Title:

Study of isotropic turbulence

The prime difficulty in finding a solution to the problem of high Reynolds' number turbulence is the large number of strongly interacting degrees of freedom. The most natural method of tackling the problem is to perform a Renormalisation Group transformation which removes from consideration the small scale degrees of freedom. The historical development of this approach is charted and the importance of the universal nature of the inertial range behaviour is noted. The problem tackled initially is that of homogeneous, isotropic, stationary turbulence. The velocity field is split into explicit (large) and implicit (small) scale modes and then the implicit modes are split into two parts, which correspond to the correlated and uncorrelated behaviour of the implicit modes with regard to the explicit modes. A new averaging procedure is introduced where the realisations of the ensemble are those where the explicit modes are held constant while the implicit ones explore the possible solutions of the NavierStokes equation (NSE). The equation for the modes averaged away yields a term which acts as an increment to the viscosity in the equation for the explicit modes. The form of this term is arrived at by considering the uncorrelated part of the implicit field as the leading contribution. It is argued that the time scales of the explicit scales will be much greater than those of the implicit scales facilitating a Markovian approximation. Further, the sum of the moment hierarchy in the uncorrelated part can be shown to be approximated by its first term within a plausible model. Hence, a form for the increment to viscosity is obtained. The renormalisation group iteration yields a fixed point in the effective viscosity. It is argued that this fixed point will have captured the essential physics of the inertial range. In order to make calculations, an ansatz is made for the uncorrelated part of the implicit field. By taking a truncated Taylor series in the implicit scales about the dissipation cutoff, itis found that the ansatz approximates the uncorrelated field. The fixed point for the effective viscosity is calculated numerically. From this, a value for the Kolmogorov constant of 1.60 is calculated. This constant shows the expected behaviour for variation of the width of the band of wavenumbers averaged out at each iteration. There is a region of parameter space where the constant shows no dependence on the output parameters. This region is taken to produce valid results, since a real quantity cannot depend on how the average is performed.
