Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Premnath, S. M. Jason
Master of Applied Science - MASc
University of British Columbia
A solution to the current problem of wind tunnel wall interference could be achieved by ventilating the test section and thereby controlling the flow pattern around the model. The motivation for the slotted wall test section arises from the fact that a fully open jet and a fully closed jet introduce corrections of opposite sign to the wind tunnel data. This current work is limited to axisymmetric wind tunnels and solid blockage corrections. Such a tolerant axisymmetric wind tunnel (TAWT), which does not need any correction to the measured flow quantities and which is also independent of the test model shape and size would find wide application in the field of industrial aerodynamics. A numerical model based on a surface singularity potential flow method showed that at 70% OAR (open area ratio) for models of size up to 25% blockage and for three different shapes the tunnel design would yield results (coefficient of pressure) with less than 2% error while such models might need up to 75% data correction if tested in a solid wall wind tunnel. Experiments indicated good agreement with the numerical investigation and at 60% OAR the TAWT gave results close to free air results for all the models tested (up to 25% blockage).
A T O L E R A N T A X I S Y M M E T R I C W I N D T U N N E L By S. M. Jason Premnath B. E . University of Madras, India A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE 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 1988 © S. M. Jason Premnath , 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstrac t A solution to the current problem of wind tunnel wall interference could be achieved by ventilating the test section and thereby controlling the flow pattern around the model. The motivation for the slotted wall test section arises from the fact that a fully open jet and a fully closed jet introduce corrections of opposite sign to the wind tunnel data. This current work is limited to axisymmetric wind tunnels and solid blockage corrections. Such a tolerant axisymmetric wind tunnel(TAWT), which does not need any cor-rection to the measured flow quantities and which is also independent of the test model shape and size would find wide application in the field of industrial aerodynamics. A numerical model based on a surface singularity potential flow method showed that at 70% OAR(open area ratio) for models of size up to 25% blockage and for three different shapes the tunnel design would yield results (coefficient of pressure) with less than 2% error while such models might need up to 75% data correction if tested in a solid wall wind tunnel. Experiments indicated good agreement with the numerical investigation and at 60% OAR the TAWT gave results close to free air results for all the models tested(up to 25% blockage). ii Table of Contents Abstract ii List of Figures vi List of Symbols xii Acknowledgement xv 1 Introduction 1 1.1 Basics of wind tunnel wall interference 1 1.2 Literature review 2 1.3 Purpose of the present research 4 2 Design of the Axisymmetric Test Section 6 2.1 Introduction 6 2.2 Test section dimensions 6 3 Numerical Investigation 9 3.1 Introduction 9 3.2 Overview of axisymmetric methods 9 3.3 Surface singularity method 11 3.3.1 Evaluation of potential and velocity at any point 11 3.3.2 Approximation of the body by ring elements 12 iii 3.3.3 Singular subelement 14 3.3.4 Solving for velocity and pressure distribution 16 3.4 Results and discussion 18 3.4.1 Accuracy of the method 18 3.4.2 Solid wall confined flow 18 3.4.3 Slotted wall 19 3.4.4 Streamlined axisymmetric body 21 4 Experimental Investigation 23 4.1 Introduction 23 4.2 Experimental setup 23 4.3 Models tested 24 4.4 Test procedure 25 4.5 Results and discussion 26 5 Concluding Remarks 29 Bibliography 31 Appendices A Design of Axisymmetric Contracting Cone 33 B Regression Method of Comparing Pressure Coefficient Data Sets 35 B.l Blockage correction factor(CF) 35 B. 2 RMS error 36 C Models Tested 37 C. l Selection of the streamlined body 37 iv Figures 3g v List of Figures Figure 1.1 Conventional method of applying correction factor to the data from solid wall wind tunnel. Comparison of free air, corrected and uncorrected pressure distribution on an ellipsoid of slenderness ratio 2.0 and blockage ratio 25% 39 Figure 2.1 Variation of Cp at the model midpoint with wall length Modehellipsoid, Blockage-25%, 20%, 10% 40 Figure 2.2 Variation of Cp at the model midpoint with wall length Model-Rankine Ovoid, Blockage-25%, 20%, 10% 41 Figure 2.3 Assembly drawing of solid wall axisymmetric wind tunnel 42 Figure 2.4 Assembly drawing of slotted wall axisymmetric wind tunnel 43 Figure 3.1 A ring source of constant strength lying in the plane x — b 44 Figure 3.2 Integration over a line segment 45 Figure 3.3 A singular subelement. 46 Figure 3.4 Approximation of body by discrete panels 47 Figure 3.5 Numerical model of axisymmetric wind tunnel OAR=90% 48 vi Figure 3.6 Comparison of results between surface singularity method and analyt-ical method. Models:ellipsoid, Rankine ovoid; Slenderness ratio:2.0 49 Figure 3.7 Cp distribution for solid wall confined flow. Modehellipsoid 25% blockage 50 Figure 3.8 Cp distribution for solid wall confined flow. Modehellipsoid 20% blockage 51 Figure 3.9 Cp distribution for solid wall confined flow. Modehellipsoid 10% blockage 52 Figure 3.10 Error induced in the model pressure due to the presence of solid walls Modehellipsoid 53 Figure 3.11 Blockage correction factor based on minimum error method Models:ellipsoid, Rankine ovoid 54 Figure 3.12 Correction factor applied to ellipsoid 25% 55 Figure 3.13 Correction factor applied to Rankine ovoid 25% 56 Figure 3.14 Cp ,OAR=30%; Modehellipsoid 57 Figure 3.15 Cp ,OAR=40%; Modehellipsoid 58 Figure 3.16 Cp ,OAR=50%; Modehellipsoid 59 Figure 3.17 Cp ,OAR=60%; Modehellipsoid 60 Figure 3.18 Cp ,OAR=70%; Modehellipsoid. 61 vii Figure 3.19 Cp ,OAR=80%; Modehellipsoid 62 Figure 3.20 Cp ,OAR=90%; Modehellipsoid 63 Figure 3.21 Cp ,OAR=100%; Modehellipsoid 64 Figure 3.22 Cp at model midpoint with different OAR Modehellipsoid; Blockage ratio:25%,20%,10% 65 Figure 3.23 Cp at model midpoint with different OAR ModehRankine ovoid; Blockage ratio:25%,20%,10% 66 Figure 3.24 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 25%; Slat width .2ft 67 Figure 3.25 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 20%; Slat width .2ft 68 Figure 3.26 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 10%; Slat width .2ft 69 Figure 3.27 Numerical configuration at 70%OAR 70 Figure 3.28 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 25%; Slat width .1ft 71 Figure 3.29 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 20%; Slat width .1ft 72 Figure 3.30 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 10%; Slat width .1ft 73 viii Figure 3.31 Comparison of Cp at 70% OAR, free air and solid wall Modehstreamlined body; Blockage ratio 25%; Slat width .2ft 74 Figure 3.32 Comparison of Cp corrected using correction factor, free air and solid wall Modehstreamlined body; Blockage ratio 25%; Slat width .2ft 75 Figure 4.1 Velocity distribution in the test section 76 Figure 4.2 Intensity of turbulence in the test section 77 Figure 4.3 Comparison of Cp between solid wall numerical and experimental anal-ysis. Modehellipsoid; Blockage ratio 25% 78 Figure 4.4 Comparison of Cp between solid wall numerical and experimental anal-ysis. Modehellipsoid; Blockage ratio 20% 79 Figure 4.5 Comparison of Cp between solid wall numerical and experimental anal-ysis. Modehellipsoid; Blockage ratio 10% 80 Figure 4.6 Comparison of Cp between OAR 30%, 40% numerical and experimental analysis. Modehellipsoid; Blockage ratio 25% 81 Figure 4.7 Comparison of Cp between OAR 50%, 60% numerical and experimental analysis. Modehellipsoid; Blockage ratio 25% 82 Figure 4.8 Comparison of Cp between OAR 70%, 80% numerical and experimental analysis. Modehellipsoid; Blockage ratio 25% 83 Figure 4.9 Comparison of Cp between OAR 90%, 100% numerical and experimen-tal analysis. Modehellipsoid; Blockage ratio 25% 84 ix Figure 4.10 Comparison of Cp between free air (numerical) and OARs 30 - 100% (experimental). Modehellipsoid; Blockage ratio 25% 85 Figure 4.11 Comparison of Cp between free air (numerical) and OARs 30 - 100% (experimental). Modehellipsoid; Blockage ratio 20% 86 Figure 4.12 Comparison of Cp between free air (numerical) and OARs 30 - 100% (experimental). Modehellipsoid; Blockage ratio 10% 87 Figure 4.13 Comparison of Cp between free air (numerical), solid wall (numerical) and OAR 60% (experimental). Modehellipsoid; Blockage ratio 25% 88 Figure 4.14 Comparison of Cp between free air (numerical), solid wall (numerical) and OAR 60% (experimental). Modehellipsoid; Blockage ratio 20% 89 Figure 4.15 Comparison of Cp between free air (numerical), solid wall (numerical) and OAR 60% (experimental). Modehellipsoid; Blockage ratio 10%. . . 