Flow Turbulence Combust (2008) 81:221–234 DOI 10.1007/s10494-007-9115-1

Measurements of the Statistical Distribution of the Scalar Dissipation Rate in Turbulent Axisymmetric Plumes C. N. Markides · E. Mastorakos

Received: 18 February 2007 / Accepted: 5 October 2007 / Published online: 15 November 2007 © Springer Science + Business Media B.V. 2007

Abstract In this paper we reconsider the high resolution Planar Laser-Induced Fluorescence (PLIF) mixing measurements that were presented in Markides and Mastorakos (Chem. Eng. Sci. 61:2835–2842, 2006). The PLIF experiments were performed in flows created by the continuous injection, with various velocity ratios, of a passive scalar (acetone) from a finite sized round nozzle, into uniform turbulent co-flows confined within a cylindrical tube with a range of turbulent Reynolds numbers. Here, we extend our study of these flows to include an investigation of the Probability Density Functions (PDFs) of the scalar dissipation rate (χ ), as well as of the scalar dissipation rate conditional on mixture fraction (χ |ξ ). To this effect, we have further processed the resulting data from the earlier work, including an additional correction for density variations in the flow. The results were then reprocessed for the accurate recovery of the spatial gradients of the normalized scalar concentration (ξ ), which resulted in direct measurements of χ . All results that are presented in this paper involve the two-dimensional scalar dissipation rate (χ2D ) evaluated in the plane of imaging, and furthermore, results are only shown and discussed along the centreline of our axisymmetric configuration. The PDFs of χ were calculated and found to deviate slightly from a lognormal distribution, consistent with other published work. Moreover, the PDFs of χ |ξ followed a similar distribution, and showed a similar deviation. This deviation was more pronounced closer to the source and decayed downstream. The ratio of the standard deviation over the mean of χ and χ |ξ increased with streamwise distance and reached values of 2.8 and 1.2 respectively.

C. N. Markides (B) · E. Mastorakos Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK e-mail: [email protected] E. Mastorakos e-mail: [email protected]

222

Flow Turbulence Combust (2008) 81:221–234

Keywords Turbulent mixing · Scalar dissipation rate · Probability density function · Small-scale · Intermittency

1 Introduction In turbulent flows, the transport of scalar energy through the fluctuations of the scalar concentration, or mixture fraction (ξ ), is determined by the scalar dissipation rate (χ ), χ ≡ 2D

∂ξ ∂ξ ∂ xi ∂ xi

(1)

where D is the molecular diffusivity. The scalar dissipation appears as a sink and describes the rate at which scalar fluctuations are destroyed at the smallest scales by molecular diffusion. The precise mechanism by which scalar fluctuations at the large scales are coupled with their dissipation at the small scales is essential to the evolution of the scalar field [2]. In turbulent reacting flows χ is also central to our understanding of the effects of turbulent mixing on the evolution of the chemistry [3]. In addition, the fluctuations of χ can affect practically important combustion phenomena including pollutant emission, flame extinction and/or re-ignition and autoignition. Consequently, the formulation of advanced theories and models for both non-reacting and reacting turbulent flows [4, 5] requires detailed knowledge of the statistical properties of ξ , χ and χ conditional on a value of ξ (χ |ξ ) [6]. Relevant studies on scalar mixing must include an investigation of the necessary P(χ ) and P(χ |ξ ), with P denoting the Probability Density Function (PDF). The possibility of a lognormal χ was first demonstrated experimentally by Gurvich [7] as an extension to the studies into turbulent dissipation (), whereas the lognormality of χ |ξ was first assumed by Peters [8] in the context of flamelet modelling for combustion. Since then, the statistical distribution of χ has been examined with experiments in non-reacting round [9–12] and planar [13] jets, boundary layers [14, 15] and both two- [16] and three-dimensional [17, 18] numerical simulations of passive scalar mixing in stationary (forced), homogeneous, isotropic turbulence. These studies have provided some evidence to support the approximately lognormal nature of P(χ ), with deviations from the theorized lognormality that are attached with great theoretical importance because they are seen as indicators of the smallscale behaviour in turbulent flows. However, even recent studies continue to report almost exactly lognormal P(χ ) [19]. It is generally acknowledged that measurements of χ are associated with noise issues and of correctly resolving and evaluating the spatial gradients of ξ [20], while reporting on PDFs is associated with significant convergence uncertainty near their tails, where the aforementioned deviations are most pronounced. Hence, more data can assist to confirm, or not, these previous findings. On the other hand, P(χ |ξ ) has not been systematically examined. Investigators have considered experimentally the statistical correlation between the scalar and its dissipation in uniformly sheared turbulent flows [21]. There, P(ξ ) was essentially Gaussian and little correlation was found between ξ and χ . In such flows, exhibiting closeness to idealized local isotropy, one can expect conformity between P(χ ) and

