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A new slotted-wall method for producing low boundary corrections in two-dimensional airfoil testing Williams, Christopher Dwight 1975-02-06

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A NEW SLOTTED-WALL METHOD FOR PRODUCING LOW BOUNDARY CORRECTIONS IN TWO-DIMENSIONAL AIRFOIL TESTING by CHRISTOPHER DWIGHT WILLIAMS B.A.Sc., University of British Columbia 1967 M.A.Sc., University of British Columbia 1973 THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1975 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my writ ten pe rm i ss ion . Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date t.]> Qdi9kf 1916 SUPERVISOR: Dr. G.V. Parkinson ii AB3 TRACT This thesis deals with a new approaca to reducing wiadtunnai wall corrections in airfoil testing, by employing a transversely-slotted wall opposite the suction side of the airfoil, and a solid wall opposite the pressure side. The solid elements of the slotted wail are symmetrical airfoils at zero incidence. This geometry permits the flow to assuaa closely the streamline pattern for unconfined flow, without degrading the flow quality through shear layer mixing near the test airfoil. The theory uses the potential flow surface source-element method, with Kutta conditions satisfied on the test airfoil and the wall slats. In experiments using a range of sizes of airfoils of three different profiles, good agreement with the predictions of the theory has been obtained. It appears that uncorrected lift coefficients and pressure distributions, accurate to within one percent, can be obtained for a wide range of airfoil shapes, sizes, and lift coefficients, using a slotted wall of open-area ratio between 60 and 70 percent. iii Cette these d_crit une nouvelle inethode en vue de diminuer les corrections da parois en soufflerie aux essais des ailes. Cette methods einploie un taur a. fente transversale, en face du flanc de depression de l'aile d'essai; et un mux solide, en face du flanc de pression. Les elements solides du' mur a, fente sont de profils aerodynamiques et syraetriques a 1'angle d'incidence zero. Cette figure geotuetrique permet l'ecoulement d'air, suivie de pres par les lignes de courant. de l'ecoulement libre; ce resultat est obtenu sans diiainuer la gualite de l'ecoulement par le melange de la couche de discontinuite, pres da l'aile d'essai- La theorie demande d'utiliser l'ecoulement a potentiel des elements de la source de surface, satisfaisant les conditions Kutta a l'aile d'essai et aux ailes murales. Les experiences ont ete farts en eaployant. des ailes de corde de trbis profils differents. Les theories obtenuas respectent assez bien les hypotheses praetablies. Les coefficients portances non verifies, et les distributions de pressions sont exacts a un pourcent pres; et ils peuvant etre obtenus pour une qrande variete de profils, de dimensions at de coefficients portances; en utilisant un mur a fente, de quotient entre la paroi et la surface totale, de soixante a soixante-dix pourcent. IV ACKNOWLEDGEMENT This research was carried out under the supervision of Dr. G. V. Parkinson, whose expert advice and guidance is gratefully acknowledged. In the design and construction of the viadtunnel models and equipment, the work done and the advice given by the technicians of the Mechanical Engineering Department was extreaely valuable. All the computing was done at the U.B.C. Computing Center. This research was supported by the University of British Columbia and the Defence Research Board of Canada. Encouragement and support were provided by F.M.W. , who contributed more than her share in our joint effort. Table of Contents Abstract Resume Acknowledgement List of Figures List of Plates List of Tables Notation Introduction Survey of Windtunnel Wall Correction Methods §2.1 Conventional Linear Theories §2.2 Results of Conventional Linear Theories §2.3 Low Correction Test Configurations A New Slotted-Wall Theory §3.1 A Physical Basis for the New Theory §3.2 Formulation of an Exact Numerical Theory §3.3 Other Airfoil-Wall Configurations Examined Methods of Numerical Solution §4.1 Assembling the Equations §4.2 Solving the Equations Results of the New Theory Experiments to Verify the New Theory §6.1 Testsection Design §6.2 Airfoil Models Tested §6.3 Test Procedures vi 7. Experimental Results 54 8. Extensions to the New Theory 60 §8.1 Potential Flow Considerations of Viscous Effects 62 §8.2 The Flow in the Plenum: The Bounding Shear Layer 69 §8.3 Summary 76 9. Conclusions 7 Appendix 1. The Integration of a Three-Dimensional Point 79 Source to a Two-Dimensional Flat Distributed Source Element Appendix 2. A Procedure for Block Computation of 84 Matrices A, B, and C. Appendix 3. Two Methods of Solving the Systems of 91 Simultaneous Linear Algebraic Equations Appendix 4. A Streamline Tracking Algorithm 96 Appendix 5. Design of the Two-Dimensional Nozzle Insert 99 Appendix 6. An Analytic Representation of a Lifting 10 3 Vortex Between a Solid, a Transversely-Slotted and a Constant Pressure Boundary Appendix 7. Standard Solid Wall Corrections 111 Appendix 8. A Reduced Airfoil Circulation Determined 113 from the Measured Lift Appendix 9. A Reduced Airfoil Circulation Determined 116 by Modifying the Profile Appendix 10. The Computer Program for the Exact 118 Numerical Theory Appendix 11. List of Equipment Used 152 Figures 153 Plates Tables References viii List of Figures Figure 2.1 Figure 2.2 Figure 2.3 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 5.1 Figure 5.2 Figure 5.3 Figure 6.1 Porosity parameter as a function of wall open-area ratio for longitudinal slots [12] Ratio of airfoil lift-curve slopes for longitudinally slotted walls: Experiment[12] Variation of pressure coefficient along a straight boundary: Theory An airfoil between transversely-slotted upper and solid lower walls: Theory Wall effect on airfoil pressure coefficient: Theory Geometry and notation of two-dimensional source elements for Smith's method Surface velocity variations for a two-dimensional source element Effect of airfoil size on ratio of lift coefficients: Theory Comparison of airfoil pressure coefficients: Theory Variation of pressure coefficient along a straight boundary for a two-dimensional airfoil with zero lift-correction: Theory U.B.C. Mechanical Engineering low-speed closed-circuit windtunnel Page 153 154 155 156 157 158 159 160 161 162 163 Figure 6.2 Variation of mean windspeed in two-dimensional testsection insert on vertical pitotstatic traverse 164 Figure 6.3 Velocity profile in floor boundary layer in two-dimensional testsection insert 165 Figure 6.4 Effect of endplate loadings on lift, drag and pitching moment coefficients for two-dimensional airfoil tests 166 Figure 6.5 Calibration of nozzle and testsection dynamic pressures 167 Figure 6.6 Error bars for measured airfoil lift coefficients 168 Figure 6.7 Variation of measured airfoil lift coefficients on three consecutive runs 169 Figure 7.1 Variation of airfoil lift coefficient with slotted-wall open-area ratio: Experiment 170 Figure 7.2 Effect of slotted-wall open-area ratio on ratio of lift-curve slopes for Clark-Y airfoil 171 Figure 7.3 Effect of slotted-wall open-area ratio on ratio of lift-curve slopes for NACA-0015 airfoil 172 Figure 7.4 Effect of airfoil size on lift-curve slope for NACA-0015 airfoil: Experiment 173 Figure 7.5 Effect of airfoil size on lift-curve slope for Clark-Y airfoil: Experiment 174 X Figure 7.6 Figure 7.7 Figure 7.8 Figure 7.9 Figure 8.1 Figure 8.2 Figure 8.3 Figure 8.4 Figure 8.5 Figure 8.6 Figure 8.7 Effect of slotted wall on airfoil pressure coefficient: Experiment Comparison of airfoil pressure coefficients: Experiment Variation of airfoil midchord pitching moment coefficient with slotted-wall open-area ratio: Experiment Variation of airfoil drag coefficient with slotted-wall open-area ratio: Experiment Effect of reduced circulation on airfoil pressure coefficient: Theory (Appendix 8) Modification of airfoil profile to reduce theoretical circulation to measured value: Theory (Appendix 9) Effect of modified profile on airfoil pressure coefficient An airfoil between a slotted upper and a solid lower wall with a plenum chamber The shear layer in the plenum chamber surrounding the slotted wall Effect of different types of wall boundaries on ratio of lift coefficients: Theory Variation of pressure coefficient along a streamline in plenum chamber: Theory 175 176 177 178 179 180 181 182 183 184 185 xi Figure 8.8 Effect on airfoil lift coefficients of assumed pressure coefficients on a streamline representing the plenum shear layer: Theory 186 Figure Al.l Geometry for integration of a point source 187 Figure A5.1 The two-dimensional nozzle insert 188 Figure A6.1 A lifting vortex between a solid, a slotted, and a constant pressure boundary: Theory 189 Figure A6.2 The image system for a lifting vortex between a solid and a constant pressure . boundary: Theory 190 Figure A10.1 Notation for the computer program of Appendix 10 191 XXX LIST OF PLATES Plate la. Plate lb. Plate 2. Plate 3. Plate 4. Plate 5. Plate 6. Page The U.B.C. Mechanical Engineering low-speed closed-circuit windtunnel The octagonal testsection in the windtunnel The airfoil-shaped wall slats The wall slats in the side wall frame The 616mm NACA-0015 airfoil The 354mm Clark-Y airfoil fitted with endplates 194 The 616mm NACA-0015 airfoil in the testsection 195 192 192 193 193 194 xixi List of Tables Page Table 1. Airfoil profile coordinates 196 Table 2. Free air airfoil coefficients: Theory 197 Table 3. Airfoil and wall configurations examined 198 theoretically Table 4. Airfoil and endplate loadings 203 Table 5. Windtunnel balance results - Clark-Y airfoils 204 Table 6. Windtunnel balance results - NACA-0015 219 airfoils Table 7. Windtunnel balance results - Joukowsky 234 airfoil Table 8. Quantities derived from balance results 238 Table 9. Pressure coefficients for NACA-0015 airfoil 243 Table 10. Pressure coefficients for Joukowsky airfoil 246 Table A10 Equations for the computer program of 249 Appendix 10 xiv Notation A.., B.., C.. matrices of disturbance velocities. Di Di Di c airfoil chord c. . element of the matrix C.. Ji 31 C_ = D/qc drag coefficient C = L/qc lift coefficient C = Mo/qc2 midchord pitching moment Mo CM = Mc/qc2 quarterchord pitching moment t ' * Cn = measured pressure coefficient v q C_ average pressure coefficient on streamline C = 1-(vt /u)2 calculated pressure coefficient i d. element of column vector of approach flow 1 boundary conditions ds_.,dXj,dy^. surface element length differentials H windtunnel testsection height (or total head) K(s) porosity parameter for longitudinally slotted walls m = lift-curve slope it outward surface normal OAR transversely-slotted wall open area ratio p local static pressure P(s) porosity parameter for porous or perforated walls q = -ipU2 dynamic pressure r. .,r(PQ) distance between surface elements Re Reynolds number XV s distance along the surface As. length of surface element 3 U magnitude of approach flow velocity V.,V ,V. ,V ,V : velocity induced by a surface element ln.'t.x.y. 2 * . _^ x x 11 approach flow velocity, magnitude U x.,y. Cartesian axes fixed to the j-th surface, element 3 3 X,Y • wind axes, with X-axis in the flow direction x aerodynamic center distance ac x0 midchord distance a airfoil incidence r circulation Y ,v , ;•• surface vortex element strength density °yVy surface source element strength density <j> disturbance velocity potential 8 inclination of surface element w.r.t. X-axis 3 p fluid density dC T = ^-Mq midchord pitching moment curve slope \\) stream function Subscripts: F free air value S solid wall value L lower surface t tangential direction n normal direction T windtunnel value s streamwise direction U upper surface oo upstream- undisturbed- flow condition —•-1 __. Introduction. In the subsonic windtunnel tastin.g oi airfoil sections, the existing theory for corrections to the measured data for the effects of windtunnel wall constraints is satisfactory for windtunnels with solid walls in which the test airfoils are small relative to the testsection crosssections, and develop relatively small lift coefficients. However, current research on high-lift airfoil sections requires testing at very high lift coefficients, and the use of relatively large models to give sufficiently high Reynolds numbers. Under these conditions, the wall corrections in windtunnels with solid walls may become unacceptably large, unless windtunnels with very large test crosssections are available- Such windtunnels are expensive to build and operate, so a method of modification of existing smaller windtunnels, that would reduce or eliminate these large wall corrections, would be most desirable. Since the major corrections to measured data in windtunnels with solid walls are of opposite sign from those for windtunnels with open jets, an obvious possibility to explore is the use of partly solid-partly open walls, to produce cancelling effects. Two such forms of windtunnel wall have been considered in.recent years for this purpose, walls with narrow longitudinal slots, and walls with a pattern of small holes. Theories have been presented for prediction of the corresponding wall corrections. Unfortunately, experiments have shown that the existing 2 theory for walls with longitudinal slots is useless for the present purposes, and the theory for porous walls is impractical to apply. An empirical porosity factor is needed, which depends on the wall geometry and on the test model. In the present thesis, a two-dimensional potential flow theory is developed for a different partly solid-partly open wall system, and the results of experiments designed to test the theory are presented. 3 2-. Survey, of Windtunnel Wall Correction Met hod s.. 2^1_ Conventional Linear Theories. In windtunnel testing of two-dimensional airfoil sections at subsonic speeds , the windtunnel wall constraint influences the measured forces, moments, and pressure distributions. The current theories for the corrections to be applied to the measured values to account for wall effects are satisfactory only when the test airfoils are small relative to the windtunnel crosssection, and develop small lift coefficients. Current practice for such cases is well summarized in a report by Garner et al [ 1 ]. The essential feature of current wall correction theories is the selection of an appropriate system of source, vortex and doublet singularities, together with images of the singularities in the windtunnel boundaries, such that the flow conditions at the boundaries are satisfied. The lift-producing characteristics of the airfoil (incidence, camber), the airfoil thickness, and the airfoil wake, are associated with distributions of vortices, doublets and sources respectively. The characteristics of the windtunnel walls are then simulated by associating an appropriate set of images with each singularity in .the field. For example, a solid wall boundary condition requires zero disturbance velocity normal to the boundary. This condition may be simulated by an infinite set of images in the windtunnel walls; sources, and doublets oriented in the streamwise direction, have images of the same sign; vortices have images of alternating sign. For further details, see Allen and Vincenti 4 [2], and Goldstein [3]. When the net effect of the systems of singularities and images has been calculated, the velocities thereby induced at the airfoil may be determined. For simplicity both in interpretation and application, it is useful if the net effect of all the systems is the direct sum of the induced effects of each of the individual systems. In terms of the corresponding field equations, a superposition is possible if the exact boundary conditions may be linearized. The linear approximations in turn are valid only for small induced velocities, which implies a small, thin, slightly cambered airfoil at low incidence. An alternative to the images technique, which yields similar results, is the conforraal mapping technique developed by Woods [4], The problem . of finding velocity, potentials in a complex domain bounded by an airfoil and windtunnel walls is transformed, by. conformal mappings, to an equivalent, but geometrically simpler, boundary value problem- The problem is thus reduced to the determination of,, an analytic function on a transformed domain whose real and/or imaginary parts are prescribed on the boundary. Woods' technique results in integral equations for the mapping functions, and usually numerical methods are required for their solution. A linearized form of his theory agrees with the results to be found in Garner et al [1]. For the case of a thin flat plate between parallel boundaries, the solutions of Havelock [5] and Tomotika [6] are available uses that Havelock uses the method of conformal transformations. of images while The results are 5 Tomotika similar. In recent publications, de Jager and van de Vooren [7], and de Vries and Schipholt [8], use image methods for thin airfoils with hinged flaps, between solid windtunnel walls. For porous or perforated windtunnel walls, Baldwin et al [9] propose an "equivalent homogeneous wall boundary condition". dary condition is a combination of the for solid walls and for an open jet. ndary conditions ar e expressed in terms of a oci ty potential which for incompressible ace's equation. T he general linear wall ay be written: «(»>!!•+ »!.>!!• cowsS - o. For a solid boundary where there is zero disturbance flow normal to the boundary, equation (2.1) has the form !*• = 0 • . (2.2) 8n . For an open jet boundary it is assumed that the disturbance from the test airfoil at the jet boundary is small. The linearized condition of constant pressure can be interpreted, using Bernouilli's equation, as requiring zero streamwise disturbance velocity. Hence for an open jet boundary, condition (2.1) becomes ' 6 3* „ ' For porous or perforated walls the pressure drop across the wall due to the cross-flow is assumed to be proportional to the normal disturbance velocity at the wall. The resulting linear relation between the cross-flow and streamwise disturbance velocity components requires A similar expression, but with a different value of P{s), is used for a wall where the porosity consists of transverse slats. For an infinite length testsection wall of closely spaced transverse slats, Maeder and Hood [10], and Woods [4] deduce a constant value for P (s) : P = tang) . (2.5) •2d In equation (2.5), »a' is the slot width and 'd» is the slat spacing; a/d is the open area ratio (OAB). For walls with longitudinal slots, a potential flow model of the cross-flow through the slots requires the disturbance potential to vanish and the pressure to be constant, in the slots. Hence 7 |i + K(s)^ =• 0 . • •'• • (2.6). 8s 3s8n ' Maeder and Wood [10] give, for an infinite length testsection wall of uniformly spaced longitudinal slots, the value K = £ log csc(|§] , ' (2.7) where a/d is again the OAS. 8 2.Z.Z. E^sults of Convantional Linear Tneories_. In the theoretical determination of homogeneous boundary conditions, the details of the slot or hole geometry can be, used to construct an exact boundary condition by, for example, the application of Kutta conditions to the edges of slots or holes. Then by examining the flow through the wall from a point many slot (hole) widths away from the walls the flow details due to the wall geometry are not felt, but only some "averaged" flow field is detected. The exact boundary condition is thus replaced by a linearized, averaged boundary condition. In applying this boundary condition, the wall is regarded as being geometrically homogeneous. The advantage is that a single averaged boundary condition can be applied uniformly over.the plane of the wall so that it is not necessary to have separate boundary conditions applied in slots (holes) and on solid sections. This averaging effect explains why the wall boundary condition for porous, perforated and transverse slotted walls are similar. For details see Maeder and Wood [ 10 ]. The same "averaging" effect, if applied to longitudinal slots, leads to erroneous predictions. In fact, the longitudinal slots render,the flow three-dimensional by imposing spanwise variations on the basic two-dimensional flow conditions. On the assumption that, the flow is quasi-plane, that is the spanwise variations are only a small perturbation of the basic two-dimensional flow, an averaged.boundary condition for the basic two-dimensional flow can be deduced. For details see Woods [4]. In the use of such geometrically homogeneous linear 9 boundary conditions, all details of slot or perforation geometry, are lost, in particular, their orientation (longitudinal or transverse). Only the effects of bulk properties such as -the porosity or-CAE are retained. Wood [11] shows, for example that the OAR for longitudinal slots would need to be less than 1%, to achieve a boundary condition appreciably different from.-;, the open. jet case. In practice, at such a small OAR, real fluid effects would be important, so a potential flow model for the cross-flow would not be valid. Moreover, Wood's analysis of this boundary condition indicates that only cross-flow velocities of less than 0.5% of the mean flow would be in keeping with the arguments for the linearization of terms involved in the derivation of this boundary condition. Investigations by Parkinson and Lim [12], and Mokry [13] have found that the "porosity parameter" P(s) is not simply an empirical function of wall OAR, but must be determined empirically for each airfoil under test. The usual procedure is to choose a value of P(s) to match lift or pressure data taken at a particular incidence and for a particular size of airfoil and then to try to use this same value of P(s) to calculate the wall effect at other incidences and for other sizes of airfoil. Generally, the results are that P(s) depends on the wall OAR and the particular airfoil under test, an impossible situation for the practical use of such linear porous wall boundary conditions. Figure 2.1 from [12], for example, shows that for two different airfoil profiles tested, there are two completely different variations of "porosity parameter" with OAR, and neither agrees with the theoretical variation of relation (2-5). 10 Other results by Parkinson.and Lim [12], Parker [14], and Tsen [15], have shown that the theory for the longitudinal wall slot parameter K(s) is not useable. Figure 2.2, from [12], for example, shows that for four airfoils of different size, of the same profile, the theoretical wall interference curves corresponding to the values of wall OAR tested, are closely grouped as though all of the wall configurations were effectively open. Catherall [16] states that "the usefulness of the method is limited by the doubts about this (linear homogenous wall boundary) condition". This is clearly the case for measurements on high lift devices where the predicted wall corrections are of the same order as the measured values themselves. The great disadvantage of more physically appropriate nonlinear wall boundary conditions is that the mathematical solutions for most boundary problems depend on such boundary conditions being linear as, for example, in complex variable theory. Wood [11] has developed a nonlinear boundary condition for a two-dimensional Helmholtz jet issuing from longitudinal slots, where the "porosity" is a function of the cross-flow velocity. His analysis is for a nonlifting airfoil; the extension to the case of a lifting airfoil does not appear to have been made. Sears [17] comments that: "Even in those (flow) regimes where the flow perturbations due to tunnel boundaries can be estimated, there is a basic flaw in the idea of "correcting" 11 measured aerodynamic data, because such correction requires that the effects of such perturbations be known. If the field of extraneous velocities is other than a uniform change of incidence, than in some of the most important technical cases these effects are not known and cannot .be calculated". Figure 2.2, from [12], also shows that the theory for solid walls gives excellent agreement with the data. Therefore in the absence of improvements to the theory for slotted- or perforated-wall corrections, it seems advisable to carry out low-speed two-dimensional airfoil tests, even for large models developing high lift coefficients, in conventional windtunnels with solid walls. 12 2^.3 Low Correction Test Configurations. An alternative approach is' to modify the walls of the windtunnel to provide flow conditions as close as possible to a free-air (unconfined) test environment. Thus the wall', corrections would automatically be small. One approach is that of the "self-correcting" windtunnel [17 ] whereby an array of sensors (located on a convenient "control surface" inside the tunnel but not in the wall boundary layers) measure, say, the flow speed and inclination there. A calculation is performed to determine if these measured values are compatible with previously calculated values of the same variables for an imaginary infinite inviscid flow field about the test airfoil. If not, adjustments are made, iteratively, until such conditions are met. This could be achieved through the use of flexible walls, and/or wall sections of variable porosity. Obvious disadvantages are the cost of "on-line" computing facilities plus the large number of pressure transducers required to extract and process the flow measurements. Whatever type of testsection is chosen, the "porosity" variation must produce results like those of Figure 2.3, in order to simulate correctly the free-air flow field. How this is accomplished mechanically is up to the windtunnel designer. 13 Is. h New Soltted-Sall Theory. 3.1 A Physical Basis for the New Theory. One reason for the lack of success of the longitudinal-slot and porous-wall theories is the occurrence experimentally of separated flows in the slots and holes. Such flows are not accounted for in the theories, primarily as they add undesirable nonlinearities to the theories. In addition, these flow separations seriously degrade the main flow in the vicinity of the walls. The approach here (see Figure 3.1), uses transverse wall slots, with symmetrical airfoil-shaped solid slats. The flow inclinations near the wall will be small even for a nearly unconfined flow field. Hence all the wall slats will operate within their unstalled incidence range, so that flows near the wall will be free of separated wakes. Moreover, only the wall opposite the negative pressure side of the test airfoil is slotted. One reason for this choice can be seen from Figure 3.2 which compares theoretical pressure distributions (calculated by the methods of section 3.2) between solid walls and in free air, for a 14% Clark-Y airfoil. : The ratio of airfoil chord 'c' to windtunnel testsection size *H,-is large at 0.72, and the incidence * a* is extreme at 20 degrees, to create a large wall effect. These pressure distributions show that almost all of the wall effect is to increase the magnitude of the negative pressure on the upper surface of the airfoil. The effect on the underside pressure is so small as to be negligible, even in this rather extreme case. 14 Another reason for using only one slotted wall is to simplify the flow field opposite the pressure side of the test airfoil. A slotted lower wall would allow an inflow of low energy air from within a plenum chamber (surrounding the slotted wall) to enter the testsection upstream of the airfoil. This inflow would consist of a shear layer and its associated turbulent mixing and would degrade the quality of the, main flow, in the vicinity of the airfoil lower surface. There will be a corresponding outflow from the test section back into the plenum downstraam of the test airfoil. On the other hand, on the wall opposite the negative pressure side, upstream of the test airfoil, there will be an outflow from the testsection into the plenum. The shear layer so formed and its associated turbulent mixing will be shielded from the test airfoil by the presence of the airfoil-shaped wall slats with their boundary conditions impressed on the flow. If this shear layer were idealized as a constant-pressure free streamline, any incorrectness in pressure or location in such a representation of this streamline should have only secondary effects on the test airfoil. The plenum air will enter the testsection downstream of the test airfoil; however, its effect on the test airfoil by entering there will be much smaller than for any air entering the testsection upstream of the test airfoil. As with irrotationa 1 for low-speed most windtunnel wall correction theories, flow is assumed, and, since the method is designed, high-lift testing, an incompressible potential 15 flow method can be used. The test airfoil (and its component flaps) and the airfoil-shaped wall slats are all treated as lifting airfoils. Hence the flow satisfies the usual tangent-velocity and trailing-edge Kutta conditions. The flow ' past the solid wall sections satisfies the tangent-velocity boundary condition. 16 3.2.2 Formulation of an Exact Numerical Theory... The formulation here is a two-dimensional potential, flow theory, based, on the surface singularity distribution method of A.H.0. Smith and his colleagues [18]. In this method, the surfaces of the solid walls, the airfoil-shaped slats in the slotted wall, and the test airfoil with its component flaps, are represented by a distribution of source and vortex elements. A normal-velocity boundary condition will prescribe either zero normal velocity, on solid surfaces, or non-zero normal velocity, for suction or blowing there. Source elements are therefore distributed over any surface on which a normal-velocity boundary condition is specified. Vortex elements are used to set the net circulation about a closed lifting body. Therefore vortex elements are distributed over any surface on which a tangent-velocity boundary condition is specified. Hence source elements only are distributed over the solid wall sections, while both source and vortex elements are distributed over the surfaces of the airfoil-shaped wall slats and the test airfoil and its flaps. The velocities at any point in the flow field due to all such sources and vortices are calculated directly. The usual flow boundary condition of~ zero normal velocity is applied at all solid surfaces. In addition, a finite-velocity Kutta condition is applied at the trailing edges of the wall slats and test airfoil, including flaps. • Again <j> is the disturbance velocity potential, which 17 .es Laplace's equation, vanishes at infinity, and. satisfies the above boundary conditions. The potential at a point P due to a single three-dimensional, point source singularity at a point Q is where m is the volume flow rate of fluid emitted .by the source and r (PQ) is the distance between the points P and Q. The total potential due to all such sources distributed over a single surface S is 4>(P) 0(Q) dS, (3.2) s r(PQ) where o(Q) is the source strength density, including the factor 1/4IT , of the source element at Q. Since the disturbance velocity is the gradient of the velocity potential, the normal-velocity boundary condition at a surface can be expressed as 8© •> -> 9n = ~ V^-n + F, (3.3) where n is the outward surface normal,, and » the undisturbed oo flow at upstream infinity. The function F denotes the value the normal velocity must take at the airfoil surface. ? is zero for a solid (impermeable) surface, but non-zero for suction or blowing there. 18 Analysis (Hess and Smith [18]) shows that the normal velocity at a point P on a surface S, due to a source strength density distribution a (Q) on S, consists of two parts. The "local" contribution is 2fro"(p) due to the source element a(P) at P. The "far field" contribution is f _d_ dn 1 ) |r(PQ)J a(Q) dS (3.4) due to the summation- of the effects of all other source elements a(Q) at points Q on S. The resulting expression of this boundary condition 2ira(P) - 9h ir(PQ)J a(Q) dS = -V -n + F oo (3.5) produces an integral equation for the unknown source strength density distribution function c(Q). This equation is a Fredholm integral equation of the second kind. Existence and uniqueness theorems for such equations are well known. The surface S may be disjoint, but the outward normal vector must be a continuous function of position. For a discussion of difficulties associated with functional singularities in such source distributions at edges or corners, such as at an airfoil trailing-edge or an unfaired wing-body junction, see Craggs et al [ 19 ]. In practice, the surfaces of the solid wall, the airfoil-shaped slats, and the test airfoil and flaps, are replaced by polygonal elements. The continuous distribution of sources 19 thereby becomes a succession of finite d.istr ibuted-source each of these finite elements.was flat and of. constant uniform strength. Successful refinements of the method have used higher-order polynomial curves fitted to sections of the body surface with the source strength density varying in a linear or parabolic way along these curved elements. For examples see Henshaw [20,21], or Hess [22]. The higher-order element shapes are needed for internal flow calculations such as in ducts, but for external flow problems the flat elements give accurate results provided a large enough number of elements, is used and their disposition on the body shape is chosen carefully. Each velocity boundary condition is applied at a single "control point" on each element. For flat elements a convenient choice is the center of each element.. .Thus the exact integral equation for a continuous distribution function may be reduced to a set of N simultaneous linear algebraic equations whose N unknowns are the strengths of the finite surface elements. The above approximations become exact in the limit as N -»- °°. The method is described as numerically exact in the sense that any degree of. accuracy may be obtained. By defining the linear operator elements. In the original method of Smith and his colleagues. J JS. 3 _9_ dn • - as., 3^ XJJ J (3.6) 20 the boundary condition (3.5) applied at the.i-th control point becomes N I A.. j = l a . = -V •n. +.F . . j 00 i 1 (3.7) This indicates that A_^is the normal velocity induced at a control point 'i' by a unit strength source element . located at another point *j'. Hence the "local" normal velocity, ^±±' ^s 2TT for ail i= 1, 2, 3, . . . N. . For the purposes of this problem, the three-dimensional point source of equation (3.1) must be integrated into a flat two-dimensional distributed-sburce element; for the details see Appendix 1.. The units of a (Q) are therefore: volume flow rate per unit arc length along the contour per unit length in the spanwise direction. With respect to Cartesian axes x and y fixed to the j-th j j element (Figure 3.3), the velocity components induced at a point 'i' by a source element at point ' j* are V = log Ky%)2 + -yj) = 2 log R2 (3.8) and V 2 jtan -1 J 2" - tan -1 ( As, 1 y. = 2£2, (3.9) where x. and y. are the distances from the j-th to the i-th J J element; the j-th element has length As . The velocity fields j about a single source element are shown in Figure 3.'4. The directions of Vv and 7„ at 'i1 are parallel and normal to the x. y. 3 3 direction of the element at 'j', respectively. The inverse tangents in (3.9) are to be evaluated in the range f-Tr/2/ +Tr/2) . The two inverse, tangents may. be combined by means of the tangent law into the alternative expression V =2 tan 1  Y3 Y. As. _J i. lxj+y?