A compatible high-order meshless method for the Stokes equations with applications to suspension flows
This work provides a stable, high-order meshless solver for Stokes flow, addressing a long-standing compatibility issue, which is important for simulating dense suspensions and complex geometries.
The authors developed a compatible high-order meshless method for Stokes equations that achieves equal-order convergence for velocity and pressure on curvilinear geometries, and applied it to dense suspension flows without needing sub-grid lubrication models.
A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a discretization that couples a staggered scheme for pressure approximation with a divergence-free velocity reconstruction to obtain an adaptive, high-order, finite difference-like discretization that can be efficiently solved with conventional algebraic multigrid techniques. We use analytic benchmarks to demonstrate equal-order convergence for both velocity and pressure when solving problems with curvilinear geometries. In order to study problems in dense suspensions, we couple the solution for the flow to the equations of motion for freely suspended particles in an implicit monolithic scheme. The combination of high-order accuracy with fully-implicit schemes allows the accurate resolution of stiff lubrication forces directly from the solution of the Stokes problem without the need to introduce sub-grid lubrication models.