An Investigation of the Ejector-Powered Jet-Flap Citation Sidor, Laurent Bernard (1974) An Investigation of the Ejector-Powered Jet-Flap. Engineer's thesis, California Institute of Technology. doi:10.7907/D5X4-X160. https://resolver.caltech.edu/CaltechTHESIS:09232010-143232435 Abstract The inviscid and incompressible potential flow aspects associated with a two-dimensional ejector-powered jet-flap configuration are investigated. The energy addition due to the mixing process in the ejector results in a non-homogeneous flow and is represented by an actuator disk located between the lifting surfaces and a powered wake. A set of singularities is developed to represent the lifting surfaces, to include camber and flap deflection, and the powered wake. A numerical procedure is used to compute the total system vorticity needed to satisfy exactly all the prescribed boundary conditions. The lift and moment coefficients are evaluated as a function of the head change prescribed across the actuator disk. A simple interpretation of the resulting lift curves is proposed in terms of analytical results obtained for single-airfoil configurations - a flat-plate airfoil and sink system, and a conventional single element jet flap. The sink effect of the actuator accounts for a finite lift at zero angle of attack, and the lift increment due to angle of attack only can be predicted by using Spence's theory of the jet flap. Item Type: Thesis (Engineer's thesis) Subject Keywords: (Aeronautical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Stewart, Homer Joseph (advisor) Harris, Gordon L. (advisor) Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 10 September 1973 Record Number: CaltechTHESIS:09232010-143232435 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09232010-143232435 DOI: 10.7907/D5X4-X160 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6044 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 23 Sep 2010 22:40 Last Modified: 30 Jul 2024 21:23 Thesis Files Preview PDF - Final Version See Usage Policy. 35MB Repository Staff Only: item control page

AN INVESTIGATION OF THE EJIECTOR-POWERED JET-FPLAP Thesis by Laurent B. Sidor In Partial Fulfillment of the Requirements For the Degree of Aeronautical Engineer California Institute of Technology Pasadena, California Lot (Submitted September 10, 1973) ~ii- ACKNOWLEDGEMENTS The author expresses his gratitude to Dr. Gordon L. Harris for suggesting the topic and providing guidance in the early part of the work. Also, he is grateful to Dr. Homer J. Stewart for his interest and his guidance in the completion of this work. In addition, the financial support of the National Aeronautics and Space Adminis- tration, Ames Research Center is deeply appreciated. And finally, he thanks Mrs, Elizabeth Fox for the typing of the thesis. ~iii- ABSTRACT The inviscid and incompressible potential flow aspects asso- ciated with a two-dimensional ejector-powered jet-flap configuration are investigated. The energy addition due to the mixing process in the ejector results in a non-homogeneous flow and is represented by an actuator disk located between the lifting surfaces and a powered wake. A set of singularities is developed to represent the lifting surfaces, to include camber and flap deflection, and the powered wake. A numerical procedure is used to compute the total system vorticity needed to satisfy exactly all the prescribed boundary con- ditions. The lift and moment coefficients are evaluated as a function of the head change prescribed across the actuator disk. A simple Suite etalon of the resulting lift curves is proposed in terms of analytical results obtained for single-airfoil configurations - a flat- plate airfoil and sink system, anda conventional single element jet flap. The sink effect of the actuator accounts for a finite lift at zero angle of attack, and the lift increment due to angle of attack only can be predicted by using Spence's theory of the jet flap. -“iV= TABLE OF CoNnTEeNntTs PART oi le I INTRODUCTION 1. General Background 2. Representation of the Effects of the Mixing Zone on the Lifting Surfaces 3. Solutions to the Lifting Problem for Multi- energy Flows al POTENTIAL FLOW PROBLEM 1, Formulation of the Problem (a) Field Equation and Elementary Solutions (b) Boundary Conditions 2. Vortex Distributions to Represent the Airfoil System (a) Leading Edge Singularity (b) Trapezoidal Distribution of Vorticity (c) Total Chordwise Distribution of Vorticity and Downwash (d) Endpoints and Control Points on the Airfoil System 3. Vortex Model of the Wake (a) General (b) Representation of the Flow Field at Down- stream Infinity 4, Description of the Iteration Procedure (a) Updating the Airfoil Vorticity PAGE On DD SH LL 12 13 15 5 15 16 16 “Ve TABLE OF CONTENTS (Cont'd) PART PSY aen (b) Updating the Wake (i) Position (ii) Vorticity 5. Computation of the Aerodynamic Coefficients (a) Forces (i) Leading Edge Suction (ii) Pressure Integral Across Camberline (iii) Pressure Integral on Actuator Disk (b) Moments (i) Main Airfoil (ii) Moment Due to Shroud (iii) Moment Due to Actuator 6. Performance of Numerical Scheme iil COMPARISON WITH JET-FLAP THEORY 1. General Characteristics of the Computed Lift Curves 2. Effect of a Sink on a Single-element Airfoil 3. Comparison with the Computed Lift Curves ata= 0°: The Actuator Disk as a Sink 4, Comparison of the Lift Increments Due to Angle of Attack with Spence's Theory a. Relation Between Head Change and Momentum Coefficient b. Comparison of the Lift Increments 5. Comparison of the Moment Curves PAGE 17 18 19 19 i9 22 23 24 25 25 ef 27 28 29 30 31 33 33 -~Vie- TABLE OF CONTENTS (Cont'd) PART TITLE PAGE 6. Conclusions 34 APPENDIX A. Field of the Trapezoidal Vorticity 35 B. Field of the Leading Edge Singularity 37 References 39 _— i, INTRODUCTION 1. General Background The past few years have seen 3 epidlenad interest in the area of V/STOL flight, spurred by potentially attractive civilian and military applications. These applications establish as a foremost requirement for the lifting system a high wing loading, necessary for range and good ride quality. Toward the satisfaction of this requirement, new aerodynamic configurations have been developed to exploit directly the power available from the propulsion system for the generation of lift by blowing on the lifting surfaces. Various schemes have been investi- gated, and are shown in Fig. 1, They are: The externally blown flap (a — review of configurations of this type is in Ref. 29) and over the wing blowing (Ref. 28) configura- tions use the jet engine exhaust directly over the upper side of the flap or wing surface. These configurations are characterized by the simplicity of the mechanical devices used to direct the exhaust