-(^j)2J (3.10) where this single inverse tangent is to be evaluated in the range (-1T, + Tr) by faking into account the individual signs of the numerator and denominator of its argument. When calculating flow quantities at off-surface points which are closer to the origin of the element than As/2, the first expression must be used. With respect to Cartesian "wind axes" X and Y, (X is in the wind direction), the g-th source element is inclined at an angle 8. to the X-axis. Thus, A.. = V cos 9.-8. - V sin(8.-8. (3.11 ii y. i j x. it v ' and B>;L = V cos (6.-9.) +V sin (9.-8.) .- (3.12) 3 3 1 2 Y3 1 :1 are the normal and tangential velocities respectively induced at element 'i' due to a unit strength density source element at a point 'j'. -The "local" normal velocity JL^ is 2TT ; the "local" 22 tangential velocity 8^ is zero. The directions of and at ' i» are normal (positive outward) and parallel (positive clockwise) respectively to the direction of the element at 'i'. Hence for the exterior flow about a single closed contour, the source and vortex elements are labelled for computations in a clockwise order about the contour. In order to fix the circulation about a lifting body, the usual equal-velocity Kutta condition is applied. This is accomplished by adding finite distributed vortex elements to the body surface, all of the same vortex strength density. The Kutta condition then implies that the tangential velocities established at the control points on the upper and lower surfaces, adjacent to the trailing edge, must be. equal in magnitude, and both . directed toward the trailing edge. Since the velocity due to a vortex is simply that due to a source, but rotated 90 degrees,expressions for the velocity components for distributed-vortex elements of circulation strength density Y(Q) can be written, corresponding fo those of (3. 8, 3. 9) . For a distributed vortex element of unit strength density at point 'j* the corresponding normal and tangential velocities induced at element 'i' are found to be -B-. and A., respectively. The number and size of vortex elements is arbitrary, since they all have the same strength density. It is convenient to use the same number of vortex as source elements and to have source and vortex elements located to coincide exactly. The velocities A . . and B.. computed for the source 2 3 elements are then immediately useable for the vortex elements. Hence the normal and tangential velocities induced at the control point on element 'i' due to a system of H coincident source and vortex elements immersed in an infinite uniform approach flow U (parallel to the X-direction) , are N N V n « . IN i 3=1 J J k=l and N N Vt. = JVBjiaj + JnAkiYk + UcosV (3.14) x j=l J J k=l Since all the vortex elements on a single closed lifting body are of equal strength Y , the description of the flow field about an N-sided polygonal body is complete when the N+1 quantities Oi,o2,.. . ,o , and Y are known. For zero F^, the normal-flow boundary conditions at each of the N elements provides N equations Vn = 0. (3.15) i while the . finite-velocity Kutta condition at the two control points adjacent to the trailing edge provides the single (N+1)st equation (3. 16) 24 For the configuration of airfoil-shaped wall slats, solid walls and test airfoil plus flaps of Figure 3.1, with a total of N source elements and a lifting bodies, there are N source and M vortex strength densities to be determined. The zero normal-velocity condition applied on each element on the airfoil, flaps, solid walls and wall slats, yields the N equations N M R(k) I Ad. - I y I B = Usin9., i=l,2,...N. (3.17) j=l 3i 3 k=1 Km=1 nu i A Kutta condition applied to each of the M bodies (airfoil, flaps, wall slats) yields the M equations N M R(k) •VB3U +BJL >°l\lv*l (Amu +AmL > = "U (cos9D +cos6L ),(3.18) j-1 J r J r . k=l m=l r r r r r=l,2,...M. The subscripts U and L indicate the control points adjacent to the trailing edge"on the upper and lower surfaces respectively of the r-th lifting body; R(k) is the number of source (and vortex) elements on this body. In summary there are: - a total of N source elements distributed over the test airfoil, its flaps, the airfoil-shaped wall slats, and the solid wall sections. - a total of M bodies requiring Kutta conditions. M a total of J R (k) vortex elements distributed over the k=l lifting bodies; there are R (k) source elements and R (k) equal-strength density vortex elements distributed over the k-th body. - N unknown source strength densities a j 25 - M unknown vortex strength densities - M+N equations in the M+N unknowns ai, a2,...a , y i / y2, YM" 26 3^3 Other Airfoil-Wall Configurations Examined.- . ' Obvious simplifications of the above general equations (.3.17,3.18) are for a test airfoil (a) in an unbounded stream (free air), (b) in the proximity of a single solid lower surface (ground effect), (c) between two solid walls, and. (d) between a solid lower boundary and an upper boundary consisting of single-sided transverse slats with no Kutta conditions applied. In each of -the cases (a) - (d) a total of N source elements are distributed over the test airfoil, its flaps, the transverse wall slats, and the solid wall sections. In addition there is an unknown vortex strength density on the test airfoil and on each of its flaps. For example, for a single test airfoil (no flaps), in free air, equations (3.17,3.18) reduce to, respectively: N N •I A..a. - y I B. . = Usin8., i=l,2,...N, (3.19) j=l 31 3 k=l Kl 1 and J <V+VAJ + YJ1(AkU+AkL) = " U(coseu+coseL) . (3.20) j=l k=l For a single test airfoil in (b) , (c) , or (d) above, the equations (3. 17,3. 13) reduce to, respectively: N NA I A .a. - y I B, . = UsinG , i=l,2,...N (3.21) j = l 3 3 k=l K1 1 27 and N NA * (BDU+BjL)aJ + \^(AkU+AkL} = " U(cos0u+cos0L). (3.22) j=l J J k=l Here there are NA source (and vortex) elements on the test airfoil and (N-NA) source elements on the appropriate solid wall sections. The subscripts 0 and L indicate the control points adjacent to the trailing edge on the upper and lower surfaces respectively of the single test airfoil. The representation of an airfoil in ground effect is different here than in the method used by. others, for example Jacob and Steinbach [23] and Mavriplis [24]. Their approach is to use a second airfoil in the "image" position so that the lower straight solid boundary is a "reflection plane". Hence the boundary condition of zero flow normal to this straight solid boundary is represented exactly. If there are N source and N vortex elements distributed over the test airfoil and its flaps, the number of equations to be solved is exactly twice the number to be solved for the same airfoil/flap configuration in an unbounded stream (free air). Hence there is a saving by the present method where the solid lower boundary can be represented by less than N source elements. The shape of the solid lower boundary is arbitrary in the present method; there is no inherent requirement that the boundary be straight as is the case for a "reflection plane". 28 iii H§thods of Numerical Solution Assembling the Jguations.. A computer program is used to construct the matrices A and B. This program is written in FORTRAN for the OBC IBM 370/168 system. The program inputs are the coordinates, lengths and orientations of the source and vortex elements on the airfoil, its flaps, the airfoil-shaped wall slats, and the solid wall sections. These matrices A and B are then used to assemble the coefficients c^ of unknowns in the N+M equations (3.17,3.18). Typically N+M is about 400, hence the matrices A and B each contain more than 150,000 non-zero nonsymmetric entries. In fact the matrices A and B are used to assemble a third matrix C such that the system of equations to be solved is written C(a,y)=d; thus all three matrices (or parts thereof) must reside in memory simultaneously. The actual computational capacity (useable memory) is about 250,000 entries, hence such large matrices must be partitioned for computation in blocks and temporary storage on peripheral devices such as magnetic discs. The matrices A and B describe the relative geometry of the source and vortex elements, that is, their relative position and orientation. These matrices must be recalculated completely for each change in relative geometry, for example, a change in airfoil incidence, size or wall OAR. In the original method of Hess and Smith [18] and in similar methods in present use (Jacob and Steinbach [23], Mavr.iplis [24], Labrujere [25], Henshaw [20,21]), the solution of (3.17,3.18) uses numerical superposition of three "basic 29 flow" solutions. This is possible for isolated airfoils, as there is no change in relative geometry when the airfoil incidence is changed. This is also possible for an airfoil with flaps as long as the airfoil-flaps combination changes its attitude as a whole, that is, without a change in relative geometry. The three "basic flow" solutions are as follows. The first is due to a uniform stream onset flow parallel -to the airfoil chord. The second is due to a uniform stream onset flow perpendicular to the airfoil chord. The third is due to a pure circulatory onset flow that corresponds to pure circulation about the airfoil. If more than one airfoil is present (i.e. a flap), there is more than one independent circulatory onset flow. The three "basic flows" for each airfoil are then linearly combined to satisfy the Kutta condition at the trailing edge and to give a prescribed incidence or lift coefficient. Henshaw [20,21], for example, linearly combines the second and third solutions to yield a fourth solution whereby a Kutta condition is satisfied at the airfoil trailing edge. Linear combinations of the first and fourth solutions yield the flow at any incidence with a Kutta condition automatically satisfied. This procedure is not possible with an airfoil between solid walls. Since the solid walls are not closed lifting bodies, they do not have vortex elements distributed over their surfaces. Hence there is no circulatory flow associated with the solid walls. For a change in airfoil incidence, size or wall. OAR, the matrices A and B must be calculated afresh. 30 Referring to equations (3.17,3.18), the system of eguations to be solved is written below. The subscripts • j,i' of the matrix elements of A, B, and C are not of standard notation, but are reversed. This is deliberate, as increased efficiency of storage and summation of matrix elements is then possible under the FORTRAN compilers. With FORTRAN compilers, in conventional form, the elements of a matrix are stored column by column. That is, the elements of a matrix are assigned sequentially to storage beginning with element (1,1) and proceeding through all values of the (left most) first subscript, then increasing the second subscript by 1, and repeating. If A, B and C are small matrices, they may all reside simultaneously in real memory. In this situation, row wise access of the elements of a matrix entails accessing elements which are already in memory. However, if the matrices are large, only a portion of each matrix will reside in real memory at any one instant. This is due to the IBM virtual memory operating system which employs some kind of paging system for dynamic storage allocation. For more details,, see Moler [26], Thus accessing of elements in a row requires a transfer in and out of real memory of blocks of each matrix, block by block, until the accessing is completed. This is not efficient. Hence a matrix must be stored row by row. This is accomplished by reversing the subscripts of the elements of a matrix so for example, the elements of the first row are then (1,1), (2,1), (3,1),... (N,1). This subscripting is evident in eguations (4. 1) . 31 The P.L/1 compilers store matrices row by row, so this problem does not arise when accessing elements in the same row of a matrix, but would if it were necessary to access elements in the same column. The system of equations to be solved is written: Cl,iai + C2,1Q2 +-'-+ CN,laN + CN+l,lYl +'--+ CN+M,1YM = dl  Cl,2ai + C2,2a2 +--'+ CN,2aN + CN+l,2Yl +"-+ CN+M,2YM = dz Cl,Nai + C2,Na2 +--+ CN,NaN + CN+l,NYl + "- + CN+M,NYM = dN^-1> C1,N+Iai+C2,N+Ia2+-*•+CN,N+iaN+CN+l,N+lYl+-'•+CN+M,N+lYM=dN+l C1,N+M01+C2,N+Ma2 + ' '*+CN,N+MaN+CN+l,N+MYl + * *-+CN+M,N+MYM_dN+M' where the matrix C and the column vector d in the system C(0,Y)-d are assembled from the matrices A and B by means of equations (3.17,3-13), that is, Cji Aj:L j = 1, 2 , . . . N; i=l,2,...N-R(k) I " I B . k=l,2,...M; j=N+k; i=l,2,'...N , m=l 32 B . j = l,2 t ' • • N;r=l,2,...M; i=N+r (4.2) c. R(k) I' (A_ +A ) k=l,2,...M;j=N+k;r=l,2,...M;i=N+r m= 1 r r and Usin6. i=l,2 t ' ' ' N [ -ucose -ucose r=l,2 , . . . M; i=N+r (4.3) r Thus the computation of the matrix C requires access to both matrices A and B. When these matrices are large, C must be assembled in blocks. For details see the FORTRAN program in Appendix 2. The alternative is: calculate A and. B, write B into peripheral storage, de-allocate the memory assigned to B, allocate memory for C, calculate all parts of C that involve A, de-allocate the memory assigned to A, allocate memory for B, read B into memory from peripheral storage and calculate all parts of C that involve B. This alternative is simpler to program and appears in Appendix 10. The large number of summations of matrix elements in the same row is evident in equations (4.2); this is the main source of inefficiencies under the present paging systeo. 33 4_-_2 Solving the Eguations-_ Usually the solution of the complete system of N + M equations is obtained directly by Gauss-elimination methods. For a FORTRAN Gauss-elimination subroutine that takes account of the above mentioned paging system, see Moler [26] or Appendix 3. Another direct method, used by Hess and Smith [18] is the successive row vector orthogohalization process of Purcell [27]. In this method an augmented matrix is treated row by row such that a series of vectors orthogonal to each row vector of the augmented matrix is constructed. The right-hand side vector 'd* is used to construct a set of N+M vectors in (N+M +1)-dimensional space, (Gli' C2i' C3iCN+M,i' -di} . i=l,2,...N+M. The solution vector (a,y) °f equations (4. 1) is such that the vector Cai, a2, a3f..., o^, yi, Y2,..-, yM, 1) (**-5) is orthogonal to all the vectors of (4.4). The process of solving equations (4.1) is equivalent to determining an (N + M+1)-dimensional vector orthogonal to the N+M vectors of (4.4) with unity as its (N+M+1)-th component. Each row of the coefficient matrix C is used at only one stage of the process, and is not needed before or after that 34 stage. Thus C is transferred from virtual to real memory a row at a time, with each row occupying the same storage location as that of the previous row. Thus an insignificant amount of storage is required for storage of a single row. However, the components of all the orthogonal vectors will tend to be in real memory, since they are used repetitively. The maximum total number of components occurs when the process is about half-finished; the total number of memory locations required is approximately (N + M)2/4. Thus the number of equations which can be solved by this process (for a given computational capacity) is about twice that for a Gauss-elimination process. For a FORTRAN subroutine based on this procedure, see Appendix 3. Indirect iterative methods such as successive-over-relaxation (SOB) are also possible. For a discussion of such methods, see Hess and Smith [18]. Unsuccessful attempts we're made to use SOR; the method was abandoned. In general, the matrix C is diagonally dominant, that is, the diagonal elements, 2TT, are the largest in the matrix. However, by examining the relations (4.2), it is seen that the summations over B .. , A „ and A _ provide large elements in the mi mu mL (last) (N+M)-th column of C. In general, the, sum of all the diagonal entries is approximately equal to the sum of all the off-diagonal entries. In essentially all cases,, the matrix is non-singular except for very thin bodies, such as cusped trailing edges. Here the source and vortex elements on the two surfaces are almost coincident so matrix singularities (rows linearly dependent)- can creep in. For a discussion of such 35 problems, see Hess and Smith [18]. The summations N M R(k) v n. I A. . a . - I y. Y B k=l m=l mi UsinS. l (4.6) and N y B . . a j=l 11 M R(k) k=l m=l , ii/ " • + Ucos6 DID i.ii-, k_J:-, mi 1 (4.7) provide the net normal and tangential velocities at control points »i' due to all source and vortex elements •j' and ' m' respectively, and the uniform onset flow U. The solution to the set of eguations (4.1) is checked by computing the velocities Vn ,Vt at each point of application of the zero normal-velocity i i or Kutta boundary conditions. At all control points on solid surfaces, V is zero, and the local pressure coefficient C is 1 calculated from : = 1 -V. U (4.8) The resulting values of Cp are integrated numerically (by trapezoidal rule or the fitting of cubic splines) around the test airfoil and flap contours to determine the lift, drag and nose-up midchord and guarterchord pitching moment coefficients, from the expressions 1 c c|>C dx. PJ 3 1 c 4>C dy. 36 CM. = 772 <|>C (x.dx.+y.dy.) , (4.9) 'MoT C where dx =ds.cos9. and dy = ds sin6 (4.10) 3 3 3 3 3 3 and integrations are performed clockwise around the polygonal contours. From a calculation of the net circulation about a lifting body, represented by NA source and vortex elements. f.y ^ t NA r = Av-<U = ov. ds. = I V, As., (4.11) J Jt. i . ^, t. I X 1 = 1 X I and since the lift coefficient C is related to the total circulation r by C is given by ~ NA °L = Uc" ^ Vt/Si * ^-13> 1=1 l By substitution of (4.7), it can be shown that (4.13) reduces to C = ^JH-(perimeter of the body) , (4.14) Ii UC-where y is the vortex strength density for the body under consideration. The C values calculated from (4.9) and (4.13) are equivalent only for an isolated airfoil. If a second body or a 37 boundary is present the two values are not equal. The calculation of the circulation about a particular body then depends on the size of the contour of integration. The integral (4.11) reduces to the correct value only as the contour size shrinks to zero. In the following pages, all C values quoted are calculated from expression (4.9). It must be emphasized that on the body surface elements, the flow field solution is valid only at the control points; for "off-center" points any calculated surface velocities are meaningless. Figure 3.4 shows how velocities vary with position, on a surface element. On a given surface element, if the normal velocity is prescribed as zero, it is in general non-zero at all points of the element except the control point. At the edges of the surface elements the tangential velocity approaches infinity because of the singularity in the expression (3.8) and/or the discontinuity in surface slope. However, the flow field solution is uniformly valid at all off-surface points. Hence at a field point ' i' the velocity components .parallel and perpendicular to the streamwise direction can be computed. The local flow . direction can be calculated and, by stepping from point to point, a particular streamline can be tracked. The algorithm is given in Appendix 4. Alternatively the expression for the stream function given in Appendix 1 can be solved iteratively for say the y-coordinate, at a given x-coordinate, to locate points on a particular streamline, that is, a locus of points (x,y) can be found along 33 which the computed value of the stream function is a constant. 39 5.. Results of the New Theory. The use of this type of two-dimensional surface singularity distribution method, to calculate pressure distributions on isolated bodies, is well established. For a comparison of pressure distributions obtained from this method, with pressure distributions derived from experiments and other two-dimensional potential flow theories, see Hess and Smith [18]. For purposes of comparison here, the slope of the curve of lift coefficient CL as a function of airfoil incidence will be used. As an example of the present theoretical method, free-air loadings were calculated for the NACA-0015 airfoil, using 50 control points to represent the profile of the airfoil. The coordinates of the 50 control points used are given in Table 1. The values of the lift coefficients so obtained are listed in Table 3, as. a function of the incidence a . By exact curve-fitting a polynomial of order 5 through the 6 points at 0,2,3,5,8, and 10 degrees incidence, the lift-curve slope at zero degrees was found to be 0.1193 The corresponding value at +3 degrees is 0.1229. These values would be slightly higher if a larger number of control points was used. From thin airfoil theory (see Pope [28]) this lift-curve slope m for symmetrical airfoils is given by (per radian) m = 2TT 1 + .773-c U + (»773|) 2J (5.1) where t/c is the maximum-thickness to chord ratio. For the NACA-0015 airfoil, the value of is is 6.919 per radian or 0.1208 per 40 degree. The agreement between the two theories is good. Strictly the expression (5.1) gives the theoretical lift-cnrve slope only at zero degrees, since an underlying assumption is that m is independent of incidence. The present theory represents a uniform flow of infinite extent, past a set of multiple airfoils, and flat surfaces aligned with the direction of the undisturbed flow. Sith this configuration (Figure 3.1), loadings were calculated for different test airfoils developing high lift coefficients in the presence of the above mentioned wall configuration. It consists of a solid lower wall in conjunction with a transversely-slotted upper wall, with various upper wall OAEs. This configuration will be referred to by the abbreviation TSOSL., meaning transversely-slotted upper, solid lower. By comparing the lift coefficient in the windtunnel, CT , to the free-air value, CT , T F the results indicated that a transversely-slotted wall of about 70%OAR gave very small lift corrections (C -C ) for all the LT LF airfoils considered, at all lift coefficients, and up to airfoil sizes, c/H, as large as unity. Figure 5.1 shows the calculated ratio of lift coefficients as a function of airfoil size, c/H, for three airfoils. The first airfoil is a 14% thick, 4.6% caaber Clark-Y (represented by 50 control points) at zero and 20 degrees incidence (the fact that the actual airfoil would be stalled at this incidence is of no consequence for the present purpose). The second airfoil is an NACA-23012 at 8 degrees incidence, with a 25.6% (overall chord length) slotted flap deflected 20 degrees (represented by 41 46 control points on the main airfoil and 35 on the flap). The third airfoil is an NACA-0015 at 3 degrees incidence-This ratio of lift coefficients is shown -for two. wall configurations. One is with two solid walls, and the second is a 70%OAH TSUSL wall configuration. It is seen that whereas the lift correction for an airfoil tested between solid walls, could be mora thau 50% of the true, free-air value, the present theory for one solid wall and one slotted wall of OAS near .701 predicts lift corrections of less than 1%, for airfoil size c/H less than 1.0. Indeed, it appears that a slightly lower value of OAS would shift the curves up slightly, and give lift corrections of less than about <A% for c/fl less than 0.8. For the details of the number of wall slats, number of control points used, slat size and spacings, for any airfoil-wall configuration tested theoretically, and referred to herein, see Table 3. A principal reason for two-dimensional airfoil testing . is to obtain pressure distributions for use in subsequent aerodynamic analysis. Figure 5.2 shows a comparison of the pressure distributions calculated by .the present theoretical method on the Clark-Y airfoil at 20 degrees incidence in the presence of a 70%OA.R TSUSL wall configuration, and in free air. The lift coefficients are the same, that is, there is zero lift-correction. The figure is plotted in terms of the usual non-dimensional pressure coefficient Cp. The figure shows that the distortion of the pressure.distribution on the. airfoil in the presence of this TSUSL wall configuration is small, even at the 42 high C of 3.09. The pressure distribution for free air shows LF larger negative pressures over the forward section of the airfoil and a lower negative pressure over the after section, than in the pressure distribution for the TSUS.L wall configuration. The resulting midchord nose-up pitching moment coefficient for free air is 0.603, which is therefore larger than the corresponding value of 0.573 for the TSOSL wall configuration. The ratio of midchord pitching moment coefficients in the tunnel, C , to the free-air value C , is MoT Mo p 0.95 here, which suggests that this zero-lift-correction TSU5L wall configuration might also find use as a low moment-correction test configuration. A flow field comparison of a TSUSL wall configuration and free air was made as follows. The flow speed and direction were computed at points on flat surfaces parallel to the plane of the testsection wails, and slightly inside the walls. A similar computation was performed for the free-air case, for two flat surfaces with the same corresponding positions relative to the test airfoil. The results are shown in Figure 5.3, for the same zero lift-correction case described above, in terms of the variation of C along these surfaces. Qualitatively the flow fields are similar in that a detailed calculation using pressures and momentum fluxes for a rectangular control volume about the. test airfoil confirms the value of the airfoil lift coefficient within 5%. 43 Experiments to Ver ify_ the New Theory. J2il Testsection DesJ.g_ru The success of ,the proposed transversely-slotted upper and solid lower wall (TSOSL) test configuration depends on the experimental verification of the present theory. Initial experiments were performed in the octagonal testsection of an existing low-speed closed-circuit windtunnel (Figure 6.1, Plate 1). This tunnel has a testsection 915mm wide by 686mm deep, over a length of 2.59m, and produces a very uniform flow, with a turbulence level less than 0.1%, over a windspeed range of zero to 50m/s. The testsection has 152 by 152mm corner fillets (actually tapered downstream to compensate for boundary layer growth), so that the octagonal crosssection area is 0.582m2. As these initial experiments were encouraging, the existing testsection was modified to accept a two-dimensional testsection insert. This insert is 915mm wide by 388mm deep in crosssection, and 2.59m long. Test airfoils are mounted vertically on the yaw-turntable of a six-component windtunnel balance, at the midpoint of the testsection, and spanned the 388mm depth. One side-wall was surrounded by a 0.39 by 0.30 by 2.44m plenum, and could be fitted with airfoil-shaped slats of NACA-0015 section (Plates 2,3) and chords of 46 or 92mra, at zero incidence. A full range of wall open area ratios (OAR) could be tested, as the slats were fitted with metal sliders which in turn were separated by wooden spacers in an aluminum channel recessed in the side-wall frame. . Modifications to the existing windtunnel consisted of an inserted nozzle and diffuser section (2.6m length) in addition to the 388 by 915mm testsection. The design of the theoretical shape of converging sections of circular crosssection to produce spatially uniform flow conditions at exit is well established; for example,.see Smith and Wang [29]. The approach here was to use the "same crosssectional area variation (in the streamwise direction) for an equivalent rectangular crosssection as for the theoretical circular crosssection. For the details of the design, see Appendix 5. The testsection floor and ceiling are parallel and solid; no structural or mechanical compensation for boundary layer growth is attempted. Pitotstatic tube traverses in the empty testsection (with two solid walls) where a test airfoil would be mounted, indicate that the testsection windspeed is spatially uniform to within 0.3% in the central "core" flow, outside the wall boundary layers. Figure 6.2 shows a typical "core" flow pitotstatic tube windspeed traverse. Boundary layer pitotstatic tube measurements in the empty testsection, again where the test airfoil would be mounted, indicate a displacement thickness of the order of 12mm, over a range of windspeeds covering the range of Reynolds numbers required for various test airfoils. Figure 6.3 shows a typical boundary layer pitotstatic tube windspeed traverse. Initial experiments without the plenum gave inconsistent trends in the data taken as functions of the wall OAR, with a slotted-wall of OAR 10% or greater. The flow exiting from the U5 testsection through the slotted wall, upstream of the test model, at large OAR was not constrained to re-enter the testsection through the slotted wall downstream of the test airfoil. Conservation of mass flow was preserved by a large influx of air into the diffuser section through a breather slot between the testsection exit and diffuser section entrance. This problem was cured by using a plenum surrounding the transversely-slotted wall. Consistent trends in data taken as functions of wall OAR are now achievable with any OAR whatsoever. 4 6 6.i2 Airfoil Models Tested. Altogether nine different airfoils were tested, with lift, drag and pitching moment data taken for each airfoil. In addition, surface pressure measurements were made on two of the airfoils. Four airfoils of NACA-0015 section, 383mm span and 153, 307, 462, and 616mm chord (model size c/H of 0.17, 0.34, 0.51 and 0.67 respectively) , were machined from .. solid aluminum billets to close tolerances, on a numerically-controlled milling machine. Each airfoil was mounted on a circular spar which passed through a circular hole in the testsection floor with 3mm clearance all around between the circular hole and the mounting spar (Plate 4)- The gaps between the tip of the test airfoil and the floor or ceiling were less than 2.5mm on all tests. Four laminated wood airfoils of 14% thickness, 4.6% camber Clark-Y section, 683mm span and 227, 354, 481 . and 608mm chord (airfoil sizes c/H of 0-25, 0.39, 0.53 and 0.66 respectively) were also tested. Since these airfoils extended outside the testsection, they were fitted with large (716mm diameter) circular aluminum endplates 3mm thick (Plate 5)- The endplates fit flush with the testsection floor and ceiling into circular stepped recesses, of 724mm diameter and 6ram depth..Thus the test airfoil incidence could be varied by rotating the test airfoil-endplate combination about the vertical axis, without allowing the endplates to touch the testsection floor or ceiling. A single laminated wood airfoil of 11% thickness, 2-3% 47 camber Joukowsky section, 683mm span and 307mm chord (airfoil size c/H of 0.34) was also tested. It was similarly fitted with circular endplates. The trailing edge region was thickened to 3.8mm to allow pressure taps to be located there. Table 1 contains the theoretical and modified profile coordinates for this airfoil. For the five airfoils fitted with endplates, it was necessary to measure the loadings on the endplates themselves. While under test, depending on the test airfoil incidence, flow could be detected out of the test section through the gap between the circular endplates and the circular recesses in the testsection floor and ceiling-.This flow would certainly affect the loadings on the test airfoil-endplate combination, particularly the loading on an endplate. Thus there would be a loading on an endplate due to the outflow through the gap in addition to the expected skin-friction drag force in the streamwise direction. The loading on a single endplate was therefore determined by mounting the endplate on the windtunnel balance with the upper surface flush with the testsection floor. Each.of the five airfoils was then suspended vertically above the endplate to establish the correct flow field over the endplate at all airfoil incidences. The gap between the tip of the test airfoil and the endplate was adjusted by raising or lowering the airfoil. Measurements of the loading on the endplate were made for each airfoil over a complete range of test airfoil incidence and for varying tip-endplate gaps. The results were extrapolated 48 to zero gap to deduce that portion of the loadings on the test airfoil/endplate combination which was due to the two endplates. The net effect on test airfoil lift and midchord pitching moment was typically less than 2%; the effect on test airfoil drag was large, typically 30%. Table 4 and Figure 6.4 show typical endplate loadings and the corresponding uncorrected and corrected airfoil loadings for the Joukowsky airfoil. The fact that the loading on the endplate did not vary markedly for a gap less than that corresponding to a gap to chord ratio of about 0.005, supports the claim that the loading on a two-dimensional test airfoil should not be too sensitive to the tip-wall gap, provided the gap is less than a certain minimum value. The 0.67-NACA-0015 airfoil was fitted with 65 center-span pressure taps (Plate 6). The Joukowsky airfoil has a 75mm wide aluminum center-span section housing 37 pressure taps. All pressure taps are surface flush and have 0.5mm diameter orifices in metal. Plastic tubes of 1.6am inside diameter and approxim ately 1.0m length transmit the surface pressures via the mounting spar to a location external to the testsection. Pressures were measured using a 48 port "Scanivalve" manual-scan pressure transducer. The tubes were disconnected from the pressure transducer and tied to the windtunnel balance "turntable during balance measurements to eliminate any effect of tension in the plastic tubes. 