UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Experimental investigation of the instability of a trailing vortex pair Eliason, Brent G 1974

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1974_A7 E45.pdf [ 6.58MB ]
Metadata
JSON: 831-1.0093233.json
JSON-LD: 831-1.0093233-ld.json
RDF/XML (Pretty): 831-1.0093233-rdf.xml
RDF/JSON: 831-1.0093233-rdf.json
Turtle: 831-1.0093233-turtle.txt
N-Triples: 831-1.0093233-rdf-ntriples.txt
Original Record: 831-1.0093233-source.json
Full Text
831-1.0093233-fulltext.txt
Citation
831-1.0093233.ris

Full Text

AN EXPERIMENTAL INVESTIGATION OF THE INSTABILITY OF A TRAILING VORTEX PAIR by BRENT G. ELIASON B.Sc. University of Alberta, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTMKOP APPLIED' SCIENCE in the Department t of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1974 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h C olumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada i i ABSTRACT The mutual i n s t a b i l i t y o f a t r a i l i n g vortex p a i r has been s t u d i e d i n the Department's l a r g e wind t u n n e l . The v o r t i c e s were v i s u a l i z e d using helium f i l l e d soap bubbles and the cores probed with a hot wire anemometer. Measurements were made to permit c a l c u l a t i o n of the wing l i f t , the c i r c u l a t i o n o f the t r a i l i n g v o r t i c e s , the vortex s e p a r a t i o n , the diameter of the vortex cores and the wavelength and plane o f o s c i l l a t i o n of the d i s t u r b e d v o r t i c e s . The r e s u l t s show t h a t the l i n e a r t h e o r i e s of Crow and Parks c l o s e l y p r e d i c t the i n i t i a l growth o f the i n s t a b i l i t y . i i i TABLE OF CONTENTS CHAPTER PAGE 1. INTRODUCTION . . . . . . . . 1 2. STABILITY THEORY ..... 3 3. EXPERIMENTAL SET-UP . . . . 10 4. RESULTS • 13 5. DISCUSSION . 19 6. CONCLUSION 21 REFERENCES 22 APPENDIX I - C o r r e c t i o n Theory . 24 APPENDIX II - C a l c u l a t i o n o f T h e o r e t i c a l Vortex S e p a r a t i o n 27 LIST OF FIGURES Figure Page 1. Geometrical quantities entering the analysis . . . . 30 2. Relation between an arc-length, dL, and a displacement, e dx, down the longitudinal axis ? •. 30 3. Unstable modes showing the plane of osc i l la t ion . . 31 4. Smoke used i n i t i a l l y to visualize a t ra i l ing vortex 32 5. Bubble head schematic 33 6. Schematic of the experimental set-up 34 7. L i f t and drag curves for model with flap spacings and deflections to produce maximum l i f t 35 8. Photo taken at test section exit looking upstream along bubble t r a i l s . The uniform sink rate produces the straight l ine of bubbles seen in the photo 36 9a. Overhead view of the flow over the wing . . . . . . 36 9b. Side view showing the formation of the t ra i l ing vortices 37 10. Typical anemometer traverse 32 chords downstream of wing showing two methods of defining core diameter. The traverse is perpendicular to the span and begins from the le f t of the figure . . . . 38 11. Typical anemometer traverse 4 chords downstream. The scale of the ordinate is different but the direction of traverse is the same as that in Figure 10 39 12. Effect on the core diameter of varying . . . . . 40 13. Theoretical non-dimensional amplification rates for the Crow theory for two d e f i n i t i o n s of core diameter. The S and A refer to the symmetric and antisymmetric modes, respectively . 41 V Figure Page 14. Theoretical non-dimensional amplification rates for the Parks theory for the same two definit ions of core diameter as in Figure 13 42 15. Photos from above, looking upstream, showing the filament ins tab i l i t y 43 16. Theoretical non-dimensional amplification rates for the symmetric mode for the two theories using two core diameter definit ions. The histo-gram shows the relative frequency of occurrence of 84 experimentally measured symmetric waveforms . . . 44 17. Photo from above looking upstream with exciter in operation 45 18. Photo taken at 45° to the horizontal; one bubble t r a i l appears as a straight l ine indicating that the plane of osc i l la t ion is close to 45° . . . . . . . 45 A l . The fractional decrease in wing l i f t with the dummy support at various spanwise positions . . . . 46 A2. L i f t correction for the flap configuration shown in Figure 7 46 A3. The tare and the effect of interference of the model on the support drag 47 A4. The fractional increase in wing drag with the dummy support at various spanwise positions . . . . 47 A5. Drag correction due to support interference for the flap configuration shown in Figure 7 48 ACKNOWLEDGEMENTS The author would l i k e to thank Dr. G.V. Parkinson and Dr. I.S. Gartshore f o r t h e i r a d v i c e and encouragement d u r i n g the course o f the r e s e a r c h . T h i s r e s e a r c h was supported by the Nati o n a l Research C o u n c i l o f Canada. NOMENCLATURE r a t e of displacement a m p l i f i c a t i o n wing aspect r a t i o o r i g i n a l vortex s e p a r a t i o n vortex core diameter wing drag c o e f f i c i e n t wing l i f t c o e f f i c i e n t c u t o f f d i s t a n c e i n Crow's s e l f - i n d u c t i o n i n t e g r a l , p r o p o r t i o n a l to c l o n g i t u d i n a l u n i t v e c t o r l a t e r a l u n i t v e c t o r v e r t i c a l u n i t v e c t o r l i n e a r i z e d anemometer output p e r t u r b a t i o n wavenumber, — l i n e increment d i r e c t e d along vortex wing l i f t r a d i a l d i s t a n c e from vortex c e n t e r displacement e i g e n v e c t o r with components (y,z) r a d i a l vortex displacement with components (y,z) s e p a r a t i o n between p o i n t s on v o r t i c e s wingspan p o s i t i o n o f unperturbed vortex n along e^ a x i s time time f o r d i s t u r b a n c e amplitude to grow by a f a c t o r time to l i n k i n g o f v o r t i c e s p e r t u r b a t i o n v e l o c i t y a t vortex with components (u,v,w) t o t a l induced v e l o c i t y , u - e z(rQ/2Tfb) a i r c r a f t speed c o o r d i n a t e along unperturbed vortex angle of a t t a c k non-dimensional a m p l i f i c a t i o n r a t e o f the symmetric mode, a non-dimensional wavenumber, k b k 2 K c i r c u l a t i o n around a vortex c i r c u l a t i o n a t mid-span