ENAE482 – Lecture 4 Page 1 Lecture 4. Wing Design What we commonly refer to as “the wing” is the primary lifting surface of the airplane. Other airplane surfaces, such as the horizontal stabilizer and vertical stabilizer, are categorized as wings from an aerodynamic point of view, and embody similar design considerations as the primary lifting surface. Wing Definition The characteristics that define wings are as follows: Theoretical Wing Area (denoted by an uppercase S) = the total area of the plane bounded by the leading edge, trailing edge and wing tips. The portion of the wing that may in fact be covered by the fuselage is included in the theoretical wing area. Wing Span (denoted by a lowercase b) = the distance between the wing tips, measured perpendicular to the plane of symmetry of the wing. Aspect Ratio (denoted by an uppercase A) = the square of the wing span divided by the theoretical wing area (b 2 /S). Chord length (denoted by a lowercase c) = the distance between the leading and trailing edges, measured parallel to the plane of symmetry, at any span-wise location. Taper Ratio (denoted by ) = the ratio of the wing tip chord length to the theoretical chord length at the plane of symmetry, called the “root” chord. Sweep Angle (denoted by ) = the angle between a line perpendicular to the plane of symmetry and any line that connects the theoretical root chord and the tip chord at the same percentage of each (see Figure 5). A subscript is used to denote the fractional chord location. At zero percentage of chord the sweep angle is called the leading edge sweep (LE ) . At 100% of the chord the sweep angle is called the trailing edge sweep (TE ) . The sweep angle at the ¼ chord (25%) commonly appears in aerodynamic calculations and is called the quarter chord sweep (c/4 ). The sweep angle at the chord-wise location of maximum thickness is also used in some calculations. This is denoted by (t/c)max . Airfoil Section = the cross-sectional shape of the outside mold line (OML) obtained when the wing is cut by a plane parallel to the plane of symmetry at any span-wise location. Airfoil sections are characterized by their thickness ratio, the chord wise location of maximum thickness, the percentage of camber and chord wise location of maximum camber (see Figure 1). Thickness ratio is the maximum thickness of the airfoil section divided by the chord length. Camber is the percentage, referenced to the chord length, of the maximum difference in height between the airfoil mean line and the chord line.
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ENAE482 – Lecture 4 Page 1
Lecture 4. Wing Design What we commonly refer to as “the wing” is the primary lifting surface of the airplane. Other airplane surfaces, such as the horizontal stabilizer and vertical stabilizer, are categorized as wings from an aerodynamic point of view, and embody similar design considerations as the primary lifting surface.
Wing Definition The characteristics that define wings are as follows: Theoretical Wing Area (denoted by an uppercase S) = the total area of the plane bounded by the leading edge, trailing edge and wing tips. The portion of the wing that may in fact be covered by the fuselage is included in the theoretical wing area. Wing Span (denoted by a lowercase b) = the distance between the wing tips, measured perpendicular to the plane of symmetry of the wing. Aspect Ratio (denoted by an uppercase A) = the square of the wing span divided by the theoretical wing area (b2/S). Chord length (denoted by a lowercase c) = the distance between the leading and trailing edges, measured parallel to the plane of symmetry, at any span-wise location.
Taper Ratio (denoted by ) = the ratio of the wing tip chord length to the theoretical chord length at the plane of symmetry, called the “root” chord.
Sweep Angle (denoted by ) = the angle between a line perpendicular to the plane of symmetry and any line that connects the theoretical root chord and the tip chord at the same percentage of each (see Figure 5). A subscript is used to denote the fractional chord location. At zero
percentage of chord the sweep angle is called the leading edge sweep (LE) . At 100% of the
chord the sweep angle is called the trailing edge sweep (TE) . The sweep angle at the ¼ chord (25%) commonly appears in aerodynamic calculations and is called the quarter chord sweep
(c/4). The sweep angle at the chord-wise location of maximum thickness is also used in some
calculations. This is denoted by (t/c)max.
Airfoil Section = the cross-sectional shape of the outside mold line (OML) obtained when the wing is cut by a plane parallel to the plane of symmetry at any span-wise location. Airfoil sections are characterized by their thickness ratio, the chord wise location of maximum thickness, the percentage of camber and chord wise location of maximum camber (see Figure 1). Thickness ratio is the maximum thickness of the airfoil section divided by the chord length. Camber is the percentage, referenced to the chord length, of the maximum difference in height between the airfoil mean line and the chord line.
ENAE482 – Lecture 4 Page 2
fractional chord location of maximum camber
maximum camber
fractional chord location of maximum thickness
Leading edge radius
Camber LineThickness form
Final airfoil shape
Chord line
Figure 1. Airfoil Parameters
The NACA 4-digit, modified 4-digit, 5-digit, modified 5-digit, 63-, 64- 65- 66- and 63A, 64A and 65A series of airfoils are constructed using families of camber lines and thickness forms that are combined by superimposing a thickness form over a camber line, as shown in Figure 1. The chord line of the airfoil is always, by convention, the reference line for the airfoil angle of attack (AoA). A wing does not necessarily have to use the same airfoil section from root to tip. For example, many airplane wings use thinner sections (smaller thickness ratio) at the tip than at the root to reduce parasite drag. Twist: It is sometimes beneficial to rotate the airfoil sections of the wing at different span-wise distances to different angles with respect to the root chord line. This is called twisting the wing.
Twist angle is denoted by the symbol , and is positive when the twisted section is at higher angle than the root and negative when it is at a smaller angle than the root, as shown in Figure 2. Twisting the wing such that the number of degrees of twist per unit span is a constant is called linear twist. Positive twist is called wash-in and negative twist is called wash-out.
