48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
4 - 7 January 2010, Orlando, Florida
AIAA 2010-1374
Compact Representation of Large Eddy Simulations of
the Atmospheric Boundary Layer Using Proper
Orthogonal Decomposition
Manjinder S. Saini ∗, Jonathan W. Naughton†
University of Wyoming, Laramie, WY 82071, USA
Edward Patton ‡, Peter Sullivan§
National Center for Atmospheric Research, Boulder,CO 80307, USA
Studies have been conducted to assess the use of Proper Orthogonal Decomposition to
create a database for a low order, compact representation of Large Eddy Simulations of
atmospheric boundary layers. In particular, POD analysis of simulations of two extreme
cases: a day-time unstable and a night-time stable boundary layer, were conducted. Most
notable advantage of using LES, as opposed to spectral based turbulence simulation codes
available today, is the ability to model the true flow features, and POD potentially allows
to capture these features in a compact manner. The presence of energy containing, large
coherent structures in the unstable case allow the decomposition technique to perform
efficiently. On the other hand, the absence of these coherent structures in a stable case
leads to poor performance of POD, thus, requiring higher number of modes for accurate
representation of boundary layer. Nonetheless, the excellent performance of POD for the
unstable case is encouraging and POD may prove to be useful for low order representation
of other possible boundary layers.
I.
Introduction and Background
With growing interest in harnessing wind energy, models capable of representing wind inflow conditions
with reasonable accuracy are needed for performance prediction as well as for other applications. A class
of the available models that are based on spectral methods fail to capture the coherent structures and the
coupling between velocity components that are inherent to the atmospheric boundary layer and are important
for wind turbine loading. In its basic implementation, Turbsim1, 2 is one such computational code. More
recently, the capability to superimpose coherent structures on a flow has been introduced in Turbsim.2
However, the close coupling of these coherent turbulent structures with other turbulent structures remains
unaddressed in such codes. Another open source turbulence inflow code, IEC turbulence simulator, an
implementation of Mann’s work,3 is based on rapid distortion theory. It models the atmospheric turbulence
using an isotropic spectral tensor that is deformed due to the shear in the mean wind profile. Nonetheless,
this code also does not provide an realistic simulation of the atmospheric boundary layer due to certain
assumptions, such as, Taylor’s frozen turbulence hypothesis. With growing computational capabilities, it is
possible to capture this realistic behavior using Large Eddy Simulations (LES). LES can be used to capture
important behaviour of atmospheric boundary layer over a range of stability conditions. Simulations of
atmospheric boundary layer using LES, though complicated, have been obtained in a number of previous
studies.4–6 However, use of such simulations requires large storage capacities that are not easily portable.
Proper Orthogonal Decomposition (POD), a low-order modeling technique, can address some of these issues
and allow to capture the realistic atmospheric boundary layer behavior in a compact manner. POD has
been used for wind application7, 8 as well as other applications.9, 10 Experimental or computational data is
∗ Post
Doctoral Researcher, Mechanical Engineering Department, Dept 3295, Member AIAA.
Professor, Mechanical Engineering Department, Dept 3295, Associate Fellow AIAA.