air- flow. Other configurations use an extensive internal ducting sys-~ tem, driven by the high pressure stage of the engine compressor, to distribute the blowing over a major spanwise portion of the wing. The conventional jet-flap (Refs. 30-31) and augmentor-type wings operate on this principle. In a conventional jet-flap, the jet sheet created by the blowing action mixes with the surrounding air producing a secondary effect on the lift by fact rather than design; it will be discussed later in this Sat section, On the other hand, ‘the name "augmentor wings" is reserved for configurations intended to exploit this mixing effect to "augment" the primary airstream with the entrained secondary in installations known as "ejectors.'"' The specific purpose for which the ejector is designed separates augmentor wings into two categories. (a) The ejector can develop extra thrust on its inlet lip and sidewalls, thus supplementing the direct thrust generated by the primary for a vertical lift system. A family of ejectors designed toward this end has been under development at the Aerospace Research Laboratories, Wright Patterson AFB (Refs. 19,20). It is specifically tailored for the needs of vertical take-off and landing operations. (b) The other type of augmentor wing is the ejector-powered jet-flap, which is the subject of the present investigation. The pur- pose of the ejector in this case is to augment the primary jet momentum with the entrained secondary, the intent being to produce an increase in circulation around the wing as compared to a conven- tional jet-flap design, other things being equal. In addition, the design is purported to offer a potential remedy to the noise problem associated with conventional jet-flaps. The ejector-powered jet-flap is aimed at producing the high lift coefficients needed for sustaining flight at low forward speeds: this is the domain of STOL. The Large-Scale Aerodynamics facility at NASA/Ames has tested several full-scale versions of this scheme (Refs. 26,27), The present work is concerned with the analysis of some aspects of the flow field around a two-dimensional section of this -3- type. Let us describe now in more specific terms the flow field and how the conventional modelling techniques can be extended to the present problem. 2. Representation of the Fffects of the Mixing Zone on the Lifting Surfaces . The effects of the mixing zone are represented by the inflow velocity associated with the jet entrainment, and a net momentum flow downstream resulting in a powered wake. The inflow velocity can in turn be represented as an equivalent distribution of sinks lo- cated on the jet centerline (Ref. 33), the strength of these sinks being equal to the rate of entrainment. The net momentum flux in the powered wake is representative of the mixing efficiency of the ejector, The specific parameters used depend on the wake model used and are discussed in the next section. 3. Solutions to the Lifting Problem for Multi~energy Flows A wide variety of multi-energy potential flows have been investigated over the past fifteen years or so, and are quite intimately associated to — developments in lifting systems. The flow models can be classified into two families: thin wake models and thick wake models. The independent parameter used to describe the energy addition will depend on which wake model is used, Sereaucss, in his analysis of jet-flap configurations (Refs. 5, 6, 13), used a thin wake model, in which the real powered wake is replaced by a single vortex sheet. In addition the boundary conditions are linearized. This model is valid in the limit of a jet of negligibly small mass flux but finite momentum fluy (it is the independent =4e parameter energy addition due to the jet), His analysis was extended by Wygnanski and Newman (Ref. 7) to include the effects of entrainment by adding sinks on the airfoil chordline. The strength of the sinks, for a given jet momentum, is related to the entrainment as mentioned previously. It is found that although significant, the contribution of the sinks to the over-all lift increment is quite small, and possibly dependent on the particular mixing model used. Thick wake work has been largely associated with the predic- tion of propeller configurations, with the exception of C. A. Sholl- enberger's investigation of the two-dimensional propulsive biplane (Ref. 8). The independent parameter, in this case, is the head AS ; it results from the Hy zeU . action of an actuator disk which raises the head of entrained fluid change AH normalized in the form C from He to Ho + AH. Efforts have been'directed toward the solution of the flow field generated by a heavily loaded actuator disk at zero angle of attack and to include the effects of non-uniform loading (Ref. 3), and at a finite angle of attack (Ref. 12). es directly relevant to our problem, Chaplin (Ref. 2) has investigated the performance of shrouded propellers at zero angle of attack and no forward speed. His treatment of the wake is quite similar to the one of the present work, described in Section II]. A review of shrouded propeller work can be found in Weissinger and Maas' article (Ref. 9). The emphasis of this work being toward propellers, the configurations investigated are always two-dimensional axisymmetric, making the results of ~5- use for our particulary configuration as general reference only. The potential flow model proposed in Section IJ is an extension of C. A. Shollenberger's analysis of the two-dimensional propulsive © wing with rectilinear lifting surfaces. It accounts for camber, and in particular for flap deflection. This will be described in the next section. Jil. POTENTIAL FLOW PROBLEM 1. Formulation cf the Problem This section provides the theoretical background for a numer- ical calculation of the potential flow around the lifting section repre- sented in Fig. 2. It was found that the sinewlestte eyeten proposed in Ref. 8 for rectilinear chordlines can be extended to this more general geometry. (a) Field Equation and Elementary Solutions In this section, we will formulate the problem of the incom- pressible, inviscid homogeneous and irrotational flow about a system of lifting surfaces of specified shape in which the amount of energy - addition is specified through the head change AH. The lifting surfaces will be represented by cambered thin lines, in the usual thin airfoil theory formalism. We will assume that the powered waké fluid does not substan- tially mix with the surrounding fluid, so that it can be assumed to be bounded by two infinitely thin vortex sheets. This assumption will model the jump in velocity between inside and outside of the wake. In the conditions stated above, the stream function Ww of the system, and its derivatives, the