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An investigation of the theoretical and experimental aerodynamic characteristics of a low-correction… Malek, Ahmed Fouad 1983

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c ,1 AN INVESTIGATION OF THE THEORETICAL AND EXPERIMENTAL AERODYNAMIC CHARACTERISTICS OF A LOW-CORRECTION WIND TUNNEL WALL CONFIGURATION FOR AIRFOIL TESTING BY AHMED FOUAD MALEK B.Sc.(Engineering), Alexandria University, 1970 B.Sc.(Mathematics), Alexandria University, 1974 M.A.Sc.(Mechanical Engineering), University of British Columbia, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1983 (Q) Ahmed Fouad Malek, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia,,I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. The University of B r i t i s h Columbi 1956 Main Mall Vancouver, Canada V6T 1Y3 x-n Department of DE-6 (3/81) - i i -ABSTRACT This thesis deals with a new approach to reduce wall corrections in h i g h - l i f t a i r f o i l testing, by employing two similar non-uniform trans-versely slotted walls. The solid elements of the slotted wall are symmetrical a i r f o i l s at zero incidence, and the spaces between the slats are non-uniform, increasing linearly towards the rear. This wall configuration provides the flow conditions close to the free a i r test environment which lead to negligible or small wall cor-rections. The theory uses the potential flow surface vortex-element method, with "Full Load" Kutta Conditions satisfied on the test a i r f o i l and wall slats. This method is very well supported by physical evidence and i t is simple to use. The surface velocities can be calculated dir-ectly and the aerodynamic l i f t and pitching moment are determined by numerical integration of the calculated pressure distributions around the a i r f o i l contour. This method can be developed to include a simulation of the flow in the plenum chambers in the analysis. •Also, the performance of this new wall configuration was examined experimentally. Two different sizes of NACA-0015 a i r f o i l were tested in the existing low speed wind tunnel after modifying both the configura-tion of the side walls and the test section to accommodate the new test. Pressure distributions about the test a i r f o i l s were measured using pressure taps around their contours. Also the l i f t s and the pitch-ing moments were obtained by integrating the measured surface pressures. The experimental,results show that the use of the new wall configuration - i i i -with AOAR = 59% would produce wind tunnel test data very close to the free air values. - iv -TABLE OF CONTENTS P a g e Abstract i i Table of Contents . iv List of Figures v i List of Tables ix Symbols x Acknowledgement .. x i I. INTRODUCTION 1 II. POTENTIAL FLOW ANALYSIS 6 11.1 Introduction 6 11.2 Surface Singularity Theory 7 II. 3 The Kutta Condition 10 III. ASSEMBLING AND SOLVING THE BASIC EQUATIONS 13 IV. RESULTS OF THEORETICAL INVESTIGATION 19 V. EXPERIMENTAL INVESTIGATION 24 VI. EXPERIMENTAL RESULTS 30 VII. CONCLUSIONS AND RECOMMENDATIONS 33 REFERENCES 35 - v -(Continued) Page APPENDIX 1 - AN ANALYTICAL REPRESENTATION OF A SURFACE VORTICITY 37 APPENDIX 2 - EVALUATION OF THE INTEGRAL IN EQUATION (5). THEORY: REF. [10] 40 APPENDIX 3 - CALCULATION OF THE VELOCITY COMPONENT INDUCED AT A POINT IN THE FIELD OF VORTEX DISTRIBUTION 45 APPENDIX 4 - AN ANALYTICAL APPROACH TO THE FLOW IN THE WIND TUNNEL PLENUM 48 A. Introduction 48 B. The Induced Tangential and Normal Velocities due to Vorticity Distribution 48 C. Simulated Shear Layer Tracking 50 APPENDIX 5 - LIST OF EQUIPMENT USED IN THE EXPERIMENTAL INVESTIGATION 53 FIGURES 54 TABLES 91 - v i -LIST OF FIGURES Page FIGURE (1) COMPARISON OF AIRFOIL PRESSURE COEFFICIENTS: THEORY KEF. [9] 54 FIGURE (2) VORTEX REPRESENTATION OF TWO COMPONENT AIRFOIL: REF. [10] 55 FIGURE (3) NOTATION USED TO CALCULATE INFLUENCE COEFFICIENTS: REF. [10] 56 FIGURE (4) STREAMLINES CONTOURS AROUND AN AIRFOIL: REF. [12] FIGURE (5) LOCATION OF TRAILING CONTROL POINT: REF. [10] 57 58 FIGURE (6) AN AIRFOIL INSIDE TUNNEL TEST SECTION WITH DOUBLE SLOTTED WALL: APP. 4 59 FIGURE. (7) NOTATION USED TO CALCULATE INDUCED VELOCITIES: REF. [10] 60 FIGURE (8) LOCATION OF ELEMENTS ON AIRFOIL SURFACE: REF. [10] FIGURE (9) COMPARISON OF HIGHER ORDER METHODS: REF. [10] 61 62 FIGURE (10) COMPARISON OF AIRFOIL VELOCITY DISTRIBUTIONS: REF. [10] 63 FIGURE (11) GEOMETRY FOR CALCULATION OF HIGHER ORDER TERMS: APP. 2 64 FIGURE (12) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION 65 FIGURE (13) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-23012 AIRFOIL IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION 66 FIGURE (14) COMPARISON OF PRESSURE COEFFICIENTS FOR CLARK-Y 14% AIRFOIL IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION 67, FIGURE (15) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN SINGLE UNIFORMLY SLOTTED WALL TEST SECTION 68 FIGURE (16) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN DOUBLE UNIFORMLY SLOTTED WALL TEST SECTION 69 - V l l -(Continued) Page FIGURE (17) COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN LIFT COEFFICIENTS 70 FIGURE (18) COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN PITCHING MOMENT COEFFICIENTS 71 FIGURE (19) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NON-UNIFORMLY SLOTTED WALL TEST SECTION 72 FIGURE (20) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-23012 AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NON-UNIFORMLY SLOTTED WALL TEST SECTION 73 FIGURE (21) COMPARISON OF PRESSURE COEFFICIENTS FOR CLARK-Y 14% AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NON-UNIFORMLY SLOTTED WALL TEST SECTION 74 FIGURE (22) COMPARISON OF THE EFFECT OF TWO DIFFERENT ANGLES OF ATTACK ON THE RELATIVE ERROR IN LIFT COEFFICIENTS FOR NACA-0015 AIRFOIL 75 FIGURE (23) GEOMETRY OF THE IDEALIZED FLOW IN THE PLENUM CHAMBER FOR NACA-0015 AIRFOIL 76 FIGURE (24) COMPARISON OF NACA-0015 AIRFOIL PRESSURE COEFFICIENTS: FLOW IN WIND TUNNEL PLENUM IS INCLUDED IN THEORETICAL ANALYSIS 77 FIGURE (25) CALIBRATION OF NOZZLE AND TEST SECTION DYNAMIC PRESSURES 78 FIGURE (26) WALL EFFECT ON THE PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.34: EXPERIMENT 79 FIGURE (27) WALL EFFECT ON THE PRESSURE' COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.67: EXPERIMENT 80 FIGURE (28) EFFECT OF AOAR OF SLOTTED WALLS ON PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.34: EXPERIMENT 81 FIGURE (29) EFFECT OF AOAR OF SLOTTED WALLS ON PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.67: EXPERIMENT 82 FIGURE (30) COMPARISON OF NACA-0015 AIRFOIL PRESSURE COEFFICIENTS: EXPERIMENT 83' - v i i i -(Continued) FIGURE (31) FIGURE (32) FIGURE (33) FIGURE (34) FIGURE (35) FIGURE (36) FIGURE (37) Page E F F E C T OF WALL CONFIGURATION ON L I F T C O E F F I C I E N T FOR NACA-0015 A I R F O I L , C/H=.34: EXPERIMENT 84 E F F E C T OF WALL CONFIGURATION ON L I F T C O E F F I C I E N T FOR NACA-0015 A I R F O I L , C/H=.67: EXPERIMENT 85 EFFECT OF WALL CONFIGURATION ON P I T C H I N G MOMENT COEFFICIENT FOR NACA-0015 A I R F O I L OF SIZE C/H=.34: EXPERIMENT 86 EFFECT OF WALL CONFIGURATION ON P I T C H I N G MOMENT COEFFICIENT FOR NACA-0015 A I R F O I L OF SIZE C/H=.67: EXPERIMENT 87 EFFECT OF A I R F O I L S I Z E ON E ^ FOR NACA-0015: COMPARISON OF EXPERIMENT AND THEORY 88 A I R F O I L S ECTION C H A R A C T E R I S T I C S AS A F F E C T E D BY V A R I A T I O N S OF THE REYNOLDS NUMBER: REF. [17] 89 U.B.C. MECHANICAL E N G I N E E R I N G LOW-SPEED C L O S E D - C I R C U I T WIND TUNNEL 90 LIST OF TABLES TABLE 1 AERODYNAMIC CHARACTERISTICS OF AIRFOILS EXAMINED THEORETICALLY TABLE II EFFECT OF THE PRESSURE ALONG THE SHEAR LAYERS ON THE NEW WIND TUNNEL THEORETICAL DATA TABLE III LOCATIONS OF THE PRESSURE TAPS TABLE IV THE EXPERIMENTAL AERODYNAMIC CHARACTERISTICS OF NACA-0015 AIRFOIL TABLE V THE LIFT-CURVE SLOPES OBTAINED EXPERIMENTALLY FOR NACA-0015 AIRFOIL SYMBOLS Definition Open Area Ratio; OAR = Slot size/(slat size + slot size). Uniform Open Area Ratio. Average Open Area Ratio. A i r f o i l chord: tunnel test section height ratio. Uniform flow velocity. Stream function. Vortex density or induced surface velocity. Total head. Static pressure head. Dynamic pressure head. Pressure coefficient The average pressure coefficient along the upper constant pressure streamline. The average pressure coefficient along the lower constant pressure streamline. L i f t coefficient. Leading edge pitching moment coefficient. Quarter chord pitching moment coefficient. L i ft-curve slope. Free a i r data. Wind tunnel data. Relative error in l i f t coefficient; Slat chord: a i r f o i l chord ratio. * 100 = (m - m T F Relative error in pitching moment coefficients; M - C, M * 100 c/4 - x i -ACKNOWLEDGMENT This is to acknowledge Dr. G.V. Parkinson's expert supervision, invaluable advice and his academic as well as financial support to me. These were very important factors in helping me carry out this research in particular and achieving a successful graduate work in general. Also, I acknowledge the help I received from the staff of the Department of Mechanical Engineering at U.B.C. during the time I was working on my re-search. I especially acknowledge my wife's moral support and her wonderful effort in typing my thesis. - 1 -CHAPTER I INTRODUCTION In recent years, much attention has been devoted to a variety of problems associated with wind tunnels as simulators of f l i g h t . It is im-portant to test the largest possible model in any given f a c i l i t y , not only to maximize Reynolds number but to improve the accuracy of the model i t s e l f and the forces and moments measured in the tunnel. Today's aircraft designers are demanding that wind tunnels should accommodate testing of a new breed of h i g h - l i f t wings using large models which produce r e a l i s t i c a l l y high Reynolds numbers, and that wind tunnel data should be measured with great accuracy. Aerodynamicists are then pushed towards testing large models and that often involves large flow distortions in a flow regime where wall effects cannot be accurately estimated and where their effects on the measured aerodynamic data cannot safely be accounted for by the standard linear corrections. An obvious solution to the above problem could be achieved by using wind tunnels with very large test cross-sections. However, this approach is not always possible since the cost of construction of wind tunnels of sufficient size is very high, and few such large tunnels are available. Another solution has been suggested by Sears [1] and others [2], in which the test section of the conventional wind tunnel is modified by installing a two-dimensional test section with perforated top and bottom walls, and providing for auxiliary blowing and suction at the top and bottom walls. The model flow f i e l d inside the test section i s detected by means of pressure sensors, then i t is adjusted by auxiliary suction (or blowing) through the control valves un t i l the flow f i e l d agrees with a desired unconfined value. - 2 -Vidal and others [3] have introduced some modifications to Sear's concept of the self correcting wind tunnel. The basis idea i s to approxi^ mate a continuous distribution of velocities, in a stepwise fashion, by segmenting the plenum surrounding the porous walls, and controlling the flow through the walls by applying suction or pressure to the plenum seg-ments. In this way i t should be possible to approximate the unconfined disturbance flow f i e l d about a two-dimensional a i r f o i l . Unfortunately, a l l the above self correcting wind tunnel arrangements need a complex system of sensors and expensive computer hard ware as well as the auxiliary equipment such as pumps or blowers. Pollock [4] has introduced the concept of the self streamlining wind tunnel, in which a set of tensioned membranes replace the solid walls and under the action of the model pressure f i e l d , deform in the same way as a streamline in an in f i n i t e flow f i e l d . In order to prevent the col-lapse of the membrane towards the model i t i s then equipped with a number of equally spaced pressure tappings which communicate with flexible bellows with r i g i d connecting tubes. The volume behind the wall membrane and containing the flexible bellows is subject to a backing pressure which is