‡ Scientist, Boundary Layer and Turbulence Group
§ Scientist, Boundary Layer and Turbulence Group
† Associate
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Copyright © 2010 by Jonathan Naughton. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
used in POD to build the low-dimensional models that can easily be integrated into turbine performance
analysis codes in use today. Spitler et al.7 demonstrated the low order modelling capability of POD using
a simulated data set from Turbsim. Saranyasoontorn et al.8 also used a data set obtained using Turbsim
to describe careful considerations required in selecting the flow reconstruction modes for use in turbine load
analysis. However, use of LES of atmospheric boundary layers for turbine load and performance analysis has
been very limited. For example, Kelley et al. used a large eddy simulation of Kelvin-Helmholtz billows11 on
a three-bladed, 1.5 MW turbine to compare and improve the inflow wind field generated by SNLWIND-3D.12
Being a low order modeling technique, the power of POD lies in its ability to capture the energy containing
structures in the flow with potentially few modes, thus, providing a compact way of representing the flow. The
current study explores the ability of POD in compactly representing two atmospheric boundary layers, with
the long term goal of developing a database that would provide inflow velocity fields for a range of typical
wind conditions in a compact manner. Such a database can then be tied into existing models that are
available for performance analysis of wind turbines as well as other studies requiring compact representation
of the atmospheric boundary layer. In order to overcome the lack of availability of field data sets and
to provide a rigorous test bed for the proposed modeling, Large Eddy Simulations (LES) of atmospheric
boundary layers are used in this study. POD is used here to decompose the velocity field, in planes that are
extracted from 3D volumes, to determine the orthogonal basis functions, a fraction of which are then used
for reconstruction to obtain a low order model. The results indicate that the performance of POD largely
depends on the type of boundary layer. The presence or absence of coherent structures is a strong factor
in determining how efficiently the POD technique captures the energy in the flow and thus the number of
modes required for flow field reconstruction.
In the following discussion, a brief introduction of POD is provided followed by relevant features of the
simulated data sets used for this study. The results are discussed and some important conclusions are drawn
from this study.
II.
Proper Orthogonal Decomposition
As in any flow, coherent flow structures are an important feature of turbulent flow dynamics in atmosphere. POD is one of the techniques that can be used to isolate these coherent structures. Two different
implementations of POD are in wide usage today: Conventional/Classical POD13 and snapshot POD.14–16
Both of the methods were tested on the available data set and yielded essentially the same results. The results
presented here were obtained using Snapshot POD, which was selected because the large eddy simulations
used in this study are comparatively more resolved in space than in time. This allows Snapshot POD to
capture the flow in fewer modes, compared to the classical POD method, resulting in reduced computation
time. Moreover, the tests revealed that a large computation capability would be required for performing
classical POD on the current dataset with full spatial resolution. For this study, the modes have been optimized in terms of turbulent kinetic energy. The eigen-value problem, as determined by using snapshot POD,
is given by,
(1)
C(t, t )an (t )dt = λn an (t),
where, λn , are the eigenvalues. The temporal correlation matrix, C, contains information about temporal
correlations of the velocity components and is defined as,
1
ui (x1 , x2 , t)ui (x1 , x2 , t )dx1 dx2 .
(2)
C(t, t ) =
N x 1 Nx 2
x2 x1
In equation 2, the repeated index i represents summation over all three velocity components and thus
maintains coupling between these velocity components. The solution to equation 1 provides the time varying
coefficients, an (t). To obtain the eigenmodes (φn (x1 , x2 )), which contain the spatial variation information,
the time varying coefficients are projected onto the velocity field, i.e.,
φni (x1 , x2 ) = (an (t), ui (x1 , x2 , t)),
(3)
where (.,.) represents the inner product between the two quantities, and i represents the velocity com-
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ponent. A reduced order reconstruction of the flow field is obtained by
ui (x1 , x2 , t) =
N
an (t)φni (x1 , x2 ).
(4)
n=1
In equation 4, N ≤ Nmax , Nmax being the total number of modes and is equal to number of time steps
used for the snapshot decomposition. For low order models of flows with significant coherent structures,
N Nmax . This feature allows POD to represent the atmospheric boundary layer in a compact manner.
III.
Relevant features of the LES data sets
z (km)
The results discussed in this paper were obtained by implementing POD on a day time unstable and a night time stable
boundary layers. While these two cases are not sufficient to
represent the full range of possible atmospheric boundary conditions experienced by the wind turbines, they represent the
extremes of the expected range. Thus, these two cases will
help in establishing the usefulness of POD for the proposed
work.
