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 Navier-Stokes equation is solved by a 643 pseudospectral code for constant-density homogeneous isotropic turbulence, which is made statistically stationary by low-wavenumber forcing. The Taylor-scale 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 mean-square 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 area-weighted expectation of R is found to be 12, where is the Kolmogorov length scale. For moderate and small radii (less than 10) the probability density function (pdf) of R is approximately uniform, there being about 5% probability of R being less than . 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.
