Flow Solvers: Three Approaches to the Lid-Driven Cavity Problem
The lid-driven cavity problem is one of the most studied benchmarks in computational fluid dynamics — deceptively simple in geometry yet rich in flow physics. A square cavity is sealed on three sides by stationary no-slip walls, while the top wall moves at a constant velocity, inducing a recirculating vortex inside. This project implements and compares three distinct numerical methods for solving the incompressible Navier–Stokes equations in this setting, and validates them against the classical benchmark data of Ghia, Ghia, and Shin (1982).
The Artificial Compressibility Method [Click here ....]
The incompressibility constraint is enforced via a pseudo-time pressure evolution equation on a staggered MAC grid. An explicit pseudo-time-marching scheme advances pressure and momentum until the continuity residual falls below the tolerance. Cases at Re = 100 and Re = 400 are presented, along with grid-convergence analysis across four mesh sizes.
The Pressure Projection Method [Click here ....]
A fractional-step approach on a staggered grid: momentum is advanced without pressure to yield an intermediate velocity, a pressure Poisson equation is then solved via Gauss–Seidel SOR to project that field onto a divergence-free space, and the velocity is corrected. Results at Re = 100 and Re = 400 are validated against benchmark centerline profiles, with spatial convergence studies confirming second-order accuracy.
The Lattice Boltzmann Method [Click here ....]
A mesoscopic kinetic approach using the D2Q9 lattice with BGK single-relaxation-time collision. Rather than solving the Navier–Stokes equations directly, particle distribution functions are evolved through collision and streaming steps, recovering macroscopic flow quantities at each lattice node. Results are presented at Re = 100, 400, and 1000, with additional studies on the effect of relaxation time and Mach number on stability and accuracy.
Key Objectives and Outcomes:
All three solvers are benchmarked against Ghia et al.'s centerline velocity data and exhibit monotonic convergence with grid refinement, consistent with second-order spatial accuracy. The artificial compressibility method offers conceptual simplicity for steady-state problems; the projection method provides physically consistent pressure–velocity coupling suitable for transient flows; and the Lattice Boltzmann method stands apart as a parallelization-friendly mesoscopic alternative with natural extension to complex geometries and transient low-Mach-number flows.