The curvature of material surfaces in isotropic turbulence
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Direct numerical simulation is used to study the curvature of material surfaces in isotropic turbulence. The NavierStokes equation is solved by a 643 pseudospectral code for constantdensity homogeneous isotropic turbulence, which is made statistically stationary by lowwavenumber forcing. The Taylorscale Reynolds number is 39. An ensemble of 8192 infinitesimal material surface elements is tracked through the turbulence. For each element, a set of exact ordinary differential equations is integrated in time to determine, primarily, the two principal curvatures k1 and k 2. Statistics are then deduced of the meansquare curvature M = 1/2(k12 + k 22), and of the mean radius of curvature R = (k12 + k22) 1/2. Curvature statistics attain an essentially stationary state after about 15 Kolmogorov time scales. Then the areaweighted expectation of R is found to be 12η, where η is the Kolmogorov length scale. For moderate and small radii (less than 10η) the probability density function (pdf) of R is approximately uniform, there being about 5% probability of R being less than η. The uniformity of the pdf of R, for small R, implies that the expectation of M is infinite. It is found that the surface elements with large curvatures are nearly cylindrical in shape (i.e., k1 ≫ k 2 or k2 ≫ k1), consistent with the folding of the surface along nearly straight lines. Nevertheless the variance of the Gauss curvature K = k1k2 is infinite. © 1989 American Institute of Physics.
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Pope, S. B., Yeung, P. K., & Girimaji, S. S.
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