aerodynamics of wings with leadingedge flow separation in supersonic regime

8/13/2019 Aerodynamics of Wings With Leadingedge Flow Separation in Supersonic Regime

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AERODYNAMICS OF WINGS WITHLEADING EDGE FLOW SEPARATION IN

SUPERSONIC REGIME

by

SYLV IN PIERRE

Department of Mechanical EngineeringMcGill UniversityMontreal Canada.

A Thesis submitted to the Faculty of Graduate Studies and Researchin partial fulfillment of th requirements for the degree of

Master of Engineering.March 31 1992

Sylvain Pierre 1992


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Abstract

The flowfleld over delta wings at angle of attack u been of constant interestto the scientific community in the last decades. Interest has been stimulated as areslllt of the applications of low aspect ratio wings to missiles, supersonic .i ansportaircra{ts and aircraft fighters. The rolling-up of the vortex sheet over the leewardside of the delta wing creates suction peaks which favourably increases the lift andthere{ore permits extended manoeuvrability at higher angles of attack.

Analytical rnethods solving for the separated flow over delta wings have been basedon the slender-body theory, which applies for wing geometries of smalt aspect ratio.The slender-body solutions are independent of the Mach number, and therefore, {orsnpersonic flow, cannot appropriately take into accon nt the directional propagationof disturbances. The slender-body model is attractively simple for the pressure distribution and lift prediction, but in the supersonic regime it can lead to large errorswhen the wing geometry is not very slender with respect to the Mach cone.

ln the treatment of attached flow over arbit:-ary wing-body cornbinations in su-personic flow regime, the metho


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ommaire

La communaut scientifique s 'intresse au champ d'coulement au-dessus de l'ailedelta . angle d'incidence depuis les quarantes dernires annes. L'application de l'ailt'trs effil aux missiles, aux avions de transport supersoniqhc et aux aVIOns dt' chasse su maintenir cet intrt. L'apparition de la nappe en cornet sur l'extrados engendrede puissantes aspirations qui u ~ m e n t e n t avantageusement la porte et permet ainsiune plus grande manoeuvrabilit des angles d'incidencc plus lev >.

Les mthodes de solution analytiques de l'coulement dtach sur l'extrados sontfonds sur la thorie des corps effils. Cette thorie s'applique la gomtrie des ailestrs effiles. La solution apporte par la thorie des corps effils cst indpendante dunombre de Mach, et donc ne peut tenir compte de la propagation direct.ionnelle desperturbations en rgime supersonique. La simplicit e la thorie des corps dlilps enfait un modle attrayant pour le calcul de la pression ct de la porte; toutefOIS, pourdes gomtries moins effils ce calcul peut induire de larges erreurs.

Dans le traitement de l'coulement attach pour des combinaisons arbItraires ailefuselage en rgime supersonique, la mthode des SingularIts de Vitesse base sur lathorie es mouvements coniques amne une solution plus correcte que la thoriedes corps effils car elle considre les conditions limites appropries de propagationdirectionnelle l'extrieur des ailes. Pour les ailes de moyenne forte envergure, leserreurs engendres par la mthode des corps effils peut atteindre plus de 30 que lasolution obtenue l'aide de la mthode des Singularits de Vitesse.

Cette thse prsente une extension de la mthode des Singularits de VItesse pourl analyse de l'coulement dtach sur l extrados de l'aile delta. La solution analytiqueainsi obtenue est compare avec la solution de la thorie des corps effils et les donnesexprimentales disponibles.

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cknowledgements

1 would like to thank my thesis supervisor Professor Dan F. Mateescu for hispatience and guidance throughout the course of this work.

1am aiso grateful to my parents Fritz and Lonie my brother and sisters StphaneMarie-Jose and Isabelle and also to my companion Sylvie Varin for their sustainedhelp and support.


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ontents

2

bstractSommaireAcknowledgementsTable of contentsList of figuresList of tablesNomenclature

IntroductionSupersonic Flows Past Delta Wings2 1 General Considerations2 2 Conical Motions in Supersonic Flow

2 2 1 Governing Equation2 2 2 Compatibility Relations

2 3 ttached Flow Solution of Delta Wings in Supersonic Flow based onthe Method of Velocity Singularities2 3 1 Type of Singularities2 3 2 Boundary Conditions2 3 3 Solution for a Thin Delta Wing2 3 4 Generalization of the Solution through the Velocity Singularity

I l

1tI

viVIII

IX

X

7

1113

1414

1516

A pproach 18

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,

3 Analysis of Delta Wings with Leading Edge Flow Separation in Supersonic Regime 313 1 BrieC Introduction of Previous Solutions 313 2 A Simplified Conical Solution for Thin Delta Wings at Tncidence . 343.2.1 Determination of the generic potential function . . . . 34

3.2.2 Determination of constants based on the Boundary Conditions 373.2.3 Analytical Expressions for the PerturbatIOn Velocities 433.2.4 TheoreticaJ Solution for the Pressure i s t r i b u ~ i o n p and the

Lift Coefficient C3.2.5 Numerical Details.

3.3 An improved Solution for Delta Wings with Leading Edge Flow Separation3.3.1 Importance of the Boundary Condition3.3.2 Additional Distribution of Supersonic Ridgt:s to Satis{y the

Boundary Conditions on the Mach Cone . . . . .3.3.3 Improved Solution Including the Additional Ridge Distribution

4343

4545

4648

4 Comparison Between the Theoretical Solutions and ExperimentalResults4 1 Lift Variation with Angle of Attack4.2 Coefficient of Pressure p Distribution along the Span .

4.2.1 Variations of the Coefficient of Pressure for a Not So Slender

596061

Wing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.2 Variations of the Coefficient of Pressure for a Slender Wing . 63

4.3 Discussion . . 645 CONCLUSIONS 71Bibliography 73A Separated Flow Slender Body Models. A l

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A 1 Legendre s Model .A.2 Brown and Michael s Model

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A-lA 4


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ist of igures2 1 Schematic drawings of separated flow over slender d lta wing2.2 Typical pressule and lift distribution over delta wing2.3 Separated flow in subsonic and super lonic flow2 4 Types of flows over delta wings in snpersonic regime .2.5 Uses of a vortex ftap .

212223

2425

2.6 Wmg geometry in the physical space 262.7 Transformation from the physical plane to the llew plane z = y + z 72.8 Wing geometry in the physical space with the associated boundary

conditions . . . . . . . . . . . . . . . .2 9 Geometry in the x plane and the X plane.2.10 Pressure coefficient cornparison between the slender-body theory and

the Velocity Singularity method in the case of supersonic attached flow(Mec = 1.9, A = 73, BI 0.493 and E k) = 1.205).

3 1 Wing geometry and boundary conditions in the physical space3.2 Geometry in the x plane and the X plane. . .3.3 Variations of the perturbation velocity cornponents on the Mach cone,

289

30

505

for BI=0.43 and a = 4 . . . . 523 4 Variations of the perturbation velocity components on the Mach cone,

for Bl=0.43 and a = 14 . . . . . . . . . 533.5 Variations of the perturbat ion velocity cornponents on the Mach cone,

for Bl=0.67 and a = 12 54vu


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3.6 Variations of the perturbatiun velvrity C()Ill l ' )lIt l t ~ l n the \t.lfh \,)1\("fr Bl--=O 67 and Cl := 20 . 55

3.7 Variations of thl perturbation vcl"Clty ul t.n the \ta\ h {1l1lC', frBI -067 and a 20 5638 Variatic.ns of the perturhtion "C' ority v Uro ( n the \L1Ch C\'IlI ' , fl rBl::::O 67 al,d a := 20 . . . . . . . . . . 57

3.9 Variatins of the perturbati0n \docity u U nn thl' \LI('h ( 1111( , fnrBl=O 67 and a 20 58

4 1 Variation of 11ft for Bl=0.67 654 2 Variation of lIft for Bl=0.43 6643 Variations of the pressure coefficient on a delta ",ing )\)th fluw separa-

tion for B1= O 67 and a 12 674.4 Variations of the pressure cocffici('nt on a delta "'Ing wlth flow separa-

tion for Bl=O 67 and a 20 684.5 Variations of the pressure coefficient on a delta w n ~ wlth fluw s('para-

tion for Blo=O 21 and a 35 . . . . . . . . . . 694.6 Variations of the pressure coefficient on a delta wing wiH flow separa-

tion for Bl=O 21 and Q = 7 5 . . . . . . . . 70

Vlll


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isto Tables

3.1 Values on the Mach cone By=l) r the perturbation velocity components 464.1 Table of cases examined for the lift coefficient Cl 614 2 Table of summary of lift resuIts . . . . . 614 3 Table of cases examined for the coefficient of pressure p 624 4 Not so slender wing parameters for the coefficient of pressure p 624 5 Slender wing parameters for the coefficient of pressure p . . 64

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Al

BC,CpD(0 1),1(0 1)

E(k),n(p,k)FrlAMPQRelmsS(Zl X2, xa0 1ZX(Zl, Z2, X3)XlyZ

U,v,w

omenclature

attached flow constantangle of incidence= VMoo2 1lift coefficientpressure coefficientIntegraIs dependent upon vortex position, defincd by equations3.25 and 3.41, respectivelycomplete elliptic integrals of the second and third kind resp.complex potential functionvortex core strengthsemi span of wing in the plane Xl = 1semi sweep angle of wingMach numberconstant defined in equation 3.36constant defined in equation 3.12real partimaginary partridge positionequation of the wing surface= 91 ih l , vortex position in the x plane

y z, complex plane defined in equation 2.9Y Z, complex auxiliary plane

cartesan coordinates= Y Zt, vortex position in the X plane= Z2 /Xl non-dimensional spanwise coordinatel :3/Z1, non-dimensional coordinate normal to the wingperturbation velo city component

