This study investigates the Reynolds-number dependence of turbulence
enhancement on the collision growth of cloud droplets. The Onishi turbulent
coagulation kernel proposed in

Several mechanisms have been proposed to explain the rapid growth
of cloud droplets, which often result in fast rain initiation
in the early stages of cloud development. Examples of these mechanisms
include the turbulence-enhanced collision rate of cloud droplets (

One direction taken by the research in this area is the simulation
of collisional growth by solving the stochastic collision–coalescence
equation (SCE). Such research relies on accurate collision–coalescence
models, which consist of models for the collision kernel

One serious problem is that the Reynolds-number dependence of turbulent
collisions has not yet been clarified. In fact, many authors ignore
the Reynolds-number dependence and assume a constant collision kernel
regardless of the Reynolds number (e.g.,

Recently,

This study, therefore, aims to compare the Ayala–Wang and Onishi kernels by
focusing on their

The geometric collision frequency per unit volume between particles with radius

The gravitational collision kernel describes the collisions due to
the settling velocity difference in the form of

It had been difficult to confidently discuss the collision efficiency in a
turbulent flow until

Droplet deformation and coalescence efficiency, which this study ignores,
affect the collision growth of droplets with

By following the expression by

In addition to the empirical

Using

If we limited the discussion for the autoconversion regime, i.e.,

Hence, this study modifies the parameterization in the original Onishi kernel
to obtain better overall matching for a wider range of

To determine the clustering effect for bidisperse systems, the empirical
formulation proposed by

Stokes number against the particle radius for various energy dissipation rates.

The gravitational settling affects the clustering effect for large

Parameter values for

The above simple treatment is not yet complete.

The turbulent collision kernel formulated from the above

Following

Enhancement factor for the Pinsky collision efficiency (PKS01),

We now solve the three-dimensional continuity and Navier–Stokes equations for
incompressible flows:

Under the limit of a large ratio of the density of the particle material
to that of the fluid (

The second-order Runge–Kutta method is used for the time integration. The
flow velocity at the droplet position

After the background airflow has reached a statistically stationary
state, monodispersed water droplets are introduced into the flow.
After a period exceeding 3 times the eddy-turnover time

The detailed description of the procedures for calculating collision
statistics can be found in

Case configurations and typical turbulence statistics.

Case configurations and typical turbulence statistics.

To obtain reference data regarding droplet collisional growth, we tracked the
growth of droplets that initially had the following exponential size
distribution (e.g.,

Table

To quantify the influence of intermittence on

The intermittency is measured by the flatness factor

Radial distribution function (RDF) at the contact of monodisperse
particles with

Figure

The updated parameterization leads to improvement, particularly for the

Radial distribution function at the contact of monodisperse
particles,

Figure

The Ayala–Wang model shows a local maximum around

Non-dimensionalized coagulation kernels for

Figure

Ratio of the turbulent coagulation kernel to the Hall kernel in the
turbulent flow with

Ratio of the coagulation kernel for

Figure

Figure

Ratio of the clustering effect

Ratio of the radial relative velocity at contact separation

Figure

The Reynolds-number dependence of the clustering effect is larger than that
of the radial relative velocity, and the contour shape of
Fig.

Note that the Fortran 90 code used to calculate the present Onishi kernel is provided in the Supplement.

We investigated the turbulence enhancement on the autoconversion rate, which
is the conversion rate from the cloud category (

Following

Turbulence enhancement factors for

Figure

Figure

In summary, both Figs.

Turbulence enhancement factors for

As noted in

Figures

This study investigated the Reynolds-number dependence of turbulence
enhancement on the collision growth of cloud droplets. The Onishi turbulent
coagulation kernel proposed in

We confirmed that the updated

The present Onishi coagulation kernel was compared with the Ayala–Wang kernel
(

We also compared the stochastic collision–coalescence equation (SCE)
simulations for both kernels; one with the Ayala–Wang kernel (SCE-Ayala) and
the other with the present Onishi kernel (SCE-Onishi). Lagrangian Cloud
Simulator (LCS;

Data for the present graphs are available from the corresponding author upon request.

Part of the presented simulations were performed on the supercomputer Earth
Simulator at the Japan Agency for Marine-Earth Science and Technology. The
large-size simulations for collision statistics for