Title:

A study of renormalizationgroup formulations for turbulence

Homogeneous, isotropic turbulence in fluids is a highly complicated phenomenon, involving the interaction of many degrees of freedom. A brief statement of the problem is made, with an outline of historical solutions. The renormalization group is introduced and the technical problems of its application to turbulence detailed. A realspace formulation is presented of the theory of iterative averaging (W.D. McComb and A.G. Watt, Phys. Rev. A 46, 4797 (1992)). The turbulent velocity field is iteratively filtered using a sharp filter. At each stage, the subgrid field is averaged and its mean effect incorporated into an eddyviscosity term, in which the viscosity is convolved with the velocity field. A twofield decomposition of the subgrid field is used to perform the averaging. After a brief summary of the kspace derivation of the iterativeaveraging method, results of numerically calculating the theory are present. A value for the Kolmogorov spectral constant of α = 1.6 is found. This value is independent of input parameters over a significant range. Two analytical results are derived, which provide useful checks on the computation. A new renormalizationgroup formulation, loosely based on the prescription of Forster, Nelson and Stephen (Phys. Rev. A 16, 732 (1977)) for stirred hydrodynamics, is presented. A fictional force is introduced to model the stirring of the small scales by the large eddies. The definition of the force is based on the twofield decomposition of the subgrid velocity field. Higherorder nonlinearities in the velocity field are produced by the elimination procedure, and are treated as part of a modified equation of motion. When the theory is numerically iterated, a value for the Kolmogorov constant of α = 1.71 is found, but only after a large error term, due to the absence of a spectral gap between subgrid and explicitscale modes, is neglected. A comparison is made of this theory with iterative averaging, with particular reference to the role of the higherorder nonlinearities, and the use of conditional averaging as opposed to filtered ensemble averaging. The behaviour of the two theories in the continuum limit of the renormalization group process is analysed. Both theories are found to break down in some way in this region. The behaviour of other formulations in the limit is also briefly examined, and it is argued that the nonexistence of the continuum limit is a property of the NavierStokes equations. Some of the above ideas are extended to the case of passive scalar convection. Results of calculating iterative averaging for this case are shown, with a value for the ObukhovCorrsin spectral constant of β = 1.0. The perturbative formulation is adapted for the passive scalar equation. The results for the ObukhovCorrsin constant do not display any independence of input parameters in this case. It is found that 0.69 ≤ β ≤ 0.76.