Flow Turbulence Combust (2008) 81:221–234

223

P(χ |ξ ), and flatness in the mean conditional scalar dissipation, χ |ξ  [17]. However, differences between the two PDFs can arise if ξ and χ are not statistically independent, which may be the case for short mixing times, close to the scalar source, where P(ξ ) is strongly non-Gaussian [22]. This is a further interesting issue worth exploring. The mixing field we are concerned with consists of a turbulent plume generated by fluid injected into a wall-bounded co-flow of approximately homogenous turbulence, through a finite sized nozzle with mean bulk injection velocity (U inj) that can be approximately equal to, or higher than that of the co-flow (U co ). This simple geometry is also practically relevant, for example to various combustion engines and industrial burners. Here, we re-examine experimental data that were generated in such a field by Planar Laser-Induced Fluorescence (PLIF) measurements in the work described in Markides and Mastorakos [1], which investigated the applicability of the timescale ratio (C D ) model. The original data are further corrected for density variations in the flow, reprocessed to provide information on P(χ ) and P(χ |ξ ), and consequently used to examine the suitability of the lognormal approximations for χ and χ |ξ . Note that all results in this paper involve the two-dimensional,    2  ∂ξ ∂ξ 2 χ2D ≡ 2D + (2) ∂r ∂z evaluated in the plane of imaging, and are presented along the centreline of our axisymmetric configuration. The mean (μ) and standard deviation (σ ) of χ and χ |ξ are quantified and the important ratios σχ /μχ and σχ|ξ /μχ|ξ are inspected.

2 Experimental Methods Recently, high resolution, two-dimensional measurements of χ with PLIF were performed in the mixing field formed by the continuous axisymmetric injection of a gaseous fluorescent tracer (acetone) into a turbulent co-flow of air, confined within an optically polished quartz tube of 34.0 mm inner diameter (D) [1]. Two injector nozzles were used, with 1.1 and 2.2 mm inner diameters (d); both with a 3.0 mm outer diameter. The apparatus and optical measurement method used in Markides and Mastorakos [1] are presented in Fig. 1 (left). The mass flow rate in the co-flow was set directly by a Mass Flow Controller (MFC) that allowed improved accuracy and reproducibility in the mean (timeaveraged) bulk (average volumetric flow rate over the cross-sectional flow area) co-flow velocity, U co , which in these experiments ranged between 3.1 and 7.4 m/s. The fluid originating from the injector, that will be referred to as ‘injected flow’, consisted of various mixtures of nitrogen-diluted gaseous fuel (either acetylene or hydrogen) and acetone vapour. The injected flow rate was adjusted to achieve a mean bulk flow velocity, U inj, that was either approximately equal to U co , in an ‘equal velocity’ scenario with U inj/U co = 1.0±0.2, or higher than U co , in a ‘jet-like’ scenario with U inj up to 5 times U co . We denote the radial distance away from the centreline by r and the axial distance downstream of the injector by z. The background (d/2 < r < D/2, z = 0) turbulence in the co-flow was enhanced by a perforated plate with 3 mm diameter circular holes and 44% solidity, positioned 63 mm upstream of the nozzle to allow the turbulence

