scholarly journals Coagulation‐Fragmentation Equations with Multiplicative Coagulation Kernel and Constant Fragmentation Kernel

2021 ◽  
Author(s):  
Hung V. Tran ◽  
Truong‐Son Van
Keyword(s):  
Fragmentation Kernel ◽  
Coagulation Kernel

A new weighted fraction Monte Carlo method for particle coagulation

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Xiao Jiang ◽  
Tat Leung Chan
Keyword(s):  
Monte Carlo ◽  
Design Methodology ◽  
Computational Cost ◽  
Particle Coagulation ◽  
Content Type ◽  
Aerosol Dynamics ◽  
Stochastic Error ◽  
Adjustable Weight ◽  
Initial Distributions ◽  
Coagulation Kernel

Purpose The purpose of this study is to investigate the aerosol dynamics of the particle coagulation process using a newly developed weighted fraction Monte Carlo (WFMC) method. Design/methodology/approach The weighted numerical particles are adopted in a similar manner to the multi-Monte Carlo (MMC) method, with the addition of a new fraction function (α). Probabilistic removal is also introduced to maintain a constant number scheme. Findings Three typical cases with constant kernel, free-molecular coagulation kernel and different initial distributions for particle coagulation are simulated and validated. The results show an excellent agreement between the Monte Carlo (MC) method and the corresponding analytical solutions or sectional method results. Further numerical results show that the critical stochastic error in the newly proposed WFMC method is significantly reduced when compared with the traditional MMC method for higher-order moments with only a slight increase in computational cost. The particle size distribution is also found to extend for the larger size regime with the WFMC method, which is traditionally insufficient in the classical direct simulation MC and MMC methods. The effects of different fraction functions on the weight function are also investigated. Originality Value Stochastic error is inevitable in MC simulations of aerosol dynamics. To minimize this critical stochastic error, many algorithms, such as MMC method, have been proposed. However, the weight of the numerical particles is not adjustable. This newly developed algorithm with an adjustable weight of the numerical particles can provide improved stochastic error reduction.

scholarly journals Reynolds-number dependence of turbulence enhancement on collision growth

2016 ◽  
Vol 16 (19) ◽  
pp. 12441-12455 ◽  
Author(s):  
Ryo Onishi ◽  
Axel Seifert
Keyword(s):  
Reynolds Number ◽  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
Stokes Number ◽  
Homogeneous Isotropic Turbulence ◽  
Collision Efficiency ◽  
Cloud Droplets ◽  
Clustering Effect ◽  
Reynolds Number Dependence ◽  
Coagulation Kernel

Abstract. This study investigates the Reynolds-number dependence of turbulence enhancement on the collision growth of cloud droplets. The Onishi turbulent coagulation kernel proposed in Onishi et al. (2015) is updated by using the direct numerical simulation (DNS) results for the Taylor-microscale-based Reynolds number (Reλ) up to 1140. The DNS results for particles with a small Stokes number (St) show a consistent Reynolds-number dependence of the so-called clustering effect with the locality theory proposed by Onishi et al. (2015). It is confirmed that the present Onishi kernel is more robust for a wider St range and has better agreement with the Reynolds-number dependence shown by the DNS results. The present Onishi kernel is then compared with the Ayala–Wang kernel (Ayala et al., 2008a; Wang et al., 2008). At low and moderate Reynolds numbers, both kernels show similar values except for r2 ∼ r1, for which the Ayala–Wang kernel shows much larger values due to its large turbulence enhancement on collision efficiency. A large difference is observed for the Reynolds-number dependences between the two kernels. The Ayala–Wang kernel increases for the autoconversion region (r1, r2 < 40 µm) and for the accretion region (r1 < 40 and r2 > 40 µm; r1 > 40 and r2 < 40 µm) as Reλ increases. In contrast, the Onishi kernel decreases for the autoconversion region and increases for the rain–rain self-collection region (r1, r2 > 40 µm). Stochastic collision–coalescence equation (SCE) simulations are also conducted to investigate the turbulence enhancement on particle size evolutions. The SCE with the Ayala–Wang kernel (SCE-Ayala) and that with the present Onishi kernel (SCE-Onishi) are compared with results from the Lagrangian Cloud Simulator (LCS; Onishi et al., 2015), which tracks individual particle motions and size evolutions in homogeneous isotropic turbulence. The SCE-Ayala and SCE-Onishi kernels show consistent results with the LCS results for small Reλ. The two SCE simulations, however, show different Reynolds-number dependences, indicating possible large differences in atmospheric turbulent clouds with large Reλ.