adjusted so that the sum of the normal wall force components at the two ends of the wall is equal to zero. One of the basic problems with such conformable walls i s that the membrane tends to bulge between the pressure tappings under the action of the pressures acting on either side; also i t i s d i f f i c u l t to manufacture. - 3 -An alternative to the above arrangements seems to be needed for modern subsonic testing. Such a method should enable us to modify our existing wind tunnels with wall arrangements which are not only reliable and cheap to operate, but also can answer the c a l l for r e a l i s t i c measure-ments and accurate corrections. Contributions to such a method are pre-sented in this thesis. It i s well known that most corrections to data in wind tunnels with open jets are opposite in sign to those in wind tunnels whose walls are solid, Ref. [5]. These opposing effects suggest the strategy of employing partly solid, partly open walls in pursuit of cancelling the corrective effects of the two types of wall. Recently, two such designs for this purpose have been considered. One has walls with narrow longitudinal slots. The other has walls patterned with small holes. Using the linear theory to investigate these two types of wall configurations, Parkinson and Lim [6] and others have found that there is a lack of agreement between the experimental results and those which are predicted by the theory for the longitudinal slotted wall. Also they have found that the "porosity parameter" is not simply an empirical function of the open area ratio but i t must be determined empirically for each a i r f o i l under test, an impossible situation for the practical use of porous wall configurations. - 4 -Parkinson and Lim [6] have attributed the lack of success of the longitudinal slots and porous wall theories to the occurrence experimental-ly of separated flows in the slots and holes. Such flows either are not accounted for in the theories, primarily as they add undesirable nonlineari-ties to them, or are dealt with semi-empirically. In addition, these flow separations may seriously degrade the main flow in the v i c i n i t y of the walls. Williams [7], using a surface source method, has investigated another type of wind tunnel, one which has an upper transversely slotted wall, with uniform gaps and airfoil-shaped slats, and a lower solid wall. His theoretical analysis shows that using this type of wall configuration w i l l improve the performance of the tunnel test section. The analysis predicts that such a test section w i l l produce l i f t data within a small percentage error of the free-air values, while the pressure distribution and the pitching moment data are of lower accuracy, Fig. (1). Also his experimental investigation shows that such a wall configuration provides a flow free of separation on the transverse slats. In the present thesis, a different surface singularity analysis, using a surface vorticity method, is used to investigate a wind tunnel with both walls transversely slotted as another approach to modify the wind tunnel wall configuration in order to recreate the free-air streamline patterns about the test a i r f o i l , which would then experience the corresponding free-air loading. - 5 -The approach here uses symmetrically transversely-slotted upper and lower walls, with symmetrical airfoil-shaped solid slats at zero in-cidence. The flow inclinations near the wall w i l l be small, for a l l practical cases envisaged. Hence a l l the wall slats w i l l operate within their unstalled incidence range, so that the air which flows near the wall w i l l be free of separated wakes. A uniform spacing of the wall slats shows that (see Fig. (16)) the upper surface of the a i r f o i l section in the presence of the slotted wall tends to experience a slightly lower negative pressure distribution than that of the free-air conditions near the leading edge and slightly high-er further aft. The effects tend to cancel for l i f t but lead to appreci-able errors in pitching moment. A solution to this problem is to use graded, narrow gaps upstream of the test a i r f o i l and wider ones downstream, rather than uniform spacing between the wall slats. Also the surface singularity analysis has been used here to simu-late the shear layers which enter or leave the test section from or to the upper and lower plenum, respectively, in order to represent the flows in there. The success of the above wall configuration depends on the experi-mental verification of the performance of the wind tunnel which is equipped with the proposed wall arrangements. The'results of such an experimental investigation are presented here. - 6 -CHAPTER II POTENTIAL FLOW ANALYSIS II.1 Introduction An efficient, reliable method for calculating the velocity d i s t r i -bution on the surface of a i r f o i l sections i s required. Conformal trans-formation methods such as that of Theodorsen [8] can analyze sections of arbitrary shape. These methods are based on the theorem which states that i t " i s always possible to transform the potential f i e l d around any closed contour into the potential f i e l d around a c i r c l e . Such methods are not . simple and, as there is no such theorem for transforming the potential f i e l d around multi-component sections, one looks to surface singularity methods of analysis. These methods replace the potential flow f i e l d out-side the a i r f o i l contour with that about a set of singularities, sources or vortices, which satisfy the same boundary conditions. The surface singularity methods can deal easily with multi-component sections and are of comparable accuracy with conformal transformation methods for single-component cases. The most widely used surface singularity method [9] employs sources and sinks on the surface of the a i r f o i l section combined with a vorticity distribution to generate circulation. Williams [7], among others, has employed this to investigate his wind tunnel model. The technique has, however, some drawbacks. One particular problem arises from the application of the Kutta Condition, in the form of equal velocity magni-. tudes at the control points of the upper and lower t r a i l i n g edge elements (U = ~U )• This can be called the NO LOAD Kutta Condition since t _ t 1 u - 7 -i t eliminates any l i f t from the a i r f o i l near the t r a i l i n g edge which is in conflict with the aim of examining sections with large rear loading. To overcome this problem a better method must be developed. Recently a different surface singularity method was developed by Kennedy [10]. This method uses a distribution of vortices on the surface of the a i r f o i l section. The vortex density, which i s determined directly, i s equal to the surface velocity (this i s shown in Appendix 1). The boundary condition which i s applied here i s that a l l solid surfaces are streamlines on which the stream function i s required to be constant. II.2 Surface Singularity Theory In two dimensional, incompressible, irrotational flow the stream function must satisfy Laplace's equation, i 5 l + i i = 0 ( 1 ) 2 2 9x 9y For the flow over a i r f o i l sections there can be no normal velocities at the solid surfaces, and thus each surface is a streamline of the flow. Since the stream functions Y (S=l,2,....,M) on the surfaces of M com-s ponents on a multi-component section are constants, the boundary condition for equation (1) can be written as, f = ¥ , on the surface S (2) The stream function for a uniform stream incident to the positive X axis at an angle cc i s given by, m = y cos a - x sin a (3) - 8 -which satisfies equation (1). This equation, and a l l subsequent equations are in dimensionless form. The distances are dimensionless with respect to the chord length C, the velocities with respect to the free system velocity u<» and the stream functions with respect to the product U^C. The point vortex of strength T, located at (X q, Y ) has the stream function, Y = - r_ £n (r), ( 4 ) 1/2 where r = <(x-x )^ + (y-y )^ i \ o | equation (4) also satisfies (1) , except at r=*0. Because of the linearity of equation (1) any collection of point vortices or any continuous dis-tribution of them as in Fig. (2), that li e s on the a i r f o i l surface, S, w i l l satisfy equation (1) in the region outside of S. Then the stream function at a general point P due to vorti c i t y having a density y(S') at S' and continuously distributed over the a i r f o i l surface S, is given by, T p = ^ ^Y(S') In r (P , S-') d S' (5) s Applying the boundary condition, equation (2), to the combined flow due to a uniform stream plus the above distribution of the vorticity, one obtains, ¥ = y cos a - x sin a - i - lY(S') In r(S,S') d S' (6) S S S 2TT / - 9 -The a i r f o i l surface i s divided up in some manner into N small sur-face elements. On each of these there i s a control point, C^, located at (x^, Y^), at which the boundary condition, equation (6), is made to apply. Each element j has vorticity of density Y(Sj) distributed on i t s surface. The integral in equation (6), over the whole surface S, i s then replaced by a summation of N integrals over the N surface elements. Applying equation (6) at the control point, C^, one obtains, N r ^ + E -T- H(S\) In r (C, S\) d S! = y.cos a - x. sin a (7) j=l £ ' ^ 1 3 3 j The results required of an a i r f o i l analysis method are the sur-face velocities. Kennedy [10] shows that the tangential velocity at the interior of the solid surfaces has to be zero so that these surfaces be-come streamlines, which also results in the discontinuity in the tangential velocity across a vortex sheet being equal to the density of the vortex sheet. Thus Y (S^) is equal to the surface velocity. in solving the equation (7) one therefore solves directly for the velocities on the air-f o i l surfaces. At this point i t is necessary to make some assumption about the section geometry, the location of the control points and the form of Y (S_.) over each element j . The simplest approximation i s to assume that the elements are straight lines with control points at the element mid-points and Y (S.) i s a constant over each element. 3 Using the above approximation and applying equation (7) at each control point yields the system of equations, - 10 -N ¥ + I K.. Y. = R- > (i = N) (8) S " j=l 1 3 3 1 where K . is the influence coefficient of the element j on the control point i , R. i s the right hand side of equation (7) evaluated at control point i and Y is the stream function for the a i r f o i l component. Using the notation of Fig. (3), the BASIC influence coefficients can be written, K. . = * ' {(b+A) Jin (r?) - (b-A) In (rj?) + 2a tan" 1 ( , 2 a ^ ,) - 4A} ip 4TT 1 2 a +b -A (9) The details of the calculation of this equation are provided in Appendix 2, which is extracted from Ref. [10]. The K.. and R. are purely functions of the geometry of the surface elements and the angle of attack. The system of equations (8) is a set of N equations for the N unknown Y- and M unknown ¥ , where there are M com-ponents. The M additional equations required for a solution to this problem are termed the Kutta Conditions and there is one for each component in the test a i r f o i l section, and each a i r f o i l slat in the wall, for the cases to be considered later. II.3 The Kutta Condition Kutta and Joukowski were concerned with a i r f o i l sections whose geo-metries are calculated by a conformal transformation technique which maps the flow over a circular cylinder into the flow over an a i r f o i l section with a cusped t r a i l i n g edge. These sections have two stagnation points, - 11 -one located near the leading edge and the other near the t r a i l i n g edge. Also the velocity at the t r a i l i n g edge w i l l be, in general, i n f i n i t e . They both proposed that the circulation around the circular cy-linder be adjusted so that one of the stagnation points in that flow be located at the point which w i l l map into the a i r f o i l t r a i l i n g edge. In this case the in f i n i t e velocity and the stagnation point, occurring to-gether at the t r a i l i n g edge, cancel and yield a f i n i t e , non-zero velocity there. It has been shown by Milne-Thomson [11] that a consequence of this assumption is that the stagnation streamline leaves the cusped t r a i l i n g