358 Aerodynamics for Engineering Students Fig. 6.47 and integrating gives after substituting for E; and E; Now, by geometry, and since EO is small, EO = 2(t/c), giving The lift/drag ratio is a maximum when, by division, D/L = a + [;(t/~)~l/a] is a minimum, and this occurs when Then 1 c 0.433 =- a [;I,,= - a2+a2 2a 4 t t/c For a 10% thick section (LID),, = 44 at a = 6.5" Moment coefficient and kcp Directly from previous work, i.e. taking the moment of SL about the leading edge: (6.157) Compressible flow 359 0.9 - - 4 8 12 16 20 24 Fig. 8.48 and the centre of pressure coefficient = -(CM/CL) = 0.5 as before. A series of results of tests on supersonic aerofoil sections published by A. Ferri* serve to compare with the theory. The set chosen here is for a symmetrical bi-convex aerofoil section of t/c = 0.1 set in an air flow of Mach number 2.13. The incidence was varied from -10" to 28" and also plotted on the graphs of Fig. 6.48 are the theoretical values of Eqns (6.156) and (6.157). Examination of Fig. 6.48 shows the close approximation of the theoretical values to the experimental results. The lift coefficient varies linearly with incidence but at some slightly smaller value than that predicted. No significant reduction in CL, as is common at high incidences in low-speed tests, was found even with incidence >20". The measured drag values are all slightly higher than predicted which is under- standable since the theory accounts for wave drag only. The difference between the two may be attributed to skin-friction drag or, more generally, to the presence of viscosity and the behaviour of the boundary layer. It is unwise, however, to expect the excellent agreement of these particular results to extend to more general aerofoil sections - or indeed to other Mach numbers for the same section, as severe limitations on the use of the theory appear at extreme Mach numbers. Nevertheless, these and other published data amply justify the continued use of the theory. * A. Ferri, Experimental results with aerofoils tested in the high-speed tunnel at Guidornia, Atti Guidornia, No. 17, September 1939. 360 Aerodynamics for Engineering Students General aerofoil section Retaining the major assumptions of the theory that aerofoil sections must be slender and sharp-edged permits the overall aerodynamic properties to be assessed as the sum of contributions due to thickness, camber and incidence. From previous sections it is known that the local pressure at any point on the surface is due to the magnitude and sense of the angular deflection of the flow from the free-stream direction. This deflection in turn can be resolved into components arising from the separate geometric quantities of the section, i.e. from the thickness, camber and chord incidence. The principle is shown figuratively in the sketch, Fig. 6.49, where the pressure p acting on the aerofoil at a point where the flow deflection from the free stream is E may be considered as the sum ofpt + pc $-pa. If, as is more convenient, the pressure coefficient is considered, care must be taken to evaluate the sum algebraically. With the notation shown in Fig. 6.49; CP = CPt + CPC + Ch (6.158) or (6.159) Lvt The lift coefficient due to the element of surface is SX -2 (Et+Ec+E,)- sc - C L-dFT which is made up of terms due to thickness, camber and incidence. On integrating round the surface of the aerofoil the contributions due to thickness and camber vanish leaving only that due to incidence. This can be easily shown by isolating the contribution due to camber, say, for the upper surface. From Eqn (6.148) D Symmetrical + \ section r- contributing thickness in Incidence contribution Fig. 6.49 Compressible flow 361 but ~c~cdx=~~(~)cdx=~cdyc= k];=O Therefore CLCamber = 0 Similar treatment of the lower surface gives the same result, as does consideration of the contribution to the lift due to the thickness. This result