Gelation time in the discrete coagulation–fragmentation equations with a bilinear coagulation kernel

2007 ◽  
Vol 40 (39) ◽  
pp. 11749-11764 ◽  
Author(s):  
Éric Brunelle ◽  
Robert G Owens ◽  
Henry J van Roessel
Keyword(s):  
Gelation Time ◽  
Coagulation Kernel

scholarly journals Estimating the division kernel of a size-structured population

2017 ◽  
Vol 21 ◽  
pp. 275-302
Author(s):  
Van Ha Hoang
Keyword(s):  
Bandwidth Selection ◽  
Random Measure ◽  
Time Interval ◽  
Constant Growth Rate ◽  
Adaptive Estimator ◽  
Daughter Cells ◽  
Structured Population ◽  
Symmetric Density ◽  
Constant Rate ◽  
Fragmentation Kernel

We consider a size-structured model describing a population of cells proliferating by division. Each cell contain a quantity of toxicity which grows linearly according to a constant growth rate α. At division, the cells divide at a constant rate R and share their content between the two daughter cells into fractions Γ and 1 − Γ where Γ has a symmetric density h on [ 0,1 ], since the daughter cells are exchangeable. We describe the cell population by a random measure and observe the cells on the time interval [ 0,T ] with fixed T. We address here the problem of estimating the division kernel h (or fragmentation kernel) when the division tree is completely observed. An adaptive estimator of h is constructed based on a kernel function K with a fully data-driven bandwidth selection method. We obtain an oracle inequality and an exponential convergence rate, for which optimality is considered.

Droplets to Drops by Turbulent Coagulation

2005 ◽  
Vol 62 (6) ◽  
pp. 1962-1975 ◽  
Author(s):  
N. Riemer ◽  
A. S. Wexler
Keyword(s):  
Cloud Droplet ◽  
Reynolds Numbers ◽  
Cloud Microphysics ◽  
Atmospheric Conditions ◽  
Droplet Concentration ◽  
Warm Rain ◽  
Rapid Production ◽  
Thermodynamic Effects ◽  
Simulation Time ◽  
Coagulation Kernel

Abstract This study addresses two central problems in cloud microphysics. The first is the source of large droplets, which initiates the rapid production of warm rain. The second is the broadening of the cloud droplet spectrum at both tails of the spectrum. The study explores how in-cloud turbulence can help to close the gaps in our understanding. With box model simulations, the development of cloud droplet spectra is calculated using a coagulation kernel that recently has been derived from direct numerical simulations. This kernel includes both the effect of turbulence on the relative velocities of the droplets and on the local increases in droplet concentration, the so-called accumulation effect. Under the assumption that this kernel can be extrapolated to atmospheric Reynolds numbers, the results show that for typical atmospheric conditions, the turbulent coagulation kernel is several orders of magnitude larger than the sedimentation kernel for droplets smaller then 100 μm. While for calm air after 30-min simulation time, only 7% of the total mass is found in droplets with sizes over 100 μm, this increases to 79% for a dissipation rate of 100 cm2 s−3 and 96% for 300 cm2 s−3 if a combined sedimentation and turbulent kernel is employed that assumes that the sedimentation and turbulent kernel can be added. Hence, moderate turbulence can enhance significantly the formation of large droplets. Furthermore, a time-scale analysis shows that broadening at the upper end of the spectrum is caused by turbulent coagulation whereas thermodynamic effects are responsible for broadening at the lower end.

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