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A potential flow model for separated flow around airfoils Brun, Sarah K. 1993

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A POTENTIAL FLOW MODEL FOR SEPARATED FLOW AROUND AIRFOILSBySarah K. BrunDipl. Ing. E.N.S.M.A. - Poitiers (France), 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESMECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1993© Sarah K. Brun, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department of ^Mektrcal Ey) 0:1 oets- ,044JThe University of British ColumbiaVancouver, CanadaDate^Au3u.s1- 2-) )313DE-6 (2/88)AbstractContinuing the study carried out by Dr. W.W.H. Yeung, a former Ph.D. student in thedepartment of Mechanical Engineering at U.B.C., some new work has been done on thewake source model, a potential flow model for steady separated flow. This work includessome modifications to the model and applications to new geometries. In the wake sourcemodel, the body of interest is conformally mapped to a circle. In the transform plane,singularities (most often sources) are used to represent the wake by creating free stream-lines simulating the separating shear layers. The empirical data required by the modelare the positions of separation points and the specification of the pressure coefficient atthese points.Applications of this model include the case of flow normal to a flat plate, a separationbubble at the leading edge of a flat plate airfoil, and an airfoil fitted with a Kriiger flap.The case of the normal flat plate was a test case in which the original model wasmodified in an attempt to specify a constant pressure on the free streamline after separa-tion. This has been achieved to a limited extent by adding sources to the original model.Future work should include the application of the same method to a spoiler and slottedflap airfoil configuration.An earlier model using a doublet as singularity was developed by Yeung for the sep-aration bubble case. It showed very good agreement with experiments in terms of thepressure distribution on the wetted surface of the plate but a high suction peak appearedon the bubble boundary, as well as an unsatisfactory bubble shape. In the present study,no improvement has been obtained by using more singularities, unlike the case for thenormal flat plate.For an airfoil with a Kriiger flap, some satisfactory results can be obtained with amodel using two doublets, especially at high angles of attack. Results could be improvedat moderate incidence with a better estimate of the bubble shape that forms betweenthe flap and the airfoil lower surface. Thus, it is recommended that flow visualizationstudies be conducted in the future to evaluate this shape. Experiments should also becarefully repeated in order to assess the accuracy of the model.111Table of ContentsAbstract^ iiList of Figures^ viiNOMENCLATURE^ ixAcknowledgement^ xiii1 Introduction 11.1 Motivation ^ 11.2 A brief review of the relevant literature ^ 21.3 Purpose and scope of the work ^ 52 Mathematical models 73 Normal flat plate 133.1 Introduction ^ 133.2 Models 163.3 Results and discussion ^ 224 Separation bubble 354.1 Introduction ^ 354.2 Models and results ^ 384.3 Concluding remarks 39iv5 Kriiger flap^ 425.1 Introduction  ^425.2 Theodorsen transformation ^  455.3 1-source model  ^495.3.1 Mapping ^  495.3.2 Flow model  545.3.3 Results and discussion  ^565.4 Doublet models ^  635.4.1 Mapping  635.4.2 Flow models ^  675.4.3 Results and discussion  ^705.5 Concluding remarks ^  816 Conclusion^ 836.1 Summary of results ^  836.2 Recommendations for future work ^  85Bibliography^ 86Appendices^ 89A Numerical results for normal flat plate^ 89B Inverse Joukowsky transformation^ 94C Determination of Theodorsen coefficients^ 97D Calculation of limits for derivative of mapping function^103VE Numerical results for Kruger flap: 1-source model^ 105F Numerical results for Kruger flap: doublet models^ 107viList of Figures2.1 Physical plane and transform plane  ^93.1 Comparison of Roshko's model [20], wake source model [17] and experi-ments [5] for a normal flat plate (pressure on the front of the plate) . . .^143.2 Comparison of Roshko's model [20], wake source model [17] and experi-ments [5] for a normal flat plate (separation streamline)  ^153.3 Physical plane and transform plane  ^173.4 Illustration of the principle of the new model  ^193.5 Results for model 1: Cp on separation streamline ^  263.6 Results for model 1: separation streamline and Cp on the front of the plate 273.7 Model 1: Cp on separation streamline with several sets of specification (5and 6 sources) ^  283.8 Model 1: Cp on separation streamline with several sets of specification (7sources)  ^293.9 Model 2: Cp on separation streamline^  303.10 Model 2: separation streamline and Cp on the front of the plate ^ 313.11 Model 2: improved results, 5 and 6 sources  ^323.12 Model 2: improved results, 7 sources  ^333.13 Comparison of results for model 2 (Cp on separation streamline), g =-0.75,0.7,0.65  ^344.1 Results for flat plate airfoil at 5.85° incidence: Cp distribution and shapeof separation bubble [28]  ^37vii4.2 Examples of Cp distributions for flat plate airfoil at 5.85° incidence: modelusing 3 and 4 doublets ^  405.1 Upper and lower surface Kriiger flaps on an airfoil [7]  ^435.2 Successive planes for Theodorsen transformation ^  475.3 Mapping sequence for Kruger flap, 1-source model  505.4 Mapping of the airfoil and its flap (/ = 0.73, Oft = 159°)  ^575.5 1-source model: comparison of Cp distributions  ^595.6 1-source model: Cp distributions (a = 110 and 14.6°) compared withexperiments [11]  ^605.7 1-source model: shape of free streamlines (a = 11°)  ^615.8 Mapping sequence for doublet models ^  645.9 True airfoil+flap and mapped contour  715.10 1-doublet model: Cp distributions (a = 13.78° and a = 12.4°) comparedwith experiments [11]  ^735.11 1-doublet model: Cp distributions (a = 8.6° and a = 6°) compared withexperiments [11]  ^745.12 1-doublet model: separating streamlines, a = 13.78° and a = 8.6° . . .  ^ 755.13 Second contour: Cp distribution with 1-doublet model (a = 13.78°) com-pared with experiments [11]  ^775.14 Comparison of pressure distributions a = 13.78° and a = 8.6°  ^785.15 2-doublet model: Cp distributions (a = 13.78° and a = 8.6°) comparedwith experiments [11]  ^795.16 2-doublet model: separating streamlines (a = 13.78° and a = 8.6°) . . .  ^ 80NOMENCLATUREfirst ith Theodorsen coefficient1133^second ith Theodorsen coefficientplate or airfoil chordCL,„„^maximum lift coefficientCp pressure coefficientCit^specified pressure coefficientf(S) a function ofF (1)^complex potentialcommon ratio of a geometric progression9(6)^a function ofplate widthh(8)^a function ofkthgk component of a multi component functionh(C)^mapping functionVET1^flap length in Z3 planenumber of sources, ornumber of Theodorsen coefficientsN'^number of specification points on the free streamlinepressure4, 41, 42^source or doublet strength (scaled: q = Q/7r)Q1) Q2 source strengthixradial coordinatecircle radius in final planeR3^ circle radius in Z3 planeRv radial position of vortex in final planecircle radius in Z4 planeX component of velocitymagnitude of free stream velocity in physical planeY component of velocityV^magnitude of free stream velocity in final planew u — iv^complex velocityX^first coordinate in the physical complex planefirst coordinate in the Z, complex plane, oran unknownX* = Xlh^scaled coordinatedownstream limit of X* after which Cp is not constantsecond coordinate in the physical complex planesecond coordinate in the Z3 complex planeZ = X -1-iY^point in the physical complex planeZ3^intermediate complex planeZio centroid of a figure in the Zi planeGreek charactersangle of attack in physical planeao^ angle of attack in final plane or rotation angle for mappingvortex strength (scaled: y = P/27r)vortex strength8, 81, 82, 83 , 8v^angular positions of singularities in final (circle) planeAo^angular spacing between source 1 and n axis6.2 angular spacing between sources j and j 1AX^increment in unknown X for one iterationLXCLm increase in maximum lift coefficient between airfoil with flapand plain airfoilconvergence criterionpoint in the final (circle) planesecond coordinate in the final (circle) plane0^angular coordinate in final (circle) planeAx, Ay coordinates of point at infinity in Z5 planeangular coordinate in 21 planefirst coordinate in the final (circle) planedensityangular coordinate in the unit circle plane01, 0)2^angular positions of slit in Z2 planeangle, mapping parameterflow potentialço^angular coordinate on the near circle= log r^first parametric coordinate in Z1 planestream function7-1, T2^angles, parameters of mappingxiSubscriptsb^base value (in the wake)fl flap baser^radial componentR reattachment pointS^separation pointT tip of the flapTB^trailing edgeV vortex0^tangential component0 value on the free streamlineoo^value at infinity (free stream value)xiiAcknowledgementFirst of all, I would like to thank Dr. G.V. Parkinson for providing me with the opportu-nity to carry out this research. His continuous interest and insightful advice has provedto be invaluable towards the completion of this study.Dr. W.W.H. Yeung, who offered his support and suggestions throughout the courseof this work, is also appreciated for his help.Yinghu Piao, graduate student, helped familiarize me with the necessary software.His help, as well as his personal support are gratefully acknowledged.Financial support for this work has been provided by the Natural Sciences and Engi-neering Research Council of Canada, grant no. 5-80586.Chapter 1Introduction1.1 MotivationThere are many engineering applications of low speed steady flows, many of them relatedto airfoils, such as airplane wings, helicopter or turbomachine blades and propellers. Inthese cases, the flow is most often separated at sufficiently high angles of attack. Sepa-ration is of great practical interest because its occurrence may bring a dramatic changein the aerodynamic behavior of the object affected. Thus, separation is sometimes pro-voked, an example being found with the case of the spoilers on a wing upper surface, butmost often, it is avoided as much as possible, as in the case of airfoil stall.Usually, accurate calculations of the flow around complex bodies are done numeri-cally using an iterative scheme in which the flow is calculated in two different regions: anouter region, where the flow is considered inviscid (and potential for 2D incompressibleflows), and the boundary layer, close to the body, where viscous effects are considered.Convergence is achieved when boundary conditions are matched on the boundary be-tween the two regions. When separated flows are considered, the process involves thedetermination of the wake boundaries and the position of separation when it is boundarylayer controlled. Of course, because all of the phenomena involved in the wake are notfully understood yet, computations always need some empirical input. Clearly, if onecould find a simple outer flow model that gives a good initial estimate of all parameters1Chapter 1. Introduction^ 2involved (including the shape of the wake), convergence could be obtained rapidly andcomputational time reduced considerably. This achievement would be significant sincenumerical computations can still take a considerable amount of time, in spite of the re-cent progress in numerical methods and the development of high speed computers. Thus,the search for a good, simple analytical model would be justified, if only for that reason.However, there is more than one advantage to such a model. If reliable, it can alsoprovide a quick picture of the flow and pressure distribution around a given body, whichcan be convenient at an early stage of design, before getting into long and detailed com-putations. A lesser advantage, but not insignificant, is that such a model implies a closerlook at physical boundary conditions (which help construct the model) and may con-tribute to a better understanding of the real flow.Following these considerations, the work reported here deals with simple potentialflow models that aim to represent the separated flow around airfoils as faithfully as pos-sible without introducing too much empiricism.1.2 A brief review of the relevant literatureAnalytical models for separated flow have already been devised in the past. In 1954,Roshko [20] improved Kirchoff's method for separated flow past a flat plate. In that po-tential flow model, the separating shear layers are simulated by free streamlines and thebase pressure is assumed to be constant and equal to the pressure at separation. Startingwith a representation of the flow in the hodograph plane, a series of conformal transfor-mations of the complex velocity are carried out up to the plane of the complex potential,Chapter 1. Introduction^ 3which is the fundamental variable. This model represents a complete boundary-valueproblem, in the sense that the velocity is specified along the free streamlines, either inmagnitude- for the first part- or in direction- the free streamlines become parallel to theflow further downstream. Good agreement with experimental data is obtained in terms ofpressure distribution, drag coefficient and initial shape of the free streamlines. The samemethod also gives good results when applied to a wedge but less satisfactory agreementwith experiments in the case of a circular cylinder in the region near separation, due toa failure in locating the