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Modelling stalled airfoils Yeung, William Wai-Hung 1990

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M O D E L L I N G S T A L L E D A I R F O I L S By William Wai-Hung Yeung B.A.Sc, The University of British Columbia, 1983 M.A.Sc, The University of British Columbia, 1985 A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILLMENT OF FOR THE DEGREE OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1990 © William Wai-Hung Yeung, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The thesis deals with some new applications of the wake source model, a two-dimensional incompressible potential flow model used for bodies experiencing flow separation. The body contour is conformally mapped to a circle, for which the flow problem is solved using source singularities to create free streamlines simulating the separating shear lay-ers. In common with other inviscid theories, it generally requires the pressure in the separated flow region, and the location of separation if boundary-layer controlled. Different mapping ^ sequences and flow models have been constructed for the follow-ing five problems, 1. the trailing-edge stall for single element airfoils, 2. flat plates with separation bubbles, 3. separation bubbles upstream of spoilers with downstream wakes, 4. spoiler/slotted flap combinations, at which the spoiler inclination is arbitrary, and 5. two-element airfoils near (trailing-edge) stall. Predictions of pressure distribution are compared with wind tunnel measurements, and good agreement is found in cases 1 and 5. The initial shape of the separation streamlines also appears to be satisfactory. Results in cases 2 and 3 are promising although more work is needed to improve the bubble shapes and their pressure dis-tributions. Partial success has been achieved on spoiler/ slotted flap configurations, depending on the spoiler inclination. For strong wake effect on the flap (e. g. 6 = ii 90° ), the model predicts a very high suction peak over it. Whereas the experimental data resemble a stalled distribution even though flow visualization indicates the flap to be unstalled. This may be related to a limitation of the method, also noted in the separation-bubble problems, that it cannot specify a complete boundary condition on a free streamline. This discrepancy diminishes as the spoiler angle becomes smaller (e. g. 6 = 30° ) in the cases of higher incidences so that the wake boundary tugs away from the flap sooner. iii Table of Contents Abstract ii List of Figures vi Nomenclature be Acknowledgement xiii 1 Introduction 1 2 Review of Literature 4 2.1 Airfoil Stall 4 2.2 Models for Trailing-Edge Stall 5 2.3 Models for Separation Bubbles 7 2.4 Spoilers 7 2.5 Models for Spoilers 7 3 Mathematical Models 11 4 Two Models for Trailing-Edge Stall 17 4.1 Vanishing Spoiler 17 4.2 Joukowsky Arc 28 5 Separation Bubble on a Flat Plate 40 6 Separation Bubble Upstream of Spoiler 48 iv 7 Multi-Element Airfoils 72 7.1 Williams' Exact Method 72 7.1.1 The Streaming Flow 72 7.1.2 The Circulating Flow 75 7.1.3 The Combined Flow 78 7.2 Adaptation to Joukowsky Profiles 81 8 Spoiler/Slotted Flap Configurations 89 8.1 Normal Spoiler 89 8.2 Inclined Spoiler 103 9 Multi-Element Airfoil Near Stall 112 10 Experiments 126 11 Discussion 129 Bibliography 132 Appendix A Curvature 135 Appendix B Pressure Gradient 138 Appendix C Finite Pressure Gradient 141 Appendix D Order of Magnitude Analysis 143 Appendix E Magnification of Angles 147 v List of Figures 2.1 Multi-Element High-Lift Configuration 10 3.1 Physical and Basic Transform Planes 14 3.2 Pressure Distribution on a Flat Plate 15 3.3 Pressure Distribution for an Airfoil with a Split Flap 16 4.1 Mapping Sequence for Vanishing Spoiler 24-25 4.2 Pressure Distributions for Joukowsky Airfoil near Trailing-Edge Stall . . 26 4.3 Separation Streamlines for Vanishing Spoiler 27 4.4 Mapping Sequence for Joukowsky Arc 31-32 4.5 Pressure Distributions for Joukowsky Airfoil near Trailing-Edge Stall (XR = .95) 33 4.6 Separation Streamlines for Joukowsky Arc (XR = .95) 34 4.7 Pressure Distributions for Joukowsky Airfoil near Trailing-Edge Stall {XR = .99) 35 4.8 Separation Streamlines for Joukowsky Arc (XR = .99) 36 4.9 Probable Streamline Patterns at Trailing Edge (XR = 1) 37 4.10 Separation Streamlines for Joukowsky Arc (XR = .75) 38 4.11 Pressure Distributions for Joukowsky Airfoil near Trailing-Edge Stall (XR = .75) 39 5.1 Mapping Sequence for Leading-Edge Separation Bubble 43 5.2 Pressure Distributions for Flat Plate (a = 5.85°) 44 vi 5.3 Comparison of Shapes of Separation Bubble 45 5.4 Comparison of Shapes of Separation Bubble 46 5.5 Pressure Distributions for Flat Plate (a = 5.85°) 47 6.1 Pressure Distributions for Joukowsky Airfoil with Normal Spoiler . . . 54 6.2 Shape of Separation Bubble Upstream of Spoiler . 55 6.3 Comparison of Pressure Distributions for Airfoil near Trailing-Edge Stall and Airfoil with Spoiler 56 6.4 Pressure Distributions for Joukowsky Airfoil 57 6.5 Pressure Distributions for Joukowsky Airfoil 58 6.6 Pressure Distributions for Joukowsky Airfoil 59 6.7 Pressure Distributions for Joukowsky Airfoil 60 6.8 Pressure Distributions for Joukowsky Airfoil 61 6.9 Shape of Separation Streamlines 62 6.10 Shape of Separation Streamlines 63 6.11 Shape of Separation Streamlines 64 6.12 Mapping Sequence for Separation Bubble Upstream of Spoiler . . . 65-66 6.13 Pressure Distributions for Joukowsky Airfoil 67 6.14 Pressure Distributions for Joukowsky Airfoil 68 6.15 Pressure Distributions for Joukowsky Airfoil 69 6.16 Separation Streamlines with Separation Bubble 70 6.17 Separation Streamlines with Separation Bubble 71 7.1 Geometry of Two Circles with Doublet Images 79 7.2 Geometry of Two Circles with Vortex Images 80 7.3 Mapping Sequence for Two-Element Airfoil 85-86 7.4 Pressure Distributions for Two Near-Joukowsky Airfoil Profiles 87 vii 7.5 Comparison of Airfoil Profiles 88 8.1 Mapping Sequence for Two-Element Airfoil with Normal Spoiler . . 95-96 8.2 Comparison of Airfoil Profiles 97 8.3 Geometry of Two Circles with Source Images 98 8.4 Comparison of Pressure Distributions from 1-Source and 2-Source Mod-els 99 8.5 Comparison of Pressure Distributions with and without Spoilers . . . . 100 8.6 Pressure Distributions with Normal Spoiler 101 8.7 Shape of Separation Streamlines (6 = 90°) 102 8.8 Mapping Sequence for Two-Element Airfoil with Inclined Spoiler . .106-108 8.9 Comparison of Airfoil Profiles 109 8.10 Pressure Distributions for Two-Element Airfoil with Inclined Spoiler (5 = 30°) 110 8.11 Shape of Separation Streamlines (6 = 30°) I l l 9.1 Mapping Sequence for Stalled Flap 117-119 9.2 Geometry of Two Circles with Doublet Images 120 9.3 Pressure Distributions for Stalled Flap (a = -6 .9°) 121 9.4 Pressure Distributions for Stalled Flap (a = 3.16°) 122 9.5 Comparison of Airfoil Profiles • 123 9.6 Separation Streamline for Stalled Flap (a = -6 .9°) 124 9.7 Separation Streamline for Stalled Flap (a = 3.16°) 125 10.1 Sketch of the Model in Wind-Tunnel Test Section 128 viii Nomenclature a Radius of circle J in the f-plane an Coefficients of expansion of X, near separation A, B, • • • Coefficients of expansion of curvature near separation A\,A% Mapping coefficients b Radius of circle II in the f-plane bn Coefficients of expansion of Xt near separation Bu B2 Mapping coefficients c Chord; distance between the centers of the circles in the f-plane c n Coefficients of expansion of gt near separation C Cross-sectional area of the wind tunnel Ci , C 2 Mapping coefficients CD Profile drag coefficient CP Pressure coefficient dn Coefficients of expansion of gt near separation dj Locations of singularity images Dn Locations of singularity images Di,D2 Mapping coefficients E Spoiler chordwise location / Flap length h(6) mt)\ Ml) \dZ/ds\ ix /,-($•) Complex potentials of the singularity images F($) Complex potential g(s,t) Streamline in the f-plane —i Complex potentials of the singularity images h'y h, h Spoiler length in various planes Complex potential of a uniform stream h0($) Complex potential of a doublet Jj Strengths of doublet images of J7y(?) Kj Strengths of doublet images of /,•(?) Lj Locations of singularity images Li Scaling coefficient 2 (1 - | ) p Pressure Poo Free stream pressure Pj{$) Complex potentials of vortex images <k(f) Complex potentials of vortex images qu q2 Strengths of sources qo Strength of a doublet R, Ri, R Radii of circles R2 Distance between spoiler tip and center of a circle s Real part of £ S Representative area over which Co is based on t Imaginary part of f; thickness of an airfoil u Horizontal velocity component x U Free stream velocity in the Z-plane Ur, U$ Normal and tangential velocity components V Free stream velocity in the f-plane W Complex velocity X , y Constants denned in the section of spoiler/slotted flap X(s,t),Y(s,t) Streamline variables in the Z-plane Xs Location of flow separation XR Location of flow reattachment XL Length of a segment defined in the section of leading-edge bubble Z Complex variable defining the Z-plane Zi , • • •, Z 8 Intermediate transform planes ZJQ Center of a (near) circle in the Zy-plane a Angle of attack in the Z-plane a, /?, • • • Coefficients of expansion of pressure gradient near separation a 0 , ctyfig, 60 Angles defined in various planes -7 Strength of a vortex at f = 0 10 Angle denned in the mappings of an inclined spoiler 6,6 Inclinations of spoiler in various planes 61,62 Angular locations of sources qi and q2 60 Angular location of a doublet tangent to a circle in the f-plane € Coefficient of the blockage term (—e,n) Center of a circle ( — C e n t e r of a (near) circle (—£2,1*2) Center of a (near) circle xi Complex variable in the f-plane Flap deflection e Angular variable in the f-plane Angle defining the spoiler location Angular location of A in the Z\-plane K Curvature of a streamline Pressure gradient i R2I R\ p Density T Segment of an airfoil surface in the Z-plane <f> Angular Variable defined in various planes T Constant defined in the section of spoiler/slotted flap » ( • ' 0 Imaginary part of (• • •) Real part of (• • •) Acknowledgement I would like to thank Professor G.V. Parkinson for providing the author the opportunity to carry out this research. His enthusiastic and invaluable guidance, and continuous interest in this area during the course of the work are much appreciated. The author would also like to thank Professor I.S. Gartshore for his insightful sug-gestions on the experiments. Special thanks to Mr. E . Abell and other technicians who built the experimental equipment used. This work was partly supported by the Natural Sciences and Engineering Research Council of Canada. xiii Chapter 1 Introduction The computation of low speed steady flow around airfoils experiencing partial sepa-ration at high angles of attack is important for many applications in fluid mechanics such as the design of wings, helicopter and propeller blades, wind turbines, cascades and many other engineering configurations. References [1,2,3,4] are just a few of many publications of the work in this area carried out recently in aircraft companies as well as research institutes. In addition, there is continuing interest in spoilers [6,13,14,19] , which are used widely on aircraft as aerodynamic control devices, as effective air brakes, as lift dumpers during landing, and recently in transonic flow. In both types of flow separation the body is exposed to a broad wake of nearly constant time-averaged pres-sure bounded by shear layers emanating from either the point of surface flow separation or the tip of the spoiler, and the trailing edge of the airfoil. An accurate prediction of the surface pressure and the shape of these free shear layers usually requires an outer potential flow model for the wetted surface of the airfoil including the boundary of the wake, and an inner boundary-layer computation to locate the position of separation. The two calculations are iteratively adjusted with respect to each other until the shape of the airfoil plus the wake does not alter considerably or some other convergence criterion is met. This type of iterative procedure is usually referred to as a v'scid-inviscid interaction scheme. A number of these schemes reported in the literature show satisfactory prediction; e. g., Henderson [1] and Maskew and Dvorak [3] . They have become standard tools for practising aerodynamicists due to 1 Chapter 1. Introduction 2 the development of high speed computers. Obviously, the rate of convergence of these schemes is sensitive to how the afore-mentioned inviscid flow model including the wake boundary is simulated. However, ^nost of these models either require information from detailed and complex experimen-tation or many empirical parameters and assumptions. For example, in Bhateley and Bradley's model [2] , the shape of the separated wake was measured experimentally, while both Henderson's and Maskew and Dvorak's models require the assumption of a sophisticated wake geometry. One objective of this thesis is to construct simple, analytical and better models which take as little empiricism as possible and yet provide a reliable initial estimate of the real flow without going through lengthy and tedious computations. By incor-porating these models into the viscid- inviscid schemes, the number of iterations may be reduced. Moreover, studying these models provides us a chance to re-examine some physical boundary conditions which may have been overlooked in the past but are im-portant in model construction in general. Another objective is to use these models to