is also borne out by the values of CL found in the previous examples, Le. Now (upper surface) = -a and (lower surface) = +a 4a CL = m (6.160) Drug (wave) The drag coefficient due to the element of surface shown in Fig. 6.49 is which, on putting E = + + E~ etc., becomes On integrating this expression round the contour to find the overall drag, only the integration of the squared terms contributes, since integration of other products vanishes for the same reason as given above for the development leading to Eqn (6.160). Thus (6.161) Now 2 ~idx = 4a2c f 362 Aerodynamics for Engineering Students and for a particular section and 2 E:&= kcP2c ! so that for a given aerofoil profile the drag coefficient becomes in general (6.16 1 a) where t/c and P are the thickness chord ratio and camber, respectively, and kt, k, are geometric constants. Lift/wave drag ratio It follows from Eqns (6.160) and (6.161) that D kt(t/c)’ + ktP2 -=a+ 4a L which is a minimum when kt(t/C)2 + kcP2 4 a= Moment coefficient and centre of pressure coefjcient Once again the moment about the leading edge is generated from the normal contribution and for the general element of surface x from the leading edge 6cM=-( 2 )-&- x dx JmC c x dx CM = h?=-l cc Now is zero for the general symmetrical thickness, since the pressure distribution due to the section (which, by definition, is symmetrical about the chord) provides neither lift nor moment, i.e. the net lift at any chordwise station is zero. However, the effect of camber is not zero in general, although the overall lift is zero (since the integral of the slope is zero) and the influence of camber is to exert a pitching moment that is negative (nose down for positive camber), i.e. concave downwards. Thus Compressible flow 363 The centre of pressure coefficient follows from 1 and this is no longer independent of incidence, although it is still independent of Mach number. Aerofoil section made up of unequal circular arcs A convenient aerofoil section to consider as a first example is the biconvex aerofoil used by Stanton* in some early work on aerofoils at speeds near the speed of sound. In his experimental work he used a conventional, i.e. round-nosed, aerofoil (RAF 31a) in addition to the biconvex sharp-edged section at subsonic as well as supersonic speeds, but the only results used for comparison here will be those for the biconvex section at the supersonic speed M = 1.12. Example 6.11 Made up of two unequal circular arcs a profile has the dimensions shown in Fig. 6.50. The exercise here is to compare the values of lift, drag, moment and centre of pressure coefficients found by Stanton* with those predicted by Ackeret's theory. From the geometric data given, the tangent angles at leading and trailing edges are 16" = 0.28 radians and 7" = 0.12radians for upper and lower surfaces respectively. Then, measuring x from the leading edge, the local deflections from the free-stream direction are ~~~0.28 1-2- -a ( :> and &L=0.12 1-2- +a ( :> for upper and lower surfaces respectively. M = I .72 Fig. 6.50 Stanton's biconvex aerofoil section t/c = 0.1 * T.E. Stanton, A high-speed wind channel for tests on aerofoils, ARCR and M, 1130, January 1928. 364 Aerodynamics for Engineering Students L$t coefficient 4a CL =- drn For M = 1.72 CL = 2.86a Drag (wave) coefficient /' [ (0.28 (1 - $) - a)2+(0.12(1- 2q + a) 2 ] dx cD=cm 0 C (4aZ + 0.0619) m CD = For M = 1.72 (as in Stanton's case), Co = 2.86~~' + 0.044 Moment coefficient (about leading edge) or 2 CM, = dm [a + 0.02711 For M = 1.72 = 1.43~~ + 0.039 Centre-of-pressure coefficient -CM~ 2a + 0.054 ke= - CL 4a 0.5 + 0.0135 a kcp = L$t/drag ratio 4a L - m= a D 4a2 + 0.0619 CY' - 0.0155 dm This is a maximum when a = dm = 0.125rads. = 8.4" giving (LID), = 4.05. Compressible flow 365 0.4 c, 0.2 t// / &served Ca, /- 2.5 5.0 7.5 a0 Ob 1 I I L I / /I I I I 0 2.5 5.0 7.5 a0 1 0.15 0.05 I I I I o1 2.5 