> a decade of atmospheric stability research

DTU Wind Energy22 January 2020 M.Kelly / 10 years of stability research

> a decade of atmospheric stability research: From p(L–1) to a(Iu , z/L) to proxies to entrainment…

(…“what’s it good for?” )

Dr. Mark KellyRisø Lab/Campus, DTU Wind Energy

for VindKraftNet, 22 Jan. 20201


> a decade of atmospheric stability research: From p(L–1) to a(Iu , z/L) to proxies to entrainment…

(…where stability is useful, and where it isn’t?)

Dr. Mark KellyRisø Lab/Campus, DTU Wind Energy

for VindKraftNet, 22 Jan. 20202

Risø DTU, Technical University of DenmarkRisø DTU, Technical University of Denmark

Motivation…

• (Wind energy):

3 17 May 2010From M-O theory to long-term mean wind profiles

real, ‘physical’ parameters

representative winds, Power

Better describe the long-term profile, § over different ranges of conditions § up to larger heights

(2010)


Modelling and issues

• Microscale (CFD): • large-scale effects missing:

• Realistic forcing (varies in x,y,z,t)• Phenomena from above • Two-way interaction among scales

• Non-stationarity

• Mesoscale: • Lots of physics is unresolved – sub-grid parameterization• PBL schemes have problems

• Un-representative stability / surface interaction• Give improper distributions of stability, other parameters…• Not optimized for wind• ABL top mis-represented

16 March 2011Mark Kelly: "Tall" wind and siting issues4

model/obs. frequency of shear exponent


Motivation…

• Fundamental concept: for nonlinear functions,

Þ Mean form of Monin-Obukhov stability correction cannot simply use a mean Obukhov length !

→ For long-term profiles, need something else/more…


( )( )f x f x¹z zL L

y yæ öæ öç ÷ç ÷ ç ÷è ø è ø

¹


Long-term adaptation of M-O theory

M-O profile:

Make it long-term:


Need the probability distribution of stability, P(L−1)


A general form for stability distributions:


211

12

( ) exp(1 )

LCP L n Ca

a

ss

-- ± ±

±±

±

é ùæ öê úç ÷= -ê úç ÷G + è øë û

where

with scale parameter

1 ,3

a =

à analytically obtain profile0/U u*

3, 1C C- += =

Risø DTU, Technical University of DenmarkRisø DTU, Technical University of Denmark8

…general form of stability distributions P(L−1)

17 May 2010From M-O theory to long-term mean wind profiles


…using stability distribution model P(L−1) :


(Høvsøre)

à Need to account for ABL depth h


Generalize “tall profile” theory: (we adapt Gryning et al. 2007)

Make it long-term:


Use effective h, G (or Lmid) in lieu of P(h) , P(G)


Long-term ASL and “tall profiles” via P(L−1)

Høvsøre (land sectors)

Hamburg (residential sectors)


DTU Wind Energy, Technical University of Denmark, Risø campus

How well does it extrapolate?

Mark Kelly EWEA (Vienna)12 5 Feb 2013

Abs.%Error in wind speed, vs. relative extrapolation distance:

2012


How well does it extrapolate?


Abs.Error in power density, vs. relative extrapolation distance:


How well does it extrapolate, with added flux info?


%Error in wind speed, vs. relative extrapolation distance:


… stability affects k-profilesCorrected/adapted Weiringa89-like formulation

5 Feb 2013 Mark Kelly EWEA (Vienna) 15

10

1 10

1 ( ) ( /1 p) x )( er

hr

rz z z zkz z z z

k zz k f zké ùæ ö- -+ -ê úç ÷- -

+ -øë û

=è

2012-13

DTU Wind Energy, Technical University of Denmark16

buoyancy in RANS (with forests): add B

( )

0dpS

jj i E i S

E E K EU P Bt x x xj e

s-

é ùæ ö¶ ¶ ¶ ¶ ê ú+ - = - + +ç ÷ ê ú¶ ¶ ¶ ¶è ø ê úë û

!"#

1/ 212 ( )dd C cS A z U Eµ= (Sogachev and Panferov, 2006 )

(Sogachev, Kelly, LeClerc 2012 )

dpS US E= µ (Seginer et al., 1976)

( )*1 3 4 52j d d

j i i

KU P C B S St x x

Cx

CEC

ECj

jj j j j

j j j j jesæ ö¶ ¶ ¶ ¶ é ù+ - = - + + -ç ÷ ë ûç ÷¶ ¶ ¶ ¶è ø

( )1 2i iC C Cj j j ja g= - +

( ) ( )23

2 11 1 ( / ) for 0, 1

1 / for 0, 1MY

MY

C C C Ri B

Ri Pj j j jg e

ae

ì é ù- + - - £ ®ï ë û= í- > ®ïî

! !

