R.H.A. IJzermans

A Lagrangian approach to droplet condensation in atmospheric clouds

The condensation of microdroplets in model systems, reminiscent of atmospheric clouds, is investigated numerically and analytically. Droplets have been followed through a synthetic turbulent flow field composed of 200 random Fourier modes, with wave numbers ranging from the integral scales [O(102 m)] to the Kolmogorov scales [O(10−3 m)]. As the influence of all turbulence scales is investigated, direct numerical simulation is not practicable, making kinematic simulation the only viable alternative. Two fully Lagrangian droplet growth models are proposed: a one-way coupled model in which only adiabatic cooling of a rising air parcel is considered, and a two-way coupled model which also accounts for the effects of local vapor depletion and latent heat release. The simulations with the simplified model show that the droplet size distribution becomes broader in the course of time and resembles a Gaussian distribution. This result is supported by a theoretical analysis which relates the droplet surface-area distribution to the dispersion of droplets in the turbulent flow. Although the droplet growth is stabilized by vapor depletion and latent heat release in the two-way coupled model, the calculated droplet size distributions are still very broad. The present results may provide an explanation for the rapid growth of droplets in the coalescence stage of rain formation, as broad size distributions are likely to lead to enhanced collision rates between droplets.

Solution of the general dynamic equation along approximate fluid trajectories generated by the method of moments

We consider condensing flow with droplets that nucleate and grow, but do not slip with respect to the surrounding gas phase. To compute the local droplet size distribution, one could solve the general dynamic equation and the fluid dynamics equations simultaneously. To reduce the overall computational effort of this procedure by roughly an order of magnitude, we propose an alternative procedure, in which the general dynamic equation is initially replaced by moment equations complemented with a closure assumption. The key notion is that the flow field obtained from this so-called method of moments, i.e., solving the moment equations and the fluid dynamics equations simultaneously, approximately accommodates the thermodynamic effects of condensation. Instead of estimating the droplet size distribution from the obtained moments by making assumptions about its shape, we subsequently solve the exact general dynamic equation along a number of selected fluid trajectories, keeping the flow field fixed. This alternative procedure leads to fairly accurate size distribution estimates at low cost, and it eliminates the need for assumptions on the distribution shape. Furthermore, it leads to the exact size distribution whenever the closure of the moment equations is exact.

Accumulation of heavy particles around a helical vortex filament

The motion of small heavy particles near a helical vortex filament in incompressible flow is investigated. Both the configurations of a helical vortex filament in free space and a helical vortex filament in a concentric pipe are considered, and the corresponding helically symmetric velocity fields are expressed in terms of a stream function. Particle motion is assumed to be driven by Stokes drag, and the flow fields are assumed to be independent from the motion of particles. Numerical results show that heavy particles may be attracted to helical trajectories. The stability of these attraction trajectories is demonstrated by linear stability analysis. In addition, the correlation between the attraction trajectories and the streamline topologies is investigated

Accumulation of heavy particles in N-vortex flow on a disk

The motion of heavy particles in potential vortex flows on the unit disk is investigated theoretically and numerically. Configurations with one vortex and with two vortices are considered. In both cases, each vortex follows a regular path on the disk. In the one-vortex case, it is shown that small, heavy particles may accumulate in elliptic regions of the flow, counter-rotating with respect to the vortex. When the particle Stokes number exceeds a threshold depending on the vortex configuration, all particles are expelled from the circular domain. A stability criterion for particle accumulation is derived analytically and verified by numerical results. In the two-vortex case, heavy particles are shown to accumulate in elliptic islands of regular motion. Again, this result is explained by a stability analysis. The results may be useful in the design of gas-particle separators containing a helical vortex filament.

Accumulation of Heavy Particles in Bounded Vortex Flow

Much research has been done on the motion of heavy particles in simple vortex flows. In most of this work, particle motion is investigated under the influence of fixed vortices. In the context of astrophysics, the motion of heavy particles in rotating two-dimensional flows has been investigated; the rotation follows from the laws of Kepler. In the present paper, the motion of heavy particles in potential vortex flow in a circular domain is investigated. The vortex describes a circular trajectory due to the presence of the boundary, so that a steadily rotating flow is obtained. In order to isolate the effect of particle inertia, only Stokes drag is taken into account in the equation of motion. The numerical simulations are based on a oneway coupling. They show that small heavy particles accumulate in an ellitpic region of the flow, counterrotating with respect to the vortex. When the particle Stokes number exceeds a threshold, depending on the vortex configuration, particles are expelled from the circular domain. A stability criterion for this particle accumulation is derived analytically. These results are qualitatively comparable to those obtained by others in astrophysics.

Calculation of Droplet Size Distribution in Condensing Flow

Condensing flows can be found in a large variety of industrial machinery such as steam turbines and supersonic gas conditioners. In many of these applications, it is very important to predict the droplet size distributions accurately. In the present research, the droplet size distribution in condensing flows is investigated numerically. We consider condensing flows with droplets that nucleate and grow, but do not slip with respect to the surrounding gas phase. To compute the coupling between the condensed phase and the carrier flow, one could solve the general dynamic equation and the fluid dynamics equations simultaneously. In order to reduce the overall computational effort of this procedure by roughly an order of magnitude, we use an alternative procedure, in which the general dynamic equation is initially replaced by moment equations complemented with a closure assumption. This closure assumption is based on Hill’s approximation of the droplet growth law. The method thus obtained, the so-called Method of Moments, is assumed to approximately accommodate the thermodynamic effects of condensation, such as the temperature, pressure and velocity field of the carrier flow. We use the Method of Moments as a basis for the calculation of the droplet size distribution function. We propose to solve the general dynamic equation a posteriori along a number of selected fluid trajectories, keeping the flow field fixed. This procedure, called Phase Path Analysis [1], leads to accurate size distribution estimates, at a far lower computational cost than solving the general dynamic equation and the fluid dynamics equations simultaneously. In the present paper, we investigate the effect of a variation in the liquid mass density on the droplet size distribution, using the proposed method. In case of a varying liquid mass density, both the equation for the dropltet growth rate and the moment equations are modified. This modified form coincides with the usual form of the moment equations in the event that the variation in liquid density is negligible. This research is relevant for condensation in flows where large temperature differences may occur which lead to significant variations in the liquid mass density. We show that the implementation of a variable liquid mass density in the Method of Moments and the Phase Path Analysis results in a higher extremum in the droplet size distribution, whereas the skewed shape of the distribution function is nearly similar to that obtained in the constant liquid density case.Copyright