Large Eddy Simulation of Rotating Turbulence

Hao Lu, Christopher J. Rutland and Leslie M. Smith

Sponsored by NSF

Project SummaryProject Summary

Research Objectives: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: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 turbulenceForced 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 S

pectrumEne

rgy

Spect

rum

Description of structure modelsDescription of structure models

ERC 3 UW - Madison

Dynamic structure model:

Consistent dynamic structure models for rotating flow:

2 , where are modified Leonard terms.ijDSMij sgs ij

mm

Lk L

L

2 , where are gradient terms.

2 , where are modeling of .

ijGCDSMij sgs ij

mm

ijSCDSMij sgs ij ij ij

mm

Gk G

G

k L C

Scatter plot of Scatter plot of 1111 by SCDSM by SCDSM

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

x

y

1.000E-4

3.339E-4

0.001115

0.003721

0.01242

0.04147

0.1385

0.4622

1.543

5.152

17.20

DNS11

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

x

y

1.000E-4

3.339E-4

0.001115

0.003721

0.01242

0.04147

0.1385

0.4622

1.543

5.152

17.20

SCDSM110 2 4 6 8 10 12 14 16

0

2

4

6

8

10

12

14

16

SCDSM

11

DNS11

variance is described by:

slope=

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.

2 2

2 2 2 2

Assume: . Ideally, want 1, 1

Regression:

Correlation:

b a

ab a b

a aab a b

a a b b

A-prioriA-priori test test

ERC 5 UW - Madison

0 20 40 60 80 100 120 140

0.2

0.4

0.6

0.8

1.0

DSM GCDSM SCDSM SSM GM

33

k=/

0 20 40 60 80 100 120 1400.2

0.4

0.6

0.8

1.0

DSM GCDSM SCDSM SSM GM

3

k=/

[Case description]Rotating flow forced at large scales withrotating at z-direction at rate of 12.[Tested models]SSM: scale-similarity modelGM: gradient model (Clark model)

DSM: dynamic structure modelGCDSM: gradient type consistent DSMSCDSM: similarity type consistent DSM[Tested quantities]Regression coefficients of τ33 and ∂τ3i /∂xi

ConclusionConclusion

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 workOngoing work