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Analytical and experimental studies of wing tip vortices Duan, Shizhong 1995

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ANALYTICAL AND EXPERIMENTAL STUDIESOF WING TIP VORTICESByShizhong DuanB. Sc. Tsinghua University, Beijing, 1984M. Sc. Beijing University of Aeronautics and Astronautics, Beijing, 1987A THESIS SUBMIITED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESMECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardtLTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1995©Shizhong Duan, 1995In presenting this thesis in partial fulfillment of the requirements for an advanced degreeat the University of British Columbia, I agree that the Library shall make it freelyavailable for reference and study. I further agree that permission for extensive copying ofthis thesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Mechanical EngineeringThe University of British Columbia2324 Main MallVancouver, CanadaV6T 1Z4Date:I M5AbstractWing tip vortices, and their relationship to wing tip geometry and wing total drag, areinvestigated here both analytically and experimentally. The purpose of the analysis is toanswer the basic question— what are the effects of wing drag on tip vortex structure?The purpose of the experiments is to determine the effect of wing tip geometry on wingtip vortices (tip vortex cavitation) and wing drag (vortex induced drag).In the analytical work, a new method has been developed for wing tip vortices. Thisapproach is referred to as the “quasi-similarity” method. This new method combines apolynomial solution with a similarity variable technique. A non-linear analytical tipvortex model is achieved for the first time. The first order (linear) velocity componentsand pressure of the polynomial solution for the new model can be expressed in completefunction form. Higher order (non-linear) velocity components and pressure can beobtained analytically or numerically. The most important feature of this new tip vortexmodel is that the predicted wing drag due to such a single vortex is finite. To the author’sknowledge, no other vortex model has this property. At distances fairly far downstream ofthe wing, tip vortex structure is found to be an explicit function of wing total drag, vortexcirculation, freestream velocity, downstream distance and fluid properties.It is verified that the first order tangential velocity component and pressure of the tipvortex model are exactly the same as that of the linear wing tip vortex model proposed byBatchelor (1964). The decay of the axial and tangential velocity components predicted bythis theory compare well with experimental measurements.In the experimental work, a novel ducted tip device was tested in a wind tunnel and awater tunnel. The ducted tip consists of a hollowed duct attached to the tip of arectangular untwisted wing. This novel tip device was found to improve the Lift/Dragratio by up to 6% at elevated angles of attack compared with a conventional round tipconfiguration with the same span. The wing tip vortex cavitation was substantiallydelayed by the ducted tip device. In view of its superior cavitation characteristics andaerodynamic performance, the ducted tip has potential application to marine propellers.HThe ducted tip is effective because it redistributes the shed vorticity in the transverseplane (Trefftz plane) behind the wing. The shed vorticity is distributed along the wing andduct trailing edge for the ducted tip configuration, rather than solely along the wingtrailing edge, which would be the case for a conventional wing tip configuration.illTable of ContentsAbstract iiList of Tables viiList of Figures viiiNomenclature xiAcknowledgements XivChapter 1 Introduction 11.1 Motivation for tip vortex research 21.2 Previous measurements on tip vortex structure 41.3 Previously developed mathematical models of vortices 51.4 Scope of the present work 11Chapter 2 Quasi-Similarity Method for Vortices in a Freestream 142.1 Overview of quasi-similarity method 142.2 The governing equations 152.2.1 General governing equations 152.2.2 Non-dimensional governing equations 172.2.3 Structure of the solutions 192.2.4 Governing equations of the first order 202.2.5 Governing equations of the second order 212.2.6 Governing equations of the arbitrary i-th order 212.3 The first order problem 222.3.1 Derivation of the governing quasi-similarity equations 222.3.2 Boundary conditions on the quasi-similarity equations 252.4 The second order problem 272.5 The i-th order problem 30iv2.6 The solutions in complete function form .352.7 Discussions of assumptions 362.8 Conclusions 38Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 393.1 The first order solutions 393.1.1 General first order solution in complete function form 393.1.2 Tangential velocity and pressure 423.1.3 Effects of the constant 13 on the axial similarity functions 433.2 Drag due to a single quasi-similar vortex in viscous flow 443.3 The second order solutions 483.3.1 Second order tangential velocity and pressure 493.3.2 Second order axial and radial velocity components 513.4 The procedure for i-th order solution (i >2) 533.4.1 Conservation of the vortex circulation F 553.4.2 The i-th order drag 583.5 The axial velocity on the vortex centerline 603.6 Comparison with Batchelor’s vortex 603.7 Comparison with experimental data 643.7.1 Axial velocity decay on the vortex centerline 643.7.2 Maximum tangential velocity decay 653.8 Conclusions 66Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 684.1 Previous work on tip modification 694.2 The ducted tip wing geometry 704.3 Experimental apparatus and procedures 72V4.4 Results and discussion .784.4.1 Flow visualization 784.4.2 Cavitation characteristics 854.4.3 Aerodynamic performance 924.4.3.1 The partial chord ducted tip 924.4.3.2 The hi-wing tip 1084.5 Experimental errors and applicability of the tip device 1094.6 Conclusions 110Chapter 5 Summary and Conclusions 1125.1 Summary and conclusions 1125.1.1 Analytical method 1125.1.2 Tip vortex structure 1135.1.3 Wing tip modification 1145.2 Suggestions for future work 115Bibliography 117viList of Tables4.1 Lift coefficients of the ducted-tip wing and the conventional wing 924.2 Drag coefficients of the ducted-tip wing and the conventional wing 934.3 Comparison of the hi-wing tip to the conventional tip at 8° angle of attack 1084.4 Comparison of the bi-wing tip to the conventional tip at ct=14° 109viiLists of Figures1.1 Helmholtz vortex laws interpretation of tip vortices 11.2 Shed vorticity from a finite wing 11.3 Rolling-up of a trailing vortex sheet 21.4 Induced drag acting on a finite wing 31.5 Burgers vortex model 62.1 Vortex in a freestream 193.1 The first order quasi-similarity functions. 13=0.361 423.2 Axial velocity excess in wing tip vortices 433.3 Axial velocity deficit in wing tip vortices 433.4 Control volume around the wing tip 443.5 The effects of wing total drag on the axial velocity 473.6 The effects of wing total drag on the radial velocity 473.7 The second order axial quasi-similarity function. 13<0 523.8 The second order axial quasi-similarity function. 13>0 523.9 The second order radial quasi-similarity function. 13<0 533.10 The second order radial quasi-similarity function. 3>0 533.11 Axial velocity decay with downstream distance in a wing tip vortex 643.12 Maximum tangential velocity decay with downstream distance in a wing tipvortex 654.1 Shed vorticity in the Trefftz plane from (a) conventional and (b) ducted tipwings 714.2 Schematics of wing tip devices. Flow is right to left. (a) Ring wing tip. (b) Biwing tip 724.3 The Low Speed Wind Tunnel at the University of British Columbia 764.4 The data acquisition system for lift and drag measurement in the LSWT 764.5 The data acquisition system for pressure measurements in the LSWT 774.6 Tip vortex rollup in the Trefftz plane 784.7 Surface flow visualization on the round tip geometry 79, 80VIII4.8 Surface flow visualization on the ducted tip wing 82, 834.9 Separated flow near the entrance of the ducted tip. a=l2° 844.10 Developed trailing vortex cavitation behind the rounded tip geometry. a=7°,Re=l.3x106.a=l.8, DAC=5.4ppm 884.11 Cavitation inception index versus angle of attack for the rounded tip geometry.Rel.4xl06,DAC=5.4 ppm. (Uncertainty in a=±0.2°, and in =±lO% at the95% confidence level) 894.12 Ducted tip vortex made visible by migration of cavitation nuclei, generated atthe duct leading edge, into the vortex. a=7°, Re=l.2x106. a=1.5,DAC=7.Oppm 904.13 Cavitation inception index versus angle of attack for the ducted tip geometry.Re1.4xl06,DAC=7.0 ppm 914.14 Drag coefficients of rounded and ducted tip wings. Re=7.lxlO5 954.15 A simple model for the reduction in induced drag of the ducted tip geometry 954.16 Lift/Drag ratio improvement of the ducted wing relative to the conventionaltip 964.17 Pressure taps in the surfaces of the basic wing 1004.18 Pressure distribution on the pressure side of the wing with the conventionaltip. a=8°. (a) the 3-D view, (b) the tip view 1014.19 Pressure distribution on the pressure side of the wing with the ducted tip.x=8°. (a) the 3-D view, (b) the tip view 1024.20 The difference between the ducted and conventional tip pressure coefficientson the pressure side of the basic wing. a=8° 1034.21 Pressure distribution on the suction side of the wing with the conventional tip,c=8°. (a) the 3-D view, (b) the leading edge view, (c) view from the wingroot 1044.22 Pressure distribution on the suction side of the wing with the ducted tip, a=8°.(a) the 3-D view, (b) the leading edge view, (c) view from the wing root 105ix4.23 The difference between the ducted and conventional tip pressure coefficientson the suction side of the basic wing, c8° 1064.24 Pressure distribution on the suction side of the basic wing without a tip device,107xNomenclaturea A constant for the i-th order problemb Wing spanb1 A constant for the i-th order problemc Wing chordCD Drag coefficientCf, A constant for the i-th order problemC/i1 A constant for the i-th order problemc, A constant for the i-th order problemc, Pressure coefficientCL Lift coefficientD Drag due to a single tip vortexDAC Dissolved air content [ppm]D Wing dragf(q) An arbitrary functionf(z) An arbitrary functionF1(i) i-th order similarity function of axial velocityOi) An arbitrary functionG1(i1) i-th order similarity function of pressureH1(ri) i-th order similarity function of function jiK(x) Vortex circulationL A constant in Batchelor’s vortexL0 Operator for ordinary differential equationsL Operator for partial differential equationsp PressurePv Water vapour pressurepr,,. Pressure at infinityP(z) i-th order amplitude function of pressureP() Similarity function for pressure in Batchelor’s vortexxiPf Reference pressureQ A similarity function in Batchelor’s vortexQ2 A similarity function in Batchelor’s vortexR Radius of the control volume around a tip vortexr The radial distance from the vortex centerlineR1 The radius of a tip vortexRe Reynolds numberR•(i) i-th order similarity function of radial velocitys Distance between two trailing vorticesT1(i) i-th order similarity function of tangential velocityU Freestream velocityU(z) i-th order amplitude function of axial velocityV1(z) i-th order amplitude function of radial velocityz The axial distance along the vortexGreek SymbolsA constant related to wing dragA parameter defined as tangential velocity over the radius, 19/rA positive number not far from unityAngle of attackp Fluid densityv Kinematic viscositySimilarity variableTI Similarity variableSimilarity variableStream function8 The azimuthal coordinateF Vortex circulationxii1e Tangential velocityp(z,r) An arbitrary functionTo Bound circulation at mid-span of a wingA constant for the i-th order problema A constant for the i-th order problema1 Cavitation inception index, a1 (pt,.— p1, )/o. 5puP(z) i-th order amplitude function of parameter Nhr Radial velocityi Axial velocityco The axial vorticityXIIIAcknowledgementsI would like to express my gratitude to my supervisor, Professor S. I. Green. Hisguidance throughout this program has been invaluable in finishing the work successfully.I wish to express my sincere appreciation to Professor I. S. Gartshore, Professor P. G.Hill, Professor S. M. Causal and Professor G. W. Bluman for their insightful discussionsand suggestions. Thanks are also due to A. Schreinders, T. Besic, D. Camp, D. Bysouthand other technicians in the Mechanical Engineering Workshop for their skillful technicalassistance in design and construction of the instruments and test models.I wish also to thank Professor G. S. Schajer for proof-reading my thesis.Financial support for this work by the Natural Science and Engineering ResearchCouncil of Canada and the University of British Columbia is gratefully acknowledged.I would especially like to thank my parents, and my wife Belinda for their patience,love, faith and encouragement. It is to them that I dedicate this work.xivChapter 1 IntroductionChapter 1 IntroductionTrailing vortices occur whenever a lifting surface terminates in a fluid. A simpleexplanation of the existence of trailing vortices derives from the application of Helmholtzvortex laws. Consider a finite length -wing impulsively started from rest. For Flowthe simplest case, a lift line (with aconstant circulation along the wingspan) must exist in order for the wingto generate lift force. Kelvin’sFigure 1.1 Helmholtz vortex lawsTheorem demands that this circulation interpretation of tip matched by an equal and oppositeshed circulation (the “starting vortex”). Because vortex lines can never end in a fluid,these two vortices must be connected by tip vortices, as illustrated in Figure 1.1.A more realistic description involves a number of lifting lines distributed along thewing chord and span, as depicted in Figure 1.2. As a result of this distribution of liftinglines, a sheet of vorticity is shed by the wing. Such a vortex sheet is not stable; roll-up ofthe sheet occurs. The free edge of the sheet curls over, under the influence of the inducedvelocity field of the vortex sheet, and takes the form of a spiral with a continuallyincreasing number of turns in the downstream direction (Figure 1.3). It has now been wellestablished that trailing vorticesgenerated by wings roll upexceptionally quickly. In general, rollup is complete, i.e. vortex circulationis virtually independent of downstreamdistance, 1-2 chords downstream of thetrailing edge of an airfoil (Shekarriz etal. 1992, Stinebring et a!. 1991, Greenand Acosta 1991). Contraction of vorticity in a plane transverse to the freestream duringroll-up results in an axial velocity in the vortex core. After the shed vorticity is fullyChapter 1 Introduction 2rolled up, there are only two concentrated trailing vortices behind a finite wing —consistent, coincidentally, with the conclusion that resulted from the simplestinterpretation of Helmholtz vortex laws.1.1 Motivation for Tip Vortex ResearchThe wing tip vortex flow is one ofgreat importance because it occurs oftenin practical problems. These problemsinclude: landing separation distances foraircraft at runways — one airplane mayexperience a dangerous rolling momentif it accidentally flies into the tip wakeFigure 1.3 Rolling-up of a trailingof another (Donaldson 1971, Kantha et vortex 1971, Snedeker 1972, Barber et al. 1975); blade/vortex interactions on helicopterblades, which is an undesirable source of noise and vibration (Martin et al. 1984, Summa1982, Mosher and Peterson 1983, Lewy and Caplot 1982, Widnall and Wolf 1980); andpropeller cavitation on ships due to the relatively low pressure in the vortex core (Ligneuland Latorre 1993, Arndt and Keller 1992, Chahine et al. 1993, Arndt et al. 1991).In the following airplane problem, the tip vortices generated by one aircraft may besufficiently strong that a following plane accidentally entering into one of the vorticesexperiences a loss of control. This accidental interaction is most likely to occur duringaerial refueling and over airport runways. Critchley (1991) has documented that seriousincidents involving such interactions occur at an average rate of 9 per year at Heathrowairport! The cost of maintaining sufficient plane separation distances on landing to limitthe frequency of such hazardous interactions is about $10 million annually for everymajor airport (Winter 1991).Helicopter rotor blades generate strong tip vortices. If the helicopter is hovering,following rotor blades will pass through the tip vortex generated by leading blades. Inpassing through the tip vortex, the following blade experiences large fluctuating forcesChapter 1 Introduction 3(due to the unsteady aerodynamic loading) that can cause premature rotor blade fatigueand excessive blade noise (Poling et al. 1989).Finally, there is a tendency for tip vortices generated in marine applications to cavitateeven at elevated inception indices (refer to section 1.4 for the definition of cavitationinception). The reason for the high values of cavitation inception index, , may be tracedto the fact that the vortical motion is particularly strong (large 0e / U) in tip vortices. Inorder to sustain these large tangential velocities (and hence extremely high centripetalaccelerations), there exists a large radial pressure gradient from the vortex core to thesurrounding fluid. If the freestream pressure is low enough, or the tangential velocity ishigh, the pressure in the vortex core will locally fall below the water vapour pressure andcavitation may occur.In every application, tip vortices act Induced dragas lifting surface inefficiencies. Flowdownwash caused by the vortices -decreases the effective incidence angleof the wing. This tilting of the localincoming flow causes the lift force,Figure 1.4 Induced drag acting onwhich is perpendicular to the incoming a finite wing.flow direction, also to be tilted. Thecomponent of the lift force in the undisturbed freestream direction is the wing induceddrag (Figure 1.4). Induced drag on wings is a particularly salient issue. Approximately35% of the drag on a typical aircraft is lift-induced drag (Webber and Dansby 1983). Dueto the small aspect ratio of marine propellers, induced drag on propellers is likely to be agreater percentage of the overall drag than in aeronautical applications. The potentialsavings to be reaped by reducing airplane induced drag, even fractionally, are staggering— roughly $100 million/year worldwide for each 1% induced drag reduction. Tipclearance flow in axial flow fans (Ruden 1974) and compressors (Raines 1954) decreasestheir efficiency. An understanding of the tip vortex flow is also pertinent to the design ofpropfan blades (Vaczy and McCormick 1987). Tip flow even plays an important role inthe design of America’s Cup yacht keels (Devoss 1986).DownwashChapter 1 Introduction 41.2 Previous Measurements on Tip Vortex StructureAttempts have been made to measure the tangential and axial velocity distributionsaround trailing vortices using Laser Doppler Velocimetry (LDV) (Orloff and Grant 1973,Baker et al. 1974, Higuchi et al. 1986a), hot wire anemometers (Corsiglia et al. 1973,Chigier and Corsiglia 1972, Zalay 1976), five-hole or seven-hole probes (McCormick1968, Logan 1971, Chow et al., 1993), and Particle Image Velocimetry (PIV) (Green1987, Shekarriz et al. 1992, 1993). Any intrusive method used to measure tip vortices issubject to large experimental errors because tip vortices are extremely sensitive. LDV isaccurate locally, but there is a risk, due to its small measurement volume and thetendency of trailing vortices to meander, that the measurement volume may miss thetarget. This unsteady movement of trailing vortices may lead to a large overallmeasurement error when LDV is employed. Particle Image Velocimetry (PIV) has thedistinct advantage that particle tracers move with the vortex. Although the local accuracyof Ply is not as high as LDV, the overall error of PIV is possibly the lowest among all themethods discussed above. Jn view of the sensitivity of vortices to disturbances and theirtendency to meander, one must view with suspicion many of the early experimentalstudies of tip vortices.More recent studies, though, have produced some reliable and interesting results. Forexample, Pauchet et al. (1993) have measured vortex core axial velocities as big as 2.5times the freestream velocity within two chord-lengths downstream of a particularhydrofoil. This axial velocity increases with the angle of attack, cx. Typical vortex coreradii (the radius at which the tangential velocity is a maximum) are approximately 0.01 ,where is