! !

4 0a = 5 1a =

( γφ = {1, 0} for φ = {ε, ω})

3PS Uµ

In consistent way ‘extra’ coefficients appearing in the supplementary equationshave to be presented as

( )1 1 2*1 ( ) / MYC C CC j j jj = + - -! ! 1 MYa = -! ! 2 0a =Compare with

2012

Risø DTU, Technical University of DenmarkRisø DTU, Technical University of Denmark 17

Improving k-e/k-w (RANS) with stability

( )*

11 PrgB p p d d

B BRiP Y P B S Sa a a a-

= =+ + + + +

( )1 2i BC C Cj j j ja g= - +

( ) ( )2 2 11 1 ( / ) for 0, 1

1 / for 0, 1MY

B

MY

C C C Ri B

Ri Pj j j jg e

ae

ì é ù- + - - £ ®ï ë û= í- > ®ïî

! !

! !

( γφ = {1, 0} for φ = {ε, ω})

Stable limit

( ) ( ) ( )* 11 1 2 1 1 2 1 1 2 1

, ,

2 1.35Pr

1 1.35g g B g

B gg cr g cr B g

Ri Ri RiC C C C Ri C C C C C C

Ri Ri Rij j j j j j j j j j

aa

a-

é ù ×¢ º + - = + - + = + -ê ú

+ê úë û

*, 1

1Prg cr

B

Ria -º 1

, Prg g cr B gRi Ri Ria -= 1*

1.35Pr1 1.35B gRia

- =+ *

,

1.351 1.35

g B g

g cr B g

Ri RiRi Ri

aa

=+

Convective limit treat T later…more research…

,/ MY g g crRi Ri®! !

( )/ 1 ?MY P B TP® - + +! !

2012-3make consistent with length-scale limitation: for ABL flows (coefficient aB on B)


unsteady (diurnal) RANS result (Sogachev & Kelly, 2012)

Aug 2012Flow Center update18


Turbulence, shear, and stability

A “decomposition” of the problem

Balance of TKE (Turbulent Kinetic Energy) !

• Surface-layer limit

where

using

21-Oct 2014

*0 02 2

0 *0 *0

( )/

u gz w TU u U u U z

qa

k q- -!

0de dUuw B Tdt dz

e= = - + + -

( )z B TU uw

ea - -® =

-

0( / )B g wq q=

1Uz TIL e

é ù» + -ê úë û*02.5U us »

( )2*0 0u uwº -

19

2014


Stability and shear

ASL (10-40m)

• Can map stability to shear in ASL

20

11 1 5( )

5 | |( )

1zP L

zP La

a a+

--

+

++ ++

+

+»

-

June 2014 / Torque from Wind

P(L-1)

2014


Stability and shear

ASL (10-40m) above ASL (60-160m)

• Can map stability to shear in ASL; NOT at current hub heights L ...

21 June 2014 / Torque from Wind


A “decomposition” of the problem

Balance of TKE (Turbulent Kinetic Energy) !

• Surface-layer limit

where

using

21-Oct 2014

*0 02 2

0 *0 *0

( )/

u gz w TU u U u U z

qa

k q- -!

0de dUuw B Tdt dz

e= = - + + -

( )z B TU uw

ea - -® =

-

0( / )B g wq q=

1Uz TIL e

é ù» + -ê úë û*02.5U us »

( )2*0 0u uwº -

22


(mean) TI ↔ shear

21-Oct 2014

Extended ASL theory (stability-modified profile, TKE): where

0

01 ( )II

ca a a=

+ -0 0 * 0( / )u UI a uka k s a= =

23


2014-15 shear/TI Summary

• Above ASL, can get <I> estimate from <α>, with IEC Iref

– Depends on effective roughness (z0 + hills); stability

• Variability: σα ~ 1/U ;

• TI is more important for input to loads (Dimitrov et al 2014)

– P(α) significant for fatigue loads in low-TI conditions

– P(α) can drive blade-tip deflection for extreme-TI

• P(α) è power curve modification

Mark Kelly May 201524


buoyant vs. ‘old’ Mann model

A.Chougule / Mark Kelly25 2015

2015stability included in RDT (extra equation), old: 3 parameters

2 extra parameters (Ri, 𝜖") (L, 𝜖, Γ)


when is stability “useful” [in Wind Energy] ?

