SPH and Lagrangian methods.

Numerical simulation has increasingly become prominent to solve various problems inherent to fluid dynamics. Two approaches have evolved to model fluid motion.

Eulerian methods emerged in the late 1960s. Those techniques are based on geometric, fixed or adaptive grids. Flow properties (velocity, pressure for example) are then estimated at each mesh element via numerical formulations such as finite differences, finite elements or finite volumes. These approaches are very successful to simulate flow driven by turbulence for instance, bounded flow or combustion reaction. Despite the success of such methods, the use of meshes results in major difficulties, particularly for the free surface treatment, movable interfaces or large deformations.

To address those cases, Gingold and Monaghan (1977) and Lucy (1977) introduced a Lagrangian approach: the SPH method (for Smoothed Particle Hydrodynamics). This approach relies on a fluid discretization into moving elementary bodies referred to as particles, the simulated flows being then represented by their movement. Physical quantities are evaluated at the positions of these particles. This evaluation considers the neighbor particles contribution by using a so-called kernel function, giving the smoothing characteristic of the formulation.

Thanks to its Lagrangian nature, the SPH method can be very easily adapted to complex and mobile geometries. Many researches have been carried out in recent years on the numerical aspects of precision and stability (Groenenboom et al., 2019; Shadloo et al., 2016; Violeau and Rogers, 2016), allowing the use of SPH for industrial problems in multiple fluid mechanics application fields, such as the automotive, naval and aeronautical industries.

Future posts will highlight how specific physics are handled in the SPH method, or how specific application cases are addressed with the SPH method.

References

Gingold, R. A. and Monaghan, J. J. (1977). Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly notices of the royal astronomical society, 181(3):375–389.

Groenenboom, P., Cartwright, B., McGuckin, D., Amoignon, O., Mettichi, M., Gargouri, Y., and Kamoulakos, A. (2019). Numerical studies and industrial applications of the hybrid SPH-FE method. Computers & Fluids, 184:40–63.

Lucy, L. B. (1977). A numerical approach to the testing of the fission hypothesis. The astronomical journal, 82:1013–1024.

Shadloo, M., Oger, G., and Le Touzé, D. (2016). Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: Motivations, current state, and challenges. Computers & Fluids, 136:11–34.

Violeau, D. and Rogers, B. D. (2016). Smoothed particle hydrodynamics (SPH) for free-surface flows: past, present and future. Journal of Hydraulic Research, 54(1):1–26.