velocity components uy, and u_, satisfy the Laplace equation throughout the field (including the wake) except on the boundaries where a jump in tangential velocity occurs, Pear i We will work with the velocity components u,, and By, because the boundary conditions are more readily expressed in terms of uy. and a, rather than w. Using the jump condition across the vortex sheet, we may decompose the tangential] velocity field into a regular part, Usther(*)» due to all the other singularities in the field, and a single-valued part, representing the self-induced velocity field due to the local vorticity: lim u(x,y) = y~>+0 4 thes ™ Mand (1) where us, = 5 y(x) for two-dimensional vortex sheets. Let ='s (0) specify the position of a vortex sheet (Fig. 3) of intensity ly (2) |. Then the velocity field Au(r) due to the vortex elements between oy and Py at any point away from the discontinuity is given by: Oo —- — = ae 2 y(o)dox(r-s) 1 = The total velocity field at that point, u(t), is found by superposing the contribution from all the vortex elements in the field. The choice of specific vortex elements is dictated by the particular prob- lem treated. Eachis referred to as an"elementary vortex" and, within this context, (2) is an elementary solution to the field equation (Poisson solution to the Laplace equation), We have found a general expression for the solution to the field equation, expressed in terms of the singularities in that field. The next step is to find a suitable set of singularities. These are imposed.by the boundary conditions, -8- (b) Boundary Conditions The boundary conditions are imposed both by the airfoil sur- faces and the powered wake. These can be separated into two distinct classes: those that are linear in the velocity field and those that are not. (i) The Kinematic or Streamline Condition: Let a be the local slope of the streamline, referred to the freestream direction (local angle of attack), Then we must have the following relationship between the velocity components: Vv waa J = tana | (3) co x Since there is no flow through the surfaces representing the airfoils, this relates the geometry of the system to the velocity field, the standard problem of thin airfoil theory. In addition, the relation must be true in particular at the vortex sheets hounding the wake from the sherds fluid. However, in this case, both a and the velocity components are a priori unknown. Also, the presence of the powered wake imposes finite velocity perturbations at downstream infinity. We will show how this boundary condition is satisfied by the wake model we will choose, (ii) The Pressure or Dynamic Boundary Condition in the Wake: The mixing process occurring in the ejector raises the total pressure of the entrained fluid from atmospheric head He to He + AH. Applying the Bernoulli equation on the upper and lower side of the wake vortex sheets (Fig. 2), we get olen 2 pul outside wake ae iT ae) ats tole HtAH = Py a + eal 1 inside wake Continuity of static pressure across the vortex sheet requires Py =P] AH = 2 p(w, tu, )(u,-u,) , (4) where AH is specified for this problem. Using (1), we can directly translate this into a condition for the vorticity: Pa ce * y(x) = AH (5) The difficulty of satisfying this boundary condition is two-fold--it te non-linear in the vediacticy field, and the location at which it should be applied is a priori unknown. | The solution to the problem is completely determined, in principle, given the representation (2) of the velocity field and the set of boundary conditions (3) and (5). We must now find suitable elementary solutions and a practical scheme to actually solve the problem. 2. Vortex Distributions to Represent the Airfoil System The chordwise distribution of vorticity in the presence of discontinuities in the shape of the boundary has been discussed extensively in Refs. 5, 6 and 16. These discontinuities occur at the leading edge, at the sharp corner representing the deflection of a flap, and possibly at the trailing edge. A singularity in the vor- ticity distribution at the trailing edge means flow around the trailing ° wi fia edge due to jet discharge angle. This is known as "singular blowing" (Ref. 30). In our problem, the jet discharges parallcl to the flap line, thus removing this condition ("regular blowing’), In the final analysis, these singularities are of two types: square root at the leading edge and logarithmic for the other two. The different nature of these singularities shows in the integrated effect: the square root singularity produces a net thrust in addition to the normal force, while the logarithmic singularity is too "weak'! to produce a thrust term, It is these criteria that dictate the choice of a particular type of vortex distribution. (a) The Leading Edge Singularity A well-known result of thin airfoil theory is the square-root leading edge singularity of the sneha distribution of a flat plate airfoil at an angle of attack: Ty ’ Yg(x)~ > as x0 (6) D- The flow around the leading edge of the thin section is interpreted as the limit of the flow about the round nose section of finite thickness, In both cases, this singularity produces a finite suction force S = 20 a, directed along the slope of the carnberline at the leading edge. The Kutta-Joukowsky flow about a lifting flat plate has the required behavior at the leading edge. It is described by the follow- ing analytic function: al bs tt u-iv il, fi - y2=4] (7) w(z) Hy In particular, it gives a constant downwash v = I’, on the chordline. Further properties of this flow are listed in Appendix 1. The chordwise distribution of vorticity for a flat plate due to the nose singuiarity becomes: yx) = TD, Ya (8) od where I; = sina for an isolated flat plate and is otherwise unknown for an arbitrary system (see Appendix B, page 38). - (b) The Trapezoidal Distribution of Vorticity The logarithmic singularity occurring at flap deflection can also be handled in a similar fashion by finding the flow function due to a sharp break in the airfoil chordline. Ref. 17 has taken advantage of this approach. In the present investigation, we found that the singularity could be adequately handled by assuming a trapezoidal distribution of vorticity between neighboring points: yx) = T, + (U,)-%)> a (9) Provided the step size 0; is chosen small enough, the agree- ment in overall lift with thin airfoil theory has been found to be excellent. The pressure distribution is of course quite different in the neighborhood of the flap hinge, but then the sharp break in alZ= camberline is not a realistic representation of the flap deflection anyway. The field due to (9) can be calculated at any point using (2). The details of the calculation and properties of the ficld are listed in Appendix 