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l U V W

U; oSubscriptsnv

complex velocity potential associated with th perturbationvelocity component u v w resp.{ree stream velocity

normal to th leading-edgewithout th vortex influenceunperturbed

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... .

hapterIntroduction

Previous Investigations on the Flow over Thin DeltaWingst was early recognized experimentally that delta wings of small aspect ratio with

sharp leading-edges provided stable and controllable attached flow conditions at Iowangles of incidence in supersonic regimes [1 2]. Classicallinearized lifting surface theory have been used (for attached flow throughout the flight conditions) and simplifiedfor small aspect rat io wings under the assumption of linear slender-wing thcory to ob-tain an analytical expression for the wing loading [3 4]. With the assumptions usedto linearize the potential flow equation, i.e. steady state inviscid irrotational flow 1followed the decomposition of the velocity potential into a free stream and a smallperturbation potential: this implied that the theory would apply to thin wings withsmall camber, twist and angle of attack which reprebent conditions of aerodynamicslenderness. Slender-wing theory goes one step further by requiring that the variations in velocities in the longitudinal direction of motion are negligible compared tothose in the cross-flow plane (geometric slenderness), therefore reducing the problemto an equivalent 2D incompressible flow where the powerful techniques o conCormaItransformation can be used. he solution thus C btained is independent of the Mach

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number and becomes inadequate for not-so-slender wings.Several attached flow theories, based on small disturbances, were developed by

various authors [5 6, 7] for supersonic flow. One of these methods is based on theassumption of conical motion. A flowfield is conical i the velocity and other flowvariables do not vary along rays issuing {rom a certain vertex. First defined byBusemann [8J conical superonic flow theory wu developed by various authors [9,10],and generalized by Carafoli, Mateescu and Nastase [11] for arbitrary wing plan orm,higher order conical motions, the direct and indirect problem. Mateescu furtherextended the application of this method to cruciform wings and tails [12 13] andrecently tn arbitrary wing and conical body of arbitrary cross-section [14].

Up to the end of the 1970's, most aerodynamicists used linear theory based methods for the design of twisted and cambered wings for efficient low-lift supersonic flight[16 17]. The idea was to avoid flow separation at the leading-edges [18]: at one designcondition, the flow should be attached to the whole length of the leading-edges, toavoid the possibility that vorticity from one edge rolls-up on the uppp.r surface andvorticity from another part sheds on the lower surface [19 26]. Attached ftow thenwas thought of being the best way to obtain the lowest drag and the highest lift todrag ratio [25].

The flow about slender-wings with steady motion separates from the edges atmoderate or even small angles of incidence. A free shear layer, or vortex sheet,separates from the wing along each leading-edge, and then proceeds to roll-up intospiral vortices, lying ab ove the wing and inboard of the leading-edges. The flowdescription is detailed in chapter 2.

Smith [43 44, 45] amongst others [42 47] has reviewed the theoretical methodsused to predict flow properties for the flow separation of thin delta wings. To understand the topological nature of such a flow a clear understanding of the conceptsrelated to 3D flow separations is needed. These concepts were first defined by Maskell[15] who identified two basic components of the structure of the viscous region, eachof which being characterized by a particular form of surface ftow pattern the bubble

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and the free-vortex layer. His approach considerably simplified the construction ofthree-dimensional skeletons of complex flow patterns. Based on slender-body andconicaI motion assumptions, two approaches have been used to ~ v e l o p expressionsfor the prediction of the wing loading and the drag for separated flow over thin deltawings : non-tinear surface theories and detached flow rnethods The detached Dow

m e ~ h o d s are discussed in Appendix A.Kchemann [46] developed a non-linear sIender wing conieal flow metbod by as

suming a model consisting of a vortex sheet that incorporates part-span vortices,who se height above the wing is unimportant . The spanwise position of the vortices isgiven empiricaIly, and it is assumed that the flow separates smoothly at tbe leadingedges. Squire [48] extended the work to supersonic flow by incorporating rt'sultsobtained by Mangler and Smith [49] ( numerical solution for the strcngth and position of the vortex core together with the shape of the vortex sheet and the total lift )for the spanwise position. By itself, this approach do not daim to compute the truespanwise position and cannot give any information as to the strength or position ofthe vortex core.

Legendre [50] developed the first model for separated flow over a tbin fiat deltawing by replacing the vortex sheets with two discrete line vortices originating from thewing apex. The position of the vortex core is obtained by requiring the vortex-coreto lie along a streamline of the three-dimensional flow.

Brmvn and Michael [52] improved the model by including a feeding vortex sheet(mathematically, a cut) [51] joining the leading edges to the vortex cores, thus makingthe problem determinate.

The Brown and Michael mode has been applied to severaI problems: slenderwing-body combinations (conical and non-conical) [53, 54J, second-order slender wingtheory [55]; in subsonic flow: leading-edge flaps [56], vortex breakdown [57], coniealaugmentor delta wing integration [58], leading-cdge blowing [59].

Mangler and Smith [60] did an analysis (in incompressible flow to determine theshape and flow field near the centre core of the spiral vortex through asymptotic

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expansion. They assumed that the vortex sheets are shed tangentially at the leadingedges and roll-up as spiral vortices. Each spiral vortex is split into an inner partsmwl in size and remote from the wing) and an outer part comprising a finite

vortex ,;heet, joining the leading-edge to the inner part). The inner part. is rt.,lacedby a concentrated vortex joined to the outer spiral by a cut. he boundary conditionof zero pressure discontinuity across the separated vortex sheet is modified in theinner region to one of zero totalload on the line vortex and eut. Smith [61] relaxedthe restrictions on the u t ~ r part of the vortex sheet and used iterative procedures toobtain a solution. Jones [62] extended the procedure to arbitrary non-conical wingplanform.

Several methods have been explored, using CFD, to solve the problem of the flowseparation over thin delta wings and we will mention here a few. Mook and Maddox[63] extended the vortex lattice method to include leading-edge separation. Weber,Brune, Johnson et als. extended the panel method to include leading-edge separation subsonic). In order to include compressibility effects, Vigevano [65] coupled thenon-linear potential governing equation with the Brown and Michael model to get afinite difference scheme. Hitzel and Schmidt [66] compared the application of timedependent Euler schemes to potential flow methods: whereas panel and vor tex latticemethods are restricted to particular regime, and can demand significant modifications and manipulation for analysis of complex geometries when the flow structurecannot be described in easy terms [66]), Euler methods are applicable to any speedregimes without any particular initial assumptions. Oh and Tavella [67] applied thevortex-cloud method to flow separation of slender flapped delta wing in subsonic flow:this method gives a more detailed definition of the free-shear layer or vortex sheet)over the lee-side of a delta wing than can be obtained with Euler, Navier-Stokes orpanel solvers, and with greater speed of computation. Kandil and Chuang [68] solvedthe unsteady supersol1ic flow around a rigid sharp-edged delta wing using the unsteady Euler and thin-layer Navier-Stokes equation: they concluded that for accurateprediction of distributed aerodynamics characteristics, the Navier-Stokes equations

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are required at least in the vortical shock interactions region), but the prediction ofthe total loads based on the Euler equations have been found sufficiently accurateprovided that the leading edge is sharp.

Thesis Objectives and OrganizationThe problem o flow separation Crom a delta wing has been adressed analytically inthe frame of the slender-body theory . This model is attractive1y simple, but can leadto large errors when it is applied to delta wings of large aspect ratios. In a.ddition,the slender-body model doe not take into account the directional propagation ofdisturbances, as weIl as the effects o compressibility for supersonic flow.

In the treatment of the attacht::d flow over delta wings in supersonic regimc, themethod of Velocity Singularities provides a more accurate alternative over slenderbody models by taking into account the directional propagation of disturbances andthe compressibility effects, within linearized supersonic potential flow methods. Bcingin very good agreement with experimental data for attached flow, this analyticalmethod treats geometries o arbitrary wing-body combinations. For wings of largeaspect ratios, the slender-body model can overshoot the vrediction of the method ofVelo city Singularities by more than 30 .

The first objective of the present work 8 to develop an analytical tool that solvesthe problem of the flow separation over delta wings in supersonic fiow through the useof conical motion. The strength o the method of Vewcity Singularities in the caseof attached flow cornes from the jUdlCious placement of the singularities on the wing,which accurately satisey the boundary conditions, including those Oil the Mach cone.This cannot be done in the case of flow separation, in which singularities appear in thefiuid domain above the wing. The first objective is thereCore to adapt th(; method ofVelocity Singularities to separated flow, by considering the appropriate singulariticsabove the actual wing, and see what gains are obtained by the use of the properphysical boundary condition on the Mach cone and at infinity.

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The second objective of the thesis is to lay the ground for future development tosolve for an exact boundary condition on the Mach cone. This would then completelysolve the problem for supersonic flow

In thc prcsent thesis the treatment of the leading-edge vortex singularity is madethrough the use of conical motion in supersonic flow In Chapter 2 the method ofVelocity Singularity in attached flow is presented. In Chapter 3 a Simplified conicalsolution is first prellented for solving the problem of fiow separation by neglectingthe influence of the Mach cone on the solution. An extension of the method is thendevej Jped in the second part of Chapter 3 to take into account the correct boundarycondition on the Mach cone. he theoretical solutions derived in Chapter 3 elrevalidated in Chapter 4 by comparison with experimental data and other results.