Root airfoil section
Outboard airfoil section
Figure 2. Wing Twist (negative twist = washout is shown)
ENAE482 – Lecture 4 Page 3
When a wing is twisted, the airfoil sections are almost always rotated about the same constant percentage of chord line at each span-wise station. Aerodynamic Twist: Using different airfoil sections at the root and tip can have the same effect as the geometric twist described above. If the tip airfoil has a smaller value of zero-lift angle of attack (i.e., a more negative value) the effect is the same as washing out the wing with a twist angle equal to the difference in zero-lift angles-of-attack, denoted α0L , of the root and tip. The amount of camber determines the magnitude of α0L; i.e, more camber yields a larger negative value. Dihedral Angle = the angle between the horizontal plane (xy plane of the aircraft in aircraft fixed coordinates) and the projection of the quarter chord sweep line. Dihedral is denoted using the
symbol and is positive as depicted in Figure 5 (wing tip chord higher than the wing root chord). Negative dihedral angle is also referred to as anhedral. Aileron = a moveable portion of the wing trailing edge that is used primarily for control of the aircraft roll attitude. The aileron pivots about a hinge line that is usually at constant sweep angle. The hinge line is typically at the 65% to 80% chord location.
Deceleron = a type of aileron, developed in the late 1940s by Northrop originally for use on the F-89 Scorpion fighter aircraft. It is a two-part aileron that can be deflected as a unit to provide roll control, or split open to act as an air brake. Decelerons are also used on the Fairchild-Republic A-10, and the Northrop's B-2 Spirit bomber aircraft.
High lift device = an addition to the wing that increases its maximum lift coefficient. Examples are leading and trailing edge flaps, slats and slots. The elevator on a horizontal stabilizer and the rudder on a vertical stabilizer fall into the category of a particular type of flap known as a "plain" flap. The Appendix to this lecture describes the most commonly used types of high lift devices.
Trim tab = an adjustable small portion of a larger control surface (rudder, elevator or aileron, as shown in the illustration below) used to trim the controls, i.e. to counteract aerodynamic or other unbalanced forces in the desired attitude without the need for the pilot to constantly apply a control force. This is accomplished by adjusting the angle of the tab relative to the larger surface.
Changing the setting of a trim tab adjusts the neutral or resting position of a control surface. As the desired position of a control surface changes (corresponding mainly to different speeds or migration of the airplane center of gravity), an adjustable trim tab will allow the operator to reduce the manual force required to maintain that position (to zero, if used correctly). The trim
tab angle relative to the main control surface stays locked in position (until it is adjusted again) even when the control surface is deflected.
Flap = a moveable portion of the wing leading or trailing edge that increases the wing's maximum lift coefficient. Flaperon = A nearly full-span trailing edge flap that functions both as a high-lift device and a roll control device (aileron). Spoiler = a device intended to reduce lift. Spoilers are a portion of the top surface of a wing that can pivot upward into the airflow. By so doing, the spoiler creates a controlled stall over the portion of the wing behind it, greatly reducing the lift of that section of the wing. Spoilers differ from airbrakes in that airbrakes are designed to increase drag without regard to the affect on lift, while spoilers are intended to reduce lift (but will also increase drag). Spoilers are used by nearly every glider (sailplane) to control their rate of descent and to achieve a controlled landing at a desired spot. Spoilers are also used on large airliners to augment low speed roll control and provide additional drag during the ground roll after landing. Wing Glove = a tapered extension of the wing root that is added to the wing in close proximity to the fuselage (see Figure 3). Wing gloves are sometimes incorporated in a design to provide additional volume for landing gear retraction or fuel, or to provide additional structural strength for main landing gear attachment in the case of a tail dragger (e.g., the Clutton-Tabenor EC 2). Since the need for a glove will not be evidenced until other design considerations are addressed it is generally not included in the initial layout of the baseline configuration. Hence, the glove is not included in the theoretical wing area.
Wing Glove
Figure 3. Wing Glove
Yehudi = an extension of a section of the inboard wing trailing edge sometimes employed on moderately swept wings to provide additional structure for main landing gear attachment on low wing, tricycle gear configurations. This is illustrated in Figure 4.
ENAE482 – Lecture 4 Page 5
Yehudi
Figure 4. The Yehudi
Many commercial jet transports employ the Yehudi in order to achieve the required main landing gear track (i.e., distance between the main landing gear wheels). The Boeing 747, Lockheed L-1011 and McDonnel Douglas MD-80 are just a few examples. Since the need for a Yehudi will not be evidenced until the landing gear is integrated with the design it is generally not included in the initial layout of the baseline configuration. Hence, the Yehudi is not included in the theoretical wing area. Mean Aerodynamic Chord (MAC, denoted by c ) = the chord length of an "equivalent" rectangular (constant chord) plan form that produces the same pitching moment as a wing with a plan form of any arbitrary shape.
dyccS
1Cwheredycc
SC
1c
2/b
2/b
lL
2/b
2/b
2
l
L
Wing tip = an aerodynamic fairing that provides a smooth, streamlined transition from the upper to lower surfaces of the wing at the tip. Wing tips will always incorporate the right-of-way running lights, a red light on the port side and a green light on the starboard side (this is the same international convention used for boats and ships). The volume afforded by the wing tip can also be used for other equipment if required. Wing tips can also take the form of “winglets” which are sections of wing that extend vertically upwards (see for example the Gulfstream II). Winglets can improve the span efficiency (reduce induced drag) but they also increase the zero lift drag. Winglet design requires analysis using CFD codes and/or wind tunnel testing for optimization. Wing Incidence = the angle subtended by the wing root chord line and the aircraft horizontal reference line (HRL). The HRL will be established by the designer when laying out the general arrangement drawing of the airplane. It is the line to which the airplane AoA is measured and is always parallel to the x-axis of the airplane-fixed (stability) axes system. Wing incidence is usually incorporated to minimize the total airplane drag during the portion of the mission profile that exhibits the highest fuel consumption (e.g., the cruise leg of a transport airplane's mission).
ENAE482 – Lecture 4 Page 6
Wing Geometries
Straight Tapered Wings: Straight tapered wings have a symmetric plan form (top view) that has continuous straight leading and trailing edges such that each half-panel forms a trapezoid, as shown in Figure 5. This type of wing plan form is exhibited by the vast majority of aircraft flying today.