o f the wing non-dimensional c u t - o f f d i s t a n c e , k d ra t e o f t u r b u l e n t d i s s i p a t i o n angle o f r from the h o r i z o n t a l wavelength o f the d i s t u r b a n c e d e n s i t y o f a i r f i r s t m u t u a l - i n d u c t i o n f u n c t i o n , .00 cos Bg d £ 0 ( g 2 + i ) 3 / 2 second m u t u a l - i n d u c t i o n f u n c t i o n , °° cos Bg+Bg s i n Bg ^ 0 (g 2 + 1 ) 3 / 2 Crow's s e l f - i n d u c t i o n f u n c t i o n , .oo cos g+g s i n g-1 ^ k C3 1 1. INTRODUCTION The problem o f wake t u r b u l e n c e , the p e r s i s t e n c e o f the p a i r o f t r a i l i n g v o r t i c e s behind l a r g e a i r c r a f t , has been r e c e i v i n g wide a t t e n t i o n i n the l a s t few y e a r s . S t a t i s t i c s r e v e a l ^ t h a t there have been about 160 a i r c r a f t a c c i d e n t s b e l i e v e d caused by wake tu r b u l e n c e i n the U n i t e d S t a t e s and Canada d u r i n g the past e i g h t y e a r s . The a n t i c i p a t e d c o n t i n u i n g growth o f a i r t r a n s p o r t w i l l a f f e c t the problem o f wake t u r b u l e n c e i n two ways: i n c r e a s e d a i r t r a f f i c d e n s i t y means a g r e a t e r p r o b a b i l i t y o f encountering wake turbulence and l a r g e r a i r -c r a f t w i t h h i g h e r weights w i l l generate more i n t e n s e and p e r s i s t e n t v o r t i c e s . There i s a c o n t i n u i n g need t h e n , f o r b e t t e r understanding o f the processes o f formation and d i s s i p a t i o n o f these t r a i l i n g v o r t i c e s . There a r e t h r e e b a s i c processes f o r the break-up o f a t r a i l i n g y ortex p a i r . Two of these, which may be c l o s e l y r e l a t e d to one another, a p p l y to a s i n g l e vortex and do not require; mutual e f f e c t s o f 2 the p a i r . • They are decay by t u r b u l e n t d i f f u s i o n and vortex 3 b u r s t i n g . The t h i r d process i s a mutual i n s t a b i l i t y o f the p a i r o f vortex f i l a m e n t s . T h i s i n s t a b i l i t y p r o c e s s , i n which the v o r t i c e s develop a s i n u s o i d a l waveform whose amplitude grows u n t i l the two v o r t i c e s l i n k and form vortex r i n g s , has been observed on numerous 4 o c c a s i o n s i n a c t u a l a i r c r a f t f l i g h t and i t presents an i n t e r e s t i n g p o s s i b i l i t y f o r hastening v o r t e x break-up. It i s the s u b j e c t o f the present i n v e s t i g a t i o n . 2 The fundamental t h e o r e t i c a l model o f the i n s t a b i l i t y process 5 was developed by Crow. He presented a l i n e a r s t a b i l i t y theory i n which the v o r t i c e s are i d e a l i z e d as p o t e n t i a l l i n e v o r t i c e s and the e f f e c t s o f the a c t u a l f i n i t e core diameter are accounted f o r by a c u t -o f f i n the i n t e g r a l r e p r e s e n t i n g the s e l f - i n d u c e d v e l o c i t y o f a fi l a m e n t . Subsequent refinements by o t h e r s 6 , 7 have accounted d i r e c t l y f o r the e f f e c t s o f f i n i t e c o r e s i z e , v o r t i c i t y d i s t r i b u t i o n and an a x i a l v e l o c i t y g r a d i e n t . However, experimental v e r i f i c a t i o n o f the theory has been slow i n appearing. F l i g h t t e s t s , such as those d e s c r i b e d i n Reference 4, have provided good q u a l i t a t i v e v e r i f i c a t i o n o f some o f the p r e d i c t e d phenomena, but such t e s t s a re expensive and d i f f i c u l t to make q u a n t i t a t i v e l a r g e l y because of u n p r e d i c t a b l e atmospheric e f f e c t s . Conventional wind tu n n e l s with s h o r t t e s t s e c t i o n s do not len d them-s e l v e s t o model s t u d i e s o f a t r a i l i n g v o r t e x system. There have been a few flow v i s u a l i z a t i o n model s t u d i e s , one using a p i s t o n generated vo r t e x p a i r i n a s p e c i a l water t a n k , 7 s e v e r a l using a w a t e r - f i l l e d towing t a n k ° ^ ' ^ and others towing a l i f t i n g wing through the a i r . ^ ' ^ The present i n v e s t i g a t i o n made use of the 80 f o o t long t e s t s e c t i o n o f the l a r g e wind tunnel o f the mechanical e n g i n e e r i n g department at the U n i v e r s i t y o f B r i t i s h Columbia. The t r a i l i n g v o r t i c e s of a l i f t i n g wing mounted a t the upstream end were v i s u a l i z e d and measure-ments made to permit c a l c u l a t i o n of the wing l i f t , the c i r c u l a t i o n o f the t r a i l i n g v o r t i c e s , the diameter of the vortex cores, : the vortex separation and the wavelength and plane of o s c i l l a t i o n of the disturbed v o r t i c e s . E f f e c t s o f a r t i f i c i a l e x c i t a t i o n o f the i n s t a b i l i t y were s t u d i e d . 3 2. STABILITY THEORY Crow idealized the t r a i l i n g vortex pair as a pair of in f in i te vortex l ines , using the Biot-Savart law to describe the kinematic relation between vort ic i ty and velocity: 2 r R x dL r m n m . . . .(1) m=l 4TT J Rmr, 1 3 ' mn' where n takes the values 1 and 2. Un i s the velocity at a point on the .th vortex in terms of the relative posit ion, R m n > the length, dLm , and the strength, r , of a l l the vortex elements in the flow. The geometry.of the model is i l lust rated in Figure 1. The vector distance from an element of vortex n, to another of vortex m, is given by: R = e (x'-x ) + e (s -s ) + ( r ' - r ) . . . .(2) mn x v m ir y m n' x m n' where the f i r s t two terms on the right involve the locations of the unperturbed vortex elements and the third term represents radial d i s -placement from their nominal positions. s m and s n are the positions of the undisturbed vortices along the e^ axis . The primes distinguish points lying on the same vortex in case m = n. r is given by: r n = V n ( V t } + V n < x n ' t } ' " ' " - ( 3 ) 4 where t is time. The relationship between dL^ and a displacement e dx is apparent from the detail sketch of Figure 2: A l l dL = (e + Sr /3x ) dx . . . . .