• i.e. when is it a valuable ‘metric’ or input?• often ‘blamed’, but might not need to be accounted for directly…

• a few applications where stability is (less/more) helpful –turbulence length scale, loads inputs–turbulence estimation from {shear, 1/L} in simple places –RANS simulations for wind

»ensemble / statistically-driven–vertical extrapolation (for some/depends) –top-down (entrainment) effects…–LOTS of indirect stuff (wakes, blockage, terrain)

26

DTU Wind Energy, Technical University of Denmark

Stability & TI: obs. vs. IEC”-1” (+Mann-model)

• Affects the shear/turbulence strength, and turb.length scale (L) – G, L (from spectral fits to sonic data) deviate from IEC…– IEC-prescribed s1 (pink) also bad in non-neutral

27 Mark Kelly May 2016


Mann-model / for IEC turbulence


from Sathe+Mann et al. (2013)

measured su , L , per {U,stability class}

Wind Energy Dept., Technical University of Denmark

…finding Mann-model parameters (without spectra)

• Also account for stability effects, without flux obs., – Using basic/neutral turbulence model

• Peña et al (2010) tried mixing-length concept: • ℓ∗ =

)∗⁄+, +- , with 𝐿 = 1.7 ℓ∗

but note that )∗,

depends on stability (ℓ∗ →3-

4 ⁄- 5in ASL limit)

• For 𝑎) ≡ ⁄𝜎) 𝑢∗ and ⁄𝑐; ≡ 𝐿 ℓ∗ get𝐿 = <=

>?

@?⁄+, +-

– Find 𝑐; ≃ 𝑎) ≃ 2.3 , 𝐿 ≃ @?

⁄+, +-= 𝑧 EFGH

IFGHfor typical range of Γ

– ASL: 𝐿|KLM → 𝑐;𝜅𝑧 = ⁄3𝛼 2 ⁄P Q𝜅𝑧 <=@?,FGHR.S>?@THF

⁄P U

→ 𝜎),VWX= 𝜎YXV ⁄𝑐; 1.1 ⁄U P

but 𝜎),VWX ∝ Γ𝜎YXV…

14 Mar 2018Mark Kelly / RAM29

2018


…finding turbulence length-scale, without spectra

P(LMM)P(1.7u* /|dU/dz|)P(2.3u* /|dU/dz|)P(σu /|dU/dz|)

��

��

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30 Mark Kelly / RAM 14 Mar 2018

𝐿 ≃ @?⁄+, +-

= 𝑧 EFGHIFGH

(for ABL range of Γ) à don’t need stability; this info is

contained within 𝜎) , 𝑑𝑈/𝑑𝑧


…finding Obukhov length (without sonics or ∂T/dz ?!?)

• note that )∗, depends on stability (ℓ∗ →3-

4 ⁄- 5 in M-O limit)

ignoring turb.transport (𝑇/𝜖): à

– recall

– Stability without a sonic anem. or heat flux obs.!

• Get 𝜓 ⁄𝑧 𝐿 → ℓ in terms of obs. 𝑈, 𝜎,, 𝛼– à modified 𝜓/correction above ASL

• but not so accurate above ASL, non-ideal places– …need to deal with Turb.transport (𝑇/𝜖)

dimensionless TKE balance : (κ au) α ≈ IU [1 + z /L - cT T /ϵ]

May 2016Mark Kelly / RAM31

-0.010 -0.005 0.000 0.005 0.010

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

LMO-1 (m-1 )

z-1 (α/I u

-1)(m

-1)

PDF

2016



– )∗,

depends on stability

• Limited success without 1/L, even at simple site:



0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

u* (m/s)

σ U/2.3

z=80m, land

0.0 0.2 0.4 0.6 0.8 1.0-0.2

0.0

0.2

0.4

0.6

0.8

1.0

u*

σ U2.3

1-(κα)

-1T

ϵ

z=80m, land



– )∗,


• Limited success without 1/L, even at simple site:



0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

u* (m/s)

σ U/2.3

z=80m, land

0.0 0.2 0.4 0.6 0.8 1.0-0.2

0.0

0.2

0.4

0.6

0.8

1.0

u*

σ U2.3

1-(κα)

-1T

ϵ

z=80m, land

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

LMO-1 (m-1 )

L MO

-1(α,I u,z,TT MS)(m-1

)

PDF

right: modification and inclusion of Turb.transport (𝑇/𝜖)

improves our 1/L estimate !



– )∗,


• Limited utility, even at simple site:



0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

u* (m/s)

σ U/2.3

z=80m, land

0.0 0.2 0.4 0.6 0.8 1.0-0.2

0.0

0.2

0.4

0.6

0.8

1.0

u*

σ U2.3

1-(κα)

-1T

ϵ

z=80m, land

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

LMO-1 (m-1 )

L MO

-1(α,I u,z,TT MS)(m-1

)

PDF

recall 2014 result:0

01 ( )II

ca a a=

+ -


Probabilistic ABL modelling via RANS

• First try: ensemble of atmospheric-stability states – stability most influences flow field

• shear/profile • wake

– We know how to model stability (𝐿aR) and its PDF; • 𝑃(𝐿aR) has universal shape [Kelly+Gryning 2010]

