Quantifying the role of turbulence on droplet growth by condensation

Transcription

1 Quantifying the role of turbulence on droplet growth by condensation Gaetano Sardina Department of Meteorology, SeRC, Stockholm University, Sweden In collaborabon with F. Picano, L. Brandt and R. Caballero FLOWING MATTER 2016 CONFERENCE Porto 14th January 2016

2 Warm Clouds Rain produced in warm clouds 30% of the total planet rainfall 70% of total rain in tropics Important for Earth climate and climate changes Droplet size distribution affects albedo and cloud fractionà substantial climate effects

3 Rain forma0on Activation of Cloud Condensation Nuclei (CCN) Growth by condensation Growth by mixing/collision due to turbulence Growth by gravitational collision

4 The bo5leneck problem A broad droplet size distribution necessary for rain formation From CCN to final distribution in few minutes (15/20 minutes) - droplet growth should take hours according to current theories - nobody knows how rains drops form so quickly, a problem known as the condensation-coalescence bottleneck

5 Complicated Mul0scale Problem Freezing and ice particle growth Evaporation Ice melting Collisional growth Condensational growth CCN activation DNS 100 µm 1 cm 1 m LES GLOBAL CIRCULATION MODELS 100 m 10 km 1000 km Pictures taken from Bodenschatz et al., Science 2010 Direct numerical simulations: resolving all the details of the flow Limited value of Reynolds number also with massive supercomputers Largest simulation so far: 512 3, 10 8 droplets (cube of 70 cm), Lanotte et al 2009 Ideal simulation of homogeneous cloud: , droplets (100 m)

6 Turbulence and droplet Condensa0on DNS: a Brief Excursus turbulence has been indicated as the key missing link to solve condensabon/ collision coalescence problem first DNS of turbulence/cloud interacbons done by Vaillancourt et al Domain 10cm, resolubon 80 3 grid points, droplets 50000à Conclusion: negligible effect of the small-scale turbulence on droplet spectra broadening Celani et al. 2007, resolving large-scale fluctuabons 2D cloudà Conclusion: dramabc increase in the width of the droplet spectrum is qualitabvely found although the dynamics of the small scales is not resolved Paolo & Sharif, 2009, same conclusions but 3D simulabon obtained adding an arbitrary large-scale forcing on the supersaturabon equabon field. Current state of the art: Lano_e et al 2009, 3D DNS simulabons increasing size of the cloud up to 70 cmà turbulence affects droplet spectra broadening mechanism by increasing the cloud size.

7 Our objec0ves 1) Has Droplet spectra variance in warm cloud been well approximated so far? 2) Metodologies: -Direct Numerical Simulation DNS Large Eddy Simulation LES 3) Current simulation time seconds/2 minutesà up to 20 minutes

8 Turbulence and condensa0on: mathema0cal model Eulerian framework: Navier-Stokes + supersaturation field s à s>0 condensation s<0 t u + u ru = 1 rp + r2 u + f, r u t s + u rs = appler 2 s + A 1 w Lagrangian framework: droplet dynamics dv i (t) = V i v i [X i (t),t] + gz dt d dx i (t) = V i (t) dt dr i (t) s[x i (t),t] = A 3 dt R i (t) s s 1 s = 4 wa 2 A 3 V NX i=1 Phase relaxation time scale Droplet modeled as point particles Force acting on droplets: Stokes drag and gravity R i

9 R Simula0ons parameters L box v rms T L T 0 Label N 3 ½mŠ ½m=sŠ ½sŠ ½sŠ Re λ N d DNS A1= DNS B1= DNS C1= DNS D DNS E LES F Dissipation rate: Kolmogorov scale: Kolmogorov time: Initial Radius: 13 µm (1) 5 µm (2) " = 10 3 m 2 s 3 =1 mm =0.1 s Phase relaxation time: 2.5 s (1) 7 s (2) DNS E1: state of the art with grid point resolution corresponding to a domain length of 3 meters with 10 9 droplets evolved. First DNS with cloud size order meter. E1, C2 and D1 do not reach 20 minutes of simulation Typical value found in stratocumuli St =3.5E 2 5E 3 C = 130/cm 3

10 Conserva0ve hypothesis We assume <s>=0 à all is due to s fluctuabons No mean updrad Consequently <R 2 >=R 0 2 Entrainment effects are not considered (Kumar, Schumacher and Shaw, JAS 2014) AdiabaBc approximabonà small temperature fluctuabons (DNS D1 T_rms=0.005 K)à A 1, A 2, A 3 costants Effects due to inhomogeneity are not captured Collisions not included Our results represent a lower limit on droplet growth

