An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

We present an immersed interface method for the vorticity-velocity form of the 2D Navier Stokes equations that directly addresses challenges posed by multiply connected domains, nonconvex obstacles, and the calculation of force distributions on immersed surfaces. The immersed interface method is re-interpreted as a polynomial extrapolation of flow quantities and boundary conditions into the obstacle, reducing its computational and implementation complexity. In the flow, the vorticity transport equation is discretized using a conservative finite difference scheme and explicit Runge-Kutta time integration. The velocity reconstruction problem is transformed to a scalar Poisson equation that is discretized with conservative finite differences, and solved using an FFT-accelerated iterative algorithm. The use of conservative differencing throughout leads to exact enforcement of a discrete Kelvin's theorem, which provides the key to simulations with multiply connected domains and outflow boundaries. The method achieves second order spatial accuracy and third order temporal accuracy, and is validated on a variety of 2D flows in internal and free-space domains

Immersed interface interpolation schemes for particle–mesh methods

The sharp and high-order treatment of arbitrary boundaries immersed in the computational domain remains a challenge to particle methods. While several techniques have been proposed to modify numerical stencils, e.g. Finite Difference ones, near the walls, the particle–mesh interpolation component of particle methods also has to be modified. This operation, mapping fields from the grid to the particles and vice-versa, has to be performed several times per computational step in the framework of particle–mesh methods. The present paper proposes an extension of classical particle–mesh interpolation approaches by computing high-order ghost fields based on the information about the solution behavior at the wall. This approach is further shown to be especially interesting when combined with a dimension-splitting Immersed Interface method to correct the spatial differential operators. Indeed, the associated corrections are computed at the intersection between the interface and the grid lines, making the necessary information for the ghost construction readily available. The mesh-to-particles and particles-to-mesh interpolation schemes are validated individually in convergence studies and, finally, both are applied to the advection–diffusion of a passive tracer past 2D objects

Vortex particle-mesh methods: accurate and efficient handling of solid boundaries

Vortex methods have been through many developments over the past decades, yet the efficient and accurate treatment of solid boundaries has remained a challenge. In this talk, we will discuss recent efforts aimed towards that goal, and in the context of a vortex particle-mesh (VPM) method. The VPM method is a state-of-the-art variant, which relies on a dual discretization: the particles handle the advection of vorticity, while the mesh is used, not only for the “remeshing” (i.e., the “particle redistribution” step, required in Lagrangian methods to maintain time accuracy), but also for the efficient evaluation of the differential operators (diffusion, vortex stretching) and the efficient solution of the elliptic problems (Biot- Savart law, re-projection step when in 3D). Specifically, this presentation will focus on the handling of solid boundaries, first through penalization methods, then through Immersed Interface techniques. We discuss the porting of such techniques to the context of vortex methods and some specific developments to ensure their efficiency [3, 5, 4]. Additional results for the handling of fluid-structure interaction problems will also be presented [2, 1]. References [1] Caroline Bernier, Mattia Gazzola, Renaud Ronsse, and Philippe Chatelain. Coupling a vortex particle-mesh method to a multi-body system solver for the simulation of articulated swimmers. In 7th International Conference on Vortex Flows and Vortex Models (ICVFM 2016), Sep 2016. [2] Mattia Gazzola, Philippe Chatelain, Wim M. van Rees, and Petros Koumoutsakos. Simulations of single and multiple swimmers with non-divergence free deforming geometries. Journal of Computa- tional Physics, 230(19):7093–7114, 8 2011. [3] Thomas Gillis, Gr ́egoire Winckelmans, and Philippe Chatelain. An efficient iterative penalization method using recycled krylov subspaces and its application to impulsively started flows. Journal of Computational Physics, 347:490–505, 2017. [4] Thomas Gillis, Gr ́egoire Winckelmans, and Philippe Chatelain. Fast immersed interface poisson solver for 3d unbounded problems around arbitrary geometries. Journal of Computational Physics, 354:403–416, 2018. [5] Yves Marichal, Philippe Chatelain, and Gr ́egoire Winckelmans. Immersed interface interpolation schemes for particle-mesh methods. Journal of Computational Physics, 326:947–972, Dec 2016