90 Figure 4.16 Comparison of Cp between free air (numerical), solid wall (numerical) and OAR 60% (experimental). Modehstreamlined axisymmetric body; Block-age ratio 25% 91 Figure A . l Contour of streamlines with constant stream function 92 Figure A.2 Contour of the curve selected in the design of a contraction cone. 93 Figure C. l Location of pressure taps on the ellipsoid model. Blockage=25%. Slen-derness ratio = 2.0 94 Figure C.2 Location of pressure taps on the ellipsoid model. Blockage=20%. Slen-derness ratio = 2.0 95 x Figure C.3 Location of pressure taps on the ellipsoid model. Blockage=10%. Slen-derness ratio = 2.0 96 Figure C.4 Location of pressure taps on the streamlined axisymmetric model. Blockage=25% 97 xi L i s t o f S y m b o l s O A R Open Area Ratio R M S Root Mean Square Value of Error SD Standard Deviation C F Blockage Correction Factor SL Slotted Test Section Length A C Airfoil Chord Ai,A2 Constants Cp Coefficient of Pressure CPc Corrected Pressure Coefficient r Distance R,r0 Radius / Model Length Pn Legendre Polynomial ft Normal Velocity Vector Voo Free Stream Velocity xii V, v,u Velocity L\U Disturbance Velocity V Normalized Velocity v Local Velocity Vc Corrected Velocity VR Reference Velocity N Number of Panels or Elements Nt Number of Pressure Taps X , Y , Z Cartesian Coordinates x,y,z Cartesian Coordinates Aij Influence Matr ix ip Angle tp Stream Function a Radius of a Ring Source 6, B Distance <t> Potential T Vortex Strength K, E Elliptic Integrals of 1st and 2nd Kind k,Ki,K2 Argument of Elliptic Integrals y 0 Y Distance of the Element Midpoint s Distance Along the Element d Half Width of the Singular Sub-element 6 Inclination of the Element As Width of an Element Vt Tangential Velocity A cknow ledgement I owe an enormous debt of gratitude to Dr. G. V. Parkinson for providing me the opportunity to pursue my graduate studies in Canada. His continual guidance and expert supervision enabled me to finish this project successfully. I would also like to express my thanks for his financial support. I wish to record my sincere appreciation to Mr. Leonard Drakes, technician in the Mechanical Engineering workshop, for skillfully making the models, ring airfoil slats, and other wind tunnel equipment. xv Chapter 1 Introduction 1.1 Basics of wind tunnel wall interference Wind tunnels have been an indispensable means for testing the models before building the prototype. Most wind tunnel tests are aimed at predicting some of the forces, pressures, moments and other parameters that will be experienced by the full scale structure in an unconfined flow. However in reality wind tunnel boundaries impose constraints on the flow around the model so that the measured values of these parameters differ from the required free air values and need correction to obtain the free air values. There has been an increasing need for testing the largest available model not only to obtain the highest Reynolds number but also to have a more accurate shape of the model. To decrease the wall effects the model must be made small which is undesirable due to the loss of advantages mentioned above or a larger test section has to be provided which demands more space and higher tunnel construction and operating costs. This persistent problem of wall interference is attacked from several angles. One method is applying empirical correction factors to the wind tunnel data. Often empiri-cal methods are model shape dependent and valid only in a particular facility. Another method is the application of computers to determine the free air flow parameters which 1 Chapter 1. Introduction 2 completely eliminates the need for wind tunnels. Though computers are gradually taking a larger share in the simulation of flow past bodies they can only be used to supplement but never reliably to substitute for a wind tunnel experiment. The third method involves the modification of the existing wind tunnel walls in such a way that the correction is negligible even for bigger models in smaller wind tunnels. Interference-free boundaries can be created by l)controlling the flow field through suction and blowing of air out of or into the test section using porous walls 2)using a flexible wall which is self stream lining or 3)using a partly open and partly closed test section which would provide free air conditions. This work deals with the third method of using a slotted-wall test section. 1.2 Literature review In this section literature relating to both 2D and 3D wind tunnel wall interference corrections is described. Low speed wind tunnel testing[l] explains wind tunnel wall interference and cor-rection techniques, both in 2D and 3D, using empirical and theoretical approaches. However, many of them are valid only if the blockage ratio is very small. Figl. l shows the application of the correction formula suggested in eqn6.30 of ref[l] to the pressure distribution on an ellipsoid of slenderness ratio 2.0 creating a blockage of 25%. The RMS error[Appendix A] is calculated as 28% between the corrected and the free air data. The solid wall and free air pressure distributions were obtained from the present numerical method. A brief review of wall interference and correction methods is given in ref[2|. The adaptive wall concept was first suggested by Sears[3] and implemented at Calspan Cor-poration, Buffalo, and was later on investigated by R.J. Vidal et al.[4]. The intention Chapter 1. Introduction 3 of the adaptive wall method is to provide interference-free testing with the aid of online computers that can continuously monitor the tunnel flow and control the cross flow at the ventilated wall. Adaptive wall methods can be used for both 2D and 3D testing. Apart from the ventilated adaptive wall, solid compliant walls, which do not require any ventilation, with contour control to generate a flow pattern close to the free air pattern were studied by Goodyer[5]. Though both kinds of adaptive wall techniques are claimed to give good results the feasibility of adopting these at places other than where they were developed is still not clear due to complicated operating system, initial and maintenance costs and the flexibility of the use of the tunnel once installed. The determination of interference corrections from boundary measurements is ex-plained in AGARD-281 publication[6]. The concept of a 2D tolerant wind tunnel was first introduced by Williams and Parkinson[7]. After a series of numerical and experimental tests on airfoils at UBC, Williams[8] (1975) showed that satisfactory uncorrected pressure and lift coefficients could be obtained with a transversely slotted(suction side only, using airfoils as the solid elements) test section at 60 - 70% OAR. Malek's[8](l983) work later on showed that for a test section with both sides slotted 59% average OAR would give good results. The double side slotted wall technique was extended, using Parkinson's wake source model, to bluff bodies with large wakes like the circular cylinder, the flat plate etc., by Hameury[9](l987) who predicted that the optimum OAR for bluff bodies is 45% using a numerical method and 60% using experiments. Slotted wall wind tunnels have already made advances in the automotive industry in testing cars and trucks. R.G.J. Flay et al. [10] have shown that slotted wall test sections (longitudinal) can be designed to produce acceptable airflow simulation for models with blockage levels up to 21%. The success of the 2D tolerant wind tunnels is the obvious reason for the moti-vation to venture into 3D wind tunnel applications. An axisymmetric wind tunnel, Chapter 1. Introduction 4 though treated as 2D in numerical investigation, is more 3D in nature for experimental purposes. 1.3 Purpose of the present research This thesis deals specifically with an axisymmetric slotted wall wind tunnel. An ax-isymmetric wind tunnel has essentially a round test section and the model is mounted along the axis of the tunnel. The model has to be a solid of revolution. The flow parameters in the tunnel can vary along the radius and along the axis but not with the angle. An axisymmetric wind tunnel can be used for testing rockets, bombs, airplane fuselages, airship models etc. The present study is aimed at evaluating the solid wall effects on the model and determining the best way to compensate for these effects. An attempt was made at defining a general correction factor which could be applied to any solid wall wind tunnel observation. Ideally, in using a wind tunnel facility one would like to concentrate on the details of the model and on interpreting the data which is the primary purpose rather than being concerned with the blockage effects of the body and correcting the data. Especially in industrial testing, where the importance of the wall interference is played down on the assumption that it is negligible or due to lack of proper application procedures, a correction free wind tunnel would be very much welcome. The fact that the adaptive wall technology is complex and costly to implement is the main incentive to look for a less expensive passive wall configuration. A tolerant wind tunnel is the ideal solution for industries and University laboratories. Since the corrections for open jet and closed jet test sections are of opposite sign, the existence of an optimum open area ratio(the fraction of test section open to the outside) Chapter 1. Introduction 5 which would provide an interference free situation, is investigated. The problem was analysed both numerically and experimentally. If such an OAR exists is it independent of the model shape and size? Chapter 2 Design of the Axisymmetric Test Section 2.1 Introduction The axisymmetric test section is designed to be used in the experimental and numerical analysis. There is no axisymmetric wind tunnel here at UBC and therefore such a test section had to be fabricated and attached to one end of an existing circular blower of 1.5ft diameter for experimental purposes. 