Effective diffusion coefficient for steady two-dimensional convective flow

We use the homogenization method to obtain an expression for the effective diffusion coefficient $D^{\mathrm{*}}$ for convective flows. We evaluate this expression numerically by a finite-mode truncation for two-dimensional laminar flows with arbitrary Péclet numbers. Our results allow us to establish the range of validity of the small- and large-Péclet-number asymptotic evaluations of $D^{\mathrm{*}}$. In particular, we find that for the case of rigid boundary conditions corrections to the asymptotic scaling decay only very slowly for large Péclet numbers and are still about 5% for Péclet numbers as high as ${10}^6$.