Geometric Alignments of SGS closures for Large Eddy Simulations in the Atmospheric Boundary Layer

Chad Higgins
PhD Thesis, The Johns Hopkins University
January 2007
Baltimore MD

ABSTRACT: Field experimental data of velocity and temperature measurements from sonic anemometer arrays in the atmospheric surface layer are analyzed using tools from statistical geometry. In the context of large eddy simulations (LES), these arrays permit the spatial filtering needed to separate large from small scales. Time series of various quantities relevant to LES are evaluated, and based on these data, we show: 1) the most likely orientation of the subgrid-scale (SGS) heat flux lies within a plane that is spanned by the temperature gradient vector and its product with the velocity gradient tensor, i.e. within the geometric span of the so called mixed tensor eddy diffusivity model. 2) The preferred filtered fluid deformation is axisymmetric extension and the
preferred subgrid stress state is axisymmetric contraction. 3)The filtered fluctuating vorticity is preferentially aligned with the mean vorticity, the streamwise direction, and the intermediate strain-rate eigenvector. 4) The alignment between eigenvectors of the SGS stress and filtered strain rate are in qualitative agreement with prior laboratory measurements at much lower Reynolds numbers. A bimodal distribution is observed, which can be reduced to co-alignment using the mixed model. 5) A geometric interpretation of energy dissipation that explicitly reveals the dependence of SGS dissipation on alignment angles between the eigendirections of the SGS stress and the filtered strain rate tensors, and use it to illuminate the recent field and laboratory observations.

The experimental setup and data collected facilitated studies of the SGS heat flux, the filtered strain rate, SGS stress, and dissipation. However, measurements were only taken and filtered in two-dimensional planes and array geometries did not permit analysis of the divergence of the SGS stress (the so-called SGS force vector), which is the term that enters directly in the LES momentum equations. The question of whether 2-D filtering will yield accurate results in subgrid-scale (SGS) models is addressed by analyzing data from a new field experiment. Comparisons are given between two dimensional and three dimensional filtering of the field measurements based on probability density functions (PDFs) and energy spectra of the SGS stress elements. Geometric alignment analyses are redone for the SGS heat flux SGS stress and filtered strain rate for the cases of stable, near neutral, and unstable atmospheric stability. Under unstable and near neutral atmospheric stability two dimensional filtering yields acceptable results; however under stable atmospheric stability, we recommend and delineate a new approach. Finally, the vector alignment of the SGS force with the divergence of the Smagorinsky and nonlinear models is investigated under near neutral, unstable, and stable atmospheric stabilities. It is shown that the nonlinear formulation has a high probability of alignment with the SGS force across all stabilities, while the alignment of the Smagorinsky model depends on atmospheric stability.

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