The Structure of Nocturnal Urban Boundary Layer: Study with Nonlocal - PowerPoint PPT Presentation

The Structure of Nocturnal Urban Boundary Layer: Study with Nonlocal Mesoscale Model Albert F. Kurbatskiy Institute of Theoretical and Applied Mechanics of Russian Academy of Sciences, Siberian Branch and Novosibirsk State University, Russia Ludmila I.Kurbatskaya Institute of Computational Mathematics and Mathematical Geophysics of Russian Academy of Sciences, Siberian Branch , Russia ENVIROMIS_ 2008, Tomsk

O U T L I N E: � Introduction � Improved turbulence model for stratified turbulent flows � Modeling and Simulation of Stratified Boundary Layer over urban-like roughness: Œ Structure features of turbulent transport in an thermal unstable stratified boundary layer Œ Structure features of turbulent transport in a stable thermal stratified boundary layer � Conclusion ENVIROMIS_ 2008, Tomsk

RANS ( Reynolds Average Navier-Stokes ) approach for Turbulence Modeling ENVIROMIS_ 2008, Tomsk

GOVERNING EQUATIONS FOR TURBULENT STRATIFIED FLOWS ∂ ∂ DU 1 P = − τ − − − ε Ω i g 2 U ; ∂ ρ ∂ ij i ijk j k Dt x x j i Θ ∂ D = − ∂ h , j Dt x j . ≡ 〈 〉 τ u u are the Reynolds stresses ij i j ≡ 〈 〉 u θ a re the hea t f lu xes h i i ENVIROMIS_ 2008, Tomsk

‘ Traditional’ turbulent models for stratified boundary layers Œ E-L model: К -theory for all (!) turbulent fluxes: ( ) −〈 〉 = ∂ ∂ = wa K A / z ; K E; l z z = 〈 〉 ⇒ (E u u / 2 TK E) i i ε = ε 2 K c E / Œ E- model: µ z →  0,09 x direction =  c µ →  the function of Ri number z direction f Ri - Richardson flux number f ENVIROMIS_ 2008, Tomsk

‘ Traditional’ turbulent models for stratified boundary layers Usually used the semi-empirical modification of K- theory, with an additional countergradient term that incorporates the contribution of large scale eddies to total flux: ( ) ′ ′ = − φ ∂ φ ∂ − γ γ w K / z , where is the countergradient term φ C C This non-local flux correction is applied only to temperature for convective mixing case, but not for the stable case! θ ' 2 g γ = ≈ × − − 5 0 1 0 . 5 10 K cm n Deardorff (1972): Θ C ' 2 w 0 ( φ ' ' w ) γ = − = u Φ = 0 1 b w S b 7 . 8 n The MRF scheme: C * w h S ENVIROMIS_ 2008, Tomsk

Turbulence equations • τ ≡〈 〉 Reynolds stresses, u u ij i j D Dt τ + = +β +β − − ε П D P h h ij ij ij i j j i ij ij ∂ ∂ ∂ ∂   ∂ p p 2 U U = 〈 〉 + 〈 〉 − δ 〈 〉  j  П = − τ + τ i u u pu P   ∂ ∂ ∂ ij i j ij k ij ik ∂ jk ∂ x x 3 x   x x k k j i k ∂ ∂ ∂   u 2 u 2 ≡ 〈 〉 + δ 〈 〉  ε = ν 〈 j 〉 = δ ε  β ≡ β i D u u u pu 2 g ij ∂ i j k ij k ij ∂ ∂ ij   i i x 3 x x 3 k k k • ≡〈 〉 u θ Heat fluxes, h i i ∂ ∂ Θ Dh U + = − − τ +β 〈θ 〉 − П , h 2 i i D h θ ∂ ∂ i j ij i i Dt x x j j ∂ ∂ p ≡ 〈 θ 〉 = 〈 θ 〉 П h , D u u θ i ∂ ∂ i i j x x i j ENVIROMIS_ 2008, Tomsk

Full Explicit Algebraic Models for Reynolds Stresses and Scalar Fluxes ( ) D 4 + = = − − Σ + Ζ + − Ï b D 0 E S B ij ij ij ij ij ij ij Dt 3 ∂ ∂ Θ Dh U + = = − − τ + β 〈 θ 〉 − Ï h 2 , i D 0 h i ∂ ∂ θ i j ij i i Dt x x j j 〈 〉 ≡ 〈 θ 〉 u i u h u : Coupled algebraic system equations for and i i j ( ) • = − α τ − α Σ + Ζ + α b E S B ij 1 ij 2 ij ij 3 ij ∂ Θ   2 • = − τ + δ + α τ β δ 〈 θ 〉 2   A h b E g ∂ ij j ij ij 4 i 3   3 x j ENVIROMIS_ 2008, Tomsk

Improved Full Explicit Algebraic Models for Reynolds Stresses and Scalar Fluxes : 2D case = τ ∂ ∂   U V K E S ( ) E < > < > = − M M τ = uw vw K   , , ε ∂ ∂ M = τ   z z K E S H H ∂Θ   1 2 < θ >= − + γ γ = + α + α τβ 〈θ 〉 w K  2  2 G s G g 1 ( ) ∂ H c c M H  2 6  5 z D 3 ∂Θ ∂ 2 ∂ 2 ( ) ( )     U V ≡ τ 2 ≡ τ 2 = β 2 ≡ + G N G S N g 2     S ∂ H M ∂ ∂     z z z [ ] ( )   + − + ×   1 2 1 1 s 1 s G s s G s s ( ) 1 = +   = 0 1 H 2 3 H 4 5 S s G   ( ) S ( ) H 6 H M × + τ β 〈 θ 〉   2 3 D c θ D   1 s G g / E 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 ENVIROMIS_ 2008, Tomsk

