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Effect of wall confinement on the aerodynamics of bluff bodies El-Sherbiny, Saad el-Sayed 1972

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EFFECT OF WALL CONFINEMENT ON THE AERODYNAMICS OF BLUFF BODIES by SAAD EL-SAYED EL-SHERBINY B.Sc, Ain Shams University, Egypt, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF ' DOCTOR OF PHILOSOPHY in the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1972 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s under-stood that publication, in part or in whole, or the copying of this thesis for financial gain shall not be allowed without my written permission. SAAD EL-SHERBINY Department of Mechanical Engineering The University of British Columbia, Vancouver 8, Canada Date flnl 27 . / ? 7 Z ABSTRACT The effect of wall confinement on the aerodynamics of a set of stationary circular cylinders and f l a t plates, representing the blockage ratio range of 3 - 35.5%, i s investigated experimentally to obtain data on mean and un-steady pressure distributions, Strouhal number, and wake geometry. In general, the influence of the Reynolds number 4 4 in the range of 10 - 1 2 x 1 0 was found to be confined to the mean pressure distribution at the higher blockage (circular cylinder only) and the unsteady surface loading. The results showed the base pressure to decrease and consequently the drag coefficient to increase with bluff-ness; however, the pressure distribution in the potential flow region remained relatively unaffected by the confinement. The wake geometry does not change appreciably under constraint and thus leads to a similarity of the several flow parameters. Variation i n the vortex shedding frequency was found to be essentially proportional to the increase of separation velocity in accordance with Roshko's universal Strouhal number. The shape of the fluctuating surface pressure distribution curves also remains unaffected by the constraint. However, the pressure intensity increases with an increase in bluffness and shows considerable dependence on the Reynolds number and the three-dimensionality of the flow. i i i V alidity of the correction methods due to Glauert and Maskell for the mean drag coefficient i s checked in the lig h t of the experimental results. They were found to be inadequate particularly at higher blockage ratios. However, modification of these methods through inclusion of the higher order terms improved their applicability. Least square f i t of the results showed linear variation [e.g., C D = C D c + Const (C D S/C)] of the parameters. A potential flow model i s developed for two-dimensional symmetrical bluff bodies under wall confinement. It provides a procedure for predicting the mean surface loading on a bluff body over a range of blockage ratios. Experimental results with normal f l a t plates and circular cylinders for blockage ratios up to 3 5 . 5 % substantiate the validit y of the approach. TABLE OF CONTENTS Chapter Page 1. INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 Brief Review of the Literature 3 1.3 Purpose and Scope of the Investigation 9 2. EXPERIMENTAL PROCEDURE . 11 2.1 Models and Supporting System . H 2.1.1 Circular Cylinder Models 12 2.1.2 Flat Plate Models 15 2.2 Wind Tunnel 20 2.3 Instrumentation and Calibration 22 2.4 Test Procedure 28 2.4.1 Mean Static Pressure on the Model Surface 28 2.4.2 Vortex Shedding Frequency . . . . 30 2.4.3 Fluctuating Static Pressure on the Model Surface 30 2.4.4 Wake Survey 30 3. EXPERIMENTAL RESULTS AND DISCUSSION 34 3.1 Mean Pressure Distribution and Forces . . . . . 34 3.1.1 Circular Cylinder 34 3.1.2 Flat Plate 44 3.1.3 A Comparative Study 58 V Chapter Page 3.2 Strouhal Number 62 3.2.1 Circular Cylinder 63 3.2.2 Flat Plate 63 3.2.3 Universal Strouhal Number . . . . 66 3.3 Unsteady Surface Loading 71 3.3.1 Circular Cylinder 72 3.3.2 Flat Plate 80 3.3.3 Observations on the Influence of Three-Dimensionality of the Flow and Vortex Formation on the Unsteady Surface Loading . . . 9 2 3.4 Wake Geometry 95 3.4.1 Circular Cylinder 97 3.4.2 Flat Plate 108 4. ANALYTICAL PROCEDURE FOR WALL CONFINEMENT CORRECTION 121 4.1 Evaluation of the Reported Theories . . . . . . . . . . 121 4.1.1 Glauert's Correction Method . . . 