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Abstract:
The statistical description of fully developed turbulence up to today remains a central
open issue of classical physics. Apart from the fact that turbulence plays a key role
in many natural and engineering environments, the solution of the problem is also
of interest on a conceptual level. Hydrodynamical turbulence may be regarded as a
paradigmatic example for a strongly interacting system with a high number of degrees
of freedom out of equilibrium, for which a comprehensive statistical mechanics is yet to
be formulated.
The statistical formulation of turbulent flows can either be approached from a
phenomenological side or by deriving statistical relations right from the basic equations
of motion. While phenomenological theories often lead to a good description of a variety
of statistical quantities, the amount of physical insights to be possibly gained depends
heavily on the validity of the assumptions made. On the contrary, statistical theories
based on first principles have to face the famous closure problem of turbulence, which
prevents a straightforward solution of the statistical problem.
The present thesis aims at the investigation of a statistical theory of turbulence in
terms of probability density functions (PDFs) based on first principles. To this end
we make use of the statistical framework of the Lundgren-Monin-Novikov hierarchy,
which allows to derive evolution equations for probability density functions right from
the equations of fluid motion. The arising unclosed terms are estimated from highly
resolved direct numerical simulations of fully developed turbulence, which allows to
make a connection between basic dynamical features of turbulence and the observed
statistics.
As a technical prerequisite, a parallel pseudospectral code for the direct numerical
simulation (DNS) of fully developed turbulence has been developed and tested within
this thesis. A number of standard statistical evaluations are presented with the purpose
both to benchmark the numerical results as well as to characterize the statistical features
of turbulence.
Studying the PDF equations, a comprehensive treatment of the single-point velocity
and vorticity statistics is achieved within the current work. By making use of statistical
symmetries present in the case of homogeneous isotropic turbulence, exact expressions
for, e.g., the stationary PDF are derived in terms of correlations between the turbulent
field and various quantities determining the dynamics of the field. The joint numerical
and analytical investigations eventually lead to an explanation of the slightly subGaussian tails for the velocity statistics and the highly non-Gaussian vorticity statistics
with pronounced tails. To contribute to the characterization of the multi-point statistics
of turbulence, the two-point enstrophy statistics is investigated. The results quantify the interaction of different spatial scales and give insights into the spatial structure
of the vorticity field. Along the lines of preceding works in this context the local
conditional structure of the vorticity field and its relation to the multi-point statistics of
the vorticity field is discussed and applied to the two-point enstrophy statistics. Finally,
the closure problem of turbulence is treated on a more conceptual level by pursuing the
question how to establish a model for the two-point PDF which is consistent with the
single-point evolution equation and a number of statistical constraints to be imposed on
probability density functions. A simple analytical model for the joint PDF is developed
and improvements in the context of maximum entropy methods are discussed. Both
models are compared to results from DNS.
Altogether, the results of the current thesis help to establish a connection between
the flow topology, dynamical quantities that determine the temporal evolution of the
turbulent fields and the resulting statistics. Beyond the characterization and explanation
of these statistical quantities this provides new insights for future modeling and closure
strategies.
