Fractal Dimension of Turbulent Premixed Flames Alan R. Kerstein a a

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1 This article was downloaded by: [University of Southern California] On: 18 March 2011 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Combustion Science and Technology Publication details, including instructions for authors and subscription information: Fractal Dimension of Turbulent Premixed Flames Alan R. Kerstein a a Combustion Research Facility. Sandia National Laboratories, Livermore, CA, USA To cite this Article Kerstein, Alan R.(1988) 'Fractal Dimension of Turbulent Premixed Flames', Combustion Science and Technology, 60: 4, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Combust. Sci. and Tech., 1988, Vol. 60, pp Photocopying permitted by license only Gordon and Breach Science Publishers, Inc. Printed in Great Britain Fractal Dimension of Turbulent Premixed Flames ALAN R. KERSTEIN Combustion Research Facility. Sandia National Laboratories, Livermore, CA USA (Received June 8, /988) Abstract-Modification ofa previous (Peters, 1988)analysis of the fractal dimension ofturbu!enl premixed flames gives the result D = 7/3. Though this result is nol much different from the previous result D = 2.35, the present analysis has novel dynamical implications. Namely, it is proposed that the flame front is not equivalent to a passively convected interface, explaining why fractal flames are observed in flow fields which do nol appear to be fractal. INTRODUCTION Evidence is accumulating that the wrinkled flame fronts observed in turbulent premixed combustion experiments are fractals, Measured values 2.31 (North, 1988) and 2.36 (Mantzaras et al., 1988) of the fractal dimension D of the flame front at high turbulence intensity have been obtained, Analyses to date (Gouldin, 1987; Peters, 1988) which exploit this fractal property have been based on the assumption that the flame front is effectively a passive interface within the fractal range of length scales. Therefore D has been compared to measurements or theoretical estimates of the fractal dimension of passive interfaces in turbulence (Hentschel and Procaccia, 1984). Here it is proposed that the assumption that the fractal flame front is effectively passive is not physically plausible. An alternative analysis is presented which indicates that a dynamical balance between the wrinkling effect of turbulent convection and the smoothing effect of local flame propagation is established at all length scales within the fractal range. The analysis is based only on the Kolmogorov cascade picture of fully developed turbulence, and involves no assumptions concerning fractal properties of the turbulent flow field. The result is therefore consistent with the recent (North, 1988) observation of fractal flames in flow fields which do not appear to be fractal. The predicted fractal dimension, D = 7/3, is consistent with measured values. The analysis is based on the fractal picture of Gouldin (1987) and Peters (1988), though with an important modification. As previously, the flame front is treated as a zero-thickness interface which propagates into the unburned gas at a constant local speed U L, i.e., stretch effects are omitted. ANALYSIS To motivate the fractal picture of a wrinkled turbulent flame front, an instantaneous snapshot of the wrinkled interface may be considered. In strong turbulence, the interface may be so convoluted that a streamwise ray (i.e., a ray in the mean propagation direction) may intersect the interface many times. The average distance 0T from the first (downstream) to the last (upstream) intersection point is adopted as a measure of the thickness of the turbulent flame brush. Note the distinction between 0T and other length scales characterizing the wrinkled flame front. The flame front is an interface propagating locally normal to itself, toward the unburned gas, at a fixed speed U L relative to the local fluid velocity v. 441

3 442 A. R. KERSTEIN The principal radius ofcurvature r at any point on the interface is typically much less than tj r. From dimensional considerations, the typical value ofr scales with the mean distance between successive intersections of the aforementioned streamwise ray, so if the streamwise ray typically intersects the interface n times, then tj r = nr, where n ~ I implies tj r ~ r. (Here and below, numerical prefactors are omitted.) The fractal picture addresses the distribution of r values rather than a typical value of r, in order to predict the composite influence of all scales of wrinkling on the turbulent flamespeed. Before introducing this picture, general aspects of flame geometry and dynamics are considered further. tj T as formulated above is a global scale rather than a local scale of the flame front. An expression for tj T is obtained by considering a macroscopic (coarse-grained) picture of the propagating flame. In a frame of reference moving upstream at the turbulent flamespeed Ur, the streamwise variation of flame structure is statistically stationary. tjr may be regarded as the streamwise span ofa reaction zone in which the volume fractions of both burned and unburned gas are of order unity at any streamwise location. Viewed as a diffusive-reactive zone analogous to a laminar premixed flame, its width can be expressed as where the turbulent diffusivity has been expressed as the turbulent velocity fluctuation u' times the integral scale L. lit' which is equal to the laminar flamespeed ill times the flame surface area per unit transverse projected area, can be expressed as where A is the flame surface area per unit volume within the turbulent flame brush. A has units of inverse length, and based on dimensional considerations, it roughly corresponds to the inverse of the quantity r defined earlier. Equations (I) and (2) can be solved for UT and tj r, yielding and _ (U'L )'/2 tj r - -- ula These expressions follow from two assumptions, namely the diffusion-reaction analogy leading to Eq. (I) and the assumption that the surface area per unit volume within the turbulent flame brush can be represented by a characteristic value A, neglecting streamwise variation within the brush. [Equation (2) follows immediately from the latter assumption, and from the earlier assumption that U L is constant.] The fractal picture is introduced in this context in order.to express A in terms of UL' u', and L. For this purpose, the formulation of Peters (1988) is adopted. The fractal property of the flame front is assumed to be a manifestation of the dynamical self-similarity of the inertial range of high Reynolds number turbulence. Therefore the upper cutoff is taken to be the integral scale L. It was initially (Gouldin, 1987) proposed that the lower cutoff should be the Kolmogorov microscale L K, but Peters showed that a new scale, the Gibson scale L G, is a more appropriate choice. L G is the length scale of (I) (2) (3) (4)