122 4.1.2 Application of Maskell's Theory 125 4.2 Empirical Correction Formulae 130 4.3 Free-Streamline Model 137 4.3.1 Analytical Development 138 4.3.2 Application of the Theory . . . . 146 4.3.2.1 Normal Flat Plate . . . . 146 4.3.2.2 Circular Cylinder . . . . 149 v i Chapter Page 4.3.3 Discussion of Results 154 4.3.3.1 Normal Flat Plate . . . . 154 4.3.3.2 Circular Cylinder . . . . 159 4.3.4 Concluding Comments 162 5. CLOSING COMMENTS I 6 6 5.1 Concluding Remarks 166 5.2 Recommendation for Future Work 170 BIBLIOGRAPHY 175 LIST OF TABLES Table Page 2-1 Physical Properties of Circular Cylinder Models and Regions of Wake Survey 13 2- 2 Physical Properties of Flat Plate Models and Regions of Wake Survey 20 * 3- 1 S for Circular Cylinders under Constraint . . 70 3-2 S for Inclined Flat Plates 70 3-3 Maximum Fluctuating Pressure in the Wake of Circular Cylinders 100 LIST OF FIGURES Figure Page 2-1 A typical circular cylinder model with notation for the model and wake geometries . . . . . . 14 2-2 Details of the microphone model 16 2-3 Sectional geometry of the f l a t plate models showing distribution of the pressure taps 18 2-4 Constructional details of a typical f l a t plate model . . . . . . . . . . 19 2-5 Schematic diagram of the low speed wind tunnel used in the test program 21 2-6 Calibration plots for Barocel pressure transducer with damping bottle 24 2-7 A schematic diagram of the fluctuating pressure measuring set-up using Barocel pressure transducer . . . . . 25 2-8 The arrangement for measuring fluctuating pressure using condenser microphone 27 2-9 Disc probe dimensions with the mean pres-sure calibration plots showing i t s relative insensitivity to pitch and yaw . 29 2-10 Instrumentation lay-out for wake survey . . . . 32 3-1 Effect of the Reynolds number on the mean static pressure distribution around circular cylinders of different blockage: (a) S/C = 4.5%;. 36 (b) S/C = 20.5%; 36 (c) S/C = 35.5%. 37 3-2 Mean static pressure distribution on c i r -cular cylinders showing the effect of wall confinement: (a) R = 1.5 x 10 4; 39 (b) R = 5 x 10 4; 40 (c) R = 10 x 10 4 41 ix Figure Page 3-3 Effect of the Reynolds number on C at 180° for circular cylinders of ^ different blockage ratios: (a) present results; . 42 (b) comparison with the available data 43 3-4 Effect of wall confinement and the Reynolds number on: (a) the average base pressure; 45 (b) the drag coefficient 45 3-5 Independence of the pressure distribution on a f l a t plate from the Reynolds number: (a) S/C = 9%, a = 30°; 47 (b) S/C = 35.5%, a = 90° 47 3-6 Variation of the mean pressure distribu-tion with plate orientation for the blockage ratios of J (a) S/C = 9%; 48 (b) S/C = 20.5%; . . . 48 (c) S/C = 35.5%. . . . 49 3-7 Surface loading over a f l a t plate as affected by wall constraint: (a) a = 30°; 51 (b) a = 60°; 51 (c) 0t = 90^e e> © © © a © • • © © © • © • • • • • 52 3-8 Base pressure coefficient for f l a t plates as a function of: (a) plate orientation; 53 (b) blockage ratio . 53 3-9 Plots showing independence of the f l a t plate force coefficients from the Reynolds number: (a) S/C = 9%; 55 (b) S/C = 35.5%. . . . 56 X Figure Page 3-10 Variation of the force coefficients with plate orientation and blockage ratio: (a) the drag coefficient as a function of a; . . 57 (b) the drag coefficient as a function of(S/C) N; 57 (c) the l i f t coefficient as a function of a. . . . . . . * 59 3-11 Illustrations of bluff body extension into the wake affecting base pressure distribution 61 3-12 The Strouhal number for circular cylinders of different blockage as affected by the Reynolds number 64 3-13 Typical plots for inclined f l a t plates showing the Strouhal number's independence of the Reynolds number 65 3-14 Effect of wall constraint and plate attitude on the Strouhal number: (a) based on plate chord; 67 (b) based on projected width. . . . . . . . . . 67 3-15 Dependence of the unsteady pressure distribution on the Reynolds number for circular cylinders of blockage ratios: (a) S/C = 3%; . . . . . 73 (b) S/C = 9%; . . . . . . . . . . 73 (c) S/C = 20.5%; 74 (d) S/C = 35.5% . . . 