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Measurements of scalar dissipation in a turbulent plume with planar laser-induced fluorescence of acetone
Chemical Engineering Science 61 (2006) 2835 – 2842 www.elsevier.com/locate/ces
Measurements of scalar dissipation in a turbulent plume with planar laser-induced fluorescence of acetone C.N. Markides∗ , E. Mastorakos Hopkinson Laboratory, Department of Engineering, University of Cambridge, UK Received 21 July 2005; received in revised form 4 October 2005; accepted 6 October 2005 Available online 22 December 2005
Abstract In order to gain a better understanding of the scalar dissipation rate in turbulent flows and to test available models for this quantity, highresolution two-dimensional planar laser-induced fluorescence measurements were undertaken in the mixing field formed by the axisymmetric injection of a fluorescent tracer (acetone) into a confined turbulent co-flow of air, with emphasis on the less explored early region close to the nozzle and on the spatial resolution and level of image denoising necessary for the correct measurement of . In the mean, the resulting plumes had Gaussian profiles and axial decay as expected from previous investigations. It was found that, with Kolmogorov lengthscale resolution and careful image processing prior to the calculation of the scalar gradients, the measured satisfied global conservation of scalar energy across the plume to within 20%. The estimated mean three-dimensional scalar dissipation rate was used to calculate CD (twice the timescale ratio) that was found to decrease from values higher than 10 adjacent to the nozzle, to approximately 2 at an axial distance of 2–3 nozzle diameters (corresponding to residence times of 0.1–0.2 turbulent timescales) and retaining this value further downstream. The data can assist the validation of models for . 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Dispersion; Fluid mechanics; Imaging; Isothermal; Mixing; Turbulence
1. Introduction The rate at which scalar fluctuations decay in turbulent flows is determined by molecular mixing, quantified by the scalar dissipation rate defined as = 2D∇ · ∇, where D is the molecular diffusivity and the scalar. The scalar dissipation is of prime importance for understanding mixing and the effects of turbulence on the evolution of chemical reactions. In theoretical models for turbulent combustion, for example, the mean and conditionally averaged | scalar dissipation are essential ingredients wherever it is necessary to predict phenomena affected by micromixing, such as pollutant emission, flame extinction and autoignition. The mean dissipation is the rate of decay of the scalar energy and appears as a sink in the transport equation for 2 . An appropriate model is hence = 2 /mix , where mix is a scalar mixing timescale usually taken as proportional to the mechanical (kinetic) turbulent timescale k/, with k the turbulent kinetic energy (=ui ui /2) and its mean viscous dissipation. ∗ Corresponding author. Tel.: +44 1223 366289; fax: +44 1223 332662.
E-mail address: [email protected] (C.N. Markides). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.10.040
This results in = CD 2 /(k/), with CD /2 the ratio of the timescale of mechanical energy dissipation to the timescale of scalar dissipation. Although a wide range of values have been reported for this ratio, CD = 2 is used as standard in CFD packages, taken from the generally accepted value obtained in the far-field of jets and other thin shear flows (Chomiak et al., 1991; Newman et al., 1981). In fact, theory and experiment suggest that CD depends on the ratio between the velocity and scalar integral lengthscales (Corrsin, 1964; Eswaran and Pope, 1988; Warhaft and Lumley, 1978), which in turn depend on the injection method (e.g. source size). Thus, close to nozzles and, in general, in early regions of mixing, the value of CD needed in the model for may not be equal to 2. This has significant implications for the accuracy of current CFD practices for engines or chemical mixers that make use of the standard CD = 2 model, and has motivated the development of more advanced approaches based on modelled transport equations for (Jones and Musonge, 1988; Newman et al., 1981). According to Batchelor (1952), the smallest scalar structure occurs at the Batchelor scale B = K Sc−1/2 , where K is the Kolmogorov lengthscale and Sc the Schmidt number. The
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typical dissipation layer thickness has been generally described −3/4 by D ≡ Re Sc−1/2 , with an outer scale and Re the Reynolds number (Buch and Dahm, 1998). For axisymmetric (including co-flowing) jets, one can relate the outer to the inner scales, obtaining D ≈ (/2)K Sc−1/2 (Su and Clemens, 2003). Experiments have shown that ranges from 7.8 (Tsurikov and Clemens, 2002) to 11.2 (Buch and Dahm, 1998), while the cold wire measurements of Antonia and Mi (1993) suggest that a 3K resolution is sufficient to capture . For Sc ≈ 1, as here, the smallest scalar lengthscale and dissipation layer thickness should be about one K and in the range 3–6K , respectively. A consensus remains to be reached concerning the resolution necessary (and the level of filtering allowable) for the correct measurement of , but a resolution of K should be sufficient for both and (George and Hussein, 1991). In the experiments referred to in this paper we achieved this resolution and, in addition, we follow the suggestion of Bilger (2004) that global conservation principles must be demonstrated before the measurement of is considered reliable. Scalar mixing behind line and point sources in grid turbulence has been measured extensively, especially in the far-field regions (Li and Bilger, 1996; Stapountzis et al., 1986; Nakamura et al., 1987). Direct numerical simulations (DNS) have also provided insight into micromixing processes and the early stages of mixing (Eswaran and Pope, 1988; Brethouwer, 2000). Jets have been studied extensively with planar laser-induced fluorescence (PLIF) and Rayleigh scattering (Buch and Dahm, 1998; Dowling, 1991; Su and Clemens, 2003) providing considerable information on the large- and small-scale structures. Here, we use PLIF to examine the early mixing field of axisymmetric plumes in (almost) homogeneous decaying turbulence, formed by the injection of a tracer-laden flow from a finitesized source into a pipe-confined uniform co-flow downstream of a grid. This simplified geometry, which simulates the mixing in premix gas turbine fuel–air passages, can provide data to validate models for . In addition, the present data can assist the interpretation of autoignition observations in the same apparatus (Markides and Mastorakos, 2005), where the possibility of delayed autoignition due to high was conjectured. The specific objectives of this paper are: (i) to investigate the spatial resolution and correct processing required for the accurate measurement of , and, (ii) to study the applicability of the CD model for . In the next section the experimental apparatus and measurement techniques are described, followed by a presentation and discussion of the results. This begins by demonstrating the processing needed to achieve accurate instantaneous measurements (images) of and resulting and culminates in results concerning the primary variable, . We close with a summary of the most important conclusions. 2. Method 2.1. Apparatus Fig. 1 shows the apparatus. The synthetic silica tube confining the co-flow of air was polished to optical quality. It had an
Laser Sheet Fuel Injection Uinj, Tinj z r Uair, Tair
Acetone Seeded Fuel
Figure 2 Window Injector Quartz Tube
Air from MFC Grid
Fig. 1. Apparatus: mixing field and laser sheet schematic.
inner diameter of D = 33.96 mm and its downstream end was open to atmosphere. A convergent section was used to create favourable pressure gradients and thus maintain thin boundary layers leading into the tube. Turbulence in the co-flow was ensured by a grid with M = 3.0 mm diameter holes and 44% solidity. The injected stream consisted of nitrogen (N2 )-diluted acetylene (C2 H2 ) or hydrogen (H2 ), with the dilution described by the mass fraction of fuel, Yfuel . The fuel flowed continuously through a ‘bubbler’-type seeder containing the fluorescent tracer (liquid acetone), before being injected axially and concentrically through a stainless steel tube of inner diameter d = 2.24 mm and outer diameter do = 2.96 mm. The injector exit (r = 0, z = 0), where r is the radial and z the axial direction (see Fig. 1), was located 63 mm downstream of the grid to allow the turbulence to develop. The co-flow had bulk velocities Uair in the range 3–7 m/s, while the fuel bulk injection velocities Uinj ranged between 3 and 27 m/s. The fuel stream density inj is the sum of the partial densities of the pure fuel, N2 and acetone vapour. The value of inj was increased by about 20% due to acetone. The flow is described by a turbulent Reynolds number Returb = uLturb / air , with air the kinematic viscosity of air, the injection to co-flow velocity ratio inj = Uinj /Uair and the injection to co-flow density ratio inj = inj / air . By varying Uair we examined the effect of Returb on the mixing field. By changing inj from near unity (termed ‘equal velocity’) to values of 2.5, we simulated mixing patterns both akin to diffusion downstream of a low-momentum release and to a jet in co-flow. For C2 H2 /N2 /acetone injected into air inj = 1.2, whereas for H2 /N2 /acetone plumes inj = 0.7–0.8. Due to the high momentum of the co-flow, buoyancy is not expected to play a role in the present measurements. Table 1 summarizes the investigated conditions.