4 SHORT COMMUNICATION 443 eddies whose characteristic velocity is equal to the laminar flamespeed U L. Eddies of size L K < 1 < L have characteristic velocity (01)'/3, where the energy dissipation e is independent of 1 according to the Kolmogorov cascade picture. Therefore setting e = u,3/l and 1 = L G, the result (5) is obtained. For length scales much smaller than L G, the flame crosses an eddy before the eddy evolves significantly, so the flow field is effectively stationary from the viewpoint of flame front propagation. Far above L G, laminar flame propagation is slow compared to the characteristic eddy velocity, leading to the inference (Peters, 1988) that the flame front is effectively a passively convected interface. (This inference is reconsidered shortly.) Thus, L G marks a transition between propagation regimes, and as such is adopted as the lower cutoff of the fractal regime. According to the fractal picture (Mandelbrot, 1982; Gouldin, 1987; Peters, 1988) the surface area per unit volume associated with length scales greater than I, where L G < 1 < L, is A(/) = (//L)2-D/L, (6) where D is the fractal dimension of the surface and L has been adopted as the upper cutoff. This dependence is typically manifested as a linear regime on a plot of In A(/) versus In I, where I is the length scale of a smoothing procedure applied to the surface. (Smaller 1corresponds to less smoothing, and therefore more surface area.) As 1falls below L G, In A(/) rolls off and converges in the limit I to A(O) = A, the surface area per unit volume of the original, unsmoothed flame front. As previously (Peters, 1988), the approximation is adopted that A(O) = A(L G ). Namely, L G is taken to be a sharp cutoff rather than a demarcation of a gradual transition. Setting A = A(Ld in Eqs. (3) and (4), substitution of Eqs. (5) and (6) yields, respectively, and (7) (8) In previous formulations, the fractal dimension D has either been regarded as a free parameter, or it has been assigned the empirically or analytically estimated value of the fractal dimension ofa passively convected interface. Here it is shown first, that the flame front is not effectively a passive interface in the fractal range of length scales, and second, that the value of D is determined by a simple scaling governing the turbulent flame brush. The previous inference that the flame front is a passive interface for length scales 1 ~ L G was based on the observation that the time I/UL for a laminar flame to cross a size-i eddy (a measure of the eddy burning time) is much greater than the eddy turnover time 1/(01)'/3 = (//u')(l/i)i/3 for 1 ~ L G (Compare the derivation of Eq. (5).) Thus, the impact of laminar burning on the geometrical structure of the flame front at such scales was deemed negligible.