74 3-16 Effect of wall confinement on the un-steady pressure distribution over circular cylinders: (a) R = 4 x 10 4; 76 (b) R = 10 x 10 4 76 x i Figure Page 3-17 Variation of the maximum fluctuating pressure coefficient with the Reynolds number and blockage ratio: (a) present results; 78 (b) Gerrard's results; 78 (c) comparison between the present results and Gerrard's data 79 3-18 Fluctuating l i f t coefficient for circular cylinders as affected by the Reynolds number and wall confinement: (a) present results; 81 (b) comparison with the reported data 81 3-19 Unsteady pressure and phase angle d i s t r i -butions over the surface of f l a t plates as functions of the blockage ratio and plate orientation: (a) S/C = 9%; 84 (b) S/C = 20.5%; 85 (c) S/C = 35.5% 86 3-20 Schematic representation of stagnation streamlines and separated shear layers near an inclined f l a t plate 87 3-21 Variation of the maximum-fluctuating pressure with the Reynolds number for several plate orientations and blockage ratios 89 3-22 Dependence of the maximum fluctuating pressure on the wall constraint and the Reynolds number for several plate orientations: (a) a = 90°, 75°, 60"; 90 (b) a = 45°, 30°- 91 3-23 Lateral variation of the fluctuating pressure amplitude in the wake of circular cylinders of blockage ratios: (a) S/C = 3, 4.5, 6 and 9%; 98 (b) S/C = 14.8, 20.5, 26.5 and 35.5% 99 x i i Figure Page 3-24 Decay of the peak fluctuating pressure in the wake of the confined circular cylinders: (a) near wake; 101 (b) intermediate wake 101 3-25 Growth of the wake in the downstream direction and i t s independence of the cylinder blockage 104 3-26 Longitudinal variation of phase angle in the wake of circular cylinders under constraint 105 3-27 Streamwise variation of: (a) longitudinal vortex spacing; 107 (b) wake geometry ratio for circular cylinders 107 3-28 Fluctuating pressure traverses in the wake of inclined f l a t plates of blockage ratios: (a) S/C = 4.5%; 109 (b) S/C = 9%; HO (c) S/C = 20.5%; H I (d) S/C = 35.5% 112 3-29 Downstream decay of the unsteady pressure in the wake of inclined f l a t plates under constraint 114 3-30 Streamwise position of the vortex rows in the wake of inclined f l a t plates of different blockage ratios 116 3-31 Lateral spacing of vortices in the wake as affected by plate orientation and blockage ratio 117 3-32 Phase angle distribution in the wake of inclined f l a t plates 119 x i i i Figure Page 3- 33 Downstream variation of: (a) longitudinal vortex spacing; 120 (b) wake geometry ratio ; for inclined f l a t plates 120 4- 1 Application of Glauert's correction method to circular cylinders: (a) direct application of Glauert's formula; . 124 (b) use of experimentally determined n; . . . . 124 (c) modified expression for the drag .124 4-2 Correction of the mean drag for normal f l a t plates using Glauert's method: (a) n = 1, as determined by Glauert; <> 126 (b) n = 0.94, as evaluated from the present data 126 4-3 Maskell's correction for circular c y l i n -ders : (a) direct application of the formula; 129 (b) use of higher order terms 129 4-4 Corrected mean drag coefficient of i n -clined f l a t plates using the modified Maskell's formula 131 4-5 Empirical correction formulae for circular cylinder aerodynamics: (a) mean drag coefficient; . (b) unsteady l i f t coefficient; (c) the Strouhal number. . . . . . . . . . 4-6 Representation of a two-dimensional symmetrical body i n : (a) the physical plane; . 140 (b) the intermediate transform plane 140 4-7 Mapping of the body and the walls in the C - plane, and the positions of singular-i t i e s 142 . 133 . 134 . 135 xiv Figure Page 4-8 (a) Flat plate geometry; 147 (b) Mapping of the f l a t plate into a circular arc 147 4-9 (a) Circular cylinder in confined flow; . . . . 151 (b) Transformation of the cylinder into a pseudo-circular arc 151 4-10 Pressure distribution over a f l a t plate: (a) comparison of the present theory with the reported analyses and experimental data for S/C = 7.15%; 156 (b) comparison between results for d i f f e r -ent blockage ratios as given by the present theory and the experiment 157 4-11 Variation with blockage ratio of C p and C D for a f l a t plate . b 158 4-12 Variation of and C D with blockage ratio for a circular cylinder 161 4-13 Comparison of the theoretical and ex-perimental pressure distributions over circular cylinders: (a) S/C = 6 and 14.8%; 163 (b) S/C = 20.5, 26.5 and 35.5% 164 AKNOWLEDGEMENT The author wishes to express his sincere thanks and appreciation to Dr. V.J. Modi for the guidance given throughout the research programme and assistance during the preparation of the thesis. His