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concentrated vortex cores above the wing (Figure 2.2): the singularities at the leadingedges are transported within the flowfield along the vortex sheets (from the wingleading-edges) up to their vortex cores, where most of the vorticity is c o n c e n t r a t ~ JFor rounded leading-edges, the separation is milder.

The variations in velocity distribution affect quantities of interests to the aerodynamicist, like the pressure distribution, the wing loading and the drag: as comparedwith an attached flow the lift increases non-linearly with the angle of incidence forseparated flow over delta wings. Two useful parameters are used to relate the strengthof flow separation with geometry and aerodynamics: BI and ail. The flrst one, BI,relates the wing geometry to its cone of directional propagation of disturbances ( BIrepresents the ratio of the semi-wing span 1to the distance to the Mach cone l B ina given plane). When this ratio is greatel than unity the leading edges are calledsupersonic, which me an that the delta wing protrudes outside its Mach cone of directional propagation disturbances and the flow velocity normal to the leading-edgeis supersonicj flow separation occurs when the leading-edges are subsonic, i.e. withintheir Mach cone. The oUler parameter is a/l which relate the angle of incidence towing semi-span: for a specified angle of attack, for which there is flow separation, anincrease in wing span will decrease the effects of ftow separation on pressure distribution and total lift (geometry effect); in order for the flow separation to be againimportant for this larger aspect ratio wing, a corresponding increase in incidence isrequired. Thus : >r tbe same value of a/l a larger aspect ratio wing will need a greaterangle of attack Cor the phenomenon to have the same importance than for a slendcrdelta wing. As a result, tbe flow separation phenomenon is more acute for sIen derwings than for moderate to large aspect ratio wings.

The flow separation for slender delta wings at incidence occurs at low speed, aswell at high speed, is schematically shown in Figure 2.3. The basic Ceatures of theflow remain qualitatively the same, changing quantitatively with changes in angles ofattack and Mach number.

In recent years, many researchers [21, 22, 23, 24, 35, 38, 40] devoted their efforts

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at the available data for the pressure distribution on the upper and lower sides of adelta wing, Wood showed that the upper side contribution to the lift reduces with theincrease in the Mach number and with the decrease in the wing aspect ratio. n thesupersonic regirne with cl ssic l flow sep r tion without shock waves), the pressuredistribution on a delta wing, as well as the perturbation velocities are conical, i.e.they are constant along any radius issued from the wing vertex. This property ofthe conical flow with separation at the leading-edges can be used to deterrnine thernathematical solution of the flow problem.

The investigations performed by various authors were clearly dirned at using thebenefits of flow separation over sien der delta wings in aircraft fighters designs [42], ina controllable and predictable manner, to improve upon high-Hft capability and poststa11 maneuvring Figure 2.5). Polhamus [30J reviewed the history of the developmentof slender-wing aircrafts. Lamar [25J reviewed the control of vortical separated flowsin various speed regirnes through the use of fixed or moving devices like strakes,leading-edge extensions, vortex flaps [27J or spanwise blowing [28 29J.

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2 2 Conical Motions in Supersonic FlowIn this section, the geometrical transformations lcadlllg to conical motion and thereduction of the equation of motion of the' flow to a Laplace equatlOn are shown.The geometry in the physical plane and in the transformed plane is dl'scribed. Thegeneral relationships between the perturbation velocity components art' shown.

2 2 1 Governing EquationConsider a thin delta wing at incidence Cl in an inviscid steady-state irrotationalsupersonic flow regime (Figure 2.6). Let the origtn 0 of the axis system Xl , X2, xa)corresponds with the wing apex, with the axis 0;:;1 p


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.1;

2.3)In irrotational flow the velocity vector V can be expressed as the gradient of

a velocity p o t e n t i a l ~ Under the assumption of smaU perturbations, the velo citypotential can be decomposed into a free stream velocity potential and a perturbationvelocity potential 4J such that the corresponding perturbation velocities u, v and wcan be defined by the relations

/P/lJz Uoo +4J/2:1 Uoo +u ,4?/8z 2 = l/J/8z 2 - v 2.4)4? /8za 8N8z3 w .The linear governing equation for a steady state inviscid irrotational flow can be

expressed in terms of the perturbation velocity potential 4J

28 24J 82l J 824J- B 8 2 + -8 + 2 = 0 k=1,2,3. 2.5)Z l 2: 2 Zaf the equation 2.5) is sequentially differentiated with respect to Zl Z2 and Z3

one can obtain

2.6)where ql = U, q2 = v, qa = w are the perturbation velocity components, along orparaUel to the respective coordinate axis Zl , Z2 za). The equations (2.6) obtainedare in a form which permits their simplification through the use of conical motions.

In a conical motion, the flow properties such as \ elocity components, pressure anddcnsity do not vary along rays issuing from the wing vertex. Defining the followingnon-dimensional coordinates

y Z2- ZlZ Z3 2.7),Zl

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the perturbation velocity components will become functions only of y and Z , sudthat the equation (2.6) reduces to the two dimensional form

This equation can be further reduced to the Laplace equation using the geometricaltransformation

{y -z - z (2.8)

82q1e 82qle _ 08y2 8z2 --- 2.9)

In the new plane a: = y z, the wing trace remains unchanged (y = y , z = 0)and the Mach cone trace in the physical plane (Gy z ) is represented by the semi-axesy E (-00, } )U(iJ,oo) (Figure 2.7).

2 2 2 Compatibility RelationsThe solution to the Laplace equation consists of harmonie functions. Defining thecomplex plane x = y iz, the perturbation velocities can be expressed as the realpart of associated complex functions,

u(y, z) - Re[U x)],v(y, z) Re V x)),w(y, z) - Re[W x)),

U = u uV = v vW = w +iw

(2.10)

The complex functions U V W, associated to the perturbation velocities u, v, wrespectively are related by the compatibility relations 11]

lJU xdV dW1 1 - B2z2 (2.11)which are obtained from the irrotationality conditions of the motion.

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f

2 3 Attached Flow Solution of Delta Wings inSupersonic Flow based on the Method ofVelocitySingularities

In this section the generaI method of velocity singularity approach {or a conical flowand the main results obtained in reference [11 69] in the case of attached flow fora thin delta wing with subsonic leading-edge flaps at angle of attack a are outlined.he influence of the boundary condition on the Mach cone on the solution is shown.

he general features of the method of velocity singularity can be summarized as{ollows:

No restrictions are made regarding the supersonic regime relation to theleading-edge sweep angle A the present method being able to handle bothsubsonic and supersonic leading edges.

The present method applies equally to the case of nonsymmetricalleading edges.t can also deal with trapezoidal and polygonal wings.

The effect of the wing thickness on the perturbation velocities around the wingcan be determined using the same method of solution.

t can deal with wing and conical body of arbitrary cross section [14].In what follows the solution of a thin delta wing with subsonic leading edges and

flaps will be presented.

2 3 1 Type of Singularitieshe wing un der consideration is shown in Figure 2.8. The wing has a semispan

length with ridges situated in the plane :1:1 1. A ridge RI is defined as aline separating wing regions of dHferent incidences. The central part of the wing is

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assumed to be at a constant incidence ao and the leading edge flaps at a constantincidence a l

The analysis of attached flow near the leading edge in [11] bas shown that thevelocity perturbations u, v present the singularities 1 y ft/2. Similarly, the sud-den change of incidence long the ridges ORl and OR correspond to a source-typesingularity of the form In(y 8).

The governing equation (2.9) being linear, the complex velocity functioll V willbe determined as the sum of the contributions oC each singularity,

(2.12)where the two first terms represent the contribution of the leading-edges, and the lasttwo the contribution of the ridges.

The problem is thus reduced to the determination of the complex function orV W ) presenting the singularities at the points corresponding to the leading edgesand ridges and satisfying the boundary conditions. Note that once is known, Vand W can be determined using the compati bility relations.

2.3.2 Boundary ConditionsThe wing is a streamsurface of the flow. Defining the surface equation oC the thindelta wing by S ~ l , Z 2 , Z 3 ) = Z - Z Z t , ~ 2 ) , the boundary condition becomes insteady :flow equivalent to

vs V = os ' sw = (Uoo + ~ = r;-UOQ

V ~ l VZlwhere the velocity vedor V is given byv= Uno +u, v,w).

The perturbation velocities can only propagate inside the Mach cone,15

(2.13)(2.14)

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u = v = w =0 for y E 00, - ~ U , 0) . 2.16)The perturbation veJocities u and v are antisymmetric with regard to the windward

and lccward side of a thin wing, i.e. the Z3 = : = 0 plane. Since the velocity musthe continuous on the portion of the z axis between the leading edge and the Machcone, it follows that the perturbation velocities u, v must be set to 0 50 that theantisymmetrical nature of the flow is accounted for.

To summarize the discussion, the boundary conditions on the wing and outside itin its plane are dcfined as

W -ooUoo fory E -I ,-s)U s,l) on : =0W = -o,Uoo for yi E -lI,s) on Z = 0u =:: v=O forY E(-CXl,-I)U(I,oo).

2.17)

Using the compatihility equations 2.11), the boundary conditions 2.17) expressedin complex form in the complex plane z = 11 + z

Im[U]z o - 0 on y E - l , -8)Im[U).:o - C on 11 E 1,8)Re[U]z o 0 on y E -00, -1) U 1,00),

whcre C is a real constant which will he determined in the following.

