AbstractUpwind and central difference schemes for laminar and turbulent flows over a step in a two‐dimensional channel are compared with each other and with experiment. Vorticity ω and stream function ψ are used as dependent variables and it is shown that an upwind difference method can give predictions which agree with experiment for high Reynolds number flows. The numerical implementation of the boundary conditions is found critically to determine the solutions obtained. Explicit prescription of ψ and ω at the inlet leads to incorrect values of the inlet velocity component perpendicular to the flow and a solution that does not agree with experiment. Experimental evidence is not available at low Reynolds numbers, but it is found that upwind differences give recirculation zones 8 per cent shorter and 8 per cent less intense than the more conventional conditionally stable central difference method.The distribution of the false diffusion effect in the upwind scheme is considered, and it is shown by use of a simple example that previous statements as to possible minimization of this effect are not generally true.The difficulty of determining the pressure distribution from the vorticity and stream function model is analysed and illustrated.Further, turbulent separated flows are seen to contain regions where two‐dimensional time‐dependent flow does not exist, and conventional theories do not give goo