C.N. Markides, E. Mastorakos / Chemical Engineering Science 61 (2006) 2835 – 2842 Table 1 Investigated experimental envelope Run (fuel)
Uair & Returb m/s & —
Yfuel & inj —&—
Uinj & inj m/s & —
1 2 3 4 5 8 10
3.1 3.2 3.9 4.1 4.3 4.3 5.7
0.73 0.73 0.73 0.73 0.73 0.14 0.15
3.5 & 1.1 3.7 & 1.2 4.5 & 1.2 4.7 & 1.1 4.7 & 1.1 10.2 & 2.4 13.7 & 2.4
(C2 H2 ) (C2 H2 ) (C2 H2 ) (C2 H2 ) (C2 H2 ) (H2 ) (H2 )
& & & & & & &
48 51 60 64 67 67 108
& & & & & & &
1.2 1.2 1.2 1.2 1.2 0.8 0.8
A Dantec constant temperature anemometer system with a 1.25 mm long, 5 m diameter hot wire was used to measure axial velocities, with and without injected flow. Power spectra and probability density functions showed that the co-flow was turbulent and near-homogeneous, as expected. Away from the injector and tube walls (0.08 < r/D < 0.42), K was found to be uniform (within 10%), having a value of 0.2 mm at the injection plane (z = 0) and 0.3 mm at 42 mm downstream of the injector (z = 42). It was obtained from K = ( 3 /)1/4 , with measured approximately by assuming isotropic turbulence and applying Taylor’s hypothesis, i.e., =15 (ju /jt)2 /U 2 , where U is the mean and u the turbulent fluctuation of the local axial velocity. The integral (or outer) turbulent lengthscale Lturb was obtained by integrating over the normalized autocorrelation function of u and the use of Taylor’s hypothesis. At (r/D ≈ 0.25, z = 0) it was found to be 3–4 mm, i.e., of the order of the grid hole size, increasing by 1 mm in the first 42 mm downstream. From measurements of U and the root mean square (rms) of u , u = u2 1/2 , the normalized rms intensity u/U was 0.14–0.15 at (r/D ≈ 0.25, z = 0), decaying to 0.10 by 42 mm downstream. The ratio U /Uair was about 1.15 across most of the tube, except at the boundary layers. The air, C2 H2 and H2 flow rates were measured with mass flow controllers, ensuring high accuracy (±0.5% of reading plus ±0.1% of full scale) and precision (< 0.2% of reading). N2 flow rates were set with suitably ranged, precalibrated rotameters (±1.25% of reading). The C2 H2 or H2 and N2 flows were mixed upstream before passing through the seeder that was immersed in a stirred, isothermal bath of hot water. This ensured that the fuel stream was always saturated with a steady concentration of acetone, thus maximizing the fluorescence signal. Both the volumetric and mass flow rates were constant, with a worst case deviation of 5% at high flow rates. To avoid potential problems with acetone condensation, both the coflow air and seeded fuel flows were preheated to 200 ◦ C by actively controlled electrical heaters, with a drift of less than ±1 K. The inlet air temperature Tair was measured with a bare, butt-welded R-type (Pt/Pt&13%Rh) thermocouple of diameter 0.20 mm, placed 25 mm upstream of injection, whereas the fuel injection temperature Tinj was measured inside the injector at a location 2 mm upstream of the nozzle exit with a 0.25 mm diameter mineral insulated K-type (Cr/Al) thermocouple. The random (indeterminate) error associated with Tair and Tinj is ±1 K, while the systematic (determinate) uncertainty is ±1% at
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most. The temperature in the entire measurement domain was measured with a finer (76 m diameter) K-type thermocouple and found to be uniform to within 12 K. Consequently, the fluorescence intensity variations due to temperature are less than 4–5% based on a 0.36%/K decay for 266 nm (Thurber, 1999). 2.2. Optical measurement arrangement Fluorophore PLIF for concentration measurements is a popular technique and has been used in various applications, with acetone the most widely employed tracer. For a detailed account of the photo-physics of laser-exited acetone fluorescence see Thurber (1999). Briefly, for a given laser wavelength and isobaric, non-reacting flows with small temperature variations, such as here, the fluorescence signal is only a function of the local laser energy and tracer concentration. Given that there is only a small drop of power in the laser sheet (radial) direction, after correction for the axial laser power variation the signal is only a function of tracer concentration. We define the local conserved scalar as the molar concentration of the tracer normalized by its injected concentration (r, z) =
nfuel (r, z) , nfuel (r = 0, z = 0)
(1)
where n are molar concentrations. Given inj , the volume fraction-based can be converted into a mass fraction by −1 . The difference between = [1 − (1 − −1 )−1 ] and can inj be up to 0.045 for C2 H2 and 0.090 for H2 . The laser beam was taken from a Continuum Surelite II Nd:Yag solid state laser at the fourth harmonic (266 nm). The beam power was measured to be 80 mJ/pulse at the measurement location. Aiming to improve the fluorescence signal across the domain, the beam height was set slightly higher than the mean height of the imaging window to take advantage of the higher acetone concentrations close to the injector. A cylindrical lens of focal length 40 mm and a spherical lens of focal length 500 mm were used to form a sheet of height 60 mm and measured 1/e2 waist thickness t = 0.10 ± 0.03 mm at the focusing line along the axis of the flow. The detector was a 12-bit LaVision NanoStar intensified charge coupled device (ICCD) camera, used with a 13 mm autoextension tube and a Nikkor AF lens at an aperture of f/2.8. The final imaging area after cropping was 1280 × 480 pixels and the spatial pixel resolution was p = 0.050–0.055 mm/pixel (18–20 pixels/mm). The actual resolution of the optical system, q, will also depend on the orientation of the scalar gradients relative to the laser sheet (St˚arner et al., 1996). Considering the combined effects of the collection optics pixel resolution and the sheet-making optics effectiveness, we estimate the overall ensemble resolution q = t/2 + p/4 as 0.09 mm. In another approach (Tsurikov and Clemens, 2002) the imaging system performance is described in terms of the modulation transfer function (MTF) that expresses the resolution as an ability of transferring contrast information. The MTF of the complete optical arrangement used in this work, from object to saved image file, has been quantified. It was found that 4 to 6p (or 0.3 mm) were enough to
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resolve 70–80% of the spatial detail. With a measured K in the range 0.2–0.3 mm, we conclude that irrespective of the definition chosen, the smallest physical scale is resolved. Following the injection of seeded fuel, some time was allowed for the flow and temperature to reach equilibrium, after which the laser was used to trigger the ICCD camera. The two-dimensional planar fluorescence signal was recorded in N = 200 images at 10 Hz (or every 0.1 s; much longer than Lturb /U < 2 ms) and with 0.4 s exposure times (much shorter than K /U > 0.02 ms). For certain conditions two or three batches of 200 images were taken. The resulting statistical uncertainty in is 5–15%, whereas in 2 and it is 10–20% (e.g. using 2N −1/2 2 1/2 /).
