Experimental Characterisation of the a-parameter in Turbulent Scalar Flux for Premixed Combustion

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1 Combust. Sci. and Tech., Vol pp Reprints available directly from the publisher Photocopying permitted by license only 2000 OPA (Overseas Publishers A\\ociation) N.V. Published by license under tbe Gordon and Breach Science Publishers imprint. Primed in Malaysia Experimental Characterisation of the a-parameter in Turbulent Scalar Flux for Premixed Combustion PETER A.M. KALT' and ROBERT W. BLGER Department of Mechanical and Mechatronic Engineering University of Sydney Sydney, NSW2006 AUSTRALA (Received September 21, 1999) The a-parameter is a variable appearing in the formulation of the Bray number, a dimensionless tcrm that can be used to predict the behaviour of the turbulent scalar flux as counter-gradient or gradient in premixed combustion. However, the successful application of the Bray number is hampered by poor characterisation of the a-parameter. The results of previous laser-imaging studies of conditional mean velocities in turbulent premixed methane/air and propane/air Bunsen-flames are used to formulate an empirical description of a. The resulting description of a suggests that the behaviour of the scalar flux is dependent on the heat release parameter and the length scale ratio, lsv rather than the velocity ratio, U'/SL' Keywords: NTRODUCTON Turbulent Scalar Flux; Premixed Combustion The turbulent scalar flux, pu" c", is a term appearing in the turbulent transport equation of c in premixed combustion. Here e is a reaction progress variable defined by Bilger (1993) in terms of sensible enthalpy and diacritical tilde represents the Favre- (or mass-weighted) average such that c= pelp. Accurate modelling of the scalar flux is necessary to describe the macroscopic flame brush structure (Mantel et al. (1993).) Consequently, a great deal of research effort has gone into accurately describing the scalar flux term. Recent laser-imaging studies of turbulent premixed piloted Bunsen-style flames (Frank et al. (1996, 1999), * Corresponding Author Peter Kalt Department of Mechanical and Mcchatronic Engineering BUilding 107 University of Sydney ph: fax: pkalt@mech.cng.usyd.cdu.au 213

2 214 PETER A.M. KALT and ROBERT W. BLGER Kalt et al. (1998» have shown a transition from counter-gradient (pu-;;z" > 0) to gradient (pu"c" < 0) behaviour of the scalar flux as the relative turbulence, l'/sv increases. The transition from counter-gradient to gradient diffusion depends on l'/sl but is not fully described by this parameter, with important effects due to heat release and turbulence scale not considered. Veynante et al. (1997) use the close correspondence between the transport of c and r to define a simple expression for the scalar flux: where a is an order-unity efficiency function that describes the ability of various sized eddies to convolute the flame front (Guider and Smallwood (1995)), 1: is the heat release parameter, ' is the turbulent rms velocity and SL is the laminar flame speed. Using Equation, Veynante et al. (1997) propose a dimensionless parameter known as the Bray number, NB' which can be used to predict the nature of the scalar flux in the gradient or counter-gradient sense. For free flames, the Bray number is defined as N B = TSL. 2au' Values of N B < indicate gradient diffusion and N B > indicate counter-gradient diffusion. The boundary, N B = is the case of no scalar flux. This parameter has been found by Kalt (1998) to be successful in qualitatively describing the transition from counter-gradient to gradient diffusion of C as turbulence increases. However, difficulties arise in the attempt to use the Bray number as a predictive tool due to poor characterisation of the a-parameter. n this paper, the experimental data of Frank et al. (1996, 1999) and Kalt et al. (1998) are used to formulate an empirical description of the a-parameter based on parameters describing the turbulent flame. (1) (2) BRAY NUMBER OF EXPERMENTAL FLAMES Frank et al. (1996, 1999) used a simultaneous PV/OH PUF technique to determine the conditional mean product and reactant velocities of turbulent premixed methane/air flames. Similar measurements on the same burner have been conducted in turbulent premixed propane/air flames by Kalt et al. (1998). The behaviour of "free flames" on the Bunsen-style burner for methane/air and propane/air flames have been observed from the experimentally determined conditionalmean velocities and the Bray-Moss-Libby (BML) formalism (Bray (1980),

