Ch a p t e r 3 D r a g Po l a r Prof.E.G.Tulapurkara
1
3 .1 . 1 . I n t r o d u ct ct i on
As mentioned in chapter 1, to evaluate the performance of an airplane we need to know as to what will be the drag coefficient of the airplane (CD) when the lift coefficient (CL) and Mach number are given. The relationship between the drag coefficient and the lift coefficient is called drag polar. The usual method to estimate the drag of an airplane is to add the drags of the major components of the airplane and then apply correction for the interference effects. 2
The major components of the airplane which contribute to drag are wing, fuselage, horizontal tail, vertical tail, nacelles and landing gear. Thus, D = Dwing + Dfuse + Dht + Dvt + Dnac + Detc + Dint
Dlg + (3.1)
where Dwing, Dfuse , Dht, Dvt and Dlg denote drag due to wing, fuselage, horizontal tail, vertical tail and landing gear respectively. Detc includes the drag of items like external fuel tanks, bombs, struts etc.. 3
Dint is the drag due to interference. This arises due to the following reasons. While estimating the drag of wing, fuselage and other components we consider the drag of the component when it is free from the influence of any other components. Whereas in an airplane the wing, fuselage, and tails lie in close proximity of each other and flow past one component is influenced by that past the other. As an illustration let us consider an airfoil kept in a stream of velocity V∞. Let the drag be 5 N. Now consider a small plate whose drag at the same speed of be 2 N. 4
Then the drag of the airfoil and the plate as a combination (Fig. 3.1) would, in general, be higher than the sum of individual drags. i.e. D
airfoil+plate>
(5+2)=(5+2)+Dint
It is evident that Dint will also depend on the place where the plate is located on the airfoil. Remarks i ) W a y s t o r e d u c e i n t e r f e r e n ce ce d r a g
A large number of studies have been carried out on interference drag and it is found that Dint can be brought down to 5 to 10% of the sum of the drags of all components, by giving proper fillets at the junctions of wing and fuselage and tails 5 and fuselage ( Fig 3.2 ).
Fig 3.1 Interference drag
6
Fig 3.2 Reduction of interference drag using fillets (Adapted from Ref.3.1, pp. 181) 7
i i ) F a v o r a b l e i n t e r f e r e n c e e f f e ct ct
The interference effects need not always increase the drag . The drag of the airfoil plus the plate can be lower than the drag of the airfoil when a thin plate is attached to the trailing edge of the airfoil which is called splitter plate. The birds flying in formation flight experience lower drag than when flying individually. ( i i i ) Co n t r i b u t i o n s t o a i r p l a n e l i f t
The main contribution to the lift comes from wing-fuselage combination and a small contribution from the horizontal tail i.e. : L = Lwing wing
+ fu fuse sela lag ge
+ Lht
(3.2) 8
For airplanes with wings having aspect ratio greater than six, the lift due to the wing-fuselage combination is roughly equal to the lift produced by the gross wing area. The gross wing area (S) is the planform area of the wing, extended into the fuselage, up to the plane of the symmetry. i v ) Co n t r i b u t i o n s t o a i r p l a n e p i t c h i n g m o m e n t
The pitching moment of the airplane is taken about its center of gravity and denoted by Mcg. Main contributions to Mcg are from wing, fuselage, nacelle and horizontal tail i.e. Mcg = Mwing + Mfuselage + Mht + Mnac
(3.3) 9
( v ) N o n - d i m e n s i o n a l q u a n t i t i es es
To obtain the non-dimensional quantities namely drag coefficient (CD), lift coefficient (CL) and pitching moment coefficient (Cmcg) the reference quantities are the free stream dynamic pressure (½ ρV∞2) ,the gross wing area (S) and the mean aerodynamic chord of the _ wing ( c ). Consequently , M cg D L CD = ; CL = ; Cmcg = (3.4) 2 2 2 1 1 1 ρV∞ S ρ V∞ S ρ V∞ S c 2 2 2 However, the drag coefficient and lift coefficient of the individual components are based on their own reference areas i.e. 10
(a) (a) For wing, hori orizon onttal tai aill and vert ertical tail the reference area is their planform area. (b) For fusela fuselage, ge, nacel nacelle, le, fuel fuel tanks, tanks, bombs bombs and and such other bodies the reference area is either the wetted area or the frontal area. The wetted area is the area of the surface of the body in contact with the fluid. The frontal area is the maximum cross-sectional area of the body. (c) For other components like landing gear the reference area is given along with the definition of CD. 11
Note: (I) The reference reference area, area, on which the CD and CL of an individual component is based, is also called proper area and denoted by Sπ; the drag coefficient based on Sπ is denoted by CDπ. (II)The reference areas for different components are different for the following reasons. The aim of using non-dimensional quantities like CD is to be able to predict characteristics of many similar shapes by conducting computations or tests on a few models. For this to be effective, the phenomena causing the drag must be taken into account. In this context the drag of streamline shapes like wing and slender bodies is mainly due12
to skin friction and depends on the wetted area. Whereas the drag of bluff bodies like the fuselage of a piston-engined airplane , is ma mainly th the pressure drag and depends on the frontal area. It may be added that for wings, the usual practice is to take the reference area as the planform area because it is proportional to the wetted area. (III) At this stage the reader is advised to the revise the background on aerodynamics (see for examples references 1.7 & 1.8 ). Follow Following ing the above above remarks remarks we can can expre express ss the the total drag of the airplane as : 13
2
D = 12 ρ V∞ S C Dwing + +
+
1 2
ρ V∞2 S fuseC D fuse
1 2
ρ V∞2 Snac CDnac
1 2
ρ V∞2C Dlg Slg + 12 ρ V∞2 Setc CDetc
+
1 2
ρV∞2 S ht CDht + 12 ρ V∞2 Svt CD v t +
(3.5)
Dint
It may be recalled that Setc and CDetc referred to areas and drag coefficients of other items like external fuel tanks , bombs , struts etc.. D Or C D = 2 1 ρ V S ∞ 2 =
+
C D w i n g + C D f u s e
C D n a c
S nac S
+
C D lg
S fu f u s e
+
S S l g S
+
C D ht
C D e tc
S ht
+
S S e t c S
+
C D v t
C D in t
S v t S
(3.6) 14
The data on drag lift and pitching moment, compiled from various sources, is available in references 1.7,1.8,1.9 and 3.1a to 3.7.