Day-time surface heating typically generates convectivelydriven turbulence, leading to formation of large coherent structures, usually appearing in the form of thermal plumes. The
daytime unstable dataset used here has an imposed geostrophic
wind of 1m/s. The domain of the simulated data set is 5.12 km
(x) x 5.12 km (y) x 2.05 km (z). Where, x is the streamwise Figure 1: Domain of Large Eddy Simuladirection, y is the spanwise direction and z is the altitude, as tions.
depicted in figure 1. The resolution is dx = dy = 20m, and dz
= 8m, with 256 grid point available in each direction. Three
planes in each direction, discussed here, have been extracted such that z=0.1 km for the x − y plane, x=
1.28 for the y − z plane and y =1.28 km for the x − z plane. More details of the simulations are provided by
Sullivan et al..6
A night-time stable boundary layer is characterized by statically stable air near the ground with weak and sporadic tur0.4
bulence.17 However, in such cases winds away from the ground
may accelerate to form a low-level jet. As an example a mean
velocity profile for the night-time simulations is shown in fig0.3
ure 2. These two process act to suppress and enhance (due to
increased shear) the turbulence, respectively, causing the tur0.2
bulence to occur in short bursts. The night-time stable simulations were obtained with geostrophic wind of 8 m/s. The 3-D
domain (see figure 1) covers a 400m (x) × 400m (y) × 400m
0.1
(z) region with a resolution of dx = dy ≈ dz = 2m. Thus, the
velocity data is available at 200 points in x, y directions, and
192 points in z direction. The results are presented for three
0
0
5
10
planes: x − y plane at z = 157 m, y − z plane at x = 2m, and
|U| (m/s)
x − z plane at y = 2m. Further details of this simulation can
be found in reference 5.
Figure 2: Time averaged velocity profile as
a function of altitude at x= 320 m for the
stably stratified LES.
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IV.
Results and Discussion
Snapshot POD was used to decompose the velocity fields obtained from large eddy simulations of the
two atmospheric boundary layers mentioned previously. In the following discussion, results are presented
first for the daytime unstable boundary layer followed by those for night-time stable boundary layer.
A.
Daytime Unstable Atmospheric Boundary Layer
Due to the presence of the large energy containing coherent structures in a daytime unstable boundary layer,
POD can be anticipated to perform efficiently. The eigen values (λn ) obtained by solving equation 1 are
representative of the energy contained in each mode. The fewer the number of eigenvalues containing high
amounts of energy, the fewer the number of modes required to capture the flow without losing its essential
features. The energy in each of the modes, captured by performing POD separately on each of the planes, is
shown in figures 3, 4, and 5. The cumulative energy captured by the modes is summarized in table 1. The
total number of modes obtained from the snapshot method is 2500. For all the planes, just 1% (25 modes)
of the total modes capture a significant amount (≥ 85%) of the energy in the flow. Such flows are ideal for
a low-dimensional model.
30
25
25
20
n
λ /Σ(λ )
15
n
λn/Σ(λn)
20
10
10
5
5
0
15
0
10
20
30
40
Mode number (n)
0
50
Figure 3: Percentage energy captured by individual modes for x − z plane.
0
10
20
30
40
Mode number (n)
50
Figure 4: Percentage energy captured by individual modes for y − z plane.
25
n
n
λ /Σ(λ )
20
No. of Modes
(N)
10
15
25
50
15
10
5
0
0
10
20
30
40
Mode number (n)
50
Cumulative Energy (%)
x−z y−z
x−y
73
72
69
81
80
78
89
88
87
96
96
96
Table 1: Cumulative energy captured by N
modes for three planes
Figure 5: Percentage Energy Captured by
individual modes for x − y plane.
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The eigenmodes and the time varying coefficients obtained from the POD analysis capture the spatial
and temporal flow variation respectively. Figure 6 shows the first two eigenmodes for u, v, and w velocity
components in the y − z plane. The modes show that most of the energy in the flow is due to thermal plumes
because these are the most significant coherent structures in the flow. The time varying coefficients also show
an interesting behavior as is depicted in figure 7. As can be observed, the time varying coefficients show a
Figure 6: First two modes for y − z plane for u,v, and w velocity components.