help and encouragement have been invaluable. Thanks are also due to the Department of Mechanical Engineering for use of their f a c i l i t i e s and to Mr. E. Abell for his valuable help during the planning and construction of the wind tunnel models. The investigation reported in this thesis was supported (in part) by the National Research Council of Canada, Grant Number A-2181. LIST OF SYMBOLS cross-sectional area of the wake in Maskell's theory cross-sectional area of a wind tunnel test-section, H£ in the theoretical analysis sectional mean drag coefficient based on circular cylinder diameter (d) or f l a t plate chord (h) average amplitude of sectional fluctuating l i f t coefficient at the fundamental frequency mean static pressure coefficient, (p-p n)/ [d/2)pv;i u base pressure coefficient, average mean pressure coefficient over the portion of the body extend-ing into the wake average amplitude of the fluctuating pressure coefficient, p '/ [ (l/2)p V Q \ , at the fundamental frequency sectional pressure drag ratio of the average amplitude of the fluctuating pressure at a given position in the wake and the absolute maximum wake pressure, p'/p* „ c w rw,max test-section width base pressure parameter, (1-C^ )l/2 the N^h increment where correlation function reaches a locally algebraic maximum value, implies two signals with time delay of N(T/100) in phase strength of a wake source Reynolds number, Vd/v or Vh/v reference area of a model, dZ for a circular cylinder and h£ for a f l a t plate X V I 1 S l / S 2 separation points S/C blockage ratio , bluffness (S/C) N blockage ratio based on a model projected area S, ,S, ,SM Strouhal numbers based on circular cylinder's diameter, f l a t plate's chord and model's pro-jected width, respectively * S Roshko's universal Strouhal number V Q measured free stream velocity far upstream of a model V, downstream f l u i d velocity at i n f i n i t y i n the theoretical model V streamwise vortex velocity, a f v J v Vg separation velocity, KVg a longitudinal spacing between vortices b lateral spacing between vortices c speed of sound d diameter of a circular cylinder d f c inside diameter of polyethylene tubing f natural frequency of Helmholtz resonator, (C/2TT) ( s/£ Hv)l/2 f v frequency of vortex shedding h chord length of a f l a t plate, or normal distance between separation points for the analytical model h 1 normal distance between parallel separated shear layers h.^  transverse distance between free-streamlines at i n f i n i t y I length of a model I, length of opening in Helmholtz resonator X V 1 1 X .£ length of polyethylene tubing m ratio of cross-sectional areas of wake and body in Maskell's theory, B/S P mean static pressure P Q static pressure far upstream of a model p' fluctuating static pressure about the mean at the fundamental frequency p.' average amplitude of the fluctuating static pressure, based on rms value, at the fundamental frequency, /~2" p ^ q relative wake source strength, 2 Q / ( H V Q ) q n dynamic head of the undisturbed stream, (1/2) P V§ r, radius of the mapped c i r c l e in £ - plane Figure 4-7 s cross-sectional area of opening in Helmholtz resonator t f l a t plate model's thickness v cavity volume in Helmholtz resonator x,y reference coordinate system with origin on the model axis at the central station, Figure 2-1 z physical complex plane, z = x + iy a angle of incidence 3 angular position of a separation point i n the c, - plane, Figure 4-7 y c i r c l e in the z, - plane, Figure 4-7 6 angular position of a wake source in the z, - plane, Figure 4-7 z, complex transform plane, z, = £ + in xix 8 angular position of a point on the surface of a circular cylinder as measured from the front stagnation point ri/X non-dimensional parameters in Glauert's correction formula \i angular position of a separation point in the w - plane. Figure 4-6 v kinematic viscosity p density of f l u i d a angular position of a point in the £ - plane/ Figure 4-7 T time delay in correlation function, i.e., total computational period <J> • average phase angle between fluctuating pressure signals ij> stream function to intermediate transform plane, w = u + i v Subscripts c corrected value of a parameter max maximum rms root mean square value w value of a parameter in the wake 1 INTRODUCTION 1.1 Preliminary Remarks Fluid dynamic