Pressure-Velocity Coupling in Incompressible Flows: Finite Difference Method

This chapter presents numerical methods for enforcing pressure–velocity coupling in incompressible flows using finite difference techniques. It examines the challenge of maintaining mass conservation without an explicit pressure evolution equation and introduces classical algorithms such as MAC, SMAC, HSMAC, Fractional-Step. The predictor–corrector structure of projection and fractional-step methods is discussed in detail, with emphasis on velocity prediction, Poisson-based pressure correction, and velocity update to ensure incompressibility. The role of staggered grids in preventing pressure–velocity decoupling is analyzed, and direction-splitting approaches for collocated grids are introduced. Additional approaches such as the artificial compressibility method and the pseudo-transient pressure gradient method are also discussed. The chapter concludes with the vorticity–streamfunction formulation and benchmark studies like the lid-driven cavity problem is presented as a case study.