the wing average chord (Green and Acosta 1991). This radius increases onlyslowly with downstream distance.Further downstream, viscosity leads to a slow diffusive increase of the core diameter.The gradual slowing-down of the azimuthal motion by viscous action leads to anincreased pressure at the axis (there being less centrifugal force then), and so to an axialdeceleration of the core fluid (Batchelor 1964). It is found experimentally that theChapter 1 Introduction 5tangential velocity decays slowly with downstream distance, whereas the axial velocitydecays much faster. (Green 1991, Pauchet et al. 1993, Shekarriz et al. 1993). Maximumtangential velocities around the core in excess of (Arndt et al. 1991) have beenmeasured.Much further downstream, trailing vortices typically “demise” after undergoing theCrow instability (Crow 1970). This instability takes the form of long wavelengthsinusoidal disturbances on the vortex in fixed planes 48° from the wing planform.Eliason, Gartshore and Parkinson (1975) have verified experimentally the Crowinstability. The amplitude of the disturbance grows in time. Following growth of thesinusoidal instability, the trailing vortices “link” in the regions where the two vorticesapproach most closely. Linking of vortices is followed closely by the formation of vortexrings. Vortex rings usually form hundreds or thousands of chord-lengths downstream of awing; once vortex ring formation has occurred, the trailing vortex dissipates rapidly(Sarpkaya and Daly 1987).The wing tip vortices are also known to be highly unsteady. The axial velocity on thecenterline can fluctuate with an r.m.s amplitude up to 0.2U, (Green and Acosta 1991).Wing tip or strake vortices generated by delta wings may undergo an unusualphenomenon known as vortex “breakdown” or “bursting” (Peckham and Atkinson 1957).A stagnation point appears suddenly in the vortex core as vortex breakdown happens(normally at high angles of attack) and the flow downstream immediately becomesunsteady and irregular. The lift drops drastically and an asymmetrical rolling momentappears which may result in a loss of control.1.3 Previously Developed Mathematical Models of VorticesMathematical models of vortex behaviour have become increasingly moresophisticated and realistic with the passage of time. The earliest and simplest model of anisolated vortex is the “Rankine vortex.” The Rankine vortex model is a two region modelof a vortex. The vortex core moves in solid body rotation (r t), and the region outsideChapter 1 Introduction 6the core is irrotational. The vortex is purely two-dimensional and axisymmetric, with onlya tangential velocity:ii.,(z,8,r)=O0TrI2itr2 rr“-ye (z,O ,r) =F/2tr r>rpF2 pf’2.rrr4t2r 8it2r4p—p(z,8,r)= CpF2r>r8it2r CThe Rankine vortex does not satisfy the Navier-Stokes equations; at r = i the velocitygradient is discontinuous. Nonetheless, the pressure field result for a Rankine vortex isinstructive— the pressure at the centerline falls with 1/r2. The pressure (and thuscavitation characteristics) of a Rankine vortex is very sensitive to r.The diffusion of a two-dimensional axisymmetric vortex was analyzed byOseen(1912). At time t = 0 an irrotational line vortex (‘us = F / 2itr) is placed in a viscousfluid. Viscosity causes the vorticity to diffuse radially in time:= 0r(Z,0,r,t) = 0(z,O ,r,t) = (F122tr) [1_exp(_r2 /4vt)]The “Lamb-Oseen” vortex is an exact solution to the Navier-Stokes equation. Thesolution shows that viscous growth of a vortex is very slow; the core size varies asThe Rankine and Lamb-Oseen models are both vortex models for which the axialvelocity is zero. More complex vortex models‘lizthat incorporate axial flow in the vortex have ‘0rbeen developed. The “Burgers Vortex” (Burgers1948) models the flow around a vortex that isstretched in the axial direction. The threevelocity components in the Burgers vortex are Figure 1.5 Burgers vortex model.liI0 zChapter 1 Introduction 7(refer to Figure 1.5):u(z,r,t) = 2Czi(z,r,t) = —Crf Fi. (z,r,t) =___.[l_exP_i-JjThis velocity field has associated vorticity distribution:Ir j-rexwhere 2 +62 _-JexP(_2Ct).The Burgers vortex solution reveals that axial stretching of vortex lines (e.g. as occursimmediately downstream of a wing) decreases the growth in vortex core size that arisesfrom viscosity.Long (1961) extended the work of Burgers by considering a viscous vortex in aninfinite liquid. Similarity arguments lead to a solution for this viscous vortex,=r = f(where = F r__, and 0 is the stream function. The similarity functions f, k, and ssatisfy a set of ordinary differential equations. Long integrated these ODEs numerically,and found that the self-similar profiles have different characteristic shapes, depending onthe value of the non-dimensional momentum transfer, M, given by:M =--. J (1 + . 2itr drT0p jBatchelor (1964) has deduced the dynamical necessity of axial flow in a tip vortexcore. In order to maintain the centripetal acceleration around the vortex, the pressure inthe core of a trailing vortex is lower than that upstream of the wing. Thus, in the absenceof viscous effects, fluid particles on a streamline originating ahead of the wing andentering the core are accelerated in the downstream direction (based on Bernoulli’sChapter 1 Introduction 8equation, flow velocity increases on a favorable-pressure-gradient streamline). Relative tothe fluid at rest at infinity, the fluid in the core moves in a direction opposite to thedirection of flight. In a coordinate system fixed to the wing, the component of velocity inthe direction of the flow at infinity is increased in the core. Batchelor showed that aRankine vortex with maximum tangential velocity u = ku0,, has maximum axial velocityin the vortex core (1), )mac = U,, -.Ji + 2k . In Batchelor’s paper, a partly-linear theory wasdeveloped to predict the axial flow in a wing tip vortex core. The axial velocity deficit orexcess was considered to be much smaller than the freestream velocity. Then, thetangential momentum equation could be linearized, and the following asymptotic form ofthe tangential velocity was obtained:c0 (u_.r21— ;vz ]where C0 is a constant, U,,, is the freestream velocity, and v is the kinematic viscosity.By integrating all terms of the linearized axial momentum equation, Batchelorsuggested the following asymptotic solution for the axial velocity in a wing tip vortex.C2 2 1720 ZU00 U,.,8vz v 8vz 8vzwhere = U r2 and L is a constant related to any initial velocity defect which may be4vzindependent of the circulation.Batchelor studied the vortex wake far downstream, where it decays under the actionof viscosity. The vortex radius is O(vz I U,., Batchelor found that viscous effectsproduce an axial velocity deficit in the far field.The principal limitations of Batchelor’s theory are that it is only valid at largedistances downstream of the vortex-generating wing, and that the theory requires that theaxial velocity deficit or excess be much smaller than the freestream velocity. As describedpreviously, the latter assumption is often violated by the flow around real wing tipvortices.Moore and Saffman (1973) considered the structure of laminar trailing vorticesrolling-up behind a wing. Their vortex structure consists of an inviscid roll-up region andChapter 1 Introduction 9a viscous vortex core. For an elliptical loading case, the wing is replaced by a boundvortex along the wing span, 0 <x < b, of strengthK(x) = 2.. [x(b — x)]Kaden (1931) had remarked that roll up starts at the wing tips so that in the initialstages of the process, the rolled up portions at each end cannot significantly interact. Thusone can usefully consider a semi-infinite lifting line of circulation K(x) = 2y x, where= F0 b. The vortex sheet rolls up into a spiral with an infinite number of turns and thevelocity field near the centre of this spiral can be obtained by a simple argument — thevelocity field, relative to the centre of the spiral, can be characterized by a distribution ofcirculationF(r) = 27[. r]where 2 is a constant. Thus if ij is the transverse velocity relative to the centre of thespiral,4t rThe pressure in the roll-up region is found by the radial equilibrium equation and isused to calculate the axial velocity, which is produced by streamwise pressure gradients.Moore assumed further that the tip loading is where n is a positive constant and0 <n < 1. The case n = 1/2 represents elliptic loading. The axial velocity excess is thengiven byw=(1 —U ) .(±_l.r2n atr02Un )provided w << U,. ic is a constant. Note that w > 0, in agreement with Batchelor’sgeneral argument. However w —* oo as r — 0. Therefore, the effect of viscosity has to beconsidered. Upon consideration of the viscosity, Moore and Saffman deduced that roll-upis 90% complete only tens of chords downstream of a wing, a result that differs fromexperiments (Shekarriz et al. 1992, Green and Acosta 1991). The discrepancy betweentheory and experiments may be due to their mathematical requirement that I U IU, which virtually never occurs in practice.Chapter 1 Introduction 10A class of self-similar solutions of the steady, axisymmetric Navier-Stokes equations,representing the flows in slender vortices was reported by Mayer and Powell (1992a).Such self-similar vortices will have constant axial velocity in the vortex core if theexternal axial flow is a constant (i.e. In a wing tip vortex, the axial velocity in thevortex core decays along the downstream distance (based on experimental observations)while the freestream velocity is a constant, and hence wing tip vortices are not part of thefamily of self-similar vortices.The inviscid and viscous stability of a vortex, which has the asymptotic structureobtained by Batchelor (1964), was first studied in the papers of Lessen et al. (l974a) andLessen & Paillet (1974b). They used a Runge-Kutta-type scheme along with theasymptotic behavior of the solutions at large and small radial distance to integrate thedisturbance equations. A spectral collocation and matrix eigenvalue method was used byMayer and Powell (1992b) to study the linear stability of Batchelor’s trailing line vortex.Their paper yielded the first global pictures of inviscid instabilities of the trailing vortex.Gartshore (1963) studied the quasi-cylindrical approximation of a trailing vortex using anintegral approach. He found that a singularity may appear in the course of integration ofthe equations and the solution cannot be continued beyond the singular point. Hesuggested that these singularities may be mathematical indications of vortex breakdown.This work was continued by Mager (1972) who also used the integral approach to solvethe quasi-cylindrical approximation equation. He made an attempt to connect thesolutions he obtained with the concept of the breakdown as a supercritical-subcriticaljump (suggested by Benjamin (1962)). Trigub et al. (1994) performed a consistentasymptotic study of steady axisymmetrical trailing vortices by using the quasi-cylindricalapproximation. They presented numerical solutions for unbounded vortex breakdownparabolically expanding far downstream.As discussed previously, pairs of trailing vortices do not decay by simple diffusion.Usually they undergo a symmetric and nearly sinusoidal instability, until eventually theyjoin at intervals to form a train of vortex rings. Crow (1970) developed a theory thatdescribes the early stage of this instability. The equation relating induced velocity tovortex displacement gives rise to an eigenvalue problem for the growth rate of sinusoidalChapter 1 Introduction 11perturbations. The perturbation that grows most rapidly is a sinusoid of wavelengthapproximately 6 wing spans, which makes an angle of 48° to the wing planform.Most practical wing tip vortex flows are turbulent. The mechanisms involved inturbulent flows may be entirely different from those of laminar flows. Hoffmann et a!.(1963) predicted by theory, and confirmed by experiment, that the circulation in aturbulent vortex is proportional to the logarithm of the radius under certain conditions.They described a universal distribution of circulation in the inner region of the vortex,analogous to the turbulent boundary layer.Govindaraju and Saffman (1971) showed that a turbulent vortex must develop anovershoot of circulation— the circulation rises above F,. for a finite r and then falls backto F,. as r —4 °°. They stressed this prediction is independent of any hypothesis about theReynolds stress. Any numerical calculation involving any closure approximation shoulddemonstrate an overshoot of circulation, provided the closure approximation is self-consistent with the conservation of angular momentum. The overshoot was foundindependently by Donaldson and Sullivan (1970), who studied the turbulent vortexnumerically with a particular closure approximation. However, experiments do notindicate the presence of any significant overshoot in the circulation profile.1.4 Scope of the Present WorkThe decay of wing tip vortices due to viscosity is investigated analytically in thisthesis. As previously mentioned, Batchelor’s linear vortex model is the most advancedwing tip vortex model so far, and is valid far downstream of the wing where the axialvelocity excess or deficit is much less than the freestream velocity. This downstreamdistance is estimated to be thousands of wing chords for a practical wing tip vortex(Moore and Saffman 1973). To understand the vortex structure fairly close to the vortexgenerating wing, non-linear effects must be considered.An analytical quasi-similarity method is developed for wing tip vortices. This newapproach incorporates non-linear effects and velocity decay in a 3-D tip vortex. Thecentral idea of the quasi-similarity method is that the solution consists of a polynomial inChapter 1 Introduction 12which each term looks like a self-similar solution in complete function form— anamplitude function times the similarity function. The amplitude function is a function ofdownstream distance only, and the similarity function depends on a similarity variable.The similarity variable in turn is a non-dimensional combination of the downstreamdistance and radius coordinates. The limitation of this new method is that the axialvelocity deficit or excess must not be significantly greater than the freestream velocity.This is much less restrictive than the limitation on Batchelor’s linear theory.The geometric details of the wing that generates a tip vortex has a substantialinfluence on the vortex flow. Changing simply the airfoil section shape (e.g. from anNACA 0020 to an NACA 16020), while keeping the wing planform constant, candramatically change the tip vortex flow (Pauchet et al. 1993). Similarly, changing thewing planform has a marked influence on the tip vortex (Fruman et al. 1995). Forexample, elliptic wing planforms tend to behave quite differently from rectangularplanforms (e.g. compare Figure 4 of Arndt et al. 1991 with Figure 1 of Green and Acosta1991). Changing the wing tip shape and even roughness (Green et al. 1988, Stinebring etal. 1991) can also dramatically alter the tip vortex.The search for a new wing geometry that avoids cavitation and/or reducesaerodynamic drag has been attractive to researchers. A novel ducted tip device, consistingof a small diameter flow-through duct, was attached to a rectangular untwisted wing andtested experimentally. Because the wing tip vortex generated by a ducted tipconfiguration would be expected to have a larger core than the tip vortex of aconventional tip, the pressure in the core would be higher than that of a conventional tipconfiguration. Therefore one might expect, and the expectation has been confirmedexperimentally (Green et al. 1988), that the cavitation inception could be delayed byaddition of a ducted tip device. A ducted tip is believed to reduce the induced drag of awing as well, although it may cause a parasitic drag increase due to the extra wetted areaat the tip.Experiments on two kinds of ducted tip devices mounted on a NACA 66-209rectangular planform wing have been done in a wind tunnel. The two tip devices studiedwere a ring-wing tip and a ducted bi-wing tip. Several configurations were considered forChapter 1 Introduction 13each specific device. The lift and drag dependence on angle of attack, cc and pressuredistribution on the wing surface were measured. All the results were compared with thebaseline case of a rounded tip wing of identical span.The cavitation characteristics of the ring-wing tip device were investigated in a watertunnel. With the hydrofoil and tip device in the tunnel, gradually U was increased anddecreased. At the moment of cavitation inception (as evidenced by the appearance of atleast one macroscopic bubble per second in the trailing vortex core under stroboscopicillumination) IL. and p. were measured. The cavitation performance is characterized by aparameter called the cavitation inception index, which is defined as (pa, — p)/O.5pU,where p, is the water vapor pressure. The smaller the index, the better the cavitationperformance.In the next chapter, the quasi-similarity method for vortices in a freestream (e.g. wingtip vortices) is discussed. Chapter 3 contains quasi-similarity solutions for wing tip vortexstructure. The cavitation and aerodynamic performance of the novel ducted tip device isdocumented in chapter 4. Chapter 5 presents a summary and the main conclusions of thepresent work.Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 14Chapter 2 Quasi-Similarity Method for Vortices in a FreestreamIn this chapter, a quasi-similarity method for 3-D viscous vortices in a freestream (e.g.wing tip vortices) is discussed. Polynomial solutions for the velocity components andpressure distribution in such a vortex are obtained. The analysis shows that the tangentialvelocity in a tip vortex decays much more slowly along the downstream direction than theaxial velocity, an observation that is consistent with experimental findings.2.1 Overview of Quasi-Similarity MethodThe main idea of the quasi-similarity method is described briefly below. We start withthe Navier-Stokes (N-S) equations in cylindrical coordinates. Some assumptions are madeto simplify the N-S equations. The simplified governing equations, including thecontinuity equation, are a nonlinear system, which can be represented mathematically as(2.1.1)L is an operator for partial differential equations, and (2.1.1) can be either the continuityor momentum equations.The following polynomials are found to be the solution of (2.1.1), provided thesummations are convergent.)=[1+.())] (2.1.2a)z(2.1.2b)1=1r(Zñ (—5.R11) (2.l.2c)p(z,r)-p =pU.-L.G(1l)J (2.1.2d)1=IHere 1 = r is a “similarity variable” and the coefficients denoted by capital C areconstants (we will verify that only one constant is independent in the first order of thepolynomial, and two constants are independent in each higher order of the polynomials).F is the circulation of the vortex, v is the kinematic viscosity, U0, is the freestreamChapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 15velocity, z is the axial distance along the vortex, r is the radial distance from the vortexcenterline, and p,,, is the pressure at infinity. Note that this is not a true similarity solutionbecause the velocity components and pressure cannot be expressed solely in terms of i.Substitution of (2.1.2) into (2.1.1) yields,L0[F(r),7(ri),R1(1),G]= 0 (2.1.3)1=1 Zwhere L0 is the operator for an ordinary differential equation.Because z is an arbitrary positive variable, for each of i = 1,2,3,..., the followingordinary differential equation (ODE) must hold:L0[F(ri), 7(rI), J? (i), G, (i)] = 0 (2.1.4)The ODE (2.1.4) can be solved with proper boundary conditions.Only one constant is involved in the first order problem. It will be shown in the nextchapter that this constant in the first order problem (i = 1) is related to the total drag of awing. The constants in each of the higher order problems (i 2) will be discussed later. Itwill be shown in Chapter 3 that the quasi-similarity solutions meet the following tworequirements. One requirement is the conservation of the vortex circulation. The otherrequirement is the independence of wing total drag on the downstream distance.With this overview of the quasi-similarity method completed, we proceed in the nextsections to a detailed discussion thereof.2.2 The Governing Equations2.2.1 General governing equationsThe incompressible Navier-Stokes equations in a cylindrical coordinate system are:Continuity equation:(2.2.1)r r ao azChapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 16Tangential momentum equation:- 1.) 