2.3.3 Solution for a Thin Delta Wing

2.18)

\\Then the leading-edges are symmetric, and that there are no ridges, the problem canbe solved in the conformai auxiliary plane ...;zr- 12, as shown in Figure 2.9. In thisnew plane, the solution is of the form

U .A . A,X=:: z - 2 2.19)The generic {orm satisSes the boundary condition on the wing and on the Mach

cone. The constant is determined through the use o{ the compatibility re lations ainee16


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the perturbation velocity w component is known on the wing and on the Mach cone.Integrating from a point on the wing Yw to the Mach cone z = 1/B ,

1/B _ll B VI -- B2z2dW . dU.1/_ Il. IZ 2.20)The integration is independent of the path. An alternate integrating Iimits wouldyield:

LDO iOO vI - B2Z2dW= . dU.o 0 IZOn the L ~ the integrat gives l Uoo On the RHS,

manipulations becomes

The complete solution on the wing z = O,z = y = y ) is

u = CtoUooE(k) 1JI - y/I)2

2.21)the integral aCter sorne

2.22)

2.23)where E k) is the complete elliptic integral of the second kind, k being the modulusle = .JI - B212 . The solution obtained in the case of slenderbody is similar

JI - (y/l)2US.B. = CloU 1 2.24)However, the difference between the two methods can be very large and is due to

the consideration of the boundary conditions on the Mach cone. The elliptie integratfactor, E Ie), becomes important when the wing semi-span is not negligible with respect to the Mach cone radius 1/B In Figure 2.10, the pressure distribution obtainedwith slender-body and the conical method is compared for a wing of moderate aspectratio, BL = 0.493; in this case the sIender body solution is shown to overshoot theconical solution by 20 .

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2.3.4 Generalization of the Solution through the VelocityS ingu larity A pproach

To deaJ with arbitrary wing pJanforms, more versatile transformations are required,so that the influence of cach singuJarity can he isolated more ea:.iJy. The methodof Ve10city Singularity separates the influence of the left-hl.nd side singularities as-sociatcd with the Jeading-edge from the right-hand side l e a a . i n g - e d ~ , e singularities toobtain the general solution. The constants are then obfained through the use of thecompatibility relations, as was shown in the previous subsection.

Complete expressions are obtained in another auxiliary plane, X 2 = : ~ : for thesinguJarities at Ah RI, 2 = for the singularities at A2 R 2 These are:

UA, = Al V I+z)/ I-z)UAJ = A2 V I- z)/ I+z) 2.25)UR, -- 2/1r Cl cosh-1V I- z) l + ,))/ 21 . - z))uRJ = 2/1r C2 cosh-tlf{z + :.:) 1 + ))/ 21 .. + z

These expressions satisfy the boundary conditions on the wing. To fully solvethe probJem, the constants Al, A2, as weil as Cl and Cz must be determined. Themethod is detailed in [11]. The ridge constants, Cl and C2 , are each ohtained byintegrating the compatibility relation on a semi-circle of a very smaU radius aroundthe ridge points; the leading-edge constants, t and A2, require more work: first werelate one to the other using the compatibility relation, and specifying that

2.26)in order to avojd infinite perturbat ion velocity w at the origin. Then, using again thecompati bility relation, we integrate from the wing leading-edge to the Mach cone todetermine the value of the A constants.

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-

Attached ftow solution for a thin delta wing of constantincidence o oIn this case, there is no need to consider the ridge singularities. The complete solutionon the wing z = O,z = y = yi) is

2.27)where the constant ao is given by

J l Bld 1 BI,)ao = a Uoo 1112 2 281 Bl I ) l + BI2)E(ka) - B 1 1 + 1,)K(ka) .where K ka) and E ka) are the complete elliptic integral of the first and second kindresp., ka being the modulus (ka = J[(1 - B1d(1 - BI 2)l [(1 Bl.)(1 in2 ] .Attached ftow solution for a thin delta wing of constant incidence with leading edge ftaps of constant incidence 1The complete solution on the wing is

0.0 2 [ -1 (1- y) l 8 cosh-1 1 y) l 8)]u = ..jz2 _ y2 ;- C cosh 21( _ y) 21(8 y) 2.29)where the constants C and ao are

{C = s ao - QI )Uoo 1.../1 - B 820.0 = aoUco I2)/E(k) - 2/1r C B 212y'12 - 8 2 / E k n p,k). 2.30)

In the a.bove relations, E k) and I1 p, k) are the complete elliptic integral of thesecond and third kind, with the argument p = 8 2.,,2 - 1 and modulus k. The casewhere the leading-edges are not symetrie can be round in [11].

An interesting feature of the solution is that a condition to obtain finite velocityconditions at the leading-edges can be determined by setting the value of the constant

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o to o. The resulting fiap de8ection a in terms of th geometry and th wingincidence o is then obtained as

2.31)

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0) Assumed flow field b) Approllmoted flow field.

Figure 2 1: Schematic drawings of separated flow over sIen der delta wmg

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'

1.2

.8L

.4

Attach( d flow

Non-lincar separated flow

Non-linear solution. 1with flow separation effec

' tLinear solutionattached flow)10 2 30a deg

Figure 2.2: Typical pressure and lift distribution over delta wing

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1 SU SONIC

Pararncfcrs invol\ cd: l n/

SUPERSONICt

M CH EFFECTSHIGH EFFECTS

igure 2 3: Separated ftow in subsonic and supersonic fJow

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2.00

1.00

o

Shock withno separation

10

l. .f _ ~ 1 t l, 1: . ~' L ~Shock induced Separationseparation bubble

20

... l, .. ,.'4 i ...

with shoc

crassical vortex

30 40

Figure 2.4: Types of lIows over delta wings n supersonic regime

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.

Controlled Vrt :x Force P ~ e n t i a l Benefits Approach trnsonic sus1ained maneu'.er l DI C ~ ~ -- . t r..- ~ Increase a QW supersonlc~ 1ncrease takeoff and insfanfaneous

/ ..ra . ~ maneuver lif1~ _ r .MP t fv'aintain subsonic c r ~ i s efficiency;increase supersonic ~ p a b i l i t y

Increase lift and drag for fanding1

4 5 ; ~ Provide W i l ~ r Erreet 1 high drag i ' r rallo Provid simple variabre geometry

Figure 2.5: Uses of a vortex flap

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l

. ~ /

igure 2 6: Wing geometry in th physical space

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z

WIN

,, MACH CONEWIN

----; 1 / - --:-;:.-- ; --- - -- .,Al Dl Yr 1 /B l i

Figure 2 7: Trans{ormation {rom the physical plane to th new plane z == y z

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U v w oMach cone)

. .

i Bz

w = s ina UooWing) y

Figure 2.8: Wing geometry in the physical space with the associated boundary con-ditions

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l

z X :::: y. 1::

- - e - - -- -I

,.< - < [2- X z..Y ::::: + iZ

-: = .=. = = = .::: _ _..

r Figure 2.9: Geometry in th x plane and the X plane29


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o 6

:, Slcnder- Lod)' 13]0 Conical solutlOn [l1,14J

c

c.

1 1i/

SIen dcr-body theory VVV'// \r - -0.2

0.1

Yelocity singularity ffiethod

o1 1 1 1

O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O S 0.9yl

fIgure 2 10 Pressurf coefficient comparison betwecn the slender-body theory andthl' Yelocity Singularity Illetlwd in the case of supersonic attached fle\\ Moo = 1.9,.\ - 73, BI = 0493 and E k) = 1 205).

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.

-

ChapterAnalysis of Delta Wings withLeading Edge Flow Separation inSupersonic RegimeIn thi s chapter, the solution to the separated flow past a delta wing in supersonic flowis presented. Two solutions are derived for the flow with leading-edge separation: i) aSimplified solution which takes into account the vortex system formed bove the wingdue to flow separation, and ii) an Improved solution which satisfies more accuratelythe boundary conditions on the Mach cone.

These theoretical solutions represent definite improvements over the slender.bodysolution, especially t slightly higher flight Mach number, at which the wing geometrybecomes less sIender with respect to the Mach cane. A numerical comparison withthe experimental resuIts is presented in the following chapter.

3.1 Brief Introduction of Previous SolutionsFlow separation over a delta wing occurs when the leading dges of the wing lie insidethe Mach cone and re subsonic BI < 1).

When the leading-edges are subsonic, t a given incidence and Mach number, the3


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1

(

pressure distribution over the thin delta wing is strongly affected by flow separation.The flow separation effects are stronger for low aspect ratio delta wings and highangles of attack than for higher aspect ratio wings and low to moderate angles ofincidence (effect of BI and 01./1 .

The first model of flow separation over slender delta wings which replaced thevortex sheets with two discrete line vortices wu done by Legendre [50]. The positionof the vortex core was obtained by requiring the vortex-core to lie along a streamlineof the three-dimcnsional flow. The model was improved by Brown and Michael [52]who included a feeding vortex sheet (mathematically a eut) [51] joining the leadingedges to the vortex cores, thus making the problem determinate. The eut permitsto satisfy Kelvil."s theorem since the vortex sheet i n c r e a s ~ s in strength with distancefrom the apex ~ o n i c a l flow , the increase being fed via the eut. Along this eut thepotential function and the pressure are discontinuous. To obtain the vortex-coreposition, Brown and Michael allowed the line vortex to be inclined at a small angleto the local velocity vector so that the force generated on the vortex-core exactlybalances the force on the eut. This force on the eut also implies that there is a finiteloading at the leading edges. Both of these methods are using the slender-body theorywith the addditional assumption of conical flow behaviour near the wing ape7.

However, the analytical solutions based on slender-body theory do not satisfythe condition that the velocity perturbations propagate only inside the Mach coneoriginating from the wing vertex, O.