• We know e.g. limits of effect on shear [Kelly et al 2014]

• summing 𝐿aR regimes: works for modeling 𝑈(𝑧) [Kelly+Troen 2015,16]

• Turbine noise propagation application

ΔSPL(𝑟, 𝑓) = ∑l 𝑎l𝑃 𝐿laR ΔSPL(𝑟, 𝑓|𝐿laR)

Cross Cutting Activity: Wind Turbine Noise35 6 Jan.2017

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ABL modelling : flow fields

• ScaDis: RANS w/2-eqn. turbulence model [Sogachev et al 2002…]

– Advanced stability treatment ; satisfies M-O theory [Sogachev+Kelly 2012]

– Captures ABL ’top’ (T-inversion)

– Radiation/clouds also

– Mean fields or diurnal cycles

– Actuator disc à turbulent wake


U(m/s)

2.5

5.0

7.5

10.0

12.5

15.0

U(m/s)

2

4

6

8


ABL modelling : flow fields

• ScaDis: RANS w/2-eqn. turbulence model [Sogachev et al 2002…]

– Advanced stability treatment ; satisfies M-O theory [Sogachev+Kelly 2012]

– Captures ABL ’top’ (T-inversion)

– Radiation/clouds also

– Mean fields or diurnal cycles

– Actuator disc à turbulent wake


U(m/s)

2.5

5.0

7.5

10.0

12.5

15.0

U(m/s)

2

4

6

8

U(m/s)

2

4

6

8


Application: turbine noise propagation

• Parabolic Equation (PE): 2-D spectral solver – Frequency-dependent propagation (refractive) – Sound-speed profile (from 𝑈(𝑧) and 𝑇(𝑧))– Acoustic ground impedance [grass]– Input: ScaDis mean flow fields– Source:

• Distributed / lumped: 3 heights (46m, 80m, 114m) • Source spectrum

– Plus geometrical spreading, molec.absorption:

SPL 𝑓, 𝑟 = LW 𝑓 − 10 ln 4𝜋𝑟U − 𝛼 𝑓 𝑟 + Δ𝐿(𝑓, 𝑟)


50 100 500 100092

93

94

95

f (Hz)

LW(dB)

Turbine source level


results…stable cases

• stable cases

• Colored lines: – with/out T(z) in PE

– Very stable case (blue lines): • less sound, more loss… à loss reduced by including T(z)

– Weakly stable case (red/orange): • Not much change by including T(z) in PE calcs


500 1000 1500 2000 2500 300030

35

40

45

50

r(m)

SPL(dB

)

Stable Cases, 2 with T(z) [ExpMAv]


Results…

• Re-weighting SPL’s for other sites/climatologies


500 1000 1500 2000 2500 3000

35

40

45

50

r(m)

SPL(dB

)

weighted-mean for different P(1/L)-�� -�� -�� -��

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Top-down stability effect• Strength of capping inversion (Nc), and ABL depth

– LES confirm

41

0 2 4 6 8 10 12

S[m/s]

0

0.2

0.4

0.6

0.8

1

1.2

z/z i[−

]

log-lawnew formLES

-10 0 10 20 30

E% log−law[%]

0

0.2

0.4

0.6

0.8

1

1.2

z/z i[−

]

ABCDFGHIJKLMNOPQR

-20 -10 0 10 20

E%newform[%]

0

0.2

0.4

0.6

0.8

1

1.2

z/z i[−

]

ABCDFGHIJKLMNOPQR


Top-down stability effect• Strength of capping inversion (Nc), and ABL depth

– LES confirm

42

0 2 4 6 8 10 12

S[m/s]

0

0.2

0.4

0.6

0.8

1

1.2

z/z i[−

]

log-lawnew formLES

-20 -10 0 10 20

E%newform[%]

0

0.2

0.4

0.6

0.8

1

1.2

z/z i[−

]

ABCDFGHIJKLMNOPQR

à need to measure stability from above …1/L from surface is part of the story


when is stability “useful” [in Wind Energy] ?

• i.e. when is it a valuable ‘metric’ or input?• often ‘blamed’; sometimes no need to be account for it directly…

• a few applications where stability is (less/more) helpful/usable input–turbulence length scale, loads inputs–mean turbulence estimation {shear, 1/L} in simple terrain–RANS simulations for wind

»statistically-driven ensembles using P(1/L) not yet over complex terrain…

–vertical extrapolation (at large z2/z1 where shear fails) –top-down (entrainment) effects…–LOTS of indirect stuff (wakes, blockage, terrain)

43


EXTRA Slides

44


Mann-model implied s1 (via shear)

• Eddy-lifetime has implicit shear• IEC-recommended constants inconsistent with shear-independence Here:

Using obs. s1(L,G should deviate…)


> a decade of atmospheric stability research

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