15
3 . 2 . Es t i m a t i o n o f D r a g Po P o l a r – L o w S p e ed e d Ca Ca se se
As mentioned in the previous section, the drag polar of an airplane can be obtained by summing-up the drags of individual components and then adding 5 to 10% for interference drag. This exercise has to be done at different angles of attack. A few remarks are mentioned before obtaining the drag polar. Remarks
i) A n g l e s o f A t t a ck ck : For defining the angle of attack of an airplane, the fuselage reference line is taken as the airplane reference line (Figs. 1.9,3.3). However the angles of attack of the wing and tail are not the same as that16
of the fuselage. The wing is fixed on the fuselage such that it makes an angle, iw, to the fuselage reference line (Fig 3.3). The angle iw is generally chosen such that during the cruising flight the wing can produce enough lift when fuselage is at zero angle of attack. This is done because the fuselage produces least drag when it is at zero angle of attack and that is what one would like to have during cruising flight, i.e. during cruise the wing produces the lift required to balance the weight whereas the fuselage being at zero angle of attack produces least drag. The tail is set on fuselage at an angle it (Fig. 3.3) 17 such that during cruise the lift required from the tail,
Fig 3.3 Wing setting and tail setting 18
to make the airplane pitching moment zero, is produced by the tail without elevator deflection. This is because, the drag, at low angles of attack, is least when the required lift is produced without elevator deflection. c o e f f i ci c i en en t o f w i n g ii) D r a g co
The drag coefficient of a wing consist of the (a) profile drag due to airfoil (Cd) and (b) the induced drag due to the finite aspect ratio of the wing (CDi). The profile drag of the airfoil consists of the skin friction drag and the pressure drag. It may be recalled that an element of airfoil in a flow experiences shears stress tangential to the surface 19 and pressure normal to it . The shear stress
multiplied by the area of the element gives the tangential force. The component of this tangential force in the free stream direction when integrated over the profile gives the skin friction drag. Similarly the pressure distribution results in normal force on the element whose component in the free stream direction, integrated over the profile gives the pressure drag. The pressure drag is also called form drag. The sum of the skin friction drag and the pressure drag is called profile drag. The profile drag depends on the airfoil shape, Reynolds number, angle of attack and surface roughness. 20
The chord of the wing varies along the span and further the shapes of the profiles may also change along it (span). Hence for the purpose of calculation of profile drag of the wing , a representative airfoil may be chosen with chord S equal to the average chord (C avg = ); where S is b the wing area and b is the wing span. As regards the generation of induced drag it may be recalled that a wing has a finite span. This results in a system of trailing vertices and induced angle due to these vertices tilts the aerodynamic force rearwards. This results in a component in the free stream direction which is called induced drag. The induced drag 21
coefficient is given by :
C L2 (1+ δ ) C Di = π × A
(3.7)
Where A is the wing aspect ratio (A=b2 /S) and δ is a factor which depends on wing aspect ratio, taper ratio and sweep. When a flap is deflected, there will be increments in lift and both profile drag and induced drag. A similar procedure can be used to estimate drags of horizontal and vertical tails. However contributions to induced drag can be neglected for the tail surfaces. 22
iii)
D r a g co c o e f f i c i en e n t o f f u s e l ag ag e
The drag coefficient of a fuselage (CDf ) consists of the drag or the fuselage at zero angle of attack (CD0)f plus drag due to angle of attack. It can be expressed as : CDf =(CD0)f +K(α)2
(3.8)
For a streamlined body (CD0)f is mainly skin friction drag and depends on (a) Reynolds number, based on length of fuselage (lf ),(b) surface roughness and (c) fineness ratio (Af ). The fineness ratio is defined as: Af =lf /de
(3.9) 23
where de is the equivalent diameter given by: (π /4)de2 = Afmax where Afmax equals the area of the maximum crosssection of fuselage. When the fineness ratio of the fuselage is small for e.g. in case of general aviation airplanes , the fuselage is treated as a bluff body. In such cases the drag is mainly pressure drag and the drag coefficient is based on the frontal area (Afmax). The drag coefficients of other bodies like engine nacelle, external fuel tanks, bombs can also be estimated in a similar manner. iv) The drag coefficients of other components like landing gear are based on areas specific to the
24
component. They should be obtain from the sources of drag data mentioned earlier. 3.2.1 Drag polar
To obtain the drag polar by adding the drag coefficients of individual components at corresponding angles of attack , needs a large amount of detailed data about the airplane geometry and drag coefficients. A typical drag polar obtained by such a procedure or by experiments on a model of the airplane appears as shown in Fig. 3.4a. When this curve is replotted as CD vs. CL2 (Fig.3.4b), it is found that over a wide range of CL the curve is a straight line and one could write. 25
CD=CD0 + KCL2
(3.10)
Fig 3.4a 3.4a Typical Typical drag polar polar of a piston piston – engined airplane airplane 26
