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(4.517)
However, suitable engineering methods are not available for estimating (Cm r)B
at supersonic speeds for arbitrary bodies. Approximate values of (Cma,)B can be
obtained by approximating the given fuselage either as a tangent-ogive or a cone-
cylinder, whichever is closer, and using the following lon:
(Cmu)B=(CNrr)Bl-f_::)a'ion (4.518)
where (CNa)B and Xcp can be estimated using Fig. 3.10 or 3.11.
Estimation of CLn. This derivative is a measure of the unsteady or time-lag
effects in airflow on the lift coefficient when the angle of attack is changing with
time, as in the case of an airplane oscillating in pitch. In such situations, the entire
fiow over the airplaneis unsteady. As a result, the aerodynamic coe:fficients become
functions of time. The derivative CLcr is a measure of the unsteady flow effects on
lift coefficient. Obviously, for steady-state flight condition, CLa iS zero.
The time-lag effects in fluid flow becomcimportantin cases involving fiow sep-
aration or vortex shedding. Consider an oscillating wing in a wind tunnel. Here,
the angle of attack varies continuously with time. Let us assume that this wing has
a delta planform with sharp leading edges and sheds leading-edge vortices. If the
wing was held at a fixed angle of attack, the vortex strength and vortex breakdown
locations will be constant with respect to time. However, on an oscillating wing,
because of flow inertia, the vortex strength and breakdown location cannot keep
pace with the angle of attack. As aresult, they lag bchind,i.e., at any given time, the
vortex strength and breakdown locations will not be corresponding to the instanta-
neous wing angle of attack cr(t) but will correspond to cr(t - At), an angle of attack
at an earlier time t - At. Similarly, if the tlow separates over the upper or lower
surface of an oscillating round leading-edge wing, the location of the separation
point will lag behind. Because Lhe lift coefficient depends on the vortex strength,
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 399
vortex breakdown position, or the location of fiow separation point, Lhe derivative
CLd will be nonzero. On the other hand, if the fiow was completely attached with
no vortex shedding or fiow separation from either the upper or lower surface, then
there is no time-lag effect in the fiow field, and the derivative CLa will be zero.
For an oscillating airplane, the situation is even more complex because the
horizontal tail operates in the downwash field of the wing. Even if the wing flow
is completely attached and has no time-lag effects, the horizontal tail experiences
time-lag effects. These time-lag effects arise because the wing downwash field
takes a certain firute amount of time to reach the horizontal tail surface. During
this time interval, the wing angle attack will have assumed a new value. As a result,
the horizontal tail flow field lags behind that of the wing.
The instantaneous angle of attack of the horizontal tail is given by
at(t) : aw(t) - /w + it - e(t)
Noting that aw = ty, the down wash angle e(t) can be expressed as
e(t) = ~ u(t - At)
(4.519)
(4.520)
where At is time lag or the time taken by a fiuid element to travel from wing
surface to the horizontal tail surface.
Using Taylor series expansion and considering only first-order time-lag effects,
we have
a(t - At) : a(t) - aAt
(4.521)
Approximately, we can assume At = lr] Uo. Thus, the larger the tail length /r is,
the longer will be the time delay. With this assumption,
ctr(t) : ct(t) - iw +/, - :1 (a(t) - U- ) (4.522)
and
C~r(t) = at SS rlt [,(t) - iw +,, - :1 (a(t) - U- )]
The derivative CLd is defined as
CL
C" = a~U~
(4.523)
(4.524)
Differentiating both sides of Eq. (4.523) with respect Lo ct and simplifying,
(C~d)r = 2a, V,,l, (~
(4.525)
The horizontal tail lift-curve slope ar and the downwash gradient (de/dcr)
at the horizontal tail location can be estimated using the methods discussed in
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