fluid mechanics and turbulence in the wind-turbine array …chertkov/smartergrids/talks/... ·...
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LANL-2010
Fluid Mechanics and Turbulence in theWind-Turbine Array Boundary Layer
Charles Meneveau,
Mechanical Engineering & CEAFM,
Johns Hopkins University
Mechanical Engineering
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Collaboration with
• Prof. Raúl B. Cal (Portland State Univ.) - exp
• Prof. Luciano Castillo (RPI) - exp
• Marc Calaf (JHU & EPFL) - LES
• José Lebrón-Torres (RPI) - exp
• Dr. Hyung-Suk Kang (JHU) - exp
• Mr. Max Gibson (Portland State Univ.) - exp
• Prof. Johan Meyers (Univ. Leuven) - LES
Funding: NSF CBET-0730922 (Energy for Sustainability)
Simulations: NCAR allocation (NSF)
Mechanical Engineering
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Shell's Rock River windfarm in Carbon County, Wyoming, USASource: http://www.the-eic.com/News/Archive/2005/May/Article503.htm
Land-based HAWT Horns Rev HAWTCopyright ELSAM/AS
Arrays are getting big:
When L >> 10 H (H: height of PBL),
starts to approach “fully developed”
The windturbine-array boundary layer (WTABL)
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Photo by Uni-Fly A/S (Wind turbine maintenance company)
The windturbine-array boundary layer (WTABL)
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Long downstream effects:
Horns Rev wind farm:
Christiansen & Hasager: “Wakeeffects of large wind farms identifiedfrom satellite synthetic aperture radar(SAR)”, Remote Sensing Env. 98, 251(2005)
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Modelling and measurements of wakes in large wind farmsBarthelemie, Rathmann, Frandsen, Hansen et al…J. Physics Conf. Series 75 (2007), 012049
Power extraction at Horns Rev wind farm:
Long downstream effects:
10H ! 10km : 10km7D
=10km
0.56km= 18
?
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Importance of wake fluid dynamics and turbulence
Elkinton, Manwell & McGowan (2006)Offshore Wind Farm Layout Optimization (OWFLO) Project: Preliminary Results AIAA paper AIAA-2006-998
Total cost optimization for various layouts.
PARK model for wake velocity reduction (wake loss):
Fluid dynamics and details of wake structure very important all the way to the economic analysis
Sensitivity of cost to variouscomponents included in
their cost analysis:
U(x) =U0 1! a1
1+"(x / D)#$%
&'(
2)
*++
,
-..
Entrainment parameters depend on turbulence structure…
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Sample of works on modelling and measurements of wind turbine wakes and arrays:
• Lissaman PBS. Energy effectiveness of arbitrary arrays of wind turbines. In: AIAAPaper 79-0114. p. 1-7, 1979
• Voutsinas SG, Rados KG, Zervos A. On the analysis of wake effects in wind parks.Journal of Wind Eng Ind Aerodyn 14, 204-219, 1990.
• S Frandsen: wind speed reduction… J. Wind Eng & Ind Appl 39, 1992.• Crespo, Hernandez & Frandsen: Survey of modeling methods… Wind Energy 2, 1-24,
1999.• Magnusson M, Smedman AS. Air flow behind wind turbines. J Wind Eng Ind Aerodyn
80, 169- 89, 1999.• Frandsen & Thogersen: Integrated fatigue loading for wind turbines in wind farms by
combining ambient turbulence and wakes. J Wind Engin 23, 327-339, 1999.• Ebert & Wood: Renewable Energy, detailed single wake measurements.. 1999.• Vermeer, Sorensen & Crespo: Wind turbine wake aerodynamics. Prog Aerospace Sci
39, 467-510, 2003.• Corten, Schaak & Hegberg: Turbine interaction in large offshore wind farms, Report
ECN-C-04-048, 2004.• Medici & Alfredsson: Measurement on a wind turbine wake: 3D effects and bluff body
vortex shedding. Wind Energy 9, 219-236, 2006.• Jakob Mann: Simulation of Turbulence, Gusts and Wakes for Load Calculations, Wind
EnergyProceedings of the Euromech Colloquium, 2007.• Chamorro & Porté-Agel: A wind-tunnel investigation of wind-turbine wakes: Boundary-
layer turbulence effects Wind Energy, 2009 (in press)• …..
