TY - JOUR
T1 - A collocated method for the incompressible Navier-Stokes equations inspired by the Box scheme
AU - Ranjan, R.
AU - Pantano, C.
N1 - Funding Information:
This work was funded in part by the National Science Foundation under grant DMS-0908491 and by the ASC program of the Department of Energy under subcontract B523297 and the California Institute of Technology. Computational resources were provided by the Computational Science and Engineering center at the University of Illinois through an allocation in the Taub supercomputer.
PY - 2013/1/1
Y1 - 2013/1/1
N2 - We present a new finite-difference numerical method to solve the incompressible Navier-Stokes equations using a collocated discretization in space on a logically Cartesian grid. The method shares some common aspects with, and it was inspired by, the Box scheme. It uses centered second-order-accurate finite-difference approximations for the spatial derivatives combined with semi-implicit time integration. The proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity-pressure form of the equations. The continuity equation is enforced exactly (to machine accuracy) at the collocated locations, whereas the momentum equations are evaluated in a staggered manner. This formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure. The method shows uniform order of accuracy, both in space and time, for velocity and pressure. In addition, the skew-symmetric form of the non-linear advection term of the Navier-Stokes equations improves discrete conservation of kinetic energy in the inviscid limit, to within the order of the truncation error of the time integrator. The method has been formulated to accommodate different types of boundary conditions; fully periodic, periodic channel, inflow-outflow and lid-driven cavity; always ensuring global mass conservation. A novel aspect of this finite-difference formulation is the derivation of the discretization near boundaries using the weak form of the equations, as in the finite element method. The method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver. To assess the applicability of the new method presented in this paper, four realistic flow problems have been simulated and results are compared with those in the literature. These cases include a lid-driven cavity, backward-facing step, Kovasznay flow, and fully developed turbulent channel.
AB - We present a new finite-difference numerical method to solve the incompressible Navier-Stokes equations using a collocated discretization in space on a logically Cartesian grid. The method shares some common aspects with, and it was inspired by, the Box scheme. It uses centered second-order-accurate finite-difference approximations for the spatial derivatives combined with semi-implicit time integration. The proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity-pressure form of the equations. The continuity equation is enforced exactly (to machine accuracy) at the collocated locations, whereas the momentum equations are evaluated in a staggered manner. This formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure. The method shows uniform order of accuracy, both in space and time, for velocity and pressure. In addition, the skew-symmetric form of the non-linear advection term of the Navier-Stokes equations improves discrete conservation of kinetic energy in the inviscid limit, to within the order of the truncation error of the time integrator. The method has been formulated to accommodate different types of boundary conditions; fully periodic, periodic channel, inflow-outflow and lid-driven cavity; always ensuring global mass conservation. A novel aspect of this finite-difference formulation is the derivation of the discretization near boundaries using the weak form of the equations, as in the finite element method. The method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver. To assess the applicability of the new method presented in this paper, four realistic flow problems have been simulated and results are compared with those in the literature. These cases include a lid-driven cavity, backward-facing step, Kovasznay flow, and fully developed turbulent channel.
KW - Box scheme
KW - Centered finite differences
KW - Discrete mass conservation
KW - Incompressible flow
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U2 - 10.1016/j.jcp.2012.08.021
DO - 10.1016/j.jcp.2012.08.021
M3 - Article
AN - SCOPUS:84868456858
SN - 0021-9991
VL - 232
SP - 346
EP - 382
JO - Journal of Computational Physics
JF - Journal of Computational Physics
IS - 1
ER -