0.05
2
1
3
a
1
a
n
a2
0
a
3
a
4
−0.05
0
1000
2000
1000
2000
time step
1000
2000
Figure 7: First four time varying coefficients for (1) x − y, (2) y − z, and (3) x − z planes.
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Figure 8: Comparison of reconstructed (using 25, or 1% modes) and original velocity components for x − z
plane at arbitrary time. The top figure is for u component, followed by v and w, respectively. Subscript 1
represents original velocities and 2 represents reconstructed velocities.
highly periodic pattern in all three planes due to the presence of coherent structures. Though no tests were
conducted using Fourier decomposition, the highly periodic pattern indicates that a Fourier decomposition
(in time) of this flow field may also work efficiently.
To determine how well the decomposition of the flow worked, the velocity field can be reconstructed
using a limited number of modes and compared against the original field. Figures 8, 9, and 10 compare
the reconstructed flow in x − z, y − z, and x − y planes using only 25 modes (1% of 2500) to the original
flow. As can be seen, the essential features have been captured with very good accuracy and very little
smoothing. Though not shown here, further reducing the number of modes, used in reconstruction, results
in a loss of fine structures and increased smoothing. On the other hand, increasing the number of modes for
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reconstruction allows for the capture of finer structures with less filtering. Thus, depending on the needs of
an application, the number of modes can be selected to capture the desired level of detail.
Figure 9: Comparison of reconstructed (using 25, or 1% modes) and original velocity components for y − z
plane at arbitrary time. The top figure is for u component, followed by v and w, respectively. Subscript 1
represents original velocities and 2 represents reconstructed velocities.
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Figure 10: Comparison of reconstructed (using 25, or 1% modes) and original velocity components for
xy plane at arbitrary. The top figure is for u component, followed by v and w, respectively. Subscript 1
represents original velocities and 2 represents reconstructed velocities.
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B.
Night-time Stable Atmospheric Boundary Layer
Largely due to buoyant inhibition, turbulence in stably stratified flows typically exhibits limited large-scale
organization, and thus POD is not expected to perform as efficiently as was observed for the unstable case.
Snapshot POD analysis of the night-time stable data set provided 560 modes, and the cumulative energy
distribution of these modes in three planes are shown in figure 11 and summarized in table 2. As observed,
the energy is widely distributed among all the modes, irrespective of the plane, and thus a low-dimensional
model of the studied flow will not capture the flow with high accuracy. Due to flow homogeneity in x and
y directions, these results are not affected much by locations of specific x − z and y − z planes. Though
not shown here, POD performs more efficiently in x − y planes at higher altitudes where the turbulence is
not affected much by the calm air near ground and is thus influenced only by shear. Nonetheless, the POD
performance at heights ≤ 200 m, which is of primary interest for wind turbines, remains poor.
Cummulative Energy (%)
100
80
Energy %
60
40
x−z
x−y
y−z
20
0
0
200
400
No. of Modes
25
50
75
90
Modes Required
(% of total modes)
x−z
x−y
y−z
22 (4)
36 (6)
32 (5)
81 (14) 110 (20) 102 (18)
212 (38) 246 (44) 237 (42)
367 (66) 392 (70) 384 (69)
Table 2: Modes required to capture a given
amount of energy in three planes
Figure 11: Cumulative energy captured by
modes for x − z, x − y, and y − z planes.
The energy distribution in eigenmodes reveals that using only a few modes for flow reconstruction will
introduce high error. Examples of flow reconstruction in y − z plane, which represents the velocity field that
a turbine would see, are shown in figures 12 and 13, using 56 (10%) and 280 (50%) modes, respectively.
The results indicate that POD fails to capture fine features of the flow when low number of modes are used.
Increasing the number of modes allows these fine features to be captured. However, errors due to smoothing
can still be observed with number of modes as high as 50%. Thus, the compactness realized for the daytime
unstable boundary layer is not possible here. Nonetheless, by retaining a significant number of modes, the
flow may still be accurately represented using POD.