provide exact comparison with and assess the accuracy of numerical methods. There are five general configurations considered, some with partial success. They are listed as follows, 1. trailing-edge stall for single element airfoils, 2. a separation bubble at the leading edge of a fiat plate, 3. a separation bubble upstream of a spoiler with a downstream wake, 4. spoiler/slotted flap combinations, and 5. trailing-edge stall for two-element airfoils. Chapter 1. Introduction 3 The types of model used throughout the thesis were extensions of the wake source model developed originally by Parkinson and Jandali [5] for symmetrical bodies, and the one later modified by Parkinson and Yeung [6] for lifting bodies. This model offers the advantage that it is simpler than other models such as those proposed by Woods [7] but as powerful. Predictions from this model are always validated by comparing with experimental data. In case 1, new experiments were carried out in our low speed, low turbulence wind tunnel (see Chapter 10). In the other cases, experimental data for comparison were obtained either from previous investigations in this laboratory [13,15,23] or from the literature [8,21,26]. Chapter 2 Review of Literature 2.1 Airfoil Stall Depending on the Reynolds number and sectional profile, significant differences may occur in the behavior of an airfoil at stall, the condition which follows the first lift-curve peak. Three basic types of stall were classified by McCullough and Gault [8] , supported by detailed experiments. • Trailing-edge stall is usually associated with blunt leading-edge sections of thick-ness approximately 15% chord and greater. It is preceded by the movement of the turbulent separation point upstream of the trailing edge with increasing angle of attack. The stall angle for a NACA 633-018 airfoil is about 14° . The following two types of stall are associated with laminar boundary-Jayer separa-tion, caused by severe adverse pressure gradient at the airfoil leading edge. The laminar shear layer after separation is so unstable that transition follows immediately. The sub-sequent turbulent shear layer then increases the entrainment from the surrounding fluid and may cause a reattachment. This results in the formation of a recirculating zone, usually alluded to as a laminar separation bubble. A separation bubble comprises four regions: laminar flow upstream of separation, turbulent flow downstream of reattach-ment, viscous recirculating flow within the bubble, and the inviscid external flow. • Leading-edge stall takes place on moderately thick airfoils (.09 <t/c < .15). Its main characteristic is an abrupt flow separation near the leading edge generally 4 Chapter 2. Review of Literature 5 without subsequent reattachment. For a NACA 63x-012 section, stall angle is about 13° . • Thin-airfoil stall occurs on sharp-nosed sections of thickness about 9% chord or less. The bubble formed at the leading edge is a long one, occupying 2 to Z% of the chord at small incidence. This type of stall is preceded by flow separation at the leading edge with reattachment at a point which moves progressively rearward with increasing angle of attack. The maximum lift occurs at much higher angle of attack than the initial stall angle, which is usually at 4° - 5° . The above classifications were based on a test Reynolds number of 5.8(10)6. As emphasized by the authors, the stalling characteristics are closely allied to a change of Reynolds number and other parameters which may influence the growth of the bound-ary layer, and eventually cause the stall of an airfoil to switch from one classification to another. 2.2 Models for Trailing-Edge Stall Schmieden [9] proposed an analytical method to model flow past a flat plate at stall, requiring tangential separation from the suction side as well as from the trailing edge to form an infinite wake. This method allows a continuous variation from the Kirchhoff model when the separation takes place at the leading edge to the Kutta-Joukowsky model when it coalesces to that from the trailing edge. Another model of similar type was formulated by Ockendon and Tayler [10] for a cambered wing of zero thickness in terms of some singular integrals. It differs from Schmieden's method by permitting the separating free streamline to have an arbitrary slope at separation. These models are, however, unrealistic in that an infinite velocity results at the leading edge of the plate, and the pressure on the free streamlines is equal to the upstream undisturbed value. Chapter 2. Review of Literature 6 Numerical modelling of trailing-edge stall was pioneered by Jacob [4], and was later extended to multi-element airfoil systems, wings, and wings with ground effect. The method combines potential-flow and boundary-layer calculations, a type of viscid-"inviscid iteration scheme. A surface singularity method using vortex distribution gives the potential-flow model around the airfoil. The separated region is simulated by a linear distribution of sources on the rear part of the upper surface of the airfoil, and a sink in the flow field downstream to form a closed dead air region. The point of separation is found by iteration postulating that the foremost point of the wake has to coincide with the point of separation. An attempt to empirically define the separated-wake shape in Bhateley and Bradley's model [2] was proved to give unsatisfactory results. That led to a free-streamline ap-proach, which is mathematically identical to the analysis of an airfoil with a finite-thickness trailing edge. In contrast to Jacob's model, this model produces an open wake created by a source embedded inside the airfoil. The model was also extended to multi-element systems. In Henderson's model [1] a surface singularity method is used on the airfoil upstream of separation as well as on the boundary of the wake, divided into the separated and the trailing components, which are determined by different boundary conditions. A common feature of the above models is that at separation the flow is required to leave the surface tangentially to avoid a cornerflow, which will create a stagnation point. A detailed initial wake geometry involving its length, height, shape of the bound-aries, and other parameters is described in Maskew and Dvorak's model [3] to improve the rate of convergence in their scheme. The simulation of a non-tangential separating streamline without evoking a stagnation point at separation is an interesting feature. No explanation is given, however, as to how this can be achieved. Chapter 2. Review of Literature 7 2.3 Models for Separation Bubbles Some recent theoretical analyses for leading-edge separation bubbles on airfoils can be found in Crimi and Reeves [11] . Again, the use of viscid-inviscid interaction schemes is the general procedure. The inviscid flow over the airfoil without separation is given by conventional surface singularity methods. Because the bubble is generally very thin, a perturbation velocity due to the displacement effect of the viscous flow can be formulated. 2.4 Spoilers A spoiler is a bluff projection which reduces the lift of an airfoil by causing flow sep-aration. Figure 2.1 is representative of the Boeing 727 wing section from Bertin and Smith [12] showing it equipped with a spoiler and other devices such as a leading-edge slat, Krueger leading-edge flap, and triple-slotted flaps. The flow around an erected spoiler includes boundary- layer separation induced by the adverse pressure gradient generated by the presence of the spoiler. Subsequent reattachment on the spoiler results in the formation of a closed bubble of circulating air of constant pressure. The flow separates again at the tip of the spoiler creating a broad wake of constant pressure but at a different value. A flow-visualization picture of the bubble taken in a smoke tunnel at a low Reynolds number can be found in Yeung [13] . 2.5 Models for Spoilers A recent paper by Tou and Hancock [14] gave a thorough review of the theoretical work on the two-dimensional airfoil-spoiler problem. Woods' authoritative linearized theory [7] and its subsequent modification by Barnes (see [15] ), might be expected to Chapter 2. Review of Literature 8 predict reasonable overall lift and moment but not necessarily realistic surface pressure distributions. Moreover, the extension of Woods' theory to multi-element systems is not obvious. - Jandali's contribution [15] was on a normal spoiler on an arbitrary airfoil by using a wake-singularity model modified from that developed earlier by Parkinson and Jandali [5] . The unfortunate restriction on the spoiler inclination ( 6 = 90° ) arises from the dependence of the method on the conformal mapping sequence. Brown and Parkinson [16] developed yet another linearized theory for both steady and unsteady spoilers by considering the wake as a cavity of uniform pressure. Brown [17] also incorporated the wake-singularity model into Smith's surface singularity method for two-element airfoil-spoiler configurations. The results from Parkinson's group have shown good agreement with experimental data. The numerical work reported by the Boeing research group associated with Hen-derson [l] was briefly discussed. Pfeiffer and Zumwalt's numerical model [18] using a turbulent jet mixing analysis to simulate the wake was considered an important ad-vance. Tou and Hancock [14] proposed another surface singularity method to predict two-dimensional characteristics at low speeds. The airfoil, the spoiler and the separation streamlines are represented by piecewise continuous elements of vorticity. The sepa-rated region, which is closed by two counter-rotating discrete vortices located within at a finite distance downstream, is assumed to have a different uniform total head from the outer potential flow region. By using this model and its extensions, they provided a fairly complete theoretical study of the characteristics of oscillatory airfoil/ spoiler, airfoil-spoiler/flap and the rapidly deployed spoiler, among other topics. Some of their results, however, are to be confirmed with experiments. Chapter 2. Review of Literature 9 The limitation on the spoiler angle in [15] was removed by introducing a new map-ping sequence described in Parkinson and Yeung [6] , in which additional physical boundary conditions were also exploited for the application of the model to airfoils fitted with lower-surface split flaps. Bearman et ol [19] made use of the mapping sequence and the standard (point) vortex method, simulating the unsteady separated flow, to study particularly the phe-nomenon of short duration adverse lift induced by rapid spoiler deployment. Compar-isons with the findings from [14] are excellent. Figure 2.1 Multi-Element High-Lift Configuration [12] o Chapter 3 Mathematical Models In the original analytical wake-source model by Parkinson and Jandali [5] and its subse-quent extensions by Parkinson and Yeung [6], the portion of the body surface exposed to the separated wake is assumed at constant pressure, and the shear layers bounding the wake are simulated by free streamlines. With these assumptions, solutions are achieved to the boundary-value problems resulting from the use of conformal mapping and the powerful properties of singularities in creating flow models. The contour to be mapped is the wetted surface plus a contour in the wake providing a slit or a cusp at each separation point, Figure 3.1. The part of the original contour exposed to the wake is ignored unless it conforms to the above requirement. The resulting contour is then mapped to a circle by a set of transformations for which the overall derivative of the mapping function has simple zeros at the flow separation points. In the transform plane the flow model consists of a uniform flow plus a doublet for the basic circle, two sources on the wake portion of the contour and their image sink at the centre, and a vortex at the centre for the circulation in lifting configurations. The source and vortex strengths and the source angular positions are five unknowns and four of these are determined by conditions at separation. Two conditions are that the separation points in the physical plane become flow stagnation points in the transform plane, thus ensuring tangential separation of the physical streamlines, since angles are doubled there. The other two conditions come from specification of the separation velocity given by the base pressure on the body, empirically determined as 11 Chapter 3. Mathematical Models 12 in all such flow models. For bluff sections with a continuously curved contour so that flow separation is boundary-layer controlled the position of the separation points is also usually specified empirically. The fifth condition is related to an additional condition at separation to be discussed later. It may be noted that the model does not represent the solution of a complete boundary value problem, since conditions along the free streamline boundaries are not specified except at the separation points. As a result, the use of sources is not exclusive, and other types of singularity such as doublets, source-sink combinations, vortices or their combinations, represent possible choices, depending on the flow situation. In fact, for spoiler/slotted flap configurations, a more realistic model requires Borne type of singularity distribution that creates separation streamlines of constant pressure. Even though separation streamlines detach from the body tangentially in the phys-ical plane, there the streamline curvature and streamwise pressure gradient, which