5.0 7.5 G llo 1, 0 I 2.5 5.0 7.5 I ao t I Incidence degrees 0 2.5 5.0 7.5 CL calculated observed CD calculated observed - CM calculated observed kCP calculated observed LID calculated observed 0 0.1 25 -0.064 0.096 0-044 0.0495 0.052 0.054 0.039 0.101 -0.002 0.068 m 0.81 0.03 0-69 0 2.5 -1.2 1 -8 0.25 0.203 0.066 0.070 0-1 64 0.1 14 0.65 0.54 3.8 2.9 0-375 0.342 0.093 0-093 0.226 0.1 78 0.60 0.49 4.0 3.5 Fig. 6.51 It will be noted again that the calculated and observed values are close in shape but the latter are lower in value, Fig. 6.51. The differences between theory and experiment are probably explained by the fact that viscous drag is neglected in the theory. Double wedge aerofoil section Example 6.12 Using Ackeret's theory obtain expressions for the lift and drag coefficients of the cambered double-wedge aerofoil shown in Fig. 6.52. Hence show that the minimum lift-drag ratio for the uncambered doublewedge aerofoil is fi times that for a cambered one with h = t/2. Sketch the flow patterns and pressure distributions around both aerofoils at the incidence for (L/D),,,ax. (u of L) 366 Aetudynamics for Engineering Students Fig. 6.52 Lift Previous work, Eqn (6.160) has shown that Drag (wave) From Eqn (6.161) on the general aerofoil Here, as before: 2 &;-=4azC f: For the given geometry i.e. one equal contribution from each of four flat surfaces, and Le. one equal contribution from each of four flat surfaces. Therefore Lift-drag ratio L Cr a I D=G= [ a’+ (a’ - +4 (3’1 - For the uncambered aerofoil h = 0: For the cambered section, given h = t/c: hprsssible flow 367 No camber Upper surface Rear f a= for [$Im c Lower surface Fig. 6.53 Flow patterns and pressure distributions around both aerofoils at incidence of [L/D],,, 6.8.4 Other aspects of supersonic wings The shock-expansion approximation The supersonic linearized theory has the advantage of giving relatively simple for- mulae for the aerodynamic characteristics of aerofoils. However, as shown below in Example 6.13 the exact pressure distribution can be readily found for a double-wedge aerofoil. Hence the coefficients of lift and drag can be obtained. Fixample 6.13 Consider a symmetrical double-wedge aerofoil at zero incidence, similar in shape to that in Fig. 6.44 above, except that the semi-wedge angle EO = 10". Sketch the wave pattern for M, = 2.0, calculate the Mach number and pressure on each face of the aerofoil, and hence determine Co. Compare the results with those obtained using the linear theory. Assume the free-stream stagnation pressure, porn = 1 bar. The wave pattern is sketched in Fig. 6.54a. The flow properties in the various regions can be obtained using isentropic flow and oblique shock tables.* In region 1 M = M, = 2.0 and ph = 1 bar. From the isentropic flow tables pol/pl = 7.83 leading to p1 = 0.1277 bar. In region 2 the oblique shock-wave tables give p2/p1 = 1.7084 (leading to p2 = 0.2182 bar), M2 = 1.6395 and shock angle = 39.33". Therefore (0.2182/0.1277) - 1 = 0.253 - - 0.5 x 1.4 x 22 e.g. E.L. Houghton and A.E. Brock, Tables for the Compressible Flow of Dry Air, 3rd Edn., Edward Arnold, 1975. [...]... tranform the independent variables so that =O =aJx'lLoJp From Eqns (7. 28a), (7. 32) and (7. 34) we obtain Thus if the form Eqn (7. 31) is to hold we must require ab = Uw and m = 1/2, therefore (7. 35) (7. 36) Substituting Eqn (7. 35) into (7. 33) gives (7. 37) 392 Aerodynamics for Engineering Students Likewise, if we replace $ by “lay obtain in Eqn (7. 33) and make use of Eqn (7. 36), we Similarly using Eqn (7. 34)... below EO f tan0 e n e 0.0 0.1 0.2 0.3 0.5 0 .7 0.8 0.9 1.0 0.2 0.16 0.12 0.08 0.0 -0 .08 -0 .12 -0 .16 -0 .20 11.31" 9.09" 6.84" 4. 57" 0.0 -4 . 57" -6 .84" -9 .09" -1 1.31" 2.22" 4. 47" 6 .74 " 11.31' 15.88" 18.15" 20.40" 22.62" 0" V M h 14.54" 16 .76 " 19.01' 21.28" 25.85" 30.42' 32.69" 34.94" 37. 16" 1.59 1.666 1 .74 2 1.820 1.983 2.153 2.240 2.330 2.421 4.193 4.695 5.265 5.930 7. 