Figure 5. The Straight Tapered Wing
Straight tapered wing formulas
Taper Ratio:
lengthchordtipthecandlengthchordrootthecc
ctr
r
t ,
Wing Area:
b)1(cb)cc(Sr2
1
tr21
Aspect Ratio:
S
bA
2
Mean aerodynamic chord: For preliminary airplane layouts the mean aerodynamic chord is calculated under the assumption that cl = CL across the entire span.
11
3
2112/
2/
2
2/
2/
2 r
b
b
b
b
l
L
cdyc
Sdycc
SCc
Span-wise location of the MAC:
ENAE482 – Lecture 4 Page 7
2)1(3
21 byMAC
For preliminary design the MAC and its span wise location are useful for a first cut estimate of the wing’s aerodynamic center location. For straight tapered wings the chord length at yMAC will be equal to the MAC. A first approximation to the location of the aerodynamic center of the wing is the point at 25% of the MAC, as shown in Figure 6.
Figure 6. First Estimate of the Wing Aerodynamic Center
Average chord:
)1()(21
21 rrtave ccc
b
Sc
Chord at any fractional span location:
)2//(11)( bywherecc r
Sweep Conversion Formula:
12)1(
)1(4tantan
12pp
App
where p1 = the fractional chord location of p1 and p2= the fractional chord location of p2. This is a useful formula, since sweep angles at various percentages of the chord will appear in formulas used to calculate aerodynamic coefficients.
Combined Straight Tapered Wings
Combined straight tapered wings, also called “cranked” wings, consist of two straight taper wings joined at a common span-wise location but having different taper ratios and/or sweep angles, as shown in Figure 7.
ENAE482 – Lecture 4 Page 8
Figure 7. Cranked Wing
Cranked wings are generally employed for subsonic-supersonic aerodynamic center management or, in the case of a canard configuration, to achieve “constructive interference” between the canard’s downwash and the wing’s upwash (e.g., the Saab Viggin).
Cranked planforms that exhibit a rectangular inboard section and a tapered section beyond some span-wise location are often incorporated when additional volume is required for fuel and/or to achieve better structural characteristics (thickness) at the wing root. The A-10 is an example in which both additional strength and volume were required. There are many utility aircraft that also exhibit such a planform (e.g., the Cessna Skylane and Robin R3140).
Combined Straight tapered wing formulas The equations provided above for straight tapered wings are valid for each section of a combined tapered wing. The total wing area is the sum of the areas of each section(S = S1 + S2). The combined MAC is calculated using the following formula:
S
SMACSMACMAC 2211 )()(
The span-wise location of the MAC is given by;
S
SySyy 2211
MGC
where y is the distance to the MAC measured from the plane of symmetry of the wing. Cranked wings are not characterized by a single taper ratio or sweep angle. The MAC will line up exactly with the leading and trailing edges at the span-wise location yMAC for a straight tapered wing. Hence, the MAC can be located precisely with respect to the wing planform and the wing aerodynamic center can be determined at 25% of the MAC, as indicated above. This is not true of cranked wings. The MAC will not line up exactly with the leading and trailing edges of the wing at the calculated value of yMAC. The location of the aerodynamic center location must be calculated. The formula is:
S
SxSxx acac
ac2211 )()(
For convenience, values of xac are referenced to the wing vertex (i.e., at the LE of the inboard
ENAE482 – Lecture 4 Page 9
root chord). (xac)1 is the distance from the wing vertex to 25% of MAC1 and (xac)2 is the distance from the wing vertex to 25% of MAC2. Thus, if d is the distance from the plane of symmetry to the end of the first panel:
22211
1111
25.0)tan(])[()tan()(
25.0)tan()()(
MACdydx
MACyx
LEMACLEac
LEMACac
Curved Wings
Curved wings exhibit a planform having curvilinear leading and/or trailing edges. An elliptic wing planform is one example of a curved wing. Curved wings are generally only used to achieve optimum area ruling for supersonic cruise aircraft (e.g., the Concorde). The wing area and MAC are evaluated using the integral relationships given above. For an elliptic planform the integrals can be analytically evaluated yielding MAC = 0.85cr and yMAC = 0.53b/2 where cr is the minor axis and b is the major axis. The aerodynamic center of a curved wing can be approximated by breaking the wing down into trapezoidal elements such that the leading and trailing edges of each trapezoid are closely aligned with the actual curved leading and trailing edges. An example is shown below.
The equations for a cranked wing are then generalized for the multiple panels thus obtained. Hence;
SSxx
SSyy
SSMACMAC
SS
n
i
iacac
n
i
iMACMAC
n
i
ii
n
i
i
i
i
/
/
/
1
1
1
1
ENAE482 – Lecture 4 Page 10
Wing Planform Design At the conclusion of the first conceptual design phase, the wing is only characterized by its area and aspect ratio. All other characteristics need to be defined for baseline configuration layout. The step-by-step process used to determine those characteristics is described in the following text.
1. If the airplane is a flying wing then the wing must have sufficient volume to carry all fixed equipment, the propulsion system, fuel and payload. This will have a significant impact on the wing layout and should be considered first. For example, flying wings will usually require the use of fairly thick airfoil sections (20 to 25%) near the root with a moderate to high taper ratio (0.6 to 0.3) to increase the root chord length. A moderate amount of sweep is also required for longitudinal stability and control purposes.
2. If the airplane is not a flying wing, then decide on the wing/fuselage arrangement, i.e.; high, mid or low wing. The operational scenario will generally have the most significant influence on this decision and usually takes priority over all other considerations, as discussed previously. Refer to the sketches generated at the end of the conceptual design phase.
3. Decide whether or not the wing will be braced. Bracing the wing will lead to a lighter wing structure and may allow one to use thinner airfoil sections, but bracing also results in increased parasite and interference drag. Hence, braced wings are primarily used on low speed airplanes. The tradeoff between wing weight and the additional drag incurred with bracing is usually favorable below about 200 knots.
