A robust face-averaged gradient method for viscous flux computation on unstructured grids

In the cell-centered unstructured finite volume method for solving the Navier–Stokes equations, the viscous flux is discretized by applying Gauss's theorem, which converts the volume-integrated viscous flux into a surface integral over cell faces. In classical methods, the computation of viscous flux on a face requires both the basic variables and their gradients at the face center. The basic variables are reconstructed from cell-centered values, and the face-centered gradients are typically computed using the gradients from adjacent cell centers. However, when grid quality is poor, this gradient computation method, which strongly depends on the geometric relationship between neighboring cells, often leads to reduced gradient accuracy and numerical instability. To overcome this limitation, the present study proposes a face-averaged gradient method for viscous flux computation. In this method, cell-centered gradients are first interpolated to grid nodes using an inverse distance weighting scheme. The nodal gradients are then averaged back to the target face center through a second inverse distance weighting interpolation. This two-step process provides a more accurate gradient at the face center, which improves the stability of viscous flux calculations. Numerical simulations show that, compared to classical methods, the proposed approach significantly reduces sensitivity to grid quality, while maintaining high accuracy and convergence performance in complex geometries and unstructured grids.