2.2 Test section dimensions The important dimensions to be decided are the diameter of the tunnel, total length of the tunnel and the length of the tunnel(test section) to be slotted. The diameter of the tunnel is limited by the blower to 1ft to have a high Reynolds number of 400,000 experimentally (Reynolds number does not have any significance in the numerical anal-ysis). Further reduction in diameter to increase the Reynolds number would render the model mounting and experimental procedure difficult. The wind tunnel wall length was selected such a way that the portion of wall beyond the design length would not have any appreciable effect on the model. The coefficient 6 Chapter 2. Design of the Axisymmetric Test Section 7 of pressure on the model for various wall lengths could be found out using the surface singularity method. Figs 2.1,2 show the variation of Cp at the midpoint of various sizes of ellipsoid and Rankine ovoid. It could be noted that the wall portion on either side beyond the middle 6.0ft section has only negligible effect on the model. Therefore the total length of the tunnel was taken as 6ft. One set of experiments was done in the solid wall setup. Also from the figures 2.1,2 it could be noticed that the central 4.0 ft wall portion has the most influence on the flow around the model. This portion is replaced by a slotted wall for the next set of experiments. The slotted wall is made of ring airfoils and open spaces. The airfoils have NACA 0015 section and .2ft chord. The selection of airfoil type and chord is arbitrary. A long chord may have an adverse effect on the test model while too short a chord is not easily machined. In the experimental set up a contraction cone is required between the blower duct(1.5ft dia.) and the axisymmetric wind tunnel(1.0ft dia.). The contraction cone should deliver a uniform flow at the test section inlet. The design is based on the method by Smith and Wang[ll]. Their method makes use of the exact analogy between the magnetic field that is created by two coaxial and parallel coils carrying electrical current and the velocity that is created by two ring vortices. Experimentally a high uniformity is noticed when the coils are well arranged. A corresponding uniformity of flow field over the same core area is obtained by simply translating the electromagnetic solution in to an aerodynamic solution. Then one of the stream surfaces can be chosen as the flow boundary. The variation at the outlet of these cones can be made less than .1%. Fig A . l shows a family of such curves. Out of these one curve suitable to fit between 1.5ft dia. and 1.0ft dia. is selected to form the contour of the contraction cone. For more details see the appendixfA]. The contour selected is shown in fig A.2 Chapter 2. Design of the Axisymmetric Test Section 8 The experimental setup of solid wall and slotted wall wind tunnels are shown in fig 2.3 and fig 2.4 respectively. The plenum, the contraction cone and the model sting were not included in the numerical scheme. C h a p t e r 3 N u m e r i c a l I n v e s t i g a t i o n 3.1 Introduction Few methods are available to solve the axisymmetric flows. In axisymmetric flow the velocity of the fluid does not vary with angle. In this chapter other methods are compared with the surface singularity method which is used in this work. The accuracy of the surface singularity method and the results of the numerical modelling of the axisymmetric wind tunnel are also discussed. 3.2 Overview of axisymmetric methods In axisymmetric potential flow, the direct problem can be solved analytically by the technique of separation of variables. However, except for the general ellipsoid and its specializations the solutions are tedious and often unavailable. Exact analytic solution for axisymmetric flow is available from a few indirect methods as well. First suggested by Rankine in 1871, a set of known singularities like point sources, line sources, doublets and vortices are placed in a uniform onset flow. This procedure does not solve the direct problem of potential flow, because it does not begin with a prescribed boundary surface but instead accepts whatever boundary results from the singularity distribution. The approximate solution for the flow about an arbitrary body of revolution was first 9 Chapter 3. Numerical Investigation 10 introduced by Theodore von Karman[12]. He determined the solution by superposing a uniform stream on a system of line sources distributed along the axis of the body. The strengths of the sources were determined so that the zero streamline passed through the given coordinates of the body, the number of coordinates being equal to the number of sources. For a body to be well represented by such a scheme, the body must be slender and there must be no discontinuities in the body surface slope. But Oberkampf[13] investigated this method and he found out that although the zero streamline passed through all the specified points the body generated had many 'holes' in its surface. Therefore the zero streamline for a system of line sources can not follow the contour of the body. Oberkampf concludes that Karman's method is not reliable and does not always produce good results. Axial singularity methods are not suitable for internal flows. Other numerical methods that can be applied to axisymmetric flows are the finite difference method and finite element method. In these network methods there is no constraint that the flow has to be potential. However this advantage is offset by the fact that the whole flow field has to be modelled and therefore the number of grid points to be solved becomes too many and is not computationally efficient. Further, network methods are not suitable for complex geometries. In the light of the difficulties present in the other methods the surface singularity method is chosen as the most suitable one. Though holes are present on the body surface they do not affect the solution in any way. There is no restriction that the body has to be slender. Also internal flows can be modelled without much difficulty. Since only the surface of the body rather than the whole flow field is modelled the number of elements is kept to the minimum. Chapter 3. Numerical Investigation 11 3.3 Surface singularity method 3.3.1 Evaluation of potential and velocity at any point The surface singularity method, otherwise called the panel method, was developed by J. L. Hess and A. M. O. Smith[14] of Douglas Aircraft Company. In axisymmetric flow the body is replaced by ring sources of unknown strengths. Ring sources give rise to a velocity and a potential at a point in space that may be expressed in terms of complete elliptic integrals. These expressions cannot be integrated analytically over a line segment and must be integrated numerically. The case i = j , that is the velocity and the potential induced by the element on its own control point does require special handling to numerically calculate the principal value of the relevant integral. The Fredholm integral from basic potential flow theory is given by In this equation denotes the differentiation in the direction of the outward normal to the surface S at a point p and n(p) is the unit outward normal vector at p. o{p) is the source density distribution. The approach adopted by Hess and Smith[l4] consists of approximating the left hand side of eqn 3.1 by a set of algebraic equations. It is accomplished by replacing the surface S about which the flow is to be computed by a large number of surface source elements of constant strength whose characteristic dimensions are small compared to those of the body. Now the task is reduced to determining a finite number of cr, one for each of the elements. (3.1) Chapter 3. Numerical Investigation 12 The left hand side of the eqn 3.1 can be written as dSj (3.2) N = Y,AHai » = l ,2 , . . . iV (3.3) Here iV is the total number of surface source elements replacing the body. _4.,-y, a linear operator, is defined as Therefore eqn 3.1 becomes N 52 Aijaj = * = 1,2,..JV (3.5) is the normal velocity induced at the control point V by a source of unit strength located at another point ,j\ The matrix Aij depends only on the geometry of the body surface and is independent of