2.3. Image processing and data analysis The raw instantaneous images were first corrected for reflections from the tube by subtraction of a ‘background’ image taken with the laser sheet in place, but without the tracer and cropping. Following this, the beam profile signals were used to correct for the profile non-uniformity in the axial direction. At this stage the image corrections were complete, because the beam profiles were also a means of taking into account the shot-by-shot variation in laser power. In the next step, a suitable intensity level was sought to normalize each image and obtain (Eq. (1)). In order to lessen the normalization error, the image was divided by the mean intensity inside a window directly downstream and adjacent to the injector nozzle outlet, where only unmixed fluid is expected. After the attainment of instantaneous corrected , each image was processed to remove noise. As shown in Markides (2005) this was critical for the correct evaluation of (and thus ) that is susceptible to noise, although not so for (and thus and 2 ). For each condition, a single parameter was estimated a priori and used to shape the entire denoising procedure. This parameter will be referred to as the ‘mixing window’ and is a square characterized by the largest mixing lengthscale at which processing was allowed. It is the two-dimensional image equivalent of the Nyquist frequency of a one-dimensional time-series. As such, the ‘mixing window’ had sides conservatively set to K . Based on the characteristics of the imaging system and the nature of the images themselves the complete denoising procedure involved the following three steps. Firstly, ‘salt-and-pepper’ or shot noise mostly caused by the intensifier was removed by median smoothing, generally accepted as being an efficient approach for removing this type of noise when edge preservation is necessary (Chan et al., 2005). It was employed inside the ‘mixing window’. Secondly, signal content smaller than the ‘mixing window’ was removed by Wiener low-pass filtering in the frequency domain as implemented by Matlab’s wiener2 function. The image obtained at this stage had incurred little processing and is termed the ‘first mixture fraction image’, 1 . Finally, noise was further removed from within the ‘mixing window’ by a custom wavelet technique based on non-blind ‘Bayesian least squares–Gaussian scale mixture’ (BLS–GSM) Matlab algorithms (Portilla et al., 2003), chosen for their supe-
rior performance. At the end of the post-processing we obtained the ‘second mixture fraction image’, 2 . Markides (2005) gives more details and justification for the above steps. The instantaneous 2D was evaluated from 2 j 2 j 2D = 2D (2) + jr jz by second-order central finite differences. The molecular diffusion coefficient of acetone into the multi-component mixtures, D, was calculated analytically for each run and at each point in the flow following Poling et al. (2001), with typical values 2.3–3.3 × 10−5 m2 /s. To investigate the effect of processing, Eq. (2) was used on both the first and second versions of the mixture fraction, resulting in first and second versions of twodimensional scalar dissipation, 1 and 2 , respectively. At a final stage, all instantaneous 2 images were further processed over a wider ‘mixing window’, based on the expectation of D being larger than K by a factor of 4. Therefore, 3 is effectively a frequency domain optimally filtered version of 2 , ‘cut-off’ at a frequency 4K . Finally, the investigation of CD involves recovery of the mean three-dimensional scalar dissipation, 3D , from the two-dimensional measurement. This requires assumptions on the magnitude of the azimuthal gradient (j/j ), but by symmetry, along the centreline the mean squared azimuthal gradient must be equal to the radial one and 3D = 2D + 2D(j/jr)2 . 3. Results and discussion 3.1. Instantaneous mixture fraction and scalar dissipation Fig. 2 shows instantaneous and 2D taken at equal velocity conditions inside a 100 × 100 pixel interrogation window about 5d from the injector and on the centreline (see Fig. 1) at successive stages of processing. The plume is broken apart relatively early by the intense background turbulence. This finding is in contrast to typical jet flows, where uncontaminated fluid first appears at the centreline after about 10 or 20 jet diameters. Fig. 2(a) shows the raw and Fig. 2(b) the 2D calculated from Fig. 2(a). The ‘first mixture fraction’, 1 , and corresponding 1 are shown in Figs. 2(c) and (d). The ‘second mixture fraction’, 2 , and corresponding 2 are shown in Figs. 2(e) and (f). It is evident that the raw cannot be used. However, 1 and 2 show the characteristic filament-type structures observed in previous work. For the conditions shown in these figures, K is 0.3 mm, or 6 pixels. The existence of filaments of width smaller than this in 1 is attributed to noise. By the 2 filtering stage, the typical width of the filaments is not smaller than K . Note that, even though 1 and 2 are similar, 1 and 2 are not. It is apparent that the filtering is affecting the thickness of the structures and (to a lesser extent) the magnitude of the calculated . The complete optical measurementtechnique (from the laser sheet, to the cameras, to the resolution of the resulting images) has been shown of being capable of resolving K . Even so, the importance of correct image processing for the accurate determination of cannot be overstated.
C.N. Markides, E. Mastorakos / Chemical Engineering Science 61 (2006) 2835 – 2842 100 90 80
70 60 50 40 30
70 60 50 40 30
20 10
20 10 10 20 30 40 50 60 70 80 90 100
z/d = 5 -2
10 -2
10 20 30 40 50 60 70 80 90 100
100 90 80
100 90 80
70 60 50 40 30
70 60 50 40 30
20 10
20 10
10 -4 10 -2
10 0 z/d (-)
102
Fig. 3. Logarithmic (base-10) plots of centreline decay of for ‘Runs 1–5’ in Table 1. 10 20 30 40 50 60 70 80 90 100
100 90 80
100 90 80
70 60 50 40 30
70 60 50 40 30
20 10
20 10
0.15 Run 1: Re = 405, υ = 1.1 Run 5: Re = 560, υ = 1.1 Run 8: Re = 560, υ = 2.4 Run 10: Re = 745, υ = 2.4
0.1 〈 ξ'2 〉 (-)
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70 80 90 100
10 0
〈 ξ 〉 (-)
100 90 80
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10 20 30 40 50 60 70 80 90 100
0.05
Fig. 2. (a) Raw , (b) raw , (c) 1 , (d) 1 , (e) 2 and (f) 2 . Detailed conditions: ‘Run 1’ in Table 1.
3.2. Mixture fraction mean and variance The effect of processing on these results is small, since no appreciable differences were observed between 1 and 2 2 , or 2 1 and 2 for any run. Assuming isotropic turbulence, uniform mean velocity, turbulence intensity and turbulent lengthscales, and relating the ‘source strength’ to the flow rate from the nozzle, the analytical result for the nearfield centreline decay of a plume from a point source becomes (r =0, z)=( 18 ) inj [(z/d)(u/U )]−2 (Csanady, 1973). Fig. 3 shows that, as expected, the centreline decay of the plume in equal velocity conditions is unaffected by the changes in Returb (or Uair ) because Uinj was suitably adjusted to set inj ≈ 1 and because changes in u/U were only marginal. The plume decay is also unaffected by the fuel (and hence inj ), collapsing onto a straight line, marked as ‘−2’, for z/d > 5. This is interpreted as confirmation that the analytical solution is a good description of the mean behaviour for z/d > 5. Radial profiles (not shown) are Gaussian and become selfsimilar, as expected. These findings build confidence that the plume is not affected by the confinement, at least in its early development, and that the co-flow turbulence dominates the mixing process.
0
0
5
10 z /d (-)
15
20
Fig. 4. Centreline decay of 2 for indicated runs of Table 1.
Fig. 4 shows the centreline evolution of 2 2 for a variety of conditions. Qualitatively, the results are as expected, with 2 increasing to a maximum and decaying downstream. For the equal velocity case, an increase in Returb causes a strong increase in 2 up to about z/d =3. The maximum increases and shifts closer to the injector, but 2 decays faster downstream of the maximum indicating enhanced mixing. The increase in inj decreases and shifts the maximum downstream, while the variance remain higher than the equal velocity case for most of the imaged region. There seems to be a Reynolds number independence of 2 for inj = 2.4. The measurements of Fig. 4 will be used later to calculate CD . 3.3. Mean scalar dissipation Markides (2005) shows contour plots and axial and radial profiles of and concludes that, unlike for and 2 , the effect of processing is pronounced for this quantity, with
C.N. Markides, E. Mastorakos / Chemical Engineering Science 61 (2006) 2835 – 2842
the processing between 1 and 2 being the most severe. However, he also concludes that 3 was almost identical to 2 , which signifies that this quantity is a good indicator of the true scalar dissipation rate in this flow. Bilger (2004) judges that many measurements of in the literature are of doubtful quality, with questionable spatial resolution and not validated by reference to conservation constraints. He suggests that the data must satisfy the conservation of the scalar and of its energy d I1 ≡ (U + u ) dA = 0, (3) dz A d [U (2 + 2 ) + 2u I2 ≡ dz A 2 2 2 + u − D∇( + )] dA (4) + dA = 0.
