Erc Poster 2007

Large eddy simulation of rotating turbulence Hao Lu, Christopher Rutland, Leslie Smith Force low energy data by Gaussian white-noise scheme

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, and anisotropic features of rotating turbulence.

Development of new models  Dynamic structure model [5]  Lij   ijDSM   2k sgs , where Lij are modified Leonard terms. L  mm   Consistent dynamic structure models for rotating flow  Gij  GCDSM  ij  2ksgs , where Gij are gradient terms. G  mm 

 ij    2k sgs , where ij are modeling of Lij  Cij .   mm  2 ' 3 4  Mixed modeling,  ij  u  S ij , u  Ck  k sgs   SCDSM ij

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3D Spectra @t=0.3 k

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slope= β=1.003

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variance is described by: ρ=0.999

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A-posteriori test of large scale forced rotating turbulence using MixGCDSM Time sequence

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[Case description] Rotating flow forced at large scales with rotation rate of 12 [rad/s]. [Tested quantities] Regression coefficients of τ 33 and ∂τ 3i /∂x i

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Quasi 2D flow at large scales

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Cyclonical vortices

Reverse energy transfers from small scales to large scales Computational efficient (less CPU cost) using MixGCDSM (spectral code, on Intel P4 3.4GHz Linux Cluster, 1cpu, 323) TSM : TMixGCDSM : TMixSCDSM : TDynSM = 

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Literature cited

[4] C. G. Speziale. Subgrid scale stress models for the large-eddy simulation of rotating turbulent flows. Geophys. Astrophys. Fluid Dynamics, 33:199–222,1985.

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[Traditional models] SSM: Scale Similarity Model GM: Gradient Model DSM: Dynamic Structure Model [New models] GCDSM: Gradient type Consistent DSM SCDSM: Similarity type Consistent DSM

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[3] L. M. Smith and Y. Lee. On near resonances and symmetry breaking in forced rotating flows at moderate Rossby number. J. Fluid Mech., 535:111–142, 2005.

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Comparison of contour plots of SGS stress τ11 (left) and similarity type consistent dynamic structure modeled stress τ11SCDSM (right) at z=0 layer. Flow is the small scale forced case ( (c) at slide 3). Cutoff wave-number: k=11.6.

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[2] C. Cambon, N. N. Mansour, and F. S. Godeferd. Energy transfer in rotating turbulence. J. Fluid Mech., 337:303–332, 1997.

17.20 2.306 0.3093 0.04147 0.005561 7.457E-4 1.000E-4

17.20 3 2.306 0.3093 0.04147 0.005561 7.457E-42 1.000E-4

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[1] C. Cambon, N. N. Mansour, and K. D. Squires. Anisotropic structure of homogeneous turbulence subjected to uniform rotation. Proceeding of the Summer Program, pages 397–420, 1994.

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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.

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Assume: b    a   . Ideally, expect   1,   1  ab    a  b  Regression:    a 2    a 2  ab    a  b  Correlation:   2  a    a 2  b2    b 2

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 New models able to duplicate important features of forced rotating turbulence

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A-posteriori test of intermediate scale forced rotating turbulence using MixGCDSM Time sequence

Scatter plot of τ 11 by SCDSM

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 New models more accurate

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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.

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Researchers have used SGS models to simulate rotating turbulence. For rotating turbulence, however, there still exists a need for a better understanding of SGS models [4] because anisotropic characteristics may influence LES modeling and comparative studies of model performance in rotating turbulence are insufficient.

Decay at 1283. And filter the initial data into 323 for LES

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Many features are well known. The trends toward 2D [1, 2], the cyclonic/anti-cyclonic asymmetry in favor of cyclones [3], and the influence of the background rotation on the energy transfers [2], are among the most challenging issues for rotating turbulence.

A-posteriori test of decaying flows

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Turbulence subjected to system rotation provides a simple configuration for studying the characteristic features of both anisotropic turbulent flows and model performances in anisotropic turbulent flows. It is of great importance in engineering and geophysics. The most important application in the first case is the development and the design of turbo-machinery. Here one has to take into account the detailed properties of the turbulent fluids, which pass through the device and are rotated (e.g., by the motion of the turbine blades). A detailed understanding of rotational effects on flow characteristics is essential for an advanced layout of these machines. Second, the whole field of geophysics is crucially determined by planetary rotation, which influences both atmospheric flows and oceanic flows and affects global climate as well as short-term weather forecasting. Understanding the fundamental processes in these fluid layer forms the basis for a detailed analysis of complex phenomena such as the development of climate anomalies (such as El Niño), the formation of hurricanes and tidal waves, the spreading of pollutants, and the oceanic circulation of nutrients.

Forced rotating turbulence – by DNS

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Introduction

Sponsors: NSF, SCREMS

[5] E. Pomraning and C. J. Rutland. Dynamic one-equation nonviscosity large eddy simulation model. AIAA Journal, 40(4):689–701, April 2002.

Acknowledgement This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 0500056 and by the NSF Scientific Computing Research Environments in the Mathematical Sciences (SCREMS) under Grant No. DMS-0532085. We would like to thank Dr. Yun-Liang Wang for helpful discussions and the Engine Research Center at the University of Wisconsin - Madison for providing computing resources.

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For further information

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Please contact [email protected]. More information on this and related projects can be obtained at http://homepages.cae.wisc/~luh

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University of Wisconsin –– Engine Research Center