Turbulent Diffusion - Modeling - Lagrangian Approach

Lagrangian Approach

The Lagrangian model to turbulent diffusion uses a moving frame of reference to follow the trajectories and displacements of the species as they move and follows the statistics of each particle individually. Initially, the particle sits at a location x’ (x₁, x₂, x₃) at time t’. The motion of the particle is described by its probability of existing in a specific volume element at

time t, that is described by Ψ(x₁, x₂, x₃, t) dx₁ dx₂ dx₃ = Ψ(x,t)dx which follows the probability density function (pdf) such that:

Where function Q is the probably density for particle transition.

The concentration of particles at a location x and time t can then be calculated by summing the probabilities of the number of particles observed as follows:

Which is then evaluated by returning to the pdf integral

Thus, this approach is used to evaluate the position and velocity of particles relative to their neighbors and environment, and approximates the random concentrations and velocities associated with turbulent diffusion in the statistics of their motion.