Parameterization of the atmospheric boundary layer for offshore wind resource assessment with a

limited-length-scale k-ε model

Javier Sanz RodrigoDaniel Cabezón

Ignacio Martí(CENER)

19-03-2009

Introduction

1. Motivation.2. On ABL parameterization: mixing length vs k-ε modeling.3. Mixing length parameterizations.4. Verification of ABL solver on the Leipzig wind profile.5. Validation with FINO-1 test case.6. Conclusions.7. Outlook.

Motivation

OFFSHORE wind resource assessment characterized by: Lack of measurements. Very low roughness important influence of thermal effects in the

structure of the ABL.

Surface similarity theory (log-laws) appear to be usable above the surface layer only in neutral and unstable conditions.

STABLE atmospheric conditions are very frequent in the offshore environment, producing: Extreme wind speed and direction shear non-uniform wind

loading. Very low turbulence long lasting wind turbine wakes.

It is necessary to model the entire ABL to consider stable conditions: from surface to geostrophic wind levels.

On ABL parameterization

Assumptions: Horizontally homogeneous conditions: d/dx0, d/dy0 Hydrostatic:

1D Momentum and Energy equations:

( )

( )

c g

c g

uwUf V V

t zvwV

f U Ut z

w

t z

1, ,g g

c

P PU V

f y x

2 sincf

Coriolis parameter

Geostrophic Wind:

Kinematic momentum and heat fluxes

, ,t

t

t

U Vuw vw

z z

wz

On ABL parameterization

Closure problem: turbulent viscosity formulation

Two unknowns: turbulent kinetic energy (k) and mixing length (lm) Transport equation for k:

k-lm: parameterization of mixing length from ABL scales (u*,L,etc) k-ε: mixing length indirectly obtained from turbulent dissipation rate

ε, assuming ld=lm

1

2t ml C k

t

k

k kG

t z z

2

1 2tC G C

t k k z z

3/ 2

d m

C kl l

2

t

kC

22 1C C C ABL flows:

Mixing Length Parameterizations (k-lm)

Prognostic equation for the mixing length based on ABL scales: u*

0, u*: Friction velocity (surface or local) z0: Roughness length L0, L: Monin-Obukhov (surface or local) zi: Boundary layer depth

Blackadar (1997):

Apsley and Castro (1971):

Delage (1972):

Gryning (2007):

Mahrt and Vickers (2003):

max

1 1 1

ml z l 0

max

1 1m

m

zL

l z l

max 0l Lb

max

1 1 1

m

b

l z l L

1 1 1

( )m

m MBL il z l z z

max 0.00027g

c

Ul

f

(Stable)

(Stable)

01

01

1

Lz

zz

Lzz

Lz

zz

Lz

Lzz

lp

im

p

imm

m

Mixing Length Parameterizations (k-lm)

Comparison: NEUTRAL ABL

Agreement with surface layer scaling only in the first 5%

Main differences in the upper part of the ABL

Blackadar=Apsley Mahrt and Delage models

agree Gryning model produces the

largest mixing length

Mixing Length Parameterizations (k-lm)

Comparison: UNSTABLE ABL

Agreement with surface layer scaling up to the first 20-30%

Blackadar=Apsley=Delage (neutral formulation)

Mahrt model between ‘neutral’ and Gryning formulation

Gryning model produces the largest mixing length

Mixing Length Parameterizations (k-lm)

Comparison: STABLE ABL

Agreement with surface layer scaling only in the first 3%

Blackadar with stability function

Apsley with local L has similar performance to Mahrt and Delage models

Gryning model produces the largest mixing length with deeper influence of lmax

Two equation closure (k-ε)

Standard k-ε model:

Produces a mixing length proportional to the height above the ground, i.e it is only valid in the surface boundary layer.

Too deep boundary layer due to too much mixing (turbulent viscosity) in the upper part of the ABL

Modifications to ε production term:

Detering and Etling (1985):

Apsley and Castro (1997):

Weng and Taylor (2003):

*1 ;d

i hi

l uC G h ch k f

2

1 2tC G C

t k k z z

1 2 1max

mlC C C Gl k

3/ 2 2

1 2m

C kC

l

Verification of ABL solver: Leipzig profile

Leipzig wind profile: Neutral ABL (z0=0.3m, u*=0.65m, Ug=17.5m) Standard k-ε model produces a very deep ABL Mixing-length and limited-length-scale k-ε model perform well

Validation: FINO-1 (North Sea)

FINO-1: Meteorological Instrumentation

Stability distribution

Very low roughness: z0~0.2mm

Predominant non-neutral conditions. STABLE ABL at all relevant wind speeds.

5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

Stability/Velocity distribution [%]

U [m/s]

Fre

q [

%]

119

542

1045

1620

1911

2308

3108

3395

3622

4324

5010

4936

4595

4056

3507

2957

2248

1798

1430

1188

755

368

295

209

113

52 46 28 9 12 10 5 4 2 0

N SU VU VS SS N

#Samples

Data screening

Objective: Clustering of wind profiles for stability classification according to flux M-O length at 80m.

Filtering of registers: Open sea Without mast distortions: 190º-250º Velocity > 3m/s Stationary test

Stability Class L # profilesVery Stable vs 10<L<50 109

Stable s 50<L<200 314Slightly Stable ss 200<L<500 145

Neutral n |L|>500 306Slightly Unstable su -500<L<-200 182

Unstable u -200<L<-50 158Very Unstable vu -50<L<-10 28

Total 1242

Preliminary Results: k-ε (Apsley and Castro)

Fitting with u*,Ug,L and lmax.

Too deep boundary layers require very low lmax (<10m) to increase wind shear and reduce turbulence intensity Local scaling?

max

1 1 1

ml z l

Mixing length parameterization

Mixing length Mahrt and Vickers (2003) parameterization. Stability function depends on local M-O length. Low dependency with height. FINO1: best fitting with b=7

*

1 0

( / )

1 0pm

z zbL Lz u

z Lu z z z

aL L

3/ 42 2uw vw

Lg w

Mixing length parameterization

Mixing length Mahrt and Vickers (2003) parameterization. Measured mixing length

FINO1: best fitting with p=3.5

01

01

1

Lz

zz

Lzz

Lz

zz

Lz

Lzz

lp

im

p

imm

m

*m

ul

dUdz

ci f

uz *12.0

Conclusions and Outlook

ABL modeling requires parameterization of mixing length.Preliminary results in offshore conditions indicate the K-ε model produces too deep boundary layers.k-lm model with Mahrt and Vickers parameterization looks promising.

Outlook: Benchmarking of mixing length parameterizations. Meso-CFD coupling: Study mesoscale databases, where geostropic

winds and boundary layer depth are readily available.