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Chapter 2
Aerodynamic Background
Flight dynamics deals principally with the response of aerospace vehicles to perturbations in their flight environments and to control inputs. In order to understand this response, it is necessary to characterize the aerodynamic and propulsive forces and moments acting on the vehicle, and the dependence of these forces and moments on the flight variables, including airspeed and vehicle orientation. These notes provide a simplified summary of important results from aerodynamics that can be used to characterize these dependencies.
2.1 Introduction
Flight dynamics deals with the response of aerospace vehicles to perturbations in their flight environ- ments and to control inputs. Since it is changes in orientation (or attitude) that are most important, these responses are dominated by the generated aerodynamic and propulsive moments. For most aerospace vehicles, these moments are due largely to changes in the lifting forces on the vehicle (as opposed to the drag forces that are important in determining performance). Thus, in some ways, the prediction of flight stability and control is easier than the prediction of performance, since these lifting forces can often be predicted to within sufficient accuracy using inviscid, linear theories.
In these notes, I attempt to provide a uniform background in the aerodynamic theories that can be used to analyze the stability and control of flight vehicles. This background is equivalent to that usually covered in an introductory aeronautics course, such as one that might use the text by Shevell [6]. This material is often reviewed in flight dynamics texts; the material presented here is derived, in part, from the material in Chapter 1 of the text by Seckel [5], supplemented with some of the material from Appendix B of the text by Etkin & Reid [3]. The theoretical basis for these linear theories can be found in the book by Ashley & Landahl [2].
12 CHAPTER 2. AERODYNAMIC BACKGROUND
c(y)
b/
y
x
ctip
Λ 0
croot
Figure 2.1: Planform geometry of a typical lifting surface (wing).
2.2 Lifting surface geometry and nomenclature
We begin by considering the geometrical parameters describing a lifting surface, such as a wing or horizontal tail plane. The projection of the wing geometry onto the x-y plane is called the wing planform. A typical wing planform is sketched in Fig. 2.1. As shown in the sketch, the maximum lateral extent of the planform is called the wing span b, and the area of the planform S is called the wing area.
The wing area can be computed if the spanwise distribution of local section chord c(y) is known using
S =
∫ (^) b/ 2
−b/ 2
c(y) dy = 2
∫ (^) b/ 2
0
c(y) dy, (2.1)
where the latter form assumes bi-lateral symmetry for the wing (the usual case). While the span characterizes the lateral extent of the aerodynamic forces acting on the wing, the mean aerodynamic chord ¯c characterizes the axial extent of these forces. The mean aerodynamic chord is usually approximated (to good accuracy) by the mean geometric chord
¯c =
S
∫ (^) b/ 2
0
c^2 dy (2.2)
The dimensionless ratio of the span to the mean chord is also an important parameter, but instead of using the ratio b/¯c the aspect ratio of the planform is defined as
AR ≡
b^2 S
Note that this definition reduces to the ratio b/c for the simple case of a wing of rectangular planform (having constant chord c).
14 CHAPTER 2. AERODYNAMIC BACKGROUND
camber (mean) line
chord line V
−α α
0
chord, c
zero−lift line
Figure 2.2: Geometry of a typical airfoil section.
The local chord is then given as a function of the span variable by
c = croot
[
1 − (1 − λ)
2 y b
]
and substitution of this into Eq. (2.2) and carrying out the integration gives
¯c =
2(1 + λ + λ^2 ) 3(1 + λ)
croot (2.10)
The sweep angle of any constant-chord fraction line can be related to that of the leading-edge sweep angle by
AR tan Λn = AR tan Λ 0 − 4 n
1 − λ 1 + λ
where 0 ≤ n ≤ 1 is the chord fraction (e.g., 0 for the leading edge, 1/4 for the quarter-chord line, etc.). Finally, the location of any chord-fraction point on the mean aerodynamic chord, relative to the wing apex, can be determined as
x¯n =
S
∫ (^) b/ 2
0
xnc dy =
S
∫ (^) b/ 2
0
(ncroot + y tan Λn) dy
3(1 + λ)¯c 2(1 + λ + λ^2 )
n +
1 + 2λ 12
AR tan Λn
Alternatively, we can use Eq. (2.11) to express this result in terms of the leading-edge sweep as
x¯n c ¯
= n + (1 + λ)(1 + 2λ) 8(1 + λ + λ^2 )
AR tan Λ 0 (2.13)
Substitution of n = 0 (or n = 1/4) into either Eq. (2.12) or Eq. (2.13) gives the axial location of the leading edge (or quarter-chord point) of the mean aerodynamic chord relative to the wing apex.
