Micro scalar timescales in premixed turbulent combustion

Proceedings of the Combustion Institute, Volume 28, 2000/pp. 351–358

MICRO SCALAR TIMESCALES IN PREMIXED TURBULENT COMBUSTION IAN SHEPHERD,1 LAURENT GAGNEPAIN2 and ISKENDER GOKALP2 1

Advanced Energy Technologies Department EETD Lawrence Berkeley National Laboratory 1 Cyclotron Road Berkeley, CA 94720, USA 2 Laboratoire de Combustion et Syste`mes Re´actifs Centre National de la Recherche Scientifique 45071 Orle´ans Cedex 2, France

Time series measurements of gas density have been obtained in lean (␾ ⳱ 0.65–0.8) premixed turbulent flames stabilized on a Bunsen-type burner (U0 ⳱ 2.3, 3.5 m/s). Two Eulerian scalar timescales were obtained from spectral analyses of these time series. The first, Tˆ, characterizes the flame front wrinkling process, describable by random telegraph signal statistics, and the second is a small scalar timescale, sD. This was shown to be due to the transitions of the flame fronts as they pass through the probe volume. The ratio between these two Eulerian scales, sD/Tˆ ⳱ 0.4 in the present flames, is a measure of the scale range of scalar fluctuations and is potentially an interesting factor at high turbulent Reynolds numbers when broadening of the preheat zone by turbulent mixing affects this ratio directly by broadening the “dissipation range” in the spectrum. A further timescale, a direct measure of the flame front transit time, was also determined from these data. Estimates of the local mean scalar dissipation were obtained from this data using flamelet assumptions and an interior distribution of the scalar, tested by fitting to numerical simulations of laminar flames, for the local scalar gradients. A simple expression was derived which related the mean scalar dissipation to the burning fraction of the scalar probability density function, c, and a chemical time characteristic of the flame front. It was found that the dissipation rate is affected more by the local scalar gradients than by c. A dissipation time scale was deduced from this rate and was found to be proportional to the ratio of the scalar integral timescale and the flame transit time.

Introduction The prediction of premixed turbulent flames is difficult. When integrating the model equations, a primary difficulty is the modeling of the chemical source terms and timescales of the scalar fluctuations. The chemical source term couples the flame front chemistry to the turbulence field through the various timescales controlling these processes. The familiar eddy break up model [1] was an early approach to modeling the mean reaction rate in turbulent flames. With suitable empirical input it has been moderately successful for high Reynolds number flows when it is assumed that the scalar dissipation rates are proportional to those of the velocity field. The introduction [2] of flamelet concepts into models of premixed combustion has shown that the scalar dissipation rate, N, is also dependent on the chemical reactions which generate the very large flame front scalar gradients. Models have been proposed in which the scalar dissipation plays a vital role [3]. Mantel and Borghi [4] have developed a premixed flame propagation model based on the scalar dissipation, and Bilger [5] 351

has extended his conditional moment closure model to premixed combustion. The experimental determination of scalar gradients in turbulent premixed flames remains a challenging problem due to the fine spatial resolution necessary and the difficulties of making three-dimensional flame structure measurements. O’Young and Bilger [6] have, however, reported some interesting experimental measurements of scalar gradients by a two-laser sheet configuration. Their experiments were performed in very high Reynolds number flows, and they found a significant broadening of the local flame thickness, indicating that N can be very much lower than in a laminar flame. The direct measurement by Rayleigh scattering of scalar timescales in lean methane/air turbulent premixed flames has been presented by Gangepain et al. [7], who compared the time scales which characterized the dynamic and scalar fields and concluded that conditional dynamic scales should be used to determine the relevant combustion regime. In this paper, scalar microtimescales are investigated in two ways: first, by the measurement and analysis of gas density time series obtained by Rayleigh scattering in lean premixed turbulent flames to

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TABLE 1 Experimental conditions and non-dimensional numbers Lu TL tg d tc Re u⬘ SL (m/s) (m/s) (mm) (ms) (ms) (mm) (ms) 0.15 0.15 0.17 0.23 0.25 0.26