0.2
0
I1
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-0.2 Plume Reaches Tube / Image Edge -0.4
0
20
30
40
50
z (mm)
A
x 10-4
2
0
I2
To establish the quality of the current measurements these two integrals were evaluated by employing the following assumptions: (i) that all axial turbulent fluxes of and 2 (in both equations) and the term involving the molecular flux of 2 (in Eq. (4)) are negligible, (ii) that the average density is constant, and (iii) that the local velocity is approximately uniform and 15% higher than the bulk, i.e., U (r, z) ≈ 1.15Uair = constant. In making the first assumption we follow Bilger (2004) directly. For injection of C2 H2 with inj ≈ 1 into the co-flow the rest of the assumptions are reasonable, with the density of the injected stream only 20% higher than the co-flow and U (r, z)/Uair = 1.10–1.15 for 70% of the tube (from the separate hot wire measurements). Fig. 5(a) shows that the integral in Eq. (3) is conserved, remaining within ±15% from the injector to 40 mm downstream. At this length the plume reaches the image edges, which have been cropped in order to eliminate the presence of laser reflections on the tube and the integrals are no longer converged. Results corresponding to Eq. (4) are indicated in Fig. 5(b) for a range of inj ≈ 1 conditions with C2 H2 . The deviation of the integrals from zero is smaller than 5×10−5 from a distance of 10 mm from the injector. This deviation from zero is marginal, corresponds to an (area integrated) of 0.1 s−1 , which is 20% of the value of the measured at that location and suggests that has been measured to within 20% (i.e., close to the statistical convergence of that was ±10.20% at 95% confidence level). From this outcome we infer that has probably been adequately resolved. Consider now 2 . Fig. 6 shows that 2D increases initially and then decreases quickly after about 5d. The peak 2D is located at the same z/d as the peak 2 (see Fig. 4). It increases with Returb , although there seems to be a Reynolds number independence for z/d > 5 in the case of equal velocity plumes and z/d > 10 for jets. Finally, in Fig. 7 we plot CD from 3D /2 · (k/) against the residence time along the centreline, res =z/U , normalized by the initial (at z/d = 0) turbulent timescale (k/)0 . For the local k/ we have used the relation (k/)/turb = 1.6 ± 0.1 (Pope, 2000) that was checked in a few cases with hot wire data with good agreement (within ± 20%). The value of turb (=Lturb /u)
10
(a)
-2
-4 (b)
0
20
40
60
z (mm)
Fig. 5. (a) Total scalar conservation from Eq. (3). (b) Conservation of scalar energy from Eq. (4).
has been calculated from the hot wire measurements of Lturb and u at various axial locations and similar conditions. It was also found that turb increases by about 40% in the first 42 mm from the injector. After a residence time of about 0.1–0.2 turbulent times (or 2–3d), CD reaches the value of 2 for all conditions, agreeing with previous studies (Chomiak et al., 1991; Dowling, 1991; Li and Bilger, 1996; Newman et al., 1981). The high initial values of CD are fully consistent with the models of Corrsin (1964) and Chomiak et al. (1991), who predict that CD scales with (Lturb /L )2/3 , where L is a characteristic scalar integral lengthscale. Here, initially L would scale with d (or perhaps with the initial laminar mixing layer thickness at the nozzle), which implies Lturb > L and hence CD > 2. At axial distances where CD ≈ 2, the model is also valid in the radial region, but only close to the centreline and away from the edges of the plume. Specifically, CD remained in the range 2–3 for r/d < 1.
C.N. Markides, E. Mastorakos / Chemical Engineering Science 61 (2006) 2835 – 2842
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also that careful data processing is necessary for scalar dissipation measurements. The mean mixture fraction was consistent with the Gaussian plume equation and the variance of the mixture fraction fluctuations increased to a maximum and then decayed downstream, as expected. The mean three-dimensional scalar dissipation was estimated along the centreline and related to the mixture fraction variance and local turbulent kinetic energy timescale. The results show that CD decreases from values higher than 10 near the nozzle to about 2 for axial distances greater than 2–3 nozzle diameters (or residence times greater than 0.1–0.2 turbulent timescales).
Run 1: Re = 405, υ = 1.1 Run 5: Re = 560, υ = 1.1 Run 8: Re = 560, υ = 2.4
60
Run 10: Re = 745, υ = 2.4 〈 χ2D 〉 (1 /s)
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40
20
Acknowledgements 0
0
5
10
15
20
25
z /d(-) Fig. 6. Centreline evolution of 2D for indicated runs of Table 1.
We thank Dr. R. Balachandran for his assistance with the PLIF measurements and Dr. J. Portilla for making available the BLS–GSM algorithms in Matlab format. This work has been funded by the EPSRC. References
10
〈χ3D〉/〈ξ′2〉.(k /ε)
8
6
4
2
0
0
0.1
0.2 0.3 τres /(k /ε)0(-)
0.4
0.5
Fig. 7. The local calculated CD as a function of (z/U )/(k/)z=0 for ‘Runs 1–5’ in Table 1.
4. Conclusion PLIF measurements were made of the concentration of a passive tracer in the axisymmetric mixing field formed by injection from a finite-sized nozzle into a confined turbulent co-flow. The spatial resolution was equal to the Kolmogorov lengthscale. The mean and variance of the normalized conserved scalar were not found to depend strongly on the level of denoising, even for smoothing to scales four times larger than the Kolmogorov lengthscale. In contrast, the scalar dissipation rate and the thickness of the dissipation filaments were affected to various degrees by the level and chosen method of denoising, suggesting that not only must the measurement resolution be adequate, but
Antonia, R.A., Mi, J., 1993. Temperature dissipation in a turbulent round jet. Journal of Fluid Mechanics 250, 531–551. Batchelor, G.K., 1952. The effect of homogeneous turbulence on material lines and surfaces. Proceedings of the Royal Society of London A, 213, 349–366. Bilger, R.W., 2004. Some aspects of scalar dissipation. Flow, Turbulence and Combustion 72, 93–114. Brethouwer, G., 2000. Mixing of passive and reactive scalars in turbulent flows. Ph.D. Thesis, Delft University of Technology. Buch, K.A., Dahm, W.J.A., 1998. Experimental study of the fine-scale structure of conserved scalar mixing in turbulent shear flows. Part 2. Sc ≈ 1. Journal of Fluid Mechanics 364, 1–29. Chan, R.H., Ho, C.-W., Nikolova, M., 2005. Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization. IEEE Transactions on Image Processing 18 (10), 1479–1485. Chomiak, J., Hakberg, B., Olsson, E., 1991. A note on the scalar dissipation rates in turbulent flows. Combustion and Flame 83 (3–4), 412–414. Corrsin, S., 1964. The isotropic turbulent mixer: part II. Arbitrary Schmidt number. A.I.Ch.E. Journal 10 (6), 870–877. Csanady, G.T., 1973. Turbulent Diffusion in the Environment. Reidel, Dordrecht. Dowling, D.R., 1991. The estimated scalar dissipation rate in gas-phase turbulent jets. Physics of Fluids A 3 (9), 2229–2246. Eswaran, V., Pope, S.B., 1988. Direct numerical simulations of the turbulent mixing of a passive scalar. Physics of Fluids 31 (3), 506–520. George, W.K., Hussein, H.J., 1991. Locally axisymmetric turbulence. Journal of Fluid Mechanics 233, 1–23. Jones, W.P., Musonge, P., 1988. Closure of the Reynolds stress and scalar flux equations. Physics of Fluids 31 (12), 3589–3604. Li, J.D., Bilger, R.W., 1996. The diffusion of conserved and reactive scalars behind line sources in homogeneous turbulence. Journal of Fluid Mechanics 318, 339–372. Markides, C.N., 2005. Autoignition in turbulent flows. Ph.D. Thesis, University of Cambridge. Markides, C.N., Mastorakos, E., 2005. An experimental study of hydrogen autoignition in a turbulent co-flow of heated air. Proceedings of the Combustion Institute 30 (1), 883–891. Nakamura, I., Sakai, Y., Miyata, M., 1987. Diffusion of matter by a nonbuoyant plume in grid generated turbulence. Journal of Fluid Mechanics 178, 379–403. Newman, G.R., Launder, B.E., Lumley, J.L., 1981. Modelling the behaviour of homogeneous scalar turbulence. Journal of Fluid Mechanics 111, 217–232.