4. Select the quarter-chord sweep angle and the airfoil thickness ratio. Quarter-chord sweep and airfoil thickness ratio can have a significant impact on the drag rise associated with high subsonic and supersonic cruise speeds. If the airplane will cruise above Mach 0.65, the wing will require moderate to high quarter-chord sweep and as
thin an airfoil as is practical. The cruise lift coefficient ( CruiseL
C ) is also an important
factor. Use the value of CruiseL
C determined during conceptual design. Figures 8a-c should
be used to determine the quarter-chord sweep required for the selected thickness ratio and a critical Mach number that is at least 5% greater than the cruise Mach number. Calculation of the airfoil critical Mach number when experimental airfoil data are available is further elaborated upon in the Addenda to this lecture. If the wing has an aspect ratio less than 6, use Figures 9a and 9b to correct the critical Mach number obtained from Figures 8a-c. The drag divergence Mach number will be roughly 0.05 higher than the critical Mach number if conventional airfoils are used. If a supercritical airfoil is used, the increment to the critical Mach number can be obtained from Figure 10. Historical data on airfoil thickness ratio and aspect ratio by airplane category is presented in Table 1.
Table 1. Typical Wing Average Thickness Ratio and Aspect Ratio
Figure 8. Critical Mach No. vs. Quarter-chord Sweep and Airfoil Thickness Ratio
20
30
45
0.04
0.03
0.02
0.01
02 3 4
A
5 6
= 10(c/4)
CRM LC = 0.2
(9a) CL=0.2
2 3 4 50
0.01
0.02
0.03
A
= 10
L
20
45
C = 0.4
(c/4)
MCR
(9b) CL=0.4
Figure 9. Critical Mach No. correction for A<6
ENAE482 – Lecture 4 Page 13
MDD
C = 0.5
0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.180.64
0.68
0.72
0.76
0.80
0.84
0.88
ConventionalAirfoils
State-of-the-ArtSupercriticalAirfoils
G 8050-046
64A406
66-210
64A211
64A412
65-215
NA RockwellNASANACANorthrop
(t/c)
L
Figure 10. Drag Divergence Mach Number for Conventional and Supercritical Airfoils
5. Decide which airfoil to use. There are literally an infinite number of airfoils to choose from. An excellent reference for NACA sections is a book by I.H. Abbot and A.E. von Doenhoff entitled “Theory of Wing Sections”. A spreadsheet entitled "Airfoil Coordinates Calculator" will calculate the coordinates of airfoils discussed in that reference has been provided with the course material. In addition, there are modern computational airfoil design tools (such as XFOIL) that can be used to design an airfoil for a specific application. In selecting or designing an airfoil, the designer must consider the following Important section characteristics:
a. Section drag coefficient at the section design lift coefficient b. Section critical Mach number. As indicated above, the wing critical Mach
number will be less than the section critical Mach number when the wing is swept. The values in Figures 8a-c at zero quarter-chord sweep are the section critical Mach number.
c. Section pitching moment coefficient about the quarter-chord. In general, the more camber an airfoil has the higher the pitching moment will be. Higher values of airfoil pitching moment will result in higher zero lift drag coefficient when the airplane is trimmed longitudinally (i.e., the pitching moment is zeroed out). This increment to the drag coefficient is therefore known as the "trim drag".
In addition, the airfoil selected for the wing should exhibit characteristics that are consistent with the airplane’s operational scenario and mission profile. Airfoil characteristics should be evaluated at the Reynolds and Mach numbers consistent with the speed and altitude that are driving the design (i.e., mission segments that consume the most fuel). Table 2 is a guide to the desired airfoil characteristics based upon the design driver.
ENAE482 – Lecture 4 Page 14
Table 2. Guide to Airfoil Selection
Design Driver Desired Airfoil Characteristic Suggested for initial selection High aerobatic capability Low drag, symmetrical lift curve Symmetrical 5-digit or 6-series
Long Range - Propeller driven
High L/D
Moderate camber 4 or 5-digit or 6-series
Long Range - Jet High L1/2
/D Low camber 4- or 5-digit or 6-series
Long Endurance – Propeller driven
High L3/2
/D
High camber 4-digit or 6-series
Long Endurance – Jet
High L/D Moderate camber 4-or 5-digit or 6-series
High Speed Low zero lift drag coef. Thin section with little or no camber
High Maneuverability Low zero lift drag coef. High max. lift coef.
6-series with moderate camber
6. Decide on the wing taper ratio. The primary considerations are span (Oswald’s) efficiency, the span wise location at which the wing begins to stall and the wing's structural weight. Span efficiency can be improved with taper down to a value of 0.35 or less, depending on the amount of sweep. From a structural point of view, consider the span loadings shown in Figure 11. Note that taper causes the wing loading to shift toward the root, which is beneficial since this reduces the root bending moment and the amount of structure required at the root.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.2 0.4 0.6 0.8 1 1.2
c lc/
c ave
C L
y/(b/2)
Taper=1.0
Taper=0.3
Figure 11. Effect of taper on wing loading
On the negative side, more taper results in lower tip Reynold’s number that could result in the flow near the wing tip stalling first. This is very undesirable since this is where the aileron, the primary roll control surface, will be located. Figure 12 illustrates the effect of taper on the span wise distribution of lift coefficient. Note that taper causes the local lift coefficient to peak out at around 80% of the span. Hence, this is where the wing will begin to stall first. This location is likely to be in the vicinity of the aileron and the onset of stall at this location is therefore highly undesirable.
ENAE482 – Lecture 4 Page 15
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.2 0.4 0.6 0.8 1 1.2
c l
y/(b/2)
Taper=1.0
Taper=0.3
Figure 12. Effect of taper on the spanwise distribution of lift coefficient
One can favorably augment the span loading and the distribution of lift coefficient by twisting the wing so that the tip section is at lower incidence than the root (negative twist angle = wash out). This is illustrated in Figure 13.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1 1.2
c lc/
c ave
CL
y/(b/2)
Effect of Washout on Wing Loading
Washout=0
Washout=3
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
c l
y/(b/2)
Effect of Washoiut on Local Lift Coefficient
Washout=0
Washout=3
Figure 13. Effect of washout on span loading and lift coefficient
Washout further shifts the wing loading toward the root and shifts the location of maximum local lift coefficient toward the root (away from the aileron) as well. Note however that wash out also increases the wing’s zero lift drag. The data in Table 3 can be used as a guide for the initial selection of taper ratio.