the onset flow. 3.3.2 Approximation of the body by ring elements The surface of the axisymmetric body can be approximated by N ring elements of constant source strength. The X - axis is taken as the axis of symmetry. For a ring(fig 3.1) source of unit strength and radius 'a' lying in the plane x = b, the potential (p induced by this ring source at a point (x,y,0) is (3.4) (3.6) (3.7) Chapter 3. Numerical Investigation 13 Since V. = - f f and Vv = - f j = 2a r Jo = 2a /" Jo (x — b)dip \{x - b)2 + (y 2 + a 2 - 2aycosip)\* (y — acosip)dxp [(x - b)2 + (y 2 + a 2 - 2ayeosxf>)]' (3.8) (3.9) By a series of substitutions and algebraic manipulations[15], these formulae may be expressed in terms of complete elliptic integrals. The results are AaK[k) y/{y + a) 2 + ( i - by 4a{x-b)E{k) [(y - a) 2 + (x - b)2} yj{y + a)2 + (x - b)2) 2a y^/{y + a)2 + {x-by K(1c\ + y 2 - a 2 + (x - bf (3.10) where i f (A;) and E(k) are the complete elliptic integrals of the first and second kind respectively and the argument k is given by k2 = 4ay (y + a)2 + (x- b)2 (3.11) A typical line segment is shown in fig 3.2. The calculations are performed in the reference coordinate system. It is convenient to translate the origin of the coordinates to Chapter 3. Numerical Investigation 14 the point on the axis of symmetry that lies directly below the midpoint of the segment. The slope angle of the segment j with respect to the positive x-axis is denoted by Oj. The distance from the midpoint is denoted by 5 and the length of the segment is As. The y coordinate of the midpoint of the segment is designated by y0. For a point along the line segment parameters of the ring sources are a = Vo + ssinOj (3-12) b = scosOj (3.13) The potential and the velocity induced by the element at the point (x,y, 0) are obtained by substituting a and b in eqn 3.10 and integrating the results with respect to 5 from —As/2 to +As/2. Since the integration is performed in terms of s, no special handling is required for vertical elements having 6, = 7r/2. The required integration is performed numerically by Simpson's rule. The line segment is divided into subelements. The number of subelements is taken as 16As/r m ,„ rounded to the nearest even integer, where r m ,„ is the distance from the point (x, y,0) to the nearer of the end points of the line segment. Therefore the farther the point in question lies from the element the fewer the number of subelements used in the calculation. 3.3.3 S ingu la r subelement The eqn 3.10 is not valid for calculating the effect of the element at its own midpoint i.e. when x = b and y = a. In the translated coordinate system this occurs at s = 0. This is a singularity of the form l/s which is not integrable and the evaluation is carried out differently as explained in [14]. This complication does not arise in the two dimensional form of surface singularity method. A certain distance d is selected. The portion of the line segment within the distance Chapter 3. Numerical Investigation 15 d of the midpoint is designated as the singular subelement. d is always less than As/2. The eqn 3.10 is expanded in terms of sjy0 and integrated from 5 = —d to s — +d to obtain the effect of the singular subelement on the midpoint. The contributions of the subelement to the potential and velocity at the midpoint are +3(1 + 2sm20J)/n(-^-)] + ....}, v'x = -sin20j{-){i + -l-(±)2{i3 + 6sin26j y0 144 y0 +«»(^")] + -•} °y0 Vi = - 2 ( i { [ ^ + ln(A)]-I ( ^[ 3 c o ^. y0 °y 0 48 y0 -2sin40j + 3/n(-^-)] + ....} (3.14) 8y0 d is taken as d = .08yo if .08yo < As/2 and d - As/2 if.08yo > As/2. The rule for determining d was developed by trial and error [14] to minimize truncation error in eqn 3.14 and numerical integration error in eqn 3.10. The contributions of the ends of the element, that is, the portion farther from the midpoint than d are evaluated as if they were separate elements using the eqn 3.10. In the above calculation the midpoint is assumed from the outset to lie on the element. The limiting process of approaching the surface gives rise to a velocity of magnitude 27r(for unit value of the local source strength) with direction normal to the local surface. The x and y components from this velocity are Vx" = -2-KsinQi Chapter 3. Numerical Investigation 16 V v" = 2-ncosOj (3.15) Thus in general the velocity components induced by an element at its own midpoint consist of the sum of the three contributions: the numerical integration of eqn 3.10 over the ends of the element, series eqn 3.14 for the effect of the singular subelement and the components eqn 3.15 arising from the limiting process of approaching the surface. 3.3.4 Solving for velocity and pressure distribution In sections 3.3.2 and 3.3.3 velocity induced at the control point (midpoint) of an element % due to a panel j of unit strength has been found out. Since the coordinate system based on the element is not used, the above formulae give the X and Y velocities directly. Now the velocity induced at i due to all the elements j = 1,2, ...JV each having a source strength o~j has to be found out. This would yield a matrix of size N X JV where JV is the total number of panels approximating the body. Appropriate values of the velocities must be substituted in eqn 3.5 to get a set of JV linear equations with JV unknowns. Since A^ represents the normal velocity at i induced by another element j. If Vx.. and VVi. represent the x and y velocities induced at the control point * by another element j then Aij = -Vx..8in0i + VViicos6i (3.16) -ni.Voo = |Voo|sm0; (3.17) Solving the JV X JV simultaneous equations by Gauss elimination technique would give the values of unknown source strengths Oj. |Voo| is taken as 1.0 for the calculations. Chapter 3. Numerical Investigation 17 The tangential velocity at the control point of the ith element is given by N VU = Y\v*ucos9i + Vy..8in9I](TJ + \Vn\cos9i (3.18) Now the pressure distribution on the model surface can be calculated as below CPi = 1 - V* (3.19) Due to axial symmetry, it is sufficient to model only one half of the wind tunnel geome-try. Therefore only the upper half is modelled. The contour of the test model(ellipsoid, Rankine ovoid etc.) is replaced by 81 elements which would give results with reason-able accuracy. The NACA 0015 profile is represented by 20 elements. 10 elements/foot length is used in the case of solid walls. More elements are used (in test models and air-foil slats) near the ends than in the middle applying the cosine rule to define them more precisely. Fig 3.5 gives a graphical representation (The circles are replaced by polygons in this figure to obtain a 3D view of the wind tunnel. But in the numerical calculations the polygonal approximation is done in the X — Y plane only and no approximation is introduced in the planes perpendicular to X axis.) of the numerical wind tunnel model when the OAR is 90%. The whole structure is immersed in a uniform flow of unit velocity . A FORTRAN program was written to solve the problem. The program was run on VAX/750 and also on the Array Processor. In VAX, 100 elements took 14 CPU minutes and 400 elements 2 hours. It took 15 minutes for 400 elements on the Array Processor. Chapter 3. Numerical Investigation 18 3.4 Results and discussion 3.4.1 Accuracy of the method Before applying it to complicated geometries, the accuracy of the surface singularity method and fidelity of the computer program written based on the panel method were evaluated. As mentioned in section 3.2 an analytical solution is available for simple models like the ellipsoid and the Rankine ovoid. Panel methods are sensitive to the number of panels used. The geometry must not only be well represented but the panels must be as small as possible to produce accurate results. The program was tried on test models without slats and solid wall, i.e., under free air or unconfined flow conditions The fig 3.6 shows the comparison between the pressure distribution on the ellipsoid and the Rankine ovoid of slenderness ratio 2. Numerical and analytical results agree very well. The table below shows the effect of number of elements on the coefficient of pressure at the midpoint of an ellipsoid of slenderness ratio 2. The error is expressed as the percentage of the peak model pressure which is a characteristic of a model to enable better understanding of its magnitude. No of Elements Num. Method Analt. Method Max Error 11 -.43 7.2% 21 -.451 -.464 2.8% 41 -.457 1.5% 81 -.462 .5% 3.4.2 Solid wall confined flow In this configuration, the model is placed inside a circular tube of 1.0ft dia and 6.0ft length and treated as an interior flow problem. Figs 3.7-3.9 show the Cp distribution Chapter 3. Numerical Investigation 19 on the ellipsoid models of blockage ratios (ratio of tunnel section area to the max. model section area) 25%, 20%, and 10%. Fig3.10 shows the percentage of error(when compared to the free air) introduced due to the presence of the solid wall. In the case of 25% blockage ratio the error is 74% which is quite high. Even in the case of 5% blockage the error involved is as high as 9% which is not negligible. A possible correction method based on minimum error technique is explained in appendix B. A set of correction factors is evaluated comparing the reference data(a'e free air) with the present data(solid wall). The free stream velocity when divided by the correction factor yields the right result. The blockage correction factor curve(fig 3.11) is obtained from ellipsoid and Rankine ovoid data. The curve is close to a straight line which suggests that the blockage correction factor is proportional to the blockage ratio. The value of the correction factor is 1.0 when there is no wall present. The application of correction factor to the data obtained from a solid wall wind tunnel test may not necessarily ensure an error-free solution(figs 3.12,13) but would yield better results. 