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Tumble Dispersal Simulations

(Project description)
4/2/2002

LES Description:

A Large Eddy Simulation (LES) software package is used to model the air flow in a three-dimensional, rectangular domain. The wind velocity is prescribed at one end of the domain and boundary conditions are specified at all others. Generally, no-slip boundary conditions are specified at the ground and free-flow conditions are specified at all the others. In addition, the LES software allows for obstacles to be placed in the domain. Obstacles are constrained by the discretization and are allowed to have either rounded or square corners.

In this version of the LES software, seeds are injected into the domain from a point source and the flight trajectory is modeled by the stochastic differential equations:

where { Y1,t , Y2,t , Y3,t } is the position of the seed, { V1,t , V2,t , V3,t } is the seed velocity and { Uw , Vw , Ww } is the wind velocity, as computed by the LES. In order to correlate the velocity of the seed over the discrete time step Dt, the velocity of the seed at time t is:

where dwt is a Weiner process, i.e., a Gaussian random variable with mean zero and variance Dt, and { Ru , Rv , Rw } are the correlation constants:

over the time scales { tu , tv , tw }. In general, it is assumed that tu = tv = 300 and tw = 100.

Experimental Setup:

For these simulations, the physical domain is defined to be 64 m long, 24 m wide and 10 m high. The corresponding computational domain is defined as 128 cells in the x-direction, 48 cells in the y-direction and 20 cells in the z-direction, i.e., each cell is about 0.5m X 0.5m X 0.5m. Six obstructions of size 2m X 2m X 3m high were placed in the domain. The domain is illustrated below:

Note that, the ground (z = 0) is denoted by the dark green plane and the obstructions are denoted by the rectangular yellow cubes.

To generate air movement within the domain, the wind velocity in the vertical plane x = 0 and the vertical plane y = -12 are specified. At each plane, the tangential velocities are specified to be zero, and the normal component is shown below for the entire duration of the simulation (1000 seconds):

Note that, the wind velocity at x = 0 is shown in the left graph and the wind velocity at y = -12 is shown in the right graph. Also, a negative wind velocity indicates wind flow out of the domain.

In the figures below, the wind velocity is shown as a vector field near time t = 260. In the left graph, the red arrows show a strong wind flow through the obstructions, but disappears as the wind flow changes directions (right graph).

In the graphs below, the wind velocity is shown as a color map. The upper graphs show the wind velocity at time t = 260 (left graph depicts Uw, or the wind velocity in the x-direction, and the right graph depicts Vw, or the wind velocity in the y-direction). At this time, the wind has a strong component in the positive y-direction. The lower graphs show the velocity at time t = 780, when the wind is again changing directions.

Note the turbulent wind profile in the downwind direction of the obstructions.

Seed Release:

For the duration of the simulation, four seeds were released from the point (x=1m, y=0m, z=2.5m) every two seconds. The location where the seeds first hit the ground were recorded. In the figures below, the cyan points denote the location were the seeds were deposited on the ground.

The hitting locations were plotted as 2-D histogram (upper left) to describe the dispersal kernel. The graph on the upper right is a vertical cross-section of the dispersal kernel at y=0, i.e., inline with the release point. The graph in the lower left shows the pdf for the dispersal distance. Note, this data does not account for seeds that leave the domain.

References:

- McGratten, K. B., Baum, H. R., Walton, W. D., Trelles, J. J. (1997) "Smoke Plume Trajectory From In Situ Burning of Crude Oil in Alaska Field Experiments and Modeling of Complex Terrain", NISTIR 5958.