Scalar dissipation rate statistics in turbulent swirling jets

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The scalar dissipation rate statistics were measured in an isothermal flow formed by discharging a central jet in an annular stream of swirling air flow. This is a typical geometry used in swirl-stabilised burners, where the central jet is the fuel. The flow Reynolds number was 29000, based on the area-averaged velocity of 8.46 m/s at the exit and the diameter of 50.8 mm . The scalar dissipation rate and its statistics were computed from two-dimensional imaging of the mixture fraction fields obtained with planar laser induced fluorescence of acetone. Three swirl numbers, S, of 0.3, 0.58 and 1.07 of the annular swirling stream were considered. The influence of the swirl number on scalar mixing, unconditional and conditional scalar dissipation rate statistics were quantified. A procedure, based on a Wiener filter approach was used to de-noise the raw mixture fraction images. The filtering errors on the scalar dissipation rate measurements were up to 15%, depending on downstream positions from the burner exit. The maximum of instantaneous scalar dissipation rate was found to be up to 35 s −1 , while the mean dissipation rate was 10 times smaller. The probability density functions of the logarithm of the scalar dissipation rate fluctuations were found to be slightly negatively skewed at low swirl numbers and almost symmetrical when the swirl number increased. The assumption of statistical independence between scalar and its dissipation rate was valid for higher swirl numbers at locations with low scalar fluctuations and less valid for low swirl numbers. The deviations from the assumption of statistical independence were quantified. The conditional mean of the scalar dissipation rate, the standard deviation of the scalar dissipation rate fluctuations, the weighted probability of occurrence of the mean conditional scalar dissipation rate and the conditional probability are reported.

I. INTRODUCTION
The dissipation rate, which is identified as a characteristic diffusion time scale imposed by the mixing field 1 , requires modelling in essentially all computational models for nonpremixed combustion. The scalar dissipation rate can be used for instance in a PDF-flamelet approach in which species mass fraction, mean reaction rate, temperature etc. are precomputed as a function of two variables, namely mixture fraction and the scalar dissipation rate and stored in a library. The scalar dissipation rate was also identified as the criterion that was able to predict the local extinction phenomenon of diffusion flames. It was pointed out that local extinction of diffusion flames occurred where the conditional dissipation rate exceeded a critical quenching value and global extinction took place when the cumulative probability of the conditional dissipation rate exceeded a critical threshold [1][2][3][4] .
In the context of combustion modelling that is based on ensemble-averaged (RANS) or spatially-averaged (LES) equations describing the fluid motion, the probability distribution of the conditional scalar dissipation rate and the corresponding mean value must also be known. A common practice is based on the assumption of statistical independence between the scalar and its dissipation rate, which means that their joint probability density function is equal to the product of their individual probability density functions 4,5 . The probability density function of the conditional scalar dissipation rate was typically assumed to be lognormally distributed, as it was firstly suggested by Kolmogorov 6 . A detailed knowledge of the statistical properties of mixture fraction fluctuations and their conditional dissipation rate is also required in conventional flamelet methods, advanced flamelet and transported PDF methods, which include the fluctuations of the scalar dissipation rate 7-9 , stochastic well-stirred reactor models 10 and conditional moment closure models 11 . Measurements of scalar dissipation rate and its statistics (conditional and unconditional) are rare despite their importance for a wide range of turbulent flow applications, especially in complex flows.
Assumptions of, e.g. statistical independence in simple jet flows may not be valid in complex (swirling) flows. Moreover, flows with high and low swirl numbers may exhibit different behaviour in terms of statistics of scalar dissipation rate and mixture fraction.
In addition, transition from low to high swirl number, and wise versa, can qualitatively change assumptions that are adopted in computational fluid dynamics. Indeed, it will be demonstrated later that statistical dependence is valid for flows with high swirl number. In this context, the swirl number is the key variable that influences the flow behaviour and hence, all the statistics should be linked to this single parameter.