(4) n v x n n' n r n and s n are given as follows: r 1 5 - r , = +r, s 1 = -b/2, s 2 = b/2. Because the vortices d r i f t downwards with a velocity 1727Tb, the coordinates of Figures 1 and 2 must be envisaged as moving downward at that rate as wel l . The vort ic i ty transport theorem ( i . e . , in an inviscid and neutrally buoyant f l u i d , elements of a vortex l ine move with f lu id par t ic les ) , provides the f inal equation describing the system dynamics: a r n/8t + u n ( 9 ? n / 8 x n ) where (u n ,v n ,w n ) are the components of U n + ez(r/2Trb) which is the velocity of convection with respect to the downward moving coordinates. The Crow model runs into d i f f i c u l t i e s , however, when the self-induced velocity term of Equation (1) is evaluated. This integral diverges logarithmically around |R | = 0 Crow overcomes this problem by cutting this integral off a certain distance, d, on either side of the point where li^ is being evaluated. This cut-off distance is proportional to core diameter. By considering small perturbations in the vortex positions, so that displacements remain small in comparison to the original vortex V n + ezwn .(5) separation, b, and their slopes remain small compared with 1: | r n | / b « l and [ 3 r n / 3 x n | « 1 , Crow l inearizes Equation (1): ~ 2 r r«» n m=l W Z (s -s ) dx' v m n' m » [(x' -x ) 2 + (s -s ) 2 ] 3 / 2  L V m n' v m n' J + e. <" ' s m - s n » 3 z : / ; " ' ; ' d x i ; - IK-*/* <v s„>2J3 / 2 + e. .oo [ (z ' - z ) -(x' -x ) ( 3 z ' / 9 x ' ) ] d x ' ' L V m rr v m n ; v nr rtr J m [(x' -x )2+ (s - s n ) 2 ] 3 / 2  L V m n y v m n y J + e. '°° ( 3 ( V S n } ( y m - y n } - t ( x ; - x n ) 2 + ( s m - s n ) 2 3 5 / 2 [ (y ; -y n)-(^-x n ) (ayy^)3 ^ x m " x n ) 2 + ( V s n ) ^ 3 / 2 ' m • -(6) where cut-offs in the integrals m = n are understood. Substituting this into a linearized Equation (5) y ie lds : 6 3r n 3t 2 I m=l 4TT i e y [ ( z ' - z )-(x'-x ) ( 8 z 7 3 x ' ) ] L V m i r v m n y v m m / J [ ( x : - x j 2 + ( s - s j 2 ] 3 / z dx' m m n m n' + e. 3<V sn> (-[ ( y ^ - y n ) - ( x ; - x n ) O y ; / a x ; ) ] m .(7) Equation (7) admits s o l u t i o n s f o r the displacement amplitude o f e x p o n e n t i a l form: r (x ,t) = r e n , and t h i s s u b s t i t u t i o n n n n leads to a s e t of a l g e b r a i c equations f o r the constant v e c t o r s r , which, w r i t t e n i n d i m e n s i o n l e s s form, i s : 2 ~ ay 1 = -z-j + 4iz 2 - 6 wZj 2 -az-] = -y-, + xy2 + & ay2 = z 2 - C*z-j + B WZ2 az. y 2 - xy-| - 6 w y 2 . . '.(8) 2rrb£ where a is the non-dimensional amplification rate, —y,— a , (a multiplied by the time i t takes the vortices to d r i f t downward a distance equal to their separation), 3 is the dimensionless wavenumber, kb, and X(B) = r°° cos 3? 0 (C+l) 11KB) f°° cos 3£+3£ sin 35 ? 5 ^ d£ = 3 K n(3) +3K, (3) 0 (c+ir^ 0 ' to («) °° cos sin C-1 d 5 (cos6-1) + sins _ c i ( 6 ) ] 6 is the dimensionless cut-off distance, kd, Ci(6) is the integral cosine and Kg(3) and K^(3) are modified Bessel functions of the second kind. Crow combines these eigenvectors into two independent modes: symmetric and antisymmetric (Figure 3). This results i n : c*s = [ ( l -^+3 2 co ) ( l+X-B 2 u)] 1 / 2 . . . .(9) a A = [ ( l+^+3 2 u ) ) ( l - X -3 2 u)) ] 1 / 2 • • • . (10) where the subscripts S and A refer to the symmetric and antisymmetric modes respectively. If a is real the disturbance is unstable and i f 8 i t is imaginary the vortices undergo neutrally stable osc i l la t ions . The disturbances osc i l late in a plane (Figure 3) whose angle to the horizontal, 9, is given by: Tan 9 S = [(l+x-e2o))/(l-^+32w)]1/2 Tan 9 A = [(l-x-32co)/(l+^+32w)]1/2 . In order to determine 6, Crow applied the l ine vortex model to two problems whose solutions are known by other means: that of a wave travel l ing around a colummar vortex and the speed of a vortex r ing. Assuming a uniform vort ic i ty distr ibution for these two cases he found the cut -off model agreed with the previous results providing that 6/8 = .321 c/b. This allows a to be calculated providing that the core diameter, c , is known, g Parks considered the effect of f i n i te core diameter and arbitrary vor t ic i ty distr ibution by assuming the vort ic i ty to be d is -tributed over the core. The velocity of a point near the vortex was calculated by using the Biot-Savart law and integrating over the core area. After l ineariz ing and assuming a uniform vort ic i ty d istr ibut ion, the self - induction function, w, became: W(B,Y) = \ Y 2 r°° cos vg+Y? sin yE.-~\ 2 • dC = ^ (I|>(Y)-U o o n 2 ) 3 7 2 Y r 2 where y = -|k and ty{y) = y KQ(y)+ yK^{y) as before. Solving the characteristic equation in the same manner as before yielded the non-dimensional amplification rates: 2 2 a s = [(1-4(0)+ K [*(Y)-1])(1+X(P)-- S2 [*(Y ) -13)] 1 / 2 Y Y 2 2 a A = [(1+^(6)+ K C*(Y)-i:>(l-x(3)- \ C * ( Y ) - 1 ] ) ] 1 / 2 Y Y where ty and x a r e a s given previously. In this case, as with Crow's analysis, a l l that is required is a knowledge of the core diameter in order to determine the wavelengths, X, and the orientation of the most unstable waves. 10 3. EXPERIMENTAL SET-UP Acc o r d i n g to these l i n e a r t h e o r i e s , the i n s t a b i l i t y grows e x p o n e n t i a l l y and the wave amplitude i n c r e a s e s by a f a c t o r e i n time, D2 to k Ta —p , or i n terms of a i r c r a f t parameters Ta p •' y - where b i s i L L v 0 somewhat l e s s than the wing span, r i s the c i r c u l a t i o n o f the i n d i v i d u a l v o r t i c e s , f& i s the wing aspect r a t i o , i s the wing l i f t c o e f f i c i e n t and V Q i s the a i r c r a f t speed. In order to hasten the growth o f the i n s t a b i l i t y i n the present experiment, was chosen as l a r g e as p o s s i b l e , the wing span, S, was chosen to be small and /R was chosen to be small but s t i l l not u n r e a l i s t i c . The a i r f o i l s e c t i o n chosen was an N.A.C.A. 23021 with 40 percent chord double s l o t t e d f l a p s . T h i s g i v e s a s e c t i o n a l l i f t c o e f f i c i e n t i n excess o f 3.4 a t high Reynolds 13 number. The span and chord were chosen to be 12 inches and 3 inches r e s p e c t i v e l y g i v i n g an aspect r a t i o o f 4. The tunnel t e s t s e c t i o n dimensions are 5'3" x 8' x 80* so wall c o r r e c t i o n s were unnecessary. The model endplates and templates were machined on an N.C. m i l l i n g machine and' the