. Durbin, P.A., 1980. A stochastic model of two-particle dispersion and concentration fluctuations in homogeneous turbulence. J. Fluid Mech. 100 (2), 279–302. Fackrell, J.E., Robins, A.G., 1982. Concentration fluctuations and fluxes in plumes from point sources in a turbulent boundary layer. J. Fluid Mech. 117 (1), 26. Ganapathisubramani, B., Hutchins, N., Hambleton, W.T., Longmire, E.K., Marusic, I., 2005. Investigation of large-scale coherence in a turbulent boundary layer using twopoint correlations. J. Fluid Mech. 524, 57–80. Ganapathisubramani, B., Longmire, E.K., Marusic, I., 2003. Characteristics of vortex packets in turbulent boundary layers. J. Fluid Mech. 478, 35–46. Gifford, F., 1959. Statistical properties of a fluctuating plume dispersion model. Adv. Geophys. 6, 117–137. Hanna, S.R., Insley, E.M., 1989. Time series analyses of concentration and wind fluctuations. Boundary Layer Studies and Applications. Springer, pp. 131–147. Head, M.R., Bandyopadhyay, P., 1981. New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297–338. Hutchins, N., Chauhan, K., Marusic, I., Monty, J., Klewicki, J., 2012. Towards reconciling the large-scale structure of turbulent boundary layers in the atmosphere and laboratory. Bound. Layer Meteorol. 145 (2), 273–306. Hutchins, N., Hambleton, W.T., Marusic, I., 2005. Inclined cross-stream stereo particle image velocimetry measurements in turbulent boundary layers. J. Fluid Mech. 541, 21–54. Hutchins, N., Monty, J.P., Ganapathisubramani, B., Ng, H.C.H., Marusic, I., 2011. Three-
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