Considering turbulent clouds containing small inertial particles, we investigate the effect of particle collision, in particular collision–coagulation, on particle clustering and particle relative motion. We perform direct numerical simulation (DNS) of coagulating particles in isotropic turbulent flow in the regime of small Stokes number (

The motion and interactions of small particles in turbulence have fundamental implications for atmospheric clouds; specifically, they are relevant for the timescale of rain formation, particularly in warm clouds

The quest for a theory of particle collision in turbulence started in 1956 when

Here, we first present results on RDF and MRV for particles undergoing collision–coagulation

Coagulation is, in a sense, the simplest outcome of collision. In the companion paper we shall argue that the major qualitative conclusions of our work also apply to cases with other collisional outcomes.

. The data are obtained via direct numerical simulation (DNS), which is the gold-standard computational method in terms of accuracy and completeness for solving the most challenging fluid dynamics problem i.e., turbulent flows. It is worth noting that the focus of our work is on the fundamental relationship between collision, RDF and MRV and on highlighting differences from the case with non-colliding particlesAnalysis of the DNS results is followed by a theoretical account of the relations between collision rate, RDF and MRV, which includes mean-field contributions

In the collision-less case, the asymmetry is much weaker and is related to viscous dissipation of energy in turbulence

To observe how particle collision–coagulation affects RDF and MRV, we performed direct numerical simulation (DNS) of steady-state isotropic turbulence embedded with particles of finite but sub-Kolmogorov size. We solve the incompressible Navier–Stokes equations (Eq.

Values of the parameters in the DNS. (Note that dm denotes decimeters.) From the left, we have the Taylor-scale Reynolds number, kinematic viscosity of the fluid, root-mean-square of fluid velocity, kinetic energy dissipation rate, Kolmogorov length scale and timescale, length of the simulation cube and particle diameters considered. We have introduced

Particles in the simulations are advected via a viscous Stokes drag force

In this context, the particle Stokes number, defined as

Values of key parameters of the DNS are given in Table

As described in

From this, one could derive an equation for

Finally we note that since particles are allowed to collide–coagulate in our theory, we use the conventional definition of MRV:

We compute the RDF via

RDFs

Figure

The strong effect of particle collision on the RDF (also on MRV as we shall see later) challenges the validity of the separation paradigm. We note that similar fall-off of RDF was previously observed

Another observation is that in the power-law regime (

To theoretically account for the new findings, we make some derivations that are partially similar to the ones in

In other words, particles may approach each other (and collide), but they can not be created at contact and then separate.

, while with increasingWe then expand

Here a positive

The term

By definition,

Simple analytical solutions
to Eq. (

The ansatz has the form

From a given DNS-produced RDF

Next, we numerically evaluate the integral in the first term of Eq. (

Mean radial component of relative velocity (MRV) for particles of specific Stokes numbers and some theoretic numerical predictions.

Comparison of the predicted

Alternatively, Eq. (

This is true in the relatively idealized system simulated but may not apply to the general problem that includes other effects

.Here we provide a simple, first-order model for

A simple phenomenological model for

We now discuss an important but precarious theoretical issue.

One advantage of Eq. (

Thus far, we have not considered the effects of gravity on the particles. Here we provide a glimpse into the role of gravity (a detailed analysis is beyond the scope of the present study). In keeping with the scope of current work, we restrict ourselves to the case of monodisperse particles only. For this, we rerun the DNS cases of

RDFs of particles (

As mentioned, the fundamental focus of our work precludes the DNS and theory from considering a number of complexities relevant to some applications. As a result, this limits the direct quantitative applicability of our results to some realistic problems (e.g., in clouds). Besides gravity, another neglected factor is the hydrodynamic inter-particle force (HDI). Recent works, e.g.,

Also neglected is the influence of temperature, humidity and the vapor–liquid phase transition, which are important in the atmospheric clouds. These factors have a substantial impact on the polydispersity of small droplets (see, e.g.,

One limitation of the theory stems from the assumption of

To conclude, we observed that collision strongly affects the RDF and MRV and imposes strong coupling between them

This statement also holds for other types of collisional outcomes (not only for collision–coagulation), but the details of the specific outcomes should be different from the current case.

. This challenges the efficacy of a “separation paradigm” and suggests that results from any studies that preclude particle collision have limited relevance for predicting collision statistics. We have presented a theory for particle collision–coagulation in turbulence (based on a Fokker–Planck framework) that explains the above observations and verified its accuracy by showing thatThe data (model codes) may be obtained from the corresponding author upon request.

The supplement related to this article is available online at:

EWS oversaw the conception and execution of the project. EWS did the theoretical derivations in collaboration with XM. XM and EWS conducted the numerical simulation and data analysis. EWS and XM wrote the article.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the National Natural Science Foundation of China (grant 11872382) and by the Thousand Young Talents Program of China. We thank Jialei Song for their help. We thank Wai Chi Cheng, Jianhua Lv, Liubin Pan and Raymond A. Shaw for discussion and suggestions.

This research has been supported by the National Natural Science Foundation of China (grant no. 11872382).

This paper was edited by Graham Feingold and reviewed by two anonymous referees.