High-resolution simulation results of kinematic and dynamic collision statistics of cloud droplets Bogdan Rosa (bogdan.rosa@imgw.pl) Institute of Meteorology and Water Management National Research Institute Hossein Parishani, Orlando Ayala, Lian-Ping Wang University of Delaware Wojciech W. Grabowski National Center for Atmospheric Research Multiphase Turbulent Flows in the Atmosphere and Ocean Boulder, USA, August 13-17, 2012
Motivation Understanding and quantitative prediction of warm rain formation Evaluate the effect of air turbulence (R λ ) on droplet statistics - clustering (RDF) - collision rate (RDF, radial relative velocity) Evaluate the effect of gravity on collision-coalescence of cloud droplets Evaluate the effect of forcing method (in DNS) on collision statistics and sedimentation velocity
Particle motion Y(k) (t), V(k) (t) Background flow U(x, t) P 1 U = U ω + U 2 +ν 2 U + f (x, t ) t ρ 2 U = 0 dv ( k ) (t ) V ( k ) (t ) U(Y ( k ) (t ), t ) u(y ( k ) (t ), t ) = +g dt τ (pk ) dy ( k ) (t ) = V ( k ) (t ) dt Forcing term g Incompressible turbulence Pseudo-spectral method (DNS) Forcings deterministic - Rosa et al. 2011 J. Phys.: Conf. Ser. 318 072016 stochastic - Eswaran and Pope1988 Comput. Fluids 16 Periodic BC in a cube 3 Dimensional Parallel FFT: Ayala and Wang Parallel Computing, (2012) Particles are governed by Stokes drag, gravity & inertia Periodic BC 2D domain decomposition Droplets modeled as solid particles
R λ vs. grid size Scalability of the HDNS code ~500 1D R λ Int. HDI 2D Coll. det. N Simulated flow Taylor microscale Reynolds number as a function of grid resolution. Simulations have been performed using two different forcing schemes i.e. stochastic and deterministic. Scalability of the major tasks in the HDNS code in terms of the total execution time. The timing measurements are carried out for a benchmark problem of 2 10 6 droplets of radii 20 and 40 microns at 512 3 grid resolution
Computational resources Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw Poland NCAR (National Center for Atmospheric Research) United States Bluefire - Power 575, p6 4.7 GHz, Infiniband / 2008 IBM Boreasz - Power 775, POWER7 8C 3.836GHz, Custom / 2011 IBM Cores # Rmax TFlops/s Rpeak TFlops/s Power KW Bluefire 4064 59.68 76.40 646.0 Boreasz 2560 64.33 78.56 156.7 Rmax - Maximum LINPACK performance achieved Rpeak - Theoretical peak performance
Normalized 1D energy spectra of the simulated flows with different R λ S Stochastic forcing scheme D Deterministic forcing scheme Comte-Bellot G. and Corrsin S. 1971 J. Fluid Mech. 48 273 337 Wells M. R. and Stock D. E. 1983 J. Fluid Mech. 136 31 62
Average settling velocity TURBULENT FLOW SUSTAINED BY STOCHASTIC FORCING TURBULENT FLOW SUSTAINED BY STOCHASTIC (dashed lines) & DETERMINISTIC (solid lines) FORCINGS
RDF and radial relative velocity as a function of separation distance a) b) wr11 (r = R) r exp β vk R
Radial distribution function l DETERMINISTIC FORCING l SEDIMENTING DROPLETS MONODISPERSE RDF FOR SEDIMENTING AND NONSEDIMENTING DROPLETS a) b) NO GRAVITY WITH GRAVITY S & D Presence of gravity enhances clustering of droplets with radii greater than 45 µm
Radial distribution function comparison with published results NO GRAVITY WITH GRAVITY
Radial distribution function power law exponent a) c b) 1 η g11 ( r) = c0 r WITH GRAVITY g 11 ( r) c η = c0 r 1
Radial distribution function power law exponent g 11 ( r) c η = c0 r 1 WITH GRAVITY
Radial relative velocity SEDIMENTING DROPLETS RADIAL RELATIVE VELOCITY FOR SEDIMENTING AND NONSEDIMENTING DROPLETS a) b) NONSEDIMENTING DROPLETS WITH GRAVITY
Collision kernel Dynamic collision kernel Γ = n c V N 1 N 2 Δt = n c 4V N 2 Δt Kinematic collision kernel Γ = 2π R 2 w r g(r)
Collision kernel
Settling velocity, RDF and radial relative velocity as a function of R λ a) b) c)
Settling velocity, RDF and radial relative velocity as a function of R λ a) b) c) 16 th ICCP, Leipzig, Germany, July 30 August 03, 2012
Problem with stochastic forcing scheme Two parameters determining the stochastic forcing scheme: acceleration variance σ 2 the forcing time scale tf If tf << Te then the dissipation rate of the simulated flow is completely determined as In the code, tf was set to 0.038 while σ 2 = 447.31 In the runs, Te 0.1 Expected dissipation rate is ~5439. We obtained 3500 to 3000. The condition of tf << Te was not satisfied leading to a smaller realized dissipation rate Additionally tf/τk effectively increased with R λ. This had an effect of increasing the eddy life time and as such increasing the RDF.
How the problem has been solved? We should preserve the ratio of tf/τk If we assume energy dissipation rate is for example 3600
Conclusions A 1. Developed a new parallel implementation of hybrid DNS, based on 2D domain decomposition 2. Performed high-resolution, hybrid DNS with flow field solved at grid resolution up to 1024 3 while simultaneously track up to 5*10 7. 3. Showed that the statistics of the background turbulent flow agree well both with previous DNS and experimental data. 4. Computed radial distribution function and relative velocity for nearly touching particles for wide range of R λ. The results extend previous simulations and the theoretical predictions.
Conclusions B 5. Gravity decreases RDF of particles with radii ranging from 25 to 45 µm but increases RDF of particles with radii larger than 45 µm. 6. Gravity significantly reduces radial relative velocity for droplets larger than 35 µm in radius. 7. Confirmed that the kinematic collision kernel is consistent with the dynamic collision kernel. 8. Collision kernel of nonsedimenting particles is larger than sedimenting particles (60µm 3 times, 40µm 2 times). 9. The large-scale forcing scheme affects collision statistics. We observed saturation of the RDF and radial relative velocity for increasing R λ.