3.4.3 Slotted wall The solid wall section of 4.0ft length in the middle is replaced by ring slats. The num-ber of slats (Na) determines the open area ratio (OAR) OAR = [ SL-Na* AC SL •]100 Na = 1,2, ...20 (3.20) SL is the slotted wall length(4.0ft) and AC is the airfoil chord which is .2ft. Fig 3.5 shows the numerical model of the slotted wall assembly. The whole assembly is immersed in a uniform flow. For solid wall elements interior flow conditions are imposed Chapter 3. Numerical Investigation 20 while for other elements exterior flow boundary conditions are imposed. The figs 3.14-21 show the model pressure distribution on an ellipsoid of 25% blockage ratio for OAR varying from 30% to 100%. Up to 70% OAR the Cp curves are on one side of the free air curve. After 70% the curves overlap. The models of other blockage ratios also exhibited similar characteristics. This behavior is, to some extent, expected since for open jet and closed jet testing the corrections are of opposite sign. However the degree of open jet effect predicted by the surface singularity method is very low because the open jet boundary condition was not imposed in the present numerical scheme. The open jet flow pattern could be simulated by satisfying the constant pressure boundary condition on the streamline leaving the upstream wall. The figs 3.22,23 show the plot of the Cp at the midpoint of the ellipsoid models and Rankine ovoid models of different sizes. Though cross over from closed jet to open jet occurs slightly lower than 70% for 10% and 20% models while slightly over 70% for 25% models yet it is reasonable to infer that 70% is the optimum OAR which would yield results close to free air values. The error in Cp, the standard deviation(Appendix B), compared to free air result is 4.9% which is quite acceptable for a model of blockage ratio 25%. The optimum OAR i.e. 70% OAR is obtained from only two kinds of models namely ellipsoid and Rankine ovoid and there is no guarantee that it is applicable to any axisymmetric body unless tested. Although 100% OAR produced good results(fig 3.21), it is not acceptable as the best OAR since the apt boundary condition was not present at the higher OAR in the numerical scheme to create the real flow situation. The flow at the higher OAR can be modelled better by imposing the constant pressure boundary condition on the streamline leaving the upstream solid wall. A new streamline is obtained every time the constant pressure boundary condition is applied on the old streamline. This is a tedious iterative process and was not attempted in the present research. Chapter 3. Numerical Investigation 21 The streamlines are not straight in the test section and therefore the airfoil slats are always at a small angle of attack with the onset flow. Though the net lift on a single slat is zero, there is a local lift developed at every section which necessitates satisfying the Kutta condition at the trailing edge of the airfoil slats. The error involved in not satisfying the Kutta condition is expected to be small and therefore Kutta condition was not included in the present numerical scheme. The local effect of slats on the models can be noticed in figs 3.14-21. For the mid range OARs 50%, 60% and 70%, the individual effect of the slats on the model is quite evident which is undesirable. The bumps in the Cp curve are caused by the two slats which are closer to the model. Fig 3.27 shows the numerical model when the OAR is 70%. For lower OAR 30% and 40%, the slats are evenly dispersed in the vicinity of the model and the model experiences a collective effect of the slats. At higher OARs, 80%, 90% and 100%, the slats are too far away to have any remarkable effect on the model. This problem can be solved by keeping the OAR constant but increasing the number of slats which means using slats of smaller chord. Figs 3.28-30 show the improvement of the quality of result as the consequence of changing the slat size. The table below compares the results for slat width .2ft and .1ft. The disturbances induced on the model by the slats in the near vicinity of the model are quite prominent and deform the Cp curve. The selection of slat size is a matter of just trial and error. The RMS error in the Cp when compared to the free air result is tabulated below. Chapter 3. Numerical Investigation 22 Error in Cp Blockage Slat width .2ft Slat width .1ft 25% 4.9% 1.7% 20% 3.9% 1.4% 10% 2.0% .8% 3.4.4 Streamlined axisymmetric body To verify the above prediction, that at 70% OAR results close to unconfined flow can be obtained, a streamlined axisymmetric body which is not symmetrical about the mid chord section, was selected. This model was generated according to ref[l6] to give minimum drag. First the free air and confined flow results were obtained using the present numerical method. Then the model flow was calculated at 70% OAR to see if results close to free air values could be obtained. Fig 3.31 shows the pressure distribution for free air, solid wall and 70% OAR slotted wall. The error is calculated as 1.7% even with .2ft slats. The above discussion indicates that any model of any size and shape (excluding bluff bodies which have not been tested) can be tested at 70% OAR to obtain results close to free air. The validity of the correction factor developed in section 3.4.3 was checked applying it on the streamlined body. The corrected and the free air values agree near the nose and midchord region. But comparing fig3.31 and fig 3.32 it can be said that testing a model at 70% OAR would give better results than applying any correction factor. Chapter 4 Experimental Investigation 4.1 Introduction The main goals of the experimental analysis are to study the real flow in the tolerant axisymmetric wind tunnel and to explore the possibility of using such a wind tunnel for practical purposes. The experimental setup is explained in the next section. The models tested and the test procedure are outlined in the subsequent sections. Finally the results from the experiments are discussed and compared with numerical predictions. 4.2 Experimental setup The axisymmetric test section, 1.0ft dia. and 6.0ft total length, is attached to the downstream end of the duct of a blower in the Mechanical Engineering Laboratory. The test section consists of two solid wall sections of 1.0ft dia. one at the downstream end and the other at the upstream end of the tunnel. The wall portion 4.0ft in between these solid walls can be either a solid wall of 1.0ft dia. or a slotted wall of 1.0ft dia. Ring airfoil slats and open spaces constitute the slotted wall. In the case of the slotted wall, a plenum of 2.0ft dia. is provided to ensure mass conservation inside the tunnel. The plenum is co-axial with the slotted wall. The plenum has a transparent removable window for easy access to the model and the ring 23 Chapter 4. Experimental Investigation 24 slats and also for flow visualization. The solid walls and the plenum are rolled from steel sheets .065in thick. The airfoil slats of NACA 0015 section and .2ft chord were machined out of alu-minium rings 11.5in inner dia., .5in thickness and 2.75in width. Aluminium is easy to work with on lathes and is light for handling inside the plenum. Two struts of lin width, .25in thick and 6in long are attached to each ring slat so that the ring slats can be supported by the plenum. The struts slide in two grooves one at the top and the other at the bottom of the plenum. The model is inserted into the test section using a 5.0ft long hollow sting of .75in outer dia. and .33in thickness. The sting in turn is fixed to a rigid stand which can move up and down to adjust the height of the model. The slats can be added or removed from the test section through the window without disturbing the model. The slotted wall section can hold a maximum of 20 slats. During initial tests, vibration of high amplitude and low frequency of the model and the sting was noticed. Therefore four additional .125in dia. supports were provided 1.0ft away from the model to arrest the model vibration. These supports also help in aligning the model along the tunnel axis. The velocity of air in the tunnel was 23m/s approximately and could not be altered. The four static pressure taps are mounted on the circumference of the solid wall portion 2.2ft away upstream of the model. The velocity of air is measured by inserting a pitot tube through one of the holes in the solid wall. The pressure tap tubes from the model are conveyed through the sting and are hooked up to a multitube manometer. 