Three-parametric turbulence model = 〈 〉 Turbulent kinetic energy E (1/2) u u i i ∂ DE 1 U + = − + − τ β h ε , i D ∂ ii ij i i Dt 2 x j , ε TKE dissipation 2 D ε ε + = − Ψ , D ε Dt E 〈θ 〉 2 Temperature variance, 〈 〉 ∂ 2 D θ Θ + = − − 2 ε , D 2h θ 2 i ∂ θ Dt x i ENVIROMIS_ 2008, Tomsk

Thermal Stratified Boundary Layer over Flat Terrain T he potential temperature θ and velocity U are shown for the convective and stable boundary layers. ENVIROMIS_ 2008, Tomsk

Boundary Layer over Urban-Like Roughness ENVIROMIS_ 2008, Tomsk

Parameterization of Urban-Like Roughness ˘ Numerical models of urban ˘ Because is not possible to boundary layer must be able to resolve city structure in detail resolve two main scales: (buildings or blocks), the ‘urban’ and ’meso’. effects of urban surfaces must be parameterized . Horizontal dimensions of ˘ � domain are on order of meso’scale (~100 km) and for keeping number of grid points compatible with CPU time cost, the horizontal grid resolution of such (meso’scale) models ranges between several v hundreds of meters and a few kilometers. ENVIROMIS_ 2008, Tomsk

Governing Equations for SBL 2D case: + = U W 0, x z 1 + U +UU +WU = - P - wu +fV D , t x z x u ρ z − + V +UV + WV = - wv fU D , t x z v z 1 + + = − − + βΘ 2 W UW WW P w g , ρ t x z z z 0 Θ + Θ + Θ = −〈 θ 〉 − θ + U W u w D . θ t x z x z ENVIROMIS_ 2008, Tomsk

Parameterization of Aerodynamical Roughness Effects The extra terms D A in the Governing Equations are: D U = turbulent momentum flux (horizontal surfaces) + drag (vertical surfaces) D θ = turbulent heat fluxes from horizontal surfaces, and the temperature fluxes from the vertical surfaces D E = increasing of mean kinetic energy conversion into the TKE [for example, Raupach and Shaw (1982), Raupach et al.(1991), Raupach (1992)] 1/2 E ˆ = L ε D =c “ a ‘second’ dissipation linked with the ( c 0.7) ε p ε p ε turbulence lengthscale L, and induced by the presence of the roughness elements ENVIROMIS_ 2008, Tomsk

Thermal Effects of Urbanized Area : ‘Heat Island‘ Œ In this modeling, the UHI effect was specified by an urban-environs temperature difference. Œ The ground temperature was specified as Θ = ⋅ π (x,0,t) 6 sin( t/43200) This is the only nonstationary boundary condition of the problem, which models the 24-hour cycle of solar heating of the Earth's surface. Œ At the ground turbulent fluxes are computing using the MOST according to the non-iterative formulation. ENVIROMIS_ 2008, Tomsk

Convective Mixing Case ENVIROMIS_ 2008, Tomsk

Thermal circulation above the roughness area 2.5 1 - 4 0 . 0 . 1.5 0 1 . 1 2 . . 1 0 . 0 0 0.6 1 1.0 1.0 -1 . U=3 mc 12:00 5 1.0 - 2 8 0 12:00 . . 0 4 0.0 -1 -1.0 U G =1 ms 1 . 5 1.2 2.0 -1.5 -0.5 1 3.0 2.5 . 1.0 0 1.4 1.0 1 1.0 1.5 -0.5 0 1 . - 1.0 0.4 1 Z, km . 0 0 . 1 0 -0.2 . 1.0 1 Z / Z i 6 . 0 1.8 1.2 0.6 1.0 1 0 0 . 1.0 -0.2 1.4 - 0 . 5 1.6 2 1.5 0.5 . 1 -1.0 1.0 0.5 2 0.4 . -1.5 0 2.0 - 0.8 4 - . 2 0 . 0 0.0 2.0 -2.5 1 0 . 5 40 roughness area 60 X, km 0 45 50 55 X, km ENVIROMIS_ 2008, Tomsk

Vertical profiles of local friction velocity 7 Real scale data 6 Oikawa and Meng Rotach M. Feigenwinter C. 5 Computation -1 U G =3 ms 1 2 -1 U G =5 ms 4 Z / Z H 3 2 1 0 -1 0 1 2 3 u * / u *max ENVIROMIS_ 2008, Tomsk

Vertical profiles of ratio u * /U 7 Computation: U G =3ms -1 6 -1 U G =5ms Real scale data from Roth (2000) 5 1 2 4 Z / Z H 3 2 1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 u * / U ENVIROMIS_ 2008, Tomsk

Vertical velocity variance 7 Simulation: -1 1- U=3ms 6 -1 2- U=5ms Experimental data from Roth's review Q. J. R. Meteor. Soc. 5 2000. Vol.126, 941-990. 2 1 4 Z / Z H 3 2 1 1 1.5 2 2.5 2 >) 1/2 /u *max (<w ENVIROMIS_ 2008, Tomsk