626 9.938 11.385 13.104 15.102 CP P 0.233... and skin-friction drag varies with ReL Using the order-of-magnitude estimate (7. 3) it can be seen that But, by definition, ReL = pU,L/p, so the above becomes (7. 26) It therefore follows from Eqns (7. 20) and (7. 25) that the relationships between the coefficients of skin-friction and skin-friction drag and Reynolds number are identical and given by 1 Cf cx - d E and CD,c c - 1 a (7. 27) Example 71 Some... Fig 7. 2 that the net streamwise force acting on a small fluid element within the boundary layer is 87 -& y ay ap ax &x When the pressure decreases (and, correspondingly, the velocity along the edge of the boundary layer increases) with passage along the surface the external pressure 380 Aerodynamics for Engineering Students 1. 0- - Flat plate Favourable pressure gradient 0. 8- 0. 6- '\ 0.4 - 0. 2- 0 -0 .2... 0.086 0. 075 0.065 WP)li7l 0.294 0.225 0.163 0.104 0.0008 -0 .0831 -0 .1166 -0 .1 474 -0 . 175 4 0.228 0.183 0.138 0.092 0 -0 .098 -0 .138 -0 .183 -0 .228 Wings of finite span When the component of the free-stream velocity perpendicular to the leading edge is greater than the local speed of sound the wing is said to have a supersonic leading edge In this case, as illustrated in Fig 6.56, there is two-dimensional... is involved, we can use Eqns (7. 11) and (7. 27) From Eqn (7. 1 1) =3 x 6 x 75 000 (=) 'I2 = 0 .73 5 mm = 73 5 pm and from Eqn (7. 27) But Df = i p U 2 S C ~ fSo, if we assume that skin-frictiondrag is the dominant type of drag and that it scales in the same way as the total drag, the prototype drag is given by = 0.00022N = 220 pN 7. 3.4 Solution of the boundary-layer equations for a flat plate There are a... total skin-friction force I ; on the surface under consideration This force is obtained by integrating the skin-friction stress over the surface For a two-dimensional flow, the force F per unit width of surface may be evaluated, with reference to Fig 7. 9, as follows The skin-fiction force per unit width on an elemental length (Sx) of surface is SF = rwSx Therefore the total skin-friction force per... (7. 34) and (7. 36) (7. 39) (7. 40) We now substitute Eqns (7. 36H7.40) into Eqn (7. 29) to obtain After cancelling like terms, this simplifies to Then cancelling common factors and rearranging leads to -+ -f-=O U d3f , d2f dn3 2va2 dn2 ‘ V (7. 41) ‘ =1, say As suggested, if we wish to obtain the simplest universal (i.e independent of the , values of U and v) form of Eqn (7. 41), we should set -= un r 1 implying... implying a = ha2 (7. 42) So that Eqn (7. 41) reduces to d3f d2f -+ f 0; d713 d.12 (7. 43) The boundary conditions (7. 15) become f =-= o df atV=O; f + l as q + o o (7. 44) d V The ordinary differential equation can be solved numerically for f The velocity profile df /dV thus obtained is plotted in Fig 7. 10 (see also Fig 7. 1 1) From this solution the various boundary-layer thicknesses given in Section 7. 3.2 can be... by writing u, du -= -ay- 6 u, 1 L E" 382 Aerodynamics for Engineering Students E D //////////////////////////////, B Fig 7. 7 Although this is plainly very rough, it does have the merit of remaining valid as the Reynolds number becomes very high This is recognized by using a special symbol for the rough approximation and writing For the more general case of a streamlined body (e.g Fig 7 I), we use x to . 0.0 0 .7 -0 .08 0.8 -0 .12 0.9 -0 .16 1.0 -0 .20 e 11.31" 9.09" 6.84" 4. 57& quot; 0.0 -4 . 57& quot; -6 .84" -9 .09" -1 1.31" ne 0" 2.22" 4. 47& quot;. 0.104 0.0008 -0 .0831 -0 .1166 -0 .1 474 -0 . 175 4 WP)li7l 0.228 0.183 0.138 0.092 0 -0 .098 -0 .138 -0 .183 -0 .228 Wings of finite span When the component of the free-stream velocity. 0. 070 2 bar. Thus = -0 .161 (0. 070 2/0.1 277 ) - 1 cp3= 0 .7 x 22 (Using the linear theory, Eqn (6.145) gives 2E -2 x (lO?T/180) c- = -0 .202) p 3-4 T5T= dK-i There is an oblique