Table 3. Taper Ratio Distribution by Airplane Category
Airplane Category % with Taper Ratio in the range
0 to .2 .21 to .4 .41 to .6 .61 to 1.0
Homebuilt 0 0 38 62
Single engine prop 0 0 33 67
Twin engine prop 0 35 24 41
Business Jets 0 82 18 0
Regional Turboprops 0 20 70 10
Jet transports 12 88 0 0
Military Trainers 6 18 71 6
Fighters 24 53 12 12
Military Bomber/transport 0 67 33 0
Flying Boat/Amphibious 0 31 31 38
Supersonic Cruise 67 33 0 0
ENAE482 – Lecture 4 Page 16
7. The wing planform can now be laid out. Calculate the wing span and the root and tip
chords using the equations presented previously. Also calculate the leading edge sweep using the sweep conversion formula with p2 = 0 and p1 = .25:
A)1(
)1(tanarctan)25.00
A)1(
)1(4tanarctan
)4/c()4/c(LE
The procedure for laying out the wing planform is illustrated in Figure 14, where lines are numbered to indicate the sequence.
1) Draw a vertical line equal in length to the root chord. 2) Draw a line parallel to the root chord at a distance b/2 from it. 3) Draw a line at the leading edge sweep angle starting at the leading edge of the
root chord line. Alternatively one can locate the point on line 2 that is a distance (b/2)tanΛLE aft of the root leading edge and connect the root leading edge and this point.
4) Locate a point on the second line a distance equal the tip chord aft from the intersection of the leading edge (line 3) and line 2.
5) Draw the final line (the trailing edge) by connecting the trailing edge of the tip chord and the trailing edge of the root chord.
6) Trim line 2 at the leading and trailing edges. 7) Mirror the lines about the root chord.
Figure 14. Wing Layout Illustration
8. Verify the clean wing maximum lift coefficient. The sizing analysis performed during conceptual design required that the clean airplane CLmax be specified. Now that the wing geometry and airfoil section have been selected the clean wing value of the maximum lift coefficient can be calculated.
ENAE482 – Lecture 4 Page 17
The maximum lift coefficient of the unswept wing can be approximated using the following equation:
2
cckC
tiplrootl
UnsweptWL
maxmax
max
kλ = 0.95 for λ = 1 and 0.88 for tapered wings. Note that the lower case “c” in the above formula implies airfoil section maximum lift coefficient. Airfoil section maximum lift coefficient is strongly dependent upon the Reynolds number. Unfortunately, there is no analytic or empirical method that will reliably predict an airfoil’s maximum lift coefficient. Hence, experimental data is the only truly reliable source for this parameter. If airfoil data is available at the root and tip Reynold’s numbers, then that data should be used. If data is not available, Figures 15 and 16 can be used (maximum lift coefficients for NACA airfoils up to 22% thick at Reynolds numbers between 3 and 9 million).
86 10 12 14 16 18 20 220.8
1.2
1.6
1.8
1.4
1.0
Rn=3MRn=6M
Rn=9M
Symmetrical Airfoils
=6-series=4 and 5 digit
Rn=3M
Rn=6MRn=9M
cl max
(t/c)
Figure 15. Airfoil Maximum Lift Coefficient vs. Reynold's No. and (t/c) for Symmetrical Airfoils
ENAE482 – Lecture 4 Page 18
6 8 10 12 14 16 18 20 221.0
1.2
1.4
1.6
1.8
2.0
Rn=3M
Rn=6MRn=9M
Rn=3M
Rn=6M
Rn=9M
cl max
(t/c)
Cambered Airfoils
=4 and 5 digit=6-series
Figure 16. Airfoil Maximum Lift Coefficient vs. Reynold's No. and (t/c) for Cambered Airfoils
If the airfoil is not in one of the families corresponding or Figures 15 and 16 or the Reynolds number is outside the specified range, CFD can be used to estimate clmax. A program known as XFOIL, developed at MIT by Dr. Dreyla, is one of the better codes available for predicting airfoil characteristics. For swept wings having a quarter-chord sweep angle between 0 and 35° the effect of sweep on the maximum lift coefficient can be calculate using:
4/cosmaxmax cUnsweptWLSweptWL CC
Finally, since the total airplane CLmax represents a trimmed value and to achieve trim a conventional tail will produce a down-load, the wing must be capable of a higher maximum lift coefficient than the total airplane. In the case of a canard configuration, the interference effect of the canard up-load and the requirement that the canard stall before the wing will also require a higher wing maximum lift coefficient than the total airplane. It is conservative in the early phases of preliminary design to assume:
1.1/1.1maxmaxmaxmax WLLLWL
CCCC
The calculated value of CLmax should now be compared to the value used for conceptual sizing calculations at the takeoff and landing Reynolds numbers and any flight condition that requires flight at CLsafe = CLmax/1.44. If the calculated value is less than the assumed value it could be necessary to resize the airplane using the calculated value. Before deciding to do this, however, the flap design should be carried out. Flaps will be designed after the ailerons are sized and incorporated in the wing design.
9. Determine the amount of twist. Precisely how much washout will be required to insure
ENAE482 – Lecture 4 Page 19
that the wing stalls inboard of the aileron requires detailed analysis that is beyond the scope of this course. More highly tapered wings will require some amount of washout and if the wing is swept additional washout may be necessary, although that determination would require aero elastic analysis. For initial design it is advisable to incorporate some washout based upon taper ratio and rely on future detailed design to determine the final value. Use the following formula for initial layout:
degrees)1(4
10. Determine the aileron size and location. The following data is presented as an aid in
selecting a value of aileron area/wing area (Sa/S) for the initial layout.
