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 of 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 Turbulence equations 1) Traceless Reynolds stress tensor b u u (2 / 3) E = 〈 〉 − δ ij i j ij D b 4 D ES Z B П + = − − Σ − + − ij ij ij ij ij ij ij Dt 3 2) Turbulent kinetic energy 2 E u / 2 = 〈 〉 i DE 1 U ∂ i D h + = − τ + β − ε ii ij i i Dt 2 x ∂ j

¡ ¡ ¡ Modeling of turbulent stresses and turbulent heat fluxes 3) Turbulent heat fluxes h u = 〈 θ 〉 i i D h U ∂ ∂ Θ h 2 i θ D h П + = − − τ + β θ 〈 〉 − i i j ij i i Dt x x ∂ ∂ j j 2 4) Temperature variance 〈 Θ 〉 D ∂ Θ 2 D 2 h 2 〈 θ 〉 + = − − ε i θ θ Dt x ∂ i

The other tensors are defined as follows: U 1 U ⎛ ⎞ ∂ ∂ j S i = + ⎜ ⎟ ij ⎜ ⎟ 2 x x ∂ ∂ ⎝ ⎠ j i U 1 ⎛ U ⎞ ∂ ∂ j R i = − ⎜ ⎟ ij ⎜ ⎟ 2 x x ∂ ∂ ⎝ ⎠ j i 2 b S S b b S Σ = + − δ ij ik kj ik kj ij km mk 3 Z R b b R = − ij ik kj ik kj 2 B h h h = β + β − δ β ij i j j i ij k k 3 p p 2 ∂ ∂ П u u pu ≡ 〈 〉 + 〈 〉 − δ 〈 〉 ij i j ij k x x 3 ∂ ∂ j i ∂ ( ) D u u ( 1 / 3 )u u u ≡ 〈 − δ 〉 ij i j i i i j k x ∂ k

The pressure-shear /scalar correlations The parameterization of ‘slow’ terms 2 П u p , u p , u p , = 〈 〉 + 〈 〉 − δ 〈 〉 ij i j j i ij k k 3 θ П p , = 〈 θ 〉 i i (1) (1) : : θ П b / , П h / τ τ ij ij i i p θ E / τ = ε c ∂ θ (1) 1 θ П p h θ ≡ 〈 〉 ≅ − i i x ∂ τ i p θ : τ τ p θ

New dependence for the pressure correlation in the stably stratified turbulence θ p Π Π = θ θ i , i u θ θ p i Π Π = θ θ Relaxation linear model for the slow term : � i , i τ p θ ‘Standard’ the SOC models usually assume, that 2 E τ p τ = � θ ε Such closure may not necessarily apply to the stably stratified flows! Because we use the original theoretical work of Weinstock (1989), τ p pointed out that the time scale must include a buoyancy damping θ factor τ τ = ‘Weinstock’s damping factor ’ p θ 2 2 1 a N + τ

RAN ANS-a S-appro roach ch for r turb rbule lent st stra ratif ifie ied flo lows U W ∂ ∂ 0, + = x z ∂ ∂ 2 U U U 1 P u uw ∂ ∂ ∂ ∂ ∂〈 〉 ∂〈 〉 U W D , + + = − − − + u t x z x x z ∂ ∂ ∂ ρ ∂ ∂ ∂ 0 W W W 1 P U W ∂ ∂ ∂ ∂ ∂ ∂ 0, + = U W x z + + = − − ∂ ∂ t x z z ∂ ∂ ∂ ρ ∂ 0 2 uw w ∂〈 〉 ∂〈 〉 g , − + β Θ x z ∂ ∂ u w ∂Θ ∂Θ ∂Θ ∂〈 θ 〉 ∂〈 θ 〉 U W . + + = − − t x z x z ∂ ∂ ∂ ∂ ∂

Three parameter turbulence model E E E E U ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ i U c u u u u g u + = 〈 〉 −〈 〉 + β δ 〈 θ〉 − ε ⎜ ⎟ k E k k i k i 3 i t x x x x ∂ ∂ ∂ ε ∂ ∂ ⎝ ⎠ k k k k 2 E U ⎛ ⎞ ⎛ ⎞ ∂ε ∂ε ∂ ∂ε ε ∂ ε i U c u u c u u g u c + = 〈 〉 + −〈 〉 + β δ 〈 θ〉 − ⎜ ⎟ ⎜ ⎟ k k k 1 i k i 3 i 2 ε ε ε t x x x E x E ∂ ∂ ∂ ε ∂ ∂ ⎝ ⎠ ⎝ ⎠ k k k k 2 2 2 E 1 ⎛ ⎞ ∂〈θ 〉 ∂〈θ 〉 ∂ ∂〈θ 〉 ∂Θ ε 2 U c u u u + = 〈 〉 −〈 θ〉 − 〈θ 〉 ⎜ ⎟ k 2 k k k θ t x x x x R E ∂ ∂ ∂ ε ∂ ∂ k k ⎝ k ⎠ k c 0 22 , ,c 0 18 , ,c 1 40 , ,c 1 90 , ,c 0 22 , ,R 0 6 , ( ) = = = = = = E 1 2 2 ε ε ε θ

