Unsteady Flow Past Thin Cambered Plates: Forces and Moment

The lift and pitching moment are easily obtained from the pressure contributions in the z-direction, see Fig. 5.2.

From the figure, the elementary lift and moment contributions per unit span are dL’= (p– p+)dx, dM’, o = —(p– p+)xdx (5.13)

Hence, the global results

or

L ‘ = p

Here we have made use of the linearized Bernoulli equation and of the relationship between Г and the jump in ф as

Г = ф(х, 0+, t) — ф(х, 0 , t) =< ф(х, t) >

= u(x, 0+, t) – u(x, 0 , t) =< u(x, t) > (5.17)

d x

The elementary drag contribution is seen to be

— dz ^

p ) dx
dx

Upon integrating, one obtains

It was indicated in Sect.3.5.4 that the result of pressure integration does not include the suction force that exists at the singular leading edge of a thin plate in steady flow. The same situation exists here, in unsteady flow, and can be handled in the same manner by replacing the slope of the profile, d'(x) — a(t), from the tangency condition as

pc	дГ дГ		d a	dh
D’ = —-	+ U		w(x, 0, t) — (xn — x)
U J0	д-t дx		dt	dt

dx (5.20)

where w(x, 0, t) is made of two parts:
, 0 ^ 1 Г д г(і, t) d і, 0 „ 1 д г(і, t) d і wb(x, 0, t) = — , ww(x, 0, t) = — 2n 0 dx x — і 2n c dx x — і (5.21) the first part, wb, which is to be taken in the “principal value” sense, is due to the bound vorticity and the second part, ww, is the contribution of the “wake” or shed vorticity. Substitution of these contributions in the drag evaluation yields
дГ(л, t) дr(x, t)
д t	д x
d a dh Wb (x, 0, t) + Ww(x, 0, t) — (xQ — x) — dt dt
(5.22)
dx
(5.23) (5.24)
d3 = —
dx
(5.25)
(5.26)
(5.27)
(5.28)

showing the antisymmetry of the kernel.

D’3 cannot be simplified much further. In summary, the drag reduces to D’ = D[ + D3, that is

D

(5.29)

We note that, in the limit of steady flow, D1 = D’3 = 0 and D = 0.