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Stretching and Bending of Material Surfaces in Turbulence

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Technical Report

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Cornell University Ithaca United States

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In the study of mixing and reaction in turbulent flows, there are several phenomena that can be usefully described in terms of surfaces. Examples are turbulent flames and the turbulent mixing of different liquids. The most fundamental type of surface is the material surface which, by definition, moves with the fluid. Because of the fluids turbulent motion and deformation, the surface is continually stretched and bent. In this study numerical simulations have been performed to understand and to quantify these processes.A pseudo-spectral method is used to solve the Navier-Stokes equations which governthe motion of the fluid. These equations are solved on a 1283 grid for the simplest possible turbulent flow - statistically stationary, homogeneous, isotropic turbulence. As the results show, the direct numerical representation of a material surface is not feasible forthe surface area grows exponentially by a factor ofl017 over the duration of the simulationsand radii of curvature less than a millionth of the grid spacing arise. Instead anindirect method is used in which ensembles 4-8,000 of infmitesimal surface elements are followed. Statistics of interest are obtained from the stretching and curvatures of these elements.For the first time, the mean rate of stretching has been determined. It is found that the surface area doubles every 2 12 Kolmogorov time scales. The Kolmogorov time scaleis the smallest physical time scale in turbulence. While this is certainly rapid growth, it is only 40 of theoretical estimates, for reasons that are explained. Hitherto, little has been known about the curvature of material surfaces. The results show that extremely small radii of curvatures arise, as small as I0-8 of a Kolmogorov length scale the smallest turbulent scale. These highly curved elements are found to be almost perfectly cylindrical in shape. Many other more refined statistics have been obtained.

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