An improved dual mesh control domain formulation for the unsteady flow of viscous incompressible fluids

The dual mesh control domain method (DMCDM) is applied, in conjunction with the reduced integration penalty formulation, to solve transient isochoric flow problems of Newtonian fluids, particularly at high Reynolds numbers. The DMCDM is a physically more intuitive method in that the global conservation laws are directly used (akin to the finite volume method), as opposed to the finite element method (FEM), where the weighted integral form of the governing equations is satisfied—an approach that bears little or no physical meaning in the context of fluid flow. However, the DMCDM exploits ideas from the FEM related to interpolation, duality, assembly, and numerical quadrature, allowing for the robust and efficient implementation of the former. In this paper, an element-based, parallelizable formulation for unsteady isochoric flows is investigated, in which the Navier–Stokes equations are semi-discretized using the DMCDM, achieving pressure–velocity coupling through the quadratic penalty method. Unexplored previously in the DMCDM literature, time-dependent problems are addressed, with the time dependence resolved using the alpha-family of approximations to arrive at a fully discretized system of nonlinear algebraic equations. This paper provides insights into implementation aspects, including numerical quadrature and the linearization of the system of equations using Newton's method—both of which have not been presented in detail previously. The ability of the method to capture fine-scale flow features, such as the formation of sub-eddies, is assessed using the lid-driven cavity problem at a high Reynolds number of 10 000. Additionally, the outflow boundary condition is examined using the flow over a backward-facing step problem, and finally, the method's ability to resolve instabilities is tested using the flow past a cylinder problem.