224

Flow Turbulence Combust (2008) 81:221–234

Fig. 1 Left: Apparatus schematic, showing seeding method and laser sheet positioning for optical measurements of injected fuel concentration. Right: Calculation of P(χ ) and P(χ |ξ ) inside a small window at a location downstream of the injector and along the centreline

to develop. The axial velocity (υ) was measured with a Constant Temperature Anemometry (CTA) system comprising a 1.25 mm long and 5 μm diameter (hot) wire, and using sampling rates of up to 100 kHz. For an extensive description of the PLIF system, the mixing experiments and resulting data on the mixing field in this configuration see Markides and Mastorakos [1], where the resolution achieved and the need for correct and adequate filtering of the resulting PLIF images are discussed at length. Particular attention was paid in accomplishing a spatial resolution at the Kolmogorov (inner) lengthscale (η K ), which was estimated from the hot wire data by using Taylor’s ‘frozen’ turbu  2 lence hypothesis to calculate the mean turbulent dissipation,  = 15ν ∂u , through ∂x the mean spatial gradients of the axial velocity fluctuations (u ),  ηK ≡

ν3 

1/4

 ,

 ≈ 15ν

∂u ∂t

2

1 U2

(3)

where the local mean axial velocity (U) and u were obtained from the hot wire measurements, U ≡ υ and u ≡ υ − U, respectively. In the present paper ξ has been additionally corrected for density variations in the flow and is thus the true mass fraction of the fluid in the injected flow. Table 1 summarizes the investigated conditions that can also be found in the same reference. Briefly, measurements of the local fuel concentration were suitably normalized to obtain the ξ field from close lengths to the source to a downstream distance of about 60–70 mm. Markides and Mastorakos [1] describes a post processing method that ensures quantitatively reliable measurements of (the two-dimensional) χ . As a result, quality checks showed that μχ was measured to within ±20%. Here, the data from Markides and Mastorakos [1] are used to obtain P(χ ) and P(χ |ξ ) at various locations

Flow Turbulence Combust (2008) 81:221–234

225

Table 1 Selected run conditions from investigated experimental envelope Run (fuel)

U co & Returb m/s & —

Y f uel & δinj —&—

U inj & U inj /U co m/s & —

A (C2 H2 ) C (C2 H2 ) D (C2 H2 ) E (C2 H2 ) G, H (H2 ) I, J (H2 ) K, L (H2 )

3.1 & 48 3.9 & 60 4.1 & 64 4.3 & 67 4.3 & 67 5.7 & 89 6.9 & 108

0.73 & 1.2 0.73 & 1.2 0.73 & 1.2 0.73 & 1.2 0.14 & 0.7 0.15 & 0.7 0.13 & 0.8

3.5 & 1.1 4.5 & 1.2 4.7 & 1.1 4.7 & 1.2 5.3, 10.1 & 1.2, 2.4 7.2, 13.7 & 1.3, 2.4 29.8, 34.0 & 4.3, 5.0

in the flow and for various conditions. For P(χ |ξ ) we also investigate the choice of ξ at which the PDF is calculated. Each PDF was evaluated at a point in space and over a number of images of ξ and (two-dimensional) χ , only at locations with mean mixture fraction, ξ  > 0.01, and hence with a strong PLIF signal, which restricted the results to z < 31 mm. Data for each PDF was compiled by considering the local (inside a window of axial length 2η K and radial length 10η K , amounting to 500–600 points) ξ and χ over 200 images, resulting in more than 110,000 data points for that spatial location. The effect of spatial broadening introduced by the finite 2 × 10η K windows was investigated by compiling similar PDFs over windows down to 1 × 1η K , while accepting their associated reduced convergence. Conditional quantities were obtained by assigning each value of χ to its corresponding value of ξ at the same location (pixel position) and time (image frame), with ξ split into 30 boxes spanning the range from the minimum to the maximum measured value of ξ during a run. This method is illustrated in Fig. 1 (right). PDFs of the resulting conditional χ |ξ , P(χ |ξ ), were generated only if the number of data points within the box of a particular value of ξ was over 2,000 points.

