Wings in Transonic Flows

Wings in Transonic Flows Подпись: (6.162)

The governing equations in the transonic regime are nonlinear and of mixed type, to represent both locally subsonic and locally supersonic flow regions. The full potential equation is given by

where p is obtained via Bernoulli’s law in terms of V2 = (Vф)2. Together with the tangency boundary condition on a solid surface and imposing the right behavior of ф in the far field, the formulation is complete. Again, a special treatment is required to handle the wake, where ф is discontinuous.

The far field behavior, for subsonic incoming Mach number, is given by Klunker. Following Klunker, the integral equation for ф is analyzed to obtain the far field solution. The governing equation for Ыж < 1 can be written in the form


Подпись: Ф Подпись: 1 1 4n /(x - X1)2 + p2(y - У1)2 + e2(z - Z1 )2 Подпись: (6.164)

where ф is the perturbation potential. Using

Подпись: Ф = Подпись: (6.165)

Green’s second identity leads to

Wings in Transonic Flows

Asymptotically, ф behaves as a doublet in 3-D and a horseshoe distribution over the wing, see Klunker [17].

+———————————————————– 3-D (6.166)

4n [((x – ф)2 + P2(y – q)2 + P2(z – C)2)]2 where Г = (1ф-(, andD = f f f# M2^ (дф) d^dndC,-Wingvolume.