AERODYNAMICS I LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTING-LINE THEORY

Transcription

1 LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTING-LINE THEORY

2 The Bot-Savart Law The velocty nduced by the sngular vortex lne wth the crculaton can be determned by means of the Bot- Savart formula υx ( ) dl( xξ) 4 3 VL x ξ Specal case nducton of the straght vortex lne: dl de, x ξ ( x ) e ye x x y dl( x ξ) y dex ex y de z 3 x ξ ( x) y 3/

3 From the Bot-Savart formula one gets AERODYNAMICS I where υ( x, y,) y 4 [( x ) y ] 3 x y d e y s ( x) y s d ds 3 3 ds d / y y ( s ) y x y s [( ) ] x y x x y ( x ) y ( x ) y z x y x y Case nducton of the nfnte vortex lne (equvalent to the D pont vortex!) x x υ( xy,,) lm e 4 z e y ( x ) y ( x ) y y z

4 Case nducton of the sem-nfnte vortex lne segment [, ) x x x υ( xy,,) lm e 4 z e y 4 x y ( x ) y y x y z x then υ(, y,) If e 4 y z

5 Flow past a fnte-span wng physcal propertes

6 Lftng-lne model of a fnte-span wng Flow past a wng s modeled by the superposton of the unform free stream and the velocty nduced by a plane vortex sheet pretendng to be the cortex wave behnd the wng. The vortex sheet behnd the wng s woven from contnuum of nfntesmally weak horseshoe vortces. These vortces are attached to the lftng lne leadng to a contnuous dstrbuton of crculaton along the wng span.

7 The vortex sheet nduces vortcty all around. The dea s to calculate the calculate the velocty nduced by ths sheet on ts front edge,.e., along the lftng lne. Next, t s assumed that each nfntely thn slce of the wng generates the (dfferental) contrbuton to the total aerodynamc force as t were a two-dmensonal arfol. Each slce senses ts ndvdual drecton of free stream, whch results from the real free stream vector V and the vertcal (normal to the vortex sheet) velocty nduces at the lftng lne n the pont correspondng to the poston of the wng slce. Accordng to the Bot-Savart formula, the nfntesmal contrbuton to the velocty nduces along the lftng lne at the pont (, y,) s dw ( y) dy 4 ( y y) The total velocty nduces at ths pont s obtaned by ntegraton wy ( ) b/ ( y) dy 4 y y b/

8 Due to (generally) non-unform dstrbuton of the nduced velocty along the wng span, the effectve angle of attack has an ndvdual value of each wng secton see fgure below. The drecton of flow sensed by the wng secton at y y s rotated clockwse by the nduces angle ( y ) atan[ w( y ) V ] For small angles ( y ) b/ wy ( ) ( y) dy V 4V y y b/ Clearly, an effectve angle ( ). eff eff y

9 For small angles one can assume that the local lft coeffcent changes lnearly wth the (local) angle. Hence c ( y ) a [ ( y ) ( y )] L eff Here, a denotes the slope of the lft characterstcs for the wng secton, s the angle of attack correspondng to the zero lft. Note that a f the thn-arfol theory s used. Note also that n general the angle ( y ). Next, we assume that the spanwse densty of the lft force developed on the wng can be computed from the Kutta-Joukovsk formula, namely where L ( y ) V c ( y ) c( y ) V ( y ) L cy ( ) s local chord of the wng secton ( y ) ( y) V c( y ) Hence, the local lft coeffcent s c L

10 Assumng that a, the local effectve angle of attack s eff ( y ) ( y ) ( y ) V c( y) Fnally, the sum of the two local angles: ( y) and ( y) s equal to the geometrc eff angle of attack. If the wng has geometrc twst, ths angle also depends of the spanwse locaton,.e., ( y). Hence, we have obtaned the followng ntegro-dfferental equaton for the spanwse dstrbuton of the crculaton b/ y ( ) 4 b/ ( ) ( y) dy ( y) ( y) V c y V y y

11 One ths equaton s solved, then the spanwse dstrbuton of the crculaton s known. The lft force developed on the wng can be calculated as follows b/ b/ L L( y) dy V ( y) dy b/ b/ The (global) lft coeffcent s C L b/ L q S V S b/ ( y) dy The local contrbuton to the drag force s The nduced drag force s equal D Lsn L b/ b/ D L( y) ( y) dy V ( y) ( y) dy b/ b/

12 Thus, the coeffcent of the nduced drag s equal b/ D CD ( y) ( ) y dy q S V S b/ Important case - ellptcal dstrbuton of the crculaton Assume y ( y) y meanng that L ( y) V We have 4 y ( y) b 4 y / b b b Hence wy ( ) b/ b/ ( y) dy ydy 4 y b/ y b b/ ( y y) 4 y / b