edge tangent to i t and photographs of flow visualization studies of Prandtl and Tietjens [12] show this effect clearly, see Fig. (4). This condition can be modelled by providing an additional control point just off the tr a i l i n g edge. Such a Kutta Condition was used success-fu l l y by Bhateley and Bradley [13]. The bisector of the tr a i l i n g edge is extended into the free stream and a control point placed a small fraction of chord downstream of the t r a i l i n g edge, as i t is shown in Fig. (5). It is then assumed that the streamlines through the other control points of that component also pass through this control point. Equation (8) then applies to these t r a i l i n g control points, C , and the Kutta Condition can be written as, N YS + . E 1 \ p ,j Y j = \ p , (m = S=1,2,....M) (10) m i*=l "-m m There are M such t r a i l i n g control points, one for each component, and hence M Kutta Condition equations. - 12 -Thus the problem of potential flow over an a i r f o i l section has been reduced to that of solving (N+M) equations, prescribed by equations (8) and (10), simultaneously to get N vortex densities, y , and M stream functions - 13 -CHAPTER III ASSEMBLING AND SOLVING THE BASIC EQUATIONS OF THE POTENTIAL THEORY The f i r s t step in the solution is to define the elements which describe the a i r f o i l surface. One obvious method of doing this is to l e t the supplied co ordinates be the end points of the surface elements. This has the disadvantage that there may be insufficient co-ordinates available or that they may be irregularly spaced. To overcome these problems the a i r f o i l i s divided up, from i t s leading edge at x=0 to i t s t r a i l i n g edge at x=l. The end points of the surface elements are located, as shown in Fig. (8), at x co-ordinates given by, x 0 = 1 (l-cos<J>,J , (£ = 0,1,2, ,N) (11) where 4>„ = 2IT*. Here N must be an even number in order that the end point be located at the a i r f o i l t r a i l i n g edge. This distribution of points pro-vides, in general, a more accurate solution because i t concentrates the control points near the leading edge and t r a i l i n g edge where the largest velocity gradients generally occur. The corresponding co-ordinates y^ of the element end points are determined by interpolation on the given a i r f o i l data. The use of a cubic spline function has been found to be the most reliable method, since i t gives smooth curves through the given points and can be easily and effi c i e n t l y computed. Here the U.B.C. com-puter subroutine SAINT has been used for interpolation. - 14 -The control points are taken as the mid-points of each surface element, as shown in Fig. (3), then the bisector of the t r a i l i n g edge i s extended, as shown in Fig. (5), and the control point is located on this extension a distance O.Olt from the t r a i l i n g edge. This distance was found to give the most reliable results for a wide range of a i r f o i l sections. The a i r f o i l system of axes x-y should be rotated clockwise an angle ot, the angle of attack, then the co-ordinates of the element end points with respect to the wind system of axes X-Y w i l l be given by, X. = x. cos a+ y. sin a 1 1 i (12) Y. = y. cos a - x. sin a i i i Having determined the co-ordinates of the element end points and control points one can proceed to calculate the BASIC influence coefficient K.., which is given by equation (9), and R,, which as a result of the rotation of the axis should be given by, R± = Y ± (13) As the vortex densities are identical to the surface velocities, counter-clockwise about the a i r f o i l section, the co-ordinates of element end points and control points should be taken in that order around the polygonal contour. For a single-component a i r f o i l in free a i r , the system of equa-tions (8) and (10) can be written in the matrix form as, - 15 -i=l i=N i=N+l K 1,1 K, 1 1,N ^,1 • • • • Kutta Condition N Y l R l Y R N N R 1 tp (14) The above system of equations i s then solved for the unknown N vortex densities Yj a n <^ the stream function When this technique i s extended to multi-component a i r f o i l sections the point distribution, given by equation (11) is f i r s t scaled to the chord of each individual component before being applied. - It is then nec-essary to move each component to i t s correct location. This i s done by specifying the amounts by which the leading edge of the component is trans-lated and the angle through which the component is rotated. With the geometry thus defined, one can calculate the K^ ^ and R^  from equations (9) and (13). The multi-component case gives rise to a different stream function for each component and each component has i t s own Kutta Condition. A two-component a i r f o i l with N elements on each component gives rise to a system of equations which can be written: - 16 -i=l K • • 1,1 K1,2N 1 0 -Y l R 1 i=N *Sj,2N X 0 YN i=N+l V i , i V l , 2 N ° 1 Y N+l = V i i=2N K2N,1 '• * ' K2N,2N ° 1 Y 2N R2N Kutta Condition, component 1 1 l=2N+2 Kutta Condition, component 2 ^2 R t P2 (15) The U.B.C. computer subroutine FSLE has been used here to solve the system of equations (14) and (15) . The solutions are the dimension-less surface velocities at the control points and the dimensionless stream functions of each component. The pressure distribution, l i f t coefficient and the leading-edge pitching moment can be calculated from the velocities as follows: 2 C = 1-Y. P i N C = I C Ax. (16) L . p. l 1=1 1 N C„ = - Z C (x. Ax, + y. Ay.) - 17 -and summations are performed counter clockwise around the polygonal con-tours (as i t is shown in Appendix 1). The technique described so far makes the simplifying assumptions of straight line elements and constant vortex density on each element, which is referred to here as the BASIC method. Kennedy [10] however, has studied the effects of including the higher order terms due to surface curvature and a linearly varying vortex density on each element. The re-sults are shown in Fig. (9). These results show that the inclusion of element curvature raises the velocities while including the linear vortex density decreases the velocities. Also from Appendix 1, one can notice that the two terms which introduce the linear velocity distribution and surface curvature into the influence coefficients are of the same magnitude but of opposite signs, thus their effects tend to cancel when they are com-bined. It is therefore recommended that only the BASIC method with straight line elements and constant vortex density be used. Kennedy's investigation [10], has shown that the BASIC method with "Full Load" Kutta Condition gives accurate results for most a i r f o i l s . An example of a section with a f a i r l y sharp peak in the velocity distribution is given in Fig. (10). This is a Joukowski a i r f o i l with a cusped t r a i l i n g edge for which 40 elements were used to describe the section, and there i s excellent agreement with the exact solution. In the current investigation the single-component a i r f o i l sections, NACA-0015 and CLARK-Y 14% have been represented by 40 elements while the two-component a i r f o i l , NACA-23012 with 25% slotted flap deflected 20 degrees, - 18 -has been represented by 70 elements (40 elements for the main a i r f o i l and 30 elements for the flap). The wind tunnel with solid walls i s represented by 128 elements while the wind tunnel with slotted walls i s represented by 64 elements for the solid parts of the wall and 8 elements for each of the 16 slats (8 slats for each wall, with c/C = .15). The earlier part of this research was reported as a University of British Columbia M.A.Sc. Thesis (Ref. [15]). It was in this theoreti-cal study that the optimum AOAR of 59% was established by considering values ranging from 50% to 70% for the same three a i r f o i l profiles con-sidered here. Also in this earlier study an attempt was made to simulate the shear layers bounding the plenum flows. This model was extended in the present investigation, and i t is described in Appendix 4. To include the flow in the wind tunnel plenums in the current analysis, each bounding streamline (two of them are considered, one for each plenum, with two different prescribed constant pressure coefficients) is then divided into 8 elements, which adds 16 vortex elements to the pre-vious 232 elements in the case where NACA-0015 is the test a i r f o i l . In a l l of the above numerical analyses, the total extent of the wind tunnel wall has been taken as four times the test a i r f o i l chord, C. Also the gaps in the non-uniform slotted walls, with AOAR=59%, are increasing linearly towards the rear of the test a i r f o i l , with 2% increment. The test a i r f o i l was in each case located with i t s mid-chord point at the tunnel center-line, and rotation for angle of attack was about this point. Finally, FSLE i s a subroutine which uses the Gaussian Elimination method to solve a general real matrix (with partial pivoting, forward and backward substitution). - 19 -CHAPTER IV RESULTS OF-THE THEORETICAL INVESTIGATION The theoretical curves presented next were calculated by the method of Ref.[10]. Figures (12), (13) and (14) show comparisons of theoretical pressure distributions for the a i r f o i l s , of chord C, NACA-0015 at a =10? NACA-23012 at a = 8°with 25.66% slotted flap at 5 = 20°and CLARK-Y 14% at a = 10? respectively, in free air and between the solid walls of a conventional wind tunnel test section of height h, with C/h = .8. The tunnel l i f t coefficients C are 1.6873, 3.6448 and 3.0131 which are 37.66%, T 30.36% and 41.11% higher than the free air l i f t coefficients C_ of 1.2257, L F 2.7959 and 2.1353 respectively. The very large errors in the a i r f o i l l i f t coefficients developed in the wind tunnel with solid walls shown by the above results, prompted a search of a wall configuration that would exhibit the known cancelling ef-fects of partly open, partly closed walls, and which would therefore provide negligible or small errors. From the above results i t can be seen that nearly a l l of the in-creased values in the tunnel l i f t coefficients are due to the greater suction over the top surface of the a i r f o i l . Therefore, the f i r s t step towards the modification of the conventional wind tunnel was to change the wall confi-guration of the upper solid wall, and this was done here by using a uniformly transversely-slotted upper wall. The pressure distribution for the a i r f o i l NACA-0015 at a = 10° for such a wall configuration of 59% UOAR, with solid lower walls, appears in Fig. (15) along with the corresponding pressure - 20 -distribution for free a i r . This figure shows some improvement in the pressure distribution opposite to the upper uniformly slotted wall, but i t also shows that there are differences in the lower surface pressure distributions, and i t suggests that the flow there i s experiencing lower induced velocities in the presence of the lower solid wall than in free air, and a possible solution to this problem is to use a slotted lower wall as well. Accordingly, the wind tunnel with double uniformly transversely slotted wall has been investigated here and the results, as in Fig. (16), show that the pressure distributions for NACA-0015 at a = 10°and of C/H = .8, opposite to the lower slotted wall is closer to the free air values than before, while the upper surface suctions s t i l l tend to be low near the lead-ing edge and slightly high near the t r a i l i n g edge, for the same UOAR 59%. A _ solution to this problem would appear to be to use graded, wider gaps rear-ward and narrow ones forward, rather than uniform spacing between the wall slats. A l l the above support the motivation for using the double slotted wall test section with nonuniform spacing, Fig. (6a). Several average OAR have been examined to look for the most suitable wall configuration which would develop the least errors in the tunnel l i f t and pitching moment co-efficients for the three different a i r f o i l s , mentioned before, and of dif -ferent sizes. It has been found that AOAR=59% i s suitable for this wide range of a i r f o i l shapes and sizes. Figure (17)shows comparisons of the effect on the theoretical l i f t - .21 -coefficients of the ratio - for different a i r f o i l s and test section wall H