Incorporating the vertical velocity in a coupled Lagrangian–Eulerian approach for particle transport in shallow flows

This work presents a method to incorporate vertical velocity into a two-dimensional depth-averaged Shallow Water Equation (2DH SWE) model, thereby improving the calculation of particle trajectories in a Lagrangian Particle Tracking (LPT) framework. The resulting formulation couples Eulerian and Lagrangian approaches. The vertical velocity is also used to modify the dispersion terms in the LPT model. The proposed approximation is first validated—without particle transport—by comparison with Hyperbolic–Elliptic and Hyperbolic-Relaxed Non-Hydrostatic Pressure (NHP) models. The differences between models are minor, confirming the suitability of the vertical velocity approximation for shallow flow problems. Subsequently, the method is applied to particle transport scenarios, demonstrating that including vertical velocity yields more realistic particle trajectories in complex flow situations. • An efficient approach for vertical velocity in 2D shallow models is proposed. • Vertical velocity approach improves Lagrangian particle transport in shallow flows. • Validated against non-hydrostatic models with good agreement in steady cases. • Reduces particle buildup and improves vertical transport realism in 2D flows.