49 6 ._3 Test Procedures. The testsection windspeed was deduced from a pitotstatic tube mounted on the flow centerline in the tunnel nozzle midway between the settling chamber exit annd the testsection entrance. Located thus, the pitotstatic tube would be far enough upstream to be relatively unaffected by test model "blockage" effects^ Also the pitotstatic tube would be sufficiently far downstream in the nozzle that the flow speed would produce numerically large pitotstatic tube readings (mm of water). Thus a sufficiently accurate reading, either on a "Betz" micromanometer.. scale or as a "Barocel" output voltage, could be obtained. • The nozzle pitotstatic tube was calibrated against a-second pitotstatic tube mounted in the empty solid walled testsection, on the flow centerline, where the test airfoils would be located. . The pitotstatic tubes were connected to "Barocel" pressure transducers. During tests, the nozzle pitot reading is simultaneously monitored on a "Betz" micromanometer. During pressure tests, the total head in the nozzle is measured (with the same pitotstatic tube), and used as a calibration pressure for the "Scanivalve" pressure transducer. The. reference windspeed and static pressure used to reduce balance and pressure measurements are determined as follows. Since the total head at the nozzle pitotstatic tube (when there is a test airfoil in place, at incidence) is essentially the same as the total head in the testsection in the vicinity of the test airfoil, the reference windspeed and static pressure can be deduced from the nozzle pitotstatic tube measurements of total 50 head and dynamic pressure (q). This equality of total heads can only be verified in the empty testsection (solid walls) with no test airfoil in place; such measurements indicated that the testsection total head was lower than the nozzle total head by .1 part in 200. Let H, p and q be the total head, static pressure';, and dynamic pressure respectively, and let the subscripts 1 and 2 refer to the nozzle and testsection respectively. In the empty testsection (solid walls), according to the above, Hi = Pl + qi = H2 = p2 + qz - (6.1) If k is the empty test section (solid walls) calibrated ratio of q! to q2, then since only Hi and gj are measured while under test, the equivalent empty testsection (solid walls) dynamic pressure and static pressure are kqx and (H.j-kqi) respectively.. Thus the reference windspeed and static pressure used to reduce data taken for any test airfoil, at any incidence, in the presence of any slotted-wall OAB, are the equivalent values in the undisturbed uniform stream conditions that would actually occur in the empty (solid walls) testsection corresponding to that measured Hx and q x. in the nozzle. Hence the reference windspeed "so "deduced is not the actual flow speed past the test airfoil nor is the reference static pressure the actual static pressure in the testsection, while under test. Figure 6.5 shows a typical nozzle-testsection windspeed calibration curve using the two pitotstatic tubes. The ratio k 51 is determined from a straight line least-squares curve.fitted through the origin. The six-component windtunnel balance has six independent four-arm strain gauge load cells so that in principle, three forces and three moments can be measured and read independently and simultaneously. In practice, only two strain gauge amplifiers, nulling and readout units are available simultaneously. As these amplifiers are of the vacuum tube type, they are left switched 'ON' at all times to minimize the drift in the zero readings on the readout units. The standard procedure for windtunnel balance readings is to shut off the windspeed every 20 to 30 minutes and record the drifts in the zero readings. These zero drifts are then applied to each reading in a proportion, assuming that these zero drifts are linear in time and that the time between readings is the same. All pressure transducer and strain gauge voltages.are amplified and time-ayeraged to produce essentially steady 3 or 4 significant, digit readings on digital voltmeter displays. The relative accuracy on a single windtunnel balance measurement is indicated as follows. Typical values and maximum possible errors are, for example, for the Joukowsky airfoil at +3 degrees incidence (including endplate effects), Reynolds number (Re) 0.5(106): C =0.713±0.004 (0.6%) ; C =-0.0625±0.0005 (0. 8%); L Mc_ 4 and C=0.0286±0.0004 (1.4%) (6.2) At this incidence and Re the net lift, quarterchord pitching moment, drag and windspeed can ba measured, to within 0.2,0.4,1.0 and 0.2%, respectively. Similarly a value of lift coefficient determined by integration of a measured pressure distribution has C =0. 709±0. 006 (0. 9%) . (6.3) In comparing coefficient values taken at different angles of incidence there is a maximum possible error of ±0.005 degrees in the measured incidence due to slack in the turntable worm gear mechanism. Care was taken to always rotate the turntable in the same direction to minimize the effect of backlash in the worm gear mechanism. This possible error in a is the largest of the possible errors which arise in computing the slope of the C (a) curves. Figure 6.6 shows a typical C (a) curve, with error bars. L The vertical error bars indicate the maximum possible cumulative error obtainable on a single measurement as described above. The horizontal error bars indicate the maximum possible error in the measured incidence. . The accuracy obtainable on a single measurement is good for the type of measurements made here. Figure 6.7 shows three C (ct) curves and indicates the L • repeatability obtainable in measurements taken in three consecutive runs under identical test conditions and procedures. The degree of repeatability obtainable gives confidence in measurements taken on a single run (which is not repeated) , for the type of measurements made here. The level of turbulence in the two-dimensional testsection insert was not. measured. It is assumed that.this level might be as high as in the unmodified , octagonal crosssection, that is 0.1%. It is probably less due to the expected reduction in any streamwise fluctuations due to the increased contraction ratio, now 11.8 versus 7 for the octagonal crosssection. 54 Zi. ExjBsrimental Results. The results of the experiments using the nine airfoils are listed in Tables 5, 6 and 7. The results are tabulated as coefficients of lift, drag, midchord and quarterchord pitching moments, according to test airfoil size (c/H) and test Reynolds number (Re), as functions of test airfoil incidence (a) . The lift-curve slope dc /da is calculated from a least-square fit straight line on an interval of 10 degrees incidence, for measurements in 1 degree increments from 2 degrees below the angle of zero lift to 8 degrees above the zero lift angle. For the NACA-0015 airfoils with a zero-lift angle of 0<>, m is calculated on (-2°,8°). For the Clark-Y airfoils with a zero-lift angle of about -6.3 degrees, m is calculated on (-8°,2°). For the Joukowsky airfoil with a zero-lift angle of about -3.8 degrees, m is calculated on '•(-6°,4°). From the fitted straight line the zero-lift angle can be determined as the a-intercept when CL is zero. In a similar manner the slope of the midchord pitching moment curve, dc /da, can be determined. The position Mo of the aerodynamic center xac/c with respect to the airfoil leading edge is x a C _ Xo _ _T /7 1\ c c m where x0, m, c, and x are the distance from the leading edge to the axis of measurement of the moment coefficient C , the lift-Mo curve slope, the chord and the midchord pitching moment-curve slope, respectively. Table 8 contains values of the lift-curve' slope, the zero-lift angle and the position of the aerodynamic 55 center for the nine airfoils and the various wall configurations tested. Figure 7.1 shows a typical variation of C (a) with upper L wall open-area ratio (OAR). The straight line plotted corresponds to established experimental free-air lift-curve slopes (Jacobs and Sherman [31], Riegels [32]). The zero-correction OAR appears to lie between zero and 40%; this fact is also apparent in Figure 7.5 and will discussed there. The resulting values of the lift-curve slopes, m, for any of the airfoils are, to the degree of accuracy required here, quite sensitive to the choice of range of incidence over which the least-squares straight line curve fitting is done. For example, for the 0. 67-N AC.A-00 15 airfoil between two solid walls, the value of m computed on (-2°,8°) is 0.1114, while the value on (0-o,10O) is 0.1156 and the value on (00,12°) is 0.1138. This is particularly noticeable for this airfoil as there is a pronounced jog in the Ci,(a) curve in this range of Reynolds numbers. This jog is attributed to (see Tani [30]) the formation of a laminar separation bubble by the separation of the laminar boundary layer near the leading edge and subsequent reattachment downstream. As the Re is increased, the jog becomes progressively less pronounced. Figure 7.2 shows a comparison of two sets of measured values of lift-curve slope m, with the corresponding slopes from the present theory, for a 14% 0.53-Clark-Y airfoil, as a function of the upper wall OAR. The large wall slats (92mm) were used for one set of measured values, and the small slats (46am) 56 for the other set. The ordinates are normalized by the value of the lift-curve slope, m , in the presence of two solid walls (zero OAB). The test Re in both sets of measurements was 0.5(10*), based on the test airfoil chord. The theoretical values of lift-curve slope m are determined from a least-squares straight line fit through three values of lift coefficient C> computed at -8, -3 and +2 degrees incidence. Figure 7.3 shows similar results for the 0.67-NACA-00 15 airfoil tested at a Re of 1.0 (106). The measured lift-curve slopes are shown for tests with both the large and small wall slats. The theoretical values of lift-curve slope m are determined from a least-squares straight line fit through three values of lift coefficient C , computed at -2, +3 and +8 degrees incidence. Figures 7.2 and 7.3 show that the theoretical values of m/m are higher than the experimental values, for both s airfoils, and for all OABs. This difference is about 2.8%, at 70%OAR, and will be accounted for by two extensions to the present theory in §§8.1 and 8.2. Figure 7.4 shows experimental values of lift-curve slope, m, (per degree) for four sizes of NACA-0015 airfoil in the presence of walls of different OAR, 0, 60, 70, and 80S, using the large slats. All tests for the three larger airfoils (c/H of 0.67, 0.51, and 0.34) were run at a Re of .0.5(10*). This Re could not be reached for the smallest airfoil (c/H of 0.17), which was tested at a Re of 0.3(10*). The data for this airfoil were then adjusted to correspond to the 0.5(10*) Re, using published m (Re) data for the NACA-0015 airfoil (see Jacobs and 57 Sherman [31]). The adjusted data are the flagged points in figure 7.4. The results show a convergence toward a free air (zero c/H) lift-curve slope value of 0.093, in good agreement with [31]. The results indicate zero lift-corrections for an upper wall OAB between 60 and 70%, in agreement with the , predictions of the present theory. Figure 7.5 shows the corresponding experimental values of m for four sizes of Clark-Y airfoil in the presence of walls of different OAR, using the large slats.: All tests for the three larger airfoils (c/H of 0.66, 0.53, and 0.39) were run at a Re of 0.5 (10*). This He could not be reached for the smallest (c/H of 0.25) airfoil, which was tested at a Re of 0.45(10*) - The data for this airfoil were not adjusted to the 0.5 (10*) Re as a (Sa)- information for the 14% thick Clark-Y section is scarce. : The results show a convergence toward a free air (zero c/fi) lift-curve slope value of 0.096, which agrees favourably with the information that is available (Silverstein [33]). An extrapolation of the curve of m (He) of Figure 11 of [ 33 ], for an 11.7% thick Clark-Y section of aspect ratio 6 gives a value of ra 0.071 at a Re of 0.5(10*). By using the theoretical relation (5.1) to estimate an equivalent value for 14% thickness, then correcting to infinite aspect ratio, the estimated value of m for the 14% thickness at a Re of 0.5(10*) is 0.096. Here zero lift-corrections are indicated for an upper wall OAR less than 60%. Experimentally, from both Figure 7.1 and 7.5, the zero-correction OAR appears to ±>e somewhat less than 60%, based on a 58 free air lift-curve slope of 0.096- If a lower value such as 0.092 were chosen, the zero-correction OAR would be about 60%. The m-values for the smallest airfoil should be lowered due to the difference in the test Re. Previous tests £36] with the smallest airfoil have been unreliable. It appears that this airfoil has a relatively smaller nose radius than the larger airfoils, which might account for the higher m-values. Figure 7.6 gives a comparison of experimental pressure distributions on the Joukowsky airfoil, at +3 degrees incidence in the presence of two solid walls, and the 70%OAR, TSOSL wall configuration, using the large wall slats. The comparison supports the theoretical prediction that the chief effect of the slotted wall is to lower the negative pressures over the upper surface of the airfoil, without appreciably modifying the positive pressures on the undersurface, or generally distorting the distribution. The close agreement of C values from balance measurements with those obtained by integration of the pressure coefficient, C , indicates satisfactory two-dimensional flow P conditions. Cubic-spline polynomials are used to curve-fit the distribution of C for integration. P In Figure 7.7,. the pressure data from Figure 7.6 for two solid walls are corrected to equivalent free-air conditions by conventional wall-correction theory {Pankhurst and Holder '£34]). The actual correction formulae used are recorded in Appendix 7. These corrected data are compared with the 70% OAR pressure data as taken. The 70%OAR data are seen to agree quite closely with the corrected solid wall data. The corrected solid wall value of 59 C is 0.675, while the 70%OAR value of CT is 0.651; this again indicates that zero lift-correction test conditions will occur at an upper wall OAR between 60 and 70%. The hump in the pressure distributions toward the rear upper surface is caused by the thicker trailing edge of the experimental airfoil replacing the theoretical trailing edge cusp. Por completeness, to show how the use: of such a TSUSL wall configuration affects other aerodynamic data, some typical curves of pitching moment and drag coefficients are shown, as functions of airfoil incidence-Figures 7.8 and 7.9 show the midchord pitching moment coefficient c and drag,coefficient C_ respectively for the Mo D 0.53-Clark-Y airfoil, as a function of airfoil incidence, and upper wall OAR. The values at 70%OAR (near free-air conditions) for C and C_ agree well with corresponding values from D . M0 mxn Figure 68 of Pinkerton and Greenberg [35], and Lim [36]. The values of at a given incidence initially increase and then decrease, with increasing OAR. This same behaviour of CD was observed for the other airfoils tested, both with and without endplates, and appears to be a property of this particular test configuration. 60 Si. Extensions to the New Theory., What is available at present is a potential flow theory for a transversely-slotted upper and solid lower wall configuration (TSOSL), that, for an upper wall OAB of between 60 and 70% predicts low lift-corrections for a variety of airfoils, sizes and incidences. However, experimentally, the OAB at which zero lift-correction occurs is less than the OAR predicted theoretically. Thus the slotted wall behaves experimentally as if it were operating at a larger OAB. The result,, at a given OAR, is that the ratios, of the theoretical values of lift-curve slope m or lift coefficient CL, normalized by their corresponding theoretical free-air values, are too high. This is because the TSOSL wall configuration theory is a potential flow representation of a viscous flow field. For example, because of viscosity, the wall slats are not developing their full circulations predicted by the potential flow theory since they are operating in a Re range of 37,500 to 180,000 (with respect to their chord lengths). , Thus the net circulation on the slotted wall is less than the circulation predicted theoretically. Hence that portion of the measured, lift at a given OAR which is due to the circulation on the wall slats, will be much less than that portion of the lift predicted theoretically. In other words, in order to predict theoretically the lift that would actually be measured experimentally on an airfoil in 61 this TSUSL wall configuration, there are two viscous flow fields to be accounted for. The first is the usual free air viscous flow field about the isolated airfoil. Now a second viscous flow field has been introduced, that of the flow through the TSUSL wall configuration. Thus a complete solution would require the application of viscous theory to the test airfoil, the wall slats and any shear layers which develop in the plenum. Let C ( ;E) and C { ;T) denote lift coefficients obtained from experiment and theory respectively, and let C (OAR; ) and C (F; ) denote lift coefficients from OAR and free air test configurations respectively. For a particular OAS, it is reasonable to assume that the ratio C (OAB;E) :C (OAB;T) will be equal to the ratio C (F; E) : C (F;T) , particularly in a low-Li Li correction test environment. In this case the viscous effects on each ratio will be the same. Thus the ratio of C (E) to C (T) will be the same for free-air tests or for tests at whatever OAR produces near free-air test conditions. Therefore it is still useful to compare the ratio of C (OAS;E):C (F;E) , with the ratio of C (OAR; T) ;C (F; T) . L» Li . What is then desired is a potential flow calculation that will account for the viscous effects present experimentally which are due only to the TSUSL wall configuration. That is, the usual viscous flow analysis for the test airfoil will still be required. 62 8.1 Potential Flow Considerations of Viscous Effects. There are two ways to extend the present theory toward such a potential flow calculation. One way is to account for the effect of viscosity on the test airfoil and wall slats, through the formation of viscous boundary layers on these streamlined shapes. The second way is to make the geometry of the flow representation more like that which actually occurs experimentally in the testsection, with the plenum surrounding the slotted wall. The following discussion applies to both the test airfoil and the airfoil-shaped wall slats. For boundary layer effects, only completely attached flows are considered, that is, boundary layer flows which separate from an airfoil surface are not considered. Thus, since the flow leaving the airfoil trailing edge is attached, the velocities and pressures at the upper and lower surfaces adjacent to the trailing edge, must be equal. This statement is true on a time-average, since the formation of a vortex street in the airfoil wake requires the shedding of consecutive trailing vortices of alternate sign. For a given airfoil shape and incidence, there is a unique circulation that will set equal the trailing edge velocities on the upper and lower surfaces. This is the physical reasoning behind the theoretical relation known as the Kutta condition, that is, that the velocities at the upper and lower 'surfaces adjacent to the trailing edge (of any body which possesses a sharp trailing edge) must be equal. Experimentally the measured value of lift (which is 63 proportional to the circulation) is always lower than the lift predicted by the usual potential flow theories; the ratio of these lifts might be, for example, k. What then is required is a procedure to reduce the usual theoretical value of the circulation r to a fraction k of r. • From the uniqueness of the value of the circulation for a given airfoil shape and incidence, the only possibility of reducing the circulation, if the Kutta condition is retained, is to reduce the incidence, and/or alter the shape of the profile-. The reduction of incidence approach is often used in the comparison of the theoretical and experimental pressure distributions. The theoretical pressures are calculated at a reduced incidence, such that the theoretical lift (determined by integration of the pressure distributions) at the reduced incidence is the same as the measured lift at the measured incidence.•, This procedure is not completely satisfactory as the magnitude of the negative pressure . peaks in the theoretical pressure distributions (near the airfoil leading edge on the upper surface) may not equal the magnitude of the peaks actually measured.: Physically the thin laminar boundary layer, which forms beginning at the forward stagnation point on the underside of the airfoil, grows in thickness as it rounds the leading edge and passes through the negative pressure peaks. The ' effect of this thin boundary layer is to increase the radius of curvature of the airfoil leading edge so that the flow velocities there are much lower than those predicted by theory. Thus the effect of the thin boundary layer over the leading edge is to reduce 64 the magnitude of the negative pressure peaks below the magnitudes predicted theoretically. Reducing the airfoil incidence for purposes of the theoretical calculation will reduce the magnitude of these negative pressure peaks, but not to the same degree as actually occurs due to the presence of the boundary layer. However, if the circulation is not fixed by the Kutta condition, it is possible to specify the circulation otherwise. The circulation may be determined for the theoretical calculation from the measured lift. The theoretical pressures are then calculated at the measured incidence. The velocities at the trailing edge will no longer be equal, but the theoretical lift obtained from the integration of the pressure distribution can be made to equal the experimentally measured lift. The procedure for this calculation is presented in Appendix 8. Figure 8.1 compares the resulting theoretical and measured pressure distributions for the 0.67-NACA-0015 airfoil at 10 degrees incidence in the presence of two solid walls. The theoretical results by this method are physically unsatisfactory in the vicinity of the airfoil trailing edge as such large negative pressures are in practice not found there. It is possible to lower the circulation developed by altering the shape of the airfoil profile. This procedure is justifiable physically if the alterations are in keeping with those which occur when the boundary layer effectively modifies the profile shape. Recall that for an airfoil, the effects of viscosity are 65 confined to the thin boundary layer adjacent to the airfoil surface, and the flow outside the boundary layer can be considered as irrotational. Thus the boundary layer modifies the profile shape and the potential flow is calculated about this modified shape, using the usual equal-velocity Kutta condition-The following procedure follows and extends that of Pinkerton [37] whose analysis is for two-dimensional airfoil sections to be mapped conforraally onto a modified circle by means of the classical Theodorsen method. Figure 8.2 shows the original profile of the NACA-0015 airfoil and the resulting modified profile which results from the present procedure when this airfoil is at 10 degrees incidence between two solid walls. This is the modified profile used in the calculation of the theoretical pressure distribution in Figure 8.3. with respect to the direction of the approach flow, the modified profile is effectively at a slightly lower incidence and has slightly less camber than the original profile. The actual profile (original profile swollen by the addition of the boundary layer displacement thickness) about which a potential flow calculation might be considered, would be blunt at the trailing edge and would have the thickness of the wake at that point. The thickness of the boundary layer on the upper surface is greater than the thickness on the lower surface, except for a symmetrical airfoil in free air or between two solid walls, at zero lift. Therefore if the trailing edge was imagined to be taken (vertically) as the midpoint of the wake, and the after portion of the profile were faired to that point, the resulting shape would be similar to the effective profile of Figure 8.2-66 The airfoil is supposed to be at incidence a - Points on the profile are to be rotated about the profile leading edge in proportion to their distance from the profile leading edge- The direction of rotation is such to reduce the effective incidence, of the profile- As not all points are rotated by the same amount, this is not a rigid body rotation about the profile leading edge. Thus the effective camber of the profile is reduced as the trailing edge.is "raised" more than other points on the profile. The details of this "raised" trailing edge procedure are presented in Appendix 9- Figure 8.3 shows a comparison of the resulting theoretical and measured pressure distributions for the 0.67-NACA-0015 airfoil at 10 degrees incidence in the presence of two solid walls. The theoretical results by this procedure are quite satisfactory except for the negative pressure peak near the airfoil leading edge. The magnitude of the theoretical peaks is still larger than the magnitude actually measured, and will always be so for any procedure that does not increase the radius of curvature of the airfoil leading edge to the same degree that actually occurs due to the boundary layer there. The above procedure was applied to the airfoil-shaped wall slats to see what the effect would be of reducing the circulations there while maintaining the full Kutta circulation on the test airfoil. For the case of the 0.66-Clark-Y airfoil at 20 degrees incidence in the presence of a 70%OAH TSOSL wall configuration, the test airfoil lift coefficient was reduced 67 from 3.010 to 2.935, that is, by about 2.5%. This calculation used a value of k of 0.80 on all the wall slats. Thus the effect on the test airfoil of the neglect of viscous effects on the wall slats alone is small in comparison with the neglect of viscous effects on the test airfoil itself. A complete accounting for the boundary layer effects even for completely attached flows must involve the calculations of the laminar boundary layer growth, transition to a turbulent boundary layer, and the growth of the wake. Seebohm [38] has performed such calculations and found good agreement with experiments. His procedure is as follows. The usual potential flow calculations are performed for the airfoil profile, the growth of the boundary layer and wake are calculated. The airfoil profile is swollen by the boundary layer displacement thickness and is extended downstream to represent the wake. Seebohm develops a condition to fix the circulation (which reduces to the equal-velocity Kutta condition for no boundary layer and no wake) based on the pressure difference across the wake, calculated at two points (vertically) above and below the airfoil trailing edge, at the outer edges of the boundary layers. The last two steps are iterated on until the pressure difference across the wake no longer changes on iteration. Such calculations could be included in the present theory, on the wall slats. A method of handling the separated flow region over the upper surface of an airfoil at high incidence, approaching maximum lift, has been developed by Jacob and Steinbach [23]. 68 The separating streamline departs tangentially from the upper surface since the observed pressures correspond to a smooth non zero velocity distribution around the, separation point. The separating streamline has ah approximately constant pressure variation along its length, near the airfoil, to represent the approximately constant pressure in the real dead air region in the separated wake. This is accomplished by requiring equal pressures at a point at the trailing edge, and at a second point above the trailing edge, on the separating streamline. The circulation is thus fixed, and the method is seen to be.similar to the procedure of Seebohm. Again the boundary layer calculations must be performed to calculate the position of the separation point, and the pressure on the separating streamline. Again such a procedure could be incorporated into the present method, for the wall slats, if desired. 69 8.2 The Flow in the Plenum: The Bounding Shear Layer. The second extension of the present theory is to make the geometry of the flow representation more like that which actually occurs experimentally in the testsection, with the plenum surrounding the slotted wall- Figure 8-4 compares the flow representation of the present theory with the physical flow which actually occurs in the testsection. The present theory represents a uniform flow of infinite extent, past a set of multiple airfoils, and flat surfaces aligned with the direction of the undisturbed flow. Hence the energy levels of the. flows inside and outside the testsection are the same. Figure 8.5 shows a shear layer formed inside the plenum chamber surrounding the upper slotted wall. This shear layer is formed as the outflowing air from the testsection, upstream of the test airfoil, mixes with and entrains the otherwise stagnant air in the plenum. This shear layer therefore divides the two flows, the high-energy flow which exists in the testsection, and the zero-energy stagnant flow of the plenum. In a potential flow theory, this shearlayer could be idealized as a constant-pressure free streamline which leaves the test section at the upstream end of the slotted wall, enters the plenum and reattaches to the solid wall section at the downstream end of the slotted wall. The position of, and the pressure variation along such a streamline are initially unknown. The initial and terminal positions, inclinations and pressures could be estimated from a flow field calculation which omits representation of this streamline entirely. 70 An iterative procedure could be developed to calculate the position of this streamline in a segmented step-by-step approach. , The position of a segment would be assumed- The resulting flow inclinations would be calculated in. the vicinity of the supposed streamline and compared with the direction the streamline is taxing there. These steps could be iterated on until a, satisfactory streamline position was found. Such procedures are used for locating streamlines in vortex wakes, but are computationally time-consuming. As the approach to a zero lift-correction wall configuration is made, the overall flow field will approach closely that of the test airfoil in a free-air test environment. An obvious position to take for such a streamline is therefore the position that the particular streamline in the free-air case occupies which passes through the two points defined by the ends of the solid wall sections at the beginning and end of the slotted wall. In order to investigate the effect of including a constant-pressure free streamline in the present theory, an analytic two-dimensional potential flow representation was developed- The test airfoil was represented by a single vortex, and the wall slats by a set of vortices near the constant pressure boundary-Figure A6.2 shows the multiply-infinite set of vortex images required to satisfy the boundary conditions. On the solid lower wall, the boundary condition is zero disturbance velocity normal to the wall. On the constant pressure boundary, the linearized condition of constant pressure can be expressed via Bernouilli's 71 equation as requiring zero disturbance velocity in the stream wise' direction. Thus in solid boundaries, the appropriate image of a vortex is a vortex of equal but opposite circulation; the image in a constant pressure boundary has identical circulation. The details of the image system and the eguations involved are in Appendix 6. The results shown in Figure 8.6 compare -the present analytic image representation with the analysis of Havelock [5], for a flat plate (a) between two solid walls, (b) in ground effect, (c) between a solid lower boundary and a constant pressure upper boundary, and (d) between two constant pressure boundaries (open jet). The image representation is unsatisfactory as it predicts a lower lift than for either the ground effect or the open jet case; the experiments indicate lifts above the ground effect values. That the lift is so low is a result, of the constant pressure boundary condition which requires the tangential disturbance velocity to be zero there. The corresponding value of the pressure coefficient, Cp, is zero. By tracking a streamline (in the TSOSL wall configuration theory) which leaves the testsection through the slotted wall upstream of the test airfoil, and re-enters the testsection downstream of the test airfoil, the theoretical variation of pressure along- such a streamline is known. Figure 8.7 shows such a pressure variation, and, excluding the large negative pressure excursions as the flow.accelerates when in the- vicinity of the wall slats, the average . value-of.the pressure coefficient on this streamline is 72 about -0.25. Hence any representation which uses a zero C value P is incorrect.\ The TSUSL wall configuration theory indicates that the wall slats collectively have a net counter-clockwise circulation, since there are a larger number of slats immersed in the re-entering flow than in the exiting flow. However the. image representation correctly predicts the effect on the airfoil lift of including the circulation on the wall slats via the set of vortices adjacent to the constant pressure boundary. For example, for a flat plate at 26.3° incidence (see Appendix 6 and Figure 8.6), the image representation predicts that the lift is depressed 25% below the free air value; the contribution from the wall slat vortices is 2% and that from the constant pressure boundary is .23%. The comparison would be better for the flat plate at 20° incidence, but the calculations use wall slat circulation values from a Clark-Y airfoil at 20° incidence and 7 0%OAR. However, any single potential flow free-streamline representation cannot model the division of the two flows of very different energy level, that is, total head, correctly. Suppose that the position of the streamline representing this shear layer is known. The flow along this streamline satisfies the tangent-velocity boundary condition. In addition, the pressure variation along this streamline must reflect the fact that the flow energy level on one side of this streamline (in the plenum) is zero. On the other side (in the testsection) the flow energy level is that of the Uniform undisturbed approach flow. This shear layer could be modelled analytically by a vortex sheet, but this • modelling was not attempted here. 7 3 Physically the pressure variation in the shear layer or on a representative streamline is a result of the manner in which the high energy testsection flow leaves the solid wall section, and of the low pressure region of recirculating flow so formed in the plenum. To represent the shear layer by the present theoretical method, coincident source and vortex elements are distributed along a representative streamline. Source elements are used in the usual manner to ensure that the flow is tangential to the surface. Vortex elements, all of different strength densities, are used to set prescribed values of the tangential velocity at each control point on the surface. Thus if the shear layer is represented by, say, S control points, there are an additional unknown S source and S vortex strength densities. Thus . there are S additional zero normal-velocity equations, and S additional prescribad-tangential-velocity eguations to be solved. Since it is usual to prescribe the pressure variation on the surface rather than the tangential velocity variation, the tangential velocity boundary condition equations have the form, from (3. 