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Large Eddy Simulation of Stable Boundary Layers with a prognostic subgrid TKE equation

8th Annual Meeting of the EMS, Amsterdam, 2008

Stephan R. de Roode and Vincent Perrin

Clouds, Climate and Air Quality, Dept. of Applied Sciences,

Delft University of Technology, Delft, Netherlands

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Contents

Problem/question

- Dutch LES model: Stable boundary layer simulation dominated by subgrid contributions

Strategy

- Analysis of subgrid prognostic TKE model

LES results

- subgrid vs resolved

- similarity relations

- high resolution results

Conclusions

8th Annual Meeting of the EMS, Amsterdam, 2008

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Prognostic subgrid TKE equation (Deardorff 1980)

subgrid fluxes ,

eddy diffusivity

length scale

subgrid TKE

€

u j"ψ" = −Kh∂ψ∂x j

€

u i"u j" = −Km

∂u j

∂x i

+∂u i

∂x j

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Km,h = cm,hλe1/2

€

λ =min Δ,cne1/2

NBV

⎛

⎝ ⎜

⎞

⎠ ⎟

€

∂e∂t

+u j∂e∂x j

=gθ0

w"θv"−u i"u j"∂u i

∂x j

−∂u j" e+p"/ρ( )

∂x j

−ε

8th Annual Meeting of the EMS, Amsterdam, 2008

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GABLS SBL intercomparison case

Neutral layer becomes stable due to a prescribed surface cooling (-

0.25 K/h)

Original set up according to Beare et al. (2003): x=y=z=6.25 m

Length scale correction turned off: λ==(x y z)1/3

ch=cm(ch,1+ch,2,λ) = cm(ch,1+ch,2)

cm=0.12, ch,1=1, ch,2=2

8th Annual Meeting of the EMS, Amsterdam, 2008

LES results: Examples taken from the 5th hour

Turbulent fluxes dominated by subgrid contribution

0

50

100

150

200

264 264.5 265 265.5 266

height (m)

potential temperature (K)

0 2 4 6 8 100

50

100

150

200

U

V

(m/s)

height (m)

-0.006 -0.004 -0.002 0 0.0020

50

100

150

200

subgridresolvedtotal

w'θ ( / )' m Ks

( )height m

-0.02 -0.015 -0.01 -0.005 00

50

100

150

200

subgridresolvedtotal

u'w' (m2/s

2)

height (m)

Solution close to Smagorinsky model's solution

Smagorinsky subgrid TKE solution:

LES subgrid constants: cf=2.5 cm=0.12, c=0.76

corresponding Smagorinsky constant: cs=0.22 €

e =12

cm

cε

Δ2S2 1−chRig( ) =12

cf2

4π 2Δ2S2 1−chRig( )

0 0.02 0.04 0.06 0.08 0.10

0.02

0.04

0.06

0.08

0.1

LES subgrid TKE (m2/s

2)

Smagorinsky subgrid TKE solution (m

2/s

2)

Changing the filter constant cf=2.52

Less filtering more resolved motions

263 264 265 266 267 268

0

50

100

150

200

250

300

potential temperature (K)

height (m)

-0.008 -0.006 -0.004 -0.002 0 0.0020

50

100

150

200

250

300

subgrid

resolved

total

w'θ ( / )' m Ks

( )height m

0 2 4 6 8 10 12

0

50

100

150

200

250

300

U

V

(m/s)

height (m)

-0.03 -0.02 -0.01 0 0.010

50

100

150

200

250

300

subgrid

resolved

total

u'w' (m2/s

2)

height (m)

8

Subgrid constants cm and ch

cm more mixing of hor. winds

c h m

ore

mix

ing o

f pot.

te

mp.

€

Rig =

gθ

∂θv

∂z∂U∂z

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

+∂V∂z

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

Similarity relations

0

10

20

30

40

50

0 2 4 6 8 10

sub>ressub<res1+5z/Λ

/z Λ

0

10

20

30

40

50

0 2 4 6 8 10

sub>ressub<res1+5z/Λ

/z Λ

Solution if solution is 100% subgrid

(Baas et al., 2008)

€

φh

φm

=cm

ch

= PrT =13

Similarity relations: cf=2 (cm=0.096)

0

10

20

30

40

50

0 2 4 6 8 10

sub>ressub<res1+5z/Λ

/z Λ

0

10

20

30

40

50

0 2 4 6 8 10

sub>ressub<res1+5z/Λ

/z Λ

DNS Van der Wiel et al. (2008)

High resolution: x=y=z=1.5626m

0

10

20

30

40

50

0 2 4 6 8 10

sub>res

sub<res

1+5z/Λ

/z Λ

0

10

20

30

40

50

0 2 4 6 8 10

sub>ressub<res1+5z/Λ

/z Λ

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Conclusions

1. =6.25 m resolution not enough

- Solution dictated by Smagorinsky subgrid TKE solution

- too much dependency on subgrid constants: bad simulation

- recommendation: refine grid resolution (smaller )

2. High resolution simulation

- smaller gradient for m and h compared to observations and DNS

€

e =12

cm

cε

Δ2S2 1−chRig( ) =12

cf2

4π 2Δ2S2 1−chRig( )