separation point correctly. In principle, the method could beapplicable to any shape. Practically, it is difficult to apply to any configuration exceptvery simple ones such as those previously mentioned, because of the difficulty in mappingany hodograph plane into the right complex potential plane.Woods' theory, developed in 1955 and reported in detail in reference [26], shares somecommon features with Roshko's method. Although not using the hodograph plane, ituses conformal transformations to reach an intermediate plane where the equations of theproblem can be solved, with the complex potential as independent variable. The theoryrequires the pressure to be specified along the free streamlines bounding the wake, withthe condition that it reaches the free stream value at infinity and starts with the valuegiven from experiments at separation. For continuously curved bodies, where separa-tion is boundary layer controlled, another empirical input is the location of separation.However, Woods suggested (see references [25, 26]) a criterion that would determine thisposition, based on the radius of curvature of the free streamline at separation: it shouldequal the radius of curvature of the body at that point, in order to achieve "smooth"separation. This criterion produced good results when applied to a circular cylinderwithin the framework of Woods' theory, but it could also be used in other models tosuppress the empirical specification of the separation points. The main advantage of thisChapter 1. Introduction^ 4theory is that it is very general and, in fact, is valid for compressible and even unsteadyflows, separated or not. It also has a broad range of practical applications, some ofwhich are reviewed in reference [26]. However, even for steady incompressible flow, theequations describing the problem, although simplified, still remain quite complicated andultimately, the problem breaks down to solving a difficult integral equation, which canbe done only numerically.In 1970, Parkinson and Jandali [17] presented a new potential flow model for sepa-rated flow, the wake source model. This model has the advantage that it is simpler anddoes not require a higher degree of empiricism than other existing models. Combiningconformal mapping and the use of source singularities to create the separating stream-lines, the model offers both versatility in its range of applications and simplicity. A majordifference with the models previously introduced is that it does not define a completeboundary-value problem since no condition is imposed along the free streamline, exceptat separation and infinity. At separation, the pressure is specified with the experimentalbase pressure value as in the other models and, of course, it tends asymptotically tothe free stream value at infinity. Originally developed for symmetric bluff bodies, themodel has, since then, been adapted to more complex configurations, particularly air-foils equipped with several different kinds of devices such as spoilers (Jandali, 1970 [9]and Yeung, 1985 [27, 28]) and split flaps (Yeung, 1985 [27]). Most recently, Yeung andParkinson [28, 29] applied the same type of model to five different configurations: (i)trailing edge stall, (ii) separation bubble on inclined flat plate, (iii) separation bubbleupstream of spoiler, (iv) spoiler/slotted flap combination and (v) two-element airfoil nearstall. Good results were achieved in some cases ((i) and (v)), and partially good resultsin the other cases. A significant outcome of these results was the fact that the lack ofChapter 1. Introduction^ 5boundary conditions on the separating streamlines might be responsible for incorrect the-oretical pressure distributions and/or wake or bubble shape. In the case of the separationbubble on the inclined flat plate, the pressure distribution was found to agree very wellwith experiments on the wetted surface of the plate, whereas agreement was poor on thebubble boundary. For the spoiler/slotted flap configuration, good results were obtainedon the main airfoil while a suction peak appeared on the flap upper surface where ex-periments suggest a uniform loading with lower suction. The free streamline emanatingfrom the main airfoil trailing edge was also found to lie very close to the flap uppersurface. Clearly, if the model could impose a constant pressure on this free streamline,the problem might be resolved. The separation bubble case might also benefit from animposed boundary condition on the separated streamline (bubble boundary). These lastcomments form the basis of the present thesis.1.3 Purpose and scope of the workThe study undertaken here is a continuation of Yeung's work [28], which has just beendescribed in the previous section. Its purpose is to modify his models in an attempt toachieve better results in some of the aforementioned cases, but also to apply it to someconfigurations which have not been investigated yet. In these applications, theoreticalresults will be compared with experimental data available in the literature.Bearing in mind the results obtained by Yeung, a first configuration was tested,namely the case of a flat plate in a flow normal to its plane. The aim was to modifythe original wake source model to make it possible to specify the pressure on the freestreamlines, with the intention to apply the new version of the model to a spoiler/slottedChapter 1. Introduction^ 6flap configuration and a separation bubble. Then the case of a flat plate airfoil (inclinedflat plate) with a separation bubble, studied by Yeung, has been reexamined with theidea developed in the previous part. Finally, the case of an airfoil fitted with a Kriigerflap has been studied.Chapter 2Mathematical modelsThe aim of the present chapter is to present the models used in this project and give abrief description of the basic principles and calculations which have been carried out toobtain the results. The models are extensions of the wake source model, a potential flowmodel originally developed by Parkinson and Jandali [17] for symmetric bluff bodies,and its subsequent extensions by Jandali [9] and Parkinson and Yeung [18, 27, 28, 29]for lifting bodies and particularly airfoils.In these models, the flow is two dimensional, which is experimentally verified in eachsection of a long span airfoil except near the tips, inviscid (outside the boundary layer)and incompressible (a good approximation at low speeds), so that potential flow theorycan be applied. Furthermore, only steady flows are considered.According to potential flow theory, the flow can be described by a complex potential,which is an analytic function that must satisfy all the boundary conditions of the problem:F(Z) = (1,(Z)d- itli(Z)(Z = X + iY represents a point in the complex plane)From this complex potential, a complex velocity can be derived:w(Z) = u — iv =dZdF7Chapter 2. Mathematical models^ 8(u, v are the X and Y components of the velocity vector)or in cylindrical coordinates:w(Z)eze =V. — ive(v7. and ye are the radial and tangential components of the velocity vector)The main features of the models are:1. The wetted surface of the body to be considered is conformally mapped from theappropriate part of a circle (the remaining part corresponding to the part of thebody lying in the wake). Separation points are made critical points, i.e. the deriva-tive h'() of the mapping function Z = h(C) has simple zeros at these points sothat angles are doubled there. This implies that separation points must be cuspsin the physical plane since an angle of 1800 on the circle, a continuous curve, willbe magnified to an angle of 360°. It also implies that tangential separation in thephysical plane is represented by separation at 90° in the circle plane.2. The part of the body lying in the wake is ignored, since the type of model usedcannot represent all the complex phenomena occurring in the wake. This allowsany separation point to be made a cusp in the physical plane since any non-wettedportion of the body can be modified for that purpose. As an example, one canrefer to [17] where a circular cylinder experiencing separation is represented by acircular slit between both separation points (wetted surface).3. The separating shear layers are simulated by free streamlines which determine theboundaries of the wake. This is relevant close to the body, where the separatingshear layers are still quite thin, but irrelevant further downstream, where theythicken very quickly: it then becomes impossible to define a clear boundary for thewake.freestreamlinehPmapping functionPhysical planewakeTransform planewakeChapter 2. Mathematical models^ 9Figure 2.1: Physical plane and transform plane4. All calculations to model the flow are performed in the transform plane (circleplane), because it is much easier to calculate the flow around a cylinder than arounda body of an arbitrary shape. Once everything has been solved in the circle plane,then one can go back to the physical plane through the mapping function (see Fig.2.1)The model uses singularities to create the flow. In the transform plane, it consists of:• uniform flow + a doublet at the centre of the circle to represent a uniform flow pasta circle.• sources (and their images to satisfy Milne-Thomson's theorem [141) placed on thecircle, in the wake. These sources are the addition to the basic uniform flow pastthe circle, in order to "create" the wake.Chapter 2. Mathematical models^ 10• a vortex at the centre of the circle for lifting configurations, to account for thecirculation.Thus the complex potential has the following form:F(C) V(Ce-"" Tea())1-F—[21Q1 log(C — Rez6')+ 2C2210g(C — Rez52) — (Qi + Q2)109()12wlog( C)271-With that complex potential, the basic boundary conditions of uniform flow at infin-ity and tangential velocity on the circle (no flow-through condition) are satisfied sinceimage sources and sinks are added according to Milne Thomson's theorem. The sources'strengths (Qi, Q2), their locations (81,6.2) and the vortex strength (r) are 5 unknownswhich are determined by applying the following boundary conditions:• the separation points are stagnation points in the transform plane: w(C) = 0 atthese 2 points (2)Note: w(() = 0^vo = 0 since v7. = 0 is already satisfied.• the pressure coefficient is specified at these points: Cp Cpb (wake value) (2)• the fifth boundary condition is devised so that it represents a realistic constraint,such as a finite pressure gradient at a separation point [18, 27] or a specified valueof the circulation in the wake [18, 27] for example.A first comment on that type of model is that the use of sources is not exclusive anddifferent types of singularities such as doublets, vortices,.., can be used instead, depend-ing on the problem considered. So a more general name for the model could be wakeChapter 2. Mathematical models^ 11singularity model. Also, the number of singularities can be changed, the only require-ment being to match the number of unknowns with the number of boundary conditions.The condition w(C) = 0 at critical points remains compulsory to avoid infinite velocityand Cp there (Kutta condition).Once the set of equations representing the boundary conditions has been solved, theflow is totally determined outside the wake boundaries.The velocity in the transform plane is:w(C) = —dF = u — ivd(In the physical plane:dF dF c1( w(() w(C)^w(Z) —^—dZ^dC dZ^:1- 1-^h/(C)The pressure coefficient is then found through Bernoulli's equation (losses are neglected):P+ jw(Z)^+I2 Poo U2= —1)^2^p^22P — Poo^111)(42CP = 1^=1^=1i pU2 U2w(C) u IfChapter 2. Mathematical models^ 12The shape of the separated region (wake) is found by calculating the coordinates ofthe separation streamlines, i.e.:T(C) = Ts, and k1f(0 Ts2in the transform plane. Their coordinates in the physical plane are then found by map-ping Z = h(C).Note that through the mapping process, a uniform flow of magnitude U and angleof incidence a at infinity is transformed into a uniform flow of magnitude V and angleao at infinity in the circle plane. The relationship betwen these quantities can be easilyderived from:W(0100 W(Z)10,, = dzdC l°°V=U dZdCU =^ <4dC I 00^ ao a — arg (t).Critical points At critical points (total derivative dZ/d( = 0), w(Z) = w(C)/di is an indeterminateform (w(Cs) = 0, dZ/das = 0). Most of the time, it can easily be determined by usingL'HOpital's rule:dw(C)W(Z)^dCd2 Zso the second derivative of the mapping function has to be calculated at critical points.Of course, w(Z) liinc,cs(wM/c-) can be determined by other means, if it is found todcbe simpler.Chapter 3Normal flat plate3.1 IntroductionAs mentioned previously, one of the reasons why the wake singularity model in its origi-nal form fails to give good results in some cases is that it does not allow for specificationof pressure on a free streamline. Among all the cases tested by Yeung [28], this was par-ticularly obvious for the separation bubble on a fiat plate and the airfoil with a spoilerand a slotted flap. If one was able to impose a constant Cp on the separation streamlinefrom the main airfoil, the pressure on the upper surface of the flap might match theexperimental Cp distribution. For the separation bubble as well, the experimental Cp isconstant on a large part of the bubble (laminar part).In order to overcome this problem, the original model could be modified to specifyCp at some points on the separation streamline. Since some specifications are added,more singularities have to be used to match the number of