1. (c) Total Chordwise Distribution of Vorticity and Downwash We may now superpose the contributions (8) and (9) to the vortex distribution which becomes: _ l-x ais OS v(x; < x < (x,+o,))= 2) ya + tT +(T47-%) S, (10) where x is referred to the mean chordline of each airfoil. The resulting downwash distribution on the camberline is - 1 3 v(x; <x< (x, +0,))=Ib+ > E Log lx ze & + 4-4) (-1 u o; = Aan lcm )| * Yother alg The Ty and [ ] terms represent the downwash induced by the ith element on itself, while Vother TePresents the field generated by all the other vortex elements in the lifting system (including those on its own airfoil). The constant term r in the downwash corresponds to a thin airfoil approximation, but it should be emphasized that the downwash component is computed at the actual location of the boundary condi- tion on the camberline, and in the local coordinate system defined by the slope of the element at that point. In contrast, thin airfoil theories handling the downwash distribution in terms of integrals we, Bx of the vortex distribution are of necessity limited to computing the downwash as it the boundary condition were imposed at the mean- chord. Regardless of whether the approxirnation is justified or not, the present numerical scheme obviates the need for making it in the first place. Having the expression of the downwash as a function of the unknown vortex distributions, we may use boundary condition (3) and solve for the distributions. This is the subject of the next section. (d) Endpoints and Control Points on the Airfoil System Let n be the number of segments chosen to represent the camberline for the lower airfoil, and m for the upper one. The set of unknowns representing the vorticity of the airfoil system is: (12) G= [Tyo Ty eeeee Ey Thy eee Fadl It contains the two leading edge singularity terms, MN, and Dew and the trapezoidal terms, We have omitted the trapezoidal terms at the leading edge of each airfoil, They may as well be taken to be zero, because the square root singularity dominates anyway. At the trailing edge, the vorticity is specified by the Kutta condition, so that V4 = ) Dtmt2 = 0, of course, in the unpowered case. In the powered case, the vorticity is specified otherwise; we will return to this point later. -]4- Thus, the total number of unknown vortex intensities is the same as the total number of segments representing the system, We may select in particular the midpoints of these segments to com- pute the downwash, whence it takes the particularly simple form o i me “= = 2 = *e r — * Vother Aa} We call these midpoints "control points", We — now solve for the vortex distribution as a Seeectidionn of the slope of the camberline. Boundary condition (3) involves velocity components parallel and perpendicular to free-stream, Writing (13) for all (ntm) control points, and rotating to the proper coordinate system, we can write symbolically vy, =U yt Uj oint+.. ‘ +U, mint int fa cee Ua Diam = U* G for control point #1, and similarly a =V: GQ for all control points (14) U and V are the induction functions by which given a vorticity distri- bution G the velocity can be calculated at any point. Conversely, given boundary condition (3) the vorticity distribution may be com- puted as follows: Define H = V - U tana Then (3) must hold on all control points so that H+ Ges [US tana] which can be solved for G: ol Sx G= i [U., tana] Solution to the unpowered (15) —_ . problem The matrix Se is referred to as the matrix of influence coefficients and plays an essential role in the following parts. 3. Vortex Model of the Wake (a) General We have seen that the difficulty of the problem arises not only from the nonlinearity of boundary condition (5), but also in the fact that the location of that boundary condition is unknown. To obviate this difficulty, a number of investigators (Refs. 8,2) have successfully used an iteration technique yielding a pro- gressively better ssa adiest een of boundary conditions (3) and (5). The wake is modelled by trapezoidal segments, (similar to the previous section), the position of which varies during the iteration, It must, however, be truncated after a certain downstream position. This truncation must be compatible with the boundary condition at downstream infinity mentioned previously. (b) Representation of the Flow Field at Downstream Infinity: The numerical scheme we are proposing involves the calcula- tion of the downwash at arbitrary points in the field. Thus, itisa matter of practical need that we choose a representation of the flow field at downstream infinity so as to get a smooth distribution of downwash even in the last wake endpoints, We do know that the far wake must consist of two parallel vortex sheets of vorticity + =] -J1+ » Which is found b Y Yoo y ~16- applying i»« jump condition at p = Poo? i.e., where 4. = U_+4y c oo) The sepe: on of these sheets is a priori unknown otherwise, The direction is known: the far wake vortex sheets must lie along the freestream direction. The velocity ficid induced by two line voriices, starting at a downstream distance Xo and separated by a distance noo can then be directly computed using (2). ; -1 x7 aa -x 4 aru, = Yes E an Fa,” - Tan * | (16) i (x5-x)*#H(y-h)92 ar, a. Log 7) 5 (17) (x -x) -(y-h) where hoo varies during the iteration. This far wake model was also used by Shollenberger, 4. Description of the Iteration Procedure: (a) Updating the Airfoil Vorticity: We can conveniently separate the field components due to the wake and due to the airfoil: vWrY (el sg (we (18) ul =u (iy iw) x x x then, we may rewrite boundary condition (3) as (a) x tana(x) = Uy tana(x) + 0 ranatx)-w" (19) The field at the control points is expressed as: H* G, where Gis to be determined. Then, the amount of vorticity imbedded in the ml. ts airfoil required to satisfy the boundary condition in the presence of the wake-induced velocity components will be Gen, [U_tanat+u OW ieee = cmd (20) —= oo x y Hu is independent of the iteration and is calculated once and for all for Cry = 0; (b) Updating the Wake (i) Updating the Position. To find the slope of the vortex sheet at the control point, we observe that if indeed this slope were actually known, then it would be continuous across the vortex sheet, Le Ge = = tana(x) (21) (1 and 0 referring to the velocities inside and outside the wake), “x1 “x0 So that since 1]+4+—-—=1+4+—— Vv Vv yl yo a | ss and tana(x) = a = (22) Vv yo yl yo Now, we want to use the jump condition across the vortex so we must transform to a system parallel to the vortex. wu =u cosa(x) - v sina(x) for u_andu ° 1 * =u sina(x) + v cosa(x) where ~]§8a so that =v oi ae other ‘s.i. 2; sinat(v_. +v_.Jcosa tanats) = other other si __ (23) — “ 2u_,, _cosa~(v