4.3 M o d e l s t e s t e d The total number of models tested was 4. There were 3 ellipsoids ( slenderness ratio 2) of sizes as per 25%, 20% and 10% blockage ratios The fourth model was a streamlined Chapter 4. Experimental Investigation 25 axisymmetric body of 25% blockage ratio. The models were machined out of birch wood. Wooden models are light and aid in minimizing the sagging of the sting. The number of pressure taps on the model is restricted by the size of the sting. A sting with larger dia. may affect the model pressure characteristics, but a sting with smaller dia. will limit the number of pressure tubes. The shape of the streamlined body is given in appendix C 4.4 Test procedure In the experiments the effect of Reynolds number was ignored. The most difficult part of the experiments was the axisymmetric mounting of the body. Due to the weight of the model the sting has a tendency to sag. First the model and the sting are made to lie in the exact vertical plane passing through the axis of the tunnel. This is done by traversing a disc with a vertical slot, from the center to the periphery, along the sting. Then the disc is removed and the sting is made horizontal by adjusting the vertical support screws. The level is verified using a spirit level. Before commencing the real experiments, a few preliminary tests were performed to estimate the quality of flow in the test section. The velocity of flow was measured at the exit of the contraction cone using a pitot tube in three radial directions and it was found that the core flow velocity variation was within 3% of the center line velocity[fig 4.1]. The jet was consistent within 1% between the symmetric planes. The turbulence was measured using a hot wire anemometer and the intensity of turbulence was found to be about 5%[fig 4.2] The first model test was carried out in the test section with solid wall. The pressure readings were noted from the multitube manometer. The manometer was allowed to settle down for 10 minutes since oscillations of the alcohol in the tubes were noticed Chapter 4. Experimental Investigation 26 initially when the fan was started. The experiment was repeated after one hour to see if consistent readings were obtained. Then the model was changed and the above mentioned process was repeated. The second set of experiments was done using the slotted wall section. The solid wall portion of 4.0ft in the middle was replaced by the slotted wall and the plenum. The open area was changed by adding or removing ring airfoil slats. If there is no slat then the OAR is 100 % and if the whole test section is filled with slats then the OAR is 0 %. The slats can be inserted or removed without disrupting the model. After the model was mounted the pressure measurements for OAR ranging from 30 to 100 % were recorded. Then the model was withdrawn from the test section, another model was inserted and the pressure measurement procedure was repeated. Since there was a variation of the velocity in the test section the mean velocity(used in the calculation of Cp) was taken as centerline (measured) velocity multiplied by 98%. The multitube manometer is marked to .lin and the measurements were accurate upto .05in which would cause an uncertainty of upto 4% in the Cp measurements. 4.5 Results and discussion The solid wall experimental results showed good agreement with the numerical results. Figs 4.3-4.5 compare the experimental and the numerical pressure distributions for ellipsoid models of blockage ratios 25%, 20% and 10%. At the downstream end of the models there was flow separation and therefore the pressure distribution is flat. The slotted wall experiments also produced results agreeing to some extent with the numerical predictions. For the middle range of OAR i.e. from 50-80%[figs 4.7-4.8] theory seems to predict the real flow well but at higher OAR 90% to 100%[fig 4.9] the agreement is poor and the theory appears to underestimate the open jet effects for the Chapter 4. Experimental Investigation 27 reason discussed in section 3.4.3. Figs 4.10-4.12 have the Cp plots for free air and open area ratios varying from 30-100% for the ellipsoids of blockage ratio 25%, 20% and 10%. From these curves it could be noticed that at 60% OAR for 25% and 20% models and at 70% OAR for the 10% model results close to free air results are obtained. At 60% OAR the 10% model still gives reasonable results. Therefore 60% is selected as the optimum OAR. The pressure distributions at the optimum OAR in experiments ,i.e, 60% for all three models are shown in Figs 4.13-15. A bump in the Cp curve is noticeable in the case of the bigger models. It is more prominent at lower OAR than at higher OAR. The swelling in the Cp curve is not found on both sides of the model in the experimental results contrary to the numerical predictions. There are two explanations possible to describe the cause of the bump. It could have been a consequence of the individual effects of the slats nearer to the model. But this fails to explain why a small bump is visible in the solid wall results where there are no slats. The second speculation is that it is caused by the presence of a small separation bubble immediately after the midsection of the ellipsoid models. The magnitude of this bump can be minimized by using smaller slats (as in the numerical case) or by testing the model at a higher Reynolds number. As in the numerical case, the streamlined axisymmetric body was tested to check that the optimum OAR indicated by the experiments on the ellipsoid was valid for other bodies. The model selected has a fineness ratio approximately 2.0. It has a peak pressure very close to the nose and then slowly falling pressure distribution to the tail. For such a body the flow separation occurs very close to the tail and therefore the drag is low. This model was tested at different OAR and it was found that at 60% OAR the data would not need a large correction[fig4.16]. This is a clear indication that the 60% OAR is quite valid for all bodies of revolution of any size and shape (again excluding Chapter 4. Experimental Investigation 28 bluff bodies). Chapter 5 Concluding Remarks A tolerant axisymmetric wind tunnel has been developed and tested using both nu-merical and experimental methods. 70% OAR in the case of numerical analysis and 60% OAR in experiments have been found to provide results close to free air results. The encouraging performance of the experimental setup would be a stimulus to further investigate a few more details that would enhance the results. The possibility of nullifying the bump in the pressure curve(experiment) can be attempted by using smaller slats. The use of the smaller slats achieved better results in the numerical investigation. The bump can be eliminated by uneven spacing of the slats also. Since the numerical modelling is effective in providing a good perception of the flow pattern the use of the numerical scheme is recommended before any experimental investigation. The effect of Reynolds number on the optimum OAR should also be investigated. Present investigation has assessed that axisymmetric bodies of any shape and size(up to 25%) can be tested with only a small error at the optimum OAR. It is quite possible that the three dimensional bodies which vary only slightly from being axisymmetric(e.g. bus, truck, front portion of an aircraft, submarines etc.) could be tested without ap-preciable error. The test section of any round wind tunnel can easily modified into a tolerant test section. 29 Chapter 5. Concluding Remarks 30 It is quite likely that the present numerical scheme can be extended to bluff body testing by placing an extra source at the tip or by adding an extra ring source at any section aft of the model. B ibliography [l] Rae jr.,W. H. and Pope, A., Low speed wind tunnel testing. John Wiley & Sons, Second edition, 1984. [2] Garner, H. C. et al. Subsonic wind tunnel wall corrections, AGARDograph 109, Oct 1966. [3] Sears, W. R., Self correcting wind tunnels, Aeronautical Journal, Feb/Mar 1974. [4] Vidal, R. J. et al., Experiments with a self correcting wind tunnel, AGARDograph Cp-174, Oct 1975 [5] Goodyer, M . J., A low speed self streamlining wind tunnel, AGARDograph Cp-174, Oct 1975 [6] Two dimensional wind tunnel wall interference, AGARDograph No 281,1983 [7] Williams, C. D. and Parkinson G.V., A low correction wall configuration for airfoil testing, AGARD Cp-174, 1975. [8] Williams, C. D. , A new slotted wall method for producing low boundary correc-tions in two dimensional airfoil testing. Ph. D Thesis, UBC, Oct 1975. [9] Hameury M. , Development of the tolerant wind tunnel for bluff body testing, Ph. D Thesis, UBC, Apr 1987. [10] Flay R. G. J. et al. Slotted wall test section for automotive aerodynamic tests at yaw, SAE international congress, Feb/Mar 1983. Paper 830302. 