Direct and Large Eddy Simulation of stably stratified turbulent Ekman layers

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1 Direct and Large Eddy Simulation of stably stratified turbulent Ekman layers Stimit Shah, Elie Bou-Zeid Princeton University 64 th APS DFD Baltimore, Maryland Nov 21, 211

2 Effect of Stability on Atmospheric Boundary Layer ABL: lowest layer that interacts with the Earth s surface, affected by heterogeneities, terrain and diurnal cycle of solar radiation Daytime ABL (Unstable): Surface temperature higher than air temperature, strong convective turbulence, larger eddies and strong mixing. Kaimal and Finnigan 1994

3 Effect of Stability on Atmospheric Boundary Layer ABL: lowest layer that interacts with the Earth s surface, affected by heterogeneities, terrain and diurnal cycle of solar radiation Nighttime ABL (Stable): Complicated by weak & intermittent turbulence, shallow BL, laminarization, anisotropy, waves, jets, smaller eddies. Kaimal and Finnigan 1994

4 Motivation: Effects Of Stability Stability causes drastic reduction in turbulent fluxes and heat transfer Large scale models based on MOST (the surface layer is assumed to have either constant vertical heat flux) or a modified form that uses local heat flux Vertical extent over which these models are valid shrinks with increasing stability King et. al. compared 4 such models in simulations over Antarctic & found variation in surface heat flux of over 2 W/m 2 - shows lack of understanding of SABLs F.T.M Nieuwstadt JAS-1984; L. Mahrt,Theoret. & Computat. Fluid Dynamics-1998; O. Williams & A. J. Smits TSFP-211, King et. al. Q.J.R. Meteorol. Soc. 21, Smith et. al. JGR 1983

5 Direct Numerical Simulation and Large Eddy Simulation Experiments - Capture effects of all interacting features Computations - To study effect of stability on idealized flows Today: Validation of the DNS code and the new LES model for scalars, with some basic analysis showing the effect of stability on low Re turbulent Ekman layers.

6 Governing Equations: DNS u i t + u ui j u j x j x i u i = x i «= 1 ρ p x i + 2 ɛ ijk Ω j (G k u k ) + δ i3 gφ + ν 2 u i 2 x j x j Φ t + (u iφ) = κ 2 Φ x i 2 x j x j P = 2 ρ ɛ ijk Ω j G k + δ i3 ρ g x i Φ = (T T )/T where, Ω is the angular velocity, G is geostrophic wind and p is the deviation from the imposed uniform pressure field P. ν are κ are kinematic viscosity and thermal diffusivity. u = at z =, u G as z Φ = Φ at z =, Φ as z Coleman et. al., Direct simulation of stably stratified turbulent Ekman layer, JFM 92

7 Filtered LES Equations eu = Z G(r, x) u(x r, t) d 3 r eu i t + eu eui j eu j x j x i eu i = x i «= 1 ep τ ij + 2 ɛ ijk Ω j (G k eu k ) ρ x i x j + δ i3 g e Φ + ν 2 eu i 2 x j x j Φ e t + eu Φ e i = σ i + κ 2 Φ e x i x i 2 x j x j where ep = ep ρτ kk ρeu j eu j τ ij = gu i u j eu i eu j σ i = g u i Φ eu i e Φ LES Modeling: Parameterize τ ij and σ i in terms of resolved variables

8 Lagrangian-Averaged Scale Dependent Dynamic Model Model is based on relation between resolved stress/fluxes at different scales Lij = Tij τ ij = u ei u ej u ei u ej Ki = Qij = b b Γij Tbij = u ed ej u ei u ej iu κi = e e u Σi σ i = u ei Φ ei Φ d b b bi = u e u e Si Σ ei Φ ei Φ Use Smagorinsky model and assume power law dependence as a function of scale of coefficient. Some sort of averaging is required to stabilize the coefficient. 2 Cs, = 4 Cs,α 2 Cs,α 2 (e), ( ) and (b) correspond to filtering at scales, α and α2 Germano et al., Meneveau et al., Porte-Agel et al., Bou-Zeid et al., Stoll & Porte-Agel