are linked, can be positive infinite or negative infinite, with an intermediate special case of finite values for both. The latter is the classical Brillouin-Villat condition, see Wu [20] . When this condition is satisfied, the streamline curvature is automatically equal to that of the body surface at separation, so for this reason it is also called "smooth separation" in the literature. Negative infinite curvature (or pressure gradient) is possible only for separation at sharp edges, as clearly shown in the pressure distribution on a flat plate, Figure 3.2 from [20]. The corresponding streamlines are convex as viewed from outside the wake. If the curvature is positive infinite, so is the streamline pressure gradient. This condition was tried successfully in [5] for flow around a circular cylinder. Its additional implication is that the streamlines would not cut into the body downstream of separation. The finite alternative is of particular interest since it appears to be the most natural possibility in separation from a continuous surface. Indeed, this is well substantiated in Chapter 3. Mathematical Models 13 the pressure distribution over the upper surface near the trailing edge of an airfoil with a split flap, Figure 3.3 from [12], and it has been studied recently as an option in airfoil design [27] to obtain improvement both in the accuracy of potential- flow simulation and ultimately in airfoil performance. Moreover, this criterion can be used as an extra boundary condition in the model to eliminate the specification of the position of separation if it is boundary-layer controlled, as in the flow around a circular cylinder, or to determine the associated circulation, as in the case of split flaps. The degree of success which is indicated in [13] encourages the exploration of further applications to the aerodynamics of airfoils. Appendices A and B are devoted to the derivations of the curvature and pressure gradient conditions suitable for models of this type. o FAGE 8 JOHANSEN (1927) > -1.0 • Figure 3.2 Pressure Distribution on a Flat Plate Chapter 4 Two Models for Trailing-Edge Stall For some airfoil profiles, as discussed in section 2.1, stall develops by flow separation starting at the trailing edge and moving forward with increasing angle of attack. The Joukowsky profile used in the previous study by Yeung [13] (camber = 2.4%, thickness = 11%) has this characteristic and is used here as the prototype airfoil for a wake source model of the partially developed stall. The objective, given the separation point and the surface wake pressure coefficient CP4, assumed constant, is to predict the resulting airfoil pressure distribution and the shape of the separating streamlines. The model could then serve as a useful partner with a boundary-layer method in an iterative solution of the stall problem. In fact, two different wake singularity models have been developed to simulate the partially stalled flow, using two different mapping sequences and two different flow models. 4.1 Vanishing Spoiler This model is essentially a modification of the spoiler model by Parkinson and Yeung [6]. Here the spoiler is made very small and its angle 6 is set at zero so that the spoiler tip represents the point of tangential separation of the partially stalled flow. The fol-lowing sequence of transformations takes a cambered Joukowsky airfoil fitted with a spoiler in the Z-plane into a unit circle in the f-plane, 17 Chapter 4. Two Models for Trailing-Edge Stall 18 Z = A1{Z1 + ^-) + B1 Z2 = {Zx - Z10 - Reito)e-*to-V z - - 2 R {Zi + lnZ4) 2R 1 Z s = n ( T - P - * Z4 — Xx Xy where £ = 7 Zio = (-c ,M) R = \l + Z10\ 1 ~LX Ax = = r - Bi = Li = 1 + 2c + 2 + Lx 2 + Li 1 + 2e and />, a as defined in Figure 4.1, which shows the sequence of the boundaries generated. The point at infinity (Z = oo and subsequently Z\ — oo,Z2 = oo and Z$ = 0) is represented by Z4oo { = Xx + iXv). Using the relation between Z$ and Z4 and separating the real and imaginary parts, two equations are obtained from which Xx and Xv can be determined iteratively, and Note that as h— •- 0, i | A . + iln(Aj + Aj)] = ^ + i i |A„ + tan-'(^)] = l -7T A x 2ti\R Chapter 4. Two Models for Trailing-Edge Stall and Ay —• 7T. The overall derivative is dZ dZ dZ\ dZy, dZ% dZ\ dZ$ dZ\ dZi dZz dZ+ dZ% d$ _ A^yZJ Zf~l Z4 + l \2 i(0o + «O) 4nR { Z\ n Zi ) [ % £ h ) Because Ue~,ct —• at infinity, after some manipulations, one gets and where and U Xv\l + Z4oo\ ct0 = a — 7r — <f>i + <j>2 — 0O h = tan"^^) At the trailing edge (point E) of the airfoil, Z\ = 1, where ^ = 2 i E e - ^ ) s i n ( ^ ± ^ ) 4>0 = t a n - ^ Y ^ ) _ c o t { ^ ) = I ( s < + l n Z ( ) _ ( « ? _ ! , Chapter 4. Two Models for Trailing-Edge Stall 20 At the tip (point C) of the spoiler, Z 4 = — 1 , - \ A l / Z ? ~ 1 \ Z * r A v( > : ~^s) 2 l 2 | 'dc ld<r;i ~ 1 IT 1 Z* J2J2 l 2* J 1 where Z 2 = /i Zl = Z10 + Reit' + he'lt'-V Z , = - ( — ^ ) . Ay Arc CDi£ in the f-plane represents the only appropriate region where surface singu-larities can be put in order to create stagnation points at C and E. It is, however, important to realize that the spoiler is needed even though its actual size is only a small fraction of the airfoil chord in the calculation. If h = 0, the intermediate mapping sequence degenerates to f = ( i f - l)e— to give — - A.(Z* ~ 1)Rci(9o+a°) ds ~ l [ z\ ) K e Chapter 4. Two Models for Trailing-Edge Stall 21 The above process shows that the tip of the spoiler becomes an ordinary point on the airfoil surface at which ^~ is some finite value other than zero. In other words, it is no longer a critical point to provide tangential separation but merely a stagnation point in the flow model. The flow model in the f-plane is the 2-source version [6] whose complex velocity is where f = eie , ft = e'\ and ft = e'*'a. 1 is the strength of the vortex located at f = 0. qi and <ft represent the strengths of the two sources which are situated on arc CDE of the circle at angular positions 6i and 62 respectively. The pressure coefficient Cp is defined as where p and Poo are the local and upstream undisturbed pressures and \pU2 is the dynamic head. Applying Bernoulli's equation along a streamline, p \W(Z)\> poo V* p + 2 " p + 2 ' The pressure coefficient can then be written as The boundary conditions used to solve for qu <?2> 7> #i and 02 are 1. W(f) = 0 at f = ei9° and f = e*'B , where 0c, and 0^  are angular locations of C and E defined in Figure 4.1, 2. Cp ~ Cpb at these two locations, where Cv\, is the corresponding pressure in the wake, and 3. finite Cp gradient at separation; that is, Chapter 4. Two Models for Trailing-Edge Stall 22 /sr. - nfx=o at f = ei9° where fx = and /j = | ^ | , and the derivative is taken with respect to 0. The equation of finite pressure gradient is derived in Appendix C. Because both W($) and ^ vanish at the critical points, their pressure coefficients have to be evaluated by using l'Hopital's rule, 'p\C,E = 1 ins) „ u d{\ d( ) After some calculations, it was found that the 1-source version [6] generally provided unrealistic results by creating an extra stagnation point either upstream of the spoiler or the trailing edge on the lower surface, depending on whether the singularity was a source or sink. This version was not pursued further because in general Cp at the spoiler tip cannot be matched to that at the trailing edge. In the 2-source version, a simple analysis of the order of magnitude of terms reveals that if Cp at the tip of the spoiler is finite, then ( ^ ) ~ o(h>) and qx ~ 0[h?). The details are given in Appendix D. Physically, it means that a source of strength 0(h2) is located very near the tip of the spoiler to allow tangential separation. In Appendix D, it is also shown that the criterion of finite pressure gradient at separation is compatible with the above. By choosing the length of the spoiler equal to .25 % chord, the predicted Cp distribution provides a reasonable correlation with the experimental measurements on the airfoil at a = 14° , as shown in Figure 4.2, where separation is Chapter 4. Two Models for Trailing-Edge Stall 23 assumed at 40 % c and Cp = —.5 in the wake. Calculations show that the predicted Cp is insensitive to even smaller but finite values of the spoiler length. However, the effect of shrinking the spoiler is reflected through the strength of the source located near the spoiler tip, usually of several orders of magnitude smaller than that of another one near the trailing edge. Hence, the streamline emanating from the tip of the spoiler lies unrealistically close to the airfoil surface until it reaches the vicinity of the other source, as depicted in Figure 4.3. By increasing the length of the spoiler from .25 % to 2.5 %, this streamline tugs away sooner but the resulting pressure distribution deviates more from the experimental data. Figure 4.1 Mapping Sequence for Vanishing Spoiler to Figure 4.2 Pressure Distributions for Joukowsky Airfoil near Trailing-Edge Stall > 0.4 . • Upper Surface Separation Streamline 0.2 . Trailing-edge Separation Streamline 0 . - 0 . 2 _ - 0 . 4 1 1 1 1 1 1 0 0.2 0.4 0.6 0.8 1 Figure 4.3 Separation Streamlines for Vanishing Spoiler s •8 CO (D ED-S' ts CD CO «-& to Chapter 4. Two Models for Trailing-Edge Stall 28 4.2 Joukowsky Arc This model was devised to exploit a simpler mapping sequence, as given in Figure 4.4. It avoids the artificiality of a small zero-inclination spoiler simulating the separation region, and gives a better initial shape of the separating streamline. Using the fact that the portion of the body exposed to the wake is ignored in the flow model, part of it is truncated to create tangential separation at point A. The contour in the Z\-plane is a circular-arc slit, centered in the second quadrant. A standard Joukowsky transformation maps it into a Joukowsky airfoil profile slit omitting the upper surface between points A and B. A is the upper-surface separation point in the stall problem and B is the end point of a trailing-edge portion of the upper surface lying in the wake region of the flow. A translation and rotation to the ^-plane then re-creates the circular-arc slit of the original wake source model for the circular cylinder flow, and a second Joukowsky transformation maps the slit into a circle. Finally, this is rotated to the f-plane (so that the approach velocity is in the direction of the real axis), where the flow problem is solved. The transformation equations are Z = Al{Zl + ±-) + Bl Zi = CXZ2 + £>: Zi = Z* - cot a - ———— Zz — cot a where Ax = — ^ - S i = L l ~ L i = 1 + 2c + 2 + Lt 2 + L x l + 2c Ci = — c n » ;sin( ) Di = ReiK * ;cos( ) + Z10 2 2 it Q <t>A~ <f>B * ~ P* h „ ~ p, = IT a = —-— R = esc a Chapter 4. Two Models for Trailing-Edge Stall 29 R = \l + Z10\ Z 1 0 = (-«,/*). The overall derivative is dZ dZ dZ\ dZi dZ$ ~* d$ dZ\ dZ2 dZ% d$ By considering the flow far away from the airfoil, it is found that ^ a T>\r> I ,$A + 4>B\ - = A1R\Cl\ ct0 = a-[ ) where and <j>s are defined in Figure 4.4. At the trailing edge (point C) of the airfoil, Z\ — \, | - , £ , H M l f f | 0 l n + ^ _ ^ . where 1-DX = Zs — cot a — C\ Z$ — cot a At the separation point (A), Z$ = cot a + i, i | ( f ) i = 2 ^ i C l ( ^ ) i ! where Z x = Z 1 0 + iZe'*-* , and Although the flow near and the pressure distribution along the arc BC predicted by the model are not of interest since the arc lies in the wake region, it was necessary to create the arc in the mapping to separate points B and C, which are critical points of different mappings in the sequence. The length XR in Figure 4.4 is an adjustable parameter chosen to give the best compromise between good agreement with measured Chapter 4. Two Models for Trailing-Edge Stall 30 airfoil pressure distribution and satisfactory separating streamline shape. The five boundary conditions are the same as in the first model, but the source closest to the upper surface separation point is replaced by a doublet tangent to the circle, which was found to be effective. The complex velocity in the f-plane is ^) = v{(1-I) + .