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Poling, B.E., Prausnitz, J.M., O’Connell, J.P., 2001. The Properties of Gases and Liquids. fifth ed. McGraw-Hill, New York. Pope, S.B., 2000. Turbulent Flows. Cambridge University Press, Cambridge. Portilla, J., Strela, V., Wainwright, M.J., Simoncelli, E.P., 2003. Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Transactions on Image Processing 12 (11), 1338–1351. Stapountzis, H., Sawford, B.L., Hunt, J.C.R., Britter, R.E., 1986. Structure of the temperature field downwind of a line source in grid turbulence. Journal of Fluid Mechanics 165, 401–424. St˚arner, S.H., Bilger, R.W., Frank, J.H., Marran, D.F., Long, M.B., 1996. Mixture fraction imaging in a lifted methane jet flame. Combustion and Flame 107 (3), 307–313.
Su, L.K., Clemens, N.T., 2003. The structure of fine-scale scalar mixing in gas-phase planar turbulent jets. Journal of Fluid Mechanics 488, 1–29. Thurber, M.C., 1999. Acetone laser-induced fluorescence for temperature and multiparameter imaging in gaseous flows. Ph.D. Thesis, Stanford University. Tsurikov, M.S., Clemens, N.T., 2002. The structure of dissipative scales in axisymmetric turbulent gas-phase jets. AIAA Paper, vol. 164. 40th Aerospace Sciences Meeting and Exhibit, Reno, NV. Warhaft, Z., Lumley, J.L., 1978. An experimental study of the decay of temperature fluctuations in grid-generated turbulence. Journal of Fluid Mechanics 88, 659–684.
Measurements of scalar dissipation in a turbulent plume with planar laser-induced fluorescence of acetone C.N. Markides∗ , E. Mastorakos Hopkinson Laboratory, Department of Engineering, University of Cambridge, UK Received 21 July 2005; received in revised form 4 October 2005; accepted 6 October 2005 Available online 22 December 2005
Abstract In order to gain a better understanding of the scalar dissipation rate in turbulent flows and to test available models for this quantity, highresolution two-dimensional planar laser-induced fluorescence measurements were undertaken in the mixing field formed by the axisymmetric injection of a fluorescent tracer (acetone) into a confined turbulent co-flow of air, with emphasis on the less explored early region close to the nozzle and on the spatial resolution and level of image denoising necessary for the correct measurement of . In the mean, the resulting plumes had Gaussian profiles and axial decay as expected from previous investigations. It was found that, with Kolmogorov lengthscale resolution and careful image processing prior to the calculation of the scalar gradients, the measured satisfied global conservation of scalar energy across the plume to within 20%. The estimated mean three-dimensional scalar dissipation rate was used to calculate CD (twice the timescale ratio) that was found to decrease from values higher than 10 adjacent to the nozzle, to approximately 2 at an axial distance of 2–3 nozzle diameters (corresponding to residence times of 0.1–0.2 turbulent timescales) and retaining this value further downstream. The data can assist the validation of models for . 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Dispersion; Fluid mechanics; Imaging; Isothermal; Mixing; Turbulence
1. Introduction The rate at which scalar fluctuations decay in turbulent flows is determined by molecular mixing, quantified by the scalar dissipation rate defined as = 2D∇ · ∇, where D is the molecular diffusivity and the scalar. The scalar dissipation is of prime importance for understanding mixing and the effects of turbulence on the evolution of chemical reactions. In theoretical models for turbulent combustion, for example, the mean and conditionally averaged | scalar dissipation are essential ingredients wherever it is necessary to predict phenomena affected by micromixing, such as pollutant emission, flame extinction and autoignition. The mean dissipation is the rate of decay of the scalar energy and appears as a sink in the transport equation for 2 . An appropriate model is hence = 2 /mix , where mix is a scalar mixing timescale usually taken as proportional to the mechanical (kinetic) turbulent timescale k/, with k the turbulent kinetic energy (=ui ui /2) and its mean viscous dissipation. ∗ Corresponding author. Tel.: +44 1223 366289; fax: +44 1223 332662.
E-mail address: [email protected] (C.N. Markides). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.10.040
This results in = CD 2 /(k/), with CD /2 the ratio of the timescale of mechanical energy dissipation to the timescale of scalar dissipation. Although a wide range of values have been reported for this ratio, CD = 2 is used as standard in CFD packages, taken from the generally accepted value obtained in the far-field of jets and other thin shear flows (Chomiak et al., 1991; Newman et al., 1981). In fact, theory and experiment suggest that CD depends on the ratio between the velocity and scalar integral lengthscales (Corrsin, 1964; Eswaran and Pope, 1988; Warhaft and Lumley, 1978), which in turn depend on the injection method (e.g. source size). Thus, close to nozzles and, in general, in early regions of mixing, the value of CD needed in the model for may not be equal to 2. This has significant implications for the accuracy of current CFD practices for engines or chemical mixers that make use of the standard CD = 2 model, and has motivated the development of more advanced approaches based on modelled transport equations for (Jones and Musonge, 1988; Newman et al., 1981). According to Batchelor (1952), the smallest scalar structure occurs at the Batchelor scale B = K Sc−1/2 , where K is the Kolmogorov lengthscale and Sc the Schmidt number. The
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typical dissipation layer thickness has been generally described −3/4 by D ≡ Re Sc−1/2 , with an outer scale and Re the Reynolds number (Buch and Dahm, 1998). For axisymmetric (including co-flowing) jets, one can relate the outer to the inner scales, obtaining D ≈ (/2)K Sc−1/2 (Su and Clemens, 2003). Experiments have shown that ranges from 7.8 (Tsurikov and Clemens, 2002) to 11.2 (Buch and Dahm, 1998), while the cold wire measurements of Antonia and Mi (1993) suggest that a 3K resolution is sufficient to capture . For Sc ≈ 1, as here, the smallest scalar lengthscale and dissipation layer thickness should be about one K and in the range 3–6K , respectively. A consensus remains to be reached concerning the resolution necessary (and the level of filtering allowable) for the correct measurement of , but a resolution of K should be sufficient for both and (George and Hussein, 1991). In the experiments referred to in this paper we achieved this resolution and, in addition, we follow the suggestion of Bilger (2004) that global conservation principles must be demonstrated before the measurement of is considered reliable. Scalar mixing behind line and point sources in grid turbulence has been measured extensively, especially in the far-field regions (Li and Bilger, 1996; Stapountzis et al., 1986; Nakamura et al., 1987). Direct numerical simulations (DNS) have also provided insight into micromixing processes and the early stages of mixing (Eswaran and Pope, 1988; Brethouwer, 2000). Jets have been studied extensively with planar laser-induced fluorescence (PLIF) and Rayleigh scattering (Buch and Dahm, 1998; Dowling, 1991; Su and Clemens, 2003) providing considerable information on the large- and small-scale structures. Here, we use PLIF to examine the early mixing field of axisymmetric plumes in (almost) homogeneous decaying turbulence, formed by the injection of a tracer-laden flow from a finitesized source into a pipe-confined uniform co-flow downstream of a grid. This simplified geometry, which simulates the mixing in premix gas turbine fuel–air passages, can provide data to validate models for . In addition, the present data can assist the interpretation of autoignition observations in the same apparatus (Markides and Mastorakos, 2005), where the possibility of delayed autoignition due to high was conjectured. The specific objectives of this paper are: (i) to investigate the spatial resolution and correct processing required for the accurate measurement of , and, (ii) to study the applicability of the CD model for . In the next section the experimental apparatus and measurement techniques are described, followed by a presentation and discussion of the results. This begins by demonstrating the processing needed to achieve accurate instantaneous measurements (images) of and resulting and culminates in results concerning the primary variable, . We close with a summary of the most important conclusions. 2. Method 2.1. Apparatus Fig. 1 shows the apparatus. The synthetic silica tube confining the co-flow of air was polished to optical quality. It had an
Laser Sheet Fuel Injection Uinj, Tinj z r Uair, Tair
Acetone Seeded Fuel
Figure 2 Window Injector Quartz Tube
Air from MFC Grid
Fig. 1. Apparatus: mixing field and laser sheet schematic.