coefficients used in engineering design of a wide variety of bodies, e.g., aircrafts and their com-ponents, bridges, t a l l buildings, submarines, etc., are normally obtained through model tests in wind or water tunnels. Ideally one would like to conduct the tests in an environment which simulates that of the actual system. A precise re-production of the geometrical details, provision of adequate structural strength and a room for instrumentation frequently lead to relatively large size models. On the other hand, high capital and operational costs tend to r e s t r i c t the size of a tunnel. One, therefore, encounters situations where models occupy a substantial portion of the test-section. This causes the test conditions to depart from free flow as the ri g i d boundaries r e s t r i c t the lateral displacement of the streamlines, leading to higher velocities in the neighborhood of the body. As indicated by Pankhurst and Holder^", the interfer-ence from wind tunnel walls during steady flow conditions may be divided into: (i) solid blockage, referring to a change of axial velocity past the model owing to the partial blocking of the flow in the presence of boundary constraint; 2 (ii) wake blockage, corresponding to a similar effect due to the reduction of speed within the wake of the model which implies increase in the vel-ocity outside the wake in accordance with continuity of mass flow; ( i i i ) l i f t effect due to the constraint imposed on the velocity f i e l d of the bound and t r a i l i n g vortices; (iv) wall boundary layer interference on the body spanning a closed tunnel introducing a departure from the two-dimensional conditions; (v) interference caused by the static pressure gradient arising from the growth of the boundary layer and interaction of the wake with r i g i d tunnel walls. This results in the model being tested under accelerated flow condition. In the study of a symmetrical body the l i f t effect ( i i i ) does not exist. For the 36 i n . x 27 i n . wind tunnel at the University of British Columbia, the wall boundary layer thickness in the test-section i s relatively small due to the f i l l e t e d corners which par t i a l l y compensate for boundary layer growth. Furthermore, the two-dimensionality 2 of the flow in the tunnel being well established by Slater , 3 Dikshit , etc., through correlation between the force measured by the balance and the integrated pressure results, (iv) can be taken to be negligible. So far as the acceler-ation effect due to boundary layer growth in the test section 3 is concerned, i t was observed to be small for the same reason of the f i l l e t e d corners. However, the interference due to the static pressure gradient in the wake as quoted by Pankhurst and Holder would amount to the square root of the solid blockage correction. Therefore, solid and wake blockages together with pressure gradient effect in the wake represent the major interference in the mean aerodynamic study. 1.2 Brief Review of the Literature 1 4 - 1 1 e t a l There i s a vast body of literature ' ' on wind tunnel interference for stationary streamline bodies in steady flow. In general, the approach has been theoretical and/or semi-empirical in character with wall effects accounted through equivalent increase in the free stream velocity. Correction factors are given as a series in ascending powers of the blockage ratio. In most cases, the analyses have been confined to the f i r s t or second order terms due to low blockage conditions. These corrections have been found satisfactory in most situations and are applied i n practice with some measure of confidence. Rogers^ and Garner"^ have presented excellent reviews of the development in this area. For stationary bluff cylinders, there i s some experi-mental information available on wall confinement effect, but the theoretical analysis i s relatively less complete. Pankhurst 4 1 7 and Holder and Durand suggested extrapolating streamline body analysis to bluff cylinders, but their applicability 12 13 i s rather limited. Bearman and Roshko applied the 4 analysis of Allen and Vmcenti to evaluate correction factors for circular cylinders in the c r i t i c a l and the * t r a n s c r i t i c a l 1 Reynolds number regimes, respectively, where the drag coeffic-ient drops to that of a slender body. More