-l +I ^ p + , l l F i r - i ] > where qr> and 6D are respectively the strength and angular location (on |f | = l) of the doublet. Figure 4.5 depicts the calculated pressure distribution compared with experimental measurement, using XR — 0.95. The agreement is as good as that from the first model. The separating streamline does indeed pull away much earlier, as shown in Figure 4.6. The two symbols represent the locations of the doublet and the source on the slit. Varying the value of XR has the effect of changing the overall lift and the shape of the streamlines. For XR = .99, the corresponding Cp distribution and the streamlines are still acceptable, as shown in Figures 4.7 and 4.8. Certainly, for extreme values of XR, the model may not be valid. For instance, if XR = 1, the flow around the trailing edge is complicated by the fact that dZ „ d ,dZs „ d . d ,dZss _ * - ° « < * > - ° 3 ? ( 5 ( * M - ° preventing the streamline from leaving the trailing edge smoothly. See Appendix E for detailed discussions and Figure 4.9 for the probable streamline shapes. For smaller values, e.g. XR = .75, the free streamline from A reattaches on the segment BC and pulls away again when approaching the trailing edge (see Figure 4.10), although the Cp distribution is still close to measurements, as shown in Figure 4.11. It is therefore recommended to choose .85 < XR < 1. Chapter 4. Two Models for Trailing-Edge Stall 34 Chapter 4. Two Models for Trailing-Edge Stall 36 Chapter 5 Separation Bubble on a Flat Plate An inviscid flow model for a body with a separation bubble presents a difficult problem because of the requirements of tangential separation and tangential reattachment with a finite pressure gradient. Therefore the problem has been studied in its simplest form, that of the leading-edge separation bubble which forms on a flat plate at small values of a. For this problem the mappings, shown in Figure 5.1, represent a special case of the mappings of Figure 4.1, in which the circular-arc slit is centered at the origin in the Zx-plane. The critical points of the Joukowsky transformation are at B and C, so that the transformation produces a flat plate in the Z-plane with the portion DA of its upper surface omitted. In this case it is the portion CD of the upper surface, lying inside the separation bubble, that is needed to keep critical points C and D of different mappings apart. The length of CD, Xi, is a free parameter here. The transformation equations are the same as those given for the Joukowsky Arc except Zio = (0,0) (3, = 7r a0 = a — -R i{±n+±i.) . ,<f>D-<f>A\ n j, ii*J2+±A.) ,<j>D — <t>AS Ci = — en ^ >sin( ) Di = Re,( > 'cos( ) where <j>u and 4>A a r e defined in Figure 5.1. At the leading edge (point C) of the plate, Z\ = —1, | l ( f , | = 2 A l { f i | C l [ 1 + ^ - i ^ ] | } ' 40 Chapter 5. Separation Bubble on a Flat Plate 41 where - 1 - D „ „ 1 = Z% — cot a . C 3 Z 3 - c o t a At the trailing edge (point B) of the plate, Zx = 1, | l ( f ) i = 2 M * | C l l i + p - i ^ ] i > ' where 1-D = Zs — cot a — C Zs — cot OL At the reattachment point (A), Z% = cot a — t, where Zx = e**A , and | - ( | ) | - ^ | 0 l ( ^ l ) | Different flow models may be devised to meet the boundary conditions of the prob-lem. The three basic conditions are tangential separation from the lower surface leading edge at C and the trailing edge at B, and tangential reattachment on the upper surface at A. In addition, for a satisfactory prediction of the pressure distribution over the wet-ted surface of the plate at least the location of A must be specified from experiments. The simplest flow model in the f-plane employs a doublet tangent to the part of the circle inside the bubble plus a vortex at the centre. The strength and location of the doublet and the vortex strength create three unknowns which are solved for using the three basic boundary conditions. The value of XL is adjusted to satisfy the criterion of finite pressure gradient at reattachment. Figure 5.2 shows the model applied to a flat plate at a = 5.85° .The calculated Cp distribution is compared with experimental data by Fage and Johansen [21]. Their Chapter 5. Separation Bubble on a Flat Plate 42 reattachment point is not measured but based on flow visualization results from similar experiments by McCullough and Gault [8] it is assumed to be at X = 60% chord. The theoretical Cp distribution over the wetted surface is seen to be in good agreement with the experimental data. It is interesting to note that even though the pressure at reattachment is not specified, the predicted value is not far from the measurement. The corresponding streamline Cp distribution is, however, unlike the experimental dis-tribution on the plate surface under the bubble. In this region, similar to a wake, the viscous effects dominate so that inviscid modelling is not suitable. The predicted shape of the bubble is compared with the measurements from [8], as shown in Figure 5.3. The shape of the bubble can be improved and made of about the right thickness ratio if the doublet-flow model is replaced by a source and a sink of equal strength on the arc AD, The three basic boundary conditions are the same but the Cv value is specified at reattachment and Xi = .56, Figure 5.4. Unfortunately, the Cp distribution, Figure 5.5, is not as good as shown in Figure 5.2, especially with an infinite pressure gradient at reattachment, and an undesirably high suction peak at the leading edge. -4 . - 3 . CP -2 J O Experiment [21] Theory, XR = .6 + 1 ; 0 o o o \ o -e e ©-0.2 0.4 0.6 0.8 X/C Figure 5.2 Pressure Distributions for Flat Plate (a = 5.85°) 0.4 . 0.2 . O Experiment [21] Theory, XR = .6 Y/C 0 _ 6-6 o 6 ob-o-o-o oo-oo a o . g . o o o O O n n « 0 - 0 . 2 . - 0 . 4 1 1 1 1 1 1 0 0.2 0.4 0.6 0.8 1 x / c Figure 5.3 Comparison of Shapes of Separation Bubble 8 •8 (6 05 •8 5 r to c cr cr § CO 3 « ox 0.4 0.2 Y/C A Experiment [21] Theory, XR = .6, XL = .56 s •8 Co •8 s to C cr cr (T S 2 - 0 . 2 - 0 .4 0 0.2 I I 0.4 0.6 X/C 0.8 Figure 5.4 Comparison of Shapes of Separation Bubble Cp -4 -3 . -2 -1 • A Experiment [21] Theory, XR = .6, XL = .56 A A A , A A 8P •8 s § to c c r c r § Co CD Co «-CD 0 . f l 1 r~ 0.2 0.4 0.6 x/c 0.8 Figure 5.5 Pressure Distributions for Flat Plate (a = 5.85°) Chapter 6 Separation Bubble Upstream of Spoiler A two-dimensional inviscid flow theory was proposed by Jandali [15] for upper surface normal spoilers fitted on Joukowsky airfoils. Figure 6.1 from [15] is a typical comparsion between theoretical and experimental Cp distributions, showing excellent correlations. An inevitable discrepancy is evident in the region just upstream of the spoiler. A stagnation point is predicted at the base of the spoiler so that Cp = 1. Actually, the adverse pressure gradient upstream of the spoiler causes boundary layer separation from the airfoil, and the flow reattaches on the spoiler to form a closed region of recirculating flow. The constant pressure separation bubble can be clearly identified in Figure 6.1. An attempt to give an estimate of the shape and Cp distribution of the bubble is made by some inviscid flow models. Flow visualization suggests that the most reason-able shape would be a concave curve joining the separation point A on the airfoil and the reattachment point H on the spoiler, as sketched in Figure 6.2. At H the curve should be tangent to the spoiler so that the flow does not stagnate there. Certainly, the flow will separate again after it reaches the tip E of the spoiler. Work along this line has been tried but unfortunately, it is as yet unsuccessful. However, in comparing the Cp distributions from the bubble-spoiler combination with those from the trailing-edge stall, it is noted that they are quite similar except the former has a pressure discontinuity upstream and downstream of the spoiler whereas the latter has a smooth pressure gradient at separation, see Figure 6.3. In other words, the physical presence of the spoiler may not be too important to the Cp distribution 48 Chapter 6. Separation Bubble Upstream of Spoiler 49 as long as such a jump can somehow be created. Indeed, the Joukowsky Arc model, if adapted appropriately, has been found to be quite suitable. Modifications of it include the following : 1. demanding the location of the separation point (Xs) to coincide with the foremost point of the bubble, 2. equating the Cp value at separation to that within the bubble, Cp [bubble], and 3. equating the Cp value at the trailing edge to that in the wake, Cp[wake]. It is assumed that the pressure is constant within the bubble, and Xs, Cp[bubble], and Cp[wake] are specified from experiments. Figure 6.4 is obtained from the above modified boundary conditions with empirical data (Xs = .585, Cp [bubble] = —.05, and Cp[wake] = —.50) and is compared with Jandali's measurements for a normal spoiler (6 = 90°) of height (h/c) equal to 5% located at (E/c) 70% at a = 11°. XR is kept to .95 and the flow model is the usual doublet-source combination. In fact, this model is not restricted to a normal spoiler since its effects are reflected through the empirical data. A test case corresponding to the same geometric configuration but with 6 — 45° and a = 12° from Yeung [13] is presented in Figure 6.5, where Xs = .535, Cp[bubble] = —.158, Cp[wake] = —.5 and XR = .95 . Another case from Jandali [15], Figure 6.6, is presented for E/c = 50%,/i/c = 10%,* = 90° and a = 13° with Xs = .335 , Cp[bubble] = .05 and Cp[wake] = —.50, and XR = .70. The use of the 2-source flow model has been found to be more effective here. From these Cp comparisons,, especially for 6 = 90°, the upper surface suction and the lower surface pressures are under-estimated by the model in the region X/c < 0.2. This is probably due to inaccurate predictions of the frontal stagnation point which Chapter 6. Separation Bubble Upstream of Spoiler 50 resulted from deleting the spoiler from the mappings. Improvement is found in Figures 6.7 and 6.8, which are obtained by modifying the angle of attack, thus producing this frontal stagnation point correctly. The streamline plots, Figures 6.9,6.10, and 6.11 reveal that the separation streamline intersects the spoiler, when it is inserted back. The portion of the separation streamline upstream of the spoiler cannot accurately represent the bubble because it is too close to the airfoil surface. In addition, modelling the free streamline from the tip of the spoiler is impossible here. Another model, which incorporates Jandali's mapping sequence with that of the Joukowsky Arc, has been constructed to improve the bubble contour. The separation streamline from A, as demonstrated in Figure 6.12, is made to reattach to the spoiler surface at H at 90° so that H is also a stagnation point. The resulting bubble contour is a convex curve as shown in Figure 6.16 and 6.17. The condition corresponding to tangential reattachment has been found inappropriate in this model. The mapping equations connecting the eight consecutive planes are ct(theor.) = ct(expt.) + 2° Z = A1{Z1 + ^-) + B1 Z2 — (Zi — Zi0)e Z4 = CXZZ + Di Za = A%Z% + 2?2 Z6 = RZ7 - cot a - — 1 RZj — cot a Chapter 6. Separation Bubble Upstream of Spoiler t = She-4-where Ai = -X^ Bx = L \ L1 = l+2e+ 1 2 + 2 + Li l + 2c Zio = (-c,//) Z 2 A = flxe'^-'o) ^2^. = ^ i E ^ l l + Zwl J22 1 1 ZSE = R2 + -^ ZSG = ~2Ri ZsA + — = 2cos<f>A Z6F + = 2 cos ^ i t 2 ^ B F 4 4ZSP 4* Cx = „ „ Dx = 2 - „ ** A2 = Z%E — Z%G ZZE — ZsG ZZA — Z$F B2 = 2t-- — R = esc a 0, = TT a = — - — ^5A — "5F * * The overall derivative is dZ Z \ - \ e"- 1 , A rfF " l l ~ z T j ( I T S ) CxA2 [~zJ~)lK + {R^S - cot a)« J ' Therefore, a0 = a - 0O - ( ) C \CxA2\ where 4>A and <fo? are denned in Figure 6.12. At the trailing edge (point C) of the airfoil, Zx = 1, ' ^ O = 2 A l | ( 1 _ 1 g ) C 1 A 2 ( ^ r ) [ ^ + (Ee- f -cota)^ 1 ' 2 3 where Z2 = (1 — Zxo)e~i9°, and Z 5 and c; are found from the mapping equations. At the separation point (A), Re%a°$ — cot a = i, Chapter 6. Separation Bubble Upstream of Spoiler 52 where Zx — Z10 + Rxei9*,Zi = t{*A and 6'A is defined in Figure 6.12. At the tip (point E) of the spoiler, Z$ = 1, af Zf (i _ ^ Ci .A2 (iEc ,a<>f - cot a) 2 where Z\ = Zio + -Rie'*0 and Z% and f are determined from the transform equations. The flow model in the f-plane consists of a doublet tangent to the arc AF simulating the bubble contour, two sources located on arc CE producing separation streamlines at E and C. The corresponding boundary conditions are 1. stagnation flow conditions at A,H,E and C, 2. Cp[bubble] = Cp at A, and 3. Cp[wake] = Cp at E and C. The complex velocity is wu) = v\(i-±) + il+ q v i e i S D + - J J - + - g ? - - ( g l + g 2 ) - ] . M l * j i J + ' f + ( f _e«i>)» + f - f t + f _ f t * 2 V Note that the condition of finite C p gradient at separation cannot be always satisfied so that a steep Cp gradient is expected locally. The position of reattachment point H should be obtained either from flow visualization or Cp measurements. For convenience, it is chosen to be 60% of the spoiler length away from the tip to give the best correlations throughout. The experimental Cp distribution from Figure 6.4 is repeated in Figure 6.13 and compared with the theoretical prediction. The model requires Xs = .485, smaller than that used in Figure 6.4 because of the nature of the solutions. Very steep Cp gradients at separation and near the trailing edge are found. If the finite Cp gradient condition is satisfied at separation, Xs and Cp[bubble] have to be altered to .245 and —.945 Chapter 6. Separation Bubble Upstream of Spoiler 53 respectively, as shown in Figure 6.14. Another comparison is presented in Figure 6.15 for the same spoiler as in Figure 6.6. Xs is changed from .335 to .265 while other input parameters are the same. --, The steep Cp gradient at separation and the creation of a stagnation point on the spoiler are believed to be the reasons for causing the poor predictions of Cp from this model. However, the streamline plots depicted in Figures 6.16 and 6.17 at least indicate an improvement of the bubble contour and a correct pattern of the streamlines from the tip of the spoiler as well as the trailing edge. A restriction of this model is that the spoiler has to be normal to the airfoil surface. Figure 6.1 Pressure Distributions for Joukowsky Airfoil with Normal Spoiler [15] Figure 6.2 Shape of Separation Bubble Upstream of Spoiler Cn Chapter 6. Separation Bubble Upstream of Spoiler 56 Airfoil with Spoiler -3 -2 -• Figure 6.3 Comparison of Pressure Distributions for Airfoil near Trailing-Edge Stall and Airfoil with Spoiler Chapter 6. Separation Bubble Upstream of Spoiler 59 0 0.2 0.4 0.6 0.8 1 X/C Figure 6.8 Pressure Distributions for Joukowsky Airfoil - 0 .4 i i r i 1 r 0 0.2 0.4 0.6 0.8 1 Figure 6.10 Shape of Separation Streamlines •8 en-CD "I 0.4 _ 0.2 . E/c = 70% h/c = 5% 6 = 45° a = 12° •8 fa 3 o' tD C cr cr <5" "8 CD - 0 .2 CD CO 0.4 0.2 0 _ E/c = 50% h/c =10% 6 = 90° a = 13° 8 •8 CO •8 (0 3 § to c e'-er (6 f e-f-o CD -0.2 •0.4 0 0.2 0.4 0 .6 0.8 Figure 6.11 Shape of Separation Streamlines .8 E/c = 70% h/c = 5% 6 = 90° a = 11° Figure 6.13 Pressure Distributions for Joukowsky Airfoil 0 0.2 0.4 0.6 0.8 x / c Figure 6.14 Pressure Distributions for Joukowsky Airfoil OS 00 E/c = 50% h/c =10% 6 = 90° a = 13° O Experiment [15] Theory ~r 1 1 1 1 r 0 0.2 0.4 0.6 0.8 1 X/C Figure 6.15 Pressure Distributions for Joukowsky Airfoil - 0 . 