Curvature statistics of turbulence

Abstract

Turbulence is present in many different flow types across natural phenomena and industrial applications but still raises open questions due to its complexity and multi-scale nature which defers most direct analytical approaches. However, a statistical approach has proven itself as a powerful tool to help with the understanding and modelling of turbulence, overcoming some limitations of analytical approaches. Within this thesis, I analyse datasets of four different flow types, three of experimental nature and one of numerical nature to study the statistics of geometric quantities such as curvature and torsion, in particular concerning the effect of anisotropy. For the experimental datasets, the focus is on Lagrangian trajectories and for the numerical datasets on stream lines and, in case of a conducting fluid, magnetic field lines. These curves can be described as three-dimensional space curves and therefore curvature and torsion can give a full description of these. Curvature and torsion are particularly interesting measures of turbulence as a salient feature of turbulence is the presence of different spatial and temporal scales, which is captured by the different derivatives involved. The curvature vector is introduced as a new measure of the effect of anisotropies, originating from different mechanisms like a temperature gradient, mean flow or magnetic background field, on the geometry of Lagrangian trajectories and either stream lines or, in case of a conducting fluid, magnetic field lines. The first study includes Lagrangian experimental datasets of a turbulent von Kármán flow, Rayleigh-Bénard convection at two different Rayleigh numbers and the logarithmic layer of a turbulent zero-pressure-gradient boundary layer over a flat plate. The curvature and torsion statistics of von Kármán flow and Rayleigh-Bénard convection agree well with previously reported experimental and numerical results for the curvature and with numerical simulations of homogeneous and isotropic turbulence for the torsion. For the logarithmic region of the boundary layer, the form of the torsion probability density function (PDF) also agrees with these results. The curvature PDF retains the same heavy tail as curvature PDFs measured for the aforementioned flow types for small curvature events, however, high curvature events are suppressed by the mean flow. Studying the curvature vector and connecting it with velocity fluctuations shows the effect of anisotropy and allows a geometric interpretation of large-scale motion in terms of the structure of trajectories. In the second study, another type of anisotropy is introduced by a magnetic background field for a flow with electrically conducting fluid in magnetohydrodynamic turbulence. Utilising high resolution data obtained by direct numerical simulation of magnetohydrodynamic turbulence across a range of Reynolds numbers, the curvature PDFs of stream lines and magnetic field lines compare well with the PDFs derived based on Gaussian statistics for the Lagrangian trajectories. For the PDFs of the curvature vector components, the magnetic field lines tend to be less curved in the direction of the magnetic background field, which can be connected to the suppression of magnetic field fluctuations parallel to the magnetic background field and the partial two-dimensionalisation of the flow. The curvature vector PDFs of stream lines show that the effect of a magnetic background field on the geometry of stream lines is less profound compared to the effect on the magnetic field lines. The models mentioned above are based on Gaussian statistics. However, turbulence is more complex and intermittency plays a crucial role. As an improvement of these models, a model for the curvature PDF for von Kármán flow and Rayleigh-Bénard convection is derived, which takes spatio-temporal intermittency into account. Using a decomposition into Gaussian sub-ensembles, where within each the curvature PDF is known exactly, an exact model expression and a closed-form approximation for the curvature PDF can be derived. These PDFs agree qualitatively and quantitatively with the measured curvature PDFs of turbulent von Kármán flow and Rayleigh-Bénard convection