0.30 0.23 0.19 0.30 0.23 0.19

8 10 11 5 6 7

57 65 68 20 24 26

0.32 0.34 0.26 0.25 0.23 0.23

0.51 0.65 0.81 0.51 0.65 0.81

Da Ka

0.58 80 99 1.8 0.94 97 69 2.8 1.36 128 50 5.2 0.58 70 35 2.4 0.94 95 25 4.1 1.36 112 19 5.9

(see Table 1), and so it can be assumed that, in the mean, the local scalar gradients can be represented by the unstrained laminar profiles. Second, Borghi [8] and Anand and Pope [9] have demonstrated that when c K 1 (c is the fraction of the burning mode in the scalar probability density function [pdf]) then “out of flamelets” contributions to the scalar dissipation cannot be ignored because, although the gradients are very small, they occupy a very large volume [3]. For the present flames, c at 具c典 ⳱ 0.5 is always greater than approximately 20%, and so the flamelet contribution is expected to dominate. The scalar dissipation, using conventional tensor notation, is N ⳱ h(⳵c/⳵xj)2 (1) where h is the thermal diffusivity and c is the instantaneous progress variable defined by c ⳱ (qu ⳮ q)/ (qu ⳮ qb). The mean scalar dissipation, 具N典, is obtained in the usual way by 具N典 ⳱

1

冮

0

N(c)P(c)dc

(2)

where P(c) is the pdf of the progress variable given by

Fig. 1. Progress variable profile for laminar premixed methane/air, ␾ ⳱ 0.65, from numerical simulations. Solid line is a fit of equation 4. Dashed lines show definition of flame zone thickness (Table 1).

obtain Eulerian scalar timescales, and second, by deduction of the mean scalar dissipation rate and timescale from the probability statistics of the same data. Where possible, these scales are related to each other. Scalar Dissipation The scalar microtimescales (those associated with the instantaneous flame front) and scalar statistics were obtained in lean premixed turbulent methane/ air flames. The latter quantities are related to the scalar dissipation in this section. Two points should be made at the outset. First, O’Young and Bilger [6] have shown that at high turbulent Reynolds number, scalar gradients in the flame front can be much lower than in the unstretched laminar flame. The flames here, however, are in relatively low turbulence flows

(3) Pc(c) ⳱ ␣d(c) Ⳮ bd(1 ⳮ c) Ⳮ cf(c) where ␣ is the unburned fraction, b is the burned fraction, and c is the burning fraction. The interior distribution of c within the flamelet is f(c) ⳱ 1/d (dc/ dz)ⳮ1, where d is the laminar flame thickness defined by the maximum gradient of the progress variable. To evaluate f(c), an expression which relates the progress variable to physical space is necessary. Fig. 1 shows the result of a numerical simulation of a onedimensional premixed methane/air flame (␾ ⳱ 0.65) using the PREMIX/CHEMKIN code and the GRI-Mech kinetic scheme. The gas density is converted to a progress variable and the resulting profile is satisfactorily fit by the expression c ⳱ (1 Ⳮ exp(ⳮ4z/d))ⳮ1 (4) which can be used to provide the scalar gradients. Also marked on Fig. 1 is the thermal laminar flame thickness, d (characteristic of the preheat zone), which is relevant to the scalar dissipation. The normal scalar gradient, where z is the coordinate normal to the flame front, can now be determined: (dc/dz) ⳱ d((1 Ⳮ exp(ⳮ4z/d))ⳮ1)/dz ⳱ 4c(1 ⳮ c)/d which on substitution into equation 2 gives 具N典 ⳱

1

冮

0

h(dc/dz)2 (␣d(c)

Ⳮ bd(1 ⳮ c) Ⳮ cf(c))dc (5) When c → 0, 1, then N → 0, and the mean scalar dissipation is

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flamelets, that is, that the mean laminar flame thickness is that of the unstrained laminar flame. Experimental Details

Fig. 2. Time series of Rayleigh signal, ␾ ⳱ 0.8, U ⳱ 2.3, c¯ ⳱ 0.4. Solid lines show threshold values used for smoothing.