t+v_.)sina other other si If the boundary condition is exactly satisfied, then Y= Tatas Vow which represents the total velocity normal to the vortex sheet, will be zero, so that (23) is satisfied identically, If the boundary condition is not exactly satisfied, then Yu* 0° and a better estimate of the slope can be found by using (23) as: (new) . tet OD, Amen) 5, ee cosalot4) . fnew) _ ather other si (24) eng © By tiew) cosa Old) _( (new) (new), singold) YVother other ‘si Having executed that for all the control points in the wake, we can find the new wake position by direct integration: x = >t t Vol) = ¥pp + J tana(x' )dx (25) TE (ii) Updating the Vorticity: The updating of the vorticity at the control points is readily obtained by using boundary condition (5). The vorticity at the endpoints is then obtained by averaging the vorticity between neighboring endpoints. The exceptions to this rule wi Ge are: (1) at the trailing edge of the airfoil, the vorticity is linearly extrapolated from the first wake control point. The con- tinuity of vorticity at this point complies with the Kutta condition for regular blowing. (2) the vorticity of the two last wake endpoints is set to tYo0° 5. Computation of the Aerodynamic Coefficients (a) Forces Definitions: We will call "lift" any force component perpendicular to the freestream direction, and "thrust" anny component parallel but oppo- site to the freestream direction (i.e., thrust is measured by negative numbers and drag by positive numbers), (i) The leading edge suction term can be found explicitly by integrating the flat plate vorticity term around a lip of finite radius and then letting the radius go to zero, The fee edge suc- tion is then Cc. = 27 ia for each airfoil , (26) and acts tangentially to the leading edge element. | This thrust, of course, does not produce any moment about the leading edge. (ii) The Pressure Integral Across the Camberline. This is the only place where the actual location of the actuator disk enters ? since the Bernoulli equation involves explicitly the head of the flow. =20— For convenience, we assume that the actuator disk is located at the trailing edge of the system, so that the fluid is assumed to be at atmospheric head No up to that point. This certainly does not affect the value of the total lift on the system, since it depends only on the total amount of vorticity shed by the actuator disk, but will affect the distribution of lift between the two airfoils and the load. However, to represent these realistically would require detailed knowledge of the mixing zone, unavailable at the present time. To find the | pressure difference acting between x, and X54, 07 the element of the +] camberline, we apply the Bernoulli equation on its upper and lower side, ~1,77% (2 112 PLP = PUG ~ 2OL Uy e)-0'] ey ak. 2 Py Pog = 2 PUG, - FOL Mees) + 0] (27) where u' = -5 y(x); thus the load is Ap = } p(4 (28) u ! othe r* ). The elementary normal force acting on the element is: e541 Ni; = f Ap dx x 1 *i4] #0 fo peg ely be)dx See 1 Uother!*) is the contribution of all the other vortices in the field, and does not vary much over segment i. It is most convenient to compute the velocity at the control point between x5 and x. Thus, normal- a izing as usual: maZ, | se c~ "iy Ph N,, = + 20 ( x, +>) [9 V(x) dx This component contributes both to the overall thrust and lift of the system, It is more convenient to refer this component to the mean chord coordinates, Let e,= slope of the camberline in these coor- dinates, Then: N. = COS €E. N . 1 1° ti ealaieg fa (29) Using the representation of the vorticity (10), we can integrate the vorticity term *i+] J y (x)dx = I [(6,,,-8,)+(sind, |) -sind.)] i ms ape 4 By, , + zit..,+ i] Ax, (30) where @ = Arc cos( -1), Including the leading edge suction term, (26), the total normal and tangential referred to the airfoil mean line become: N = -20f sinB + » N ae nat Sw T = -2n Dy cos * 2, T; . (31) (B is the slope of the camberline at the leading edge fp = ey: ) To obtain the lift and thrust coefficients we rotate to the freestream coordinates: -22= C, =N- cos(atp) ~ { sin(atB) L om =N- sin(aiB) + uy cos(atf) (32) (iii) The Pressure Integral on the Actuator Disk, The forces due to the actuator disk (for the case of uniform loading) (re- ferring to Fig. 8) are simply its normal thrust referred to the thrust and lift directions defined previously: Cc =C.,+ h-: sin(até) C,, =+C,,* h+ cos(até) (33) So that finally we obtain the total lift and thrust coefficient acting on the system: Ee =C,, +C,, +C . (34) It is these coefficients characteristic of the total vorticity in the field (airfoil + wake) that are significant in the comparison with jet-flap theory. (b) Moments We refer all our moments to the leading edge of the main air- foil. In this definition the nose thrust of the main airfoil does not contribute to the moment of the system, but the nose thrust of the shroud does, «236 (i) Main Airfoil. Referring to Fig. 8, the contribution of the ith element to the moment about the nose is . it) Vita M, = f xal _-f ydT, (35) a Yi where dN. = cose.« aN. 1 1 tl dt, = ~sine 5 dN,; (36) and y is related to x by the local slope of the camberline: yix) = xtane, + Y, ~ x; tane Fe Making the same approximation as in (a) concerning the velocity, we get, after some manipulation: ~(i) oy i . mt a Mi’ = +2U (x, t-)° [cose ,+sine tane,] ° f xy (x)dx | i ~ (i = (y,~x,tane .) pl ) (37) Writing the vorticity integral explicitly: *ith t es ; J xy (x)dx = 4 T,[0,, 179,72 (sin20,, )-sin20,)] + 1 am ,2 2). “itil 3 {13 4g Fath Oia) tee, [sin FOX, 2 al (38) The total moment acting about the nose due to the lower airfoil be- comes: 48 Gc. =) 4) (39) oy pote " ~ ~24~ (ii} Moment Due to the Shroud Moment due to the normal force on the shroud. We assume that the maximum camber of the shroud airfoil is small enough so that we may neglect the local slope in computing the mo~ ment about the leading edge of the shroud, i.e. (referring to Fig. 8) *2 dN M, = fox: (Sor) dx" (40) x 1 and then the moment transferred to the leading edge of the main airfoil becomes: x y 2 2 M =f xd y -f ydF (41) 1 1 using Yor¥y taney, a ee (42) 2 dF =dN cose : M (43) dF = +dN sin aN we finally get: a = M(N) = M,(N) + [x,° cose, ty): sine, | : [N| (44) where (x) > y,) is the position of the nose of the shroud in the main CM airfoil coordinates, We must add to this the moment due to the leading edge suction, which is readily found to be ~2 : Mo(51 5) = aude [y, cose, ,-x, sine, ,] (45) Lm (iii) Moment Due to Actuator. Referring to Fig. 8, the contribution to the moment due to the actuator is found to be independ- ent of the position of the actuator and Cy, = Cy hed (46) act where dis the lever arm of the actuator. The total moment can be found by simply adding the various contributions: Cu = Su Fou, TM ole) t Ong (47) tot 1 2 act 6. Performance of Numerical Scheme The numerical calculations were performed on the IBM/System 370 available at the CIT. Computing Center. 