In the context of combustion calculations three parameters are usually required; the mean scalar dissipation rate conditional on the stoichiometric mixture fraction, the weighted probability of occurrence of the mean conditional scalar dissipation rate and the standard deviation of conditional dissipation rate. The weighted probability of occurrence of the mean conditional scalar dissipation rate represents the molecular diffusion in scalar space in the transport equation for the scalar p.d.f. and is proportional to the mean reaction rate 12 . The conditional scalar dissipation rate is also proportional to the mean reaction rate 13 . Mastorakos 14 demonstrated that the probability of autoignition was proportional to conditional scalar dissipation rate. The maximum of conditional scalar dissipation rate was not the stoichiometric mixture fraction as it would have been intuitively expected. In addition, the extinction of a diffusion flame occurs when the conditional scalar dissipation rate exceeds a certain value 14 .
The conditional mean, the weighted probability of occurrence and the standard deviation of fluctuations of conditional dissipation rate are computed as follows: where χ is scalar dissipation rate, z st is stoichiometric mixture fraction, P is probability density function, χ|z st = z * is mean scalar dissipation rate conditional on the stoichiometric mixture fraction, E χ|z is weighted probability of occurrence of the mean conditional scalar dissipation rate, χ |z st = z * is standard deviation of conditional dissipation rate, N st is number of scalar dissipation rate samples corresponding to the stoichiometric mixture fraction. The averaging denoted by the angular brackets is only carried out for events that satisfy the conditions to the right of the vertical rule.
Measurements of the scalar dissipation rate are not straightforward due to temporal and spatial resolution requirements. Such measurements require high spatial resolution, which is comparable to Batchelor or Kolmogorov length scale (for Schmidt number equal to one).
These requirements are not easily achieved in practice due to optical system limitations, e.g. laser sheet thickness, optical aberrations etc. Effective resolution and noise-induced apparent dissipation, i.e. artificial over-or underestimation of scalar dissipation rate are mainly determined by the system transfer function 15 , 16 . The system transfer function will be dependent on optics quality, experimental system alignment and random fluctuations of noise in the measured data and is discussed later in the manuscript. Appropriate de-noising techniques must be used in order to obtain meaningful statistics of scalar dissipation rate.
A description of de-noising technique, which is employed in this work to obtain 'true' scalar dissipation rates, will be discussed in subsequent sections.
Measurements of scalar dissipation rate have been demonstrated in a number of papers.
Dibble 17 presented the first laser-based measurements of scalar dissipation rate in turbulent non-premixed, non-reacting jets and the first laser-based or otherwise in reacting jet flames by using an optical multichannel analyser. Scalar dissipation rate measurements and associated issues were also reported in a number of papers [18][19][20][21][22][23][24][25][26][27][28] . Even though the scalar dissipation rate and structural information about the scalar fields was addressed in a number of papers, the statistics were limited to either counterflow geometry or to non-swirling jets.
The scalar dissipation rate statistics in turbulent swirling flows have not been examined to the best of our knowledge. Our aim is, therefore, to extend knowledge to configurations relevant to practical combustors, which are typically based on swirl-stabilized burners. In this context, the primary goal of the present work was to provide the statistical information, relationships between scalar fluctuations and their dissipation rate, as well as to present both unconditional and conditional scalar dissipation rate statistics.
Though the computation of scalar dissipation rate generally involves the computation of 3D gradient of the mixture fraction, it is practically hard to measure the 3D fields of mixture fraction. In homogeneous and isotropic turbulent fields, the three-dimensional scalar dissipation rate can be easily calculated, because the statistics of the mixture fraction fluctuations are the same in all three directions. In this case, one of the components can be measured and the calculation of the scalar dissipation rate is quite straightforward. The scalar dissipation rate, like other variables in turbulent flows, fluctuates in time and space and its formal definition is given by the following formula, where the diffusivity for the scalar (acetone vapour), D a , is assumed to be constant and equals to 0.124 cm 2 /s.
where z is mixture fraction, D a is acetone diffusivity in air and x and y are spatial coordinates correspondingly The remaining paper is structured as follows: The next section describes the swirling air flow burner and the optical instrumentation used to measure the scalar (mixture fraction), which was then used to evaluate the mixture fraction scalar dissipation rate. The spatial requirements and de-noising technique for the dissipation rate measurements are also briefly discussed. The last section describes the results and discusses the findings. The paper ends with a summary of the main conclusions.

A. Experimental setup
The flow section (item 15 in Fig. 1) consisted of two concentric pipes with the annulus supplying swirling air and the central pipe delivering air seeded with acetone vapour (measured scalar quantity) 29,30 . The central pipe (fuel) had an inner diameter D f of 15 mm and an outer diameter of 18 mm, was 0.75 m long and was located concentrically in the outer pipe of inner diameter, D of 50.8 mm and centred within it by three screws at 25 mm upstream of the burner exit. The flow development section was 0.264 m long.