14,3.18): N M R(k) S 7 (B .U.+A.V.) % v ri l ri i-* r=l (8.1) = -Ucose. ± /(l-C ), i=l,2 1 p. , . . . s. Here y and v are the source and vortex strength densities respectively on the streamline representing the shear layer. For 74 calculations in the same sense as the flow direction, the squareroot term is positive if the source and vortex elements are distributed sequentially. The initial and terminal positions of this streamline are known, and the corresponding inclinations and pressures could be estimated from the flow conditions on the solid wall sections. It remains only to specify the variation of pressure (between two known end values) along the streamline. Values of spreading coefficients for unbounded shear layers are to be found in the literature, (for example, [40]) and if estimates could be made of the effect of confinement which occurs here in the plenum, such estimates could then be used to specify the position of a streamline representing the shear layer. It is proposed that the pressure variation along the streamline representing the shear layer is a free parameter in the analysis here. That is, several pressure variations were assumed and specified, and the computations performed. The streamline tracking procedure of Appendix 4 was used to compute the variation of pressure along the streamlines of similar position and shape as are required for specification of the position of the shear layer. This pressure variation is shown on Figure 8.7. This streamline leaves the upstream upper solid wall section and re enters the testsection downstream of the test airfoil, thus apparantly "entraining" some of the exterior flow. It would be 75 expected, from continuity, that this streamline should end on the downstream upper solid wall section. Hess [22] experienced similar "leakage" flows in the calculation of the interior flow in a right angle bend. At first it appeared that there were too few source elements on the solid wall sections, and that the solid wall sections were too short. These sections were lengthened from two test airfoil chord lengths to ten, and the number of source elements on a section was increased from 20 to 50. The "entrainment" effect was still present. This apparantly is a common problem for interior flow calculations using flat distributed source elements with constant uniform source strength densities; the problem is eliminated by using curved source elements with linear or parabolic variations of source strength densities over the elements. The "entrainment'•• flow rate here is about 5% of the net flow rate; in Hsss's example the "leakage" flow rate was 12%. With the end values of pressure specified, several pressure variations were tried such that the average pressure along the streamline representing the shear layer was similar to the average pressure on the tracked streamlines. The results in Figure 8.8 show how the airfoil lift coefficient varies with the assumed value of C . This representation of the shear layer is P used in the following section to compare the theoretical and measured values of lift-curve slope for two airfoils. 75 8^2 Summary... A comparison of the curves of Figures 7.2 and 7.3 indicates that at an upper wall OAS of 70%, the experimental value of the ratio of the lift-curve slopes ra/m is about 2.8% lower than the s theoretical value, for both the Clark-Y and the NACA-0015 airfoils. Thus there is a small residual difference of 2.8% to be accounted for by the extensions to the theory outlined in §8.1 and 8.2. Assuming that there are curves similar to the curves of Figure 8.8 of the ratio of lift coefficients C /C for the LT LF present case, the residual 2.8% could be accounted for by an increase in C of about +6%, for example from -0.33 to -0.31. A P value of C of -0.31 corresponds to the pressure level P established by the recirculating flow in the plenum. F.ecall that an adjustment of the same order could be accounted for with the procedure of §8.1, that is, by reducing the circulations on the wall slats. Thus the difference between the theoretical and experimental curves of Figures 7.2 and 7.3 at any OAR can be accounted for in the theory by a combination of (i) the reduction of the circulations on the wall slats by modifying their effective profiles, and (ii) by the representation.of the shear layer in the plenum by a single streamline of assumed position and streamwise pressure variation. These extensions are preliminary; further work is required to establish a satisfactory theory. 77 9j. Conclusions. A two-dimensional theory which predicts a satisfactorily correction-free windtunnel test configuration has been developed. The theory is an extension of the two-dimensional potential flow theory based on the method of distributed surface singularities. The extended theory takes into consideration not only a wide range of airfoil sizes and shapes, but also the effect on the airfoil loadings of different windtunnel wall configurations. The results of the theoretical study indicate that for two-dimensional airfoil testing, a windtunnel consisting of a solid wall opposite the pressure side of the airfoil, and a transversely slotted wall, the solid portions of which are symmetrical airfoil-shaped slats at zero incidence, with open area ratio between 60 and 70 percent, opposite the suction side of the airfoil, will yield uncorrected pressure distributions and lift coefficients which are within a few percent of the free air values. The theory predicts that this low-correction wall configuration will remain relatively correction-free for a wide range of airfoil sizes and angles of incidence. Experiments carried out on a number of airfoils for a Reynolds number range of 300,000 to 1 million (based, on. the airfoil chord), in a two-dimensional test configuration support the predictions of the theory. Experimental work showed that the correction-free test configuration could be achieved with a slotted wall consisting of symmetric airfoil shaped slats at zero incidence, when the slotted section was surrounded by a 78 plenum chamber. The above theory was then extended to account for viscous effects on the wall slats, and the effect of the shear layer which forms in the plenum chamber. Measurements taken with the correction-free wall configuration of the lift, drag and pitching moments for nine different airfoils which ranged in size (chord to height ratio) from-0.17 to 0.67, showed good agreement with established free air values. Furthermore, measurements of the pressure distribution on two airfoils, with the correction-free wall configuration, showed good agreement with pressure distributions measured in solid wall configurations and corrected by standard methods. The low-correction test configuration theory which has been tested and verified in the work reported in this thesis can be developed to provide a reliable means of testing high lift airfoils in existing windtunnels which can be modified to achieve the low-correction wall configuration. Such tests would otherwise require elaborate test facilities or complex correction procedures. Appendix 1. The Integration of a Three Dimensional Point Source to a Two  Dimensional Flat Distributed Source Element The potential cj> at a point P due to a source strength density distribution a(Q) over a surface S is, from (3.2), <MP) q(Q) ,r (PQ) dS , (Al.l) where r(PQ) is the distance from the point P to the point Q. From Figure Al.l, the potential at P(x,y,0), for unit strength density a, is As <}>(x,y,0) = As 2 d? _„ /(x-a2+y2+c2 (A1.2) The inner integration sums over all infinitesimal source . elements d£d£; to produce that part of the potential at P due to a line source element of width d£. The outer integration sums over all such line source elements to produce the potential at P due to a flat distributed source element of width As, and of constant uniform strength density. The velocity components in the x and y directions respec tively, induced at P by a source element of width As, are: 80 V = x V 34) 3x 3(j) 3y + + As 2 (x-C)I(x,?/y) d£, As 2 As 2 yl(x,5,y) d£ , As 2 (A1.3) (A1.4) where I(x,£,y) = !«»• ((x-?) 2+y2+?2) 3/2 (A1..5) Now _3 ((x-o2+y2)/((x-?)2+y2+c2)j . ((x-£)2+y2+s2)^ (A1.6) therefore I(x,£,y) = ((x-02+y2)/((x-£)2+y2+C2) -~ (x-?)2+y2 (A1.7) Also ^(-log((x-?)2+y2)) = +2(X"S) , 95 (x-?)2+y2 (A1.8) (tan"1 ^1) = 3 5 C-x i y J (5-x)2+y2 (A1.9) Therefore V As •log( (x-5) 2+y2) As ' 2 +log [(x-^) 2+y2J (Al.10) 81 V = 2tan Y 5-x t y As r ,As X+-— As ' 2 2 (tan -1 l y J - tan -1 x— As y J ). (Al.ll) In a two dimensional incompressible, irrotational flow, the stream function ty and the potential ty are conjugate harmonic functions. Thus the Cauchy-Riemann equations require dty _ 3<j> 3y 3x' Hi 3x 9(j) 3y (A1.12) Hence ty may be expressed as ijj(x,y) - o (|i)dy + f (x) -j(||)dx + g(y) , (A1.13) where f and g are arbitrary functions and ty0 is an arbitrary constant. Some useful identities are: log(y2+a2) = -^(ylog (y2+a2) - 2y + 2atan 1 ), (A1.14) tan -1 (xtan 1 — - ^-log (x2+a2 ) ) , 3x' (A1.15) tan ''"A ± tan XB = tan x (j^g) • 1 rA±B (A1.16) Therefore, from Al.10, H = sff"YA + xB + c)< (A1.-17> where A = log (x+%.2 + y2 ^ As {(x~)2 + y2J C = Astan B = 2tan 2xy -1 yAs [x2+y2-(^)2J (A1.18) lx2"Y2-(f)2J According to equations (Al.13) and (A1.17), the stream function for a flat distributed source element of width As and unit strength density is iMxfy) = tyo + A - B + C. (A1.19) As in equation (Al.13), the potential <j> may be expressed as (J)(x,y) - <J)o = - (|i)dy + f(x) = (||)dx.+ g(y), (A1.20) where f and g are arbitrary functions, and (J>0 is an arbitrary constant. The application of the three relations (Al.14), (A1.15), and (A1.16) to (Al.19) results in 83 (A1.21) D = flog((x^)V)[(x-^)V!^ Therefore the potential function for a flat distributed source element of width As and unit strength density is (J)(x,y) =cf>o+xA + yB + D- 2As . (A1.22) The corresponding results for a unit strength density flat vortex element can be written immediately since in a two-dimensional incompressible irrotational flow 4> (vortex) = -^(source), i> (vortex) = +cj) (source) . (A1.23) 84 Appendix 2._ h Procedure for Block Commutation of Matrices Aj_ 8 and C__ When the matrices A., B and C are large, C might be assembled from A and B by partitioning C into blocks as follows. Memory is allocated for A, B and C according to the size "of the largest block. *j = l, 2, . . .N* j=N+l,N+2,-. .N+M* i=l,2,..NL4 i=NSUl,NSU2 i=NKA,M r A. . Di R(k) r - y B . * L, mi . m=l { i  -j •; A . . ]l R(k) c - y B . * L -, mi m=l ' 'B . +B.T r r f 1 R(k) >• y (A TT +A _ L, mil mL m=l r r ::B. +B . nU jL • r r R(k) 'I (A +A )^ ^, mU mL . m=l r r test airfoil, flaps and solid walls slats r-th slat r-th airfoil or flap The subscript 'i' refers to the row number in the matrix C. The meanings for the variables NL4, NSD1, NSB2, NK1, NK2 and NKA are given in the programs which follow. Subroutine SUB1 calculates A and B in blocks as needed to assemble the blocks of C. SUB 1 is called twice to set up the zero normal-velocity boundary condition eguations on all solid surfaces. It is called first for the test airfoil and flaps, and all solid wall sections and a second time for the wall slats. Subroutine SUB2 uses A and B to set up the Kutta condition 85 equations on the wall slats. Subroutine SUB3 does the. saiae for the test airfoil and flaps. The FORTRAN coded versions of SUB1, SUB2 and SUB3 follow. The subroutines READER (and WRITER) read (write) a matrix from (onto) a peripheral storage device, such as a magnetic disc. 1 C SUB1 CALCULATES A £ B IN BLOCKS 6 SETS UP -EON'S ZERO NORM VEL ON ALL SOLID 2 C SURFACES 3 SUBROUTINE SUB1(A,B,C,XX,YY,DS,CS,SI,N,L,MfNA,NAF2,L1,L2,NSPS, 4 1 NSLATtNS-Ul tNKl) 5 C A,B - MATRICES OF INFLUENCE CCEFFS FOR SOURCE £ VORTEX ELEMENTS 6 c C- MATRIX FOR SYSTEM OF EQN1S C*SIG=D 7 c SIG(M) - UNKNOWNS IN SYSTEM G*SIG=D - PART IS GAM 8 c GAM(NAFt-NSLAT) - VORTEX STRENGTHS ON KUTTA BODY. 9 c M - TOTAL U UNKNCWS= TOTAL M EQUATIONS 10 c NAF = #TE.ST AIRFOILS I FLAPS WITH KUTTA CONDITIONS APPLIED 11 c NSU1,NSU2 - SET OF EQN'S FOR ZERO NORMAL VELOCITY ON SLATS. 12 c NK1,NK2 - SET OF EQN'S FOR KUTTA CONDITIONS ON WALL SLATS. 13 c XX,YY - CONTROL POINT COORDINATES; DS - ELEMENT LENGTH 14 c NSLAT - •# OF SLATS WITH KUTTA CONDITIONS; NSPS - U ELEM'S PER SLAT 15 c CS SI - SIN,CCS CF ELEMENT INCLINATION 16 c NA(KJ - RANGE OF CONTROL POINT #'S FOR K-TH TEST AIRFOIL OR FLAP I.E. 1,50 17 c N - TOTAL # CONTROL PTS WHERE NORM VEL IS ZERO ON ALL SOLID SURFACES 18 c L - BLOCK SIZE 19 c L1,L2 - RANGE OF CONTROL PT H'S FOR BLOCK - SET OF EQN #S ALSO 2 0 REAL A(N,L),B(N,L),C(M,L),XX(N),YY(N),DS(N),CS{N),SI(N) 21 INTEGER NA(NAF2\ 22 NSU2=NSU1+NSPS*NSLAT-1 23 NK2=NSU2=NSLAT 24 NAF=NAF2/2 25 c CALCULATE A AND B. 26 DO .1 I=L1,L2 27 K=I-L1+1 28 DO 1 J=1,N 29 IF{J.NE.I) GO TO 2 30 AIJtK)=6.2831927 31 eu,K j=o. 32 GO TO 1 33 2 DXJ = XX(I }-XX{J) 34 DYJ= YY{I )-YY(J ) 35 XJ=DXJ*CS(J)+DYJ*SI(J) 3 6 YJ=DYJ*CStJJ-DXJ*SI(J) 3 7 DSJ2=0S( J )/2. 38 Y JS=YJ*YJ 39 S=XJ-rDSJ2 40 T=XJ-DSJ2 IQ ft CM I joo I3 |>-4-CO >-+ •H-oo —> + •»• 00 o o _J II X — -) -5 -J OO 00 —• oo ' 00 l_) Q I + CM •— — Z 00 oo < u o < — — CM >-« 00 II OO <_} >- I) II —<—>—> zz -> a CM co -J->i- <j- <r 00 LU OO * >-»—* t—t <J X 00 a. _t il X LU CO C£ 3 Z Z < CO cC CC LU LU ai oi 3: ?. ro LJ O lT\ O Is--4- -4-CO QT- O -J- vf lA OO LU o < U-r> 00 a oo o oo LU > cc o z a 0£ LU M cc a 00 2 -3 CJ II LU -3 a. <J-ZZ> a LU OO on 00 < 00 a CD CC a H s: o a e> 2: r-— CC < OO-I • LL OO Of 2 LU 00 •• • - —I f-.Z II < Of 00 —I LU ^ lA o Q LU 00 CM COivf LA 43 ir\ Lolin in ir\ I 00 id •«• 00 .-t a. I 00 00 Z Q. + O0 Z ZD > OO z 11 11 ^ _J •XL M co 0* LO lT\ ii w ii —-II CO I » co O ^ 00 II II co a .co 00 o 00 r-i CM s0 O 00 a. < 1 u. oj OO o cc 00 —I LU > 2: ex a z a a: LU M or a a. a. < z 00 •• CO - 1-1 cfl Z l| I! OZ — LU »—4 » Q. CO -3 3 — a 01—0 LU OO m u r-ro <r on -0 O z ii I <r> co ii ii < < z z II II —1 CM ii ii r- co ii + U-<t z t CM ii o O II CO o 00 a o 00 LU II ZD — Z -3 Z w a o o cr« co co CM 87 LL. < a 2: * LJ — a s: r- O CX >»• <t Qi O- LU r-OO >—1 >- ai X ^ z r- Qi _l ZZ) HI J h y— < LU >-i o CX oi 3 r— r«- r> f>- f— CO 79 80 93 94 95 96 97 98 99 100 101 102 103 104 105 106 10 7_ 108 109 110 111 112 113 114 115 116 SUB2 SETS UP KUTTA EQN'S DN WALL SLATS.' SUBROUTINE SUB2(A,B,C,NA,N,L,MtNAF2,NSLAT,NSPStNSU1,NK1) Xf ~~~ REAL A ( N , L ) , B (fl7TT7C(KTNSnm 82 INTEGER P » Qt NA(NAF2I 83 C A» B - MATRICES -OF INFLUENCE COEFFS FOR SOURCE £ VORTEX ELEMENTS 84 C C- MATRIX FOR SYSTEM OF EQN'S C*SIG=D 85 C M - TOTAL ti UNKN0WS= TOTAL # EQUATIONS 86 C NAF=#TEST AIRFOILS £ FLAPS WITH KUTTA CONDITIONS APPLIED ~87 C NSU1,NSU2 - SET OF EQN'S FOR ZERO NORMAL VELOCITY ON SLATS. 88 C NK1,NK2 — SET OF EQN'S FOR KUTTA CONDITIONS ON WALL SLATS. 89 C NSLAT - # OF SLATS M Jj-_ J_UJ_T A_ CONDITIONS; NSPS - # ELEM'S PER SLAT ~9Q~~- C _l^Tin—:nRANGE~OF CONTROL POINT #»S FOR K—TH TEST AIRFOIL OR FLAP I.E. 1,50 91 C.N- TOTAL # CONTROL PTS WHERE NORM VEL IS ZERO ON ALL SOLID SURFACES 92 C L=NSPS*NSLAT - BLOCK SIZE ; : : NAF=NAF2/2 NSU2=NSU1+NSPS*NSLAT-1 NK1=NSU2+1 NK2=NSU2-fNSLAT C READ IN THAT PART OF A £B WHICH CONTAINS THE INFLUENCE COEFFS FOR VELOCITIES C INDUCED AT THE CONTROL POINTS ON THE WALL SLATS. CALL READER(A,N,L ) CALL READER IB,N,L) SET UP EQNS F_Q_R KUTTA CN WALL SLA_TS_ ' "_TQ i i=i, N SLA T P=1+NSPS*(1-1) Q=P+NSPS-1 C P,Q - T.E. CONTROL PT #S FOR SLATS C 2 LOOP - TANG'L VELS DUE TO ALL SOURCE ELEM'S. ' DO 2 J=L ,N •  2 C C(J, I ) = B(J,P)+BtJ,Q) 4 LOOP - TANG'L VELS DUE TO VORTEX ELEM'S ON SLATS DO 4 KS=1,NSLAT . .  J=NK1+KS-1 KK=NSU1+NSPS*(KS-l) KL=KK+NSPS-i SA = 0. DO 3 K=K'K , KL SA=SA+A(K,P)+A(K,Q) 117 4 C(J,I)=SA 118 C 6 LOOP - TANG'L VELS DUE TO VORTEX ELEM'S ON AIRFOILS £ FLAPS. -co co 119_ _10_ DO 6 KN- 1, NAF . 120 " K3=2*(NAF+1-KN)-1 121 K1=NAIK33 122 K2=NMK3 + 1) : ; __ 123 J-=N-NAF + KN 124 SA=0. 125 DO 5 K=K1, K2 ; - . . 126 5 SA = SA + A(K,P)+A{K,Q ) 12 7 6 CU»I) = SA 128 1 CONTINUE _129 C WRITE THIS PORTION OF C INTO A FILE. 130 CALL WRITER(C,M,NSLAT) 131 . RE TURN - ; 132 ' END 133 C SUB3 SETS UP KUTTA EQN'S ON TEST AIRFOILS & FLAPS. 134 SUBROUTINE SUB3(A,B,C,NA,NTE,N,M,NL4,NAF,NAF2,NSPStNSLAT,NSU1,NK1 )  135 REAL A(N,NL4),B(N ,NL4) ,C(M,NAF) 136 INTEGER NA{NAF2>,NTECNAF» 1.3 7 C AtB - MATRICES OF IN FLU EN CE COEF'FS FOR SOURCE & VORTEX ELEMENTS ____ 138 C C- MATRIX FOR SYSTEM OF EQN'S C*S.IG = D 139 C SIG(M) - UNKNOWNS IN SYSTEM C*SIG=D - PART IS GAM 140 C GAM ( NA F + NS LAT ) - VORTEX STRENGTHS ON KUTTA BODY. _ 141 C NL4 - M OF CONTROL POINTS ON TEST AIRFOIL, FLAPS L SOLID WALL SECTIONS 142 C M - TOTAL # UNKNOWS= TOTAL EQUATIONS 143 C NA F-ffTEST AIRFOILS £ FLAPS WITH KUTTA CONDITIONS APPLIED 144 C NSLAT - # OF SLATS WITH KUTTA CONDITIONS; NSPS - # ELEM'S PER SLAT 145 G NA{K) - RANGE OF CONTROL POINT #'S FOR K-TH TEST AIRFOIL OR FLAP I.E. 1,50 146 C N - TOTAL H CONTROL PTS WHERE NORM VEL IS ZERO ON ALL SOLID SURFACES  147 - C NT E{K I - CONTROL PT # FOR UPPER T.E. ON K-TH TEST AIRFOIL OR FLAP 148 hSU2=NSUl+NSPS*NSLAT-l !_ 9_ M<l = NSU2+._ . ' ; . 150'" C READ IN THAT PART" OF A &B WHICH CONTAINS THE INFLUENCE COEFFS FOR THE 151 C CONTROL POINTS ON THE TEST AIRFOILS I FLAPS. 152 CALL READER ( A ,N ,NL4) :  153 CALL READER{B,N,NL4J 154 C SET UP EGN'S FOR KUTTA GN AIRFOILS AND FLAPS 15_5 '_„__.___ JJOJL J<|_f i_NAF ; ' _ _ _ '156"" " ' I=KN" " . ' " • 157 MT=NTE(NAF+l-KN) 15 8 MTE = MT+1 . • o CTi 15g c MT t MT E ARE #S FDR T.E. CONTROL PTS. ON TEST AIRFOILS t FLAPS. 160 C 2 LCGP - TANG'L VELS DUE TO ALL SOURCE ELEM'S. 16 i DO 2 J=1,N „ . 162 2 C(Jf n = B(J,Mf)+B(J»MTE) 163 164 c IF{NSLAT.EG.0) GO TO 5 4 LOOP - TANG'L VELS DUE TO VORTEX ELEM'S ON WALL SLATS. i u • 165 DG 4 KS=1,NSLAT 166 167 J=NK1+KS-1 KK=NSU1-NSPS*IKS-1) Jt, W I 168 KL=KK+NSPS-1 169 L70 SA = /. DO 3 K=KK,KL i 1 w 171 3 SA=SA+A{K,MT)+A(K,MTE) 172 173 4 c C(J,U=SA 9 LOOP - TANG'L VELS DUE TO VORTEX ELEM'S ON TEST AIRFOILS & FLAPS. 174 5 DO 9 KM=1 »NAF 175 K3=iNAF-rl-KM)-l 176 K1=NA(K3) 177 K2=NA{K3+1) 178 J=M-NAF-rKM 179 SA^O. ,—, . 180 181 ; 8 DO 8 K=K1tK2 SA=SA^A(K,MT)-AIK,MTE) 18 2 9 C(J» I ) = SA 183 184 1 C CONTINUE WRITE THIS LAST PART OF C INTO A FILE. 185 CALL WRIT ER{C,M» NAF) _ 186 Hfcll 18 7 END. O 9i A£_endix 3_ Two Methods of Solvin_ the Systems of Simultaneous Lil^eaE liaebjcaic Ej_uationj__ ___ Gaussian Elimination. This PORTRAN coded subroutine provides for significant savings when solving a large number of equations under a virtual memory operating system which employs some kind of paging system for dynamic storage allocation. To illustrate the reversed subscript notation, the system of equations is written: ci io-. +c2\02 +c3 i.a3=d1. Ci20i+C2 20"2+C32a3 = d3 (A3. 1) Ci 3a i +c2 3a2+c3 3a3=d3 The printout of the subroutine ATXB follows. 92 3__2 Successive Row Vector Qrtho_onali_ Process. The matrix C for the system C(a,y)=d is augmented by the right hand side vector »d' to form the eguations C i \0 i +C2 lO"2+C 3 iO 3~d lt=0 c i 20" i +C2 20"2+c 3 2cr 3-d 2 t=0 (A3.2) C 1 3CT1+C2 3a2+C 3 3^3-d 3t = 0 A set of N vectors in (N+1)-dimensional space is formed: (Cli'C2i'C3i'***CNi '~di) i=1,2,...N (A3.3) The solution vector of . equations (A3.3) is such that the vector {olfa2,a3,..,o^t 1) (A3. 4) is orthogonal to all the vectors of (A3.3). The process of solving equations (A3.2) is equivalent to determining an (N+1)-dimensional vector orthogonal to the N vectors of (A3.3) and having unity as its (N+1)st component. Let U. denote the j-th row of the augmented matrix. At each 3 stage j, (j=1# 2, . . . N+ 1) , a set of vectors for ±= 1, 2, . . . N + 2- j is constructed. All (N+2-j) vectors in this j-th set are constructed to be orthogonal to the (j-1)th row of the augmented matrix, that is, a 1-. In fact, all (N + 2-j) vectors in'this j-th set are orthogonal to all of the first (j-1) rows of the augmented matrix, that is", the vectors U^,.. . U Thus vil. (the last member of the (present) j-th set of N+2-j vectors V"? ) , is a vector which is orthogonal (by construction) to __A or fhe first (j-1) vectors 0^, 0 ... U ^, and thus is 93 orthogonal to all of the first rows of the augmented matrix. N+1 Ultimately there is a single vector which is orthogonal to all M rows of the augmented matrix. This is the solution to the system of equations, as the (N+1)th (last) N+1 component of will be unity. In fact, at each stage ' j' the last component of V^+2_j is unity. To actually construct the set of vectors at the j-th stage, scalar multipliers c^ 1 must be calculated such that the set of vectors vi=ci"lvr1+vi+i (A3.5) is orthogonal to the (j-1)th row of the augmented matrix, that is OJ_-L" Thus c^ is defined by the scalar product Vi*Uj-l = 0" (A3.6) Hence c~! 1 is calculated from i ci~1=-(Viii*uj-i)/(Vi~1*Dj-i) (a3-7) To initiate the process, the set of vectors V1 is chosen to be the set of "unit" vectors (0, 0 ,. . . 1. . . 0, 0) , where 1 is the i-th component. This procedure is best illustrated by an example. For the system of equations x+y-z=2 2x+y+z=1 (A3.8) x + y-t- 2z=- 1 The augmented matrix is 94 1 1-1-2 2 1 1-1 112 1 (A3. 9) and 0= (1,1,-1.-2) r U =(2,1,1,-1) and 0.= (1,1,2,1) . (A3-. 10) For j=1, N + 2-j is 4, so there' are 4 "unit" vectors v-., so vj-=(1, 0,0,0), vj=(0,1,0,0) , ^=(0,0,1,0) , V^(0,0,0,1) . (A3.11) 1 z j 4 For j=2, N+2-j is 3, so there are 3 vectors V2 and 3 multipliers c . Using the reversed subscript notation, i •V-+*U =a 1111 and r± *U =a i+1 1 1+1,1 therefore i I+I,1 11 and ^=-1, c^=1, c1=2. 1 ' 2 3 Therefore V2 = c1V1 + V1 i i 1 i+1 so V2-(-1,1,0,0) , ¥^=(1,0,1,0) and ^=(2,0,0,1). 2 All three V are seen to be orthogonal to 0 . i 1 (A3. 12) (A3. 13) (A3. 14) (A3. 15) (A3. 16) For j=3, N + 2-j is 2, so there are 2 vectors and multipliers c.. Here • I Vl*°2=-1 and Ci'Vi+l*D2 2 2 therefore c_=3 and c =3. Therefore ^22 2 Vf = cfvf + VT. . 1 X 1 x+1 so V3= (-2,3,1,0) and V3 = (-1, 3, 0 ,1) . Both V3 are seen to be orthogonal to U and II , i 12 (A3.17) (A3. 18) (A3. 1 9) (A3. 20) 3 For j=4, N+2-j is 1, so there is a single multxplxer c^ and vector V . Here V3*0 =3, V3*U =3, so c3=1. 1 3 2 3 1 (A3. 2 1) 95 Therefore V^- (T , 0,-1,1) is the solution vector for the system (A3.8) as .x=1, y=0, and z=-1. The FORTRAN programs for reading and writing matrices onto peripheral storage devices are highly system-dependent, so the program for this vector method is not given here. 96 AE_.§____ __ A __________ ________ _________ This algorithm is used to track streamlines from a specified starting point, given an increment in the x-direction. The method uses the subroutine THETA to calculate the flow direction, velocity components, velocity magnitude, and pressure coefficient at each point. From a starting point (x_,y_), the flow direction 81 is calculated, and used to locate the next point (xi+Ax,yi+Axtan8!). The flow direction 92 is calculated there. The two flow directions are averaged to give 8, and the y-coordinate is changed so that the next point is now (xi+Ax,yi+Axtan0). The flow direction 92 is calculated there, and so on. An outline of the FORTRAN coded subroutine THETA follows. •s-fr*: z' _ w cc >< - z nj •-CD 0 . •«-*• Z > 0 0 CO z ir-X-X CS 0 to 0 *—V z _1 z zz *—1 •«* IO —* «* 0 _ >v r-l. _ •• * —* >•*_ • _;••<-« &' il •. ""^ til _> If f-f ZD < 0 fe: * _ • + !-« cc _ _ _ _ w CO ^ _5 -rr CD -*r 11 1 H .<£ »—> _: i II ZD _! io. _. 0 II t_ II _ O l» CO £_ _ z to->-+ a: • • BR _3 _•' < il II z Ml _S Z . II CDl Z. t_ H r-l Hjj-T H NV <r«i ir- CO 'TJ''"-* — — CD 03| _ II -3 -l> ; £_ I! 2lH II ft':. 11: 11 _. «"» at _ o _|! -—'.: or. <: _ il w V T X .XI • .XI ^ jr—t *" to X • 'ir* •»! Z X —•_-*] «-*• O _l <C| *- 1^. s >-* a: o £_ o o 3: :t_: z Ir-t .;—• CO 00 ^ M - < •"; ;: X • y-i —1 i rr —* w fc— cc zz <; OH 2: i— -t> Z5 •r -^^tal E— Q O ' 1_ Z O X QL. Ui . " •04-* 188 SUBROUTINE THETA {X,Y,THE,CP,XX,YY,DS,CS,S1,N,VNT,VTT,VM,A,B,SIG, 189 1 GAM,NA,GAMM,M) 190 REAL XX(N), YYIN), DSINJ* CS(N), SMN), A < N) , B (N) , S IG (N i , GAM { N) 19 1 INTEGER P, Q 19 2 COMMONOAO NA,NL4,NSLAT,NSPS 193 C IX,Y) - CALCULATING FLOW PARAM * S HERE. 19 4 C XX,YY - CONTROL POINT COORDS FOR ALL SOURCE £ VORTEX ELEMS: DS IS ELEM LENGT 195 c SI,CS - SIN,COS ELEM INCLINATION 196 c N - TOTAL 3 CONTROL POINTS 197 c A,B - INFLUENCE COEFFS FOR VELOCITIES INDUCED 2 <X,Y) DUE TO ALL SOURCE £ 198 c VORTEX ELEM'S. 199 DO 1 J= 1 i N 200 DXJ = X - XXJJ) 201 DYJ = Y - YYTJ) 202 XJ = DXJ*CSIJ) + DYJ4SUJ) 203 YJ=DYJ*CSIJ)-DXJ*SIIJ) 204 DS J2 = DS(J ) /2 . 20 5 YJS=YJ*YJ 206 S=XJ+DSJ2 20 7 T = XJ-DSJ2 208 PHIX-=ALOG( (S*S + YJS)/(T*T + YJSJ ) 209 PHI Y=2.*ATAN2((DS1J)*YJ),(XJ*XJ+YJS-DSJ2*DSJ2)) 210 A(JJ =. PFIY*CS<J) + PHIX*SI(J) 211 1 BUI = PHIX*CS(J) - PHIY*SIU) 212 c SIG - SOURCE £ VORTEX STRENGTHS 213 c GAM - VORTEX STRENGTHS FOR TEST AIRFOIL £ SLATS 214 c GA MM - TEST AIRFOIL VORTEX STRENGTH 215 c VNSTiVTST - TOTAL NORMAL £ TANG'L VELS 3 tX,Y) DUE TO SOURCE ELEMS. 216 VNST =0 217 VTST = 0 218 DO 2 J= I ,N 219 VNST = VNST + MJJ*SIG1J) 220 2 VTST = VTST t eU)*S IG(J ) 221 AS=0. 222 BS = 0. 223 C NSALT,NSPS - # SLATS,# SOURCE ELEMS PER SLAT 224 IF(NSLAT.EQ.O) GO TO 3 225 CO 4 J=l ,NSLAT 226 C NL4 - # OF CONTROL POINTS ON TEST AIRFOIL, FLAPS 6 SOLID WALL SECTIONS 227 P=NL4+NSPS*J CO 228 C=P-NSPS+1 229 C P,Q - 1ST £ LAST CONTROL PTS ON A SLAT. 230 C AP,8P - NORM £ TANG VELS 3 (X,Y) DUE TO VORTEX ELEMS ON SLATS 231 AP = 0. 232 BP = 0. 233 • DO 5 M=P,Q 234 A P = A P + A { M } 235 5 BP = BP+B-(M1 23 6 AP=AP*GAM-l J ) 23 7 BP = BP*GAV.{ J) 238 AS=ASfAP 239 4 BS=BS+BP 240 C . ATiBT - NORM £ TANG VELS 3 (X,Y.) DUE TO VORTEX ELEMS ON TEST AIRFOIL. 241 AT=0. 242 B7-0. 243 C NA - # CONTROL PTS ON TEST AIRFOIL. 244 ' DO 6. J-liNA 245 A T = A T +A f J) 246 6 BT=BT+B1J) 247 AT=AT*GAMM 248 8T=BT*GAMM 249 C VNVTrVTVT - TOTAL NORM £ TANG VEL 2. (X,Y) DUE TO VORTEX ELEMS. 250 VNVT=-BS-BT 251 VTVT=AS+AT 252 C VNOT.VTVT - NORM £ TANG UNIFORM ONSET FLOW VELS. 253 VNOT=0. 254 VTOT=U 255 C VNTiVTT - TOTAL NORM £ TANG VEL 3(X,Y). 256 VNT=VNOT+VNVT+VNST 257 VTT=VTOT+VTVT+VTST 258 vS=vTT*VTT+vN.T*VN259 C THE,CP,VM - FLOW DI RECTI ON,PRESSURE, MAG(VELOC) 260 P = .1.-VS _ 261 VM= SGRT(VS ) ~ " ~ 262 THE = ATAN2(VNT , VTT) 263 RETURN 264 ENC a 99 De_i__ of the Two-Dimensional Nozzle Insert. The design- of the rectangular contracting section insert was based on the method of Smith and Wang [ 29 J. The. solution given makes use of the exact analogy between the magnetic field that is created by two circular coaxial and parallel coils carrying an electric current, and the velocity field that is created by two analagous ring vorxices. Experimentally, the high uniformity of the magnetic field over a core area (when the coils are suitably arranged) is well known. Essentially the same uniformity in throat speed will occur in the case of real airflows, provided the normally favourable pressure gradient along the contracting surface is not disturbed. A precise application would require that the contracting surfaces be swollen by the boundary layer displacement thickness. In practice,, such boundary layers, are quite/thin and the thickness increases slowly. Normally, no appreciable error is made in neglecting the boundary layer displacement thickness. Let T be the strength of a ring vortex of ring radius 'a1, with the plane of the vortex ring normal to the z-axis (Figure A5.1), The vortex ring is centered on the z-axis, a distance ' b' from the origin. The (axisymmetric) stream function is then -, . . Fra iMr,z) • = -2pr 2TT COS8 _, - , - r- ... d0 , (A5- 1) 0 PQ where 100 PQ 2 = (z-b)2 + a2 + r2 - 2ar cos9 (A5.2) = ((z-b)2+(a+r)2) (l-k2cos2(f)}, 4ar ((z-b)2+(a+r)2) ' (AS: 3) Then (r, z) ar 4V( (z-b) 2+(a+r) 2) -4 • (1,-2) k2j0 2TT / (l-k2cos2a) da da 0 /(l-k2cos2a) (A5.4) aF/r r k SE/_(<I-_>*-"> where a = - (A5.5) Non-dimensionalizing, let Z = f, R = f, B = K i> cl cl - cl _n_ ar ' (A5.6) and let F(k) = (1-|2)K - E) , (A5.7) with 101 ;2 = iR . (A5_8) (Z-B)2 + (1+R)2 Tnen $(R,Z) = /R F(k) , (A5.9) where K and E are the complete elliptic integrals of the first and second kind, respectively. To generate the uniform flow at the exit of the contracting section, the streamlines must be parallel there. Hence an identical ring vortex must be centered at (0,0,-b). The resulting stream function is therefore u>(R,Z) = /R (F (k J ) + F(k2)) , (AS. 10) where i 2 4R ,2 4R ,, c „ ki =• , k2 = . (A5.11) (Z-B)2 + (1+R)2 (Z+B)2 + (1+R)2 Reference [29] states that when b is 0.46936a the velocity distribution over a flow core area of radius 0.42241a in the median plane between the vortex rings will be uniform to within 1 part in 500. The elliptic integrals can be evaluated simply from polynomial approximations 17.3.34 and 17.3.36 of [41]. A program was written to search for the value of R which, for a given 102 value of Z gives the same value of ty as through a starting point (E0,Zo)- In this way the coordinates (R,Z) of the stream surface were generated, using a value of ty of 0.10510. R0 is chosen to be 0.30 (see Figure 6 of [40]) to obtain a throat flow uniformity within 0.23. Thus ro is 0.30a. The required testsection entrance area is 0. 34 8m2. Hence ur2} is 0.348m2. Therefore r0 is 333mm and 'a' is 1110mm. The nozzle length is chosen to be 1.52m, due to physical restrictions in the existing converging section, hence the nozzle entrance area Tix^ is fixed at 1.068m2. Hence r, is 583mm, or 1.751r0 or 0.525a. Therefore Si is 0.525, zi is 4.53ro or 1.37a. Hence Zi is 1.37. The table of nozzle coordinates follows. Z, z, R, r are as above, and w and h are the existing width of the contracting section and height of the new nozzle insert respectively. A~7T r2 z . z R r = w h w h 0.0 0 mm .3 00 333mm 0.34 84m2 9 1 4mm 381mm 0. 1 111 .300 3 33 .3484 915 381 0.2 222 .300 333 . 3484 916 381 0. 3 333 .301 334 . 3507 917 383 0.4 444 .304 337 . 3575 920 389 0.5 555 .309 343 .3706 927 400 0.6 666 .318 353 . 3924 934 421 0.7 777 .331 367 . 4248 942 452 0. 8 888 .343 386 ' . 4700 952 • 496 0.9 999 .370 411 .5298 962 554 1.0 1100 .395 440 .6065 979 626 1.1 1221 .4 26 473 .7023 1002 7 10 1.2 1332 .460 511 .8199 103 6 803 1.3 1443 .498 55 3 .9617 1082 902 1.37 1524 .525 583 1.068 1 123 951 103 AH An.aly_t.ic Representation of a Lifting Vortex between a Solid a XEiLHSversexy-Slgtted and a Constant Pressure Boundary.. The following describes an analytic two-dimensional potential flow "method of images" model for a lifting airfoil between a solid lower boundary and a transversely-slotted upper boundary consisting of airfoil-shaped slats. A constant pressure boundary outside the slotted wall represents a free streamline that"divides the testsection flow from the plenum flow. The airfoil ana the wall slats are represented by point vortices which are "imaged" appropriately. The image of a vortex, in a solid boundary is a vortex of equal but opposite circulation; the image in a constant pressure boundary is a vortex of identical circulation. From [41], the complex potential for an infinite vertical row of point vortices of the same sign, spaced a distance 'd' apart, is where the "central" vortex is at zo, and the strength K is related to the circulation r (positive clockwise) by with reference to Figures A6.1, A6.2, the image system for F(z) = Klog sinh-r(z-z0) (A6. 1) (A6.2) a single vortex between the solid and constant pressure 104 boundaries is the sura of four sets of images. Using the notation of Figure A6.2, two sets are of positive circulation, "centered" with z0 values of ai and (a+2b)i. The other two sets are of negative circulation, "centered" at -ai and - (a + 2b) i. All four sets have the same spacing, 4 (a + b) . The complete system of images for the single vortex immersed in a uniform flow U (from left to right) has the complex potential F(z) = Uz + K log sinhA(z-ai) + K log sinhA(z-(a+2b)i) (A6.3) - K log sinhA(z+ai) — K log sinhA(z+ (a+2b) i) ., where A = irS+bT • (A6*4) The complex velocity w(z) is the derivative of F(z) with respect to z. Hence for the single vortex, w(z) = U + KA(cothA(z-ia) + cothA(z-i(a+2b)) (A6. 5) - cothA(z+ia) - cothA(z+i (a+2b) )) ». To calculate the force on the vortex representing the airfoil, due to the effect of the two boundaries, the Blasius relation {[41]) is used, 105 D - iL = w2 (z) dz, (A6.6) where D and L are the forces in the X and Y directions., respectively. Integration is performed about a contour enclosing only the airfoil vortex. To evaluate this integral using residues, with the. airfoil vortex at ih, the coefficient in the Laurent series expansion of w2(z) of the term in 1/(z-ih) is required. The Laurent series expansion for the coth function about ih is _„ / -„\ 1 . (z-ih) z-ih , ,,r ns coth(z-xh) - r—- + -.—- - -—j-=—- + ... (A6.7) z-xh 3 45 The required coefficient is 2UK + 2K2A(cothA(z-i(a+2b)) - cothA(z+ia) - cothA(z+i(a+2b))). . (A6.8) The residue at ih is 2UK + 2K2Ai(+cot(2Ab) + cot(2Aa) + cot(2A(a+b))), (A6.9) since coth (iz) =-icot (z) . (A6. 10) Therefore \ • D - il. = (^)2iri (Residue (ih) ) . (Ab.11) Substituting for K in terms of. r, D is zero, and 106 L - L0(l - jj^csc (ka)csc(kb)) , (A6.12) w here k - 2TaTbT ' (46-13) ana L0 = pur (A5. 14) is the tunnel lift. This is the expression for the reduction in lift experienced by a single point vortex between a solid lower boundary and a constant pressure upper boundary. If the vortex is midway between the two boundaries, Lo = 1 ~ 4UH ' <A6*15> Using (A6.14) and CT = T —— , (A6.16) ° |PU2c where •c' is the airfoil chord, Now consider the vortices Y representing the airfoil-n 107 shaped wall slats. From equations (A8.4), (A6.5) using . the notation of Figure A6.1, CO w(z) = J k B (cothB (z-2ih-nr) + cothB (z-2i(h+£ )-nr) ^ n n ^ n n n n=-oo (A6.-18) - cothB (z+2ih-nr) - cothB (z+2i(h+e )-nr)1 n n n. J is the additional complex velocity due to the four sets of images corresponding to 'n* infinite vertical rows of point vortices y spaced a horizontal distance 'r' apart. Here k = , B = TTOT-I r . (A6.19) n 2TT n 4 (2h+e ) n and *r' is related to the slotted wall open-area ratio. Hence the complete complex velocity field for the vortex r representing the test airfoil, the vortices y representing the wall slats, and the uniform flow U, n is . w(z) - U + KA(cothA(z-ih) + cothA(z-i(3h+26)) - cothA(z+ih) - cothA(z+i(3h+26))) (A6.20) CO + Y ' k B (cothB (z-2ih-nr) + cothB (z-2i(h+e )-nr) L n nv n n n n=-°° - cothB (z+2ih-nr) - cothB (z + 2i (h+e .) -nr) ) n n n } where 6 and e are the distance of r and y respectively from 108 the constant pressure boundary. ' Following the previous procedure to calculate the coefficient of. 1/(z-ih) in the Laurent series expansion for w2 (z), the required coefficient is, using (A6.9), 2UK + 2K2Ai(cot(2A(h+6)) + cot(2Ah) + cot(2A(2h+6))) + 2K T k B (cothB -xh-nr) + cothB -x(h+2e )-nr) L n nA n n n n=-co (A6.21) - cothB (+3ih-nr) - cothB (+i(3h+2e )-nr)K n n n . wnere A = T(2h+6T* (A6*22> Mow using coth(x+iy) = coth(x)csc2(y)-icsch2(x)cot(y) (A6.23) coth2(x)+cot2(y) to calculate the residue of w2 (z) at ih, and using (A6.12) and (A6.10) , " " L = pur - £|^-(cot (Bh) + cot(8(h+S))) - •— y y B csch2(B nr)F(B ,h,e ,n,r), (A6.24) 2TT L 1 n n n n n n=-°° and 109 D pr Y y B coth(B nr)E(B ,h,_ ,n,r), (A6.25) " rt y~\ Y\ n n 2TT ^ 'n n n=-°° nwhere TT 2(2h+S)' Bn 4(2h+£ ) ' n (A6.26) E(B ,h,e ,n,r) .