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The “fully developed” WTABL:
z
U(z) = u (x, y, z) xy
U0
x
y
Assume flow driven by imposed (unaffected) dpdx(depends on Ug, and we disregard “turning” effects)
I = V/R
Models for R so that we may evaluate flow (I) for a given driving force (V)
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• Momentum theory (time averaged + “canopy average”):
0 = !1"dp#dx
+ddz
! u 'w 'xy! u "w" xy( ) + fx xy
The “fully developed” WTABL:
z
U(z) = u (x, y, z) xy
Horizontal averageof turbulent Reynolds stresses
We must include correlationsbetween mean velocity deviations
from their spatial mean(Raupach et al. Appl Mech Rev 44, 1991, Finnigan,
Annu Rev Fluid Mech 32, 2000)
u " = u ! u xy
U0
x
y
Assume flow driven by imposed (unaffected) dpdx(depends on Ug, and we disregard “turning” effects)
thrust force due to WT
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0 = !1"dp#dx
+ddz
! u 'w 'xy! u "w" xy( ) + fx xy
The fully developed WTABL: momentum theory
If top of WT canopy still
falls in the “surface layer”, where
and if wakes have “diffused” so that
! u 'w 'xy(ztop ) " ! u 'w '
xy(zbottom ) +
12CT
AdiskAxy
UR2
u "w" xy ! 0
dpdxz ! 0
Sten Frandsen, J. Wind Eng & Ind
Appl 39, 1992):
Horizontally averaged variables
u*2hi ! u*
2lo +
12CT
"4sxsy
UR2
Integrate in z-direction:
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The fully developed WTABL: momentum theory
u*2hi ! u*
2lo +
12CT
"4sxsy
UR2
Frandsen 1992: postulated the existence of 2 log laws with u*hi and u*lo
u xy = u*hi1!log z
z0,hi
"
#$%
&'
u xy = u* lo1!log z
z0ground
"
#$%
&'
u xy (z)
log(z)zh
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The zeroth-order question:
Flow resistance as effective roughness length
S. Frandsen, 1992:
u xy = u*hi1!log z
z0,hi
"
#$%
&'
This is of the form “I = V/R”
I = velocity (output),
V=u* is applied driving force, and
R=resistance:
If z0,hi is known, we can relate u*
and mean velocity u(z)
Note: for Geostrophic forcing, u* must
Further be related to geostrophic wind
(straight-forward if z0,hi is known)
R =1!log z
z0,hi
"
#$%
&'"
#$
%
&'
(1
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The fully developed WTABL concept and effective roughness: sample application
Keith et al. “The influence of large-scale wind power on climate” PNAS (2004)
Barrie & Kirk-Davidoff: “Weather response to management of large Wind turbine array”, Atmos. Chem. Phys. Discuss. 9, 2917–2931, 2009
(under review for ACP)
Use z0,hi ~ 0.8 m - assumes
rough-wall log-law (MO)
Grid-spacings 100’s of km,
first vertical point ~ 80m
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The zeroth-order question:
Flow resistance as effective roughness length
u*2hi ! u*
2lo +
12CT
"4sxsy
UR2
S. Frandsen 1992, 2006:
UR = u*hi1!log zh
z0,hi
"
#$%
&'3 unknowns: z0,hi , UR , u* lo
Knowns: u*hi , z0,ground , CT , sx , sy
u*hi1!log zh
z0,hi
"
#$%
&'= u* lo
1!log zh
z0,ground
"
#$%
&'
Solve for effectiveroughness:
z0,hi = zh exp !" #CT
8sxsy+
"ln(zh / z0,ground )
$
%&'
()*
+,,
-
.//
!1/2$
%&&
'
())
u xy = u*hi1!log z
z0,hi
"
#$%
&'
u xy = u* lo1!log z
z0ground
"
#$%
&'
u xy (z)
log(z)zh
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Another important question: Fluxes of kinetic energy
c
d
a
b
VV (1! 2a)
For single wind turbine, extracted power = difference in front and back fluxes of kinetic energy
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Another important question: Fluxes of kinetic energy
For multiple (∞ ) wind turbines in fully developed WTABL, extracted power = must be brought to wind turbine
by vertical fluxes of kinetic energy:
U0
x
y
d 12 u xy
2
dt= !"turb ! "canop !
ddz
u 'w 'xyu xy + u "w" xy u xy( ) ! u xy
1#dp$dx
! PT (z)
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Objectives:
• Measure profiles of horizontally averaged momentum fluxes & mean velocity for various wind turbine arrangements and conditions (neutral for now)
• Verify existence of double log-law structure, check Frandsen formula for z0,hi, develop improvements?