45 segments were chosen to represent the camberline of the test configuration. The total computing time for the ee con- figuration, including the inversion of the influence coefficients matrix, was about 20 seconds. The step size of the segments selected to represent the wake must be smaller than or equal to the exit height of the ejector, 2 ‘ For a fixed downstream wake cutoff distance, low 2 ratios translate into a larger number of segments to insure convergence, resulting in an increase of the required computing time. Thus, for at = 15% and wake cutoff at 5 chords downstream, 50 segments were used. The number of iterations required for a given accuracy in the aerodynamic coefficients appear to be directly proportional to Crp and sensitive to a lesser degree to'an increase in the angle of 4 hw fon ! attack and/or flap deflection. For the case of high Cry and low a the total amount of com- puting time for a cormplete set of iterations is about 2 minutes, -27- Il, COMPARISON WITH JET-FLAP THEORY In a well-known paper (Ref..6), Spence developed a linearized theory for the two-dimensional potential flow about a single element jet-flap airfoil and computed the lift increment due to blowing in the form of lift derivatives tabulated versus the jet momentum coefficient of the primary Cy. In the previous section, we proposed a numerical scheme to compute the lift coefficient on a multi-element airfoil, given the head change C The purpose of this section is to establish Hw a technique for the comparison between the lift increments obtained by the two different methods. 1. General Characteristics of the Computed Lift Curves For a constant flap deflection, the functional dependence of the lift coefficient can be written as: Gy = Gy, (ap Gel . (48) so that ac ac, Cy 0 ba Ic. ° + GG la’ Sw (49) L (a = 0°,C =O) + L H (neglecting cross-terms) We may regroup oC reste ° as = o _ L « CG, (a = 0°, C,,) = Cy (a = 0°, Cy, = 0) + aC, Co (50) so that the lift increment due to angle of attack only becomes AC, = -axI6 1 QO ! Q i Qa 1 8 QO -28- The presence of a finite lift at zero angle of attack in the powered case results from the asymmetry in the lifting surfaces. In the early part of the work, some difficulty was encountered in the inter- pretation of the lift curves, because a finite lift at a = 0° results in lift derivatives (as computed directly from (48)) —e infinite at that angle, thus precluding any meaningful coxmnparison with Spence's theory. Dr. H. J. Stewart recognized the similarity of this behavior with that of a wing equipped with suction-type boundary-layer control, of which he previously made the theory for a single element airfoil (Ref, 18). This theory will be repeated below, and affords a direct. interpretation of the lift curves ata = 0°, Having established that, it will then be shown that the lift increment due to angle of attack only can be directly related to Spence's increment, having suitably related the head change Cry to the total momentum flowing in the wake. 2. Effect of a Sink on a Single-element Airfoil The suction effect of a boundary layer control device can be represented by a sink on the upper side of the airfoil. (Fig. 10) The analysis of this effect was done in Ref. 18 and is repeated here, as it provides the theoretical background necessary to interpret the presence in our configuration of finite lift at a = 0°. In the trans- formed plane, the combination of sources and sinks necessary to maintain the circle a streamline is also shown in Fig. 10. Let 0 be the angle between the direction of the sinks and the rear stagnation point. Let the distance between the sinks go to zero. Then, the ce horizontal velocity component generated by the sink of strength ~20 is cancelled by the velocity due to the source +Q at the origin. The net vertical velocity component is given by: 2Q Y, 3 aereS te Ge 3) (52) Y 2n(2R sin zi Q @ = San cot (5) “4 (53) Thus, to maintain the stagnation point at the trailing edge, an extra amount AD. Ad of vorticity at the center of the circle is necessary, directed counter-clockwise so that its induced velocity component exactly cancels (53): OD sad _ Q 2a7mR 2T7rR ‘cot(S) SAT, 4 = Qcot(S) (54) The resulting lift increment is pul at, The total lift ind’ becomes: —— 6 L2 TpUcsing + puL Q cot(s) (55) ° and in particular is nonzeroas@=0., 3. Comparison with the Computed Lift Curves at a@ = 0°: the Actuator as a Sink To relate this to our configuration, we must first estimate Q as a function of Cy The strength of the sink is the difference in mass flux between the power-on and power-off configuration, which can be estimated as: «tie -Q = - hu. Q hvu, ay i =hu, fb - (1+ C,,)7] (56) Also, we recognize that the cot(s) factor is valid only fora single element airfoil. For a multiple element airfoil, we expect that the factor should be somewhat different. In particular, if the two chords are equal, then the symmetry of the problem dictates that there can be no liftata@=0 . Thus, the estimate of the lift increment becomes: c h AC, = 2k, 2) [1 + C,) ind L 2 ~ i (57) c where the factor kf, 2) must be determined for the particular geometry considered. The lift slope computed numerically in Fig. 11 corresponds s to a system having a chord ratio a = 40% and an ejector height h = = 15%. The numerical value of coefficient k was determined by a best fit of the curves C, (a = O°, C,;) and AC , which was obtained Lag for k = .1625. The agreement between the two curves validates the interpretation proposed for the mechanism of generation of lift at a@a=0. 