An Assessment of Renormalization Methods in the Statistical Theory of Isotropic Turbulence
For the latter half of the last century renormalization methods have played an important part in tackling problems in fundamental physics and in providing a deeper understanding of systems with many interacting scales or degrees of freedom with strong coupling. The study of turbulence is no exception, and this thesis presents an investigation of renormalization techniques available in the study of the statistical theory of homogeneous and isotropic turbulence. The thesis consists of two parts which assess the two main renormalization approaches available in modeling turbulence. In particular we will be focusing on the renormalization procedures developed by McComb and others. The first part of this thesis will discuss Renormalization Group (RG) approaches to turbulence, with a focus on applications to reduce the degrees of freedom in a large-eddy simulation. The RG methods as applied to classical dynamical systems will be reviewed in the context of the Navier-Stokes equations describing fluid flow. This will be followed by introducing a functional based formalism of a conditional average first introduced by McComb, Roberts and Watt [Phys. Rev A 45, 3507 (1992)] as a tool for averaging out degrees of freedom needed in an RG calculation. This conditional average is then used in a formal RG calculation applied to the Navier-Stokes equations, originally done by McComb and Watt [Phys. Rev. A 46, 4797 (1992)], and later revised by Mc- Comb and Johnston [Physica A 292, 346 (2001)]. A correction to the summing of the time-integral detailed in the latter work is shown to introduce an extra viscous life-time term to the denominator of the increment to the renormalized viscosity and is shown to have a negligible effect in the numerical calculations. We follow this study by outlining some problems with the previous approach. In particular it is shown that a cross-term representing the interaction between high and low wavenumber modes which was neglected in the previous studies on the grounds that it does not contribute to energy dissipation, does in fact contribute significantly. A heuristic method is then put forward to include the effects of this term in the RG calculation. This leads to results which agree qualitatively with numerical calculations of eddy-viscosities. We finish this part of the thesis with an application of the RG method to the modeling of a passive scalar advected by a turbulent velocity field. The second part of this thesis will begin by reviewing Eulerian renormalized perturbation theory attempts in closing the infinite moment hierarchy introduced by averaging the Navier-Stokes equations. This is followed by presenting a new formulation of the local energy transfer theory (LET) of McComb et. al. [J. Fluid Mech. 245, 279 (1992)] which resolves some problems of previous derivations. In particular we show by the introduction of time-ordering that some previous problems with the exponential representation of the correlator can be overcome. Furthermore, we show that the singularity in the LET propagator equation cancels by way of a counter-term. We end this study by introducing a single-time Markovian closure based on LET which, unlike other Markovian closures, does not rely on any arbitrary parameters being introduced in the theory.