configurations. The a i r f o i l s and their angles of attack are those used before, and the test section walls are either solid or double uniformly slotted with AOAR=59%. It can be seen that with the solid walls, for the calculated range of -, the corrections can exceed 40% of the free air values, while with a suitable AOAR of the slotted test section the pre-C dieted errors can be kept within 4% for the three a i r f o i l s , and for -.sr .8. Also, Fig. (18) shows comparisons of the effect on the theoretical C quarter chord pitching moment coefficients of the ratio - for the same a i r -H f o i l s and test section wall configurations. It can be seen that with the solid walls the corrections can exceed 3% of the free air values, while with the slotted test section of AOAR=59% the predicted errors can be kept with-C in 1.2% for the three a i r f o i l s , and for -«r.8. Figures (19), (20) and (21) show comparisons of the theoretical pressure distributions, for the above a i r f o i l s and angles of attack, in free a i r and in the double slotted test section of AOAR=59%, again with C_=».8. It can be seen that there is quite H good agreement between the two distributions for the three different a i r -f o i l s . Figure (22) shows comparisons of the effect on the theoretical l i f t C coefficient of the ratio - for NACA-0015 a i r f o i l at two different angles of H o o attack, a = 10 and a = 20 , and with two different wall configurations; solid walls and double nonuniformly slotted walls of AOAR=59%. The two wall configurations exhibit the known blockage effect of increasing the error in l i f t coefficients as the angle of attack increases, however the effect i s not very appreciable when the slotted walls are used. Also, Fig. (22) shows that the slotted walls with AOAR=59% s t i l l i s the most suitable wall configuration - 22 -not only for a wide range of a i r f o i l sizes and shapes but also for a wide range of angles of attack. For a more accurate representation of the flow in this double slotted wall test section the limiting upper and lower streamlines AB and CD, as shown in Fig. (6)a, should be considered. Physically, the stream-lines are shear layers and they could be idealized as streamlines at constant pressure. Using the method which has been developed here (see Appendix 4) for tracking streamlines, their geometries can be described and consequently they can be included in the present analysis. Figure (23) shows the geometry of the idealized flow in the plenum chambers, the two bounding streamlines of a test section with the new wall o c configuration of AOAR=59%, and the test a i r f o i l NACA-0015 at a = 10 , - = .8. It also shows the values of the stream function, Y , and the average pressure s coefficient C ; for each streamline. Note that Y > V and > Y„ because p u 1 Z 2 of the approximate numerical procedure which has been used in determining the geometry of these streamlines. Figure (24) shows comparisons of the theoreti-cal pressure distributions, for NACA-0015 a i r f o i l at angle of attack a = 10°, in free air and in the double slotted test section of AOAR=59% with - = .8, H and the flow in the plenum is included; the average pressure coefficient along the upper streamline is C =-.2 while the average pressure coefficient along the lower streamline is C = .11. It can be seen from a comparison of Figures (19) and (24) , that the proposed analytical approach to the flow in the wind tunnel plenum does not produce a test a i r f o i l pressure distribution in as good agreement with free air values as does the simpler model of Figure (19). Hence the plenum model analysis is relegated to Appendix 4, as useful background information. - 23 -Table I contains a l l the theoretical data, l i f t and pitching moment coefficients, presented in this chapter. Also, Table II has the theoretical l i f t and pitching moment coefficients for NACA-0015 a i r f o i l , tested in the double non-uniform transversely slotted walls when the flow in the plenum i s included in the analysis (the two shear layers in the upper and lower plenum are simulated by two constant pressure streamlines). - 24 -CHAPTER V EXPERIMENTAL INVESTIGATION The success of the proposed transversely slotted upper and lower walls with the non-uniform gaps, depends on the experimental examination of the performance of a wind tunnel test section devised with the new wall configuration. Such experimental verification i s described as follows. V.1 Test Section Specifications The experiments were performed in a two-dimensional test-section insert designed and built for an existing low-speed closed-circuit wind tunnel. This insert i s 915-mm wide by 388-mm deep in cross-section, and 2.59-m long. The test a i r f o i l s were mounted at the midpoint of the test section and spanned the 388-mm depth. The side walls (two of them) were surrounded by .39 by .3 by 2.44-m plenums and they could be fitte d with a i r f o i l -shaped slats of NACA-0015 section and chords of 92-mm, at zero incidence (8 slats were used for each side). A f u l l range of wall average open area ratio (AOAR) could be tested, as the slats were fitt e d with metal sliders which in turn were separated by wooden spacers in an aluminum channel re-cessed in the side wall foam. Modifications to the existing wind tunnel consisted of an inserted nozzle and diffuser section in addition to the 388 by 915-mm test section. The test section walls are parallel and solid (or slotted); no provision for compensation for boundary layer growth is attempted. Williams' [7] experimental research, using the same wind tunnel and test section showed that the test section wind speed i s spatially uniform to within 0.3% in the central "core" flow outside the wall boundary layers. Also, boundary layer pitot-stati.c tube measurements in the empty test section (over the range of wind speeds covering the range of Reynolds numbers for a l l a i r f o i l tests) indicated a displacement thickness of the order of 8-mm, where the test a i r f o i l s would be mounted. Pankhurst and Holder [5—ii3 show that a wind tunnel test section, with the above wind quality, would produce a i r f o i l pressure measurements (taken at the mid-span) with very small boundary layer effects. They indicate that a wall boundary layer, as the one specified before, w i l l not cause any contamination of the area around the a i r f o i l pressure taps. V.2 A i r f o i l Models Tested Two different a i r f o i l s have been tested. The surface pressure measurements were made on the two a i r f o i l s , in addition the l i f t and pitching moment data were obtained by the integration of the pressure coefficients. The a i r f o i l s for which results are reported here are two a i r f o i l s of NACA-0015 section, 383-mm span and 307 and 616-mm chord, which were machined from solid aluminum b i l l e t s to close tolerances. Each a i r f o i l was mounted on a circular spar which passed through a circular hole in the test section floor with 3-mm clearance a l l around. The floor and ceiling t i p clearances were less than 2.5-mm on a l l tests. The test a i r f o i l s were mounted ver-t i c a l l y on the turntable of a 6-component balance, which was not used in obtaining the l i f t s and the moments in the present work; the reason w i l l be explained later in the chapter. Both a i r f o i l sections were fi t t e d with a number of center span pressure taps. A l l pressure taps are flush mounted and have 5-mm diameter - 26 -or i f i c e s . Plastic tubes of 1.6-mm inside diameter and approximately 1-m length transmit the surface pressures through the mounting spar to a loca-tion external to the test section. Pressures were measured via a 48 port "Scanivalve" manual scan pressure transducer. The signal coming out of the "Scanivalve" was fed to a data acquisition system "NEFF", which was hooked to the computer PDP-11. With the help of a computer program, along with the "NEFF" system, the surface pressures were measured, then the pres-sure coefficients were calculated and recorded instantaneously in a computer f i l e which could be called any time later. V. 3 Test Procedures The test section wind speed was deduced from a pitot-static tube mounted on the flow centerline in the tunnel nozzle midway between the settling chamber exit and the test section entrance. Located thus, the pitot-static tube would be far enough up-stream to be relatively unaf-fected by the test model "blockage" effects. Also the pitot-static tube would be sufficiently far downstream in the nozzle that the flow speed would produce numerically large pitot-static tube readings (mm of water). Thus a sufficiently accurate reading on the "Betz" micromanometer could be obtained. The nozzle pitot-static tube was calibrated against a second pitot-static tube mounted in the empty solid walled test section, on the flow centerline, where the test a i r f o i l s would be located. The second pitot-static tube was connected to a "Barocel" pressure transducer; i t s output is read using a d i g i t a l volt meter. During tests, the total head in the nozzle is measured (with the same pitot-static tube) using a - 27 -Barocel Electronic Manometer; i t s output signal was fed to the data acqui-sition system "NEFF". The total head was used as a calibration pressure for the "Scanivalve" pressure transducer. The reference wind speed and static pressure used to reduce the pressure measurements are determined as follows: Since the total head at the nozzle pitot-static tube (when there i s a test a i r f o i l in place, at incidence) i s essentially the same as the total head in the test section in the vic i n i t y of the test a i r f o i l , the reference wind speed and static pressure can be deduced from the nozzle pitot-static tube measurements of the total head and dynamic pressure. This equality of total heads can only be verified in the empty test section (solid walls) with no test a i r f o i l in place; such measurements indicated that the test section 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 head and dynamic pressure head respectively, and let the subscripts "N" and "T" refer to the nozzle and test section respectively. In the empty test section (solid walls), according to the above, H = P + q = H = P + q . (17) N N N T T T v If K is the empty test section (solid walls) calibrated ratio of q N to q^, then since only and ^  are measured while under test, the equivalent empty test section (solid walls) dynamic pressure head and static pressure head are respectively as follows: < 1 8 ) - 28 -and PT " \ - K % l l 9 ) Thus the reference wind speed and static pressure used to reduce data taken for any test a i r f o i l , at any incidence, in the presence of any slotted-walls AOAR, are the equivalent values in the undisturbed uniform stream conditions that would actually occur in the empty (solid walls) test section corresponding to that measured and q N in the nozzle. To evaluate the pressure coefficient, C , of the test a i r f o i l at any point x along the P x a i r f o i l center-span contour (where there i s a pressure tap), the following was considered; C = <V PT ) / C*T ( 2 0 ) x Using equations (18) and (19) along with equation (20); therefore Cp = { P x " < V K V } / K * N ( 2 1 ) x Where P^ is the test a i r f o i l surface pressure head, i t was measured directly by means of the "Scanivalve" and the "NEFF" systems. Figure (25) shows a typical nozzle test-section wind speed calibration curve using the two pitot-static tubes. The ratio K i s determined from a straight line least-squares curve f i t t e d through the origin. In the i n i t i a l experiments, the l i f t and moment data were taken from the 6-component balance, but that data was much less than the expected one, for the same test a i r f o i l and angle of attack. Sugiyama [16] attributed this trend to the existence of a clearance flow, between the wind tunnel - 29 -ceiling and floor and the a i r f o i l tips, this flow causes a t r a i l i n g vortex sheet i n the wake of the a i r f o i l , and the l i f t of the a i r f o i l i s then der-creased through the downwash induced by the t r a i l i n g vortices. In later experiments the clearance between the test a i r f o i l tips and the wind tunnel ceiling and floor was sealed off completely by ad-hesive tape to eliminate the