13 Let us apply the followng change of coordnates Thus w( ) y bcos, dy bsn d cos d b cos cos b Concluson: for the ellptcal dstrbuton of the crculaton, the downwash velocty s constant! w V bv The nduced angle s The lft force b/ 4 b sn 4 b/ L V y b dy V d V b Thus, the maxmal crculaton s 4L Vb

14 On the other hand L V SC L Hence V SCL b and the nduced angle s V SCL SC bv b bv b L We defne the aspect rato of the wng b S Then, the alternatve form of the formula for the nduced angle for the ellptcal dstrbuton of vortcty s C L

15 The coeffcent of the nduced drag s calculated as follows b/ b b b CL V SC L ( ) sn b/ CD y dy d V S V S V S V S b Thus, we have obtaned the formula C D C L Consder the wng wth no geometrcal or aerodynamc twst. Then, both and are constant along the wng span. For the ellptcal load dstrbuton the angle s also constant, hence the effectve angle of attack and the lft coeffcent L c a ( ) are also constant along the wng span. eff eff Snce L( y) c q c( y) then L cy ( ) L( y) cq. L

16 Concluson: the spanwse varaton of the wng chord follows the varaton of the aerodynamc load. Hence, the planform of such wng s also ellptcal!

17 General lft dstrbuton Agan, we use the transformaton y bcos, [, ] The ellptc dstrbuton s expressed now as ( ) cos sn Generalzaton: ( ) V b An snn n We wll need the dervatve d d d d dy d dy dy V b nan cosn n The central equaton of the lftng-lne theory takes the form b cosn ( ) ( ) A snn na d n n c( ) n n cos cos

18 cos n sn n The Glauert ntegral appears agan d cos cos sn Hence, the man equaton s transformed to the algebrac form b ( ) ( ) sn sn n An n nan c( ) n n sn In order to fnd approxmate soluton, we frst truncate the nfnte seres b ( ) ( ) sn sn n N N L An n nan c( ) n n sn and make use of the collocaton method,.e., requre fulfllment of ths equaton at N dfferent pont m [, ], m,.., N. Ths way, the lnear algebrac system s obtaned for the unknown coeffcents { A, A,..., A }, whch can be solved, e.g., by the Gauss Elmnaton Method. N

19 Once ( ) s known, one can calculate all aerodynamc characterstcs of the wng. We have b/ N b CL ( y) dy An snn sn d V S S b/ We use the orthogonalty of the Fourer modes and obtan the formula n CL A b A S, n sn n snd, n We see that only the frst coeffcent of the Fourer seres s needed to calculate the lft force coeffcent!

20 The calculaton of the nduced drag s more complcated We have b/ N b CD ( y) ( ) ( )sn sn y dy An n d V S S b/ We need expresson for the nduced angle, namely b/ N N y dy n snn nan d na n b/ n n ( ) cos 4V y y cos cos sn n Hence, the formula for the C D can be transformed as follows N N b sn k sn k S k sn n N C ka A sn n d D b S ka A snk snn d k n kn, n

21 Agan, usng the orthogonalty property of the Fourer modes sn k sn n d, k m, k m the formula for the nduced drag coeffcent smplfes to the form b A CD na na A na A n S N N N N ( ) n n n n n n n n A We can wrte shortly N n n n A C D CL C ( ) L e A where and e ( ) (Oswald aerodynamc effcency parameter). Note that, hence C D ellptc wng C (optmalty!) D any wng

22 The most famous arplane wth the ellptcal wng: Trapezodal wngs are easer to construct and to buld. Theoretcally, they are nearly as good as ellptc ones f only the taper rato (.e., c / c ) s near the optmal value. In the wde range of aspect ratos, ( 4 ), the smallest values of are acheved when the taper rato s close to.3. Other factors, lke the stall pattern also matters! tp root

23 LEFT: Spanwse lft dstrbuton for trapezodal wng wth dfferent taper ratos. RIGHT: Stall patterns for dfferent planforms.

24 Reducton of the lft slope The fnte span not only leads to the appearance of the nduced drag t also changes (reduces) the slope of the lft vs angle of attack characterstc. Denote: dc a L - slope of the lft characterstc for the D wng secton (equvalent to ) d dc a L - slope of the lft characterstc for the 3D wng. d C a ( ) const L Due to appearance of the nduced angle, the 3D wng acheves the same value of the lft coeffcent at larger geometrc angle of attack see fgure. We have

25 C Hence, for the ellptc wng C ( L L a ) const Thus For other planforms dcl a d a a a a ( a )( ) Correcton factor typcally ranges from.5 to.5. The value of ths factor can be expressed by the coeffcents { A, A,..., A N}