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that the dynamic vortex positions differ appreciably from the corresponding static
positions.ln addition, the dynamic vortex position for a given roll angle depends
on the direction in which the wing is rolling. For example, consider all the vortex
locations when 4 = +15 deg. The 'vortex locations are quite different depending
on whether the modelis rolling in the positive or negative direction, i.e., whether
4 > O or 4 < O. A larger vertical displacement occurs when the wing is rolling in
a direction away from the vortex, and a lower displacement occurs when the wing
is moving towards the vortex. From Fig. 8.37b, we observe that the time lag exists
only in the normal direction and not in the spanwise direction.
To quantify the effects of the normal time lag,let Az - Zr - Zl denote normal
asymmetry in vortex location as shown in Fig. 8.38. Let us assume that the strength
STABILITY AND CONTROL PROBLEMS AT HIGH ANGLES OF ATTACK 707
60
/,o
20
*o o
- 20
-/o
- 60
og3oo oaa o~o oo6bo
┏━━━━━━━━━━━━━━━━━━━┓
┃oo ao oo, r ┃
┃_ooooooo ┃
┃D O O O O O o Time, ┃
┣━━━━━━━━━━━━━━━━━━━┫
┃-oooooooo9pooooqooo o1-5o o 2 ┃
┃ . - r. i O i n ┃
┗━━━━━━━━━━━━━━━━━━━┛
Time History
~ ~~
*= o(-)
t =t7
9= o (+)
t - t,&: to
5o (+)
3
5 (-)
I
t*5'(- )
t = tjo
Fig. 837a Sketch of vortex trajectories in one cycle of wing rock.30
iso (+)
/ 5o (+)
I.
~
~ -. o (-)
t=t7
~/
+= o (*)
t =to
708 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
LEFT SIOE RIGHT SIDE
,v
┏━━━━━━┳━━━━━━━━━━━┓
┃ -z,s ┃ h-Dynamic hd ┃
┃ ┃ ┃
┃ ┃ o-Static Mea ┃
┃ ┃-o /:.,,.. ' ┃
┃ .tt ┃ IIl ┃
┗━━━━━━┻━━━━━━━━━━━┛
-1-6 -1-2 .- 0-8 -0-4 O 0-4 0-8 1-2 1-6
y/s
suremc:nts
tments
Fig. 837b Static and dynamic vortex Ioc:ations for a = 30 deg, zr/cr = -0.75.3o
of both of the vortices is identical. rlhen, Az > 0 gives rise to a positive rolling
moment and vice versa Thus, when ~ is increasing or 4 > O,if AzP> O,itleads to
a destabilizing effect. In other word~s, a4la Az > 0 corresponds to a destabilizing
effect and a#/aAz < 0 to a stable effecL
In Fig. 8.39, the static and dynamic variations of Az during a wing rock cycle
are presented. We observe that a4/8Az < 0 for 20 deg S 4 < 45 deg during
positive half cycle and for -45 deg < ~ S -20 deg during the negative half cycle.
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