effect of the clearance flow mentioned before. This way the 6-component balance was not used at a l l to get any of the l i f t and the moment data; these data were obtained then by the numerical integra-tion of the pressure coefficients around the test a i r f o i l contour. The following equations describe how these data were obtained numerically. .4. V N P c p d x ( 2 2 ) T and Cm = -(h>cp (x d x + y d y) (23) Where CT and C, are the l i f t and the pitching moment coefficients, re-LT M0 . spectively. The U.B.C. computer subroutine QINT4P has been used in the evalua-tion of equations (22) and (23) . This subroutine f i t s the best polynominal to the "n" experimental data points (C^ vs the pressure tap coordinates) and then i t integrates to get the C_ and the C . The error was in the T M0 —4 order of 1.9 x 10 on the average, "n" i s the number of the pressure taps at the center-span of the test a i r f o i l . There were 48 taps for the a i r f o i l with the 616-mm chord and 47 taps for the other a i r f o i l with the 307-mm chord, their locations are l i s t e d i n Table III. - 30 -CHAPTER VI EXPERIMENTAL RESULTS In this chapter the experimental results of testing two different sizes of NACA-0015 a i r f o i l section are presented. The two sizes were C/H = .3' and C/H = .67, where C/H i s the test a i r f o i l chord to the wind tunnel height ratio. In a l l tests the Reynolds number "Re" was 500,000, while the wind tunnel test section was devised with two different wall configurations; the solid walls configuration and the double non-uniform transversely slotted wall configuration. Three AOAR were examined, AOAR = 50%, 55% and 59%, because i t was f e l t that the experimental optimum AOAR might be lower than the theoreti-cal one. Also a wide range of angle of attacks were used. Figures (26) and (27) show comparisons of pressure distributions for two NACA-0015 a i r f o i l sections, C/H = .34 and C/H = .67, respectively, at a = 10° and tested in the wind tunnel with two different wall configura-tions; the conventional solid walls and the new double non-uniform trans-versely slotted walls. These two figures il l u s t r a t e the expected greater suction over the top surface of the test a i r f o i l when the wind tunnel is devised with the conventional solid walls. Figures (28) and (29) show comparisons of the pressure distributions for the two different sizes, C/H = .34 and C/H = .67, respectively, of NACA-0015 a i r f o i l at the same angle of attack a =10°. Each size was tested in the wind tunnel with the proposed double non-uniform transversely slotted walls, but with three different average open area ratios; AOAR = 50%, 55% and 59%. Again, i t can be seen from these two figures that the smaller the AOAR the greater the suction over the top surface of the test a i r f o i l . - 31 -Figure (30) shows the pressure distributions for the two different sizes of NACA-0.0.15 a i r f o i l , C/H: = *34 and C/H = .67, at a = 10°; both sections were tested in the wind tunnel with the same nonconventional wall configuration, the double non-uniform transversely slotted wall with AOAR => 59%. Because the data are collapsing together, this figure indicates that the new wall configuration is very much independent of the size of the test a i r f o i l . Figures (31) and (32) show the C - a curves for the two different tested sizes of the NACA-0015 a i r f o i l , C/H = .34 and C/H = .67, respectively. Both figures show the effects of the wind tunnel conventional solid walls and the nonconventional ones, the double non-uniform transversely slotted walls with AOAR = 50%, 55% and 59%, on the a i r f o i l l i f t coefficients C LT As expected, the conventional solid walls cause more blockage, thus the wind tunnel l i f t coefficients with this wall configuration are higher than the wind tunnel l i f t coefficients with the new proposed wall configuration, also, the smaller the AOAR the more the blockage effect i t has. Similarly, Figures (33) and (34) show the C a curves for c/4 the same two a i r f o i l sizes and the same wall configurations as mentioned in the previous two figures. Figure (35) shows the corresponding experi-mental values of E for the tested sizes of NACA-0015 a i r f o i l , C/H = .34 L and C/H = .67, in the presence of the conventional solid walls and the double non-uniform transversely slotted walls of different AOAR; 50% 55% and 59%, and in comparison with the corresponding theoretical curves from Figure (17). A l l tests were run at the same Reynolds number of 500,000. The experimental value of m^  used in determining E t was taken as 0.0915 - 32 -from Figure (36), extracted from Jacobs and Eastman [17]. Agreement i s seen to be quite good, except for an apparently high experimental value for C/H = .34 in the presence of solid walls. Table IV contains a l l the experimental data, l i f t and pitching moment coefficients, presented in this chapter. Also, Table V contains the values of the lift-curve slopes, m, for the two NACA^0015 a i r f o i l sections, C/H = .34 and C/H = .67, tested in the low^speed wind tunnel with the solid walls and the proposed double non^-uniform transversely slotted walls, with the three examined AOAR; AOAR = 50%, 55% and 59%. The values of m were ob-o tained by f i t t i n g least-squares straight lines to the C - a data between LT a = (0°, 10°). Appendix 5 l i s t s the equipment which has been used in the experimental investigation. Figure (37) shows an outline of the U.B.C. low-speed wind tunnel which was used in the experimental work presented here. - 33 -CHAPTER VII CONCLUSIONS AND RECOMMENDATIONS A two-dimensional theory which predicts a satisfactorily correction-free wind tunnel test configuration has been developed. The theory i s an ap-plication of the two-dimensional potential flow theory based on the method of distributed surface singularities. The extended theory takes into con-sideration not only a wide range of a i r f o i l sizes and shapes, but also the effect on the a i r f o i l loadings of different wind tunnel wall configurations. The above potential flow theory was then developed to account for the flow in the plenum chambers, in order to represent the flow in the test section as close as possible to the physical situation. The results of the theoretical study indicate that for a i r f o i l testing, a wind tunnel consisting of two symmetrically non-uniform trans-versely slotted walls, with the gaps between the slats linearly increased downstream, and the slats symmetrical airfoil-shaped at zero incidence with average open area ratio 59%, w i l l yield uncorrected pressure d i s t r i -butions , l i f t coefficients and pitching moment coefficients which are within a few percent of the free a i r values. The theory predicts that this low correction wall configuration w i l l remain relatively correction-free for a wide range of a i r f o i l sizes and shapes. In the present analysis, the flow in the plenum chambers i s idealized by two constant pressure streamlines, but actually there are two shear layers. Each one divides two flows, the high-energy flow which - 34 -exists in the test section, and the low-energy stagnant flow outside the plenum. The model as developed did not lead to a prediction of better agree-ment of test a i r f o i l loadings with free air values. . Experiments carried out on two NACA-0015 a i r f o i l sections for Reynolds number of 500,000, in a two-dimensional test section support very much the predictions of the theory. Experimental work showed that the correction-free test configuration could be achieved with double non-uniform trans-versely slotted walls, with AOAR = 59%, consisting of symmetric a i r f o i l shaped slats at zero incidence, when the slotted sections were surrounded by plenum chambers. Measurements taken with the correction-free wall configuration of pressure distributions, l i f t s and pitching moments for the NACA-0015 a i r f o i l of two different sizes, C/H = .34 and C/H = .67 (chord to height ratio), showed good agreement with established free air values. The low-correction wall configuration concept which has been examined theoretically and verified experimentally in the present thesis can be de-veloped to provide a reliable means of testing high l i f t a i r f o i l s with high Reynolds numbers in the existing wind tunnels. Such a test would otherwise require expensive and complex test f a c i l i t i e s . It would be desirable to con-tinue the experimental program by testing a variety of a i r f o i l profiles in the new test section, including higher-lift profiles such as the NACA-23012 with slotted flap, examined theoretically in Chapter IV. - 35 -REFERENCES [1] Sears, W.R., "Self Correcting Wind Tunnels." Aeronautical Journal, February/March, 1974. [2] Sears, W.R. , Vidal, R.J., Erickson, J.C., Jr. and Ritter, A., "Interference-Free Wind Tunnel Flows by Adoptive-Wall Technology." J. of Aircraft, November 1977. Vol. 14, No. 11. [3] Vidal, R.J., Erickson, J.C., Jr. and Catlin, P.A., "Experiments with a Self-Correcting Wind Tunnel." AGARD No. 174, Wind Tunnel Design and Testing Techniques 1978. [4] Pollock, N., "Self Streamlining Wind Tunnels Without Computers." 7th Australasian Hydraulic and Fluid Mechanics Conference, Brisbane, 18-22 August, 1980. [5] i . Pope and Harper, "Low Speed Wind Tunnel Testing", Wiley, 1966. i i . Pankhurst and Holder, "Wind Tunnel Technique", Pitman, 1952. [6] Parkinson, G.V. and Lim, A.K., "On the Use of Slotted Walls in Two-Dimensional Testing of Low-speed A i r f o i l s " , CASI Trans. 4, Sept. 1971. [7] Williams, CD., "A New Slotted-Wall Method for Producing Low Boundary Corrections in Two-Dimensional A i r f o i l Testing", Ph.D. Thesis, Oct. 1975, University of British Columbia. [8] Theodorsen, T., "Theory of Wing Sections of Arbitrary Shape", NACA Report No. 411,1931. [9] Hess, J.L. and Smith, A.M.O., "Calculation of Potential Flow about Arbitrary Bodies", Prog, in Aero. Sci., 8, Pergamon, 1966. [10] Kennedy, J.L., "The Design and Analysis of A i r f o i l Sections", Ph.D. Thesis, 1977, University of Alberta. [11] Milne-Thomson, L.M., "Theoretical Aerodynamics", Dover, 1973. [12] Prandtl, L. and Tietjens, O.G., "Applied Hydro- and Aerodynamics", Dover, 1957. [13] Bhateley, T.C. and Bradley, R.G., "A Simplified Mathematical Model for the Analysis of Multi Element A i r f o i l s near the S t a l l " , AGARD-CP-102, 1972. [14] Hess, J.L., "Higher Order Numerical Solutions of the Integral Equation for the two-dimensional Neumann Problem", Computer Methods in Applied Mechanics, Vol. 2, pp. 1-15, 1973. - 36 -[15] Malek, A., "A Theoretical Investigation of a Low-Correction Wind Tunnel Wall Configuration for A i r f o i l Testing", M.A.Sc. Thesis, August, 1979, University of British Columbia. [16] Sugiyama, Y., "Aerodynamic Characteristics of a Rectangular Wing with a Tip Clearance in a Channel." Journal of Applied Machanics, Dec, 1977. [17] Jacobs, E.N, and Sherman, A., " A i r f o i l Section Characteristics as Affected by Variations of The Reynolds Number". Report No. 586, National Advisory Committee for Aeronautics, 1937. - 37 -APPENDIX 1 AN ANALYTICAL REPRESENTATION OF A SURFACE VORTICITY The fo l lowing descr ibes an ana l y t i ca l two-dimensional approach to the surface v o r t i c i t y problem using the complex ana l y s i s : The contour C-| represents an a i r f o i l su r face , and ZQ i s any point in the f i e l d of a constant complex v e l o c i t y W,. Let {W(Z)=dF(Z)/dZ} be the complex surface v e l o c i t y at the contour C-|. A small c i r c l e i s drawn around the point ZQ at which the complex v e l -o c i t y W ( Z Q ) i s to be found, The funct ion { W ( Z ) / ( Z - Z Q ) > i s ana l y t i c in the region enclosed by the contour {Cn,(C1+Cr,)}t --38 -the re fo re ; r r r L o L l u 2 Consider the 1st term of the in tegra l I as Z approaches ° ° , then W(Z) = W-j the f ree stream complex v e l o c i t y , which i s constant , thus: h = W ( Z ) J>TTT = W l 2 l T i ( 2 ) * r ^ while in C l and therefore equation (1) becomes, I = 2TH W] - Jjffi-) dZ - 2TTi W(ZQ) = 0 (5) C l or * (3) h*fr£h ^ = W(ZQ) 2 T r i (4)* C 2 * * Now i f we replace the a i r f o i l surface by vortex d i s t r i b u t i o n , with dens i ty Y U ) » t n e complex v e l o c i t y at the point Z Q would be W ( Z Q ) , and i t i s wr i t ten as , - 39 -where i s the complex v e l o c i t y of the onset f low. Compare equations (6) and ( 7 ) , we get W (Z ) a t p = . Y ( Z ) . (8) * Therefore the complex v e l o c i t y W(Z) at the a i r f o i l surfaces i s equal to the vortex dens i ty there , and th i s i s only true i f the in tegra t ion in equation * (7) i s done in the counter c lockwise d i r e c t i o n . - 40 -APPENDIX 2 EVALUATION OF THE INTEGRAL IN EQUATION (5) The most straightforward method of evaluating the i n t e g r a l i n equation (5) i s to do so using a numerical i n t e g r a t i o n procedure. The surface element i s chosen to be curved and the v e l o c i t y d i s t r i b u t i o n can vary over the element. The influence of t h i s d i s t r i b u t i o n on one element on the control point of another element i s c a l c u l a t e d by the i n t e g r a l S2 V fc^ >- I n C\ Ac ^ 1 = 2 T / Y(S)£n r ( C ± , S) dS. ** SI A coordinate system ( n ) i s set up with o r i g i n at the control point of the influ e n c i n g element as shown i n F i g . (11). The influenced control point i s located at (b,a)in t h i s coordinate system. The surface element i s defined by n=n_(£) . In the neighbour-hood of the o r i g i n a power serie s expansion i s used, n = c K2 + e C 3 + . . . ( 2 f * The i n t e g r a l i s taken over the surface distance and i t i s convenient to use: dS / , , d l ^ -.