unknowns with the number ofequations to be solved. The idea, suggested by Yeung 1, has been tried first in the caseof a flat plate at 900 incidence, because it is a relatively easy case to study, involving avery simple transformation. It is also one of the test cases presented by Parkinson andJandali [17] to introduce the wake source model in 1970. In 1954, Roshko [20] presentedthe notched hodograph model, an analytical model in which Cp is specified on a portionof the free streamline. The original wake source model and Roshko's model have beenlprivate communication13Chapter 3. Normal flat plate^ 14Figure 3.1: Comparison of Roshko's model [20], wake source model [17] and experiments[5] for a normal flat plate (pressure on the front of the plate)Chapter 3. Normal flat plate^ 15Figure 3.2: Comparison of Roshko's model [20], wake source model [17] and experiments[5] for a normal flat plate (separation streamline)Chapter 3. Normal flat plate^ 16compared with experimental data of Fage and Johansen [5] and the results are presentedon Fig. 3.1 and 3.2 for the pressure distribution on the front of the plate, the pressuredistribution on the separation streamline and the shape of the separation streamline re-spectively. It can be seen that Roshko's model provides relatively good results, with aconstant Cp distribution over a large part of the separation streamline, which matchesexperimental results better than the wake source model.The aim of the work presented in this section is to devise a simple model that couldgive a fairly constant Cp on the first part of the separation streamline, which correspondsto the experimental results, and still give a good prediction of the pressure distributionon the front of the plate and of the shape of the separation streamline.3.2 ModelsThe models presented here are a variation of the original wake source model. The plateis transformed into a circle very simply using a Joukowsky transformation (see Figure3.3)Instead of using two sources (one on the upper part of the circle + its symmetric onthe lower part) located at angles 8 and —8 as in the original model to represent the flow,one can use several sources and the complex potential can be writtenR2 NF(c) U((13+ E q,[log(C — Re') log( — Re-'63) — log()] 3=-1uniform flow + doublet at centre + sources at 6.1^+ image sinksand -53 (symmetry)^at centreChapter 3. Normal flat plate^ 17Figure 3.3: Physical plane and transform planeChapter 3. Normal flat plate^ 18where:=^represents the strengths of the sources7C53 their angular positions on the circle2N sources are used in total (N on lower half and N on upper half by symmetry)With the number of sources being increased, more equations are required to matchthe number of unknowns with the number of boundary conditions. This will be achievedby specifying Cp at N' arbitrary locations on the separation streamline. The angularposition of these locations is chosen arbitrarily and the corresponding radial position hasto be found i.e.Cp is specified at (k (k from 1 to N') where (k = rkelem,Ok being given, rk to be determined(see Figure 3.4)The angular positions (83) of the sources are chosen arbitrarily so that the unknownsof the problem are• The strengths of the sources q3^(N unknowns)• The radial coordinates of the points on the free streamline where Cp is specified(rk)^ (N' unknowns)Total number of unknowns: N N'The boundary conditions used to solve for this set of unknowns are:• separation point=stagnation point in (-plane: w(e's) = 0^(1 condition)• Cp^Cpb at ( = e19.9 (separation)^ (1 condition)(Os = in the case of a normal flat plate)Q.; Chapter 3. Normal flat plate^ 19Figure 3.4: Illustration of the principle of the new modelChapter 3. Normal Bat plate^ 20• Cp — Cpb at (ik^ (N' conditions)• (k's lie on the free streamline (i.e. stream function at (k = stream function atseparation): 4'((k) =^ (N' conditions)Total number of equations to be solved : 2N' + 2In order to be able to solve this system of equations, one must satisfyN N' + 2 (number of sources = number of specifications+2)In summary, there are 2N' + 2 equations to be solved, for the set of 2N' + 2 unknownsql^• • • OP-{-2) rl r2 • • • rN,These equations are:• Cp(rkethk ) — C Pb 0• xli(rkethk) — 7r Ellv=i qj = 0• EN^— 2UR = 031 collo;(w(eies) = 0) (1) to (N')(N' 1) to (2N')(2N' + 1)• El..'" qj^2URO - Cpb = 03=1 cos2(C p(ei") CPb)(2N' + 2)This set of nonlinear equations (only the last two are linear) is solved numerically bya solver NDINVT, using the generalized secant method. The user must provide an initialguess and the solution is calculated by iterations until a convergence criterion specifiedby the user is reached. NDINVT as it is programmed solves the system of M equationsgk(Xi, X2 XM), k from 1 to M, for the M unknowns Xj and considers convergence isreached when(E AX(k)21112 < ERRk=1Chapter 3. Normal flat plate^ 21AX(k) being the last increment in X(k)ERR being the convergence criterion.As mentioned earlier, the angular positions of the sources, 83, have to be chosen ar-bitrarily. Two different ways of placing the sources on the circular arc have been tried:1) The sources are evenly distributed on the arc lying in the wake i.e.53 ^7i^,; ^If 3^2^2N+1The corresponding model will be referred to as model 1ii) The sources are placed on the arc according to a geometric distribution (moredense near the .-axis) i.e.A, = 81+1 —Sj = g.A,_i where g is the common ratio of the geometric progressionwhich leads toAo^— g)/(1 — 0.5g' — 0.5gN+1)Chapter 3. Normal flat plate^ 22The corresponding model will be referred to as model 2.3.3 Results and discussionFor all the calculations, the base pressure coefficient has been chosen as the experimentalone [51: Cpb —1.38. All calculations have been made using a unit circle (R 1), andunit velocity (U =1 m/s). It can be seen easily that the results will not be affected if weintroduce a multiplying factor for the velocity since the parameters of the flow model q,will be directly proportional to U. The first part of the free streamline will be consideredas being at constant pressure if 1.36 < —Cp < 1.4 in the range 0 < X* < Xj whereX* X/h.Model 1 has been tried first, by specifying Cp = Cpb every 5° in the C plane i.e:Cp = Cpb at (k = rkelek, Ok = 85°, 800, 75°, ... The results are presented in Fig. 3.5 and3.6 from 1 specification (3 sources) to 5 specifications (7 sources) on the free streamline(i.e. : Cp^Cpb from 91 = 85° to Ok = 85°,80°, 75°, 70°, 65°). It can be seen on Fig.Chapter 3. Normal Bat plate^ 233.5 that the part of the separation streamline over which Cp is nearly constant can beincreased by adding sources. For 7 sources, one can get as far as X0* 0.3, though anundesirable suction peak appears near the tip of the plate. It can also be noted that thepressure distribution on the front of the plate is still very good and the initial shape ofthe separation streamline does not vary much, even though the downstream spacing isdecreased significantly when more sources are added. The results for solving the systemof equations are given in appendix A for reference.It has been tried to investigate the effect of altering the position of the points whereCp Cpb by trying several different sets of 0k, in order to see if the range over whichCp is nearly constant (X) can be extended but, as can be seen on Figures 3.7 and 3.8,no satisfactory result has been obtained. Specifying Cp at a point below 65° by addingsources has also been tried but no convergence can be obtained with more than 7 sources.Based on the observation that the strength of the sources (or sinks if q3 happens to benegative) close to the separation point is always very small compared to the strength ofthose close to the -axis, model 2 has been tried, with a common ratio g = 0.75. Resultsare again presented in Figures 3.9 and 3.10 for Ok = 85°,80°, 75°,... Numerical outputsare given in appendix A. Cp Cpb is achieved up to X* = 0.34 with 7 sources, but themost noticeable result is that the overshoot near the tip of the plate has disappeared.Systematically adjusting the Ok's can also improve the results slightly, as shown on Figures3.11 and 3.12, even though slight oscillations in the solution can be observed. Note someof the following results using this method:• with 5 sources and 01 85°, 02 = 75°, 03 = 65°, the desired range can be extendedto Xt7 0.28 (as compared to X0' 0.18 for 85°, 80° and 75°).Chapter 3. Normal flat plate^ 24• with 6 sources and 01^85°, 02 = 750, 83 = 65° and 04 = 600 , it reaches .X,';^0.38(as compared to X(7,^0.25 for 85°,80°, 75° and 70°).• with 7 sources and 01 = 85°, 02 = 80°, 03 = 70°, 04 = 65° and 05 = 60°, it reaches0.4 (as compared to X,1`^0.34 for 85°,80°, 75°, 70° and 65°) and there isalmost no oscillation.Whatever the number of sources, the attempt to specify Cp^Cp6 below 60° hasbeen unsuccessful so far, because of a failure in converging to a solution, as mentionedpreviously for model 1. This shows a limitation of the method.Since the introduction of a geometric distribution of the sources has improved theresults significantly, one can hope that lowering the value of g would lead to anotherimprovement (raising the value of g would provide a model closer to the previous onewith the limiting case of g = 1 corresponding to model 1). This has been checked with7 sources and Ok = 85°,80°,75°,70°,65° for g =0.75,0.7 and 0.65. Figure 3.13 showsthat the results are not significantly affected for such a minor change in g. However,decreasing g further leads to problems in converging, probably due to the fact that thetwo sources (or sinks) closest to the axis on both sides become very close to each otherhence causing numerical problems.A practical comment on both models is that convergence is very sensitive to the initialguess and one must be very careful in assigning initial values. For the radial positionsrk's, it is not very difficult since these do not change much (for the same Ok's) over thetrials, but guessing the q3's may be a more involved task. Often, setting q3=0 leads toconvergence, but sometimes one must assign different values to each q3. Furthermore,the convergence criterion as it is originally set in NDINVT may become inappropriateChapter 3. Normal flat plate^ 25when the strengths of the sources take up a wide range of values, because the convergencecriterion is an absolute one. Setting a relative convergence criterion such asM AX(k) 1,,[E(^y,t,,, < ERRk=1 X(k)as well as examining the residues of the equations gk(Xi, X2, .., Xm) is sometimes a morerealistic approach.In summary, the results obtained in the case of the flat plate are encouraging, eventhough the range over which Cp'._' Cpb on the free streamline is limited, and the ideamay be applied to more sophisticated configurations.1.5original model3 sources4 sources5 sources6 sources7 sources1.0-Cp0.50.00.00 0.25 0.50 0.75^1.00 1.25 1.50 1.75 2.00X/hGlobal view-1Chapter 3. Normal Bat plate^ 26Figure 3.5: Results for model 1: CF on separation streamline_Ioriginal model3 sources4 sources5 sources6 sources7 sources_-------------- -I_ -_-_^ original model 3 sources4 sources5 sources^ 6 sources7 sources--I 1^ 1-Chapter 3. Normal flat plate^ 271.000.900.80Y/h0.700.600.00^0.25^0.50^0.75^1.00^1.25^1.50^1.75X/hSeparation streamline0.00^0.10^020^0.30^0.40^0.50Y/hCp distribution on the front of the plate1.00.50.0Cp-0.5-1.0Figure 3.6: Results for model 1: separation streamline and Cp on the front of the plate1.751.50Specification (0k):^ 85°,80°,75°800,750,700— 75°,70°,65°— 85°,75°,65°-Cp1.251.000.00 0.10 0.20^0.30 0.40 0.50X/hChapter 3. Normal flat plate^ 28Figure 3.7: Model 1: Cp on separation streamline with several sets of specification (5and 6 sources)0.00 0.10 0.20^0.30X/h0.40 0.501.60Specification (0,):^ 800,750,700,650,600^- - - - - 850,750,700,650,600 850,800,700,650,600 85°,80°,75°,65°,55°1.501.301.20Chapter 3. Normal flat plate^ 29Figure 3.8: Model 1: Cp on separation streamline with several sets of specification (7sources)Chapter 3. Normal fiat plate^ 30Figure 3.9: Model 2: Cp on separation streamlineChapter 3. Normal flat plate^ 31Figure 3.10: Model 2: separation streamline and Cp on the front of the plateChapter 3. Normal fiat plate^ 32Figure 3.11: Model 2: improved results, 5 and 6 sourcesSpecification (Ok):^ original model 85°,80°,75°,70°,60°85°,80°,70°,65°,60°^ 80°,75°,70°,65°,60°[7 sources]0.40 0.500.200.10 0.600.30X/h0.00^0.20^0.40^0.60^0.80^1.00^1.20^1.40^1.60^1.80X/h1.501.401.30Specification (0,):original model^ 85°,800,750,700,600850,800,700,650,600^ 80°,75°,70°,65°,60°-Cp1. 