Analogy and incongruity between classical and quantum turbulence in isotropic and anisotropic turbulent systems : a numerical study

This thesis is a study of quantum turbulence in superfluid helium. Superfluid he- lium consists of two interpenetrating fluids, a viscous normal fluid and an inviscid superfluid, coupled by a mutual friction. The thesis is divided in two parts. The first part deals with fully developed turbulence. In analogy to classical turbulence theory, I develop a two-fluid shell model to study superfluid turbulence. I investigate the energy spectra and the balance of fluxes between the two fluids as a function of temperature in continuously forced turbulence and show how both fluids gener- ate the classical k−5/3 Kolmogorov scaling law within the inertial subrange, whilst simultaneously exhibiting deviations from this law outside the subrange due to the mutual friction force. I furthermore investigate the decay of turbulence in the ab- sence of forcing. I compare my results with experiments and existing calculations. I find that, at sufficiently low temperatures a build-up of energy develops at high wavenumbers suggesting the need for a further dissipative effect, such as the Kelvin wave cascade and phonon emission. The second part of this thesis is concerned with complex vortex flows. It is well known that two coaxial vortex rings can leapfrog about each other. By direct numerical simulation, I show that in superfluid helium the effect can be generalised to a large number of vortex rings, which form a toroidal bundle. The bundle is shown to be robust, travelling a significant distance compared to its diameter, whilst at the same time becoming linked and turbulent. I also discuss the effects of friction at non-zero temperatures, and show how in this case the presence of normal fluid rotation is necessary for the stability of the bundle. Although I am unable to model numerically the number of rings realised in experiments, I compare my results with those of experiments both qualitatively and by extending the equations for a single quantised vortex ring.