= G(B h) + G(B (h+2e )) n n n n n + G(3B_h) + G(By.(3h+2e_)) , n n n F(B ,h,e ,n,r) = H(B h) + H (B (h+2e )) nn n n n H(3B h) - H(B (3h+2e )) n n n In the above expressions (A6.27) (A6.28) G(u) = esc (u) coth2(B nr)+cot2(u) n (A6.29) and H(u) = cot (u) coth2(B nr)+cot2(u) n (A6.30) When all of the y are of identical strength, and all of the e (distance of Y from the constant pressure boundary) are n n 110 equal, the .drag force D is zero, since coth (B_nr) is an odd function of n. Numerically ' it is satisfactory to take n greater than 10, that is, 21 or more vortices y . For the case of a flat plate at 26.3° incidence, and average values of y £ taken from streamline, calculations for a 0.66-Clark-i airfoil at. 20° incidence in the presence of a 70%OAR.TSOSL wall configuration, the values of the expressions are .0625,1=0.1, £ = 0.67, | = 0.67. (16.31) H ence ~ - 1 " (0.168)(I ) - (-0.317)) c uc s = 1 - 0. 235 - 0. 020 = 0. 745 , (A6.32) and the drag force D is zero. Hence the effect of the wall slat circulation y on the test airfoil is small compared to the effect of the constant pressure boundary on the test .airfoil. Another possible analytic representation of the bounding shear layer, is a vortex sheet, across which there is a jump in tangential velocity and total head. This was not attempted here. r U, = 1.40, Ill Appendix 7. Standard Solid Wall Corrections The following seven expressions are reproduced from page 382 of [34], and are the expressions used for the calculations herein. They are written for the incidence correction applied as an equivalent change in lift. The corrections are to be applied at the measured incidence. The subscripts T and F imply measured and equivalent free-air values respectively. The first five expressions are, respectively, the correc tions to be added (regardless of sign) to the measured values of windspeed, incidence and lift, quarterchord pitching moment, and drag coefficients. AU = UF - UT = eUT (A7.1) Aa = a_ - a_ = 0. (A7.2) ACL " = -2EVKV 2¥(V4CMCT>£?" <"-3> ACHC = CM= "Sic = -2eCMc +!V^(W '^f' (A7-4) 4 4F 4T 4T 4T ACD = CD -CDT = -2£CDT + 27(CL +4CMc > (CL ~ IfD> ' <A7-5>" -T T In the above equations, e is composed of the corrections 112 for wake and solid blockage, and is given by 1 rc^ e = ffe)CD_ + AK' (A7.6) where K = ^(|)\ (A7.7) and c/H is the model size. The quantity A is obtained from Fig.6.8 of [39]. In practice, the a-derivative values are determined graphically. Otherwise the following values may be used: = 27T, = 0, _M^=0, ^=.2. (A7.8) The following correction is applied to the measured pressure distributions (at the measured incidence ct^) : 113 _E___S__ __ _ _______ _______ _____i___2_ __________ ____ ___ ________ Lift. In this method, the usual Kutta condition, of equal velocities on the upper and lower surfaces of a lifting body, adjacent to the trailing edge is abandoned. The circulation is to be determined from the measured lift. To simplify the equations, consider a single test airfoil between solid walls. The airfoil is represented by N control points, and the two plane solid walls by M-N additional control points. Thus the total number of control points is li. There are therefore Pi. unknown source strength densities o and a single unknown vortex strength density y on the airfoil. Thus H+1 equations are required to determine the M+1 unknowns. There are M control points at which the norma 1-velocity boundary condition must be satisfied; these yield M equations. In the notation of §3.2, the tangential velocity at a point •i' is The (M+1)st equation usually contains the Kutta condition at the . airfoil trailing edge. The usual Kutta condition is expressed as M N (A8. 1) = -V (A8. 2) x i + 1 114 which results in the equation M N |1(Bji+Bji+l^0j.+.YkI1(Aki+Aki+l) = -U(cos0i+cos9i+1) (A8.3) for the control points 'i' and 'i+1' adjacent to the trailing edge. The resulting full Kutta circulation ?o about the airfoil can be calculated from the definition N = 4 V-d£ = I V As. JC i=l \ 1 (A8.h) To determine the circulation r from the measured lift, the Kutta-Joukowsky law expresses the measured lift force, L, on the airfoil as L = PUP . (A8.5) The measured lift coefficient CL is defined as CT = T-V" ' U8-6) pU2c 2 where 'c1 is the airfoil chord. Therefore T = fUcCL - (A8.7) The left.side of equation (A8-7) is therefore written N N M N IV As. = I I B a + Y I A + UcosG I As. , i=l ti 1 i=lAj = l 31 3 k=l Kl ^ 1 (A8.3) and (A3. 7) become; M C N r N r N ^ y B ..AS. • i 31 x a . + 3 I li=l 2 Aki As . l J (A3.9) = -U N J cos6.As. i=l 1 1 + ^CCL The last row of the matrix C in the system of equations C ( o, y)-d, will now be N 'j ,M+1 = I B ±Asi , j=l,2,...M, (A8.10) i=l N r N = y y A 'M+1, M+1 . L, L -. ki ' i=l <-k=l As. = constant, l (A8. 1 1) and the last component of the right hand side vector will be M+1 -U N J cosO.As. iii 1 x + |UCCL (A8. 12) The expressions (A8.10), (A8. 11) and (A8.12) replace the corresponding expressions in (4.2) and (4.3). 116 ________ __ _ _______ _______ _i_______2_ __________ ki _2____i__ ___ Profile For a two-dimensional airfoil profile which can be mapped conformally • onto a. circle of radius R, the full Kurt a circulation T0 is [37] (A9 1) F o = 4TTRU sin(a-cto) , where U, a and a0 are the flowspeed, incidence, and zero-lift angle respectively. If the circulation is reduced to a value kr0, then T = kf0 = 4TrRUksin (a-a0) •= 4TrRUsin (a-a 0-Aa) . (A9.2) Hence Aa = (a-a0) - arcsin (ksin (a-a 0)) (A.9.3). is the effective reduction in incidence required to reduce the circulation (and hence the lift) to the fraction »k» of the full Kutta value. In' order to achieve a' reduction in the "effective camber, the profile shape is modified by "raising" the trailing - edge. Points on the profile are rotated about the profile leading edge .through an angle which is proportional to the distance from the leading edge. The direction of rotation is such to reduce the 117 effective incidence of the profile. For an origin of profile coordinates (x,y) at raid-chord, and flow from left to right, the expression 0(x) = ^-(1 + ~) , (A9.4) where *c' is the airfoil chord, will assign a zero rotation to the point at the leading edge and the full rotation Act to the point at the trailing edge. The modified profile coordinates are thus x' = xcos0(x) - ysin6(x) (A9.5) yy = ycos9(x) + xsin9(x). The effective reduction in camber is roughly proportional to. the amount of lift (or circulation) being developed. The zero-lift angle a0 for the profile will not change, since when a equals a0 , Aa will be zero. 118 Appendix 10. The Computer Program for the Exact Numerical Theory The program contains a subroutine MAIN1, which calls all of the following subroutines: CALCAB, ASSEMA, ASSEMB, ASSEMD, CPS, FORCES, and MODPRO. In addition, subroutines RE, WR, and WRD are required for reading and writing matricies from/into peripheral storage. The system of equations is solved by the subroutine ATXB, described in Appendix 3. The U.B.C. (system dependent) subroutines GSPACE and FSPACE allocate and deallo cate, respectively, blocks of real memory required for the matricies A, B, and C. The U.B.C. subroutine CALLER is used to call subroutines which use matrices that have memory . allocated by GSPACE. A subroutine WALLCO is used to create the control.point coordinates for all test airfoils, flaps, solid wall sections and wall slats. The control point coordinates XX and, YY, along with DX, DY, DS, CS, and SI, are written into peripheral storage so that all coordinates may be checked before further calculation. The definitions of the variables used are described by comment statements within the subroutines. The control points (XSOLSL,YSOLSL) for an arbitrary solid surface such as the plenum boundary are read in at execution 119 time by the program which calls WALLCO. The control point coordinates (XM,YM), and the velocity VTI, on the streamline representing the shear layer are read in by the program which calls MAIN1. This program also reads in the coordinates of the slat leading edges (XLE), centers ((XCENT,YCENT))., and trailing edges (XTE), the flow angle that each slat sees (ALF), and the fraction of the full circulation (Kl) required. As shown, this program handles only a single test airfoil; a similar program is used for a flapped airfoil. The notation for the enumeration of the control points is shown in Figure A10.1. The layout of the system of equations to be solved is shown in Table A10.1; the numbers in paren- . theses indicate the particular DC—loop in the program which assembles the corresponding coefficients for the unknowns in the equations. The equation numbers (rows in the matrix C) are indicated by 'E*. Across the bottom of Table A10.1, the range of the index of summation (j, k, p or q) for each column of the matrix C is specified. A complete sample run with the required calling programs follows. The sample is shown for the Clark-Y airfoil at 20° 120 incidence, in a 70% OAR TSUSL wall configuration. The shear layer is modelled by a streamline which is represented by 20 control points, that is 20 source and 20 vortex elements. The airfoil is represented by 50, the upper solid end-wall sections each by 20, and the solid lower wall by 80 control points. There are 8 airfoil-shaped slats, each represented by 9 control points. Thus there are 262 control points where the zero normal-velocity boundary condition is specified, and 20 control points where the tangential velocities are prescribed. This leads to 262 unknown source strength densities and 29 unknown vortex strength densities. The result is 291 equations in 291 unknowns. The velocity distribution on this streamline representing the shear layer is specified, and corresponds to a pressure coefficient with the average value of -0.35, and is similar to the mean variation of pressure shown in Figure 8.7. The profiles of the wall slats are modified to reduce their circulation to 0.8 times their full circulation. The output from the program.includes the pressure . distributions on the walls, the wall slats, and the airfoil. Also printed are the lift, drag, and pitching moment coefficients for the airfoil and the wall slats. MICHIGAN TERMINAL SYSTEM FORTRAN G(41336) MAIN 10-22-75 12159113 PAGE P001 0001 000? 0003 OO01 0005 0006 0007 0008 0009 0010 00 11 0012 0013 0011 0015 0016 0017 0018 PC19 0030 0021 0022 0023 0021 0025 0026 0027 0028 0029 0030 0031 0032 0033 0031 0035 0036 0037 REAL XX(262),YY(262),DX(262),DY(262),DS(262),CSC262),SI(262) REAL SlG(291),VTT(262),CP(262),GAM(9),MU(20),GNU<20) REAL VTI(20),XM(21),YM(21) REAL AXX(2a2),AYY(242),ADXC2«2),A0Y(24?),AD3C242),ACS(242) REAL ASK2H2) C XM.YM •. PROFILE COORDS FOR STREAMLINE REPRESENTING SHEAR LAYER C XX,YY - CONTROL POINT COORD3> DX.OY.OS - ELEMENT LENGTHS C SIG - SOURCE STRENGTH OENSITlES (ALSO USED AS SOLUTION VECTOR IN C SYSTEM ) C (GAM,MU,GNU APE PART OF SIG) C VTT.CP - TANG VEL, PRESSURE COEFF'. REAL XQ(10),YQ<10),XH(10),YR(10) C XQ,YG,XR,YR - MODIFIED SLAT PROFILE COORDS REAL XCENT(fl),XLE(8),XTE(8),ALF(8),KH6) REAL YCcNT(ft),DTHICK(8) C AXX,AYY ETC - CONTROL PTS TO BE READ IN FROM FILE (PUT THERE BY C WALLCO) EQUIVALENCE (AXX,XX),(AYY,YY),(ADX,DX),CADY,DY),(ADS,OS) EQUIVALENCE (AC3.CS),(ASI.SI) INTEGER CALAH,CALCD,WRAR,WRCD,SDLV,GAUSS,ITER,CALCP,C»LCL,HSIG COMMON/ei/ NW5,NSLAT,N3U1,NKA,NM2,MS V,NA,NSPS.NTEU,NTEL COMM0M/A2/ U,CH C0HM0M/B3/NUl,NWuT,NU3,NWU2,NLl,NKLl,NL3,NWL2iNS0Ll,NS0LSL,NP1, 1 NFLAT,N3PF,Ni1 C0MM0H/H4/CALAB,CALCD,WRAB,wRCD,S0l.V,GAU3S,ITER,CALCP>CALCL,HSIG C0Mn0H/n5> NPi,MP2,NL« REAL TITLE(20) REAI)C5,303) TITLE 303 FORMAT C20A/I) WRITEr6,304) 301 FORMAT(IH1) WRITE(h,305) TITLE 305 FORMAT(iX,20A4) RE AD(5,25) CALAB,CALCD,WRAB,WRCO,SOLV,GAUSS,ITER,CALCP, CALCL » HSI G 25 Fn.7.MAT(20I«) WRITE(6,1) CALAB,CALCD,WRAB,WRCD,SOLV,GAUSS,ITCR,CALCP,CALCL,H3IG 1 FORMAT ( 'CALABr I ,I2,2X, ICALCO" 1 , 12, 2X, 'WRABa M2.2X, 'WRCO=M2,2X, { 'SOLVr',12,2x,'GAUSS=',I2.2X,'ITERai,I2,2X,'CALCP',I2,2X, 2 "CALCL"',12.2X,'HSIG=',12) RE A.i (5.30) NA,NWUl,NHU2,NwLi,NWL2,NS0LSL,NFLAT,NSPF,NSLAT,NSPS, t MSV 30 FORMAT(20I4) WRITE(6,31) NA,NWUl,NWU2,NKLl,NWL2iNS0LSL.NFLAT,NSPF,NSLAT,NSPS» 1 MSV 31 FORMAT<'NAs<,13,2X,•NWU1a•,13,2X,1NwU2a',13,2X,'NWL1a 1,13,2X, 1 INWL2=',IS,2X,'NSOLSLs',I3,2X,•NFLATt>,IS,2X,INSPFs',13,2X, 2 INSLAT=',I3,2X,iNSPSai,I3,2X,iMSVat,13) READ(5,32) NfEU,NTEL,CH,U 32 F0RKAT(2I1,2Ffl'.3) WRITE(6,400) NTEU,NTEL,CH,U 400 FORMAT ('NTEU= I ,I3,2X, 'NTELa' ,I3,2X, • CH= ', F8'. 3, 2X, 'U= ' ,F6,1) C READ COCROS FOR AIRFOIL, WALL & WALL SLATS FROM WALLCO FILE (WALLS & C SLATS HAVE YY = 0'.) READ(2) AXX,AYY,ADX,AOY,AOS,ACS,ASI YWal'8'. . 1.000 2.000 3.000 4.000 5,000 6.000 7.000 8,000 8.000 9.000 10.000 11.000 12.000 13.000 14.000 15.000 15.000 16.000 17.000 18,000 19.000 20.000 21.0o0 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 33.000 34.000 35.000 36.000 37.000 38.000 39.000 40.000 41.000 12.000 43,000 44,000 45.000 46.000 47,000 48,000 49,000 50.000 51.000 52.000 • 53.000 MICHIGAN TERMINAL SYSTEM FORTRAN G(H336) MAIN I0-22-73 003B 0039 0010 OOH' COM2 0013 0011 eo-'i5 0016 0017 ooie 0019 0050 0051 0052 0053 0051 0055 0056 0057 0058 0059 ooto 0061 0062 0063 0061 0065 0066 0067 0068 0 06? 0070 0071 0072 0073 0071 0075 101 33 31 35 C NUleNA+J NU3 = NuUNWUl MU'lsNu3 + NWU2-I WLUNU3 + NWU2 NL3=NLl+NWLl NL'l = NL3 + NwL2-i MS0LlrNL1+l IFCNSOLSL'.EO'.O) NSOLl=NLfl NF1=N|.1+N30LSL*1 IFCNFLAT'.EO'.O;. NFI*NL4+NSOLSL NF1 « 1ST CON PT ON 1ST FLAT SLAT NSUi=NL1+NS0LSL*NFLAT*NSPF*i IF CN3LAT.E0'.0) NSU1=NLU+N30LSL*NFLAT*NSPF NSIin = NSUl + NSLAT*NSPS»l IF(NSLAT.EO'.O) NSU2=NSU1 NSU2 - L*ST CON PT ON LAST SLAT Nil - 1ST CONTROL PT ON STREAMLINE FOR SHEAR LAYER Mil=NSU2+l IF CMSV.LQ'.O) NilsNSU2 WRITER,101) NU1,N';3,NL1,NL3»NS0L1.NF1,NSUI,NSU2,NI1 FORMATC'NU1««,I3,2X,INU3»',13,2X,"NL1"',I3,2X,• NLS»',13,2X, 1 iNSOLi"'#i3,2X,INF 1 = 1,I3,2X.'NSU1s',13,2X,'NSU2=1,13,2X,I Nils•, 2 13) NWALL - TOTAL * OF CON PTS ON ALL FLAT SOLID WALL SECTIONS NWALL=NWU1+NWU2+NWL1+NWL2 NW3 - TOTAL * CONTROL POINTS ON AIRFOIL SOLID WALL' SECTIONS & SLATS MWS=NA+NWALL+NSPS*NSLAT+NSOLSL+NSPF*NFLAT NKA « » OF EON FOR KUTTA ON TEST AIRFOIL NKA=NSU2+NSLAT*1 NSVT - TOTAL » SOURCE & VORTEX ELEMS NSVf=NWS+MSV NM2 «• » OF LAST EON FOR ZERO NORM VEL ON INNER EDGE OF S,L« NM2sNKA*M3V NUN - TOTAL « UNKNOWNS NUN=NtaS+2*MSV»NSLAT*l NFL - TOTAL « CON PTS ON FLAT SLATS (NO KUTTA) NFLsNSPF*NFLAT NSL - TOTAL * CON PTS ON ALL AIRFblLSHAPED SLATS NSL=NSPS*N3LAT . WRITE (b.33) NA,NWALL,NSOLSL,NFL,NSL,NWS FORMAT('NA=',13,2X,'NWALL"',13,2X,INSOLSLo',13,2X,INFL"',13,2X, I iNSLs'.njZX^NwSst.IS) WRITE(6,31) MSV,N3VT FORMAT('»3V=',l3,2X,'N3VT»",IS) WRITE(6,35) NKA,NM2,NUN FORMAT('NKA=',13,2X,'NM2s•,13,2X#'NUN«I,13) NA « a CON PTS ON SINGLE TEST AIRFOIL NlaNA+1 N2=NA+NWU1+NWU2 SET TESTSECTION WALL HEIGHT DO 2 I=Nl,N2 YW - Y-C00RO FOR UPPER AND LOWER WALLS YY(i)=YW . N3=N2tl N«3N2*NwH + NKiL2 12159113 PAGE P002 51.000 55,000 56,000 57,000 58,000 59,000 60,000 61.000 62.000 63.000 61,000 65.000 66,000 67,000 68.000 69.000 70.000 71.000 72.000 73.000 71.000 75.000 76.000 77.000 78.000 79.000 80,000 81.000 82,000 83.000 81,000 85.000 86,000 87.000 88,000 89.000 90.000 91.000 92.000 93.000 91.000 95.000 96.000 97.000 98,000 99.000 100.000 101.000 102.000 103,000 101,000 105,000 106.000 107.000 108.000 MICHIGAN TERMINAL SY3TCM FORTRAN G(fll336) MAIN 0076 OO 3 i=N3,Nfl 0077 3 YY(I)=-YW 0078 IF(NSLAT.EO'.O) GO TO 500 0079 DO 1 I=NSU1,N3U2 0080 fl YY(i)=YY(I)+YW 0081 500 CONTINUE 0032 IF(NSLAT'.EO'.O) GO TO 15 C XCENT.YCENT - CENTER OF SLATS 0083 REA0C5,16) (XCENT(K), Kai,NSLAT) OOSfl WRITEC6.308) 0085 308 FORMAT('XCENT') 0086 WRUE(6,1B) (XCENT(K), Kai,NSLAT) 0087 REA0C5,'i6) CYCfNT(K), K»l,NSLAT) 0038 WRITEC6.309) C089 309 FORMAT('YCENT') 0090 KRITE(6,18) (YCENT(K), K«l,NSLAT) C MODIFY SLAT PROFILES FOR REDUCED CIRCULATION 009{ REAn(S,50) MOD 0092 50 FORMAT(12) 0093 wnifecft.si) MOD Oo9f| 51 FORMAT<'M0DPR0a',J2) C MOOiFY PROFILES.IF IMOD' NOT ZERO 0095 IF(MOD,CO'.0> GO TO 15 C XLE.XTE - X-COOROS OF SLAT LEADING & TRAILING EDGES 0096 READ(5,16) (XLECK), K=l,NSLAT) 0097 16 F0RMAT(i3F6'.l) 0098 WRITE(6,3060099 306 FORMAT(* XLE1) 0100 WRITE(6,18) CXLE(K), Kn1.NSLAT) 0101 18 F0RMATUX,13F6'.l) 0102 REA0(5,16) (XTE(K), Kai,NSLAT) 0103 WRITE(6,307) 0101 307 FORMAT( 'XTE *) 0105 WRlfEt6,1B) CXTECK), K=l,NSLAT) C ALF « FLOW ANGLE AT EACH SLAT 0!06 READ(5»16) (ALFCK3), KS=1,NSLAT) 0107 WRlfEC6,310) ClOB 310 FORMAT('ALFI) 0109 WRITEC6.18) CALF(K), Kai,NSLAT) C Ki « FRACTION OF CIRCULATION 011 0 REAO(5.17) (Ki'(K). Kai,NSLAT) 0111 «7 F0RMAT(16F5'.3) 0112 WRITE.6.311CMS 311 FORMAT (' K 1 ') 0111 WRITE(6,19) (KICK), K»l,NSLAT) 0115 19 F0RMAT(1X,16F5.3) C DTHiCK . SYMMETRIC DISPLACEMENT THICKNESS 0116 READ(5.17) (DTHICK (K), Ka1,NSLAT) 0117 WRlfE(6,3l2) 0118 312 FORMAJCDTHICKI ) 0119 WRITE{6,19) CDTHICK(K), KoJ,NSLAT) 0120 NPS=((NSP3-l)/2)-i 0121 NSP=NSPS»1 0122 MPsNSPS+0123 DO 11 KS=1,NSLAT 12)59113 109.000 110,000 111,060 112.000 113.000 111.000 115.000 116.000 117.000 118.000 119.000 120.000 121.000 122.000 123.000 121.000 125.000 126.000 127,000 12e,000 129.000 130.000 131.000 132.000 133.000 131.000 135.000 136,000 137,000 138.000 139.000 110.000 111.000 112.000 113.000 111,0Q0 115.000 116.000 117.000 118,000 119.000 150,000 151,000 152.000 153.000 151.Opo 155.000 156,000 157.000 158,000 159.000 160,000 161.000 162.000 163.000 PAGE P003 h 00 M.CHIOAN Tt£RM_NAl. SY3TEM FORTRAN 6(11336) MAIN 10-22-75 12I59US PAGE P001 0121 0J25 0126 0127 oi'2B 0129 01*9 0131 0132 0133 0131 0133 0 136 0137 0138 0139 oiiq 0111 0112 0]13 0111 0115 0116 0117 0118 0119 ©ISO 0151 0152 0153 0.51 0155 0156 0157 0158 0159 0160 0161 0162 0163 0|61 0165 0166 0167 0168 0169 0170 11 15 16 17 18 15 C 22 23 36 NT»NSllUNSP3»(KS-l) 161.000 MTsNT+NSP 165,00MlnMT.NPS 166.00NLaNT+NPS 167,00ALZ - ZfRO LIFT ANGLE FOR SLATS (0015) 168.000 ALZaO. t 169,00CALL MODPRO(XX,Vy,DX,OY,03,CS,SI,NSVT,XR,YR,MP,NT,MT,ALF(KS),ALZ, 170.000 1 Kl(KS),0THICK(K3),XCENT(KS),YCENT(KS),ML,MT,NL,NT,XO,YQ,XLE(KS), 171.000 2. XfECKS)) 172. 000 CONTINUE 173.00CONTINUE 171.00MVS=MSV+1 175.00iF(MSV.EQ'.O) GO TO 36 176.00READ COORDS FOR STREAMLINE FOR SHEAR LAYER 177,000 READ(5»16) XM 178.00REAOC5.16) YM 179.00F0RMAT(l'2F6'.l) 180 . 000 WRITEC6.17) XM 181.00WRITE'C6,18) YM 182,00•F0RMAT('XMi,10F7",2:-' 183.00FORMAT<'YMi,10F7.2) 181.00DO 13 K=1,M3V 185.00I=NwD+K 186,00J=K+1 . . , 187.000 XX(I)a(XM(K)+XM(J))/2, 188.00YYCJ)=(YMCK)+YM(J))/2. 189,00DXCI)=XM(J).XM(K) 190.00OY(i)=YH(J)«YM.K) 191.00D3{n=S0RTlDX(I)*0X(I)*0Y'(I)*DY(I)) 192.000 CS(I)=BX(I>/nS(I) 193.00* Si(I)=DY(I)/DS(I) 191,00M1,M2 * RANGE OF CONTROL PTS *IS ON SHEAR LAYER 195.000 MlaNW9+i 196.00M2=NWS+MSV . 197,000 KRITE(6,7) 198,00FORMAT('STREAMLINE FOR SHEAR LAYER•) 199,00WRITE(6,6) 200.00FORMAJ(7X,'XX',6X.'YY',6X,«0X',6X,<DY',6X»'D3',6X,'C3',6X,'SI') 201.000 WRITE(6,5) (XXCI),YY(I),DX(I),DY(I),DS(I),CS(I),SI(I), IaMl,M2) 202.000 F0RMATC1X,7F8'.3) 203.00READ VELOCITY DISTRIBUTION ON STREAMLINE FOR S.L, 201,00REA0C5/22) (VTI(I), Ial,M3Y) 205,00FORMA7(i0F8'.3) 206.00WRITE.6,23) VTI 207.00F0RMAT('VTI',i0F8.3) 208.000 CONTINUE 209,00NrNSVT. 210.00MsNlJN 211,00NG - a VORTEX STRENGTH DEN'S ON SLATS i TEST AIRFOIL 212,090 NG=NSLAT+1 213,00NMaMSV 211,00NLSsNSLAT 215,00CALl. MAiNlCXX,YY,0X,DY,D3,CS,SI.N,SIG,M,VTT»CP,GAM,NG,MU#GNU,NM, 216.0 00 l" VTI.XCENT.YCENT.NLS) 217.00STOP 218.00MICHIGAN TERMINAL SYSTEM FORTRAN G'(HS36) MAIN 0171 END •OPTIONS IN EFFECT* 10,EBCDIC,SOURCE,N0LI3T,NODECK,LOAD,NOMAP •OPTIONS IN EFFECT* NAME = MAIN , LINECNT = S7 •STATISTICS* SOURCE STATEMENTS a 171,PROGRAM SIZE n •STATISTICS* NO DiAGNoSTICS GENERATED NO ERRORS IN MAIN NO STATEMENTS FLAGGED IN THE ABOVE COMPILATIONS', EXECUTiON TERMINATED SR "L0AD+0UJBSL3+ATXB 2 = FlL'E«<l7 3 = «A «o-B 9«*DUMMY* EXECUTION BEGINS 10-22-75 12I59U3 PAGE POOS 219.000 260 ft* CY A«20 N»50 70XTSUSL C/H".66 MSVs20 •** CALAH" t CALCDB 1 WRABa 0 WRCD" 0 SOLV" J GAUSS" 1 NAB 50 NWU1= 20 NWU2e 20 NWLl" 10 NWL2" 40 NS0L3L" NTTU» 25 NTELB 26 CH= 23.940 U" l'.O NU1S 51 NU3 = 71 NLt" 91 NL3*1J1 NSOLlal70 NWALL=120 NSVT=J62 NM2=271 NSOLSL« SI.2 ie*. o i NUNI291 io'.2 8.0 • I'.B .1S'.8 •25*.fl i's'.o I'S'.O IB'.O 32.a 20.1 8.1 NA« 50 MSV = 20 NK A = 251 XCF.NJ 46.2 YCF/JJ 18.0 XLE , <i'i.<i XTE , ie.O 36.0 21.0 ALF -2.7 -4.8 -8.8 K> . • , .... O.BOOOjBOOO.8000.8000.8000.8000.8000,SOC DTHIC* o'.o o'.o o'.o o'.o o DEl.CPST = -u'.00943(RAD) 0Fi.EPST = -O^0l678C«AO) OELCPST=-0.03089(RAD) NFL" 0 NSL" 72 NWS.2D2 ITER" 0 CALCP 1 CALCl" 1 HSIGa 0 0 NFLATa 0 NSPF" 0 NSLATa 8 NSPSa 9 MSV" 20 NF1«IT0 NSUl«17i NSU2e2(|2 Nll"24S 12.0 15'. 2 -3'. 6 0.0 '15".0 -15.6 • 12*.0 27.6 21'.0 • S7.8 IB'.O • 39*.6 -36*. 0 5.5 11.0 10. T DELCPST = -o'.o5324(RAn) On.CPST= o'01924(RAO) 0ELCPST= 0 03B7KRAO) DELEPSTa O.03767CRAO) XM x« YM Y M Y" 00 00 -A.e'oo 22 60 18.00 STREAMLINE XX ' 49' 500 a'l 500 39 500 5";500 29 500 21,500 19 500 I'I,500 9 500 '1,500 -0,500 .5,500 .'JO,500 .15,500 -20,500 -25,500 -30,500 -35,500 -40,500 -45.500 47,00 • 3.00 18 20 22.90 42,00 -8.00 0 0 0.0 -0 54(REG> -0,96(REG) -1,77(DEG) -3,09(DEG) -3,05(DEG> 1 10C0EG) 2,22<l)EG) 2.16C0EG) 37.00 32*. -13.00 -18.00 0,0 Kao',80000 Kao',80000 K = o'.80000 K 0,80000 K=0.80000 Kao K = 0 Kao 00 BLOISPLTHICKB BLDISPLTHICKB BLOISPLTHICKB BLDISPLTHICKB BLDISPLTHICKB BLDISPLTHICKS BLDISPLTHICKB BLDISPLTHICKB 27^00 22 00 17.00 -23.00 -28.00 -33,00 -,80000 ,80000 ,80000 0,0 0.0 o;o 0,0 0,0 °f.° 0.0 00 00 18 40 22.90 18.70 22.60 22.10 19, 21'. 50 40 20,10 .70 20. 20.70 19,70 21,40 19.20 FOR SHEAR YY 100 300 550 900 300 800 400 050 750 350 750 900 750 350 750 050 200 450 850 250 LAYER DX ,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000 000 ,000 ,000 ,000 ,000 ,000 ,000 ^.000 ,000 ,000 ,000 -5.000 DY 0'.200 0'.200 0,300 0.400 0'.400 0'600 0.600 0'700 0.700 O'.SOO 0'300 0,0 .0.300 .0'.500 .0'.700 -0'.700 -f.000 -0',500 .0.700 -0'.500 DS 5,004 5,004 5,009 5.016 5,016 5.036 5.036 5.049 5.049 5.025 5,009 5.000 5,009 5,025 5.049 5.049 5.099 5.025 5.049 5.025 -0.999 • o!.999 -0,998 -0,997 .0.997 -O'993 -0.993 »0'.990 -0J.990 .0.995 -0',998 .1,000 -0,998 -0,995 -0.990 990 981 .0,995 •0,990 •0.995 SI 0*.040 0'. 0 4 0 0' 060 0.080 O'.OBO 0'. 119 0'. 119 0.139 0' 139 0.100 0'.060 O'.O .0,060 .0.100 -0' 139 -0.139 -0.196 -0'. 100 -0.139 -O'.lOO 7', 00 •43,00 22.10 18'.50 nj AJ Aj AJ AJ —#rvj — -* AJ oj —• — ooooooooooco G o tiitiirttiit • • UlUJUJU-UJUIUJUJuJaJUJUJ Ul o- j cc o SKI o- offN^ooao-aooLno-irir^ co j)ctC'-Kir-NF«TLnJin-0- — — =a ~ cr KI OJ K1^0NMiTSQD^Of\J-£0'aj3 o-GP-rvinc-Kio-o o o o • • • I UJ tu UJ f\l o cc CMC =r =r oj o —• cr r-~ — — — r— o-(C — — Kl — ^1 O CT — — oooooooo 000000000000000000000 sir * • » i » tiiiiiirti i t I I i • • UJ Ul UJ UJ Ul DJ Ul Ul Ul UJ Ul l Ul Ul Ul Ul Ul cc —* -O =3 ^ m ^ cr r\i1,",t\j^)-'K'io£^) — Nf\J3ui-'C'CC(r -o fu (\J-JD30JC007AJO(Miaj 50NN30J3«-*>^',-'vf-^Cl!'' t^O^N^N-..CinKl—'-OcJAjmcj CO =3' •^0'J5NfVltCN.'Oa3r-3 Nnionjrvj-"4(C?r • -~ KI oj — o in *-o.-*tO -*« m -« o-ro ooooooooooo OOOOOO I ooooooooooo cr c PJ o • tri t t ' 111 Ul UJ uJ Li UJ l wina-Kioeooa, crjXJL -O'COiociO'r-ocoirt .©-OeG-OCrKlcjr— O— Klf Aj o» AJ r- co e>. o in C- o —• « AJAIKIAJOJ — CJO- cr r- »-* * oooooooooooooooooo • » t » t UJ Ui UJ Ul uJ rj ID oo ^3 3 o =r o o «-• a -C in ffi ^ » - ru —« CT -O —• • =3 in i to nj i KI AJ tu UJ UJ UJ in O -O CO <" Mfueorj (O^NCN-^fU-OCJ • stitrri Ul U UJ UJ UJ (\j ry f CO CC : KI r~- L i-- o • -c -4 -O- =r CT PU to oj • t I UJ Ul UJ C 7 C o rj co —• oj (U C 5 ^ r» o o- C -« -4 ^) *H » t • r • Ul UJ UJ UJ Ui cMOOhff -< in r~ ii LTccoo-aioo-omKi—. s -c w o -• AI c r- r*- =r cr ooooooooooo II r i — —• rj — —. — O O O O O O O r • • r » ii » ui ui tu ui UJ ui ui rTVJKll/lO^IliTCfflCMT- • fj co c K> ai KI o rj -o m a BO^O-^MKKM/lini-" <; in o- —•-csfMy ooooooooooooooooooooooooooo rvi —. 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V. *- «. V • J— >C*000 — -40 < ft; ft t CO 0',0?7 -0,279 0.156 -0.247 SLAT »216 VNST -o',44t -0,463 -0,136 0-'130 1,251 0,703 0,268 0 101 0.397 221 VNVT 0',397 0,3o7 0.183 o',024 -0,251 -0,259 -0.221 -0'.228 -0'.386 0,183 0.091 VNOT 0* Oil 0,156 -0,0t7 -0 454 -1,000 -0,444 -0,047 0,126 .0.010 0,000 0.000 VNT • 0'. 000 • O'.OOO -o'.ooo -o'.ooo -o'.ooo o'.ooo o'.ooo 0.000 o'.ooo •0.078 .0.333 VTST 0.001 -0.315 -0.175 0.369 1 .I'll 0.825 0.331 -0.136 -0'.265 0,560 0.511 VTVT •01203 .0,150 -0.170 •ol190 -0.010 0',265 0,375 0,162 0.466 0.996 i',207 22.870 VTOT .0'.999 -0.988 .0'.999 .0'.891 0'.002 0.896 0' 999 0.992 I'.OOO VTT -1.201 -1.452 -f.343 -0.712 T.433 1.986 1.704 1.318 1.201 . ASUM 3,358 8,175 24.983 23^049 20,332 20,432 21.680 -t' 173 18,10 -1,26 •1.146 0^084 10.022 17.92 -0,18 •0,456 0,562 BSUM YY XX CP SIG -8' 002 18.03 -12.18 •0,442 0.599 1,149 17,88 -13,26 -1.108 0.099 7.307 17,76 -14,69 -0.804 0.110 i2'.89« 17.88 -15,40 0.493 0,172 8. 348 18.00 -15.58 -1.054 0.134 -5.410 18.12 -15,40 -2.945 .0.039 • fe', 44 1 18,23 -14,69 -1.905 .0,100 -3'.587 18.14 -13,27 -0.737 -0,134 7'. 7 01 18,03 -12,18 .0.442 .0,602 SLAT »225 VIJST -o',414 -0,3l'9 -0,004 0,468 0,969 0,424 0,1Q4 0,036 0.379 SLAT «234 VNST -O'163 0,141 0,529 0,863 0,245. •0,057 0.134 STRCAMLINC VNST 0',089 0,830 0,476 0,397 0,342 0,328 0,156 0,074 -0,096 -0,274 -0,375 -0,460 .0,487 -0 421 -0,407 -0,331 -0.262 233 VNVT 0^343 0.148 0,015 •0,057 -0.147 •0'.342 24 2 VNVT 0'.094 VNOT O'071 0,171 -0,047 -0,459 -1,000 -0,439 -0,047 0,111 -0.038 V NOT O' 069 0,171 -0.047 ".0*, '158 -1,000 -0,439 -0,047 0,112 -0.036 •0,071 0,137 0.194 OJ105 0.014 • 0'.098 FOR SHCAR LAYF.R VNVT -0'.049 -0'.790 -0J416 -0.317 • 0',262 -0,209 -0.037 0,235 0,373 0.435 O',460 VNOT -0^,040 -0,040 -0,060 -0,080 -0,080 -0,119 -0 119 -0 139 -0,139 100 0,019 0.087 0.322 0'.268 0J192 0,066 0,031 -0,158 »0'. 186 -0,060 -0,0 0,060 0,100 0,139 0 139 0 196 0,100 0,139 0.100 VNT VTST -o'.ooo .0.255 -o'.ooo -0.540 -0.000 -0.252 -0^000 0.452 -0.000 1.910 o'.ooo i. o'.ooo 0 .484 0' 000 -0.099 0.000 -0.203 VNT VTST .o'.ooo -0.326 .o'.ooo -0.558 -o'.ooo •0.454 -0,000 -0.113 -0.000 0.793 o'.ooo 0.777 o'.ooo 0.597 o'.ooo 0.224 -o'.ooo 0.035 I VNT VTST -o'.ooo •0.849 o'.ooo 0.298 • o'.ooo 0.281 -o'.ooo 0.204 -o'.ooo 0.392 -0,000 0.457 -o.ooo 0.580 -o'.ooo 0 .588 -0,000 0.582 0.000 0.5f'3 o'.ooo 0.349 o'.ooo 0.215 0' 000 0.059 o.ooo -0.071 o'.ooo •0.115 0'000 •0.202 0 000 -0.273 -0.000 -0.269 -o'.ooo -0.256 -o'.ooo -0.171 VTVT 8* 188 0,209 0,199 0 J209 0,277 0,192 0,201 0.256 0'.268 VTVT 0,267 0,255 0,266 0,313 0,320 0.067 • o'000 0,006 0.022 VTVT •0,319 •0,309 -0.261 • 0'.479 -0^573 -0,732 -0,781 -0.600 iO'470 -0.316 -o'.ies -0'. 136 -0'.033 0' 054 0,098 0,111 0.072 VTOT .0' 997 .0.985 .0".999 .0'889 0.002 0'.899 0.999 0".9?q 0.999 VTOT »0'.998 -0'.985 • 0'.999 .0.889 0' 002 0.898 0'.999 0'.994 0'.999 VTOT -0'.999 • 0'.999 • 0'.998 • 0'.997 • 0'.997 -0'.993 -0'.993 0'.808 -0.990 0'.759 -0'.995 • 0'.998 • I'.OOO -0.998 • 0'.995 »0'.990 • 0'.990 -0'.9B1 -0'.995 -0'.990 -0'.995 VTT •1.065 •1^317 -1.052 -0'.228 2.190 2.219 1,684 1,151 1.065 VTT -f.057 -1.289 -1.187 -0.688 1.116 1.742 1.595 1.224 l'.057 VTT •1,015 •1.020 •1.026 -1.054 -1.084 •l'. 109 -1.145 -1.183 -1.217 -l'.24I -T.249 -1.255 -1.255 -1.251 • l'.241 -l'.225 -T.200 -l'. 166 -1.136 -1.095 ASUM 5^666 4,791 5,430 10,191 25.486 21^553 18,379 18 335 19.559 ASUM 6^772 5,808 6,381 11,032 25.567 20),770 17,431 17,290 18.449 ASUM -1 f,696 •12,560 -6,478 -15.433 -7' 946 -ll,5(/j -11,110 -11,112 -13.140 -ll'.917 -13; i io •12.416 -12', 223 -12,'123 -10.616 -12', 381 -9 316 -10,026 -9 276 -5.801 BSUM .7^097 2,175 7,784 12,464 6.486 -6,815 -7.185 .4'. 044 6'.869 BSUM -6^934 2.417 7,799 12.113 5'.579 -7'.291 -7'.315 • 4'.009 6'.742 YY XX CP SIG 18,06 -24,18 -0.133 l'.068 17.89 -25, • 26 •0.734 0,172 17.76 -26, ,69 .0.107 0.164 17.87 -27 ,40 0.948 0.218 18.00 -27 ,58 -3.794 0,124 18.12 -27 ,40 -3.925 -0.099 18.23 -26 ,69 -1.835 •0.157 18,16 -25 • 27 -0.325 -0,204 18,06 -24 .18 -0.133 -1,070 YY 18,06 17.89 17.76 17.87 18.00 18.12 18.23 18.16 18,06 XX •36,18 •37,26 •38,69 •39,40 •39,58 •39,40 •38.69 '37,27 •36,18 CP •0.117 -0.661 •0.409 0.526 .0,245 -2.033 -1.545 •0.499 -0.117 SIG 0.666 0,095 0.086 0.138 0,121 -0.019 -0,076 -0.128 -0.689 BSUM YY XX CP SIG -4',866 16.10 49,50 -0,030 0.024 8,277 18,30 44,50 -0.040 .0,(106 -3.238 18.55 39,50 -0.053 -0.010 -3.127 18.90 34,50 •0.111 •0.016 •1.312 19,30 29,50 -0.175 -0.009 -4'.578 19.80 24.50 -0.230 0.010 0^351 20,40 19,50 •0.311 0.018 -1,842 21.05 14,50 -0.399 0,011 0.351 21.75 9,50 •0.461 0.061 0,326 22.35 4,50 •0.540 0,020 0.467 22.75 -0.50 •0.560 0.197 t',591 22.90 -5,50 -0.575 0,080 0.869 22.75 -10.50 -0.575 0,0*>6 2.284 22.35 -15,50 •0.565 0.059 1.748 21.75 -20.50 -0.540 0.050 1^846 21.05 -25,50 .0.501 0,028 3,213 20.20 -30.50 -0.440 0.005 -0.417 19.45 -35.50 -0.360 •0.019 7^470 16,85 -40,50 -0.291 •0.052 2.238 18.25 -45,50 -0.199 -0.0>>8 CAM" 0'.23810E-Ol-0'.O26ilC-02-0'.1015hE-Ol-0'.16088E-0l.0',,9:»665E-62 0'.999(I1E"02 0'.17789E"01 o'.1 1430E-GAM"» 0'.61ia5E-01 Mu< ' ' MU: GNU = V<U=> 0.-71097 VKL = -0'.7l0S0 FORCES ON BODY # i, 50 CENTER AT C O'.O , o'.O ) CU3 3',013I'1 COT= 0'.06599 CMOs 0'.5703B CM« = CIRC« 37'.0638f CLCs 3'.o9640 PERIMs 49,298 FORCER ON BODY #171.179 CENTER AT ( 46'.20, le'.OO) ilT- 2'.07780 COr= O'.4o096 CMOs -0'.3S347 CM4 = Cl«C= l'.R8B9i CLCs l'.0'l911 PERIMs 7'.3775 FORCES ON HODY #iflo,l8A CENTER AT ( 3l'.20, IB'.OO) CLT = -o',24827 COT= -0*.0o750 CMOs -0:. 10270 CM4* ClRCs -0'.44013 CLCs -o'.24452 PCRIMs 7'.3776 FORCES ON BO')Y #189, 197 CENTER AT ( 22'.20, 18'.00) CLT3 .o'.flasflo CDT = -o'.05425 CMOs -0.20537 CM4 = ClRCs -0'.7112B CLCs -0'.111B2 PERIMs 7'.3776 FORCES ON IIPOY #198,206 CENTER AT ( 10'.20, lfl',00) CLT= -0',7569l CRTs -o'.14861 CMOs -0'.36254 CM4 = ClRCs -1 . 17703 CLCs -o'.65391 PER I Ma 7'.3778 FORCES ON BUOY #207,215 CENTER AT ( -f.80, IB'.OO) CLT= -O',5066S CDT= -o'.02944 CMOs -o'.23201 CM4s ClRCs -0.6*213 CLC= -o'.37S96 PERIMs 7'.3778 ON tfnDY #210,224 CENTER AT C -13'.80, IB'.OO) 0',47677 COT= -0".04066 CMOs 0'.24933 CM4 = 0.72990 CICs 0.10550 PERIMs 7'.3776 ON BODY #225,233 CENTER AT < -25'.80, IB'.OO) 0',7C."66 CDl's -O'. 15769 CMOs 0'.35735 CMIs 1'.28502 CLCs 0.71390 PERIMs 7'.3777 ON tinOY #234,242 CENTER AT C -37'.80, lfl',00) 0',50513 CPTs -o'.0{722 CMOs o'.20048 CM4» 0'.785lj CLCs 0'.43634 PERIMs 7'.3777 01 FORCES CLT = ClRCs FORCES f LT = ClRCs FORCES CLT = C IRCS -0'. 18295 -1.02292 »0.04064 P0'.09392 -0'.17332 •0'.10S37 0.13014 0'.16144 0'.07419 I' EXECUTION TERMINATED isiG S LO xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx^ MICHIGAN TERMINAL SY3TEM FORTRAN 0(41336) MAIN1 10*22-75 10119136 PAGE POOl 0001 0002 0003 9001 0005 0006 0007 OOOB 0009 0010 0011 0012 0013 0014 0015 0016 SUBROUTINE MAiNlCXX.YY,DX,0Y,D3,CS,S!.N»S!G,M,VTT.CP,GAM,NG.MU, 1.000 i GNU,NM,VTI,XC,YC,NLS) 2.00REAL XX'(N),YYiN),DX(N),DY'(N),DS(N),CS(M),SI(N) 3.00XX,YY > CONTROL POINT COORDSf DX,DY,D3 » ELEMENT LENGTHS 4.000 CS,SI - COS,SIN OF ELEMENT INCLINATION 5.00REAL XC(NLS),YC(NLS) 6.00XC,YC - CENTERS OF WALL SLATS 7.00N i TnTLA * CONTROL POINTS 8.00REAL Slr,(M),VTT(N),CP(N),r.AM(NC),MU(NM),GNU(NM) • •• 9.000 SIG - SOURCE STRENGTH DENSITIES (ALSO USEO AS SOLUTION VECTOR IN 10.000 SYSTEM ) 10.00(GAM,MU,GNU,' ARE PART OF SIG) 11.00M- TOTAL * UNKNOWNS IN SYSTEM C*SIG=D 12.00VTT,CP - TANG VEL, PRESSURE COEFF'. 13.000 GAM - VORTEX STRENGTH DENSITIES ON TEST AIRFOIL t SLATS 14.00MIJ.GNU - SOURCE 8 VORTEX STRENGTH DEN'S ON STREAMLINE REPRESENTING 15.000 S'.L'. 15.00REAL VTI(NM) 16.00VTI - PRESCRIBED TANG'L VEL ON SHEAR LAYER STREAMLINE 17.000 CALAB,CALCD,CALCP,CALCL - IF NONZERO CALCULATE A,B,C,D,CP,CL 18.00K- R A B, w R C D - IF MONZr-IP WRITE A,B,C,D INTO FILES 19.00SOLV - IF NONZERO SOLVE SYSTEM OF EONS C*SIG=0 . 20.000 GAUSS - IF NONZERO USE GAUSS-ELIMINATION 21.00ITER - IF NONZERO USE ITERATIVE METHOD 22.00HSIC - IF NONZERO ALREADY HAVE SIG IN FILE FROM PREVIOUS RUN 23.000 INTEGFR CALAB,CALCD,WRAB,WRCD,SOLV,GAUSS,ITER,CALCP,CALCL,H3IG 24.0 0 0 COMMON/81/ N»S,NSLAT,NSU1,NKA,N«<2,MSV,NA,NSPS,NTEU,NTEL 25.000 N*3 - TOTAL # CONTROL POINTS ON AIRFOIL SOLID WALL SECTIONS & SLATS 26.000 NSLAT'.NSPS - aSLATS,((CONTROL PTS /SLAT 27. 000 flSU 1 - 1ST CONTROL ON 1ST SLAT 28.00NKA - EON M FOR KUTTA CONO'N ON TEST AIRFOIL 29,00MSV - » CONTROL PTS ON STREAMLINE FOR SHEAR LAYER (MU,GNU) 30.000 NA - ((CONTROL PTS ON SINGLE TEST AIRFOIL 31.00NTEU, MTFL - CONTROL PT * » TEST AIRFOIL TU'. (U'.L) 32.0QNM2-- « OF LAST EON FOR ZERO NORM VEL ON INNER EDGE OF S.L'. 33.000 C0MH0N/B2/ U,CH 34.00U - UNIFORM ONSET STREAM SPEED 35.00CH - SINGLE TEST AIRFOIL CHORO 36.00COHMON/P.3/NU1,NWIII(NU3,MWU2,NL1.NWL1,NL3,NWL2»NSOL1,NSOLSL,NF1, 37.000 1 NFLAT,N3PF,NI1 38,00NU1,NU3.NL1,NL3 - 1ST CON PT ON EACH FLAT SOLID WALL SECTION 39.000 NWIJ1,NWU2,NWL1 ,NWL2 - « CON PTS ON EACH FLAT 30LIO WALL SECTION 40 . 000 N30LSL - * CON PTS ON ARBITRARY SHAPED SOLID SURFACE E'.G', PLENUM 41.000 BOUNDARY 41,00NSOL1 - 1ST CON PT ON ARBITRARY SOLID SURFACE 42.00NFLAT,NSPF - « FLAT SLATS, « CON PTS OFLAT SLAT - NO KUTTA APPLIED .43.000 NF1 - 1ST CON PT ON 1ST FLAT SLAT 44.00Nil - 1ST CON PT ON SHEAR LAYER 45.00COMMON/B4/CALAB,CALCD,WRAB,KRCD,SOLV,GAUSS,ITER,CALCP.CALCL.HSIG 46.00 0 EXTERNAL CALCAB,ASSEMA,AS3EMB,ASSEM0,RE,WR,WRD,ATXB,CPS,FORCES 47.000 N3VT=N " 4 8.000 NUNsM 49.00NDA=4*N3VT*NSVT 50.00NDBsNDA 51.00NDCs4*NUN*NUN 52.00I—' 4s> MICHIGAN TERMINAL SY3TEM FORTRAN G(41336) ..AINl 10-22-73 tOt19136 PAGE P002 0017 CALL GSPACE(A,NDA.0,&30n 53.0Q0 ooia CALL GSPACE(B,NDH,0,&302) 54.000 0019 iF(CALAB.Nc'.O) GO TO 200 55.000 0020 LA = 3 56.000 0 021 MsNSVf 57.000 0022 HaNSVT 58.000 0023 CALL CALLER(RE,AfiPTR(N),IPTR(M),IPTR(LA)) 59.000 0021 LB = 4 60.000 0023 N=NSVf - 61.000 0026 M=NSVT 62.000 0027 CALL CALLER(RE,B,iPTR(N),IPTR(M),IPTR(LB)J 63.000 oo2e GO TO 2ol 64,000 0029 200 N=NSVf 65.000 0030 MrNSVT 66.000 0031 CALL CALLER{CALCAB,A,B,IPTR'(N),IPTR(M),IPTR(XX),IPTR(YY5,IPTR(0X), 67.000 1 IPTR(0Y),IPTR(05), IPTR(CS),IPTR(SI» IF'(HSIG),NC,0) GO TO 201 68.0 00 0032 69,000 O033 IF(WRAB.NE'.O) GO TO 202 70.000 0034 IFCCALCD'.EO'.O) GO TO 203 71.000 0035 GO T0 204 72,000 0036 202 MsNSVT. 73.000 0 0 37 M=NSVT 74.000 0036 LA = 3 75.000 0039 CALL CALLER(WR,A,IPTR(N),IPTR(M),IPTR(LA)) 76.000 C01« N=NSVT 77.000 ooii M=NSVT 78.000 0042 LB = 4 79.000 0013 CALL CALLER(WR,B,iPTR(N),IPTR(M)»IPTR(LB)) 80.000 OOH IF(cALCO'.eo'.O) GO TO 205 81.000 0045 204 IFC^RAH'.NE.O) GO TO 206 82,000 0016 N=NSvf 83.000 0017 MsNSVT 84,000 0016 LB = 4 85.000 0019 CALL CALLER(WR, B, JPTR (N) , IPTR (M) , IPTR(LB)) 86.000 0050 206 CALL FSPACE(B,»303) 87.000 C051 CALL r,SPACE(C,NDC,0,&304) 88.000 0052 M=NSVT 89.000 0053 MBNIJN 90.000 0051 CALL CAi LERitASSEMA,A,C,IPTR(N),iPTR(M)) 91.000 0055 IFCWRAR'.NE.O) GO TO 207 92.000 0056 LA-:: 93.000 0057 MsNSVT 94.000 0058 MzNSVT 95.000 O059 CALL CALLER(WR,A,iPTR(N),IPTR(M)fIPTR(LA)) 96.000 0060 CALL FSPACE(A,*305) 97.000 0061 N0B=4*"SVT*NSVT 98.000 0062 CALL GSPACEitB,NOB,0,8306) 99.000 0063 LB = 4 100.000 0061 N=NSVT 101.000 0065 MsNRVT 102.000 0066 CALL CALLER(RE,B,iPTR(N),IPTR(M),IPTR(LB)> 103,000 0067 207 N=N3VT 104.000 006e M = NIJN 103,000 0069 CALL CALLER(A3SEMB,B,C,IPTR(N),IPTR(M)) 106.000 0070 ND0=4*NUN 107.000 Ul MICHIGAN TERMINAL SYStEM FORTRAN G(«13S6) MAIN1 10-22-73 10119136 0071 CALL r,SPACE(D,N0D.0,i307) 108.000 0072 NsNSVT 109.000073 M=NUN 110.000071 N i = M 51V 111.000075 IFtMSV.CO'.O) Nfal 112.000076 N2=NXE - 113.000 0077 IF(MXE.EQ'.O) N2 = l 111.000078 CALL CALLER(A.1SEMr),0,IPTR(M),IPTR(CS),IPTR(SI),IPTR(N),IPTR(VTn, 115.000 l" 1BTR(N1).IPTR(VT0),IPTR(N2)) 116.000079 iF(SOLV'.LQ.O) GO TO 208 1 17 .000 0080 IF(r,All33'.NE'.0) GO TO 209 lie.000031 IF(ITER.NE.O) GO TO 210 119.000082 208 IF(WPCO.I:0',0) GO TO 21 1 120.000033 210 LC=7 121.000081 NsHi.iN -• 122.000 0085 MsNUN 123.000086 CALL CALLER(WR,C,iPTR(N),iPTR(M),IPTR(LO) 124.000087 LD=8 125.0000?8 MrNUN 126.000089 CALL CALLERC<.Rn,D,IPTRCN),IPTR<LD)) 127.000 0090 IF (GAUSS.EO'.O) GO \»i 212 128.000091 209 LU=6 129.00C092 MsNlIN 130.000093 CALL CALLER(ATXB,iPTR(M),C,iPTR(SIG),0,IPTR(LU)) 131.000 C091 KRITEJ9) SIG 132.000095 wniTE(6,900) M 133.000096 900 FORMATCSIGCIS.')') • 134.000 0097 wRITE(h,3) GIG 135.000098 CALL F5PACECC&308) 136.000^99 IF(CALCP'.EO'.O) GO TO 216 137.000100 201 iF(HSlG'.L'O'.O) GO TO 214 138.00010f READ(9) SIG 139.000102 WRITE(6,3) SIG 140.000 0103 3 FORMATdX, 10C12.5) I'll.000104 214 IF{HSlG>E.O) CO TO 215 112.000105 NOAs'l*NSVT*NSVT 143.000106 CALL r,SPACE(A,NDA,0,*309) 144.000107 LA=3 . 