1 ap ( 2 h “1-+V.V1)9 rr. (2.2.2)Radial momentum equation:-. ia 1 2”+V•V1)r ‘1V=1)r (2.2.3)Axial momentum equation:(2.2.4)at- pazwhere,- a ua aV’V=u —+-—-—+1)—ar r O za1 a( a 1 a2 a2V2 =—.—I r—r ar ar1 ra8 azIn the above equations p is the density,1.L the axial velocity, the tangential velocity,l)r the radial velocity, p the pressure, z the axial distance along the vortex, r the radialdistance from the vortex centerline, and 9 the azimuthal coordinate. We can simplify theabove equations by making the following assumptions for a vortex in a freestream:(1) The flow is incompressible, steady, and laminar(2) The flow is axisymmetric(3) r <<‘09a2u a2u(4) 2 << 9______<<az ar- az- arThen, the governing equations can be simplified and rewritten as:!. a(-r)= 0 (2.2.5)r ar aza9 a (a29 ia92 (2.2.6)ar r c’z cir rcir rChapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 17(227)par“ar =•+ (2.2.8)ar àz paz ar rarIt is convenient to define a new variable N = i I r. With this definition of N, (2.2.6)and (2.2.7) become:(2.2.9)NJ2r=P (2.2.10)The function i has a definite physical meaning. For an axisymmetric vortex flow,or r=O r=Owhere c is the axial vorticity. ji(r = 0) is thus half the value of the vorticity at r = 0.Because the vorticity is non-zero in a real vortex at r =0, so too is N• (In fact, becausethe central core of a vortex is forced by the action of viscosity into solid body rotation, thevorticity on the axis of a vortex is 22, where 2 is the angular frequency of the rotation.)This form of i is useful because it makes the tangential momentum and axial momentumequations take a similar form. It is therefore convenient in numerical calculations.2.2.2 Non-dimensional governing equationsThe dimensional information in the governing equations (2.2.5)-(2.2.8) can beexpressed as(2.2.11)r zrO ccv.- (2.2.12)r z r(2.2.13)r prChapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 18lii)ccZZ locV- (2.2.14)r z pz rThe non-repeated information is listed below(2.2.15)r z(2.2.16)p cc p’u (2.2.17)pocp1) (2.2.18)Now we assume that the circulation of the vortex, f, is a constant along the axis ofthe vortex. Thusu0.rccF (2.2.19)Furthermore, it is reasonable to nondimensionize the axial velocity by the freestreamvelocity (2.2.20)Therefore, the variables describing a viscous vortex in a freestream can be nondimensionized using the following characteristic physical parameters — the freestreamvelocity U, the circulation of the vortex F, the fluid density p. and the kinematicviscosity v:F2z= (2.2.21)U•vr = —. (2.2.22)Uo= (2.2.23)U•v= (2.2.24)rp=pU • (2.2.25)= or = (2.2.26)where the variables with a symbol “—“ are non-dimensional parameters. Thedimensionless governing equations are therefore:Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 191a(Ir)z0 (2.2.27)r arr= aN! (2.2.28)= (2.2.29)a- a— a2m-2-- (2.2.30)or ciz ciz or rcirThe boundary conditions for the above equations will be discussed later.2.2.3 Structure of the solutionsIn the derivation that follows, we willassume that a “quasi-similarity” solution UOOdescribes the vortex in a freestream. Thephrase “quasi-similarity” is here used todescribe a polynomial solution in which each___________________________________term of the polynomial is a power product of Figure 2.1 Vortex in afreestreamsome function of the axial distance (the amplitude function), and a different function ofthe similarity variable (the similarity function). If a quasi-similarity solution exists for theflow in such a vortex in a freestream, one might expect the similarity variable to takemuch the same form as that of a free jet or boundary layer, i.e.:(2.2.31)With this choice of similarity variable, the quasi-similarity solution for the three velocitycomponents and pressure is then posited to take the form:= l+U1().FO1) l+i (2.2.32)r( )=V (i). R (ii) = 1)r(j) (2.2.33)Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 20= ‘P)•H1(il) = j’J(1) (2.2.34)= = P(j) (2.2.35)where the amplitude functions (those to be functions of only ) will be verified insection 2.5 to take the form:V()= C• , P1()=-’, F()=2& (2.2.36)zThe coefficients denoted by capital C are constants. Substitution of (2.2.32) and (2.2.33)into (2.2.27), yieldsji+L0[F(ii),R11)]=0where L0 is an ODE operator. L[F(r),R1(ri)j=0 must hold because is arbitrary.Similarly, substituting (2.2.33)-(2.2.36) into (2.2.28)-(2.2.30) results in three otherordinary differential equations for each order (i.e. --, i = 1,2,3,.”) of the polynomialsolution.2.2.4 Governing equations of the first orderThe first order partial differential equations consist of all the terms inverselyproportional to the distance that occur when (2.2.32)-(2.2.36) are substituted into(2.2.27)-(2.2.30):=0 (2.2.37a)ar r aza2 (2.2.37b)a a2• F = (2.2.37c)= _La2(i) +!. a(i3), (2.2.37d)F FNote that F2/ can be replaced by the similarity variable 12 in the above equations.Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 212.2.5 Governing equations of the second orderLike the first order differential equations, the second order partial differentialequations consist of all the terms inversely proportional to the square of the distance :a(i5r)2 aQ)2O (2.2.38a)} (22.38b)(2.2.38c)=P a2() OZ)2 } (2.2.38d)Similarly, 2/ can be replaced by the similarity variable r2 in the above equations.2.2.6 Governing equations of the arbitrary i-th orderSubstitution of (2.2.32)-(2.2.36) into (2.2.27)-(2.2.30), and retention of all the termsproportional to --, yields the governing equations for the ith order problem:+ += 0 (2.2.39a)ra2 +— J. )z j(2.2.39b)+‘{(i3r)i.[T’-i + 2]}=.(fr (2.2.39c)Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 222() +La)1 a(i),_L+’f( a()1_a2 — Z J(2.2.39d)i-Il a(i):.+1(’Or).•ZJIn like fashion to the first order problem, the combination j2/ can be replaced by thesinlilarity variable r2 in the above equations.2.3 The First Order Problem2.3.1 Derivation of the Governing Quasi-Similarity EquationsTo simplify subsequent mathematical manipulations, we state below some simpleidentities. First, we recognize that the following fundamental relations hold for thesimilarity variable defined by r =ai1 1 iar z r(2.3.2)3 2Next, considering an arbitrary function p(, ) f() . g(r), the following differentialrelations are found to be valid:= { 2 f’() . g(i) -1. (2.3.3)z 2z f(z)=g’(i) (2.3.4)r= !Lf. g”(r) (2.3.5)Making use of the above relations, the continuity equation (2.2.37a) for the first orderproblem can be rewritten as2Ji Ro)+ RiO1)1 2U) (-l’(i) = 0 (2.3.6)U1(z) 1 j 11(z)Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 23A “quasi-similarity” solution can exist for the continuity equation (Equation 2.3.6) if theequation can be made to be solely a function of the independent variable, Equation2.3.6 is a function of 17 alone provided:1z (2.3.7)-.IzU1(z) = C1 Z_al (2.3.8)where C1, a1 and K are constants. The constant K does not affect the computed velocities,and is set to 1 for simplicity. Using equations (2.3.7) and (2.3.8), the continuity equationcan be rewritten as211R( )+2R(11) = 12. (11)— 2a1 Th (1) (2.3.9)The tangential momentum equation (2.2.37b) can similarly be simplified to read2 ‘“ H (ii) -iH(1) = 2H1)$ H(1) (2.3.10)‘P1(z) iiA quasi-similarity solution exists for the tangential momentum equation if and only if(2.3.11)where C2 and b are constants. The simplified ODE of the tangential momentum equationis thus,21H (1)+[12+6].Hj(fl)_2bl.1.Hi(1)=0 (2.3.12)In similar fashion to the derivation of (2.3.6) above, the ODE form of the radialmomentum equation (2.2.37c) can be shown to be—G1(11)=Z lZ.l.H2(l) (2.3.13)F(z)for which a quasi-similarity solution exists only if(2.3.14)Finally, the axial momentum equation (2.2.27d) can be rewritten as21.F(1)+[12 +2].1 (11)—2a i(i)P() (2.3.15)= i- .[2(1+2b)i.G(i) .G’(l)jU1(z)The following relation must hold for a quasi-similarity solution of (2.3.15) to exist,Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 24F(z)= (2.3.16)U1()where 3 is a constant.The ODE form of the axial momentum equation is then,2i I (11)+[12 +2j.I (11)_2a •11I(1i) (2.3.17)=[2(1+2b)1.G(1)_.T2 .G(r)]Combining (2.3.8) with (2.3.11), (2.3.14) and (2.3.16), one obtainsF(!)_i()_1+l_aj (2.3.18)U1(z) U1(z) C1Because is a constant, we havea1=2b+1 and (2.3.19)The constants b1 and C2 can also be determined by applying the condition that thecirculation around the vortex far from the vortex axis is independent of downstreamdistance (a consequence of Kelvin’s Theorem):= constant (2.3.20)The dimensionless form of (2.3.20) is (F•). =——, thus( = ( = =1 (2.3.2 1)It can be verified that for the second or higher order, (2 ., ) [12 H1 (1])] 0(refer to section 4.1 in Chapter 3). Thus, (2.3.21) can be rewritten asC2 .[12 .H1(ii)j =- (2.3.22)and therefore, b1 = —1 and C2 = 1 (2.3.23a)21t[1]2 H1 (ii)]thus a1 =2b+l=—1, C1 =---= 2 (2.3.23b)f3 47t213[12.H1(1)]To summarize, the following amplitude functions have been obtained for the firstorder problem:Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 25U1()= 1 2 (2.3.24a)42[2 H (n)]TJI—’_ Iv1Zj,—Jz(2.3.24c)= 21t[ H1()] .= 24112Hj (n)] U1 () (2.3.24d)where f3 is a constant.2.3.2 Boundary Conditions on the Quasi-Similarity EquationsWith all the unknown constants (except 6) evaluated, we may rewrite the governingODEs that determine the first order similarity functions:2iiR1(i )+2R1(1) = 2 IOi)÷ 2i1• F(11) (2.3.25a),, I2rj•H1 (‘ri)+0i2+6)•H1 (rj)+21•H1rj)=O (2.3.25b)G1(i) = i•H(11) (2.3.25c)21•()+(fl +2) H(i)] (2.3.25d)Equations 2.3.25a-d are four coupled ordinary differential equations in the four unknownfunctions R1, F1 ,G1 , and H1. The equations are first order in R1 and G1, and second orderin F1 and H1, and consequently require one boundary condition on R and G1, and twoeach on the F1 and H1. The boundary conditions that apply to these equations are:R1(O)=O (2.3.26a)H1 (0) = 1, H(0) = 0 (2.3.26b)(2.3.26c)G1(oo)=0 (2.3.26d)(2.3.26a) must hold to ensure the flow is symmetrical about the vortex centerline. H1(0) isset equal to one for simplicity (2.3.26b); a constant different from one will not affect thecalculated tangential velocity. The second equation of (2.3.26b) ensures the secondChapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 26derivative at the origin has a finite value for symmetrical flow = 0). The firstequation of (2.3.26c) holds because f3 in the amplitude function U1(z) is a free constant.A non-unity value of the constant F1(O) will not affect the final results. The secondcondition of (2.3.26c) ensures the second derivative exists at the origin for a symmetricalflow (-- = 0). (2.3.26d) arises from the freestream velocity r=OThe ODEs given by Equations (2.3.25a-d) form a decoupled system. We can solve(2.3.25b) for H1(rI) first, then (2.3.25c) for G1(rj), then (2.3.25d) for fj(11), and finally(2.3.25a) for R1(rI), as will be discussed in Chapter 3.The limit on 112 . H1 (i) in the amplitude functions (2.3.24a-d) can be evaluated. Thenon-singular solution of (2.3.25b) with the boundary conditions (2.3.26b) is4( -H1(rl)——--I 1—e (2.3.27)JThus,[12. H1 (TI)] =4 (2.3.28)The amplitude functions (2.3.24a-d) for the first order problem can therefore besimplified and rewritten as:(2.3.32a)64it zlzV (z) = (2.3.32b)..JzP() —_____=.U1() (2.3.32c)64ic.JJ() 1=8i3.U() (2.3.32d)8it. zwhere 3 is a constant. There is only one constant, 13, which is undetermined in the firstorder problem. We will show in the next chapter that this constant is closely related to thetotal drag of the wing that generates the wing tip vortices.Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 272.4 The Second Order ProblemThe ODE form of the second order continuity equation (2.2.38a) is similar to the formof the first order continuity equation:2iR(i)+2R(fl =12. F’(1)—2a 11F(1i) (2.4.1)z ,(z)where a., = (2.4.2)- U7()A quasi-similarity solution exists for (2.4.1) if and only ifV2(z)= ,— (2.4.3)qzThe tangential momentum equation for the second order problem (2.2.38b) can besimplified to:2P)H,(1)-1H(11) = 2H’(1)$H(1)1 (244- U1((i) {[ii+ 4H1 (11)]R (1)- [21H, (1) + 12H(1)]1 (1)}The following relation has to hold for a quasi-similarity solution of (2.4.4) to exist.U1()P)=a (2.4.5)‘P,(z) 2where 2 is a constant.thus •W() •U1’() b =—2 (2.4.6)‘P2(z) U(z) I’1(z)The tangential momentum equation thus can be rewritten as21•H’(1)+(1 +6)•H(1)+41•H2(1)(2.4.7)=c f{2iiH(1) + 4IJ (n)f R1 (1) _[12 H() + 21H (1)] F (ii)}The radial momentum equation (2.2.38c) can be rewritten as the following ODE:G(i) = 21H(1)•1) (2.4.8)Quasi-similarity solutions exist for (2.4.8) as long as)=P1()2) ±.p().U() (2.4.9)Finally, the axial momentum equation (2.2.38d) can be rewritten asChapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 28211F ()÷O.2 + 2)F’(q)—2aflF(ri) =—‘ [4o) + 2ii3H1(11) •112 (i)] (2.4.10)U2(z)U,+ (1)1’(ii) - iiI (ia) [2I (11) + ii1’(ii)j}U2(z)A quasi-similarity solution exists only if the following relations hold(2.4.11)U2(z)U)=c2 (2.4.12)U-,(z)where J6 and c2 are constants.ZUC) 2.U1’() (2.4.13)thus a2— =U2() U1()By applying the relations (2.4.9), (2.4.11), (2.4.12) and (2.3.16), one findsF) c2F(z)-U1( ) C213 (2.4.14)Comparing (2.4.11) with (2.4.14), one obtainsC= (2.4.15)f2To summarize, the following relations must be satisfied in order for quasi-similaritysolutions to exist:__________1 (2.4. 16a)U2()=±.U=(6C2( 1 1 (2.4.16b)C2___1 2 2 (2.4.16c)2 .P().U()=)2.C13 C21 2 8ic2 (2.4.16d)12 uJ•c2 I(U)=(M2) •2cNotice that there are only two independent constants in the second order problem (exceptfi. It is permissible to select c2 = 1 and a = 1, and let 112(0) = Ch2 and F2(0) = Cf2,Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 29without loss of generality. From equation (2.4.15), /3 = /3. Therefore, the amplitudefunctions can be expressed asU2() =U() (2.4.17a)V2()=—1•U) (2.4.l7b)41ZP,(z) = 3•U(z) (2.4.l7c)P2 (z) = 8t13 U1 (z) (2.4. l7d)The second order quasi-similarity solutions of the vortex flow in a freestream can beobtained by solving the following ODEs with their boundary conditions:2r1R(1)+2R( )=2 .F’(r)+4rj.F2(ri) (2.4.18a)2r1•H,”(1)+(T1+6)• ,’(1)+4T1.H((2.4.1 8b)= {2riH(i) +4H1(‘rj)j. R1 (ii) _{12 H1(1) + 2rH1(ii)]. F (rj)G(i) 211H( ) ) (2.4.18c)21 F, (i) + (2 + 2) . ‘(ii) +4 F ()=-_.[4r.G(rj)+2i,3.H1(ri).H2] (2.4.18d)+ 2rR1( ) . j7’()— rF (ri).[2F (i) +where /3 is the constant that appeared in the first order problem. The boundary conditionsof this second order problem areR2(O)=O (2.4.19a)H2(O)= C/i1 = constant, H(O)=O (2.4.19b)1(O) = Cf, = constant, F’(O) = 0 (2.4.19c)G2(oo)=0 (2.4.19d)Analogously to the first order boundary conditions, (2.4. 19a) must hold to ensure the flowis symmetrical about the vortex centerline. The constant in the first equation of (2.4. 19b)is related to the second order vorticity at the origin r = 0. The second equation of(2.4. 19b) ensures the second derivative at the origin has a finite value for symmetricalflow = 0). The constant in the first equation of (2.4.19c) represents the amplitudear r=Oof the second order axial velocity on the vortex centerline. The second condition ofChapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 30(2.4.1 9c) ensures the second derivative exists at the origin for a symmetrical flow(- = 0). (2.4. 19d) arises from the freestream velocity condition.r=OThere are two constants, Ch2 and Cf2 (in addition to /), involved in the second orderproblem. They will be discussed in the next chapter.2.5 The i-th Order Problem (i = 2,3,4...)We will prove by mathematical induction the proposition that the i-th order quasi-similarity solution of the tip vortex can be obtained by solving the following ODEs withtheir boundary conditions:2riR’+2R1 12 •F’+2i•rj•1 ([’+ (112 + 6)II’+ 2111Fl1‘— 1 1 (2.5. ib)=+ 4fl )R -(12w + 2(i- j)11n )iG1=i(H.H+) ( +2)’+2i1 = _.[2iflGi +11(H .H1+)]j=1 (2.5.ld)+{2iR -l.+‘j}where /3 is the constant that was established in the first order problem solution. Like thesecond order problem, the boundary conditions of this i-th order problem are:R(0) = 0 (2.5.2a)H1(O)=ch, H,’(O)=O (2.5.2b)I (0) = cf1, F’(O) =0 (2.5.2c)G,(oo)=O (2.5.2d)The following relations have to be satisfied if the quasi-similarity solutions exist:= U,’() (2.5.3a)Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 31l’ () = =. U () (2.5.3b)(2.5.3c)‘{ () = 8it• U () (2.5.3d)We begin our proof by first substituting the quasi-similarity parameters into thegoverning equations of the i-th order problem (2.2.39), yielding:• (2R[+2R.)+U1). z 0 (2.5.4a)U1(z)2.’I’().11H]-‘ I 2P.() 1•11—Ti2H .FJ (2.5.4b)+4H1].R}(2.5.4c)U().[22 +6)•’- 2•U)[2’()G2G]P()(2.5 .4d)F 2.R]}Now, we proceed with the proof by induction by checking that the proposition is correctfor i = 2.For i =2, we have shown in section 2.4 thatU2() = U12() (2.5.5a)V,() =—=•U() (2.5.5b)Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 32P,(!) = f3.U) (2.5.5c)2 () = 8itf3. U () (2.5.5d)while the quasi-similarity functions satisfy the following ODEs,2rjR+2R 2F’+4iF (2.5.6a)2ri• H’+ (2 +6) H + 4ri F!2(2.5.6b)=[211H1’+4j.R,_[12 .H[+21,Hj.FG = 21 H1 H2 (2.5.6c)211F”+(’ri+2)•F’+4 i.F =—13.[411•G2+2 i3.H.Hj(2.5.6d)+2rR1 •F’—llF .{2F +iiF’jThe proposition is obviously correct for i =2.Next, we assume the proposition is correct for i = 1, 1 k. This means that for the 1-th order problem:U, () = U () (2.5.7a)V1() =_J=.U(C) (2.5.7b)p () = f3. U1’ () (2.5.7c)P, () = 8ic3• U’ () (2.5.7d)and the quasi-similarity functions satisfy2TjR[+2R, =12 F’-f-21•1F (2.5.8a)21H[’+ (rj2 + 6)H[+ 2liJl,= {(2TIH + 4H,)R — (112H + 2(1—j)i,H,)F} (2.5.8b)G=n.(H .H,÷1) (2.5.8c)21F’- (12 + 2)J’+ 211F = [3 .[2liGi + ii(H H1÷i)](2.5.8d).- .{2(l_j) +TI’j}Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 33Now, we prove the proposition is correct for i = k + 1.The general governing equations for (k÷ 1)-st order problem (2.5.4), replacing i withk+ 1, can be written as• () (2R÷ + 2Rk÷ ) + Uk+l () [2 Uk’ (z) _12. (2.5.9a)Uk÷l()k+1 () [21H1+(12 + 6) ‘ 2 . :+ () .lHk+l]k+I— w‘k÷I()k÷1—112. H+ii]. } (2.5.9b)= {U .P().[ ‘Pk+1J(z)j=Ik÷1—l+ ( k+lJ(Zi.[2h1’k÷lJ +4Hk÷1J]•R1}j=lk÷1() G÷1 = {pj () k+l-j+1 () . Hk+l 11k+l-j+l j} (2.5.9c)j=lUk÷l . [21 + (12 + 6) ‘ 2 U’÷1 ()k÷l—Uk+1 (z)•11•Gk+j 12L +() (2.5 .9d)k÷1—l E2 .U÷1() ] F }+ UJ().Uk+lf().[ Uk+IJ()L.I—I+ {. ).uk÷lJ().[21F;+lJ .R]}By using (2.5.7), it is apparent the following equations must be satisfied for quasisimilarity solutions to exist,Uk+l() (2.5.lOa)Ck +1= k+I(2.5. lOb)CChapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 34() = • () U () (2.5. lOc)f3 c‘k+l(Z) =—---Y1(!)•U)= ‘ (2.5.lOd)k+Iwhere Ck+l , ,6+ and a’k÷l are constants. ,8 is the constant appearing in the first orderproblem. Like the second order problem, there are only two independent constants(except /5j. It is permissible to select Ck+I=l and ak+=l, and at the same time letH1(O)=ChandF1(O)=Cf without loss of generality. Thus fk÷1 = ---- = J3. Therefore the amplitudek÷1functions (2.5. lOa-d) can be rewritten asUk+l() =U’() (2.5.lla)+I () = u1() (2.5.1 ib)=U1() (, (2.5.9) can be rewritten as:2TjR÷1+2Rk+l 1 •F’÷1 +2(k+1)1F+ (2.5.12a)2i1H1 + [2 + 6]H÷1 + 2(k + 1)ilHk÷lk+1—1 1 (2.5.12b)=(2iI-I+_ +4Hk+l_J)RJ [12Hz +2(k+1—j)’flHk+{JIFJIG÷1 = ‘ri (H . Hk÷I_J÷,) (2.5. 12c)Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 352i1F1 +(r2 +2)Fk’+I +2(k+1)1F,+1=. + l)1G÷ + ( H÷1i+)] (2.5. 12d)k+1—I+-iF .[2(k + 1- j)F +where j3 is the constant of the first order problem.The proposition is correct for i = k + 1, provided that it is correct for i =1, 1 k.Since the proposition is correct for i=2, this concludes the proof by mathematicalinduction.2.6 The Solutions in Complete Function FormCombining all the results obtained previously, the quasi-similarity solution of vortexflow in a freestream is given by:(z,r)=U .[1+U(z).(i)] (2.6.1)1)r(z,r) = .U((z).R,(i) (2.6.2)W(z,r)=8U(z)H1(i1) (2.6.3)or 8(z,r) = 8•U VZ U(z)() (2.6.4)where 7(ii)=iiH1(ri)p(z, r)—p. = pU13 U (z) . G1 (11) (2.6.5)where r = r!—-, andV VzU1(z)=642Uv•z (2.6.6)Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 36where 3 is a constant. The solutions (2.6. 