A Low Dissipation Riemann SPH formulation for Multiphase Flows

Multiphase flows with large density ratios, such as violent air-water interactions during wave breaking, remain particularly challenging for numerical modelling. Smoothed particle hydrodynamics (SPH) is an attractive scheme for these problems because its Lagrangian, mesh free nature can naturally handle large deformations and complex topology changes. However, multiphase SPH formulations still face challenges such as pressure noise and acoustic time-step restrictions, and often require additional stabilisation mechanisms. These difficulties are closely related to sharp discontinuities in density and other physical properties across the interface. Riemann-based SPH schemes provide a physically consistent treatment of shocks and discontinuities by integrating approximate Riemann solvers into the SPH particle-particle interactions. However, their application to multiphase flows remains limited because of their inherent excessive numerical dissipation, particularly at low Mach numbers. Moreover, many existing Riemann-based multiphase models do not incorporate a physically consistent surface tension model. Surface tension is a natural interfacial force that generates the capillary pressure jump across curved interfaces and strongly influences interface deformation, breakup, and coalescence. This limits their applicability to surface tension dominated problems such as rising bubble, square droplet relaxation, and capillary waves. This thesis develops and evaluates an efficient Riemann-based SPH framework for large density ratio multiphase flows and implements it within the open-source code DualSPHysics. A novel formulation is proposed to reduce the excessive numerical dissipation in Riemann-based SPH schemes. Specifically, the diffusive term in the star region is scaled with the local Mach number, thereby ensuring physically consistent dissipation across different flow regimes. The proposed formulation is independent of tuning parameters and can be used regardless of the specific approximate Riemann solver adopted in the scheme. In addition, MUSCL reconstruction is introduced as a complementary technique to further reduce dissipation in the Riemann solver. To accurately handle fluid-structure interaction, a novel boundary treatment is proposed. Solid boundaries are represented using the fixed particles representing the walls as in the mDBC approach of DualSPHysics, while the fluid-boundary interaction is treated as a partial Riemann problem. The proposed formulation is validated for a range of single phase benchmarks test cases, such as Poiseuille flow, Taylor Green vortex, lid-driven cavity, and impact of rectangular water jets, with and without MUSCL reconstruction, and compared against existing dissipation coefficients used in Riemann-based formulations. Numerical results demonstrate that the proposed dissipation coefficient formulation increases accuracy, significantly reduces excessive numerical dissipation, and maintains an accurate pressure field even in highly dynamic impact flows. To accurately handle the multiphase flow with a sharp interface, a continuum surface tension model is incorporated into the low dissipation Riemann solver. This links interface regularisation to physical surface tension force instead of using artificial techniques to stabilize the interface and provides stable pressure field even under high impact, large density ratio conditions. The resulting multiphase solver is validated against a broad range of multiphase benchmark test cases, from bubble dynamics to high impact multiphase flows. The simulations show sharp interfaces with no significant void formation, accurate pressure jumps and impact loads, and good agreement with experimental or reference numerical data. A single artificial speed of sound that is defined for both phases, making the proposed scheme computationally efficient. Overall, the proposed Riemann based SPH framework provides a computationally efficient tool for the simulation of violent, large density ratio multiphase flows, with reduced reliance on empirical tuning parameters.