Most numerical studies on solute mixing rely on mesh-based methods, and complicated schemes have been developed to enhance numerical stability and reduce artificial diffusion. This paper systematically studies the depth-averaged random walk scheme, which is a meshfree method with the merits of being highly robust and free of numerical diffusion. First, the model is used to solve instantaneous release problems in uniform flows. Extensive parametric studies are carried out to investigate the influences of the number of particles and the size of time steps. The predictions are shown to be independent of time steps but are sensitive to the particle numbers. Second, the model is applied to the solute transport problem along an estuary subject to extensive wetting and drying during tidal oscillations. Finally, the model is used to investigate the wind-induced chaotic mixing in a shallow basin. The effect of diffusion on the chaotic mixing is investigated. This study proposes a generic sampling method to interpret the output of the random walk method and highlights the importance of accurately taking diffusion into account in analysing the transport phenomena. The sampling technique also offers a guideline for estimating the total number of particles needed in the application.

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