CD0 is the intercept of the straight line and is called zero lift drag coefficient or parasite drag coefficient (Fig.3.4b). The term KCL2 is called induced drag coefficient or more appropriately lift dependent drag coefficient. K is written as: K
1
=
π
e
(3.11)
where e, called Oswald efficiency factor, includes the changes in drag due to angle of attack of the wing, the fuselage and other components (Refs.1.9, Chapter 2 & 3.3, Chapter 2). 27
Fig Fig 3.4( 3.4(b) b) Drag Drag pol polarar- CD vrs.CL2
28
It may be added that in the original definition of Oswald efficiency factor only the contribution of wing was included. Remarks:
i) The reason why an expression like Eq.(3.10) fits the drag polar is because the lift dependent drags of wing and fuselage are proportional to the square of the angle of attack. ii) The drag polar given by Eq.(3.10) is called parabolic drag polar. iii) It found that CD0 is roughly equal to the sum of the minimum drag coefficients of various components plus the correction for interference . 29
iv) P a r a s i t e d r a g a r e a a n d e q u i v a l e n t s k i n f r i ct c t i o n c o e f f i ci ci en t The product CD0 x S is called parasite drag area. For streamlined airplanes the parasite drag is mostly skin friction drag plus a small pressure drag. The skin friction drag depends on the wetted area of the surface. The wetted area of the entire airplane is denoted by Swet and the equivalent skin friction coefficient (Cfe) is defined as : CD0 x S = Cfe x Swet or
CD0
=
C fe
S w et S 30
Reference 3.7 , Chapter 12 gives values of Cfe for different types of airplanes. v) The The factor factor ‘e’ ‘e’ lies lies between between 0.8 to 0.9 0.9 for airplan airplanes es with with unswep unsweptt wings wings and betw between een 0.6 0.6 to 0.8 for those with swept wings. See Refs.3.3 & 3.4 for estimating CD0 and K. vi) The parabolic polar is an approximation . It is inaccurate near CL =0 and CL= CLmax (Fig.3.4b). It is should not be used beyond CLmax . A quick estimate of the drag polar is carried out in example 3.1.
31
Ex a m p l e 3 . 1
An airplane has a wing of planform area 51.22 m2 and span 20 m. It has a fuselage of frontal area 3.72 m2 and two nacelles having a total frontal area of 3.25 m2. The total planform area of horizontal and vertical tails is 18.6 m 2 . Obtain a rough estimate of the drag polar in a flight at a speed of 430 kmph at sea level (s.l.). when landing gear is in retracted position.
32
So l u t i o n :
Flight speed is 430 kmph = 119.5 m/s. Average chord of wing = S/b = 51.22/20=2.566 m. Reynolds number (Re) based on average chord is: 1 1 9 .5 × 2 .5 6 6 1 4 .6 × 1 0 −6
=
21
×
106
Assuming a 12% thick airfoil the (CDmin)wing at this Re would be 0.0054 (See Reference 3.4). Since the frontal area is specified, the fuselage is treated as a bluff body; (CDmin)fuselage can be taken as 0.08 (Ref.3.4). 33
The nacelle generally has a lower fineness ratio and (CDmin)nac can be taken as 0.10. (CDmin)tail for the tail surfaces is taken as 0.006, which is slightly higher than that for wing as the Re for tail surfaces would be smaller. The results are presented in Table 3.1. Part Sπ (m2) CDπSπ (m2) CDπ Wing
51.22
0.0054
0.279
Fuselage
3.72
0.080
0.300
Nacelles
3.25
0.1
0.325
Tail surfaces
18.6
0.006
0.112
Total
1.013
Table 3.1 Rough estimate of CD0
34
Adding 10% for interference effects, the total parasite drag area (CDπSπ ) is: 1.013 + 0.1013 = 1.1143 m 2. Hence CD0= 1.1143/51.22 = 0.0216 Wing aspect ratio is 202/51.22=7.8 Taking e = 0.83 (see reference 3.4, page A119 for details) we get the drag polar as C D
or
=
0.0216 +
1 π X 7 .8 X 0 .8 3
C L2
CD = 0.0216 + 0.049 C L2 35
Remarks:
i) A detail detailed ed estim estimati ation on of the drag drag polar polar of Piper Piper Cherokee airplane is presented in appendix A. ii) Typical values of CD0 , A, e and the polar for subsonic airplanes are given in Table 3.2.
36
Type of airplane
CD0
A
e
Typical polar
0.8 to 0.9
0.025 + 0.055C L2
Low speed (M <0.3)
0.022 to 6 to 8 0.04
Medium speed (M around 0.5)
0.018 to 10 to 12 0.85 0.020 to 0.9
High subsonic (M around 0.8, Swept wing)
0.015 to 6 to 8 0.017
0.6 to 0.75
0.019 + 0.04CL2
0.016 +0.06CL2
Table 3.2 Typical values of CD0, A ,e and polar
37
Note:
Table 3.2 shows that for low speed airplanes CD0 is higher than in other cases. This is because these airplanes have exposed landing gear, bluff fuselage and struts. They also have only moderate aspect ratio (6 to 8) so that wing-span is not large and the hanger-space needed for parking the plane is not excessive. The CD0 for high subsonic airplanes is low due to smooth surfaces, thin wings and slender fuselage. It may be added that during the design process, the values of airfoil thickness ratio, aspect ratio and angle of sweep for the wing are obtained from considerations of optimum performance. 38
3 . 3 D r a g p o l a r a t h i g h s p ee d s
At this stage the reader is advised to revise background on compressible aerodynamics and gas dynamics (see Refs.1.7 & 1.8). Some important aspects are brought out in the following remarks. (1) (1) When When the the Mach Mach numb number er rough roughly ly exc exceed eeds sa value of 0.3, the changes in the fluid density within the flow field become significant and the flow needs to be treated as compressible. (2) (2) In a comp compres ressi sibl ble e flow flow the the chan changes ges of temperature in the flow field may be large and R T ) may vary hence the speed of sound (a= γ RT from point to point.