In this chapter, the method of Velocity Singularities is applied to the problem offlow separation over delta wings in order to obtain a conical flow solution withoutresorting to the sIender-body assumption. The problem is solved in two steps: (i)a simplified method, and (ii) an improvement over the simplified method. In thesimplified treatment, the effects of the flow separation on a delta wing is considered byderiving the conica solution based on the vortex-type singularities situated above thewing as a result of the l e a d i r ~ g - e d g e flow separation. This solution is then improyedby using a distribution of singularities on the Mach cone in order to satisfy more

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accurately the corresponding boundary conditions.


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3 2 A Simplifled Conical Solution for Thin DeltaWings at Incidence

A simplified method of solution to the problem of the separated ftow over a thindelta wing is presented in this section. This step was considered necessary sincethe method of Velocity Singularities has never been applied to a problem where theflow singularities do not reside on the wing. The method presented here will onlyconsider the conical motions in supersonic flow, and therefore removed the geometricallimitations related to slender-body, i.e. a theory limited to low aspect-ratio wings.

3 2 1 Determination of the generic potential functionConsider a thin delta wing at incidence a with flow separation t both subsonicleading-edges, in the physical plane (Figure 3.1). The vorticity is concentrated in thevortex cores, about 90 , white the rest (about 10 ) is disseminated in the vortexsheets issuing at the leading-edges. The representation of the physical phenomenonin a tractable equivalent analytical model is done in the following way: the physicalvortex sheet which rolls-up above the wing to a vortex core is represented by a vortexline of strength r issuing t the wing apex and joined to the wing leading-edge bya mathematical eut. rhis is a simplified representation in which a pair of vortices isused to represent the complex system of vortices associated to the leading-edge flowseparation. This model keeps the general features of the flow and it has been used byother researchers in conjunction with the slender-body theory for this flow separationproblem [52, 53, 56J. In the present anaIysis, this model will be used to derive aconical flow solution, without resorting to the slender-body assumption, which canaIso be va1id for larger values of BI In this model, the vortex-li ne is connected to theleading-edge by a eut to ensure th t the analytical solution remains single-valuedj acontour integral around the delta wing must include the vortex-lines for the integralsolution in order to avoid situations where you could have a contour excluding thevortex-lines, making the potential solution multiple-valued. In subsonic flow, Smith

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[52] has generalized the approach by using multiple line-vortices originating at thewing apex, interconnected with cuts, 80 that the definition of the eut vortex-li necontour matches doser the vortex sheet shape in the physical plane. (For the workpresented, such a treatment of the singularities would be outside the scope of thethesis objectives.)

In the plane z y z, the vortex-line position is given as Tt = 9t ht. Thegeneric potent ial function is obtained in the auxiliary plane X = Y Z definedby the conformaI transformation (Figure 3.2)

(3.1)In this new plane, the geometry, i.e. the wing trace and the Mach cone are

represented by the segments

{y E -i l, +il) the wing trace,y E -oo,-vt/ 2) u J -i A ,-2 , 0) the Mach cone trace. (3.2)

The vortex position in the plane X = Y Z is given by

(3.3)The vortex sheets springing from the leading edge to the vortex core join, in the

new plane X = Y Z, the origin of the X plane to eaeh vortex.The main diff erence, in the determination of the generie potentia) functton F X),

hetween attached flow and separated fiow, is that the singularities previously put atthe wing leading-edges to give infinite perturbation velocities ( U, v w) in at tached flow[11, 69], have now to be removed from the wing leading-edges and positioned abovethe wing on the line-vortices. The perturbation velocities ( U, v w) are then finiteat the leading-edges. Because at low incidences, the flow is aUaehed, the generiepotentiaI function F X) should tend towards the attached flow type of solution, inthat limit. One way to ensure that type of behaviour is to build the required complex

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potential function in the plane X = Y + Z to be the combination of an attachedflow singularity and a vortex-li ne potentiallingularity, and to adjust the strength oftheir respective constants to get fini te velocities at both leading-edges. Bence, theproposed generie potential function F X) in the plane X = Y + Z is of the form

F(X) = - iAX + r ln X - Xt} - ln X +Xl , 3.4)wherc X = Jz2 _1 2. The generie potential function is linked to the harmonieperturbation veloeities U, V,W) through differentiation. Using a property of conicalmotion, the generie potential function F can be expressed in terms of the variablesy and z

and the perturbation velocities beeome

Re U u - 8F/8z1 - - y v - z wRe V = v - 8F/8z 2 = 8F/8y = 8F/8yRe W w - 8F/8z3 8F/8z = 8Fj8z.

3.5)

3.6)

To solve the problem with the harmonie function V is, in this case, the simplestway, because it can be linked directly to in the eomplex plane z = y + z.

V 8F dF 3.7)z dzdFdX 3.8)X dz~ [ i A + re 1 1 3.9)X +X l ]X X X l

V is related to the other perturbation velocities U and W through the compatibilityrelations.

IZdU = -zdV = dVl- B 2 Z 236


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.

In the case where there is no flow separation, r = 0 the velocity potential isof the correct form for attached flow. The harmonie funetion does not beeome nullon the Mach cone physical boundary.

Since the perturbation velocities propagate inside the Mach cone, in supersonicflow they are null on and outside the Mach cone. The velocity potential does notcompletely satisfy this requirement in the present form. This requirement is addressedin section 3.2 . But for relatively low-aspect ratio wings BI < 0.1), the contributionof the line-vortex singularity on the Mach cone is small: the harmonie funetiontends to

which satis y the boundary condition on the Maeh cone since the real part of velocitypotentia is zero. In this sense, the velocity potential funetion derived from thegeneric potential function is a simplified velocity potential function which is nowdeveloped in the following subsection.

3 2 2 Determination of constants based on the BoundaryConditions

The link hetween the constants in the velocity potential is obtained by ensuringthat the velocity potentia remains finite at the leading-edges. The delta wing beinga stream-surface of the flow the perturbation veloeity component w can be expressedin terms of the free-stream velocity U and the angle of incidence a. Using thecompatibility relations, the eorresponding harmonie function W is related to V andtherefore, the equation linking the vortex position to the harmonie potential functionis obtained. The vortex position is then obtained by ensuring that the line-vortex sa conica streamline of the flow. The details to each of these boundary conditions willnow he done.

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.

Finite Perturbation Velocities at the Leading-EdgesIn separated flow, the perturbation velocities have finite values at both leading-edges.To satisfy the boundary condition, the constants of the attached flow singularityand the vortex-li ne singularity r must be linked so that

1X Xl)] ' finite at X = 0 3.10)1.e. - iA reX Xl 1-x-+--=x=;) = 0 at X = o. 3.11)

Renee we have a relationship between the constants and r, which is the constantQ = XIXI/(XI Xl), and the generic potential function F and the velo city potentialV reduce to

r = -AQ, 3.12)X - X lF = - iA[X Qln X Xl)] 3.13)

V A X X I -X l 3.14)1 Z (X - XI)(X Xl)Determination of the constant AThe equation of the delta wing surface is defined by the function S Zl, Z2, :1:3) Underthe assumption of small disturbances, it is uauaI to write the wing planform equationin the form [11):

3.15)where S C Zl, :1:2) is the approximate wing planform surface function in the Zl, :1:2)plane. In steady state flow, the no-flow boundary condition through the wing surfacecan be written , with the use of the material derivative D/Dt as

S 8 ..-=- VSV=ODt 8t 3.16)38


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.... as as(Uoo u -a v-a = wZl Z (3.17)Keeping first order terms, we obtained the following boundary condition on the

Wtllg

w Uoo as = aUoo (3.18)OZlUsing the compatibility relations (2.11), the preceding boundary condition can he

applied to the velocity potential V in the following manner: integrating the compatibility equation (2.11)from the wing leading-edge to infinitYI we obtain

(3.19)The velocity potentials are single-valued funetions. The integratioll is independent

of the path, and therefore the integrals depend only on their end points. Dy havingone of the points ol integration at infinity, where aU perturbation ve10cities arc nullby definition, the real parts on both sides of the equation (3.19) are takell, so thatthe lelt-hand side of the equation becomes

Re 1 00 dW =w 00) - w l) = sin a Uoo , (3.20)where the boundary condition for perturbation velocity w is neglected on the Machcone. Defining V(z) =V(z)/A, equation 3.20) becomes

sin a Uoo = Re 100 i ;1 - B2 ;2 dV dzl= A Re] oo i ;1 - B2 ;2 d l,

and performing the integration along the axis z = iz,

sina Uoo39

(3.21)(3.22)(3.23)

(3.24)


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r

AD

where the integral D is defined as

or:

D(D l) =

3.25)3.26)

Re 1 ~ I I B2Z2 { 12 VI2:- z2 - 2 Zl) 2z2 112 Zl)(v 12 2 - 2 Zl)}o 12 z2 l? J12 Z - Zt}2 Y? V12 z2 - Zd 2)2VB2 Z2 { 12(JI 2 Z2 - 2 zZt} 2z2t;2 Zn VI 2 Z2 - zZt} })d1 2 Z2 }?Z2 (VI 12 z2 - ZZl)2 (1 ;2z2 (VI 12 z2 _ ZZl)2)2 Z

The value of the integral D is only dependent upon the position of the line-vortex(in the term V). Once the position of the line-vortex in the complex plane z = y zis known, the constants A and r can be determined. In the case where the line-vortexstrength becomes negligible, the integral tends towards the elliptic integral solutionE ka) for attached fiow.

The term D is a function of the position D t, or D =D(D d. The final form of thesecond boundary condition links the constant A to the position D 1.

