Kinematics

In Section 13.1 (Eq. (13.12)) it was shown that the continuity equation in the moving frame of reference x, z remains as:

V2<D = 0 (13.42)

where Ф is the equivalent of the steady-state perturbation potential.

The time-dependent version of the boundary condition requiring no
normal flow across the surface is given by Eq. (13.13a):

(УФ – V0 – vrel – Я X r) • n = 0 (13.43)

In this section the instantaneous shape of the airfoil is given by rj(x, t) and therefore the vector n normal to the surface is

(13.44)

To establish the kinematic relations for the airfoil’s motion (according to Eq. (13.8)) the instantaneous velocity and orientation of the x-z system can be described by a flight velocity U(t) and a rotation 6 about the у coordinate. Note that the x coordinate was selected such that the instantaneous velocity of the origin ( )o (of Eq. (13.9a)) resolved into the directions of the x-z coordinate system is

Vo = [-t/(f), 0,0] (13.45)

The instantaneous rotation is then

Si = [0,0(0,0] (13.46)

Also, by allowing a relative motion of the camberline within the coordinate system x-z the relative velocity of Eq. (13.9c) becomes

At this point, it is convenient to divide the velocity potential Ф into an airfoil potential Фв and to a wake potential Ф№ (for example, if a time­stepping numeric approach is used, then the strengths of the wake singularities are assumed to be known and only the airfoil’s singularity distribution Фв must be obtained):

Ф = Фв + Фи, (13.48)

Evaluating the product

Si X r = (6z, 0, — Qx)

and substituting Eqs. (13.45)-(13.47) into the boundary condition (Eq. (13.43)) results in

^ ox ox dz dz

This can be rearranged in terms of the boundary condition for the unknown

potential Фв:

The main advantage of this formulation lies in the previous assumption that if the wake potential is known (and it is usually known from the previous time steps, when a time-stepping solution is applied) then the solution can be reduced to solving an equivalent steady-state flow problem, at each time step. For example, if this model is compared to the thin, lifting airfoil of Section 5.2 then the boundary condition of Eq. (13.50) is equivalent to the steady-state boundary condition of Eq. (5.29). Therefore, by exchanging the local downwash W(x, t) with the right-hand side of Eq. (5.29), the methods of solution developed in Chapter 5 can be applied at each moment. Also, note that the boundary condition (Eq. (13.50)) is not reduced to its small disturbance approximation yet and can be specified (numerically) on the airfoil’s surface and not on the z = 0 plane.