The annular air stream was split into two separately metered streams named 'swirling' and 'axial' air ( Fig. 1). The swirling stream was created by passing air through a static swirler containing six milled tangential slots to impart angular momentum. The static swirler was located in a plenum chamber in which the swirling air was combined with the second stream that delivered 'axial air'. Metal plates were installed in the axial and tangential air sections of the plenum chamber to ensure that the axial and tangential air streams were distributed uniformly upstream of the inlets into the annular air supply stream of the burner, where they were combined to control the strength of swirl at the burner exit. The tangential and axial air flowrates were metered by flowmeters after correction to atmospheric pressure and temperature. The axial direction, parallel to the flow propagation, was denoted as 'y' and the radial direction, perpendicular to the main direction of the flow, as 'x'. The swirl number used as the primary variable was computed by using the velocity profiles obtained by Laser Doppler Anemometry 30 . The swirl number is defined as follows: where r i is radius of the inner pipe, ρ is density, U is axial velocity component, W is tangential velocity component, G Θ is axial flux of angular momentum, G z is axial flux of axial momentum, S is swirl number, R is radius of the outer pipe, D is diameter of the outer pipe.
All swirling flows can generally be split into two groups i.e. weak (S < 0.6) and strong In addition, the same burner will be used to study reacting swirling flows, flame stabilisation and thermal dissipation rates. The flame stabilisation is affected by the swirl number and, therefore, both low and high swirl numbers have to be considered. . It is also interesting to note that the Batchelor scale obtained from the spectrum that was computed from radial-derivative is smaller than that from the spectrum computed from axial-derivative. This suggests that turbulence is more developed and intensive in radial direction, which is also expected in swirling flows. An Imager Intense R CCD camera from LaVision Inc. was used to record the images (item 12 in Fig. 1). This camera was equipped with a 50 mm f1.4 Nikkor lens. A 13 mm extension ring was also used after the camera lens for closer focusing. A BG3 bandpass Schott filter (item 13 in Fig. 1) was used in front of the camera lens in order to block any remaining 532 nm light that might interfere with the acquired signal. The optical magnification was determined by using a calibration target plate and was found to be 0.0263 mm/pixel.
The laser beam waist was measured directly by using a CCD camera technique. The positive cylindrical lens was rotated to 90 • , so the laser sheet thickness could be captured by measuring the intensity of acetone vapour fluorescence. Averaging of 50 images minimized laser beam intensity fluctuations. The beam waist was defined as the diameter where the beam irradiance was 1/e 2 or 0.135 times its maximum value and found to be 0.158 mm. The average laser sheet intensity distribution used in the correction procedure was obtained by supplying acetone vapour through the axial fuel nozzle without air coflow and subsequent measuring the intensity at the jet potential core. In order to eliminate the dark noise, a set of images was recorded, averaged and then simply subtracted from each image obtained during the experiments. No background image was recorded during experiments, because incident background light was blocked by the BG3 filter.
The synchronization of the laser flashlamp, the laser Q-switch and the camera was ensured by using a programmable PC-based timing unit (PTU) using TTL pulses. The laser flashlamp was fired continuously at 10 Hz. The laser Q-switch was triggered every time when an image was recorded. The camera was triggered at the same time as the Q-switch trigger (TTL pulse was sent to Q-switch and to the camera simultaneously) and camera exposure time was set to 1 µs. The camera shutter opened before the start of the laser pulse and conditions and, therefore, has been widely applied to study various aspects of mixing in non-reacting jets.
Acetone has been used as a tracer for many years. The choice of acetone is simple due to its outstanding properties and in many ways can be regarded as an ideal tracer for the PLIF technique due to its linear relationship between the fluorescent intensity, laser power and acetone concentration in the flow. Acetone has also an accessible absorption spectrum at the UV. The fluorescent signal is emitted in the visual range of light spectrum 350-600 nm and the corresponding peak of the fluorescent emission spectrum is between 450 to 500 nm. Fluorescent signal can be recorder onto a digital sensor of a CCD camera and after which can be post processed as required. Moreover, the signal from acetone fluorescence is relatively high and its interpretation in actual fuel concentration is in fact straightforward.
