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Unformatted text preview: Particle Tracking Project, Phase 1 This project is the starting point for the Vortex Method project and the pollution prediction projects on the sphere. The first phase is to develop the infrac-structure needed to track a number of particles in a flow field where the velocity is prescribed for all time and space. 1 Mathematics of Particle Trajectories The position of the particle can be obtained by integrating the following ordinary differential equation d ~x i d t = ~u i ( ~x,t ) (1) where the subscript i refers to the particle tag. Notice that the equation is non- linear since the velocity depend on position and time, hence numerical solutions are required. One of the time-stepping scheme discussed in class can be adopted for the particle trajectory calculation, e.g. Runge-Kutta 4. The details of the equation varies according to the spatial dimensions of the problem, and whether we are dealing with a cartesian plane or a spherical domain: • 2D Cartesian plane d x i d t = u i ( ~x,t ) (2) d y i d t = v i ( ~x,t ) (3) • 2D Spherical surface d λ i d t = u i a cos θ (4) d θ i d t = v i a (5) where u and v are the zonal and meridional velocity components, ( λ,θ ) are the longitude and latitude, and a is the Earth radius. 2 Programming Considerations An outline of the program is shown in 1. The position of the particles, wether on a Cartesian plane or a sphere, will be given by a two-dimensional vector, and hence the most efficient way to store the information is in a 2D matrix of the form pos(ndim,nparticles) where the spatial dimensional and the number of particles 1 program partrajecs implicit none integer, parameter :: ndim=2 ! number of spatial dimensions integer, parameter :: particles=10 ! number of particles real*8 :: pos(ndim,nparticles) ! position of particles!...
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This note was uploaded on 01/08/2012 for the course MSC 321 taught by Professor Staff during the Fall '08 term at University of Miami.
- Fall '08