A simple Eulerian-Lagrangian weakly compressible smoothed particle hydrodynamics method for fluid flow and heat transfer

In this article, we propose a simple, consistent, and robust Eulerian-Lagrangian weakly compressible smoothed particle hydrodynamics (EL-WCSPH) for the simulation of hydrodynamics and convection heat transfer problems. First, the basic governing equation is formulated using the ALE-SPH framework. The universal formulation is similar to the standard WCSPH, but the degree of Eulerian and Lagrangian schemes can be controlled by inserting a pre-defined arbitrary parameter alpha is an element of[0,1]$$ \upalpha \in \left[0,1\right] $$. To alleviate potential numerical errors and inconsistency, particle mass correction is proposed along with particle disorder correction. Then, an algorithm is introduced, in which only a slight modification is necessary compared to the standard WCSPH. Finally, extensive benchmark simulations are conducted to validate the proposed method. As a result, the proposed method is found to perform well in hydrodynamics problems with various flow conditions (pipe flow, lid-driven problem, Taylor-Green vortex, vortex shedding problem, and water sloshing) and thermal-hydraulics problems (natural convection in a square cavity).