systematic attempts in this direction were initi a t e d by investigators at the Aeronautical Research Council i n England during the period 1928 to 1933. Fage^ conducted experimental investigation on bodies of different shapes, e.g., Joukowski a i r f o i l s , Rankine oval, ellipse of eccentricity 0.984 and circular cylinder. From the analysis of the Rankine oval and the corresponding experimental data, he obtained g empirical correction factors for the bodies. Lock modified Fage's correction procedure by considering the blockage effect as equivalent to an increase in the incident velocity. The increase corresponded to the induced velocity at the position of the body due to the presence of the walls, estimated using the image technique. This was later referred to as the solid blockage. However, Lock's correction failed to agree with Fage's experimental data, hence an empirical factor was i n -14 troduced to account for the discrepancy. Glauert attributed this deviation to the wake blockage and presented a modified expression to replace Lock's empirical factor. This correction formula has been i n use extensively and considered satisfactory 5 for small blockage. In a l l of the above investigations, determination of the mean drag was the main objective. A word of caution would be appropriate at this stage. It may be pointed out that Lock's and Glauert's correction procedures failed to agree with the experimental results for Rankine ovals and circular cylinders as obtained by Fage. No attempt was made to check the experimental results a l -though Glauert doubted them. A comparison with the present data tends to substantiate Glauert's assertion. Existence of the pressure gradient drag was f i r s t 15 realized by Pannell and Campbell , however, i t s detailed 16 17 analysis was pioneered by Glauert and Taylor , who assumed the flow condition to be similar to that in a convergent 4 channel. Allen and Vincenti , using a source to represent the wake, found a more reliable expression for pressure gradient in terms of the measured drag coefficient at the uncorrected tunnel speed. The correction to C D involved a shape factor which depended on the geometry and thickness to chord r a t i o . 18 Glauert derived an expression for the drag coefficient for a body forming a wake of alternate vortices in a channel of f i n i t e width. Closely following Kantian's"^ clas s i c a l analysis and using the image technique, he developed a re-lation involving an additional term to account for the increase in drag due to confinement. As in Karman's theory, i t i s 6 necessary to know, a p r i o r i , the values of the vortex speed and the wake geometry ratio b/a. He also presented a method for the determination of these parameters in a confined flow when the corresponding values in an i n f i n i t e stream are known. The results showed good agreement with the experimental data for f l a t plates. However, the correlation became poorer with increase in slenderness of the body due to the question-able v a l i d i t y of the assumptions inherent to the analysis. 20 Rosenhead and Schwabe performed experiments in an open water channel, and used photographic technique to study constraint effect on the wake geometry of a circular cylinder. By varying the distance between the channel walls they were able to obtain blockage ratios from 6% to 66% in the Reynolds number range of 50 to 800. The lateral and the longitudinal spacings of the vortices were found to decrease with increasing blockage at such a rate as to maintain the wake geometry ratio b/a essentially constant at 0.32 for blockages up to 33%. However, the ratio b/a attained a much higher value of 0.46 for the blockage of 66%. There appears to be a wide gap between the flurry of activity on the subject in the twenties, which was reviewed 21 comprehensively by Glauert , and revival of interest in the early si x t i e s . Except for the paper by Allen and Vincenti mentioned before, the period i s consipicuous for the lack of any significant contribution in the f i e l d . 