145.000108 NsNSVT ... 146.000 0109 MsNSVf 147,000110 CALL CALLER(RE,A,iPTR(N),iPTR(M),IPTR(LA)) 118.00C!l{ 215 NrNSVf 119.000112 MrNUN 150.00' 0113 KsNJLAT 151.00C114 IF (NSLAT.EO'.O) KSJ - 152.0Q0 0115 LsMsV 153.000116 iF(MSV.EO'.O) Lsl , 154 .000 0117 CALL CALLER(CPS,IPTR(CP),IPTR(VTT)nPTR{XX),IPTR(YY),IPTR(C3)» 155.000 1 IPTR(SI),JPTR(N),IPTRCSIG),IPTR(M),IPTR(GAM),IPYR(K),IPTR(MU)r 156.000 2 iPTR(GNU),jPTR(L),A,B) 157.00Olie 213 IFCCALCL'.EO.O) GO TO 216 158.000J19. CALL FfiPACE(A,&310) 159.000120 CALL FSPACE(B,&311) 160,000121 . NsNSVT 161.000122 Nl=i 162.00•liCHIGAN TERMINAL SYSTCM FORTRAN G<41336) MAIN! 10-22-75 10119136 PAGE P004 0123 0124 0125 0126 0127 0126 0129 0130 C 13 i 0132 0133 0131 N2SNA XA--0'. YA = o'. CALL CALLER(FORCES,IPTR(CP),IPTR(XX),IPTR(YY),IPTR(DX).IPTR(DY), 1 IPTP(OS),IPTR(VTT),IPTR(N),IPTR(U),IPTR(CM),IPTR(NI),IPTR(NJ), 2 IPTR(XA),1PTR(YA)) IF(NaLAT'.Ed'.O) GO TO 2 00 { K = 1,NSL AT Nl = MSlll + NSPS*(K-l) N2=M1+NSFS-1 CHrj'.h xs=xc'rK) YS=i«'. CALL CALLER(FORCES,IPTR(CP),IPTR(XX),IPTR(YY),IPTR(OX).IPTR(DY), 1 IPTR (IIS), IPTR (VTT), IPTR (N), IPTR (U), IPTR (CH), IPTR (NI), IPTR (N2), 2 IPTR(XS),IPTR(YS)) 0135 2 CONTINUE 0136 IF(HRIG'.NE.O) GO TO 217 0137 216 CONTINUE 0138 GO TO I 99 0139 203 STOP 203 oiio 205 3T0P 205 ciii 211 STOP 211 0142 212 STOP 212 0113 217 STOP 217 0111 301 STOP 30 i 0145 302 3T0P 302 0146 303 STOP 303 0147 304 STOP 301 0148 305 STOP 305 0149 306 STOP 306 0150 307 STOP 307 0151 308 STOP 308 0152 309 STOP 309 0153 . 310 STOP 310 .0154 311 STOP 311 0155 99 RETURN 0156 END •OPTIONS •OPTIONS • STATISTICS •STATISTICS* IN IN EFFECT • IO,EnCDIC,SOURCII,N0LI3T,NODECK,LOAO,NOMAP EFFECT^ NAME = MAINl , LINECNT a . 37 3J.J,:LE STATEMENTS = 156,PROGRAM SIZE s NO DIAGNOSTICS GENERATED 69S2 163.000 164.000 165.000 166.000 167.000 168.000 169.000 170.000 171.000 .172.000 173.000 174.000 175.000 176.000 177.000 ne.ooo 179.000 180.000 181.000 182.000 183,000 184.000 185,000 186.000 187.000 188.000 189.000 190.000 191.000 192.000 193.000 194.000 195.000 196.000 197.000 198, 000 199.000 200,000 NO ERRORS IN MAINl LO MICHIGAN TERMINAL SYSTEM FORTRAN G(41536> CALCAB 10-22-73 0001 SUBROUTINE CALCAB(A,B,N,M,XX#YY,DX,DY,DS,C3,SI) C CALCAB CALCULATES MATRICES A, B OF INFLUENCE COEFFICIENTS 0002 REAL XXCN),YYCN),DX(N),DY(N),OS(N),CS(N),SI(N) 0005 REAL A(N,M),B(N,M) 0 0 01 COHMOM/Bl/ NWS,NSLAT,NSU1,NKA,NM2,MSV,NA,NSPS.NTEU,NTEL 0005 DO ? I = i,M 0006 O 2 J*1,N 0007 IFCl'.EQ'.J) GO TO J 0008 OXJ=XX(I)-XX(J) 0009 DYJiYYd)-YY(JC X.I.YJ - DIST. riCE OF 'II TO 'J' IN 'J< COORD'. SYSTEM 0010 XJ = DX.!*CS (J)+DYJ*SI (J) 0011 YJ=DYJ*C5(J)-DXJ*3I(J> 0012 l)SJ2 = r>r>CJ)/2'. 00 1 S D3J4 = DS.I2*D5J2 0011 XJS = X.I*XJ 0015 YJS = Y.r*YJ 0016 XP = X.J + DSJ2 0017 XM=XJ-DSJ0016 XP3=XP*XP 0019 XMS=XM*XM C XJ IS ZERO IF ELEMENTS VERTICALLY ABOVE EACH OTHER 0020 IFCXJ'.EQ.O.) GO TO 140 C PHIX IS VELOCITY IN *IND DIRECTION 0021 PHIX=ALOG((XPS+YJS)/(XMS*YJS)) 0022 GO TO 141 0021 140 PHiXrO'. C YJ IS ZERO IF ELEMENTS ARE ON SAME FLAT HALL SECTION 0024 141 IFfYJ'.EO'.O'.) GO TO 142 C PHIY IS VELOCITY PERP*. TO WIND RIRN 0 025 PHIY=2.*ATAN2((0S(J)*YJ),(XJS+YJS-DSJ4)) 0026 GO TO 143 <027 142 PHlYao. 0028 143 IFCSI '(.D'.EO'.O'.) GO TO 144 0 029 31•J = Sl(i)«CS(J)-CSCI)*SI(J) 0030 coj=cs(i)*cs(j)tsi(i)*si(JC A is NORMAL VEL IN 'I' COORD SYSTEM c 8 is TANG'L VEL IN W COORD SYSTEM 0031 A(J,I)=PHIY«COJ-PHIX*SIJ 0032 B(J,I)=PHIX*COJ+PHIY*SI0033 GO TO 2 0034 3 A(TJ,I) = 6'.283l05 0035 B(J,I) = o'. 0036 GO TO 2 0037 144 3ij=SI(I)*CS(J) 0038 COJ=CSCI)*CS(J0039 A(J,I)=PHIY*C0.1-PHIX*SIJ 0040 B(J,I)=PHIX*COJ*PHIY*SI0041 2 CONTINUE 0.042 RETURN 0013 END ': • OPTIONS IN EFFECT* ID,EBCDIC,SOURCE, NOLIST, NODECK»LOAD, NOM AP • OPTIONS IN EFFECTii NAME = CALCAB , LINECNT » 57 •STATISTICS* SOURCE STATEMENTS s 43,PROGRAM SIZE B. 1700 •STATISTICS* NO DiAGNOSTTCS GENERATED NQ ERRORS IN CALCAB 10119137 PAGE P001 201.000 202.000 203.000 204.000 205.000 206.000 207.000 208.000 209.000 210.000 211.000 212.000 213.000 214.000 215.000 216.000 217,000 2ie.o6o 219,000 220.000 221.000 222.000 223.000 224.000 225.000 226.000 227.000 22e,000 229,000 230.000 231.000 232.000 233.000 231,000 235.000 236.000 237.000 23e.000 239,000 240.000 241.000 242.000 243.000 244.000 245.000 246.000 247.000 248,000 249.000 250.000 251.000 U) CO MICHIGAN TERMINAL SY3TEM FORTRAN G(11336) \SSEMA 10-22-T5 10119|37 PAGE P00I 0001 0002 0003 0001 0005 0 0 06 0007 0008 0009 0010 0011 0012 0013 0011 0015 0016 (.017 0018 0019 0020 0021 0022 0023 0021 0025 0 026 0027 0028 0029 003Q C031 0032 0033 0031 0035 C036 0037 0038 0039 ooio 0011 0012 SUBROUTINE AS3EMA(A,C,N,M) c c is MATRTX FOR SYSTEM C*SIG»D C ASSC.MA ASSEMBLES THOSE PARTS OF C THAT DEPEND ON A'. REAL A(N#N).C(M,M) INTEGER i:,P.O C0MMON/B1/ NWS,NSLAT,NSUt,NKA,NM2,MSV,NA,NSPS,NTEU,NTEL • COMMON/B2/ U,CH NSPsNSPS-1 NN2=NM2+MSV NWSV=MWS+M8V C»**«« LOOP 1 - ASSEMBLE NORMAL VEL EQNS FOR ALL NWS CONTROL PTS ***** DO 19 1=1,NWS C E iS EOUATION « E = I C L00P2 - NORM VELS AT ALL NWS C, P'. DUE TO ALL NWS SOURCE ELEMS DO 2 .1=1, NWS 2 C(J,E)=A(J,i) iF(MSV.EB'.O) GO TO 19 C LOOP 8 - NORM vELS AT ALL NWS CON PTS DUE TO SOURCE ELEMS (MU) ON C S'.L DO 8 K=i,HSV J = N K A + K M=NWS+K 8 C(J,E)=ACM,i5 19 CONTINUE IFCNSL AT'.EO'.O) GO TO 12 C****«LOOP '[O - ASSEMBLE KUTTA EONS FOR AIRFOIL-SHAPED SLATS ***** DO 52 KS=1,NSLAT KL = NSUl+N3PS*(K3-'l) • KU=KL+NSP E = N*'S + KS C LOOP 13 - TANG VELS AT T.E*, 00 13 Ks'l, NSLAT J=NwS+K P = NSU'f*NSPS*<K-l) Q = P+NSP ... SA = o'. DO 11 M=P,Q 11 SA=SA+A(M,KL)+ACM,KU) 13 C(J.E)=SA SA = o'. C LOOP 15 - TANG VELS AT T'.E', OF SLATS DUE TO VORTEX ELEMS ON TEST C AIRFOIL DO 15 K=1,NA 15 SA=3A+A(K,KL)+A(K,KU) C(NKA,E)=3A IF(MSV.EO'.O) GO TO 52 C LOOP l"8 - TANG VELS AT T'.E', OF SLATS DUE TO VORTEX ELEMS ON (GNU) C S'.L DO 18 Kil.MSV J=NM2+K M=NKS+K 18 CCJ,E)=A(M,KL)*A(M,KU) 52 CONTINUE 12 IFtNSLAT.EO'.O) GO TO 20 OF SLATS DUE TO VORTEX ELEMS ON SLAT3 252.000 253.000 251.000 255.000 256.000 257.000 258.000 259.000 260.000 261.000 262.000 263.000. 261.000 265.000 266 .000 267.000 268,000 269.000 270.000 270.000 271 272 000 000 273.000 271.000 275 276, 000 000 277,000 278.000 279.000 280.000 281.000 282.000 283.000 281.000 285.000 286,000 287.000 28e.000 289,000 290.000 291 .000 292.000 292.000 293.000 291.000 295.000 296.000 297.0Q0 297.000 298.000 299.000 300.000 301 ,000 302.000 303.000 CO MICHIGAN TERMJNAL SYSTEM FORTRAN GCH336) ASSEMA 10-22-T5 10ll9|37 PAGE P002 0013 0011 0015 0016 0017 00 IB 0 0 '19 0050 oo5i O032 0 053 0051 0055 0056 0057 0058 0059 0060 0 061 0062 0063 0061 0065 0066 0067 0066 0069 0070 0071 C072 0073 0071 0075 0076 0077 C07B 0079 0080 008 i 0082 0083 22 21 20 C C 23 26 21 C*****ASSCMfiLC KUTTA EONS FOR TEST AIRFOIL ***** 301.000 C LOOP 2{ - TANG VELS AT TEST AIRFOIL T.E-. DIT'. VORTEX ELEMS ON SLATS 305.000 00 21 Kai,NSLAT 306.00PBU3U1+NSPS*(K-1) 307.00OsPtNsP 308,00JoNWS+K 309,003A = u'. 310.00DO 22 MaP.Q 311.000 SA=SA+A(M,NTEU>+ACH,NTEL) 312.OpO C(J,NKA)=SA 313,003A = o'. 311.00LOOP 23 - TANG VELS AT TE3T AIRFOIL T'.E*. O'.T'. VORTEX ELEMS ON TEST 315.OoO AIRFOIL 315.00DO 23 Kil.NA 316,000 3A = SA + A'CK,NTEU)+A(K,NTEL) 317.00C(NKA,HKA)=SA 318,00iFCMSV'.EQ'.O) GO TO 21 319.0QLOOP 26 - TANG VELS AT TEST AIRFOIL T.E*. o'.T*. VORTEX ELEMS (GNU) ON 320.000 DO 26 K = 1,MSV 321.OOO J = M<2 + K 322.00M=NwS+K 323.00C(J,NKA)=A(M,NTEU)*A(M,NTEL) 321.00IF(MSV.EO'.O) GO TO 61 325.000 C«****ASSEMf)LE NORMAL VELOCITY EONS FOR MSV CON PTS ON S,L'. 326.00DO 27 KM=1,MSV 327.00IsMwS+KM 328.00E=NKA+KM 329.00C LOOP 28 - NORM VELS AT MSV CON PTS O'.T'. ALL NWS SOURCE ELEMS 330,000 00 28 Jal,NWS 331.00C(J.E)=A(J,I) 332.00LOOP 33 - NORM VELS AT MSV CON PTS O'.T'. ALL MSV SOURCE ELEMS (MU) 333.000 ON INNER 333.00DO 33 Kel.MSV . 331.Ogo J = Nk'AtK 335.00MBNHS+K 336.00C(J,E)=A(M,i) 337.00CONTINUE 33e.«o***»*ASSCMBLC TANG'L VEL EONS FOR MSV CONTROL POINTS ON INNER EDGE OF 339.000 SIL*.***** 339,00DO 35 KM=1,MSV 310,00I=NWS+KM 311,00E=NM2+KH 312,00IFCMSLAT'.En'.O) GO TO 37 313,000 LOOP 38 - TANG VELS AT MSV CON PfS D'.T'.ALL VORTEX EiEMS ON SLATS 311.000 DO 38 Kc1/NSLAT 315,00PsMSUl+NSP3*CK-l) 316.00OsPtNRP 317.00J=NwS+K 318.003A = o'. 319.00DO 39 M=P,0 350.000 3A = SA + A(M,I) • • - • 351 .000 C(J,E)=3A 352.00SA = o'. 353.00LOOP 10 » TANG VELS AT MSV CON PTS D.T'.ALL VORTEX CLEMS ON TEST 351.000 351.000 28 33 27 C C 39 38 37 C C AIRFOIL O MfCHICAN TERMINAL SY3TCM FORTRAN C(H336) A9SEMA t0-22"75 lot 10137 PA6E POOS 0081 0035 0036 DO 40 K«1,NA 10 3AaSA + A'(K,n C(NKA,E)a3A LOOP 42 • TANO VELS AT MSV CON PTS D'.T.ALL VORTEX ELEHS (GNU) ON C . C s L" 0097 * *00 42 Kil.MSV 0088 JeNM2+K 0039 MsNwS+OO'Q 12 C(J.E)=A(M,i) t'O'l 35 CONTINUE 0092 61 CO'iTlNUE 0093 RETURN 0094 END •0PTJOM3 IN EFFECT* ID , EHCDIC , SOURCE,NOLI3T,NODECK#t.OAD#NOMAP •OPTIONS IN EFFECT* NAME = ASSEMA , LINECNT • . 57 •STATISTICS* SOURCE STATEMENTS s 94,PROGRAM SIZE a •STATISTICS* NO DIAGNOSTICS GENERATED NO CRRORS I" ASSEMA 353.000 356,000 357.000 35e.00O 358.000 359.000 360 , 000 361.000 362.000 362.500 363.000 364.000 365.000 2980 MICHIGAN TENMjNAL SYSTEM FORTRAN 0(11336) 'SSEMB 10»22»75 10U9I38 PAGE P001 0001 0002 0003 0001 0005 0006 0007 0008 0009 ooio ecu 0012 0013 con O0I5 O0I6 0017 tioie 0019 0020 0021 0022 0023 0021 0025 C026 0027 0028 0029 0030 0031 0032 0C33 C031 0035 0036 0037 0038 0039 ooio 001 i 0012 0013 SUBROUTINE ASSEMBCB,C,N,M) 366.000 C A3SEMB ASSEMBLES THOSE PARTS OF C THAT DEPEND ON B*. 367.00REAL B<N,N) ,C (M,M) 368.00INTEGER C.P.O 369.00COMMON/BIZ NW3,NSLAT,NSU1,NKA,NM2,MSV,NA,NSPS»NTEU,NTEL 370.000 COMMON/B2/ U,CH — 371.00NSP=NSPS-1 372.00MN2=NM2+MSV 373.OQNWSV=NHS+MSV 371,00C*****ASSEMRLC NORMAL VEL EQNS FOR ALL NWS CON PTS ***** 375.OgO 00 i|9 1 = 1,NWS 376.000 E=I 377.00IF (NSLAT'.EO'.O) GO TO 3 378.00C LOOP i\ - NORM VELS » ALL NWS CON PTS D'.T'. VORTEX ELEMS ON SLATS 379.000 DO 1 K=i,NSLAT 380.00J=NWS+K , 381.000 P=NSUi+NSRS*(K-l) 382.000=P+NSP  383.000 3B = o'. 381.0000 5 "=P,Q 385.005 Se=SS-B(M,I) 386.00I CCJ,E)=SB 387.003 3B = o'. 388.00C LOOP 6 - NORM VELS » ALL NW3 CON PTS o'.T'. VORTEX ELEMS ON TEST 389.000 C AIRFOIL 389.00DO 6 K=i,NA 390.006 3B=SR-B(K,I) 391,00C(NKA,E)=OB 392.00IF(MSV.CQ'.O) GO TO 19 393.000 C LOOP 9 - NORM VELS • ALL NWS CON PTS D'.T'. VORTEX ELEMS (GNU) ON 391 .000 C S'.L'. 391.0000 9 K = i",MSV 395.00J=NM2+K 396,00M=NWS+K 397.009 C(JiE)=-B(M,I) 398.0019 CONTINUE 399.00iF(NSLAT.t'O'.O) GO TO 12 100.000 C*****A3SrMP.LC KUTTA EONS FO AIRFOIL-SHAPED SLATS ***** 101.00DO 52 KS=t,NSLAT -- - 102.000 KL = NSlll+N3PS*CKS-l) 103.00KU-KL+NSP IOI.OOO E=NwS+KS 105.00C LOOP il - TANG VELS # T'.E'. OF SLATS D'.T', ALL NWS SOURCE ELEMS • 106.OOO 00 11 Jsl.NWS 107.00II C(J,E)=B(J,KL)+B(J,KU) • 108.000 IFCMSV.EO'.O) GO TO 52 109,00C LOOP 17 - TANG VELS • T'.E', OF SLATS D'.T. MSV SOURCE ELEMS (MU) ON 110,000 C S'.L. 110.00DO 17 K=1,MSV 111.00J = NK A + K 112.00MsNWStK 113.0017 C(J,E)=B(M,KL)+B(M,KU) 111.000 52 CONTINUC 115.00C LOOP 19 - TANG VELS « TEST AIRFOIL T'.E'. O'.T', ALL NWS SOURCE ELEMS 116.000 12 DO 19 Jal,NWS 117.00to MICHIGAN TERMINAL SY3TEM FORTRAN G(41336) A33EMB 10-22-75 10110)38 PAGE P002 0001 9015 0016 0047 0048 004") 0 050 0051 0052 0053 0054 0055 0056 0057 0058 0059 0060 0061 0 062 0063 0064 0065 0066 0067 0068 0 0 69 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 0081* 0082 0083 0084 0085 •OPTIONS 19 C(J,NKA;aB(J,NTEU)+B(J,NTEL) IF (MSV.CO',0) GO TO 24 C LOOP 25 - TANG VELS * TEST AIRFOIL T.E*. o'.T. MSV SOURCE ELEMS(MU) C ON INNER . 1)0 23 Ksl.MSV JsNK A + K M=NWS+K 25 C(J.NkA)nB(M,NTEU)+B(M,NTEL) 24 IF (MSV .Efl'.O) GO TO 61 C**«**ASSEMBLC NORMAL VEL EQNS FOR MSV CON PT3 ON SHEAR LAYER***** 00 27 KMaijMSV ISNKS+KM E=NKA+KM IF(NSLAT'.Eu'.O) GO TO 29 C - LOOP 30 - NORM VELS • MSV CON PTS D,T'. VORTEX ELEMS ON SLATS 00 30 Ks1,NSLAT JrNWS+K P=N3U1+NSPS*(K-1) 0=P+NSP SB = 0'. DO 31 M=P,0 31 3BsSB-B'(M,l) 30 C(J.E)»SB 29 3B = o'. ^. • C - LOOP 32 - NORM VELS » MSV CON PTS D.T. VORTEX ELEMS ON TEST C AIRFOIL DO 32 K=l,NA. 32. 3B=SB-B(K,I) C(t«A,E) = SB C - LOOP 34 - NORM VEL3 • MSV CON PTS O'.T'. MSV VORTEX ELEMS (GNU} ON C s' L' DO 34 Ksl.MSV J=NM2,K M=NW8+K 34 C(J,E)=-B(M,I) 27 CONTINUE ... C*****AS3CMRLC TANG'L VEL EONS FOR MSV CON PTS ON SHEAR LAYER***** DO 33 KM=1,MSV I=NW5*KM E=NM2+KM C LOOP 36 - TANG VELS P MSV CON PTS D.T, ALL NW3 SOURCE ELEMS DO 36 Jrl.NWS 36 C S.L C(J,E)=B(J,I) LOOP 41 - TANG VELS » MSV CON PTS O'.T". MSV SOURCE F! EMS (MU) ON DO 41 Ksl.MSV JrNKA-tK M=NwS+K 41 C(J.E)=H(M,i) 35 CONTINUE 61 CONTINUE 64 CONTINUE RETURN END IN EFFECT* ID,EBCDiCSOURCE,NOLIST,NODECK,LOAD,NOMAP 418.000 419.000 420.000 420.000 421.000 422.000 423.000 424.000 425.000 426.000 427, 42e, ooo ooo 429.000 430.000 431.000 432.000 433.000 434.000 435.000 436.000 437.000 438.000 439.000 440.000 4 41.000 4 41,000 442.000 • 443.000 444,000 445,000. 445.000 446,000 447.000 448.000 449,000 450.000 451.000 452.000 453.000 454.000 455.000 456.000 457.000 458.000 458.000 459.000 460.000 461 .000 462,000 462.500 463.000 464.000 465.000 466.000 MICHIGAN TERMINAL SY3TEM FORTRAN G(11336) SSEMO 10-22-75 10H9I3B PAGE PO01 0001 0002 OC03 oooo '0005 0006 0007 OOOB 0009 0010 0011 0012 00 13 C011 0015 0016 0017 0018 0019 0020 0021 0022 0023 0021 0025 0026 0027 J02C 0 029 *OPTJONS *OPTIONS •STATISTICS* •STATISTICS* SUBROUTINE ASSEM0C0,M/C3,SI,N,VTI.N1,VT0,N2) 167.000 A3SEMD ASSEMBLES R'.H'.S. VECTOR FOR SYSTEM C*SIG«0 16e.00REAL 0<M),CS(N),Sl(N),VTI(Ni),VT0(N2) 169.00INTEGER P.O 170.00C0MM0M/H1/ IJ*S,NSLAT,NSU 1 ,NKA,NM2,MSV,NA,NSPS,NTEU,NTEL 171.000 C0MM0N/P2/ U,CH 172.00NSP=NSPS-1 173.00NWSV=MWS+MSV 171,00NN2=NM2+H3V 175.00LOOP 13 - NORMAL ONSET FLOW VEL AT ALL NWS CON PTS 176.000 00 13 1=1,NWS 177.00niti)=u*stm i7e.o6o IF(wSl.AT'.EO.O) GO TO 11 179.00LOOP 15 - TANG'L ONSET FLOW VEL3 # T'.E*. OF SLATS 180.000 DO 13 Kst,NSLAT 181.00I=NnS+K 182.00P = lvSUl+MSPS*(K-l) 183.000=P+NRP 181,00n'(I)a.U*CCS'(P)+CS(D)) 185.00LOOP 1/| - TANG'L ONSET FLOW VELS • T'.E. OF TEST AIRFOIL 186,000 D(NKA) = -0*(CS(NTEU)->CS(NTEL)) 187.00IF(MSV.CO'.O) GO TO 16 188.00LOOP 17 - NORMAL ONSET FLOW VEL • ALL MSV CON PTS 6.N SHEAR l'. 189.000 DO 17 K=1,M3V 190.00JsNWS+K 191.00I=NKA+K 192.00D(I)=H*SrCJ) 193.00LOOP 18 - TANG'L ONSET FLOW VEL • ALL MSV CON PTS ON SHEAR L'. 19a.000 * PRESCRIBED TAN 'L VEL THERE*. 195.00DO 18 K=1,MSV . • • • 196.000 JshwS+K 197.00I=NM2+K . 19e.00O D(I)=-U*C3(J)+VTI(K) 199.00CONTINUE 500.00RETURN 501.00END 502.00IN EFFECT* ID,EBCDIC,SOURCE,NOLI ST,NODECK,LOADiNOMAP IN EFFECT* NAME = ASSEMD , LINECNT » 5T SOURCE STATEMENTS = 29,PROGRAM SIZE n 1191 - • NO DIAGNOSTICS GENERATED 13 15 C 11 17 C 48 16 u NO CRRORS IN ASSEMD 4^ MICHIGAN TERMINAL SYSTEM rORTRAN G(41336) CPS 10-22-.73 10119t 36 PAGE P001 ooof SUBROUTINE CP3<CP,VTT,XX,YY.CS,SI,N1,3TC,N2,GAM,N3,MU.GNU,N0,A,B) 503.000 c " CPS CALCULATES VEL, PRESSURE » ALL CON PTS'. 504.000 ooo? REAL A(Nl,Ni),H(Nl,Nl) 503.000 0003 REAL SIG(NJ),GAM(M3),MU(N4),GNU(N4) 506.000 0001 REAL VTT(Nn,CP<Nl),XX(Nl),YY(Nl),CS(Nl),SI(N15 507.000 0005 INTEGER P,0 ; 508.000 OC06 COPMOM/Hl / NWS,NS1.AT,NSU1,NKA,NM2,M3V,NA,NSPS,NTEU,NTEL 509.000 0007 coMMON/nn/ U,CH 510.000 OOOB C0MMOH/B3/NUl,NWUl,NU3,NWU2,NLl,NWLi,NL3,NHL2,NS0Ll.NSOLSL,Nri, i NFLAT,N3PF,NII 511.000 512.000 0009 NN2sNM2tMSV 513.000 oo l o NWSV=M«3+MSV 514.000 9011 NVNsMwS+MSV 515.000 0012 NSU2=NSU1+MSP3*NSLAT-1 516.000 0013 NSP=NSPS-1 517.000 0011 iF(N3LAT'.En'.0) GO TO 2 518.000 0015 110 1 K = i,N3LAT 519.000 0016 I=NWS+K 520.000 C GAM . SLAT VORTE* STRENGTH DENSITIES 521.000 0017 1 nAM(K)=srGm 522.000 c GAMM - TEST AIRFO VORTEX STRENGTH DENSITY 523,000 0018 2 GAMMsSIG(NKA) IFCMSV.CQ'.O) GO TO 4 524.000 0019 525.000 ;>o2o 00 3 K=1,M8V 526.000 0 021 IsNKA+K 527.000 0022 J=NM2+K 528.000 c • MU - S'.L'. SOURCE STRENGTH DENSITIES 529.000 0023 MUCK)=SiG(I) 530.000 c GNU - VORTEX STRENGTH DENSITIES FOR SHEAR LAYER 531.000 0021 3 GNUCK)=SIG(J) 532.000 C025 4 LL»0 533.000 0026 DO }2 lel.NVN 534.000 0327 li=I+N3P " 535.000 002B LK="SultLL*NSPS 536.000 0029 IFCI.Cf".', 1) WRITE(6,52) 537.000 0030 ' 52 FOIiMATClHl) . , • •• • • 538.000 C VNSf.vTST - TOTAL NORMAL & TANG'L VELS DUE TO SOURCE ELEMS 539,000 C'031 VN3f=0, 540.000 C032 VT3T=0. 541,000 0033 ASM=ol 542,000 0031 B3MiO. 543.000 0035 DO 5 .i=i#Nws • - 544.000 0036 VNSf=VNRT+A(J»I)*SIG(J) 545.000 0037 VT3T=VTST+BCJ,I)*SIG(J) 546.000 003B ASM=A3M+A(J,I) 547.000 0039 5 BSM=BSM+B£J,I) 548.000 0040 A3 = 0. 549.000 004 1 B3 = o'. 550.000 0012 IF CNSLAT.EO'.O) GO TO 8 551.000 •t 043 DO 6 K=ltNSLAT ' • . 552.000 eon • P=N3Ui'tNSPS*(K-l) 553.000 0015 QsP+NSP 554.000 C P,Q - {ST & LA3T CON PTS OS A SLAT 555.000 c AP.BP - NORM * TANG VELS DUE TO VORTEX ELEMS ON SLATS 556,000 0046 AP = o'. 557.000 MICHIGAN TERMINAL SYSTEM FORTRAN G(11336) "PS 10-22-75 10H9I38 PAGE P002 0017 0018 0019 OcSO 0 051 0052 0053 0051 5 055 0056 0057 0058 C059 0060 0 061 0062 0063 0061 Oo65 0066 0067 0068 0069 0070 0071 0072 0073 0071 0075 0076 0077 0 078 C079 0030 0031 0032 0083 OoSI 0085 9086 0C87 0088 0039 0C90 0091 0092 0093 BP = 0 . 00 7 M=P,Q AP=AP+A(M,I) 8PaflP + r>(M,I) AP.= AP*GAM(K) nP=BP*GAH(K) A3=AS+AP PS=ns+BP AT.BT - NORM AT = 0 . BT*o'. no •> .!=! i NA AT=AT+A(J,I) BT=BT+R(J,I) AT=AT*GAMM RTsHTftGAHM AM,B« - NORM & TANG VELS DUE TO VORTEX ELEMS ON TEST AIRFOIL t. TANG VELS DUE TO MSV SOURCE ELEMS ON INNER EDGE OF NORM & TANG VEL3 DUE TO MSV VORTEX ELEMS GNU ON INNER C •C S'-L'. AM = U'. BM = u'. C . AG,BO C EDGE OF S'L'. AG = O'. BG = o'. IFCMSV.EO'.O) GO TO 11 no io K=I,MSV J=NWS+K AG=ACH A'CJ,I)*GNUCK) flG=HGtB(J,I)*GNUtK) AM = AM+A(.J,I)*MU(K). 10 RM=RM+RCJ,I)*MU(K) 11 VNST=VNST+AM VTST=vTST+BM C VNVT.VTVT - TOTAL NORM t TANG VEL DUE TO ALL VORTEX ELEMS VNVTs-HS-BT-HG VTVT=AStAT+AG C VNOT,VTOT - NORM & TANG-VEL DUE TO UNIFORM STREAM U VNOf=-U*3ICI) VTOT=U*CSCI) VNT=VMnT+VNVT+VNOT VTT'ci) = VTST + VTVT + VTOT C VKL VKU - TEST AIRFOIL T'.E*. KUTTA VELS IF(l'.r"'.NTEU) vKUrVTTCI) IF(l'.EQ'.NTEL) VKL = VTT'(I) CPCI) = 1'.-VTTCI)*VTT(I) iFti'.r's'.n GO TO ni IFUl'.EC.NUl)' AND*. CNWUl'NE',0)) GO TO 12 IFC'CI Ea'.NU3),ANDr(HWU2.NE',0)) GO TO 13 IF(CI.EO'.NLI) AND (NWH.NE.O)) GO TO 11 tFCJl,EO.NL3)lAND.CNWL2.NE'.0)> G0 TO 15 IFCCI EO'.N8OLI).AND'.<N80L3L'NE,0>) GO TO 46 IFC(I.L:(!.NF1)'.ANO'.(NFLAT'.NE.O)) GO TO IT IF((NSLAT,.EO'.0).OR".(l'.GE',NSU2)) GO TO 70 iFCCl'.Eo'.LKj'.ANn'.CLL'.LE'.NSLAT)) GO TO 18 70 CONTINUE 558.000 559.000 560.000 561 .000 562.000 563.000 561.000 565.000 566.000 567.000 56e,000 569.000 570.000 571.000 572.000 573.000 571.000 571.000 575.000 576.000 577.000 577.000 57e.000 579.000 580.000 581.000 582.000 583.000 581.0(10 585.000 586.000 587.000 588.000 589.000 590.000 591.000 592.000 593.000 591.000 595.000 596,000 597.000 598.000 59").000 600.000 601.000 602.000 603.000 601.000 605.000 606.000 607.000 608,000 609.000 610.000 MICHIGAN TERMINAL SY3TEM FORTRAN G(H336) CP9 10-22"T5 lOt t<9t3R PAGE P003 009q 0095 CO'6 0097 00<"8 C099 0100 '0131 01 32 C 1 03 0 101 0105 0106 0107 010B 0109 0110 0 1 11 Ci 12 0 113 0114 01 15 0116 oi 17 0118 0119 0!20 0121 Cl'22 0123 9'24 0123 0126 0127 0123 0129 0130 0131 0132 0133 0134 C135 0136 0137 0138 0139 ciio 0111 0142 0113 0144 •OPTIONS iF((I.EO.Nii).AND.(MSV'.HE.O)) GO TO 49 GO TO 12 00 F0RMAT.C4X, 'VNST' ,4X, l VNVT ', 4X, I VNOT t, 4Xi 'VNT ' »5X» 'vfST',4X,'VTVT•, 1 IX,'VTOT'»4X,'VTT',5X,'ASUM',4X,'BSUM',5X.'YY',5X,'XX',5X,'CP'i 2 6X,'SiG') 41 WnifE't6,30) 30 FORMATC'MAIN AIRFOIL') GO TO 51 42 WRlfE(6,31) 31 *^n.i.\,-('o.nv-.' Ri«.c-: cwilw MALL') GO fo 51 43 wRITEi'6,32) 32 FORMAT('UPPER LEFT SOLID MALL') GO fo 51 44 WRITE'c6,33) 33 FORMAT('LOWER LEFT SOLID WALL') GO TO 51 45 WRITE(6.34) -34 FOfiMATCLOUER RIGHT SOLID WALL 1 ) no fo si 46 WRITE(6,35) 35 FORMAT('G0LIO STREAMLINE WALL') GO TO 51 47 WRlfE(6,36) .. - . 36 FORMAT('FLAT SLATS/NO KUTTA') GO TO 51 48 IFd'.ME'.NSUl) GO TO 67 wRlfE(6,37) 37 FORMATPUPPER SLATS') 67 LL=LLtl 69 IF(LL',GT'.NSLAT) GO TO 12 MRITE(6f68) I,II 68 FORMAT(/,'SLAT *',I3.2X,I3) GO TO 51 49 WRlfc'(6,38) 38 FORMAJ£'STREAMLINE FOR SHEAR LAYER") 51 WRITEC6.40) 12 WRI TEC'', 13) VNST, VNVT, VMOT, VNT, VTST, VTVT, VTOT. VTT'(I), ASM, BSM, YY(I),XXci),CP(i),SIG(I) 13 FORMATdX, 10F8.3,2F7.2,2F8,3) IF(wSLAT.CO'.O) GO TO 20 WRlTi:(6iH) GAM WRlfEC6,18) GAMM FORMAH'GAMMs • ,G12-.5) IF(MSV.EQ'.O) GO TO 19 WRlfE(6,15) «U KRITEfh,16) GNU FORMAT ('GAMs ', i0G12*.5) FORMAT('MU=',10G12.5) FORMAT('GNU=' , iOG12'.5) WRlfE(fc,17) VKU,VKL FORMAT (' VKU=' ,G12'.5,2X, ' VKL» ' , Gl2'.5) RETURN END IN EFFECT* 10,EBCDIC,SOURCE,N0LI3T,NODECK,LOAO,N0MAP 20 18 14 15 16 19 17 611.000 612.000 613.000 614,000 615.000 616.000 617.000 618.000 619,oro 620 . 000 621 .000 622.000 623,000 624.000 625.000 626.000 627.000 628.000 629.000 630.000 631 .000 632,000 633.000 634.000 635.000 636.000 637.000 638,000 639.000 640,000 641 .000 642.0Q0 643.000 644,000 645.000 646.000 647.000 648.000 649.000 650.000 651 .000 652.000 653.000 654.000 655.000 656.000 657.000 658,000 659.000 660.000 661.000 662.000 663.000 664.000 MICHIGAN TERMINAL SYSTEM FORTRAN G(11336) i-ORCES 10-22-75 0001 SUBROUTINE FORCES(CP,XX,YY,DX,DY,OS,VTT,N.U.CH,Nl,N2,XC.YC) 0002 REAL CPC''0«XX(N),YY(:N),DXCN),DYCN),DS<N>,VTTCN) C XCYC - CENTER OF BODY 0003 WRITEC6.5) N1.N2.XCYC • 0001 5 FORMAT (' FORCES ON BODY *•, IS,1,',13,3X,1 CENTER AT (',F7*,2, t >.>.F-:s,n'i 0005 CLTBO 0006 COTBO*, 0007 CM0=0. 000B CiRCeO0009 PER=0*. 0010 DO 1 I=N1,N2 0011 CLT=CLT-CP(i)*OX(i) 0012 CDTBCDT+CP(t)*DY(I0013 C1RC=CI«C+VTT(I)«0S(I) C PER - ROOY PERIMETER CC11 PCRiPCfND3(i) 0015 1 CMO=CMO+CPCI)*((XX(I)-XC)»DXCI)*(YV{I).YC)*DY(I)) C CLT - TUNNEL LIFT COEFF'. 0016 CLT=CLT/CH C COT - TUNNEL DRAG COEFF*. (THEOR »Y ZERO) 0017 CDT=CDT/CH C C*0 - TUNNEL MIDCHORD PITCHING MOM, COEFF*. 0018 CMOiCMO/CH/CH C CM'I - TUNNEL OUARTERCHORD PITCHING MOM'. COEFF*. 0019 CMflsCMO-CLT/'1'. C ciRC - CIRCULATION ABOUT BODY C CLC- LIFT COEFF'. FROM CIRCULATION 0020 CLC=2,*CtRC/CH/U 0021 wRITEC6,2) CLT,CDT,CM0,CM1 0022 2 FORMATCCLTi'.Flo'.S^X.'CDTa'.Flo'.-S^X.'CMOs'.Fio'.-S^Xj'CMfl 1 F10'5). 0023 HRiTEf6,3) CIRC,CLC,PER 0021 3 FORMATCCIRCa'.FlO'.S^X.'CLCo'.FlO.SjZX.'PERIMBljGia'.S) 0025 RETURN 0026 , END •OPTJONS IN EFFECT* 10iEBCDIC,SOURCE,NOL1ST,NODECK.LOAD,NOMAP •OPTIONS IN EFFECT* NAME = FORCES , LINECNT " 57 •STATISTICS* SOURCE STATEMENTS s 26,PROGRAM SIZE B 1276 •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN FORCCS 10H9I39 PAGE P001 665.000 666.000 667.000 666.000 669.000 670.000 671.000 672.000 673.000 671.000 675.000 676.000 677.000 67e.000 679.000 680.000 681.000 682.000 683.000 681,000 685.000 686.000 687.000 688.000 689.000 690.000 691.000 692.000 693.000 691.000 695.000 696.000 697.000 698.000 699.000 700.000 14 ooooooooooooooooooooooooooooc-oooooooooooooooooooooooooo o o o o o o O O O O O O O OOO OO o o o oo o ooo o o o o o o o o o o o oo oo-o.o ooo o o o o ooo o oooooooooooooooooooo. ooooooooooooooooooooooooooooooooooo J r" n a uioVJ tr*o- rj in en* OVOJ c o - o to .•^b^o^«(^owfJloc.^o^«(^o*<',J'M^rtnc^««^o»^^JW!^^n o o o o o o o o o - - « w - - -H rj rj rj rj rj rj rj rj rj rj io ro M M M KI KI KI KI n c =r =r ~ — — — — - LO LfVLn LH LO in ru <, ~ co _» •> ULJ« »JZ CO z •»-» o »to n c >- Lu » a. Ci »- «~ X » z z X •» w 2 O 3 >- >-* U D » >- —I » — >- z *-« e_ * Z X X X Z X X ^ucx 0 UJ - -or >- ^ 01 »Z a. C h- X. O Z >- " a LJ >- >-u » * L_J X '-x ^ Z »Z CL - r-4 ac-w I H U X ^ —.•—XX U- *J s < z O *-» CJ Lu »-(D D < < 3 LJ LJ <r> cr cr cc z. «L r- o c LO •—*- • • CT 1— C" If fl u u r- cr < Lit ^ IU II U_ cr ti. •--*-« 1 N O I I CT • -K X -O LO LO C U I ru ru rj rj »-N UJ _4 -J '-J CO a a. u o u 1; II n • 1- o cr u LJ G i <: t~ a o * W . I- C. G C LJ < _J 11 n»- £ c:. h- O Cl -LJ UJ c: c •j-> I— v- I x : Z Z IU •» U U'H •J '_> X < ;< 1 II 11 It o o —<; X C ( . • o r\! e: tu x. »-i _J * Z i-t (_) c: u_ HUC X >- + -« CO *-e 1-c ru a. ww CO z z LJ LU o o z z < < H H U U » I I ru ru cr.n +. x >- x >• I Q_«—X >- II II j ro >-«!-«>-: o x » X X ru 1 o 11 '-> a. a. » x >- x >-CL w o Z LU JH' XOO+.-C ^^»» ZZ. x >• - • —• Lu '—I "_' ^ U. . • -.-. I IX. il- X < < 'J X JU II ^* < * wv^t-i-k c C 1 HMHi^jHhini 1 rj. v « ~ ^ L". _ .CUOHhHHH. MX>. «^ a o JO: c c.^*'-' a 11 n w 11 11 n -w 11 ^ • COUU.li. I 'J O Z 11 o u. u. C X >• X •-• < U« M>-HM ^ o « " »-» • XX O X X o X >• L> •H s: x >- C3 ru f\ *r in o e» o o o o o 000 o OOOO co o o-—• ro ^n *r in o*r*- 00 o 00—• — « —• —• ^ O G> O O O CT C> C> OOOO O C'C'^'OOC'C* OOOO o" —« ai KI c IP or- c o* o • —< <\r KI cr tn o>r- co cr- • ru"»n er ruruojruftJPJfurvjruruM KIKIKIMMMMK\ m crcj c c cr 00000000000 00000000 T> & cr> o cz> c> C C> C- t> i» O o c» -> oooc-oc»oo OO'OOOO MICHIGAN TERMINAL SYSTEM FORTRAN G(41336) HOOPRO 10-22-75 0045 M=MP-I C XX.YY - NOW MODIFIED CONTROL POINT COORDS. 0046 DO 5 I=L1,L2 0047 J=l-Lt+1 0048 K = .W1 0049 XXCl) = (XH(J) + XM(K))/2'f 0050 YY(I)=(YMCJ)+YM(K))/2. 0051 OX(I)=XM(K)-XM(J) 0052 DYCI)=YM(K)-YM(J0053 D3CI)=S0RTCDX(I)*0Xa)+DYCI)*DY'(I)) 0054 CS(i)sDXCI)/DS'(n 0C55 5 3i(i)=DY(I)/D3'CI) 0C56 RETURN 0057 END • OPTIONS IN EFFECT* 10,CMCDIC,SOURCE,NOLIST,NOOECK.LOAD.NOMAP •OPTIONS IN EFFECT* NAME = MOOPRO , LINECNT * S7 •STATISTICS* SOURCE STATEMENTS = 57.PROGRAM SIZE s •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN MOOPRO 2700 10119t40 PAGE P002 756.000 757.000 758.000 759.000 760.000 761.000 762.000 763.000 764.000 765.000 766.000 767.000 768.000 769.000 MICHIGAN TERMINAL SY3TEM FORTRAN G(11336) RE 10-22-75 0001* SUBROUTINE RE(A,N,M,LA) 0002 REAL A(N,M) OC03 READ(LA) A 0001 RETURN 0005 ENO •OPTlGNS IN EFFFCT* 10.EBCOIC,SOURCE,N0LIST,NOOECK,L0AO,NOMAP •OPTIONS IN EFFECT* NA«C = RE , LINECNT • 57 •STATISTICS* SOURCE STATEMENTS * 5.PROGRAM SIZE s 410 •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN RE MICHIGAN TERMINAL SY3TEM FORTRAN G(11336) WR 10-22-75 C00 1 SUBROUTINE WR(A,N,M,LA) 0002 REAL A(N,M) 0003 WRITE(LA) A 0001 RETURN 0005 END • OPTIONS IN EFFECT* In, EBCDIC, SOURCE , NOLIST, NODECK, LOAD, NOMAP •OPTIONS IN EFFECT* NAME = WR , LINECNT * 57 •STATISTICS* SOURCE STATEMENTS = 5,PROGRAM SIZE o *STAT13T1C3» NO DIAGNOSTICS GENERATED NO CRRORS IN WR 440 MICHIGAN TERMINAL SY3TLM FORTRAN G(11336) WRO 10-22-T5 0001 0002 0003 OC01 0005 SUBROUTINE WPD(D,M,LD) REAL D(M) wRITE(LO) D RETURN END • OPTIONS IN EFFECT* ID,EBCDIC,SOURCE,NOLI ST,NODECK,LOAD,NOMAP •OPTioNS IM EFFECT* NAME a wRD , LINECNT « 57 •STATISTICS* SOURCE STATEMENTS = 5,PROGRAM SIZE o •STATISTICS* NO DIAGNOSTICS GENERATED NO ERRORS IN WRD 38B NO STATEMENTS FLAGGED IN THE ABOVE COMPILATIONS', EXECUTION TERMINATED 13IG lOllRJll PAGE P001 770.000 771.000 772.000 773.000 771.000 lOllRHl PAGE P001 775.000 776.000 777.000 778.000 779.000 10ll9t11 PAGE P001 780.000 781.000 782.000 783.000 781.000 152 Appendix 11. List of Equipment Used Instrument Barocel Pressure Sensor Type 511 (10mm Hg = 10 Volts) Barocel Electronic Manometer Type 1018B Barocel Signal Conditioner Type 1015 Disa Digital Voltmeter Type 55D31 Digitec Digital Voltmeter Model 2780 Leeds & Northrup Microvolt Indicating Amplifier Model 9835-B Druck pressure transducer Model PDCR-22 Description pressure transducer for pitotstatic tube measurements 4-1/2 digit voltmeter for windspeed from pitotstatic tube. amplifier for Barocel' pressure transducers. 3- 1/2 digit voltmeter for windtunnel balance measure ments . 4- 1/2 digit voltmeter for windtunnel balance measure ments and Scanivalve pressure measurements, strain gauge amplifier for Aerolab windtunnel balance. for Scanivalve airfoil surface pressure measurements 154 157 160 161 -4- i ' 1 •14-4 1 1 i i_! I 1 i 1 1 l 1 ! i 1 I 1 1 ! i 1 1 j 1 1 i a Figure 5 2 Comparison of airfoil pressure ! 1 i 1 1 —l—:— i 1 1 ! ! ! 1 ! 1 i i 1 1 ] i ! : coeff lnifitir.s: Theory 1 1 i 1 i i i 1 1 ! 1 ! I i 1 1 > i i 1 i 1 1 > I 1 .1 _L 1 1 l : ! ! I 1 j TTT 1 > i 1 i i 11 i i 1 •< l l 1 1 1 1 ! i ! ! 1 1 I 1 I ! i i 1 l L..LM i j 1 1 ! 1 1 1 i j i 1 i 1 ! 1 1 j j i -j- i i i r i 1 1 I ! ! 1 I ! 1 > 1 i ( i ! 1 1 I i i 1 i 1 i i i I I ! ; j I 1 i I t ! 1 [ t * i I 1 i 1 '< 1 l i 1 i 4- 1 1 1 1 I 1 •H+ j j ! ! 1 1 j ' 1 -Tr 1 ! i i 1 1 1 1 1 ! ! i i i i i ...1 1 1. I i 1 i ! 1 1 1 1 i i i < i I 1 1 1 i 1 1 Ml 1 ! i 1 ! ! i -}- 1 1 i i i I 1 1 i ! ! 1 1 ! i 1 ! 1 1 1 1 ! 1 I ' ! 1 ! 1 i 1 i 1 I i t i i • 1 1 1 i ! 1 • ! ! '• 1 -H 1 - ! 1 _ . i H | i P 4-- 1 1 4 I j 1 i |! 4- 1 < X f f 4-4-- / • 4 i -\ i— I 4 - / c ¥- \ Hi 1 » 1 1 1 _ C L 111 1 1 i { 1 !' i \ HS -WW-—\-{& "-I- 4-f 1 •j- 1 C11 \ / -MM i. ...v J . 1 i i M i i i i f>°A at \- 1 I i 1 'v- -f j i 1 i i Qt/v K A. i i ! V P c r 1 1 1 -i.. i -H i 4 1 I ttl (TTT ~1 R" r- i 1 ill i 31! 1 I -44- 1 r> y a [ hfl Ii- n 6 \ t ! i i. >t _ ii.- Mil B | M i J> *T a i \ j i, i 1 1 ret r — - i Ml.! 1 1 1 -trb r-- •ff H-TU t i 1 i i i 1 i i i 1 j i'L 1 j 1 \\ 1 i i M i i i ill 1 1 1 I ) ! H i l j i 1 i 1 1  ! 1 I ! ! 1 1 I I i i ~]—j— i ,'„. i ~4"L V "1 i -j-i 1 . _ i i 1 I . i r T I i 1 I i i i i ( • i i — T~ 1 1 1 l I 1 1 >^ I 1 ! i i t! i } i 1 ! " i 1 i i V j 11 I | i 1 ! SN I— -f- ! 1 -p s i -f-- I i i r! !, r" ! i I I i i i I ! >*. i - I -J 1 i i ! I i 1 1 i i ! i 1 i 1 i i i 1 1 1 i 1 ! ! i 1 —i—\— —j— 1 i 1 : i 1 i i • j I L 1 f - i 4—|- -—j- 1 i i 1 i i i i —1~ i • i i —r -1 i 1 i i i i i i -4-r 1 I- i • I i • | -44- l II 1 i 1 i I 1 1 \ 1 i I i i 1 t I i 1 -4" i T" i |" i i —r— 1 i 1 | i ! 1 i i i i 4- i I i TT"..._I u- -i i 1 I I M 1 1 MM MM i i i 1 1 | ; 1 i I i i i i 1 i -j- i i i _ 1 1 i i "Ti I l -14-4 ,._! -'-Li . 1 ~~t~ 1 I i i i 1 ^ ; j ' -4+ 1 ! 1 _ 1 ii ~F~TT i _ 4 i i TT s i i r 1 1 1 M '" , 1 • i i 1 i i i l i i i 1 I'M i i 1 1 i i 1 1 Mi i 1 1 1 l ! i l i i i i i I ! 1 ! I 1 i i l i -j— 1 | 1 —r~ -44 —r \! i i 1 ! 1 '. : -1 1 'i 1 1 11. MM ! I .1. I i i i I i i i : i i 1 1 ! 1 ! 1 '• 1 ' | —TT ...1 1 % I -r _i_ 1 •iii i t | i r r 1 ..... I- -!-!-j * \ 4- ._!_ . ! _ !__ 1 j 1 1 i i i :< 1 [ -j \ N 1 .1 I j -|- i -U I i (1 f: R! ~ A / f I 4 •1 1 1 i X >: i i h [ ( } ii rj"l'.L t .„L. § ft 1 ! i 1 >' 3 i . 1 ( "T ~rr _L. 1 ; I t 1 >' * i ft j t i 1 ! i i \ h _ _-|_ ! 1 - !. f [ Id i / \ A _ -j.. I • • ^ r~ . - ( • r ! . ' i , l / i) \ i 1 t \ L, OH i r' "\ • t-1 -1-I.. [ "] 1 _ 11 •" - 4- • -1 " « 1 1 L 1 ! 1 y i / r ty j 1 | J "J Q Oi 1 { i L'TJrL I 1 j •1_ I 1 I ; "T ] \ j .... j / & I j ~\~ " 4 \ ~\'~ ._.}. !... 1 r i u • i V _1_ t ) J t f Li ) } I l 1 f 1 N ""] ~| C f i I i j / 4 ] f « *• \ t' << i L V • j < u La e. m i. *. .-- - - * : i_ a . »• • - - - -- - - c B - - - - --c - - - _ - * « ? - t- C t ,^ m ] I * > 1 <* « s T ! i < t ri i » i •! i i - t / c i: ( ft >v ( "I I J -1 i. oc U i 1 1 ,.;.L fh » j I ! i > 1 I 'ttt «* "~r I • I , i _ r ! 1 j i ! r ... c V 1 I £ A -|_!_ : ! - - il * e B - - J L': U ... _ - - - --- -Li. ... t sua c «< u 12S3S1 _l_ • -I.J.-r i ; i rt i _i_ ~j_ el "/ "1" L I ) ""!' ' ! i • i ! _ Variation of along for - -- -tigure • J pressure coeiiicient a stidiynt correction uuunucix y ---±t a two -•axmensxonax axrioij. wx tn zero f rneoxy - _ - -16.3m Figure 6.1 U.B.C. Mechanical Engineering low-speed closed-circuit windtunnel Figure 6.2 Variation of mean windspeed in two-dimensional testsection insert on vertical pitotstatic traverse CTi 166 Figure 6.4 Effect of endplate loadings for two-dimensional airfoil tests 167 168 Figure 6.6 Error bars for the measured airfoil rHi1 I i lift coefficients 169 -_LLL -Li-Figure 6.7 Variation of measured airfoil lift coefficients on three consecutive runs I'M ; 1 I M i J- .» > U 111 -i 1—1 • ' ! ! I_L < > ! ! ! ! Ml fHT-H i F-H Lr iij-tiJ4^4t:TT £ i i Mini'iili'ilp Ti.l: 4H-TH+H-4T-4---hr+h-4+f -r-^-H-^ =fT+ "! h4HTt+hrHTSSI 1 TfcrTjSH4-_u^ -f—r-r-M4 444^-44—- r- - !4- -L-uL. _LI 4 i I ' M 1 i I Mi ' i 4^1:44^1^4-! ff-pTZMTirxtizzz^ ^^44--g&4^g^^pfe4^-1 ' 1 . | | 1 .'-V4- -r 1 I 1 1 1 1 1 / " ~ f-Ui-U-Ui- 1 1 1 r4--U... 1 ' M /I "ITT 44-I rf-4-r 1 + 1 4-TT-4-4 4 / - = MI 1 i 1 1 f-n 1 i/|Ar—1—h T4ri^-4MX^4-r---T:T---T^ :?=5z:=:: 170 i ! .' Mil ! 1 ; 11; III " 1 J M J M i M'T lii -T 1 ' 11 1 i r. - -----MI T-tL 1" 1 d i / -A4C%- ^ -i /—' TT hr-TTTT—TMH--tt/1 " -L-Trh-j u1 d± l~ ES':--i4Zx^: V 8(^%[:AR:: _,./.. LJ-,... " 11 1 T" M L __4|_i_ MMJ L-4-L44 ^~ T i [ 1 H-1 Mi _i I ' ' "T -, S3:—| 1 - ' ; 1 1 -4- M ' h . 1 i ' -T i_ 1 L 1 _I_| X Jx:±4M--ExT-I^--rr---NT_-1 i 1 ___i --,11 , , 1 M i_ _i_ i ^-^-^1:1 , 1 j 1 izbfa: i_J L," T 1 i -J 1 pOJ j L.1-T 1! 1 11 1—1 1 i 1 i 11 1 1 1 r -n 1 | 1 —J—- ; | • 1 -L-i 1—„ M—/ I 1 1 1 U | | 1 ul -z+pz^zzPzz^zzxz^zT-^-H— 4-r rL-ff-J 1 1 —i- ~i — : 1 ! : '"• ' —r- -, ^"i-T-^r^ MI dG TT-T: 4^44^ I , I _ MIM ' • • -J i / MIM I"1"- Mi 4_ X X • 7 Mi i 1 -I-l iF ! M . _l_ ' - -4H _ Z _j_ i "TT" j_ ~ Ui. _ i i A / Ii • - •• -Hr , _j_ u TT 4 -^X—^-^-L-^-^-M-n- rX:^CT#LX-^:_tt:_i MM -4- !• • 1 i i i / i i i M M --H H—r—r ±H- tRT drxx d+q- £z± zzrzr ±44-1:1-4^ TfexE1?1^ i H jff-iU-4f4^^ ; 1111 i!1 111 11 -144 U —j U_LM_.L. ..I...I '/ 1 Ml 1 M -: 1 _]_ _ M /L •  iii M~ . L. 4. 1 1/1  1 i 1 1 -Ml MM iii' imi/i 1 •• i M 1TT rrr—rrr 11 MI 11 it/ M  1 MI -I----h —'Mr M  _j 1 1/ M 1  ir- M.I -J-Ur—i-1—M—L ii _,_ 1 'gi i M . 1 1 _i •- -1 MI 1 -r' I T" L-i—4-- 4- Mm< MI- _ M 1 x TTTT 1 1 -'-xr x " M 1 0 0 1 1 ' ' ' ' 11 1 > ' 1 M r , s - -i-L, ~rrH— T±T4-xi+imAoMHTtT^ —H^e4Hff U! TUl—1 - 1. |T' 1 1,' U4-. Mil .... M - I 11 I-l 1 -H- h"f-4-rT-TTf4^Lr-444L-4^_4p TTT---j-j-H-- Figure 7.1 Variation of airfoil lift cc ' - ~r x !_J 1 , LJ L t- —L '  1 I -Li- .4-X±EVX:XX:4±I4±-L--TX ! TT 1 1 -j- •• 1 11 - - Ti X 1 TT T-^-nUr t^~vf 1 1 ifff,-T-E 1 1 M " :x:x 4tTFiT±::: _ _ _ TT 4-1 1 X ~ TT 1 j • - ~M 1 i 1 4—J J ^-A^j XXXT^TT-T— H r—i L +J Lp-lf-L-^J 4-X-T1 1 ,1 ' TTT ± r~-^rff-^rr-rL-. 1 T ! Tfi T" "f^f M . . | I Ml 1 I|J?S l M' 1 -H—rr 44-4- -H+4- 44--^ -TTT--4±+ri r44dio!4"| H 1 H p - 4- 1 - Tj-rC^r-M., 1 . LT. H H4rrn—r T Izzzzzz. >efficient with r slotted-wall open-area ratio: Experiment 171 172 on ratio of lift-curve slopes for dlX' NACA-0015 airfoil j± — H~h~r 173 Figure 7.4 Effect of airfoil size on lift-curve slope for NACA-0015 airfoil: Experiment 174 1 I.U ! ! M i i . i MM ! -U_! .L MM 1 ! i 1 ! I t j i 1 i | i 1 ! | . i , . , .... r-7- 1 —r~ ! i i i i i _} 1 l _) j i 1 1 MM j [ i 1 i' I • i ! \ | 1 i i i • ; > ! 1 i I i 1 1 M i i i i ; i 1 ! 1 i l i ! 1 1 1 1 i ! I i M I ! 1 ! 1 i - _|_L i _ L. 1 ! | 1 1 1 1 ' 1 1 1 \ i I i i 1 I I 1 1 1 1 MM ! j 1 i 1 i I I t i ; . MM- | i ! j 1 Ll L i i i i > 1 MM' 1 1 i M MM I i 1 1 1 l l 1 i j • i 1 ', Mil ill! Ml. i 1 i I i 1 1 1! j ! i : i 1 i j i 1 1 i 1 1 1 ill! 1 ! ' ! - -4" t-j 1 U-ll i i i 1 • 1 i "] 1 ! | 1 1 -Iii ! I ' 1 i 1 l L i 1 i 1 rnr 1 i ! 