model was b u i l t o f mahogany. The wing was t e s t e d i n a s m a l l e r a e r o n a u t i c a l tunnel where l i f t and drag were measured f o r v a r i o u s f l a p angles and spacings. From these r e s u l t s the s e t t i n g s which gave maximum l i f t were chosen and used throughout the remainder of the t e s t s . The model was mounted from a s i n g l e f a i r e d support a t mid-span. It had been o r i g i n a l l y planned to v i s u a l i z e the v o r t i c e s with smoke. As a p r e l i m i n a r y t r i a l , v a r i o u s types o f smoke were i n j e c t e d through a p o r t i o n o f the blown f l a p o f a l a r g e wing mounted as a semi-wing from the tunnel f l o o r . As can be seen from the photos i n F i g u r e 4, the smoke d i f f u s e s q u i c k l y , making q u a n t i t a t i v e measurements d i f f i c u l t . I t was f e l t t h a t using smoke as a means o f flow v i s u a l i z -a t i o n f o r the s m a l l e r model would r e q u i r e a mass i n j e c t i o n r a t e so l a r g e t h a t the nature o f the wake i t s e l f might be changed. A helium f i l l e d soap bubble technique was then adopted as the means o f flow 14 v i s u a l i z a t i o n . The commercially a v a i l a b l e bubble generator produced approximately 500 bubbles per second from each head, v a r y i n g i n diameter from 1/32 i n c h to 1/8 i n c h . F i g u r e 5 shows a cut-away o f the head. The buoyancy, i n a i r , o f the bubbles ranged from s l i g h t l y n e gative to s l i g h t l y p o s i t i v e . Although the h e a v i e r bubbles tended to be thrown out o f the v o r t e x flow, the l i g h t e r ones g r a v i t a t e d to the c e n t e r s of the v o r t i c e s , r e n d e r i n g the cores v i s i b l e f o r long d i s t a n c e s . The bubbles were extremely d u r a b l e , w i t h s t a n d i n g high l e v e l s o f t u r b u l e n c e . The bubble t r a i l s , though v i s i b l e to the naked eye, proved i m p o s s i b l e to photograph without s p e c i a l l i g h t i n g . A p a i r o f 500 watt tungsten-halogen l i g h t s o u r c e s , each with a l a r g e plano-convex l e n s , were used to p r o v i d e high i n t e n s i t y l i g h t beams which c o u l d be shone down the t e s t s e c t i o n . A diagram o f the experimental set-up i s shown i n F i g u r e 6. A hot wire anemometer was used to t r a v e r s e the v o r t i c e s i n order to determine the core diameters a t two downstream p o s i t i o n s . The measured core diameters are used as an i n p u t to the s t a b i l i t y t h e o r i e s . A r t i f i c i a l d i s t u r b a n c e s a l s o could be c r e a t e d using an o s c i l l a t i n g panel mounted i n the r o o f j u s t downstream o f the model. The amplitude 12 as well as the frequency o f these d i s t u r b a n c e s could be v a r i e d . The t e s t s were run a t wind speeds o f 60 and 70 f e e t per second g i v i n g chord Reynolds numbers o f 1.24 x 10 5 and 1.44 x 10 5. 13 4. RESULTS The l i f t and drag curves f o r the model with optimum f l a p s pacing are shown i n F i g u r e 7. In o r d e r to s t a n d a r d i z e the performance o f the wing model, the curves i n F i g u r e 7 have been c o r r e c t e d to e l i m i n a t e the e f f e c t of the model support. At the o p e r a t i n g value o f C^, t h i s c o r r e c t i o n was l e s s than 2 percent. A summary o f the c o r r e c t i o n procedure i s given i n Appendix I. ^imax i s lower than t h a t p r e d i c t e d by the s e c t i o n c h a r a c t e r i s t i c s because of the f i n i t e /R and the o r d e r o f magnitude d i f f e r e n c e i n Reynolds number. The l i f t curve s l o p e , .069/°, compares f a v o u r a b l y with t h a t p r e d i c t e d t h e o r e t i c a l l y 15 f o r a r e c t a n g u l a r wing o f /R = 4 , .070/°. The sudden s t a l l c h a r a c t e r i s t i c made i t necessary to operate a t an angle o f a t t a c k 2° l e s s than t h a t o f s t a l l i . e . = 1.95. The t h e o r e t i c a l v a l u e of b f o r a r e c t a n g u l a r wing o f /R = 4 has been c a l c u l a t e d i n Appendix II and i s .854 S. The a c t u a l value o f the v o r t e x s e p a r a t i o n was measured p h o t o g r a p h i c a l l y and found to be .895 S qr 4.8 percent l a r g e r than the t h e o r e t i c a l v a l u e . The c i r c u l a t i o n , r ^ , a t midspan o f the wing c a l c u l a t e d from the measured 2 l i f t u s i n g b = .854 S, i s 17.1 f t / s e c . The a c t u a l c i r c u l a t i o n , r of the t r a i l i n g v o r t i c e s was measured by obs e r v i n g t h e i r r a t e o f downward mutual i n d u c t i o n . The v o r t i c e s had dropped 27 inches i n a d i s t a n c e o f 75 f e e t at a tunnel speed of 60 f p s . r as c a l c u l a t e d by 2 t h i s method was 9.7 f t /sec or 57 percent of TQ. T h i s measurement i s i n agreement with those o f other i n v e s t i g a t o r s ^ ' ^ 7 ' ^ ' ^ and 14 shows t h a t the measurement o f the s i n k r a t e provides a q u i c k and r e l i a b l e means of measuring c i r c u l a t i o n i n the l a b . The method has an advantage over v o r t i m e t e r or hot wire measurements i n t h a t there i s no probe modifying the flow. Once the v o r t i c e s were a few chords behind the wing, no change i n the s i n k r a t e was d e t e c t a b l e . T h i s o b s e r v a t i o n i m p l i e s t h a t the t o t a l amount o f v o r t i c i t y d i d not change a p p r e c i a b l y w i t h i n the l e n g t h o f the t e s t s e c t i o n even though the d i s t r i b u t i o n o f t h i s v o r t i c i t y may have changed. F i g u r e 8, a photo taken a t the t e s t s e c t i o n e x i t l o o k i n g upstream i n the plane of the bubble t r a i l s shows each vortex to be a s t r a i g h t l i n e thereby c o n f i r m i n g the uniform s i n k r a t e . At the t e s t s e c t i o n e x i t the d i s t a n c e from the v o r t i c e s to the tunnel f l o o r was approximately 1.8b. At t h i s p o i n t the ground e f f e c t induced v e l o c i t y l e s s than 10 percent o f the mutual i n d u c t i o n v e l o c i t y so i t i s f e l t t h a t ground e f f e c t s can be n e g l e c t e d . Photographs o f the bubble s