| A higher-order algorithm has been developed for computing steady turbulent flow over two-dimensional airfoils. The algorithm uses finite-differences applied through a generalized curvilinear coordinate transformation, applicable to single- and multi-block grids. Numerical dissipation is added using the matrix dissipation scheme. Turbulence is modeled using the Baldwin-Lomax and Spalart-Allmaras models. The various components of the spatial discretization, including the convective and viscous terms, the numerical boundary schemes, the numerical dissipation, and the integration technique used to calculate forces and moments, have all been raised to a level of accuracy consistent with third-order global accuracy. The two exceptions, both of which proved not to introduce significant numerical error, are the first-order numerical dissipation added near shocks and the first-order convective terms in the Spalart-Allmaras turbulence model. Results for several grid convergence studies show that this globally higher-order approach produces a dramatic reduction in the numerical error in drag. It can provide equivalent accuracy to a second-order algorithm on a grid with several times fewer nodes. For subsonic and transonic single-element cases, errors of less than two percent are obtained on grids with only 15,000 nodes while 4 times as many nodes are required for the second-order algorithm. Similar accuracy is obtained for a three element case on grids with only 73,000 nodes, a third of that required by the second-order algorithm. The results provide a convincing demonstration of the benefits of higher-order methods for practical flows. |