39
(3) When the Mach number exceeds unity, the flow is called supersonic. (4) When a supersonic flow decelerates, shock waves occur. The pressure, temperature, density and Mach number change discontinuously across the shock. The shocks may be normal or oblique. The Mach number behind a normal shock is subsonic; behind an oblique shock it may be subsonic or supersonic. (5) When supersonic flow encounters a concave corner, as shown in Fig 3.5 (a), the flow changes the direction across a shock. When such a flow encounters a convex corner, as shown in Fig 3.5.(b) the flow expands across a series of Mach waves. 40
(a)
(b)
Fig 3.5 Supersonic flow at corners a) Concave corner (b) Convex corner (From Ref.1.7,chapter 5)
41
(6) A typical flow past a diamond airfoil at supersonic Mach number is shown in Fig 3.6. If the Mach number is low supersonic (i.e. only slightly higher than unity) and the angle θ , as shown in Fig 3.6, is high then instead of the attached shock waves at the leading edge, a bow shock wave may occur ahead of the airfoil. A blunt-nosed airfoil can be thought of an airfoil with large value of ‘θ’ at the the lead leadin ing g edge edge and and will have a bow shock at the leading edge as shown in Fig 3.7. Behind a bow shock there is a region of subsonic flow ( Fig 3.7) . 42
Fig 3.6 Supersonic flow past a diamond airfoil (From Ref.1.9, chapter1)
43
Fig 3.7 Bow shock ahead of blunt-nosed airfoil 44 ( Adapted from Ref.1.7, chapter 5 )
(7) T r a n s o n i c f l o w This type of flow occurs when the free stream Mach number is around unity. The changes in the flow and hence in the drag occurring in this range of Mach numbers can be appreciated from the following statements. (I)
In sub subso son nic flow flow pas astt an airfoi rfoill the flow has zero velocity at the stagnation point. Then the flow accelerates, it reaches a maximum value (Vmax) and later attains the free stream velocity (V∞). The ratio Vmax /V∞ is greater than unity and depends on (a) shape of airfoil (b) thickness ratio (t/c) and ( c ) angle of attack (α) 45
(II) As (Vmax / V∞ ) is greater than unity, the ratio of the maximum Mach number on the airfoil ( M max) and free stream Mach number (M∞) would also be more than unity. However ( Mmax /M∞ ) would not be equal to (Vmax /V∞ ) as the speed of sound varies from point to point on the airfoil. (III) Critical Mach number As M∞ increases, Mmax also increases. The free stream Mach number for which the maximum Mach number on the airfoil equals unity is called critical Mach number (Mcrit). 46
(IV) When M∞ exceeds Mcrit , a region of supersonic flow occurs which is terminated by a shock wave. The changes in flow pattern are shown in Fig 3.8. (V) As free stream Mach number increases the region of supersonic flow enlarges and this region occurs on both the upper and lower surfaces of the airfoil (Figs. 3.8 c & d). (VI) At a free stream Mach number slightly higher than unity, a bow shock is seen near the leading edge of the airfoil ( Fig. 3.8 e). (VII) At still higher Mach numbers the bow shock approaches the leading edge and if the leading edge is sharp, then the shock waves attach to the 47 leading edge as shown in Fig 3.6.
Fig 3.8 Flow past airfoil near critical Mach number (Adapted from Ref. 1.9,chapter 1)
48
(VIII) Transonic flow regime When M∞ is less than Mcrit the flow every where i.e. in the free stream and on the body is subsonic. It is seen that when Mcrit < M∞ < 1, the free stream Mach number is subsonic but there are regions of supersonic flow on the airfoil ( Figs. 3.8 c & d ) . Further When M∞ is slightly more than unity i.e. free stream is supersonic, there is bow shock ahead of the airfoil resulting in subsonic flow near the leading edge. When the shock waves are attached to the leading edge ( Fig. 3.6 ) the flow is supersonic
49
every where i.e. in the free stream and on the airfoil. The above flow features permit us to classify the flow in to three regimes. (a) Sub-cri Sub-critic tical al regime regime when the Mach number number is is subsonic in the free stream as well as on the body ( M∞ < Mcrit ). (b) Transoni Transonic c regim regime e when when the regions regions of subso subsonic nic and supersonic flow are seen within the flow field. (c) Super Superso soni nic c regi regime me when when the the Mach Mach num number ber in the the free stream as well as on the airfoil is supersonic. The extent of the transonic regime is loosely stated as between 0.8 to 1.2. However the actual 50 extent is between Mcrit and the Mach number at
which the flow becomes supersonic everywhere. The extent depends on the shape of the airfoil and the angle of attack. (8) In the transonic regime the lift coefficient and drag coefficient undergo rapid changes with Mach number ( Fig.3.9). For a chosen angle of attack the drag coefficient begins to increase near Mcrit and reaches a peak around M∞ =1. D r a g d i v e r g e n c e M ac ach n u m b e r When the change in Cd with Mach number is studied experimentally, we can notice the effect of appearance of shock waves in the form of increase in drag coefficient . The beginning of the transonic region is characterized by drag 51 divergence Mach number (MD) at which the
Fig 3.9 Schematic variations of Cl and Cd of an airfoil in transonic regime. (Adapted from Ref. 1.9, chapter1)
52
increase in the drag coefficient is 0.002 over the value of Cd at sub-critical Mach numbers. It may be added that for a chosen angle of attack the value of Cd remains almost constant at subcritical Mach numbers. As mentioned earlier the increase in the drag coefficient in the transonic region is due to appearance of shock waves and hence it is also called wave drag. The drag divergence Mach number of an airfoil depends on its shape, thickness ratio and the angle of attack. (9) The drag divergence Mach number of a wing depends on the drag divergence Mach number of the airfoil used, and the aspect ratio. It can be 53 Λ increased b incor oratin swee to the
wing. The geometrical parameters of the wing are shown in Fig.3.10. The beneficial effects of sweep on (a) increasing MD , (b) (b) decrea decreasi sing ng peak value of wave drag coefficient (CDpeak) and (c ) increasing Mach number at which CDpeak occurs are shown in Fig.3.11.