9 Specifications Of The Code and Simulations Carried Out Code uses pseudospectral approach in horizontal directions, 2 nd order central difference with staggered grid in vertical Time integration is second order Adams-Bashforth Horizontal BCs - periodic, vertical BC - no slip at surface, stress free BC at top and imposed constant temperature at the surface with stable temperature profile above, similar to Coleman et. al. All the simulations are over smooth walls Parameters DNS Re f = 4 DNS Re f = 6 LES Re f = 6 u /U g δ t = u /f c (m) L x x L y x L z 2δ t x 2δ t x 1.5δ t 2δ t x 2δ t x 1.5δ t 2δ t x 2δ t x 1.5δ t n x x n y x n z 128 x 128 x x 256 x x 128 x 384 δx + x δy + x δz x 5.28 x x 4.92 x x 1.26 x 2.23 dt (s) period (tf c) 2π 2π 2π Re δt = Ugδ t ν 5, 2 1, 65 1, 65 Re f = U g ( 1 2 νfc)1/2 f c = 2 Ω v sinφ and φ = 9

10 Mean Velocity Profile Present DNS Re f = 4 Present LES Re f = 6 Present DNS Re f = 6 Morris et. al. Re = 4 f Shingai, Kawamura Re f = 6 Shingai, Kawamura Re f = Q Q + = (u2 +v 2 ) 1/2, z u + = zu ; Shingai,Kawamura JoT 23, Morries et. al. JoT ν 211, Dotted line is Q + = 1.41 ln(z+ ) z +

11 Root Mean Square Fluctuations of Velocity, Re f = Total streamwise spanwise vertical Coleman et. al. total Coleman et. al. streamwise Coleman et. al. spanwise Coleman et. al. vertical q/u * z/δ t Profiles of r.m.s. fluctuations of velocity components q 1 = ( u u ) 1/2, q 2 = ( v v ) 1/2, q 3 = ( w w ) 1/2 in comparison with Coleman et. al. simulations JFM 1992

12 TKE Budget, Neutral, Re f = 4 Loss Gain T ii /2 P ii /2 Π ii /2 D ii /2 ε ii /2.5.1 y/δ t Terms in turbulent kinetic energy (k) budget equation non-dimensionalized by u 4 /ν. T ii /2 = w k/ z, P ii /2 = u w U/ z v w V/ z, D ii /2 = ν 2 k/ 2 z 2, Π ii /2 = w p / z, and ɛ ii /2; Dotted lines is the residue

13 TKE Budget, Neutral & Stable, Re f = 4 Loss Gain T ii /2 P ii /2 D ii /2 ε ii /2 B ii /2.5.1 y/δ t Terms in turbulent kinetic energy (k) budget equation non-dimensionalized by u 4 /ν. T ii /2 = w k/ z, P ii /2 = u w U/ z v w V/ z, D ii /2 = ν 2 k/ 2 z 2, Π ii /2 = w p / z, ɛ ii /2 and B ii /2 = g w Φ ; : Neutral, : Stable

14 Temperature Profiles: Stable case, Re f = 4 z/l z DNS tf = z/l z SMAG (a) g δ T / U 2 x 1 3 t x,y g (b) g δ T / U 2 x 1 3 t x,y g 1.8 LASD z/l z.6.4 z/l z (c) g δ t T x,y / U g x 1 3 (d) g δ t T x,y / U g 2 x 1 3 (c): Result with the newly developed LASD model for scalars; (d) From Coleman et. al. simulations for comparison

15 Reynolds Shear Stress and TKE, Re f = 4 z/δ t tf = z/δ t τ / ρ / U 2 x 1 4 g q 2 /U g Reynolds shear stress magnitude, τ/ρ = ( u w 2 + v w 2 ) 1/2 profiles with time; and profiles of twice the TKE q 2 = ( u u + v v + w w )

16 DNS Velocity and Temperature Spectra, Re f = slope = 5/3 inertial subrange 1 5 slope = 5/3 E u (k x ) U g 2 z z/δ t = E * T (k x ) T 2 o z k x z k x z One dimensional spectra of velocity and temperature in the streamwise direction: E T (kx) = gδte T (k x)/u 2 15

