Controlled variation scheme in a sequential solver for recirculating flows, part I: Theory and formulation

The formalism of the total variation diminishing (TVD) schemes is utilized to design a convection scheme for incompressible recirculating flows. The scheme has been named the controlled variation scheme (CVS). Even though the CVS does not possess the TVD property for sequential solution algorithms, due to the appearance of source terms, the concept of controlled variation fluxes can be effective in suppressing spurious oscillations that commonly occur in convection-dominated viscous flows, by injecting a nonlinear numerical diffusion, similar to the original TVD schemes, into the central difference scheme. This is demonstrated by using the one-dimensional linear convection-diffusion equation with and without a source term as model problems. The formulation and an efficient implementation of the CVS in a sequential pressure-based solver for incompressible steady-state Navier-Stokes equations is presented in this work. The applications of the CVS for two-dimensional laminar and turbulent flows is presented in Part II of the present work.