具N典 ⳱

1

冮

0

(c/d)(dc/dz)dc

(6)

To integrate equation 6, an expression for the temperature dependency of the thermal diffusivity is needed. The numerical simulations indicate that it can be well represented by a quadratic of the form h ⳱ h0 Ⳮ ac2 Ⳮ bc

(7)

where h0 is the thermal diffusivity in the reactants. Substituting this expression into Eq. 6 and integrating gives 具N典 ⳱

冢

冢

冣冣

c 2 1 a b Ⳮ Ⳮ sd 3 h0 5 3

(8)

where sd ⳱ d2/h0 is a timescale based on the preheat zone residence time. Equation 8 is similar to an expression obtained by Libby and Bray [2]. The second term in the right-hand side parentheses arises from the thermal diffusivity temperature dependence. Typical values for the quadratic fit of h(c) of a ⳱ 2.1, b ⳱ 3.6, and h0 ⳱ 0.16 give 具N典 ⳱ 10.8 c/sd, which relates the scalar dissipation to the burning fraction. The time scale, sd, can be converted to a more familiar chemical reaction time obtained from the laminar burning velocity and laminar flame thickness, sR ⳱ dL/SL, assuming that d ⬃ 5dL [10]. The mean scalar dissipation is therefore 具N典 艑 0.4c/sR

A detailed description of the burner is available elsewhere [7], but essentially, the flames were stabilized on a 25 mm diameter Bunsen type burner by a co-annular pilot flame. Isotropic turbulence (7.3% at an integral length scale of 2.9 mm at the exit nozzle) was generated by a perforated plate 50 mm upstream of the burner exit. The turbulence characteristics were determined by laser Doppler velocimetry. Lean methane/air flames at two exit flow velocities (U0 ⳱ 2.3 and 3.5 m/s) and three equivalence ratios (␾ ⳱ 0.65, 0.7, 0.8), using clean bottled gases, were investigated. The flow and reaction parameters are given in Table 1. The Reynolds number is Re ⳱ Luu⬘/m, where Lu is the integral length scale, u⬘ is the root mean square of the axial fluctuating velocity, and m is the kinematic viscosity. The Damkohler number is defined by Da ⳱ TL/tc, where TL is the turbulence timescale, tc is the chemical timescale, and SL is the laminar flame velocity. The Karlovitz number is Ka ⳱ tc/tg, where tg is the Kolmogorov timescale. The flame thicknesses were obtained from the numerical simulations and the laminar flame velocity from Jarosinski [10]. The characteristics presented here were obtained by velocity measurements at the cold boundary of the flame brush to account for their modification between the burner exit and the flame zone, in particular when the flame heights differ significantly. These non-dimensional numbers indicate that these low turbulence flames are in the wrinkled laminar flame regime. Laser-induced Rayleigh light scattering was used to characterize the flames’ scalar structure by measuring the local instantaneous gas density. The 488 nm line of a 25 W Spectra Physics cw argon-ion laser was used as the light source. The scattered light was collected at a right angle by two planoconvex lenses (f4 200 mm) and an interference filter (centered at 488 nm); it was then focused onto a Hamamatsu R647-04 photomultiplier tube. A 150 lm pinhole determined the probe volume length, and its height corresponded to the beam waist (40 lm). The Rayleigh scattering signal was amplified, low-pass filtered at 10 kHz, and then sampled at 20 kHz. 100,000 samples were collected at each measurement point. Results and Discussion

(9)

which uses a definition for the laminar flame thickness, dL ⳱ h0/SL. The above analysis assumes that the local flame structure is that of unstrained laminar

Scalar Time Series Analysis Figure 2 shows a typical Rayleigh times series for the ␾ ⳱ 0.8, U0 ⳱ 2.3 flame at 具c典 ⳱ 0.4. The