The PLIF method is based on molecule excitation to a higher electronic energy level by absorbing the energy of a laser beam. The molecule at this energy state is unstable and will go from upper energy level to the lower energy level emitting photon of the same energy as the original absorbed photon. The emitted photon can be detected and used to measure the light intensity, which is proportional to a number of molecules, or simply concentration of a tracer. The measured acetone fluorescent signal (the total number of photons) at each pixel position on the CMOS sensor (x, y) can be written as follows: where η ef is transmission efficiency of collection optics, Ω is solid angle collected by imag- ing optics, f 1 (T ) is fractional population of lower laser-coupled state in the absence of the laser field, χ m is the mole fraction of the absorbing species, n is the total gas number density, V c is collection volume imaged onto photodetector element, B 12 is Einstein B coefficient for single-photon laser stimulated process, Q 21 is the collision transfer coefficient, E v is the spectral fluence of the laser, A 21 is Einstein A coefficient for spontaneous emission. If the reference position is at the exit of the fuel nozzle in a potential core, where no mixing with ambient air occurs, then the above mentioned ratio is directly related to the mixture fraction z(x, y).
The raw images that were recorded during PLIF experiments I raw (x, y) were corrected by using the following formula. This formula takes into account several different sources of noise.
where I cor is corrected image of acetone fluorescence intensity, I bgr (x, y) is background image, image formed by a photodetector in absence of light, L sh is average laser sheet intensity distribution.
The background image I bgr (x, y) is usually acquired with the same camera settings and with the laser firing but without flow in the test volume. The background image that was acquired during experiments was blank, i.e. no photons were detected. This was achieved through BG3 filter that blocked ambient light. In addition, no reflections from the burner were observed. Reflections from the dust particles (due to 532 nm light wavelength component in the laser beam) that were present in the test volume were also blocked by the BG3 filter. The dark image was acquired with a cap placed on the lens and then was simultaneously subtracted from recorded images during imaging. The average laser sheet intensity distribution was obtained by supplying acetone vapour through the axial fuel nozzle without air co-flow and subsequent measuring the intensity in the jet potential core. The laser sheet profile measured in this way is negligibly different from the laser sheet profile that can be obtained from the dye cells 24 . The variations of the profile from laser shot-to-shot were negligible.
The reference position (fluorescence intensity) was at the exit of the fuel jet in a potential core, where no mixing with ambient air occurs (without swirling co-flow). In order to reduce uncertainties in acetone fluorescent intensity at a reference position of known acetone concentration, the reference measurements were obtained over a small square region (1×1 mm) within the potential core of the acetone vapour jet without co-flow of air and for air flow rate of 40 l/min passing through the acetone seeder. The acetone seeder (item 9 in where N is the number of acquired images. The fluctuations of the acquired acetone intensity are mainly due to the fluctuations in laser power from pulse to pulse. The fluctuations in laser power from pulse to pulse were minimised by using longer warm-up time (typically longer than 30 min) and were measured to be less than 3%, which is less than filtering errors of 15% (due to Wiener filtering procedure). It is, therefore, assumed that fluctuations in laser power from pulse to pulse should not significantly affect the absolute values of scalar dissipation rate.

B. Data processing
The most commonly used cameras in imaging experiments are currently based on CCD image sensors and the image quality is the most important characteristic. The image quality is related to the actual spatial resolution and the ability to record the contrast of a real object to the camera detector. The magnification (scale factor) of any optical system is typically determined by using a calibrating target with known spatial resolution, e.g. 1951 USAF resolution test chart. However, in several applications, the nominal spatial resolution (or magnification) is not a limiting factor and differs from the so-called 'real' spatial resolution.
For instance, in mixture fraction gradient measurements (or scalar dissipation rate measurements) the point spread function (PSF), which is defined as the intensity distribution at an image plane, produced by imaging an infinitesimally small portion of light can represent the actual optical resolution 16 .