2 _ 0 0.2 0.4 0.6 0.8 1 Figure 6.16 Separation Streamlines with Separation Bubble o - 0 . 2 _ 0 0.2 0.4 0 .6 0.8 1 Figure 6.17 Separation Streamlines with Separation Bubble Chapter 7 Multi-Element Airfoils 7.1 Williams' Exact Method An exact analytical method was proposed by Williams [22] to calculate the plane in-compressible potential flow about two lifting airfoils of adjustable configurations. The potential flow about two lifting circles is determined, and the two circles are then mapped conformally into two airfoils by Karman-Trefftz transformations. The method of calculating the potential flow about two circles can be very complex, involving the use of elliptic functions. Here, the flow which is systematically established by the method of images based on Milne-Thomson's circle theorem, is represented simply by a sequence of three components: a streaming flow past both circles, a flow with unit circulation around the first circle, and one around the second circle. 7.1.1 The Streaming Flow Consider a uniform stream of unit speed at an angle of incidence 60 to the line joining the centers of the two circles (Figure 7.1). Its complex potential in the f-plane is The circle theorem states that if a circle, |f | = a, is introduced into a flow, represented by the complex potential, F = /(f), then the total complex potential becomes F(s)=f(s)+J(j) 72 Chapter 7. Multi-Element Airfoils 73 where the overbar denotes the complex conjugate. Therefore, inserting a circle, |f| = a into a uniform flow gives rise to an additional term in the potential, where Ke = a2eis° D0 = 0. Now the second circle, |f — c| = 6, is introduced into the flow. The circle theorem is again applied to all singularities lying outside the second circle. The additional terms are * M - ^ A M = ^ where b2 b2 J0 = b2eiS° L0 = c Hi = - iTo( - ) Dx = e . c2 c <7o(f) can be interpreted as a doublet at f = L0 of strength J e , and /1(f) as the image °f o n the second circle. Now the first circle is not a streamline, thus the circle theorem is again applied to all the singularities outside the first circle, giving Jl r 1 .\ %i ft(f) = T - V Ms) = where _ , a2. a2 , a2 » a2 These two new doublet images are the reflections of the images produced in the second circle by the previous step. This process is repeated and after each step either the first or the second circle is a streamline. Each reflection entails the addition of two image doublets. The following general formulas are found to generate these images efficiently. Chapter 7. Multi-Element Airfoils 74 With Jo,Lo,K0 and D0 as denned above, for j = 1,3,5,••• «fe> = ^ A M = ^ "where J n ^ - T~ K i - ~T. F T ~ H D i ~ c LU 1 Li-i 3 ( * - A - i ) 2 ' ( c - A - i ) For j = 2,4,6,"-where _ J,-:6 2 _ fe2 „ * i - i ° 2 D a 2 The strength of the doublets, which are added after each reflection, is monotonically decreasing and approaches zero. After several reflections, the new image doublets will only slightly change the complex potential of the system. This establishes a necessary condition for the series to converge to the complex potential for the streaming flow past two circles. Detailed proofs for convergence are given in [22], The complex potential for the streaming flow of strength V is nf)=v[Mf)+f;wf)+/i(f)}]-;=0 The horizontal (u) and vertical (v) velocity components in the f-plane containing the circles are given by from which the normal velocity (Ur) and tangential velocity (Ue) components at a point on the surface of a circle are given by Ur = u cos 0 + v sin 0 Chapter 7. Multi-Element Airfoib 75 Ue = v cos 0 — u sin 0 where 0 denotes the angular co-ordinate of this particular circle with respect to its own centre. ~" If sufficient terms are included in the series of F($), then the normal velocities on the surfaces of the circles should be close to zero. In this present work, the series is truncated once the accuracy set for the normal velocity components has reached 10"7. The values of Ug at all points on the circles are used in the calculation of Cp distributions. 7.1.2 The Circulating Flow The complex potential, = £ Inf. represents a vortex flow of unit strength around the first circle (Figure 7.2). The second circle is now introduced into the flow and is made into a streamline by the use of the circle theorem which gives the additional terms in complex potential, Pi(f) = ^ l n ( f - L i ) p2(f) = - J L l n ( t - L a ) where b2 L\ — c JJI — c . c Pi(f) and Pi{c) are the image system consisting of a vortex of strength +1 at the centre of the second circle, and a vortex of strength —1 at f = L2, which is sometimes called the inverse of the centre of the first circle. The overall circulation around the second circle is zero. Now the first circle is not a streamline and the circle theorem is applied again to Chapter 7. Multi-Element Airfoils 76 all singularities outside the first circle, resulting in where a2 a2 Ls = — If4 = -=-. C X/ 2 This image system consists of a vortex of strength — 1 at f = L3, and a vortex of strength +1 at f = L\. This process is repeated, alternately making the first and second circles streamlines. In general, these image functions can be easily generated by the following procedure. With p 0 , p i , p 2 , L 0 , L i and L 2 3 8 defined above, for i = 3,7,l l , .--where a2 . a2 Lj = - Lj+1 = Lj-2 Lj-i For j = 5,9,13, • • • PiM = ^ ln(f - A ) p , + 1 (f) = ~ - iy+i) where 62 b2 Lj = c- -— Lj+i = c - — . C — Lij-i C — -L/j-i Unlike the strengths of the doublets, the strengths of vortices do not diminish after each reflection. However, each reflection entails the addition of the two vortices of opposite sense at inverse points in one of the circles. In [22] it is proved that this set of inverse points in a circle converges on to a complementary inverse point. Thus the image system which is added after each reflection approaches a vortex doublet for which Chapter 7. Multi-Element Airfoils 77 the complex potential is zero. This constitutes a necessary condition for the complex potential of the above system to converge to the complex potential for a flow with unit circulation around the first circle. A proof of sufficiency can be found in [22]. To determine the complex potential for a flow with unit circulation around the second circle a similar method is used. With 9o(<r) = ^ in(r - d0) 9l(<r) = JL in(<r - dx) <fe(<r) = ~ Ht -where Forj =3,7,11, a 2 d0 = c d\ = 0 di = —. c where 62 62 dj = c - — d]+i = c — . c — fly-2 c — <*y—l For j = 5,9,13,---<7/(f) = j£p - dy) <7y+i(?) = ln(f - di+l) where a 2 . a 2 dy = — dy+i = dy-2 ' dy_i The complex potentials for the flow around the first and second circles are i^(c) = f>y(<r) *"•(?) = £ ; * ( f ) . y=o y=o Again, the normal and tangential velocity components can be found by using the same method as described for the streaming flow. To be consistent, the tolerance for the normal velocity components is set to 10 - 7. Chapter 7. Multi-Element Airfoib 78 7.1.3 The Combined Flow The tangential velocity of the complete flow at a point P on the surface of a circle can be expressed as U,{P) = V[Ut'{P) + 7 l • U*>{P) +12 • Up(P)] where Ug is the component from the streaming, Ugl is the component due to unit circulation around the first circle, Ug2 is the component due to unit circulation around the second circle, 71 • V is the vortex strength around the first circle, and 7 2 • V is the vortex strength around the second circle. 7i and 72 are determined by considering velocities at G and F, where G and F axe the points on the two circles, which transform on to the trailing edges of the two airfoils, Figure 7.3. In order to achieve finite velocities at the trailing edges, G and F must be stagnation points. Thus, the strength of these two vortices may be found by solving the two linear equations, WO) + 71 • USl{G) + 72 • U?{G) = 0, U;{F) + 71 • US1 [F) + 72 • ir,a{F) = 0. Chapter 7. Multi-Element Airfoite 80 < c Figure 7.2 Geometry of Two Circles with Vortex Images Chapter 7. Multi-Element Airfoils 81 7.2 Adaptation to Joukowsky Profiles In [22], an idealized airfoil with an external flap is mapped from two circles. The Karman-Trefftz transformation is used twice, resulting in both the main foil and the flap having near-Karman-Trefftz profiles. The relative sizes and positions of the two circles, and the locations and orientations of the coordinate axes in the transformations determine the geometric parameters of the resulting two-element airfoil. In this present model Joukowsky transformations are used instead of Karman-Trefftz, and they are used twice as indicated in Figure 7.3. Using the following transformations, a uniform flow of speed V at an angle of attack 60 + ct0, with respect to the line of centers of the two circles in the f-plane corresponds to a uniform flow of speed U at an incidence a , with respect to the real axis of the Z"-plane. ZA = Zi0 + RxZhtia" Zs = ZA + ±-Z2 = A\Z% + f?i Zi = z2 + - i -z 2 Z = CXZX + A where R\ = |1 + Z^o\, Z^o = (—ei,/xi). Complex constants A\ and B\ are obtained by setting points E and F to (—Cj,^) and (1>0) * n t n e ^2-plane. C\ and A are to adjust the orientation of the main foil, and the total chord length of the assembly to be of one unit. The overall derivative is Chapter 7. Multi-Element Airfoils 82 a„ and JJ are found by considering the flow far away from the two foils, V a0 = a-(4>1 + fa) - = Bi|i4iCi| where The values of a0,60ib,6p and c, as defined in Figure 7.3, are uniquely determined by specifying a, 77 (flap deflection), / (flap length), gap size and overlap of the assembly. At the trailing edge (point G) of the main foil, Z4 = 1. Therefore, | - £ ( | ) | = 2 * ^ ( 1 - ^ ) 1 where Z2 - 2AX + Bx. For the trailing edge (point F) of the flap, Z2 = l, | l ( f ) | = 2 ^ , ( 1 - ^ 1 where ( = c + betfF, and Z4 is determined from the transform equations. The C p of a point in the Z-plane of the two airfoils, except at the trailing edges, is given by c - i • u> " 'P -x UTTdZ\ At the trailing edges, c = 1 _ r U(»)l l ^ ( f ) | J The Cp distribution for a single airfoil can be calculated using the same formulation, by moving the flap to a great distance from the main airfoil. This also provides a check on this model since the exact result for a single airfoil is well known. Indeed, the Cp distribution for a single element airfoil is recovered on each component once the gap size is about 10 times the total chord, which is defined as the sum of the chord lengths of the main airfoil and its slotted flap. Chapter 7. Multi-Element Airfoils 83 Calculated Cp distributions are compared in Figure 7A with distributions measured in the same wind tunnel and with the same main airfoil used in the experiments of [13]. For the flap a Joukowsky profile of suitable camber and thickness was chosen and the flap model and its mounting were constructed so that the flap chord was 30% of the overall chord and the flap deflection, gap, and overlap could be changed. The vertical mounting of the airfoil system on the wind tunnel balance (used here only for support and angle-of-attack control) was by the strut used previously for the main airfoil at its quarter- chord point. This had the disadvantage that the strut was so far forward for the two-element airfoil system that the system experienced torsional deflection under the aerodynamic loading, making the actual a appreciably less than the nominal value. The test Reynolds number was 7(10)6 based on the overall chord. In the experiments for which results are presented here the airfoil configuration had 2.2% gap and 1% overlap, and flap deflection r\ = 20° . In addition to the measurement of Cp distributions surface flow visualization using tufts was carried out. In Figure 7.4 the theoretical curves are calculated for a = 0°, whereas the exper-imental data are for a nominal a = 4° to compensate for the deflection mentioned above. The deflection was not measured precisely and it is unlikely that it was as large as 4°, but the good agreement in Figure 7.4 indicates that the theoretical model is reproducing the effects of shape correctly, such as the high trailing-edge loading on the main foil caused by the presence of the flap. Moreover, the flap Cp distribution is found to be insensitive to a for a given configuration of main foil and flap, so the good agreement there is independent of the issue of actual and nominal a. The largest discrepancy between theory and experiment for the flap is on the upper surface near the leading edge, and this is because the true Joukowsky profile of the experiment and the near-Joukowsky profile of the theory show their largest differences in this region as can be seen in Figure 7.5. Finally, the relatively large discrepancy on the main foil Chapter 7. Multi-Element Airfoils 84 upper surface near the trailing edge is probably caused by the artificial thickening of the trailing-edge portion of the experimental model discussed in [15]. Chapter 7. Multi-Element Airfoils Figure 7.3 Mapping Sequence for Two-Element Airfoil Chapter 7. Multi-Element Airfoils 86 Figure 7.3 Mapping Sequence for Two-Element Airfoil 0.4 0.2 f/c = 30% v = 20° Near-Joukowsky - J"- - True-Joukowsky 8 •8 Cb c I i s 6r - 0 . 2 - 0 . 