1)-(2.6.5) are convergent provided that thesimilarity functions are of the order of unity andU1 (z) <1 (2.6.7)The quasi-similarity functions of arbitrary i-th order H(rI), G1(), F(r1) and R1(1)can be solved one by one from lower order to higher order by solving the ODEs (2.5.1)with the boundary conditions (2.5.2). Details will be discussed in the next chapter.2.7 Discussion Of AssumptionsFive assumptions have been made to derive the governing equations (2.2.5)- (2.2.9)used in this analysis. Those assumptions are:(1) incompressible, steady, laminar flow(2) axisymmetric flow(3) 1•)(4) U v U U vaz2 z2(5) the circulation of the vortex is independent of the downstream distanceThe first two assumptions are straightforward, although they will be violated by anymechanism that causes vortex instability. Consider now the third and fourth assumptions.Using (2.6.2) and (2.6.4), one findsIU . I r 1 v—°i . oc—.U(z) (2.7.1)V z 8iqU.vzJ FAlso, be:ause T=rifZ, (2.3.3) together with (2.3.5) yields(2 2 (2 i(2U (z) (2.7.2)a2 j-’j a2 j’} 2 j’} 4• r’}2 —‘ —2ar ar- rOne can verify that the terms in the governing equations neglected due to assumptions (3)and (4) are proportional to 3.) U1 (z). In contrast, other terms in the governingChapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 37equations are of order unity. Assumptions (3) and (4) are justified provided that thevortex circulation is much greater than the kinematic viscosity of the fluid. Because thekinematic viscosity is commonly very small (V lO m2 / s for the laminar flow of water,v 105m2Is for the laminar flow of air), these assumptions will almost always bejustified for real wing tip vortex flows.The fifth assumption — that the circulation of the tip vortex is independent ofdownstream distance — is a consequence of the Helmholtz vortex law requirement thatvortex lines not terminate in a fluid. The assumption could be violated if vortex linesdoubled back on themselves to form vortex rings, but this behaviour does not occur in thequasi-similarity solution. Alternatively, it can be viewed as a consequence of Kelvin’sTheorem (DF I Dt = 0). Kelvin’s Theorem applies here because on a material loop farfrom the vortex axis there is negligible effect of viscosity, and this flow is incompressibleand without body forces.The assumptions made in the derivation of the quasi-similarity solutions are thus notparticularly restrictive. Rather, convergence problems of the quasi-similarity solutionsmore significantly limit their application. In particular, these limitations can berepresented mathematically by,IUt(z)I< 1 (2.7.3)The first order axial velocity excess or deficit in the vortex in a freestream is thusconstrained to being less than the magnitude of the freestream velocity.Finally, let us discuss the quasi-similarity solutions. The quasi-similarity expressions(2.6.l)-(2.6.5) can be verified to be the solution of the governing system (2.2.5)-(2.2.8)consisting of the continuity equation and simplified N-S equation. Substitution of(2.6.l)-(2.6.5) into (2.2.5)-(2.2.8) yields:(2.7.4a){u; ()• M (I1 ,f3)} 0 (2.7.4b)2z(2.7.4c)Chapter 2 Quasi-Similarity Methodfor Vortices in a Freestream 38±.{u().M,G1,H,)}O (2.7.4d)provided U1(< 1, because•C01(R1,F)=M1(H,f3)= MS,(G,I1) = M(F,G,H3) O, i=l,2,3,...where C0, represents the continuity similarity equation (2.5.1 a), M the tangentialmomentum similarity equation (2.5. ib), M the radial momentum similarity equation(, and M1 the axial momentum similarity equation (2.5. id).Therefore, the first order terms in the polynomial are the approximate solution of thegoverning equation with accuracy of O( U (i)) (order U ()). The residue for the secondorder solution is O(U()). To summarize, the quasi-similarity solution exists providedU1( < 1. The more terms in the polynomial we obtain, the more accurate is the solution.2.8 ConclusionsA quasi-similarity method for 3-D vortices in a freestream is discussed in this chapter.The general solution in polynomial form has been found. The quasi-similarity solutions1 F2are divergent when z < 2 (or U1 (z) > 1). Thus, these solutions are most64it f U, vaccurate far downstream. The quasi-similarity solutions have some interesting features.All three velocity components decrease with increasing downstream distance. The radialvelocity diminishes mostly rapidly, followed by the axial velocity. The tangential velocitychanges most slowly. For example, for the first order problem, the tangential velocitydecays as one over the square root of the downstream distance, while the axial velocity isproportional to one over the downstream distance, and the radial velocity is proportionalto one over the downstream distance to the power of 1.5.Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 39Chapter 3 Quasi-Similarity Solutions for Wing Tip VorticesIn the previous chapter, the quasi-similarity method was discussed. The solutions inpolynomial form of the nonlinear governing equations have been presented. Each term inthe polynomial consists of the product of two functions. One function is called theamplitude function, and is a function of downstream distance only. The other function,called the similarity function, is a function of a similarity variable that is a combination ofthe radial coordinate and the axial coordinate. The amplitude functions of eachpolynomial term have been obtained in the previous chapter. The ODEs and boundaryconditions for the similarity functions in each order of the polynomial have also beenlisted. There is one constant, fi, involved in the first order problem, whereas twoconstants, C’h1 and Cf. are involved in the higher order problems (i 2). In this chapter,we solve the first and second order problems numerically and analytically. It is found thatthe constant ,6 is closely related to the total wing drag. The two constants Ch and C’f(i 2) can be evaluated as well. The quasi-similarity solutions meet the requirements thatthe vortex circulation and wing total drag are independent of the downstream distance.Fortunately, first order velocity components and pressure in completed functional formcan be obtained. The tangential velocity component and pressure in the second orderproblem can be presented in completed functional form as well. Second order axialvelocity and radial velocity similarity functions can only be calculated numerically. Thenumerical procedure to be used for solving higher order problems will also be mentionedbriefly.3.1 The First Order Solutions3.1.1 General first order solution in complete function formThe first order problem consists of four ODEs for the similarity functions, along withthe proper boundary conditions:2iiR1(1)+2Ri1)=12 .I (1)+21 1(11) (3.1.1)Chapter 3 Quasi-Similarity Solutionsfor Wing Tip Vortices 40211• H1 (11)+(12 +6) H1 (1)+21H()= 0 (3.1.2)G(i1)=11•H(i1) (3.1.3)211.J(1)+(1,2 +2).) (ii)+2m1(ii)=—f3.{2ii.GOi +i3.HO1)] (3.1.4)where is a constant. The boundary conditions of this first order problem areR1(O)=O (3.1.5)H(0)=LO, H(O)=O (3.1.6)F(O)=1.0, F’(O)=O (3.1.7)G1(oo)0 (3.1.8)(3.1.5) must hold to ensure the flow is symmetrical about the vortex centerline. We applythe first equation of (3.1.6) because the choice of value ofH1(O) does not affect the finalresults (one of the characteristics of the similarity method). The second equation of(3.1.6) ensures the second derivative at the origin has a finite value for an axisymmetricflow. The first equation of (3.1.7) holds because J3is a free constant, the second conditionensures the second derivative exists at the origin for a symmetrical flow. (3.1.8) arisesfrom the freestream velocity condition.The following relations must hold if quasi-similarity solutions exist:U1()= (3.1.9)64it f3z(3.1.10)(3.1.11)1=83U() (3.1.12)8itzThe ODEs (3.1.1 )-(3. 1.4) are decoupled. The non-singular solution of (3.1.2) with theboundary conditions (3.1.6) is4( .iiH1(rI)=—- 1—e (3.1.13)}Thus, the solution of (3.1.3) with boundary condition (3.1.8) isChapter 3 Quasi-Similarity Solutionsfor Wing Tip Vortices 41G1 (i) —8+ 16e —8e2 (3.1.14)where Ei(n,x) = $----.dt is the exponential integral function. Substituting (3.1.13) and(3.1.14) into (3.1.4), one finds that the solution of (3.1.4) with the boundary conditions(3.1.7) is(i) =4e .ç{e (3.1.15)The solution of (3.1.1) with the boundary condition (3.1.5) is(3.1.16)where F1(i) is given by (3.1.15).By combining the amplitude functions and similarity functions obtained above, onecan get the first order solutions in completed function form.= F•Ie4.J!e4EEi12,_2Ei12,Y_+1l.dy+±.e4 (3.1.17)z() l6it2vz y [ 2 ) 4) J 413 j= i._L_.11e4__.11e4 (3.1.18)vz 27tT ) 2tr )r(1)= 322U.•s.{ie(3.1.19)P(l)=16’-e+ Ei[l]_ Ei[1} (3.1.20)Equations (3.1.1 7)-(3. 1.19) show that the radial velocity in a tip vortex falls most rapidly(as z’5) with downstream distance. The axial velocity changes more slowly with z(roughly as z1), and the tangential velocity varies most slowly with downstream distance(as z°5).Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 42It is interesting to note that this first order tangential velocity distribution is identicalto that of the Lamb-Oseen vortex provided one replaces the time variable in the LambOseen problem with its equivalent, in the tip vortex problem.The Runge-Kutta method was also used to solve the first order problem. The RungeKutta results agree to within numerical error with the exact analytical results given by(3.l.13)-(3.1.16). The results corresponding to 3 = 0.361 (this is the case when the secondderivative of the axial velocity component in the radial direction at the origin is zero) areshown in Figure 3.1. 7 (rj) = i H1 (Tj) is the similarity function for the tangential velocitycomponent in the tip vortex.3.1.2 Tangential velocity and pressureNotice that the first order tangentialFirst Order Similarity Functionsvelocity component and pressure1STdistribution are independent of the constantT1(q)1 We note in addition thatG1(0)—2.773 (3.1.21)and this result is also independent of .Furthermore, 7 (‘ii) = ‘qH1 (ri), the2function describing the radial distributionFigure 3.1 The first order quasi-similarityof tangential velocity, is a maximum when functions. I = 0.361= 2.25, and its maximum value is= 2.25) = 1. 2763 (3.1.22)Thus, to first order, the radius of a tip vortex (R1) is given byR. 2.25R1 =2.25W I—, or —-= (3.1.23)vUo zThe similarity between the growth of a tip vortex and the growth of a laminar, zeropressure-gradient boundary layer (for which /x 5/.J) is apparent.Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 43The first order vortex radius is independent of the constant 1. This independenceimplies that the radius of the wing tip vortex far downstream (where the first ordersolution is a good approximation) increases as the square root of the downstreamdistance.3.1.3 Effects of the constant [3 on the axial similarity functionsThe first order axial velocity consistsof an amplitude function multiplied by asimilarity function. The amplitudefunction is always positive if the constant13 is positive. The similarity functionscorresponding to several positive valuesof the constant j3 are plotted in Figure3.2. When [3 equals 0.361, the secondorder derivative at the origin is zero. For8 >0.361 the peak in the axial velocitydistribution occurs away from the vortexaxis. This finding is consistent withobservations by Mason and Marchman(1972) of a real wing tip vortex. For 0<fl<0.36 1 there is still an axial velocityexcess in the tip vortex, although the axialvelocity decreases monotonically from thecenterline. Green and Acosta (1991) haveobserved similar behaviour.*When [3 is negative, the first orderFigure 3.2 Axial velocity excess in wing tip vortices.(The curves from top to bottom correspond with thevalues ofJ3 listed in the column to the right)Figure 3.3 Axial velocity deficit in wing tip vortices.(The curves from top to bottom correspond with thevalues of/i listed in the column to the right)* One must be careful with these comparisons. As we shall see, 3>O corresponds with negative wing drag,which was not the case for either of the two experiments mentioned. In addition, both experiments werecarried out for small z, where this theory likely does not apply.1.4Effect of BETA on Axial Similarity Functions0.50.36 0.70o ‘f so ‘ c ‘t- C’ t 00 V- - —TIEffect of BETA on Axial Similarity FunctionsF ( ) -0.1-0.361-0.5-0.7-0.4-0.6-0.800— — so —Chapter 3 Quasi-Similarity Solutionsfor Wing Tip Vortices 44amplitude function is always negative. The first order similarity function is shown inFigure 3.3. Because the axial velocity is the product of these two functions, the behaviourshown in Figure 3.3 implies a centerline axial velocity deficit combined with a small coreedge velocity excess. This kind of axial velocity distribution was reported by Logan(1971) and Thompson(1975).Baker (1974) and Green and Acosta (1991) also observed an axial velocity deficit byusing LDV and PIV respectively. However, they did not report a core edge velocityexcess, possibly because of the small downstream distance in their experiments, or theircomparatively high (and therefore turbulent) vortex Reynolds number.3.2 Drag due to a single quasi-similar vortex in viscous flowA single trailing vortex is considered inthis section. The axial momentum equationwill be applied to a control surfaceenclosing the wing tip region. The controlsurface has the form of a right cylinder withgenerator parallel to the z-axis, whichmarks the centerline of the wing tip vortex, Figure 3.4 Control volume around theand the area A at each end face (Figure wing tip.3.4). The upstream end face and the curved surface of the cylinder are both at a largedistance from the wing tip (compared with the vortex radius ), so that conditions thereare approximately as in the freestream. Then, in the usual way, we findz-momentum flux outwards across curved surface = pU., J(u— .z-momentum flux outwards across end face= pJ — u) . dAresultant normal force on end faces=(p—p) dA*Tip vortex radius is just a few percent of the chord (Green and Acosta 1991, Fruman et a!. 1995), so it ispossible to select a radius that is large relative to the vortex yet small relative to the wing. Batchelor (1964)used the same control volume.pzDRAGChapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 45where i and p are, respectively, the z-component of the velocity and the pressure at thedownstream end face, a distance z from the wing tip. One can verify that the ratio of theshearing force on the curved surface of the control volume over the force due to pressuregradient and momentum exchange is proportional to /R2. The viscous stress at thecontrol surface is therefore small and can be neglected, when .& approaches infinity. Thedrag D on the wing due to this single vortex is thus given byD =J{p. —p+p .(u —i)}.dA (3.2.1)The integral of (3.2.1) can be represented as a polynomial by using the previously derivedformulae for pressure and axial velocity components in the vortex.—p+pu, •(U—1)r i-I 1 (3.2.2)Substituting (3.2.2) into (3.2.1), the drag of this single vortex can be represented asD =-2p• u G10i) + F1 (1) + (i) . F1iOi)] 1• di}orD=-2pF2{2].4J[G.i+F+F1].d}(3.2.3)Because the wing drag is independent of downstream distance , (3.2.3) impliesD = urn f[Gi(ii)+ i •dii, for i =1 (3.2.4)32Jt1R-and fori2 (3.2.5)At first glance, G1 (rj) oc -4- at large radius, and therefore the integration in (3.2.4)11seems to be logarithmically divergent in the outer field (which is why the limit has beenChapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 46taken in that integration). However, we will verify that the integration is finite because‘—- for large radii too, and fortuitously the 1/112 terms cancel.13 TNow let us consider the integration in the first order drag. BecauseIurn J’ ii’di,= urn13 o [ [3 j= 8 limf{J i] .dY}.d[e4J+= urn $.EEjI2,—2Ei1,+1R. di4 (3.2.6)ROOO11L 2) 4) jj 13Thus,urn [G1(i)÷ l)]lldllflR0 [3UR I 2N r / 7 2\1 F / / 2= urn $ —i l—e I +4rI Eu l,— 1—Eu i,— 1 I+--I Eu 2,— 1—2EiI 2,— i+i j!>.dTj+r ) [ t 2) 4,)J ru[ 2) 4) Jj 13(3.2.7)The integration on the right side of the above equation was proved to be zero using MapleV. a symbol manipulation software package. The drag due to a single quasi-similar wingtip vortex is therefore inversely proportional to the constant [3:D=— lim J[G1(ii)+ l1idi,=— (3.2.8)32it-0L 13 j l6itf3It has been assumed in the analysis of Chapter 2 that the distance between the twowing tip vortices is much larger than the vortex radius, and therefore that interferencebetween the two vortices (e.g. the Crow instability) is negligible. The total drag on thewing is thus two times the drag due to a single vortex:(3.2.9a)8t[3112Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 47Thus, = — pF2 (3.2.9b)8tDIn Figure 3.5 the non-dimensionalaxial velocity (defined as the axial7velocity divided by the term 2l6it V•Zis plotted against the non-dimensional2it•Ddrag (defined as ). The larger thepFdrag, the greater the deficit of the axialvelocity on the vortex centerline. In a Figure 3.5 The effects of wing total drag on the axialreal trailing vortex an axial velocity velocity. (The different curves from top to bottomcorrespond with the different non-dimensional dragsexcess is often observed just tabulated to the right)downstream of the trailing edge (Green and Acosta 1991). This velocity excess occurs asa result of the wing tip roll up. After the shed vorticity has been fully rolled-up into thewing tip vortices, usually far downstream of the wing, only an axial velocity deficit canexist for a wing experiencing a positive drag force. An axial velocity excess can exist fardownstream of a wing generating thrust (e.g. with tip mounted engines).Figure 3.6 shows the non-dimensional radial velocity (defined asthe radial velocity divided by the termF2 ) as a function of the16it2.JiJc.z’5non-dimensional drag. The amplitude ofradial velocity is small_____Figure 3.6 The effects of wing total drag on the radialrc—..Ju1() oc —), and it decays velocity. (The different curves from top to bottom‘Uz(I) F F correspond with the different non-dimensional dragstabulated to the right)more rapidly along z than does the axialvelocity. Despite its small value, the radial velocity has important effect on the secondorder terms. As will be mentioned in section 3.3, the existence of a radial velocity leadsEffects of Dntg on Axial VelocityNon-dm2nsional Drag= -‘-5-0.69-0.5-0.36= 2• 0‘> -I0z ---30.360.50.692.5Effects of Drag on Radial VelocityNon-rjinnsionaI Drag— -2.5-4169-0.5-0.36—t0.360.5(1692.5Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 48to one important conclusion of this chapter — the second order tangential velocity mustbe zero.3.3 The second order solutionsSubstitution of the first order solution (3.1.16) into ODEs (2.4.18a-d), which specifythe second order problem, yields2rR(ri)+2R,(rj) = rj2 F,’(rI)+4r. F,(rj) (,.H’(11)+(1+6).H(rI)+4rI.H,(1)=O ( HrI) (•(11)+(fl+2).F,’(1 )+4fl(1) (33 id)where fi is the constant that was established in the first order problem. The boundaryconditions of this second order problem areR2(O)=O (3.3.2a)H2(O) = Ch., = constant, H(O) = 0 (3.3.2b)F, (0) Cf, = constant, F,’(O) = 0 (3.3.2c)G2(co)=0 (3.3.2d)(3.3.2a) must hold to ensure the flow is symmetrical about the vortex centerline. Theconstant in the first equation of (3.3.2b) is related to the value of second order vorticity atr=0. The second equation of (3.3.2b) ensures the second derivative at the origin has afinite value for symmetrical flow. The constant in the first equation of (3.3.2c) is relatedto the second order axial velocity on the vortex centerline, the second condition ensuresthe second derivative exists at the origin for a symmetrical flow. (3.3.2d) arises from thefreestream velocity condition.The following relations have to be satisfied in order for the quasi-similarity solutionsto exist,(3.3 .3a)Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 49(3.3.3b)(3.3.3c)= 8iU() (3.3.3d)There are two constants, Ch2 and Cf2. in this second order problem. Two conditionsmust be satisfied by the second order solution— the downstream independence of boththe vortex circulation and the total drag of the wing. These conditions are representedmathematically bylim[ri2.H(i)J=O (3.3.4a)$[2 (3.3.4b)3.3.1 Second order tangential velocity and pressureThe non-singular solution of ODE (3.3. ib) with boundary condition (3.3.2b) is= Cli, e (3.3.5)The solution of (3.3. ic) therefore isG2(i) = Ch .[4Ei(1,1)_4Ei(1,)j (3.3.6)The tangential velocity component and pressure distribution are therefore= U, •Ffr(2)(z,T1) = 8iti3U(z) H2(ij) (3.3.7)= pU .f3.U(z)G,(ri) (3.3.8)Ch2•F3thus, 1)e(,)(z,T1)=11e (3.3.9)512ic3J3U.(vz)= Ch2pT2 .[Ei(1,c)... Ei(1,!)] (3.3.10)1024ic (vz)Now let us consider the constraints (3.3.4a). Sincelim[i2. 112(11)] = Ck li(112 .e-5) = 0 (3.3.11)Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 50(3.3.4a) is satisfied without placing any constraint on the constant Ch2. (3.3.4b) can beevaluated by integrating all the terms in (3.3. id), yieldingF (1)+(12(3.3.12)=J{.