The PSF is related to an optical transfer function (OTF) via a modulation transfer function (MTF). The OTF is defined as the ability to transfer the contrast of a real object  In this work, the Wiener-Kolmogorov filtering procedure (Wiener filter for simplicity) was used to reduce noise that was present in the measured data. This filtering procedure is based on a signal comparison between noiseless (estimated) and noise corrupted signal (measured) 24,32,33 . In order to be able to compute the Wiener filter, the power spectral densities of both the noise-free image and the noise must be known, which are typically not known in advance. It was suggested 33,34 that a recorded set of images (measured mixture fraction) can be used to estimate the general trend of the true signal spectral density and the corresponding variations of the noise density. If the general trend of the true signal is known, then it will be possible to model the true signal for the wave numbers, where the noise contributes significantly. A detailed description of the Wiener filter with application to scalar dissipation rate measurements is given by Soulopoulos 24,32 . In this research, the estimated scalar dissipation rates after filtering were accurate to within 15% depending on shear layer and centreline. Five spatial locations were considered and are shown in Figure   3. All probability density functions of the scalar dissipation rate of the mixture fraction are presented here in terms of P (C), where P denotes probability density function and C is defined as follows: where σ ln(χ) is standard deviation of natural logarithm of scalar dissipation rate It should be noted that the orientation of the scalar dissipation rate vector may be away This slightly asymmetrical scalar distribution may be due to the weak internal recirculation zone that always exists in swirling flows with sufficiently high swirl numbers. This non-symmetrical distribution was also reported by Milosavljevic 30 and could be linked to the presence of the recirculation zone, which was also supported by the velocity measurements, which were performed by Milosavljevic 30 . It is hardly unlikely that this non-symmetrical scalar distribution is due to wakes downstream three locking screws retaining the central pipe delivering the acetone vapour jet. Note that slightly asymmetrical scalar distribution may be due to the weak internal recirculation zone that always exists in swirling flows with sufficiently high swirl numbers. Note that slightly asymmetrical scalar distribution may be due to the weak internal recirculation zone that always exists in swirling flows with sufficiently high swirl numbers.

B. Instantaneous and mean scalar dissipation rate
An example of instantaneous spatially distributed raw scalar dissipation rate (computed from raw mixture fraction image) and the result of the application of the proposed denoising procedure is shown in Figure 13. It is clearly seen that the scalar dissipation rate, which is computed from the raw mixture fraction distribution without denoising does not contain meaningful information.  This negative skewness was also reported in a number of previous publications 19,24,25,[36][37][38] .
The assumption of a log-normal distribution of the scalar dissipation rate is usually employed in modelling of turbulent reacting flows and was also experimentally confirmed in turbulent flows 39 . Nevertheless, this assumption might be questionable and the deviations from the log-normal distribution can be observed.
We report that the probability density functions of the scalar dissipation rate are nega- In general, the correlation coefficient can serve as an initial estimator of statistical independence and further investigation by comparing the joint p.d.f. to the product of the individual p.d.fs. is usually required.
Pearson's correlation coefficients are shown in Figure 23 as a function of axial distances and window locations for two swirl numbers of 0.58 and 1.07. Non-zero correlation coefficients are direct indicators that the scalar and its dissipation rate are statistically related.
However, the relationship between the scalar and its dissipation rate is not linear and is also x w z , χ = z χ pdf z , χ It is also possible to assess the relative importance of the axial, radial or both components of the scalar dissipation rate to the overall correlation by computing the weighted integrands from single components of the scalar dissipation rate (axial or radial). This approach provides a rapid overview of identifying the origin of the correlation between the scalar and its dissipation rate. The weighted integrands computed from both axial and radial components of the joint p.d.f. and radial component only are presented in Figure 26 for S=1.07 at y/D f = 1 computed from window 2. The weighted integrands from both components (radial and axial) of the scalar dissipation rate demonstrated that correlations were primarily from large negative scalar dissipation rate fluctuations, which tend to be rare events but have very large contribution. Figure 27 shows the weighted integrands computed from axial and radial components of the scalar dissipation rate for window 2 by using the joint p.d.f. at y/D f = 1 for S=0. For the flow conditions without swirling motion, i.e. for S=0 (acetone vapour jet only), the radial and axial components contribute almost equally to the overall correlation. The contribution of radial component increases, as the level of mixing increases, which is illustrated in Figure   28. A general conclusion is that in swirling flows the contribution from the axial and the radial component of the scalar dissipation rate is not equal and is directly related to the degree of mixing, i.e. the swirl number. It could be assumed that for a certain swirl number (0<S<1.07), the contribution from axial and radial components to the overall correlation would be equal.