Chapter 3 Lecture 7. Drag polar 2. Topics. Chapter-3

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Tobias Roy Daniels
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1 hapter 3 eture 7 Drag polar Topis 3..3 Summary of lift oeffiient, drag oeffiient, pithing moment oeffiient, entre of pressure and aerodynami entre of an airfoil 3..4 Examples of pressure oeffiient distributions 3..5 Introdution to boundary layer theory 3..6 Boundary layer over a flat plate height of boundary layer, displaement thikness and skin frition drag 3..3 Summary of the lift oeffiient, drag oeffiient, pressure oeffiient, pithing moment oeffiient, entre of pressure and aerodynami entre of an airfoil In order to understand the dependene of pressure drag and skin frition drag on various fators, it is appropriate, at this stage, to present brief disussions on (I) generation of lift, drag and pithing moment from the distributions of pressure (p) and shear stress ( ) and (II) outline of boundary layer theory. These and the related topis are overed in this subsetion and in the subsetions 3..4 to In subsetions to the airfoil harateristis and their nomenlature are dealt with. Subsequently, the estimation of the drags of wing, fuselage and the entire airplane at subsoni speeds are disussed(setions to 3..1). Figure 3.7 shows an airfoil at an angle of attak ( α)kept in a stream of veloity V. The resultant aerodynami fore (R) is produed due to the distributions of the shear stress( ) and the pressure (p). The distributions also produe a pithing moment (M). By definition, the omponent of R perpendiular to the free stream diretion is alled lift () and the omponent along the free stream diretion is alled drag (D). The resulant aerodynami fore (R) an also Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 1

2 be resolved along and perpendiular to the hord of the airfoil. These omponents an be denoted by and N respetively(fig.3.7). From the subsequent disussion in this setion, it will be evident that it is more onvenient to evaluate N and from the distributions of shear stress ( ) and pressure (p) and then evaluate and D. Fig.3.7 Aerodynami fores and moment on an airfoil From Fig.3.7 it an be dedued that : =Nosα-sinα (3.7) D=Nsinα+osα (3.8) Figure 3.8 shows elements of length and ds u and ds l at points P u and P l on the upper and lower surfaes of the airfoil respetively. The artesian oordinates of points P u and P l are (x u,y u ) and ( x l,y l ) respetively. Whereas s u and s l are respetively the distanes along the airfoil surfae, of the points P u and P l measured from the stagnation point (Fig.3.8). Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras

3 Fig.3.8 Pressure and shear stress at typial points on upper and lower surfaes of an airfoil To obtain the fores at points P u and P l, the loal values of p and are multiplied by the loal area. Sine the flow past an airfoil is treated as twodimensional, the span of the airfoil an be taken as unity without loss of generality. Hene, the loal area is (ds x 1) and the quantities,, D, N and, on the airfoil, are the fores per unit span. Keeping these in mind, the loal ontributions, dn u and d u, to N and respetively, from the element at point P u are obtained as: dn u = -pudsuosθu - udsusinθ u (3.9) d = -p ds sinθ + ds osθ (3.10) u u u u u u u Note that the suffix u denotes quantities at point P u and the positive diretion of the angle θu is as shown in Fig.3.8. Expressions similar to Eqs.(3.9) and (3.10) an be written down for the ontributions to N and from element at point P l. Integrating over the entire airfoil yields : u u u u u l l l l l (3.11) upper surfae lower surfae N=- p osθ + sinθ ds + p osθ - sinθ ds u u u u u l l l l l (3.1) upper surfae lower surfae = -p sinθ + osθ ds + p sinθ + osθ ds Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 3

4 Proeeding in a similar manner, it an be shown that M le, the pithing moment about leading edge of the airfoil, per unit span, is : M = p osθ + sinθ x - p sinθ - osθ y ds le u u u u u u u u u u u upper surfae lower surfae + -plosθ l+ l sinθl x+psinθ l l l+ losθl yl dsl (3.13) Note: One N and are known, the lift per unit span () and drag per unit span (D) of the airfoil an be obtained using Eqs.(3.7) and (3.8). It is onvenient to work in terms of lift oeffiient ( l ) and drag oeffiient ( d ). The definitions of these may be realled as : and l = 1 ρv D d = 1ρV It may be pointed out, that integration of a onstant pressure, say p, around the body would not give any resultant fore i.e. (3.14) (3.15) pds=0 (3.16) Hene, instead of p the quantity p-p an be used in Eqs.(3.11), (3.1) and (3.13). At this stage the following quantities are also defined. pressure oeffiient : skin frition drag oeffiient : p-p p = 1ρV f = 1ρV (3.17) (3.18) Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 4