4 0.2 0.4 0.6 0.8 Figure 7.5 Comparison of Airfoil Profiles 00 00 Chapter 8 Spoiler/Slotted Flap Configurations A spoiler in the presence of a slotted flap can be useful in flight for roll or flight-path control. In the version of this system simulated here the main airfoil and the flap are near- Joukowsky profiles and an inclined spoiler is erected on the upper surface of the main foil near its trailing edge. 8.1 Normal Spoiler In this model Joukowsky transformations are used four times as indicated in Figure 8.1. The first use maps the main circle to a flat plate, the second maps the flat plate to a circle with radial fence, and the third maps this circle into a Joukowsky airfoil with normal spoiler. Finally, the fourth use of the Joukowsky transformation maps the near-circle which the second circle has become into a near-Joukowsky profile for the flap, while the main foil becomes a near-Joukowsky profile with a normal spoiler. ' (Intermediate translations and rotations are of course required to properly orient the coordinate axes). Figure 8.2 shows the resulting configuration for which results are presented. It can be seen that the near-Joukowsky profiles are very close to true Joukowsky profiles. The mapping equations are ei{$o-a0) Z 6 = fcx + fe[Z6e'<-"'-> + —^ —} _ Z 4 — Z 4Q Rie,s°  6 ~ Rxtiia Z4-Z4Q 89 Chapter 8. Spoiler/Slotted Flap Configurations 90 Zs = ZA + ±-"4 Z% — A\Z% + B\ Zi = Z2 + -^-Z = Ctfx + Dx where 1 £ 2 -f 1 l + i £ = ~ i2i = | l + Z 4 0 | Z40 = {-euiil) and U, V, a, and ct0 are defined in Figure 8.1. A\, B\, C\ and D\ are complex constants for adjusting the orientation of the trailing edges and the total chord length. The overall derivative is f = C l ( l - i , ) A l ( l - - k , t i e f i l ( , J [ ^ -a0, and ^ are found to be where V a0 = a - <f>i - <f>2 — = Ei|AiCi|A;2 The values of ao,6o,b,0F,c, £, and 0O, as shown in Figure 8.1, are uniquely determined by specifying a, rj (flap deflection), / (flap length), gap size, overlap, h/c (spoiler height) and E/c (spoiler location) of the assembly. ..{.I the trailing edge (point F) of the flap, Z2 = 1, )| - 2| C l { A,(l - ±), , 1 ^ - ' - ^ H > ' l Chapter 8. Spoiler/Slotted Flap Configurations 91 where ~ 1 1 - Bi and C = c + bei$F. At the trailing edge (point A) of the main foil, Z4 = 1, and 1^(^)1 - 2|c x(i - ^ M i { f i _ s a o l * - — J ) I where Ze = e,$A, Z2 = 2A\ + B\ and 6A is the angular location of A on \ZQ\ = 1, Figure 8.1. At the tip (point C) of the spoiler, where Z 6 = e~^a°-9°\ l | ( f ) l = 21^(1 - £ ) ( i - ^ ) [ , ]| where Z4 = Z 4 0 + -foe'*0, ZS = Z 4 + Z 2 = A 1 Z 3 + B x . Points A, C and F are critical points of the mappings at which angles are doubled, so that if they are made stagnation points of the flow model in the circle plane they become the required tangential separation points in the airfoil plane. In Williams' flow model the doublet and the vortex needed to provide shape and circulation for a single circular cylinder become infinite sequences of doublets and vortices to satisfy the boundary conditions for two circles. A similar process occurs in the present flow model for the wake sources and their image sinks added to the appropriate region of the circle which becomes the main foil with spoiler after transformation. The complex potential of a unit source and its images in the two circles (Figure 8.3) can be written as m = +-M<-ti+t tow+*(*)] 2 ? r ,=1 where ft is the location of the source on the first circle, and ft = etSl. Chapter 8. Spoiler/Slotted Flap Configurations 92 The terms in the summation are where Forj = 3,7,11, Mt) = ~ H( ~ In) fi[{) = ± Ht ~ I*) gi{t) = ~ ln( f - Di) gt{c) = ± ln( f - D») a 2 b2 Li = 0 L2 = — Di = c D2 = c-) = ln(f - Lt) fi+l({) = i - ln(? - Li+l) 2w x* " " T 4 W / 2TT where 62 b2 _ a 2 a 2 Lj = c- -—=— L , + i = c - - — = — Dj = ==— = C - Ly_ 2 C — Ir,-_i £>y_2 F o r i = 5,9,13,---^ = " 2 ^ l n ( f " L y ) = h l n ( f ~ 5 y W = ~ h l n ( i r " ^ 9 i + i ® = 2 ^ l n ( s r " D j + i ) where a 2 a 2 62 b2 Li = =— Li+1 = =— Dj = c- -—==— Dj+1 = c - -—===—. Lj-2 Jjj-1 C — Uj-2 C — Uj-\ It can be shown using the method suggested in [22] that the series is also a conver-gent one. Pictorially, these images in each circle are located in a set of inverse points which converges on to a complementary inverse point. They are pairs of sources and sinks of equal strength which cancel each other after sufficient terms are taken. The same tolerance, 10"7, is set for truncating the series for the normal velocity components. Chapter 8. Spoiler/Slotted Flap Configurations 93 The total tangential velocity for the 1-source model at a point P on the surface of a circle is U9(P) = V[U;{P) + 71 • W(P) + 72 • U?{P) + q i • Use>(P)] where the first three terms on the right hand side are the same as those discussed in section of multi-element airfoil, Ugl is the component from a unit source on both circles, and qi and 6\ are the strength and angular location of the source. The 1-source model allows the pressure to be stipulated on the tip of the spoiler. 71, 7 2 , qi and 6\ can be determined by the usual boundary conditions. For the 2-source model an extra term <72?//* is added to Ug(P) leading to six unknowns in total. The pressure at the trailing edge of the main foil can now be specified, increasing the number of conditions to five. Contriving an additional boundary condition, which is physically sound and mathe-matically admissible, is not trivial. The zero lift incidence method used by Jandali [15] as the extra condition is not applicable here. For a sharp discontinuous surface such as the tip of a spoiler, the condition of finite Cp gradient is not physically realistic. In this current work it is proposed not to search for another condition suitable for this model, but merely to demonstrate how successfully this simple theory works for a rather com-plicated flow configuration. The second source is therefore fixed on the airfoil surface in a position that minimizes the difference in the Cp distributions obtained from the 1-source and 2-source models. The result is shown in Figure 8.4 for a 10% spoiler at 60% of the overall chord on the main foil at a = 4°. The effect of the spoiler as a lift dumper on the main foil is clearly demonstrated through the theoretical pressure distributions as shown in Figure 8.5. The theoretical Cp curves in Figure 8,6 are calculated for a = 4° while the experi-mental data are for a nominal a = 8°, for the reasons given previously. Agreement is seen to be good for the main foil except for the usual effect of the separation bubble Chapter 8. Spoiler/Slotted Flap Configurations 94 upstream of the spoiler. However, for the flap the theoretical and experimental values are completely different. Whereas the theoretical curves show an increase in maximum suction from about Cp = —2 for the configuration without the spoiler (Figure 7.4) to Tip = —7 with spoiler the experimental distribution shows a drop to a nearly constant suction about Cp = —1. Similar results have also been reported in Brown [17], in which the maximum suction peak increases from —1.5 to —3.85, and in Tou and Hancock [14], from —3.7 to —14.62, although for different geometric configurations. The obvious ex-planation of course is that the erection of the spoiler created a flow that caused the flap to stall because of the high positive pressure gradient on the flap surface. However, the flow visualization mentioned previously indicated that the flap was not stalled, since the tufts on the flap upper surface lay flat in the stream direction instead of fluttering and reversing as they would in a stalled flow. Moreover, the same behaviour was observed in the experiments with spoiler erected through a considerable range of values of a and flap deflection 77 from a = 0°, q = 0° to a = 8°, r; = 40°. This anomaly will be discussed later. The theoretical initial shape of the separation streamlines corresponding to the same configuration as Figure 8.6 is shown in Figure 8.7. Chapter 8. Spoiler/Slotted Flap Configurations 95 Z Figure 8.1 Mapping Sequence for Two-Element Airfoil with Normal Spoiler Chapter 8. Spoiler/Slotted Flap Configurations 96 Figure 8.1 Mapping Sequence for Two-Element Airfoil with Normal Spoiler 0.4 . 0.2 E/c = 60% h/c =10% 6 = 90° rj = 20° Near-Joukowsky True-Joukowsky 8 -8 oo 2. n Co o n •8 to c 0) - 0 . 2 - 0 . 4 0.2 0.4 0.6 0.8 Figure 8.2 Comparison of Airfoil Profiles CO - 8 Cp - 6 _ - 4 _ - 2 _ 0 _ E/c = 60% h/c = 10% 6 = 90° r; = 20° O Experiment, ajvoM = 8° [23] Theory, a = 4°, Cpb = -1.0 ° ° o o oo o o \ o °ooooooo 0 .2 0 .4 x / c 0 . 6 0 .8 Figure 8.6 Pressure Distributions with Normal Spoiler Chapter 8. Spoiler/Slotted Flap Configurations 102 Chapter 8. Spoiler/Slotted Flap Configurations 103 8.2 Inclined Spoiler The theory for a single airfoil fitted with an upper surface spoiler of arbitrary inclination is available [6], and its extension to two-element systems should be straightforward, simply by adding another circle in the f-plane, and following the mapping sequence as suggested in [6,13] with an extra Joukowsky transformation to transform the near circle in the z"2-plane to a near-Joukowsky flap in the i^-plane, Figure 8.8. The transformations are f _ . - * . ( * ! £ ) t — £i% Z6 = -\{2 - n) + ih - l-[nln(| + 1) + (2 - ») l n ( ^ - 1)] Z6 = iR s'm^ cot(~-) Z, = (Z 4 - A)e^ z3 = z4 + ±-Z4 Z2 — A1Z3 + B\ Z! = Z2 + ^r Z = CXZX + Di where T _ n = 2(1 - -) A = Z 4 0 + Rcos 6ei[Bo+fo) R = |1 + Z40\ 7T „ T „ 1 \ .1 r2sin6 + ft. lo = - - e o - 6 Zi0 = {-euHi) h' = ln[ ] and Ut V, h', h, h, a, a c are defined in Figure 8.8. The details of the mapping sequence from the plane to the Z"3-plane can be found in [13]. Ai,Bi,Ci and D\ are complex constants for orientation adjustment as well as scaling. Chapter 8. Spoiler/Slotted Flap Configurations 104 The two-element system obtained from the mapping sequence is compared with the true profiles in Figure 8.9. The overall derivative is dZ Z \ - \ Zl-l •„ -iRsmf • Ze Z7Xy(i - Z8)2e<°° T< - ^ — Z T ^ - Z T ^ I - — 1C S C ^Y)[(Z7 + n)(Z7-2 + n)]-The relation between the angles of attack in the Z-plane and the Zs-plane is ct0 = cc — <f>i — fa where and V . A _ .JRsin£ 77 = ^ l C l [(AI + „)2 + A2][(A,-2 + n)2 + A2] 1 1 1 1 Av \l (A' + A>) where Z700 = Xx + t'A,,, see Figure 8.8. The values of a0,60,b,dF (the angular location of point F in the f-plane),c, h',60 and 6, as defined in Figure 8.8, are uniquely deter-mined by specifying a,rj (flap deflection), / (flap length), gap size, overlap, ~E (spoiler inclination), h/c (spoiler height) and E/c (spoiler location) of the system. At the trailing edge (point F) of the flap, Z2 = 1, I d (dZ\\ -n\r S A ,Zl-\ Rsm6y 2 f Z 6 l Z7Xv(i - Z6)2 , l ^ ( - ^ ) | - 2 | C 1 { A 1 ( - i r ) [ - 1 - l c s c ( T ) [ ( Z 7 + n ) ( Z 7 _ 2 + n )]} I where $ = c + bei9p. At the trailing edge (point B) of the main foil, £ 4 = 1, l d / d Z M _ o l 4 ^ tZ* ~ 1 A r r J E s i n ^ i 2^eA Z7Xv{i-Z8)2 l 2 , | - ( - ) | _ 2 | A 1 c 1 ( - ^ - ) { [ - r - ] c s c ( y ) ( Z 7 + n ) ( Z 7 _ 2 + n )} I where z * = - | ( 2 - n) + »9(Z 6 ) 3(Z6) = - i m[|^ ±Il] Chapter 8. Spoiler/Slotted Flap Configurations 105 X = cos(-y0 — <j>0) — sin 6 Y = sin(70 — <f>0) — cos 7 T = cos^o — <f>0) + sin? <t>0 = t a n - ^ - i M = h! - \[n l n (^ + 1) + (2 - n) l n ( - ^ - - 1)] Z% — 2 Zi = 1A\ + J5j. At the tip (point A) of the spoiler, where Zj = 0, l5<f>l = l*<M. Z\-\^(Z\-\^ Rain? ] c s c 2 ( f ) [ ^ ^ n Z\ n Z\ n2n{n-2) where Z% — — Aj./Ay ZQ = iti ZA = Z40 + Reiie + he-*0 and Z% is determined by the mapping equations. Figure 8.10 gives the C p distributions for the same geometric configuration as in Figure 8.6 except that the inclination of the spoiler is changed from 90° to 30° measured with respect to the surface of the main airfoil, and the flap angle r\ is reduced from 20° to 0°. The theoretical curves are calculated for a = 8° whereas the experimental data correspond to a nominal a = 12°. The agreement is quite satisfactory over most of the main foil. A separation bubble at the leading edge probably explains the reduced suction peak. The predicted Cp on the spoiler is also shown. However, the spoiler is too short to make measurements and the symbols in this region represent the constant wake pressure. Over the flap, the prediction is quite good on the lower surface. The wake influence as pointed out previously is still strong on the upper surface as shown, but the high suction peak disappears. Figure 8.11 shows the initial shape of the predicted separation streamlines for the same geometrical configuration as Figure 8.10. Chapter 9. Multi-Element Airfoil Near Stall 106 Z2 Figure 8.8 Mapping Sequence for Two-Element Airfoil with Inclined Spoiler Figure 8.8 Mapping Sequence for Two-Element Airfoil with Inclined Spoiler Chapter 8. Spoiler/Slotted Flap Configurations 108 Figure 8.8 Mapping Sequence for Two-Element Airfoil with Inclined Spoiler 0.4 J 0.3 0.2 J 0.1 0 0.1 J 0.2 0.3 J 0.4 E/c = 60% h/c = 10% 6 = 30° ; v = 0° Near-Joukowsky True-Joukowsky 0.2 0.4 0.6 0.8 Figure 8.9 Comparison of Airfoil Profiles 0 0.2 0.4 0.6 0.8 1 X / C Figure 8.10 Pressure Distributions for Two-Element Airfoil with Inclined Spoiler Chapter 8. Spoiler/Slotted Flap Configurations 111 Chapter 0 Multi-Element Airfoil Near Stall Combining what has been constructed for single-element airfoils experiencing trailing-edge stall with Williams' model for flow around two-element airfoils, a model is pre-sented here for two- element systems near stall. This model can be used for stalled flow on either element. However, the strong adverse pressure gradient caused by the rela-tively large deflection of the flap, would usually induce it to stall earlier. The following section is devoted to this situation. As in the Joukowsky Arc model, a stalled airfoil is created in the Z3-plane from a circle (II) in the f-plane, Figure 9.1. Another circle (I) upstream of (II), which would be a near-circle in the Zg-plane, undergoes a Joukowsky transformation in the •Z2-plane to become a near-Joukowsky airfoil. In the Z-plane, the two profiles, one complete and another one trailing behind with its upper surface partly omitted are both near-Joukowsky. The details of the mapping equations are ? = Z7eiS° ZQ = C2Z1 + D2 „ 1 ZB = Z 6 - cot a - — — Z6 — cot a Z4 = A2Z1 + B2 ZS = ZA + ^-Z2 = C\Z% + D\ 112 Chapter 9. Multi-Element Airfoil JVear Stall 113 Zx = Z2 + \ -Z = AiZx + Bi Complex constants Ax, Bi,Ci, Di, A2, B2tC2 and D2 are for the purpose of orientation and scaling. U, V, a, and a 0 are defined in Figure 9.1. The overall derivative is OLQ and ^  are found to be ao = a - (4>i + fa + 4>z + fa) where * l = t M l to5 * 2 ( ^ ) } * s ( ^ ) } ^ 4 = t a n <S(Ctf> a = - 2 — ^ — where </>s and <J>R are defined in Figure 9.1, and ^ = l A x d ^ d l . The values of ao,6o,b,0T (the angular location of point T on |f — c| = 6 ), c and ^ 5 , as depicted in Figure 9.1, are uniquely determined by specifying a, T) (flap deflection), / (flap length), gap size, overlap and Xs (location of flow separation on the flap). At the trailing edge (point T) of the flap, Z4 = 1 , where f = c -f 6e'*T , and Z2 and Z$ are determined by the mapping equations. At the trailing edge (point B) of the main foil, Z2 = 1 , Chapter 9. Multi-Element Airfoil Near Stall 114 where f = ei9B, Z 4 and Z 7 are found through the transform equations and 6B is the angular location of point B on |f| = 1, Figure 9.1. At the separation point (5) on the flap, where Ze = cot a + t, where f = c + bei$st and 0$ is the angular location of point 5 on |f — c| = 6 in Figure 9.1, and The usual boundary conditions for a stalled airfoil are used. In addition, tangential separation from the trailing edge of the main airfoil is enforced. The flow model in the f-plane consists of the usual streaming and circulating components around the two circles. A surface source and a doublet tangent to circle (77) with their appropriate image systems for both circles are adopted. The general procedure to generate the complex potentials of the doublet and its images is similar to that of the streaming flow. For a doublet of unit strength tangent to a circle at £ = , the complex potential is where $D = c + beiS°. Its images and their corresponding reflections are given by, Figure 9.2, for j = 1,3,5,' where Ki - T.¥J \i Ui ~ C T f \ J> ~ (T. \2 3 T a2 (c-D^)2 3 {c-D,.x) 3 (L,_i) 2 ' Ly-i Chapter 9. Multi-Element Airfoil Near Stall 115 Forj = 2,4,6,---MS) = where with J0 = -ieiS° , L0 = $DiK0 = -ieiS° and D0 = ft> c — (c-X,-x) ' Therefore, the total complex potential is 00 m = ho(!)+Z \fiis)+Oi{t)) The proofs of the necessary and sufficient conditions for convergence of the series for the streaming flow can be used here. The tolerance set for truncating the series for the normal velocity components is 10~7 as before. The tangential velocity at a point P on the surface of either circle is where the first three terms within the bracket on the right hand side are the same as those discussed in the section of multi-element airfoils. The fourth term is the component due to a source of strength qs located at f = c + 6e'*s. Finally, the last term is the contribution of the doublet of strength qp at f = c + bt%ir>. The experiments carried out in [23] do not include the situation of a stalled flap. However, some Cp distributions over a NACA 23012 airfoil with a NACA 23012 external-airfoil flap are reported in [26]. Two representative cases corresponding to a = —6.9° (Figure 9.3), in which Xs = 25% chord (of the flap) , Cp(wake) = -.545, and a = 3.16° (Figure 9.4), in which Xs = 5%, Cp(wake) = -.571 are presented with XR = .90 cho-sen for both cases. Surprisingly good agreement between theory and experiments is obtained on both elements. The discrepancy near the leading edge of the main airfoil U9{P) = V\Ul{P) + 7i • WiP) + 12 • U?(P) + qs - Up{P) + qD - £#>(P)] Chapter 9. Multi-Element Airfoil Near Stall 116 is probably due to the inevitable differences in the airfoil shapes as shown in Figure 9.5. Relatively high suction peaks predicted by the theory at the leading edge of the flap, Cp = —11 for a = —6.9° , and Cp = —8.6 for a = 3.16°, are probably caused by the 'narrow channel' bounded by the streamline from the trailing edge of the main foil and the leading edge of the flap, as shown in the streamline plots, Figures 9.6 and 9.7. In principle, the model in the previous section can easily be modified for the sit-uation in which stall develops on the trailing edge of the main foil and attached flow on the flap. Experience from the spoiler/slotted flap studies, however, suggests that modelling an attached flow in the close proximity of a wake, using this type of wake singularity theory, would be inappropriate. Thus, no further work is pursued. Chapter 9. Multi-Element Airfoil Near Stall 117 Figure 9.1 Mapping Sequence for Stalled Flap Figure 9.1 Mapping Sequence for Stalled Flap Chapter 9. Multi-Element Airfoil Near Stall 119 Figure 9.1 Mapping Sequence for Stalled Flap -6 -5 _ f/e = 20% V = 40° a = -6 .9° | -4 . • A Experiment [26] : Theory Cp - 3 _ X, = .25 \ XR = .90 -2 . Cp[wake] = —.545 !«? -1 _ A A \ \ \ \ A ' \ V n 1-1 r . 0 _ fjfl A" A" A ^ - j! v_ D _g.^ /! cj + 1 , i 1 0 0.2 1 1 1 0.4 0.6 0.8 X / C i i 1 1.2 Figure 9.3 Pressure Distributions for Stalled Flap 8 •8 C i 1 § ? to CO EL to 0.6 0.4 . 0.2 . " Z NACA Profiles Near-Joukowsky 0 . 0.2 0.4 0.6 - 0 . 2 0 0.2 0.4 0.6 0.8 1.2 1.4 Figure 9.5 Comparison of Airfoil Profiles 0.4 J - 0 . 2 0 0.2 0.4 0.6 0.8 1.2 1.4 Figure 9.7 Separation Streamline for Stalled Flap Chapter 10 Experiments The experiment for the trailing-edge stall developed on the Joukowsky airfoil was car-ried out in the Low Speed Aeronautical Wind Tunnel in the Department of Mechanical Engineering at the University of British Columbia. The facility is a conventional closed-circuit wind tunnel with low turbulence level, less than 0.1% over a wind speed range of 0 — 45m/s, operating at slightly less than atmospheric pressure. The model, which has a constant chord of .31m, is a Joukowsky section constructed mainly of mahogany of 12% thickness and 2.4% camber, and spans the height (,69m) of the tunnel test section with a small clearance (less than 2mm) at the ceiling and the floor. It was mounted at its quarter-chord point, about which it pivots in the pitch plane, on the tunnel center line. A set of 37 static pressure orifices of which 24 are on the upper surface is located in an aluminium mid-span section. An artificial thickening at the trailing edge is required for structural purposes, as reported in [15]. Figure 10.1 shows the experimental arrangement. Surface-tufts studies indicated that the natural flow on the model was very close to being two-dimensional up to an incidence of 12° where separation was about 10% chord. When tested at angles beyond 15° , the model was found to have strongly three-dimensional flow over the rear part of its upper surface through the visual observation of complicated transient phenomena exhibited by the tufts. The region of acceptable flow quality could be extended to 14° by adding judiciously-placed, streamwise rectangular fences. They are a pair of endplates placed on either side of the midspan and 1.25 126 Chapter 10. Experiments 127 chords apart. It was considered that the small gaps mentioned above may be partly responsible for the development of the three-dimensionality in the flow. As a result, the ends were sealed with masking tape. The pressure measurement shown in Figures 4^.2, 4.5, 4.7 and 4.11 corresponds to an incidence of 14° and without the surface tufts so that the flow would be steadier. The free stream speed was 22m/s, resulting in the Reynolds number based on the chord equal to 5(10)5 . Apart from the main tests, there were subsidiary tests at lower angles of attack to define the airfoil's lift curve for the calculations of wind tunnel boundary correction. The location of the flow separation Xs was not measured directly from the exper-iments. It was interpolated from the chordwise pressure distribution as the position downstream of which the pressure becomes constant, identified as C p i . The wind tunnel correction technique employed was established by Pope and Harper [24]. For the controversial wake blockage term, eCo(S/C) with e = 1 was adopted from Maskell [25] for stalled airfoils, where S is the representative area over which the profile drag coefficient Co is based, and C is the cross-sectional area of the tunnel. The details of the overall procedure were previously discussed in [13]. No similar corrections are made to the two-element experiments with and without spoiler, conducted by Allan [23]. The reason is that it is not known exactly how much torsional deflection the airfoil experienced in the streaming flow. Further details of Allan's experiments are given in [23]. Figure 10.1 Sketch of the Model in Wind-Tunnel Test Section 8 •8 n a ill I Front View a Side View I CD a. S — 1 • • • • • . « • • • k to 00 Chapter 11 Discussion Of the five problems considered, the applications of the wake singularity model to trailing-edge stall for both single- and two-element airfoils are the most successful, giv-ing good prediction of CV distribution and a satisfactory initial shape to the separating streamlines. The value of the free parameter XR has been found to be most effective if it lies in the range .85 < XR < 1.0. The separation characteristics of a given profile may be different for various Reynolds numbers. For instance, the Joukowsky airfoil is set at 14°. Reducing the Reynolds num-ber from 5(10)6 would cause the upper surface separation point to shift towards the leading edge and lead to a less prominent suction peak. A further reduction of the Reynolds number would eliminate this suction peak because of a laminar separation bubble. This phenomenon is the consequence of lesser energy being available to sustain the adverse pressure gradient. On the other hand, if the Reynolds number is increased from 5(10)5, then the region of the separated flow over the upper surface will diminish. The Reynolds number used in full scale tests are usually an order of magnitude larger than 5(10)B as reported in [1,2,3,4]. In order to achieve that, the wind-tunnel speed has to be increased beyond the operating range. Besides, the effects of compressibility may become important. Therefore, further tests along this direction were prohibited. The pressure distributions quoted in these references, however, exhibit a very similar shape to those found in our experiments. In general, the pressure in the separated region is about the same order of magnitude but the stall angle is larger and the separation 129 Chapter 11. Discussion 130 point is closer to the trailing edge at full scale Reynolds numbers. The leading-edge separation bubble is unlikely to set in. As a result, the stall models developed here are certainly suitable for predicting the pressure distributions at full scale Reynolds numbers. The results of applying the method to the flat plate separation bubble are promising, since the Cp distribution on the wetted surface is quite accurately predicted, but more work on the problem is needed to improve the predicted bubble shape and its Cp distribution, particularly near the separation and reattachment points. Progress is made towards predicting the characteristics of a bubble upstream of a spoiler. The first model provides quite reliable Cp predictions but the bubble streamline is unrealistic. The second model is more physically and mathematically correct because separation streamlines are simulated reasonably. However, steep pressure gradients at separation and the stagnation point on the spoiler are not supported by experiments. Further improvement can be achieved if the tangential reattachment of the bubble can somehow be established in order to avoid this awkward stagnation point. Partial success has been achieved on the spoiler/slotted flap configuration. The theoretical Cp distribution over the main element fitted with the spoiler correlates well with experimental data. Nevertheless, the general agreement on the flap is poor. The resolution of the problem created by the failure of the model to predict the flap Cp distribution depends on whether or not the flap was stalled - the measured Cp distribution suggests it was, the tufts indicate it wasn't. If the flap was stalled it should be possible to model the flow by combining and adapting the mappings of Figures 4.4 and 8.1 to produce an airfoil with spoiler in the presence of a stalled flap. If, however, the flap was not stalled, and inspection of the Cp distribution over the flap upper surface in Figure 8.6 indicates that it is not a typical stall pattern, then another explanation suggests itself. If the low-energy Chapter 11. Discussion 131 wake of the spoiler is at essentially constant pressure so is the boundary shear layer from the main foil trailing edge. This shear layer, represented by a free streamline in the flow model, lies close to the flap upper surface if the flap is unstalled (see Figure 8.7), and by providing a constant-pressure boundary condition forces the flap upper-surface pressure to be nearly constant but at a slightly higher suction because of the curvature of the unseparated stream. This explanation is consistent with the observed Cp distribution and with the evidence of the tufts. Moreover, it is further substantiated by the improvement in agreement found in Figure 8.10 for larger incidences (e.g. a = 12° ) and smaller spoiler inclinations (e.g. 6 = 30°), because the interaction between the wake and the flap is believed to be lessened. A more detailed experimental study on how the pressure over the upper surface of the flap varies with the gap size is recommended to further support the above argument. However, it does not lead to an improved solution by the wake source model, which cannot require a free streamline to be at constant pressure. Indeed, the incorrect theoretical result for the flap in Figure 8.6 arises from the lack of such a boundary condition so that the solid boundaries determine the pressure distribution. Bibliography [1] Henderson, M.L. (1978) A solution to the 8-D separated Wake Modeling Problem and Its use to Predict Cmux of Arbitrary Airfoil Sections. AIAA paper 78-156. [2] Bhately, I.C. and Bradley, R.G. (1972) A Simplified Mathematical Model for the Analysis of Multi-element Airfoils Near Stall. AGARD-CP-102. [3] Maskew, B. and Dvorak, F.A. (1978) The Prediction of C L ^ X Using a Separated Flow Model. Journal of American Helicopter Society, Vol.23 pp 2-8. [4] Jacob, K. (1987) Advanced Method for Computing Flow Around Wings with Rear Separation and Ground Effect. Journal of Aircraft, Vol.24, no. 2, pp 126-128. [5] 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. [6] Parkinson, G.V. and Yeung, W. (1987) A Wake Source Model for Airfoils with Separated Flow. Journal Fluid Mechanics, Vol. 179, pp 41-57. [7] Woods, L.C. (1961) The Theory of Subsonic Plane Flow. Cambridge University Press. [8] McCullough, G.B. and Gault, D.E. (1951) Examples of Three Representative Types of Airfoil-Section Stall at Low Speed. NACA T.N. No. 2502. [9] Schmieden, C. (1940) Flow Around Wings Accompanied by Separation of Vortices. NACA T.M. No.961. 