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
is the induced drag related to the mechanism of generation and shedding of wing-tip vortices. This effect is usually modeled as a quadratic function of CL, hence the parabolic dependence of CD on α. The typical parabolic bucket shape of CD is quite clearly visible in Figure 2.3. Lower aspect ratio and a larger deviation from the ideal elliptical span-wise lift distribution
2.5
2 Cm
CL CD
1.5 1 0.5 0 –0.5 –1
–1.5–10 0 10 20 30 40 50 60 70
Angle of attack, α (deg)
80 90
FIGURE 2.2 Longitudinal aerodynamic coefficients as function of α for a typical military airplane.
2.5
1.5
0.5
–0.5 –1
–1.5 –10 0 10
Angle of attack (deg) 0
1
2 Cm
CL CD
FIGURE 2.3 Zoom-in of the longitudinal aerodynamic coefficients as function of α for low angles of attack.
are two factors that tend to increase the induced drag. For the standard trapezoidal wing planform, a near-elliptical span-wise lift distribution can be obtained by judicious choice of the wing twist and the wing taper ratio (the ratio of tip chord to root chord), thereby minimizing the induced drag.
For airplanes, such as gliders, for which maximizing endurance is a key objective, use of a large aspect ratio wing to cut down induced drag as far as possible is standard. However, for military airplanes, there are usually conflicting objectives and other requirements often that take precedence over CD reduction in the selection of the aspect ratio. At higher angles of attack in Figure 2.2, CD continues to increase, though not as sharply, before peaking at α ≈ 90 deg.
The lift coefficient CL is very close to linear at low angles of attack, as exemplified in Figure 2.3 with CL≈ 0 at zero α. As a rough estimate, one can assume a CL of 0.08–0.1 per degree of α at low angles of attack. Thus, for example, at α ≈ 10 deg, one may expect a value of CL around 0.8–1.0.
The CL per degree of α (called the wing lift curve slope) for a given airfoil section depends on the wing sweepback angle and aspect ratio. In gen- eral, the lift curve slope decreases for higher sweepback angle and lower aspect ratio, though in case of some low aspect ratio wings, there may be nonlinear effects that provide a sharp increase of the lift curve slope over some ranges of α. For larger α in Figure 2.2, note that the lift coefficient continues to increase, though with a smaller value of the slope. The peak value of CL, approximately 1.8, occurring at α ≈ 35 deg corresponds to stall. However, in case of the airplane model in Figure 2.2, stall does not lead to a precipitous fall in CL; rather, CL decreases gradually and contin- ues to be positive all the way to α ≈ 90 deg.
Coming to the pitching moment coefficient evaluated at the airplane CG, there is a small (usually positive) contribution to Cm from the fuselage and other nonlifting components of the airplane. Otherwise, a large pro- portion of Cm arises from the lifting components (wing and tail). A lifting force that acts at a point (center of pressure) separated from the airplane CG creates a pitching moment about the airplane CG. Thus, the variation of Cm with α is similar (nearly linear at low angles of attack) to that of the lift coefficient CL. However, there may also be a smaller effect on Cm due to the shift in the location of the center of pressure relative to the airplane CG. In Figure 2.3, a positive CL acting at a point aft of (behind) the air- plane CG can be seen to produce a negative Cm at the airplane CG (and vice versa). As a result, if the slope of CL with α is positive, then the slope of Cm with α is negative. At higher angles of attack in Figure 2.2, the value of
Cm remains largely negative though the slope fluctuates and even becomes positive over a small stretch of α around75–80 deg.