inner diameter of D = 33.96 mm and its downstream end was open to atmosphere. A convergent section was used to create favourable pressure gradients and thus maintain thin boundary layers leading into the tube. Turbulence in the co-flow was ensured by a grid with M = 3.0 mm diameter holes and 44% solidity. The injected stream consisted of nitrogen (N2 )-diluted acetylene (C2 H2 ) or hydrogen (H2 ), with the dilution described by the mass fraction of fuel, Yfuel . The fuel flowed continuously through a ‘bubbler’-type seeder containing the fluorescent tracer (liquid acetone), before being injected axially and concentrically through a stainless steel tube of inner diameter d = 2.24 mm and outer diameter do = 2.96 mm. The injector exit (r = 0, z = 0), where r is the radial and z the axial direction (see Fig. 1), was located 63 mm downstream of the grid to allow the turbulence to develop. The co-flow had bulk velocities Uair in the range 3–7 m/s, while the fuel bulk injection velocities Uinj ranged between 3 and 27 m/s. The fuel stream density inj is the sum of the partial densities of the pure fuel, N2 and acetone vapour. The value of inj was increased by about 20% due to acetone. The flow is described by a turbulent Reynolds number Returb = uLturb / air , with air the kinematic viscosity of air, the injection to co-flow velocity ratio inj = Uinj /Uair and the injection to co-flow density ratio inj = inj / air . By varying Uair we examined the effect of Returb on the mixing field. By changing inj from near unity (termed ‘equal velocity’) to values of 2.5, we simulated mixing patterns both akin to diffusion downstream of a low-momentum release and to a jet in co-flow. For C2 H2 /N2 /acetone injected into air inj = 1.2, whereas for H2 /N2 /acetone plumes inj = 0.7–0.8. Due to the high momentum of the co-flow, buoyancy is not expected to play a role in the present measurements. Table 1 summarizes the investigated conditions.
C.N. Markides, E. Mastorakos / Chemical Engineering Science 61 (2006) 2835 – 2842 Table 1 Investigated experimental envelope Run (fuel)
Uair & Returb m/s & —
Yfuel & inj —&—
Uinj & inj m/s & —
1 2 3 4 5 8 10
3.1 3.2 3.9 4.1 4.3 4.3 5.7
0.73 0.73 0.73 0.73 0.73 0.14 0.15
3.5 & 1.1 3.7 & 1.2 4.5 & 1.2 4.7 & 1.1 4.7 & 1.1 10.2 & 2.4 13.7 & 2.4
(C2 H2 ) (C2 H2 ) (C2 H2 ) (C2 H2 ) (C2 H2 ) (H2 ) (H2 )
& & & & & & &
48 51 60 64 67 67 108
& & & & & & &
1.2 1.2 1.2 1.2 1.2 0.8 0.8
A Dantec constant temperature anemometer system with a 1.25 mm long, 5 m diameter hot wire was used to measure axial velocities, with and without injected flow. Power spectra and probability density functions showed that the co-flow was turbulent and near-homogeneous, as expected. Away from the injector and tube walls (0.08 < r/D < 0.42), K was found to be uniform (within 10%), having a value of 0.2 mm at the injection plane (z = 0) and 0.3 mm at 42 mm downstream of the injector (z = 42). It was obtained from K = ( 3 /)1/4 , with measured approximately by assuming isotropic turbulence and applying Taylor’s hypothesis, i.e., =15 (ju /jt)2 /U 2 , where U is the mean and u the turbulent fluctuation of the local axial velocity. The integral (or outer) turbulent lengthscale Lturb was obtained by integrating over the normalized autocorrelation function of u and the use of Taylor’s hypothesis. At (r/D ≈ 0.25, z = 0) it was found to be 3–4 mm, i.e., of the order of the grid hole size, increasing by 1 mm in the first 42 mm downstream. From measurements of U and the root mean square (rms) of u , u = u2 1/2 , the normalized rms intensity u/U was 0.14–0.15 at (r/D ≈ 0.25, z = 0), decaying to 0.10 by 42 mm downstream. The ratio U /Uair was about 1.15 across most of the tube, except at the boundary layers. The air, C2 H2 and H2 flow rates were measured with mass flow controllers, ensuring high accuracy (±0.5% of reading plus ±0.1% of full scale) and precision (< 0.2% of reading). N2 flow rates were set with suitably ranged, precalibrated rotameters (±1.25% of reading). The C2 H2 or H2 and N2 flows were mixed upstream before passing through the seeder that was immersed in a stirred, isothermal bath of hot water. This ensured that the fuel stream was always saturated with a steady concentration of acetone, thus maximizing the fluorescence signal. Both the volumetric and mass flow rates were constant, with a worst case deviation of 5% at high flow rates. To avoid potential problems with acetone condensation, both the coflow air and seeded fuel flows were preheated to 200 ◦ C by actively controlled electrical heaters, with a drift of less than ±1 K. The inlet air temperature Tair was measured with a bare, butt-welded R-type (Pt/Pt&13%Rh) thermocouple of diameter 0.20 mm, placed 25 mm upstream of injection, whereas the fuel injection temperature Tinj was measured inside the injector at a location 2 mm upstream of the nozzle exit with a 0.25 mm diameter mineral insulated K-type (Cr/Al) thermocouple. The random (indeterminate) error associated with Tair and Tinj is ±1 K, while the systematic (determinate) uncertainty is ±1% at
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most. The temperature in the entire measurement domain was measured with a finer (76 m diameter) K-type thermocouple and found to be uniform to within 12 K. Consequently, the fluorescence intensity variations due to temperature are less than 4–5% based on a 0.36%/K decay for 266 nm (Thurber, 1999). 2.2. Optical measurement arrangement Fluorophore PLIF for concentration measurements is a popular technique and has been used in various applications, with acetone the most widely employed tracer. For a detailed account of the photo-physics of laser-exited acetone fluorescence see Thurber (1999). Briefly, for a given laser wavelength and isobaric, non-reacting flows with small temperature variations, such as here, the fluorescence signal is only a function of the local laser energy and tracer concentration. Given that there is only a small drop of power in the laser sheet (radial) direction, after correction for the axial laser power variation the signal is only a function of tracer concentration. We define the local conserved scalar as the molar concentration of the tracer normalized by its injected concentration (r, z) =
nfuel (r, z) , nfuel (r = 0, z = 0)
(1)
where n are molar concentrations. Given inj , the volume fraction-based can be converted into a mass fraction by −1 . The difference between = [1 − (1 − −1 )−1 ] and can inj be up to 0.045 for C2 H2 and 0.090 for H2 . The laser beam was taken from a Continuum Surelite II Nd:Yag solid state laser at the fourth harmonic (266 nm). The beam power was measured to be 80 mJ/pulse at the measurement location. Aiming to improve the fluorescence signal across the domain, the beam height was set slightly higher than the mean height of the imaging window to take advantage of the higher acetone concentrations close to the injector. A cylindrical lens of focal length 40 mm and a spherical lens of focal length 500 mm were used to form a sheet of height 60 mm and measured 1/e2 waist thickness t = 0.10 ± 0.03 mm at the focusing line along the axis of the flow. The detector was a 12-bit LaVision NanoStar intensified charge coupled device (ICCD) camera, used with a 13 mm autoextension tube and a Nikkor AF lens at an aperture of f/2.8. The final imaging area after cropping was 1280 × 480 pixels and the spatial pixel resolution was p = 0.050–0.055 mm/pixel (18–20 pixels/mm). The actual resolution of the optical system, q, will also depend on the orientation of the scalar gradients relative to the laser sheet (St˚arner et al., 1996). Considering the combined effects of the collection optics pixel resolution and the sheet-making optics effectiveness, we estimate the overall ensemble resolution q = t/2 + p/4 as 0.09 mm. In another approach (Tsurikov and Clemens, 2002) the imaging system performance is described in terms of the modulation transfer function (MTF) that expresses the resolution as an ability of transferring contrast information. The MTF of the complete optical arrangement used in this work, from object to saved image file, has been quantified. It was found that 4 to 6p (or 0.3 mm) were enough to
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resolve 70–80% of the spatial detail. With a measured K in the range 0.2–0.3 mm, we conclude that irrespective of the definition chosen, the smallest physical scale is resolved. Following the injection of seeded fuel, some time was allowed for the flow and temperature to reach equilibrium, after which the laser was used to trigger the ICCD camera. The two-dimensional planar fluorescence signal was recorded in N = 200 images at 10 Hz (or every 0.1 s; much longer than Lturb /U < 2 ms) and with 0.4 s exposure times (much shorter than K /U > 0.02 ms). For certain conditions two or three batches of 200 images were taken. The resulting statistical uncertainty in is 5–15%, whereas in 2 and it is 10–20% (e.g. using 2N −1/2 2 1/2 /).
