The paper describes the development and application of a finite volume scheme for the solution of the Favre-averaged Navier–Stokes equations on mixed-element grids, consisting of triangles and quadrilaterals in 2D, and of tetrahedra, pyramids, triangular prisms and hexahedra in 3D. The important features of the present approach are the discretization of the domain via a single, unified edge-data structure for mixed-element meshes and the use of Laplacian weights to calculate the viscous fluxes. The Laplacian weights are evaluated using an approximation of the Galerkin finite element method and the formulation results in nearest-neighbour stencils. Transonic, turbulent flow over a turbine blade was studied as a validation case. It was shown that the proposed viscous flux discretization could not only handle significantly distorted meshes but also allow higher CFL numbers than standard finite volume viscous flux discretization methods. Copyright © 2000 John Wiley & Sons, Ltd.