New numerical results on scalar pair dispersion through an inertial range spanning many decades are presented here. These results are achieved through a new Monte Carlo algorithm for synthetic turbulent velocity fields, which has been developed and validated recently by the authors ͓J. Comput. Phys. 117, 146 ͑1995͔͒; this algorithm is capable of accurate simulation of a Gaussian incompressible random field with the Kolmogoroff spectrum over 12-15 decades of scaling behavior with low variance. The numerical results for pair dispersion reported here are within the context of random velocity fields satisfying Taylor's hypothesis for two-dimensional incompressible flow fields. For the Kolmogoroff spectrum, Richardson's t 3 scaling law is confirmed over a range of pair separation distances spanning eight decades with a Richardson constant with the value 0.031Ϯ0.004 over nearly eight decades of pair separation, provided that the longitudinal component of the velocity structure tensor is normalized to unity. Remarkably, in appropriate units this constant agrees with the one calculated by Tatarski's experiment from 1960 within the stated error bars. Other effects on pair dispersion of varying the energy spectrum of the velocity field and the degree of isotropy, as well as the importance of rare events in pair separation statistics, are also developed here within the context of synthetic turbulence satisfying Taylor's hypothesis.
Monte Carlo methods for computing various statistical aspects of turbulent diffusion with long range correlated and even fractal random velocity fields are described here. A simple explicit exactly solvable model with complex regimes of scaling behavior including trapping, subdiffusion, and superdiffusion is utilized to compare and contrast the capabilities of conventional Monte Carlo procedures such as the Fourier method and the moving average method; explicit numerical examples are presented which demonstrate the poor convergence of these conventional methods in various regimes with long range velocity correlations. A new method for computing fractal random fields involving wavelets and random plane waves developed recently by two of the authors [J. Comput. Phys. 117, 146 (1995)] is applied to compute pair dispersion over many decades for systematic families of anisotropic fractal velocity fields with the Kolmogorov spectrum. The important associated preconstant for pair dispersion in the Richardson law in these anisotropic settings is compared with the one obtained over many decades recently by two of the authors [Phys. Fluids 8, 1052 (1996)] for an isotropic fractal field with the Kolmogorov spectrum. (c) 1997 American Institute of Physics.
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