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Fig. 5. Normalized frequency spectra from the time series shown in Fig. 2, ␾ ⳱ 0.8, U ⳱ 2.3, c¯ ⳱ 0.4. Fig. 3. Probability density function of the Rayleigh signal presented in Fig. 2. Note the characteristic Poisson noise broadening. Dashed lines are Gaussian fits through the noise-broadened burned and unburned peaks. The solid vertical lines are the threshold values of Fig. 2.

data was performed in two parts: (1) a statistical analysis of the scalar field and (2) a spectral analysis of the time series. Statistical analysis The statistical properties of these time series were obtained from their pdfs. Fig. 3 shows the pdf of the time series shown in Fig. 2. The pdf has three parts: the unburned, ␣, burned, b, and burning fraction, c, where

␣ⳭbⳭc⳱1

(10)

These fractions are determined by least mean square fitting two Gaussian functions through each of the bimodal peaks, which are centered at the unburned and burned scattering intensities, Fig. 3. The areas under these fits as fractions of the total area are ␣ and b, respectively, and from equation 10, c is obtained. Fig. 4 presents the distribution of ␣, b, and c for the these data as a function of mean progress variable, 具c典. The latter quantity is calculated directly from the pdfs as the fractional probability greater than the midpoint value between the bimodal peaks, marked on Figs. 2 and 3. Fig. 4. Distribution of the scalar probability fractions, ␣, b, and c derived from flame ␾ ⳱ 0.8, U ⳱ 2.3.

intensity of scattered light is proportional to the density of the gas, and so the upper level derives from the unburned gas and the lower from the burned gas. Transitions occur between these two states when a flame crosses the probe volume. In Rayleigh scattering, the variance of the noise is proportional to the mean signal, and hence the signal-to-noise ratio is better in the unburned gas. Analysis of these

Spectral analysis The spectral characteristics of the time series were obtained by fast Fourier transform (FFT) analysis. The spectrum is determined by dividing the time series into 4096 point segments, applying the FFT to each segment, and averaging. The normalized spectrum of the time series presented in Fig. 2 is shown in Fig. 5. This spectrum can be analyzed into three components: (1) a noise spectrum, (2) a spectrum associated with the flame crossings, and (3) a spectrum derived from the flame front transitions.

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timescale, sD, is obtained [11] for this region from equation 11: (sD)ⳮ2 ⳱ 2p2

Fig. 6. Frequency-weighted spectrum for the smoothed spectrum in Fig. 5 used to calculate the dissipative timescale, sD.

This is done by progressively “smoothing” the time series and then calculating the spectrum of the resulting data. To eliminate the shot noise in the burned and unburned scattering signals, the two Gaussian-broadened peaks in Fig. 3, all values above and below the thresholds shown in Figs. 2 and 3 were set to constant values. The time series now consists of the unsmoothed flame front transitions separating regions of constant value. The spectral affect of this, Fig. 5, is to move the impact of the noise spectrum from 2 to 5 kHz. In the next stage, the time series is reduced to a random telegraph signal (RTS). This treats the scalar field as two states with infinitely fast chemistry at the flame front. The RTS time series is generated by using the midpoint threshold (used to calculate 具c典) and setting all values to either 0 or 1, unburned or burned respectively. The resulting spectrum, Fig. 5, is characteristic of a RTS where, at high frequencies, the spectral energy is proportional to (Ef)RTS ⬀ 1/f 2, where f is the frequency. Comparison at frequencies below ⬃400 Hz reveals that the spectra are identical, because this region, due to the large-scale wrinkling of the flame front by the turbulent field, is unaffected by the smoothing. It is characterized by the scalar integral timescale, Tˆ, which is calculated in the usual way by integrating the autocorrelation function obtained from the inverse transform of the normalized power spectra, Ef. At frequencies higher than ⬃400 Hz, the unsmoothed and noise-reduced spectra diverge from the RTS spectrum because of the flame front transitions. This region may be described by analogy to the turbulence spectrum as a “dissipation” region. A