Improved Full Explicit Algebraic Models for Reynolds Stresses and Scalar Fluxes : 2D case K E S U V = τ ∂ ∂ ⎛ ⎞ E M M uw , vw K , ( ) < > < > = − τ = ⎜ ⎟ M z z K E S ε ∂ ∂ = τ ⎝ ⎠ H H ∂Θ 1 2 w K ⎧ ⎫ < θ >= − + γ 2 2 1 G s G ( g ) γ = + α + α τβ 〈θ 〉 ⎨ ⎬ H c c 2 M 6 H 5 z D 3 ∂ ⎩ ⎭ 2 2 ∂Θ U V 2 2 ∂ ∂ ⎛ ⎞ ⎛ ⎞ 2 G N G S N g ( ) ( ) 2 ≡ τ ≡ τ = β S ≡ + ⎜ ⎟ ⎜ ⎟ H M z ∂ z z ∂ ∂ ⎝ ⎠ ⎝ ⎠ s 1 s G s s G s s 1 2 1 1 ( ) ⎧ ⎫ + − + × ⎧ ⎫ ⎡ ⎤ 1 ⎣ ⎦ ⎪ ⎪ 0 1 H 2 3 H 4 5 S s G ( ) = + S = ⎨ ⎬ ⎨ ⎬ H 6 H * M D 3 c θ D ( 2 ) 1 s G g / E ( ) × + τβ 〈θ 〉 ⎩ ⎭ ⎪ ⎪ 1 6 H ⎩ ⎭ 2 2 D 1 d G d G d G G d G ( d G d G G ) G = + + + + + − 1 M 2 H 3 M H 4 H 5 H 6 M H H

Vertical profiles of mean temperature

Vertical profiles of mean U velocity 0.7 0.6 0.5 0.4 Z (m) Ohya's data 0.3 S1 (U=1.50 m/s) Ohya's data S2 (U=1.20 m/s) 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -1 ) U (ms

Vertical profiles of velocity fluctuations

Turbulent Prandtl number as function of Richardson number 1.5 2 1.4 1.3 1 1.2 1.1 Pr T 1 0.9 0.8 0.7 0.6 -1 0 1 2 10 10 10 10 Ri g

Time history of gradient Richardson number ∂Θ g z β ∂ é Ri = g 2 U ⎛ ⎞ ∂ ⎜ ⎟ z ∂ ê ⎝ ⎠

Thermal Stratified Boundary Layer over Flat Terrain T he potential temperature θ and velocity U are shown for the convective and stable boundary layers.

Velocity profile in SBL with Low-Level Jet 0.4 -1 U G =8 ms 0.35 simulation 0.3 LES data 0.25 (Beare et al. 2005) z ,km 0.2 0.15 0.1 0.05 0 4 5 6 7 8 9 10 -1 ) U (m s

The potential temperature in the SSBL 0.4 The ¡surface ¡temperature ¡(265 ¡K ¡ ini5ally) ¡decreasing ¡at ¡a ¡constant ¡ initial profile rate ¡of ¡0.05 ¡K/h. ¡ ¡Such ¡a ¡profile ¡ 0.3 Measurements: developed ¡into ¡the ¡observed ¡ BASE data z (km) simulation profile ¡(square ¡symbols ¡at ¡the ¡ leD ¡on ¡a ¡figure) ¡aDer ¡8 ¡h ¡of ¡ 0.2 simula5on. ¡ ¡ ¡ The ¡elevated ¡inversion ¡layer ¡within ¡the ¡ 0.1 SBL, ¡similar ¡to ¡the ¡ones ¡here, ¡have ¡been ¡ found ¡by ¡Kosovic ¡and ¡Carry ¡(2000) ¡on ¡ the ¡Arc5c ¡sea ¡in ¡their ¡LES ¡simula5ons . 0 260 265 270 [K] Θ

Model results for total horizontal wind speed 400 Time variation of the total horizontal wind speed . 12:00 ê The ground temperature was 300 24:00 specified as (x,0,t) 6 sin( t/43200) Θ = ⋅ π z m 200 This is the only nonstationary boundary condition of the problem, which models the 12- hour cycle of solar heating of the 100 Earth's surface with decreasing at a constant rate of 0.6 K/h. 3 4 5 6 7 8 9 10 U m/sec

Time variation of total horizontal wind speed • Observations data • Modeling results 10 10 9 9 8 8 123 M 121 M 7 7 -1 -1 U , ms U , ms 6 6 5 5 4 4 2 M 3.125 M 3 3 2 2 0 5 10 15 20 25 5 10 15 20 25 Time, hours Time, hours

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