. Governing Equations for Unsteady Potential Flow

Figure 7.1: Positions and velocities seen by an Earth-based observer, and by an onboard observer.

The body’s motion is defined by the velocity U(t) of some chosen reference point on the body, and by the body’s rotation rate O(t). An arbitrary point P on the body has location rp relative to the reference point, and Rp relative to the ground observer. This point’s velocity relative to the ground observer is then

Up = -^ = U + Пхгр (7.11)

which is the same as equation (6.3) considered earlier. The fluid velocity as seen by the ground observer is Vp as defined previously. The local fluid velocity seen in the local body frame at point r is then obtained by subtracting the body frame’s local velocity.

Подпись: (7.12)Vrei(r, t) = Vp — (U + Ox r)

The field equation and boundary conditions for the perturbation potential p(r, t) are

Подпись: 92p dx2 <92p <92p dy2 ' dz2 = V2p = 0 Solid-body BC: Vrel ■ n = 0 or equivalently: Vp ■ n = (Ufflxr)-n Freestream BC: <p -A 0 (7.13)

which must all instantaneously hold for each instant in time. The time dependence arises through the body motion U(t) and O(t), and possibly also through atmospheric motion and body deformation as will be described later.