Low-diffusivity scalar transport using a WENO scheme and dual meshing

Interfacial mass transfer of low-diffusive substances in an unsteady flow environment is marked by a very thin boundary layer at the interface and other regions with steep concentration gradients. A numerical scheme capable of resolving accurately most details of this process is presented. In this scheme, the fifth-order accurate WENO method developed by Liu et al. [13] was implemented on a non-uniform staggered mesh to discretize the scalar convection while for the scalar diffusion a fourth-order accurate central discretization was employed. The discretization of the scalar convection-diffusion equation was combined with a fourth-order Navier-Stokes solver which solves the incompressible flow. A dual meshing strategy was employed, in which the scalar was solved on a finer mesh than the incompressible flow. The order of accuracy of the solver for one-dimensional scalar transport was tested on both stretched and uniform grids. Compared to the fifth-order WENO implementation of Jiang and Shu [10], the Liu et al. [13] method was found to be superior on very coarse meshes. The solver was further tested by performing a number of two-dimensional simulations. At first a grid refinement test was performed at zero viscosity with shear acting on an initially axisymmetric scalar distribution. A second refinement test was conducted for an unstably stratified flow with low diffusivity scalar transport. The unstable stratification led to buoyant convection which was modelled using a Boussinesq approximation with a linear relationship between flow temperature and density. The results show that for the method presented a relatively coarse mesh is sufficient to accurately describe the fluid flow, while the use of a refined dual mesh for the low-diffusive scalars is found to be beneficial in order to obtain a highly accurate resolution with negligible numerical diffusion.