Accurate and efficient solutions of unsteady viscous flows

This paper describes two accurate and efficient numerical methods for computing unsteady viscous flows. The first one solves the incompressible Navier‐Stokes equations in their vorticity‐velocity formulation, using a staggered‐grid finite‐volume spatial discretization to provide second‐order accuracy on arbitrary grids, and combines effectively an alternating direction implicit scheme for the vorticity transport equation and a multigrid line‐Gauss‐Seidel relaxation for the velocity equations. The second method solves the compressible Reynolds‐averaged Navier‐Stokes equations in strong conservation form, with a k−ω turbulence closure model. The equations are discretized in time using an implicit three‐time‐level scheme, combined with a dual time stepping approach, so that the residual at every physical time step is annihilated using an efficient multigrid Runge‐Kutta iteration with variable time stepping and implicit residual smoothing. The space discretization uses a Roe’s flux difference splitting for the convective terms and standard central differences for the diffusive ones. A turbulent unsteady cascade flow is used to demonstrate the accuracy and efficiency of the method. The authors are currently working towards extending the two approaches described in this paper to three space dimensions.