4, Comparison of the Lift Increments Due to Angle of Attack with spence's Theory The lift increment due to angle of attack computed numerically by Spence in Ref. 6 can be approximated by ~31- iL ACy = 2nu[1+.151 C,? +.219 C;] a. (58) (no lift at a = 0°) where Cy is the momentum flowing in the powered wake. Fora jet flap, this is just equal to the momentum of the primary Cs =C , as H this model neglects any entrainment in that region. To relate AC Lyp to the lift increments computed in our configuration, we must relate Cy to Cry a. Relation Between Head Change and Momentum Coefficient Referring to Fig. 2, the total momentum flow in the far wake is: 2 M =p a ho where 4 is related to the head change by + pun +AH =p. +d pus Peo j so that ul =oU + 2ZAH P j Pas and 2 M = ho [pUS + 2AH] . (59) For the type of geometries representative of an ejector- h powered jet-flap, the exit height to chord ratio — will be-of the order of 10%. Under these conditions, ‘the results of the numerical ° 32. scheme proposed in Section II show no appreciable slipstream con- h. traction, i.e., a ~ 1. This allows us to neglect the pressure 4 integral on the wake and identify (59) with the total momentum flow at the exit of the ejector: M_ =h_ [pU2 + 2AH] s g LPM oo so that the jet coefficient that will be used in the comparison with Spence's theory will be Mt C,=--—5 # 3pUL¢ h h 1 zpU h. me [1+C,,] (60) To relate Cs to the conventional primary jet momentum C 2 we note that Cry is the force, i.e., change in momentum, exerted per unit height by the actuator disk on the pumped fluid, so that @ « "8g (61) be c -H and Cy = 2 — +2 _- (62) A further implication of this choice is clear when we consider the ejector at zero forward speed Ven = so that (51) => M; = an M and 9 ypu =2 =~ 33— This ideal augmentation coefficient can be achieved only by an ideal mixing device (Ref, 21). b, Comparison of the Lift Increments We are now in a position to compare the lift increments due to angle of attack computed by the two methods. This has been done in Figures 11 and 12 by comparing the lift curve obtained nume rically and the curve C, =C_(a=0°, C3y) FAC, ; where C is ) Lo ~L IF L H also computed numerically, for the following configurations (the (a=0°,C geometries are in Fig. 2-A): Fig. 11. System of unflapped flat-plate airfoils, with height a = 15% and chord ratio -oi = 40% Fig. 12. System of flapped cambered sections corresponding to a configuration tested in the 10-foot tunnel at GALCIT and described in Ref. 34. It is seen that for these widely different configurations the lift increments computed by both methods agree quite well, In addi- tion, the distribution of lift between the two airfoils has been plotted in Figs. 15 and 16 for an actuator disk effectively located at the trailing edge of the system. Both lift slopes are negative, indicating the presence of the upper airfoil stagnation point on its upper sur- face. This again illustrates the sink effect of the actuator on the lifting surfaces, 5. Comparison of the Moment Curves A similar comparison was made for the moment curves of the same configurations as above, the moment increment being also obtained from Ref. 6, =~ 34. For the flat-plate system (Fig. 13), the curves have differ- ent slopes and cross for a value of Cry high enough. For the cambered system (Fig. 14), the magnitude of the moment increment as well as the slopes agree with the moment in- crement computed from Ref. 6. To the extent of this comparison, this indicates that the moment increment is sensitive to the particular geometry consid- ered and cannot in general be reliably computed from the jet flap tmeremient, 6. Conclusions The method of superposition that has been just discussed allows a simple interpretation of the lift curves computed numeri- cally in terrhs of analytical results for single~airfoil configurations. While it does not remove the need to compute numerically the lift ata = 0°, it lends to the numerical solution a flexibility that is inherently lacking in this type of approach. ~35- APPEND A. The Trapezoidal Vorticity 1. Expression for the Flow Field Let o. be the mesh size o. =x. -~ Xe i 1 it] i The trapezoidal vorticity is a linear distribution of vorticity between x. and x. ,. 1 it] yx) = + (BH): (-1) 1 r 1 We want to compute the induced velocity using the Biot Savart law: as: dur = YEJoe fie [oy anu =f =VEVY ax (1-3) 6] L G1. amu = f = “. y(x)dx (1¥4) oO r E Carrying out the integration: a SO, 1 anu. =iI,* [ran (jo )~Tan @| + x i y y i y ra — lie Lx i -x) ty t (ieS] 2. 2 ic 2mrao“2i. 10g] 4 "| + yY (o,-«)“+y i x x+y" ° Vv he -Lx +(T )-): ate Log Sz) -Z-|Tan “( )-Tan “(=) 1 (0, -x) +y i y y = 36s 2. Limit Forms of (1-5) and (1-6) as yo 0 x~O, 1 i (i)x<Oorx>o Tan “( +2 wl XM T T )-Tan i ge = 0 (1-7) .'. atu =0 x i _ — 4 bss aPyJ.1 4% x ih I, Log recs = ag YT) 1+ om Log De | Same holds forx > Oo, (ii) O<x<o, 1x 1 *7 9% T T Tan i Dens rs Jrtgt (ts) sit So that u(0<x<o,, y=+0)= + 3 yx) (1-8) (iii) Expression for the downwash In computing the downwash, we require “= 0 so that - -™ wh jen eee aol one, = hy DOs ee |+ sag ee -1 + o, cei lg-x | and in particular (1-9) evaluated at the midpoint yields: 1 oF it = = a = 3. Comment It should not be concluded that the field at the end of these segments x = 0 is inherently singular, since the downwash is com- puted as v = ¥. tv although, should one need the downwash at i other that particular point to interface with a complete integral model of the turbulent flow outside the ejector additional numerical sophisti- cation would be needed. oe B. Field of the Leading Edge Singularity The Kutta~-Joukowsky flow about a lifting flat plate is repre- sented by w(z) = u-iv if 0 ~ ch (1-11) Z where _ = sina for an isolated flat plate. Using the angles in Fig. 5, this can be decomposed into real and imaginary parts Ro = (l-x)* + . a” ag? 4g” (1-12) Then ai 2 u = ae [—— siny] (1-13) r2 y20 i ve BS [1~ 7 cosy] (1-14) re 0-0 where y = _ 4 The downwash on the chordline y = 0, 0S x € 1, is constant: v e Tes Outside this interval, on the line through the two branch points, we have ad x-1 vel: [2 - |, x<Q and x21 ~38 - In addition, the vorticity distribution can be deduced from (11) and is, = ]-x yp(0 <x <1) = 215° -/ (1-15) with the square root singularity at the leading edge. The complex function (1-ll) represents the leading edge singularity for any thin airfoil as well as for a flat glans, provided the branch cut is moved to coincide with the camberline. In the particular scheme we used for the numerical solution, the branch cut was not moved, so that the contribution of the leading edge singularity to the total downwash on its own airfoil, given in equation (11), is just Tye and the vorticity is (1-15). These two results are linearized approx- imations. The nonlinear results could have been incorporated ata slight increase in the complexity of the calculation. A consistency. check was made by computing the velocity components directly from the computed vorticity distribution equation (15), and they satisfied the imposed boundary condition within three significant digits. Thus, in this particular example, this approximation was well justified. N 10. BOs REFERENCES Herold, Alan C., A Two-dimensional, Iterative Solution for the Jet Flap, NASA CR-2190. Chaplin, H. R., A Method for the Numerical Calculation of Slipstream Contraction of a Shrouded — Disk in the Static Case, DIMB Rept. 1857, June 1964. Greenberg, Michael D., Powers, Stephen R., Non-Linear Actuator Disk Theory and Flow Field Calculations, Including Non Uniform Loading, NASA CR-1672., von Karman, Th., Burgers, J. M., Aerodynamic Theory, Durand, ed., Vol. Il, Perfect Fluids, Section 10. Spence, D. A., The Lift Coefficient of a Thin, Jet Flapped Wing, Proc. Royal Soc. A 238, 1956. Spence, D. A., Some Simple Results for 2-D Jet Flap Aero- foils., Aero Quarterly, Nov. 1958. Wyegnanski, IL, Newman, B. G., The Effect of Jet Entrain- ment on Lift and Moment for a Thin Airfoil with Blowing, Aero Quarterly, May 1964. Shollenberger, Carl A., An Investigation of a 2-D Propulsive System, Calif, Inst. of Technology, Ph.D. Thesis, 1971. Weissinger, Johannes, Maas, Dieter, Theory ofthe Ducted Propeller - A Review. Article in 7th Symposium on Naval Hydrodynamics, Aug. 25-30, 1968. Yen, K. T., On the Thrust Hypothesis for the Jet Flap, Including Jet-Mixing Effects, Journal of the Aerospace Sciences, Aug. 1960. 