Position of the Line-Vortex in the z planeThe position of the vortex in the plane Zl = I is described by the position vectorRwhich has for components the origin of the Zl , Z2 :1:3) system axis and the pointZl = 1,91, hl) The position of the vorteJC lies along the line-vortex which is a

strearnline of the fiow. The strearnlines of a potential flow are aligned with the

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........

velocity vedor V and therefore the veetors il and V are paraUel. Taking their crossproduct CV 1\ il = 0) will relate the position to the velocity, and wc will obtain thefollowing equations

decomposing into

91 Uoosina+w) =zl Uoosina+w) - Uoo+u,91 Uoo cos a u V.

These equations combine to give

{gt/zd Uoo ~ v,ht/zd UCX) w + sina Uoo

(3.28)

(3.29)

(3.30)

The position of the vortex is stationary. This stems from the the faet that the linevortex has no self-induced velocity [44]. So the correct expression to he used for \tand w at the vortex position 0 1 is obtained by subtracting the influence of the vortexsingularity at Ul from the complete solution. Let us denote thcse funetions by thesubscript v. The generie potential function F heeomes

F = F - irln z - ud (3.31)- iA[X + Q ln X - Xl) -ln X Xt} - l n ~ - O d ,at = O t. (3.32)

Using the identity X - Xt}(:I: + Ul) = :1: - ut) X + Xx), we get at = 0 1

Ft . + 0 1 at z = 0 1 (3.33),A[X + Q ID X + Xt} X XI)]V" dF (3.34)dz

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r

3.35)3.36)

Then, we get Wu using the compatibil ity relations 2.11), integrating {rom a pointYw on the wing to the vortex position.

ri 1C1 IJ dWv = iv 1 - Z2 vv. I lTaking the real part on both sides, the left-hand side becomes

Re 1rl dWv = w(at} sinaUoo ,I lwhile the right-hand sideRe LV iv l - B Z dVv _

Il .

3.37)

(3.38)

3.39)3.40)

The final boundary conditions for the line-vortex position is given by the followingtwo equations, which are applied at Zl = l,

{91/Zt = Uao Re Vv,ht/Zl = rL Re Wu sina Uoo

Or, uaing the equation (2.1) relating the wing leading-edge sweep angle A to thesemi-span length n the plane 1 we get

91 sina 1 R Vv- ecotA D at} 3.41)ht sina 1 R 1(0 1)- ecot A D(O t) (3.42)The last two equations are solved numerically by iterating on the line-vortex

position. Then the constants are obtained by back substitution.42


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3.2.3 Analytical Expressions for the Perturbation Velocities

For points on the wings X = iZ), using the position of the vortex Xl = YI + Z"and B the contribution of the irnaginary part of the potential F), the expressionsfor the perturbation velocity components V and U = F - :cV can be simplified to:

VIA 2Z1 z 3.43)z Z _ Zl)2 + i2(AB) 1A = lm F 3.44)

UIA = yl2 + 2 2Z1 - zZ + 2Yl AB :1 : (Z _ Zl)2 +Yl 3.45)For points on the Mach cone X = Y), the analytical expressions are:

V/A - z y2 _ y? _ Zl)2 +4ZlY2 3.46)U/A 1 ;2 + ~ B 2 2Z1 Y? t n- Z+ 21 1 :1 : (Y2_}?-Zl)2t4ZlY2' 3.47)

3.2.4 Theoretical Solution for the Pressure Distribution Cpand the Lift Coefficient C,

The pressure coefficient defined as Cp = 2 plPoo - 1)/hM ) can be calculated infunetion o the perturbation velocity components from the equation [52

u t 2 +w2C 2 2 2p = Sin Q - U cos Q - U2 cos a,00

3.48)The lift coefficient C, is obtained by integrating the pressure distribution over the

wing.3.2.5 N umerical DetailsBeCore proceeding to the next chapter, sorne of the details of the nurnerical calculationwill be given: these adress the vortex-line position in the plane :1:1 = 1 and the

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l iteration procedure.The equations for the vortex position previously obtained are:9t s no l ReV1 cotAD(tTl)ht sin 11 = cotAD(tTl)Rel(tTl).

To solve for 91 and hl a nested iteration has been performed using a 2 step shoot-ing method. As uming 2 starting values for hl, say ha and hb for the correspondingvalues for 91, say 9a and 9b are solved {or using equation 3.41. 1.'hen the new valu" sfor 91 are substitute back in equation 3.42 to predict correspondillg values for the h's.The difFerence between initial and iterated h values is then used to obtain hn , andanother cycle of computation starts then. The calculation ends when the difFerencebetween 2 successive predicted values of h is less than the desired accuracy, whichis specified at the beginning of the numerical procedure. The convergence of theiteration depends highly on the choice of the initia l position.

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1

>

3.3 An improved Solution for Delta Wings withLeading Edge Flow Separation

The introduction of the line-vortex singularity to model flow separation sat.isfies theboundary conditions on the wing, but violates the condition of no perturbation propagation outside the Mach cone.

In this section, the last boundary condition is satisfied through the addition ofa distribution of singularities or ridges) on the Mach cone, which will force theperturbation velocities to be zero on the Mach cone.

3.3.1 Importance of the Boundary onditionThe magnitude of the ratio of the perturbation velocity components to a given freestream velocity increases for delta wings of intermediate to large aspect ratio atmoderate-to-high angles of attack. The value at the Mach cone of the pert.urbationvelocities therefore is larger.

In Table 3.1, the values of each perturbation velocit.y components at the Machcone are given. Whereas for slender wing 1.1 : Uoo and BI this could be agood assumption, for not-so-slender wings the slender-body assumption is no longercorrect. In Figures 3.3 to 3.6, the variation for the velocity components u,v,w) areshown for BI = 0.43 and BI =0.67.

The perturbation velocity component w is ohtained by integration of the compatibility relations:

> 12>1dW = hl1 B z dVv 3.49)00 = D2z) 3.50)

= w z > l /A 3.51 )which, setting z =1/ , is equal to

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BI a ujU v/Uoo w/Uoo0.14 2 -0.00004 0.00004 0 000040 14 4 -0.00015 0.00016 0 000160.43 4 -0.00108 0.00122 0 001090 43 14 -0.01032 0.01156 0 010660.67 12 -0.01827 0.02230 0 016800 67 20 -0.04032 0.04941 0 03856

Table 3.1: Values on the Mach cone (By=l) of the perturbation velocity components

3.3.2 Additional Distribution of Supersonic Ridges to Sat-isfy the Boundary Conditions on the Mach Cone

Let 8 def1ne the {unction Vt as the sum of the velocity potential V def1ned in theprevious chapters and a distribution of supersonic ridges type, Vr def1ned in theCollowing manner:

Vt = V + Vr 3.53)where

V ~ ~ c . { l jB-z) ,+ I _ ~ l I B + Z ) ( i + I ) }r - L..- arccos IB ) arccos jB ) .i=1 11 2 i - z 2 i +z 3.54)The method to obtain the form of the singularity used in the distribution has been

developed by Carafoli and Mateescu [11] By using a distribution of ridges l outside46


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1 the Mach cone , the coefficients Ci are obtained by Corcing V, to be zero outside theMach cone.The real part of the arc-cosine functions can be obtained as

Re arccos l /B -- z)(' lIB) _2/B( i-:Z:) -

and

Re arccos 1/ B Z:)( i 1/B2/B( i+:Z:) -

1r/2 for:z: E (1/B, s,)arccos f iB-z)( . , t l / B ) for z E (--l/B, 1/11)V 2/B( -2)o {orzE (-oo,-l/B)U(s oo),

(3.55)

-1r 2 fo rzE -s , , - l jB)arccos 1/B+z)(.,+1/B) {or z E (- 1/B 1/H2IB .,+:II) ,o {or z E (-00, -s,) u 1,00).

(3.56)For ease oC manipulation, the real part of the arc-cosine {unctions will be denoled

as

?-l1(:Z:, Si, 1/B) Re arccos l iB - z) s, /B)2/B si - z)l i7B + z) s, 1/H)?-l2 Z, i l 1/E) Re arccos -2 / 8 s . z) -,

and the function V,. can be rewritten asi=n 2V,. =E Ci {1t1:z:, .,1/B) - ?-l2 Z, Si, 1/Eni=1 1r

The derivative of V,., which is needed further, can be expressed as

47

(3.57)

(3.58)

(3.59)

(3.60)


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l The generic form for the velocity potential U, for Ut = +U,. is obtained throughthe integration of the compatibility relations.3.61)

Similarly, through the integration of the compatibility relations, W,. is obtainedas

The constant C. is obtained by forcing V to be zero at the Mach cone and at all theridges . The other constants Ei and Fi are obtained SiW;l1.;ly for Ur and W,..

3 3 3 Improved Solution Including the dditional RidgeDistribution

Through the use of the distribution of supersonic ridges, the boundary conditionon the Mach cone is satisfied at a number of discrete points from the Mach coneto in'inity. In Figures 3.7 to 3.9, the resulting behaviour obtained for Bl=.67 ata = 2 deg is shown.