2.3. Image processing and data analysis The raw instantaneous images were first corrected for reflections from the tube by subtraction of a ‘background’ image taken with the laser sheet in place, but without the tracer and cropping. Following this, the beam profile signals were used to correct for the profile non-uniformity in the axial direction. At this stage the image corrections were complete, because the beam profiles were also a means of taking into account the shot-by-shot variation in laser power. In the next step, a suitable intensity level was sought to normalize each image and obtain (Eq. (1)). In order to lessen the normalization error, the image was divided by the mean intensity inside a window directly downstream and adjacent to the injector nozzle outlet, where only unmixed fluid is expected. After the attainment of instantaneous corrected , each image was processed to remove noise. As shown in Markides (2005) this was critical for the correct evaluation of (and thus ) that is susceptible to noise, although not so for (and thus and 2 ). For each condition, a single parameter was estimated a priori and used to shape the entire denoising procedure. This parameter will be referred to as the ‘mixing window’ and is a square characterized by the largest mixing lengthscale at which processing was allowed. It is the two-dimensional image equivalent of the Nyquist frequency of a one-dimensional time-series. As such, the ‘mixing window’ had sides conservatively set to K . Based on the characteristics of the imaging system and the nature of the images themselves the complete denoising procedure involved the following three steps. Firstly, ‘salt-and-pepper’ or shot noise mostly caused by the intensifier was removed by median smoothing, generally accepted as being an efficient approach for removing this type of noise when edge preservation is necessary (Chan et al., 2005). It was employed inside the ‘mixing window’. Secondly, signal content smaller than the ‘mixing window’ was removed by Wiener low-pass filtering in the frequency domain as implemented by Matlab’s wiener2 function. The image obtained at this stage had incurred little processing and is termed the ‘first mixture fraction image’, 1 . Finally, noise was further removed from within the ‘mixing window’ by a custom wavelet technique based on non-blind ‘Bayesian least squares–Gaussian scale mixture’ (BLS–GSM) Matlab algorithms (Portilla et al., 2003), chosen for their supe-
rior performance. At the end of the post-processing we obtained the ‘second mixture fraction image’, 2 . Markides (2005) gives more details and justification for the above steps. The instantaneous 2D was evaluated from 2 j 2 j 2D = 2D (2) + jr jz by second-order central finite differences. The molecular diffusion coefficient of acetone into the multi-component mixtures, D, was calculated analytically for each run and at each point in the flow following Poling et al. (2001), with typical values 2.3–3.3 × 10−5 m2 /s. To investigate the effect of processing, Eq. (2) was used on both the first and second versions of the mixture fraction, resulting in first and second versions of twodimensional scalar dissipation, 1 and 2 , respectively. At a final stage, all instantaneous 2 images were further processed over a wider ‘mixing window’, based on the expectation of D being larger than K by a factor of 4. Therefore, 3 is effectively a frequency domain optimally filtered version of 2 , ‘cut-off’ at a frequency 4K . Finally, the investigation of CD involves recovery of the mean three-dimensional scalar dissipation, 3D , from the two-dimensional measurement. This requires assumptions on the magnitude of the azimuthal gradient (j/j ), but by symmetry, along the centreline the mean squared azimuthal gradient must be equal to the radial one and 3D = 2D + 2D(j/jr)2 . 3. Results and discussion 3.1. Instantaneous mixture fraction and scalar dissipation Fig. 2 shows instantaneous and 2D taken at equal velocity conditions inside a 100 × 100 pixel interrogation window about 5d from the injector and on the centreline (see Fig. 1) at successive stages of processing. The plume is broken apart relatively early by the intense background turbulence. This finding is in contrast to typical jet flows, where uncontaminated fluid first appears at the centreline after about 10 or 20 jet diameters. Fig. 2(a) shows the raw and Fig. 2(b) the 2D calculated from Fig. 2(a). The ‘first mixture fraction’, 1 , and corresponding 1 are shown in Figs. 2(c) and (d). The ‘second mixture fraction’, 2 , and corresponding 2 are shown in Figs. 2(e) and (f). It is evident that the raw cannot be used. However, 1 and 2 show the characteristic filament-type structures observed in previous work. For the conditions shown in these figures, K is 0.3 mm, or 6 pixels. The existence of filaments of width smaller than this in 1 is attributed to noise. By the 2 filtering stage, the typical width of the filaments is not smaller than K . Note that, even though 1 and 2 are similar, 1 and 2 are not. It is apparent that the filtering is affecting the thickness of the structures and (to a lesser extent) the magnitude of the calculated . The complete optical measurementtechnique (from the laser sheet, to the cameras, to the resolution of the resulting images) has been shown of being capable of resolving K . Even so, the importance of correct image processing for the accurate determination of cannot be overstated.
C.N. Markides, E. Mastorakos / Chemical Engineering Science 61 (2006) 2835 – 2842 100 90 80
70 60 50 40 30
70 60 50 40 30
20 10
20 10 10 20 30 40 50 60 70 80 90 100
z/d = 5 -2
10 -2
10 20 30 40 50 60 70 80 90 100
100 90 80
100 90 80
70 60 50 40 30
70 60 50 40 30
20 10
20 10
10 -4 10 -2
10 0 z/d (-)
102
Fig. 3. Logarithmic (base-10) plots of centreline decay of for ‘Runs 1–5’ in Table 1. 10 20 30 40 50 60 70 80 90 100
100 90 80
100 90 80
70 60 50 40 30
70 60 50 40 30
20 10
20 10
0.15 Run 1: Re = 405, υ = 1.1 Run 5: Re = 560, υ = 1.1 Run 8: Re = 560, υ = 2.4 Run 10: Re = 745, υ = 2.4
0.1 〈 ξ'2 〉 (-)
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70 80 90 100
10 0
〈 ξ 〉 (-)
100 90 80
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10 20 30 40 50 60 70 80 90 100
0.05
Fig. 2. (a) Raw , (b) raw , (c) 1 , (d) 1 , (e) 2 and (f) 2 . Detailed conditions: ‘Run 1’ in Table 1.
