A new Navier-Stokes procedure for the calculation of the complex turbulent flow about a two-dimensional airfoil has been developed. The solution to the Reynolds-averaged Navier-Stikes equations are sought in the domain that encompasses the entire airfoil using the CELS (Coupled Equation Line Solver) algorithm.
Major elements of the calculation have been critically examined and improved whenever possible. Firstly, the grid is made orthogonal. Using conformal mapping, the potential-and stream-lines for an arbitrary airfoil at incidence are generated. The equations written in this coordinate system are expected to be least prone to the error due to numerical diffusion. Secondly, to accurately predict the turbulence quantities and thereby the entire flow field, a modified k-$\varepsilon$ turbulence model has been devised the new model is basically a combination of the model by Hanjalic & Launder, which accounts for the irrotational strain, and improves the performance in the adverse pressure gradients region, and the model by Pope, which takes the vortex stretching (or squeezing) effects into account. The Reynolds normal stresses, which are required for the closure of the model, are obtained by incorporating Yoshizawa's anisotropic model rather than using experimentally observed relations of these stresses to the turbulence kinetic energy as was done by Hanjalic & Launder. The new model gives a markedly improved result over other models when applied to the plane-of-symmetry boundary layer of a body of revolution. Thirdly, the transport equations for the turbulence quantities, k and $\varepsilon$, are solved for the entire domain to ensure the smooth transition to turbulent flow. In the laminar region, however, the eddy viscosity is set equal to zero to suppress the growth of turbulence.
By comparing the result with that obtained in the non-streamline coordinates, it has been found that the solution can be obtained more efficiently in the streamline coordinates. The code is then applied to predict the turbulent flows about NACA airfoil sections, 0012 and 4412. The results are in much better agreement with the experiments than those of other methods. The pressure and the lift are well predicted even past the stall angle. The distribution of pressure, turbulent kinetic energy and eddy viscosity indicate that the transition is smooth. Furthermore, as the predicted turbulent kinetic energy in the laminar region is almost nil, the present approach of handling the laminar region appears physically sound. The mean velocities and the turbulence quantities are also compared with experiments and show reasonable agreement. The Reynolds normal stresses are better predicted by the present model than the standard k-$\varepsilon$ model. The discrepancies observed in these quantities may be due to the insufficient grid distribution near the surface; a further investigation is necessary to improve the results.
