Effect of spatial dimensionality on the chaotic properties of turbulent flow

Abstract

The phenomenon of fluid turbulence is found almost universally in the world around us, however, there is still much that is not understood about the underlying physics. What is known about turbulent fluid flows is that their dynamics can be vastly different depending on the spatial dimension; these differences are particularly stark between two and three dimensions. Additionally, it is known that turbulent flows exhibit deterministic chaos, which manifests itself as an extreme sensitivity to initial conditions. This has important consequences for real world predictability, where finite measurement precision eventually leads to a total loss of predictability. This work is focussed on the effect of the spatial dimension on the chaotic properties of turbulent flows.

The computational cost of performing fully resolved simulations, known as direct numerical simulation (DNS), of turbulent flows at even modest Reynolds numbers can be enormous. This cost increases rapidly with both the Reynolds number and the spatial dimension. Compounding this issue, in order to measure chaotic properties, for example Lyapunov exponents, in such simulations requires the concurrent evolution of many velocity fields. As a result, only nowis it beginning to become feasible to performsystematic measurements of these chaotic properties of turbulent flows, albeit only in the idealised case of homogenous and isotropic turbulence (HIT). It should be noted that these fully resolved studies are entirely unfeasible beyond three spatial dimensions at present, and will remain so for the foreseeable future. As such, it is necessary to turn to models where the degrees of freedom are reduced, in our case a popular two point closure: the eddy damped quasi normalMarkovian (EDQNM) approximation. To this end, a parallel d-dimensional EDQNM code has been developed for this thesis in order to study predictability in higher spatial dimensions. Included here is some discussion of the details needed to write such a code.

This thesis presents the results of such studies in both two and three spatial dimensions, with a focus on the scaling properties of the Kolmogorov-Sinai entropy and the attractor dimension. In three dimensions simple dimensional analysis and the Kolmogorov 1941 (K41) theory predicts that this scaling will be determined entirely by the Reynolds number of the flow. In our results it is seen that this is true, but the rate of scaling is not entirely consistent with K41, nor popular intermittency models. However, in two dimensions it is found that these quantities have a dependence on not only the Reynolds number, but also on the systemsize and the length scale at which energy is injected. Thiswas not predicted by simple dimensional arguments and provides further evidence of non-universal behaviour in two dimensional HIT.

It has long been observed that features of both two and three dimensional turbulence coexist in the Earth’s atmosphere, largely as a result of the geometry of the system. This geometry is best described as a thin layer, and in previous experiments and simulations a transition between two and three dimensional phenomenology has been observed as the layer height is varied. By performing DNS of thin layer turbulence, we find the Lyapunov exponents can be used as an indicator of this transition. The predictability times either side of the transition are different, which may have consequences for atmospheric forecasting. This co-existence of two and three dimensional dynamics is also found in stratified systems, those undergoing rotation, and those under the influence of strong magnetic fields. Hence, these results may be applicable to a wider range of situations. Additionally, we also present results of non-integer dimensional turbulence using the EDQNMapproximation to allow us to disentangle the role of the cascade in the predictability transition found in the thin-layer case.

Anomalous scaling in the structure functions of HIT has drawn numerous comparisons with critical phenomena in the literature; in particular with the idea of an upper critical dimension for turbulence. Using the EDQNM model, we have performed a numerical study of HIT in higher dimensions. Here we find an enhanced forward energy cascade with increasing dimension evidenced by greater velocity derivative skewness and dimensionless dissipation rate. Despite these changes, in general the statistical picture seems to be very similar as the spatial dimension increases from three. However, between five and six dimensions the chaotic properties show a dramatic phase transition to a non-chaotic regime which we relate to the energy cascade as a function of spatial dimension.