3. Normal flat plate^ 331.401.201.000.80-Cp0.600.400.20Figure 3.12: Model 2: improved results, 7 sourcesChapter 3. Normal fiat plate^ 34Figure 3.13: Comparison of results for model 2 (Cp on separation streamline),9 = 0.75,0.7,0.65Chapter 4Separation bubble4.1 IntroductionOne phenomenon commonly observed in airfoil flow at low subsonic speeds is the oc-currence of a separation bubble. In this case, the flow separates and reattaches furtherdownstream, after the boundary layer has become turbulent, creating a region of recircu-lating flow between the body and the separated boundary layer. The separation bubbleforms near the leading edge on the upper surface of thin and moderately thick (9% to18%) airfoils at small angles of incidence. The occurrence and subsequent behavior of theseparation bubble affect the stalling characteristics of an airfoil, making it a phenomenonof considerable practical interest. For that reason it has been and is still being studied atlength, both experimentally [2, 6, 16] and theoretically [3, 4, 15, 19, 24]. A good reviewof the phenomenon can be found in Tani [22]. One classification of separation bubblescan be made according to their size: a "short" bubble will typically occupy about 1% ofthe airfoil chord, whereas the length of a "long" bubble will be at least several percentof the chord. It can then grow with increasing angle of attack until it extends over theentire chord, at which stage its maximum thickness is about 3%. Short bubbles affect theflow only locally and have little effect on global loads on the airfoil, as long as they havenot "burst" (failure to reattach). The presence of a long bubble, however, will be morenoticeable in the pressure distribution since it reduces the leading edge suction peak andreplaces it with a plateau-like distribution.35Chapter 4. Separation bubble^ 36In an attempt to calculate the flow in the vicinity of the bubble, several theoreticalmodels have been proposed, like those of Crimi and Reeves (1976, [4]), Roberts (1980,[19]), Vatsa and Carter (1984, [24]) and recently Choi and Kang (1991, [3]). These mod-els always include some empirical data, like that of Choi and Kang, that can providevery good results if the location of transition inside the separated shear layer is givenprecisely. All these models take into account viscous effects and indeed have to be solvednumerically using the traditional viscous-inviscid scheme. A more simple, potential flowmodel using singularities has been proposed by Newman and Tse (1992, [15]) for a sepa-ration bubble on a flat plate. It provides good results in terms of global quantities (lift,drag, position of reattachment, momentum thickness) but cannot capture the details ofthe flow locally. In particular, it does not define the shape of the bubble.The application of the wake singularity model to the separation bubble has been triedby Yeung [28]. The goal was, as usual for this model, to define the boundaries of theseparated flow, i.e. the shape of the bubble in this case, and the pressure distributionon the airfoil. Yeung studied the simple case of a flat plate airfoil and his results areshown on Figure 4.1 for an incidence of 5.85°. They are compared with experimentaldata from Fage and Johansen [5] for the pressure distribution. McCullough and Gault'sexperiments [13] have determined the shape of the bubble which is used for comparison.It can be seen that the pressure distribution compares very well with experiments onthe wetted surface but a large suction peak appears on the bubble boundary, instead ofthe plateau typically observed for long bubbles. Also, the shape of the bubble does notmatch the experimental results. An attempt has been made here to improve these results,but has not been successful. Therefore, the different models used will be presented onlybriefly in the next section, followed by a discussion about the results.Chapter 4. Separation bubble^ 370.0500.000Y/c-0.050-0.100,^.' A A A'A A; A^. A A- .^A ^ bubble (calculated)^ plateA^bubble (experiments) [131-0.00^0.25^0.50^0.75^1.00X/CX/cFigure 4.1: Results for flat plate airfoil at 5.85° incidence: Cp distribution and shape ofseparation bubble [28]Chapter 4. Separation bubble^ 384.2 Models and resultsThe mapping sequence used for all models can be found in Yeung [28]. The plate is a slittruncated on its upper surface to create the critical point necessary at reattachment toallow tangential flow. It is mapped from a circular slit by a Joukowsky transformation.After a proper rotation and translation, this circular slit is transformed into a circle byanother Joukowsky transformation. The position of reattachment is specified from ex-periments.The most successful model devised by Yeung, that leads to the results shown in Fig-ure 4.1, uses a doublet tangent to the circle, on the portion covered by the bubble. Thecorresponding complex potential is1iqe26.F(C) = V(C + —c) + i-y log C c ez5The use of a source and sink of equal strength does not lead to better results. Onemodel, which has not been reported by Yeung but which also creates a "closed" sepa-ration streamline, uses a vortex in the plane (i.e. not on the circle, it would otherwisecancel with its image). This model, whose complex potential is:eth vF(C) = 17(C + —1) + log — i-yv [log (C — Rye') — log (C Rv ) + log C]and the doublet model produce very similar results.The idea of a multi-singularity model, previously applied to the normal flat plate, hasbeen tried for the separation bubble, in order to avoid the peak in the Cp distribution.Several doublets (instead of one) evenly distributed have been used, so that Cp could bespecified on the bubble boundary. Figure 4.2 shows some results with 3 and 4 doubletsChapter 4. Separation bubble^ 39(Cp specified at 1 and 2 locations). The elimination of the suction peak at the locationof specification results in a suction peak at another location, so that the results are notimproved. Also, no solution can be found with 5 or more doublets. Another model of thatkind, in which the shape of the bubble is specified beforehand, also leads to unrealisticresults. Similar unsatisfactory results occur for a model that uses two doublets of equalstrength and imposes a finite pressure gradient at reattachment.As a last remark, it may be noted that a flat plate airfoil has a sharp leading edgewhich makes it quite different from usual airfoils. One may wonder if that characteristicis responsible for the high suction peak. Thus, the original model (doublet tangent tothe circle) has been tried on a Joukowsky airfoil. This also resulted in a high leadingedge suction peak.4.3 Concluding remarksFrom all the results which have just been mentioned, it follows that a wake singularitymodel does not seem suitable for configurations with a leading edge separation bubble.It can be observed that this configuration exhibits some particularities not found in othertypes of separated flow treated by the wake singularity model:• here the flow is only partially separated since it reattaches. That may be a reasonwhy the multi-singularity approach that was successful in the case of a normal flatplate is not appropriate for a separation bubble.• because the bubble is very thin, its boundary may come very close to the singular-ities, and this could result in "numerical" problems. However, it should be pointedout that a non-separated potential flow model (with no singularity on the circle)also shows a leading edge suction peak. It would be therefore erroneous to think-0.00 0.25 0.50X/c0.75 1.005.0-Cp 2.5platebubblespecificationexperiments [5]0.0-0.00 0.25 0.50 0.75 1.005.0-Cp 2.5platebubblespecificationexperiments [5]0.0XicChapter 4. Separation bubble^ 40Figure 4.2: Examples of Cp distributions for flat plate airfoil at 5.85° incidence: modelusing 3 and 4 doubletsChapter 4. Separation bubble^ 41that the presence of singularities alone is responsible for the occurrence of the suc-tion peak. Furthermore, when an attempt was made to specify a bubble shape anda thicker bubble was chosen, no improvement appeared.• the bubble appears near the leading edge, in a region of rapid flow changes. Thepressure on the bubble boundary is therefore very sensitive to its shape, whichitself is determined by the complex phenomena happening inside the bubble. Inparticular, viscous effects are predominant in its vicinity, which makes a potentialflow model inappropriate, as already pointed out in Yeung [28]. The case of thenormal flat plate has shown that conditions on the pressure can be imposed on thefree streamline, but only over a limited range. This was satisfactory in that casebecause practically, the region of interest is limited to some distance behind theplate. For the separation bubble however, one needs a correct pressure distributionover the totality of the separating streamline.Following these observations, it should be concluded that in the presence of a separa-tion bubble one may have to resort to more sophisticated models like the ones previouslymentioned and, indeed, to numerical solutions. For short bubbles, however, a more sim-ple (but less accurate) approach is to adopt a simplified bubble shape and a simplifiedpressure distribution on the bubble, as described in Tani ([22], pp.92 and 100), the restof the flow remaining relatively unchanged, as compared with the non-separated case.If the study reported here did not provide any improvement in the results, it made itpossible to define clearly some limitations of the wake singularity model when it comes toits range of applications. It emphasizes the need for an alternate, more complete modelin some specific cases.Chapter 5Kruger flap5.1 IntroductionThis chapter reports the application of wake singularity models to an airfoil with a KrUgerflap. KrUger flaps are simple high-lift devices located at the leading edge of airplane wingsand used during take-off and landing. There are two types of Kriiger flaps: upper surfaceflaps that slide out from the upper surface of the wing and lower surface flaps that aredeflected about a hinge. Figure 5.1 shows pictures of these flaps, taken from a NACAreport [7]. The pictures also show split flaps at the rear of the airfoil, although in thepresent study, only KrUger flaps are considered 1. The main characteristic of these flaps isthat they increase the maximum lift coefficient. This is particularly noticeable for sharpedged airfoil sections, where this increase can reach ACL,„,„ 0.7. This is achievedmainly due to an increase in the stalling angle (by 8° to 100). Another interesting featureis the improvement in the quality of stall that appears with the use of a lower surfaceflap. The lift curve is rather flat at its peak, so that there is no sudden loss of lift afterstall. All these characteristics depend on several parameters such as the flap length,its deflection angle, the rounding of the flap tip,etc. The influence of these parametershas been investigated to some extent. A few examples can be found in references [10, 11].The first investigations of these flaps have been carried out by Kruger (thus theiriKriiger flaps are sometimes referred to as nose flaps. "Nose flap" is a more general term also includingflaps that are simply a part of the airfoil front end which is deflected mechanically.42Chapter 5. Kruger flap^ 43Figure 5.1: Upper and lower surface Kruger flaps on an airfoil [7]Chapter 5. Kriiger flap^ 44name) as early as 1944 in Germany, and continued interest in their use has been shownthroughout the late 40's and 50's. Nowadays, they tend to be replaced by leading edgeslats on most large airplanes. However, they are still in use on some recent commercialairplanes such as the Airbus Super Transporter (A300-600R), the A310, and the Boeing737 and 747. Furthermore, their simplicity and associated low cost make them moreappealing for smaller airplanes. Therefore, they are still of practical interest and it maybe advantageous to devise a reliable model for them.The Kriiger flap investigated in this chapter is an upper surface flap, fitted on a Ger-man profile designated as "Mustang 2". Experimental results are taken from reference[11J. In this particular case, the flap (which is 10% chord long), provides an increase inthe maximum lift coefficient from 1.13 to 1.43. At this point, it may be useful to point outthat all data, including the airfoil coordinates, have been taken "by hand" from graphsand figures of reference [11]. This provides some uncertainty, added to the uncertainty ofthe measurements, for which no estimate is given in [11]. Furthermore, these experimentswere carried out on a finite span rectangular wing (aspect ratio 5.4) so that this wingwas subjected to three dimensional effects. Correction for the angle of attack has beenperformed according to lifting line theory to obtain the effective two dimensional angleof attack, but this correction itself is an approximation, as noted by the author in [11].The induced angle of attack has been determined using the 2D lift curve slope of theplain profile, instead of the profile with the flap. The resulting inaccuracy of the effectiveangle of attack appears clearly in a comparison of the direct measurement of the liftforce with the lift resulting from integration of the pressure distribution. It is thereforeobvious that the actual effective angle of attack is smaller than the one calculated in [11].Following these comments, one should not expect perfect agreement of theoretical resultswith experimental data, because the latter are only approximate.Chapter 5. Kriiger flap^ 45Two different types of models are presented here, depending on the assumption thatthe flow reattaches on the lower surface or not. No indication on that matter is givenin [11]. First, it was assumed that the flow separates at the tip of the flap and at thetrailing edge to form a broad wake beneath the airfoil. A model, using one source inthat wake, has been devised and is referred to as a 1-source model. However, even if noflow visualization study has been carried out, pressure distributions suggest that the flowreattaches on the lower surface of the airfoil after separating from the flap. Therefore, asecond set of models has been devised to account for this and is referred to as doubletmodels.5.2 Theodorsen transformationFor both models presented in this chapter, the mapping sequence will require, as a firststep, the conformal transformation of an arbitrary airfoil shape into a circle. This willbe done using an updated version of the well known Theodorsen transformation, whichis briefly presented in this section. Although the basic idea is still that of Theodorsen[23], as described in [1], the procedure has been changed to make calculations easier andfollows the lines of Ives [8].The mapping of an arbitrary airfoil from an exact circle requires three basic steps:1. The airfoil is mapped from a near-circle through a Karman-Trefftz or a Joukowskytransformation. For simplicity, and because it provides the necessary cusp at thetrailing edge, a Joukowsky transform has been used here. The transformation is:Z = 21+1/Z1, the Z plane being the airfoil plane. In that plane, the airfoil trailingedge is located at Z = +2, and the point which is half way between the leading edgeChapter 5. Kriiger flap^ 46and its center of curvature is located at Z = —2. This first step removes corners(trailing edge) and transforms the airfoil into a smooth contour. If the airfoil wasan exact Joukowsky airfoil, this step would produce an exact circle. Since airfoilsare usually quite similar in shape, the resulting curve in the general case will differonly slightly from an exact circle.2. The near circle is translated to place its centroid at the origin: Z =^— Z10 whereZ10 is the centroid in the Z1 plane.3. The near circle is transformed into an exact unit circle, centered at the origin,through the transformation:2 expA iB •)= Z^E  3 . 