7 With increasing attention towards wind e f f e c t s on buildings and structures, and use of large models to maintain the geometrical d e t a i l s and aerodynamic s i m i l a r i t y , the need for correction factors for steady and unsteady aerodynamics 22 becomes quite apparent. Maskell developed a semi-empirical theory for wake blockage using an approximate r e l a t i o n describing the momentum balance i n the flow outside the wake. He obtained a simple correction formula for the measured drag i n terms of the base pressure c o e f f i c i e n t . V a l i d i t y of the analysis was assessed by comparison with his own t e s t data for square plates of blockage up to 4.5%. The success 23-27 et a l of Maskell's theory has led many investigators ' to apply modified forms of his expression to a va r i e t y of flow conditions often unrelated to the sharp edge separation with uniform base pressure, for which the theory was developed. 23 Cowdrey substituted the base pressure by the average pressure over the separated flow region for bodies that ex-tended into the wake. The approach was found s a t i s f a c t o r y for a b o i l e r model with the blockage r a t i o of 18%. As against Maskell's correction formula, which represents a s t r a i g h t 24 l i n e [C D vs C D (S/C)], Gould proposed the use of higher order polynomial to account for large blockage r a t i o s . The c o e f f i c i e n t s of the polynomial were obtained by c u r v e - f i t t i n g of the experimental r e s u l t s . A second order polynomial was used for rectangular plates with blockage r a t i o s up to 30%. He also applied the theory to models mounted at d i f f e r e n t 8 locations i n the t e s t section, and to two models tested at the same time i n the wind tunnel but each outside the wake of the other, with s a t i s f a c t o r y r e s u l t s . Sprosson and 25 Brown extended Gould's approach to square plates with blockage r a t i o s up to 50%. A cubic expression was suggested to f i t the experimental data. The same method when applied 26 to s o l i d prisms of square section, by Boak and Buckle , was 27 found to break down for blockage r a t i o s above 17%. Vickery's analysis showed Maskell's correction to be applicable to the Strouhal number and the f l u c t u a t i n g l i f t force. When applied to a square cylinder, the r a t i o C j i/C D was found to vary by less than 2%, thus substantiating the v a l i d i t y of t h i s approach. A comment concerning the use of s l o t t e d walls with a view to eliminate wall confinement e f f e c t s would be appropriate here. As the e f f e c t s of closed and open t e s t sections on aerodynamic c o e f f i c i e n t s tend to be opposite i n character, there e x i s t s a p o s s i b i l i t y of choosing the correct proportion of porous walls to e s t a b l i s h a condition i d e n t i c a l to that of free flow. Pioneering work i n t h i s area was undertaken 28 29 by Theodorsen . Lim , who c a r r i e d out extensive tests on Clark - Y a i r f o i l i n a porous te s t section, has presented a concise review of the l i t e r a t u r e i n t h i s f i e l d . As against the problem discussed above, there are s i t u a t i o n s , usually i n the f i e l d of i n t e r n a l aerodynamics (flow through turbine, compressor, carburetor, e t c . ) , where 9 a s t r u c t u r a l element normally operates under a highly confined configuration. Here again, a r e l i a b l e method for evaluating the e f f e c t of wall constraint would help i n r e l a t i n g the desired aerodynamic parameters to the e x i s t i n g t h e o r e t i c a l or wind tunnel r e s u l t s for models i n i n f i n i t e streams or under low confinement conditions. 1.3 Purpose and Scope of the Investigation The aerodynamics and dynamics of b l u f f bodies have been a subject of considerable i n v e s t i g a t i o n since the early 30 31 32 twenties of t h i s century. Rosenhead , Wille , Morris , 33 34 3 5 * * 36 37 Ross , Morkovin , Roshko , Kuchemann Mair and Maull and others have reviewed various aspects of the a v a i l a b l e l i t e r a t u r e at some length. In t h i s department, the problem of a e r o e l a s t i c i n s t a b i l i t y of two dimensional b l u f f cylinders has been a c t i v e l y studied, both experimentally and theoretic-a l l y , since 1958. Reviews of the progress made have been 38 39 40 reported i n several survey papers ' ' . The systematic study has contributed to our understanding of wind e f f e c t s on buildings and structures. The i n v e s t i g a t i o n described here forms a part of t h i s