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Chapter 3 Lecture 10 Drag Polar Flight dynamics-I Prof. E.G. Tulapurkara Chapter-3 Chapter 3 Lecture 10 Drag polar – 5 Topics 3.2.18 Parasite drag area and equivalent skin friction coefficient 3.2.19 A note on estimation of minimum drag coefficients of wings and bodies 3.2.20 Typical values of CDO, A, e and subsonic drag polar 3.2.21 Winglets and their effect on induced drag 3.3 Drag polar at high subsonic, transonic and supersonic speeds 3.3.1 Some aspects of supersonic flow - shock wave, expansion fan and bow shock 3.3.2 Drag at supersonic speeds 3.3.3 Transonic flow regime - critical Mach number and drag divergence Mach number of airfoils, wings and fuselage 3.2.18 Parasite drag area and equivalent skin friction coefficient As mentioned in remark (ii) of the previous subsection, the product C x S is Do called the parasite drag area. For a streamlined airplane the parasite drag is mostly skin friction drag. Further, the skin friction drag depends on the wetted area which is the area of surface in contact with the fluid. The wetted area of the entire airplane is denoted by S . In this background the term ‘Equivalent skin wet friction coefficient (Cfe)’ is defined as: C x S = C x S Do fe wet S S Hence, C C x and C= C wet (3.47) fe = Do DO f e Swet S Reference 3.9, Chapter 12 gives values of Cfe for different types of airplanes. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 1 Flight dynamics-I Prof. E.G. Tulapurkara Chapter-3 Example 3. 3 A quick estimate of the drag polar of a subsonic airplane is presented in this example which is based Ref.3.7, section 14.8. However, modifications have been incorporated, keeping in view the present treatment of the drag polar. 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 m2. Obtain a rough estimate of the drag polar in a flight at a speed of 430 kmph at sea level, when the landing gear is in retracted position. Solution: Flight speed is 430 kmph = 119.5 m/s. Average chord of wing( cwing ) = S / b = 51.22/20 = 2.566 m. The value of kinematic viscosity ( ) at sea level = 14.6× 10-6 m2 Reynolds number (Re) based on average chord is: 119.5× 2.566 =21×106 14.6× 10-6 It is assumed that NACA 23012 airfoil is used on the wing. From Ref.3.14, 6 Appendix IV, the minimum drag coefficient, (Cd)min, of this airfoil at Re = 9 x 10 is 0.006. However, the value of drag coefficient is required at Re = 21× 106 . 1 - 7 Assuming the flow to be turbulent (Cd)min can be taken proportional to Re (Eq. 6 3.35). Thus, Cdmin at Rew = 21 x 10 would roughly be equal to: 11 0.006 21 106677 / 9 10 =0.0053 As regards the fuselage and nacelle, the frontal areas are specified. Hence, they are treated as a bluff bodies. The value of (C ) can be taken as 0.08 Dmin fuselage (Ref.3.4). The nacelle generally has a lower fineness ratio and (C ) can be Dmin nac taken as 0.10. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 2 Flight dynamics-I Prof. E.G. Tulapurkara Chapter-3 As regards the horizontal and vertical tails, the Reynolds number based on their average chords (Retail) can be calculated if the areas and spans of these were given. The following is suggested to obtain a rough estimate of Retail. 2 SS18.6/29.3mht vt . Then cS tail tail cSwing wing Re c 9.3 and tail tail =0.426 Rew cwing 51.22 666 Hence, Re tail 21×10 ×0.426 = 8.95×10 9×10 At this Reynolds number (Cdmin)tail can be assumed to be 0.006 The calculation of the parasite drag coefficient (CDo) is presented in Table E 3.3. Part 2 C 2 S (m ) D C S (m ) D Wing 51.22 0.0053 0.271 Fuselage 3.72 0.080 0.298 Nacelles 3.25 0.1 0.325 Tail surfaces 18.6 0.006 0.112 Total 1.006 Table E3.3 Rough estimate of CDo Adding 10% for interference effects, the total parasite drag area (C S ) is: D 1.006 + 0.1006 = 1.1066 m2. Hence,- C = 1.1066/51.22 = 0.0216 D0 Wing aspect ratio is 202 / 51.22 = 7.8 To obtain Oswald efficiency factor for the airplane (e) , the quantities ewing, efuselage and eother are obtained below. Equations (3.42) and (3.43) give : Dept. of Aerospace Engg., Indian Institute of Technology, Madras 3 Flight dynamics-I Prof. E.G. Tulapurkara Chapter-3 1.1 CLαW /A e=Wing CLαW R+1-R A 2A C=LαW tan2Λ A22β 1 24 1+2 + 2 2 β Here A = 7.8, M = V/a = 119.5/340 = 0.351 Hence, β = 1-M22 = 1-0.351 = 0.936 For the purpose of calculating ewing, the taper ratio ( λ ), the quarter chord sweep ( 1 ) and the quantity , are taken as 0.4, 0 and 1 respectively. 