3 Results and Discussion 3.1 Velocity field The hot wire measurements presented in Fig. 2 concern the case of equal velocity between the injected flow and the co-flow in the entire range of U inj ≈ U co = 2.5– 3.8 m/s, corresponding to Returb between 36 and 54. The local mean axial velocity, U, normalized by the mean bulk co-flow velocity (U/U co ) was about 1.12±0.02 in the radial region away from the injector and quartz tube wall boundary layers (0.1D < r < 0.4D), and over the axial extent z = 0 – 42 mm. Also, U decayed along the centreline (r = 0) from a value of 1.80±0.05 close to the injector nozzle (z = 3 mm), towards a value of 1.15±0.02 at z = 42 mm. The root mean square (rms) of the fluctuations of the local axial velocity, u , i.e. u ≡ (u2 )1/2 , normalized by the local mean axial velocity (u/U) was 0.125±0.010 at z = 0 over a similar radial extent away from the injector and quartz tube walls, decaying to 0.105±0.015 by 42 mm downstream. Along the centreline u/U tended to a value of 0.090±0.005 at z = 42 mm. The longitudinal (axial) integral (outer) lengthscale Lzz was obtained by integrating over the normalized autocorrelation function of the turbulent velocity

226

Flow Turbulence Combust (2008) 81:221–234

Fig. 2 Clockwise from top-left: U/U co as a function of r/D and spanning 0 < z < 42 mm, U/U co as a function of z along the centreline (r = 0), u/U as a function of z along the centreline (r = 0), and, u/U as a function of r/D and spanning 0 < z < 42 mm. All results are with the 2.2 mm nozzle and for ‘equal velocity’ conditions, with U inj ≈ U co in the range 2.5–3.8 m/s, corresponding to Returb between 36 and 54

fluctuations and the use of Taylor’s hypothesis. At 0.2D < r < 0.4D and z = 0 it was found to be 3.5±1.0 mm, and hence of the order of the grid hole size, increasing by about 1.0±0.5 mm in the first 42 mm downstream. The background turbulent Reynolds number (Returb ) based on the rms of the velocity fluctuations in the co-flow (uco ) and Lzz ,

Returb ≡

uco Lzz ν

(4)

was in the range 48–108. We chose Returb to be based on Lzz rather than the Taylor microscale in order to be consistent with our previous work, but also because Lzz is the most energy-containing eddy size, as is evident in Fig. 3a (right) and discussed at the end of this section. For comparison with other work, the corresponding Taylor scale Reynolds number (Reλ ) in our range of conditions can be estimated from [23],

Reλ ≈ (10Returb )1/2

(5)

giving values in the range 22–34. Equation 5 was checked with data from the hot wire measurements and found be accurate within ±20%. In addition, the velocity is chosen to be that of the co-flow, since the turbulent mixing is not set up in the large

Flow Turbulence Combust (2008) 81:221–234

227

Fig. 3 (a) Left: Normalized turbulence spectral power, and, Right: energy content spectra with U inj ≈ U co in the range 2.5–3.8 m/s, corresponding to Returb between 36 and 54. (b) Turbulence PDF with the local mean velocity at U = 2.6 m/s and Returb = 33

scales by shear in the jet, but is rather a result of the background turbulence in the co-flow, as discussed at the end of the section that deals with the moments of χ . In fact, this is the reason behind our referring to the low momentum release of the scalar in this configuration as an ‘injected flow’, rather than a ‘jet’. In any case, the co-flow velocity can be related to the injection velocity at z = 0 through the top two graphs in Fig. 2. Furthermore, the hot wire was used to measure turbulence spectra and the PDF of u at various locations and with various inlet conditions. Figure 3a is entirely consistent with the sample data for canonical wall-bounded flow in Bogard and Thole [24] over the entire range of (axial) wavenumbers (κz ), and matches well the prediction of Von Kármán’s interpolation formula derived theoretically for isotropic turbulence and ignoring viscous dissipation [25], 2π Ezz (κz ) 4 =  5/6 u2 Lzz 1 + (4/3· Lzz κz )2

(6)

