A numerical method to determine the history of particle paths is presented and its application for mixing quantification is illustrated. When more than one source exists in a flow field, the current technique can reveal the particleâ€™s identity found in any time and place in the field, by backward tracking its origin. Since the particle position at a preceding time is not known, the velocity vector is implicit. To resolve this uncertainty, a three-stage iterative procedure is developed and implemented. The current particle velocity is multiplied by the time increment and the product is used to estimate the previous time increment particle position. Two velocity matrices are generated on a mesh around the estimated position. The first matrix is the Eulerian velocity field interpolated on the mesh. The second matrix contains velocity vectors that point to the current particle position. A correlation matrix is calculated from the two velocity matrices in order to resolve the actual particle position in the previous time increment. Determination of the time increments' size is performed by checking the maximum of the correlation matrix. The new algorithm was validated using a numerical solution of the confined twin-jet flow at low Reynolds number. This flow performs a Hopf bifurcation at a Reynolds number of about 30, therefore chaotic trajectories might exist in the flow. First, the convergence of the backward particle path to the streamlines in steady flow was demonstrated. Convergence of the particle paths for various time increments was achieved also at the unsteady two-dimensional confined twin-jet flow. When the flow has more than one source, the proposed tracking method can be applied to generate complete and ordered particle source images in any desired position and time in the flow field. Maps of particle sources are used to visualize the flow patterns and the stretched interface between the two fluid sources. Those maps are demonstrated as a powerful tool for mixing quantification and can be implemented also for pollution source detection and many other fluid dynamics applications.
|Journal||Physical Review E - Statistical, Nonlinear, and Soft Matter Physics|
|State||Published - 2006|