Some computations have been made, using the test case NACA12.

In this part are shown the influence of the angle of attack of the airfoil, and the polars wich can be drawn by computing the Lift and Drag coefficients.

The hope of this study is to obtain the critic angle, wich corresponds to the physical unhooking of the wing (...when the plane dives into the earth...).

All these steady-state computations have been made with a laminar Euler model, which is commonly used in aerodynamical studies of external flows.

The mesh is the coarse one, and some artificial viscosity (0.02) is used for shock capturing. The CFL is 0.7.
This configuration allowed to get an acceptable convergence in not a too long time.

Two types of flow configurations have been calculated:


The first serie of experiments concerned a subsonic flow incoming on the NACA12 with Mach=0.85, with various incidences.

Please click on images to get a full view:

Contours of Mach Number (Mach=0.85): alpha=0 (left side) and alpha=6 degrees (right side)

Contours of Pressure (Mach=0.85): alpha=0 (left side) and alpha=6 degrees (right side)

Incidence zero: we observe as expected a fully symmetric flow around the foil. The flow is accelerated on both sides of the profile, in order to validate the mass flux conservation. The pressure decreases at the same time, as expected for a subsonic flow. There is a brutal recompression just before the end of the se profile, which forms a shock.
Incidence = 6 degrees: the flow becomes asymmetric: it is accelerated only on the extrados, and the stop point moves into the intrados. The shock is attached only on the extrados and has moved into the end of the airfoil.
Clich here to see a little animation of how the flow can change with incidence (for alpha=0 to 28 degrees).

The pressure default on the extrados sucks up the airfoil into the top: this phenomenon is called Lift.
The friction of air on wall induces a force called Drag.

The code AVBP allows to calculate the lift and drag coefficients (Cp and Cf).
Then, we can draw the variation of those coefficients with the incidence of the incoming flow: we obtain the polar of the airfoil NACA12 for Mach = 0.85.

We may observe that the drag increases with incidence. The lift increases until a critical value of the angle of attack (~16 degrees), after wich, it begins to decrease. This is called unhooking: the boundary layer unhooks itself from the extrados, because of an adverse gradient of pressure.

The next picture represents the flow at the end of the extrados, for an incidence greater than the critical one:

Please click on the image to get a full view:

Contours of U velocity (alpha=20degrees, Mach=0.85)

We observe that the shock moves upstream on the extrados. Then the area of pressure default decreases, that's why the lift decreases too. We may also see the existenz of a reverse flow after the shock.
Click here to see the unhooking in direct-live...


The next calculations concern a supersonic flow incoming on the profile with Mach=1.10.

Please click on images to get a full view:

Contours of Mach Number (Mach=1.10): alpha=0 (left side) and alpha=6 degrees (right side)

A shock is developing in front of the profile. It is not attached on the profile, because of its geometry. When the incidence increases, the shock goes away from the wing.
For the incidence equal to 6 degrees, we observe on the extrados a beautiful wave of pressure release and then a shock.

Clich here to see a little animation of how the flow can change with incidence (for alpha=0 to 18 degrees).

It is also possible to draw the polar for Mach=1.1. Nevertheless, more calculations would be needed to get the unhooking of the wing (wich should appear further than for the subsonic flow).

Note: these experiments allowed us to validate the computed lift values, wich lie on the output of the executable of AVBP (in fact, we weren't sure that those values really corresponded to lift and drag). A program dedicated to post-processing give the local pressure and Cp around the airfoil. Then, by calculating all the normal vectors, we could integrate these local values and compare the result (as  an approximation) with the computed value.
Computed lift:  1.134355
Calculated lift: 1.165692