Another effect cornes in the determination of the constant A. From integratingthe compati bili t y relations from the wing to the Mach cone (instead of infinity),

the integral can be rewritten as:

w I/B)+Uoosina)/A = j,ooiv 1_B2z2 dV_ i VI - B2Z2 dVll/B

= D(O I) D:Z(O l)48

( 63)3.64)


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Since all perturbaton velocity components are null on the Mach cone, it followsthat:

A Uaosina.D D2 3.65)The extra term 2 has a large influence on A s will be seen in the next chapter

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\

u v w o(Mach conc

w = -s ina U Dl(Wing)

Figure 3.1: Wing geometry and boundary condit ions in the physical plane

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tTl 1U V W o: _ == _== ~ _ _ A2 l_ 7

X lu v w o

. ,

z

z

1

:: :: : === _ ~= sin et

l'

y

x == y

C _= _ y

Figure 2: Geometry in the x plane and the X plane

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o. --- ~ - - - - - - - - - T - , - ' - I - ' - - I . r r ~ I ---, 1 1 1o.::: -- - ---- -----r ---------,------------,---- -- ---- -:-----1 1o - - 1 . . l

o. _ ~ : l - = - I 1r : : : ~ I .= - -----1- [ 00 -r---- ---- --------- --- -.t111

1 1~ ~ ~ ~ t --

---- , - - - - - ~ - ~i: [r - - -- - - - - - - - - - 1 1 _ ~ _ l ~ r tl j - ---1+-- = = l1 ul-- - --------- - ,-=--f----- - - - - 1 - 1 ___ P CI:

~ . . . . . . - - ;1 1 .

1/ 1- r - - - - - - - - - - ~ - - - - - ~ - - - - ~ - - - - - - + - - - ~ - - - - - - r - - - - - ~ - - - ~ - - - - ~ l -

1 -- ----- t - - - - - - - - -.----r_---r_----t-------t---- f - - ~ - - - - - - -1 i0 I ~ - - - - - j - - - - - - - -1- - -+- - - - - t - - - - - - L_- ---

o::1. ___ . ~ ______________________, _ l ~o. ( : : . - --- ~ ~ -- - - - - I - - - - - t -------___ U I:) 1~ ~ _ + - - ~ - ~ _ ~ J . ___, - - -- _____ ____ _1 ________ -_____ ____ .L_____________ '- -______ - _ __ _

-f----t----t----+------..-- -- 1

: . 5 0 9 1. 1 1 1 2 1 3 1 4 1 5 1 6 1 7 LE 1 y

Figure 3 Variatic IlS 0f the pert urbation \ elocity components on the ~ l a c h cone, forBI=O..t3 and 4

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1 r l ~ ~

. " , 1 1 1- . l l 1 11- -1---1--- --- - -. 1 ..... I [ ; ~ ~ ~ ~ i r - _ t l - - - - k - - - - -j

1 1 l 1 1 1--- 1 -r--I 1 . . 18 . ~ . :--- --1- \

1 1 1 1 _ -...- .~ l u :_+__-_ r i- - t - - - t - - - - - t - - - - . I - - - - - ~ - - - - I - - - - - ~

f I . II - - - - - --- - - - - -

: . 0 3 1 f ~ - -- - -.

: .02 l - - - - - - -< : - -+- - -+ - - -+ . - - - - -1r - - - - - - - l - - - - t .. :

0.8 0.9 1. 1.1 1.2 1.3 1.4 1.5 1.6 1.7

y

,,- .

1

Figure 3.4: Variations (If the perturbation vcJonty ( , ( > l 1 I p C l l ( , l I t ~ ()TI t llf \1 arh ( -.Ilf , f ,rBl=0.43 and Ct == 14

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. i--r---i ---: :, i -- , -- -- f - ----- - - 1

111

" ,

:o 1

111

_ 1 r 1

1

1 1 1 1 '\ l --r ------,- ~ -11 1 1

- - - - - - - r - - ~ -- ------ - - - r - - - - - f r u

- . i -- - - - - - - - - - ~ - _ _ 1 f _ _ - _ - - - _ t _ - - _ - - _ - - - _ t _ - - - t - - -

- ,,- -- - - - - - I t ; ~ f _ _

C . 11 --- - - , - j 1r I I -----41--- ---

0 9 1. 1 1 1.2 1.3 1.4 1.5 1.6 1.7 1.8y

Figure : 5 Variations of the perturbation velocity components on the Mach cone forHl -0 G a ncl Cl :; 12

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:. :s...----,.----r---r----.----i-r--------,-------- : ~ ~ ~ -d---- ~ - . = ~ - .. -----.. l r ~. ' ~ . ; . :: . --1---- ---- o --- -- f - - - - - 1 - - - - - - - - - - - - - -- ---r--L._- __ 1\

1-- - - - -1- - - - -+-- -+-- -1- - - - - - - - - - - - - -- 1 \t r -- -

~ ~ _ _ _ _ _ _ J - - - - - - - - _ ---- ..- - -j

111-----1------- 11

- . 15 r j r t t . t t / - ---

f f . f t l t -- ---

J . 2 \ t - 1 - 1----1---- ---- -

.. v ~ r - - - - ~ r - - - i _ ~ - - - - - ----O __ - -_ . _J L . . - _. - - - - -

- 1

-- --_---r---I-- - --10.8 0.9 1. 1 1 , 1- 1 3 1 4 1.5 1.6 1 7 1 Ay

FIgure 3 6: Variations of the perturb tion \ elocity cmponcnls OII the ~ t r h ((_ Ile, forBl=O.67 and Q = 20

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uUoo

0 . 0 2 ~ ~ I T T\

o

o.L --I--

Simplificd cOlllcaJ c;nlutieonSUpcrf,(JJ\ic ridge., rontriLutton(of opposite 5ign)Imprr)Vf d cOJlitaJ solution

- . 4 1 ~ <

Simplified conical solutionSupersonic ridges contribution(of opposite sign)Irnproved conical solut ion

0 03 i t I i j t . ~ 1

0.02 t

0.01 -

_ . 1 - - . L . _ ~ _ _ _L . - -0 8 0 9 1 1 1 1 2 1 3 1.4 1 5 1 6 1 7 1 8 1 9B Y

Figure 3.9 Variations of the perturbation velocity w U on the Mach cone, fornt -0.67 and 0 = 20

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1

ChapterComparison etween theTheoretical Solutions andExperimental Resultsn this chapter a comparison between the analytical methods derived in chapter 3

the slender-body model of Brown [52] and experimental data from reference [33] isdone looking at the lift and the pressure distribution.

The lift results obtained with the simplified method and the improved methodis compared to the slender-body model of Brown [521 and to experimental results[11}. The pressure distribution results obtained wlth the simplified mcthod and theimproved method is compared to the slenderbody solution obtained by Brown andto experimental results summarized by Wood in ref [33] which reCers also to otherworks. The spanwise position of the vortex-li ne will be glven as the variable gl andits relative height by kil The aspect ratio is denotcd by R and the semi-span lengthto the Mi.f.h cone radius ro ratio by BI The relation of the angle of incidence to thewing semi-span length il is used in the lift results. The other quantltles of interestsmentioned in chapter 1 are the Mach number normal to the leading-edge Mn and theincidence normal to the leading-edge an These quantities are related to the deltawing leading-edp;e sweep angle A through the Collowing equations

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Mao sin AJ + sina tan A)2,arctan ( tan a cos A )

4 1 ift Variation with Angle of Attack

4.1)(4.2)

The variation of lift with angle of attack were obtained for delta wings of moderateand large BI to measure the improvement gained from slender-body with the methodof Velocity Singularitjes for the simplified solution. Because of the discretizationchosen, there can be differences in local pressure distribution near the leading-edge,bui the integration to obtain the overall wing loading is proportion al to the vortexintensity concentrated in the line-vortex cores above the wing, so that the lift shouldrepresent the best aelOdynamic quantity to compare methods.

The comparison or the numerical results for the lift coefficient are summarizedin Table 4.1 {or two wing configurations for which previous results were available.In both cases, the lift coefficients predicted by both the simplified and improvedsolutions are in better agreement with the experiments than the slender body model.For the wing of larger aspect ratio BI = 0.67), the sien der body is seen to overshootthe predicted lift of the conical solution by more than 60 for large a/l (Table 4.2and Figure 4.1). However, the simplified model overestimates the experimentallift:the overshoot hecomes more significant as the parameter increases a/l 0.8). Theimproved solution clearly shows the benefits of incorporating the Mach cone boundarycondition. The lift is in reasonable agreements with experimental data for BI = 0.67(Figure 4.1). What is significant is the improvement of the improved conical solutionover the simplified conical solution (nearly 15 ) for an angle of attack of 20.

The improvements introduced by the present conical solutions are clearly significant for larger values of BI, where the lift coefficients are predicted with a goodaccuracy. Improvements with respect to the sien der-body solution can also be observed in the case of smaller values of BI (Figure 4.2), although they are not 50

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-

r----

BI 0./1

0.43 0.520.78

r - -0.67 0.500.84

F i ~ u e U >q;4 1 1 9 043 75.04.2 l 9 067 67 5

Table 4 1: Table of cases examined for the lift co('ffic('lIt C----- .SIen der-body

solutionC,/I'

5.239.014.989.99

------ Simphficd conical soluti ln Iml ro\ cd ( , 'Ilieal s ~ l l l t i o n

C jlz

4.817.85--3.456.18

Tmprovement C,jl Impro\'t'l1wntw.r.t

Slcndcr-bodySolution8.7314.7744.3461.65

4.657.523 13539

w r.tS1enrlcr- bodySolution

124719.8159108568

Table 4.2: Table of summary of lift resultsspectacular as for larger BI.

The lift, for the same a/l, is seen ta decrease as the parameter BI is increased;given an angle of aUack and a wing span, an increase in the Mach llllmbcr wj)J rt'ducethe effect on lift of the flow separation to eventually disappear whcn the learling-cdgcsof the delta wing become supersonic.

4 2 Coefficient of Pressure p Distribution aiongthe Span

In this section, a cornparison for slender wings and not-so-slender wing is madethrough variations of the angle of incidence l in Figure 4.5 to Figure 4.4. The

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{

Figure oo Cl ANot-so-slender wing 4.3 1 9 12 67.5

4.4 1 9 20 67.5r----Slender wing 4.5 1 9 3 5 82.54.6 1 9 7 5 82.5

Table 4 3: Table of cases examined for the coefficient of pressure Cp

Figure Moo a A I Mn n4.3 1 9 12 67.5 0.670 0.813 29.044.4 1 9 20 67.5 0.670 0.942 43.66

Table 4.4: Not-so-slender wing parameters for the coefficient of pressure CpTable 4 3 summarises the geometrical configurations considered in each case.