^  • ** ** on expanding (3) as a se r i e s about £=0, | | = 1 + 2 c 2 C 2 + 6 ce ? 3 + . . . (4)** The vortex density can also be written as a series defined by, T(S) - Y ( 0 ) + Y ( 1 ) S + Y ( 2 ) S 2 + Y ( 3 ) S 3 + ... ( 5 )** Applying (4) and (5) , Y m = T<0) + Y<D e + y ( 2 ) 5 2 + , 2 ^ ( 1 , + Y ( 3 ) , £3 + _ (6) The distance r(C^,S) from the control point to the surface i s , r ( c i f S ) = {(a-n) 2 + (b - 5 ) 2 } % (7)** At t h i s point i t i s necessary to employ the technique used by Hess [14]. Instead of expanding t h i s term d i r e c t l y a modification i s used which permits the basic, f l a t element, term to appear as the f i r s t term of the s e r i e s . By wr i t i n g 2 2 2 ** = a* + (b -£) (8) the distance to the f l a t surface n=0, the remaining terms are expanded and, 2 2 2 3 . ** r = r f - 2ac r - 2ae T + ... (9) Substituting (9) i n the logarithm term and expanding a l l 2 but the r f term about £=0 y i e l d s , 2 |_' . / 2* 2cic w2 2s.s »_3 , In r = - \ in ( r f ) ^ £ ^ £ + ** r f r f ** (10) . The i n t e g r a l (1) can now be evaluated to as high a degree of accuracy as i s desired. In t h i s case only the f i r s t few terms are retained, higher order terms being of diminishing importance. ** Equation (1) then becomes, 1= hf^ i0) *» + T U ) r f 2 5 - ^ | Y < ° > 5 2 + ... , d 5 (11)** the f i r s t term i s the s t r a i g h t element, constant v e l o c i t y case, second term introduces a l i n e a r v e l o c i t y d i s t r i b u t i o n and the t h i r d introduces the surface curvature. - 42 -** Each term i n (11) can be integrated separately, as follows: A / -A IS J £n r f 2 d£ = (b+A) In r^2 - (b-A) Hn r 2 2 - 4A + 2 a t a n " 1 ( 2 2 a ^ 2) (12) a + b -A An r f 2 5 d K = a 2 ^ 2 , * * 2 i n (^) - 2bA ** ~ , , ~1 . 2aA . •r 2 a b tan (—x ^—2~) a + b -^A A 2 5 d E. = (b 2-a 2) t a n " 1 ( 2 a A — 5 - ) + 2aA (13) 2 " s " ; uaii v 2 2 2 --.e a +b -A -A f + a b i n (—) (14) r l For the s t r a i g h t element ' j ' , which makes an angle 6.. with the X-wind axis, the 'a' and 'b' are given by, a = ( X j - x ^ sin9j + (y^-y^) cose., b = (x.-x.) cos0. + (y.-y.) sin9. ** (15) Also r 1 and can be wr i t ten as , = A2 + (b + A ) 2 r l ** and (16) r2 A2 + (b - A ) 2 - 43 -In the above equations (x . , y.) and (x . , y . ) are the coordinates of the contro l points on the elements ' i ' and ' j ' r e spec t i v e l y . In case of s t r a igh t elements with constant vortex dens i t i e s y^ the in tegra l ( l ) i s c a l l e d the BASIC in f luence c o e f f i c i e n t s K-. and i t i s it it given by equation (12) . For higher order terms invo lv ing surface curvature the constant C must be determined. In th i s case th i s was done by f i t t i n g parabolas through sets o f three adjacent element end po in t s . The curvature thus determined was assumed to be the a i r f o i l surface curvature at the centre of the three po in t s . The curvature at the control points were then found by i n t e r p o l a t i o n . The values at the elements adjacent to the t r a i l i n g edge were found by ex t rapo la t i on . Having determined the curvature of the element the loca t ion of the control point can be ca l cu la ted as th i s point i s no longer on the s t r a igh t l i n e j o i n i n g the element end po in t s . In employing va r i a t ions in the s i n g u l a r i t y strength the term y ^ i s an unknown and must be re la ted to the y ^ . Various schemes are a v a i l -able to do th i s and the technique used by Hess [14] i s fol lowed here. The de r i va t i ves of the d i s t r i b u t i o n on the j t h element are determined by as -suming a parabo l i c d i s t r i b u t i o n through the three successive values Y-j_-/°^ (0) (0) Y j ' Y j+ l• The l i nea r vortex dens i ty term, unl ike the other two terms, i s therefore comprised of terms that involve the vortex dens i t i e s of adjacent-elements. The app l i c a t i on of the higher order methods to the so lu t ion involves the c a l cu l a t i on of the extra terms (13)** and (14)**. The curvature terms -44 -are simply added to the in f luence c o e f f i c i e n t K-. ca l cu la ted f o r the basic case. The l i n ea r v e l o c i t y terms must be added to the c o e f f i c i e n t s K. • -,, K.., K. . , . Although th i s i s not d i f f i c u l t to do, the extra ca l cu l a t i ons involved do take considerable amounts of time to perform. while In summary, equations (1) and (12.) g i v e , K.. = 1_ {(b+A) Jin ( r 2 ) - (b-A) Jin ( r 2 ) + 2a t a n " 1 ( ? 2 a ^ 2 ) - 4A> ^ ^  • i ' ' • L a S b ' - A ** K.. = — {Jin A - 1} (18)** 11 TT APPENDIX 3 CALCULATION OF THE VELOCITY COMPONENT INDUCED AT A POINT IN THE FIELD OF VORTEX DISTRIBUTION: The stream function at a point P^ due to a st r a i g h t vortex element ' j ' , with constant density Yj d i s t r i b u t e d over that element, i s given by _Y -; P * * * - i where r i s the distance from the point P^ to the point Q on the surface, as i n F i g . (10a), and i t can be written as rL. = {(x^-O + yp (2) Thus —Y . A o o i« * i j = ~2% f * n { ( x j - 5 } +y- > d ? (3) A The v e l o c i t y components i n d i r e c t i o n s p a r a l l e l and normal to the element ' j ' r e s p e c t i v e l y , induced at P^ are: u ± j = = I i T dC , (4a) -Y • t - j r *1 2TT / ( x l - ^ ) 2 + y ! 2 - _ ^ Y . A - - l r (x!-£) 3x. 3 2TT / -A (x^-^2+y' 2 *** *** v.. = = ^ / L _ _ d ? ( 4 b ) Therefore *** and Y. , (x'-A) , (x'+A) U i j = 2? { t a n y T - " t a n - ^ T — } <5a> v, . = - J - ' *n{ y!^+(x!-A)^ / y ^ + t x l + A ) ^ } (5b) 13 4TT z 3 3 -*3 3 Where x^ and y^ are the co-ordinates of the point P i with respect (referring) to a system of axis i t s o r i g i n i s the control - 46 -point on the element ' j ' which has the length 2A and makes an angle 6.. with the wind axis, and they are given by *** x^ = (x i-x ;.) cos6j + (y^y-j) sine.. (6a) *** y! = (x^-x^ sine.. + ( y ^ Y j ) cos6.. (6b) To express the v e l o c i t i e s tangential and normal to the i control surface we consider the following: From F i g . (7b) and from the vector analysis of the element • j ' , the v e l o c i t y vector V.. can be written as 3 * * * V.. = u.. t . + v.. n. (7) 1 3 3-D D I D D also thus *** V. . = X. . I + y. . 3 '(8) I D I D I D X. . = V. . I *** x3 I D (9) then from equations (7)*** and (9)***, we get X ± j - U i j C O s 9 j ^ n e . (10)*** s i m i l a r l y *** Y. .' = u. . sine. + v. . cose. .(11) I D I D D I D _ D Also the v e l o c i t y vector V\ ^ , r e f e r r i n g to F i g . (7b) and from the vector analysis of the element ' i ' , can be written as *** V.. = V m t. + V n. (12) I D T ± j i N i j x thus *** - V i (13> then from equations (8) and (13) , we get *** V_ = X.. cos6. + Y.. sine. (14) T ± j I D 1 I D i s i m i l a r l y — — *** V.. = V. . n. (15) i j x3 1 = y.. cose. - x.. sine. I D i I D i - 47 -*** *** *** *** substitute (10) and (11) into equations (14) and (15) , hence *** v T = u i ; . c o s t e . - e . . ) + v i ; . s i n t e ^ O j ) ( 1 6 a y V = - u i 3 s i n f e . - e . . ) + v ± j cos ( 9 ^ ) (16b)*** V T and V N are the tange n t i a l and the normal v e l o c i t i e s , r e s p e c t i v e l y , induced at element * i ' , t angential and normal to i t , due to a s t r a i g h t vortex element ' j ' with constant density - 48 -APPENDIX 4 AN ANALYTICAL APPROACH TO THE FLOW IN THE WIND TUNNEL PLENUM A. Introduction An extension of the current analysis is to make the geometry of the flow representation more like that which actually occurs experimentally in the test-section, with the plenum surrounding the slotted walls. Figure (6) compares the flow representation of the current analysis with the phy-sical flow which actually occurs in the test-section. As shown in Fig. (6)a, two streamlines AB and CD are used to simulate the actual shear layers shown in Fig. 6(b). AB leaves the test-section upstream of the test a i r f o i l and opposite to i t s negative pressure side, while CD enters the test-section upstream of the test a i r -f o i l and opposite to i t s positive pressure side. In this section the flows in the upper and lower plenums are represented by considering these two streamlines t© be constant pressure streamlines, in the current analysis. In order to define the geometry of the constant pressure stream-lines and investigate i t s effects, the current theory should be developed to provide the induced tangential and normal velocities of the flow at any point in the f i e l d , since they are necessary terms for tracking any streamline. B. The Induced Tangential arid Normal Velocities due to Vorticity Distribution The stream function ¥.. at a control point P. whose coordinates i ] 1 with respect to the j straight element are P^ = CX! • y')., a s i n Fig. (7a), - 49 -due to vorticity having a constant density y over that element is given - Y j" In r (P_L,1 ¥ i j = 2^ I l n r ( P. '£> d ^ (1)**** A Then the velocity components can be calculated from equation (1) . Using the notation of Fig. (7) the velocity components induced at the element ' i 1 due to a vortex element ' j ' , are, Y - -, x ' .-A •, x ' .+A 1 -i , —1 n —1 1 , * * * * u. . = -=J {Tan X f-^-) - Tan X (^-)} (2a) in 2TT y! y, D y Y-; . 9 2 . 2 . . 2. -i * * * * v - . = -L zn { ( y ' / + (x ! -A ) ) / ( y ! + (x- + A ) )} ( 2b) i ] 4n : D D 3 The directions of u.. and v.. at P. are parallel and normal to the ID ID i direction of element ' j 1 , respectively. With respect to cartesian 'wind axes' X and Y (X is the wind direction), the j vortex element and the element ' i ' are included at angles 6^  and 6^ to the X-axis respectively. Thus, V = u.. cos ( 9 . - 9 . ) + v.. sin ( 9.-e.) (3a)**** ij 1^ i D I D i D ; V = v. . cos ( 9 . - 9 . ) - u. . s i n ( 9 . - 9 . ) (3b)**** I D l D I D 1 D are the tangential and the normal velocities induced at the element ' i ' due to the vortex element ' j ' of density y . The local tangential velocity . Y i D Vm, . is - — w h i l e the local normal velocity V„.. is zero. The details of T n 2 1 Nn **** **** the calculation of equations (2) and (3) are provided in Appendix 3. Therefore, the tangential and the normal velocities induced at - 50 -the control point on the element * i ' due to a system of 'N' vortex elements immersed in an i n f i n i t e uniform flow U, parallel to the X-direction, are, N V = Z {u. . c o s ( 9 . - 9 . ) + v. . sin ( 9 . - 6 . ) } + U«,cos 9 . (4 T. . n ID l D 13 1 3 1 l D=l **** a ) V „ = Z (v.. c o s ( 6 . - 9 . ) - u.. sin ( 6 - 9 )} _ n s i n 9 - (4b) Hence, at any free point in the f i e l d of the above system the tangential and the normal velocities of the flow there are given by, N **** V = V = Z {u., cos 6 . - v.. sin 9.} + U (5a) T ± X ID 3 ID D °° **** N V = V = Z {v.. cos 6.