3j=0^2This, in theory, can always be accomplished. Practically, the series is truncated toa finite number of terms, and the transformation is written:Z; = Z2 exp(N A3 + iB3)(5.1)Figure 5.2 illustrates these different steps. The determination of the coordinates on thenear circle, which amounts to an inverse Joukowsky transformation, is rather straight-forward and is explained in appendix B. Since the airfoil is not given by a continuousrepresentation but point by point, so is, in a first stage, the near circle. By approxi-mation, the centroid of the near circle is taken as the centroid of a figure obtained byconnecting successive points with straight lines. Similarly, a polynomial interpolation isperformed between adjacent points in order to define a continuous parametric represen-tation of the near circle in the 4.1 plane: Z; r(co)et. Of course, the greater the numberof points used to define the airfoil, the more accurate these approximations will be.Chapter 5. Kriiger flap^ 47Z planeTE^XZ1 plane Ze plane+2Figure 5.2: Successive planes for Theodorsen transformationChapter 5. Kriiger Rap^ 48The fundamental step of the method is to determine the Theodorsen coefficientsA3, B,, = N. The parametric representation of the centred near circle is writtenin the formZ' = r(cp)e tipwhile the representation of the centred unit circle is:Z2These expressions are substituted in equation (5.1). By taking the logarithm of both sidesand separating into real and imaginary parts, one obtains the fundamental equations ofthe problem. These will have to be solved at chosen points on the circle:log r = Ao +^A cos( j(k) + B3 sin(j)^ (5.2)3=1(,0 -= + Bo E Bi cos(jo) — Ai sin( j0)^(5.3).1=1The solution for the A3's and /33's is determined iteratively using (5.2) and (5.3) at 2Npoints with the additional condition that the trailing edge be located at i= 0. AppendixC describes in detail how this is achieved. Starting with the initial guess A3 = B, = 0,j = 1, N, iterations are performed until the resulting values for the A 's and ./33'sbecome stable. The criterion for convergence is:Aji+1 — Ai3 Bij+1 —max     < 6AU k+1where the superscript i denotes the ith iteration. In the present study, € has been fixedat e = 10-6.The method is, by principle, only approximate because a continuous analytic functionis represented by its truncated Laurent series. However, it will be seen that the airfoilChapter 5. Krtiger flap^ 49shapes considered in this chapter can be represented quite accurately using no more than25 terms (N = 25) for the most "deviant" one.5.3 1-source model5.3.1 MappingThis model assumes no reattachment on the lower surface of the airfoil. Since the flowseparates at the tip of the flap and at the trailing edge, these two points are critical pointsin the following mapping sequence, illustrated on Figure 5.3. The mapping sequence isessentially the same as the one described by Yeung [28] for the flow over a stalled airfoil("vanishing spoiler" mapping). Here a Theodorsen transformation has been added andsome slight modifications account for a different flap orientation.The mapping sequence starts with a Theodorsen transformation that transforms acircle and a tangent straight slit into the corresponding airfoil and a flap:1= Z10 + Z2 expN AAfter a rotation to place the slit on the real axis:Z2^(Z3 + i)e1(4)1I-i)the following mappings are performed to open the slit representing the flap and make ita part of a unit circle in the final plane:Chapter 5. Krüger flap^ 50Figure 5.3: Mapping sequence for Krüger flap, 1-source modelChapter 5. Kriiger flap^ 51- Z5 + log(*) 2 1 iZ4 = 1^7rZ5 As + AyZ6- e-ta° + Z6^e"°i - Z6<4.^=In this sequence, the length and position of the flap are determined by the parame-ters 1 and qji respectively. As and Ay are the coordinates in the Z5 plane of the point atinfinity in the physical plane. Since Z5,, = As + iAy corresponds to Z3 = oo and Z4-.5,- ,As and Ay are determined by solving:^0= ^log( 24-00-)^2^17r^/^7rBy separating into real and imaginary parts, one obtains:0 -1 (Ax llog(A2 + )^2 12)) - - -7r^2^v 1^7r0 1 + -1 Ay + tan-17r^AsThe above equations are solved iteratively using Newton's method. A similar procedureis used to determine the positions, in the final plane, of characteristic points such as thetrailing edge and the tip of the flap. cx0 is the angle of the last rotation that places thecircle at zero incidence relative to the incoming flow.The total derivative of the mapping function- is calculated as:dZ^dZ dZi dZ2 dZ3 dZ4 dZ5 dZ6d4-^dzi dz2 dz3 dz4 dz, dZ6 d(Chapter 5. Kruger flap^ 52(1 ,) +L6)2 et(ao+010dZ1dZ2where:A; i/ 31)exP (i=A + iI3;From this expression, it is seen that the trailing edge (Z1 = 1) and the tip of the flap(Z5 = —I) are critical points. Note that Z3 = 0 and Z6 i are not critical pointsbecause, for both points, Zi2(i — Z6)2 has a finite value.• Z3 = 0 is a singular point. It corresponds to the point where the flap attaches tothe airfoil on the upper surface. At this point, Z6 = 00 and:A2In the final plane, this point corresponds to C —e-2"°. It can be noted that Z3 = 0corresponds to another point in the final plane, given by Z5 = 0, Z6This second point lies in the wake and is therefore ignored.• Z6 = i corresponds to the point at infinity (Z3 = oo, Z oo) andz32(2 - z6)2These limits are calculated in appendix D.472Ay2 (1+ zbc.)2From the value of dZIA at infinitydZidZ2 i= 1472z32(i z6)2 _Chapter 5. Kriiger flap^ 537r (  Z5cc^eAo e i(ao^t +Bo)Ay 1 +500ooone can derive the magnitude of the free stream velocity in the final plane: .Z500V = U eA91 + Z500and the angle for the last rotation:a° = — — + T2 — BO + 7rwhere:ri = arg(Z5.) arctan (t(-)7-2 = arg(1 + Z5.) = arctan (A)7r. = arg(Ay)In order to calculate the velocity at critical points using PHOpital's rule, the secondderivative of the mapping function will be needed at these points. It is given by:( dZ dZ3az, c1( d(At the trailing edge (Z2 = 1):dZd(d2Zd(2(1 + z15•-■ dZiZ6 )4^CLLJ2d2Z^A2c/(2^87r22At the flap tip:Chapter 5. Kn'iger flap^ 54d2Z V212 (1^1 dZid(2 87r Zi2) dZ25.3.2 Flow modelThe model described here is called a 1-source model because it uses one source in thewake, combined with a vortex for the circulation.The complex potential in the circle plane is:F(C) = V (C. +^q [log(( — e) — log (1 i-y log (The corresponding complex velocity is:^w (C) = V (1 —^+ q (^ 1) +c2^C —^2(^CThe three unknowns of the problem are the source strength q, its angular location b.and the vortex strength -y. They are determined by applying the following boundaryconditions:• w(() = 0 at the trailing edge (TE) and the flap tip (T) (2 conditions)• Cp^CI; at the trailing edge or the flap tip (1 condition)CI; = CpT. or CPT depending where it is chosen to specify the pressure coeffi-cient. It must be noted that in the case considered here, CpTR and Cpr are usuallydifferent, as experimental results show.The corresponding 3 equations to solve are:^217 sin OT^cot (02'2 5)^-y = 0^(5.4)2 -Chapter 5. Kruger Bap 552V sin OTE -y = 0 (5.5)—^cot (6TE22 26)1 —dwdC (5.6)d2 Z42 T (or TE)These equations can be solved simultaneously using a solver for a system of non-linearequations, NDINVT for example. However, this requires an initial guess for q, 7 and S.With the exception of 8, which is known to lie between OT and OTE, there is no indicationa priori on the range of values that these unknowns can take up. It is therefore moreconvenient to reduce the system to one single equation with S as the only unknown. Theinitial guess is then much easier since a lower and a upper limit are known for S.By subtracting 5.5 from 5.4, q can be expressed as a function of 8:2V (Sill OT — sin OTE) [^(OT — OTE)^(OT °TE q^ COS cossin (8TR2-8T)^2 28)]^f(8)^(5.7)By substituting this expression into 5.4 (or 5.5), an expression for -y as a function of 6 isobtained:f(6) cot (OT — 6)-y = 2 c^2^2V sin OT = g(8) (5.8)Now, since all unknowns are expressed as a function of 5, so iscalculate the pressure coefficient at one of the critical points:dwh(8)So the remaining equation to solve for S is:dwdCwhich is used toChapter 5. Kruger flap^ 562=T (or TE)An initial guess such as 8 0.5(07, + OTE) has been shown to lead to convergence quiteeasily. Once 8 has been determined, substituting its values back into 5.7 and 5.8 rapidlygives q and 7. The pressure distribution on the airfoil and the flap can then be determined.The shapes of the separation streamlines from the tip of the flap and the trailing edgeare given by:TT and W TTEwhere:+TTE = i(8 — 7r)S E [0, 27r[5.3.3 Results and discussionA proper mapping of the airfoil upper surface and flap has been obtained by mappingan airfoil as shown on Figure 5.4. The airfoil leading edge has been artificially thickenedon the lower surface in order to ensure a good mapping of the flap. Without this, it wasimpossible to get a flap shape that would faithfully represent the real flap. Since thelower surface lies in the wake, the mapped airfoil shape has to match the real airfoil onthe upper surface only. It can also be noticed that the trailing edge has been thickened,on the upper side, because of the boundary layer effects there. It is felt that the outerflow boundary would match this new shape better. This has, indeed, improved the results1h(8) d2zdc2 Chapter 5. Kruger flap^ 57Figure 5.4: Mapping of the airfoil and its flap (1 0.73, Op = 159°)Chapter 5. Krtiger flap^ 58near the trailing edge. Because the airfoil has been transformed, it is important to notethat its leading edge is located below the chord line of the real airfoil. Therefore, thechord lines of the two airfoils are different and there is a 2° angle difference between them.In the following results, the given angles of incidence are relative to the chord line of thetrue airfoil, and correspond to the angles of attack given in [11], for which experimentaldata are available. Of course, in the calculations, the angle a which is entered is equalto the "real" angle of incidence minus 2°. The Theodorsen coefficients and the mappingparameters are given in appendix E.Results shown on Figures 5.5 to 5.7 have been obtained by specifying Cp at the tipof the flap, although similar results are obtained by choosing the trailing edge as a pointof specification. Obviously, results are bound to be the same if the value of CI*, at thetrailing edge is chosen to be the value there that arises from specifying Cp at the flap tip.For comparison with experiments (Fig. 5.6) calculations have been made for a 110and 14.6°. The corresponding numerical results can be found in appendix E.Figure 5.5 shows that the results are much more sensitive to the value of Cp* which ischosen than to the angle of attack a introduced in the mapping sequence. It is thereforeimportant to select an accurate value for C. This is difficult for the flap tip because ofthe infinite pressure gradient in this area. The values used for CPT in the calculationshave been chosen as the ones giving the best agreement with experiments and they fallindeed into the experimental range. From a practical point of view, this is not very usefulbecause one wants to predict good agreement starting with a given experimental valueof C;„ not vice-versa. It is felt that this problem could be solved by choosing to specifyCp at the trailing edge, where a more precise value would be given by experiments. It isfurther validated by the observation of Figure 5.5: a slight change in CPT brings a moreChapter 5. Kriiger flap^ 593.02.0-CP 1.00.0-1.0-2.5^-2.0^-1.5^-1.0^-0.5^0.0^0.5^1.0^1.5^^1.00.0-1.0-2.5^-2.0^-1.5^-1.0^-0.5^0.0^0.5^1.0^1.5^2,0Figure 5.5: 1-source model: comparison of Cp distributionsa=11°Cp-0.84 at tip^ uwer surface lower surface0^expt-upper surf.expt-lower surf.