11 DNS results σ R2 [µm 2 ] A2 B1 A B2 C1 C2 D E <s r 2 >[µm 2 ] 10-5 t[s] t[s] t 1/ dh(r 20 i )2 i dt R 2 standard deviation of he flu square radius fluctuations Standard deviation increases continuosly even if s has reached the quasi steady state Power law t 1/2 Proportional to Re λ and Larger scales are responsible for variance growth Correlation hs 0 R 20 ireaches a quasi-steady state = d 2 R 2 dt s =4A 3 hs 0 R 20 i

12 with zero mean supersaturation is Gaussian and R 2 is su cient to correctly 1-D predict stochas0c the size distribution. model/1 To quantitatively estimate this droplet growth, we propose a 1-D wi(t 0 stochastic + dt) =w model i(t) 0 to describe the velocity fluctuations and the supersaturation field T 0 for the i-th droplet: 0 0 w s 0 i(t + dt) =s 0 s 0 i i(t) dt + A 1 w 0 s i(t 0 + dt) =w 0 wi 0 i(t) (t) r dt + v 0 rms 2 dt i T idt i (t) (7) T 0 h s i dt+ 0 T 0 r + (1 Cws)hs 2 02 i 2dt r s 0 i(t + dt) =s 0 s 0 i i(t) dt + A i (t)+c r ws hs T 02 i 2dt 1 w 0 s 0 i T idt 0 h s i dt+ i (t) r 0 T i h i (1 Cws)hs 2 02 i 2dt r i (t)+c ws hs T 02 i 2dt i (t) (8) R 0 i 20 (t + dt) =Rpi 20 (t)+2a T 0 3 s 0 idt Ri 20 (t + dt) =Ri 20 (t)+2a 3 s 0 idt (9) wi 0(t) r dt + rv rms r2 dt i (t) T 0 p 0 i p h i where C ws = hw 0 s 0 i/(v rms hs 02 i) is the normalized T 0 velocity-supersaturation correlation, h s i is the supersaturation relaxation time based on the mean droplet velocity/supersaturabon auto-correlabon ra- Large eddy turn over Bme

13 h i e ects are 1-D important stochas0c and the stochastic model/2 inviscid model dhs 0 overestimates R 20 i the correct behavior. = A 1 hw 0 R 20 i +2A 3 hs 02 hs 0 R 20 i To validate the model for i a larger cloud size, we perform a Large Eddy Simulation (LES E1) of a cloud of dt h s i dhw 0 R 20 i =2A 3 hw 0 s 0 hw 0 R 20 i about 100 meters. LES can i be seen as a good model for dt T our problem since 0 dhs 02 it well resolves the larger flow scale, those relevant to i =2A droplet 1 hw 0 scondensation/evaporation. 0 i 2 hs02 i We model the small dt scales with a classic h s i Smagorinski model [24] and usedhw the 0 s 0 imethod = A 1 vrms 2 of droplet hw 0 s 0 i renormalization Steady stateà described in [19] dtto evolve a feasible h s i droplet number. The Taylor Reynolds number is hs 02 i qs The = A 2 1v time rmsh 2 evolution s i 2 of h i h i h i R 2 and hs 0 R of20 hs0 i qs R=2A 20 i are 3 A 2 depicted 1vrmsh 2 s i 2 Tin 0 =2A 3. The 3 hs 02 analytical i qs T 0 predictions from (15) and (16) well fit the numerical data as and consequently shown q R 2 = p in figure 3, thus validating q our stochastic model. Expression 8A 3 A 1 v rms (16) h s can i(t 0 be t) 1/2 formulated = 8hs 02 iin qs Aterms 3 (T 0 t) of 1/2 Kolmogorv scales since v rms ' Re 1/2 v and T 0 ' 0.06Re (16) [23]: R 2 ' 0.7A 3A 1 1/2 h s ire t 1/2 (17) Standard deviabon does not depend on dissipabon and so on small scales but is proporbonal for t = T 0 (short times) the lower limit proposed in [19] is recovered, R 2 ' to scale separabon Re3/2. From (17) we note that R 2 radius R 2 re distribution a is shown in fi higher Reyno and can be fi proposed mo To solve th pose and valid lence on the clouds. We sh droplet radiu the growth li ration, param dynamics, ru energy dissip for the impac real clouds ar homogeneous Re larger th cording to th ues of R 2, m broadening o

