The following figures show the development of the flow field around a two-dimensional NACA 0012 airfoil section in the Mach number range 0.50 - 0.90. The data was obtained with a two-dimensional Euler flow solver. Since the program solves the Euler equations, only the compressibility drag due to the presence of shock waves is accounted for. Other effects such as shock-induced separation cannot be predicted with this model.
The different shades of color represent the changing values of Mach number in the flow domain. Red represents regions of high Mach number (mostly on the upper surface where the flow is being accelerated) and blue represents regions of low Mach number (mostly at the stagnation point regions in the leading and trailing edge areas).
The sonic line (countour line where the Mach number is exactly 1.0) is shown as a faint white line when sonic flow exists. The flow is presented for the following Mach numbers:
The appearance of drag in the compressible regime is directly related to the existence of shock waves and the consequent total pressure losses and entropy creation. This image shows the entropy field for the Mach 0.80 condition. As you can see, in an inviscid calculation, entropy is created at the shock and is convected downstream with the flow. Ahead of the shock the dark blue color indicates that no entropy has been generated and that the level of entropy there is that of the free stream. In a viscous calculation, additional entropy would be generated in the boundary layer.
With an average angle of attach of 3.966 degrees for these flow solutionss, you can get an idea of the location of the crest for this airfoil. The following two figures are plots of the coefficient of drag of the airfoil vs. Mach number at two different scales. From theses plots and the images of the flow field, you should be able to get an idea of the relationships between critical Mach number, Mc, crest critical Mach number, Mcc, and divergence Mach number, Mdiv.
Notice that the scale in the following plot is quite large. Drag divergence occurs somewhere between Mach 0.65 and 0.70 for this airfoil. For carefully designed supercritical airfoils Mdiv achieves a higher value (around 0.80 - 0.85).
These results courtesy of:jjalonso@stanford.edu