1 i I i i 1 I I j i 1 Mil MM i r i 1 i lili 1 : 1 1 i i i j 1 i i j | 1 ! " iT 1 1 ! I .1 i Ml! i i i MM \ - i 1 1 i MM 1 i 1 1 1 i i 1 1 1 l i j i i J i 1 1 1 i 4. L' J j._ _L i Li Ll 1 1 i 1 1 1 | j 1 1 1 1 -1 i 1 T - P-\t ti~ <r V \ r f{ "V 1-F i i4. •f 1 r 4- 1 I 1 1 ! i 1 • • - i i l-y— MM -1 V-i U ! i i • 1 ~'~ -!- 1 1 —!— 1 i. t 1 —I— i I i ! 1 I 1 1 ' i ( C 1 I j _L 1 1 1 1 I \ i I 1 i i | i 1 1 1 i (A _ t I r V j t I-1* 1 j 1 1 1 1 1 . ! I 1 i USL ta *" J: X Jj i 1. ! ! 1 1 1 i 1 | j j 1 1 i 1 I i * 1 1 1 1 1 1 1 1 1 | 1 1 1 1 ; i i i i i I 1 1 1 1 i 1 i ! 1 l 1 i ; i . j I 1 1 i I 1 1 1 | ! i i 1 1 1 1 1 1 1 1 i 1 i j 1 i 1 i 1 ! | | I . 1 1 1 1 I 1 1 j 1 1 i 1 L i —1— I 1 1 + -t- s r< -w i ml rS lh f <- 1 -H-- 1 1 1 —r-| 1 1 J_"L MM fl J _ 1 111 i , VI 1 f 1 1 1 5r &\ -}— 1 1 ! 1 1 ! 1 1 1 i i 1 1 1 (j 1 I 1 1 1 I 1 pu 1 t 1 1 i 1 1 1 t 1 1 1 i 1 1 i 1 \ l •w-u I 1 1 1 I 1 1 > — '*! 1 O ! i 1 1 1 f 1 ! i 1 >*= t O j 1 .1 1 1 I e a. •* 1 X n I -r. 1 j I ! ' 1 i 1 1 l 1 1 - C. 1 .i.l.U ( j | 1 1 l 1 I i ! —(— 1 1 1 | 1 | r .).. 1 l 1 ! i 1 i 1 I 1 I i i 1 j 1 i 1 1 1 1 | 1 v r- i I 1 / 1 c y i i >— \ l 1 1 I | j— i 1 i 1 i i i 1 i j | 1 1 1 1 1 1 1 i 1 l 4 1 i i 1 1 ; j ' 1 Y I ) ! u ! n 1 1 i u 1 L, i 1 1 1 | i " i i • 1 t ! i 1 i i 1 i i j 1 1 1 1 I i I 1 1 1 1 i 1 1 1 i 1 i 1 i 1 1 1 i 1 i i 1 i 1 | 1 I 1 1 [ . 11 3 1 1 1 i JUJU I 1 1 i i .5 s, 1 ! 1 1 1 1 i i i ! i . 1 1 1 | 1 1 r i 1 ! 1 i s >. I 1 1 1 i i 1 1 1 | i ' i 1 1 1 j 1 1 i i I 1 I 1 1 1 I 1 •N - 1 1 i I 1 1 I 1 i i i 1 M "> 1 A I ! 1 1 ; 1 • 1 I i "I-1 1 t • 1 I | , — * _l_ 1 Olf1 | 1 1 1 i 1 1 «»» _ -IM 1 "** /a 1 t 1 i 1 1 sL 1 1 J l_L.i Mi i I I 1 _ 1 t~ I I 1 3 - -j— .7!** 1 ! i • r % , i. --. "V . I t 1 — rf\ r. 1 1 •< i I 1 ; . i , -i i llii Mil 1 1 1 j i I 1 1 1 1 -' J _ I 1 i I 1 1 1 1 i ' >v - tJVL/ AXI \ 1 1 j 1 i 1 1 1 1 i I | 1 1 1 1 1 I 1 1 i i J— i 1 1 ! i i [ i 1 I i ; I i I ; i i i 1 1 1 t i i 1 1 1 , i 1 ! I 1 1 i i I ! 1 1 L i i I 1 1 1 ! i 1 1 I i 1 1 1 1 i i 1 1 1 i 1 1 1 1 1 I i 1 1 . t— 1 . 1 i ! i 1 1 i 1 1 1 1 1 i 1 1 1 1 1 1 i ! ! i 1 1 1 i 1 ! 1 1 1 1 1 1 i I i 1 -4 p. 1 1 1 1 i I 1 1 1 ! 1 1 I 1 \ 2 \ > I 1 1 1 1 1 1 ' i I I I | 1 . 4 1 1 1 i 1 i ; 1 1 ! 1 ! | 1 1 t i i 1 1 |. 1 1 i i I 1 1 1 1 ! i 1 i i. i 1 1 1 i i 1 1 1 : ! 1 i ; i i 1 ; j 1 1 1 1 1 i 1 I 1 1 i 1 ! i 1 i | 1 i 1 i ...... 1 | j 1 1 i i 1 1 1 | i 1 ! 1 i i 1 ! 1 i 1 i 1 i 1 i ; 1 ! i i |. I i i 1 | i 1 1 1 I 1 ! 1 I 1 | 1 1 i ( 1 i 1 1 i r I ! 1 1 I i I i 1 i I ! ! 1 j i i i I I lilt i i i I >< F 1 1 I 1 ! : l I 1 1 1 1 i | 1 1 1 1 !.l i i i /L|4 | I i | i i i 1 j_ i 1 i 1 ! i 1 ! 1 l H 1 1 j 1 ~r 1 i 1 i i 1 i | • •1 1 i ! 1 ! i i IT I 1 i j ! 1 1 ! t i i i 1 ! 1 1 I •• '• i i i i ' 'M i I'll i . i i 1 i I i 1 i«LM M II i ! i i ! t i i • i i 1 i i MM i 1 i C 1 1 i i 1 (A AS l ! ! r M I A ft 1 1 | i-U-TL f ! I 1 i J 1 i 1 I Zi 1 IV. o 1 j i irXJ- 1 + 1 I i I I . 1 i 1 I Mill I I 1 i 1 I 1 "M-rr i I i i i i • 1 MM 1 ! 1 1 i 1 ! MUM 1 ! i i i 1 1 I Mil 1 j 1 i l | 1 1 1 1 1 : 1 i 1 1 ; 1 ! i 1 1 ' I 1 i i , 1 i i i 1 | 1 n 1 1 I I i ! I i i • i I 1 1 1 i p | | 1 i 1 1 i i _J I ! 1 1 i 1 i 1 1 I 1 i i ! i i ! 1 I [ I 1 1 i i 1 1 ; ; i 1 I 1 L _ 1 1 1 1 1 ! I 1 I 1 i 1 i i ! — t j- 1 i I i i 1 ! "i " 1 i ; i : ! 1 i ! 1 1 .1 T MM 4 i TT i 1 i 1 i 1 -4-1 44- i i i 44-44 j M 1 1 I ! -1444- i i i MM 1 | i i 1 -4-U 1 MM . 1 i. i 444- I Figure 7.5 Effect of airfoil size on lift-curve slope for Clark-Y airfoil: Experiment 175 177 179 ill _i i j lH-hi- ...LA I. ; | { i zhLtT _ — _ - i ! 1 {—•—-f — T -i_LI_i_ MM 1 1 -J4f 1 i 1 1 ! 1 1 1 i '!" 1 MM i j_ i t i M . f r i -i— i h f _i | " i~ 1 M M | 1 MM 1 1 : T ~4 — — 1 r T 1 1 ! 1 T 1 1 1 ' 1 i I 1, 1 1 MM _! 1 "i 1 i ! ^ I 1 1 i i i ; 1 j ! ' 1 1 MM .L ] j 1 1 1 i 1 1 ! 1 I i 1 1 ! i ; ! ! 1 1 1 1 i i i i ! ! 1 111 _J_I - 1 1 ! 1 1 1 1 i 1 i i I 1 1 1 1 1 1 1 1 I l 1 t 1 1 1 •• 1 1 1 — 4 !1 1 : 1 1 1 ; MM _U ' i T: rr I ! | 1 ' 1 -L 1 i r ! 1 1 1 1 | 1 i i - -j-i_i I 1 I 1 1 I 4 1 i 1 4 1 1 Ti 1 i i I 1 i i _Lj 1 1; ... 4- M-- I ! ! 1 ] 1 I i 1 1 i 1 i | 1 i 1 1 I"! 1 ! [ ! 1 ! I 1 . t I — 1 1 i I I i i 1 1 1 1 1 1 1 ! 1 MM 1 rr 1 1 : 1 ' i i V X 1 t i i .1 1 i 1 \ i 1 1 i 1 i 1 1 1 M_ 1 i I 1 i 1 1 1 1 1 1 : ; ! i id: i 1 1 | i i i I ! 1 -}- 1 j i —r 1 1 i I I ! 1 1 |-M 1 1 ' 1 1 | | | T 1 1 ~i—p" 1 ! ! i I I 1 1 1 I ; i ! i ! 1 i \ 1 i 1 ! 1 1 1 1 1 4+44 L 4 1 1 1 1 ! n 1 1 F i ~I 1 1 1 T i 1 i-S w < i . 1 M 1 I J 1.1 1 1 Li 1 I 1 1 i ! 1 1 v t\ _ A. Ch 1 /\ iv J 1 ! 1 1 1 i 1 —:—I—I— r 1 \ /• V SCT: I \J M i rr u 1 1 1 i 1 i 1 1 1 I i !_ l-H-i • hr 1 111 4TT 1 1 i I 1 —:—ri 1 M 1 T—1— — ! itf! | P h—4 •~W •Jfo.t - 1 1 1 1 1 i 1 1 1 I r -4-44 1 I] I -t-1 1 V i 1 444- 1 ~T 1 1 r_ t 1 hH- 1 1 c T Urf 1 1 1 i 1 1 i 1 1 I 1 _l_ 1 1 1 -4-r • 1 i T • 1 1 ~i—1—r- 1 I 1 t • \ 1 1 pi Ii ti 1 iitrr — ,.l | t 1 ! 1 1 I ! 1 t 1 1 r y u \ U TT: s 1 "1— 1 1 1 I 4—j— j 4ri i -4 I 1 1 1 1 1 1 ' I ''• j 1 —5-1V 1 1 r I 1 1 ! —1- f i 1 i L-U 1 \ i \* t ! 1 ! -~J~ t 1-i 1 1 44 I 1\ T* I i I i I 1 7- i l_ il 1 1 1 1 1 1 -p-L i i 44- 1 1 •fiV •• 1 • n0r in *4 -r IT 1— l i i T 1 J CM \ 1 MM 1X2 LI -a LV r J -V JO 1 1 IM 1 1 1 I \ \ i l 1 1 Ml. 1 1 1 1 1 1 MM 1 -i-i- 11 -t i -t =41 rr-11 1 !n 44 n 4r-i ey f-t Y~ m r+~ 1 1 . . ., —t" J ii 1 1 I 1 1 r r LI ' 1 J J L ti -+ 1 1 1 i 1 \ - ii^-1 1 1 t 1 III 1 1 1 ; 1 \ y 1 1 1 1 \ 1 ! ! i ~j Tt-- V 1 1 1 1 j 1 i 1 44—4- 1 i 1 J -V4-— 1 1 I 1 1 i • —— 1 1 1 1 1 i \ 1 ! 1 4 1 1 1 4 1 i 1 1 I i 1 1 1 1 i i t 1 1 1 \l 1 1 ... 1 | 1 1 1 .1 1 I i ' 1 1 | 1 1 1 1 1 4 ! 1 i | t -V 1 1 1 .. 1 1 1 1 i !" v\ —L 1 1 I -)-| 1 rv \ ! i j -t— 1 1 1 i i> ( 1 1 1 1 | i i 1 1 1 1 | l 1 1 1 1 1 i 1 > * - -L. 1 | 1 i 1 1 -4 —r — 4 I 1 1 j s -> i 1 j 1 1 l 1 1 1 1 1 t 1 1 1 I >wS4_ ! 1 1 1 1 J_ ! 1 1 ^ I »-4 1 1 ! 1 1 1 1 4" 1 -- 1 i I 1 | 1 1 « 1 1 -j- 1 1 1 I -- 1 1 1 i i 1 i 1 1 i ! 1 1 1 "IT 1 i I 1 1 i ! j | J i 1 i 1 T\ 1 TT T i 1 1 1 ! 1 1 i 1 1 1 j j 1 1 i 1 | 1 1 11 1 •KI 1 1 1 1 1 i 1 1 | 1 1 1 1 1 1 1 1 1 1 • 1 1 - ! 1 1 1 1 " ! 1. i 1 1 1 1 ^T 4- 1 1S> T~P 1 44 1 1 1" 1 —I— 4 1 —4 1 41- 1 1 1 1 i : 1 I 1 1 11 1 1 1 N 1 i 1 4T i i i j i -f .!.. ..!., ; 1 1 1 1 1 i 1 |. -4. i 1 -i—ra 1 i > i 1 1 ! | t 1 1 t 1 4-44 1 j 1 1 1 1 V «S II*" 444-- 1 -rV 1 1 1 1 1 i ...| ; 1 1 1 1 1 1 J+t> ^& 1 i 1 n 1 i i i -1" r 1 I 1 1 1 t -4 1 i 1 1 1 1 -4 | 1 ! " ; J__ "S \m —— .{. | I u 44 i i 1 -rf tn "7*- i -~ -1= 1 11 1 1 1 , v „,„,... II J i_ 1 1 I i 4 i 1 1 i i J 4"^ Ti 1 11 11 4-1 1 1 1 1 1 1 1 1 1 1 1 1 1 > 1 Ti i 1 1 i ' ! i -14— i i. *> »• 1 I 1 : 1 __. 1 1 1 MM nin •|" I i ! 1 1 I i 1 1 j | | 1 1 1 i i 1 M 1 v ! 1 ! *> i i 1 ; 1 1 ; 1 i ! 1' ! i i i r 1 1 ! i i 1 j •1 • 1 1 1 j ] IM "11 1 1 ! ! i \ ! i 1 1 i 1 : 1 ; 1 1 1 1 i I I 11 Mi 1 ! 1 - & i -j .... ; ' ! ' MM 1 1 i 1 1 1 • 1 1 III! 44-44 i 1 1 1 1 w i 1 i 11 i 1 I 1 1 1 -7- 1 ! ! I 1 1 T+ 1 I 1 1 1 i 1 1 l 1 j ! ! 1 11 i 1 ! 1 ; I j 11 1 1 | 1 I I 1 TI 1 ' i ! 1 lii i ' i 1 i 1 ! 1 "1 1 1 IT i I 1 1 ; 1 i 1 I i I -| i 1 1 I 1 1 I I ! i 1 1 1 ; i : 1 ! 1 i 1 1 [_ 1 I 1 I 1 1 i J_l 1 : 1 1 'i I i if i , i ! i 1 1 1 1 1 \ ! i 1 1 i j ! 1 1 1 I 1 1 I 1 1  1 -J—J— 4 1 !4f i i i TJ i J +H- 4- 1 1 1 1 1 1 1 i 1 1 i 1 1 1 I ! i 1 1 1 1 4 1 1 —i— 1 1 • T~| 1 -. L.ij !.. ill .1. i_ 1 Figure 8.1 Effect of reduced circulation on airfoil pressure coefficient: Theory (Appendix 8) Figure 8.2 Modification of airfoil profile to reduce theoretical circulation to measured value: Theory (Appendix 9) 03 O 181 (a) Theory Figure 8.4 Figure 8.5 The shear layer, in the. plenum chamber surrounding the slotted wall CO 184 Figure 8.6 Effect of different types of wall boundaries on ratio of lift coefficients: Theory 00 VJ1 186 4 1 T~ T" -!:: III! M 1 i TTTT"' rr i r -> M i L ;~T -- ...... I'M :4J ! 1 . j r-i 11 !' I ! 1 I I 1 1 11 I_I. '1 • 'i MM 4 , r ! 1 i i i ill! 1 M 1 r:TLK ! I i 1 11 11 11 11 i i -fl ! •• i ; _! Mil 'Ml 11 i i 11 TT i i i r i i TT 4 ! 1 1 ! i : i i I I 1 ! 1 i i ! ; 1 III! MM, Mil. MM +t-h-i i i 1 I 1 1 1 1 ' 1 I I i 1 ft-i i i ! 1 i 1 i i i ! 1 i I 1 1 i I 1 i ! j 1 1 r4 l 1 1 1 1 1 1 I i -TTT 1 !'" i / i f-j—1 1 i i i 1 ! -H ! 1 1 I i 1 4 i i ! 1 < 1 1 1 ! 1 1 1 1 -r-1 1 1 ! ! te Ml rr 1 1 1 ! 1 ! ! : i ! 44 1 1 ! 1 1 1 1 1 i i ! i ; 1 : i 1 1 ' 1 1 1 J-4 1 \ 1 1 1 1 ! 4M ! I 1 i i i i i i ' i i i i i i -rC ' 1 I i 1 ! i I Jk 11 WT i i t 1 1 1 | 1 ! 1 1 i I 1 +41 i i 111 Ml ! 1 I 1 ! 1 44 L.....1 i i A i i i i i : I i I i i i < tit i i i i i i i I 1 i 1 ! 1 1 1 j 1 I 1^ i L L i I i 4r , 1 ! 1 1 1 1 1 1 r4 11 11 i i ! 1J i S( 1 1 )°/ 1 i i i i i i i TA i R i i i i i i 44 i i i t X_ 1 ' 1 1 1 1 -4 i i 1 i i i i ' i M 1 1 i i i i _r i i i 4 1 r i i 4-ir fXf -hi k 11 i 1 1 i 1 t ; ' i i 1 1 1 1 1 tf 1 1 1 rf 1 -j— J f1 i -4 i i I i 1 'I i • i • i-i i •• .... 4 i !.-i i 1 1 i i 4 4-l ! I i 1 1 1 1 1 i j 44 } I ~r i 1 i I 1 ...!_ I 1 1 1 i —r 1 1 1 4 i • i i 1 Xt €T i . i i i 1 4-i ! 1 1 f [4f n i . i i T---+ i i i i i i i i i i i i i 1 1 1 -4-i i 1 I i i i ! 1 1 i l -4-1-4T i i i i ! ! 1 -444 1 i i i ! 1 1 1 1 I 1 1 j ir i i i — "A // 111 111 111 —L 1 1 ' f i 1 1 r - 1 i v 4+ 1 A // i i 4-r^E >^ i i --a V- 1 ftp a m 4^ 1 -I I 1 i 1 1 1 1 I i 1 m i I 1 1 4S —I—j Js« m _—_ I 1 I I I 1 44- 1 1 i i i i I 1 =u ::| 1 i i I i —.—L. ( 4— i— -, | | I 1 1 i i ti- i t 1 [— i — i ! i i 1 —|n Q-8€ )-* f f -i,.i 1 [ i i 1 i fr --)-1 I 1 1 i 1 1 i t • i r/—-frr\-4 t l "f l i . .j... ! 1 I I i i 1 1 i j— i i f L4 —r —1— i i i 1 1 i i i — 1 \ 1 ! | i I 1 | i f ! | I | i i • ! 1 i i i I 11 _|— i i i i 11 ! T 1 1 I 1 i 4-i i I I i I ! I 1 T44- 11 i i X. 1. I 1 11 TTT--4- i i *w i i i • i Ml' i i : T" —lT\ _XJ i ._ i i u _ 11 i i i 1 I i i 1111 i ! 1 1 | i HT- 1- 1 I i 1 ! M i I i.J ! : i ! 1 1 i I r - JU 1 i \f\ Ill" 1 1 i TTTT-Ml! f\ | 1 -MM i i i • -r\ t 54r -T 11 jr\ rrr. r 4rh rrr- - r "1 TTT i 1 1 10 1 I 1 I i 1 Mil WT- I 1 1 1 i y i . 1 1 11 1 ! i 1 m ; ' 1 4~ I -• i i I i ! 1 1 i —!—1—1 1 ! 1 1 I 1 MM MM 1 I ! !'•• 'ill 1 1 1 t 1 i : 1 I i i till !—j i —t- 1 1 i i 1 i i i11 1 —! i I M ' I • i 1 1 1 1 1 1 1 1 ' i 1 .Ml MM MM ill! MM 1 i i 1 1 i J i i 1 -— ! 1 ! ) 1 1 : i 'ill I'M IM! 1 ' —pr 1 I i i i i i 1 i i i ! 1 i i i 1 i L.| 1 1 1 1 MM MM M 1 | 'Ml i Figure 8.8 Effect on airfoil lift coefficients of assumed pressure coefficients on a streamline representing the plenum shear layer: Theory Figure Al.l Geometry for integration of a point source 188 Pfr.O.z) Axi-symmetric (b) *~ z Figure A5.1 The two-dimensional nozzle insert E-2 t h + S Constant-pressure boundary H il I 11)11 I I I Figure A6.1 A lifting vortex between a solid, a slotted, and a constant pressure boundary: Theory u> OJ Q Q + + ro cr cr I a~ ro cr I Q Q -a--o-CT *-II Q ro cr (DI <T> OJ o + cr OJ Q + cr W 1 <±) a + cr Q + cr Fiaure A6.2 The image system for a lifting vortex between a solid and a constant pressure boundary: Theory NSLAT * NSPS NSOLI NU2 NUI U NTEL ^ NWLI NLI JH f i < r i i i > i ' i ' f » > > 111 r NWL2-NL2 NL3 NL4 i i i i r'lMiVi * ' '*' > i > > • i i i ) i r } i i Ji / i i it f > i' / i/t/iii / >Hi Figure A10.1 Notation for the computer program of Appendix 10 192 Plate lb. The octagonal testsection in the windtunnel 193 Plate 3. The wall slats in the side wall frame 195 Plate 6. The 616mm NACA-0015 airfoil in the testsection. 196 Table 1. Airfoil profile coordinates. 14% Clark-Y NACA-0015 Joukowsky 11% X Y U Y L . X + Y X : y U X Y L 0. 00 4. 19 4. 19 0. 00 0. 00 0. 00 3. 92 0. 05 3. 53 0. 32 5. 15 3. 15 0. 40 1. 37 0. 02 4. 16 0. 40 2. 90 0. 96 6. 15 2. 49 1. 00 2. 13 0. 35 4. 89 1. 09 2. 31 1. 92 7. 24 1. 98 1. 90 2. 88 0. 97 5. 41 2. 11 1. 77 . 3. 20 8. 35 1. 54 3. 20 3. 65 1. 91 6. 28 3. 45 1. 29 4. 80 9. 35 1. 15 4. 80 4. 37 3. 15 6. 96 5. 14 0. 88 6. 72 10. 26 0. 84 6. 70 5. 02 4. 74 7. 62 7. 10 0. 55 8. 96 11. 14 0. 60 9. 00 5. 63 6. 63 8. 26 9. 41 0. 29 11. 52 11. 94 0. 38 11. 50 6. 15 8. 77 8. 85 11. 99 0. 12 14. 40 12. 65 0. 21 14. 40 6. 60 11. 17 9. 38 14. 87 0. 02 17. 60 13. 25 0. 09 17. 60 6. 97 13. 86 9. 85 18. 05 0. 00 21. 12 13. 70 0. 02 21. 10 7. 25 16. 76 10. 25 21. 45 0. 07 24. 96 13. 94 0. 00 25. 00 7. 43 19. 89 10. 58 25. 10 0. 20 29. 12 14. 00 0. 00 29. 10 7. 50 23. 24 10. 83 28. 95 0. 40 33. 60 13. 95 0. 00 33. 60 7. 47 26. 74 10. 99 33. 02 0. 65 38. 40 13. 74 0. 00 38. 40 7. 32 30. 44 11. 07 37. 25 0. 96 43. 52 13. 34 0. 00 43. 50 7. 07 34. 27 11. 06 41. 59 1. 29 48. 96 12. 73 0. 00 49. 00 6. 69 38. 22 10. 98 46. 06 1. 66 54. 72 11. 85 0. 00 54. 70 6. 22 42. 26 10. 80 50. 58 2. 04 60. 80 10. 80 0. 00 60. 80 5. 62 46. 39 10. 55 55. 12 2. 41 67. 20 9. 44 0. 00 67. 20 4. 91 51. 15 10. 19 59. 69 2. 77 73. 92 7. 83 0. 00 73. 90 4. 09 54. 75 9. 86 64. 19 3. 10 80. 96 5. 92 0. 00 81. 00 3. 14 58. 93 9. 43 68. 59 3. 40 88. 32 3. 86 0. 00 88. 30 2. 07 63. 05 8. 95 72. 86 3. 66 96. 00 1. 45 0. 00 96. 00 0. 84 67. 12 8. 44 76. 96 3. 86 100. 44 0. 00 0; 00 100. 59 0. 00 71. 09 7. 92 80. 83 4. 01 74. 94 7. 38 84. 43 4. 11 78. 62 6. 85 87. 73 4. 16 82. 12 6. 34 90. 71 4. 17 85. 37 5. 85 93. 29 4. 14 88. 40 5. 40 95. 50 4. 08 90. 71 5. 07 97. 27 4. 01 93. 29 4. 70 98. 61 3. 94 95. 50 4. 39 99. 50 3. 88 97. 27 4. 16 100. 0 3. 84 98. 61 4. 00 99. 50 3. 90 100. 0 3. 84 Table 1 cont1d. Main Airfoil 0.00 1.00 " 0.00 1.00 2.40 -1.10 2.50 3.61 -1.71 4.00 4.45 -2.10 7.00 5.65 -2.55 10.00 6.43 -2.92 15.00 7.19 -3.50 20.00 7.50 -3.97 25.00 7.60 -4.28 30.00 7.55 -4.46 35.00 7.43 -4.53 40.00 7.14 -4.43 45.00 6.80 -4.35 50.00 6.41 -4.17 55.00 6.00 -3.92 60.00 5.47 -3.65 65.00 4.95 -3.35 67.00 - -3.18 69.00 - -2.83 70.00 4.36 -2.51 71.00 - -1.98 72.32 - -1.02 74.57 - +0.67 75.00 3.78 77.82 - 2.30 80.00 3.08 2.67 82.70 2.64 2.64 196A NACA-23012 Flap X yu X YL 0.00 0.04 0.00 0.04 0.45 0.99 0.36 -0. 72 1.08 1.59 0.95 -1.00 2.11 2.27 1.74 -1.15 3.65 2.93 2.44 -1.21 5.17 3.33 3.44 -1.21 6.68 3.55 4.95 -1.15 7.69 3.57 6.45 -1.07 8.69 3.52 7.45 -1, 03 10.18 3. 32 8. 46 -0.99 12.66 2. 86 9.96 -0. 94 15.13 2.36 12.47 -0. 82 17.61 1. 85 14.98 -0.71 20.09 1. 35 17.49 -0.61 22.07 0.93 20.00 -0.46 24. 05 0.52 22.01 -0.33 25.54 0.21 24.02 -0.19 26.53 0.00 25.52 -0.08 26.53 0.00 Origin of flap coordinates is (78.87,-0.81) for 6=20°. Table 2. Free air airfoil coefficients: Theory. NACA-0015 a0 CL CM0 /. da 0 0.000 0.000 0.0000 0.1193 3 .365 .086 - .0050 .1229 5 .607 .143 - .0086 .1300 10 1.210 .282 - .0204 • 14% Clark-Y a0 CL CM0 CMc dCL da -6. 3 0.000 -0.087 •4 -0.087 0.1206 -3 .401 .012 - .0883 .1208 0 .763 .101 - .0901 .1203 5 1.362 .244 - .0965 198 Table 3. Airfoil and wall configurations examined theoretically. All solid walls are 4.88m long, with MWUl = MWU2 = 20, and NWL1 = NWL2 =40. The slotted wall is 2.44m long, composed of large (92mm) NACA-0015 slats with NSPS = 9. Airfoil is in the center of the testsection; NA = 50 for Clark-Y and NACA-0015; NA = 81 (46 main and 35 flap) for NACA-23012. Further notes are found at the end of this table. a CM0 CMc 4 1. Clark-Y airfoil a) Free air -8 -0.203 -0.137 -0.086 2 1.003 + .159 - .092 20 3.088 .603 - .169 b) Solid walls -8 -0.250 -0.150 -0.087 c/H = 0.53 -3 .444 + .014 - .097 2 1.140 .178 - .108 20 3.632 .711 - .197 c) 40% TSUSL -8 -0.270 -0.153 -0.086 NSLAT = 16 -3 + .376 -000 - .094 c/H = 0.53 + 2 1.012 .149 - .104 20 3.179 .610 -. .185 d) 60% TSUSL -8 -0.260 -0.151 -0.086 NSLAT =10 -3 + .377 + .003 - .092 c/H =0.53 + 2 .992 .148 - .100 20 2.986 .572 - .175 199 Table 3 (cont'd). a CM0 CMc 4 c/H Clark-Y (cont'd)(60%) 20 3.061 .579 - .186 0.66 70% TSUSL 20 2.929 0.570 -0 .162 0.25 NSLAT =8 20 2.907 . 562 - .164 .39 20 2.923 .560 - .170 .53 20 2.970 .563 - .180 .66 20 3.084 .573 - .198 .86 20 3.200 .587 - .213 1.0 -8 - .256 - .149 - .085 .53 -3 + .381 + .005 - .091 .53 + 2 .989 .149 - .098 .53 80% TSUSL -8 -0.250 -0.14.8 -0 .086 0.53 NSLAT =5 -3 + .388 + .007 - .090 .53 + 2 .994 .152 - .097 .53 20 2.888 .554 - .168 .53 NACA-0015 airfoil Free air -2 -0.243 -0.058 + 0 .003 + 8 + .970 + .227 - .015 15 1.803 .412 - .038 20 2.382 .530 - .065 Solid walls -2 -0.305 -0.068 + 0 .008 0.67 + 8 +1.223 + .272 - .034 .67 10 1.510 .333 - .045 .67 20 3.074 .675 .093 .67 + 3 .371 .087 - .006 .17 3 .388 .090 _ . 007 . 34 200 Table 3 (cont'd)-— a CM0 4 c/H 2. NACA-0015 (cont'd) b) Solid walls (cont'd) 3 .416 .094 - .010 .51 3 .453 .100 - .013 .67 3 . 546 .116 - .021 1.0 c) 40% TSUSL -2 -0.327 -0.072 +0.010 0. 67 NSLAT =16 + 3 + .364 + .081 - .010 . 67 8 1.039 . 227 - .033 .67 20 2.592 . 550 - .098 .67 d) 60% TSUSL- -2 -0.320 -0.070 +0.010 0. 67 NSLAT = 10 + 3 + .359 + .081 - .009 .67 8 1.006 .222 - .230 .67 20 2. 421 .514 - .092 .67 e) 70% TSUSL +3 0. 355 0. 084 -0.005 0.17 NSLAT =8 3 . 356 .083 - .006 .34 3 .358 .083 - .007 .51 3 . 361 .082 - .008 .67 3 .365 .082 - .009 1.0 f) 80% TSUSL -2 -0.311 -0.068 +0.010 0.67 NSLAT = 5 + 3 + .367 + .084 - .007 .67 8 1.000 .224 - .026: .67 20 2.335 . 500 - .086 .67 3. NACA-23012 airfoil a) Free air + 8 2.442 0. 320 -0.291 201 Table 3 (cont'd) a V CMc 4 c/H NACA-23012 (cont'd) 70% TSUSL +8 2.415 0. 308 -0.295 0.2 NSLAT =8 8 2. 305 .290 - .286 '.4.. . .... 8 2.318 .283 - .296 .6 8 2.296 . 280 - . 312 -8 8 2.442 .280 - .331 1.0 Clark-Y airfoil Compare C (C ) with CT (T), J-i p Li with NA = 110 • and a = 20 o L p CM0 4 cL(D i) Free air 3.117 0.548 -0.231 3.114 ii) Solid walls, c/H=.66 4.165 .744 - .298 3.742 Circulation on wall slats reduced by modifying slat profiles, NSPS =15, a = 20°, c/H = 0.66. k CL CMc 4 70% TSUSL 1.0 3.010 0.569 -0.184 NSLAT =8 .8 2.935 .556 - .178 • 7 2.610 .502 - .150 Shear layer representation . MSV = 2 0, a = 20°, c/H = 0.66, V. = /(l-C ). t p ' C P CL CM0 CMc 4 i) 60% TSUSL 0.0 2.420 0. 451 -0.155 NSLAT =10 -.12 2.686 .502 - .170 T.28 3. 305 . 572 - .187 202 Table 3 (cont'd) 4c) Clark-Y - Shear layer representation (cont'd). C P CMc 4 i) 60% TSUSL (conf d)-.35 3.188 .600 - .197 NSLAT = 10 -.44 3.390 .640 - .207 ii)70% TSUSL 0. 2. 321 .433 - .147 NSLAT = 8 -.12 2.591 .486 - .162 -.28 2.951 .558 - .180 -.35 3.101 .586 - .190 -.44 3. 308 .626 - .200 5. NACA-0015 airfoil with reduced circulation, a = 10°, solid walls; a reduced airfoil circulation determined. CT CM CM L Mo Mc _4 i) from measured lift (k = 0.741) 1.120 .335 +0.0549 ii) by modifying the profile (k=0.724) 1.120 .280 - .0003 Notes: The positions of the wall slats correspond exactly to those in the experimental setup. The slats are spaced uniformly, beginning with a slot opening at the upstream end on the sidewall. The number of large slats required for 40, 50, 60, 70 and 80% OAR is 16, 13, 10, 8, and 5 respectively. The effect of increasing the number of control points on the test airfoil is seen in 1(a) and 4(a)(i); of increasing the number of control points on the wall slats in 1(e) and 4(b) (k=l). 203 Table 4. Airfoil and endplate loadings. Joukowsky Re=.5(10)6 Solid Walls Ct CL C* Mc 4 Mc 4 c* CD -7 -.350 -.345 -.0719 -.0723 .0330 . 0194 -6 -.245 -.241 -.0727 -.0728 .0317 . 0180 -5 -.137 -.134 -.0728 -.0727 .0309 .0171 -4 -.025 -.024 -.0727 -.0723 .0308 . 0168 -3 .086 .086 -.0724 -.0717 .0314 .0172 -2 . 194 .192 -.0716 -.0706 .0323 ' .0179 -1 .299 .296 -.0712 -.0699 .0340 .0193 0 .408 .403 -.0705 -.0689 .0361 .0212 1 .512 .506 -.0693 -.0673 .0384 . 0233 2 .617 .610 -.0669 -.0645 .0410 . 0259 3 . .721 .713 -.0652 -.0625 .0441 . 0289 4 . 820 .811 -.0639 -.0607 .0482 . 0326 5 .917 .907 -.0628 -.0591 .0530 .0370 6 1. 009 .998 -.0617 -.0574 .0592 .0425 7 1.103 1. 091 -.0600 -.0551 . 0652 .0478 8 1.192 1.179 -.0581 -.0524 .0719 .0534 9 1.277 1.265 -.0557 -.0491 .0790 .0594 10 1. 340 1. 326 -.0536 -.0466 .0869 . 0672 11 1.382 1.368 -.0540 -.0465 .0972 .0773 12 1.408 1. 394 -.0620 -.0546 .1161 .0959 13 1.199 1.186 -.1611 -.1538 .2551 .2345 CL'C5c'CD loading on airfoil plus two endplates CMc'CD loading on airfoil only 4 Table 5. Windtunnel balance results -0.25-Clark-Y Pt = '.«5(lO)6 SOLID WALLS ALF CL CD CM0 CMC/'I -10'. -0'.3'l5 0'.0260 -0.0916 -.«»'. -0.251 0'.0?35 -o'. w.o -0.0907 -o'. 151 0'.0221 -0.133 r-0 . 0906 -T. -0'. 057 0'.0215 -0.107 -0.0910 -6'. O'.O'II 0'. 0 2 0 1 —0.080 -0.0908 -5". 0'. Ill C '. 0 2 0 7 -0'.053 -0.0911 -1'. o'.2'U 0'. 0205 -0.026 -0.0908 0'.313 0'.02() '1 0-002 -0.0908 "2r (>'.13S 0'.02?1 o'.032 -0.0871 o'.553 0'.0237 0.060 -0.0905 0'. (i'.665 0-.0256 0'.037 -0.0942 i'. 0.753 o'.0278 0'. 1 11 -0.0920 2'. 0-815 0'.0302 0. 110 -0.0905 i'. 0'.938 0'.0330 0'. 168 -0.0889 l'. T.030 0'. 0373 0.193 -0.0880 5'. '1'. 1 oa O'.03(i7 0'.2 18 -0.oias 6'. l'. 185 0'.0133 0.211 -0.0820 l'.256 o'.oi87 0.260 -0,0782 B'. l'.321 0'.0532 0-,286 -0.0736 r.3^6 0'.05R8 0.305 -0.07U 16'. r.'i27 0-0650 0.321 -0,0655 ir. r.'i35 0'.0716 0'.331 -0.0572 12'. 0'. 0788 0.333 -0.0513 13'. r.'i26 0'. 1 03'l 0.322 -0.0638 11'. f. 388 0-. 1218 0'.308 -0.0682 -Y airfoils CL ARKnY Rti = *.'15(10)6. 'I0ZSS+P TSU AI.F CL CO CMO CMC/1 10'. -o',330 0'. 0237 -0'. 187 ^•0.0963 -9' -0,237 0'. 0218 -0.160 -0'. 0916 -0'. I'll 0'.0195 -0-131 -0.0915 -7' -o'. 016 0'.01«9 -0'.107 -0.0938 -6' 0'. 0 •'! 8 0'.0177 -0.0 82 -0.0912 •*5' 0. I'll 0'. 0 17 6 -0-055 -0,0911 0'.2'I0 00 1 7 '1 -0.027 -0.0921 — > 0'. 5'40 0'.0176 0.000 -0.0933 -2' ()'.'I3 0 O". 01R5 0'.0 29 -0,0896 -i' 0'.533 0'. 020 1 0'.055 -0,09 16 o' 0'.6'I3 0'.0231 0,081 -0.0968 r. ()'.729 o'.0262 0.105 -0.0958 z'r 0^8 15 0'. 0 285 0.130 -0.0915 0',90l" o'.0315 0.155 -0,0933 0'.986 0-0316 0*. 181 -0.0911 5 » f.062 0'.0389 0.200 -0.0885 6 1.133 0'. 0'I26 0.227 -0,0852 7 1.201 0'.0'I62 0.2'19 -0,0826 8. l'.?68 0'. 0119 0-269 -0,0795 ?'. l'.323 0'.0550 0.288 -0,0761 10 * l'. 359 0', 0 6 0 6 0.30a -0. 06.85 11. l'.377 O'. 0675 0.316 -0 .0609 12 m r.562 0'. 0765 0.319 ^0,0589 13 • l'. 366 0'.09;i 3 0.309 -0 . 0636 11 • 1-333 0'. 1 162 0'.29« -0,0701 15. f.293 0'.1378 0'.280 -0,0718 CLARK-Y RC=.15(l"0)6 10%L5+P TSUSL ALK -10'. -< -8'. -7'. -6; -5'. -1*. -z: -?-'. -1. 0". 1'. 2\ 3 1 5 6 7 e 9 10 *h 12. 13'. 15. Cl, -(>'. 310 • 0'. 2 'i 5 •O'. 117 -()'. 052 O', 012 O'. 133 0'.237 0'.335 O-. 126 0*.52H o'.6'll 0'.725 0'.812 0'.89 7 0'. 9 31 l'. 053 132 ,203 ,267 .361 I'. 377 1-379 l'.365 1/.329 I'.293 cn 00 219 O', 0222 0'.02()0 O'. 0 196 0'.0181 0'.0182 0'.0185 0'.0192 0'.0196 ,0213 ,0212 ,0265 ,0201 0'.0319 0'.0319 03^5 0128 . 016 6 ,0502 ,0550 ,06o'l ,0666 ,0761 ,09i|8 .1165 0.1379 CHO -0'. 185 -0. 159 -0.132 -0.107 -0.081 -0.055 -0.028 -0.000 0'.023 0.051 O',073 0.103 0'. 128 0. 151 O'. 178 0'.201 0.221 0'.216 0.266 0.23 6 0'.302 0.312 0'.315 0'.307 0.290 0'.277 CMC/4 -0,0928 r-0.0921 -0.0917 -0.0921 -0,0921 -0.0919 -0.0923 ~0 . 0909 -0,0875 -0,0391 -0.0951 -0.0937 ^0.0927. -0 , 0899 -0.0888 -0.0353 -0.0827 -0.0796 .-0,0758 *0 ,0715 ^0 ,0618 -0.0586 -0.0551 rO.0591 -0.0676 -0,0715 205 Table 5 - 0.25-Clark-Y Ri: = '.45Cf0)6 50XL.S + P TSUSL CLARK-Y Rl! = '. '15 (10) 6 70ZL3+P TSUSL ALF CL CO CMO CMC/4 ALF CL CD CMO CMC/4 10'. -o'.333 0'.0239 -O'. 181 -0.0929 -IO'. -0'.324 0'.0273 -0'.133 -0.0925 -9'. -0.24U 0'.0210 -.0.158 -0.0915 -9'. -0'.228 0'.023'l -0'. 157 -0,0929 -8. -0.113 O'.02o5 -o'.131 -0,0916 -8'. -O'. 132 0'.0222 -0.130 -0.0924 -7'. -0'. 019 O'. 0131 -.0.106 -0.0915 -7'. -O'. 038 0'.0212 -0.103 -0.0922 -6'. O'. 015 0'. 0185 -0.080 -0.0915 0'.057 O'. 0 196 -0'.077 -0.0920 -5'. 0'. 110 0'.'173 -0.051 -0.09)6 ~sf. 0.153 0-.0191 -0.051 -0.0926 "If 0'.236 0'.0175 -0'.027 -0,0910 -4'. o'.246 O'.O 199 -0.021 -0,0919 *" • 0'.335 0'. 0 1 3 I 0.001 -0.0906 -3". 0'.34 2 0'.0198 o'.ooi -0,0914 0'.'I22 o'.oiaa 0'.029 -0.0870 0'.4 27 0'.0202 0'.032 -0,0372 0'.525 O'.02o7 • o'.osi -0.0898 o'.533 0'.0219 0'.058 -0,0910 0'. o'.63'l 0'.0230 0.080 -0.0912 0'.639 0'.0 239 0.083 -0.0952 t. 0'.720 0'.0251 0'. 105 -0.0932 f'. 0.722 0'.0263 0.108 -0,0939 2'. 0'.801 0'.0277 0.129 -0.0917 2'. 0'.807 0'.0284 0'.133 -0.0926 3'. 0'.S91 0'. 0 295 0.155 -0.0903 3'. 0'.892 0'.0293 0.158 -0,0913 4'. 0'.978 0'.0320 O-. 180 -0.0386 '4'. ()'.974 0'.0320 0.132 -0.0895 5'. T.051 0'.0373 0'.203 -0.0861 5'. V.05 0 0'.0354 o'.205 -0,0877 6'. .1.127 0'.0102 0.226 -0.0835 6'. 1'. 1 17 0-.0379 0.226 -0.0349 7'. l'. 195 0'.0137 0.217 -0,0301 7'. l'. 185 0'.0117 0'.24 7 -0.0326 8'. T.257 0'.0474 0'.268 -0.0762 8'. l'.243 0'.0462 0'.268 -0.0778 l'. 1'. 311 0'.0521 0.236 -0,0724 1. i'.297 0'. 0 5 1 0 0'.236 -0,0741 10'. 1.35 0 0'.0577 0'.301 -0,0671 IO'. T.330 0'.0563 0.300 -0.0637 11'. f.372 0'.0613 0.311 -0.0623 .11. 1.339 0'.0628 0'.309 -0.0613 12'. T.370 0'. 0739 0.315 -0.0575 12'. l'.34 0 0'.0732 0.310 -0,0603 13'. l'.354 0'.09;>5 0.301 -0,0639 13'. T.323 0'.0928 0'.293 -0,0679 11'. 1'. 311 0'. 1156 0.286 -0,0718 14'. T.285 O'. 1 150 0'.233 -0.0734 15'. 1.282 0'. 1352 0.275 -0,0749 15'. i',246 0'.1337 0.268 -0,0773 CLARK-Y RE = '.'I5(T0}6 60%L3+P TSUSL . C\ ARKrY Rlt = *. 45 C {0) 6 80XL3+P TSUSL ALF CL CD CMO CMC/4 ALF -10'. CL CD CMO CMC/4 -10'. -0'.328 0'.0236 -0 .183 -0.0931 -0',313 0'.0258 -O'.l 74 -0.0881 -9'. -o'.23'l O'. 0210 -0 .158 -0.0933 -9. -0.214 0'.023'l -0.118 -0 ,0886 O * -0.137 0'.0191 -0 .131 -0.0929 -8'. -0'. 123 o'.02?6 -0.122 -0,0874 -7'. -0'.013 O'.O 175 -0 .105 -0.0930 -7. -0.030 O'.02l0 -0'.095 -0.0863 -6'. 0'.051 0'.0161 -0 .079 -0,0921 -6'. 0'.063 O'.O 19 1 -0.073 -0.0900 -5'. 0'. 146 O'.O 151 -0 .053 -0.0925 -5'. O'. 155 O'.O 191 -0,048 -0,0396 -4'^ 0'.2'll O'.O 158 -0 .0 26 -0.0915 -l'. 0.251 O'.O 138 -0'.021 -0.0893 -3:. 0'.336 0'.0161 0 .001 -0.091 1 -3'. 0'.343 O'.O 191 0.006 -0,0880 -2'. 0'.425 o'.0166 0 .029 -0.0878 -?-'. 0'.131 o'.oiao 0 .034 -0.0843 -i'. 0'.530 o'.oiao 0 '.055 -0,0907 "1". 0'.53S 0'.0202 0'.060 -0,0080 o:. 0.637 0'.02()7 0 .080 -0.0955 O'. o'.oio 0'.0219 0.035 -0,0910 1'. 0'.722 0'.0229 0 .a 0 4 -0.0946 i'. 0'.725 0'.0234 o'.l 10 -0.0898 2' 0'.307 0'.0254 0 .130 -0,0927 2: 0',806 0'.0215 0.135 -0,0873 z'. 0'.893 0'.0277 0 .155 -0.0909 3'. ()'.890 0'.0270 0.160 -0,0850 1'. 0'.978 0'.0302 0 .180 -0,0895 4'. 0'.967 0'.0296 O'. 135 -0.0313 5'. 1.055 0'.0352 0 .203 -0.0873 5'. l'.042 0'.0336 0.206 -0,0300 6. 1.127 0'.0381 0 .226 -0,0810 6'. r.107 0'.0366 0-,228 -0,0761 7'. I'. 193 0'.0116 0 .247 -0.0807 7'. l'. 167 0'.0H3 o'.218 -0.0716 8'. l'.258 0'.0461 0 .267 -0.0780 8. l'.230 O'.O'ISI 0.268 -0,0692 9'. 1.310 O'. 0509 0 .286 -0,0727 <?'. l'.28l" 0'.0500 0.236 -0,0643 10'. l'.346 O'0562 0 .300 -0.0680 10". T.298 0'. 0557 0.300 -0.0548 1 r. l',359 0.0624 0 '.310 -0,0610 11'. t'. 315 O'. 0629 0.308 -0.0508 12' l'.359 0'.0726 0 .314 -0,0568 12'. 1.310 0'.0743 0'.306 -0,0510 13. 1 .345 0'.09i'l 0 .302 -0,0653 13'. i'.28 4 0'.0911 0.291 -0.0596 14'. T.307 O'.l 125 0 '.285 -0.0717 14'. i'.246 O'.l 122 0.275 -0,0650 15'. l',270 o'.1313 0 .274 -0,0739 15'. f,207 0'.1278 0.262 -0,0677 ft— ft— ft— i—* ft— I 1 i 1 I I i I J ^* IX. 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()'. 128 -4 '. 0'.233 "* i . 0.3^2 -2. 0.441 -1 . 0.546 0 '. o'.659 1. 0.753 2 . 0,8/11 3 '. 0'.933 4 '. 1.024 5 . 1.107 6 . T. 187 7 . f.265 S . i'.34i 9 . 1'. 4 1 2 10. . T.475 11 . T.522 12 . ' 1". 5 6 1 13 . T.585 n . i',598 15. T.535 16 . 1.541 CLARK'-Y ALF CL -10 . -0.360 -9 . -0.273 -8 . -()'. 176 -7 . -0.001 -6 . 0'. 0 11 -5 . o'. 11 o -1 . (,'.206 -3 . •(')'. 305 -2 . O'.IOl -1 o'.489 0 . 0'.597 i . 0'.687 2 . 0'.771 3 . 0'.852 4' . 0'.933 5' . 1'. 0 1 1 6' . 1 '.079 7' . 1.151 8' . 1 '.219 9' . T.2P8 10' . T.348 11' . T.396 1?' . 1-031 13' . i'.460 14' T.481 15' i',4 85 16'. T.442 CD 0'.02l3 0'.0216 0'. 0 19 7 0'.0183 O'.O 176 O'.O 179 0'. 0 1ft6 0'. 0196 0'.0206 0'.0218 0'. 024 0 0'.0263 0'.0290 0'.0327 0'.0368 0'.0423 0'. 0 4 7 0 O'.05J6 0'.0570 0'.0635 0'.07()5 0'.0785 0'.0837 0'.1021 O'. 12/11 0'.1952 0'.2181 CMO -()'. 192 -0. 166 - 0'. 11 0 -0.111 -o'.087 -0'.060 -0'.033 -0.001 0'.025 0'.052 0'.079 O'.l 06 0'.133 O'. 160 0'.187 0'.212 0'.236 0'.260 0'.233 0.304 0'.324 0.339 0.351 o'.359 0'.354 0.301 0'.301 CMC/4 -0.0917 -0.0923 -0.0929 -0.0933 -0 .0935 -0.0936 -0.0911 0919 0933 0936 0966 0950 0912 -0 . 0886 -0.0863 -0.0829 -0.030 1 -0.0772 -0.0735 -0.0704 -0.0672 -0.0635 -0.0615 -0.0600 -0.0675 -0.1081 -0.1102 Y RE=.5(10)6 10%L5+P TSUSL CD 0'.0254 0'. 0221 O'.02o2 O'.O 188 0'.0179 O'.O 170 O'.O 175 0'.0183 O'.O 197 0'. 0208 0'.0233 0'.0261 0'.0291 0'.0326 0'.0366 0'.0424 0'.0473 O'.O 5.2 6 0'.0580 0'. 0 61 'I 0'.0708 0'.0778 0'.OS60 0'.0917 O'.l 0 75 0'.1376 0'.1932 CMO -0. 188 -o'. 161 -0.139 -O'.l 13 -0'.088 -0.0 62 -0 .036 -0.009 0'.019 0'.015 0'.070 0'.095 0.120 O'.l 15 0'.170 0'.193 o'.211 0'.236 0.257 0'.276 o'.295 0'.309 0'.323 0.330 0'.336 ()'.330 0'.282 CMC/4 -0.0897 -0.0893 -0,0903 -0.0905 -0.0911 -0,0912 -0.0916 -0,0920 -0.0903 -0.0889 -0.0928 -0,0922 -0.0906 0876 0851 0332 0800 -0.0779 -0.0753 -0.0742 -0.0704 -0.0688 -0.0642 -0.0641 -0.0642 -0.071 1 -0.1106 ALF -10' -9 -8 -7 -6 -5 -1 -3 -2 -1 0 1 2 ii 4 5 6 7 e < io'. n'. 12'. 13'. 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CL -0.314 -0'.?52 -0'. 157 -o'.063 0.030 0.124 0'.221 0.319 0 '.409 O'. 4 99 0'.595 O'. 68 6 ()'. 769 0'.855 g'.912 T.024 1'. 1 0 1 T. 176 l'.25f 1 '. 3 1 9 T.373 1 '.4 05 1', 4 1 2 l'.4 69 l'.4«8 1 '.490 l'.4R8 "l'.4 83 l'.463 l'.O 41 1.408 CD 0'.0225 O'.O 190 0'.0162 0'. 0 11 0 O'. 0130 O'.O 123 '0122 0137 0152 0164 0191 0230 0'.0263 0'.0315 0'.0375 0'.04?7 0'.0480 0'. 0511 0'.0605 " 0688 0763 0856 0967 1 1 0 1 1279 O'. 1466 O'. 1681 0'. 1885 0'.2093 0'.2283 0'.2500 CMO CMC/1 ALF CL -0.179 -0 .0874' -10'. -0'.331 -0.151 -0 . 08'69 -9'. -0'.240 -0.129 -0.0872 -8. -0'. 143 -0.105 -0.0801 -7'. -0.053 -o'.os 1 -0.0381 -6'. 0'. 037 -0.056 -0.0086 -5'. 0'. 130 -0.031 -0.089-5 -4'. 0'.224 -0'.005 -0.0904 -3'. 0'. 3 1 6 0'.021 -0 .088 t 0.109 0.016 -0.0076 • -V. 0'. 19 <| 0.070 -0.089 1 o. 0'.587 0.095 -0.O890 r. 0.672 0.118 -0.0076 1'. 0'. 756 0.113 -0.0861 - 3. 0'.836 0.166 -0.0858 4'. 0'.924 o'. 189 -0.0851 5'. ()'.992 0.211 -0.0035 6'. {'. 068 0'.232 -0.0817 7'. l'. 137 0'.252 -0.0813 8'. l'. 198 0.272 -0.0791 • 9'. T.256 0 .289 -0.0770 10'. l'.302 0'.302 -0.0720 ['.342 0'.310 -0.0729 12. l'.37i 0.318 -0.0713 13'. i'.393 0.319 -0.0758 14'. 1'. 4 0 1 0.313 -0.0824 15'. 1'. 4 01 0'.309 -0.0364 16. '['.383 0'.300 -0.0911 17'. l'.361 0.292 -0.0976 0.279 -0.1061 CLARK-Y 1 0.265' -0.1121 CLARK-Y Rir = '.5"ci0)6 60XLS + P TSUSL ALF CL -10'. -0'.336 -9. -0'.215 -8'. -0.151 -7'. -0'. 057 -6'. O'. 0 34 -5'. 0.127 -4'. 0'.221 -3. 0.315 -2 0'. 10 9 O'.l 93 o'. O'. 589 i'. 0'.678 2'. 0'. 762 0'.813 4. 0'.932 5. l'.O05 6'. i'. 08 7 7'. !'. 151 8'. 1.223 9'. ~l'.282 10'. ['.333 ll'r 1 '.379 i'.'113 13'. '['.4 37 14'. 1 .151 15'. ['.4 53 16'. l'.l 16 17'. 1 '.131 18'. 1 '. 4 0 7 19'. 1.375 CD 0'.0216 0'.0212 0'.0l-i5 O'.O 177 0'.0171 O'.O 168 O'. 0169 0'.0175 O'.O 188 o'.02ol 0'.0228 0'.0266 0'.0310 0'.0352 0'.0402 0'.0 4S3 0'.0503 0'.055 3 o'.06()5 0'. 0677 0'.0718 o'.osn 0'.0960 0". 1078 " 1228 14 00 1583 1768 19i|9' 2145 CMO -0.179 -0'. 155 -0. 129 -o'. 104 -0.030 -0.055 -0.031 ••0'.005 0.022 0'.016 o'.071 0.095 0.118 0'. 1 4 2 O'. 165 0.188 0.209 0'.230 0.250 0'.268 o'.281 0.296 0.305 0'. 3 1 1 0'.312 0'.307 0'.300 0.289 0.281 0'.268 CMC/1 -0.0878 -0.0003 -0.0870 -0.0881 -0.0386 -0.0893 -0.0901 -0.0905 -0.0391 -0'.0H75 -0.0889 -0.0097 -0.0390 -0.0870 -0.0877 -0.0851 -0.0861 -0.0032 -0.0013 -0.0789 -0.0761 -0.O756 -0.0755 -0.0758 -0.0783 -0.0838 -0.0898 -0.0975 -0.0987 -0.1041 CD CMO CMC/4 0'.0237 -0. 176 -0.0861 0'. 020 1 -o. 151 -0.0360 O'.O 173 -o'. 125 -0.0 852 O'.O 161 -0. 101 -0.0860 0'. 0 110 -o'. 076 -0.0858 0'. 0 128 -o'. 052' -0.0866 0'. 0 129 -o'. 027 -0.0873 0.0132 -o. 003 -0.0380 O'.O 136 0. 024 -0.0065 0'.0112 o'. 018 -0.0853 O'.O 164 0. 071 -0.0872 O'.O 192 o'. 095 -0.0865 0'.0224 ()'. 118 -0.0O50 0'.0262 o'. 142 -0.CO39 0'.0299 o'. 165 -0.0R47 0'. 0343 o'. 186 -0.0819 0'.0392 o'. 207 -0.081t O'. 0159 o'. 226 -0 .0799 0'.0514 0. 215 -0.0772 0'.057l o'. 263 -0.0741 O'.O 639 o'. 278 -0.0713 0'.0712 o'. 289 -0.0708 o'.ooii o'. 297 -0.0701 o'.0952 0. 302 -0.0703 O'.l 1 il 0. 293 -0.0761 0'.1260 •o. 291 -0 .00.05 0'. 1439 o'. 286 -0.0835 0'.162t o'. 274 -0,0900 80%LS+P TSUSL ALF •10'. -9'. -6. -5'. -4'. -3'. -2'. o'. 1 . 2'. 3'. 4'. 5'. 6'. 7'. fi'. 9'. i0'r 11. 12'. 13'. 14'. 15'. 16'. CL •^0.315 -0'.224 -0'. 132 -(>'. Oil 0.047 0'. 137 0'.227 ()'. 320 0'. 4 1 1 0'. 493 0'.588 0'.671 0'.751 0'.830 0'.908 (t'.982 ['.052 1'. 11 2 l'. 168 l'. 2 1 6 f.24 3 "['.282 T.296 1 .304 T.29 0 r.265 '['.238 CD ' 0 213 0179 0157 0115 0132 0125 0 133 O'.O 137 O'.O 111 "0155 0165 0191 0214 o'.022<l 0'.0216 0'.0269 0'.0302 0'.0317 0".0331 0'.0424 0'.0477 O'.OSIO ,0623 ,0727 .0866 ,ioio ,1172 CMO -0.171 -0.116 -0.122 -0.097 -0.073 -0.050 -0.025 0.000 0.027 0.051 0'.075 0.099 0.122 0'. 1 4 6 O'.l 68 O'. 189 0.209 0.228 0.215 0'.26t 0'.272 0'.28 0 0.234 0'.2S3 0.277 0.266 0''.254 CMC/4 -0.0310 -0.0843 -0.0854 -0 .0851 -0 .0855 -0.0370 -0.0375 -0.0877 -0.0060 -0.0350 -0.0873 -0.0871 -0.0053 -0.0836 -0.0828 -0.0321 -0.0313 -0.0787 -0.0762 -0,0733 -0,0703 -0.0711 -0.0698 -0.0727 -0.0751 -0.0788 -0.0843 212 Table 5 0.53-Clark-Y CLARK-Y RE = '.5C10)6 50XSS+P TSUSL CLARK-Y RC=.5(10)6 70%SS*P TSUSL ALF CL CD CMO CMC/4 ALF CL CD CMO CMC/4 -10'. - 0.3.'17 o'. 0234 -0.181 -0 .0076 -10'. -0'.331 o',0225 -0.177 -0.0874 -9. -0.254 o'. 0 195 -0.157 -0 . 0882 -9'. -0.212 O'.O 195 -0.153 -0.0871 -s'. -0. 159 o'. 0165 -0'. 131 -0.0875 -8'. -()'. 149 0'.0173 -0.128 -0.0374 -7'. -()'. 065 o'. 0151 -0.106 -0.0885 -7'. -0.057 0'.()157 -0.102 -0.0864 -6'. ()'.028 o'. 0143 -o'.OR2 -0.0386 -6. ()'..0 31 o'. 0152 -0.078 -0.0867 -5'. O'. 122 u'. 0110 -0'.057 -0.0092 -5'. 0'. 125 0'. 0 151 -0.051 -0.0877 -4'. 0'. 21 7 o'. 0145 -0.032 -0 .0900 -4. ()'.?13 O'. 0152 -0.029 -o .oaai -3'. 0'.313 o'. 0 159 -0.006 -0.0905 ~f ' 0'. 3 1 1 O'. 0158 -0.005 -0.0389 -2'. 0'.4 07 o'. 0177 0.020 -0.0392 -2'. O'.l 06 ()'. 0 167 0'.0 22 -0 .0873 -r. 0'. 4 9 '1 o'. 0135 0.015" -0.0081 0'.489 O'.O 179 0.016 -0.0865 o'. 0'.591 o'. 0205 0'.070 -0.0397 o'. 0'.581 O'.02o3 0'.070 -0.0882 r. 0'.678 o'. 0211 0'.091 -0 .0889 1 . 0'.669 0'. 0234 0.093 -0.0382 2'. 0'.763 o'. 0280 O'.l 17 -0 .0883 2'. 0'.751 0.0270 O'.l 17 -0.0869 3'. 0'. 8 'I 8 o'. 0322 0'. 112 -0.0867 3'. ()'.834 0'. 0 3 0 7 o'.l H -0.0052 4'. 0'.936 o'. 0366 0'. 166 -0.0863 . 4'. 0'. 9 1 6 0'.0319 0'. 161 -0.0813 5'. 1'. 0 1 2 o'. 0 413 0.188 -0.0846 5'. 0.993 O'. 0 38 6 0'. 185 -0.0040 6'. T.092 o'. 0463 0'.210 -0.0839 6'. T.0 63 0'.0433 0.206 -0 .0 3.-;"4 7'. 1. 162 o'. 052'! 0.230 -0.0820 7'. l'. 130 O'.045'l 0.225 -0.0003 e'. T.235 o'. 0584 0'.252 -0.0800 8'. r. 199 O'.05l 4 0.215 -0 ,0786 9'. l'.302 o'. 0668 0.271 -0.0779 9'. l',257 0'.0577 0'.26 3 -0.0763 10'. l'.357 o'. 0758 0'.288 -0.0756 10'. • i".304 0'.0652 0'.277 -0.0743 11'. 1 '.401 o', 0860 0.30 0 -0.0741 11 '. r.3ii 0'.0738 0'.288 -0.0727 12'. l'.'HO o'. 0972 0'.3i0 -0 .0750 12'. l'.369 0'.083l o'.295 -0.0724 13'. l'.4 66 o'. 1097 0.316 -0.0753 13'. T.307 0'.0911 0'. 3 0 1 -0.0709 14'. 1.481 o'. 1262 0'.318 -0.0770 14'. 1'. 4 01 0'. 1096 0.299 -0 ,0780 15'. l'.490 o' 1454 0'.310 -0.0880 15'. 1 '.39 3 0'. 1255 0.292 -0.08 16 16'. l'.4a6 o'. 1616 0'.309 -0.0874 16. 1.38 0 0'. 1433 0'.281 -0.0867 17'. f.476 o' 1869 0'.299 -0.0961 17'. i',362 0'. 1603 0.273 -0.0928 18'. T.457 o'. .2012 0'.291 -0.0993 ie; l'.331 0'.1791 0'.262 -0.0967 CLARK-Y RII = .5C10)6 60ZSS+P TSUSL CLARK-Y RE=.5Cl0)6 80%3S+P TSUSL ALF -10. il'. -3. -7'. -6'. -5. -4'. -2. -r. 0. f. 2'. 3'. 4'. 5'. 6'. 7. ' s;. 10'. 11'. 12'. 13:. 14'. 15'. 16'. 17'. 18. CL -0'.342 -0'.247 -O'. 15.5 •o'.06J 0.030 0.124 0'.218 0'.315 0'.406 ()'.492 0'.587 ()'.676 ()'.760 ()'.812 o'^o 1', 0 01 1.035 l". i55 1 .224 1.286 1.336 i'.soi 1 .416 T.4 39 r.45i f.453 l'.4 43 T.4 29 1 '.413 CD 0.0236 0205 0103 0U.5 0155 0152 0154 0165 0174 0135 0214 0241 0277 0321 0362 040 0 454 0508 0558 0627 0708 0304 09 11 1019 1132 134 0 1533 1711 1901 7 0 CMO o'.ioo 0'. 