t r e a k s immediately behind the wing, F i g u r e 9, show t h a t there i s a w e l l d e f i n e d v o r t e x present a t l e s s than one chord behind the t r a i l i n g edge. I t should be noted t h a t the core r e g i o n cannot be d e f i n e d by such flow v i s u a l i z a t i o n methods; the bubble p r o p e r t i e s , the time o f exposure and the bubble g e n e r a t i o n r a t e determine the diameter o f the well-marked r e g i o n i n the photos. The measurement o f the vortex core diameter, c, was accomplished by a l i g n i n g the hot wire a x i s with the mean flow. As the probe was t r a v e r s e d through t h e v o r t e x , t h e e f f e c t of t h e t a n g e n t i a l v e l o c i t y 15 component c o u l d be c l e a r l y seen on the anemometer output. F i g u r e 10 shows a t y p i c a l t r a v e r s e 32 chords downstream o f the wing mid chord. The t r a v e r s e begins beneath the vortex on the l e f t o f the f i g u r e and moves v e r t i c a l l y up through the v o r t e x . Although t h i s method cannot be r e l i e d upon to g i v e any q u a n t i t a t i v e i n f o r m a t i o n on the t a n g e n t i a l v e l o c i t y because o f the p o s s i b i l i t y o f v a r y i n g a x i a l v e l o c i t i e s i n the v o r t e x , i t does d e f i n e the core r e g i o n very a c c u r a t e l y . The e f f e c t o f probe contamination on the s i g n a l , although weak, was i n c r e a s i n g l y n o t i c e a b l e as the probe passed through the c e n t e r o f the c o r e . In o r d e r to ensure t h a t t h i s was not due to vortex asymmetry, the flow was probed from the o p p o s i t e s i d e . The r e s u l t s showed the v o r t e x to be symmetrical. A l s o i n c l u d e d i n F i g u r e 10 i s the product o f the l i n e a r i z e d anemometer output, E, and the r a d i a l d i s t a n c e , r . T h i s product i s r e l a t e d to the c i r c u l a t i o n contained w i t h i n a g i v e n r a d i u s and can be used i n d e f i n i n g the core diameter. One can d e f i n e the core diameter i n a number o f ways. One way would be to l e t c be the approximate mean diameter o f the t r a n s i t i o n zone from the core s o l i d - b o d y r o t a t i o n to the outer i r r o t a t i o n a l f i e l d , as measured from the peaks o f the v e l o c i t y curve. T h i s method, how-ever, n e g l e c t s the v o r t i c i t y d i s t r i b u t e d o u t s i d e t h i s r e g i o n . A second approach would d e f i n e the core diameter to be t h a t r e g i o n i n which a l l the measurable v o r t i c i t y l a y , as determined by the p o i n t s a t which E*r becomes constant. Both d e f i n i t i o n s are used and t h e i r e f f e c t s on the t h e o r e t i c a l p r e d i c t i o n s are compared. The core diameters were measured a t two downstream p o s i t i o n s , 4 chords and 32 chords as measured from mid-chord. A t y p i c a l t r a v e r s e 16 a t 4 chords i s shown i n F i g u r e 11. At t h i s p o s i t i o n the anemometer output does not vary i n v e r s e l y with r a d i u s . T h i s may be due to strong a x i a l v e l o c i t y g r a d i e n t s or incomplete r o l l - u p . At the 32 chord p o s i t i o n E does vary as ^ ( F i g u r e 10) implying a flow f i e l d much c l o s e r to the p o t e n t i a l vortex model. The flow was not probed a t g r e a t e r downstream d i s t a n c e s due to the unsteadiness o f the flow. The magnitude o f the t a n g e n t i a l v e l o c i t i e s had decreased c o n s i d e r a b l y a t the 32 chord p o s i t i o n and a h i g h e r anemometer g a i n was necessary to achieve the same output l e v e l . The l e v e l of t u r b u l e n c e had i n c r e a s e d c o n s i d e r a b l y a t the 32 chord p o s i t i o n as can be seen by comparing F i g u r e 10 and F i g u r e 11. The core diameter, based on peak v e l o c i t y , was 1.0 inch or ,098b. T h i s v a l u e i s o n l y h a l f t h a t p r e d i c t e d by S p r e i t e r and 20 Sachs, .197b, but i s g r e a t e r than the values found by o t h e r i n v e s t i -18 21 g a t o r s using wings with much lower l i f t c o e f f i c i e n t s . ' The e f f e c t on core diameter o f v a r y i n g i s shown i n F i g u r e 12. These r e s u l t s , from data s i m i l a r to F i g u r e 11, were taken a t the 4 chord p o s i t i o n and because they c o u l d o n l y be measured from peak to peak, and are a f f e c t e d by* probe contamination, they should be accepted o n l y as a q u a l i t a t i v e i n d i c a t i o n o f the e f f e c t o f v a r y i n g C^. From the 4 chord p o s i t i o n to the 32 chord p o s i t i o n , the core diameter had grown by 2.4 p e r c e n t . D e f i n i n g c as the diameter which i n c l u d e s 100 percent o f the measureable v o r t i c i t y , as determined by the p o i n t a t which E*r becomes c o n s t a n t , ( F i g u r e 10), g i v e s c = 2.1 inch or .195b. The measurement o f the core diameter provides an input to the s t a b i l i t y t h e o r i e s . 17 In his s tab i l i t y model Crow showed ^ = .321 ^ . Using the c 6 c measured value of r- = .098 then = .031, and with T-- .195 then b 3 b -g = .0625. Crow's expression for a<, and as a function of 3 is shown in Figure 13 for the above two values of core diameters. Park's expression for a<- and for the same two values of core diameter has been plotted in Figure 14. In each case, the wave which w i l l grow most quickly, has the wave-number, 3m , v> corresponding to the points at which the curve peaks. In the symmetric mode there are long waves and short waves which are unstable but in the antisymmetric mode only the short waves are unstable. The vort ices, made v is ib le by the bubbles, were photographed from above using 35 mm s t i l l and high speed 16 mm motion pictures, Figure 15. Measurements of the wavelengths were made by photographing a graduated string in the same position and from the same position as the vortices were photographed. Only 10 percent of these photos showed measureable disturbances, and only those waves which were clearly sinuous symmetric waves were measured. The wavehumbers of 84 such waves are shown in the histogram in Figure 16. Along with the histo-gram are plotted the