31 Bibliography 32 [11] Smith, R. H. and Wang, C , Contracting cones giving uniform throat speeds, Journal of aeronautical sciences, Oct 1944. [12] Collected works of Theodore von Karman, vols. I-IV, London, Butterworths, 1957. [13] Oberkampf, W. and Watson, L. E . , Incompressible potential flow solutions for arbitrary bodies of revolution, AIAA Journal, Mar 1974. [14] Hess, J. L and Smith A. M. O, Calculation of potential flow about arbitrary bodies, Progress in aeronautical sciences, vol 8, 1966. [15] Smith, A. M. O. and Pierce J., Exact solution of Neumann problem, Douglas Aircraft Company Report No. 26988, Apr 1958. [16] Young, A. D. and Owen, P. R., A simplified theory for streamline bodies of revolu-tion and its application to the development of high-speed low-drag shapes, R.A.E Report No Aero. 1837, July 1943 Appendix A Design of Axisymmetric Contracting Cone The difference in diameters between the test section and the blower, to which the test section was attached, introduced the necessity for a contraction cone. The important design consideration was that the contraction cone should deliver a uniform flow at the entrance of the test section. Smith and Wang [11] have shown that, using the analogy between electromagnetic coils and ring vortices, such a flow field could be obtained by using one of the stream surfaces as the boundary for the contracting cone. Further the nozzle inlet in the Green wind tunnel was designed [8] based on this principle. When there are two rings of equal core radii a, strength T, and with centers at (0, b) and (0,-6) the stream function ip at a point P(x,r) is, given by(for more details see [11]) i . . . ki [(1 - ^)K2 - Et] (A.l) where k * ~ (1 + Ry + (X- BY ( A ' 2 ) = (I + RY + (X + BY (A" 3) Here K\ and Ei, which are elliptic integrals of the first and second kind respectively, correspond to argument ki and if 2 and E% correspond to k%. 2B is the distance between the two ring vortices and a equals unity. In eqn A . l R is the radius of the contraction 33 Appendix A. Design of Axisymmetric Contracting Cone 34 cone and has to be found out for different values of X keeping ip constant. The value of aT/ft is taken as 8ir, and the value of B is taken as 0.46936. The value of ip is found out by substituting for X and R. First the value of ip at different grid points (X,R), for X varying from 0.0 to 2.0 units and R varying from 0.0 to 1.0 unit, was determined. The value of ip and (X,R) were input into a contour drawing computer subroutine to look at different contour surfaces and to choose a suitable curve for the contracting cone. See fig A . l . Although the curves with lower ip provide more uniform flow, the total length will be longer than the ones with higher ip values(see ref [11]) and might not be suitable for the present experimental setup. Any curve between ip = 3.0 and ip — 4.0(fig A.l) appeared to meet the needs regarding dimensions and uniformity of flow. The shape of the curve corresponding to ip = 3.2(fig A.2) was selected to be used for the contraction cone. A computer program was written to search for the value of R for the constant value of ip i.e. 3.2 and for X varying from 0 to 3ft. The contraction cone has a smaller diameter 1.0ft, larger diameter 1.5ft and length 1.76ft. Theoretically, the uniformity of throat speed in this case is within .2%. Appendix B Regression Method of Comparing Pressure Coefficient Data Sets The correlation between two pressure data sets, e.g. free air and solid wall, can be determined by performing a linear regression analysis on a velocity basis([9],[l4]). The main constraint in this method is that the two sets of data should have an equal number of entries. B . l Blockage correction factor(CF) Let CPi be the indicated pressure coefficient(e.(7.from a solid wall test) at tap number i and v,- is the local velocity . Vi = = (1 - C p , ) 5 * = l , 2 , . . J V t ( B . l ) »oo VQO is the upstream velocity and Vi is the normalized local velocity. Nt is the total number of pressure taps. Let VR. be the normalized reference velocity (e.g. free air-numerical). The measured velocity Vj- is corrected to VCi using the formula VE. = Al + A2Vi (B.2) Here A\ and A2 are two constants to be determined through least square analysis. The sum of squares of the differences between the reference data set and the indicated data set is minimized, i.e. £ (V* - (A, + A2Vi))2 (B.3) t=i 35 Appendix B. Regression Method of Comparing Pressure Coefficient Data Sets 36 is to be minimized. Ax is set to zero on the assumption that at the stagnation point the velocity is necessarily zero in both the reference and in the measured values. This leaves only a single variable A2 to be calculated. d N< dA2 fzi E (V* ~ ^Vif = 0 (B.4) Differentiating eqn( B.4) we get The factor A2 is the blockage correction factor(CF) and is applied on the velocity. It is the factor by which free stream velocity has to be divided in order to get corrected pressure distribution. CpCi = 1 - — ^ 1 (B.6) CPi Applying this correction factor would not give an error free solution but a minimum error solution. CF = 1.0 would result in no b lockage correction. B.2 R M S error RMS error is defined as the root of the mean of the error squares. This is a measure of the standard deviation between the expected(e.<7. free air or reference) values and the obtained(e.f7. solid wall) data. RMS ^ACvRi-Cptf Nt The RMS error is expressed as the percentage of the free air peak pressure of the model which is a characteristic of the model Appendix C Models Tested The models used in the numerical investigation were ellipsoids and Rankine ovoids of b/a ratio .5. The sizes depend on the blockage ratio. For the 25% blockage models b=3in and a=6in. The model surface was replaced by straight elements using cosine distribution. The coefficient of pressure was evaluated at the midpoint of each element. Ellipsoid models corresponding to three blockage ratios, 25%, 20% and 10%, were used to facilitate the experimental investigation. There are 21 pressure taps located along a meridional line. There are 4 pressure taps on the circumference in the plane perpendicular to the axis and passing through the model centre. The figs C1-C3 show the models and the location of the pressure taps. Apart from these models, there was one more model used, both in the numerical and experimental analysis, to see if the deductions from the previous models were applicable to other bodies. A streamlined axisymmetric body with one half not the mirror image of the other half was selected, not only that it has an arbitrary shape but it also has a small wake in experiments. C . l Selection of the streamlined body A simplified theory for developing a shape for a streamlined body of revolution as per some constraints, location and relative magnitude of the peak pressure, is explained by Young and Owen [16]. Bodies of revolution having velocity distributions conforming to specified types can be produced by the method suggested. The bodies will have blunt 37 Appendix C. Models Tested 38 nose, pointed tail and a roof-top type of velocity distribution. Supposing that the velocity distribution to which the body must approximate in type is given by u = U0[l + Au] then Au can be expressed as a function of x. If r0 represents the radial distance of a point on the circumference on the model and p^ represents the x-distance(axial) then Here P n is the Legendre polynomial of the first kind. Maximum value of n is taken as 7 since the terms beyond n=7 are small. Let Hn = / 1 M l . The value of Hn for p. varying between -1 and 1 is zero. Now eqn C. l in written as The velocity distribution is characterized by four quantities a, b, c and X. a/100,6/100 and c/100 are the values of Au at the leading edge, peak point and the trailing edge respectively. The peak suction occurs at X. The values of Hn for pi varying between -1.0 and 1.0 and n varying from 1 to 7 are given in ref [16]. The values of An for different values of a, b, c, and X are also provided. The model that was selected for the present investigation has a = 1.0, b = 1.0, c = .5 and X = — A(i.e. 30% from the leading edge). The shape of the model is shown in fig C.4. The pressure tap locations on the experimental model are shown in fig C.4. It has 21 pressure taps along a meridian and 4 on the vertical plane perpendicular to the axis. One pressure tap is common to both sets . n = l (C.l) r 0 2 7 ~ 7 ^ ~ = y ] %AnHn (C.2) 1 0 . 7 5 0.5 0 . 2 5 0 - 0 . 2 5 - 0 . 5 - 0 . 7 5 -1 . 1 1 1 1 — 0.5 - 0 . 4 - 0 . 3 - 0 . 2 -0.1 0.1 0.2 0 . 3 0 . 