228

Flow Turbulence Combust (2008) 81:221–234

up to and including a short −5/3 inertial subrange region, beyond which the effect of viscosity cannot be ignored. As κz approaches zero, all spectra collapse forced by the choice of normalization to a value of about 4–5, exactly as with Bogard and Thole [24], and the value of 4 from Eq. 6. The relative contribution to the total turbulent kinetic energy in the flow is apparent in the plots of Fig. 3a (right). Eddies of the characteristic size of Lzz contribute the most to the energy. The maximum normalized energy content is around 0.3, compared to 0.3–0.4 of Bogard and Thole [24]. In Fig. 3b a small deviation can be seen from the random process, as expected. The skewness of the PDF is −0.2 and the kurtosis 3.2. The above results allow one to reason that, initially, mixing occurs in regions of relatively uniform velocity, with the plume quickly diffusing into the tube. They also confirm that the background turbulence in the co-flow is well developed, near homogeneous and isotropic, with no peaks or spikes in the spectrum associated with vortex shedding from the nozzle exits or any other large-scale coherent structure associated with shear, and hence showing a character as would be expected in decaying grid-generated turbulence, but also confined in the tube. 3.2 Mixture fraction Results for the mixing field are shown in Fig. 4. It was observed that away from the immediate vicinity of the source, this configuration resulted in self-similar distributions of the mean mixture fraction, ξ , akin to diffusion from a point source. Downstream of the nozzle, i.e. the region occupying z > 10 mm, measurements of ξ  were consistent with the Gaussian plume equation [26], 1 ξ  = 8



U inj U co



 1 r 2 exp − (s/d)2 2s

(7)

with the resulting plumes having Gaussian radial profiles adhering to a ‘short length’ spread behaviour associated with characteristic width s = (u/U)z, and a z−2 power law axial decay. Equal velocity plumes were initially relatively thin and short, with the ξ  = 0.1 iso-contour occupying radial lengths of r/d < 1.5 and axial lengths of up 20 mm. The mixture fraction variance, ξ 2 , increased to a maximum and then decayed downstream. The rms of the mixture fraction fluctuations, ξ 2 1/2 , normalized by the mean, ξ 2 1/2 /ξ , increased from zero at the source to about 2.5–4.0 downstream for the equal velocity flows, the value increasing with increasing Returb , and 0.5–1.5 for the jets, the value decreasing with increasing U inj. 3.3 Scalar dissipation moments The mean (two-dimensional) scalar dissipation, χ , and the rms of the (twodimensional) scalar dissipation fluctuations, χ 2 1/2 , along the centreline are shown in Fig. 4. The mean scalar dissipation reaches a peak at about 2–4 injector diameters, after which a quick decay begins due to mixing. The rms is always higher than the mean and both increase with the Reynolds number. In Markides and Mastorakos [1] the full, mean three-dimensional scalar dissipation, χ3D , was estimated by

Flow Turbulence Combust (2008) 81:221–234

229

2 1/2 for Runs A, C, D and E (see Table 1), as Fig. 4 (a) ξ , (b) ξ 2 1/2 , (c) χ2D , and, (d) χ2D functions of z along the centreline (r = 0). Adapted from Markides and Mastorakos [1]

symmetry, in the mean, along the centreline. This was found to be adequately described by the timescale ratio model, χ3D  = C D

ξ 2  (k/)

(8)

with C D ≈ 2.0 for axial distances corresponding to residence times greater than 0.1–0.2k/ turbulent timescales and radial distances r/d < 1, where k ≡ 3/2·u2 is the turbulence kinetic energy and  its mean dissipation. As mentioned above, it is evident from the above that mixing, especially for the equal velocity case, was mostly dominated by the background turbulence in the co-flow. 3.4 Scalar dissipation PDFs Figure 5 presents PDFs of (the two-dimensional) scalar dissipation, P(χ ), for all investigated runs, spanning U co from 3.1 to 6.9 m/s and Returb from 48 to 108. Each line corresponds to a PDF evaluated at a different downstream axial distance from the injector nozzle (z) along the centreline (r = 0). The independent axes are