a<. curves of Crow and Parks for the two definitions of core diameter. Varying the l i f t coefficient during the tests produced no noticeable effect on the measured wavelengths. There were generally fewer waves produced at lower values of C^. Exciting the vortices with the osc i l la t ing roof panel in the neighbourhood of 3 = .85 produced waves of much larger amplitude as shown in Figure 17. The exciter produced waves of larger amplitude because of the type of 18 turbulent energy added to the wake. When the exciter wavenumber is near 3 . this turbulent energy excites more strongly those waves max which grow most quickly, thus causing more rapid growth of the ins tab i l i t y . As the level of a r t i f i c i a l excitation was increased the flow visual ization became less effective,making measurements more d i f f i c u l t . The plane of osc i l la t ion of the vortices was measured with the exciter in operation in order to take advantage of the larger ampli-tude and more periodic nature of the waves. Figure 19 shows a photo taken at an angle of 45° to the horizontal showing one bubble t r a i l as a straight l ine . This and similar photos confirmed that the plane of osc i l la t ion lay between 40° and 50° to the horizontal. The result i s in good agreement with the 48° prediction of the l inear theory. 19 5. DISCUSSION In comparing the t h e o r e t i c a l and experimental r e s u l t s i n F i g u r e 17, i t should be kept i n mind t h a t the t h e o r e t i c a l o r d i n a t e s are a m p l i f i c a t i o n r a t e s , whereas the experimental histogram merely g i v e s the observed frequency o f occurrence of v a r i o u s wavenumbers. The c o n f i r m a t i o n o f the theory l i e s i n the f a c t t h a t i t p r e d i c t s the waves with maximum a m p l i f i c a t i o n r a t e s to be most probable. Comparing the two s t a b i l i t y t h e o r i e s shows t h a t Park's refinement to the Crow model g i v e s a s l i g h t l y b e t t e r f i t to the experimental data u s i n g the peak to peak d e f i n i t i o n of core diameter but F i g u r e 17 a l s o i n d i c a t e s t h a t each theory can p r e d i c t the r e s u l t s e q u a l l y w e l l by choosing an a p p r o p r i a t e core diameter. The c h o i c e o f core diameters would not be necessary i f , i n Park's t h e o r y , the a c t u a l v o r t i c i t y d i s t r i b u t i o n were used i n s t e a d o f the assumed uniform d i s t r i b u t i o n . However, the i n c r e a s e d mathematical d i f f i c u l t i e s o f c o n s i d e r i n g the a c t u a l d i s t r i -b u t i o n would not j u s t i f y the s l i g h t improvement to a l i n e a r theory. The wavelengths measured show e x c e l l e n t agreement wi t h the f u l l s c a l e 4 measurements o f C h e v a l i e r . None o f the s h o r t waves noted i n Reference 7 were observed i n the present t e s t program even though the theory p r e d i c t s the s h o r t waves to grow s l i g h t l y f a s t e r than the long waves. One p o s s i b l e reason f o r t h i s i s t h a t the t u r b u l e n t energy spectrum i n the tunnel may be such t h a t there i s a l a r g e r amount of energy a v a i l a b l e to e x c i t e the long waves. They would begin to grow f i r s t and thus dominate the i n s t a b i l i t y . The h a l f - a m p l i t u d e s o f the l a r g e s t waves w i t h o u t the e x c i t e r were o f the o rder o f ,2b a t the end o f the t e s t s e c t i o n . Assuming the wave w i l l c o n t i n u e to grow e x p o n e n t i a l l y and u s i n g a<* = . 8 1 , p r e d i c t e d by the P a r k ' s theory f o r jj-.= .195 and B = . 8 5 , one would expect the v o r t i c e s to l i n k 1 .8 second a f t e r they a re genera ted . I t i s i n t e r e s t -i n g to compare t h i s e x t r a p o l a t e d t ime w i t h a theory which p r e d i c t s l i n k up t i m e . 22 Bisgood e t a l _ . , on the b a s i s o f d imens iona l a n a l y s i s , p r o -pvQs3 posed the t ime to l i n k i n g , T^ = c ' — j - — . From t h e i r f u l l s c a l e o b s e r v a t i o n s o f v o r t e x l i n k i n g , they found the va lue o f c ' ^ 10. The t ime to l i n k - u p p r e d i c t e d by t h i s model f o r the p resent i n v e s t i g a t i o n 23 i s T^ = .68 second .Cond i t and Tracy u s i n g d i f f e r e n t f u l l s c a l e data found c ' = 14 and t h i s p r e d i c t s T^ =• .95 second f o r the p resent c a s e . One reason f o r t h i s i s t h a t the d imens iona l a n a l y s i s does not take i n t o c o n s i d e r a t i o n the i n t e n s i t y and s c a l e o f the atmospher ic t u r b u l e n c e which p r o v i d e s the i n i t i a l e x c i t a t i o n . D i f f e r e n t t u r b u l e n c e c o n d i t i o n s w i l l p rov ide d i f f e r e n t i n i t i a l ampl i tudes and hence d i f f e r e n t va lues 23 o f T^. Lissaman e t al_. have used e , the t u r b u l e n t r a t e o f d i s s i p a t i o n , to d e s c r i b e the e f f e c t s o f t u r b u l e n c e on l i n k - u p t i m e , e was not measured i n the p resent i n v e s t i g a t i o n but the t u r b u l e n t i n t e n s i t y i n the empty tunnel was low (<0.5 percent ) and t h i s may e x p l a i n the r e l a t i v e l y l a r g e va lues o f T^ a s s o c i a t e d w i t h the present i n v e s t i g a t i o n . 21 6. CONCLUSION In conclusion then, the measured wavelengths and plane of osc i l la t ion show that the l inear s tab i l i t y theory is adequate in describing the i n i t i a l growth of the filament ins tab i l i t y . Both the Crow theory and Park's modification to i t give equally good agreement with the experimental data depending on the definit ion of core diameter used in each case. Although l inking did not occur within the length of the test section, the measured amplitudes allowed the time to l inking to be estimated. This extrapolated time is consider-ably greater than the times predicted by dimensional analysis and f u l l scale observations. The effect of exciting the vortices at a frequency corresponding to maximum wave growth rate i s to cause larger amplitudes and thus decrease the time required for the vortices to l ink up. 22 REFERENCES 1. Bowie, W.R., Private Communication, Accident Investigation Divis ion, C i v i l Aeronautics Branch, Canadian Department of Transport, Oct. 1973. 