• Quantify fluxes of kinetic energy: correct magnitude? Turbulence or mean flow?
• Challenges: requires statistics at entire 3-D volume surrounding a turbine elementary volume, including in transverse direction
• LES: fully developed, but contains many modeling assumptions
• Wind tunnel: more realistic, but not fully developed and exp. uncertainties
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Large Eddy Simulation
• Closure problems:
- need a model for (Lagrangian dynamic Smag. model)
- need a model for wall stress (log-law wall function)
- need a model for
Subgrid scalestress tensor
! ij = uiu j! " "ui "u j
!wi
fTi
! !ui!t
+ !uk! !ui!xk
= "1#!!p!xi
"!!xk
$ ik "1#! !p!xi
+ fTi
• LES equations (formally filtered Navier-Stokes eqns. - here: neutral):
Wind-turbineforce model
Imposed PGforcing of flow
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Actuator disk modeling of turbines in LES
fTx = !12CT
11! a
U"#$
%&'2 (Ayz
(V= !
12
)CTU2 (Ayz
(V
CT = 0.75! a " 0.25# $CT = 1.33
Jimenez et al., J. Phys. Conf. Ser. 75 (2007) simulatedsingle turbine in LES using dynamic Smag. model
U(t) = (1! ")U(t ! dt) + "Udisk (t)
fTx = !12CTUref
2 "Ayz
"V, CT = 0.75
They used fixed reference (undisturbed) velocity:
Here we use disk-averaged and time-averaged velocity, but local at the disk (see Meyers & Meneveau 2010, 48th AIAA conf., paper)
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Simulations setup:
H = 1000 !1500m, Lx = "H ! 2"H , Ly = "H(Nx # Ny # Nz ) = 128 #128 #128
• Code: horizontal pseudo-spectral (periodic B.C.), vertical: centered 2nd order FD(Moeng 1984, Albertson & Parlange 1999, Porté-Agel et al. 2000, Bou-Zeid et al. 2005)
• Top surface: zero stress, zero w
• Bottom surface B.C.: Zero w + Wall stress: Standard wall function relating wall stress to first grid-point velocity
• Scale-dependent dynamic Lagrangian model (no adjustable parameters)
• Calaf, Meneveau & Meyers, (Phys Fluids 2010, 22)
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Simulations results:
Instantaneous stream-wise velocity contours:
top-viewside-view
front-view
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Simulations results:
Instantaneous stream-wise velocity contours:
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Simulations results:
Instantaneous power output (mean over all WT and ±rms):
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Simulations results:
Instantaneous power output (along rows and columns):
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Comparison of wake profiles, regular Smagorinsy (with wall damping) and dynamic model:
Dynamic Smagorinsky coefficient: increases in wake region, while decreases near wall
Simulations results:
cs2 1/2
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Mean velocity profile:
Simulations results: horizontally averaged profiles
u xy
Log-law without WTSame slope,higher-z intercept(is z0,hi)
Lower slope (u*,lo)
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Mean Reynolds stress profile:
Conclusions: Momentum fluxes carried by cross-stream heterogeneity are significant
Simulations results: horizontally averaged profiles
Spatial “canopy stress” profiles,! "u w ' xy ! u "w" xy
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Suite of LES cases:
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Suite of LES cases:
measure z0,hi from intercept
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Improvement to Frandsenʼs model:
z0,hi = zh 1+ D2zh
!"#
$%&
'
exp ()CT
8* 2sxsy+ ln zh
z0,ground
1( D2zh
!"#
$%&
'+
,--
.
/00
!
"##
$
%&&
(2+
,
---
.
/
000
(1/2!
"
###
$
%
&&&
where ' =1w
*
1+ 1w* , and 1w
* =1T
*u*zh , eddy viscosity due to wake
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Comparison of LES results with models:
Triangles: Lettau formula Asterisks: Frandsen (2006) formula
Circles: improved Frandsen modelCalaf, Meneveau & Meyers, (Phys. Fluids 2010, 22)
z0,hi = zh 1+ D2zh
!"#
$%&
'
exp ()CT
8* 2sxsy+ ln zh
z0,ground
1( D2zh
!"#
$%&
'+
,--
.
