Dd. Knight, A FULLY IMPLICIT NAVIER-STOKES ALGORITHM USING AN UNSTRUCTURED GRID AND FLUX-DIFFERENCE SPLITTING, Applied numerical mathematics, 16(1-2), 1994, pp. 101-128
An implicit algorithm is developed for the two-dimensional, compressib
le, laminar Navier-Stokes equations using an unstructured grid of tria
ngles. A cell-centered data structure is employed with the flow variab
les stored at the centroids of the triangles. The algorithm is based o
n Roe's flux difference split method for the inviscid fluxes, and a di
screte representation of the viscous fluxes and heat transfer using Ga
uss' Theorem. Linear reconstruction of the flow variables to the cell
faces, employed for the inviscid terms, provides second-order spatial
accuracy. Interpolation of the flow variables to the nodes is achieved
using a second-order accurate method. Temporal discretization employs
Euler, trapezoidal or 3-point backward differencing. The complete, ex
act Jacobian of the inviscid and viscous terms is derived. The algorit
hm is applied to the Riemann Shock Tube problem, a supersonic laminar
boundary layer on a flat plate, and subsonic viscous flow past an NACA
0012 airfoil. Results are in excellent agreement with theory and previ
ous computations.