Canonical impulse solutions

A solution to equations (7.42) and (7.43) have been obtained by Wagner [54] based on a unit step in h,9,9 corresponding to a sudden change of airfoil motion in heave rate, pitch, and pitch rate. Kussner [55] obtained a solution for the case of a unit step of wgust, corresponding to the airfoil flying into a uniform vertical gust field with a sharp boundary.

In the Wagner case an important quantity is the angle of attack relative to the camber line at the |-chord location, or x = c/4.

h 9c

«ЗС/4 – JJ + 9 + —

In the Kussner case the relevant quantity is the apparent angle of attack resulting from the vertical gust velocity shown in Figure 7.8, which is assumed to act everywhere on the chord.

If these quantities have step jumps of Д«3с/4 and/or Aagust at t = 0, then the lift is

V

pU‘2c

where Ф(ї) is the Wagner function, Ф(ї) is the Kussner function, and 5(t) is the unit impulse function. Their argument t is a non-dimensionalized time, which can also be interpreted as the distance that the airfoil has moved, in units of half-chord.

The lift is seen to be the sum of two parts:

1) C£q is the “quasi-steady” or “circulatory” part associated with vorticity shedding and circulation, and evolves in time according to the Wagner and Kussner functions shown in Figure 7.9.

2) ciA is an “apparent-mass” or “impulsive” part associated with the instantaneous acceleration of the fluid immediately adjacent to the airfoil. A step change in velocity or angle produces an infinite acceleration, resulting in the impulsive lift.

Both the Wagner and Kussner functions asymptote to unity in the limit t ^ <x>. Therefore, the general solution (7.50) asymptotes to the value = 2n(Aa3c/4 + Aagust), as expected from steady thin airfoil theory.

The exact Wagner and Kussner functions cannot be expressed in terms of elementary functions, but are available in tabulated form. For calculations, the following curve-fit expressions are fairly accurate and convenient.

Ф(t) = 1 – 0.165 exp(—0.045t) – 0.335 exp(—0.3t) Ф(ї) = 1 — 0.5exp(-0.137) — 0.5exp(—1.0f)