Large Eddy Simulation of Rotating Turbulence Hao Lu, Christopher J. Rutland and Leslie M. Smith Sponsored by NSF

Project Summary Research Objectives:  Direct numerical simulation (DNS) of rotating turbulence: We have small scale forced cases, large scale forced cases and decaying cases. DNS provides data for LES model development.  Developing sub-grid scale models: Model has the capability to capture small-scale turbulence properties, reverse energy transfer from small to large scales, and length scale anisotropy of rotating turbulence. ERC 1 UW - Madison Broader Impact: Broader Impact: Rotating turbulence has a wide range of application in engineering science, geo- and astrophysics. It provides a simple setup to study the characteristic properties of homogeneous but anisotropic turbulence flows. One of the most important applications is the development and design of turbo-machinery. The detailed understanding of the consequences of rotation on the flow characteristics is important for an advanced layout of these machines. The whole field of geophysics is crucially determined by our planet’s rotation, which influences both atmospheric and oceanic flows, effecting global climate as well as short-term weather forecasting. Understanding the fundamental processes forms the basis for a detailed analysis of complex phenomena such as the development of climate anomalies (El Niño), the formation of hurricanes and tidal waves, the spreading of pollutants or the oceanic circulation of nutrients. Approach:  Pseudo-spectral method, Gaussian white noise forcing scheme, and various spatial filters are used in this work. Fundamental analyses, such as the invariance of models, anisotropy of rotating turbulence, and correlation/regression studies, are employed.  A-priori test of various models.  A-posteriori test of various models.

Forced rotating turbulence ERC 2 UW - Madison Development of cyclonic two-dimensional coherent structures appearing in rotating turbulence as indicated by iso-surfaces of vorticity, contours of kinetic energy and velocity vectors: (a) initial very low energy level isotropic turbulence; (b) final state (at normalized time 3.88) of large scale forced rotating case; (c) final state (at normalized time 3.68) of small scale forced rotating case. Energy Spectrum

Description of structure models ERC 3 UW - Madison  Dynamic structure model:  Consistent dynamic structure models for rotating flow:

Scatter plot of  11 by SCDSM ERC 4 UW - Madison Comparison of contour plots of SGS stress  11 (left) and similarity type consistent dynamic structure modeled stress  11 SCDSM (right) at z=0 layer. Flow is the small scale forced case ( (c) at slide 2). Cutoff wave-number: k=11.6. Scatter plot analysis, and correlation/regression analysis. The correlation coefficient can represent the variance between the modeled and the exact terms on the scatter plot and on the PDF diagram. The regression coefficient can represent the contour level ratio between the modeled and the exact terms, the slope of the regression (scatter) line.

A-priori test ERC 5 UW - Madison [Case description] Rotating flow forced at large scales with rotating at z-direction at rate of 12. [Tested models] SSM: scale-similarity model GM: gradient model (Clark model) DSM: dynamic structure model GCDSM: gradient type consistent DSM SCDSM: similarity type consistent DSM [Tested quantities] Regression coefficients of τ 33 and ∂τ 3i /∂x i

Conclusion ERC 6 UW - Madison Models those are not consistent with MFI cannot give high correlation and regression level at rotation direction. The SGS stress tensor predicted by eddy viscosity models is uncorrelated with the stress tensor. Dynamic structure models yield very close energy flux prediction. Also, two new consistent models increase regression coefficients at all special directions when compared with other models. They improve the correlations significantly comparing with eddy viscosity models for a wide range of filter size. These results demonstrate their capabilities in capture of SGS dynamics. Grid resolution effects on LES models. A-posteriori test of LES models  Decaying isotropic and rotating turbulence.  For one-equation models, reverse energy transfers via forcing at sub-grid scales (at SGS kinetic energy equation).  Large scale forced testing. Ongoing work