2.3 Aerodynamic properties of airfoils
The basic features of a typical airfoil section are sketched in Fig. 2.2. The longest straight line from the trailing edge to a point on the leading edge of the contour defines the chord line. The length of this line is called simply the chord c. The locus of points midway between the upper and lower surfaces is called the mean line, or camber line. For a symmetric airfoil, the camber and chord lines coincide.
2.3. AERODYNAMIC PROPERTIES OF AIRFOILS 15
For low speeds (i.e., Mach numbers M << 1), and at high Reynolds numbers Re = V c/ν >> 1, the results of thin-airfoil theory predict the lifting properties of airfoils quite accurately for angles of attack not too near the stall. Thin-airfoil theory predicts a linear relationship between the section lift coefficient and the angle of attack α of the form
cℓ = a 0 (α − α 0 ) (2.14)
as shown in Fig. 2.3. The theory also predicts the value of the lift-curve slope
a 0 =
∂cℓ ∂α
= 2π (2.15)
Thickness effects (not accounted for in thin-airfoil theory) tend to increase the value of a 0 , while viscous effects (also neglected in the theory) tend to decrease the value of a 0. The value of a 0 for realistic conditions is, as a result of these counter-balancing effects, remarkably close to 2π for most practical airfoil shapes at the high Reynolds numbers of practical flight.
The angle α 0 is called the angle for zero lift, and is a function only of the shape of the camber line. Increasing (conventional, sub-sonic) camber makes the angle for zero lift α 0 increasingly negative. For camber lines of a given family (i.e., shape), the angle for zero lift is very nearly proportional to the magnitude of camber – i.e., to the maximum deviation of the camber line from the chord line.
A second important result from thin-airfoil theory concerns the location of the aerodynamic center. The aerodynamic center of an airfoil is the point about which the pitching moment, due to the distribution of aerodynamic forces acting on the airfoil surface, is independent of the angle of attack. Thin-airfoil theory tells us that the aerodynamic center is located on the chord line, one quarter of the way from the leading to the trailing edge – the so-called quarter-chord point. The value of the pitching moment about the aerodynamic center can also be determined from thin-airfoil theory, but requires a detailed calculation for each specific shape of camber line. Here, we simply note that, for a given shape of camber line the pitching moment about the aerodynamic center is proportional to the amplitude of the camber, and generally is negative for conventional subsonic (concave down) camber shapes.
It is worth emphasizing that thin-airfoil theory neglects the effects of viscosity and, therefore, cannot predict the behavior of airfoil stall, which is due to boundary layer separation at high angles of attack. Nevertheless, for the angles of attack usually encountered in controlled flight, it provides a very useful approximation for the lift.
Angle of attack,
Thin−airfoil theory
Stall
α (^0)
Lift coefficient, C
l
α
2π
Figure 2.3: Airfoil section lift coefficient as a function of angle of attack.
2.4. AERODYNAMIC PROPERTIES OF FINITE WINGS 17
2.4 Aerodynamic properties of finite wings
The vortex structures trailing downstream of a finite wing produce an induced downwash field near the wing which can be characterized by an induced angle of attack
αi =
CL
πeAR
For a straight (un-swept) wing with an elliptical spanwise loading, lifting-line theory predicts that the induced angle of attack αi is constant across the span of the wing, and the efficiency factor e = 1.0. For non-elliptical span loadings, e < 1 .0, but for most practical wings αi is still nearly constant across the span. Thus, for a finite wing lifting-line theory predicts that
CL = a 0 (α − α 0 − αi) (2.17)
where a 0 is the wing section lift-curve slope and α 0 is the angle for zero lift of the section. Substituting Eq. (2.16) and solving for the lift coefficient gives
CL =
a 0 1 + (^) πeaA^0 R (α − α 0 ) = a(α − α 0 ) (2.18)
whence the wing lift-curve slope is given by
a =
∂CL
∂α
a 0 1 + (^) πeaA^0 R
Lifting-line theory is asymptotically correct in the limit of large aspect ratio, so, in principle, Eq. (2.18) is valid only in the limit as AR → ∞. At the same time, slender-body theory is valid in the limit of vanishingly small aspect ratio, and it predicts, independently of planform shape, that the lift-curve slope is
a =
πAR 2
Note that this is one-half the value predicted by the limit of the lifting-line result, Eq. (2.19), as the aspect ratio goes to zero. We can construct a single empirical formula that contains the correct limits for both large and small aspect ratio of the form
a =
πAR
1 +
πAR a 0
A plot of this equation, and of the lifting-line and slender-body theory results, is shown in Fig. 2.5.