/00
!
"##
$
%&&
(2+
,
---
.
/
000
(1/2!
"
###
$
%
&&&
where ' =1w
*
1+ 1w* , and 1w
* =1T
*u*zh , eddy viscosity due to wake
z0,hi = zh exp !" #CT
8sxsy+
"ln(zh / z0,ground )
$
%&'
()*
+,,
-
.//
!1/2$
%&&
'
())
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Using the roughness model for array optimization - find s-opt:
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Using the roughness model for array optimization - find s-opt:
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Using the roughness model for array optimization - find s-opt:
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Using the roughness model for array optimization - find s-opt:
• Lease of land, yearly payout per wind turbine is around US$ 5,000 for present typical spacings of 500x500m (e.g. http://www.windustry.org/how-much-do-farmers-get-paid- to-host-wind-turbines).
• So over a 20-year lease: pL ≈ 0.4 US$/m2.
• Land purchase e.g. in Texas ~ US$ 1,000 per acre (e.g. http://recenter.tamu.edu/data/agp),
or costL ≈ 0.25 US$/m2
• Representative cost of a wind-turbine US$3.5x106 for a 2MW-rated wind turbine. With D= 70m: costT/A ≈ 700 US$/m2
(e.g. http://www.windustry.org/how-much-do-wind-turbines-cost).
• The corresponding parameter α is then roughly
α = (costT/A)/costL ~ 1,700 to 3,000
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Using the roughness model for array optimization - find s-opt:
At common s ~ 7D, 10-20% suboptimal possible reason for “array underperformance” ?
Meyers & Meneveau, 2010 (preprint, submitted to Wind Energy)
α ~ 2,000
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Wind-tunnel measurements: mechanics of vertical KE entrainment??
Contractionsection
CR=25:1
Corrsin Wind Tunnel (1966): Test Section (1.2m×0.9m)
Flow
Rough surface
1x
2x
0.7 m
D
2.9 m
D = 12 cmModel wind turbine
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Flow
Strakes
1x
2xD
2.9 m
Active gridKang et al. (JFM 2003)
Wind-tunnel measurements: mechanics of vertical KE entrainment??
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Wind-tunnel measurements
Flow
Strakes
1x
2xD
2.9 m
Strakes
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Wind-tunnel measurements
Flow
Strakes
1x
2x
hh=12cm
2.9 m
Wind turbine models
•Turbine Models–Scaled down 850 times from typicalreal life length scales (real diametersof 100 m scaled down to 12 cm).
• Rotors–Made from G28 galvanized sheet metal–Twisted 1.1 degrees per cm, from 15o at the root to 10o at the tip–Tip speed ratio, λ = Vtip/Uhub is 5–Rotate at 4800 RPM
• Tower– Height of 12 cm– Constructed using rapid prototyping
D=12cm
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Wind-tunnel measurements
Flow
Strakes
1x
2xD
2.9 m
optical sensorfor phase-lock and
Ω rpm measurements
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Hot-wire inlet profiles
u* = 0.48 m/s
x
y
1.4 m
u =u*!ln yy0
=0.480.4
ln y3.5 "10#4
TI∞ ~ 6%
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TSI System with:
• Double pulse Nd:YAG laser(120mJ/pulse)
– Laser sheet thickness of 1.2 mm
– Time between pulses of 50 ms
– Optical sensor external trigger for phaselock measurements
• Two high resolution cross/autocorrelation digital CCD cameras with
– a frame rate of 16 frames/sec.
