Effect of spatial dimensionality on the chaotic properties of turbulent flow
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Date
10/06/2022Author
Clark, Daniel
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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.