2. Saffman, P.G., "Structure of Turbulent Line Vortices," The Physics of Fluids, Vol. 16, No. 8, Aug. 1973, pp. 1181-1188. 3. H a l l , M.G., "Vortex Breakdown," Annual Review of Fluid Mechanics, Vol . 4, 1972, pp. 195-218. 4. Chevalier, H., "Flight Test Studies of the Formation and D iss i -pation of Trai l ing Vortices," J . A i rc ra f t , Vol. 10, No. 1, 1973, pp. 14-18, 5. Crow, S .C . , "Stabi l i ty Theory for a Pair of Trai l ing Vortices," AIAA Journal, Vol. 8, No. 12, Dec. 1970, pp. 2172-2179. 6. Parks, P . C , "A New Look at the Dynamics of Vortices with Finite Cores," Aircraft Wake Turbulence and Its Detection, Olsen, J .H. et a l , ed . , Plenum Press, New York, 1971, pp. 355-388. 7. Widnall, S . E . , B l i s s , D., Zalay, A . , "Theoretical and Experimental Study of the Stabi l i ty of a Vortex Pa i r , " Ai rcraft Wake Turbulence and Its Detection, Olsen, J . H . , et a l . , ed. Plenum Press, New York, 1971, pp. 305-338. 8. Olsen, J . H . , "Results of Trai l ing Vortex Studies in a Towing Tank," Aircraft Wake Turbulence and Its Detection, Olsen, J . H . , et a l . , ed . , Plenum Press, New York, 1971, pp. 455-472. 9. Hackdtt, J . E . , Theisen, J . G . , "Vortex Wake Development and A i r -craft Dynamics," Ai rcraft Wake Turbulence and Its Detection, Olsen, J . H . , et a l . , ed . , Plenum Press, New York, 1971, pp. 243-263. 10. Withycombe, E., "Wingtip Vortex Decay: An Experimental Investi-gation," S.B. Thesis, M.I.T., 1970, (as referenced in Ref. 7). 11. Kiang, R.L. , "Sub-Scale Modeling of Aircraft Trai l ing Vortices," Ai rcraft Wake Turbulence and Its Detection, Olsen, J . H . , et a l . , ed . , Plenum Press, New York, 1971, pp. 81-95. 12. Patterson, J.C. J r . , "Lift-Induced Wing-Tip Vortex Attenuation," AIAA Paper No. 74-38, 1974. 23 13. H a r r i s , T.A., Recant, I.G., "Wind-Tunnel I n v e s t i g a t i o n of.NACA 23012, 23021, and 23030 A i r f o i l s Equipped with 40% Chord Double S l o t t e d F l a p s , " NACA Rept., No. 723, 1941, pp. 321-347. 14. Ordway, D.E., Sage A c t i o n Incorporated, I t h a c a , New York. 15. G l a u e r t , H., Elements o f A e r o f o i l and A i r s c r e w Theory, Cambridge U n i v e r s i t y P r e s s , 1930, Ch. 11, 12. 16. McCormick, B.W., T a n g i e r , J.L., "A Study o f the Vortex Sheet Immediately Behind on A i r c r a f t Wing," Dept. o f A e r o n a u t i c a l E n g i n e e r i n g , Pennsylavnia S t a t e U n i v e r s i t y , Dec. 1965. 17. Mason, W.H., Marchman, J.R., I I I , " F a r - F i e l d S t r u c t u r e o f A i r c r a f t Wake Turbulence," J . A i r c r a f t , V o l . 10, No. 2, Feb. 1973, pp. 86-92. 18. Grow, T.L., " E f f e c t of a Wing on I t s T i p Vortex," J . A i r c r a f t , V o l . 6, No. 1, Jan-Feb., 1969, pp. 37-41. 19. Dosanjh, D.S., Gasparek, E.P., E s k i n a z i , S., "Decay o f a Viscous T r a i l i n g Vortex," Aero. Quart., May 1962, pp. 167-188. 20. S p r e i t e r , J.R., Sachs, A.H., "The R o l l i n g Up o f the T r a i l i n g Vortex Sheet and I t s E f f e c t on the Downwash Behind Wings," Jou r n a l o f the A e r o n a u t i c a l S c i e n c e s , Jan. 1951, pp. 21-32. 21. Rorke, J.B., M o f f i t t , R.C., "Wind Tunnel S i m u l a t i o n o f F u l l S c a l e V o r t i c e s , " NASA C o n t r a c t o r Report, NASA CR-2180, 1973. 22. Bisgood, P.L., Maltby, R.L., Dee, F.W., "Some Work a t the Royal A i r c r a f t E s t a b l i s h m e n t on the Behaviour of Vortex Wakes," A i r c r a f t Wake Turbulence and I t s D e t e c t i o n , O l s e n , J.H., e t a l . , ed., Plenum P r e s s , New York, 1971, pp. 171-206. 23. C o n d i t , P.M., T r a c y , P.W., " R e s u l t s o f the Boeing Company Wake Turbulence T e s t Program," A i r c r a f t Wake Turbulence and I t s D e t e c t i o n , Olsen, J.H., e t a l . , ed., Plenum P r e s s , New York, 1971, pp. 473-508. 24. Lissaman, P.B.S., Crow, S.C., MacCready, P.B. J r . , Tombach, I.H., Bate, E.R. J r . , " A i r c r a f t Vortex Wake Descent and Decay Under Real Atmospheric E f f e c t s , " Report No. FAA-RD-73-120, Oct. 1973, Aerovironment Inc., Pasadena, C a l i f o r n i a . 25. Tombach, I v a r , "Observations o f Atmospheric E f f e c t s on Vortex Wake Behaviour," J . A i r c r a f t , V o l . 10, No. 11, Nov. 1973, pp. 641-647. 24 APPENDIX I CORRECTION THEORY The supports for a model in a wind tunnel test affect force measurements in two ways: 1. Cause forces themselves which are included in the measured forces, called tare. 2. Cause the flow about the model to be altered, called interference. Ideally the supports should be shielded to minimize the tare, however, because of the angle of attack mechanism, i t was not practical to shield the model support in the present investigation. For this reason then a series of tests were made in order to separate the effects of tare and interference from the l i f t and drag measurement. Consider f i r s t the l i f t force corrections. The measured l i f t L . i s given by: meas . J Lmeas L N + *S/M + !M/s + T S where is the l i f t of the model in the normal posit ion, i . e . without the support, I j ,^ is the influence of the support on the model, 1 ^ is the interference of the model on the support and T<. is the l i f t of the support without the model. (1^/5 + T^) were measured by supporting the model with a support fixed to the tunnel wall so that 25 t h e r e was a small c l e a r a n c e between the model and the support. The f i x e d support was a s l e n d e r bar i n the spanwise d i r e c t i o n attached to one w i n g t i p . I t was f e l t t h a t t h i s f i x e d support had a n e g l i g i b l e e f f e c t on the i n f l u e n c e o f the model on the r e a l support. Measurement showed t h a t the maximum l i f t developed by the support i n the presence o f the model was l e s s than 1/2 percent o f the maximum l i f t developed by the model and thus n e g l i g i b l e . Then: LN Lmeas " TS/M where the only unknown i s 1$/^-In o r d e r t o determine f o r most types o f models, i t i s necessary to use an image system which i n v o l v e s a complicated s e t o f supports and dummy supports. In the prese n t case i t was f e l t t h a t advantage