Abstract
The validity of the ∈-expansion in the turbulence problem is discussed using the example of diffusion of a passive scalar in a random velocity field. A generalization of Wilson's rule for calculating a diagram of arbitrary order is introduced. The resulting perturbation series, while having zero radius of convergence, is summed exactly yielding amplitudes which differ from those obtained to lowest order in the ∈-expansion by only a few percent. The properties of the expansion are analyzed in detail and it is shown that various subsets of diagrams, while differing by an infinite number of terms, give close results in the vicinity of the fixed point. This indicates nontrivial compensation of high-order interactions in turbulence. The irrelevance of high-order couplings is demonstrated for arbitrary values of ∈. The quality the approximation is tested by comparison with numerical experiments on diffusion of a passive scalar in a band-limited random velocity field in the limit of infinite Peclet number.
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Orszag, S.A., Yakhot, V. Analysis of the ∈-Expansion in Turbulence Theory: Approximate Renormalization Group for Diffusion of a Passive Scalar in a Random Velocity Field. Journal of Scientific Computing 14, 147–178 (1999). https://doi.org/10.1023/A:1023228619798
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DOI: https://doi.org/10.1023/A:1023228619798