17 Instantaneous Vorticity Plots Instantaneous x-vorticity (ω x) plot in a plane YZ: Comparison between neutral and stable case; y and z non-dimensionalized by δ t, and vorticity magnitude non-dimensionalized by U g and δ t

18 Summary and Future Work Developed Lagrangian-averaged scale dependent dynamic model for active and passive scalars Carried out wall-resolved LES with LASD model and DNS at low Reynolds numbers Tested the code for neutral and stable turbulent boundary layers. The profiles of velocity, temperature, fluxes and variances and spectra respond to the change Re and stability realistically Currently working on wall-resolved LES of higher Re number turbulent Ekman layers, and in future will carry out wall-modeled simulations at even higher Re numbers Future work: Study the effect of increasing stability and with heterogeneities in surface roughness and fluxes

19 Project funded by NSF Physical and Dynamic meteorology: AGS Princeton University, Siebel Energy Grand Challenge THANK YOU

20

21 BACK-UP SLIDES

22 Neutral DNS: Friction velocity & surface shear stress angle u * /U g.4 β u * (t) Coleman et. al. <u * (t)> DNS <u * (t)> time (hrs) 5 β(t) Coleman et. al. <β(t)> DNS <β(t)> time (hrs) Re δ = 52, δ is the boundary layer height, u is friction velocity, U gorg is the geostrophic velocity, β is the surface shear stress angle with geostrophic wind

23 Wind speeds and Hodograph U/U g V/U g Coleman et. al. U/U g DNS Coleman et. al. laminar Coleman et. al. V/U g Z 1 <v/u g > Wind speed <u/u g >

24 Reynolds stress and Eddy viscosity <u w > <v w > Coleman et. al. <u w > Coleman et. al. <v w > z/δ 1 z/δ Reynolds stress/u * ν t Eddy viscosity is non-dimensionalized by δu

25 Stable DNS u * /U g.4 β tf tf u * /U g β tf tf Time history of friction velocity at the surface, u, and angle (deg.) between the geostrophic wind and surface shear stress, β. (c),(d) Coleman et. al. simulations JFM 1992, dotted lines correspond to neutral case

26 Stable DNS z/l z z/l z tf = <u>/u <v>/u g g g D <T> / U g x 1 z/l z z/l z <u>/u g <v>/u g g D <T> / U g 2 x 1 3 Mean temperature and velocity profiles with time. (a),(b) Coleman et. al. simulations JFM 1992

27 Stable DNS z/δ tf = z/δ τ / ρ / U 2 g x q 2 /U g 1.8 z/δ z/h τ / ρ / U g 2 x Ri gr Reynolds shear stress magnitude, τ/ρ = ( u w 2 + v w 2 ) 1/2 profiles with time; Profiles of r.m.s. fluctuations of velocity q 2 = ( u u + v v + w w ); figure (3) Coleman et. al. simulations JFM 1992; Ri gr = g d T /dz T (d u /dz) 2 +(d v /dz) 2

28 DNS velocity and temperature spectra E u (k x ) U g 2 z E T (k x ) T o 2 g δ u* 2 (z/δ) 1 x k x z 1 k x z

29 Log law Van Driest Cs=.16 Van Driest Cs=.65 scale invariant scale dependent DNS (1/.41)*log(z + ) Wall resolved LES Vs DNS 12 Q z+ DNS grid size with x + = y + = 5.3, z + 1 =.61, z + 2 = DNS grid of with x+ = y + = 3.5, z + 1 =.41, z+ 2 =.82 have been tried without significant difference LES grid size with x + = y + 1.5, z , z+ 2 = 2.4

30 Re f = 4: LES Vs DNS: Stable case Z DNS SMAG LASD.5 1 Wind speed (m/s) u * /U g DNS SMAG LASD 5 1 time (hrs) Ri x DNS 5 SMAG LASD tf E/DG Ri = g d T dz D2 /(T U 2 g ) DNS SMAG reso LASD reso tf

31 Instantaneous Vorticity Plots Instantaneous y-vorticity (ω y) plot in a plane XZ: Comparison between neutral and stable case; x and z non-dimensionalized by δ t, and vorticity magnitude non-dimensionalized by U g and δ t

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