^2.0-CR1.03.00.0-1.0a=14.6°Cp=0.8 at tip^ upper surface_ _ _ lower surfaceo^expt-upper surf.A^expt-lower surf.Chapter 5. Kriiger flap^ 60-2.5^-2.0^-1.5^-1.0^-0.5^0.0^0.5^1.0^1 5^2.0X3.02.0-CR1.00.0-1.0-2.5^-2.0^-1,5^-1.0^-05^0.0^0.5^1.0^1.5^2.0XFigure 5.6: 1-source model: Cp distributions (a = 110 and 14.6°) compared with exper-iments 1111Chapter 5. Kriiger flap^ 612.001.00Y0.00-1.00-2.00-3.0^-2.0^-1.0^0.0^1.0^2.0^3.0XFigure 5.7: 1-source model: shape of free streamlines (a = 11°)Chapter 5. Kriiger flap^ 62important change in Cpm so that a value of Ci*,, at the trailing edge would not have tobe as accurate as at the tip. Unfortunately, it is difficult to derive this value from thegiven experimental data, because no data is given very close to the trailing edge. It isassumed that this could be done in the general case.Good agreement with experiments is found on the upper surface of the airfoil, in-cluding the flap (Fig. 5.6). On the lower surface however, a constant value for Cp equalto that at the trailing edge does not seem to be realistic. First of all, the level of thepressure is too high as compared with the experimental values on the lower side. Second,and this is particularly obvious for high angles of attack, the experimental pressure is notconstant, even if it does not vary much for low angles of attack. Furthermore, the shapeof the separating streamline from the flap tip, shown on Figure 5.7, is totally unrealistic.One would expect a free streamline heading downstream to form the wake boundary. Itis felt that this could be achieved if the flap tip was oriented backward so that the flowwould leave the flap instead of approaching it tangentially, and head downstream. Ofcourse, a different mapping sequence would be required to obtain the necessary artificial"hook" at the tip.All these comments seem to imply that the assumption of no reattachment on thelower surface may be wrong in this case. This is further validated from the observationof the pressure distribution on the lower surface behind the flap. Experimental datasuggest a typical separation bubble distribution, with pressure constant on the laminarpart, then increasing up to the value at reattachment. For this reason, a second modelhas been devised that takes into account reattachment on the lower surface. This modelis presented in the next section.Chapter 5. Kruger flap^ 635.4 Doublet models5.4.1 MappingSince in this model the flow is assumed to reattach on the lower surface, three pointshave to be made critical points: the trailing edge, the tip of the flap and the reattach-ment point. The mapping sequence which is described here satisfies this condition and,combined with a proper flow model, has been shown to give the best results. It is basedon the observation that, since the flap is tangent to the airfoil, they can both be madepart of a single airfoil. Thus an artificial "airfoil" is constructed, whose upper surfaceconsists of the flap and the real airfoil upper surface. Its lower surface consists of thereal airfoil lower surface from the trailing edge up to the point of reattachment. Therest of the shape, between the flap tip and the reattachment point, is free, with the onlycondition that it connects smoothly with the other part. The resulting shape looks likean airfoil with a rather odd front part, but an inverse Joukowsky transformation willtransform it into a shape which is sufficiently close to a circle to be easily transformedto a circle by Theodorsen's method. In case convergence could not be achieved, or toreduce the number of iterations, more modern numerical mapping techniques like theone described by Luchini and Manzo [12] may be used. This has not been necessary inthe present study because convergence was obtained quite rapidly if enough coefficientsin the Theodorsen series were used.The mapping sequence is shown on Figure 5.8. It starts with a slit corresponding tothe wetted surface of the "airfoil". This slit is transformed into a circular arc slit througha Theodorsen transformation.Chapter 5. Kriiger flap^ 64sin -0cos 0Chapter 5. Kriiger flap^ 651Z = Zi + —ZiINN AZ.This circular arc is rotated, scaled and translated to locate its two extremities at 2i and—2i:Z10 + Z2 expZ2Z3+ cosR3(02 — 01  ) et( 9'1 V52 )2 A Joukowsky transformation opens the circular arc slit into a circle which is translatedto the origin:1Z4 — cot 0Finally, the circle is scaled and rotated to obtain a unit circle at zero incidence in thefinal plane:Z4^Ce'a°q1, 02 are the angular locations of the circular arc extremities in the Z2 plane. Thesecorrespond to the flap tip and the reattachment point. The relationships between thedifferent parameters are:Z3 = Z4 — COL 19';Chapter 5. Kriiger flap^ 66cto, the angle of the last rotation, has to be determined to place the last circle at 0 inci-dence.The total derivative of the mapping function is:dZ—Zi zoao + 01 2472  )ck." = cosCOS (1 —^(1 + ^Z?^(Z4 — COt q3)2 dZ2 ewhere:1 E.) 3^) exp2^j=A;j=1i B j)N , A^iB-At infinity (Z1 = 00 , Z4 = 00 , Z2 = 00 ):95, +2 = cos eA° ei(B°+"°+ 2)so that:{ V = U cosao = a — Bo chi 22 From the above expression for dZ 7 one can see that the trailing edge (Z1 = 1), the flapck tip (Z4 = cot 4 — i) and the reattachment point (Z4 = cot^i) are critical points, asthey should be. The second derivative of the mapping function at the trailing edge is:dZidZ2dZd( 00 2d2 Zd(2= 2 cos2dZi (1+ ^( Z4 — COt^dZ2and at the separation and reattachment points is:d2Zd(2= 21? cos(1^1^dZiZ?) dZ2Chapter 5. Kriiger flap^ 67In this sequence, the position of critical points has to be determined in the final plane.One delicate step is the transformation from the Z3 plane to the Z4 plane. This is aninverse Joukowsky transformation and the procedure is very similar to the one describedin appendix B. Another important step is to determine the angular coordinate in the Z2plane, given the coordinates in the physical plane. This can be achieved only iterativelyusing the secant method. However, for the determination of OT, 01 can be adjusted byhand. Because the extremity of the flap is rounded, the position of separation is onlyapproximate and is adjusted to provide the most reasonable results (see section 5.4.3).5.4.2 Flow modelsThe models use one or two doublets tangent to the circle in the separation bubble, whichcreate a closed separation streamline.The complex potential is:Fi(C) = V (C +^+ i-y log1for one doublet and:ez61^e 162^F2(C) = V (c +^i-y loge5'^q2 — e2for two doublets. The associated complex velocities are:ez51^-ywi(C) = V (1 —^+^"i(c et5)2'y 161^ C252^w2(C) = V (1 —^iql(c e 0502 iq2 (C — C152)2ethChapter 5. Kriiger flap^ 68There are three unknowns for the 1-doublet problem: the doublet strength q, its an-gular location 6 and the vortex strength 7. The condition of tangential separation andreattachment is applied at the three critical points, which provides the three boundaryconditions necessary to solve for q, 6 and 7. The corresponding equations are:2V sin OTE -y ^ 04 sin2(9"-5)22V sin OT -y ^ = 04 sin2(-V-)2V sin OR + ^ — 04 sin2(P-EL---5-)2(5.9)(5.10)(5.11)As discussed in section 5.3.2, it is much easier to work with one equation involving6 alone, to make the initial guess easier (OT < 8 < OR). This equation is obtained bytaking 5.9-5.10 and 5.9-5.11 and equating them. The resulting equation to solve for 6 is:sin2rez-j;\sin2(2.11-7-1) — sin2(9T)42-5)k  ^\ 2 ^sin OTE — sin OR = 0sin2(211=6-)^sin2(^— sin2( °T" )^sin OTE — sin OT2 2 2Then, q is given by:sin2(9'2-8)sin2()f (8)q 8V(sin OTE — sin OT) sin2 ( ' ) sin2(e"^2 -8)2^_ and 7 by:^ 2V sin OTE g(S)-Y^e4 sin2(_x2iiWhen two doublets are used, the five unknowns are the vortex strength -y, the doublets'42_vadCd2 ZdC21midC d2 ZdC2(5.15)(5.16)Chapter 5. Kriiger flap^ 69strengths (qi, q2) and angular positions (81, 82). The first three boundary conditions arethe same as for one doublet, and the additional two are the specification of pressure atseparation and reattachment. The equations to solve are then:Q1^Q2 2V sin T E^ = 04 sin2( GTR2-61-)^4 sin2(82-52)Q1^Q2 2V sin T -y  = 04sin2( 4 sin2(8T^qi q2 2V sin R^ 04 sin2(9R2-81 )^4 sin2( 8R -67 )22=CPT2= C pRA similar procedure as before is applied to express all quantities as a function of Si and82 only and reduce the system to 2 equations (5.15, 5.16) to solve for Si and 82.The bubble shape and the free streamline from the trailing edge are determined by:2and1= —2 (41 q2)They have been calculated using Newton's method in the appropriate range. The streamfunctions at^re'e are:Chapter 5. Kriiger flap^ 70r cos(0 — 8) — 1= V (r — —1 \ sin 0 + -y log r — q^r 1^v2 + 1 — 2r cos(0 — 8)412 V^— —1) sin 0 + -y log — r cos(0 — 81) — 1q2r cos(8 — 152) — 1r2 + 1— 2r cos(19 — 81)^r2 1— 2r cos(0 — 82)5.4.3 Results and discussionAs noted in section 5.1, the 2D angles of incidence given in [11] are probably overesti-mated, so that a further correction was necessary. On a graph showing simultaneouslythe lift versus the geometrical angle of attack from lift force measurements and the liftversus the 2D angle of attack from integration of pressure distributions, the last curveappeared to be lower than the first one. This means that the incidence in plane flowthat would produce an equivalent force is actually lower than the one given. A furthercorrection has been applied by taking the angle on the first curve that corresponds to thesame lift as on the pressure distribution curve. The resulting angles are indicated below:42D[11] 7.3° 110 14.6° 16.5°acorrected 6° 8.6° 12.4° 13.78°The highest angle of attack corresponds about to the maximum lift angle, just beforestall. At higher angles of attack the flow separates on the upper surface starting fromthe trailing edge, so that the present models are not relevant anymore.The airfoil used for mapping in this part is shown on Figure 5.9. The angle differencebetween the real airfoil chord line and the mapped airfoil chord line is 50. For betterresults, it was observed that the separation point in the model should be chosen a littleChapter 5. Kriiger flap^ 711.^-1.5^-1.0^-0.5^-0.0^0.5^1.0^1.5^2.0XFigure 5.9: True airfoil+flap and mapped contourChapter 5. Kruger flap^ 72further up the flap tip such as to produce a rounded curve before separation. Figures5.10 and 5.11 show the results for pressure distributions obtained with one doublet atseveral different angles of attack. Corresponding numerical results are given in appendixF for reference. For convenience of reading, coordinates have been given as a percentageof the true airfoil chord. Overall, good agreement with experiments is obtained on thewetted surface. The oscillations in the solution are due to the Theodorsen mapping andthe inaccuracy of the airfoil definition. Because no precise coordinates are available, itmight be fruitless to add more terms in the Theodorsen series in an attempt to imposemore airfoil coordinates. The slight discrepancy near the trailing edge is probably dueto the thickening of the boundary layer in this region, as observed in §5.3.3. This couldnaturally be resolved in the same way by artificially thickening the trailing edge region inthe mapping. It may also be noted that for the highest incidence (13.78°) the flow mayhave started separating from the upper surface in a small region near the trailing edge.Results in the leading edge region, around the flap base, are very sensitive to a changein the airfoil surface shape. This has been confirmed by comparing the results on onlyslightly different airfoil shapes. Therefore, much care should be exercised in defining theairfoil precisely in this region. For the reason indicated above, this was impossible in thepresent study.Agreement is much less satisfactory on the bubble boundary, except at high angles ofattack (12.4° and 13.78°). Even though higher incidences are of more practical interestin the use of Kriiger flaps, these still have to be deployed before stall occurs on the plainairfoil at moderate angles of attack, about 9° (corrected) for the Mustang 2. One shouldtherefore be concerned about the prominent and sharp suction peak following separationat this level of incidence. One observation is that the results are dependent on how the"free" portion of the airfoil has been defined. Because the doublet strength required to4.03.02-C^0pa=13.78°)0o^9wetted surfacebubbleexpt-upper surfaceexpt-lower surface1.00.04 A-1.0-0.00^0.25,0.50, 0.75 1.00a=12.4°0^9wetted surfacebubbleexpt-upper surfaceexpt-lower surfaceChapter 5. Kriiger flap^ 73Xlc4.03.00.0-10-Cp 2.010-0.00^0.25^0.50^075^1 00X/cFigure 5.10: 1-doublet model: Cp distributions (a^13.78° and a^12.4°) comparedwith experiments [11]0.750.500.25 1.00X/c4.0wetted surfacebubbleexpt-upper surfaceexpt-lower surface3.02.0-Cp1.00.0-1.0-0.001000.75025 0.504.0wetted surfacebubbleexpt-upper surfaceexpt-lower surface3.02.0-Cp1.00,0-1.0-0.00XlcChapter 5. Kriiger flap^ 74Figure 5.11: 1-doublet model: Cip distributions (a ------- 8.6° and a^6°) compared withexperiments [11]^a-13.78°^ wetted surface^ bubble- _._ separating streamline (TE)airfoil-0.00^0.25^0.50^0.75^1.00^125X/c^ wetted surface bubble- separating streamline (TE)- airfoilChapter 5. Krager flap^ 750.500.250.°°-0.25-0.50-0.00^0.25^0.50^0.75^1.00^1.25X/c0.500.25Y/c 0.