230

Flow Turbulence Combust (2008) 81:221–234

Fig. 5 PDFs of χ , with F(χ ) ≡ exp{−(lnχ − μ) ˜ 2 /2σ˜ 2 }/[(2π)1/2 σ˜ χ ], at z = 4.3, 17.6 and 30.8 mm, and r = 0. Runs A to F are equal velocity flows. Runs G to L are jet-like flows. Black lines indicate lognormality. For the conditions, see Table 1

scaled and normalized, exp{−(lnχ − μ) ˜ 2 /2σ˜ 2 }/[(2π )1/2 σ˜ χ ] with μ˜ and σ˜ the mean and standard deviation of ln(χ ) respectively, such that lognormality appears as a straight line. With this normalization we aim to investigate, in a global sense, whether there is agreement or not with the lognormal distribution, rather than to demonstrate a detailed comparison between the PDFs at various points. The figure indicates that the lognormal description of χ is reasonable, but that there exist non-negligible deviations at the tails, which mostly influence the higher order moments. The data include flows with approximately equal velocity between the injected flow and the co-flow, and jet-like flows with U inj being 3–5 times U co . The results of Markides and Mastorakos [1] show that due to the dominance of the background turbulence in the co-flow, large-scale mixing in this flow is adequately approximated as turbulent diffusion from a point source for both equal velocity and jet flows. The collapse of the normalized PDFs suggests that the degree to which P(χ ) is lognormal is independent of position and flow type. Figure 6 presents P(χ |ξ ) for all investigated runs. Each line corresponds to a PDF evaluated either at a different axial location (z) along the centreline (r = 0), or at a different value of ξ . Note that in Fig. 6, as in Fig. 7a, the PDFs of χ |ξ and ln(χ |ξ ) respectively are plotted at ‘all available ξ ’, meaning at those boxes of ξ at which the condition for the number of data points being greater than 2,000 is met. This figure shows that P(χ |ξ ) has shape almost identical to P(χ ), is also approximately lognormal but with similar deviations, and that these conclusions are independent of the value of ξ where it is compiled.

Flow Turbulence Combust (2008) 81:221–234

231

Fig. 6 PDFs of χ |ξ , with G(χ |ξ ) ≡ exp{−(lnχ |ξ − μ) ˜ 2 /2σ˜ 2 }/[(2π)1/2 σ˜ χ |ξ ], at ‘all available ξ ’, z = 4.3, 17.6 and 30.8 mm, and r = 0. Runs A to F are equal velocity flows. Runs G to L are jetlike flows. Black lines indicate lognormality. For the conditions, see Table 1

In more detail, Fig. 7a shows that P[ln(χ |ξ )] tended to the Gaussian with increasing z and along the centreline for the same conditions. Hence, we find that the deviation from lognormality, which is mostly encountered at the tails, decreased with increasing z. Furthermore, it seems that the tails can be described by an exponential distribution as suggested in Lee and Pope [27], which in the semi-logarithmic plots of Fig. 7a would appear as straight lines. For P[ln(χ |ξ )] the skewness increased from about −2.5 reaching −0.3 at long z, in agreement with the trend observed in a previous DNS study [17] at Reλ = 28 and 49, and matching closely the value of −0.12 found in Vedula et al. [18] for S c = 1 and Reλ = 38, where S c is the Schmidt number and Reλ is the Reynolds number based on the Taylor microscale. These values of Reλ correspond to Returb of about 80, 240 and 140 respectively from Eq. 5. The kurtosis decreased from about 11 to 2, similar to the value of 3.2 found in Eswaran and Pope [17] at long times and 2.9 stated in Vedula et al. [18]. It seems as though the results quoted in Vedula et al. [18] correspond to relatively long simulations. Finally, the ratio σχ /μχ was found to increase with z, from 1.2±0.2 at z = 1 mm, to 2.8±0.3 at z = 31 mm, whereas σχ|ξ /μχ|ξ increased from 0.8±0.1 at z = 1 mm, to 1.2±0.2 at z = 31 mm. The latter ratio was also found to be approximately independent of ξ . Actually, it should be pointed out that σχ|ξ /μχ|ξ was indeed insensitive to the choice of ξ , except in the perfectly unmixed fluid (ξ = 0) where it was found to have much higher values, and which is responsible for the difference between σχ /μχ and σχ|ξ /μχ|ξ . Both increased gradually with z as shown in Fig. 7b,