54
Fig 3.10 Geometric parameters of a wing55
Fig 3.11 Effect of wing sweep on variation of CD with Mach number. (Adapted from Ref.1.9, chapter 1)
56
(10) Drag at supersonic speeds At supersonic Mach numbers also the drag of a wing can be expressed as sum of the profile drag of the wing section plus the drag due to effect of finite aspect ratio . The profile drag consists of pressure drag plus the skin friction drag . The pressure drag results from the pressure distribution caused by the shock waves and expansion waves (Fig.3.6) and hence is called wave drag. At supersonic speed the skin friction drag is only a small fraction of the wave drag. The wave drag of a symmetrical aerofoil (Cdw) can be expressed as (Ref.1.7 , chapter 5 ): 57
Cdw
=
4 M ∞2 − 1
[α 2 + (t / c ) 2 ]
The wave drag of a finite wing at supersonic speeds can also be expressed as KCL2 ( see Ref.1.7 , chapter 5 for details). However in this case K depends on free stream Mach number (M∞ ), aspect ratio and leading edge sweep of the wing (see Ref.1.7 for details). (11) It can be imagined that the flow past a fuselage will also show that the maximum velocity (Vmax) on the fuselage is higher than V∞. Consequently, a fuselage will also have a critical Mach number (Mcritf ) which depends on the fineness 58 ratio of the fuselage. For the slender fuselage,
typical of high subsonic jet airplanes, Mcritf could be around 0.9. Above Mcritf , the drag of the fuselage will be a function of Mach number in addition to the angle of attack.
59
3 . 3 .1 . 1 D r a g p o l a r o f a t h i g h s p e ed ed s
The drag polar of an airplane, which is obtained by the summing the drag coefficients of its major components, will also undergo changes as Mach number changes from subsonic to supersonic. However it is found that the approximation of parabolic polar is still valid at transonic and supersonic speeds, but CD0 and K are now functions of Mach number i.e. : CD = CD0(M) + K(M)CL2
(3.12)
Detailed estimation of the drag polar of a subsonic jet airplane is presented in Appendix B 60
Remarks:
i) G u i d e l i n e s f o r v a r i a t i o n s o f CD 0 a n d K f o r a s u b s o n i c j e t t r a n s p o r t a ir i r p l an an e Subsonic jet airplanes are generally designed such that there is no significant wave drag up to cruise Mach number ( Mcruise ). However to calculate the maximum speed in level flight (Vmax) or the maximum Mach number (Mmax ), we need guidelines for increase in CDo and K beyond Mcruise .Towards this end we consider the data on B727-100 airplane. Reference 3.8 gives drag polars of B727-100 at M=0.7,0.76,0.82,0.84,0.86 and 0.88. Values of CD and CL corresponding to various Mach numbers were read and are shown in Fig. 3.12 by symbols. 61 Following the parabolic approximation, these polars
were fitted with Eq.(3.12) and CD0 and K were obtained using least square technique. The fitted polars are shown as curves in Fig. 3.12. The values of CD0 and K are given in Table 3.3. and presented in Figures 3.13 (a) & (b). M
CD0
K
0.7
0.01631
0.04969
0.76
0.01634
0.05257
0.82
0.01668
0.06101
0.84
0.01695
0.06807
0.86
0.01733
0.08183
0.88
0.01792
0.103
Table 3.3 Variations of CD0 and K with Mach number 62
Fig 3.12 Drag polars at different Mach numbers 63 for B727-100
Fig 3.13 (a) Parameters of drag polar -C - CD0 for B727-100
64
Fig 3.13 3.13 (b) (b) Paramete Parameters rs of drag pola polarr- K for B727B727-100 100 65
It is seen that the drag polar and hence CD0 and K are almost constant up to M=0.76. The variations of CD0 and K between M=0.76 and 0.86, when fitted with polynomial curves give the following equations (see also Figures 3.13 a & b). CD0=0.01634 =0.01634 -0.001( -0.001( M-0.76)+0 M-0.76)+0.11 .11 (M-0.76) (M-0.76)2 (3.13) K= 0.05257+ (M-0.76)2 + 20.0 (M-0.76)3
(3.14)
Note: For M≤ 0.76 , CD0= 0.01634 , K=0.05257 Based on these trends the variations of CD0 and K beyond Mcruise but up to Mcruise+0.1 +0.1 ca can n be expressed by Eqs. (3.13a) and (3.14a) and treated as guideline for calculation of Mmax and 66 range of an airplane (see also Appendix B )
CD=CD0cr -0.001 ( M-Mcruise)+0.11 (M-Mcruise)2 (3.13 a) K=Kcr+ (M-0.76)2 + 20.0(M-0.76)3 (3.14 a) Where CD0cr and Kcr are the values of CD0 and K at cruise Mach number. It may be pointed out that the value of 0.01634 in Eq.(3.13) has been replace by CD0cr in Eq.(3.13a). This has been done to permit use of the equation for different types of airplanes which may have their own values of CDcr (see Appendix B). Similar is the reason for using Kcr in Eq.(3.14a). (2) V a r i a t i o n s o f CD 0 a n d K f o r a f i g h t e r a i r p l a n e Reference 1.8 has given drag polars of F-15 fighter airplane at M=0.8,0.95,1.2,1.4 and 2.2.These are shown in Fig 3.14. These drag polars were also fitted with Eq.(3.12) and CD0 and K were calculated. The variations of CD0 and 67
K are shown in Figs.3.15 (a) & (b). It is interesting to note that CD0 has a peak and then decreases, whereas K increases monotonically with Mach number. It may be recalled that the Mach number, at which CD0 has the peak value, depends mainly on the sweep of the wing.