4 0.5 NON DIMENSIONAL MODEL LENGTH Figure 1.1 Conventional method of applying correction factor on the data from solid wall wind tunnel. Comparison of Free air, cor-rected and uncorrected pressure distribution on an ellipsoid of slenderness ratio 2.0 and blockage ratio 25% w to 40 Ellipsoid o Blockage = 25 % v Blockage - 20 % x Blockage = 10 % * - * X x X X X -V s? V ^ e o- -e o 0.0 2.0 I 1 1 1 1 12.0 14.0 16.0 18.0 20.0 4.0 6.0 8.0 10.0 Wall length in feet Figure 2.1 Variation of Cp at the model midpoint with wall length Model-ellipsoid, Blockage-25%, 20%, 10%. Rankine Ovoid Blockage = 25 % Blockage - Blockage = 10 % 41 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 Wall length in feet Figure 2.2 Variation of Cp at the model midpoint with wall length Model-^ Rankine Ovoid, Blockage-25%, 20%, 10%. Figure 2.4 Assembly drawing of slotted wall axisymmetric wind tunnel 44 Figure 3.1 A ring source of constant strength lying in the plane x — b. Figure 3.2 Integration over a line segment. Figure 3.3 A singular subelement. 47 Figure 3.4 Approximation of body by discrete panels. 49 Figure 3.6 Comparison of results between surface singularity method and analyt-ical method. Models:ellipsoid, Rankine ovoid; Slenderness ratio:2.0 1 0.75 _ i i 1 1 — i 1 1 r ~ -0 .5 -0.4 -0 .3 -0 .2 -0 .1 0 0.1 0.2 0.3 NON DIMENSIONALIZED MODEL LENGTH Figure 3.7 Cp distribution for Solid wall confined flow. Modehellipsoid 25% blockage 0.4 0.5 O 1 -1 - 0 . 5 - 0.4 - 0 . 3 - 0 . 2 - 0 . 1 0 0 .1 0 .2 0 . 3 0.4 0 .5 NON DIMENSIONALIZED MODEL LENGTH Figure 3.8 CP distribution for solid wall confined flow. Modehellipsoid 20% blockage 1 -0 .75 J -0 .5 -0.4 -0 .3 -0 .2 -0 .1 0 0.1 0.2 0.3 0.4 0.5 NON DIMENSIONALIZED MODEL LENGTH Figure 3.9 Cp distribution for solid wall confined flow. S Modehellipsoid 10% blockage c 0 R R E C T I 0 N F A C T 0 R 1.05 1.025 0.975 _ 0.95 0.925 _ 0.9 0.875 0.85 _ A — A EI l i p s o l d Rankine Ovoid 20 T" 25 I 35 40 0 5 10 15 30 BLOCKAGE RATIO X Figure 3.11 Blockage correction factor based on minimum error method Models:ellipsoid,Rankine ovoid 45 50 Ox 0 . 7 5 0 . 5 0 . 2 5 0 - 0 . 2 5 - 0 . 5 - 0 . 7 5 -1 J \\ // \\ // \ \ F R E E AIR // \ \ . SOLIDWALL- CORRECTED / / \ V , SOLIDWALL- UNCORRECTED / / / ' \ \ \ '/! \ \ \ •'/1 \ V \ \ X x ^ / \ 1 1 1 . 5 - 0 . 4 - 0 . 3 - 0 . 2 i i i i i i - 0 . 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . NON DIMENSIONAL MODEL LENGTH Figure 3.12 Correction factor applied to Ellipsoid 25% Cn Cn - 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 -0.1 0 0.1 0.2 0.3 0 . 4 0.5 Model length-Non Dimensional1zed Figure 3.13 Correction factor applied to Rankine ovoid 25% to <=> -l CO © -I © T H 1 1 1 1 1 1 1 1 1 1 -0.5 -0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Non-dim. Model Length Figure 3.14 Cp ,OAR=30%; Modehellipsoid Figure 3.15 Cp ,OAR=40%; Modehellipsoid Figure 3.16 Cp ,OAR=50%; Model:ellipsoid numerical Figure 3.17 Cp ,OAR=60%; Modehellipsoid Figure 3.18 Cp ,OAR=70%; Modehellipsoid Figure 3.19 Cp ,OAR=80%; Modehellipsoid 63 CO C 5 " © " © " © © • © • © • CO © ' I numerical 90 % OAR Free air -0.5 -0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 0.3 Non-dim. Model Length 0.4 0.5 Figure 3.20 Cp ,OAR=90%; Modehellipsoid o -i 00 I -0.5 -0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Non-dim. Model Length Figure 3.21 Cp ,OAR=100%; Modehellipsoid free air • — O — A — L E G E N D 2570 Blockage 20% Blockage 10% Blockage 1 1 1 1 1 1 1 1 1 1 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100 Open Area Ratio Figure 3.22 Cp at model midpoint with different OAR Modehellipsoid; Blockage ratio:25%,20%,10% Blockage Ratio o 25% v 20% H 10% 1 1 1 1 1 1 1 — ; — i 1 1 1 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100 Non-dim. Model Length Figure 3.23 Cp at model midpoint with different OAR ModehRankine ovoid; Blockage ratio:25%,20%,10% Figure 3.24 Comparison of Cp at 70% O A R , free air and solid wall Modehellipsoid; Blockage ratio 25%; Slat width .2ft o 7 - r - — i 1 1 1 1 1 1 i i i -0.5 -0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Non-dim. Model Length Figure 3.25 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 20%; Slat width .2ft 69 0.0 0.0 0.1 0.2 0.3 Non-dim. Model Length 0.4 0.5 Figure 3.26 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 10%; Slat width .2ft 1.75 1.5 1.25 J F t 0.75 0.5 0.25 J 0 -3 -2 -1 0 Ft Figure 3.27 Numerical configuration at 70%OAR —4 O 71 © i H 1 1 1 1 i i 1 i i i -0.5 -0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Non-dim. Model Length Figure 3.28 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 25%; Slat width .1ft © TH 1 1 1 1 1 1 1 1 1 1 -0.5 -0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Non-dim. Model Length Figure 3.29 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 20%; Slat width .1ft =0 6 -i o 7 H 1 1 1 1 1 1 i i i i -0.5 -0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Non-dim. Model Length Figure 3.30 Comparison of Cp at 70% OAR, free air and solid wall Modehellipsoid; Blockage ratio 10%; Slat width .1ft 1 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Model length-Non D i m e n s i o n a 1 i z e d Figure 3.31 Comparison of Cp at 70% OAR, free air and solid wall Modehstreamlined body; Blockage ratio 25%; Slat width .2ft —j c p 0.75 _ -0 .5 - 0 . 4 -0 .3 -0 .2 -0 .1 0 0.1 0.2 NON DIMENSIONAL MODEL LENGTH 0.3 0 . 4 0.5 Figure 3.32 Comparison of Cp corrected using correction factor ,free air and solid wall Modehstreamlined body; Blockage ratio 25%; Slat width .2ft o 0 deg v 90 deer v 270 deg Va=Centerline V e l o c i t y V t = V e l o c i t y at a p o i n t r=Radial D i s t a n c e R=Tunnel Radius o.o o.z 0.4 0.6 r/R 0.8 1.0 Figure 4.1 Velocity distribution in the test section o 0 deg x 90 deg • 270 deg v=RMS v e l o c i t y V t = V e l o c i t y at a p o i n t r=Radial D i s t a n c e R=Tunnel Radius r/R Figure 4.2 Intensity of turbulence in the test section 78 Figure 4.3 Comparison of Cp between solid wall numerical and experimental anal-ysis. Modehellipsoid; Blockage ratio 25%. 79 Figure 4.4 Comparison of Cp between Solid Wall numerical and experimental ana ysis. Modehellipsoid; Blockage ratio 20%. 80 I 6 I I I • Experiment o Numerical 1 1 1 1 1 1 1— -0.5 -0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 Non-dim. Model Length 0.3 0.4 0.5 Figure 4.5 Comparison of Cp between solid wall numerical and experimental anal-ysis. Modehellipsoid; Blockage ratio 10%. 1 1 1 1 1 1 1 1 1 1 -0.5 -0.4 -0.3 -0.1 0.0 0.0 O.I 0.1 0.3 04 OJ Non-dim. Model Length -OJ -0.4 -0.3 -o.i oo 0.0 O.I OJ Non-dim. Model Length Figure 4.6 Comparison of Cp between OAR 30%, 40% numerical and experimental analysis. Modehellipsoid; Blockage ratio 25%. Figure 4.7 Comparison of Cp between OAR 50%, 60% numerical and experimental analysis. Modehellipsoid; Blockage ratio 25%. -OJ -0.4 -0.3 -OJ 0.0 0.0 0.1 OJ 0.3 Non-dim. Model length -0.3 -OJ o.o o.o 0.1 OJ Non-dim. Model Length Figure 4.8 Comparison of Cp between OAR 70%, 80% numerical and experimental analysis. Modehellipsoid; Blockage ratio 25%. 00 CO -0.3 -0.1 0.0 0.0 0.1 0.Z 0.3 Non-dim. Model Length 0.4 03 -0.3 -O.t 0.0 0.0 0.1 0.1 0.3 Non-dim. Model length Figure 4.9 Comparison of Cp between OAR 90%, 100% numerical and experimen-tal analysis. Modehellipsoid; Blockage ratio 25%. oo 00 d -l l "1 l i I I I I I I I l -0.5 -0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Non^dim. Model Length Figure 4.10 Comparison of Cp between free air (numerical) and OARs 30 - 100% (experimental). Modehellipsoid; Blockage ratio 25%. Figure 4.11 Comparison of Cp between free air (numerical) and OARs 30 -(experimental). Modehellipsoid; Blockage ratio 20%. 100% Figure 4.12 Comparison of Cp between free air (numerical) and OARs 30 -(experimental). Modehellipsoid; Blockage ratio 10%. 100% 88 eo d to d ' d d ' d ' d • I d • I to d-CO d • © I I i I -0.5 -0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 0.3 Non-dim. Model Length 0.4 0.5 Figure 4.13 Comparison of Cp between free air (numerical), solid wall (numerical) and OAR 60% (experimental). Modehellipsoid; Blockage ratio 25%. 89 © i T H 1 1 r — — i 1 1 1 1 1 i -0.5-0.4 -0.3 -0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Non-dim. Model Length Figure 4.14 Comparison of Cp between free air (numerical), solid wall (numerical) and OAR 60% (experimental). Modehellipsoid; Blockage ratio 20%. Figure 4.15 Comparison of Cp between free air (numerical), solid wall (numerical) and OAR 60% (experimental). Modehellipsoid; Blockage ratio 10%. 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Non-dim. Model Length Figure 4.16 Comparison of Cp between free air (numerical), solid wall (numerical) and OAR 60% (experimental). Modehstreamlined axisymmetric body; Block-age ratio 25%. 4 _ UJ CD o c W • H < I X A L o c a t i o n of pressure taps X-Axis(Inches) -6 -4 - 2 . 0 . 2 4 Figure C. l Location of pressure taps on the ellipsoid model. Blockage=25%. Slen-derness ratio = 2.0 8 6 J in JC u c CO •H X < I A Location of pressure taps Figure C.2 Location of pressure taps on the ellipsoid model. Blockage=20%. Slen-derness ratio = 2.0 to Figure C.3 Location of pressure taps on the ellipsoid model. Blockage—10%. Slen-derness ratio = 2.0 CO 8 A LOCATION OF PRESSURE TAPS -4 -2 0 2 4 6 Figure C.4 Location of pressure taps on the streamlined axisymmetric Blockage=25%
Thesis/Dissertation
eng
Mechanical Engineering
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A tolerant axisymmetric wind tunnel
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