5 N Normalforeoeffiient: n = 1ρV hordwiseor axialforeoeffiient: = (3.19) 1 ρv M le Pithingmomentoeffiient: mle = 1ρ V It may be noted that dx = ds os θ and dy = -ds sin θ, where ds is an elemental length around a point P on the surfae and θ is the angle between the normal to the element and the vertial (Fig.3.8). Note that θ is measured positive in the lokwise sense. It an be shown that : 1 n = p l -pu dx+ fu dy+ fl dy 0 upper surfae lower surfae 1 = fu+fldx + pudy - pl dy 0 upper surfae lowersurfae (3.0) Following setion 10. of Ref.1.4, the expressions for n, and mle an be rewritten as: 1 dy u dy l n = pl -pudx+ fu +fl dx dx dx dy u dyl = pu -p dx + fu-f dx l l 0 dx dx 0 1 dy u dyl mle = pu -plxdx- fu +fl xdx 0 0 dx dx 1 dyu dy l + pu +fu yu dx+ -pl + fl y l dx 0 dx 0 dx (3.1) Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 5

6 Remarks: (i) From n and the lift oeffiient ( l ) and drag oeffiient ( d ) are obtained as : = osα - sinα l n (3. ) d = nsinα +osα (3.3) (ii) entre of pressure : The point on the airfoil hord through whih the resultant aerodynami fore passes is the entre of pressure. The aerodynami moment about this point is zero. It may be noted that the loation of entre of pressure depends on the angle of attak or the lift oeffiient. (iii) Aerodynami entre: As the loation of the entre of pressure depends on lift oeffiient ( l ) the pithing moment oeffiient about leading edge ( mle ) also hanges with l. However, it is found that there is a point on the airfoil hord about whih the pithing moment oeffiient is independent of the lift oeffiient. This point is alled Aerodynami entre. For inompressible flow this point is lose to the quarter hord point of the airfoil. (iv) If the distributions of p and f are obtained by analytial or omputational methods, then the pressure drag oeffiient ( dp ) and the skin frition drag oeffiient( df ) an be evaluated. In experimental work the pressure distribution on an airfoil at different angles of attak an be easily measured. However, measurement of shear stress on an airfoil surfae is diffiult.the profile drag oeffiient ( d ) of airfoil, whih is the sum of pressure drag oeffiient and skin frition drag oeffiient, is measured in experiments by Wake survey tehnique whih is desribed in hapter 9, setion f of Ref In this tehnique, the momentum loss due to the presene of the airfoil is alulated and equated to the drag (refer setion of Ref.3.11 for derivation) Examples of pressure oeffiient distributions Though the expression for lift oeffiient ( l ) involves both the pressure oeffiient ( p ) and the skin frition drag oeffiient ( f ), the ontribution of the Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 6

7 former i.e. p is predominant to deide l. On the other hand, the pressure drag oeffiient ( dp ) is determined by the distribution of p and the skin frition drag oeffiient ( df ) is deided by the distribution of shear stress. In this subsetion the distributions of P in typial ases and their impliations for l and dp are disussed. The distribution of the pressure oeffiient is generally plotted on the outer side of the surfae of the body (Fig.3.9a). The length of the arrow indiates the magnitude of p. As regards the sign onvention, an arrow pointing towards the surfae indiates that p is positive or loal pressure is more than the free stream pressure p. An arrow pointing away from the surfae indiates that p is negative i.e. the loal pressure is lower than p. (a) Ideal fluid flow (b) Real fluid flow Fig.3.9 Distribution of p around a irular ylinder Figure 3.9 shows distributions of p in ideal fluid flow and real fluid flow past a irular ylinder. It may be realled that an ideal fluid is invisid and inompressible whereas a real fluid is visous and ompressible. From the distribution of p in ideal fluid flow (Fig.3.9a) it is seen that the distribution is symmetri about X-axis and Y-axis. It is evident that in this ase, the net fores in vertial and horizontal diretions are zero. This results in l = 0, dp = 0. These results are available in books on fluid mehanis and aerodynamis. In the real Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 7