+ u.. sin 9 . } < 5 b ) N i Y j=l 1 3 3 3 3 (i = 1,2,...., N) and the pressure coefficient of the flow at this point P^ is given by, p o p **** P ± = 1 - (V + V n ) , (i = 1,2,...., N) (6) . i i C. Simulated Shear Layer Tracking The routine described in the previous chapter must be carried out to calculate N vortex densities, y , for the N elements which represent the test a i r f o i l , the wall slats and the solid walls. The values of the •k 4c "k It it it * * vortex densities can then be inserted in equations (5a) and (5b) to get the induced tangential and normal velocities, respectively, at a specified point (x., y ) in the flow f i e l d . Thus from a starting point (x., v ), 1 i 1 i which is supposed to be at the downstream edge of the solid wall but, to - 51 -avoid the singularity there is located instead very close to the edge, the flow direction 0^ i s calculated, given by, _, **** 6 = tan" 1 (V /V ) (7) 1 1 where V T and V are the induced tangential and normal velocities 1 1 at the point (x^, y^), respectively, and they are given by equation (5a) **** and (5b) Then 0.^  is used to calculate the next point (x^, y 2) where, . **** x 2 = x.^  + A x (8a) Y2 = y± + A x * tan 6 1 **** (8b) The flow direction 6 ^ i - s then calculated there in the same way as 0^. The flow directions are averaged to give 0, and the y-co-ordinate i s changed so that the next point i s now, **** x 2 = x^ ^ + A x (9a) y 2 = y x + A x * tan 0 **** (9b.) Having defined the geometry of the streamlines simulating the shear layers, which i s assumed to remain constant in the following analysis, one can proceed to include them in the present theoretical analysis as follows: Fir s t the pressure variation along each of the streamlines i s calculated, **** using Equation (6) , and the large negative pressure excursions, as the flow accelerates when in the vi c i n i t y of the wall slats, is excluded and an average pressure coefficient, C p, would be obtained for each - 52 -streamline. Then each streamline i s divided into a number of elements n, which adds n vortex elements to the previous N elements. Thus we should once more solve (N'+M) questions, (N'=N+n) elements and M components, pre-scribed by the following set of equations, N ¥ + E K. . Y. = R. , (i=l,2, N) (8) S I D ' D i and N ¥_ + E K . Y- = (m=s=l,2, M) sm tp ,] D t p , 10) m J—J- m m The additional n equations required for a solution to this problem, are the prescribed - tangential - velocity equations for streamlines, i N **** V = E {u. . c o s ( 0 . - 6 ) + vm s i n ( 6 . - 9 . ) } + U c o s 9 , , (4a) l D=1 where **** VT = ± ^ d - ^ ) / (i=l,2, n) (10) T i P i The sign of the squareroot i s positive i f the vortex elements are distributed sequentially for calculations in the same sense as the flow directions and C p is the prescribed pressure coefficient along either of the i two constant pressure streamlines (C^ could be the average theoretical - i pressure coefficient or any other experimental value). Thus the problem of including the flow in the wind tunnel plenum into the present theoretical analysis has therefore been reduced to that of solving the above (N+n+M) equations simultaneously to get (N+n) vortex densities, Y» a n d M stream functions ¥. - 53 -APPENDIX 5 LIST OF EQUIPMENT USED IN THE EXPERIMENTAL INVESTIGATION Instrument 1. Barocel Pressure Sensor Type 511 (10-mm Hg=10 v o l t s ) . Descr ip t ion Pressure transducer fo r p i t o t - s t a t i c tube measurements. 2. Barocel E l e c t ron i c Manometer Type 1018B. 4-1/2 d i g i t voltmeter fo r i nd i ca t i ng the to ta l pressure head from p i to t-s t a t i c tube. 3. Barocel Signal Condit ioner Type 1015. Amp l i f i e r fo r Barocel pressure t ransducer . 4. Betz Water Micromanometer. For p i t o t - s t a t i c tube wind speed measurements. 5. Setra Pressure Transducer Model 237. For Scanivalve a i r f o i l surface pressure measurements. 6. NEFF Data Acqu i s i t i on System. For scanning and d i g i t i z i n g the pressure transducer s i g n a l . 7. PDP-11 Computer hooked up to NEFF. For eva lua t ing , d i sp lay ing and s tor ing the surface pressure c o e f f i c i e n t s . - 54 -FIGURE (1) COMPARISON OF AIRFOIL PRESSURE COEFFICIENTS: THEORY REF. [9] - 55 -FIGURE (2) VORTEX REPRESENTATION OF TWO COMPONENT AIRFOIL: REF. - 56 -FIGURE (3) NOTATION USED TO CALCULATE INFLUENCE COEFFICIENTS: REF. [10] - 57 -—Growing of the itart ing vortex. FIGURE (4) STREAMLINES CONTOURS AROUND AN AIRFOIL: REF. [12] - 58 -FIGURE ( 5 ) LOCATION OF T R A I L I N G CONTROL POINT: REF. [10] (a) THEORY FIGURE (6) AN AIRFOIL INSIDE TUNNEL TEST SECTION WITH DOUBLE SLOTTED WALL - 6 0 ^ F I G U R E (7) N O T A T I O N U S E D TO C A L C U L A T E I N D U C E D V E L O C I T I E S - 61 -FIGURE (8) LOCATION OF ELEMENTS ON AIRFOIL SURFACE: REF. [10] - 62 -i i i i 1 1 1 r T c> V KarmanTrefftz Aerofoil a = 0° Element Shape O Straight Line • Curved V Straight Line A Curved —= Exact Analytic Vortex Density Constant Constant Linear Linear FIGURE (9) COMPARISON OF HIGHER ORDER METHODS REF. [10] - 63 -x/c FIGURE (10) COMPARISON OF AIRFOIL VELOCITY DISTRIBUTIONS REF. f lQ] - 64 -FIGURE ( 1 1 ) GEOMETRY FOR CALCULATION OF HIGHER ORDER TERMS: REF. [10] FIGURE (12) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION FIGURE (13) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-23012 AIRFOIL IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION - 67 -FIGURE (14) COMPARISON OF PRESSURE COEFFICIENTS FOR CLARK-Y 14% AIRFOIL IN FREE AIR AND BETWEEN SOLID WALLS TEST SECTION - 68 -FIGURE (15) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN SINGLE UNIFORMLY SLOTTED WALL TEST SECTION - 69 -FIGURE (16) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN DOUBLE UNIFORMLY SLOTTED WALL TEST SECTION FIGURE (17) COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN LIFT COEFFICIENTS - 71 -NACA-0015 a = 10° / NACA-23012 WITH 25.66% SLOTTED FLAP / ' 5 """S r — 5— r ~ ~ S — ' — I C/H FIGURE (18) COMPARISON OF THE EFFECT OF TEST AIRFOIL SIZE ON THE RELATIVE ERROR IN PITCHING MOMENT COEFFICIENTS - 7 2 --8. 0 -7. 0 CP -6. 0 - 5 . 0L -4. 0 - 3 . 0 © -1. 0 FREE AIR DOUBLE NON-UNIFORM SLOTTED WALLS WITH A0AR=59% NACA-0015 C/H=. 8 ALPHA= 10 DEGREES E L =0.62 % E M = _ 1 - 2 1 % 2 s - 2 . 0 i a t ^ * * A 0 . 01 . _ — — » — , — * — 6 ^ ' $ £ 6 ' © £> © & © A * * 0 - 0 0 . 5 X/H 1 b 0 FIGURE (19) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-0015 AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NON-UN I FORMLY SLOTTED WALL TEST SECTION - 7 3 --14. 0 -13. 0 12. 0 CP •1 1. 0 • FREE AIR DOUBLE NON-UNIFORM SLOTTED WALLS WITH A0AR=59% D NACA-23012 C/H=. 8 - i 0 . 0 D ALPHA= 8 GAMA= 20 I* -Q. 0 -8. 0 - 7 . 0 -6. 0 -5. 0 -4. 0 -3. 0 -2. 0 -1. 0 O -0. 0 1. 0. E L =-4.03 * E M=-0.91 % • g $ d £ £ ^ ® © © © © 0.0 0.5 1.0 FIGURE (20) COMPARISON OF PRESSURE COEFFICIENTS FOR NACA-23012 AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NON-UNIFORMLY SLOTTED HALL TEST SECTION -74 --14. 0 -13. 0 |© ^ FREE AIR ! DOUBLE NON-UNIFORM SLOTTED WALLS WITH A0AR=59% CLARK-Y 14% C/H=. 8 - i B . 4 ALPHA= 10 DEGREES 12. 01 CP 11. 01A - S . 0 - S . 0 - 7 . 0 - e . 0 - 5 . 0 -4. 0 -3. 0 -2. 0 -1. 0 - 0 . 0 E L =-0.79 % E M = ~ 1 ' 2 4 % © A © • s ffi 6 6 6 d d * 6 , i » * M • • • • • 0 . 0 0 . 5 X/H 1 , 0 FIGURE (21) COMPARISON OF PRESSURE COEFFICIENTS FOR CLARK-Y 14% AIRFOIL IN FREE AIR AND BETWEEN DOUBLE NON-UNIFORMLY SLOTTED WALL TEST SECTION - 75 -FIGURE (22) COMPARISON OF-THE EFFECT OF TWO DIFFERENT ANGLES OF ATTACK ON THE RELATIVE -ERROR IN LIFT COEFFICIENTS FOR NACA-0015 AIRFOIL C/H = .8 c/C = .15 a = 10 FIGURE (23) GEOMETRY OF THE IDEALIZED FLOW IN THE PLENUM CHAMBER FOR NACA-0015 AIRFOIL - 77 -B. 0 r 7. 0 CP 6. 0 5. 0 © FREE AIR SLOTTED WALLS A0AR=59% WITH THE FLOW IN THE PLENUM C/H=. 8 ALPHA= 10 DEGREES C. =1.2137 C M =-0.0412 L T M c/4 4 - 0PA E. =-0.98% E u=-1.82% M 3. 0 . © • A i ft i 2 . 0 L © • 1 * • O • l " * r * © A © © A 0. 0 * ° i * • • t * ' A £ ~© o " © 4" $ $ A © ~ $ $A' A © A © *A © i . 0[2© 0 . 0 0 . 5 X / H 1 " 0 FIGURE (24) COMPARISON OF NACA-0015 AIRFOIL PRESSURE COEFFICIENTS: FLOW IN WIND TUNNEL PLENUM IS INCLUDED IN THEORETICAL ANALYSIS - 78 -MM OF WATER (TEST-SECTION) 413. 0 r FIGURE (25) CALIBRATION OF NOZZLE AND TEST SECTION DYNAMIC PRESSURES - 79 -a . 0 r 7 . 0 CP 6 . 0 5 . 0 4 . 0 3 . 0 2. 0 0 . 0 e SOLID WALLS A SLOTTED WALLS A0AR=59% RE= 500,000 C/H=. 34 ALPHA= 10 DEGREES A © A ® A s © A © *©©© i . 0 t A S © * » 9 44-1. 0} 0 . 0 0 - 5 X / H 1 - 0 FIGURE (26) WALL EFFECT ON THE PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.34: EXPERIMENT - 80 -B. 0 7. 0 CP 6. 0 5 . 0 4. 0 3. 0 2. 0 © SOLID WALLS A SLOTTED WALLS A0AR=59% RE= 500,000 C/H=. 67 ALPHA= 10 DEGREES © © © • A © © A ©O '©, A © , A A © A ~ © A s a e a a g ta ft-0 . 0 * A © 6 * 1. 0{4fe 0 - 0 0 - 5 X / H 1 , 0 FIGURE (27) WALL EFFECT ON THE PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.67: EXPERIMENT - 81 -—8. 0 -7. 0 CP -6. 0 - 5 . 0 -4. 0 -3. 0 o SLOTTED WALLS AOAR-50% A SLOTTED WALLS A0AR=55% + SLOTTED WALLS A0AR=59% C/H=. 34 RE=500, 000 ALPHA = 10 DEGREES © 0 4 -2. 0 -1. 01 0. 0 1. 0 0 © 4 © 4 9 * • » * t 1 t ' • • A ' • • • ^ • • • . • • • • 0 0 . 5 X / H 1 , 0 FIGURE (28) EFFECT OF AOAR OF SLOTTED WALLS ON PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.34: EXPERIMENT - 82 -— B. 0 -7. 0 © CP -6. 0 -5 . 0 •4. 0 SLOTTED WALLS AOAR=50% SLOTTED WALLS A0AR=55% SLOTTED WALLS A0AR=59% C/H=. 67 RE=500, 000 ALPHA = 10 DEGREES I •3. 0 I + -2 . 0 - 1 . 0+ 0. 0 • * » 1. 0 ( _ 0 . 0 0 . 5 x / H 1 - 0 FIGURE (29) EFFECT OF AOAR OF SLOTTED WALLS ON PRESSURE COEFFICIENT OF NACA-0015 AIRFOIL, C/H=.67: EXPERIMENT - 83 -S. 0 7. 0 CP 6. 0 5 . 0 4. 0 3- 0& 2. 0 0. 0 o C A R 34 A C/H=. 67 SLOTTED WALLS A0AR=59% RE= 500,000 ALPHA= 10 DEGREES 0 • V 1 . C # • • A 6 6 • ft ® m 1 . 0 [ 0 . 0 0 . 5 X / H 1 - 0 FIGURE (30) COMPARISON OF NACA-0015 AIRFOIL PRESSURE COEFFICIENTS EXPERIMENT 2. 0 CL 1. 5 1. 0 0. 5 © SOLID WALLS A* SLOTTED WALLS AOAR=50% + SLOTTED WALLS AOAR-55% x SLOTTED WALLS A0AR=59% C/H=. 34 RE=500,000 - 4 . ALPHA 16. 0 FIGURE (31) EFFECT OF WALL CONFIGURATION ON LIFT COEFFICIENT FOR NACA-0015 AIRFOIL, C/H=.34: EXPERIMENT - 85 -2. 0 CL 1. 5 1. 0 © + X SOLID WALLS SLOTTED WALLS AOAR=50% SLOTTED WALLS A0AR=55% SLOTTED WALLS A0AR=59% C/H=. 67 RE=500,000 o © © © o A X A 0. 5 © X 0. 0|_ -0. 5 0 -4. 0 0. 0 4. 0 e. 0 ALPHA 12. 0 16. 0 FIGURE (32) EFFECT OF WALL CONFIGURATION ON LIFT COEFFICIENT FOR NACA-0015 AIRFOIL, C/FK67 : EXPERIMENT - 86 -0. 10 CM C/4 0. 05 0 . 00?-•0. 05 -4. 00 © A + X 0. 00 SOLID WALLS SLOTTED WALLS AOAR=50% SLOTTED WALLS A0AR=55% SLOTTED WALLS A0AR=59% C/H=. 34 RE=500, 000 0 * - 4 A 4. 00 o 6 + A s. 00 X © ALPHA 12. 00 16. 00 FIGURE (33) EFFECT OF WALL CONFIGURATION ON PITCHING MOMENT COEFFICIENT FOR NACA-0015 AIRFOIL OF SIZE C/H=.34: EXPERIMENT - 87 0. 10 CM C/4 0. 05 + 0. 0l£ © 4-X •0. 05 -4. 00 SOLID WALLS SLOTTED WALLS AOAR=50% SLOTTED WALLS A0AR=55% SLOTTED WALLS A0AR=59% C/H=. 67 RE=500, 000 ¥ 4^  X O O 0 . 0 0 4. 00 B. 00 ALPHA 12. 00 16. 00 FIGURE (34) EFFECT OF WALL CONFIGURATION ON PITCHING MOMENT COEFFICIENT FOR NACA-0015 AIRFOIL OF SIZE C/H=.67: EXPERIMENT - 88 -FIGURE (35) EFFECT OF AIRFOIL SIZE ON E,_FOR NACA-0015:. COMPARISON OF EXPERIMENT AND THEORY -89-N . A. C. A.tlrton oooo.. 0012.. 0015. Re= 500,000 m F = 0.0915 0018.. 3412.. 33012.. (millions) 8.470 8.280 6.100 8.410 1.760 .882 .446 .223 .112 8.870 R450 6.280 (.640 1.740 .871 .449 8.610 8.990 8.350 1.730 • 874 (dec) 33012-33.. 2Ril2 4406. 4412.. 4418. 