[41.G,(1)+21 .Hj(11).H,(1)J_2n12(1)}.d11Because$ + 2). F,’(l) d = (12 + 2). F, (1: - 2$ F, (1)o ° (3.3.13)=—2Cf,—2$F(ri).ridri$ 2iF, (1) . d = 2{1F’(1) -j F’(1) di}and 0 0 (3.3.14)= -2{F,O1)1} = 2Cfthus, (3.3.12) can be rewritten asj{1.F,(1+121}.d1 =_J{2l.G2(l)+[l2.G(i)]}.do o (3.3.15)= —13$ G,(1)1•dflThat is${13.G2(1)+F(1)+I2(1)}.1.d1 = 0 (3.3.16)(3.3.16) is exactly (3.3.4b). The equivalence implies that the second order drag equalszero, independent of the constants Ch2 and Cf2.To determine these two constants, let us consider the physics behind the quasi-similarity method. The first order problem Consists of the most important terms. Becausethe first order solutions include information on the vortex circulation and drag due to thevortex, they construct the first order approximation to the solution. The second orderproblem can be considered to be an ODE system with the lower order results as inputs.Higher order solutions are used to compensate for the non-linear terms (or higher orderChapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 51terms with respect to the downstream distance) in the governing equation. For the secondorder ODE system, complementary functions, which are the solutions of thehomogeneous equation, exist. The non-singular complementary functions of (3.3. ib) and(3.3.ld) are Cli, e and Cf, .(rj2 —4).e respectively. These solutions are unrelated tothe circulation of the vortex and the wing total drag, and are also not functions of thelower order solutions as well. Physically, it is reasonable to select the constants Ch2 andCf2 as zero so that only the part of the second order solution beyond the first order resultsremains:Ch2 = Cf, =0 (3.3.17)The second order tangential velocity and pressure are thus,= 0 (3.3.18)(3.3.19)Therefore the first order tangential velocity and pressure are actually the second orderapproximation of the solution. In other words, the first order solution are accurate up tothe third order, O(U(z)).3.3.2 Second order axial and radial velocity componentsThe governing equations (3.3.1 a) and (3.3.1 d) can be rewritten as2flR(1)+ 2R()=i2F2’(l)+ 411. F(1) (3.3.20)2Th F (ii)+0i2+2). F’(1)+41 F,(i) = _2112(11) (3.3.2 1)The boundary conditions of this second order problem areR2(0)=0 (3.3.22)F(0)=Cf=0, F’(0)=0 (3.3.23)The axial and radial velocity components can be obtained numerically by the RungeKutta method. Notice that because F1 is a function of the constant J3 F2 must be afunction offlas well.Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 52Second Order Similarity Function0.05-0.3011Second Order Similarity Function0.100.00-0.20 “ ‘.0 t. .0 03 .—05-0.50-0.6011Figure 3.7 The second order axial quasi- Figure 3.8 The second order axial quasi-similarity function. fi <0 similarityfunction. /3 >0The second order radial similarity function can be expressed in terms of the axialsimilarity function,R2(ii)=flF,(11)+.Jx .xdx (3.3.24)2 110Numerical values of the radial similarity function for different values of 3 are plotted inFigures 3.9 and 3.10.By using the Runge-Kutta method and the first order functions, one can obtain thesecond order functions for any value of ,& The numerical results for the axial similarityfunction are shown in Figures 3.7 and 3.8. For all values of /3, the second order axialsimilarity function is negative near 11=0, whereas the second order axial amplitudefunction is always positive. Therefore, the first order solution underpredicts the axialvelocity defect in the vortex core when /3<0 (i.e. when the wing experiences a positivedrag). For the case /3 >0 (a wing generating thrust), the first order solution overestimatesthe axial velocity excess in the vortex core.Chapter 3 Quasi-Similarity Solutionsfor Wing Tip Vortices 53Second Order Similarity Function0.00-0.fO ‘-0.20-0.40-0.30-0.50TFigure 3.9 The second order radial quasi-similarityfunction. JJ <0Second Order Similarity Function0.00:-1.50-2.0011Figure 3.10 The second order radial quasi-similarity function. fi >03.4 The procedure for i-th order solution ( i> 2)The i-th order solution can be obtained by solving the followingboundary conditions and constraints:21R[+2R=1•’+2i•TI•FODEs with the(3.4.1)where fi is the constant that was established in the first order problem solution.Like the second order problem, the boundary conditions of this i-th order problem areR(O)=O (3.4.5)H,(0)=ch1=0, (3.4.6)F(0) = = 0, (3.4.7)2iiH1’4-(i +6)H,’+2ilH,= {(2iH +4H)R _(2HF +2(i—j)i1)} (3.4.2)G = ( . (3.4.3)2iF1’(12 + 2)’+ 2i1 = — .[2i1G +1=’ (3.4.4)- .[2(i_j)F.F’(0) =0Chapter 3 Quasi-Similarity Solutionsfor Wing Tip Vortices 54G1(oo)=0 (3.4.8)There are two conditions the i-th order solutions must satisfy — conservation of thevortex circulation and the independence of the wing total drag with the downstreamdistance. Respectively,lim[12 H (ii)] 0 (3.4.9)and G1(ii)+(+(i)• F (i)}.di = 0 (3.4.10)Meanwhile, the following relations have to be satisfied if the quasi-similarity solutionsexist,U1()=U) (3.4.11)(3.4.12)I(z) =f3U() (3.4.13)‘() =8icI3•U(() (3.4.14)The following procedure could be used to solve these coupled ordinary differentialequations. Suppose that the lower order (less than i-th order) solutions have beenobtained. Then the i-tb order problem can be calculated numerically by solving ODEs(3.4.1)-(3.4.4) with the boundary conditions (3.4.5)-(3.4.8) to find R,, F1, H1, and G1.In summary, the i-th order results can be expressed as= . U (z) F (Ti) (3.4.15)= U rf(1)(z,r)= 8U VZU(z)Ti(1) (3.4.16)r(i) (z,r)= V U( (z) R1 (i) (3.4.17)p(l)(z,r)—pU,l3Ul(z)GI(T1) (3.4.18)1 1-2where U1(z)=64t213 UChapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 553.4.1 Conservation of the vortex circulation FWe have assumed in section 3.1 of Chapter 2 that, in order to conserve the vortexcirculation, the following equation must hold,{2HO)] =0, (i2) (3.4.19)Now we will verify that this relation does hold for the higher order problem. According tothe results of the first order problem, the following asymptotic characteristics can befound,H1(r) —-- (3.4.20a)11G1(rj) )—- (3.4.20b)11-- (3.4.20c)11R1(il) ‘° -- (3.4.20d)11Based on the ODEs governing the high order problems, one can prove bymathematical induction the proposition that the asymptotic results for the i-th orderproblem areR(ii) 4- (3.4.21a)11G1(ii) (3.4.21b)11I(i) °° —- (3.4.21c)11R1(1) ‘ (3.4.21d)11We begin our proof by checking the first order problem (3.4.20). Obviously theproposition (3.4.2 1) is correct when i=1.Next, we assume the proposition is correct for i=l, lk. This assumption means thatfor the 1-th order problem,Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 56H1() >—- (3.4.22a)TIG,(rl) >—- (3.4.22b)TIF(ii) °° —- (3.4.22c)TIR1(1) (3.4.22d)1-I-Now, we prove the proposition is correct for i=k+1. We are going to use L’Hopital’sRule in our proof. For an arbitrary function f(TI) if limf(TI) =0 and limf’(TI) = 0, thenone has,lim = — urn = -‘-. urn f”i) (3.4.23)‘TI °TV 2m- TIprovided these limits exist or are +00 or -oo (Barnett and Ziegler 1991). The above relationcan be rearranged aslimTI2 f’(TI)=—IimTIf(TI) (3.4.24)lim f”() = 2 urn TI f(TI) (3.4.25)Replacing i with k+1 in equation (3.4.2) and taking the limit on both sides, one canobtain the following equation using (3.4.22a-d):lim[2TIH, ] + lim[(ri + 6)H÷1] + 2(k +1). n{TIHk÷l] TI1 (3.4.26)The value 6 is small compared with 772 when 77 approaches infinity, and based on (3.4.25)TI H1 is much smaller than m Hk+I at large i Therefore, (3.4.26) can be rewritten asurn TIHk÷I (TI)or Hk÷l(TI) (3.4.27)Similarly, replacing i in (3.4.3) with (k+l), multiplying by 172 on both sides, andtaking the limit yieldslirn[TI2.G’÷1 (TI)] iiTI’Chapter 3 Quasi-Similarity Solutionsfor Wing Tip Vortices 57or lim[i (‘n)]—Therefore,Gk+j (11)>(3.4.28)We can do the same thing to equation (3.4.4) as we did to equation (3.4.2):ijm[21,Fij+iim[(i12 +2)F]+2(k+ 1) n{iiF+1]‘(k+l-j) (3.4.29)The first term on the left side of the above equation is much less than the third term. Thevalue 2 in the second term can be neglected compared with 772 at large 17. Consequently,(3.4.28) can be rewritten as1lim rF÷1 2k÷1or ÷(11) )(3.4.30)(3.4.1), along with boundary condition (3.4.5), can be solved. The solution is211R( ) = 12. ()+2(i— 1)$x(x).dx (3.4.3 1)Replacing i with k+ 1 in the above equation, and dividing by 277 on both sides, yieldsRkl (1) = — ()+Jx(x) . dx (3.4.32)2 11QTaking the limit on both sides of the above equation, one haslim Rk+I (i) = 1im[. F÷1 (ii)] + k 1im[’rF (n)j (3.4.33)Therefore,Rk÷l (1) >12(k÷I)-1 (3.4.34)We have therefore proved that the proposition is correct for k+ 1, provided that it iscorrect for i—i, lk. Since the proposition is correct for = 1, this concludes the proof bymathematical induction. Hence*The second order tangential equation happened to be a homogeneous ODE. Therefore the second ordersimilarity functions described by (3.3.5) and (3.3.6) are different from that described by (3.4.2 la-b). TheChapter 3 Quasi-Similarity Solutionsfor Wing Tip Vortices 58[2.H(1)} 0, for i2Thus, we have verified that[2H(1)] =0 (i2)3.4.2 The i-th order dragIn this sub-section, we will prove that the i-th order solutions satisfy the condition thatthe higher order drag is zero.Integrating the axial momentum equation from zero to infinity, yieldsj{2o2 +2)’+2i1} •dii = .[2iiG +ii(H .0 0 (3.4.35)+ F’ - .- j) + dj=IBecause52 () di = 2{’()I: - S ()= -2{(1)I} (3.4.36)=2CJand5(2+ 2)’Oi).dii = (2 +2). - 2J(i).idii(3.4.37)=—2Cf—2$I(1).r1dT1the left side of the equation (3.4.35) is thus,+ 2)I’+2iiI} .di = 2(i — 1).f I(ii)• i,dii (3.4.38)Furthermoredifference does not affect the validity of the assumption, it only hastens the approach of the similarityfunctions to zero as the similarity variable approaches infinity.Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 59= _{2(i_ 1)G,0 j=I 0 (3.4.39)= - {s 2(i -1). G1 () + [2G }For the higher order problem (i 2), [112G (11)] = 0. Therefore— .J[2iiGi +ii(H +1)].d —2(i — 1).JG(i).idii (3.4.40)Now, consider the last term in equation (3.4.35). We can replace R with F based onthe continuity equation (3.4.1). (3.4.1) can be rewritten as2[iR1(i)] = [12F (11)] + 2(i - 1)i (11)or 2iiR1(1) =2()+2(i— l)JxFj(x)dx (3.4.41)Thus,.1=’ 0= { - 1)• f . - 2(i i)1F } di0 0 (3.4.42)={2i1.s .d_2(ij)fflFF .d}=—2(i— 1)Ji •dj=l 0Substitution of (3.4.38), (3.4.40) and (3.4.42) into (3.4.35), yields2(i_1){J[.G()÷(1)+F1 .F()].d} = 0 (3.4.43)For the higher order problem (i 2)0, i 2 (3.4.44)Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 60This result means that the higher order solutions satisfy the requirement that the wingtotal drag is independent of the downstream distance because the higher order drags areall zero.3.5 The axial velocity on the vortex centerlineThe axial velocity on the vortex centerline is1)(z,O)=U .[l+U1(z)]= i_ D 1 (3.5.1)[ 8itpvzUJsinceF1(O)=O for i2. The result given by (3.5.1) is valid provided1U (z)I = D <1 (3.5.2)8itpvzU,,where D is the wing total drag, U, is the freestream velocity, z is the downstreamdistance from the wing tip, p is the density, and v is the laminar kinematic viscosity. Theaxial velocity component on the tip vortex centerline can be evaluated by using (3.5.1) ifthe wing total drag is known. The axial velocity deficit on the vortex centerline isproportional to the wing total drag, and inversely proportional to the downstreamdistance. At a cursory glance it would appear that the axial velocity is not related to thecirculation of the wing tip vortices. This impression is not correct. The total drag of thewing is the sum of the friction drag and the tip vortex induced drag, and because theinduced drag is a function of the wing circulation, so too is the vortex axial veloctiy.3.6 Comparison with Batchelor’s vortexBatchelor (1964) made a first effort to solve the wing tip vortex flow analytically.Batchelor’s analysis involves a linearization of the governing equations. Is there anyconnection between Batchelor’s vortex and the first order quasi-similarity solutions? Aswe shall see, the tangential velocity component and pressure distribution in Batchelor’svortex are exactly the same as our first order tangential velocity and pressure, because theChapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 61ODEs and boundary conditions for the tangential velocity and pressure are the same forboth models. In contrast, the axial velocity is different between the two models. The axialvelocity field of Batchelor’s vortex results in a single vortex with an infinite drag.Batchelor resolves this problem by invoking the (somewhat artificial) concept of a “dragassociated with the vortex core of a trailing vortex.” There is no equivalent limitation inthe quasi-similar solutions; the axial velocity distribution does not lead to infinite valuesof drag. The limitation on our quasi-similarity method is U1 (z)I < 1, which is a significantimprovement over Batchelor’s vortex, for which the constraint is equivalent to1U(z)I <<1.A summary of the procedure Batchelor used to obtain his similarity solution, alongwith commentary, follows.Far downstream, where the boundary-layer-type approximations (assumptions (3) and(4) in section 2 of Chapter 2) are supplemented by the approximationU,—<<U (3.6.1)the governing equations (2.2.5)-(2.2.8) reduce toia(!1)r)al)z0 (3.6.2a)r 3r 3z(3.6.2b)az ar r3r(3.6.2c)r par(3.6.2d)az az ar r ar)Batchelor solved the tangential and radial momentum equations by similaritymethods. The tangential velocity and pressure obtained by him are found to be exactly thesame as our first order solutions.Batchelor found1)9 =_—(i_-e) (3.6.3)Chapter 3 Quasi-Similarity Solutionsfor Wing Tip Vortices 62=- pUT2 = {(i_e+2ei()_2ei(2)} (3.6.4)Ur2 2 ..where the similarity variable = =—, and the function ei ()= J dy. It is4vz 4 yjnot difficult to show that (3.6.3) and (3.6.4) are the same as the first order quasi-similarsolutions. The axial momentum equation (3.6.2d) now becomesu ._v.1à2z E()+] (3.6.5)iz 3r r ar ) 32itvz [ d jIn order to obtain a clearer view of the asymptotic dependence of v on z, all the terms ofabove equation were integrated over a cross-sectional plane:d F2 7d(P)— i (u — ) rdr = , dZ l6ivU ‘ d (366)F2— l6it2U zsince P— as—>co.ThusF zUj(U,—ii).rdr= , log +const. (3.6.7)o l6rU,. vwhere v/U, has been used as a convenient unit of length in the logarithm.The relation (3.6.7) suggests the following form for an asymptotic solution of theaxial momentum equation:F2 .10gZUo F.Q2()—Le (3.6.8)32it VZ V 32it v 8vzwhere L is a constant with the dimensions of area, and the last term, the complementaryfunction, accounts for any initial velocity defect that may be independent of thecirculation. The similarity functions Q1 and Q2 can be obtained and expressed asQ1()=e (3.6.9)Q2()=e1[ +Pp().e].d (3.6.10)Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 63The axial velocity given by (3.6.8) is obviously different from our first order axialvelocity.Now, let us consider the drag on the wing:-= J P — P + — 2icrdr (3.6.11)p p JBatchelor found that the integral of (3.6.11) is logarithmically divergent as r—oo, andtherefore that the drag associated with his isolated trailing vortex is infinite. Batchelorargued that the divergence does not dependent on the structure of the core of the trailingvortex, and on this basis he defined the “drag associated with the core of a trailingvortex” as= limr2j — +i(U u).rdrL.log RU 1 (3.6.12)p R[ p j 4it vjD differs from half the total drag on the wing by an amount that depends only on f andthe distance between the two trailing vortices far downstream. In the case of two trailingvortices a distance s apart far downstream, it may be shown quite readily, from anevaluation of the kinetic energy of the motion in the lateral plane, that the total drag D isD =2D +2—log_- (3.6.13)W CvSubstituting the pressure and axial velocity solutions ((3.6.3) and (3.6.4)) into(3.6.12), Batchelor found(3.6.14)p 8ic 2where 2 is a positive number not far from unity. The solution (3.6.8) can now berewritten as= U *, Q2() (3.6.15)8vz ir p 4it v 32itvzIf the trailing vortex system from a wing on which the total drag D consists of twovortices with centres a distance s apart, the axial velocity can be written in the furtheralternative formChapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 64= ut,, _ --F (3.6.16)8v j,ic p 4it v v j 32it vzIn summary, Batchelor’s linearized analysis yields the same tangential velocity andpressure distribution as our first order quasi-similarity solution. Batchelor’s vortex has anaxial velocity distribution that results in a single vortex with an infinite drag, a problemthat Batchelor resolves by defining the “drag associated with the vortex core.” Our axialvelocity distribution is different from Batchelor’s and does not have infinite drag.Batchelor’s analysis is limited to UJtL <<1; by way of contrast, our analysis has the lessrestrictive limitation- U1(z)I <1.3.7 Comparison with experimental data3.7.1 Axial velocity decay on the vortex centerlineThe axial velocity defect in thecores of trailing vortices behind a Axial Velocity Decay in Tip Vortex0.4lifting airfoil of rectangular_. 0.12planform was measured using a _ °scanning laser Doppler‘•‘ I—velocimeter (Ciffone and Orloff, 20) 40) 600 .— 8(4) 10(4) 120)-0.041974). Data were obtained at -o. /Lie 111several different angles of attack_______________________________________________Figure 3.11 Axial velocity decay with downstream distance inand downstream distances ranging a wing tip vortex. (The square symbols represent theexperimental data, the solid line is the quasi-similar resultfrom 30 to 1000 chord lengths. and the dashed line is Batchelor’s result)The experiments were performed in a water tow-tank facility 61 m long, 2.44m wide, and1.7m deep. The test Reynolds number based on the wing chord was nominally 2.43x 1O.The wing used has an NACA 0015 airfoil section, an aspect ratio of 5.33, and a span of0.61m. The test was designed to obtain continuous data from the near field to the far fieldby the hydrogen bubble technique.Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 65The axial velocity defect data at 4° angIe of attack are selected for comparison. Goodagreement between the quasi-similarity theory and the experimental results occurs for zic> 300 (Figure 3.1l)* . This agreement was achieved by using a laminar viscosity v and adrag coefficient, CD = 0.0005. This drag coefficient is about twenty times smaller than atypical aircraft drag coefficient of CD = 0.01.There is a plausible explanation for the unreasonably low drag coefficient. The flowin a real wing tip vortex is probably turbulent. If one can model the effects of turbulenceby means of an eddy viscosity VT that is constant spatially, then the quasi-similaritysolution is still valid provided V is replaced by VT. Modeling the flow of Figure 3.11 inthis way, but enforcing the constraint CD = 0.01, one determines VT 20v. Although nomeasurements of turbulent Reynolds stresses have ever been made in the far field of a tipvortex, a turbulent eddy viscosity twenty times the laminar viscosity is typical of manyflows. Thus, although the quasi-similarity theory has effectively a free constant, whichhas been used here to match the theory with the experimental results, the constant (relatedto the drag coefficient) selected to give good agreement with experiment is reasonable.Batchelor’s theory gives much poorer agreement with the axial velocity results(Figure 3.11) at these small downstream distances owing to the logarithmic terms in(3.6.16). At yçjy far downstream distances Batchelor’s theory and the quasi-similartheory show the same dependence of axial velocity deficit on the downstream distance zbecause iim(logz)/z = liml/z.3.7.2 Maximum tangential velocity decayA study of aircraft trailing vortex system, involving actual flight testing, was reportedby McCormick et al.(1968). Detailed velocity measurements were made through thevortex immediately behind a test aircraft up to a distance of approximately 1000 chordlengths downstream of the aircraft. Flight testing was performed using a U.S. Army 0-1aircraft and a Piper Cherokee. Measurement of the trailing vortices of each aircraft was* The disagreement for zJc < 300 may be caused by varying eddy viscosity in the tip vortex near field andthe limitations of the theory itself.Chapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 663.8 ConclusionsThere are two major practical limitations on the quasi-similarity solution. One is theassumption that the axial velocity deficit or excess must not be significantly greater thanthe freestream velocity. Very near the wing this condition may not apply. The quasi-similarity method is limited as well by the assumption that the flow must be laminar.The limitation on the quasi-similarity solution from the convergence requirement(2.7.3) can be rewritten (assuming a rectangular wing) asZ>DC (3.8.1)l6twhere CD is the drag coefficient, Re is the Reynolds number based on the chord, and Sis the span of the wing. This distance is about a hundred meters downstream of the wingMaximum Tangential Velocity Decaymade using a vortimeter and a tuft grid. The vortimeter consisted of a vertical array of 30,1-ft-long, l/4-in.