°°-0.25-0.50Figure 5.12: 1-doublet model: separating streamlines, a = 13.78° and a = 8.6°Chapter 5. Kriiger flap^ 76satisfy the boundary conditions is very small, the bubble shape follows this airfoil por-tion very closely (Fig. 5.12). Results obtained with a different definition of the airfoil inthe bubble region (see Fig. 5.13) are indeed quite different in this region and show thesensitivity of the method to this contour definition. In fact, the whole purpose of using aseparated flow model is lost if the bubble shape has to be known beforehand, because asimple non-separated potential flow model using the bubble contour would produce verysimilar results, as shown on Figure 5.14.One way to improve the results would be to impose more conditions at separationand/or reattachment, that would not be naturally occurring from the relative position ofthese points in the flow. This is the reason why a 2-doublet model has been introduced.One "strong" additional boundary condition is the specification of the pressure at sep-aration from the flap, because this is where the discrepancy occurs. The specificationof Cp at reattachment can be considered as a "weak" boundary condition because, evenwith only one doublet, the pressure there is never very different from that of experiments.Results for Cp distributions and separating streamlines, using two doublets, are shownon Figures 5.15 and 5.16 for a = 13.78° and a = 8.6°. At high angles of attack thereis not much improvement because CPT and CpR were already quite close to experimentalvalues in previous results. For lower incidences however, the additional conditions allowreduction of the sharp peak at separation, even though a reduced peak still appears onthe bubble. It is therefore felt that a 2-doublet model combined with a reasonable defi-nition of the contour in the bubble region might help to achieve a realistic pressure levelin that region. Of course, this would require preliminary flow visualization studies on arange of airfoils at several different angles of attack to have an approximate but generalidea of the bubble contour under different conditions. This was beyond the scope of thisstudy but would be an interesting idea to explore.C 2.0-p1.0wetted surfacebubbleexpt-upper surfaceexpt-lower surface2Chapter 5. Kriiger flap^ 771.51.00.5y 0.0-0.5-1.0-1.5-2.0^-1.5^-1.0^-0.5^-0.0^0.5^1.0^1.5^2.0X4.03.00.0-1.0-0.00^0.25^0.50^0,75^1.00XJcFigure 5.13: Second contour: Cp distribution with 1-doublet model (a^13.78°) com-pared with experiments [11]-0.00 0.25 0.50X1c0.75 1.00^ non-separated^-^ wetted (separated)- - - bubble100050 0.750.25X/c4.0^ non-separated^-^ wetted (separated)- bubble3.02.0-Cp100.0-1.0-0.00Chapter 5. Kriiger flap^ 78Figure 5.14: Comparison of pressure distributions a = 13.78° and a = 8.6°a=13.78°^ wetted surface^ bubble0^expt-upper surfaceA expt-lower surface-0 .00^025a=8.6°wetted surfacebubbleexpt-upper surfaceexpt-lower surface2Chapter 5. Kriiger flap^ 794.^0.25^0.50^0.75^1.00XlcFigure 5.15: 2-doublet model: Cp distributions (a = 13.78° and a = 8.6°) comparedwith experiments [111-0.00 0.25 0.50^0.75 1.00 1.25Chapter 5. Krtiger flap0.500.25Y/C-0.25-0.5080X/c^ wetted surface bubbleseparating streamline (TE)-^- airfoil0.500.25^ wetted surface bubbleseparating streamline (TE)airfoil-0251.00-0.00 0.25 0.50 0.75 1.25X/c-0.50Figure 5.16: 2-doublet model: separating streamlines (a = 13.78° and a = 8.6°)Chapter 5. Kriiger Bap^ 815.5 Concluding remarksThe first model that has been devised, using only one source, is very simple and maybe useful in predicting Cp distributions for fully separated flows, although this needsto be confirmed with proper experimental data. This situation may occur at very lowangles of attack, which is of minor practical interest in this case, but also for differentflap parameters such as a longer flap for example. Lower surface flaps may also induceseparation without reattachment. However, the circumstances under which this mayhappen are still to be determined. One major advantage of the model, when applicable,is that it requires only one empirical input, which is the pressure at the trailing edge.One shortcoming is the impossibility to define the wake boundary.With reattached flows, one can expect only approximate results in the bubble region.At high angles of attack, just before stall, a non-separated flow model may indeed beenough to achieve that purpose. This model would include the bubble boundary in asolid contour representing the airfoil and its flap. Of course, this requires the knowledgeof the bubble boundary, but an approximate shape is enough to produce satisfactoryresults in this case. This approach is much less satisfactory at moderate incidence. Animprovement of the results may be obtained with a model using two doublets and requir-ing the specification of the pressure at separation. In the present study, the 2-doubletmodel eliminated the sharp and high suction peak appearing at separation, but anotherpeak still remains a little further up on the bubble, although at a lower level. More workstill needs to be done to assess the effectiveness of such a flow model coupled with amore correct definition of the mapped contour in the bubble region, since results havebeen proven to be very sensitive to it. Of course, this would add some empiricism to themodel because one would need a general idea, a priori, of the bubble shape, along withChapter 5. Kriiger flap^ 82the knowledge of the pressure at separation, which may be obtained through experimen-tal correlations. Another model which has not been tested because of time limitations,would use the same mapping but a source and a sink of equal strength to create theflow. This would allow the specification of the pressure at the flap tip, which is the mostsensitive point.In this study, only upper surface flaps have been considered. A similar investigationcould be carried out for a lower surface Kriiger flap. If the flow proved to be non-reattached, one example of possible mapping can be found in Rossow [21]. This mappinghas the advantage that it offers better control over the flap shape which is defined notonly by its length and position but also curvature. A one source flow model wouldprobably suit. For reattached flows, a mapping using the same basic idea and creatingcritical points at separation and reattachment may still be used. Doublets or any kindof singularity combination creating a closed streamline would create the flow.Chapter 6Conclusion6.1 Summary of resultsThe first configuration studied was the case of a flat plate in a flow perpendicular toits plane. By adding more sources to the original wake source model, it was possibleto specify the pressure at several locations along the free streamlines. With a properdistribution of sources, a fairly constant pressure was achieved on these streamlines to adistance of up to 0.4 times the plate width behind the plate. The range over which thispressure can be kept constant is limited however, and the addition of more sources provesuseless. Of course, in reality, the pressure is about constant only over a limited distancebehind the plate, but an attempt was made to see if one could have more control overthe boundary conditions in this type of model. The results show that some conditionscan indeed be imposed to a limited extent.One major outcome in this study is that configurations involving flow reattachmentare much more delicate to treat with wake singularity models. The case of the longseparation bubble on a flat plate airfoil has not benefited from the idea introduced in theprevious part. The original model developed by Yeung [28] used one doublet and showed adiscrepancy between theoretical and experimental results on the bubble boundary. Usingmore doublets and specifying the pressure at discrete points on the bubble boundary has83Chapter 6. Conclusion^ 84not improved the results. The failure of the model in this case is attributed to the non-applicability of potential flow models in regions of high viscosity, like a separation bubble.The second configuration involving flow reattachment was the airfoil fitted with anupper surface Kriiger flap. The models proposed here used one or two doublets in theseparated region. Results in terms of pressure distributions are satisfactory on the wettedsurface, although more accurate experimental data would be needed to further validatethe agreement. Agreement is poor on the separation bubble that forms under the airfoilin the area between the flap and the lower surface up to the point of reattachment, exceptfor high angles of attack. A suction peak appears on the bubble boundary immediatelyafter separation. It has also been observed that the results depend on the mapping andon a good "guess" of the bubble shape, which adds some empiricism to the method.This incapacity of the model to deal with separation bubbles has already been observedin Yeung [28] (see the separation bubble upstream of spoiler and spoiler/slotted flap),where results exhibited discrepancies with experiments in the bubble region, even if theywere globally in good agreement with it elsewhere. This suggests that when the bubbleis short enough, results can be overall quite satisfactory.A model using one source in the wake has also been devised for non-reattached flowover an airfoil with a Kriiger flap. Results seem realistic in terms of pressure distributionon the wetted surface but the lack of relevant experimental data does not allow judgementof the accuracy of the model. The main shortcoming of this model is the impossibilityto determine the wake boundary with the mapping sequence used in the present study.Chapter 6. Conclusion^ 856.2 Recommendations for future workThe results obtained in the case of the normal flat plate show that it is possible to imposea constant pressure on an open free streamline, at least over a certain range. This is whythe same kind of model should be tried in a spoiler and slotted flap airfoil configuration,which is what motivated the study in the first place. It was assumed in Yeung [28] thatthe original model suffered from the lack of such a boundary condition on the separatingsreamline issued from the trailing edge of the main airfoil.The study on the Kruger flap was only preliminary and qualitative. Much work stillremains to be done on such a configuration. First of all, experiments should be repeatedon a well defined airfoil, in order to have a set of precise and reliable data. For config-urations having reattached flow on the lower surface, a simple flow model to investigatewould be a source and a sink of equal strength. Correlations for the pressure at sepa-ration should be established, as they would be necessary as an empirical input in themodel, as well as in a two doublet model. In order to improve the results with the latter,a good estimated guess of the bubble boundary is necessary. Thus, flow visualizationstudies should be carried out to provide a general idea of this shape.Finally, some models still have to be devised for lower surface Kruger flaps, requiringa different mapping. The appropriate experiments have to be carried out first to provideexperimental data with which to compare theoretical results and to determine the natureof the flow, which may or may not be reattached.Bibliography[1] Abbot, I.H. and Von Doenhoff, A.E. (1949)"Theory of Wing sections."Dover[2] Arena, A.V. and Mueller, T.J. (1980)"Laminar Separation, Transition, and Turbulent Reattachment Near the LeadingEdge of Airfoils."AIAA Journal, Vol. 18, no. 7, pp 747-753[3] Choi, D.H. and Kang, D.J. (1991)"Calculation of Separation Bubbles Using a Partially Parabolized Navier-Stokes Pro-cedure."AIAA Journal, Vol. 29, no. 8, pp 1266-1272[4] Crimi, P. and Reeves, B.L. (1976)"Analysis of Leading Edge Separation Bubbles on Airfoils."AIAA Journal, Vol. 14, no. 9, pp 1548-1555[5] Fage, A. and Johansen, F.C. (1927)"On the Flow of Air Behind an Inclined Flat Plate of Infinite Span"Proceedings of the Royal Society of London, series A, Vol.116, pp 170-197.[6] Fitzgerald, E.J. and Mueller, T.J. (1990)"Measurements in a Separation Bubble on an Airfoil Using Laser Velocimetry."AIAA Journal, Vol. 28, no. 4, pp 584-592[7] Fullmer,F.F. (1947)"Two-dimensional Wind Tunnel Investigation of the NACA 641-012 Airfoil Equippedwith two Types of Leading Edge Flap."NACA TN 1277[8] Ives, D.C. (1976)"A Modern Look at Conformal Mapping Including Multiply Connected Regions."AIAA Journal, Vol.14, no.8, pp 1006-1011[9] Jandali, T. (1970)"A Potential Flow Theory for Airfoil Spoilers"Ph.D Thesis, The University of British Columbia.86Bibliography^ 87[10] KrUger, W. (1947)"Systematic Wind Tunnel Measurements on a Laminar Wing with Nose Flap."NACA TM 1119[11] Kruger, W. (1947)"Wind Tunnel Investigation on a Changed Mustang Profile with Nose Flap; Forceand Pressure Distribution Measurements."NACA TM 1177[12] Luchini, P. and Manzo, F. (1989)"Flow Around Simply and Multiply Connected Bodies: A new Iterative Scheme forConformal Mapping."AIAA Journal, Vol.27, no.3, pp 345-351[13] McCullough, G.B. and Gault, D.E. (1951)"Examples of Three Representative Types of Airfoil Section Stall at Low Speed."NACA TN 2502[14] Milne-Thomson (1968)"Theoretical Hydrodynamics"McMillan & Co., pp 157-159[15] Newman, B.G. and Tse, M-C. (1992)"Incompressible Flow Past a Flat Plate Aerofoil With Leading Edge SeparationBubble."Aeronautical Journal, Feb. 1992, pp 57-64[16] O'Meara, M.M. and Mueller, T.J. (1987)"Laminar Separation Bubble Characteristics on an Airfoil at Low Reynolds Num-bers."AIAA Journal, Vol.25, no. 8, pp 1033-1041[17] Parkinson, G.V. and Jandali, T. (1970)"A Wake Source Model For Bluff Body Potential Flow"Journal of Fluid Mechanics, Vol. 40, no. 3, pp 577-594.[18] Parkinson, G.V, and Yeung, W. (1987)"A Wake Source Model for Airfoils with Separated Flow"Journal of Fluid Mechanics, Vol. 179, pp 41-57.