5 444 A. R. KERSTEIN Here it is proposed, however, that the characteristic burning time ofan eddy of size I ~ Lc should be estimated based on a turbulent burning velocity u T (/ ) rather than the laminar burning velocity. Namely, the flame front in such an eddy is wrinkled by all eddies of size smaller than I, with a commensurate increase in the surface area and thus in the effective burning rate. [Consistent with the Richardson picture of relative diffusion (Monin and Yaglom, 1975), eddies larger than / are deemed to displace the size-ieddy over distances larger than I without modifying the internal structure ofthe size-i eddy.] The appropriate turbulent burning velocity is therefore obtained by replacing u' in Eq. (7) by the characteristic velocity (e/)i/j of the size-i eddy, yielding (9) The relationship between the eddy burning time and the eddy turnover time for I in the fractal range thus depends on the fractal dimension D. It is not assured that the former time will exceed the latter, so the flame front is not necessarily a passive interface in this range. In particular, comparison of the characteristic burning time l/u T (/ ) based on Eq. (9) to the eddy turnover time (//u')(l/i)'/j indicates that these times are equal for alii in the fractal range if D = 7/3. It is proposed that these times are in fact equal. This result is based on the comparison of two estimates of the burnout time of a large eddy of unburned fluid. The first estimate is obtained as follows. Consider a large (size-l) eddy ofunburned fluid immediately upstream of the turbulent flame brush. When the flame brush encounters the eddy, a length-scale breakdown process is initiated by which the eddy is broken into unburned parcels of decreasing size until the intermingling of burned and unburned fluid leads to burnout of the eddy. Irrespective of the small scale dynamics, the rate-limiting time scale governing this process is the eddy turnover time II. = Lju' of the size-l eddy. Therefore II. is the residence time of an element of unburned fluid in the turbulent flame brush. The second residence time estimate is Equating this to II.' the result D = 7/3 is obtained. DISCUSSION It is not surprising that the analysis yields a value of D at which the convection and propagation processes balance at all length scales in the fractal range, because balance principles have been applied at both the upper and lower cutoffs. In essence, the fractal picture offlame front geometry implies a form ofdynamical similarity governing all length scales within the fractal range. In fact, it may be physically more intuitive to postulate dynamical similarity at the outset and obtain the fractal property of the flame front as a consequence. To derive D = 7/3 from dynamical similarity, substitute Eqs. (3) and (4) into the assumed relation 0T/UT = IL, obtaining IL = 1/[uLA(Lc)]. Based on Eq. (6), this gives s. (!::...)2-n UL i; (10) (II)

6 SHORT COMMUNICATION 445 Dynamical similarity implies that this derivation can be generalized to 1 < L, with ut(i) defined as earlier, giving ( 12) The turnover time If ofa size-ieddy scales as 1 2 /3. Comparing powers ofion both sides of Eq. (12), D = 7/3 is obtained. This derivation does not depend on the scaling, Eq. (5), of the lower cutoff. However, to derive the dependence of UT on u' from Eq. (2), the scaling of the lower cutoff must be specified. The value D = 7/3 is indistinguishable in practice from the estimated value D = 2.35 of the fractal dimension ofa passive interface in turbulence (Hentschel and Procaccia, 1984). Furthermore, the linear dependence of U T on u' which follows from Eq. (7) with D = 7/3 is quite close to the prediction of the formulation of Peters (1988). Nevertheless, the present analysis suggests a dynamical picture quite different from that of previous analyses, in that the wrinkled flame front is identified as a geometrical entity distinct from a passively convected interface. The fractal dimensions of these two entities are not necessarily related, and in fact, the fractal property of the flame front is not crucially dependent on geometrical properties of the passive interface. As noted earlier, this inference is consistent with a recent observation of fractal flame fronts in flow fields which appear to be non fractal. Finally, it is noted that the turbulent flamespeed prediction UT - u' applies to the regime of high turbulence intensity, u' ~ UL, in which there is a wide range of length scales within the fractal regime. [See Eq. (5).] This result agrees, to leading order in In (u'/u L), with the turbulent flamespeed formula of Yak hot (1988), which gives UT = u'[ln(u'/ul)r 1/2for high turbulence intensity. In fact, the approach adopted here is the leading-order approximation to a simplified derivation of Yakhot's formula (Kerstein, 1988). The latter derivation yields a logarithmic multiplicative correction to the assumption that the eddy turnover time I L = Llu' is the relevant time scale for fuel consumption within the turbulent flame brush. ACKNOWLEDGEMENT The author would like to thank F. C. Gouldin for helpful comments. This work was supported by the Divisions ofchemical Sciences and of Engineering and Geosciences, Office of Basic Energy Sciences, U.S. Department of Energy. REFERENCES Gouldin, F. C. (1987). An application of fractals to modeling premixed turbulent names. Combust, Flame 68,249. Hentschel, H. G. E., and Procaccia, I. (1984). Relative diffusion in turbulent media: the fractal dimension of clouds. Phys. Rev. A 29, Kerstein, A. R. (1988). Simple derivation of Yakhot's turbulent premixed flamespeed formula. Combust, Sci. Technol., in press. Mandelbrot, B. B. (1982). The Fractal Geometry of Nature, Freeman, San Francisco. Mantzaras, J., Felton, P. G., and Bracco, F. V. (1988). Fractals and turbulent premixed engine names. Submitted to Combust, Flame. Monin. A. S., and Yaglom, A. M. (1975). Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 2, MIT Press. Cambridge. North, G. (1988). Personal communication. Peters, N. (1988). Laminar flamelet concepts in turbulent combustion. Twenty-First Symposium (International's on Combustion, The Combustion Institute, Pittsburgh, p Yak hot, V. (1988). Propagation velocity of premixed turbulent name. Combust, Sci. Technol., in press.