-diam cylinders. Each cylinder was supported on strain-gaged flexures sothat the aerodynamic drag on each cylinder could be measured. Hence, the velocitydistribution through the vortex could be determined as the vortex moves through thearray, by measuring the forces acting on each cylinder as a function of time.Velocity distributions through thetrailing vortices shed by the 0-1 andCherokee aircraft at several flight speedswere measured. The non-dimensionaldata are compared with our analyticalresults in Figure 3.12. For i1c < 200 theagreement is poor, likely because ofturbulent flow in the tip vortex. j Figure 3.12 Maximum tangential velocity decaywith downstream distance in a wing tip vortex. (Thecontrast, the agreement far downstream, round symbols represent the experimental data, thesolid line is the analytical result)zJc >200, is satisfactory. This agreementconfirms that the maximum tangential velocity decays as one over the square root of thedownstream distance— one of the results predicted by the quasi-similarity method.21.5l0.5..00 200 400 600 800 1000 200 140071C(IjChapter 3 Quasi-Similarity Solutions for Wing Tip Vortices 67trailing edge for a typical drag coefficient CD 0.01, Reynolds number Re 5 x anda one meter span wing (for which the flow should be laminar). We can also make thefollowing specific conclusions:1. The quasi-similarity method represents an improvement over the solution ofBatchelor. The limitation on the quasi-similarity method is 1U(z)I <1, instead of—u /U.. <<1 as required by Batchelor. Thus, the quasi-similarity method can beused in trailing vortices comparatively close to the wing.2. The tangential velocity component and pressure in the Batchelor’s vortex are the sameas the first order quasi-similarity solution. The axial velocity field of Batchelor’svortex differs from the quasi-similarity vortex. Batchelor’s vortex has an infinite dragdue to a single vortex whereas a quasi-similar vortex has a finite drag due to a singlevortex.3. The first order tangential velocity and pressure distribution have the third orderaccuracy, O(U(z)). That might explain why Batchelor’s theory is a good model ofthe tangential velocity even at fairly short distances downstream of a wing.4. The maximum tangential velocity results have been compared with experimental data.The agreement is satisfactory far downstream. The agreement confirms that thetangential velocity decays as one over the square root of the downstream distance, aspredicted by the theory.5. The axial velocity predictions on the vortex centerline has been compared withmeasurements. The agreement is good far downstream, which confirms that the axialvelocity deficit on the vortex centerline decays as one over the downstream distance.6. There are two main restrictions on the quasi-similarity method. One is the laminarflow assumption. The other is that quasi-similar solutions are only valid at fairly largedownstream distances.Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 68Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip WingA novel hydrofoil design, consisting of a small diameter flow-through duct affixed tothe tip, has been studied. The tip vortex cavitation inception index, cy., of this hydrofoilgeometry is about a factor of 2 lower than that of a conventional rounded hydrofoil tip.This inception improvement comes with little associated performance penalty. For anglesof attack greater than 8° the non-cavitating lift-drag ratio is actually superior to that of anunducted hydrofoil of equal span, although with lower wing loadings the hydrofoilperformance is diminished by application of the ducted-tip.The ducted tip is effective at reducing the tip vortex inception index because, incontrast with the rounded tip for which shed vorticity in the transverse plane behind thewing (Trefftz plane) is confined to a line, the ducted tip shed vorticity at the trailing edgeis distributed over a line and circle. Distributing the vorticity in this fashion causes thetrailing vortex to roll up less tightly, and hence have a higher core pressure and lower a1,than a conventional hydrofoil tip. It is also suspected that the interaction at the microscalelevel between the flow through the duct, and the flow around it, makes the vortex coresize larger, and therefore a1 smaller. The superior lift performance of the ducted-tip wingat elevated angles of attack results from the redistribution of the shed vorticity as well.The downwash caused by the vorticity on one portion of the ring tip in the Trefftz plane issmaller than that by the same amount of vorticity positioned in the spanwise direction.Less downwash produces less induced drag. At small angles of attack the benefit from theducted tip device of reducing the induced drag is more than offset by the increasedparasite drag due its larger wetted area. The ducted tip design has many potential marineapplications, including to ship and submarine propellers, submarine control fins, and shiprudders.The following section reviews the previous work on tip modifications. The secondsection explains the rationale for the novel ducted tip geometry. The third sectiondescribes the experimental apparatus and procedures. Flow visualization, cavitationcharacteristics and aerodynamic performance of the ducted tip configuration are presentedChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 69and discussed in section four. The experimental uncertainties are estimated in sectionfive. Finally, conclusions are presented in section six.4.1 Previous Work on Tip ModificationsThe purposes of tip modification can be grouped into three categories: attenuatingvortex strength to alleviate the rolling moment on a following aircraft, reducing thevortex induced drag, and increasing the vortex core pressure to avoid cavitation.Various kinds of tip devices have been tested to reduce the rolling moment. A leadingedge disk flow spoiler, a trailing edge disk flow spoiler and a porous wing-span extensionwere investigated by Scheiman and Shivers (1971). They achieved vortex cross-sectionalredistribution at the expense of wing lift and/or drag characteristics. “Vortex-attenuatingsplines” were tested by Hastings et al. (1975) and found to cause a considerable noiseincrease and a lift/drag ratio reduction. Snedeker (1972) measured the rolling moment ona simulated following aircraft caused by a tip vortex with and without axial injection(injection of fluid in the chordwise direction from the wing tip). He found that theinjection reduced the rolling moment by only 13% (for vortex generating wings at a6°). The applicability of such a scheme to commercial aircraft is dubious. The Whitcombwing tip (Whitcomb 1976 and Flechner et al. 1976) is the only wing tip modificationdevice that has found commercial application. This wing tip consists of a short (about 1/2chord high) lifting surface mounted almost normal to the wing at the tip. The wing isconnected to the suction surface of the wing with a smooth fillet. Some of the aircraft thathave flown with such a winglet are the MD-il, the Gulfstream III, the DC-1O and the B767 (Webber and Dansby 1983 and Devoss 1986). The Whitcomb wing tip reducesmarginally, by -1% to 2%, the drag on some aircraft (Shevell 1989). Closely related inconcept to the Whitcomb winglet is a series of winglets set at varying angles to theplanform plane, which are referred to as “wing tip sails.” Spiliman (1978) has reported upto a 29% reduction in lift dependent drag in flight tests of (small aspect ratio) wings fittedwith sails.Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 70Platzer and Souders (1979) is a good, though now somewhat dated, review of tipvortex cavitation knowledge. Cavitation inception is known to occur in the regionimmediately downstream, to as far as two chords downstream, of the generating wing(Arndt et al. 1991, Maines and Arndt 1993, Stinebring et al. 1991, Green 1991). Inceptionis generally highly unsteady, occurring first at one location and subsequently up ordownstream from it. Upon further reducing the flow cavitation number below g, a longportion of the tip vortex cavitates. These observations are consistent with the knownminimum of vortex core pressure near the hydrofoil trailing edge (owing to rapid rollup),and with the small axial pressure gradient along a tip vortex centerline (resulting from thesmall variation in i with downstream distance).Tip cavitation inception is highly dependent on small details of the flow near the tip.Stinebring et al. (1991) and McCormick (1962) have shown that merely roughening thepressure surface tip region of a hydrofoil substantially reduces c. Drilling small holes ina hydrofoil from the pressure to the suction surfaces at the tip also reduces a (Arakeri etal. 1985, Sharma et al. 1990). Recently, Chahine et al. (1993) have shown that injectionof modest amounts of Polyox solution (a viscoelastic polymer) into a propeller tip vortexcan, under optimum conditions, reduce the cavitation inception index by up to 35%. Theyattribute this reduction to a significant thickening of the vortex core caused by theviscoelasticity of the solution. The dependence of tip vortex a on small details of the tipflow is congruent with the known sensitivity of the single phase tip flow to the samedetails. The size and quantity of freestream nuclei also has a substantial impact on o.Arndt and Keller (1992) have measured a doubling of c1 when the freestream fluid ischanged from ‘strong’ water (few and small nuclei) to ‘weak’ water (many, large nuclei).4.2 The ducted tip wing geometryThis chapter documents one successful tip cavitation inception delay device — theducted tip. Before discussing our experimental studies of the ducted tip geometry, webegin with a brief explanation of the rationale for this novel geometry.Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 71As a consequence of Helmholtz vortex laws, the total shed circulation from a wing isessentially established once the wing lift is specified. Therefore, the only way to modifythe tip vortex core pressure (and thus the cavitation inception index, c ) is to redistributethis shed circulation. In particular, the less concentrated the shed circulation from a wing,the larger the tip vortex core will be, and hence the lower its inception index.A conventional planar wing sheds (in theory, see e.g. Mime-Thomson (1958)) a lineof vorticity in the Trefftz plane (Figure 4.1(a)). Any wing with an elliptical or nearelliptical loading (a desirableWing Root Wing Tipcharacteristic for maximizing L/D)has this shed vorticity +concentrated at the wing tips, (a)which accounts in part for theWing Rootsmall vortex core radius and hencehigh y, of elliptically loaded+wings*. (b) +Non-planar wings (Cone 1963)Figure 4.1 Shed vorticily in the Trefftz plane fromare not subject to the same near- (a) conventional and (b) ducted tip wingselliptical loading constraint as conventional wings, and hence do not necessarily shedconcentrated vorticity at the wing tip. They therefore potentially have much lower tipvortex cavitation inception indices and, according to Cone, less induced drag, thanconventional wings.The ducted-tip wing is one example of a non-planar wing. It consists of aconventional planar wing to which is mounted a hollow duct, approximately aligned withthe wing chord (Figure 4.2). Figure 4.1 (b) is a Trefftz plane schematic of theconfiguration. Provided that the flow through the duct has little rotation (a reasonableassumption if the entrance to the duct is near the wing leading edge), and the flow aroundthe exterior of the duct has a significant swirl (again, a reasonable assumption because the++* Real fluid effects that cause the formation of secondary vortices have only a slight impact (addition of asmall amount of countersign vorticity near the tips, and smearing of the distribution) on the shed vorticitydistribution.Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 72large pressure gradient that exists near the wing tip, from the pressure surface to thesuction surface, will cause a rapidtangential flow around the tip),there will be significant vorticityshed from the duct, as illustrated inFigure 4.1(b). Because this shedvorticity is spread over a region ofradius comparable to the (large)duct radius, it seems plausible that(a) (b)the resulting tip vortex core radius______________________________________________Figure 4.2 Schematics of wing tip devices. Flowwill also be large. This insight into is right to left. (a) Ring wing tip. (b) Bi-wing tip.the distribution of shed vorticity encouraged us to explore experimentally the ducted tipgeometry.4.3 Experimental Apparatus and ProceduresThe line-of-reasoning described in section 4.2 leads one to hope that the ducted tipwing geometry has superior tip cavitation characteristics. From a practical standpoint, oneis not merely interested in improvements in tip a., one must also know that theseimprovements are not coincident with a large lift reduction or drag increase.Consequently, the experimental program described here consisted of two discrete parts— measurement of the inception characteristics of a ducted tip wing, and measurementof the lift and drag behaviour of such a wing.The tip vortex inception measurements were carried out by my supervisor, S. I. Green,in the Low Turbulence Water Tunnel (LTWT) at Caltech. The water tunnel facility has atest section of 30.5 cm x 30.5 cm x 2.5 m long, with a freestream velocity adjustable upto 10 mIs. The freestream pressure, p, can be varied from slightly above atmosphericpressure to 25 kPa by means of a vacuum pump. By using a deaeration system anddiatomaceous earth filtration, the freestream nuclei content of the water was controlled. Avan Slyke device was used to measure the water’s total dissolved air content.Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 73A rectangular planform, untwisted, constant airfoil section (NACA 64-309 modified),hydrofoil was reflection-plane mounted in the LTWT. The chord of this hydrofoil was15.2 cm, and the semispan was 17.8 cm, resulting in an effective aspect ratio of 2.3. TheReynolds number based on chord length for all the cavitation tests was in the range 1.1 x106<Re< 1.6x 106.This basic wing was fitted with two different tips. The first tip was a duct comprisedof a 2.9 cm outside diameter (with a wall thickness of 0.2 cm) cambered brass pipe, 10.2cm long, attached flush with the hydrofoil trailing edge, with axis aligned with thecamberline (Figure 4.2(a)). The size of the duct selected for these studies is probably notoptimum. A duct was designed based on intuition guided by the following considerations:too large a duct will cause excessive parasite drag, while a duct that is too small will havereduced flow through the duct and will cause little change to the shed vorticitydistribution. The second tip, which was used for comparison purposes, was a rounded tipgeometry with approximately semicircular cross-sections perpendicular to the chordwisedirection, fitted onto the end of the basic wing (Figure 4.7).Two different procedures were used to study the flow around these wing tipgeometries. Surface flow visualization was accomplished using the paint drop technique(Green et al. 1988). With the hydrofoil out of the water tunnel, drops of oil-based paintwere dotted on its surface. The hydrofoil was then returned to the tunnel, and the LTWTwas quickly accelerated up to a set velocity, causing the paint drops to be smeared in thedirection of the local shearing stress.The second procedure involved cavitation inception measurement and cavitationphotography. The large range of inception numbers encountered precluded us frommaintaining a constant Reynolds number for all the tests. Instead, cavitation inceptiontesting consisted of setting the hydrofoil angle of attack and gradually increasing U anddecreasing p, until cavitation inception was observed under stroboscopic illumination.Hydrofoil leading edge inception was defined to occur when cavitation was observed onthe foil surface half way between the wing root and tip; trailing vortex inception when atleast one cavitation event per second was seen. When the leading edge inception index,(a1 )ie was larger than the tip vortex inception index, (a, ),. it was necessary to achieveChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 74(a1 ), quickly after (a, )k to avoid erroneous measurements due to the recirculation ofcavitation nuclei in the LTWT. Photographs of cavitation were taken by illuminating thehydrofoil and tip vortex in the spanwise direction, and recording on film the lightreflected normal to the hydrofoil planform. The cavitation behaviour of the rounded tipwas studied first, and several weeks later, following attempts with different ducted tipgeometries, testing of the final ducted tip geometry began.The aerodynamic performance of the two tip geometries was evaluated in the LowSpeed Wind Tunnel (LSWT) at the University of British Columbia (Figure 4.3). Thetruncated rectangular cross-section test section of this facility has dimensions 69 cm x 91cm and is 4 m long. The maximum velocity attainable in the test section is 40 m/s,although the velocity was maintained at 30 m/s for these experiments. A six-axis strain-gauged force balance, combined with filters, amplifiers and a data acquisition sub-system(Figure 4.4), measures lift and drag forces on models in the test section. The sub-systemconsists of a Computerscope model ISC-16 for the IBM-PC family of computers,provided on a single plug in card. The ISC-16 model has a 12 bit 1 llsec A/D conversion,accepts analog input signals between -10 Volt and +10 Volt, and has a 1 to 16 channelprogrammable input multiplexer. About 10,000 samples were taken and averaged for oneparticular lift or drag measurement. This data acquisition system was calibrated and theresults are summarized by the following relations:Drag Force [N] = 0.579472 x [Volt] + 0.029056Lift Force [N] = 10.07694 x [Volt] + 0.26435where [Volt] is the corresponding digital reading in voltage from the A/D converter.The data acquisition sub-system, together with a scanivalve/pressure transducer subsystem, which consists of four pressure transducers, a solenoid driver and a four-capscanivalve (Figure 4.5) was used for pressure distribution measurements. Each pressuresensor converts the pressure difference (in the range -5 inches to +5 inches H20) betweenits two input connectors to an analog signal of between 1 and 6 Volts. The calibrationresults of the four pressure sensors are respectively,Pt — Pref[N / m2J= 491. 6743 x[Volt]— 1730.78Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 75P2 — Pref[N / m2]= 500.3630 x[Volt]— 1738.413Prej[N/m1=484.0616><[’101tb1730.47P4 Pref[N / m2]= 495.5708 x[VoltJ— 1745.80where [Volt] is again the digital output reading in voltage from the A/D convertor.Aerodynamic testing in the LSWT was done on a rectangular planform, untwisted,constant airfoil section (NACA 66-209) basic wing. This acrylic wing is 30.5 cm in chordand 35.6 cm in semi-span, and is fitted with 92 surface pressure taps. Several differenttips were attached to the basic wing, including a series of cambered circular aluminumducts 3.8 cm in outside diameter (with a wall thickness of 0.2 cm) with lengths rangingfrom 19.8 cm to 30.5 cm (i.e. 65% of wing chord to 100% of wing chord). These ductswere aligned with the wing trailing edge and were affixed approximately parallel to thewing camberline (Figure 4.2(a)). An elliptical wing tip duct, with major axis 10.5 cm(aligned normal to the wing planform), minor axis 3.9 cm, and length 19.8 cm was alsotested. The last type of ducted tip tested was a Thi-wing” configuration.The “bi-wing” tip (Figure 4.2(b)) was constructed of two parallel NACA 0006 shortairfoils of 15 cm chord and 3.5 cm span joined together at both span ends with thinaluminum plates. The chordline-to-chordline separation of these airfoils was 3.5 cm. Thisbi-wing tip was attached to the tip of the basic wing with the short airfoils alignedparallel, and in subsequent tests ±10° from parallel, with the basic wing. All plate/wingjunctions were then faired with putty before aerodynamic testing.For comparison purposes, two other tip geometries were also wind tunnel-tested. Asemicircular tip constructed of acrylic and body filler 3.8 cm in span was oneconventional geometry tested, and a square cut tip (i.e. constant airfoil section to the tip,where the wing chord falls abruptly to zero) 3.8 cm in span was also tested.In summary, a number of different wing tip goemetries were tested in the wind tunnel,all with semi-spans of 39.4 cm, though with aspect ratios varying from 2.58 for the squarecut tip, to 2.60 for the rounded tip, to 2.67 for the 65% partial chord ducted tip. Theaerodynamic test configurations were not geometrically similar with the cavitation testmodels, but they were nearly so.Chapter 4 Cavitation and Aerodynamic Peiformance of the Ducted-Tip Wing 76Three aerodynamic measurements were made in the LSWT: the lift coefficient, CL,versus angle of attack a, the drag coefficient, C0, versus a, and the pressure distributionon both sides of the basic wing. Flow visualization of the flow field around the ducted tipin the LSWT was carried out by using tufts on the model.Figure 4.3 The Low Speed Wind Tunnel at the University of British Columbia.Amplifier60,000 GainCtJI.Amplifier3,000 GainECl)Cl)1)0Cl).