– Interrogation area of 20 cm by 20 cm
Mirror
20 cm
Laser Sheet
20 cm
3rd Row ofwind turbines
Phase-lockSensorFlow
Stereo-PIV system
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PIV data planes:
3 cm
6 cm
18cm
18cm
Top view:
Statistics:
2000 vector maps for each front plane12000 samples each back plane (6 phase-locked cases)
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Wind-tunnel measurements
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Velocity maps:
3.7D
3 D
Mean streamwise velocity
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Velocity maps:
3.7D
Mean transverse velocity
Wake angular momentum
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Velocity maps:
3.7D
3 D
Mean vertical velocity
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Velocity maps:
(negative) Reynolds shear stress
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Cross-stream maps:
(negative) Reynolds shear stressStreamwise velocity
Measured induction factor:
a =121! Uback
U front
"
#$%
&'( 0.087
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Horizontally (canopy) averaged profiles:
“copy” - assume periodic+ linear interpolation
Mean velocity(a) in front(b) in back(c) overall
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Horizontally (canopy) averaged profiles:
Horizontally averaged Reynolds stresses:
u*L ! 0.1 ! 0.32 m/s u*H ! 0.28 ! 0.53 m/s
Cal et al.: J. RenewableAnd Sustainable Energy 2, 2010
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Horizontally (canopy) averaged profiles:
u = u*L1!ln yu*L
"#$%
&'(+ 5.5 ) *U +#
$%&'(
!U + " 2.2
u*L ! 0.32 m/s
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Power extraction:
P =12!CP
"4D2UR
3
CP! ideal = 4a(1! a)2 = 4 "0.087(1! 0.087)2 # 0.29
P =12!1.2 !0.29 ! "
4!0.122 !6.243 # 0.47 W
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Direct torque measurements:
Torque
Back View
Strain Gage 1
Strain Gage 2
DC Motor
HousingHousing
Model generator(DC Motor)
Ball Bearings
Side View
Shaft
DC motor as a generator
Vary R
V
Thanks to Duane Dennis (High-Schooljunior at Baltimore Polytech Inst.)for his help with these experiments
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Direct torque measurements:
Therefore at 4800 rpmPreal = T meas !" = 0.34 W
# CP-real =Preal
12 $%R
2UR3 & 0.21
Measured torque
Telec =VI!
Tmec
Ohm
72% of ideal 0.47 W for given a
About 50% of real-life wind turbines
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Small except in back
Horizontally averaged profiles - kinetic energy terms:
d 12 u xz
2
dt= !"turb ! "canop !
ddy
u 'v 'xzu xz + u "v " xz u xz( ) ! u xz
1#dp$dx
! PT (y)
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Horizontally averaged profiles - kinetic energy terms:
d 12 u xz
2
dt= !"turb ! "canop !
ddy
u 'v 'xzu xz + u "v " xz u xz( ) ! u xz
1#dp$dx
! PT (y)
u 'v 'xzu xz (ytop ) ! u 'v '
xzu xz (ybottom ) " 1.4 W / m2
(kg / m3) #Pturb! flux = 1.4$APturb! flux = 1.4 %1.2 % (3% 0.12)(7 % 0.12)Pturb! flux = 0.51 W
Analysis consistent with view thatkinetic energy extracted by turbine(0.34W) is delivered vertically byturbulence fluxes (0.51W)(rest goes into dissipation, etc…)
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Quadrant and spectral analysis of vertical KE entrainment:
“Quadrant Analysis”:contributions from u’ > or < 0 v’> or < 0Fi = ! u 'v ' Qi u (y)
u '
v 'Q1Q2
Q4Q3
From: Gibson, Cal & Meneveau (2010)
Main conclusion:Q2 (ejections) + Q4 (sweeps) dominant
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Quadrant and spectral analysis of vertical KE entrainment:
“Co-spectral Analysis”F = ! u(y) "uv (kx , y)# dkx
From: Gibson, Cal & Meneveau (2010)
Main conclusion:Large scales dominate entrainment process (~ π/k ~ 10-30 cm = 1-3 D)
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LES: Horizontally averaged profiles - kinetic energy terms
d 12 u xy
2
dt= !"turb ! "canop !
ddz
u 'w 'xyu xy + u "w" xy u xy( ) ! u xy
1#dp$dx
! PT (z)
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• LES and data so far confirms that overall, boundary layer momentum theory(friction velocity, thickness length-scales, momentum loss, etc..) givesvaluable insight and parameterizations of dynamics.
• LES confirms existence of 2 log-laws below and above WT region
• We have extended Frandsen model for effective roughness height, withimproved agreement with LES results.
• The notion of the fully developed WTABL is a useful theoreticalnotion, otherwise mix of BL + wake, scale separation, 3-D complex flowcomplicates statistical treatment and data analysis.
• How much momentum transfer caused by mean velocity dispersive stresses?we found them small in experimentwe found them large in LES
• Energetics and profiles show that kinetic energy is delivered to turbine mostlyby turbulence (both LES and experiments confirm this).
• Expand experiments with model rotors with higher induction factor, vary λ...
• Model for effective roughness height (R) can be used to design optimal powerextraction per cost (terrain and # of WT)
Conclusions: