Modeling and simulation the stable stratified boundary layer with - - PowerPoint PPT Presentation

Modeling and simulation the stable stratified boundary layer with low-level jet: comparison with the wind tunnel data.

• L. I. Kurbatskaya

Inst stit itute of Comp mputatio ional l Ma Mathema matics ics and Ma Mathema matica ical l Geophysics ysics

• f Russia

ssian Aca Academy my of Scie Science ces, s, Sib Siberia rian Bra Branch ch, Russia ssia

Exp Experime rimental l arra rrangeme ment for r SBL SBL wit ith lo low- le leve vel l je jet

Modeling of turbulent stresses and turbulent heat fluxes

¡ ¡ ¡

2

1) (2 / 3) 4 3 2) / 2 1 2

ij i j ij ij ij ij ij ij ij ij i i ii ij i i j

Turbulence equations Traceless Reynolds stress tensor b u u E D b D ES Z B П Dt Turbulent kinetic energy E u U DE D h Dt x = 〈 〉 − + = − − − + − = 〈 〉 ∂ + = − + − ∂ δ Σ τ β ε

Modeling of turbulent stresses and turbulent heat fluxes

¡ ¡ ¡

2 2 2

3) 4) 2 2

i i h i i i j ij i i j j i i

Turbulent heat fluxes h u U D h D h П Dt x x Temperature variance D D h Dt x = 〈 〉 ∂ ∂ + = − − + 〈 〉 − ∂ ∂ 〈 〉 ∂ 〈 〉 + = − − ∂

θ θ θ

θ Θ τ β θ Θ Θ θ ε

The other tensors are defined as follows:

( )

2 1 2 1 2 2 3 2 3 1 3 3

j i ij j i j i ij j i ij ik kj ik kj ij km mk ij ik kj ik kj ij i j j i ij k k ij i j i i i ij i j ij k j i j k k

U U S x x U U R x x b S S b b S Z R b b R p p П u B h h h D u u ( / )u u u x u pu x x ⎛ ⎞ ∂ ∂ = + ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎛ ⎞ ∂ ∂ = − ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ Σ = + − δ = − ∂ ∂ ≡ 〈 〉 = β + β − δ β ∂ + 〈 ≡ 〈 − δ 〉 ∂ 〉 − δ 〈 〉 ∂ ∂

The pressure-shear /scalar correlations

The parameterization of ‘slow’ terms (1) (1) (1) 1

, / 2 , , , 3 / , / П

i i ij ij ij i j j i ij k k p i i p i i i p

П p П b П h c p h x П u p u p u p E

θ θ θ θ θ θ θ

θ τ τ θ δ τ ε τ τ τ = 〈 〉 ∂ ≡ 〈 〉 ≅ = 〈 〉 + 〈 − ∂ 〉 〉 = − 〈 : : :

New dependence for the pressure correlation in the stably stratified turbulence

Relaxation linear model for the slow term: ‘Standard’ the SOC models usually assume, that

• Such closure may not necessarily apply to the stably stratified flows!

Because we use the original theoretical work of Weinstock (1989), pointed out that the time scale must include a buoyancy damping factor ‘Weinstock’s damping factor’

=

, θ

Π θ Π θ

i i

p

=

, θ

Π θ Π θ

i i

p

θ

θ τ

i p

u

θ

τ p 2 = τ ε E

θ

τ p

2 2

1 = +

θ

τ τ τ

p

a N

RAN ANS-a S-appro roach ch for r turb rbule lent st stra ratif ifie ied flo lows

0, U W x z ∂ ∂ + = ∂ ∂

2

1 ,

u

U U U P u uw U W D t x z x x z ρ ∂ ∂ ∂ ∂ ∂〈 〉 ∂〈 〉 + + = − − − + ∂ ∂ ∂ ∂ ∂ ∂

2

1 , W W W P U W t x z z uw w g x z ρ β ∂ ∂ ∂ ∂ + + = − − ∂ ∂ ∂ ∂ ∂〈 〉 ∂〈 〉 − + Θ ∂ ∂ . u w U W t x z x z ∂Θ ∂Θ ∂Θ ∂〈 〉 ∂〈 〉 + + = − − ∂ ∂ ∂ ∂ ∂ θ θ 0, U W x z ∂ ∂ + = ∂ ∂

Three parameter turbulence model

3 i k E k k i k i i k k k k

U E E E E U c u u u u g u t x x x x ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ + = 〈 〉 −〈 〉 +β δ 〈 θ〉 − ε ⎜ ⎟ ∂ ∂ ∂ ε ∂ ∂ ⎝ ⎠

2 1 3 2 i k k k i k i i k k k k

U E U c u u c u u g u c t x x x E x E

ε ε ε

⎛ ⎞ ⎛ ⎞ ∂ ∂ε ∂ε ∂ ∂ε ε ε + = 〈 〉 + −〈 〉 +β δ 〈 θ〉 − ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ε ∂ ∂ ⎝ ⎠ ⎝ ⎠