Airplane Category Aileron area/wing area (Sa/S)
Minimum Average Maximum
Homebuilt 0.063 0.095 0.140
Single Engine Prop 0.060 0.080 0.110
Twin Engine Prop 0.044 0.062 0.087
Business Jet 0.012 0.050 0.096
Regional TBP 0.027 0.060 0.085
Jet Transport 0.021 0.036 0.051
Military Jet Trainer 0.059 0.074 0.100
Military TBP Trainer 0.063 0.092 0.130
Fighter 0.031 0.063 0.140
Military Jet Bomber/Transport
0.022 0.046 0.058
Military TBP Bomber/Transport
0.040 0.059 0.077
Supercruiser 0.014 0.057 0.120
For the initial layout, the average values can be used. Final aileron size will be determined based upon roll acceleration, which is inversely proportional to the roll inertia. For designs that will have additional masses (such as stores, tip tanks and outriggers) mounted to the wings, use a value closer to the maximum in the above table. Sa/S can be calculated for a straight taper wing using the following equation:
)()(2
11
1
)n1(2
S
Si0i0
a
n = the fractional chord location of the aileron hinge line ηo = yo/(b/2) the non-dimensional span location of the outboard end of the aileron ηi = yi/(b/2) the non-dimensional span location of the inboard end of the aileron λ = the taper ratio Typical values of n range between 0.70 and 0.80, typical values of ηi range between 0.50
ENAE482 – Lecture 4 Page 20
and 0.75, and typical values of ηo range between 0.90 and 1.00. In order to lay out the aileron, first select a value of n, then select a value of ηo and calculate the value of ηi based upon the desired Sa/S. Solving the equation for Sa/S for ηi yields:
)1(
)1()1/()/)(1()1(211 2
21
21
ooa
i
nSS
Draw the corresponding percent chord line between the corresponding aileron inboard and outboard aileron ends on the wing planform drawing.
11. Determine the increment to the wing maximum lift coefficient produced by high lift devices. A description of the various types of high lift devices and their effect of upon the airplane lift curve is presented in the Appendix to this lecture. For plain flaps assume that the flap and aileron hinge lines will lie on the same percentage chord line. For split flaps, fowler or slotted flaps assume that the flap leading edge will coincide with the same percentage chord line as the aileron leading edge. This is highly desirable from structural and fabrication points of view. The offset between the aileron hinge line and aileron leading edge can be estimated using the data of Figure 24. The increment to the wing maximum lift coefficient due to flap deflection is related to the increment to the airfoil with flap deflected using the following formula:
maxmax l
wf
WLc
S
SKC
)4/c(
75.0
)4/c(
2 cos]cos08.1[K
In the above equation Swf is the area of the portion of the wing in the region bounded by the flap, as shown in Figure 17.
S12 wf S1
2 wf
yb/2
i
o =2y /bi
=2y /b o
Figure 17. Definition of Swf
For straight tapered wings, the flap area ratio can be calculated using the following formula:
ENAE482 – Lecture 4 Page 21
1
))(1(2)(
S
Sio
io
wf
The inboard and outboard non-dimensional span stations (ηi and η0) are illustrated in Figure 17. The values of ηo and ηi should be selected to yield the largest flap possible for the first iteration. Since the fuselage will limit the value of ηi, and the fuselage design has not yet been initiated at this point, assume ηi = 0.1. The value of ηo cannot be larger than that afforded by the inboard aileron location, so assume that the flap will end at that location; i.e., (ηo)flap = (ηi)aileron. The value of Δclmax depends upon the particular type of flap. The methods used to calculate Δclmax for various flap types are presented next.
12. Compute the incremental section lift maximum lift coefficient (Δclmax) corresponding to
the type of flap, flap angle and flap chord ratio. Refer to the Appendix to this lecture for the definition of flap types and corresponding geometric parameters. The increment to the section lift coefficient and the increment to the maximum lift coefficient clmax for trailing edge flaps are illustrated in Figure 18.
c lmax
c l
c l
Figure 18. Section Lift Increment and Section Maximum Lift Increment with TE Flaps
For preliminary design it is conservative to use the following formula for this relationship:
llcKc
max
where the quantity K depends upon the type of flap and the flap chord ratio cf/c, and is determined using Figure 19.
ENAE482 – Lecture 4 Page 22
c /cf
c lmax
c l
Plain and
Split Flap
Single Slotted Flap
Fowler and
Double Slotted Flap
Figure 19. Effect of Flap Type and Chord Ratio on K
Note that the flap chord is equal to the portion of the total chord between the hinge line and the trailing edge for split and plain flaps. For fowler and slotted flaps the flap chord is the total flap length.
The magnitude of the increment to the section lift coefficient with the flap deployed depends upon the type of flap, the flap chord ratio and the flap deflection angle. For initial calculations the following deflection angles should be used for the takeoff and landing configurations:
Takeoff: Flap deflection = 12° All flap types
Landing: Flap deflection = 60° Split and plain flaps
Flap deflection = 50° Slotted flaps
Flap deflection = 40° Fowler flaps
Equations for computing the value of Δcl for four different types of flaps are presented next.
Plain Flaps: Kccfll
The value of clδ is obtained from Figure 20 and the value of K’ is obtained from Figure 21.
ENAE482 – Lecture 4 Page 23
0 0.1 0.2 0.3 0.4 0.5
2
3
4
5
6 .15
.12
.10
.08
.06
.04
.02
0
t/c
Cl
f
c /cf
Figure 20. Effect of Thickness Ratio and Flap Chord Ratio on clδ
0 20 40 60
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.3
.10
.15
.20
.25
.30
.40
.50
c /cf
K'
Flap Deflection, f
10 30 50
Figure 21. K' vs. Flap Deflection and Flap Chord Ratio
Split Flaps: 2.0)c/c(lfl f
)c(kc
Values of kf and 2.0)c/c(l f
)c(
are obtained from the charts in Figure 22.