3 TURBULENT SCALAR FLUX 215 Bray et al. (1981». The BML formalism is used to model the scalar flux from these conditional mean velocities as (3) Here, up and u r are the conditional mean velocities of the products and reactants, respectively. The nature of the scalar flux in a particular coordinate depends on the sign of the slip velocity. (Up,i - Ur,i)' Table shows the actual and predicted behaviours of the radial component of the scalar flux for the flames examined by Frank et al. (1996, 1999). ego refers to the case of counter-gradient diffusion, GO refers to gradient diffusion and a dash indicates the case of negligible scalar flux. Also shown in Table 1 is the relative radial turbulence level for each flame, v'/sl' and the heat release parameter, 'to Values for the Bray number, NB are determined from Equation 2 using these values and a =. TABLE The Bray number, N B and the predicted and actual behaviours of the radial scalar flux for methane/air names of Frank et at. (1996, 1999) Flame v's L [m/s] r N B predicted actual M CGO CGO M CGO CGO M GO M GO GO M CGO CGO Table 11 presents values of NB' v'/su t and the predicted and actual behaviour of the radial component of the scalar flux for the propane/air flames of Kalt et al. (1998). For the radial component of each of Flames M -M5 and P-P4, the Bray number is calculated and compared to the actual behaviourof 15;;;-;;".. The value of a is important in describing the behaviour of pu" C" but is difficult to get an accurate estimation for the term. Veynante et al. (1997) estimate a as a function of the lengthscale ratio, lol' based on fitting parabolas to U"C" data provided by ONS. Unfortunately, the estimate is affected by various numerical difficulties, and values have a large (20%) uncertainty. TABLE The Bray number, N B, and the predicted andactual behaviours of the radial scalar flux for propane/air names of Kalt er at. (1998) Flame v's L [m/s] r N B predicted actual P CGO CGO P CGO CGO P GO P GO GO

4 216 PETER A.M. KALT and ROBERT W. BLGER The lengthscale ratios for Flames M -M5 and P-P4 are large (j» 20), and a = for the these flames. t can be seen that uncertainty in the value of the efficiency function, a, could have a large effect on the predictions made by the Bray number. Flames M, M2, M5, P and P2 behave in a counter-gradient manner. The Bray number predictions for these flames range from 1.19 to 1.97, accurately suggesting counter-gradient behaviour. Flames M3 and P3 have a very small slip velocity and correspond to the border between gradient and counter-gradient diffusion. As such the Bray numbers for these flames should be very close to unity. The Bray numbers for these flames are 0.71 and 0.82 respectively, which incorrectly predict gradient diffusion. Flames M4 and P4 are flames with strong gradient diffusion. The Bray numbers for these flames are less than one, which is an accurate prediction of gradient behaviour. The Bray number seems to accurately capture the trends of the transition, but overpredicts the occurrence of gradient diffusion. Arbitrarily selecting a value of a '" 0.75, it is possible to reproduce the trends observed in both the methane/air and propane/air flame data. However, selection of a as a function of the lengthscale ratio, using the curve-fit data to the DNS results of Veynante et al. (1997) does not give accurate predictions of the behaviour of the scalar flux. The a parameter is a complex function that is likely to require formulation in terms of many flame parameters, including u'/sl and t. n this sense a introduces difficulties in applying the Bray number as a predictive tool. The usefulness of the Bray number, NB, to predict the behaviour of the turbulent scalar flux is critically dependent on the accurate characterisation of the efficiency function. THE a-parameter The conditional mean velocity data are the basis for an attempt to formulate an empirical description of c. Considering the radial component for the piloted Bunsen-style flames and combining Equations and 3 gives (4) which allows the characterisation of a in terms of known or measured quantities for each flame condition. Here (D' - Dr) is the radial component of the slip velocity, r is the heat release parameter, SL is the laminar flame speed and v' is the radial rms velocity fluctuation. Values for TB, - (v)' - fir) n = --=---;:'-;:----'..:.... 2v' (5)