The location of the airplane CG has a predominant impact on the value and trend of the coefficient Cm with angle of attack. For a vehicle like a hang glider, shifting of the CG by moving the flyer’s weight is usually the only way to adjust the vehicle trim. For conventional airplanes, CG loca- tion and its movement in flight is normally part of the problem. Change in CG location due to configuration change (e.g., variable-sweep wings), fuel consumption or reallocation (e.g., Concorde), or payload arrangement/
store drop can have a major impact on net Cm requiring a vehicle to be retrimmed in flight. Obviously, the tail usually being at a greater distance from the CG, changes to the tail lift have a greater impact on Cm than from the wing. Two design parameters that the designer has control over in this regard are the tail setting angle and the tail volume ratio. The latter, in case of the horizontal tail, is defined as
HTVR S l= Sct t (2.4) where HTVR stands for horizontal tail volume ratio and St and lt are respectively the tail planform area and the distance between the tail and wing aerodynamic centers.* The tail setting angle it is used to set the tail chord at an angle with respect to the body XB axis, to bias the tail lift as it were, in order to adjust the value of Cm at zero angle of attack. The HTVR is then used to decide the slope of the Cm curve with α—a larger value of HTVR should give a larger negative slope.
The trend of CD, CL, and Cm varying with α in the low-α range in Figure 2.3 is fairly typical of most conventional airplane configurations. The vari- ations at higher angles of attack, however, are particular to each airplane’s aerodynamic quirks and must be considered individually.
Homework Exercise: Many modern airplane wings feature devices such as flaps and slats, which are meant to increase the lift coefficient during particular phases of flight. Look up information on how the aero coef- ficient curves (such as those in Figure 2.2) will be altered when flaps/slats are deployed. Figure 2.4 gives one example of the lift coefficient for a DC-9 airplane. Notice the significant increase in CL between the zero-flap and deflected flap cases—the lift curve shifts upward. On the other hand,
*Alternatively from the airplane CG, but that is a variable parameter.
extending the slats shifts the CL curve to the right and upward, yielding higher CL but at a higher effective angle of attack.
2.3.2 Lateral Coefficients with Angle of Attack and Sideslip Angle The lateral static aerodynamic coefficients—CY, Cl, and Cn—are usually functions of both α and β. In many cases, the range of variation of β con- sidered for the aerodynamic model is limited and these coefficients may justifiably be modeled as
Ck( , )α β = ∂∂Cβ α βk( )⋅ =Ckβ( )α β⋅ (2.5) An example of the variation of Clβ( ) and Cα nβ( ) for a typical military α airplane is shown in Figure 2.5.
Notice for instance that Clβ( ) is negative and grows more negative α with increasing α between 5 and 15 deg of angle of attack. The main design parameter that directly correlates with Clβ( ) is the wing dihedral α angle—the angle by which the wings are canted up (or down, called nega- tive dihedral/anhedral) relative to a planar surface. Other configuration features that contribute to Clβ( ) are the wing sweepback, which gives the α variation with α referred to above, the wing position vis-à-vis the fuselage
3.0 2.5 2.0 1.5 CL
1.0 0.5
–0.5–10.0 –5.0 –0.0 –5.0 10.0 Angle of attack (deg)
0° flaps 50° flaps
Slats retracted
Slats extended DC-9-30 Tail-off lift
Mach 0.2
15.0 20.0 25.0
0.0
FIGURE 2.4 Effect of flap and slat deflection on lift coefficient, example of DC-9 airplane.
(high, low, or mid), and the vertical tail volume ratio and the location of its aerodynamic center above the body XB axis. However, the wing sweep- back and position, as well as the vertical tail size are usually determined by other factors, leaving the dihedral angle as the main weapon in the designer’s arsenal to manipulate Clβ( ). Too large a dihedral angle may α give an acceptable Clβ( ) at α α= 5 deg but too much Clβ( ) at α α= 15 deg due to the cumulative effect of the sweepback angle. On the other hand, a small dihedral angle (or even a slight anhedral) may give an acceptable value of Clβ( ) at α α= 15 deg at the cost of an undesirable value of Clβ( ) α at α= 5 deg. The variation of Clβ( ) at higher angles of attack is very con-α figuration dependent.