ENAE482 – Lecture 4 Page 24
0.10
0.12
0.14
0.16
0.18
0.20
0.22
t/c
Upper Lim
it
Flat P
late
(The
oret
ical)
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Cl
c /
c =
0.2
f
0 20 40 60
0.6
0.8
1.0
1.2
1.4
0.1 0.2 0.3 0.4
kf
f
fc /c
Figure 22. Split Flap Parameters
Single Slotted Flaps: fflappedll
)c(c
αδ is obtained from Figure 23. The flapped section lift curve slope (clα) can be obtained from;
)c/c)(c()c(lflappedl
Where clα is the lift curve slope of the airfoil with the flap retracted, obtained from airfoil data or, if not available, assume 6.0 per radian (0.104 per degree). The parameter c’/c will only be greater than one if there is some aft translation associated with deployment of the flap. This will vary from installation to installation but for initial layout a value of 1.05 can be assumed.
f
0 10 20 30 40 50 60 70 800
0.2
0.4
0.6
0.15
0.20
0.25
0.30
0.40
c /cf
Figure 23. Section Lift Effectiveness Parameter vs. Flap Deflection and Flap Chord Ratio
ENAE482 – Lecture 4 Page 25
Fowler Flaps: fflappedll
)c(c
(same equation as single slotted)
Where )c/c1)(c()c(flflappedl
assumes full translation of the flap.
The above four flap types are all trailing edge devices. In some cases it may be necessary to also employ leading edge devices as well. Wherever possible it is preferable to use experimental data to estimate the lift coefficient increment with a leading edge device. In the absence of such data, the following approximation can be used:
clmax with LE device = (c’’/c)(clmax without LE device) where c’’ is the distance between the forward-most point on the flap or slat and the airfoil trailing edge (i.e., the “effective” chord length). Use the calculated values of Δclmax for takeoff and landing configurations to calculate ΔCLmax for those configurations as described above. Finally, since there will be an additional amount of trim required to overcome the additional pitching moment generated when the flap is deployed, the corresponding increment to the airplane’s maximum lift coefficient will be about 95% of the increment to the wing’s maximum lift coefficient:
LandTOWLLandTOL
CC// maxmax
95.0
13. Decide whether or not the flap/wing design meets the assumed takeoff and landing
parameters. Note that for takeoff and landing the significant quantity is the total maximum lift coefficient with flaps deployed. Hence, even if the assumed clean CLmax is greater than that calculated in step 8, the sum of the calculated clean CLmax and calculated flap increments to CLmax for takeoff and landing could achieve or exceed the values of (CLmax)TO and (CLmax)Land that were assumed for sizing. If this is the case, there is no need to resize the airplane. If either value is not achieved, however, the airplane should be resized at this point based upon the calculated values.
14. Locate and incorporate the fore and aft spars on the planform drawing. If a torque box, semi-monocoque or monocoque structure will be used for the wing the forward spar will typically be located on the 10-15% chord line. For wings without leading edge devices use 10%. If leading edge devices are used, use 15%. The location of the aft spar will depend upon the type of flap, flap chord ratio, and airfoil thickness. For split, slotted and fowler flaps the forward spar can be very close to the flap leading edge. For plain flaps the data shown in Figure 24 can be used to estimate the fractional chord offset between the hinge line and the aft spar.
ENAE482 – Lecture 4 Page 26
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.15 0.2 0.25 0.3 0.35 0.4
Spar
Off
set
(fra
ctio
n o
f ch
ord
)Flap chord ratio (cf/c)
10%
12%
15%
18%
Airfoil t/c
Figure 24. Rear Spar Offset from Plain Flap Hinge Line
For fowler flaps the aft spar is located .005c ahead of the flap leading edge.
15. Calculate the required fuel volume. The conceptual design yields the required fuel in pounds. This must be converted to a volume requirement using fuel density. Two types of fuel must be considered, depending upon the propulsion system type. ICEs use Avgas 100LL, which has a density of 6.01 lbs/gal at STP. Turboprop and jet engines use Jet A-1 (the military equivalent of JP-8), which has a density of 6.71 lbs/gal at STP. Fuel will sometimes be pumped into the airplane at a temperature higher than standard day (59˚F=15˚C), so these densities are be decreased to take this into consideration. At 95˚F the density of Avgas is reduced to 5.89 lbs/gal and the density of Jet A-1 is 6.57 lbs/gal. Hence, Vreq'd = pounds of fuel required/5.89 U.S. gallons for Avgas Vreq'd = pounds of fuel required/6.57 U.S. gallons for Jet A-1 Multiplying the result in gallons by 231 yields cubic inches. A fuel tank will be integrated with the torque box formed by the wing skins and forward and aft spars. The volume of a section of the torque box can be approximated using the formula
ioiio AAAA
yyV
0
3
)(
where subscripts o and i correspond to the outboard and inboard locations of the ends of the tank and A0 and Ai are the cross-sectional areas at locations y0 and yi. The cross-sections are bounded by the airfoil OML and the forward and aft spars. It is necessary to allocate 4% of this volume to accommodate material thickness and any internal plumbing. Hence,
ioiioioiio
fuel AAAAyyAAAAyy
V
00 )(32.03
)(96.0
where Ao and Ai can be computed to the OML of the airfoil section and spars. Maximum wing tank volume: Fuel is not carried beyond the 85% span point. This is to minimize the probability that lightning strikes (which are most likely to hit the airplane
ENAE482 – Lecture 4 Page 27
extremities) will initiate a fire. On the inboard end, eliminate the first 10% of the span to accommodate the fuselage. Since the fuselage has not been designed at this point, some adjustment to the inboard end may be required after integrating the wing with the fuselage. The resulting maximum tank size is illustrated in Figure 25.
Figure 25. Maximum Wing Tank Fuel Volume Allocation
The maximum tank volume (includes both port and starboard tanks) can be approximated using the formula given previously:
85108510max24.0 AAAAbV fuel
A10 and A85 are the cross-sectional areas of the torque box. These areas are obtained from section cuts through the torque box parallel to the plane of symmetry. If the wing tank volume is not sufficiently large to carry all of the required fuel volume then additional tanks will have to be integrated with the fuselage when it is designed. If the wing tank capacity exceeds the required fuel volume, then the portion of the torque box used for fuel will be shortened. Determine where the outboard end of the tank should be using trial and error (the binomial search method will work just fine). When there will be engines or stores attached to the wing the fuel tank will be interrupted by a dry bay at those locations. There will therefore be several individual tanks inboard and outboard of the dry bays. The total fuel volume will be the sum of the individual tank volumes. For dry bays at engine locations the width of the dry bay is equal to the width of the engine and is centered on the engine centerline. At pylon locations used for stores or engine mounting pylons a dry bay width equal to twice the pylon width is usually sufficient.
