Traditionally, aerodynamic coefficients have been modeled in what is called a quasi-steady manner. That is, they are functions only of flight variables at that instant of time, not those at previous instants of time. The only exception is the modeling of the downwash lag effect, which we shall examine shortly. In effect, the history of the airplane maneuver—that is, how the airplane came to be in that particular state of flight—is of no con- sequence in evaluating the aerodynamic coefficients.
Without doubt, it is the relative flow incident on the airplane that is responsible for the generation of the aerodynamic forces and moments acting on the airplane, that is, the magnitude and orientation of the rela- tive velocity and the angular velocity of the airplane with respect to the relative wind. Of the factors that certainly do not influence the aerody- namic coefficients—the position and orientation of the airplane relative to the ground (body-axis Euler angles) have no effect on the aerodynamic
coefficients. Atmospheric density does change with altitude from the ground, which affects the aerodynamic forces and moments, but that is accounted for in the factor q=( )1 2/ ρV2 in Equations 2.1 and 2.2 as is the magnitude of the relative velocity V. Likewise, the angular velocity of the body-fixed axes relative to the ground has no bearing on the aerodynamic forces and moments either. It is the angular velocity of the body-fixed axes
199 ft 11 in (60.9 m)
199 ft 11 in (60.9 m) 70 ft 7.5 in (21.5 m)
70 ft 7.5 in (21.5 m) 36 ft 0 in (11.0 m)
36 ft 0 in (11.0 m)
60 ft 9 in (18.5 m)
70 ft 7.5 in (21.5 m)
60 ft 9 in (18.5 m) 70 ft 7.5 in
(21.5 m) 209 ft 1 in
(63.7 m)
242 ft 4 in (73.9 m)
Boeing 777-200
Boeing 777-300
36 ft 1 in (11.00 m) 211 ft 5 in*
(64.44 m) 231 ft 10 in
(70.66 m)
225 ft 2 in (68.63 m)
747-400
72 ft 9 in (22.17 m)
63 ft 8 in (19.41 m) 84 ft 0 in
(25.60 m)
FIGURE 2.1 3-View drawings of different airplanes (clockwise from top left):
B-707, B-777, B-747. (From www.aerospaceweb.org.)
with respect to the relative wind that matters; that is, the relative angular velocity between the body- and wind-fixed axes. The effect of airplane size on the aerodynamic forces/moments is represented by the factors S, b, c in Equations 2.1 and 2.2 and they appear “outside” the aerodynamic coef- ficients as well.
Homework Exercise: How can the reference area S and reference lengths b, c be chosen for configurations that do not have a lifting surface such as a wing? For example, a missile or an airship. Note that with different choices of S, b, or c, the values of the aerodynamic coefficients can be grossly dif- ferent, so it is important to make sure that the values reported and being compared use similar reference area and lengths.
With this background, the aerodynamic forces may be modeled as sum of four different effects, as follows:
C t C Ma C p p b
V q
k ksta
Static term
kdyn b w b
( )=( , , , )α β δ+ ( −2 ) ,( −− −
+
Vq c r r b V
C
w b w
Dynamic term
k fl
) ,( )
2 2
oo w w w
Flow curvature
p b kdow
V q c V r b
V C t
2 ,2 ,2 ( (
+ −
α ττ β), (t τ))
Downwash lag
− (2.3)
where k represents any of the aerodynamic coefficients in Equations 2.1 and 2.2, and all variables are evaluated at the same time t (except where explicitly mentioned). The static term is a function of the relative flow Mach number Ma, the aerodynamic angles α, β, and the control surface deflections δ. The dynamic term is a function of the three components of the relative angular velocity between the body- and wind-fixed axes, suitably nondimensionalized. The third term brings in the flow curva- ture effect, which arises when the airplane is flying along a curved flight path—it is a function of the nondimensional wind-axis angular velocity components, which represent the flight path curvature. The downwash lag term is the only one that is strictly not a quasi-steady effect as it represents the aerodynamic forces/moments generated at the tail due to changes in α, β at the wing at a time τ ≈ lt/V earlier, where lt is roughly the distance between the wing and tail aerodynamic centers.
The model in Equation 2.3 is fairly representative for most mod- ern airplanes though certain additional effects may need to be added in
special cases. For instance, in case of rapidly rolling vehicles, an addi- tional Magnus moment effect should be included. For extremely low- speed flight, as in the case of micro air vehicles, the Reynolds number may become an important parameter for the aerodynamic coefficients, especially CD. By and large, the quasi-steady model of Equation 2.3 is used to simulate airplane motion in both steady-state and unsteady flight con- ditions. However, certain unsteady aerodynamic modeling approaches, such as indicial functions and internal state variables, have been proposed though they are not used widely [1].
Homework Exercise: An added mass model becomes necessary for mod- eling the flight of aircraft such as airships and parachutes. Actually, a body accelerating in a fluid also displaces (accelerates) some of the fluid adja- cent to it. This added mass of fluid displaced should be accounted for as an inertial effect in the dynamics equations. For heavier-than-air vehicles such as airplanes, the amount of air so displaced is of the order of 1% of the airplane mass, and so may be ignored. But for lighter-than-air (LTA) vehicles like airships or parachutes, the added mass is a significant factor.
Find out how the added mass effect is modeled for such LTA systems.
In the following sections, we shall dwell on each of the terms in Equation 2.3 one by one, look at general trends or a specific example, and point out the main design features that impact each coefficient term.
2.3 STATIC AERODYNAMIC COEFFICIENT TERMS