4.2.1 Variations of the Coefficient of Pressure for a Not So SlendWing

The experimental resuIts used for comparison are taken from Reference [11] Thecomparisons contained in Figures 4.3 and 4.4 refer to a wing with a leading-edgesweep angle of A 67 5 where the angle of attack is increased from a = 12 toa 20 The Mach number is Moo 1.9 which corresponds to I 0.67 in thiscase.

In all cases for the simplified and the improved method the coefficient of pressureCp on the leeward side peaks to its maximum value at the spanwise vortex locationwhile remaining fairly constant on the windward side. The Table 4.4 shows thatthe two cases lie not within the region of classical separated flow of a delta wing ofFigure 2.4.

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1 The simplified and the improved mode reproduce the main qualitative (eaturesof the flow. The results for the simplified conica} solution on the windward side areunderpredicted compared to the experimental data. On the leeward, the simplifiedconical solution largely overpredicts the coefficient of pressure, while havllg the suction peak nearer to the leading-edge than predicted by the experimental data; thesituation worsens as the angle of incidence is inereased The improved ( ouieal solution is in better agreement with the experiments than the simphfied solutions: themost noticeable difference is in the height of the suction peak on the upper side wheredifFerences in magnitude are of the order of 23 at 12 and of 43% at 20 with t.hesimplified method.4.2.2 Variations of the Coefficient of Pressure for a Slender

WingThe experimental results used for comparison are taken from References [ and [791The comparisons contained in Figures 4 3 to 4 4 refer to a wing with a. lcading-edgesweep angle of A 82.5 where the angle of attack is increased from a = 3.5 toa = 7.5. The Mach number is Moo 1.9, which corresponds to I = 0.21 in thiscase.

In all cases, the coefficient of pressure p on the leeward side (which in the figuresis above the y l axis) peaks to its maximum value at the spanwise vortex location,while remaining fairly constant on the windward side (which in the figures is belowthe yI axis). Table 4.5 shows that all cases considered lie weIl within the region ofclassical separated flow of a delta wing of Figure 2.4.

The simplified mode} reproduces the main qualitative features of the flow. Theresults on the windward side shows excellent agreement with the experimcntal data.On the leeward, the simplified model overpredicts the coefficient of pressure whilehaving the suction peak nearer to the leading-edge than predicted by the experimentaldata, because a concentrated vortex is used in this analysis, instead of an aclually

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Figure ao a A I n n4.5 1 9 3.00 82.5 0.213 0.266 21.870

4.6 1 9 7.5 82.5 0.213 0.349 45.24Table 4.5: SIen der wing parameters for the coefficient of pressure Cp

distributed vortex sheet which is rolling up and forming a vortex core; the improvedsolution behaves similarly, but with slightly less amplitude. The situation do es notimprove as the angle of incidence is increased, for both cases.

4 3 Discussionhe importance of the correct implementation modelisation of the boundary condi

tions on the Mach cone has been demonstrated through the magnitude of the improvements of the improved conical solution over the solutions which restrict themselves tothe slender-body approach. At small BI the gain obtained with the method are notso important: this is to be expected since the magnitude of the perturbation velocitiescomponents are much sm aller in comparison with the {ree stream velocity ao (recallTable 3.1). With increasing BI the magnitude of the perturbation velo city components increases up to 5 for I = 0.67 at 20. Improvement on lift prediction wereseen to be as high as 85 compared to the slender-body model of Brown, whereaspeak reduction were up to 43 for the pressure coefficient distribution. Most interestingly, in comparison with the experimental results, the prediction of lift for the casewhere l = 0.67 is sufficiently accurate to be considered as a practical engineeringtool.

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-

Cl

10.r r r r - - - -/-

9.>

6Jm pro\ ( d coni ca.} sol utionS;mplificd conira} sc.}utionSlender- bodyExperi men ts

8

7. ---------- - - - -

6. ------+--

5. t - -

4.

3.

2 I - - - - - - J - - - - - ~ - - - ~ : . . . : . - _ _ . . I _ - - - l l - - - - I -

l ~ ~ ~ ~ ~ ~ ~

o. - - - - - ~ - - - - - - ~ - - - - - - ~ - - - - - - ~ - - - - - - ~ - - - -O. 0.1 0.2 0.3 0.4 0.5 0 6 0. 7l

Figure 4.1: Variation of lift for B1=O.67

65

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C2

12.

10.Impr\ ( d fUlliral sc.lutionSimplified cniral solutionSklldC r-body

t>ri Ilcn ts

,8 - - -------

6. -- -- -- ----- --1Xt--

. ~ ~ ~ . ~ . ~

o. - --------- - ______ - .1 . - ______O. 0.2 0.4 0.6 0.8

1,Figure 4 2: Variation of bft for Bl=0.43

66

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1

3 .. . . _.

2.5

2.

p

>x Improved conical solutionSimplified conical solutionSIender-bodyExperi ments

1 5 ~

1

0 5 1

-.5O 0.2 0.4 0.6yI 0.8

Figure 4.3: Variations of the pressure codflcient on a d( lta wing with flow separationfor Bl=O 67 and Q = 12

67

\\

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4 . ~ ~ ~ r ~ ~

3 5

3. -

2.5c

2.

l 5 - --

)

~Improved comcnl solutionSimplified conical solutionSIen der- bodyExpemncnts

- ---- t

1. ------ - ~ 4 _ . . ; - - ~ _ J - _ l - - - , - - - - .

O. 5 - ~ A -FA : A ~ - : - ~ = -6- A - " A - t ; - A - 1 J = ~ - : : : J - r-x::::::1Fft: ' ]J~ ~ = - : - - - t 4 = - ~ ~ ~ - - - ) ( - - - - ~ - 4 - _

o. ----- - -------- --------- ---- ) --- -*-----) - ) )

-1.O.

~ ~ t t _ M- - Il 11- 1l1J 11 11- ~ , ~ I _ _ i I _ - 6 ; - ~ r _ I t _ _ / r _ ; ~ : _ A _ - I r l h r _ _ I t _ - I r - J . I _ _ I r _ l r _ J . _ . A . . . . _ 4 _ ~ ~ ~ . . . , . . . . . . , I \ _ ~ - a

0.2 0.4 0.6 0.9y I

FIgure' 4.4 Variations of th( pressure co( fficient on a delta wing with flow separationf\\r Hl 067 I\nd Cl = 20

68

1


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' ~ .

-

O . 3 ~ - - - - - - - - - - - - r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0.25oXb,

Improved conical solutionSimplified coniral solutionSlcnder body ~ ~

C


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o ~ -

0.4o6

Irnprovcd conical solutionSimplificd conical solutionSl('ndcr-body

D


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hapterON LUSIONS

The problem of flow separation for a delta wing has previously been adresscd analytically by various authors in the frame of the slender-bdy theory . The slendcr-bodymodel is attractively simple, but it can lead to large errors when it 5 applied todelta wings of large aspect ratios. In addition, the slender-body mode} docs not takeinto account the directional propagation of disturbances, as weil as the cffed ofcompressibility for supersonic flow.

An analytical model to solve the leading-edge ftow separation over a delta wing insupersonic regime has been developed in this thesis in the framework of the conicalftow theory. This method is compared to experimental data availablc in the litcrature.The first objective of the thesis was to adapt, to the problem of separated flow themethod of Velocity Singularities, which has been applied to arbitrary wing-bodycombinations in attached flow with good results [14J.

Two models have been developed in the framework of cunical flow thery: i) asimplified model which is taking into account the vortex-type singularitics formedabove the wing as a result of the flow separation at the leading-edges, and il) an improved solution which more accurately takes lnto account the appropriate boundarycondition on the Mach cone. The simplified conical solutIOn is in much bettcr agreement with the experimental data than the slender-body solution for the predictionof the lift, even without satisfying rigourously the boundary conditions on the Mach

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,

cone, which is common to the slender-body solution. This justifies the applicationof the method of Velocity Singularities to the flow separation problems, which yieldsgood results. Th,.. improved conica solution showed further improvements for the l i tand the pressure distribution, especially for large BI In the case of large ratios ofthe semi-span to Mach cone radius, the improved solution was found to predict theh t coefficient with good accurac.y to be of engineering use (for preliminary designcstimates ).

The pressure coefficient becomes sometimes too large on the leeward side whenthe wing becomes not-so-slender. The peaks observed could be smeared out by usingmultiple line vortices originating at the wing vertex. The improved solution in thatrespect has a considerable reduction in peak magnitude for large BI over the peakmagnitude of the slender-body solutions for large BI thus showing the large influenceof the correct modelisation of the boundary condition on the Mach cone.

The first objective of the thesis, which required to solve the flow separation problem with the vortex singull.rities above the wing plane in the framework of the conicaftow theory, can be consilered as being achieved. The inclusion of additional singularities on the Mach cone to solve analytically the problem with the correct boundaryconditions has b.: en shown to be successful in the improved conical solution.

The second e,bjective of the thesis is to lay the ground for future developments tosolve for an ex ct bound ry condition on the Mach cone. This would then completelysolve the problem for supersonic flow. The long term goal is to use the powerful toolsdeveloped for attached conical flow and solve, using higher order conical flow theory,the

aerodynamics of wings with leadingedge flow separation in supersonic regime

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