Large Eddy Simulation of
Rotating Turbulence
Hao Lu, Christopher J. Rutland and Leslie M. Smith
Sponsored by NSF
Project Summary
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.
Research Objectives:


Approach:
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.

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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.
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Forced rotating turbulence
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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Description of structure models
 Dynamic structure model:

DSM
ij
 Lij 

 2ksgs , where Lij are modified Leonard terms.
 Lmm 
 Consistent dynamic structure models for rotating
flow:

GCDSM
ij
 ijSCDSM
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 Gij 

 2k sgs , where Gij are gradient terms.
 Gmm 
 ij 

 2ksgs , where ij are modeling of Lij  Cij .
  mm 
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Scatter plot of 11 by SCDSM
16
14
slope=
12
8
6
4
variance is
described by: 
2
0
1.000E-4
DNS
2
4
6
8
DNS
11
10
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.
12
14
11
16
SCDSM
11
3.339E-4
3
0.001115
0.001115
0.003721
0.003721
0.01242
0.01242
0.04147
2
0.04147
2
0.1385
0.1385
0.4622
0.4622
1.543
1
1.543
1
5.152
5.152
17.20
17.20
0
0
-1
-1
-2
-2
-3
-3
-3
-2
-1
0
x
1
2
3
-3
-2
-1
0
x
1
2
3
Comparison of contour plots of SGS stress 11 (left) and similarity type consistent dynamic structure modeled stress
11SCDSM (right) at z=0 layer. Flow is the small scale forced case ( (c) at slide 2). Cutoff wave-number: k=11.6.
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1.000E-4
3.339E-4
3
y
0
y
11
SCDSM
10
Assume: b    a   . Ideally, want   1,   1
 ab    a  b 
Regression:  
 a 2    a 2
 ab    a  b 
Correlation:  
 a 2    a 2  b2    b 2
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A-priori test
[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)
1.0
33
0.8
0.6
DSM
GCDSM
SCDSM
SSM
GM
0.4
1.0
0.2
20
40
60
k=/
80
100
120
140
0.8
3
0
DSM: dynamic structure model
GCDSM: gradient type consistent DSM 0.6
SCDSM: similarity type consistent DSM
0.4
[Tested quantities]
Regression coefficients of τ33 and
∂τ3i /∂xi
0.2
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DSM
GCDSM
SCDSM
SSM
GM
0
20
40
60
k=/
80
100
120
140
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Conclusion
•
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.
•
•
Ongoing work
•
•
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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.
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