## Title data

Wilczek, Michael:

**Statistical and numerical investigations of fluid turbulence.**

Münster
,
2011

(
Doctoral thesis,
2011, Westfälische Wilhelms-Universität Münster)

DOI: https://doi.org/10.17617/2.3075289

## Abstract in another language

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.

## Further data

Item Type: | Doctoral thesis |
---|---|

Institutions of the University: | Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Physics > Chair Theoretical Physics I > Chair Theoretical Physics I - Univ.-Prof. Dr. Michael Wilczek Profile Fields > Advanced Fields > Nonlinear Dynamics |

Result of work at the UBT: | No |

DDC Subjects: | 500 Science > 530 Physics |

Date Deposited: | 23 Feb 2022 12:36 |

Last Modified: | 23 Feb 2022 12:36 |

URI: | https://eref.uni-bayreuth.de/id/eprint/67584 |