5 TURBULENT SCALAR FLUX 217 for the experimental flames are given in Table and V. The a-parameter has a slight variation with C, because the slip velocity is largest in the centre of the flame brush. This is the region of the flame brush where the scalar flux is most pronounced and the values for a given in Tables and V are the average value over the range 0.4 < C< 0.6. The behaviour of a has been correlated against the non-dimensional parameters u'/su /O L and t. The laminar flame thickness, 0L =v/sv is a characteristic flame length where v)s the unburnt kinematic viscosity. The integral length scale, /o corresponds to the size of the energy-containing eddies which grow in size as the turbulence decays downstream. Values for the integral length scale are estimated from the size of the holes in the turbulence generating grid and the flow conditions, assuming the eddy growth rate is similar to that of an axisymmetric jet into a coflow (Antonia and Bilger (1973)). The relationship is found to give the best approximation to the experimental values. Here k a = 20 is a scaling constant. The correlation coefficient, R, in this case is R = 0.41, which indicates only reasonable correlation. Flame MS has a value of a that seems excessively low and, given the small number of experimental flames, this has a marked effect on the correlation coefficient. The correlation coefficient calculated from the remaining flames is R =0.93. Figure is a plot of experimental values of a against values of a given by Equation 6. (6) / o ' o a (experiment) FGURE Experimental values of a plotted against values from Equation 6

6 218 PETER A.M. KALT and ROBERT W. BLGER TABLE Experimentally determined values for the a-parameter for the methane/air flames of Frank e1 al. (1996, 1999) Flame M/ M2 M3 M4 M TABLE V Experimentallydetermined values for the a-parameter for the propane/air names of Kalt et al. (1998) name P/ P2 P3 P Combining Equation 6 with Equation 2 and rearranging gives a reduced Bray number, N'B, expressed as 0.67 f )-1/ N'B ==:0 (7) ( Equation 7 would suggest that the scalar transport is exclusively dependent on the heat release parameter, 1:, and the lengthscaie ratio. This seems contrary to the notion that the behaviour of the scalar flux is related to the turbulent fluctuations. The behaviours of the turbulent scalar fluxes for all the experimental flames are predicted with a high degree of accuracy. Like the original Bray number, values of Ni] > 1 correspond to counter-gradient diffusion and N'B < 1 to gradient diffusion. N'B is calculated for Flames Ml-M5 and P-P5 and presented in Table V. However, Flame P3 (N'B == 1.17) displays only very slight counter-gradient diffusion in reality and N'B is expected to be closer to unity. TABLE V The reduced Braynumber, N'B l and the predicted and actual behaviours of the radial scalar tlux for free names Flame N'B predicted actual M 1.74 CGO CGO M CGO CGO M M GO GO M CGO CGO P 1.90 CGO CGO P CGO CGO P CGO P GO GO

7 TURBULENT SCALAR FLUX 219 Equations and 6 have been used to calculate the scalar flux, rather than simply predicting the qualitative nature of its behaviour. Figures 2 and 3 show the comparison between the modelled scalar flux and the experimental values for Flames M -M5 and Flames Pl-P4, respectively. The behaviour of the scalar flux is well described in the gradient or counter-gradient sense and the calculated values of in:"e" show reasonable quantitative agreement with values determined using the conditional mean velocities and Equation j )( c: <ll OJ i' (ij l...l...l...j.-'-.l.l...l...j.-l-.l.l.-'-''-'-.l.l...l...j'-l...j o Favre-mean progress variable, C o Flame M1 0 Flame M2.. Flame M3 o Flame M4 + Flame M5 FGURE 2 Comparison between experimentally determined radial tlux, fluxes using Equations and 6 for Flames M M5 pv" c" and modelled CONCLUSONS n this paper, the available experimental data characterising turbulent premixed flames are re-examined to test the accuracy of predictions based on the Bray number. The Bray number is found to be very successful in predicting a trend from counter-gradient to gradient transport as turbulence increases, and capture the effect of heat release on promoting counter-gradient diffusion. However, shortfall is found in the general applicability of the Bray number as a predictive tool, due to the uncertainties associated with modelling constants such as a, the