155 0.130 •o'.105 '0.081 0.056 0.031 •0.005 o'.021 0.015 0'.069 0.093 0.117 O.IH 0.164 187 0'.209 0'.230 0.251 0'.270 0.235 0'.29 7 O'.305 0.312 0.312 09 0,298 0.289 0.230 0 CMC/4 -0.0377 -0.0881 -0 .0882 -0 . 0886 -0.0886 -0'.0892 -0.0399 -0'.0903 -0.0308 -0.0831 -0.0396 -0,0900 -0.0835 -0.0863 -0,0867 -0.0343 -0.0339 -0.0813 -0.0790 -0.0759 -0.0738 -0.0741 -0,0743 -0.0734 -0.0769 -0.0805 -0,0089 -0.0915 -0 ,0996 ALF CL 10. -0.321 -9'. -0.237 -8. -0'. ill -7'. -b'. 047 -6'. o'.0 39 -5'. 0'. 132 -4;. 0'.222 w3. 0'.315 -2'. 0'.4 06 -f. ()'. 493 0'. (l'.583 i. (]'.669 2'. ()'.749 f ' 0'.827 4'. 0 '.909 5. 0'.978 6'. 1'.053 7'. 1.12 0 e'. i'.182 9'. 1 '.237 10'. r.28o 11'. r. 310 12'. t'.334 13'. 1.349 14. 1.353 15'. r.34o 16'. ('.321 17. l'.279 18'. l'.236 CO 0222 0191 0169 0157 0113 0135 0139 0142 0151 0159 0173 0199 0227 0259 0 279 0305 0331 0 37 4 0'.0424 0'.0478 0539 0606 0693 0801 0934 1 0 05 1 251 0'. 1 4 1 2 0'.1557 CMO -0.173 -0-149 -0.121 -0.099 -0.0 76 -0.051 -0.027 -0.002 0'.025 0.050 0.073 0'.097 0.121 O'. 14 1 0.166 O'. 188 0'.208 0.229 0 . 2 '16 0.265 0'.277 o'„237 0'.291 0'.295 0.290 0'. 2 81 0'.271 o'.26 0 0.242 CMC/4 -0.0838 -0.0332 -0.0845 -0,0357 -0.0359 -0.0868 0874 0881 . 0864 0857 0370 0872 0.0845 0.0832 0834 0805 03 13 0782 0778 0732 0727 0699 069 1 0711 0769 0003 0879 0.0800 0 .0910 -0 . -0 . -0. -O', -0 . -0. -0 . -0. -0. -0 . -0 , -0. -0. -0 1 -0 . -0. -0 . -0 . -0 . cn H IN + cn 5-I CT O O- 0s C- OC* CP O —« O 0* ©CCC '•-CsT*JsTMf\J--(\J?N(CMffiIO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOft u o o o o > I o I OOOO I I I I o o I: I OOOO III! 0 O O O o o 1 I I I. 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Table 5 - 0.66-Clnrk-Y CLARK-Y WE=(to)6 SOLID WALLS CLARK-Y RE = '(10)lS 4 0XLS+P TSUSL ALF CL CD CMO CMC/4 -lo'. - 0'. '13 7 0 .0304 -0'.196 -0 .0843 -0'.319 0 .0270 -0'.174 -0.0927 -s'. -()'. 198 0 .0211 -O'.l 47 -0.0959 -7'. -0.085 0 .0226 -O'. 120 -0.0977 -6'. O'. 0 28 0 .0213 -O'. 0 9? -0.0986 -5'. (>'. 145 0 .0209 -d'.063 -0.0999 -4'. 0'.262 0 .0209 -o'.035 -0.10 11 -3'. 0'.379 0 .0216 -0'.005 -0.1016 — 2' o'. 196 Kl .0227 0.025 -0.10 11 -r" 0'.599 0 .0215 0.053 -0.0999 0'. 0'.697 0 .0266 o'.oao -0.0980 r. 0'.807 0 .0301 O'.l OH -0.0983 2'. 0'.931 0 .0351 0'. lit -0.0974 3'. T.031 0 .0106 0 . 169 -0.0910 4'. 1 .134 0 .0169 O'. 196 -0.0932 5'. T.233 0 .0534 0.224 -0.0905 6'. 1.339 0 .0602 o'.250 -0.0918 7'. T.4 26 0 .0692 0.276 -0,0879 fi'. 1'.529 0 .0791 0.298 -0.0911 9'. T.603 0 .0900 0.320 -0.0380 10'. T.657 0 .1026 0'.335 -0.0861 11'. "r.692 0 .1201 0.3-12 -0.0879 12'. T.730 0 .1391 0.319 -0.0907 13'. T.77 0 0 .15^5 0.352 -0.0974 11'. l'.78 6 0 .1806 0.355 -0.0932 15'. l'.826 0 .2010 0.357 -0.1070 16'. l'.837 0 .2267 0.359 -0. 1084 17'. 1 '.810 0 .2554 0.355 -0,1132 IB'. it'.846 0 .2893 0.316 -0.1251 19'. T.837 0 .3294 0.335 -0.1361 20'. l',S56 0 .3729 0.325 -0.1531 ALF CL CO CMO CMC/4 -10'. -0.417 0'.0289 -0.195 -0,0070 -9', -0'.313 0'.0252 -0.170 -0 ,0386 -s'. -0.209 0'.0?26 -0.144 -0,0089 -7'. -0'.107 O'.02o7 -0.113 -0',0399 -6'. -0'.006 O'.O 196 -0'.093 -0.0909 -5'. O'. 095 O'. 0 199 -0.067 -0.0913 -4' 0. 199 0'.0?.08 -0.040 -0.0916 -3' 0'. 30 i 0'.0218 -o'.on -0.0923 -2' . 0'.399 0'.0234 0'.0 12 -0.0918 -l' . 0'.491 0'.0259 0'.036 -0,0912 O' . 0'.581 0'.0289 0.061 -0.0902 i' . 0'.677 0'.0331 0.006 -0.0904 2' . 0'.789 0'.0386 0.115 -0.0904 3' . 0'.885 0'. 0 4 5 0 0'. 110 -0 . 0904 4' . 0'.981 O'. 0508 0.165 -0.0902 5' . l'.075 0'.0575 0.190 -0.0901 • 6 . l'. 166 . T.258 0'. 0 61 3 o'.211 •-0,0900 7 0'. 0 7 1 8 0'.237 -0.0905 8 . it'.346 O'. 0316 0.259 -0.0909 9 . 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Nr. -\J~) > ru CD •— L>J >0 t— OD >— -Cr --J >— *J1 -C: —1 o w -0 Ji J-r o Jl Ul o © -u o U4 Ul CO CO r-rO Table 6 0. 34--NACA-0P15 0015 RCsO'.sdOld 50XS3P TSUSL 0015 RE = 0'.5(t0)6 70%SSP TSUSL ALF CL cn CMO CMC/4 -0'.342 O'.Ol 99 -0 .100 -o .oi"?.n -0.251' O'.O 169 -0 .074 -0.0107 -2'. -0.16 2 0.0142 -0 .048 -0.0073 -1'. -0.074 O'.O 132 -0 '.023 -0.0040 o'. -o'.ooo 0'.0124 0 '.oot 0.0008 1'. 0.076 0'. 0 1 2 0 0 .024 0.0051 0'. 1 6 0 0.0 124 0 '.048 0,0078 0.250 O'.O 138 0 '.074 0.0109 1'. 0.337 O'.O 153 0 '.10 0. 0.0144 c r 0'. '13 'I O'.O 177 0 .125 0.0159 (>'. 0.545 0:.02l3 0 '.150 0.0 123 r. O'. 669 0.0257 0 '.174 0.0 061 8'. 0'.773 0'.030l 0 '.198 0.0045 <>'. 0.8 32 0'.0337 0 .219 0.01 OS 10'. 0 .887 0'.0379 0 .240 0.0186 n'. 0'. 9 .'.| 8 0'. 0 4 32 0 .261 0.0245 12'. 1.010 0.0498 0 .281 0 .0297 13'. T.063 0'. 0568 0 .297 0,0 329 11'. 1'. 1 0 1 0'.064 8 0 .30 7 0.034 0 15'. l'. 101 0'.0780 0 .302 0.0287 16'. l'.04() O'. 1187 0 '.249 -0 .011 0 17'. 0.721 0.234 1 0 .123 -0,0677 0015 RE = 0'.5C10)6 60%SSP TSUSL ALF Cl. CO CMO CHC/4 -4'. -0.34 1 O'.02ol -0 .101 -0.0153 "3. -0.253 O'.O 168 -0 .074 -0,0103 _ 7»' -()'. 160 0'. 0 1 4 6 -0 '.049 -0.0084 "\'. -0'.072 0.0133 -0 .024 -0.0057 o'. 0'. 0 0 4 0'. 01 1 9 0 '.002 0.0004 1. 0'. 08 0 0 '. 0 12 1 0 .025 0.0 050 2'. 0'. 166 0'.0127 0 '.051 0.0087 3'. 0.251 0'.0137 0 .075 0.0119 a'. 0'.34 1 0.0153 0 .101 0.0154 5'. 0.437 O'.O 174 0 .128 0.0 174 6!. 0.546 0'.0207 0 '.154 0.0167 7. ()'.671 0*. 0250 0 '.179 0.0101 8'. 0'.775 0.0293 0 '.203 0.0084 «». ()'.835 0'.0328 0 .224 0.0145 10'. 0'.8f,9 0'. 0370 0 '.245 0.0225 11'. 0.950 0.0420 0 .266 0.0294 12'. 1.0 16 0'. 0 4 *18 0 .286 0.0327 13'. T.066 0'. 0554 0 .303 0,0379 11'. 1'. 102 0.0634 0 '.313 0.0392 15'. 1 . 1 02 0'.0795 0 .302 0 .0287 16'. l'.038 ()'. 1206 0 .249 -0.0105 17'. 0'.702 0'.2343 0 .124 -0.0623 ALF CL CD CMO CKC/4 -()'. 34 0 O'.O 197 -0 .101 -O.0 154 _ T -0'.253 O'.Ol 66 -0 .075 -0.0 115 -()'. 158 0". 0144 -0 .049 -0,0089 -r. -0.075 O'.Ol 31 -0 .024 -0.0052 o'. 0 , 0 0 6 O'.O 125 0 .001 -0,0007, i'. O'.O 83 O'.O 120 0 .025 0 . 0 0 4 1 r>' 0'. 169 0'. 0 1 ? 1 0 '.051 0.0 0 79 3 . 0.2S7 O'.O 130 0 ',076 0.0116 <*'. O'. 346 0'. 0149 0 '.103 0.0153 5'. 0'.441 O'.O 171 0 '.129 0.0 177 6'. ()',554 0'. 0 2 o 4 0 .155 0.0 157 7'. 0'.674 0'.024 4 0 '.179 0.0094 8'. 0'.778 ()'.0284 0 .203 0.0077 9'. 0'.832 0'.0320 0 '.223 0.0148 10. 0'.889 0'.0362 0 .24 5 0,0225 H'. 0'.952 0'. 04] 8 0 '.266 0.0286 12. 1'. 0 1 4 0'.0477 0 '.285 0.0325 13'. 1'.064 0.054 6 0 .300 0.0355 H'. l'.(!95 0.0627 0 .310 0.0381 15'. 1.084 00 8o0 0 .298 0.0290 16'. 1'. 0 1 9 0'. 1217 0 .24 5 -0.010" 17'. 0.69 8 0'.2330 0 .124 -0.0617 0015 Rl-=0'.5(10)6 80ZSSP TSUSL ALF CL CD CMO CMC/4 -0'.337 0.0215 -0 .101 -0,0157 -3. -0.249 O'.Ol 81 -0 .075 -O'.O 119 - 0'. 1 6 0 0'. 0156 -y .04 8 -0,0078 -!'! -0.069 0'. 0 1 4 1 -0 .023 -0.0057 or. 0 . 0 0 6 O'.O 131 0 .001 -O.OOOI r. O'. 079 0'.0127 0 .025 0.0056 -i * <-. 0 . 166 O'.O 122 0 . 0 5 0 0'. 0 0 8 4 3. 0.254 O'.O 129 0 .075 0 , 0 1 t 3 1'. 0'.34 4 O'.O 144 0 '.101 0,0147 5'. . 0'.4 39 O'.O 164 0 '.128 0.0174 6'. 0'.553 0.0193 0 .153 0.0143 7'. 0'.672 0'.0251 0 .178 0.0090 8. 0.774 0.0267 0 .20 1 0,0068 «>'. 0.831 0'.0298 0 '.223 0.0145 10'. 0'.883 0'. 0.338 0 .24 3 0,0221 11. 0'.943 ()'. 0 383 0 '.263 0.0285 12. 1'. 0 0 8 0'. 0 4 4 6 0 '.284 0.0333 13'. T.054 0.0505 0 .297 0.0352 H'. l'.078 0.0582 0 '.304 0,0370 15'. 1.060 0'.0786 0 .282 0,0188 16'. ()'.717 0'.2075 0 '.113 -0,0753 225 Table 6 " 0.5O-HACA-OOI5 oois Rir=o'.s(i0)6 SOLID WALLS CD 0.0267 0.0229 0.0198 O'.O 178 O'.O 16*1 0',0161 0.0160 0.0174 O'.O 192 0.0226 0'.0266 0'. 0318 0.0362 0'. on io 0.0470 0.05/16 0.0625 0'.0721 0'.0f.23 .0'. 0 94 9 0". 113fl 0'. 1 '17 <7 0'. 1726 0'.1926 0'.2178 0015 RlZ = 0'.5(i"0)6 40ZI.S + P TSUSL RC = U.5C10)6 .IOZSST-P TSUSL ALF CL -0.367 -0.269 mm T» ' -0.170 -r" -0.077 0'. -o'.ooo r. (>'. 076 ?.'. 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Jl o L-J O Jl JJ L-J CP CD ft Ji c- Ji ji X: ii fi- in si ft JJ •CD CO si ft i Ji — Cc Jl ro PJ •CO - L-4 . 3- »— '•-,'J1 LS ru -= l-J l-J -1 © JJ X3 o o = © o © o o o O o © © O o o © © o © o © o n ft o © o o o © © © © O o © o o o o © o o o O ft ru o o CO -J c- CP 'Jl Ji Ji LJ L-l ru ru ft ft — ft ft ft ft PJ PJ ft X-- si • X2 Ji -J CD ft .fi -4 L-l si • ft "Jl —• .CD c- xi fi fi L<1 s| • o fi o JJ c» J> PJ CP JJ - Co •-* n ru CO l-l rj ~j o ru L-l - o ft I 1 1 1 1 O © o o o o o o o © © © © c © © © o o o © © © © n 3C CP ru ru PU ru ru ru PU rj ru ft ft ft ft ft O © o o © O o o o © o — ft ft o o o o o j; CO c* UJ ft JJ cs fi PJ o PU fi --I JJ 30 3s ru ru JJ JJ 3- Jl 1-' • CP o © JJ c- l-l -•1 'Ji —• o L-l -J ft* ~i -4 1 i i l I I 1 1 1 I 1 1 t i-C o o o o o o © o o O o © © o o o o o o O o o o o C/> m '» ft '• '* * ' -ft ft ft - 'ft 'ft •« ft • '» ft ft •ft 'ft » « •ft m CE CO O o o © © o o © o O o o © o o © o o o o o © © O + CO CO CO Si 'CP fi L-l ft IO o o ft Ift ift Ift. ift ;© o o o © o © -s • w l-l ft o O -J Ji JJ -4 L-l Jl ru 'Jl Jl ji ft CO 3- ru ft Ji CP JJ Ji CO Ji '•J CO Ji o si ru Jl CD JJ ft ru ft ft' o JJ JJ JJ Ji o sl CD co <z co 234 Table 7. Windtunnel balance results - Joukowsky airfoil JOUKOWSKY KE = 0'.5(;10)6 SOLIH WALLS JOUKOWSKY RE-O'.S 110) 6 ALF CL CD CMO CMC/4 ALF -7 . -o ,3 4 6 0'.01°5 -0.159 -0.0735 „7' -6' . -o .242 0.0182 -0.133 -0.0732 -6'. -5' . -o .134 0'. 0174 -0.107 -0.0 731 -5. -4r . -o .0 23 O'.O 167 -0.079 -0 .0729. -4. -3' . o .087 0'. 0168 -0.051 -0.0723 " • -2' 0 .193 0,.0174 -0.024 -0.0714 -2'. -1 0 .297 O'.O 138 0.003; -0.0705 -l'. o' . 0 .405 0'. 0206 0.031 -0.0691 0'. i' . 0 .511 0'.0228 0.059 -0.0675 1'. z\ 0 .615 0'.0256 0 .087 -0.0653 p' . o .715 0'. 02Q0 O.ll'l -0.0631 • 3'. 0 .815 0.0328 0.14 1 -0.0615 4'. 5. 0 .910 O'.0376 0.166 -0.0592 5. 1 .0 00 0'. 0 4 03 0'. 190 -•0.0582 <•>'. 7. 1 .096 0. 0 4ci4 0.215 -0.0559 7-'. fi'. 1 . 189 0".0506 0.24 1 -0.0530 fi'. <>'. 1 .267 0'.0567 0.263 -0'. 04c»5 9. 10. 1 .331 O'. 0645 0.231 -0.0471 10'. 11. 1 .377 0'. 074 1 0.291 -0.0473 11'. 12. 1 .393 0.0921 0.289 -0.0548 12'. 15:. 1 .176 0'.2304 0.14 2 -0.1547 13. CL •0.345 •0.24 1 •0.134 -0.024 0.'. 0 8 6 0', .92 0'. 296 0'.4 03 0.506 0'. 6 1 0 0'.71 3 o'.m i 0'.907 O', 99 3 l'. 091 1'. 1 79 1'.265 T.326 T.363 l".394 1.186 CD O'. 0194 O'.O 180 0'. 0 1 7 1 O'.O 168 O'.O 172 0'. 0 179 O'.O 193 0.0212 0'.0233 O'. 0259 0'.0289 0'.0326 0'. 0370 0'.0425 0'.04 78 0.0534 0'. 0594 0'.0 672 0'.0773 0'.0959 0'.234S CMO -0.158 -0. 133 -0.106 -0.078 -0.051 -0.023 0.003 0.031 0.0 58 0.037 0.114 0 .14 1 0. 166 0 . 1 9 0 0.215 0 .239 0 .263 0.230 0.29 0 0.289 0'. 1 4 6 SOLID WALLS CMC/4 -0.0723 -0.0728 -0.0727 -0.0723 -0.0717 -0.0706 -0.0699 -0 .0689 -0 .0673 -0.0645 -0.0625 -0.0607 -0.0591 -0.0574 -0.0551 -0.0524 -0.0491 -0.0466 -0.0465 -0.0546 -O'. 153T. JOUKOWSKY RE = 0'.5(10)6 SOLID WALLS ALF CL CD CMO CMC/4 -7 . -0.34 6 0'. 0195 -0 .159 -0.0733 -6 . -0-.24 1 O'.O 1*3 -0 .133 -0.0732 -5 . -0.13 4 O'.O 174 -0 .106 -0 .0726 -4 . -0.024 O'.O 165 -0 .078 -0.0723 . 0.086 O'.O 166 -0 .051 -0.0718 . 0,193 O'.O 179 -0 '.023 -0'.0709 -1' . 0.297 0.0192 0 .003 -0.0703 0 . 0'.4 03 O'. 0205 0 '.031 -0.0689 1 . 0.5 03 0.0223 0 .059 -0.0674 2' . 0.613 0'.0246 0 .037 -0.0651 3 . 0.713 0'.0273 0 '.114 -0.0628 4' . 0'.8 13 0". 0307 0 .141 -0.061 1 5' . 0'.90H 0'.0349 0 .166 -0.0593 6 . t'. 0 03 0'.0415 0 .191 -0.0579 7' . T.094 O'. 0469 0 .215 -0.0556 e' r. ia6 0'.0525 0 .24 1 -0.0524 9' T.266 O'. 0584 0 .263 -0.0495 10 . 1.331 O'. 0660 0 .231 -0.0467 if 1.376 0'.0770 0 .292 -0.0 469 12' . 1 '.401 0'.0943 . 0 .29 1 -0.0540 13' . f. 173 0'.2357 0 .146 -0.1518 Table 7 JOUKOWSKY KE:-0'.5(lO)6 "UO/SSS + P TSUSL ALF CL CO C! *0 r.MC/4 -7. -0.331 0". 0139 -0 .155 -0.0734 -6. -0'.227 O'.O 173 -0 .129 -0.0722 -5'. -0.125 O'.O 157 -0 .102 -0.0710 -4. -0.018 O'.O 152 -0 .076 -0.0703 -3'. 0'.085 0'. 0 1 5 7 -0 .049 -0.0694 :/. 183 0'.0161 -0 .023 -0.0683 0'.282 O'.O 169 0 .003 -0.0671 o. 0.381 O'.O I 37 0 .029 -0.0656 f. 0 . '1 7 7 0',0212 0 .054 -0.0641 2. 0'.572 0.0238 0 .08 1 -0.0614 3. 0'. 666 0'.0272 0 .106 -0.0589 0,755 0.0309 0 .130 -0.0575 5. 0'.8'I5 0,03 4 7 0 .154 -0.0554 ^. 0.932 0'. 0 4 0 0 0 .177 -0.0535 7'. 1.014 0', 0450 0 .20 0 -0.0515 8". 1.099 0'. 0507 0 • 2 2 -0.0492 9. 1.171 0'.0570 0 .24 3 -0.0463 10'. 1 .239 0'.0642 0 .262 -0 .0438 1 1 . 1.282 0'.0727 0 .273 -0.0426 12'. 1 .309 o',0855 0 .275 -0 . 04<i8 13'. 1.073 0'.2073 0 '.134 -0 . 1380 JOUKOWSKY RF=0.5(10)6 40%LS+P TSIJSL ALF CL CD CMO C M C / 4 -7. -0'.335 O'.O 195 -0 .156 -0.0733 -6. -0'.234 O'.O 178 -0 .130 -0.0723 -5. -0.131 0'. 0166 -0 . 104 -0.0717 -4. • -0.025 O'.O 162 -0 .078 -0.0711 -3. 0'.079 O'.O 153 -0 .050 -0.0698 -2. 0. 173 O'.O 165 -0 .0 25 -0.0690 -1 . ()'.277 0', 0178 0 '.001 -0.0677 0. 0.375 O'.O 188 0 .027 -0.0665 1. 0'.472 0.0217 0 .052 -0.0650 2'. 0'.570 0'. 0244 0 .079 -0.0621 7 " -> m 0.663 0'.0277 0 .105 -0.0597 <K 0.751 O'.03t4 0 .120 -0.0580 5', 0.84 1 O'. 0358 0 .153 -0.0559 6. 0 .930 0'. 0 4 o 4 0 .177 -0.0536 7'. 1,017 0'.0457 0 '.200 -0.0516 8. 1.094 0.0520 0 .221 -0.0495 9'. 1.170 0.058 1 •0 .24 2 -0.0467 10'. 1.237 0'.0652 0 .26 1 -0,0438 1 1 . 1.282 0'.07 38 0 .272 -0.0«30 12. i .516 O-. 0864 0 .277 -0.0464 13. l'. 1 0 1 0'.2018 0 .138 -0.1393 236 Table 7 JOUKOWSKY RE = 0'.5'dl0O6 50%LS+P TSUSL JOUKOWSKY Rfr.-O'.5 C10) 6 70XLS+P TSUSL ALF CL -7. -0.327 -6'. -0.227 -5. -0.125 "t'r -0.022 -3. o'.oei -2'. 0. 178 -r. 0.2 7-1 0. 0.372 i. 0.468 2' o',564 —*. 0.657 *•. 0.747 5. 0.832 6. 0'.924 7'. I '.00 6 e'. l'. 084 l'. 162 10. T.223 11. T.268 12'. 1 '.297 13'. 1 ,0 76 CD 0'. 0 1 9 1 O'.O 17-1 O'.O 163 O'.O 162 0'.0163 0'.G1.".6 0'. 0 1 8 0 O-, 0195 0'.0223 0'.0?53 O'. 0282 0'. 031-1 0'.0353 ()'. 0 38-1 0.0-135 0'. 0 '18 8 0'. 0556 0.0623 0'.07t2 0.0838 0'. 1984 CMO cuc/a ALF CL CO CMO CMC/4 -0.152 -0.0712 -7. -0.314 0'. 0189 -0.150 -0.0726 -0'.127 -0.0701 -6. -0.216 0'.()172 -0.125" -0,0717 -0.101 - 0'. 0 7 0 0 _ L • - J . -0'. 114 0'.0155 -0.099 -0.0707 -0.075 -0 .069-4 -0.009 O'.O 155 -0.072 -0.0692 - o'. o t\ a -0.0677 "3. 0.0 9). o'. 0155 -0.046 -0.0681 -0.023 -0.0670 -2'. O'. IBS 0'. 01S5 -0.020 -0,066 7 0.002 -0.0660 .-r. 0.283 0', 0165 0.005 -0.0652 0.027 -0.06U8 o'. 0.37 7 O'.O 179 0.030 -0.0639 0.053 -0.0631 r. 0 .'171 0'.02o4 0.055 -0 .0621 0.079 -0.0612 0.56 4 0'.0227 o'. o n o -0.0598 0.104 -0.0587 3. 0.655 0'. 0 25 6 0.105 -0.0572 0.128 -0.0570 0.7 '13 O'.O?.-15 0. 129 -0.0551 0.151 -0.0551 5. 0'.835 0.0321 0.154 -0.0533 0.176 -0.0533 6. 0.917 0'. 0 3 70 0.176 -0,0513 0. 198 -0.051-4 7'. 0.995 O'.O'113 0.1'18 -0 .0486 0 .219 -0.0487 8. 1'.070 O'. 0467 0.218 -0.0460 0'. 2 4 1 -0.0059 9. 1 . 1 4 0 0 '.052 '1 0.238 -0.0438 0 .258 -0.0434 10'! 1 . 1 96 0'.0585 0.253 -0.0414 0.270 -0.0424 U'. 1.239 O'.O 67 7 0.2 6/I -0.0412 0.273 -0.0/159 12'. 1'.234 0'.u787 0.257 -0.0469 0.135 -0.1362 13. 0.961 0. 199.1- 0.104 -0. 13-7 JOUKOWSKY ALF CL -7. -0'. 323 -6'. -o'.221 -5'. -()'. 119 -1'. -o'.016 -3'. O'. 0 88 -2'. 0, IB6 -r. o'.279 o. 0.375 r. 0'.470 0.564 0'.656 0'.743 5. 0.829 6. 0.914 7. 0.997 B. 1 .0 73 <?'. 1.146 10'. 1,202 11. 1 .26.", 12. 1 .296 13'. 1'.064 CO 0'. 0 1 8 6 O.0170 O'.016O 0'.0156 0',01S5 0'. 0162 O'.0t70 O'.O 188 O'.O.? 15 0'.0239 0.02<-6 0'.03()1 0'. 0338 0'. 0 395 0'. 0 4 4 6 0'. 0504 0.0562 0. 06 21". 0'.07?7 0'.0835 0'. 2 0 8 0 C10)6 60XLS+P TSUSL JOUKOWSKY RE=0.5 Cl 0)6 80%1.3+P CMO CMC/4 ALF CL CD CHO CMC/4 -0.153 -0.0733 -7. -0.304 0'.0 178 -0.147 -0.0719 -0.127 -0.0724 -6. -0.203 0'. 0153 -0.121 -0.0702 -0.101 -0.0716 -5'. -0.104 0'. 014 7 -0 .095 -.0 .0694 -0.075 -0.0704 --•'. ~ 0 ', 0 0 0 0'. 0 1/10 -0.069 -0 .0685 -0.047 -0.0690 -3. 0 . 1 0 1 0 '.0140 -0.043 -0.0674 -0.0 22 -0.0630 0. 197 o'.0 13fl -0.0 17 -0.0653 0.00 2 -0.0667 -I . o'.2?.a 0 '.0151 0.007 -0.0644 0.023 -0.0653 0. 0'.3fi5 0,0159 0.033 -0.0623 0.053 -0.0636 1'. 0'. 4,-:!) O'. 0177 0'.059 - 0 , 0 60 3 0.079 -0,0614 y' 0.572 0 .02(1 i 0.0 8 4 -0,0 5 8 0 0.104 -0.0587 z. 0 .660 0 '.0227 0.1 118 -0.055 4 0.127 -0.0568 ''' '•'.751. 0'.0.-5 6 0. 133 -0.0535 0.150 -0.0551 I." 0.6 37 O.02H7 0.156 -0.0511 0.174 -0.0528 6'" 0.91 8 (''. 0 3 1 9 il -1 7.-< -0.0493 0. 196 -0.0512 V !j'.99.-» 0.036 1 0.20 0 -0 .0-16 5 0.218 -0.0484 p ' ! . o 7 1 (i'. 0 n i '1 0.220 - 0 . 0 /14 4 0.2.37 -0,0458 •<>''. i .134 -0'.0-'l70 0.2 3*'. -0,0.114 0.25 3 -0.0 429 10. 1.188 .d'. 0 525 0.253 -0 . .'39-. 0.269 -0.0429 11'. r.224 0'. u 6 21 .0 .2/. 1 -0 . ll"/iq 0.271 -0.0474 12. 1 .237 (.'.1263 i) .25 0 -0.01 0.125 -0.1433 13'. 0.963 0.1971 0.107 -0,1369 Table 7 JOUKOWSKY RiT = 0.5ClO)6 50 ALF CL CO C MO -'!'. -0 .324 O'.Ol 78 -0 .151 -':>'. -0'.225 0'. 0 1 6 2 -0 .127 -5'. -()'. 120 O'.O 152 -0 .101 "q> -d'.O 19 O'. 0 14 9 -0 .074 ** .'j . 0;. OP.5 O'.Ol '!3 -o .04 7 0'. 1 83 0.0154 -0 .021 0'.?77 0'. 0 1 67 0 .003 o. 0'.375 O'.O 182 0 .029 i'. 0'.4 68 0'.0206 0 .053 o'.564 0.0234 0 '.030 3. 0.653 0'. 0262 0 .104 0'.746 0'.0296 0 .128 5'! 0'.832 0'.033l 0 .151 6. 0'.921 0'. 0 '10 0 0 '.176 7 . 1 '.002 0'.0'l^5 0 '.198 ?•'. l'.0P2 0.0512 0 .210 9'. {'. 156 O-. 057 7 0 .240 io'. 1 .215 0'. 0 6 4 8 0 .257 i r. \ '.26 0 0'.0739 0 .268 12'. 1 .288 0'.0863 0 .271 13'. 1.057 0.207^- 0 .130 Jf:iiSr!.!S.<Y .=? f:' =. 0' 5 ( UO 6 60 ALF CL cn c •'0 -?'. -•''.:>?7> O'.O 177 - ;l .152 - c. - 0' 22? O'.Ol 66 -0 .126 -O-. 123 o'. 0 160 - 0 .10 1 - '. -O'.O 18 0'. 0 1 4 4 -o .073 _ 7 ' 0'. 0 8 4 0'. 0 1 4 7 -0 .04 7 _J O'. 131 O'.O 155 -0 .022 -l'l 0-.276 O'.O 167 0 .003 0. 0'.371 O'.O 179 0 '.028 f. 0'.4 67 0'.0203 0 .053 2'. 0'.561 0'. 0231 0 '.079 3'. 0'.653 0.0259 0 .104 n'm 0'.7'l) 0'.029<l 0 .123 5'! 0'.830 0'.0330 0 '.152 6'. 0'.914 0>370 0 '.174 7'. 0'.993 0'.0427 n .197 (V 1'.076 0'.0433 0 .218 9'. 1'. 143 0'.0547 0 .239 io'! i'.2()5 0.0623 0 .255 i r. f.252 O'. 07 04 0 .267 12. T.277 O'.OS 15 0 .268 13'. 1.055 0'.1950 0 ,t28 •,;3.Si-P TSUSL JOUKOWSKY CMC/4 ALF CL 0.0712 -7'. -0'.325 0.0706 -6. -0.221 0.0698 — I" '* -0'. 1 19 0 .0689 "4 • - 0:. 0 1 6 0.0682 ** -> • ()'.087 0.0666 -? : 0'. 185 0.0657 0'.23J 0.064 1 0'.374 0.0623 i. 0'. 4 7 1 0.0603 *>' 0'.563 0.0579 ji, 0 .66 0 0.0566 0'.746 0.0547 5'. 0'.832 0.0526 6'. 0'.9 16 0.0506 7.. ()',998 0.0483 e'.' i'. 074 0.0454 9'. 1'. 1 4 3 0.0423 10. T.209 0.0 4 1 9 11 1.252 0.0459 12'. l'.252 0.1363 13'. 0'.995 /.SSt-?' TSUSL JOIJKOwSK Y C'-:C/4 ALF CL 0.0720 -7'. -0'.316 0.0704 -6'. -0.218 0.0700 . -5 . -(>'. 116 0.0687 -4'. -O'.O 14 0.0678 -3'. 0'. 0 8 9 0.0668 0. 185 0.0654 -r. 0'.279 0.0639 o'. ()'. 574 0.0626 r. 0'.470 0.0601 2. 0'.562 0.0576 7 ' -* • 0'.655 0'.0560 4 . (i'.74(l 0,0540 5. (i'.S27 0.0522 6'. 0 .909 0.0500 y\ 0'.991 0.0476 8'. 1.0 65 0.0442 9'. i'. 137 0.0421 io:. i"'. r->4 0.0412 1 I. ii'.230 0.0460 12'. l'.246 0.1378 13'. 0'.998 Wt" = 0.5ClOj6 70-/,-SS + P TSUSL CMO CMC/4 0'. 0182 -0 .156 -0.0751 0'. 0 1 7 0 -0 .129. -0.0733 0.0158 -0 .103 -0.0727 0'. 0 5 0 2 -0 .0 76 -0.0715 O'.O 154 -0 .04 9 -0.0699 O'.O 158 -0 '.023 -0.0690 0'. 0167 0 .0 02 -0.0677 O'.O 189 0 .026 -0.0664 0.0215 0 .053 -0,064 3 O'.0?4 1 0 .078 -0.0618 0'.0271 0 .104 -0,059a 0'. 03 0 0 0 '.128 -0.0575 0'. 0334 0 .151 -0.0553 0'.0370 0 '.174 -0'.0530 0'.0424 0 .196 -0.0511 0.0475 0 '.217 -0.0482 0].0533 0 .238 -0.0452 O-. 0 603 0 .255 -0.0432 0'. 0693 0 .266 -0.0425 0'. 0 3 0 0 0 .26 0 -0.0475 0'. 1925 0 .110 -0.1407 RF=0.5C10)6 8 05.SS+P TSUSL CO CMO CMC/4 0'. 0 170 -0 .152 -0.0734 O'.O 154 -0 .126 -0.0717 0'. 0 1 4 3 -0 .099 -0.0704 O'.O 136 -0 '.073 -0.0695 O'.O 133 -0 '.04 6 -0.0682 0.0140 -0 '.022 -0.0673 O'.O 154 0 .004 -0.0654 O'.O 167 0 .029 -0.0633 O'.O 180 0 '.055 -0.0614 0.0199 0 '.081 -0.0586 0'. 0226 0 .106 -0.0567 0.0257 0 .129 -0.0544 0'. 023 6 0 .152 -0.0525 0'.0335 ' 0 '.175 -0.0505 0.0374 0 .197- -0 .0485 0'.0426 0 '.217 -0.0456 0.0475 0 .233 -0.0427 0.0549 . 0 .254 -0.0405 0.0630 0 .262 -0.0405 0'. 0 7 4 2 0 .259 -0.0470 0'. 18 72 0 .115 -0 .1561 Table 8. Quantities derived from balance results. 0.25 Wall .. Clark-Y . m Re = ctro 0.45(10)6 X ac da a mo solid 0.1011 -6.42 0.231 0.0276 -3.12 40%LP .0970 -6.45 .233 . 0263 -2.96 50%LP .0958 -6.48 .229 . 0263 -3.00 60%LP . 0954 -6. 54 . 228 . 0263 -3.02 70%LP .0947 -6.60 . 255 .0265 -3.11 80%LP . 0940 -6.67 .229 .0259 -3.25 40%SP .0967 -6.51 .228 .0267 -2.99 50%SP .0952 -6.52 .230 . 0261 -3.03 60%SP . 0944 -6.57 .231 . 0257 -3.09 70%SP .0917 -6.36 . 277 .0245 . -2.77 80%SP .0906 -6.46 .228 .0250 -2.90 0.39 Clark-Y Re = 0.5(10)6 Wall m X dCM a ac da u mo solid 0.1032 -6.28 0. 235 0.0275 -2.86 40%LP .0956 -6.16 .229 .0261 -2.66 50%LP .0936 -6.22 .230 .0254 -2.73 60%LP .0914 -6.35 .227 .0251 -2.79 70%LP .0903 -6.40 .225 . 0250 -2.87 80%LP .0882 -6.54 .223 . 0246 -3.01 40%SP . 0952 -6.19 .227 .0261 -2. 69 50%SP .0941 -6.21 .230 . 0256 -2.72 60%SP .0923 -6.30 . 233 .0249 -2.78 70%SP .0917 -6.36 .227 .0253 -2.84 80%SP .0912 : -6.45 .222 .0255 -2.92 0.53 Clark-Y RE = 0.5 (10)6 Wall m c*i 0 X dCM a ac da u mo solid 0.1044 -6.42 0. 235 0.0278 -2.97 40%LP .0944 -6.27 .231 . 0256 -2.73 50%LP .0932 -6.35 .234 . 0250 -2.79 60%LP .0917 -6.39 .230 .0249 -2.81 70%LP .0907 -6.43 .232 . 0245 -2.90 80%LP .0889 -6.55 .226 .0246 -3. 03 40%SP .0938 -6.29 . 233 .0252 -2.73 50%SP .0928 -6.32 .233 .0250 -2.75 60%SP .0920 -6.35 .232 . 0249 -2.76 70%SP .0907 -6.39 .231 .0246 -2. 82 80%SP .0895 -6.47 .227 . 0246 — 2.96 Table 8 (cont'd). 0.66 Clark-Y Re = Wall . . m otic, solid 0.1074 -6. 33 40%LP .0947 -6. 02 50%LP .0931 -6. 14 60%LP .0909 -6. 19 70%LP .0899 -6. 16 80%LP .0884 -6. 31 40%SP .0932 -6. 05 50%SP .0930 -6. 19 60%SP .0915 -6. 11 70%SP .0902 -6. 18 80%SP .0887 -6. 17 0. 66 Clark-Y Re Wall m Ctl 0 solid 0.1124 -6. 28 40%LP .0989 -5. 95 50%LP .0974 -6. 03 60%LP .0962 -6. 09 70%LP .0955 -6. 09 80%LP .0930 -6. 21 4 0%SP .0982 -6. 00 50%SP .0976 -6. 00 60%SP .0946 -6. 13 70%SP .0934 -6. 11 80%SP .0924 -6. 16 0.5(10)6 0.234 0.0283 -2.74 .234 .0250 -2.41 .235 .0245 -2.50 . 224 . 0249 -2.50 .226 . 0244 -2.57 . 219 . 0247 -2.81 .232 .0248 -2.41 . 224 .0255 -2.53 .229 .0246 -2.46 .228 .0244 -2.57 .222 .0245 -2.68 ..0 (10) 6 dC„ X ac da " a mo 0.243 0.0287 -2. 82 .238 .0257 -2.41 .238 .0253 -2.49 .235 .0253 -2.55 .231 .0255 -2.58 .228 .0251 -2.82 .235 .0259 -2.43 .237 . 0255 -2.49 .233 .0251 -2.60 .230 . 0250 -2.62 .255 .0253 -2.76 240 Table 8 (conf d) . 0. .17 NACA--0015 Re = 0.3 (10) 6 Wall . m ct i o X a ac da m o solid 0.0982 0.03 0.230 0.0251 0.08 40%LP .0958 .04 . 231 .0245 . 04 50%LP .0957 • -0.01 .232 . 0243 - .01 60%LP . 0961 - .04 .227 .0249 .00 70%LP .0967 - .02 .233 . 0244 - .08 80%LP .0960 - .10 . 236 . 0240 - .16 40%SP .0967 .04 .232 .0245 .04 50%SP .0963 .04 .230 .0247 .03 60%SP .0954 .00 .232 . 0243 - .01 70%SP .0958 .00 . 231 .0245 .00 80%SP .0956 - .04 .229 .0246 - .05 0. 34 NACA--0015 Re = 0.5(10)5 Wall m Cti o X dCM a ac da 0 m0 solid 0.0993 0.01 0. 236 0.0266 0.02 40%LP .0945 .10 .231 .0258 . .07 50%LP .0941 .07 .233 .0255 .02 60%LP .0929 .04 .234 . 0251 - .01 70%LP .0927 .03 .233 .0251 - .04 80%LP .0927 - .07 . 234 .0251 - .13 40%SP .0929 .06 .236 .0249 . 03 50%SP .0925 .04 .237 .0247 - .03 60%SP .0924 .01 .230 . 0253 - .03 70%SP . 0928 .01 .231 .0254 - .04 80%SP .0925 - .01 .233 .0251 - .05 0. 50 NACA-•0015 Re = 0.5(10)6 Wall m X dCM a ac da u mo solid 0.1036 0.04 0.252 0.0249 0.07 40%LP .0940 .14 .250 . 0228 .07 50%LP .0935 .13 . 249 .0227 .04 60%LP .0936 .12 .249 .0227 . 03 70%LP .0924 - .01 .247 .0226 - .01 80%LP .0906 - .04 .246 .0223 - .16 40%SP .0933 .14 .250 . 0226 .07 50%SP .0929 .14 .249 .0226 .07 60%SP .0929 .12 .246 .0229 .02 70%SP .0921 .20 . 249 .0226 . 00 80%SP .0913 .06 .247 .0224 - .08 Table 8 (cont'd) 241 0. 67 NACA--0015 Re = 0.5(10)5 Wall . m • - -a io X dCM a ac da 0 m0 solid 0.1080 -0.03 0.214 0.0264 0.01 40%LP .0954 0. 25 . 219 .0228 .13 50%LP .0945 .22 .219 . 0226 . 06 60%LP . 0940 .13 . 211 .0232 - .01 70%LP .0915 . 09 . 212 .0225 - .12 80%LP .0891 - .04 . 210 . 0221 - .31 40%SP .0960 .22 .214 .0234 .10 50%SP .0935 .22 .216 . 0226 .07 60%SP . 0928 .23 . 215 . 0226 . 07 70%SP .0921 .16 . 212 .0227 .00 80%SP .0902 . 05 .211 . 0223 - .18 0. 67 NACA-0015 RE = 1.0(10)6 Wall m X ~=i—Mn a ac da 0 m0 solid 0.1114 -0. 08 0.226 0.0259 -0. 08 40%LP .0981 .29 .213 .0241 .15 50%LP .0969 .25 .208 .0242 .25 60%LP .0952 .23 .213 .0233 . 08 70%LP .0940 .19 . 213 .0230 .00 80%LP .0922 .04 . 208 . 0230 - .18 40%SP .0975 .22 .213 . 0239 . 11 50%SP .0962 .27 .211 .0237 .14 60%SP .0943 .22 . 212 . 0232 .06 70%SP . 0935 .17 .214 .0229 .00 80%SP .0935 .16 .210 .0232 - .05 Table 8 (cont'd) . Joukowsky Re = 0.5.(10) 6 Wall . m a i o • X ac ' dc7M° ' a m0 solid#l 0.1059 -3.78 0.236 0.0275 -1.14 solid#2 .1056 -3. 78 .236 . 0274 -1.14 solid#3 .1055 -3.77 .236 .0274 -1.14 40%LP . 0988 -3. 74 . 233 . 0260 -1. 02 50%-LP .0975 -3.77 .234 . 0256 -1.07 60%LP .0965 -3. 83 .232 .0255 -1.07 70%LP .0958 -3.88 .230 .0254 -1.15 80%LP .0954 -3.98 .230 .0254 -1.29 40%SP .0984 -3. 80 . 232 . 0259 -1.09 50%SP .0970 -3. 80 .233 .0255 -1.10 60%SP .0965 -3.80 .232 . 0254 -1.10 70%SP .0968 -3. 83 .231 .0257 -1. 04 80%SP .0959 -3. 85 .230 .0256 -1.14 Table 9. Pressure coefficients for NACA-0015 airfoil RErl.0 + (i0)6 SOLID WAl.L S O.feT-wACX-OOIS 0(115 AL"-'. X/C o'. 0 0.2 0. « 1. o 1'. 5 ' •I .9 T>' c 2'. 9 1'. 9 7, i .8 1 2'. H ' 1'. 1 1 7 . 3 a 1.00 c'.77J o'.5 5 i o'. 151 -1''. 0 - i 3 - o'. i o o -o,336 -0.183 29.7 ' 'J'! 7 3 9. 6 1 9 . 5 59 , 5 t«, i 7 -l 2 P, ^ u 97']o 0.? 0.3 1 '. 0 1 ^ •"V-i -? . ^ 2 .' il'. 1 (',« J«'.8 ?. it. fl' 29.7 ij 9. 5 6 O, 1 79,2 r.5 r 96'] 9 C, - o.: -'i i -o'. 5 25 -C''.5u9 - 0 . 1 H 3 -O.^-iill - 0 '. 2 f 9 -fc'.221 - 0 '. i -'I 3 - 9 '. .1 ;i c 0'. 0 9 3 Cl'. 7 60 tj'rr,?.'J (I . ! 7 2 0 '. 0 1 2 «'. 1 61 •0.271 •<-•'. 3 3 1 i.'. -'i 70 0 '. 5 7 7 o.'.593 o'.59i 'J^u9 0.357 «'.216 «'. 1 1 3 o'.o 8 8 o.« 71 0'."77 0 . il n 7 - 0 . 3 6 7 - o.L; 'i 5 -o '.i, (.-•.; -0.7 0 '.5 -0.8 -'I .'| -o .Kr-"/ -0 ,8'K| - 0 . * 2 3 -0'. 7?'. -0.765 -0.723 -0.707 -0 . (.31 -0 .1,30 - 0 . 6 ,"• 9 -0.571 •ii.'.Jl - o.:. 'i;' -0.2 3.1 - 0 '. 1 !. 3 -0.017 O.Ol.'l 0.958 0.859 0 . 55u 0'.377 0 . 2 3 0 Cl'. 1 2 0 O,0«3--0.15,-' -0.330 -0 .1'lfl -0'.-/•/3 -0.383 •0.28a •0.17a • 0 . 1 2 t • •0.053. • 0.073 0'.5?7 -0^51 -9,6'1.3 - 1 . 0 il 0 -l'. 176 -l'.275 -1.29I -l',302 - 1'. 7 5 - 1'. .-: 13 - 1 j. l as -1018 -1 .061 - 1'. 0 0 'J -O'.'.i) 1 - o'. 9 a 4 - o'. n ft 3 -0>16 -'/.769 -0'.71 1 -9'.6 13 -0'.51,1 -O'.'IOS -0'.303 - 0 '. 114 0 3fl o.°. a 932 *;<?a ,836 o'.6B9 0". 5 6 il ')'. (15 a 0.371 <>'. Iil6 - 0 c 9 a • 0.1-0<J •0'.251 •0' 556 •0.2 15 •0' 152 •0,115 •0.0 37 -0. 153 -1 .179 - 1 .6 0 il -1.113 -2,015 -2.025 -I .991. - 1 , 9 5 8 -1, n o o -1 .6-15 -1.527 - 1, '11 5 -1.312 -1 .225 -1.117 - 1.086 -1.030 - 0 .') 7 1 -0.9?,.? -0.810 -0.753 -0 , >591 - 0 . '16 1 -0.313 -0.215 -0.H71 0,06.3 0 . 765 0.039 0'. 911 0.109 0 . fl 2 1 0.73H 0 , A 6 2 0.137 0.150 0.011 -0.001 -0,112 •0.117 • •o.odt • -0.066 • •0.030 0.037 •1 , 0.001 0.201 0.06 0.427 0.674 081 ».?'. 396 aoo -3'. 021 -2'.9/ifl -2'. 907 615 -2'.71rt -2'. .306 151 -1.355 -t',,722 -l'. 591 -r. 'i87 -1.370 -r. 309 -1 '.212 -r. 155 -l'. 099 -0'.977 -0'.069 -0.696 -0'.537 - o'. 3 a v -0'.251 - 0'. 0 9 3 o'. 02-1 0'. 3 0 0 0'. 668 0'. 9 fl 1 1 '.0 05 0'.9b9 0 '. 9 1 S 0 '. 3 6 7 0', 6 6 0'. 36 1 C.203 0'. 060 O'. 021 •0'.022 •0'. 0 22 •0.011 . O'.OH • 0'. 050 • - 1 0 -2.101 -3.573 - 3 .912 -1.035 -3.H15 -3.701 -3.50 7 -3.3»8 -3,02U -2,iii -2.199 -2.03D -1 .(155 -1.732 -1.5*9 -1.191 -1,3fl9 -1.317 -1.235 -1.096 - 0 .') 7 ft -0.718 -0.5/3 -0.109 -0,250 -0, 102. -0,025 -0,2ril 0,252 0.817 0,9fl5 1 ,001 0,991 0,965 0 , 8 0 6 0.509 0 , 339 0, 160 0.111 0.032 O.OOl -0,009 •0,009 . •O.OOfl . 12 -3.302 -1>12 -5.301 -5'. 1 12 -'(.773 -l'.518 -•'1.363 -"'.235 -3.210 -2.902 -2'.579 -2.317 -2.117 -1 >3fl -1'. 7 71 -1 .671 -l'.558 "1'.'I51 -1.363 -1.199 • l'. 056 -0.810 -0',59ft - 0 .1 1 0 -0'. 25 1 -0.118 -0,102 -1,015 -0.297 0.616 0.883 0.970 1.000 1.011 0 '. 9 1 3 0.612 0.162 0.262 0.211 0.093 0'. 0 31 o'.OOh '0.015 • .O'.OflO • 11 -'I .555 -6.330 -6,573 -6,191 -5.77C. -5 ,1611 -5,262 -5.159 -3.HH9 -3,311 -2,898 -2.573 -2,35 1 -2.135 -1.911 -1.820 -1,675 - 1,567 -1.153 -1.273 -1.092 -0,803 -0,555 -0,395 -0.308 -0.256 -0,225 -1,80 1 -0.917 0.327 0,709 0.871 0,957 0.998 0,972 0.735 0.55i) 0,313 0,281 0.131 0,013 0,0 02 • •0,050 . •0,117 • - 16 -5', 567 -7.168 -7.571 -7.006 -6'.5'l3 -6'. 286 -6.081 -5.113 - 1'. 2 1 6 -3.532 -3'. 085 -2.715 -2'.132 -2'.201 -1.97 0 -I.826 -I'.636 -l'. 197 -1.371 -1.117 -0,896 -0.670 -0.618 -0.583 -0'. 562 -0,190 -0.103 -2'.576 171 Oil 55 3 77'i 903 980 005 0'. 8 0 0 0.630 0.399 0'. 322 0. 152 0'. 021 0.033 0.110 0.207 ALPHA X/C °'2 0,2 0 .1 I'.O J '.5 1'.9 2.5 2'.' 1.9 7'. 1 9'. 8 12'." 11.8 17.3 20.2 22.3 21.8 27.2 29.7 3'('.7 39.6 '19'. 5 59.5 6 9', 3 79.2 8 8'. 9 ' 97', 0 0.2 0'. 3 1.0 1'.5 2.0 2'. 1 2. 9 1'.9 9'.9 11.0 21'.8 2^'. 7 19.5 69'. 1 79.2 88 '.5 96'.9 RF=l'.0*('lO)6 2 1 lO^LS Oi» UPPER, SOLID LOKLH; PLu-^UH 0.940 1.120 1.301 1.452 1.541 1.0 02 0'. 7 38 0'.568 0'. 206 -0'.2'I9 -(.''. 126 -0,203 -0.269 -0^112 -0.179 - 0'. 5 0 9 -0.530 -0^511 -0.501 -ll'. 19 9 -0'..'I81 - 0). - I ti fi -0,118 -0,133 -0,112 -0,376 -0.3 0 5 - '•>'. 2 Z 9 -»'. 177 -o'. 1 1 1 -o'. 021 0'. 1 09 o),757 0.568 0'. 1 65 -o'.ooi -0'. 157 -o),261 -0,315 -0.163 - 0'. 5 5 0 -0'.565 •0'.509 •0'.'43« •0'.336 •ll'f\9Z •0.116 • •0'.031 • 0'. 1 0 9 0,91.'| 0'. 16 7 0'. i53 -0.218 -0,117 - 0 . 5 " 0 -0.592 -0.613 -0.720 -0'. 711 -0.725 -0'.703 -0'.68i -0.659 -0.633 -0.607 -0,576 -0.551 -0.525 -0.179 - 0'. 13 3 - 0', 315 -0.273 -0,201 -0.109 -o'.oi i 0.122 0.96'j 0.863 0.5 39 0.361 0.225 0.117 0'. 0 -10 •0.115 •0.311 •0.376 •0.371 •0,355 •0.263 •0.150 •0.093 •0.021 0.107 0.623 -0'. 061 -0'.(i2u - 0'. 8 0 6 -0'.910 -l". 012 -1'. 058 -l'.078 -T.073 -1'. 0 22 -C.960 - 0 > 1 9 -0'. 863 -0'.H16 -'j'.770 -<>'. 739 -0^703 -0.662 -f>'. (,36 -0'.575 - 0'. 5 2 3 - 0 '. '! 1 0 -0'.313 -0'. 225 -<>'r 133 -0.020 0^ 1 1 9 0.993 0'. 913 o'. n o a 0.639 o'.53S 0].127 0.315 •0', 1 2" -0,097 -0.189 -B'.211 -0'.211 -0'.205 -0'. 133 -0'.092 - 0'. 0 1 5 0'.093 0. 127 -0.78! -1,169 -1.511 -1.53 I -1'.613 -1.612 -1 .597 -l'.'nn -I.3«9 -1,251 -1.163 - 1.065 -0,098 -0.92 I -0.830 S23 781 75 1 673 59 6 .'16 7 353 •0.255 •0.152 •0.02S O.i-it 0.370 0.983 0.963. 0 . '*75 0.78 2 0.689 0.617 0.-95 0. 127 0.003 •0.085 •0.105 •0. 100 •0.061 •0.011 •0.0)3 0. 0S5 -0.027 0.143 0.327 8 10 12 11 16 -0'.663 -1.5H -2 '.556 -3,703 - a '.931 -f.7 78 -2.828 -1 '.028 -5.313 *b .6 17 -2'. 163 -3.175 -1 . 291 -5,509 -h '.7 56 -2'.'100 -3.251 .1 '. 2 28 -5,230 -0 '.232 -2'. 379 -3.151 -3 .952 -1,819 _ r '.7 79 • i?' ^r. z -3.0-2 - 3 .776 -1,611 - lj .50 2 -2'!i76 -2,899 - 3 '.623 -•I ,136 _ r . 120 -2'. 189 -2.766 - 3 ,161 -1, 31 3 - a . 651 -l'.''68 - 2 , il 2 9 _ (i '.853 -3,223 '.7 55 -l'. 717 -1.985 -2 .102 -2,753 -3.107 -f.557 -1.796 -2 .105 -2.108 m 2 ', 69n -l',397 -1,633 - 1 .891 -2.165 . ~> .10 3 - 1'. 2 9 0 -1.510 -r .7 35 -1.9^9 -2 ,156 -l'. 197 -1.382 -i' .597 -1 ,783 -1 . 9 S 6 - 1'. 1 1 5 -1.275 -r. .153 -1.603 , 761 -l'. 053 -1.209 -1; .366 -1 .520 -r . 0 6 3 -0.997 -1,132 -r, . 259 -1.112 -1' . 5 0 1 - 0'. 9 3 0 -1,050 -1', .192 -1 .308 -1 .HI -0'. 889 - 0 , 9 '.> 1 -1, , 1 20 -1.231 -1 .313 -O'. 796 -0,3«7 -0 , ,98 7 -1.076 -1 .139 - 0'. 7 0 9 -0.775 -o', , 861 -0.937 -0' , 979 -0'.519 -0.60 1 ,659 -0.715 -0; 7 "• 2 -0'. 120 -0.153 - o'. ,189 -0.529 -0, \- 1 7 - 0'. 3 0 8 -0.330 -0.351 -0.359 -0. , 112 - 0". 1 9 0 -0.192 -0'. 202 -0.201 -0'. •2 •« -O'. 056 -0,019 -o'. 059 -0,106 -c. 193 0'.C72 0,058 0.013 -0,051 -0. 157 0 '. 19 9 0.007 -o. hlB -1.386 . 2' 213 0'. 782 0 ,11 1 -0 . 012 -0.581 -1. 252 0>)97 0,906 0 . 731 0,17 2 0'. 1 2 5 0', 992 0,987 0 . 921 0. 792 0. 57 3 0'. 916 0.992 o'. '••38 0.921 c. 773 0'.890 0,962 r. 003 0.973 0 . 8 9 1 • C'. 6 2 f. 0,936 o'. 995 1 .001 0'. 96 3 C'.633 0,773 o'. 885 0.967 0 . 991 0'. 315 0,197 o'4 629 0.710 0'. 8 11 3'. 196 0.3'M 0 . iii>5 0,565 0 . t:^5 0'. 072 0,130 o'. 285 0.379 0 . 119 0.036 0.110 0. 231 0,317 0 . 388 0'. 0 0 0 0,063 0. 126 0, 183 0 . 22 8 0'. 0 0 6 0,013 0. C75 0,106 0'. 131 O'.Ol 1 0,037 o'. 05u C .069 0. 079 C.0'17 0,013 0. Oil 0,011 0. 023 0'.07H 0,063 0'. 039 O.OOS -0. 019 0.785 0.953 1. 128 1.295 1. 432 f il L Table 9. (cont'd) 0.61-NACA-oci5 0015 RE=!'.0*(10)6 50ii.S OAR UPPER/ SOLID LOWER) PLENUM 0015 RE = l'.0*(10)6 604LS -OAR UPPER, SOLID LOWER; PLENUM AL A 0 ? 1 6 ,< /c 0 , 0 1 . 0 06 0 .913 • 0 .635 0,i?2 C 1 0 .791 0 . 131 -0 .05 7 -0.764 Q , 1 0 .56 1 -0 .107 -0 ."30 -1.15 0 ; . 0 .199 -0 . 3 " 6 -0 .307 -1.4 8 0 1 c 6 " u 8 -0 .116 -0 . 9 4 2 -1,552 1 9 • 0 137 -0 .535 -1 .030 -1.603 r -(; 209 -0 .58 6 -1 .030 -1,573 -i j 9 -0 276 -0 .633 -1 .071 -1.547 , 9 -0 ,'121 -0 .715 -1,071 -1.439 7 . 1 -0 «3 - 0 .730 -1.009 - 1.325 <> .6 -0 .5 01 - 0 .7 15 -0/.5 2 -1.312 \2 . 1 - 0 .519 - 0 .7 00 -0,900 -1.135 I -!< , 8 -0 5 1 4 -0 .'.71 -0,859 -1 .047 1 7 , 3 -e 5 0 9 -6 ,648 -0 .812 -0.975 P.'J . 2 -0 .'! Q t| -0 .607 -0 .750 -0.898 P. 2 . 3 -0 6 8 3 -0 .581 -0,735 -0.846 t p. -0 13 7 -0 ,53 5 -9.68 3 -0,805 ? ~f - 0 1 4 2 -0 .535 -0 .64 7 -0.754 29 ,'7 -i' •'12 6 - 0 .509 -0 .6 16 -0,723 , 7 -0 3 8 5 -0 ,468 -0 ,554 -0,651 , 6 -0 3 6" -a .421 -0,497 -0.573 , 3 - ii 3o2 - 0 . .3 2 9 - 0 . 3!'. 8 -0.4 55 . 5 -0 -i 7 /; - 0 - 0 .295 -0.317 6 0 . 3 -<j 1 7 3 -0 . i 70 -0 .213 -0.254 7 9 -ii 101 -0 ,097 -0 1 H -0.146 8 8 , 9 -0 0 1 3 -0 . 0 01 • -0 ,006 - 0 . ll 2 2 9? , 0 0 1 ! 7 (; .13 0 0 134 0.112 0 , 2 0 779 0 ,969 a , 9 9 2 0.853 0 , J 0 •: s 7 0 .86 1 0 987 0,967 1 , 0 0 1 5 9 0 . 557 .0 8 06 0 , 956 ; , 5 0 ;KI8 0 .383 • 0 661 0.H74 -> . 0 1 3 7 0 3 ~ 8 a 54 3 0.771 , -1 -0 •j • i 3 0 0 4 39 0.688 2 , 9 -0 3 07 .0 , 0 5 2 0 356 0,621 tl ,9 -0 .4 5 3 -0 .133 0 1 4 4 0.400 9 , 9 -0' 5 •'! 5 -0 .303 -0' 083 0.132 j 8 -0 5 5 6 -() .335 -0 171 0.011 -0] 5 1 1 -0 .360 -0,333 -0,081 !7 -'/ •17 3 -0 .3'-''I -0.328 -0.100 £ - 0' 3 33 - 0 .257 -0/87 -0,094 , 1 -0' 199 -0 .138 -'•>, 1 l<> -0,058 7'/ , 2 -(/ 1 i 6 -0 . 0 '•'• 2 -0.073 -0.033 p v , 5 -0' 0 2 R -0 .0 15 : O' 000 - 0 . (10 7 96 .9 ()' 117 0 .109 0.103 0,091 c -0 .023 0 .145 0 .326 0. 530 L 8 10 12 14 16 -0'.656 -1.510 -.2.4 98 -3,630 -1' 763 -f.768 -2.80a -3.93! -5.180 -6' 367 -3.142 -3.151 -1.212 -5.392 -6' 542 -2'. 375 -3.270 -1.119 -5.107 -6' 0 25 -2'.355 -3.151 -3'.92l -1.718 -5' 637 -2'. 331 -3.021 -3'. 734 -4.490 -5 109 -2'. 251 -2.902 -3.517 -1.330 "\ 259 -2;173 -2,777 -3.417 -1,210 -4' 69 0 -1.939 -2.450 -2.610 -3.132 _ i' 6 4 1 -l'.721 -1,956 -2.316 -2.687 035 -1.513 -1.780 -2.077 -2.350 - 2' 601 -l'. 358 -1.629 -1.854 -2.080 -1? 270 -l'. 259 -1.164 -1.693 -1.873 -2' 053 -f. 17B -1.354 -1.558 -1,702 -1 8 67 -1.077 -1,260 -1.397 "1.557 670 -l', 025 -1 . 193 -1.311 -1.453 -1 . 57 7 -0'. 968 -1. UO -1.231 -1,319 -f. 153 - 0'. 9 0 6 -1,037 -1'. 1«3 -1,26! -l'. 341 -0'.864 -0,975 -l'. 080 -1.163 -1 216 -0'.766 -0,871 -0/56 -1.028 -1. 075 - 0'. 6 7 7 -0.762 -0'.82b -0,893 -0. 93d -0'.527 -0.596 -O.t.39 -0,651 -0. 67 2 -0.407 -0.415 -0.^73 -0.191 -0. 48 1 -0'. 2 98 -0,315 -0.332 -0,328 -o'. 320 -0'. 179 -0.135 -0.167 -0,183 -0. 212 - n'. 0 41 -0,040 -0,017 -0.060 -0. 160 0'. 081 0.064 0'. 0 2 6 -0.028 -0. 13<l O'. 4 96 0,012 - 0'. h 0 a -1,324 -2 * 141 0' 7 82 0.H3 -0.005 -0,551 -1 '. 189 1.000 0.910 0.737 0,4 a 0 0'. 135 fl'.995 0.999 0.9 24 0.791 0. 5 7 4 0 .999 0.99 7 0,936 • 0. 786 0'.89i 0,978 1.007 0 ,988 0'. 905 0'.B31 0.932 0 '. 9 9 3 1.009 0 . 962 0'.631 0,791 0 '. 9 0 3 0,957 0. 998 0'. 351 0,505 0'.633 0.739 0'. B23 0'. 206 0.319 0'.472 0.579 0. 667 0'. 076 0, 18a 0'.296 0.397 0'. 8 28 0'. 0 a 5 0,117 0.244 0.330 0. 4 04 0'. 0 0 3 0,074 O.HO 0.205 0 . 249 0'. 0 1 3 0,018 0'.093 0.128 0 . 150 0'.021 0,013 0.073 0.09 1 0, 104 0'. 055 0,053 .0.057 0.0 6S 0. 052 o'.oai 0.0A5 0'.057 0,024 -0, 020 0.770 0.950 1.114 1.266 1 404 ALPH V 0 2 4 6 •A/C 0'.6H 0 . 0 f. 000 0.905 0,098 0.2 0'. 793 0.4 44 -0'.079 -0.796 0.1 0'. 5 5 5 0', 1 2" -0/131 -1.200 r.o 0. 133 -0,25" -0.814 -1.510 1.5 -0'. 0 04 -0,430 -0/'43 -1,588 r.9 143 - 0 , 5 4 9 -1,026 -1.624 2'. s •u'r 221 -0.606 -1,052 -1,603