3.2. Mixture fraction mean and variance The effect of processing on these results is small, since no appreciable differences were observed between 1 and 2 2 , or 2 1 and 2 for any run. Assuming isotropic turbulence, uniform mean velocity, turbulence intensity and turbulent lengthscales, and relating the ‘source strength’ to the flow rate from the nozzle, the analytical result for the nearfield centreline decay of a plume from a point source becomes (r =0, z)=( 18 ) inj [(z/d)(u/U )]−2 (Csanady, 1973). Fig. 3 shows that, as expected, the centreline decay of the plume in equal velocity conditions is unaffected by the changes in Returb (or Uair ) because Uinj was suitably adjusted to set inj ≈ 1 and because changes in u/U were only marginal. The plume decay is also unaffected by the fuel (and hence inj ), collapsing onto a straight line, marked as ‘−2’, for z/d > 5. This is interpreted as confirmation that the analytical solution is a good description of the mean behaviour for z/d > 5. Radial profiles (not shown) are Gaussian and become selfsimilar, as expected. These findings build confidence that the plume is not affected by the confinement, at least in its early development, and that the co-flow turbulence dominates the mixing process.
0
0
5
10 z /d (-)
15
20
Fig. 4. Centreline decay of 2 for indicated runs of Table 1.
Fig. 4 shows the centreline evolution of 2 2 for a variety of conditions. Qualitatively, the results are as expected, with 2 increasing to a maximum and decaying downstream. For the equal velocity case, an increase in Returb causes a strong increase in 2 up to about z/d =3. The maximum increases and shifts closer to the injector, but 2 decays faster downstream of the maximum indicating enhanced mixing. The increase in inj decreases and shifts the maximum downstream, while the variance remain higher than the equal velocity case for most of the imaged region. There seems to be a Reynolds number independence of 2 for inj = 2.4. The measurements of Fig. 4 will be used later to calculate CD . 3.3. Mean scalar dissipation Markides (2005) shows contour plots and axial and radial profiles of and concludes that, unlike for and 2 , the effect of processing is pronounced for this quantity, with
C.N. Markides, E. Mastorakos / Chemical Engineering Science 61 (2006) 2835 – 2842
the processing between 1 and 2 being the most severe. However, he also concludes that 3 was almost identical to 2 , which signifies that this quantity is a good indicator of the true scalar dissipation rate in this flow. Bilger (2004) judges that many measurements of in the literature are of doubtful quality, with questionable spatial resolution and not validated by reference to conservation constraints. He suggests that the data must satisfy the conservation of the scalar and of its energy d I1 ≡ (U + u ) dA = 0, (3) dz A d [U (2 + 2 ) + 2u I2 ≡ dz A 2 2 2 + u − D∇( + )] dA (4) + dA = 0.
0.2
0
I1
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-0.2 Plume Reaches Tube / Image Edge -0.4
0
20
30
40
50
z (mm)
A
x 10-4
2
0
I2
To establish the quality of the current measurements these two integrals were evaluated by employing the following assumptions: (i) that all axial turbulent fluxes of and 2 (in both equations) and the term involving the molecular flux of 2 (in Eq. (4)) are negligible, (ii) that the average density is constant, and (iii) that the local velocity is approximately uniform and 15% higher than the bulk, i.e., U (r, z) ≈ 1.15Uair = constant. In making the first assumption we follow Bilger (2004) directly. For injection of C2 H2 with inj ≈ 1 into the co-flow the rest of the assumptions are reasonable, with the density of the injected stream only 20% higher than the co-flow and U (r, z)/Uair = 1.10–1.15 for 70% of the tube (from the separate hot wire measurements). Fig. 5(a) shows that the integral in Eq. (3) is conserved, remaining within ±15% from the injector to 40 mm downstream. At this length the plume reaches the image edges, which have been cropped in order to eliminate the presence of laser reflections on the tube and the integrals are no longer converged. Results corresponding to Eq. (4) are indicated in Fig. 5(b) for a range of inj ≈ 1 conditions with C2 H2 . The deviation of the integrals from zero is smaller than 5×10−5 from a distance of 10 mm from the injector. This deviation from zero is marginal, corresponds to an (area integrated) of 0.1 s−1 , which is 20% of the value of the measured at that location and suggests that has been measured to within 20% (i.e., close to the statistical convergence of that was ±10.20% at 95% confidence level). From this outcome we infer that has probably been adequately resolved. Consider now 2 . Fig. 6 shows that 2D increases initially and then decreases quickly after about 5d. The peak 2D is located at the same z/d as the peak 2 (see Fig. 4). It increases with Returb , although there seems to be a Reynolds number independence for z/d > 5 in the case of equal velocity plumes and z/d > 10 for jets. Finally, in Fig. 7 we plot CD from 3D /2 · (k/) against the residence time along the centreline, res =z/U , normalized by the initial (at z/d = 0) turbulent timescale (k/)0 . For the local k/ we have used the relation (k/)/turb = 1.6 ± 0.1 (Pope, 2000) that was checked in a few cases with hot wire data with good agreement (within ± 20%). The value of turb (=Lturb /u)
10
(a)
-2
-4 (b)
0
20
40
60
z (mm)
Fig. 5. (a) Total scalar conservation from Eq. (3). (b) Conservation of scalar energy from Eq. (4).
has been calculated from the hot wire measurements of Lturb and u at various axial locations and similar conditions. It was also found that turb increases by about 40% in the first 42 mm from the injector. After a residence time of about 0.1–0.2 turbulent times (or 2–3d), CD reaches the value of 2 for all conditions, agreeing with previous studies (Chomiak et al., 1991; Dowling, 1991; Li and Bilger, 1996; Newman et al., 1981). The high initial values of CD are fully consistent with the models of Corrsin (1964) and Chomiak et al. (1991), who predict that CD scales with (Lturb /L )2/3 , where L is a characteristic scalar integral lengthscale. Here, initially L would scale with d (or perhaps with the initial laminar mixing layer thickness at the nozzle), which implies Lturb > L and hence CD > 2. At axial distances where CD ≈ 2, the model is also valid in the radial region, but only close to the centreline and away from the edges of the plume. Specifically, CD remained in the range 2–3 for r/d < 1.
C.N. Markides, E. Mastorakos / Chemical Engineering Science 61 (2006) 2835 – 2842
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also that careful data processing is necessary for scalar dissipation measurements. The mean mixture fraction was consistent with the Gaussian plume equation and the variance of the mixture fraction fluctuations increased to a maximum and then decayed downstream, as expected. The mean three-dimensional scalar dissipation was estimated along the centreline and related to the mixture fraction variance and local turbulent kinetic energy timescale. The results show that CD decreases from values higher than 10 near the nozzle to about 2 for axial distances greater than 2–3 nozzle diameters (or residence times greater than 0.1–0.2 turbulent timescales).
Run 1: Re = 405, υ = 1.1 Run 5: Re = 560, υ = 1.1 Run 8: Re = 560, υ = 2.4
60
Run 10: Re = 745, υ = 2.4 〈 χ2D 〉 (1 /s)
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40
20
Acknowledgements 0
0
5
10
15
20
25
z /d(-) Fig. 6. Centreline evolution of 2D for indicated runs of Table 1.
We thank Dr. R. Balachandran for his assistance with the PLIF measurements and Dr. J. Portilla for making available the BLS–GSM algorithms in Matlab format. This work has been funded by the EPSRC. References
10
〈χ3D〉/〈ξ′2〉.(k /ε)
8
6
4
2
0
0
0.1
0.2 0.3 τres /(k /ε)0(-)
0.4
0.5
Fig. 7. The local calculated CD as a function of (z/U )/(k/)z=0 for ‘Runs 1–5’ in Table 1.
4. Conclusion PLIF measurements were made of the concentration of a passive tracer in the axisymmetric mixing field formed by injection from a finite-sized nozzle into a confined turbulent co-flow. The spatial resolution was equal to the Kolmogorov lengthscale. The mean and variance of the normalized conserved scalar were not found to depend strongly on the level of denoising, even for smoothing to scales four times larger than the Kolmogorov lengthscale. In contrast, the scalar dissipation rate and the thickness of the dissipation filaments were affected to various degrees by the level and chosen method of denoising, suggesting that not only must the measurement resolution be adequate, but
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