⬁

冮

0

f 2 Ef df

(11)

The reduction of spectral noise allows a satisfactory estimate of sd to be obtained (Fig. 6). Equation 11 also shows that the RTS spectrum has no associated dissipation scale because (Ef)RTS ⬀ 1/f 2. The noise above ⬃3 kHz does not have a significant affect on these integrations. This shows that, in these low-turbulence flames; the turbulent field can wrinkle the flame front down to the scale of the flame front thickness, where local scalar dissipation occurs. An alternative method of estimating this timescale [7] is to perform a Taylor expansion of the autocorrelation function at t ⳱ 0 and then fit the autocorrelation with the resulting quadratic. This method, however, was found to be very sensitive to the number of data points used in the fit, and so equation 11 has been used in all the estimates of the dissipative timescale. The spectral analysis yields two timescales, Tˆ and sD, which characterize the large- and small-scale temporal structure of the premixed turbulent flame, a wrinkling scale and a flame front scale, respectively. A further microtimescale, the mean flame front transit time, sf, may be determined directly from c and the mean flame crossing frequency, m (obtained from a zero crossing analysis of the RTS time series) by sf ⳱ c/m. In an earlier method used to obtain this parameter [12], the laminar profile is fit through each crossing in the time series with the transit time as a free parameter, followed by averaging. This method was also used here for one flame comparison, and a timescale approximately 15% shorter was found, but this was due to the definition of the transit time. As can be seen from Fig. 1, the analogous transit time must be shorter than the transit time obtained from c. The method employed here is much simpler than the earlier technique and so was adopted. The scalar integral time scale, Tˆ, familiar from the Bray-Moss-Libby (BML) model, has been measured and reported in many flame geometries and so is not discussed here; further details for the present system may be found in Ref. [7]. These timescales will be affected by the speed at which the flame zone is being convected past the probe volume, U0. This is shown in Fig. 7, which presents the transit times for all the flames studied, normalized by the transit time for a flame front which is perpendicular to the mean convection direction, sn ⳱ d/U0. The times are collapsed together and, assuming that tf ⳱ tf (U0, d, r), ¯ where r¯ is a direction cosine which in the normal case is unity, it is clear that the scalar structures for all these flames as revealed by the flame orientation are similar. The spectral analysis of the time series has shown that sD derives from flame front transitions. This is

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Fig. 9. Scalar dissipation rates for flames, U0 ⳱ 3.5 m/s. Fig. 7. Flame transit times normalized by the normal crossing time. The effect of the bulk flow velocity has been removed.

timescale for all these flames. It would be of interest to investigate the evolution of this scale ratio for flames at higher Reynolds numbers. Scalar Dissipation

Fig. 8. Dissipative timescale, sD, normalized by the scalar integral timescale.

also an Eulerian timescale (a function of convection speed) and so cannot be directly related to the scalar dissipation time. However, interesting information can be derived from this scale if it is compared with another Eulerian time scale, the scalar integral time scale, Tˆ. This comparison is presented in Fig. 8, where sD has been divided by Tˆ. This figure shows that for these low turbulent Reynolds flows, the timescale at which dissipation occurs at the flame front is approximately 40% of the scalar integral

Equation 9 relates the mean scalar dissipation to the local burning fraction, c, and the chemical time scale, sR. This result was obtained using flamelet assumptions and an interior distribution of the progress variable, which was a good fit to the numerical simulations of the one-dimensional laminar flames. Fig. 9 shows the variation of 具N典 through the flame zone of the higher-velocity flames. The distributions are similar to those of c shown in Fig. 4. Although the peak c value in the ␾ ⳱ 0.65 case is 40% higher than in the ␾ ⳱ 0.8 flame, 具N典 is significantly higher in the latter case because of steeper flame front gradients. The balance between these two effects determines 具N典. Similar results were obtained for the U0 ⳱ 2.3 m/s case. At higher turbulence levels, O’Young and Bilger [6] have found significant reductions in the scalar gradients. In terms of the present analysis, this is equivalent to an increase in the chemical time, sR, and the present estimate would then provide an upper bound to the mean scalar dissipation rate. Libby and Bray [2] have also shown that the mean scalar dissipation is related to the mean burning rate, and Borghi [8] found that it is possible to relate 具N典 to the flame surface density, R. Assuming that for unstrained laminar flamelets, R ⳱ c/dL, then with some manipulation, the present analysis gives 具N典 ⳱ 0.4SLR

(12)

which shows the relationship between the scalar dissipation and the burning rate.

