An alternative full multigrid SIMPLEC approach for the incompressible Navier–Stokes equations

Abstract

An alternative approach to solve the steady-state incompressible Navier–Stokes equations using the multigrid (MG) method is presented. The mathematical model is discretized using the finite volume method with second-order approximation schemes in a uniform collocated (nonstaggered) grid. MG is employed through a full approximation scheme-full MG algorithm based on V-cycles. Pressure-velocity coupling is ensured by means of a developed modified SIMPLEC algorithm which uses independent V-cycles for relaxing the pressure-correction and momentum equations. The coarser grids are used only internally in these cycles. All other original SIMPLEC steps can be performed only on the finest grid of the current full MG level. The model problem of the lid-driven flow in the unitary square cavity is used for the tests of the numerical model. Computational performance is measured through error and residual decays and execution times. Good performances were obtained for a wide range of Reynolds numbers, with speedups of orders as high as $O({10}^3).$ Linear relationships between execution times and grid sizes were observed for low and high Re values ($Re=0.1,1,10,2,500,3,200,5,000,$ and 7,500). For intermediate Re values ($Re=100,400,$ and 1,000), the linear trend was observed from more refined grids (512${}^2$ onwards).

Keywords:

• Finite volume method
• full-multigrid
• Navier–Stokes equations
• SIMPLEC

Disclosure statement

The authors declare that there are no conflicts of interest related to this study.

Additional information

Funding

The authors would like to thank the financial support for this work provided by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. They also acknowledge the Graduate Program in Numerical Methods in Engineering (PPGMNE) and the infrastructure of the Numerical Experimentation Laboratory (LENA) of the Federal University of Parana (UFPR). The authors declare that no specific funding was received for this study.