Exact renormalization group analysis of turbulent transport by the shear flow.

*(English)*Zbl 1292.82028Summary: The exact renormalization group (RG) method initiated by Wilson and further developed by Polchinski is used to study the shear flow model proposed by Avellaneda and Majda as a simplified model for the diffusive transport of a passive scalar by a turbulent velocity field. It is shown that this exact RG method is capable of recovering all the scaling regimes as the spectral parameters of velocity statistics vary, found by Avellaneda and Majda in their rigorous study of this model. This gives further confidence that the RG method, if implemented in the right way instead of using drastic truncations as in the Yakhot-Orszag’s approximate RG scheme, does give the correct prediction for the large scale behaviors of solutions of stochastic partial differential equations (PDE). We also derive the analog of the “large eddy simulation” models when a finite amount of small scales are eliminated from the problem.

##### MSC:

82C28 | Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics |

82C70 | Transport processes in time-dependent statistical mechanics |

##### References:

[1] | Avellaneda, M.; Majda, A.J., Mathematical models with exact renormalization for turbulent transport, Commun. Math. Phys., 131, 381-429, (1990) · Zbl 0703.76042 |

[2] | Avellaneda, M.; Majda, A.J., Approximate and exact renormalization theories for a model for turbulent transport, Phys. Fluids A, 4, 41-57, (1992) · Zbl 0850.76263 |

[3] | Eyink, G.L., The renormalization group method in statistical hydrodynamics, Phys. Fluids, 6, 3063, (1994) · Zbl 0830.76042 |

[4] | Forster, D.; Nelson, D.R.; Stephen, M.J., Large-distance and long-time properties of a randomly stirred fluid, Phys. Rev. A (3), 16, 732-749, (1977) |

[5] | Orszag, S.A.; Yakhot, V., Analysis of the \(ϵ\)-expansion in turbulence theory: approximate renormalization group for diffusion of a passive scalar in a random velocity field, J. Sci. Comput., 14, 147-195, (1999) · Zbl 0982.76051 |

[6] | Polchinski, J., Renormalization and effective Lagrangians, Nucl. Phys. B, 231, 269-295, (1984) |

[7] | Sinai, Y.G.; Yakhot, V., Limiting probability distributions of a passive scalar in a random velocity field, Phys. Rev. Lett., 63, 1962-1964, (1989) |

[8] | Smith, L.M.; Reynolds, W.C., On the yakhot-orszag renormalization group method for deriving turbulence statistics and models, Phys. Fluids A, Fluid Dyn., 4, 364, (1992) |

[9] | Wilson, K.G.; Kogut, J., The renormalization group and the \(ϵ\) expansion, Phys. Rep., 12, 75-199, (1974) |

[10] | Yakhot, V.; Orszag, S.A., Renormalization group analysis of turbulence. I. basic theory, J. Sci. Comput., 1, 3-51, (1986) · Zbl 0648.76040 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.