Numerical Simulation of Homogeneous Nucleation of Water

What is this page about?

Throughout my time as an undergraduate (Fall 2000-Spring 2003), I worked with Dr. Michael Marder at the Center for Nonlinear Dynamics at the University of Texas at Austin. Below I describe the goal of the project and the approach we took. Please contact me for my undergraduate thesis for this project.

Goal of the Project

The goal of this project was to make a computer simulation of how clouds (such as thunderclouds) form in the atmosphere.

Stage 1: Continuity Equation

Initially we model a cloud as a function g where g(R, t) is proportional to the number of water droplets in our system at time t with radius R. We first assume that the continuity equation holds:

∂g/∂t = -∂/∂R (vg)
where v(R, t) represents how quickly the size of a droplet at radius R changes at time t. We first pick some initial droplet distribution g0(R). We then pick appropriate spatial and time steps, ΔR and Δt, respectively for our simulation. Then we approximate the above equation with the new equation
Δg/Δt = -Δ(vg)/ΔR
If we write Δg = gt+1 - gt, then we can solve the above equation for gt+1. The resulting expression was used to write a computer program simulating the evolution of the distribution g over time by increasing time by Δt at each step and recomputing the new value of g for each R.

The above technique is called forward Euler differencing. We actually modified our simulation to do forward Euler differencing only for values of R that are greater than or equal to the critical radius R*. For smaller values of R, we use backward Euler differencing instead. As a general rule, droplets with radius greater than or equal to the critical radius will continue to grow larger and larger until no more water vapor is left in the system. Droplets smaller than the critical radius tend to shrink and eventually turn into water vapor.

Stage 2: Adding a Nucleation Source Term

Next, we made a simulation similar to that in Stage 1, except we added a nucleation source term to model the effect of steadily injecting water vapor into our system. The new partial differential equation (PDE) we use is:

∂g/∂t = -∂/∂R (v(g+ n(R, t)))
Then we limit the total amount of vapor present in the system and recalculate the critical radius at each step, so that droplets do not grow infinitely large.

Stage 3: Approximating the PDE with ODEs

In 1980, Langer and Schwartz1 introduced a method for approximating a PDE similar to the one above using two ordinary differential equations (ODEs). Since ODEs are much faster to simulate than PDEs, this approximation makes our simulation much faster. The approximation is not too bad an approximation as we show in our comparison of the PDE and ODEs. The two variables for which we have ODEs are φ, a number proportional to the volume of liquid water droplets in the system, and ρ, the average radius of water droplets in the system.

Stage 4: Spatial Model with Coupling

Now we construct an array of square-shaped cells in a two-dimensional grid, each with its own φ and ρ variables. We experimented with various initial values of these variables for each cell to see how the system would evolve over time. In general there would be a shell of "warm" cells surrounding any set of "cool" cells if our initial condition contained only "cool" cells surrounded by cells of some fixed "warmth". Here "warm" and "cool" refer to low and high values of the average radius ρ, respectively. This is intuitively similar in behavior to, say, the shell of relatively warm nitrogen gas that forms around a cold liquid nitrogen drop when the liquid comes into contact with some warm surrounding.

References

1 J. S. Langer, A. J. Schwartz. Kinetics of Nucleation in Near-Critical Fluids. Physical Review A Vol. 21(3), March 1980.
2 R. Rogers. A Short Course in Cloud Physics, Pergamon Press, 1979.