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Figure 12: Comparison of reconstructed (using 56, or 10% modes) and original velocity components for yz
plane at arbitrary time. The top figure is for u component, followed by v and w, respectively. Subscript 1
represents original velocities and 2 represents reconstructed velocities.
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Figure 13: Comparison of reconstructed (using 280, or 50% modes) and original velocity components for yz
plane at arbitrary time. The top figure is for u component, followed by v and w, respectively. Subscript 1
represents original velocities and 2 represents reconstructed velocities.
V.
Conclusion and Future Work
Large eddy simulations of a daytime unstable and a night-time stable boundary layers have been used
to test the usefulness of POD for compactly representing the atmospheric boundary layer. The ability of
POD to capture essential flow features with limited number of modes can be useful in developing a compact
database that can be used in conjunction with wind turbine performance analysis codes. Furthermore, this
method provides a capability to couple the velocity components, which is not available in spectral-based
techniques. Though the two cases studied here are not sufficient to represent the entire range of atmospheric
conditions that are desired to rate a wind turbine, these cases represent the extremes of the range within
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which POD can be useful.
High levels of turbulence in a daytime unstable boundary layer lead to formation of energy containing large
coherent structures. These energy containing coherent structures lead to high correlation levels for which
POD analysis works very efficiently. In this convectively-driven case, as few as 1% of the total available modes
are capable of representing the flow without any considerable loss of detail. In such cases, POD produces
a truly low-order flow representation. On the other hand, stably-stratified turbulence typically does not
contain large-scale organized structures. The lower correlations that result, lead to poor performance of
the POD. In the stable case, as many as 70% modes were required to capture 90% energy in the flow and
thus higher number of modes must be retained to capture the flow accurately. Nonetheless, since this is an
extreme case, POD can expected to perform more efficiently for cases that contain relatively more organized
turbulent structures.
Future work will concentrate on using large eddy simulations at higher wind velocities that are more
relevant to wind turbines. Also, the low order models of inflow velocity fields obtained from POD will be
tested against those obtained from a spectral method, such as Turbsim, using a turbine performance/load
analysis software such as AeroDyn18 and Fast.19 Such a comparison will establish the advantages of using
true models of atmospheric boundary layer for a wind turbine rating.
References
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500-36971, National Renewable Energy Laboratory, September 2005.
2 Jonkman, B. and Bhul, M. L., “Turbsims’s User Guide,” Tech. Rep. NREL/EL 500-36970, National Renewable Energy
Laboratory, September 2005.
3 Mann, J., “Wind Field Simulation,” Prob. Engng. Mech., Vol. 13-4, 1998, pp. 269–282.
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11 Kelley, N., Shiraz, M., Jager, D., Wilde, S., Adams, J., Buhl, M., Sullivan, P., and Patton, E., “Lamar Low-Level Jet
Project Interim Report,” Tech. Rep. TP-500-34593, NREL, January 2004.
12 Kelley, N. D., “Inflow Simulation in Natural and Wind Farm Environments Using an Expanded Version of the SNLWIND
(Veers) Turbulence Code,” Tech. Rep. TP-442-5225, NREL, 1992.
13 Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry,
Cambridge University Press, 1st ed., 1996.
14 Sirovich, L., “Turbulence and the Dynamics of Coherent Structures Part-1: Coherent Structures,” Quarterly of Applied
Mathematics, Vol. 45, No. 3, 1987, pp. 561–571.
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of Applied Mathematics, Vol. 45, No. 3, 1987, pp. 573–582.
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Mathematics, Vol. 45, No. 3, 1987, pp. 583–590.
17 Stull, R. B., An Introduction to Boundary Layer Meteorology, Kluwer Academic Publishers, 1988.
18 Laino, D. J. and Hansen, A. C., “User’s Guide to the Wind Turbine Aerodynamics Computer Software AeroDyn,” Tech.
rep., Windward Engineering, LC (Prepared for National Renewable Energy Laboratory), December 2002.
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