232

Flow Turbulence Combust (2008) 81:221–234

Fig. 7 (a) Evolution of P[ln(χ |ξ )] for equal velocity, U inj ≈ U co = 4.1 m/s, Returb = 64, at all available ξ with z along r = 0. Black curves indicate lognormal distributions. Dashed red lines indicate exponential distributions. (b) Evolution of σ/μ for χ , and χ |ξ averaged over all available ξ , with z along r = 0 for four different equal velocity runs spanning U inj ≈ U co from 3.1 to 4.1 m/s and Returb from 48 to 64

although it is possible to argue that the latter tends to level out at the value of 1.2±0.2 from as short a length as z = 10 – 15 mm. No clear evidence was found that the ratios were significantly affected by the source size (injection diameter), U co (or equivalently Returb ), at least within the tested range of conditions, or the global/large-scale nature of the flow (equal velocity vs. jets). The present final result for σχ /μχ is in excellent agreement with the experimental results of Dowling [9] and Dahm and Buch [11] of 1.4 and 1.1 respectively. On the other hand, it is considerably lower than 9.5, taken from Vedula et al. [18] for S c = 1 and Reλ = 38. It is unclear whether the numerical forcing scheme used in these simulations is responsible for part of this discrepancy. Nevertheless, given the fact that σχ /μχ does not seem to level out at long lengths in Fig. 7b, one must take into account that, as stated previously, the simulation run time perhaps covers a significantly longer spatial domain than what is available in the present study. The maximum length in Fig. 7b, i.e. z = 31 mm, corresponds to about 1.15±0.35(k/)o initial turbulent timescales. The initial (z = 0) turbulent timescales were estimated, as in Markides and Mastorakos [1], by relating these to the turbulent eddy turnover times (τzz ) and the known Returb and Reλ from Eq. 5 [23], via (k/)/τzz = 1.0±0.1, at z = 0. This expression was checked in a few cases with the hot wire data and

Flow Turbulence Combust (2008) 81:221–234

233

agreement was found to within ±20%. In turn, τzz were calculated directly from the hot wire measurements, through τzz ≡ Lzz /u. The reported results on the ratios σχ /μχ and σχ|ξ /μχ|ξ may be considered the main results of this experiment. These ratios are very important for turbulent combustion, as extinction events tend to occur at high χ , while autoignition tends to occur in low χ . The present data can assist the development of advanced models for such phenomena.

4 Conclusions The high spatial resolution PLIF measurements of Markides and Mastorakos [1], which were concerned with the mixing field formed by the axisymmetric injection of a passive scalar from a finite sized round nozzle, into a uniform turbulent co-flow confined within a cylindrical tube, with various velocity ratios (equal velocity, or jet-like), were processed for the accurate recovery of the two-dimensional scalar dissipation rate. The PDFs of both χ and χ |ξ were calculated along the centreline and found to deviate slightly from a lognormal distribution, with the deviation being more pronounced closer to the source and decaying downstream. The tails of the PDFs can be described by an exponential distribution. The ratio of the standard deviation over the mean of χ and χ |ξ increased with streamwise distance, reaching values of 2.8 and 1.2 respectively at 31 mm, or 1.15±0.35 initial turbulent timescales, from the source. The values and evolution of these ratios with axial distance from the source were not affected by the source size (injection diameter), co-flow velocity (or equivalently Re), or the global/large-scale nature of the flow (equal velocity vs. jets).