-—. Drag Force Signal—‘-——-r Lift Force SignalFigure 4.4 The data acquisition system for lift and drag measurement in the LSWT.Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 77Data AcquisitionSub-systemScanivalve/Pressure Transducer Sub-systemFigure 4.5 The data acquisition systemfor pressure measurements in the LSWTChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 784.4 Results And DiscussionThis section includes the cavitation results of the ducted tip, which were part of thework done by my supervisor at Caltech. A completed picture on the novel ducted tip willbe presented in this manner.4.4.1 Flow VisualizationExamining the surface flow over an airfoil is perhaps the easiest way to assessqualitatively the performance of different wing tip geometries. Figure 4.7(a) is a view ofthe surface flow over the pressure side ofthe rounded tip geometry. The flowdirection is nearly streamwise near thereflection-plane mount (near the bottom ofthe photograph) and acquires an increasingspanwise component as the tip isapproached. Over the majority of thesuction side of the wing (Figure 4.7(b))there is also an increasing spanwise Figure 4.6 Tip vortex rollupvelocity component (for this side, directed in the Trefftz plane.from the tip to the root) as the tip is approached. Streaklines directed toward the tip on thesuction side result from, respectively, the tip vortex rollup in the region near the tip (referto Figure 4.6), and a secondary vortex near the wing root. Note that the reflection planemount seems to affect the wing only over the bottom 20% of the semi-span. The tip flowin particular is basically unaffected by the presence of the water tunnel floor.Tip VortexChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 79(a)(b)Figure 4.7 Surface flow visualization on the rounded tip geometry. (a) Pressure side.Flow is right to left. (b) Suction side. Flow is left to right.(Photograph source: Green 1988)Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 80(c)Figure 4.7 Surface flow visualization on the rounded tip geometry.(c) Inboard view of tip. Flow is left to right. The wing pressure surface is at the top.(Photograph source: Green 1988)Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 81The surface flow over the ducted tip geometry is quite different from that over theconventional tip geometry. The spanwise velocity component at the trailing edge, on boththe suction and pressure surfaces (Figure 4.8), is substantially less than that of therounded tip wing. At one third of the spanwise distance from the reflection plane mount,for example, the angle the rounded tip streaklines make with the freestream direction isabout 50% greater than the strealdine angles of the ducted tip. This difference in spanwisevelocity component implies that the ducted tip wing sheds less bound circulation over themajority of the wing span than does the rounded wing tip. Smear lines in the duct (Figure4.8(d)) have little swirl, whereas those on the duct exterior indicate a large tangentialvelocity. This observation confirms that substantial vorticity is shed from the duct, asindicated in Figure 4.1(b).One may be concerned that the flow will separate from the ducted tip. Vibration oftufts located on the outside suction side close to the entrance of the duct implies that localflow separation occurs there at elevated angles of the attack (Figure 4.9). This flowseparation will cause elevated parasite drag.Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 82(a)(b)Figure 4.8 (a) Surface flow visualization on the pressure side of the ducted tip wing.The flow is right to left. oL=7°, Re=1.2x106.(b) Suction side view of the wing in Figure 4.8(a). Flow is left to right.(Photograph source: Green 1988)Chapter 4 Cavitation and Aerodynamic Peiformance of the Ducted-Tip Wing 83(c)(d)Figure 4.8 (c) Inboard view towards tip of the flow in Figure 4.8(a). The flow isleft to right. The pressure surface is at the top of the photograph.(d) Upstream view through the duct of the flow in Figure 4.8(a).The pressure surface is to the right.(Photograph source: Green 1988)Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 84Figure 4.9 Separatedflow near the entrance of the ducted tip. c=]2°.The flow is right to left.Chapter 4 Cavitation and Aerodynamic Peiformance of the Ducted-Tip Wing 854.4.2 Cavitation CharacteristicsFigure 4.10 is a photograph of developed tip vortex cavitation behind the rounded tipgeometry. The cavity cross-section is oval immediately downstream of the tip, andbecomes nearly circular within one chord of the trailing edge. This shape is consistentwith the known transition of the tip vortex velocity field from asymmetric at the hydrofoiltrailing edge to nearly axisymmetric a short distance downstream (Fruman et al. 1991,Green and Acosta 1991). Development of this continuous vapour-filled tip vortex cavityoccurs when the cavitation number is decreased by just 10-20% below y,. The rapidity ofthe cavitating vortex development implies that the axial pressure gradient along the coreof a tip vortex is small, in agreement with recent measurements by Green (1991).Unlike a developed trailing vortex cavity, which is a steady phenomenon, cavitationinception is highly unsteady. Cavitation inception in the rounded tip vortex is observed0.3 to 1.5 chords downstream of the hydrofoil trailing edge; the exact location of trailingvortex inception fluctuates in an apparently random manner within this range. It is notknown if the fluctuation of the inception location is due to variability in the location atwhich freestream nuclei are captured by the vortex (Ligneul and Latorre 1993), or due tothe fluctuating pressure field within the tip vortex core (Green 1991).Trailing vortex inception occurs for o = 1.8 ± 0.2 near the hydrofoil operating angleof attack, a = 7° (Figure 4.11). As a is varied about the operating angle of attack, theinception index varies much less rapidly than the (ct—c0)2 dependence predicted by asimple Rankine vortex model. Arndt and Keller (1992) have observed the same effect,which they attribute to the significant tension that can be sustained by low-nuclei-contentwater prior to cavitation inception. Hydrofoil leading edge surface cavitation inceptionoccurs for cy = 0.9 ± 0.1, substantially lower than the value for tip vortex inception. Thisobservation implies that tip vortex cavitation occurs in many situations for which surfacecavitation on the hydrofoil is not present.The inception indices plotted in Figure 4.11 were determined in ‘strong’ (i.e. lownuclei content) water. Water with more freestream nuclei exhibited much higherChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 86inception indices (greater by a factor of 2 when the dissolved air content of the water wasdoubled from 4 to 8 ppm). Arndt and Keller (1992) and Arndt et al. (1991) observed asimilarly strong dependence àf a on nuclei content.Cavitation inception from the hydrofoil ducted tip was first observed on the outsideedge of the suction surface portion of the duct. This surface cavitation is caused by theseparation (Figure 4.9) that occurs as the flow, which has a high tangential velocityaround the tip, encounters the duct leading edge. Figure 4.12 is a photograph of cavitationnuclei generated by the duct surface that have migrated into the ducted tip vortex. Notethe diffuse appearance of the vortex.True trailing vortex cavitation occurs only when o is reduced well below theinception index of both the ducted tip surface cavitation and the hydrofoil leading edgecavitation. As a consequence, cavitation nuclei generated by both the wing leading edgeand duct surface cavitation are swept into the tip vortex prior to inception there. Becausethe tip vortex is exposed to water of much higher nuclei content than the freestreamwater, the tip vortex will cavitate at relatively higher . The ducted tip vortex inceptionmeasurements are thus biased to high values.Despite this unfavourableB bias, the ducted tip vortex data (Figure 4.13) show aremarkable decrease in c5 relative to the rounded tip geometry. For an angle of attack of7°, the inception index of the ducted tip vortex is, = 0.9 ± 0.1. In contrast, at the sameangle of attack, the rounded tip vortex has, = 1.8 ± 0.2 (i.e. a of the ducted tip is 50%less). For angles of attack of 5° and 10°, the ducted tip vortex a. is 0.5 ± 0.05 and 1.8 ±0.2. These values are respectively 61% and 28% less than the rounded tip a values (1.3 ±0.1 and 2.5 ± 0.1) at the same a.As explained previously, the inception index of tip vortex cavitation is a strongfunction of the nuclei content of the water. One might therefore posit that the largereduction of a,, when the ducted tip is installed, results from a decrease in the nucleicontent of the water between the time of the rounded tip tests and the ducted tip tests. Infact, two observations point to an increase in freestream nuclei between the tests. Onesuch observation is the increase in the dissolved air content (from 5 ppm to 7 ppm with anChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 87error of ± 0.5 ppm) between the rounded and ducted tip tests. The observed increase inhydrofoil leading edge a1 (from 0.3 to 0.8 at a 5°, from 1.0 to 1.5 at a 7°, and from3.0 to 3.5 at a = 10°) between the rounded and ducted tip tests is another strong indicatorof an increase in freestream nuclei concentrations.In summary, two factors tend to bias the ducted tip a. to elevated values — thepresence of more naturally-occurring freestream nuclei during testing of that geometry,and the generation of nuclei by cavitation from the hydrofoil leading edge. Despite thesetwo influences, cavitation inception from the ducted tip geometry occurred at much lowerinception indices than from the rounded tip geometry.One may be concerned that the gains in cavitation inception improvement resultingfrom the ducted tip modification are negated by the increase in other forms of cavitation.For example, one might postulate that the ducted tip geometry would redistribute thehydrofoil loading, causing the wing leading edge a. increase referred to above. This isnot the case. Except for a small region near the wing tip, the measured pressuredistributions on the basic wing were virtually unchanged (refer to pressure distributionmeasurements shown in section 4.4.3; c the same to ±7%) with the addition of theducted tip. The unchanged pressure distribution implies an unchanged wing loading.Therefore, leading edge cavitation is not significantly affected by the addition of theducted tip, whereas tip vortex cavitation is significantly reduced by the ducted tip.Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 88Figure 4.10 Developed trailing vortex cavitation behind the rounded tip geometry.a=7°, Re=1.3x106a=1.8, DAC=5.4 ppm.(Photograph source: Green 1988)Chapter 4 Cavitation and Aerodynamic Peiformance of the Ducted-Tip Wing 89Figure 4.11 Cavitation inception index versus angle ofattackfor the rounded tipgeometry. Re 1.4x106,DAC = 5.4 ppm (uncertainty in a = .i 0.2° and in a, = 10% atthe 95% confidence level)CAVITATION INCEPTION INDEX(ROUND TIP)-10 0 10 20ATTACK ANGLES [DEGREES]• Leading Edge • Tip VortexChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 90Figure 4.12 Ducted tip vortex made visible by migration of cavitation nuclei,generated at the duct leading edge, into the°, Re=1.2x106 y=].5, DAC=7.O ppm.(Photograph source: Green 1988)Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 91CAVITATION INCEPTION INDEX(RING WING TIP)-10 -5 0 5 10 15ATTACK ANGLES [DEGREES]Tip Voex A Lding Edge ading Edge of WingFigure 4.13 Cavitation inception index versus angle of attackfor the ducted tipgeometry. Re 1.4 x ]Q6, DAC = 7.0 ppm. (Uncertainty in a = .t 0.2° and in a. = 10%at the 95% confidence level).Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 924.4.3 Aerodynamic PerformanceThese encouraging cavitation inception results prompted a study of the aerodynamicperformance of the ducted tip and other tip geometries.Wind tunnel testing demonstrated that the aerodynamic performance of the square cutand rounded tip wings was identical to within the experimental error (±1% drag and ±0.5% in lift). The partial chord circular duct tip had the best performance of any of theducted tip geometries. Consequently, the aerodynamic characteristics of the partial chordducted tip and the rounded tip — the same tip geometries whose cavitation behaviourwas studied in section 4.4.2— will be emphasized here. The partial chord ducted tipFor all elevated angles of attack (a > 100) the lift coefficient of the ducted tipgeometry (where CL is based on the planform area of the rounded tip) is about 4% lessthan that of the rounded tip (Table 4.1). However, the partial chord ducted tip geometryhas a 3.4% smaller planform area than the rounded tip, and thus the actual lift coefficientsfor the two geometries are identical to within experimental error. Surface flowvisualizations described in section 4.4.1 suggest that the ducted tip generates greater liftnear the wing root. The implication of the congruity of the two lift coefficients is that theadditional lift gained from the root of the ducted tip wing is almost precisely offset by thereduced lift attained in the vicinity of the tip.Table 4.1: Lift Coefficients of the Ducted-tip Wing and the Conventional Wing.a=6° 8° 10° 11° 12° 14° 15° 17°Conventional 0.3797 0.4864 0.5946 0.6272 0.6614 0.7158 0.7484 0.7864TipDucted Tip 0.3730 0.4828 0.5703 0.6014 0.6309 0.6832 0.718 1 0.757 1Improvement -2% -0.7% -4% -4% -4.6% -4.6% -4% -4%Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 93Unlike the virtually unchanged wing lift, a significant difference in wing drag isobserved when the ducted tip is installed (Figure 4.14). For a < 8° the ducted tipgeometry has greater drag than the rounded tip, but for larger a the ducted tip geometryhas less drag: 6% less at 100, 8% less at 12°, and fully 10% less at 14° (Table 4.2). Evenwhen allowance is made for the slightly reduced planform area of the ducted tipgeometry, there is nonetheless an up to 6% drag coefficient benefit arising from the tipmodification. This reduction in drag coefficient is somewhat surprising in view of theobviously increased parasite drag of the ducted wing (arising from both its additionalwetted area and the likely existence of interference vortices at the duct-wing junction);the only explanation is that the duct attachment to the wing must reduce substantially itsinduced drag. This conclusion is borne out by the observation that the drag reduction ofthe ducted tip is most pronounced at elevated angles of attack, for which the induced dragis a larger fraction of the overall drag.Table 4.2: Drag Coefficients of the Ducted-tip Wing and the Conventional Wing.c=6° 8° 10° 110 12° 14° 15° 170Conventional 0.0107 0.0183 0.0326 0.0488 0.0680 0.1135 0.1418 0.1932TipDuctedTip 0.0110 0.0183 0.0307 0.0449 0.0623 0.1018 0.1279 0.1770Improvement -3% +0.0% +6% +8% +8.3% +10.3% +10% +8.4%A simple model to explain the reduced induced drag of the ducted tip is shown inFigure 4.15. The downwash caused by shed circulation from the duct is less (by the factorcos2 0, where 0 is the angle between a portion of the duct and a portion of the planarwing) than the downwash would be if the same shed circulation were redistributed alongthe duct diameter, in the spanwise direction. The reduction in downwash causes aconsequent reduction in induced drag, as suggested by Cone (1963).Owing to the large parasite drag of the ducted tip geometry, the lift-drag ratio of thisgeometry is inferior to the conventional tip geometry at low angles of attack. However,for a> 8° the aerodynamic performance of the ducted tip is superior. As shown in FigureChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 944.16, the improvement in liftldrag ratio with application of the ducted tip varies from 0%to 6% for a between 8° and 15°.Figure 4.14 Drag coefficients of rounded and ducted tip wings. Re = 7.1 x105(Theuncertainty in CD is 0.0003 and in a is t 0.1°, both at the 95% confidence level).Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 95DRAG COEFFICIENT0.250.2CD 0.15-10 -5 0 5 10INCIDENT ANGLE IDEGREES115 20* Rounded tip • Ducted tip++++Figure 4.15 A simple modelfor the reduction in induced drag of the ducted tip geometryChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 96Figure 4.16 Lift/Drag ratio improvement of the ducted wing relative to the conventionaltip (The uncertainty in L/D is 1%, and in a is .i O.1°at the 95% confidence level).Lift-Drag Ratio Improvement ofDucted Tip7—6a1—1 9 10 11 12 13 14 15 16 17INCDENT ANGLE [DEGREES]Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 97Figure 4.17 shows the locations of the 92 pressure taps fitted on both the pressure andsuction sides of the basic wing. Note that there are more taps on the suction side than thepressure side, and also that the area near the tip has a dense population of taps. Fromthese measured point values of surface pressure, one can obtain the pressure distributionon both sides of the wing by interpolation and extrapolation. Because no pressure taps areinstalled on tip devices, only the pressure distributions on the basic wing can be comparedfor different tip configurations.As can be seen in Figure 4.17, the pressure taps are quite sparse in some areas of thewing surface. There were only five pressure taps at the leading edge on the suction sideand the pressure tap spacing was also very large in the centre of the wing. We haddifficulty getting reasonable interpolated/extrapolated pressure data in those regions. Forexample, even though the experimental results from the five pressure taps at the leadingedge decreased monotonically along the spanwise direction, the 2-D inter/extrapolationmethod we employed yielded a wavy pressure distribution. This difficulty was overcomedby combining l-D and 2-D inter/extrapolation techniques in this region.To give some indication of the accuracy of the interpolation procedure, the surfacepressure distribution was numerically integrated to yield the lift force. The liftcoefficients evaluated in this way at 5°, 8° and 12° angles of attack were compared withthe force balance measurements. The maximum error was 7%. The error was likely due topoor resolution of the suction surface pressure peak owing to the large pressure tapspacing near the wing leading edge.The pressure distribution on the pressure side of the basic wing with a conventionalwing tip configuration is shown in Figure 4.18. Figure 4.19 shows the pressuredistribution for the ducted tip configuration. Both of the two figures show the localreduction in pressure caused by the outboard crossflow near the tip region (fluid flowfrom high pressure to low pressure). It is interesting to observe in Figure 4.19 a pressurepeak near the 35% chord position at the tip. This location marks the entrance to the duct,and the pressure peak is likely due to the existence of a stagnation point at the entrance ofthe ducted tip. The pressure behaviour in the leading edge-tip region is obviouslydifferent (Figure 4.20) for the two configurations because the conventional tip had a tipChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 98extension all the way to the leading edge, whereas the ducted tip did not extend to theleading edge and therefore had more three-dimensional flow near the leading edge. Theaverage pressure gradient from the wing root to the tip on the pressure side is related tothe magnitude of the outboard crossflow and is characteristic of the strength of the tipvortex. We can evaluate