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should be noted that the convection speed which affects both Tˆ and sf is effectively eliminated in equation 15. Conclusions

Fig. 10. Scalar dissipation timescales from equation 14 for same conditions as in Fig. 8.

The scalar dissipation rate can also be related to a dissipation timescale, sv, by 具N典 ⳱ 具c⬘2典/sv. This timescale may be evaluated by using the scalar pdf, equation 3, from which the variance of the progress variable is 具c⬘2典 ⳱ c¯(1 ⳮ c¯) ⳮ c/4

(13)

where the first term on the right-hand side of equation 13 results from infinitely fast chemistry assumptions. By substitution into equation 9, the dissipation timescale is sv ⳱ 0.625sR (4c¯(1 ⳮ c¯)/c ⳮ 1)

(14)

This dissipation timescale, with the chemical time, sR, from Table 1, is presented in Fig. 10 for the same flames as in Fig. 9. As expected this scale is affected by the equivalence ratio: the mean value averaged between 具c典 ⳱ 0.3 and 具c典 ⳱ 0.7 varies from 1.4 ms for ␾ ⳱ 0.8 to 2.1 for ␾ ⳱ 0.65. It is also of interest to evaluate c in equation 14 in terms of the flame transit time and the mean flame crossing frequency, m, which can be expanded by use of the familiar BML expression, m ⳱ gc¯ (1 ⳮ c¯)/Tˆ. Equation 14 shows that ˆ fⳮ 1 sv/sR ⬀ KT/t

(15)

where K is a constant of order one, that is, the normalized dissipation timescale is proportional to the ratio of the scalar integral timescale and the flame front transit time. The physical picture which emerges from equation 15 is straightforward. When the scalar dissipation is dominated by flamelet contributions, it will be determined by the spacing between the flame crossings, Tˆ, and the transition time, sf, when scalar dissipation becomes significant. It

The purpose of this paper was to investigate the micro scalar timescales and scalar dissipation in turbulent premixed flames. This has been done from two perspectives: first, from spectral analysis of gas density time series and, second, by analysis of the scalar statistics. Two Eulerian scalar time scales were obtained from the spectral analysis. The first is a scale which characterizes the wrinkling process and is governed by random telegraph signal statistics. The second, a small scalar timescale, has been established by spectral analysis to derive from the transitions of the flame front as they pass through the probe volume. This scale appears as a steeper rolloff in the energy spectrum than the f ⳮ2 rate characteristic of a random telegraph signal. The ratio between these two scales is potentially an interesting factor at high turbulent Reynolds numbers when broadening of the preheat zone by turbulent mixing will affect this ratio directly by broadening the dissipation range in the spectrum. A further timescale, a direct measure of the flame front transit time was also determined from these data sets. Estimates of the local mean scalar dissipation were also obtained from these data. The analysis used flamelet assumptions and, for the local scalar gradients, an interior distribution of the scalar tested by fitting to numerical simulations of laminar flames. A simple expression was derived which related the mean scalar dissipation to the burning fraction of the scalar pdf and a chemical time characteristic of the flame front. It was found that this dissipation rate is affected more in this system by the local scalar gradients than the probability of intermediate states. A dissipation timescale was derived from the dissipation rate and was found to be proportional to the ratio of the scalar integral timescale and the flame transit time. Acknowledgment This work was supported by the Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098 (IS) and by the Centre National de la Recherche Scientifique and the Conseil Regional Centre (LG and IG).

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