References 1. Markides, C.N., Mastorakos, E.: Measurements of scalar dissipation in a turbulent plume with planar laser-induced fluorescence of acetone. Chem. Eng. Sci. 61, 2835–2842 (2006) 2. Sreenivasan, K.R.: Possible effects of small-scale intermittency in turbulent reacting flows. Flow Turbul. Combust. 72, 115–131 (2004) 3. Frank, J.H., Kaiser, S.A., Long, M.B.: Multiscalar imaging in partially premixed jet flames with argon dilution. Combust. Flame 143, 507–523 (2005) 4. Pitsch, H., Steiner, H.: Scalar mixing and dissipation rate in large-eddy simulations of non-premixed turbulent combustion. Proc. Combust. Inst. 28, 41–49 (2000) 5. Sreedhara, S., Huh, K.Y.: Assessment of closure schemes in second-order conditional moment closure against DNS with extinction and ignition. Combust. Flame 143, 386–401 (2005) 6. Bilger, R.W.: Some aspects of scalar dissipation. Flow Turbul. Combust. 72, 93–114 (2004) 7. Gurvich, A.S., Yaglom, A.M.: Breakdown of eddies and probability distributions for small-scale turbulence. Phys. Fluids 10, S59–S65 (1967) 8. Peters, N.: Local quenching due to flame stretch and non-premixed turbulent combustion. Combust. Sci. Technol. 30, 1–17 (1983) 9. Dowling, D.R.: Mixing in gas phase turbulent jets. Ph.D. thesis, California Institute of Technology (1988) 10. Effelsberg, E., Peters, N.: Scalar dissipation rates in turbulent jets and jet diffusion flames. Proc. Combust. Inst. 22, 693–700 (1988) 11. Dahm, W.J.A., Buch, K.A.: Lognormality of the scalar dissipation pdf in turbulent flows. Phys. Fluids A Fluid Dyn. 1, 1290–1293 (1989) 12. Pitts, W.M., Richards, C.D., Levenson, M.S.: Large- and small-scale structures and their interactions in an axisymmetric jet. Internal Report, National Institute of Standards and Technology (1999)

234

Flow Turbulence Combust (2008) 81:221–234

13. Su, L.K., Clemens, N.T.: The structure of fine-scale scalar mixing in gas-phase planar turbulent jets. J. Fluid Mech. 488, 1–29 (2003) 14. Sreenivasan, K.R., Antonia, R.A., Danh, H.Q.: Temperature dissipation fluctuations in a turbulent boundary layer. Phys. Fluids 20, 1238–1249 (1977) 15. Antonia, R.A., Sreenivasan, K.R.: Log-normality of temperature dissipation in a turbulent boundary layer. Phys. Fluids 20, 1800–1804 (1977) 16. Kerstein, A.R., Ashurst, W.T.: Lognormality of gradients of diffusive scalars in homogeneous, two-dimensional mixing systems. Phys. Fluids 27, 2819–2827 (1984) 17. Eswaran, V., Pope, S.B.: Direct numerical simulations of the turbulent mixing of a passive scalar. Phys. Fluids 31, 506–520 (1988) 18. Vedula, P., Yeung, P.K., Fox, R.O.: Dynamics of scalar dissipation in isotropic turbulence: a numerical and modelling study. J. Fluid Mech. 433, 29–60 (2001) 19. de Bruyn Kops, S.M., Mortensen, M.: Conditional mixing statistics in a self-similar mixing layer. Phys. Fluids 17, 095107:1–095107:11 (2005) 20. Ghandhi, J.B.: Spatial resolution and noise considerations in determining scalar dissipation rate from passive scalar image data. Exp. Fluids 40, 577–588 (2006) 21. Ferchichi, M., Tavoularis, S.: Scalar probability density function and fine structure in uniformly sheared turbulence. J. Fluid Mech. 461, 155–182 (2002) 22. Mi, J., Antonia, R.A., Anselmet, F.: Joint statistics between temperature and its dissipation rate components in a round jet. Phys. Fluids 7, 1665–1673 (1995) 23. Pope, S.B.: Turbulent Flows, pp. 242–245. Cambridge University Press, Cambridge (2000) 24. Bogard, D.G., Thole, K.A.: Wall-bounded shear flows. In: Johnson, R.W. (ed.) The Handbook of Fluid Dynamics, pp. 13:57–13:59. CRC Press, Boca Raton (1998) 25. Hinze, J.O.: Turbulence, pp. 221–259. McGraw-Hill, New York (1975) 26. Csanady, G.T.: Turbulent Diffusion in the Environment, pp. 58–65. Reidel, Amsterdam (1973) 27. Lee, Y.Y., Pope, S.B.: Nonpremixed turbulent reacting flow near extinction. Combust. Flame 101, 501–528 (1995)