8 fluid flow ase, shown in Fig.3.9b, it is seen that the flow separates from the body (see desription on boundary layer separation in setion 3..7) and the pressure oeffiient behind the ylinder is negative and nearly onstant. However, the distribution is still symmetri about horizontal axis. Thus in this ase l = 0 but dp > 0. The distributions of p over symmetrial and unsymmetrial foils at l = 0 and l > 0 are shown in Figs.3.10 a to d. Note also the loations of entre pressure and the prodution of pithing moment for the unsymmetrial airfoil. Flow visualization pitures at three angles of attak(α) are shown in Figs.3.36 a, b and. An attahed flow is seen at low angle of attak. Some separated flow is seen at moderate angle of attak and large separated flow region is seen near α lose to the stalling angle ( α stall ). It may be pointed out that theoretial alulation of skin frition drag using boundary layer theory an be done, when flow is attahed. This topi is disussed in the next subsetion. (a)distribution of pressure oeffiient on symmetrial airfoil at l = 0 and α= 0 Note : u = l Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 8

9 (b) Distribution of pressure oeffiient on symmetrial airfoil at l > 0 and α> 0 () Distribution of pressure oeffiient on ambered airfoil at l = 0, α< 0 ; Note: u and l form a ouple; entre of pressure is at infinity, ma < 0, Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 9

10 (d) Distribution of pressure oeffiient on ambered airfoil at l > 0, α> 0 Note : ma same as in Fig.() Fig.3.10 Distributions of pressure oeffiient on symmetrial and unsymmetri airfoils at l = 0 and l > Introdution to boundary layer theory Under onditions of normal temperature and pressure a fluid satisfies the No slip ondition i.e. on the surfae of a solid body the relative veloity between the fluid and the solid wall is zero. Thus, when the body is at rest the veloity of the fluid layer on the body is zero. In this and the subsequent subsetions, the body is onsidered to be at rest and the fluid moving past it. Though the veloity is zero at the surfae, a veloity of the order of free stream veloity is reahed in a very thin layer alled Boundary layer. The veloity gradient normal to the surfae U is very high in the boundary layer. Hene even if the oeffiient of y U visosityμ is small, the shear stress, μ,in the boundary layer may be y large or omparable to other stresses like pressure. Outside the boundary layer Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 10

11 the gradient U/ y is very small and visous stress an be ignored and flow treated as invisid. It may be realled from text books on fluid mehanis, that in an invisid flow the Bernoulli s equation is valid. Features of the boundary layer over the surfae of a streamlined body are shown in Fig.3.11a. On the surfae of a bluff body the boundary layer develops upto a ertain extent and then separates (Fig.3.11b). The definitions of the streamlined body and bluff body are presented at the end of this subsetion. (a) boundary layer over a streamlined body (b) Boundary layer over a bluff body Fig.3.11 Boundary layer over different shapes (not to sale) Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 11

12 The features of the flow are as follows. 1.Near the leading edge (or the nose) of the body the flow is brought to rest.this point is alled the Stagnation point. A laminar boundary layer develops on the surfae starting from that point. It may be realled, from topis on fluid mehanis, that in a steady laminar flow the fluid partiles move downstream in smooth and regular trajetories; the streamlines are invariant and the fluid properties like veloity, pressure and temperature at a point remain the same with time. In an unsteady laminar flow the fluid properties at a point may vary but are known funtions of time. In a turbulent flow, on the other hand, the fluid properties at a point are random funtions of time. However, the motion is organized in suh a way that statistial averages an be taken. In a laminar boundary layer the parameter whih mainly influene its development is the Reynolds number R =ρu x/μ ; x being distane along the surfae, from the x e stagnation point..depending on the Reynolds number (R X ), the pressure gradient and other parameters, the boundary layer may separate or beome turbulent after undergoing transition. The turbulent boundary layer may ontinue till the trailing edge of the body (Fig.3.11a) or may separate from the surfae of the body (point S in Fig 3.11b). It may be added that the stati pressure aross the boundary layer at a station x, is nearly onstant with y. Hene the pressure gradient referred here is the gradient (dp/dx) in the flow outside the boundary layer. 3.Nature of boundary layer deides the drag and the heat transfer from the body. If the boundary layer is separated, the pressure in the rear portion of the body does not reah the freestream value resulting in a large pressure drag (Fig.3.9b). Inidently a streamlined body is one in whih the major portion of drag is skin frition drag. For a bluff body the major portion of drag is pressure drag. A irular ylinder is a bluff body. An airfoil at low angle of attak is a streamlined shape. But, an airfoil at high angle of attak like α stall is a bluff body. Remark: General disussion on boundary layer is a speialised topi and the interested reader may onsult Ref.3.11 for more information. Here, the features Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 1