14 Comparisons with DNS results σ R2 [µm 2 ] <s r 2 >[µm 2 ] t[s] t[s] t 1/2 A1 A2 B1 B2 C1 C2 D1 E1 The model approximates the largest simulation Smallest simulations are influenced by viscous effects of the smallest scales In general the stochastic model tends to overestimate

15 Comparison with Large Eddy Simula0on We want to see the effects of the large scale on droplet condensabon Maximum cloud size in homogeneous condibons order 100 meters Classic Smagorinsky model for the fluid velocity and supersaturabon field Droplet number: order à unfeaseableà use of renormalization as described in Lanotte et al., 2009 Parameters " = 10 3 m 2 s 3 v rms =0.7 m/s Re = 5000

16 Large Eddy Simula0on microphysics parametriza0on We assume no sgs-model àsupersaturabon value is not evaluated at the droplet scale LES does not evolve the correct number of dropletà rescaling EquaBon for droplet radius dr 2 i dt =2A 3 (s res + s sgs ) > dh(r20 i )2 i dt From DNS results Small scale dynamics is lostà underesbmabon Now we have a lower limit at moderate cost = d 2 R 2 dt hs 0 R 20 i res >> hs 0 R 20 i sgs =4A 3 (hs 0 R 20 i res + hs 0 R 20 i sgs ) LES < DNS < Stochastic Model

17 Model vs LES σ R2 [µm 2 ] E-03 <s r 2 > [µm 2 ] 2.0E t[s] t[s] E1 MODEL Re λ =5000 MODEL Re λ =10000 Very good agreement for both standard deviation and correlation Model extension for higher Reynolds number Significant values of standard variation found after several minutes Importance to have longer simulations Impact of condensation has been underestimated in the last years

18 Droplet distribu0on 10 2 A A2 B C1-1 E1 10 MODEL Re λ =5000 MODEL Re λ = PDF PDF R 2 -<R 2 >[µm 2 ] Gaussian distribution R 2 -<R 2 >[µm 2 ] RMS is sufficient to characterize pdf More details in Sardina et al., 2015, Physical Review Letters 115 (18)

19 Conclusions and perspec0ves/1 New state of the art simulabons able to reproduce condensabon growth with Bme comparable with rain formabon Standard deviabon of square radius fluctuabons increases conbnuously in Bme according to a power law Importance of large/small scale separabon ValidaBon of a simple stochasbc model that is able to capture all the essenbal dynamics Droplet distribubon is Gaussian

20 Conclusions and perspec0ves/2 Limit of our study: smallest droplets Include nucleabon Combining condensabon+collisions Difficult but maybe more interesbng since CondensaBon >large scale Collisionà small scales and difficult to include in LES/stochasBc models

21 Numerical Methodology Combined Eulerian/Lagrangian Solver Pseudo-spectral code 2/3 rule for dealiasing Tri-linear interpolabon to evaluate fluid velocity and saturabon field at the droplet posibon Tri-linear extrapolabon to calculate droplet feedback on the saturabon field Full MPI parallelizabon for both carrier and dispersed phase ComputaBonal Bme step linearly scales up to cores à huge simulabons

22 Error es0ma0on DNS/StochasBc model comparison EsBmate of the supersaturabon fluctuabons (<s 2 > qs T -<s 2 >qs DNS )/<s 2 >qs T % L[m] DNS DATA Stochastic models overestimates but. The error tends to diminishing by increasing the large turbulent scales 20% is already a good approximation for evaluating the order of magnitude We found an upper limit at no cost

23 Dhs 2 /2i Dt {z } =0,s.s. <s 2 /τ s >/(A 1 <w s >) Why these differences? Supersaturation variance equation = appleh(rs) 2 i + A 1 hwsi h s2 i {z } {z } {z} s viscous dissipation<0 production>0 droplet sink <0 Balance between droplet sink and production L[m] DNS DATA At smallest scales viscous dissipation dominates Two different regime and supersaturation balance between small and large scales At large scale the s equation tend to the classical Twomey equation

24 Comparison with large DNS s rms DNS LES 32 3 σ R2 [µm 2 ] DNS LES E t[s] LES filter small scales effect DNS 10 days of computations in 4096 cores on the 32 nd TOP500 list supercomputer LES almost 1 hour in 1 core on my laptop t[s]