Kutta condition for lifting flows

The Wayback Machine - https://web.archive.org/web/20060504055740/http://pc.freeshell.org:80/notes/node21.html

This is an extract from
Euler solvers as an analysis tool for aircraft aerodynamics
W Schmidt and A Jameson
which appeared in "Advances in Computational Transonic", Vol. 4 in "Recent advances in numerical methods in fluids", Ed. W. G. Habashi.

In potential flow theory, Kutta conditions are needed at all surfaces where the flow is leaving the contour, e.g. at trailing edges, side edges. In two-dimensional lifting airfoil flow the classical Kutta-condition says that the static pressure at the trailing edge on upper and lower (surfaces) is equal and thus the velocity vector is equal in magnitude and direction since in isentropic flow total pressure is constant. This leads to zero velocity at the tailing edge for nonzero trailing edge angle. For three-dimensional lifting flow Mangler and Smith in Ref [1] discuss in detail possible solutions and trailing edge wakes behind wings. All standard methods in linear and nonlinear compressible potential flow theory assume the wake, and thus the lines with the jump in potential, to leave in the bisector-direction and to have constant jumps in potential along x in spanwise constant locations. The flow around the wing tip in general is neglected. This can cause at higher lift coefficients quite large deviations from the physically correct situation.

Solutions to the full compressible Euler equations do not need any explicit Kutta-condition to be unique, neither in two- nor in three-dimensional flow. Numerous examples have been presented by different authors, e.g. Ref. [2] and [3]. This might be explained on the basis of Fig. 1 and 2. Potential flow needs for uniqueness a Kutta condition, since all rear stagnation point locations are possible. As sketched in Fig. 1, for all points q=0 and static pressure p = stagnation pressure

is a solution. So this point has to be specified by an additional condition.

In the full compressible inviscid equations of motion (Euler) a flow around a sharp corner or an edge with a small radius of curvature will always cause expansion to supersonic flow. Compression to the point where the flow is leaving the surface Fig. 1 now can only happen through a shock which will cause total pressure loss and a rise in entropy.

$\includegraphics[height=5cm]{kutta/kutta2.eps}$
Figure 2: $p_{T,1} \le p_{T,2}$ ,

, $q_2 \ge q$

This will require a point on the surface with different total pressure which implies different velocities for the same static pressure, i.e. a contact discontinuity or wake. However, this solution is not stable since there exists in subsonic flow a solution without any total pressure loss, namely the one with flow leaving the trailing edge and thus not having the large expansion and the shock. In transonic lifting cases with shocks, the total pressure loss on one side is larger than on the other which will only allow the flow to stagnate at the trailing edge upper surface while the velocity at the trailing edge lower surface is finite, thus leaving the lower surface smoothly as shown in Fig. 2. The wake contact discontinuity in velocity and total pressure is captures in the fully conservative finite volume scheme. Therefore the wake shape is not fixed due to the mesh but will be a result independent of the mesh chosen.

Different two-dimensional numerical experiments on airfoils with subcritical flow proved that this phenomenon does not depend on the initial solution. Even with a fully converged potential flow solution forced to have

as starting solution the Euler time stepping method on the same mesh gave the $C_L \ne 0$ converged solution identical to the one starting from undisturbed flow. The same experiments showed no basic influence of the dissipative terms added. O or C meshes do not change the results. Flow leaving the trailing edge smoothly seems to be the only stable Euler solution without having something different enforced.

Feb	MAY	Jul
	04
2005	2006	2007