Computers Fluids Vol. 23,
No. 1, pp. 1?21, 1994
An Implicit Upwind Algorithm for Computing
Turbulent Flows on Unstructured Grids.
W. Kyle Anderson and Daryl L. Bonhaus
NASA Langley Research Center
Hampton, Virginia 23665?5225
An implicit, Navier-Stokes solution algorithm is presented for the computation of turbulent flow on unstructured grids. The inviscid fluxes are computed using an upwind algorithm and the solution is advanced in time using a backward-Euler time-stepping scheme. At each time step, the linear system of equations is approximately solved with a point-implicit relaxation scheme. This methodology provides a viable and robust algorithm for computing turbulent flows on unstructured meshes.
Results are shown for subsonic flow over a NACA 0012 airfoil and for transonic flow over a RAE 2822 airfoil exhibiting a strong upper-surface shock. In addition, results are shown for 3?element and 4?element airfoil configurations. For the calculations, two one?equation turbulence models are utilized. For the NACA 0012 airfoil, a pressure distribution and force data are compared with other computational results as well as with experiment. Comparisons of computed pressure distributions and velocity profiles with experimental data are shown for the RAE airfoil and for the 3?element configuration. For the 4?element case, comparisons of surface pressure distributions with experiment are made. In general, the agreement between the computations and the experiment is good.
For computing flows on complicated geometries such as multielement airfoils, the use of unstructured grids offers a good alternative to more traditional methods of analysis. This is primarily due to the promise of dramatically decreased time required to generate grids over complicated geometries. Also, unstructured grids offer the capability to locally adapt the grid to improve the accuracy of the computation without incurring the penalties associated with global refinement. While work remains to be done to fully realize their potential, much progress has been made in computing viscous flows on unstructured grids.
While several advances have been made for computing turbulent flow on unstructured grids (e.g.  ), probably the most mature and widely used code for computing two-dimensional turbulent viscous flow on unstructured grids is that of Mavriplis . In this reference, the solution algorithm is a Galerkin finite-element discretization and a Runge-Kutta time-stepping algorithm is used in conjunction with multigrid to obtain very efficient solutions. The turbulence model predominantly utilized in this code is that of Baldwin and Lomax  although extensions have been made to include a two-equation turbulence model . Other modifications to this code are presented in reference  in which backward-Euler time-differencing is used in conjunction with GMRES  to produce results which are competitive with multigrid for the cases considered.
The use of upwind differencing offers several advantages over a central-differencing formulation for computing viscous flows. For example, in references  and , it is clearly shown that with the flux-differencing scheme of Roe  the resolution of boundary layer details typically requires only half as many points as with a centraldifferencing code. As discussed in reference , the poor performance of the central-difference formulation is attributed to the scalar artificial dissipation formulas commonly used to damp odd-even oscillations and to provide non-linear stability.
For upwind calculations on unstructured grids, Barth  has described methodology for utilizing Roe's approximate Riemann solver  for the inviscid flux computations and a Galerkin formulation for the viscous terms. In this work, a sparse matrix solver is used in conjunction with a Runge-Kutta time-stepping algorithm for