68
Fig 3.14 Drag polars at different Mach numbers for F15 (Adapted from Ref.1.8, chapter2) Please note : The origins for polars corresponding to69 different Mach numbers are shifted.
Fig 3.15(a) Typical variations of CD0 with Mach number for fighter airplane 70
Fig 3.15(b) Typical variations of K with Mach number for a fighter airplane 71
3 .4 .4 Dr a g p o lar at h y p e r so n i c sp eed s
When the free stream Mach number exceeds five, the changes in temperature and pressure behind the shock waves are large and the treatment of a flow has to be different. Hence the flows with Mach number greater than five are termed hypersonic flow. Reference 3.8 may be referred to for details. For the purpose of flight mechanics it may be mentioned that the drag polar at hypersonic speeds is given by the following modified expression (Ref. 1.1). CD=CD0(M)+K(M)CL3/2
(3.15) 72
Note that the index of CL term is 1.5 and not 2.0 3.5 L i f t t o d r a g r a t i o The ratio CL /CD is called lift to drag ratio. It is an indicator of the aerodynamic efficiency of the design of the airplane. For parabolic polar CL /CD can be worked out as follows. CD=CD0 +KCL2 Hence CD /CL = (CD0 /CL) +KCL
(3.16)
Differentiating Eq.(3.16) with CL and equating to zero gives CLmd which corresponds to minimum of (CD /CL) or maximum of (CL /CD). 73
CLmd = (CD0 /K)1/2
(3.17)
CDmd = CD0 +K(CLmd)2= 2CD0 (3.18)
1
(L/D)max = (CLmd /CDmd) = (3.19) 2 C D 0 K Note: To show that CLmd corresponds to minimum of (CD /CL ), take second derivative of the right hand side of Eq.(3.16) and verify that it is greater than zero.
74
3 . 6 Ot Ot h e r t y p e s o f d r a g
In sections 3.1,3.2 and 3.3 we discussed the skin friction drag, pressure drag (or form drag), profile drag , interference drag , parasite drag, induced drag, lift dependent drag and wave drag. Following additional types of drags are mentioned briefly to complete the discussion on drag. I) Co o l i n g d r a g : The piston engines used in airplanes are air cooled engines. In such a situations when a part of free stream air passes over the cooling fins and accessories, some momentum is lost and this results in a drag called cooling drag.
75
II) B a se s e d r a g : If the rear end of a body terminates abruptly , the area at the rear is called a base. The abrupt ending causes flow to separate and a low pressure region exists over the base. This causes a pressure drag called base drag. III) Ex t e r n a l s t o r e s d r a g : Presence of external fuel tank, bombs, missiles etc. causes additional parasite drag which is called external stores drag. Antennas, lights etc. also cause parasite drag which is called protuberance drag. a k a g e d r a g : Air leaking into and out of gaps IV) L e ak and holes in the airplane surface causes increase in parasite called leakage drag. 76
V) T r i m d r a g : In example 1.1 it was shown that to balance the pitching moment about c.g. (Mcg), the horizontal tail produces a lift (-L t) in the downward direction. To compensate for this , the wing needs to produce a lift (LW) equal to the weight of the airplane plus the downward load (LW = W+Lt) . Hence the induced drag of the wing, which depends on Lw , would be more than that when the lift equals weight. This additional drag is called trim drag as the action of making Mcg equal to zero is referred to as trimming the airplane.
77
3 . 7 H i g h l i f t d e v i ce ce s 3 .7 . 7 .1 . 1 I n t r o d u ct ct i on From earlier discussion we know that: 1 2 L = ρ V S C L 2
(3.20)
Further for an airplane to take-off , the lift must at least be equal to the weight of the airplane , or 1 2 L = W = ρ V SC L (3.21) 2 Hence
V =
2W
ρ SC L
Since CL has a maximum value , we define stalling speed (Vs) as:
(3.22)
78
V s
=
2W
ρ SC L max
(3.23)
The take-off speed (VTo) is actually higher than the stalling speed. It is easy to imagine that the takeoff distance would be proportional V2To and in turn to Vs2. Thus to reduce the take-off distance we need to reduce Vs. Further the wing loading (W/S) is decided by other consideration like cruise. Hence CLmax should be high to reduce take-off and landing distances. The devices to increase CLmax are called high lift devices.
79
3 . 7 .2 . 2 Fa F a ct c t o r s l i m i t i n g CL m a x
Consider an airfoil at low angle of attack (α). Figure 3.16a shows a flow visualization picture of the flow field . Boundary layers are seen on the upper and lower surfaces. As the pressure gradient is low, the boundary layers are attached. The lift coefficient is nearly zero. Now consider the same airfoil at slightly higher angle of attack (Fig.3.16b). The velocity on the upper surface is higher than that on the lower surface and consequently the pressure is lower on the upper surface as compared to that on the lower surface. The airfoil develops higher lift coefficient as 80 compares to that in Fig.3.16a.