13 of the laminar and turbulent boundary layers on a flat plate are briefly desribed. While disussing separation, the boundary layer over a urved surfae is onsidered aminar boundary layer over flat plate height of boundary layer, displaement thikness and skin frition drag The equations of motion governing the flow of a visous fluid are alled Navier-Stokes (N-S) equations. For derivation of these equations refer to hapter 15 of Ref.3.1. Taking into aount the thinness of the boundary layer, Prandtl simplified the N-S equations in These equations are alled Boundary layer equations (hapter 16 of Ref.3.1). Solution of these equations, for laminar boundary layer over a flat plate with uniform external stream, was obtained by Blasius in Subsequently many others obtained the solution. The numerial solution by Howarth, presented in Ref.3.10, hapter 7, is given in Table 3.. In this table U is the loal veloity, U e is the external veloity (whih in this partiular ase is V ), and η is the non-dimensional distane from the wall defined as : η =y U e x (3.4) Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 13

14 Table 3. Non-dimensional veloity profile in a laminar boundary layer over a flat plate Height of boundary layer It is seen from table 3. that the external veloity (U e ) is attained very gradually. Hene the height at whih U/U e equals 0.99 is taken as the height of the boundary layer and denoted by δ From table 3., U/Ue 0.99 is attained at η =5. Noting the definition of η in Eq.(3.4) gives : 5=δ 0.99 Ue x δ 5 5 Ux x U x R 0.99 e Or = = ;R x = e x (3.5) Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 14

15 Figure 3.1 shows a typial non-dimensional veloity profile in a laminar boundary layer. While presenting suh a profile, it is a ommon pratie to plot U/U e on the absissa and ( y/δ 0.99 ) on the ordinate. Fig.3.1 Non-dimensional veloity profile in laminar and turbulent boundary layers on a flat plate It is seen from Eq.(3.5) that δ 0.99 grows in proportion to 1 x (see Fig.3.13). It may be added that in this speial ase of laminar boundary layer on flat plate, the veloity profiles are similar at various stations i.e. the non-dimensional profiles of U/U e vs (y/ δ 0.99 ) are same at all stations. Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 15

16 Fig.3.13 Shemati growth of boundary layer Displaement thikness and skin frition drag oeffiient The presene of boundary layer auses displaement of fluid and skin frition drag. The displaement thikness δ is defined as : U δ 1 = 1- dy U (3.6) 0 e The loal skin frition oeffiient ( or f )is defined as : u = = ; =μ ;Note: is a funtion of 'x'. wall f f wall wall 1 ρu y y=0 e f 1 (3.7) If the length of the plate is, then the skin frition drag per unit span of the plate (D f ) is : D = f 0 wall dx Hene, skin frition drag oeffiient df is given by: D f df = 1ρV (3.8) From the boundary layer profile (table 3.) it an be shown that for a flat plate of length,, the expressions for δ1 and df are: δ = R (3.9) Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 16

17 1.38 df = ; R = R V (3.30) Remark : Referene 1.11, hapter 6 may be onsulted for additional boundary layer parameters like momentum thikness ( δ ), shape parameter (H = δ 1/ δ ) and energy thikness ( δ 3 ) of a boundary layer. Example 3.1 onsider a flat plate of length 500 mm kept in an air stream of veloity 15 m/s. Obtain (a) the boundary layer thikness δ and the displaement thikness δ at the end of the plate (b) the skin drag oeffiient. Assume and the boundary layer to be laminar. Solution: -6 = 0.5 m, V = 15 m/s, =15 10 m /s -6 =15 10 m /s Hene, R = = onsequently, from Eq.(3.5): Or δ 0.99 δ = = = R = = m = 3.54mm From Eq.(3.9): Or δ = = = R δ 1 = = m = 1.17 mm From Eq.(3.30): Remark: δ df = = = R / is found to be 7.07 x Hene the assumption of the thinness of boundary layer is onfirmed by the results. Dept. of Aerospae Engg., Indian Institute of Tehnology, Madras 17

Chapter 2 Lecture 5 Longitudinal stick fixed static stability and control 2 Topics

Chapter 2 Lecture 5 Longitudinal stick fixed static stability and control 2 Topics hapter 2 eture 5 ongitudinal stik fied stati stability and ontrol 2 Topis 2.2 mg and mα as sum of the ontributions of various omponent 2.3 ontributions of ing to mg and mα 2.3.1 orretion to mα for effets