The theory of . . . lines and surfaces, for example, has long been recognized as an important branch of geometry, but in treatises on motion it was regarded as lying as much outside of the subject as the four rules of arithmetic or the binomial theorem . . . . geometry itself is part of the science of motion, and [thatl it treats, not of the relations between figures already existing in space, but of the process by which these figures are generated by the motion . . . . This method of regarding geometrical figures seems to imply that the idea of motion underlies the idea of form . . . .

Fractals and multifractals in fluid turbulence

Turbulent flame speed and thickness: phenomenology, evaluation, and application in multi-dimensional simulations

One basic effect of turbulence in turbulent premixed combustion is for the fluctuating velocity field to wrinkle the flame and greatly increase its surface area. In the flamelet theory, this effect is described by the flame surface density. An exact evolution equation for the flame surface density, called the Σ-equation, may be written, where basic physical mechanisms like production by hydrodynamic straining and destruction by propagation effects are described explicitly. Direct numerical simulation (DNS) is used in this paper to estimate the different terms appearing in the Σ-equation. The numerical configuration corresponds to three-dimensional premixed flames in isotropic turbulent flow. The simulations are performed for various mixture Lewis numbers in order to modify the strength and nature of the flame-flow coupling. The DNS-based analysis provides much information relevant to flamelet models. In particular, the flame surface density, and the source and sink terms for the flame surface density, are resolved spatially across the turbulent flame brush. The geometry as well as the dynamics of the flame differ quite significantly from one end of the reaction zone to the other. For instance, contrary to the intuitive idea that flame propagation effects merely counteract the wrinkling due to the turbulence, the role of flame propagation is not constant across the turbulent brush and switches from flame surface production at the front to flame surface dissipation at the back. Direct comparisons with flamelet models are also performed. The Bray-Moss-Libby assumption that the flame surface density is proportional to the flamelet crossing frequency, a quantity that can be measured in experiments, is found to be valid. Major uncertainties remain, however, over an appropriate description of the flamelet crossing frequency. In comparison, the coherent flame model of Marble & Broadwell achieves closure at the level of the Σ-equation and provides a more promising physically based description of the flame surface dynamics. Some areas where the model needs improvement are identified.

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The evolution equation for the flame surface density in turbulent premixed combustion

Molecular transport effects on turbulent flame propagation and structure

The growth of rough surfaces and interfaces

Two-dimensional flame structure measurements of a freely propagating premixed turbulent flame were examined for fractal character over a range of turbulent Reynolds numbers from 52 to 1431 and Damkohler numbers from 10 to 889. The fractal dimension was found to increase with u'/SL, from 2.13 at u'/SL of 0.25 to 2.32 at u'/SL of 11.9. An inner cutoff was not identified due to limited spatial resolution, however, it was shown that the inner cutoff was smaller than the Gibson scale for the conditions studied. The outer cutoff was found to occur at scales comparable to the integral scale, however, the cutoff was a gradual, rather than sharp, transition from fractal behavior. A heuristic relationship between the turbulent flame surface fractal dimension and u'/SL was developed which agrees well with the fractal dimension measurements made in this study, as well as those made in a number of different flame configurations.

The Fractal Nature of Premixed Turbulent Flames

The curvature of turbulent premixed flame fronts is an important spatial property that needs to be quantified over a range of turbulence conditions. In this study, measurements of flamelet curvature along with orientation statistics for u'/ SL = 1.42-5.71 are obtained from OH planar laser-induced fluorescence images of the flame boundary by applying a curve-tracing difference formula with interval length of the order of the inner cutoff scale. Use of this particular interval length is essential for accurate tracing of the flame boundary with implicit filtering of extraneous noise that can introduce significant errors in the curvature measurements. The distributions of flamelet curvature are found to be symmetric with respect to the zero mean, while the variance increases with increasing u'/SL. These distributions can be approximated by Gaussian distribution functions. The positive and negative mean curvatures show a nearly square-root dependence on u'/SL whereas the mean flamelet radius of curvat...

G. L. North

Papers

The Fractal Nature of Premixed Turbulent Flames

Curvature and Orientation Statistics of Turbulent Premixed Flame Fronts