3.16a Flow past an airfoil at low angle of attack. Note: The flow is from left to right (Adapted from Ref. 3.10 , chapter 3)
81
3.16b Flow past an airfoil at moderate angle of attack. Note: The flow is from right to left (Adapted from Ref. 3.11 , part 3 section II B)82
However the pressure gradient is also higher on the upper surface and the boundary layer separates ahead of the trailing edge (Fig.3.16b) . As the angle of attack approaches about 150 the separation point approaches the leading edge of the airfoil (Fig.3.16c). Then the lift coefficient begins to decrease (Fig.3.16d) and the airfoil is said to be stalled. The value of α for which Cl equals Clmax is called stalling angle (αstall). Based on these observations , delay of stalling is an important method to increase Clmax. Since stalling is due to separation of boundary layer, many methods have been suggested for boundary layer 83 control. In the suction method the airfoil
3.16c Flow past an airfoil at angle of attack near stall. Note: The flow is from left to right (Adapted from Ref. 3.10 , chapter 3) 84
3.16d Typical Cl vrs α curve 85
surface is made porous and boundary layer is sucked (Fig.3.17a) . In the blowing method, fluid is blown tangential to the surface and the low energy fluid in the boundary layer is energized (Fig.3.17b). This effect (energizing ) is achieved in a passive manner by a leading edge slot (Fig.3.17c) and a slotted flap (section 3.7.3) . See Ref.3.13, chapter 11, for other methods of boundary layer control and for further details.
86
Fig. 3.17 Boundary layer control with suction and blowing (Adapted from Ref.3.12, section 9)
87
3 . 7 . 3 . W a y s t o i n c r e as a s e Cl m a x
Beside the boundary layer control, there are two other way to increase Clmax viz. increase of camber and increase of wing area. These methods are briefly described below . I) I n cr c r e as a s e i n Cl m a x d u e t o ch c h a n g e o f ca ca m b e r It may be recalled that when camber of an airfoil increases, the zero lift angle (αol) decreases and the Cl vrs α curve shifts to the left (Fig.3.18) . It is observed that αstall does not decrease significantly due to the increase of camber and a higher Clmax is realized (Fig.3.18). However, the camber of the airfoil used on the wing is chosen such that minimum 88 drag coefficient occurs near the lift coefficient
Fig. 3.18 Increase in Clmax due to increase of camber
89
corresponding to the cruise or the design lift coefficient . One of the ways to achieve the increase in camber during take-off and landing is to use flaps. In a plain flap the rear portion of the airfoil is hinged and is deflected when Clmax is required to be increased (Fig.3.19a) . In a split flap only the lower half of the airfoil is moved down (Fig.3.19b) . To observe the change in camber brought about by a flap deflection, draw a line in-between the upper and lower surfaces of the airfoil with flap deflected. This line is approximately the camber line of the flapped airfoil. The line joining the ends of the camber line is the new chord line . The difference between the ordinates of the camber line and the chord line is a measure of 90 camber.
Fig. 3.19 Flaps, slot and slat (Adapted from Ref.3.7 , chapter 12)
91
I I ) I n cr c r e a s e i n Cl m a x d u e t o b o u n d a r y l a y e r control
In a slotted flap (Fig.3.19c) the effects of camber change and the boundary layer control are brought together. In this case when the flap is deflected a gap is created between the main surface and the flap (Fig.3.19c) . As the pressure on the lower side of airfoil is more than that on the upper side, the air from the lower side of the airfoil rushes to the upper side and energizes the boundary layer on the upper surface. This way the separation is delayed and Clmax increases (Fig.3.20). The slot is referred to as a passive boundary layer control , as no blowing by external92 source is involved in this devise.
Fig.3.20 Effects of camber change and boundary layer 93 control on CLmax
After the success of single slotted flap , the double slotted and triple slotted flaps were developed (Figs.3.19 d and e). III) I n c r e a se s e i n Cl m a x d u e t o c h a n g e i n w i n g a r e a Equation (3.20) shows that the lift can be increased when the wing area (S) is increased. An increase in wing area can be achieved if the flap, in addition to being deflected, also moves outwards and effectively increases the wing area. This is done in a Fowler flap (Fig.3.19 f) . Thus a Fowler flap incorporates three methods to increase Clmax viz viz chan change ge of ca camb mber er,, boundary layer control and increase of wing area. It may be added that while defining the Clmax in case of Fowler flap, the reference area is the original area 94
of the wing and not that of the extended wing. A zap flap is a split flap where the lower portion also moves outwards as the flap is deflected. I V ) L e ad a d i n g e d g e d e v i ce ce s
High lift devices are also used near the leading edge of the wing. A slot near the leading edge (Fig.3.19 g) also permits passive way of energizing the boundary layer. However a permanent slot has adverse effects during cruise. Hence leading edge slat as shown in Fig.3.19h is used . When deployed it produces a slot and increase Clmax by delaying separation. On high subsonic speed airplanes , both leading edge and trailing edge devices are used to increase 95 Clmax(Fig.3.2.).
