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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 infracstructure 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 timestepping scheme discussed in class can be adopted for the particle trajectory calculation, e.g. RungeKutta 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 twodimensional 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
 Staff

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