Premium
High‐order divergence‐free velocity reconstruction for free surface flows on unstructured Voronoi meshes
Author(s) -
Boscheri Walter,
Pisaturo Giuseppe Roberto,
Righetti Maurizio
Publication year - 2019
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.4723
Subject(s) - mathematics , discretization , delaunay triangulation , vector field , voronoi diagram , polygon mesh , interpolation (computer graphics) , mathematical analysis , flow (mathematics) , geometry , classical mechanics , physics , motion (physics)
Summary In this paper, we present an efficient semi‐implicit scheme for the solution of the Reynolds‐averaged Navier‐Stokes equations for the simulation of hydrostatic and nonhydrostatic free surface flow problems. A staggered unstructured mesh composed by Voronoi polygons is used to pave the horizontal domain, whereas parallel layers are adopted along the vertical direction. Pressure, velocity, and vertical viscosity terms are taken implicitly, whereas the nonlinear convective terms as well as the horizontal viscous terms are discretized explicitly by using a semi‐Lagrangian approach, which requires an interpolation of the three‐dimensional velocity field to integrate the flow trajectories backward in time. To this purpose, a high‐order reconstruction technique is proposed, which is based on a constrained least squares operator that guarantees a globally and pointwise divergence‐free velocity field. A comparison with an analogous reconstruction, which is not divergence‐free preserving, is also presented to give evidence of the new strategy. This allows the continuity equation to be satisfied up to machine precision even for high‐order spatial discretizations. The reconstructed velocity field is then used for evaluating high‐order terms of a Taylor method that is here adopted as ODE integrator for the flow trajectories. The proposed semi‐implicit scheme is validated against a set of academic test problems, and proof of convergence up to fourth‐order of accuracy in space is shown.