An adaptive multigrid finite‐volume scheme for incompressible Navier–Stokes equations

An algorithm for the solutions of the two‐dimensional incompressible Navier–Stokes equations is presented. The algorithm can be used to compute both steady‐state and time‐dependent flow problems. It is based on an artificial compressibility method and uses higher‐order upwind finite‐volume techniques for the convective terms and a second‐order finite‐volume technique for the viscous terms. Three upwind schemes for discretizing convective terms are proposed here. An interesting result is that the solutions computed by one of them is not sensitive to the value of the artificial compressibility parameter. A second‐order, two‐step Runge–Kutta integration coupling with an implicit residual smoothing and with a multigrid method is used for achieving fast convergence for both steady‐ and unsteady‐state problems. The numerical results agree well with experimental and other numerical data. A comparison with an analytically exact solution is performed to verify the space and time accuracy of the algorithm.