Computation of Pressure Fields around a Two-Dimensional Circular Cylinder Using the Vortex-In-Cell and Penalization Methods

Abstract

The vorticity-velocity formulation of the Navier-Stokes equations allows purely kinematical problems to be decoupled from the pressure term, since the pressure is eliminated by applying the curl operator. The Vortex-In-Cell (VIC) method, which is based on the vorticity-velocity formulation, offers particle-mesh algorithms to numerically simulate flows past a solid body. The penalization method is used to enforce boundary conditions at a body surface with a decoupling between body boundaries and computational grids. Its main advantage is a highly efficient implementation for solid boundaries of arbitrary complexity on Cartesian grids. We present an efficient algorithm to numerically implement the vorticity-velocity-pressure formulation including a penalty term to simulate the pressure fields around a solid body. In vorticity-based methods, pressure field can be independently computed from the solution procedure for vorticity. This clearly simplifies the implementation and reduces the computational cost. Obtaining the pressure field at any fixed time represents the most challenging goal of this study. We validate the implementation by numerical simulations of an incompressible viscous flow around an impulsively started circular cylinder in a wide range of Reynolds numbers: Re=40, 550, 3000, and 9500.

Numerical Simulation of Vortex-Dominated Flows Using the Penalized VIC Method

“…The grid for solving the Poisson equation can define a compact computational domain with non-homogeneous Dirichlet boundary conditions. The boundary condition of a domain Ω can be approximated using a Green's function approach [28], which is given by…”

“…Furthermore, this approach takes long time of computation. Alternatively, pressure field can be evaluated based on the penalized Navier-Stokes equation [28]. Satisfying the continuity, the Poisson equation for pressure can be expressed as…”

“…Another important point of concern about VMs is boundary conditions for wall-bounded flows. In recent years, the VIC method has been successfully combined with the penalization method [24], which is useful to enforce boundary conditions at a solid wall (for example, refer to [25][26][27][28][29][30][31][32][33]). The combination, hereafter called the penalized vortex-in-cell (pVIC) method, greatly simplifies an entire numerical procedure required to construct a solution algorithm, thus leading to a well-performing parallel program.…”

Numerical Investigation on Vortex Shedding from a Hydrofoil with a Beveled Trailing Edge

“…Pressure which can be evaluated in an explicit manner using the identified vorticity and velocity fields [27] is not part of the solution algorithm. For a 2D flow parallel to the -plane, the vorticity transport equation in terms of the vorticity can be expressed as…”

“…To compactly place a square FFT-domain Ω, we used nonhomogeneous Dirichlet boundary conditions in this study. Unknown stream functions at the domain boundaries ( Ω) can be directly obtained from Green's function solution [27], whereby…”