For airplanes with a vertical tail, Cnβ( ) is expected to be positive at α small values of α. In fact, this is one of the key requirements for sizing the vertical tail and the value of Cnβ( ) depends on the vertical tail volume α ratio (VTVR) defined as
VTVR S l= V VSc (2.6)
0.016 0.012 0.008 0.004 0
Cnβ
Cnβ,dyn
Clβ –0.004
–0.008 –0.012 –0.016
0 5 10 15 20
α (deg)
25 30 35 40
FIGURE 2.5 Typical variation of the lateral aerodynamic coefficients as function of α for a military airplane. (From NASA TP-1538, publ. 1979.)
where SV and lV are the vertical tail planform area and distance of its aero- dynamic center from the wing aerodynamic center,* respectively.
With increasing angle of attack, as more of the vertical tail gets blanked by the flow separating from the fuselage, the vertical tail effectiveness decreases, and Cnβ( )reduces correspondingly. This trend is clearly vis-α ible in Figure 2.5 where Cnβ( ) hits zero around α α= 30 deg and then reverses sign to go negative. This trend is fairly universal for a wide range of airplanes.
Homework Exercise: Figure 2.5 features a third curve—that labeled Cnβ,dyn, defined as follows:
Cn Cn I
I Cl
dyn zz
β, = βcosα− xx βsinα
Look up references (e.g., Reference 2) for the significance and use of Cnβ,dyn.
2.3.3 Variation with Mach Number
The typical variation of drag coefficient CD with Mach number is sketched in Figure 2.6. At low Mach numbers, there is little or no change in CD with Mach number. Beyond the critical Mach number, Macr, the drag coeffi- cient begins to gradually increase and then increases steeply beyond MDD, the drag divergence Mach number. This is due to the formation of shock waves over the airplane body, primarily the wing—the additional drag component is called wave drag, a form of pressure drag. Peak CD usually occurs somewhere in the vicinity of Mach number 1 beyond which the drag coefficient falls off sharply. This trend of CD with Mach number in the transonic regime is typical of many airplanes.
Figure 2.6 also shows the typical variation of the lift coefficient with Mach number in the subsonic and supersonic regimes. The increase in CL with Mach number in the subsonic range, and the fall with Mach num- ber in the supersonic range are both fairly standard. The variation in the transonic regime is somewhat more difficult to standardize and is overly dependent on the finer details of the airplane configuration, especially the key wing parameters such as aspect ratio, sweep angle, taper ratio, twist, airfoil shape, wing–fuselage interaction, possible leading-edge kinks, under-wing attachments, etc. However, for many airplane configurations,
*Alternatively from the airplane CG, but that is a variable parameter.
there may be a dip in CL around Ma ≈ 0.85–0.9 before a rise in CL peaking around Mach 1, as sketched in Figure 2.6.
The variation of the side-force coefficient CY with Mach number is in principle similar to that of CL. And since the moment coefficients Cm and Cn arise primarily as a consequence of CL and CY, respectively, the trend of their variation with Mach number may be expected to be similar as well. The variation of rolling moment coefficient Cl with Mach number is harder to pin down but, since much of the rolling moment in a conven- tional airplane configuration arises due to wing lift, the trend is likely to mirror the change of CL with Ma to some extent.
In practice, it is not necessary to have functional relationships for the variation of the static aerodynamic coefficients with α,β,Ma—it is fairly common and acceptable to have tabular data that are looked up and inter- polated during analysis. However, it is worthwhile and highly advisable to plot these variations and check the magnitude and trend for consistency. It may be possible to raise some red flags before even launching into a flight dynamics analysis.
Mac MaD 1.
1 Drag coefficient,
CD
CD ∝
Ma2∞ – 1
Ma Ma
1.
Lift coefficient, CL
√
CL ∝ 1
Ma2∞ – 1
√
CL ∝ 1
1 – Ma2∞
√
FIGURE 2.6 Typical variation of the drag and lift coefficients as function of Mach number.
