Geschwindigkeitsgradientenstatistik in voll entwickelter Turbulenz - Statistische Entwicklungsgleichungen, DNS Untersuchungen und Implikationen für niedrigdimensionale Modelle
Zusammenfassung der Projektergebnisse
The objective of this project was to contribute to our understanding of the small-scale statistics of fully developed turbulence. These scales are thought to exhibit universal statistical properties with strong deviations from Gaussianity. The deviations are closely related to coherent structures like strain sheets and filaments of concentrated vorticity and indicate that extreme jumps in the velocity field are very likely to occur in a turbulent field. Statistically, these observations can be comprehensively described in terms of the velocity gradients, i.e. the tensor constructed of taking spatial derivatives of all velocity components. The statistics of the velocity gradient tensor consequently was at the center of this project. The statistical properties of this quantity can be studied in terms of an evolution equation for its probability density function. This evolution equation takes into account the dynamics of the velocity gradient tensor in a statistical manner and features the local (closed) self-amplification and self-depletion terms as well as the nonlocal (unclosed) pressure and viscous contributions. To get an impression of the structure of the unclosed terms, especially the pressure Hessian term, these were evaluated analytically under the assumption of a Gaussian velocity field. While the assumption of Gaussianity is known to be insufficient to capture the intermittent characteristics of turbulent flows, these analytical calculations gave valuable hints at the complex tensorial structure of the unclosed terms. For example, it was found that the pressure Hessian contributions do not only attenuate the self-amplification effect of the local term, they also influence the geometrical alignment of the small-scale structures in turbulence. The analytical calculations were compared to Direct Numerical Simulation (DNS) data from the publicly available Johns Hopkins Turbulence Database. While it was found that the qualitative features of the analytical Gaussian closure compare quite well with the DNS data, the Gaussian closure underestimates the amplitudes of the nonlocal pressure contributions. This result underpinned very concretely the non-Gaussian nature of turbulent flows. Still, the analytical calculations served as a good starting point for further investigations. Based on a mean-field approach, we maintained the overall tensorial structure of Gaussian closure and adjusted its amplitudes, which led to a much improved agreement with the DNS data. This enhanced Gaussian closure also led to the proposal of a low-dimensional model for the evolution of the small scales of turbulence, which can be cast in terms of a stochastic differential equation (as compared to a partial integro-differential equation for the velocity gradient field). Such a model may serve as a toy problem to study the evolution of small-scale coherent structures and to qualitatively predict turbulence statistics. The research for this project stimulated a number of follow-up research questions like for example the development of closure strategies based on non-Gaussian random fields and the implementation of local and nonlocal dynamical effects into quantitative low-dimensional models for small-scale intermittency, which will be the topic of future research.
Projektbezogene Publikationen (Auswahl)
- A wavenumber-frequency spectral model for atmospheric boundary layers. J. Phys.: Conf. Ser., 524:012104, 2014
M. Wilczek, R.J.A.M. Stevens, Y. Narita, C. Meneveau
(Siehe online unter https://dx.doi.org/10.1088/1742-6596/524/1/012104) - Large-eddy simulation study of the logarithmic law for second and higher-order moments in turbulent wall-bounded flow. J. Fluid Mech 757:888, 2014
R.J.A.M. Stevens, M. Wilczek, C. Meneveau
- Pressure Hessian and viscous contributions to the velocity gradient statistics based on Gaussian random fields. J. Fluid Mech., 756:191, 2014
M. Wilczek, C. Meneveau
- Spatio-temporal spectra in the logarithmic layer of wall turbulence: large-eddy simulations and simple models submitted to J. Fluid Mech., 2014
M. Wilczek, R.J.A.M Stevens, C. Meneveau