[19] Roberts, W. B. (1980)"Calculation of Laminar Separation Bubbles and Their Effect on Airfoil Perfor-mance."AIAA Journal, Vol. 18, no. 1, pp 25-31Bibliography^ 88[201 Roshko, A. (1954)"A New Hodograph for Free Streamline Theory."NACA TN 3168[21] Rossow, V.J. (1973)"Conformal Mapping for Potential Flow about Airfoils with Attached Flap."Journal of Aircraft, vol.10, no.1, pp 60-62[22] Tani, I. (1964)"Low Speed Flows Involving Bubble Separation."Progress in Aeronautical Sciences, Vol.5, pp 70-103Pergamon Press[23] Theodorsen, T. (1932)"Theory of Wing Sections of Arbitrary Shape."NACA TR 411[24] Vatsa, V.N. and Carter, J.E. (1984)"Analysis of Airfoil Leading Edge Separation Bubbles."AIAA Journal, Vol. 22, no. 12, pp 1697-1704[25] Woods, L.C. (1955)"Two-Dimensional Flow of a Compressible Fluid Past Given Curved Obstacles WithInfinite Wakes."Proceedings of the Royal Society of London, A 227,367[26] Woods, L.C. (1961)"The Theory of Subsonic Plane Flow."Cambridge University Press[27] Yeung, W.W.H. (1985)"A Mathematical Model for Airfoils with Spoilers or Split Flaps."M.A.Sc Thesis, The University of British Columbia.[28] Yeung, W.W.H (1990)"Modelling Stalled Airfoils."Ph.D Thesis, The University of British Columbia.[29] Yeung, W.W.H and Parkinson, G.V. (1993)"A Wake Singularity Potential Flow Model for Airfoils Experiencing Trailing EdgeStall."Journal of Fluid Mechanics, vol. 251, pp 203-218Appendix ANumerical results for normal flat plateResults are presented for a unit circle (R = 1)Ns=number of specificationsN=number of sourcesQT E q3 : indicates downstream spacing of separation streamlinesNote: Numbers are roundedModel 1: evenly distributed sources1) N' = 1, N = 318k 85°rk 1.51181 2 3Si 64.29° 38.57° 12.86°qj 0.1107 2.1837 4.0220QT = 1.27242) = 2, N = 489Appendix A. Numerical results for normal flat plate^ 901^2ek^85°^80°rk 1.5146 1.80971^2^3^4Si^70.0°^50.0°^30.0°^10.0°qj 3.2732-2 0.4522 3.0582 -2.2952QT = 1.24803) N' -,-- 3, N 51^2^3Ok^85°^80°^750rk 1.5172 1.8169 2.0983^1^2^3^4^5Si^73.64°^57.27°^40.91°^24.55°^8.18°qj —3.7171-3 0.8202 -4.6091 17.703 -12.736QT = 1.17444) N' 4, N = 6k 1 2 3 4Ok 85° 80° 75° 70°rk 1.5211 1.8234 2.1085 2.3965Appendix A. Numerical results for normal flat plate^ 91j 1 2 3 4 5 6Si 76.15° 62.31° 48.46° 34.62° 20.77° 6.92°gi 9.1146-3 -0.2838 6.7615 -35.818 84.025 -53.572QT = 1.12105) N' = 5, N = 71^2^3^4^5Ok^85°^80°^75°^70°^65°rk 1.5149 1.8182 2.1061 2.3983 2.7038^1^2^3^4^5^6^7Si^78.0°^66.0°^54.0°^42.0°^30.0°^18.0°^6.0°—2.5046-2 0.9914 -11.620 80.688 -285.22 495.68 -279.44QT 1.0630Model 2: geometric distribution of sources, g = 0.751)^= 1, N = 31Ok^85°rk 1.50751^2^3Si 54.33° 27.59°^7.52°qi 0.7322 1.3415 -0.7627Appendix A. Numerical results for normal flat plate^ 92QT^1.3112) N' = 2, N = 41^2ek^85°^80°Tk 1.5168 1.81551^2^3^48, 58.89°^35.55°^18.05°^4.92°0.4256 -3.0330 22.777 -18.981QT = 1.18833) N' = 3, N = 51^2^3e,^85°^80°^75°rk 1.5191 1.8206 2.10471^2^3^4^561; 61.60° 40.30°^24.33°^12.35°^3.37°qj 0.1724 5.8321 -74.791 274.58 -204.66QT 1.13214) N' 4, N = 6Appendix A. Numerical results for normal flat plate^ 931 2 3 4Ok 85° 80° 750 70°rk 1.5208 1.8240 2.1108 2.40141 2 3 4 5 6Si 63.35° 43.36° 28.37° 17.13° 8.70° 2.37°0.1297 -1.4104 128.80 -1,277.7 3,900.4 -2,749.1QT = 1.08535) N' 5, N 7k 1 2 3 4 5Ok 85° 80° 75° 70° 65°rk 1.5224 1.8274 2.1167 2.4114 2.7208j 1 2 3 4 5 6 783 64.52° 45.42° 31.09° 20.34° 12.28° 6.23° 1.70°q3 4.2873-2 14.740 -715.83 11,901.6 -76,831.6 198,474 -132,842QT = 1.0245Xcosh 7,/, =2 cos 12sinh = .2sinp(B.1)(B.2)Appendix BInverse Joukowsky transformationThe following describes how to calculate coordinates on the near circle, given coordinateson the airfoil. It is reminded that the airfoil, in the Z plane, is defined with its trailingedge located at Z = +2 and the point half way between the leading edge and its centerof curvature is located at Z = —2.Let:e00-0-fii4Then:1Z = X + iY = Z1 + —zi  = Ob+44^= 2 cos IL cosh 1,/, + 2i sin /2 sinhso that:Since cosh2 — sinh2 t= 1, then:.X2^y24 cos 2 p,^4 sin2 /2^11<#. sin2— (1) 2 —^2 + \/1.^2 —^2 +(B.3) Equation B.3 leads to 4 possible solutions for 12. From B.1, cos /2 is of the same sign as X94Appendix B. Inverse Joukowsky transformation^ 95(since cosh > 0 always), so there are 2 possibilities left for /2. This is solved by notingthat the upper surface (resp. lower surface) of the airfoil is transformed into the upperpart (resp. lower part) of the near circle, which corresponds to a positive imaginary part,so that sin /2 > 0 (resp. sin /2 < 0). The limiting case sin /2 = 0 corresponds to either theleading edge (X < 0) or the trailing edge (X = 2 > 0). The results are summarized asfollows:For the upper surface:if X < 0, /.2 =^ar cs in N/F ( X , Y )if X < 0,^= arcsin 1/F(X,Y)For the lower surface:if X < 0,^= 7r + arcsin F(X,Y)if X < 0,^i = 27r - arcsin N/F(X., Y)where:F(X,Y) = 2-1 [1 - (-2)(-)2 (-102 + I 1 - (-2'1V-)2 - (D2 Y 2Once /2 has been determined, 1,/, is given by B.2:2=logs ^(  ^ ) in /2^2sin/2^2 sin /2This procedure gives a set of coordinates for all corresponding points Z1, given a set ofpoints Z to define the airfoil. In cartesian coordinates, these will be:{ Xi ,----- eti' cos IIXi --=--: etb sin 11Appendix B. Inverse Joukowsky transformation^ 96Then the coordinates for the translated points are easily obtained:I X; = — xio— owhere Z10 = X10 + iY10 is the centroid of the near circle in the Z1 plane.Now the polar coordinates are defined by:r \IX? + Y1.12argso that the parametric representation: Z; = r(cp)ezw can be determined.Z; = — ZioAppendix CDetermination of Theodorsen coefficientsThe following procedure has been devised to take advantage of fast Fourier techniquesand speed up the calculations. However, even with direct evaluations of the equationspresented, it still retains the advantage of avoiding the solution of a linear system ofequations, as in the original Theodorsen method [23]. The basic procedure is summarizedin Ives [8] but the detail of the calculations has been worked out here and is given below.It may be useful to point out some differences with Ives' results: yi = Bo (C.5'a) insteadof yl = B0/2, Ao yi (C.8a) instead of Ao = 2y1, AN = yN+1 (C.8c) instead ofAN = 2Re(yN+1).The fundamental equations:= + Bo E B; cos(j) — A sin(j0),=1log r = Ao^A, cos(j0) + 133 sin(j0)3=1have to be satisfied at 2N points. These 2N points are chosen to be, on the circle:27r(k — 1), j = 1,...,2N194 - 2Ni.e. they are equally spaced on the circle, starting at the trailing edge.Since Ok 0 is imposed at the trailing edge, one must have:B3 — 'PTE^ (C.1)Bo +=1971 2NY3 =^(Cak^Ok)e-1(3-1)95k2N k= 1(C.5)Appendix C. Determination of Theodorsen coefficients^ 98where cpTE is the trailing edge angular coordinate in the 4 plane .Another condition will be imposed:BN 0Thus, the equations to be satisfied for k = 1,..., 2N are:N-1(Pk - Ok = BO +^133 cos(j0k) - A sin(j0k)^(C.2)3-1(the Nth term in the series has disappeared because BN 0 and sin(N0k) = 0) and:(log r)k Ao +^A cos(j0k) 133sin(j0k)^(C.3)3=1In the iterative process that will be explained below, C.2 can be evaluated directly, butin order to use fast Fourier transforms, it can be expressed in the form:2Ny ei(j-1)C6k2Nyj271.1(3-i)(k-1) 2N (C.4).=1^j=1where:Equation C.4 is obtained by taking a discrete Fourier transform of C.2, which leads to thecoefficients yj, and then an inverse discrete Fourier transform of C.5. A simple expressionfor the 2N coefficients y3 as a function of the Al's and B3's can be derived. By reportingthe expression for col, - Ok (given by C.2) into C.5, the following expression is obtained(the Nth term has been kept in the series to have a more general expression):Bo 2N2N^e 3 kk=1^1 N^2N-2N E B1 cos(10k)C"l=1^k=1j = 0,...,2N -2NA1^sin(10k)e-z3Gbkkr-.1(C.6)Appendix C. Determination of Theodorsen coefficients^ 99The terms of the above series are calculated one by one. For these calculations, a pre-liminary result can be derived:Lemma 1 If m is an integer Iml < 2N:E12,N=1 sin(m) = 0cos(mk)^2N if m = 0 or Ind = 2N00^otherwiseproof:Let:2NCOS = E cos(mok)2NSIN =^sin(m0k)k.i2NS = Y e1m0kk=1since e'nul'k = cos(m0k) i sin(m) then:{COS = Re(S)SIN = Irn(S)From the expression for Ok:2N-1S E akare^2NS is the sum of the first 2N terms of a geometric progression of common ratio^andfirst term 1. If m = 0,2N or —2N then the common ratio is 1 and S = 2N otherwise1 — ez274S —^ 01 — ei21ratAppendix C. Determination of Theodorsen coefficients^ 100Therefore:1 COS = 2Nif m 0, 2N or —2NSIN = 0COS = SIN = 0 otherwiseNow C.6 can be calculated.1) if j^N ^1 N 2N^ N 2NBo 1yi ----, 2N 1 + —2N E Bi E cos(i0k) — 2N E Ai E sin(i0k)^k=1^1=1^k=1From lemma 1, Eri cos(l0) = 0 and Eri sin(14) = 0 since 1 = 1, ...N. Thus:yi = B02) for j = 1,...,2N — 1v2NI 2N 1--dk=1 ^= 0^E2N^x-,2N^Ek2N=1 si no ok)-^since k=1 - 2_4=1 COSUCtok)-^= 0 from lemma 1 (.j: 0, 2N or --2N).ii) j f ri B1Eri cos(10k)e-t3c6k1 \--NN p1 k2N=1COS(10k) COSU Ok) -2N L-'1=1 ' 1_, ‘-‘1‘1. 1 B1 ^cos(10k)sin(j0k)The products cos(4k)cos(jOk) and cos(/0k)sin(10k) can be expressed as:cos(/Ok ) cos(JO )^[cos((/ ))01.) + cos((1 — Abk)]cos( /0k) sin(30k) = [sin((/ j)0k) sin((j — 1)4)]From lemma 1, the sum (from k = 1, ..., 2N) of all these terms will cancel except for afinite number of values for 1:Appendix C. Determination of Theodorsen coefficients^ 101if 1 = j (only possible for j^1,^N), B1E2ki. cos(lcbk)cos(j0k)= f1;1- = NB;if / 2N—j (only possible for j N, ...,2N —1), 131E2k-111i ws(10k)cos(jOk) = B2N-jIN B2N-jTherefore:1 2NE cook) cosufkk =k=11B^j =1,...,N— 12 3BN^= N•B2N — j j 1, N — 1 2N 1=1All terms in the sine series cancel.iii) a similar procedure can be applied to the last term leading to:1 2NE sin(10k)e-3kk _=1^k=1i-}A;^,^N — 10^j = N—i12:A2N — j j 1, N — 12NTherefore, the coefficients y are:yi= Bo^ (C.5'a)YJ-1-1^-}(B; +^1, ...,N — 1^(C.5'b)YN+1 = BN = 0 (C.5'c)Y2N-j+1 = .•(B — iA;)^(C.5'd)Equation C.3 can be treated in the same way. Bo is replaced by Ao, 131 by Al and A by—B • . Therefore ifAppendix C. Determination of Theodorsen coefficients^ 102one can write:1 2NE (log r)ke-43-1)4'h2N k=i(C.7)1 y1 =^Aoyi+1 = 12-(A — i ) j = 1, ..., N — 1YN+11--- ANThus:Aj = 2Re(yi+i)^j = 1, N — 1AN = YN+1—21-m(y3+i) j = 1,...,N — 1(C.8a)(C.8b)(C.8c)(C.8d)Method of solution The iterative procedure that will give the solution for the 243's and B3's is describedbelow.1. An initial guess is chosen for A 3, Bj (most likely 0)2. cbk is evaluated either directly using C.2 or, to take advantage of fast Fourier trans-forms, C.4 and C.5'.3. The corresponding set of (log r)k is determined through the known relationshipr r(4) resulting from interpolation.4. A new set of Al, B3 is given by C.7, C.8 and Cl, and the process can be repeatedfrom step 2 until convergence.Appendix DCalculation of limits for derivative of mapping functionThe calculation of the first derivative of the mapping function requires the determinationof the following expression: Z1(i — Z6)21) At infinity, Z6 = i , Z3 = 00Z3(i -- Z6) =Using L'HOpital's rule:— Z6z,limZ3(i — Z6) = Z6) ddZ6 (ifs) 1CO1 dZ3 dZ3^8dZ3.dZ6 00dZ3 dZ3 dZ4 dZ5^4 y^1 )^ —1^--'—dZ6 dZ4 dZ5 dZ6^2 'jr^+ ZsTherefore:and:^27r^1lim Z3(i — Z6) =^Z3 -+CO^ Ay^(1 +^4R-2^Z500 )2liFF1^— Z6)2 = ^^Z3 -.CO /112^1 + Z5c.0 )2) At the flap base, Z3 = 0, Z5 = 00> Z6 = 00103Appendix D. Calculation of limits for derivative of mapping function^1042^2i–Z6-Flogl /Z6)^2^11- I(2: Z5 — x —27r (-26 + As + )yi)Ay^Ay (—Zs + log(1/Z6) +^— 1 — i7r)27rAy—1 + z6log(1/Z6)1[ ' 2ir14+ zb1Zs•^irZslog(1 /Z5 )^log Z5 z, ,00 025 Z5Jim Z3(z. — Z6) = — —27rZb^ Ayliurn Z23 (2 — Z6)2 = 472Z5^ A2Z3 — Z6 = Z3SO:and:Appendix ENumerical results for Kruger flap: 1-source modelTheodorsen coefficientsZio = -0.2148 + i 0.9396-1coefficients:0 1 2 3 4 5.41 0.1624 0.6197-2 0.3264' -0.3008-2 0.1766-2 -0.1028-2B., -0.8260-1 -0.1221-1 0.2500-1- -0.1056 0.1352-2 0.2336-2j 6 7 8 9 10AI -0.2727-2 0.1431-2 0.1379' -0.1363' -0.8112-4B, -0.7981' 0.1277-2 -0.1126-2 0.1343' 0Mapping parametersThe following parameters have been obtained with / -- 0.73, Of( = 159°:Ax = 5.7571, Ay = -2.7027= -25.15°, T2^-21.80°ao = 29.08°V = 1.19496Positions of characteristic points in the final plane:OA =- 150.92° - a (singular point)105Appendix E. Numerical results for Kruger flap: 1-source model^ 106OTE = 355.35° — a = —4.65° —OT = 194.52° — aFlow parametersa Cit 6 q -y TT TTE12.6° 0.8 254.63° —0.52564 0.43718 —1.99364 —0.342319° 0.84 294° —0.24258 0.31725 —1.00342 —0.24133Appendix FNumerical results for Kruger flap: doublet modelsTheodorsen coefficientsZio = -0.9691" +i O.1821coefficients:j 0 1 2 3 4 5A3 0.1176 -0.1660-2 0.6038-2 -0.2659-1 0.1768-1 -0.5617-3B, -0.9391 -0.3380-2 -0.8343-1 0.2337-1 -0.5854' -0.1224"j 6 7 8 9 10 11A, -0.7912-3 0.9759-2 -0.7633-2 0.1601-2 0.6465' -0.2060-2B3 0.8548-2 -0.1205-2 -0.4231-2 0.5023-2 -0.2396-2 0.1444-3j 12 13 14 15 16 17A3 0.4350' 0.4829-3 -0.1274-2 0.5535' 0.1085' -0.4042'B3 0.1102-2 -0.1081-2 -0.4352' 0.3970' -0.3784 0.1562'j 18 19 20 21 22 23A, -0.3585-3 0.2385-3 0.1425-3 -0.8228-3 0.8102-3 -0.8114-3B3 0.3256-4 0.4748-3 -0.6869' 0.5386' -0.4249-4 -0.4170'j 24 25A, 0.1981-3 -0.1371-3B, -0.4170-3 0.2488-3107Appendix F. Numerical results for Kriiger flap: doublet models^108Mappinga + 5° XR/c -- Oi. 02 00 V OTE OT OR13.78° 0.1 2100 237.05° -209.36° 1.1170 -14.32° 202.60° 216.12°12.4° 0.1 210° 237.05° -210.74° 1.1170 -12.94° 203.98° 217.50°8.6° 0.2 210° 246.37° -219.21° 1.1107 -9.32° 210.11° 228.30°6° 0.3 210° 257.33° -227.29° 1.1010 -7.01° 215.45° 239.12°One doublet a + 5° 8 q 'Y13.78° 210.88° -6.4160-3 0.550712.4° 211.99° -7.9906 0.49808.6° 220.42° -2.4640-2 0.35186° 228.67° -5.4338-2 0.2514Two doubletsa + 5° CI), CPR Si qi 62 42 'Y13.78° 0.73 0.86 204.99° 3.3367-4 209.95° -8.1316-3 0.55028.6° 0.25 0.55 215.09° 1.1421-2 218.16° -4.4716-2 0.3490


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