Homework Exercise: Predicting the variation of the aero coefficients accurately in the transonic regime is a bit of a challenge. Most theoreti- cal methods do not work well for Mach numbers around 1.0. For exam- ple, see the variation of lift curve slope for the DC-9 airplane plotted in Figure 2.7 and the comparison therein with the calculated values using the Prandtl–Glauert rule and an empirical prediction method using DATCOM.
2.3.4 Variation with Control Surface Deflection
The static aerodynamic coefficients may be modeled as a function of a con- trol surface deflection in the following manner:
Ck( )δ = ∂∂ ⋅ = ⋅Cδ δk Ckδ δ (2.7) Each Ckδ is called a control effectiveness parameter and may itself be a function of α,β,Ma. Typical values for the elevator control effectiveness parameter in case of the longitudinal coefficients have been presented in an Exercise Problem. Usually, CLδe is positive and Cmδe is negative, suggest- ing that a positive (down) elevator deflection adds to the tail lift and hence the total airplane lift, and that the added tail lift acting aft of the airplane CG induces a negative (nose-down) pitching moment at the airplane CG.
0.14 0.13
Variation of lift curve slope with Mach number DC-9-30 Tail-off
Mach number Wind tunnel
DATCOM Prandt1–Glauert 0.12
0.11
Lift curve slope (per degree)
0.10 0.09 0.08
0.070.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
FIGURE 2.7 Lift curve slope variation with Mach number and theoretical pre- dictions, example of DC-9 airplane.
Sample values of the aileron and rudder control effectiveness param- eters in case of the lateral aerodynamic coefficients are as under (control deflection angle in deg):
C C C
Y
Y
l a
r
a δ
δ
δ
α α
= − +
= − +
=
25 0 002271 0 039 30 0 002651 0 141 251
( . . )
( . . )
(( . . )
( . . )
( .
− −
= − +
=
0 00121 0 0628 30 0 0003511 0 0124 25 0 01
α
δ α
δ
C C
l
n r
a 000213 0 00128 30 0 0008041 0 0474
α
δ α
+
= −
. )
( . . )
Cnr
Positive aileron deflection is defined as right aileron deflected down and left aileron deflected up. The net aileron deflection, δa, is then defined as
δa=δaR+δaL
2 (2.8)
Positive aileron deflection increases lift on the right wing and decreases lift on the left wing, producing a net negative rolling moment. Hence, Clδa is usually negative.
Each deflected aileron creates additional drag and the drag differen- tial between the left and right wing sections produces a yawing moment, which is the primary source of Cnδa. In general, Cnδa may be either positive (called adverse yaw) or negative (proverse yaw). In the present instance, Cnδa is positive—that is, adverse yaw, which means that when the airplane rolls left due to aileron application, it also yaws right (and vice versa).
This combination is considered preferable from the point of view of pilot handling. A common way to ensure adverse yaw is to deflect the up-going and down-going ailerons unequally such that the ensuing drag differential assures a positive Cnδa. Usually, the up-going aileron is given a proportion- ately larger deflection for this purpose.
Rudder deflected to the left (as seen from the rear) is positive δr.
Usually, this gives a positive side force, that is CYδr>0, which creates a posi- tive rolling moment Clδr>0 and a negative yawing moment Cnδr<0. With increasing angle of attack, all the rudder control effectiveness parameters are generally degraded, similar to the case of vertical tail effectiveness with α as seen earlier.
Homework Exercise: Several airplanes use alternative control surfaces, either exclusively or in addition to the regular control surfaces mentioned above.
Some examples are: canards, all-moving tail-planes and elevons for pitch control, elevons, spoilers, tailerons, and flaperons for roll control. Investigate how these may be modeled and find some typical aero data for these control surface deflections. Note that most roll control devices also produce a yaw- ing moment, so they may, in principle, also be used for yaw control.