2 2 2 2 2

1

θ

⎛ ⎞ ∂〈θ 〉 ∂〈θ 〉 ∂ ∂〈θ 〉 ∂Θ ε + = 〈 〉 −〈 θ〉 − 〈θ 〉 ⎜ ⎟ ∂ ∂ ∂ ε ∂ ∂ ⎝ ⎠

k k k k k k k k

E U c u u u t x x x x R E

( )

1 2 2

0 22 0 18 1 40 1 90 0 22 0 6

ε ε ε θ

= = = = = =

E

c , ,c , ,c , ,c , ,c , ,R ,

Improved Full Explicit Algebraic Models for Reynolds Stresses and Scalar Fluxes : 2D case

( )

M

U V uw , vw K , z z ∂ ∂ ⎛ ⎞ < > < > = − ⎜ ⎟ ∂ ∂ ⎝ ⎠

H c

w K z ∂Θ < θ >= − + ∂ γ

M M

K E S τ =

H H

K E S τ =

( )

2 H

G N τ ≡

( )

2 M

G S τ ≡

2

N g z β ∂Θ = ∂

2 2 2

U V S z z ∂ ∂ ⎛ ⎞ ⎛ ⎞ ≡ + ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠

H H M H H H M H M

G G G d G d G d G G d G d G d D ) ( 1

6 2 5 2 4 3 2 1

− + + + + + =

ε τ E =

( ) ( )

( )

1 2 3 4 5 H H M 2 6 H

s 1 s s s s s 1 D 1 s g G G / G S E ⎧ ⎫ + − + × ⎡ ⎤ ⎣ ⎦ ⎪ ⎪ = ⎨ ⎬ × + τβ 〈θ 〉 ⎪ ⎪ ⎩ ⎭

( )

6 * 1

1 2 1 1 3 ⎧ ⎫ = + ⎨ ⎬ ⎩ ⎭

H H

S s G D c θ

2 2 2 6 5

1 2 1 ( ) 3 ⎧ ⎫ + α + α τβ 〈θ ⎨ ⎬ ⎩ ⎭ γ = 〉

M H c

G s G g D

Vertical profiles of mean temperature

Vertical profiles of mean U velocity

U (ms

• 1)

Z (m)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7

S2 (U=1.20 m/s) S1 (U=1.50 m/s) Ohya's data Ohya's data

Vertical profiles of velocity fluctuations

Turbulent Prandtl number as function of Richardson number

Rig PrT

10

• 1

10 10

1

10

2

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

1 2

Time history of gradient Richardson number

g 2

g z Ri U z ∂Θ β ∂ = ⎛ ⎞ ∂ ⎜ ⎟ ∂ ⎝ ⎠

ê é

Thermal Stratified Boundary Layer over Flat Terrain The potential temperature θ and velocity U are shown for

the convective and stable boundary layers.

Velocity profile in SBL with Low-Level Jet

U (m s

• 1)

z ,km

4 5 6 7 8 9 10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

UG=8 ms

• 1

simulation LES data (Beare et al. 2005)

The potential temperature in the SSBL

The ¡surface ¡temperature ¡(265 ¡K ¡ ini5ally) ¡decreasing ¡at ¡a ¡constant ¡ rate ¡of ¡0.05 ¡K/h. ¡ ¡Such ¡a ¡profile ¡ developed ¡into ¡the ¡observed ¡ profile ¡(square ¡symbols ¡at ¡the ¡ leD ¡on ¡a ¡figure) ¡aDer ¡8 ¡h ¡of ¡

• simula5on. ¡ ¡

¡

The ¡elevated ¡inversion ¡layer ¡within ¡the ¡ SBL, ¡similar ¡to ¡the ¡ones ¡here, ¡have ¡been ¡ found ¡by ¡Kosovic ¡and ¡Carry ¡(2000) ¡on ¡ the ¡Arc5c ¡sea ¡in ¡their ¡LES ¡simula5ons. Θ

z (km)

260 265 270 0.1 0.2 0.3 0.4

[K]

Measurements: BASE data initial profile simulation

Model results for total horizontal wind speed

Time variation of the total horizontal wind speed . êThe ground temperature was specified as This is the only nonstationary boundary condition of the problem, which models the 12- hour cycle of solar heating of the Earth's surface with decreasing at a constant rate of 0.6 K/h.

U m/sec z m

3 4 5 6 7 8 9 10 100 200 300 400

12:00 24:00

(x,0,t) 6 sin( t/43200) Θ = ⋅ π

Time variation of total horizontal wind speed

• Modeling results
• Observations data

Time, hours U , ms

• 1

5 10 15 20 25 2 3 4 5 6 7 8 9 10

121 M 2 M

Time, hours U , ms

• 1

5 10 15 20 25 2 3 4 5 6 7 8 9 10

123 M 3.125 M

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