c o u l d be taken o f the r e l a t i v e l y f l a t spanwise l i f t d i s t r i b u t i o n o f the r e c t a n g u l a r wing. A second dummy support was pla c e d a t v a r i o u s p o s i t i o n s along the span and the e f f e c t on the measured l i f t noted. F i g u r e Al shows t h i s e f f e c t . The c l o s e grouping f o r the c o r r e c t i o n curves f o r 25 percent, 50 percent and 75 percent o f the semispan from the ^ i n d i c a t e t h a t i t i s p o s s i b l e to determine using the second dummy support. P l a c i n g the dummy support a t the 50 percent p o s i t i o n o f midway between the wing ?. and w i n g t i p f o r the f l a p c o n f i g u r a t i o n shown i n Fi g u r e 7 produced the l i f t c o r r e c t i o n curve shown i n Fi g u r e A2. As can be seen, the c o r r e c t i o n s are l e s s than 2 percent f o r the range o f angle o f att a c k used i n the i n v e s t i g a t i o n . The drag corrections are made in an analogous manner: 26 Dmeas = DN + !S/M + !M/S + T S where the subscripts are as given previously. (^/s + T $ ) w e r e measured and are shown in Figure A3. The use of a second dummy support to measure I<-^ for the drag case is more questionable as indicated in Figure A4. The correction varies considerably with the spanwise position of the dummy support. In the angle of attack range of interest, however, this correction is less than one third of the tare correction. For the purposes of this investigation then, the 50 percent position was chosen as being representative. The correction to due to the presence of the dummy support for the above flap configuration is shown in Figure A5. The total drag correction is obtained by combining the corrections in Figures A3 and A5. 27 APPENDIX 2 CALCULATION OF THEORETICAL VORTEX SEPARATION The t h e o r e t i c a l c i r c u l a t i o n i n a p a i r o f t r a i l i n g v o r t i c e s i s equal to the c i r c u l a t i o n , r Q , a t mid-span o f the wing. The a c t u a l c i r c u l a t i o n d i s t r i b u t i o n on the wing can be r e p l a c e d by an e q u i v a l e n t horseshoe v o r t e x o f s t r e n g t h T Q and width b, somewhat l e s s than the span. That i s : L b = . . . . . ( A l ) p vo ro It i s well known that the circulat ion across the span of the a i r f o i l can be represented by the Fourier series: r = 2SVQ I A nsin n9 . . . .(A2) where A p are the Fourier coefficients and 6 = cos _ 1 ( -y/S) . It is also known that: L = 2 T T ( | ) 2 P V 0 2 A 1 - < A 3 ) Combining (Al ) , (A2) and (A3) yields k = I ' _ (A4) s 4 ( W V ' ' ' The A n ' s are functions of aspect ratio and for a rectangular wing must be calculated using the equation: Z A n sin n6 (nu + sin 8) = ua sin 9 where u = ^ ^ = .3927. We choose 4 points to satisfy the equation, 0 = 22.5°, 45°, 67°, 90°. This results in 4 equations in the 4 unknowns A-j, A 2 , Ag, A^. Solving these 4 equations and substituting into (A4) y ie lds : b = i .8148 S 4 8148-.0763+.0125-.00193 .854 . 29 Thus the theoretical vortex separation for a rectangular wing of FR = 4 is .8545 S. F i g u r e 2 R e l a t i o n between an a r c - l e n g t h dL and a displacement e xdx down the l o n g i t u d i n a l a x i s . Figure 3 Unstable modes showing the plane of oscillation. Figure 4 Smoke used initially to visualize a trailing HELIUM SOAP SOLUTION AIR Figure 5 Bubble head schematic. CO co CAMERA F i g u r e 6 Schematic of the experimental set-up. F i g u r e 7 L i f t and drag curves f o r model with f l a p spacings and d e f l e c t i o n s to produce maximum l i f t . 36 F i g u r e 8 Photo t a k e n a t t e s t s e c t i o n e x i t l o o k i n g upstream a l o n g bubble t r a i l s . The u n i f o r m s i n k r a t e p r o d u c e s the s t r a i g h t l i n e o f b u b b l e s seen i n t h e Dhoto. F i g u r e 9A Overhead view o f the f l o w o v e r the wing. Figure 9B Side view showing the formation of the t ra i l ing vortices. 7 F i g u r e 10 T y p i c a l anemometer t r a v e r s e 32 chords downstream o f wing,showing two methods of d e f i n i n g core diameter. The t r a v e r s e i s p e r p e n d i c u l a r ' CO 00 to the span and begins from the l e f t o f the f i g u r e . -— * 8 volts inches F i g u r e 11 T y p i c a l anemometer t r a v e r s e 4 chords downstream. The s c a l e of the o r d i n a t e i s d i f f e r e n t but the d i r e c t i o n of t r a v e r s e i s the same as t h a t i n F i g u r e 10. co c, B .06 (BASED ON PEAK VELOCITY) .04 ,02 1.5 1 . 6 1.7 .8 2 . 0 2 . 1 Figure 12. Effect on the core diameter of varying C O 8 .6 .2 c = . 0 9 8 b S c = l 9 5 b S 10 c=.l95b S , A c= .098b S ,A 15 20 25 30 35 F i g u r e 13 T h e o r e t i c a l non-dimensional a m p l i f i c a t i o n r a t e s f o r the Crow theory f o r two d e f i n i t i o n s o f core diameter. The S and A r e f e r to the symmetric and antisymmetric modes r e s p e c t i v e l y , f: 1.0 .8 c=-098b S -c=. l95b S .6 c = . l 9 5 b S , A 10 c = . 0 9 8 b S , A 15 20 25 3 0 35 8 F i g u r e 14 T h e o r e t i c a l non-dimensional a m p l i f i c a t i o n r a t e s f o r the Parks theory f o r the same two d e f i n i t i o n s of core diameter as i n Figure 13. PO Figure 15 Photos from above, l o o k i n g upstream, showing the f i l a m e n t i n s t a b i l i t y . B F i g u r e 16 T h e o r e t i c a l non-dimensional a m p l i f i c a t i o n r a t e s f o r the symmetric mode f o r the two t h e o r i e s using two core diameter ^ d e f i n i t i o n s . The histogram shows the r e l a t i v e frequency of occ u r r e n c e o f 84 e x p e r i m e n t a l l y measured symmetric waveforms. 45 Figure 18 Photo taken at 45° to the h o r i z o n t a l ; one bubble t r a i l appears as a s t r a i g h t l i n e i n d i c a t i n g that the plane of o s c i l l a t i o n i s c l o s e to 45°. Figure A2 L i f t correction for the flap configuration shown in figure 7. Figure A4 The fractional increase in wing drag with the dummy support at various spanwise positions. The numbers indicate the fraction of semispan from the wing centerline. 48 Figure A5 The drag correction due to support interference for the flap configuration shown in Figure 7. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0093233/manifest

Comment

Related Items