the average pressure gradient qualitatively by looking at the tipviews (Figures 4.18 (b) and 4.19 (b)). There is no significant difference between thispressure gradient on the conventional wing and on the ducted tip wing except near thewing tip. Figure 4.20 verifies this conclusion.The three dimensional flow in the tip region is also evident on examination of thepressure distribution on the wing’s suction side (Figure 4.21 and 4.22). There is nosignificant difference, on the majority of the wing surface, between the pressuredistribution of the conventional configuration and the ducted tip configuration (Figure4.23). Close to the midspan of the leading edge, the difference in between the twoconfigurations is about 10%. This difference is within the error caused by theextrapolation method we used. Consider pressure taps numbered 43 and 44 (Figure4.17(b)) for example. A 7% error in the pressure measurement at those two taps willcause a maximum 10% error in the estimated c,, at the leading edge, if a linearextrapolation method is used. There is possibly a local suction peak in the trailing edge-tip region, as shown more clearly in the leading edge view (Figures 4.21(b) and 4.22(b)).The measurements do not capture the whole local peak because, due to the add-on tip, nopressure measurements were made within 10% of the semi-span from the tip. Figure 4.24shows the pressure distribution on the suction side of the basic wing without any tipdevice installed. There is obviously a suction peak near the wing tip, near the locus of 3-D stagnation points (Point B in Figure 4.6), caused by the roll-up of the primary tipvortex. Such a suction peak has also been observed by Chow et al. (1993) in theirpressure distribution measurements on a wing with a rounded tip.In summary, no pressure distribution difference between tip geometries was observedover the majority of the wing surface. However, near the wing tips the pressuredistribution was significantly changed by the addition of tip devices. The pressuremeasurements are not accurate enough (about ±7%) to confirm the LiftlDrag ratioChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 99improvement, but they are supportive to the cavitation results— tip vortex cavitation issignificantly modified by the ducted tip, whereas leading edge cavitation is notsignificantly affected by addition of the tip device.o..ESSLccEf2.1-1.8-1.6-1.2-8.8-8,6-8.2-0.1- 4 Cajtatjon andAerodyflam. Peopma,lce ofthe Ducted Tip WingI(b)(a)Figure 4.18 Pressure distribution on the pressure side ofthe Wingwith the conventional tz, _—8° (a) the 3-D view (b) the ti view.LeadingEdge101LCoc•’J0.99 0.71 0.B4 0.iTIpChapter 4 Cavitation and Aerodynamic Peiformance of the Ducted-Tip Wing 102(a)(b)LeadingEdgeFigure 4.19 Pressure distribution on the pressure side of the wingwith the ducted tip, a=8°. (a) the 3-D view, (b) the tip view.dC-,c\jdI 0.89 0.71 0.54 0.36 0.19 0.00I TIPChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 103Uj(1)(1)0.Figure 4.20 The dzfference between the ducted and conventional tip pressure coefficientson the pressure side of the basic wing, a = 8‘0PRESSURECQE.-0.8-0.5-0.2E m-2.1-1.8-1.4-1.1Ill,.II& & 0’•7•J/7/////1//1“/5/IM”//-1:’‘— %—(...0 oPRESSURECOE.-2.1-1.8-1.4-1.1-0.8-0.5-0.2IIIIII___________________________________—i__i__i_I.PRESSIJFCcE.f2.1-1.8-1.5-1.2-9.8-8.6-9.2JTh-tic’)hD4p’9‘I’1%‘10 Iy0“\“5”\“5’&-cD & (_)-&-&rn-i0.’(DCoPRESSURECOE.-2.0-1.7-1.4-1.1-0.6-0.6-0.2ii,,,I,,.,.ii,..I&—& —4—S&5—& & —4PRESSLRECLE,j-2.9-f.7-I.4-f.f-8.8-0.6-8.2V’/,i,,,, i,,,i,,,,i,,,,—/1—/1—/I,,,‘/I/il////‘.I/I/I///////I/IIII/I/II/1((iiI I // (H1 U)) ))I/I/))—,/———.-/.—_..——7////IUCI)(DCD)C)frII Co o 5---4—S5--o .. I-’_4rI,C C ‘-3PRESSURECOE.1.4-1.1-0.8-0.5-0.2‘I’00 *7 leo0-6) 6) 6) 6) c_I-0’—I&-6) 6)-So(Do..PRESSURECOE.-e31-0.17- 1%CChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 107Lu,— Suction%‘ PeakFigure 4.24 Pressure distributionon the suction side of the basic wing without tip device, a 60Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 1084.4.3.2 The bi-wing tipIn order to improve the poor lift efficiency of the ducted wing surface, a hi-wingconfiguration was constructed, which was an improvement over the ducted tip in tworespects. One respect was the ease with which the angle of attack of the hi-wing could bevaried with respect to the wing chord. Another advantage of the hi-wing tip was that itconsists of two smooth lifting surfaces. We chose NACA 64-0006 airfoil sections to buildthe hi-wing because that airfoil section has good stall characteristics.Three variants of the hi-wing tip were studied— “underloaded”, “meanloaded” and“overloaded”. The “underloaded” bi-wing has a local angle of attack that is less than thatof the basic wing, and vice versa for the “overloaded” attachment. Experimental resultsfor two angles of attack show that the underloaded arrangement of the hi-wing tip is thebest of the three geometries.A comparison between hi-wing tip configurations and the simple extended wing tip isshown in Table 4.3. The overload attachment produces a 1.4% increase in lift coefficientdue to both the relatively higher angle of attack at the tip and the increased area of the hi-wing structure. The total drag coefficients of the three hi-wing tip configurations werebigger than that of a conventional tip at this angle of attack. At the same angle of attack,as mentioned before, the drag coefficients were virtually identical for the ring wing tipand conventional tip configurations. This difference in performance is due to theexistence of sharp edges and corners in the hi-wing tip configuration. Those sharp edgesand corners will generate interference vortices and possibly cause local flow separation.Table 4.3: Comparison of Bi-Wing Tip to Conventional Tip at 8° AngIe of Attack.CD CL CL/CDConventional Tip 0.0188 0.4699 25.00Overload Bi-Wing Tip 0.0234 +24% 0.4767 +1.4% 20.4 -18%MeanloadBi-WingTip 0.0211 +12% 0.4699 0% 22.3 -11%Underload Bi-Wing Tip 0.0202 +7% 0.464 1 -1.2% 23.0 -8%Chapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 109The total drag coefficient at a=14° was successfully reduced by 4% by using anunderloaded bi-wing tip configuration (Table 4.4). This reduction implies that there is anidentical drag reduction mechanism for both ring wing tip and bi-wing tip configurationsat elevated angles of attack. The higher lift coefficient attained by the bi-wing tips relativeto the ducted tip (at a=14°) implies that the rationale for the bi-wing tip design —diminished separation at high c — was valid.Table 4.4: Comparison of the Bi-Wing Tip to the Conventional Tip at x=14°.CD CL CL/CDConventional Tip 0.1140 0.7093 6.22Overload Bi-Wing Tip 0.1142 +0.2% 0.7 141 +0.7% 6.25 +0.5%Meanload Bi-Wing Tip 0.1141 +0.1% 0.7021 -1% 6.15 -1%Underload Bi-Wing Tip 0.1091 -4.4% 0.6949 -2% 6.37 +2.4%4.5 Experimental Errors and Applicability of the Tip DeviceThere are several factors that affect the accuracy of lift and drag force measurements.These factors are:(1). angles of attack;(2). wind speed in the test section of the wind tunnel;(3). the electronics in data acquisition system.The effect of the incident angle can be eliminated by testing different tip devices withthe angle of attack fixed. The wind speed in the wind tunnel is measured by a manometer,that reads the pressure difference between the test section and the settling chamber (referto Figure 4.3) accurately to 0.05mm alcohol. The wind speed can thus be controlled at thetesting speed, 3OmIs, with an accuracy of better than 0.1%. The data acquisition systemcollects 10,000 samples within one second. The datum for lift or drag force is the averageof those samples. In several minutes, a number of data for each set-up can be obtained. ItChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 110was found that lift force can be measured to ±0.5% and drag force to ±1% at elevatedangle of attack. All experiments were redone several weeks later to confirm therepeatability of data.The lift and drag measurements described in this chapter are of high precision (lifttypically measured to ±0.5% and drag to ±1% at elevated angle of attack), but fairly lowaccuracy because of the interaction of the wind tunnel boundary layer with the wing.However, it is known (Green and Acosta 1991) that in the experimental configuration thedomain of influence of the wing/wall interaction is confined to a region within 0.2 chordsof the wing root. Consequently, it is highly unlikely that modifying the tip geometrywould have a significant impact on this interference drag. It is therefore reasonable toargue that although the lift and drag measurements described here are likely to sufferfrom large errors for any particular wing geometry owing to wind tunnel floorinterference, comparison between different wing geometries is valid because thisinterference effect is nearly constant.The ring-wing tip modification improves the Lift/Drag performance of a small aspectratio rectangular planform wing. Therefore, it seems worthwhile to speculate on thepotential applications of this device. The most apparent immediate application of thistechnology is to marine propellers. Marine propellers, which are of small aspect ratio andoften operate at high lift coefficients, would benefit from both the superior cavitationbehaviour and the improved Lift/Drag ratio of the ring-wing tip. The potential forapplying this technology to aircraft wings is much less evident. Aircraft wings commonlyoperate at a moderate lift coefficient, and have much larger aspect ratios than studiedhere. Both these factors would mitigate against the installation of ring-wing tips onnormal aircraft wings.4.6 CONCLUSIONSA novel hydrofoil tip geometry, consisting of a flow-through duct attached to thehydrofoil tip, has been tested for both cavitation inception and lift/drag performance. Theprinciple behind the ducted tip design is to cause the Trefftz-plane shed circulation of theChapter 4 Cavitation and Aerodynamic Performance of the Ducted-Tip Wing 111hydrofoil to take the form of a line with an attached circle, rather than the simple line of aconventional hydrofoil.The tip vortex inception index of the ducted tip geometry is 50 ± 15% less than that ofa conventional tip at normal operating angles (e.g. a = 7°), and is at least 30% less for allpositive angles of attack. Because the ducted tip hydrofoil shed vorticity has beenredistributed relative to the baseline hydrofoil, the induced drag on the hydrofoil shouldalso be reduced. Such a reduction in the induced drag has been observed. For a > 8° thetotal drag of the ducted tip hydrofoil is less than that of a conventional hydrofoil, despitethe fact that the parasite drag of the conventional hydrofoil is less, owing to its muchreduced wetted area. The overall lift-drag ratio of the ducted tip hydrofoil, for a > °, isup to 6 ± 1% greater than that of a conventional tip. Thus, the much improved cavitationbehaviour of the ducted tip comes at no cost (for elevated a), and even some benefit, interms of the hydrodynamic (non-cavitating) performance of the hydrofoil.The ducted tip geometry is most advantageous when the lifting surface is highlyloaded. In practice, propellers on tugs and fishing trawlers are often highly loaded.Studies are presently underway to explore the effectiveness of the ducted tip geometry onsuch marine propellers.Chapters Summary and Conclusions 112Chapter 5 Summary and ConclusionsThis final chapter is divided into two parts. The first section summarizes the “quasisimilarity” methodology and conclusions. The second section makes some suggestionsfor future work.5.1 Summary and ConclusionsA wing tip vortex was studied both analytically and experimentally. In the analyticalwork, a new method was developed that describes the tip vortex structure from fardownstream to a downstream distance fairly close to the vortex-generating wing. Wingdrag is the key parameter affecting the tip vortex structure. In the experimental work, ahydrofoil with a novel ducted tip device was found to have superior cavitational andaerodynamic performance over a conventional rounded tip configuration. This section isdivided into three sub-sections: analytical method, tip vortex structure and wing tipmodification.5.1.1 Analytical methodA new analytical method, referred to as the “quasi-similarity” method, has beendeveloped for modelling vortices in a freestream. The new approach combines apolynomial solution with the similarity variable technique. By using this method, the nonlinear partial differential governing equations are reduced to sets of ordinary differentialequations. The fundamental idea of the “quasi-similarity” method can be applied to abroad range of non-linear problems that cannot be described by self-similar solutions.The “quasi-similarity” method is used here to model wing tip vortex flow. It providesthe first non-linear analytical wing tip vortex model. In this new method, each resultingset of ordinary differential equations describes the flow with varying degrees offaithfulness. The first order set of ODEs gives solutions that are second order accurate inthe axial variable z; higher order sets of ODEs provide higher order accuracy. Fortunately,ChapterS Summary and Conclusions 113the first order terms of the solution polynomial can be obtained analytically in completefunction form. The second order tangential velocity component and pressure distributionare also obtained in complete function form. Other higher order terms must be calculatednumerically by solving a set of ODEs with appropriate boundary conditions.There are three major assumptions of the quasi-similarity method. These assumptionsare:(1). the axial velocity deficit or excess in tip vortices is not significantly greaterthan the freestream velocity;(2). the vortex circulation remains constant with downstream distance;(3). the flow is laminar.The first limitation can be expressed mathematically (assuming a rectangular wing) asz> CRe.S/l67uwhere z is the downstream distance, CD is the wing total drag coefficient, Re is theReynolds number based on the wing chord, and S is the span of the wing. This distance isabout a hundred meters downstream of the wing trailing edge for a typical dragcoefficient CD O.O1, Reynolds number Re 5xlO, and a one meter span wing (forwhich the flow should be laminar). The second assumption is a reasonable one after thevortex roll-up is complete. Therefore, very near the wing the first two conditions may notapply. The flow in a practical trailing vortex is usually turbulent, at least near the wing,due to large Reynolds number. This theory is still valid for a turbulent flow only if aneddy viscosity model is used and the eddy viscosity is a constant everywhere in the flow.5.1.2 Tip vortex structureAll three velocity components in tip vortices decrease with increasing downstreamdistance at fairly large downstream distance of the vortex-generating wing. The radialvelocity diminishes most rapidly, followed by the axial velocity. The tangential velocitydecays most slowly. This prediction is consistent with many experimental observations.Chapters Summary and Conclusions 114At a fairly large distance downstream of the wing, only an axial velocity deficit canexist for a wing experiencing a positive drag force. An axial velocity excess can exist fardownstream of a wing generating thrust (e.g. with wing tip-mounted engines).A unique and important feature of the quasi-similar wing tip vortex model is that thepredicted drag on a vortex-generating wing due to a single quasi-similar tip vortex isfinite. To the author’s knowledge, no other vortex model has this property.The tangential velocity component and pressure of the first order quasi-similaritysolution are exactly the same as those in an advanced linear tip vortex model (Batchelor1964). The axial velocity field of the linear vortex model differs from that of the quasi-similarity model. This discrepancy in the axial velocity causes a single vortex in the linearmodel to have an infinite drag whereas the quasi-similar model indicates a finite drag.The axial velocity and tangential velocity predicted by quasi-similarity theory havebeen compared with experimental measurements. The agreement is good at largedownstream distances from a wing. The experimental results confirm that the axialvelocity deficit on the vortex centerline decays as the reciprocal of the downstreamdistance; the tangential velocity in a tip vortex is proportional to the inverse square rootof the downstream distance.5.1.3 Wing tip modificationA novel ducted tip device has been tested in a wind tunnel and a water tunnel.Aerodynamic and cavitation performance of the ducted tip has been compared with thatof a conventional rounded tip.The ducted tip has up to 6% better Lift/Drag performance than a square cut wing tip*of equal span at higher angles of attack, roughly 8° or more. The principal cause of theseperformance improvements is the greatly reduced induced drag afforded by the ducted tipat high lift coefficients, which more than offsets the increase in parasitic drag resultingfrom the increased wetted area.Experimental data show that a square cut wing has similar aerodynamic performance to a rounded tipwing. The difference between those two conventional tip configurations is within experimental error.ChapterS Summary and Conclusions 115The tip vortex cavitation inception index of the ducted tip geometry is at least 33%lower than that of a conventional rounded hydrofoil tip at all positive angles of attack.No pressure distribution difference between tip geometries was observed over themajority of the wing surface (differences from one tip geometry to another were less thanthe experimental error, which is about 7%). However, near the wing tips the pressuredistribution was significantly modified by the addition of tip devices. The pressuremeasurements are not accurate enough to confirm the Lift/Drag ratio improvement, butthey are supportive of the cavitation results— tip vortex cavitation is significantlymodified by the ducted tip, whereas leading edge cavitation is not significantly affectedby addition of the tip device.The ducted tip is effective at reducing the tip vortex inception index and the vortexinduced drag. This tip design spreads the shed vorticity in the transverse plane behind thewing (Trefftz plane) over a line and circle, while a conventional wing tip spreads thevorticity only over a line.The vastly reduced tip cavitation index, together with the improved Lift/Dragperformance, suggests there is a significant role for the ducted tip device in both militaryand civilian marine propeller design.5.2 Suggestions for future workA careful experiment should be designed in order to further verify the quasi-similarityvortex model. The wing total drag and vortex circulation should be measured accuratelyin a direct or indirect way with the Reynolds number maintained low enough that thevortex flow is laminar.Application of the “quasi-similarity” method to other non-self-similar problems couldbe possible. In fact, jets with a tangential velocity component (twisted jets) were studiedin detail by Loitsyanski (1966) and Goldshtik et al. (1979 and 1986). They employed anon-similar power series to construct their analytical solutions. However, twisted jets aredifferent from wing tip vortices. The swirling jets are characterized by the followingconservation integrals: flowrate, axial momentum and angular momentum. In contrast,Chapter 5 Summary and Conclusions 116the tip vortices are characterized by a freestream velocity, and the conservation of vortexcirculation and axial momentum. Related problems, for instance, jet flows (with orwithout swirl) in a freestreàm seem amenable to treatment using the quasi-similaritytechnique.A different set of polynomials, with an amplitude function proportional to z1 insteadof the liz’ used in the theory developed in this thesis, is a possible solution of thesimplified governing equation (2.1.1). This amplitude function has the general feature ofvortex rollup — the magnitude of velocity components increases with increasingdownstream distance. Is this a solution pertinent to the roll up of the wing tip vortex? Arethere any physical problems related to the new polynomial solution? Furtherinvestigations are needed.Instability of a trailing line vortex has attracted great interest in recent years. Some ofthe stability studies based on Batchelor’s partly-linearized theory include works byStewartson and Capell (1985), Stewartson and Brown (1985), Mayer and Powell (1992b),and Duck et al. (1992). 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