NACA 12 profile, using a fine mesh :

        In this step we used a finer mesh containing 9216 cells, taking more CPU time in order to seek for better results. The mesh is very refined especially around the wing as can be seen in the tow figures :


        Simulations were done with the same conditions than those of coarse mesh ;

                    - Incoming flow --> Mach 1.2,
                    - Euler equations.

        Different simulations was done with changing attack angle to [ 0, 7, 8, 12, 14 and 18 degres]. Convergence is considered to be reached when residuals become smaller than 10-5 as shown in the figure below.

        This is considered not to be a good convergence citerion, because better ones have to be based on evolution of lift or drag coefficients. However, time and hardware limitations makes us doing this choice.

        As it was found with coarse mesh, results given by this simulations appear to give be physically resonable. A shock-wave appears in the front side of the wing, and an under-pressure takes place on it's upper face. This is represented for an attack angle of 7 degres juste below :

        For this attack angle (7 degres),  running the "PROSTROC" procedure gives evolution of the undimensionnal pressure coefficient Cp along the wing surface. In the extrado side Cp is showen to be smaller than the intrado one.

        The air of the domain specified by curves represented above ( obtaned by integration) gives the lift of the airfoil. For attack angle of 7 degres, for exemple, the total lift is found to be equal to 0.522825.

        Lift values are evaluated for different angle of attack. The same thing is done for drag values, and the results obtainedare represented in this figure :

        It's obvious that Euler equations seems not to give correct resultas even with this second fine mesh. In fact, the airfoil wing seems to lift for big angles of attack (18 degres ). This is not realist because the airfoil stalls for angles greater than 14 degrees. This conclusion was been well predicted. For better and realist results, simulations have to be done with Navier-Stockes equations.