15. Decide on the wing dihedral angle. The primary reason for incorporating wing dihedral is to improve the lateral stability of the aircraft. Lateral stability implies that the aircraft should return to the original wings level flight condition if disturbed by a gust that rolls the airplane from level orientation. The lateral static stability is primarily represented by a stability derivative called aircraft dihedral effect, defined as the change in aircraft rolling moment coefficient (Cl) due to a change in aircraft sideslip angle (β); i.e., ∂Cl/∂β, abbreviated Clβ . If a disturbance causes an aircraft to roll away from wings level orientation the aircraft will begin to move somewhat
ENAE482 – Lecture 4 Page 28
sideways toward the lower wing. The airplane's flight path starts to move toward the left while the nose of the airplane is still pointing in the original direction. This means that the oncoming air is arriving somewhat from the left of the nose (negative sideslip angle). The airplane therefore experiences a sideslip angle in addition to the bank angle. In order to return to the wings level orientation a positive rolling moment is required. This leads to the requirement that Clβ should be negative to insure lateral stability. Incorporating dihedral in the wing increases the angle of attack of the low wing and decreases the angle of attack of the higher wing when the airplane sideslips and this yields the required negative value of this derivative. Note that at the same time the angle of sideslip is building up, the vertical fin is trying to turn the nose back into the wind, much like a weathervane, minimizing the amount of sideslip that can be present. If there is less sideslip there is less restoring rolling moment due to dihedral effect. Hence, the directional stability fights the tendency for dihedral effect to roll the wings back level by not letting as much sideslip develop. There are three possible dynamic modes that can result, depending upon the relative amounts of directional and lateral stability inherent in the design. These are illustrated in Figure 26.
Figure 26. Lateral-directional dynamic modes
With insufficient dihedral effect per unit of directional stability the airplane continues to roll in the direction of the initial disturbance. This is known as spiral instability. With just the right amounts of dihedral effect and directional stability the airplane will return to wings level and continue flying in the original direction. This is known as spiral stability. With too much dihedral or not enough directional stability the airplane will oscillate from left to right in roll and direction. This is called a Dutch roll (named after the ice skating maneuver that looks very similar). Spiral stability is not a hard requirement, and most aircraft are in fact spirally unstable. Level flight is achieved either by pilot or autopilot intervention to restore wings level attitude.
Other Factors affecting dihedral effect: Wing sweepback increases dihedral effect. This is one reason why aircraft with high sweep angle sometimes exhibit anhedral, even on low-wing aircraft such as the Tu-134 and Tu-154.
The longitudinal location of the CG is of primary importance for longitudinal stability of the aircraft, but the vertical location is also important. The vertical location of the CG with respect to the wing location changes the amount of dihedral effect. This is sometimes referred to as the pendulum or "keel" effect. On high wing airplanes the dihedral effect is increased while on low wing airplanes the result is to decrease the dihedral effect. An extreme example of this is a paraglider. Hence, low wing airplanes generally require more wing dihedral than high wing airplanes. The dihedral effect
ENAE482 – Lecture 4 Page 29
created by a very low vertical CG in combination with a highly swept wing is so significant that an appreciable amount of negative wing dihedral (anhedral) is sometimes required to achieve the correct overall dihedral effect. The data of Table 4 can be used as a guide in selecting the initial value for wing dihedral. A good rule of thumb is to start with 6˚ for low wing and 3˚ for high wing and then deduct 1˚ for every 10˚ of leading edge sweep. Future lateral-directional stability analysis will refine this initial estimate.
Table 4 Wing Dihedral Angle for Some Airplanes
Aircraft Type Wing position Dihedral (deg) Pilatus PC-9 Turboprop Trainer Low-wing 7 (outboard)
MD-11 Jet Transport Low-wing 6
Cessna 750 Citation X Business Jet Low-wing 3
Kawasaki T-4 Jet Trainer High-wing -7
Boeing 767 Jet Transport Low-wing 4o 15'
Falcon 900 B Business Jet Low-wing 0o 30'
C-130 Hercules Turboprop Cargo High-wing 2o 30'
Antonov An-74 Jet STOL Transport High-wing -10
Cessna 208 Piston Engine GA High-wing 3 Boeing 747 Jet Transport Low-wing 7
Airbus 310 Jet Transport Low-wing 11o 8'
F-16 Fighting Falcon Fighter Mid-wing 0
BAE Sea Harrier V/STOL Fighter High-wing -12
MD/BAe Harrier II V/STOL Close Support High-wing -14.6
F-15J Eagle Fighter High-wing -2.4
Fairchild SA227 Turboprop Commuter Low-wing 4.7
Fokker 50 Turboprop Transport High-wing 3.5
AVRO RJ Jet Transport High-wing -3
MIG-29 Fighter Mid-wing -2
Wing Weight Following the guidelines for weight estimates presented previously, Class I Wing weight estimates for preliminary design will rely on historical data. Historical data is presented in the following tables for specific airplanes in each category. Weights are pounds and wing areas are square feet.
Wing Structure The primary wing structure is of a “torque box” formed by the front and rear spars and the upper and lower skins (as shown in Figure 21). The torque box is semi-monocoque, having an appropriate number of internal ribs to retain the shape of the wing. To these ribs an appropriate number of stringers are attached to prevent buckling of the skins under load and to resist the internal-external pressure differential. The leading edge is attached to the torque box using “nose” ribs and a pre-formed skin. Similarly, trailing edge fairings are attached to the aft spar in the absence of flaps or ailerons. Flaps and ailerons are hinged to the aft spar. A typical wing structure (minus skins) is illustrated in Figure 22.