4 o Consequently, ΛLE =3.14 2×7.8 -1 Hence, C=Lα =5.121rad 7.822 ×0.936 2+ +4 1 From Ref.3.14, chapter 6, the leading edge radius, as a fraction of chord, for NACA 23012 airfoil is : 2 2 1.109 t = 1.019 x 0.12 = 0.016 Rle = 0.016 x c = 0.016 x 2.566 = 0.041 m Reynolds number, based leading edge radius (ReLER ), is : 0.041×119.5 R= =3.35×105 eLER 14×10-6 2 52 Hence, RcoteLERΛ LE1-M cos Λ LE = 3.35×10 ×18.22× 1-0.351 ×0.998 = 57.16 x 105 Aλ 7.8×0.4 Further, ==3.13 cosΛLE 0.998 Dept. of Aerospace Engg., Indian Institute of Technology, Madras 4 Flight dynamics-I Prof. E.G. Tulapurkara Chapter-3 2 Corresponding to the above values of (RcoteLERΛ LE1-M cos Λ LE ) and Aλ ( ), Fig 3.14 of Ref.3.6, gives R = 0.95. cosΛLE Hence, 1.1× 5.121/7.8 e= =0.925 wing 0.95 × 5.121 / 7.8 + 0.05× To obtain efuselage , it is assumed that the fuselage has a round cross section. 1 In this case, Fig.2.5 of Ref 3.6 gives: / S/S= 0.75 when A = 7.8. fuselage efuselage Consequently, 1 = 0.75×3.72/51.22 = 0.054 efuselage 1 is recommended as 0.05(Ref.3.6, section 2.2) eothers 11 1 1 1 Thus, =+ + = +0.054+0.05 = 1.185 eewing e fuselage e other 0.925 Or e = 0.844 11 Hence, = = 0.0484 Ae ×7.8×0.844 Hence, a rough estimate of the drag polar is: 2 CD = 0.0216+0.0484CL 2 Answer: A rough estimate of the drag polar is :CD = 0.0216+0.0484CL Remark: i) A detailed estimation of the drag polar of Piper Cherokee airplane is presented in appendix A. 3.2.19 Note on estimation of minimum drag coefficients of wings and bodies Remark (ii) of section 3.2.17 mentions that the parasite drag coefficient of an airplane (C ) is given by : D0 Dept. of Aerospace Engg., Indian Institute of Technology, Madras 5 Flight dynamics-I Prof. E.G. Tulapurkara Chapter-3 1 C=D0 CSD +CDint S where the values of CD represent the minimum drag coefficients of various components of the airplane. In example 3.3 the minimum drag coefficients of wing, fuselage, nacelle, horizontal tail and vertical tail were estimated using experimental data. However, the minimum drag coefficients of shapes like the wing, the horizontal tail, the vertical tail and the streamlined bodies can be estimated using the background presented in subsections 3.2.5 to 3.2.10. The procedure for such estimation are available in Ref.3.6 which in turn is based on Ref.3.5. The basis of this procedure is that the minimum drag coefficient of a streamlined shape can be taken as the skin friction drag coefficient of a flat plate of appropriate characteristic length, roughness and area. With these aspects in view, the procedure to estimate the minimum drag coefficient of the wing can be summerised as follows. It is also illustrated in the sections on drag polar in Appendices A & B. (a) The reference length (l ) is the mean aerodynamic chord of the exposed wing i.e. the portion of wing outside the fuselage. This chord is denoted by c.e Obtain roughness parameter (l /k) with ce as ‘l ’ and value of k from Table 3.3. (b) The flow is assumed to be turbulent over the entire wing. (c)Choose the flight condition. Generally this is the cruising speed (Vcr) and the cruising altitude (hcr). Obtain the Reynolds number (Re) based on Vcr, kinematic cVecr viscosity cr at hcr and the reference length as ce i.e. Re = . cr Obtain (Re)cutoff corresponding to (l /k) using Fig.3.2 of Ref.3.6. Obtain Cdf corresponding to lower of Re and (Re)cut-off. Following Ref.3.6 this value is denoted by Cfw in Appendices A & B.
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