Equation (2.21) can also be modified to account for wing sweep and the effects of compressibility. If the sweep of the quarter-chord line of the planform is Λc/ 4 , the effective section incidence is increased by the factor 1/ cos Λc/ 4 , relative to that of the wing,^1 while the dynamic pressure of the flow normal to the quarter-chord line is reduced by the factor cos^2 Λc/ 4. The section lift-curve slope is thus reduced by the factor cos Λc/ 4 , and a version of Eq. (2.21) that accounts for sweep can be written
a =
πAR
1 +
πAR a 0 cos Λc/ 4
(^1) This factor can best be understood by interpreting a change in angle of attack as a change in vertical velocity ∆w = V∞ ∆α.
18 CHAPTER 2. AERODYNAMIC BACKGROUND
0
1
0 2 4 6 8 10 12 14 16
Slender-body^ Lifting-line Empirical
Aspect ratio, b^2 /S
Normalized lift-curve slope,
a 2 π
Figure 2.5: Empirical formula for lift-curve slope of a finite wing compared with lifting-line and slender-body limits. Plot is constructed assuming a 0 = 2π.
Finally, for subcritical (M∞ < Mcrit) flows, the Prandtl-Glauert similarity law for airfoil sections gives
a2d =
a 0 √ 1 − M^2 ∞
where M∞ is the flight Mach number. The Goethert similarity rule for three-dimensional wings modifies Eq. (2.22) to the form
a =
πAR
1 +
πAR a 0 cos Λc/ 4
1 − M^2 ∞ cos^2 Λc/ 4
In Eqs. (2.22), (2.23) and (2.24), a 0 is, as earlier, the incompressible, two-dimensional value of the lift-curve slope (often approximated as a 0 = 2π). Note that, according to Eq. (2.24) the lift-curve slope increases with increasing Mach number, but not as fast as the two-dimensional Prandtl-Glauert rule suggests. Also, unlike the Prandtl-Glauert result, the transonic limit (M∞ cos Λc/ 4 → 1 .0) is finite and corresponds (correctly) to the slender-body limit.
So far we have described only the lift-curve slope a = ∂CL/∂α for the finite wing, which is its most important parameter as far as stability is concerned. To determine trim, however, it is also important to know the value of the pitching moment at zero lift (which is, of course, also equal to the pitching moment about the aerodynamic center). We first determine the angle of attack for wing zero lift. From the sketch in Fig. 2.6, we see that the angle of attack measured at the wing root corresponding to zero lift at a given section can be written
− (α 0 )root = ǫ − α 0 (2.25)
where ǫ is the geometric twist at the section, relative to the root. The wing lift coefficient can then be expressed as
CL =
S
∫ (^) b/ 2
0
a [αr − (α 0 )root] c dy =
2 a S
[
S
αr +
∫ (^) b/ 2
0
(ǫ − α 0 ) c dy
]
20 CHAPTER 2. AERODYNAMIC BACKGROUND
the forward part of the fuselage (where Sf is generally increasing), and negative lift on the rearward part (where Sf is generally decreasing), but the total lift is identically zero (since Sf (0) = Sf (ℓf ) = 0, where ℓf is the fuselage length).
Since the total lift acting on the fuselage is zero, the resulting force system is a pure couple, and the pitching moment will be the same, regardless of the reference point about which it is taken. Thus, e.g., taking the moment about the fuselage nose (x = 0), we have
Mf = −
∫ (^) ℓf
0
x dL = − 2 Qα
∫ (^) ℓf
0
x dSf = 2Qα
∫ (^) ℓf
0
Sf dx = 2QαV (2.31)
where V is the volume of the “equivalent” fuselage (i.e., the body having the same planform as the actual fuselage, but with circular cross-sections). The fuselage contribution to the vehicle pitching moment coefficient is then
Cm =
Mf QS¯c
2 V
S¯c
α (2.32)
and the corresponding pitch stiffness is
Cmα =
∂Cm ∂α
fuse
2 V
S¯c
Note that this is always positive – i.e., destabilizing.
