# Mathematical Model PARTRACE

## Short Description

The particle tracking program PARTRACE was developed at the Coastal Department of BAW in Hamburg in 1996. It simulates the movement of fluid or sediment particles in a time-dependent 2D-depth-averaged flowfield. The flowfield could have been computed ie with the Navier-Stokes-methods TRIM-2D or TELEMAC-2D. Particle methods such as PARTRACE are based on the Lagrangian description of the the flowfield, whereas Navier-Stokes and most other CFD methods rely on the Eulerian description of the flow. The goal of PARTRACE is to aid the analysis of estuarine flows, especially, it should help to gain insight into

The following physical influences acting on a particle can be taken into account:

• 2D-depth-averaged velocity field
• vertical convergence or divergence of streamlines due to inclination of the seabed and of the water-level surface
• settling velocity
• stocastic diffusion due to turbulence and dispersion due to depth-averaging of the velocity field

Particle sources as well as the properties of the particles to be seeded, such as mass density, diameter, settling velocity, diffusion properties, ect., can be specified by the user in various ways. To the end of a simulation run the history of particle locations is written to a file and can afterwards be visualized as particle paths within the flow domain.

## Method

It is assumed that a particle follows the flow without slip and that inertia forces acting on the particle can be neglected. Under this assumption the equations of motion of a particle in the three spatial directions can be written as a coupled system of three ordinary first-order differential equations in time. In PARTRACE this system of equations is solved stepwise by a standard Runge-Kutta method. On the r.h.s. of this equation system appear terms that describe the aforementioned physical effects. For example, in x- and y-direction it contains an interpolation polynom that interpolates the depth-aeraged velocity components U nad V at the nodes of the triangle where the particle is currently located in, and at time instances before and after the current simulation time, onto the location of the particle and to the current simulation time. The effect of diffusion is modeled by sampling stochastical velocity components in all 3 spatial directions using (pseudo-)random numbers. Based on a simple diffusion or turbulence model, the amplitude of these fluctuating velocity components is given as a function of the magnitude of the depth-averaged velocity and of the water height.

If during its course a particle attempts to intersect with a triangle edge, the seabed, or the water-level surface, the particle is at first moved just to that edge or surface. The step size that in this case exactly brings the particle to the intersection point with the cell boundary is computed with the aid of the Newton Iteration Method. If a particle moves into a neighboring triangle the index numbers of the nodes used for interpolation of the velocity field are updated. If a particle impinges on the water-level surface its vertical velocity component is specularly reflected at this surface. If a particle strikes the seabed it is deposited at the bed for the rest of the timestep and will continue to move from there, if a change in the vector sum of depth-averaged and stochastic velocity components permits it.

A thorough documentation of the physical modelling and implementation details of PARTRACE can be found in the Programmbeschreibung des Partikelverfahrens PARTRACE (currently in german only, sorry!).

## Proporcessors

• TRIM-2D: Navier-Stokes method for computation of synoptic 2D-depth-averaged flowfields (Finite difference method)
• TELEMAC: Navier-Stokes method for computation of synoptic 2D-depth-averaged flowfields (Finite element method)

## Postprocessors

• HVIEW2D: visualization of 2D synoptic flowfield quantities and particle tracks
• XMGR: interactive display of xy-diagrams