An efficient method for two-fluid incompressible flows appropriate for the immersed boundary method

Highlights: • Coupling the Immersed Boundary method with Fast Direct Solvers for incompressible two-fluid flows. • Proposing a two-fluid pressure-correction scheme for a constant coefficient Poisson equation. • Validated for low and high density/viscosity ratios. • Same accuracy with variable coefficient Poisson at reduced computational cost. • Achieving speed-up factors of 10–100 for density ratios of 1000–10. -- Abstract: The numerical simulation of two-fluid flows with sharp interfaces is a challenging field, not only because of their complicated physical mechanisms, but also because of increased computational cost. An efficient and robust numerical formulation for incompressible two-fluid flows is proposed. Its novelty is the consistent coupling of Fast Direct Solvers (FDS) with the Immersed Boundary (IB) method to represent solid boundaries. Such a coupling offers several advantages. First, it extends the range of applicability of the IB method. Second, it allows the simulation of practical problems in geometrically complicated domains at a significantly reduced cost. Third, it can shed light on regions of the parametric space which are considered out of reach, or even impossible today. Instead of using a conventional variable coefficient pressure Poisson equation, a pressure-correction scheme is suggested for the solution of a constant coefficient Poisson equation for the pressure difference, extending the novel work of Dodd and Ferrante [8]. The conservative Level-set (LS) method is used to track the interface between the two fluids. Appropriate schemes, based on the local directional Ghost Cell Approach (GCA) are proposed, in order to satisfy the boundary conditions (BCs) of the pressure and the LS function around the IB. The accuracy, robustness, and performance of the proposed method is demonstrated by several validations against conventional approaches and experiments. The results verify that the pressure BCs are properly recovered along the IB solid interface, while a non-smooth pressure field is also allowed across the solid obstacle. The accuracy of the method was found to be 2nd-order, both in time and space. The performance of the proposed method is compared against the conventional approach using a multigrid iterative solver. The impact of the time-step on the accuracy of the constant coefficient approach is examined. Results show that the final speed-up strongly depends on the specific physical and numerical parameters such as the density ratio or the Reynolds number. It is demonstrated that for the range of parameters examined, speed-up factors of 100–10 can be achieved for density ratios of 10–1000 respectively.