2.6 Wing-tail interference
The one interference effect we will account for is that between the wing and the horizontal tail. Because the tail operates in the downwash field of the wing (for conventional, aft-tail configurations), the effective angle of attack of the tail is reduced. The reduction in angle of attack can be estimated to be
ε = κ
CL
πeAR
where 1 < κ < 2. Note that κ = 1 corresponds to ε = αi, the induced angle of attack of the wing, while κ = 2 corresponds to the limit when the tail is far downstream of the wing. For stability considerations, it is the rate of change of tail downwash with angle of attack that is most important, and this can be estimated as dε dα
κ πeAR
(CLα)wing (2.35)
2.7 Control Surfaces
Aerodynamic control surfaces are usually trailing-edge flaps on lifting surfaces that can be deflected by control input from the pilot (or autopilot). Changes in camber line slope near the trailing edge of a lifting surface are very effective at generating lift. The lifting pressure difference due to trailing-edge flap deflection on a two-dimensional airfoil, calculated according to thin-airfoil theory, is plotted in Fig. 2.7 (a) for flap chord lengths of 10, 20, and 30 percent of the airfoil chord. The values plotted
2.7. CONTROL SURFACES 21
0
1
2
0 0.2 0.4 0.6 0.8 1
Lifting Pressure, Delta Cp/(2 pi)
Axial position, x/c
Flap = 0.10 c Flap = 0.20 c Flap = 0.30 c
0
1
0 0.1 0.2 0.3 0.4 0.
Flap effectiveness
Fraction flap chord
(a) Lifting pressure coefficient (b) Control effectiveness
Figure 2.7: Lifting pressure distribution due to flap deflection and resulting control effectiveness.
are per unit angular deflection, and normalized by 2π, so their integrals can be compared with the changes due to increments in angle of attack. Figure 2.7 (b) shows the control effectiveness
∂Cℓ ∂δ
also normalized by 2π. It is seen from this latter figure that deflection of a flap that consists of only 25 percent chord is capable of generating about 60 percent of the lift of the entire airfoil pitched through an angle of attack equal to that of the flap deflection. Actual flap effectiveness is, of course, reduced somewhat from these ideal values by the presence of viscous effects near the airfoil trailing edge, but the flap effectiveness is still nearly 50 percent of the lift-curve slope for a 25 percent chord flap for most actual flap designs.
The control forces required to change the flap angle are related to the aerodynamic moments about the hinge-line of the flap. The aerodynamic moment about the hinge line is usually expressed in terms of the dimensionless hinge moment coefficient, e.g., for the elevator hinge moment He, defined as
Che ≡
He 1 2 ρV^ (^2) Sece^ (2.37)
where Se and ce are the elevator planform area and chord length, respectively; these are based on the area of the control surface aft of the hinge line.
The most important characteristics related to the hinge moments are the restoring tendency and the floating tendency. The restoring tendency is the derivative of the hinge moment coefficient with respect to control deflection; e.g., for the elevator,
Chδe = ∂Che ∂δe
The floating tendency is the derivative of the hinge moment coefficient with respect to angle of attack; e.g., for the elevator,
Cheαt =
∂Che ∂αt
where αt is the angle of attack of the tail.
Bibliography
[1] Ira H. Abbott & Albert E. von Doenhoff, Theory of Wing Sections; Including a Summary of Data, Dover, New York, 1958.
[2] Holt Ashley & Marten Landahl, Aerodynamics of Wings and Bodies, Addison-Wesley, Reading, Massachusetts, 1965.
[3] Bernard Etkin & Lloyd D. Reid, Dynamics of Flight; Stability and Control, John Wiley & Sons, New York, Third Edition, 1998.
[4] Robert C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, New York, Second Edition, 1998.
[5] Edward Seckel, Stability and Control of Airplanes and Helicopters, Academic Press, New York, 1964.
[6] Richard Shevell, Fundamentals of Flight, Prentice Hall, Englewood Cliffs, New Jersey, Sec- ond Edition, 1989.