Remarks:
i) Se See e Refs Refs.. 1. 1.7, 7,1. 1.8, 8,1. 1.9 9 and and 3. 3.7 7 fo forr ot othe herr typ types es of high lift devices like Kruger flap, leading edge extensions, blown flap etc. ii) See Ref.1.8, Chapter 1 for historical development of flaps. iii) G u i d e l i n e s f o r v a l u e s o f CL m a x o f w i n g s w i t h v a r i o u s h i g h l i f t d e v i ce c e s. s. An estimate of the maximum lift coefficient of a wing is needed to calculate the stalling speed of the airplane. The maximum lift coefficient depends on (a) wing parameters ( aspect ratio, taper ratio and sweep) , (b) airfoil shape , 96 (c ) type of high lift devices (s),
(d) Reynolds number (e) surface finish (f) the ratio of the area of flap to the area of wing, (g) interference from nacelle, fuselage etc. Figure 3.21 can be used for initial estimate of CLmax for subsonic airplanes with wrings of aspect ratio greater than 6. Quarter chord sweep has predominant effect on CLmax and hence it is used on the abscissa in Fig.3.21. The different curves correspond to no flap case and those with different types of high lift devices. For example a wing with sweep of 300 would have a CLmax of about 1.4 with no flaps and 2.3 with Fowler flaps. With addition of leading edge slat this can go upto 2.6. 97
Fig.3.21 Guidelines for CLmax of wings with different high lift devices (Adapted from Ref.3.7, Chapter 5)
98
The following may be noted. (a) The value of CLmax shown in Fig 3.21 can be used in landing configuration. (b) The flap setting during take-off is lower than that during landing. The maximum lift coefficient during take off can be taken approximately as 80% of that during landing. (c ) Do not use these curves, curves, for supersonic supersonic airplanes with low aspect ratio wings and airfoil sections of small thickness ratio. (d) As the the Mach number number (M) incre increase ase beyon beyond d 0.5 the Clmax of the airfoil section decreases, due to the phenomenon of shock stall. Figure 3.22 give, guidelines for estimating Clmax for M > 0.5. 99
Fig.3.22 Effect of Mach number on Clmax (Adapted from Ref.3.7, Chapter 5)
100
References
3.1. Jackson , P. (Editor) ‘Jane’s all the world’s aircraft aircraft 19991999-200 2000’ 0’ Jane’s Jane’s informa informatio tion n group Ltd., Ltd., Surrey , U.K. , 1999. 3.1a. Royal Aeronautical Society data sheets – Now known as Engineering Sciences Data Unit (ESDU) 3.2. DATCOM prepared by U.S. Air Force 3.3 Roskam, J. ‘Methods for estimating drag polars of sub subso soni nic c airp airpla lane nes’ s’ Rosk Roskam am avia aviati tion on and and engineering (1973). 3.4. 3. 4. Wood Wood K.D. K.D. ‘Ae ‘Aeros rospa pace ce Vehic Vehicle le Desig Design’ n’ Vol Vol.I .I and II. Johnson publishing Co., Boulder Colarado 101 (1966) .
3.5. Torenbeek, E. ‘Synthesis of subsonic airplane design’ design’ Delft Delft Unive Universi rsity ty press press (198 (1982). 2). 3.6. 3.6. Hoerner, Hoerner, S.F. S.F. ‘Fluid ‘Fluid dynami dynamic c drag’ Publis Published hed by author (1965). 3.7. Raymer D.P. ’Aircraft Design: A conceptual Appr Approa oach ch’’ Ai AiAA AA Edu Educa cati tion onal al Ser Serie ies, s, Fou Fourt rth h Edition 2006. 3.8. 3. 8. Rosk Roskam am,, J ‘A ‘Air irpl plan ane e des desig ign’ n’ vo volu lume me I-VI I-VIII II Roskam Roskam Aviati Aviation on and and engin engineeri eering,1 ng,1990 990.. 3.9 Anderson, Jr, J.D. ‘Hypersonic and high tempera temperatur ture e gas dynami dynamics’ cs’ McGraw McGraw Hill Hill 1989. 1989. 102
3.10 3.10 Prandtl Prandtl , L. ‘Essent ‘Essential ials s of fluid fluid dynami dynamics’ cs’ Blackie and sons , London 1967. 3.11 3.11 Kaufmann Kaufmann,, W. ‘Fluid ‘Fluid mecha mechanic nics s ‘ McGraw McGraw Hill Hill , New York, 1963. 3.12 3.12 Streete Streeterr , R.L.(Edi R.L.(Editor) tor) ‘ Handboo Handbook k of fluid fluid dynamics ‘, McGraw Hill , New York, 1961. 3.13 3.13 Sch Schli lich chti ting ng , H. H. and and Gerst Gersten en , K. K. ‘Bou ‘Bound ndary ary layer th theory ory’ 8th Edition, Spinger-Verlag, Berlin Heidelberg, 2000.
103
Exercises:
3.1 Following data relate to a light airplane: W =11000 N, Wing: S = 15 m2, CD0 =0.007, a = 4.9/rad A.R. = 6.5, taper ratio (λ) = 1.0, e = 0.9. Fuselage :Has a drag of 136N at V = 160 km/hr at sea level. Hori orizon onttal ttai aill: CD0=0.006,at = 4.1/rad,St=2.4 m2 Vertical tail:
CD0
= 0.006, Sv = 2.1 m2
Other components:CD0 based on wing area = 0.003 104
Estimate the drag polar of the airplane assuming the contribution of the fuselage to the lift dependent drag as small. [ An sw er :
CD = 0 .0 . 0 1 9 3 + 0 .0 . 0 5 4 4 CL 2 ]
3.2 A drag polar is given as: CD=CD0 + KCLn Show that: CLmd = {
(CL /CD)max =
C D 0 K ( n − 1)
}
1 / n
, CDmd = n
1 n n − 1
(n
{ −
1 ) 1 / n
n −
1
C D 0
1 C
( n −1) / n D 0
K
1/ n
} 105
Verify that when n=2, the above expressions reduced to those given by eqs. (3.17),(3.18) and (3.19). 3.3 The drag polar of a hypersonic glider can be given as follows (Ref.1.1, chapter 6) CL
CD
0
0.028
0.05
0.0364
0.1
0.05
0.15
0.07
0.2
0.0907 106
Fit Eq.(3.15) to this data and obtain CD0 and K. Also obtain CLmd, CDmd and (CL /CD)max. You may use expression mentioned in exercise 3.2. [Answer: CD 0 = 0 .0 . 0 2 8 , K = 0 .7 .7 0 1 ; CD = 0 .0 . 0 2 8 + 0 .7 . 7 0 1 CL 3 / 2 CL m d = 0 .1 . 1 8 8 5 , CD m d = 0 .0 .0 8 4 , ( CL / CD ) m a x = 2 .2 .2 1 ]
107