8 220 PETER A.M. KALT and ROBERT W. BLGER 0.15 <!! o <8 e '" E 1;; <> 0.1 X :::l :;:: c: <l 0.05 <D E, Q)... > <l 0 u.. :!!! '0 <l o Favre-mean progress variable, C e Flame P1 Flame P3 o o Flame P2 Flame P4 FGURE 3 Comparison between experimentally determined radial nux, [n)" en and modelled fluxes using Equations 1 and 6 for Flames P-P4 efficiency function. The efficiency function is a parameter included in the formulation of the Bray number to account for the reduced ability of smaller turbulence scales to influence the flame surface. Analysis of the conditional mean velocity data for methane/air and propane/air piloted Bunsen-style flames collected over a range of turbulence and flame conditions indicates that the a-parameter is strongly dependent on the velocity ratio and the lengthscale ratio. Further. using this empirical description of nin the equation for the Bray number leads to the surprising result that the nature of the scalar flux is dependent only on the heat release, r, and the lengthscale ratio,!8l' t is worth noting that the modelling of a is based on the limited available experimental evidence. Equation 7 gives an accurate description of the behaviour of the scalar flux, counter-gradient or gradient, for the available flames. 1 Though the Bray number, as a single parameter, shows great potential to describe the behaviour of the turbulent scalar flux, its future application depends on accurate, experimental characterisation of the efficiency function over a spec-

9 TURBULENT SCALAR FLUX 221 trum of flow conditions beyond the data presently available in the combustion literature. This highlights the urgent need for further experimental characterisation of the scalar flux for a wider range of real flames. Acknowledgements This work is supported by the Australian Research Council. References [ Antonia, R. A. and Bilger. R. W. (1973). An experimental investigationof an axisymmetricjet in a co-flowing air stream. J. Fluid Mech.. 61 (4): 805. [2) Bilger. R. W. (1993). Turbulence and Molecular Processes in Combustion, Elsevier Science Publishers,New York, chap. Conditional MomentClosureand Advanced LaserMeasurements. pp [3J Bray. K. N. C. (1980). Topics ill Applied Physics. Springer-Verlag, Heidelberg, vol. 44, chap. Turbulent Flows with Premixed Reactants. pp f4j Bray, K. N. C, Libby. P. A.. Masuya, G. and Moss, J. B. (1981). Turbulence production in premixedturbulent flames. Combust. Sci. Tech., 25: Frank. J. H.. Kalt, P. A. M. and Bilger, R. W. (1996). Two-dimensional measurements of conditional velocities in turbulent premixed methane/air flames. n First Australian Conference Oil Lara Diagnostics and Fluid Mechanics in Combustion. University of Sydney. pp, 71-76, [6J Frank, J. H., Kalt, P. A.V!. and Bilger. R. W. (1999). Measurements of conditional velocities io turbulent premixedflames by simultaneousoh PLFand PV. Combust. Flame, 116: GOlder. O. L. and Smallwood, G. J. (1995). nner cutoff scale of flame surface wrinkliog in turbulent premixedflame. Combust. Flame, 103: 107. [8] Kalt, P. A. M. (1998). Experimental nvestigation of Turbulent Scalar Flux in Premixed Flames. Ph.D. thesis, Uoiversity of Sydney. [9J Kalt, P. A. M.. Frank, J. H. and Bilger, R. W. (1998). Laser imaging of conditional velocity in premixedpropane/air names using simultaneous OH PLFand PV. n Twenty-Seventh Symposium (tntemationut) un Combustion. The Combustionnstitute, Pittsburgh, pp [101 Mantel, T.. Borghi, R. and Picart, A. (1993). Turbulent Shear Flows 9, Springer-Verlag. [J Veynante, D.. Trouve. A.. Bray, K. N. C. and Mantel, 1'. (1997). Gradient and counter-gradient scalartransport in turbulent premixednames. J. Fluid Mech., 332: 263.