132 Bibliography 133 [10] Ockendon, H. and Tayler, A.B. (1983) Inviseid Fluid Flows. Applied Math. Sci-ences, Vol. 43, Springer-Verlag. jll] Crimi, P. and Reeves, B.L. (1976) Analysis of Leading-Edge Separation Bubbles on Airfoils. AIAA Journal, Vol. 14, no. 11, pp 1548-1555. [12] Bertin, J . and Smith, M. (1979) Aerodynamics for Engineers. Prentice-Hall. [13] Yeung, W.W.H. (1985) A Mathematical Model for Airfoils with Spoilers or Split Flaps. M.A.Sc. Thesis, University of British Columbia. [14] Tou, H.B. and Hancock, G.J. (1987) Inviseid Theory of Two-Dimensional Air-foil/Spoiler Configurations at Low Speed. Aeronautical Journal; Parts I and II: Oct., 350-366, Parts III and IV: Nov., 406-428, Parts V and VI: Dec, 479-498. [15] Jandali, T. (1970) A Potential Flow Theory for Airfoil Spoilers. Ph.D. Thesis, University of British Columbia. [16] Brown, G.P. and Parkinson, G.V. (1973) A Linearized Potential Flow Theory for Airfoils with Spoilers. Journal of Fluid Mechanics, Vol. 57, no. 4, pp 695-719. [17] Brown, G.P. (1971) Steady and Unsteady Potential Flow Methods For Airfoils with Spoilers. Ph.D. Thesis, University of British Columbia. [18] Pfeiffer, N.J. and Zumwalt, G.W. (1982) Computational Model for Low Speed Flow Past Airfoils with Spoilers. AIAA Journal, Vol. 20, no. 3, pp 376-381. [19] Bearman, P.W., Graham, J.M.R. and Kalkanis, P. (1989) Numerical Simulation of Separated Flow Due to Spoiler Deployment. Conference Proceedings of the Royal Aeronautical Society, London, pp 2.1 - 2.15. Bibliography 134 [20] Wu, T.Y. (1972) Cavity and Wake Flows. Annual Review of Fluid Mechanics, Vol. 4, pp 243-284. [21] 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 A, Vol. 116, pp 170-197. [22] Williams, B.R. (1971) An Exact Test Case for the Plane Potential Flow about Two Adjacent Lifting Aerofoils. Royal Aeronautical Establishment, T.R. No. 71197. [23] Allan, W.D.E. (1988) Ah Experimental Study of Flow about an Airfoil with Slotted Flap and Spoiler Using Joukowsky Profiles. M.A.Sc. Thesis, University of British Columbia. [24] Pope, A. and Harper, J.J. (1966) Low-Speed Wind Tunnel Testing. John Wiley & Sons. [25] Maskell, E.C. (1963) A Theory of Blockage Effects on Bluff Bodies and Stalled Wings in a Closed Wind Tunnel. Royal Aeronautical Establishment, R & M No. 3400. [26] Wenzinger, C.J. (1938) Pressure Distribution over an NACA £3012 Airfoil with an NACA SS01S External-Airfoil Flap. NACA T.R. No. 614. [27] Ormsbee, A.I. and Maughmer, M.D. (1986) A Class of Airfoils Having Finite Trailing-Edge Pressure Gradients. Journal of Aircraft, Vol. 23, no. 2, pp 97-103. Appendix A Curvature The general formula for the curvature of a curve Y = Y (X) is given by Consider a streamline g(s,t) = g0 in the f-plane, where c; = s+it. Under a conformal transformation Z — /(f), where Z = X+ iY, the streamline is represented by a set of parametric equations X = X(a,t) and Y = Y{s,t). The property of an analytic function suggests that X. = Yt -Xt = Yt Xtt = Ytt Y,t = -Xtt Ytt = -Xtt. {A2) The purpose here is to express the curvature in the Z-plane in terms of variables s and t. By the chain rule, dt dt But dg = gtds + gtdt = 0 along the streamline. Therefore, dY = y^ds_ + Y^dt_ (A3) dX dX dX dX dt dX v d s „ — = X.- + X , ( A 5 ) dY _ , Y,gt - Ytg, , dX-[X,gt-XtgJ' (A6) 135 Appendix A Curvature 136 Following (A3), J-(-) - d (dY) d s d (dY) d t dX^dX' ~ ds^dX'dX + dVdX>dX' Partial differentiation of ^ with respect to s by using (A6) gives dsKdX} (X.gt - Xtg,)*[ J' where Similarly, where Therefore, and [ ], = -X,gt{Xltgt + Xtg„) - Xtgt{-Xtgtt + Xttge) -Xtgt{Xtigt + Xtgtt) + Xtg,(Xtglt - Xttgt). d dY 1_ dVdX]~ {Xagt-Xtgs) ,1 ]. ]t = -X,gt{Xttgt + Xlgdt) + Xtgt{Xtgtt + X.tg,) +Xtgt{Xttgt - Xtg,t) + Xtgt(Xtgtt - Xttgt). d2Y ^gt{ ],-<?.[ ]t dX* {X.gt - Xtg.Y [1 + [dxn - (xegt-xt9ly Substituting (A9) and (A 10) into (Al) gives 1 K = (x» + x?) 8 "W + y?) where (A7) {AS) (A9) (A10) (All) [A] = - ( X 2 + X*)[gM - g]) - 2g.gtgtt] + Appendix A Curvature 137 (rf + g])\-9t{XtXti - XtX„) - g.{X.X„ + XtXtt)\. In the f-plane separation points coincide with stagnation points so that (g2 + g2) = 0. Also, the separation points are the critical points so that (X2 + X2) = 0. In other words, K = ±oo at separation unless [A] = 0. In the neighbourhood of separation (say 3=0) one can write X, = axs + a2s2 + a 3s 3 + • • •; Xt = bxs + b2s2 + 63s3 + • • •; g„ = cxs + c2s2 + c 3s 3 H ; gt = dxs + d2s2 + d 3s 3 H . Substituting these series into the equation for K gives _ As1 + Bs6 + Cs7 + • ••  K-[(a2 + b\)(c2 + d2)}s/2s« where A = (a\ + &i)[c2(ci - d\) + 2cxdxd2] - (c\ + d\)[cx(axa2 + bxb2) + dx(axb2 - a2bx)\ and B = 2{al + bl)[cs(c2-d2) + cx{d2 + 2dxds + c2)}-[c\ + d\)\cx(a\ + b\) + 2cx{axa3 + bxb3) + 2dx(axbs - asbx)} -(axa2 + 6162) [Scjdi — 2cxdxd2 + c2c\] — (0162 — fl2^i)[^2ci + 2cxc2dx + Sc^d2]. If A = 0 , then the curvature is finite at separation and is given by B K \{a\ + b2){c\ + «*?)]»/»• Appendix B Pressure Gradient A general formula for the pressure gradient along streamlines in the physical plane (Z = X + iY) is derived and expressed in terms of the variables in the transform plane {C = s + it). Z = fit) X 8 dr = yJ{dXy + {dY)* dCp dCp dn dr dn dr 2 dp dn pU2 dn dr' dr, = yj{ds)2 + (dty Cp = P - Poo \PU2 ' 2 . ds dt, {ds)2 + (dt)2 pU2[P'dr, +Ptdnl\\ {dX)2 + {dYY pV^'^^dsUn dx\\i + (%y 138 Appendix B Pressure Gradient 139 _ 2 1 . dt. ds ~ Wji+\%)>[p'+PtTsldx Along a streamline, p + |/>u2 = constant. Therefore, du du and Using (A6), u = \W(Z)\ = \W(()\ \gt + ig.\ | f | \X.-iXt\ \X> + X? gt + g] V + W (Xtgt-Xtgt) Substituting (B2), (B3) and (B4) into (Bl), after some manipulations, gives —2 (X,gt — Xtg,) .du ds du dt + pU2 (X* + X?) KdsdX dtdX Using (A4) and dg = 0, ds gt dt ~9B dX X,gt — Xtg3 dX X,gt — Xtga Differentiating (B3) with respect to s and t respectively, du d ds ds\ gt + gl X? + X* 2\ Xj + X*d g* + g* g? + g> ds^Xt + X** 1 where [ ]. = [X] + Xf)(gtg.t + g.g„) - [g* + g]){XtXtt + XtXtt). And similarly, du _ 1 , , 'dt ~ {X* + Xi)'/*{g* + gi)W[ U where dt + + «?)•»/»' [ ], = (X2t + X])(gt9ti - gtg„) - (rf + g?)[XtXlt - XtX„). (Bl) (B2) (B3) (54) (B5) (B6) (B7) (B8) Appendix B Pressure Gradient 140 Substituting (B6), (B7), (B8) into (B5) gives ,2 I _2 d£p _ 2 g,+gt 1 ( B Q ) dr pU^X^ + Xf^X^ + Xf^ig^ + gf)^1 1 -where [T] = - (X, 2 + X?)[glt(gl - g]) + 2g,gtgtt] + (g2t + g])[gt[XtXlt + XtX,t) - g,(XtX,t - XtXtt)\. Note the similaxity between the formulas for K and At separation where gt = 0 , gt = 0 , X, = 0 and Xt = 0 simultaneously, p. = ±oo unless [T] = 0. Using the series for X,,Xt,g, and gt from Appendix A, one gets dCp 2 (c\ + d2^ as6 + ps6 r r 2 V _ 2 i 1.2/I dr C/2 v a 2 + ft2' [(a? + 6?)(cf + d 2.)] 3/^ where a = (a2 + fc?)[^2(c2 - d\) - 2dicxc2] + (c2 + d2)[di(aia2 + 6162) - C i ( a i 6 2 - a 26i)] and /3 = 2(a2 + 6 2 ) [d 3 (c 2 -d 2 ) -d 1 (d 2 + 2c l C 3 + c2)] + (cj + dj)[di(a| + 6^ ) + 2d1(o1a3 + 6163) - 2c1{aias - asbi)} + ( d a 2 + 6i62)[3c2d2 — 2cic2di + djd2] — (0162 — 026l)[ c2^i + 2c\d\d2 + 3c 2c 2]. If a = 0, then the pressure gradient is finite at separation and is given by dCp _ 2 (c\ + d\ (3 dr " U^al + bVlial + blKcl + dl)}*/*' Appendix C Finite Pressure Gradient Equation (B9) from Appendix B is useful in showing the similarity between the pressure gradient and the curvature at separation. However, using it to derive the criterion of finite pressure gradient at separation involves a lengthy process of Taylor series expansions because its numerator has a zero of order 6. The following method is found to be more practical. The relation between a segment of the airfoil surface and its corresponding arc in the f-plane can be written as dT=\dZ\ = \^\\d<\ = ft{0)d9 where f = et9. Therefore, dCL_dCL_ 1 dCp dr ~ \dZ\ ~ f2{0) dO ' Along a streamline, or the surface of an airfoil, Therefore, dCp -2h{6) [0] dr U* /,(*) /,(*) where [n] = M ^ M . Although both fL and / 2 vanish at separation, jfc is finite and equal' to ^1 — C p 6 . If ^ is finite, then [ft] = 0. Applying l'Hopital's rule twice to [fl] since both of its numerator and denominator are zeros of order 2, it is found that the criterion of finite pressure at 141 Appendix C Finite Pressure Gradient 142 separation is where/1 = |^(f ) |and/ 2 = | f |. A sirnilar expression has been derived in [27] as well. Appendix D Order of Magnitude Analysis The strength and location of a source in the 2-source model for the vanishing spoiler are estimated in terms of h, the length of the spoiler in the Z2-plane. Then, the equation of finite pressure gradient from Appendix C is checked using these results. Recall that for the vanishing spoiler model, dZ dZ dZ\ dZ2 dZ$ dZ4 dZ$ where d$ dZ\ dZ2 dZ$ dZ4 dZ$ d$ D Z = A(1 _ J _ \ dZL = .<(».-*) dZ2 _ Z\ dZx U Zl' dZ2 dZs 2R dZs = + J _ ) d^i = \ dZf> = eia\j Z*~X*y dZ4 7r Z4 dZ§ d$ 2i Xv Therefore, , id. dZ.. . d . dZ. dZ\ dZ d . dZ\. dZ2 dZ d . dZ2. dZ% f * = 'd7 ~<k = ^Tcd~z['~dJ + dY1^dz~J~dJ + dY2^dz3^~dJ dZ d .dZ%.dZ4 dZ d .dZ4.dZ& dZ d fdZ&.. (m\ dZ$ d$ dZ4 d$ dZ4 d$ dZ§ d$ dZ§ d$ d$ At separation, Z2 = h,Z4 = —1, and = 0, see Figure 4.1. Hence, But as h —»• 0, 2Rn and Xv — • 7T. 143 Appendix D Order of Magnitude Analysis 144 Therefore, Similaxly, after some lengthy calculations, one gets : - \(dZ y ~ 2 \(dZi\S -i- J^M_^idL[<M±s , I f f ^ w ; _ Z * ~ X*\ri(a<,+$)\ ~ ]{dZ3R*Zi)( d$} + nZ!dZs ds dc{ d<rJ d$Kd$R Xy 1 As h —• 0, each term on the right hand side can be approximated by dZ , -2 wdz"4.3 2 J_ J_ rfZ3l7rZ43Arf^ ~ ' / i 6 / i 4 2 dZ dz"3 d ,dz"4, 2 1 1 1 irZfdZs df dfv df 7 / i 2 / i 3 h3 - Z 4 - A « ) e « ( « » + * ) ~ 1 . 1 = 1 . * * *v / i 2 /i >i3 It can be concluded that as h —• 0, / 2' ~ 0(/i"4). P3) Now, recall that the equation of Cp at the separation point is given by P " l ^ ( f ) l 2 * Since it is proved that >S-£(f>i ~ « C p is finite, as h —• 0, if and only if :W{i)\ ~ 0( /T 2 ) p _d_ A more convenient form for W($) for the 2-source model is, using £ = exS, \W{t)\ = W(8) = ^[-4sine-27 + 9 l co t (^ -^) + 92cot(—j--2-)] Appendix D Order of Magnitude Analysis 145 and W{$) = -V[4cos* + § c s c 2 ( ^ ) + f c s c 2 ( ^ ) ] . Therefore, W'{0) ~ 0(/r 2 ) if and only if qlC8C>{LJ±) „ 0(h-2). (D4) At separation, W{0) = 0. If every term in W(8) = 0 is of the same order, then ftcot(i^) ~ 0(1). (D5) Dividing (Dl) by (D2) gives and ?i ~ O(fc'). P7) Is the finite pressure gradient condition compatible with the above analysis ? Recall from Appendix C that tifx - An=o (ci) where fx = and f2 = | ^ | . Therefore, and /J' = 7[4sin0 + - csc2(——) cot(—2—) + — cscz(-^—) cot(—— )]• As h —• 0, using (D6) and (D7), one gets: fx ~ °(h~2) (DS) Appendix D Order of Magnitude Analysis 146 and n ~ o{h-% w By substituting (D2), (D3), (D8), and (D9) into (Cl), it is now obvious that the finite pressure gradient at separation is compatible with the above order of magnitude estimate. Appendix E Magnification of Angles Consider a conformal transformation Z = /(f). Za is the intersection of two infinitesimal segments dZx and dZ2. f0 is the transform of Za with the corresponding segments dfr and d&. dZx z = m Without loss of generality, assume that f has a zero of order n at f0. Therefore, dZx and dZ2 can be written as 1 and 4 - ( » + !)!• Dividing dZ\ by dz"2, and taking the argument, ^ = 7 - T 7 T T / n + 1 ( f o ) ( ^ 2 r + 1 Ary(dZx) - Arg{dZ2) = (n + l p r ^ ) - Ar^(dft)]. (£1) 147 Appendix E Magnification of Angles 148 In other words, angles of intersection in the Z-plane are (n -f l) times that in the f-plane. Figure 4.9 shows the probable streamline pattern from S (where n = l) and point T (where n = 3). The stagnation streamline is orthogonal to the circle at S and T in the f-plane. Using (El), it is obvious that the separation streamline is tangent to the airfoil at 5. Whereas, the one from T makes an angle of 360° so that the separation is backward. 

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