6412.. .222 .113 7.840 6.240 1.300 1.730 .866 .430 .214 .100 8.240 6.100 8.420 1.730 .870 .438 .218 .110 8.370 8.160 6.070 8.400 1.760 .884 .446 .221 .112 8.000 6.360 3.380 1.760 .600 .4M 8.370 6.810 8.540 1.770 .884 .464 8.080 4.S70 8.340 1.700 .866 .438 .218 .110 7.920 e. ioo S.270 1.680 .874 .433 .216 .111 7.920 6.280 3.340 1.730 .682 .431 .219 .110 8.210 6.020 8.350 1.700 .882 .441 .219 .110 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3.0 -2.1 -2.0 -2.1 -2.1 -2.0 -2.2 -1.3 -1.2 -1.2 -1.2 -1.2 -1.2 -1.3 -1.6 -1.4 -1.2 -1.2 -1.2 -1.2 -1.2 -1.4 —.6 - .7 - .7 - .8 - .8 - .9 -3.9 -3.6 -4.0 -4.0 -4.1 -4.1 -3.7 - Z 8 -4.0 —4.1 —4.1 -4.2 -4.3 -4.3 -4.3 - 3 6 -4.0 -4.0 —4.1 -4.2 -4.3 -4.4 -4.4 -3.1 -8.9 -8.9 -61 -6.2 —6.3 -6.2 -8.9 -8.4 m aoee .067 .097 .066 .096 .105 .117 .104 .099 .100 .097 .067 .096 .064 .098 .067 .066 .064 .063 ..092 •*ro»i .101 .134 .096 .096 .066 .065 .060 .086 .062 .114 .068 .097 .068 .066 .066 .066 .102 .100 .098 .068 .097 .066 .096 .109 .067 .096 .096 .066 .094 .066 .098 .067 .067 .065 .066 .100 .066 .066 .095 .068 .066 .097 .105 .115 .068 .096 .098 .097 .096 .094 .100 .113 .097 .065 .096 .065 .094 .066 .086 .066 .096 .067 .087 .097 .067 .106 41.89 *1.38 D.94 D.86 D.86 D.gi O.B3 ».7B 41.66 *1.65 41.62 '1.49 41.18 D.B1 C. 89 41.66 41.60 C1.48 C1.28 ci.oe o.98 ».89 O.tO 41.63 41.83 C1.42 oi.26 C1.15 41.03 o.96 D. 86 41.72 41.68 C1.83 01.S3 »1.16 »1.08 01.08 01.03 41.72 *1.67 41.83 »1.41 »1.28 »1.19 »1.15 Oi.OO •1.49 41.42 »1.26 »1.12 01.07 Ol.Ol 41.61 "1.85 "1.44 »1.28 »1.14 01.06 41.77 »1.70 C1.80 01.29 01.26 • »1.23 »1.21 »1.09 01.74 01.70 Ol.61 »t.4« B1.86 Dl.Sl 01.82 O1.20 C1.72 Ol.66 01.86 01.48 01.41 D1.35 01.84 01.84 01.82 01.75 D1.64 Ol.B4 01.48 »1.47 »1.46 Ol.4S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .14 .14 .15 .30 .22 .42 .26 .06 .06 .05 .16 .28 .12 .37 .20 .20 .10 .23 .28 .10 .40 .10 .02 .11 .23 .28 .85 .26 .26 .84 .41 .40 .65 .87 .77 .82 .22 .30 .37 .86 .61 .87 .32 .20 .33 .31 .84 .89 .46 .68 .87 .35 .38 .82 .60 .85 .70 • 0.0061 .0064 .0064 .0062 .0060 .0049 .0065 .0131 .0138 .0066 .0069 .0073 .0077 .0075 .0065 .0105 .0077 .0082 .0086 .0088 .0084 .0076 .0148 .0158 .0088 .0092 .0068 .0100 .0102 .0127 .0176 .0267 .0071 .0080 .0076 .0069 .0085 .0067 .0156 .0227 ».0071 .0070 .0078 .0080 .0060 .0084 .0066 .0179 .0182 .0071 .0075 .0076 .0071 .0084 .0096 .0073 .0078 .0077 .0077 .0073 .0118 .0073 .0080 .0077 .0084 .0080 .0097 .0066 .0189 .0082 .0085 .0067 .0065 .0061 .0106 .0164 .0276 .0090 .0093 .0064 .0069 .0103 .0123 .0168 .0366 .0061 .0096 .0099 .0104 .0066 .0129 .0205 .0160 (percent c) 0 0 0 0 0 0 0 0 0 0 0 O 0 -.043 -.043 -.045 -.045 -.054 .006 .007 .007 .012 .010 .010 .010 .011 .014 .011 .014 .005 .006 .005 .002 -.001 (percent c) .066 .068 .090 .092 .098 .088 .088 .061 .065 .067 .085 .086 .085 .090 .092 -.133 -.130 -.131 -.135 1.0 1.0 1.8 1.7 .6 .6 .8 1.0 1.1 1.3 1.1 1.3 2.4 1.5 1.7 1.6 33 3.2 3.4 1.8 .8 1.1 1.1 .9 1.8 1.2 1.3 1.3 1.4 3 0 .6 .6 1.0 .9 .9 .4 1.0 1.1 1.0 .6 1.1 .6 .7 1.0 1.1 1.4 .8 .9 1.0 1.3 1.1 1.0 1.4 1.4 1.7 1.4 .9 1.1 .8 1.0 3 1 -1 -1 -4 3 1 -1 -5 -2 FIGURE (36) AIRFOIL SECTION CHARACTERISTICS AS AFFECTED BY VARIATIONS OF THE REYNOLDS NUMER: REF. [17j FIGURE (37) U.B.C. MECHANICAL ENGINEERING LOW-SPEED CLOSED-CIRCUIT WIND TUNNEL - 91 -TABLE I AERODYNAMIC CHARACTERISTICS OF AIRFOILS EXAMINED THEORETICALLY 1.1 NACA-0015 a = 10° Free A i r Values C, = 1.2257 C M = 0.2829 C M = -.0189 L F M 0 M c/4 Wall Conf igurat ion C/H \ % CM c/4 So l id Walls 0.1 1.2335 0.2841 -0.0196 „ 0.2 1.2562 0.2841 -0.0217 „ 0.3 1.2939 0.2936 -0.0250 „ 0.4 1.3458 0.3019 -0.0294 „ 0.5 1.4115 0.3127 -0.0348 " " . .9-6 1.4906 0.3260 -0.0410 0.7 1.5825 0.3420 -0.0476 H 0.8 1.6873 0.3607 -0.0547 „ 0.9 1.8050 0.3825 -0.0619 1.0 1.9361 0.4075 -0.0692 S lotted Walls A0AR=59% 0.1 1.2341 0.2841 -0.0198 II H 0.2 1.2438 0.2863 -0.0199 - 92 -1.1 (Continued) Wall Conf igurat ion C/H cM  Mo C M C / 4 S lot ted Walls . A0AR=59% Q.3 1.2474 0.2872 -0.0199 n n 0.4 1.2443 0.2857 -0.0206 ii ii 0.5 1.2385 0.2825 . -0.0224 n H 0.6 1.2336 0.2784 -0.0253 H H 0.7 1.2320 0.2741 -0.0293 n H 0.8 1.2333 0.2699 j -0.0338 n n 0.9 1.2380 ; 0.2659 ' -0.0389 1.0 1.2452 0.2623 -0.0442 1.2 NACA-23012 a = 8 6 = 20° Free A i r Values C. = 2.7959 C M = 0.3890 C M = -0.3031 L F M 0 M c/4 Wall Conf igurat ion C/H \ M 0 M c / 4 So l id Walls 0.1 2 . 8 0 9 9 0 . 3 9 0 4 - 0 . 3 0 5 2 I I H 2 , 8 4 9 9 0 . 3 9 9 0 - 0 . 3 0 6 5 n n 0 . 3 2 . 9 1 7 0 0 . 4 1 0 7 - 0 . 3 1 1 5 - 93 -1.2 (Continued) Ka l i Conf igurat ion C/H .. . v K c / 4 • So l id Walls 0 . 4 3 . 0 1 0 6 , 0 . 4 2 5 6 - 0 . 3 1 9 7 n n 0 . 5 3 . 1 2 9 9 0 . 4 4 4 1 - 0 . 3 3 0 8 II I I 0 . 6 3 . 2 7 5 2 0 . 4 6 6 5 - 0 . 3 4 4 4 n I I 0 . 7 3 . 4 4 6 5 0 . 4 9 3 2 - 0 . 3 6 0 1 n I I 0 . 8 3 . 6 4 4 8 0 . 5 2 4 8 - 0 . 3 7 7 5 II n 0 . 9 3 . 8 7 1 1 0 . 5 6 2 0 : - 0 . 3 9 6 3 n n 1 . 0 4 . 1 2 7 6 0 . 6 0 5 7 - 0 . 4 1 6 2 S lot ted Walls A 0 A R = 5 9 % 0.1 2 . 8 0 9 2 0 . 3 9 0 2 - 0 . 3 0 5 2 II n 0 . 2 2 . 8 2 6 0 0 . 3 9 2 8 - 0 . 3 0 6 8 H I I 0 . 3 2 . 8 2 3 7 0 . 3 9 2 2 - 0 . 3 0 6 8 II I I 0 . 4 2 . 8 0 0 8 0 . 3 8 6 2 - 0 . 3 0 7 2 n II 0 . 5 2 . 7 7 0 4 0 . 3 7 6 2 - 0 . 3 0 9 7 0 . 6 2 . 7 3 6 4 0 . 3 6 3 5 - 0 . 3 1 4 0 n II 0 . 7 2 . 7 0 7 3 0 . 3 4 9 7 - 0 . 3 2 0 5 ti I I 0 . 8 2 . 6 8 3 1 0 , 3 3 5 8 - 0 . 3 2 8 5 - 94 -1.2 (Continued) Wall Conf igurat ion 1 C/H C L L T cM M 0 C M c / 4 S lo t ted Walls A0AR=59% 0.9 2.6655 0.3219 -0.3380 1.0 2.6527 0.3083 . -0.3484 1.3 CLARK-Y 14% a = 10° Free A i r Values C, = 2.1353 C M = 0.3926 C H = - 0.1331 Lp "c/4 Wall Conf igurat ion C/H C i T M 0 c ' M c / 4 So l id Walls 0.1 2 . 1 4 9 0 0 . 3 9 4 6 - 0 . 1 3 4 5 M II 0 . 2 2 . 1 8 8 3 0 . 4 0 0 6 : - 0 . 1 3 8 2 n n 0 . 3 2 . 2 5 4 6 0 . 4 1 0 8 - 0 . 1 4 4 3 n n 0 . 4 2 . 3 4 7 8 0 . 4 2 5 4 - 0 . 1 5 2 6 0 . 5 2 . 4 6 8 7 0 . 4 4 4 8 - 0 . 1 6 3 0 I I I I 0 . 6 2 . 6 1 8 4 0 . 4 6 9 6 - 0 . 1 7 5 1 II n 0 . 7 2 . 7 9 8 9 0 . 5 0 0 4 - 0 . 1 8 8 7 II n 0 . 8 3 . 0 1 3 1 0 , 5 3 8 3 - 0 . 2 0 3 5 II H ° - 9 3 . 2 6 5 2 0 , 5 8 4 5 - 0 , 2 1 9 4 H n 1 . 0 3 . 5 6 1 1 0 , 6 4 0 6 - 0 . 2 3 6 1 - 95 -1.3 (Continued) S lot ted Walls A 0 A R = 5 9 % C/H V n Q F 1 c/4 ii n 0.1 2 . 1 5 0 4 0 . 3 9 4 6 - 0 . 1 3 4 8 n n 0 . 2 2 . 1 6 5 6 0 . 3 9 7 8 - 0 . 1 3 5 4 n n 0 . 3 2 . 1 6 9 2 0 . 3 9 8 2 - 0 . 1 3 5 8 II H 0 . 4 2 . 1 5 9 5 0 . 3 9 4 5 - 0 . 1 3 7 2 n n 0 . 5 2 . 1 4 4 6 0 . 3 8 7 8 - 0 . 1 4 0 2 II I I 0 . 6 2 . 1 3 1 2 0 . 3 7 9 5 - 0 . 1 4 5 2 II M 0 . 7 2 . 1 2 2 0 0 . 3 7 0 8 - 0 . 1 5 1 7 II II 0 . 8 2 . 1 1 8 5 0 . 3 6 2 0 ; - 0 . 1 5 9 6 n H 0 . 9 2 . 1 2 0 6 0 . 3 5 3 6 - 0 . 1 6 8 5 n n 1 . 0 2 . 1 2 7 0 0 . 3 4 5 4 - 0 . 1 7 8 2 - 96 -TABLE II EFFECT OF THE PRESSURE ALONG.THE SHEAR LAYERS ON THE NEW WIND TUNNEL THEORETICAL DATA NACA-0015 AIRFOIL a = 10° C/H = .8 Free A i r Values C. =1 .2257 C M =-0.0189. L F c/4 V \ . E L% C M c / 4 -0.4 0.11 1.4038 14.53 -0.0555 -2.99 -0.3 0.11 1.3094 6.83 -0.0484 -2.42 -0.2 0.11 1.2137 -0.98 -0.0412 -1.82 -0.1 0.11 1.1164 -8.91 -0.0338 -1.22 -0.1 0.2 1.1163 -8.93 -0.041 -1.8 -0.1 0.3 1.1137 -9.14 -0.0488 -2.44 -0.1 0.4 .1,1079 -9.61 -0.0561 -3.04 - 97 -TABLE III LOCATIONS OF THE PRESSURE TAPS III.1 For C/H = .34 Sect ion Upper Surface Lower Surface x/H y/H x/H y/H 0.00346 0.01224 0.0 0.0 0.01060 0.02188 0.01800 0.02797 0.01750 0.02778 0.03190 0.03633 0.02930 0.03493 0.05410 0.04587 0.04980 0.04426 0.07880 0.05338 . 0.07520 0.05241 0.10400 0.05917 0.10200 0.05876 0.15400 0.06701 0.12700 0.06331 0.20500 0.07176 0.15200 0.06677 0.2550 0.07406 0.17800 0.06959 0.30400 0.07458 0.20200 0.07155 0.35400 0.07379 0.22800 0.07309 0.40400 0.07179 0.25200 0.073977 0.45300 0.06899 0.27600 0.07443 0.50300 0.06518 0.30300 0.07459 0.55300 0.06100 0.351000 0.07387 0.65200 0.05066 0.40100 0.07194 0.75100 0.03855 0.45300 0.06899 0.85100 0.02462 - 98 -I I I . l (continued) • Upper Surface Lower Surface x/H y/H x/H y/H 0.50200 0.06526 0.90000 0. 01714 0.55200 0.06526 0.95100 0. 00894 0.60200 0.05612 0.65200 0.05066 0.70100 0.04488 0.75100 0.03855 0.80000 0.03195 0.84900 0.02491 0.90000 0.01714 III.2 For C/H = .67 Sect ion Upper Surface Lower Surface x/H y/H x/H y/H 0.00664 0.06969 0.0 0.0 0.01808 0.00705 0.00667 0.01894 0.02847 0.06378 0.01874 0.02852 0.05345 0.05929 0.02887 0.03469 0.07914 0.05347 0.05401 0.04584 0.10457 0.04564 0.07984 0.05365 - 99 -III.2 (Continued) Upper Surface Lower Surface x/H y/H x/H y/H 0.13003 0.03466 0.10467 0.05931 0.15426 0.02803 0.13031 0.06382 0.17915 0.01890 0.15423 0.06704 0.20921 0.07204 0.17990 0.06974 0.22971 0.07317 0.20922 0.07204 0.25405 0.07403 0.22986 0.07318 0.27904 0.07447 0.25452 0.07404 0.30338 0.07459 0.27949 0.07448 0.35456 0.07377 0.30406 0.07458 0.40460 0.07177 0.35351 0.07380 0.45178 0.06907 0.40306 0.07184 0.50045 0.06538 0.45223 0.06904 0.59984 0.05635 0.50149 0.06530 0.69811 0.04523 0.55123 0.06117 0.79565 0.03255 0.60081 0.05625 0.88765 0.01907 0.69823 0.04522 0.96950 0.00567 0.79608 0.03249 0.89199 0.01839 0.97060 0.00547 - 100 -TABLE IV THE EXPERIMENTAL AERODYNAMIC CHARACTERISTICS OF NACA-0015 AIRFOIL IV.1 C/H = .34 Re = 500,000 Wall Conf igurat ion 0 a C L L T . % CM n c / 4 So l id Walls -4 - 0 . 4 3 4 9 - 0 . 1 0 5 4 0 . 0 0 3 0 2 II n -2 - 0 . 1 8 1 5 - 0 . 0 2 5 7 3 0 . 0 1 9 6 0 n n 0 - 0 . 0 0 5 1 0 . 0 0 8 6 0 . 0 0 9 8 II n 2 0 . 1 8 9 1 0 . 0 4 8 5 0 . 0 0 1 3 n II 4 0 . 4 3 4 8 0 . 1 1 8 9 0 . 0 1 0 5 n n 6 0 . 6 9 4 4 0 . 1 9 8 8 0 . 0 2 6 2 II I I 8 0.9.061 0 . 2 6 4 6 0 . 0 4 0 3 n II 1 0 1 . 0 1 7 6 0 . 2 7 6 6 0 . 0 2 6 0 H n 12 1 . 1 3 2 6 0 . 3 0 3 3 0 . 0 2 5 3 n H 14 1 . 1 9 1 5 0 . 3 3 2 2 0 . 0 4 3 1 S lot ted Walls -4 - 0 . 3 5 6 8 - 0 . 0 7 8 6 0 . 0 1 0 4 AOAR = bU/o n II -2 - 0 . 1 6 2 6 - 0 . 0 2 3 7 0 . 0 1 6 9 II n 0. .. . 0 . 0 1 0 9 0 . 0 1 2 0 0 . 0 0 9 3 - 101 -IV.1 (Continued) Wall Conf igurat ion 0 or M c / 4 S lot ted Walls AOAR = 5 0 % 2 0 . 1 6 8 6 0 . 0 4 1 6 - 0 . 0 0 0 II n 4 0 . 3 7 5 1 0 . 0 9 7 4 0 . 0 0 3 8 n II 6 0 . 5 7 4 6 0 . 1 5 4 2 0 . 0 1 1 4 n n 8 0 . 7 9 7 9 0 . 2 1 0 0 0 . 0 1 2 4 H II 1 0 0 . 9 2 0 3 0 . 2 3 4 1 0 . 0 0 7 5 II I I 12 1 . 1 0 9 6 0 . 2 9 9 5 0 . 0 2 8 2 H II 14 1 . 1 4 2 5 0 . 3 0 1 6 0 . 0 2 4 3 S lot ted Walls AOAR = 5 5 % -4 - 0 . 3 5 9 7 - 0 . 0 7 8 0 0 . 0 1 1 7 H n -2 - 0 . 1 8 2 5 - 0 . 0 2 8 4 0 . 0 1 7 2 ti H 0 - 0 . 0 1 4 4 0 . 0 0 7 1 0 . 0 1 0 7 n H 2 0 . 1 5 4 9 0 . 0 4 0 0 0 . 0 0 1 3 n I I 4 0 . 3 5 4 1 0 . 0 9 2 0 0 . 0 0 3 7 n n 6 0 . 5 5 7 0 0 . 1 4 6 0 0 . 0 0 7 5 n I I 8 0 . 7 7 8 6 0 . 2 0 7 3 0 . 0 1 4 6 n n 10 0 . 8 6 7 7 0 . 2 1 4 4 0 . 0 0 0 8 - 102 -IV.1 (Continued) Wall Conf igurat ion 0 a \ \ M c / 4 S lot ted Walls AOAR = 5 5 % 12 1 . 0 5 7 4 0 . 2 7 8 6 0 . 0 2 0 1 II n 14 1 . 1 1 2 2 0 . 2 9 1 0 0 . 0 2 1 2 S lot ted Walls AOAR = 5 9 % -4 - 0 . 3 3 7 0 - 0 . 0 7 0 5 0 . 0 1 3 6 II II -2 - 0 . 1 5 6 1 - 0 . 0 2 0 8 0 . 0 1 8 2 0 - 0 . 0 1 3 5 0 . 0 0 6 7 0 . 0 1 0 1 II II 2 0 . 1 5 4 0 0 . 0 3 8 6 0 . 0 0 0 1 II H 4 0 . 3 3 3 7 0 . 0 8 2 8 - 0 . 0 0 0 4 H I I 6 0 . 5 3 6 0 0 . 1 3 7 1 0 0 . 0 0 3 8 H II 8 0 . 7 5 8 3 0 . 2 0 1 6 0 0 . 0 1 3 9 II H 1 0 0 . 8 6 1 1 0 . 2 1 2 4 0 . 0 0 0 4 12 0 . 9 5 1 0 0 . 2 2 2 2 - 0 . 0 1 0 3 n n 14 1 . 0 9 0 0 0 . 2 8 4 1 0 . 0 1 9 7 - 103 -IV.2 C/H = .67 Re = 500,000 Wall Conf igurat ion 0 a C M c / 4 So l id Walls -4 - 0 . 4 7 4 7 - 0 . 1 1 3 9 0 . 0 0 4 4 -2 - 0 . 2 6 5 0 - 0 . 0 4 8 7 0 . 0 1 7 5 0 - 0 . 0 1 7 1 0 . 0 1 2 5 0 . 0 1 6 7 n II 2 0 . 1 9 2 7 0 . 0 6 4 3 0 . 0 1 6 1 n I I 4 0 . 4 2 2 9 0 . 1 2 2 3 0 . 0 1 6 9 6 0 . 7 0 7 8 0 . 2 0 6 6 0 . 0 3 0 6 I I n 8 0 . 9 2 3 3 0 . 2 4 2 3 0 . 0 1 3 7 I I n 1 0 1 . 1 0 5 3 0 . 2 7 8 4 s 0 . 0 0 6 3 I I I I 12 1 . 2 6 4 3 0 . 2 9 8 8 4 - 0 . 0 1 0 3 n I I 14 1 . 4 2 5 6 0 . 3 2 2 5 1 - 0 . 0 2 3 3 n n 16 1 . 4 3 7 3 0 . 3 3 2 5 - 0 . 0 1 3 0 S lot ted Walls AOAR = 5 0 % -4 - 0 . 3 2 9 9 - 0 . 0 6 7 0 0 . 0 1 5 3 I I H -2 - 0 . 1 5 9 4 - 0 . 0 2 3 3 0 . 0 1 6 6 I I II 0 0 . 0 2 5 2 0 . 0 2 3 3 0 . 0 1 7 0 ' I I I I 2 0 . 1 9 0 3 0 . 0 6 4 1 0 . 0 1 6 6 I n I I 4 0 . 3 7 9 4 0 . 1 0 9 3 0 . 0 1 4 6 - 104 -IV.2 (Continued) Wall Conf igurat ion 0 a L T . . . M 0 C M C / 4 S lot ted Walls AOAR = 5 0 % 6 0 . 5 7 8 1 0 . 1 6 7 5 0 . 0 2 3 8 II n 8 0.83.63 0 . 2 2 4 5 0 . 0 1 7 4 II n 1 0 0 . 9 6 5 8 0 . 2 5 1 2 0 . 0 1 3 4 12 1 . 1 1 9 3 0 . 2 8 1 2 0 . 0 0 7 5 II H 14 1 . 2 2 3 5 0 . 2 9 2 8 - 0 . 0 0 4 0 II I I 16 1 .3391 0 . 3 1 7 2 - 0 . 0 0 4 6 S lot ted Walls AOAR = 5 5 % -4 - 0 . 3 5 1 4 - 0 . 0 7 7 2 0 . 0 1 0 5 n II -2 - 0 . 2 0 3 9 - 0 . 0 3 0 5 0 . 0 2 0 4 0 0 . 0 1 0 7 0 . 0 2 9 1 0 . 0 2 6 4 H n 2 0 . 1 6 2 9 0 . 0 5 6 6 0 . 0 1 5 9 n n 4 0 . 3 5 6 2 0 . 1 0 3 7 0 . 0 1 4 9 6 0 . 5 9 9 0 0 . 1 7 2 8 0 . 0 2 3 9 n H 8 0 . 7 9 9 8 0 . 2 1 6 3 0 . 0 1 8 3 n II 1 0 0 . 9 4 0 4 0 . 2 4 5 9 0 . 0 1 4 4 n II 12 1 . 0 7 2 0 0 . 2 6 9 8 0 . 0 0 7 7 II n 14 1 . 1 9 2 4 0 . 2 8 4 6 - 0 . 0 0 4 7 n n 16 1 . 2 8 8 6 0 . 3 0 4 3 - 0 . 0 0 5 4 - 105 -IV.2 (Continued) Wall Conf igurat ion 0 a c 4 \ C M c / 4 S lo t ted Walls AOAR = 59% . -4 -0.3186 -0 .0612-0,0183 II H -2 -0.1616 -0.0222 0.0182 H n 0 -0.0041 0.0170 0.0181 n H 2 0.1757 0.0594 0.0155 I I n 4 0.3506 0.0996 0.0122 n H 6 0.5286 0.1539 0.0224 n I I 8 0.7719 0.2088 0.0177 n n 10 0.8982 0.2334 0.0122 n II 12 1.0287 0.2601 0.0086 n n 14 1.1657 0.2791 -0.0037 n n 16 1.2753 0.2980 -0.0085 - 106 -TABLE V THE LIFT-CURVE SLOPES OBTAINED .... EXPERIMENTALLY FOR NACA-0015 AIRFOIL Wall Conf igurat ion m ( 0 ° ^ a < 1 0 ° ) For C/H = .34 C/H = .67 So l i d Walls 0.10750 0.11560 S lot ted Walls A0AR=50% 0.09478 0.09771 S lot ted Walls A0AR=55% 0.09264 0.09717 S lo t ted Walls A0AR=59% 0.09126 0.09254 

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