2 ae Les lode 14. LB. 16. Ley 18. 19. BO. 21. we De REFERENCES (Cont'd) Stratford, B. S., Early Thoughts on the Jet Flap, Aero Quarterly 7, 1956. — Levinsky, E. s. , etal., Lifting Theory for V/STOL Aircraft in Transition and Cruise L, J. of Aircraft, Vol. 6, No. 6. Spence, D. A., The Lift Coefficient of a Thin Jet-Flapped Wing. Il, A Solution of the Integro-differential Equation for the Slope of the Jet., Proc. Roy. Soc., Series A 261, Stratford, B. S., Mixing and the Jet Flap, Aero Quarterly, Vol. VII, May 1956, Stratford, B. S., A Further Discussion on the Mixing and the Jet Flap, Jassaman, P, B. &, A Aeesense (ella for the Jet Flap in Ground Effect, Ph.D. Thesis, Calif. Inst. of Technology, 1965, Lopez, M. L., Shen, C. C., Recent Developments in Jet Flap Theory and Its Application to STOL Acrodynamic Analysis, AIAA Paper 71-578, Stewart, H. J., Private Communication. Quinn, Brian, A Wind Tunnel Investigation of the Forces Acting on an Ejector in Flight, ARL Rept. 70-0141. Fancher, R. B., Why Ejectors for Aircraft Propulsion-Lift Systems and Where We Stand, ARL Rept. 71-0140. von Karman, T., Theoretical Remarks on Thrust Augmenta- tion. Reissner Anniversary Volume, J. W. Edwards, 1949. Biles 23. Lbs 25. 26. 2 tx 28. 29. 30, 31. Bes a4]. REFERENCES (Cont'd) Steady Flow Ejector Research Program. Final Report Contract Nonr-3067(00), Dec. 1960. Curtet, Roger, Thése, Faculté des Sciences de l'Université de Grenoble, Nov. 1958, Publ. Sc. and Tech au Min de 1'Air, No. 106. Rose, James R., Ph. D. Thesis, Calif. Inst. of Technology, An Ancdpeks of the 2-D, Incompressible Jet Ejector, 1969, Hill, P. G., Turbulent Jets in Ducted Streams, JFM 22, 1965, Koenig, David G., Corsiglia, Victor R., Aerodynamic Char- acteristics of a Large Scale Model with an Unswept Wing and Augmented Jet Flap, NASA TN D-4610, 1968. Cook, A. M., Aiken, T. N., Low Speed Aerodynamic Char-~ acteristics of a Large Scale STOL Transport Model with an Augmented Jet Flap, NASA TM X62017, 1971. Flight International, Boeing's AMST Proposal, 8 Feb. 1973, p. 210, Perry, D. H., A Review of Some Published Data on the External-Flow Jet-Augmented Flap, Aeronautical Research Council Current Papers, CP No. 1194, London, 1972. Lachmann, G. V., ed., Boundary Layer and Flow Control, Vol. I, Pergamon Press, London, 1961. McCormack, Barnes W., Jr., Aerodynamics of V/STOL Flight, es Press, New York, 1967. Harris, G. L. and Lissaman, P. B. S., The Mechanics of Ejector Thrust Augmentation, CIT Ae TR 67-1, 1967. em 34, iP REFERENCES (Cont'd) Newman, B. G., The Prediction of Turbulent Jets and Wall Jets, Canadian Aeronautics and Space Journal, October 1969, GALCIT Report 926 (to be published), BS a Blowing Slots Kiera VTOL Augmentor Wing STOL Augmentor Wing (Ejector Powered Jet Flap) FIG. POWERED LIFTING SYSTEMS 44. AMLSWOS9 DISVG :-NW318G0ud MO1d WILNSLOd 2 Sls LOLDNLOY Tr NOIMLOSS Ni GSSN SSINLSNOSS OlsloOadS %-2°OlAs 926 jodey LID TVS ‘UolsDunB1yU09 JSS] JO OA BLUSSeIdsy WoskS PaIsquiDd ti, ebuly , f JQ papnso4 | ———— giz -£99 VON GIVRVOVN jojS Buimojg pnoius. PIOUS % OL volDIS sBuily dojy WasskS S1Djg 4IDIJ al =/y Ov = Yn r—s Vortex Sheet Element Of Intensity ¥ (o) \ wer —— te op S Sem x FIG.3 POISSON SOLUTION TO V#u =O ~ fe ¥ (x) h | nla) Pr; , Sm O o; Xx Downwash Distribution I (Ty-1-Tj) eT | | FIG.4 DOWNWASH OF THE TRAPEZOIDAL VORTICITY ALIYVINONIS SSON 3O G1slda HSVMNMOT G Old J y2(x) 4 KPO sOA e Ate oem aes ene Ge Ge eet Gn ee tes ae ‘ ¥ [ 2 =f PLILL LILIA L LIT IMU VIE Te lg YSDMUMOG (A°X) W (TiOSYIV YSMO071) ALIOILYOA SHL SO NOILVLNSS3ud3y TVOIWSNAN 9 ‘Sls OyDOA 404 sUI0d $}01ND) Ox a6pQ Buryoap 2Duly dp}4 ¥ | | SUI|pJoUD uDEYW sBp3 sia ciniel, \ mf t tens } Splozedbly Oy nt (UoHIPUOD DHA) AT stT 4 FEI . splozscps) \ lop Uday oe. THOAYIV NIVAN NO ONILOVY SLNSWON/S3980S 2°9ls lly ly uolons e65p3 Buipos lt e tk wt I+! M5 e6pq Buippe 4 jnoqy juswoy Uoo Moment Due To Actuator Disk Ciag yt CH hed » _ Force Due To Actuator Disk ra \ fas ait (a) Force And Moment System Due To Actuator Disk Contribution Mo(S,e) \ \ cont Edge Suction (Xp 29, ) Normal “ Force Contribution CMe (Xo Yo) (b) Moment System Due To Upper Airfoil FIG. 8 Airfoil Coordinates ¥ Flow Field Ch=0 Compute H , i” G=H" [tan a(x] ! Input CH, Wak Coefficients And Initial Guess we a Compute Velocities | Induced On Airfoil System By Present Wake Update Airfoil Vorticity G* it. [to n o(x) + ue tan a(x) —vy tan ay! y Compute Velocities Induced By Airfoil System And Old Wake At Old Wake Position \ Upadgte Wake Vorticity And Position Compute Aerodynamic “| Coefficients f & Start New Itcration FIG.9 POTENTIAL FLOW PROGRAM FLOW CHART ay Bee ee Suction Slot M “2Q ¥ G a\ ae EL MEET AE A Uo FLAT PLATE AIRFOIL WITH SUCTION SLOT -Q M .— Td Avy, Net Velocity \\s Induced By Source-Sink K Oo “\t sy stem senna Staaqnation Point} TRANSFORMED FLAT PLATE AND SINK~- SOURCE SYSTEM FIG.1IO0 EFFECT OF A SINK ON A FLAT PLATE AIRFOIL 2.2 2.0 Cy TOTAL LIFT FLAT PLATES i ~ COMPUTED O NUMERICALLY Vv ad LIFT INCREMENTS FROM SPENCE'S THEORY EQ (57) k=0.1625 2 {0 12 14 16 ! { ! 1 \ 1 Cy iQ 2.l 3.3 4.5 I j i I Cy TOTAL. LIFT, FLAT PLATE SYSTEM (M7, = 15 %) FIG. {I 7.0 6.0 5.0 “=O CL - TOTAL CAMBERED SYSTEM LIFT COMPUTED NUMERICALLY LIFT INCREMENTS FROM A a} oO $ x J SPENCE'S THEORY 6 8 10 12 14 1 1 I ! | BS] i 2.4 3.3 4.5 i 1 Lee Cy TOTAL LIFT, CAMBERED SYSTEM (M/e= 15%) FIG. 12 POSITIVE HOSE GOWN FLAT PLATES © S06" a} COMPUTED NUMERICALLY O v} LIFT INCREMENTS FROM xJ SPENCE'S THEORY ae Bee" 15 - a=0° a C fs 4 6 g io “Ht fe) j i] 1 j J J 9 2.1 33 | I | C) TOTAL MOMENT, FLAT PLATES FIG. {3 5.0 4.0 3.0 Cr - TOTAL CAMBERED SYSTEM MOMENT (POSITIVE NOSE DOWN) a=6° a=2° - at 4 | COMPUTED . NUMERICALLY LIFT INCREMENTS ¥ FROM SPENCE'S THEORY 2 4 6 8 10 i2 140 Sy I j a 1 1 J i 9 2.1 3.3 45 j i i I} TOTAL MOMENT, CAMBERED SYSTEM FIG. 14 8.0 CL LOWER AURFOIL ' FLAT PLATE 4 | comeuteo oO f NUMERICALLY -2.0- -A.0- -6.0/" CL. -gol UPPER AIRFOIL FLAT PLATE LIFT DISTRIBUTION, FLAT PLATES (ACTUATOR DISK AT TRAILING EDGE FIG. 15. 8.0 ~ 20 CL LOWER AIRFOIL CL UPPER AIRFOIL LIFT DISRIBUTION,CAMBERED SYSTEM ACTUATOR DISK AT TRAILING EDGE FIG. IG