Numerical Simulations of Inhomogeneous Quantum Turbulence

Abstract

The modelling of turbulent fluids has been an active area of research for over a century, with applications in diverse areas such as aeronautics and meteorology. The discovery and realisation of quantum fluids, such as low temperature helium and Bose-Einstein condensates (BECs), o ers an experimental and theoretical route toward insights into the dynamics of classical fluids in an ostensibly simpler, due their inviscid properties, context. However, to utilise this we need to appreciate where they are comparable and where they di er signi cantly, in terms of their material properties, their flows, and any phenomena arising from the underlying quantum mechanics. The work in this thesis aims to further this understanding by considering the problem of vorticity transport, and extending previous work concerning dissipation in quantum fluids. Turbulence in classical fluids consists of vortices of many length scales, from the size of the system down to the Kolmogorov length scale at which viscosity acts to dissipate energy; the direct numerical simulation of classical flows with even moderate Reynolds numbers is extremely computationally intensive due to the need to resolve at many scales. In contrast, vortices in quantum fluids have quantised circulation, and only those quantum vortices with the lowest circulation are stable. Two distinct regimes of turbulence have been identi ed in quantum fluids, the quasiclassical regime in which organised bundles of vortex laments are believed to emulate the range of scales and energy distribution observed in classical turbulence, and the ultraquantum regime, composed of an essentially random tangle of vortices with no large-scale structure. Despite these fundamental di erences many parallels have been observed between classical turbulence and quantum turbulence, including various hydrodynamic instabilities, the development of K arm an vortex streets in the wake of barriers in a flow, and the Kolmogorov velocity statistics and energy spectrum. The numerical investigations in this thesis can be split into those pertaining to homogeneous turbulence, and those pertaining to inhomogeneous turbulence. Many experiments probe the properties of quantum turbulence in super fluid helium through channel ow experiments, with versatile theory due to Vinen (Proc. Royal Soc. Lond. A 240, 1220:114- 127 (1957)) describing the evolution of the statistical vortex line density in terms of opposing vortex generation and dissipation. We simulate homogeneous turbulence generated by thermal counter flow using the vortex lament method (VFM) in order to quantify the balance of the generation and dissipation of vorticity with an established technique. We then use a new numerical technique to probe the dissipation of homogeneous ultraquantum turbulence, by arti cally injecting random vortices to reach statistically steady states. Other experiments generate quantum turbulence locally, producing systems that are initially inhomogeneous. The behaviour of inhomogeneous quantum turbulence is less well understood than in the homogeneous case, and we address how vorticity spreads in such systems. We rst consider the problem in systems of reduced dimensionality, relevant to BECs in which strong con nement in one direction results in quasi-2-D condensates. In these systems vortices are essentially topological point defects, since excitations along vortex lines are suppressed in the tightly con ned direction. We model the evolution of an initially con ned region of point vortices in such a geometry using the point vortex model and the Gross-Pitaevskii equation, and identify a value for the e ective viscosity as an emergent property of the spreading of the vortices. A related investigation is performed for turbulence in super fluid 4He at zero temperature in a fully three-dimensional geometry. The dynamics of quantum vortices are modelled with the VFM, and a value for the e ective viscosity is found. We compare our method to a previous study, and review the values found for the e ective viscosity at zero temperature and nite temperatures below the lamdba-point.