continuing programme. I t presents information on the influence of wall confinement on the aero-dynamic c h a r a c t e r i s t i c s of two dimensional stationary c i r c u l a r cylinders and f l a t plates, and attempts at providing a r a t i o n a l basis for t h e i r c o r r e c t i o n . The shapes selected are 10 representative of two distinct groups of bluff bodies charac-terized by the sharp edge and the boundary layer type separations. Furthermore, they represent two extreme values of eccentricity whose systematic variation can generate a large family of bluff geometries. It i s hoped that this would help arrive at general conclusions concerning the wall confinement effect on bluff body aerodynamics. In particular, the thesis attempts to assess, experimentally, the effect of wall constraint on: (i) mean static pressure distribution; (i i ) the Strouhal number; ( i i i ) fluctuating pressure distribution; and (iv) wake geometry in the blockage ratio range of 3 - 35.5%. In a l l the tests, 4 4 the Reynolds number was limited to the range of 10 - 12 x 10 . On the analytical side, the va l i d i t y of the existing 15 22 correction methods ' i s examined. The modifications of Glauert's and Maskell's theories are presented by including higher order terms to explore their s u i t a b i l i t y in wall confinement corrections, particularly at higher blockage. Finally, a potential flow model i s presented for two-dimensional symmetrical bluff bodies under wall confinement, to predict their mean aerodynamical characteristics. Validity of the approach i s examined by comparison with.the experimental data. In general, relevant results from the literature are included to help establish trends. 2 EXPERIMENTAL PROCEDURE 2.1 Models and Supporting System In planning the experimental program in a desired range of the Reynolds number and blockage ratio, one has several alternatives to select from, each having certain limitations. The f i n a l decision i s dictated by a series of compromises. In the present case the choice primarily l i e s between the following two procedures: (i) The experiment may be conducted on a set of models, of varying sizes, between the fixed walls of the wind tunnel. This would obviously require several models with a corresponding increase in cost. Larger size of the models for a given span leads to reduction in aspect ratio which would affect correlation of the unsteady pressure. For a given tunnel, a large variation in the size of the models may present problem in measuring the aerodynamic parameters over the entire range of the Reynolds number and bluffness of interest. At times one would be satisfied with only a partial overlap of the Reynolds number range to help establish trends; however, even this may not always be possible. Also, the relative downstream distance available for the wake study would reduce markedly. (ii) The alternative would be to test one model between the moving walls of a wind tunnel, thus providing a range of blockage ratio, with associated com-plexity of design and construction of the contrac-tion and test sections. Furthermore, growth of the boundary layer on the side walls would now occupy a substantial portion of the test section, particular-l y at the higher blockage ratios, thus creating an accelerated flow condition. In the present investigation, the latter choice was ruled out, the overriding factor being the av a i l a b i l i t y of the fixed wall closed c i r c u i t tunnel with uniform low turbulence flow in the test-section. 2.1.1 Circular Cylinder Models Eight circular cylinder models, 27 inch long, were designed to span the wind tunnel test section thus approximating the two dimensional flow condition. Constructed from acrylic or aluminum tubes, each model was provided at the center section with a set of pressure taps, 0.025 inch in diameter. The taps were connected to polyethylene tubings of l^ = 3 f t . and d t = 0.066 i n . which were brought out from one end of the cylinder. Each of the models was also provided, over half of i t s span, with five pressure taps spaced at 4.5, 8.5, 10.5, 12 and 12.5 inches from the central section. These helped in 13 checking the two dimensional character of the flow. The end construction of the models was such as to permit t h e i r mount-ing on the wind tunnel balance or the e x i s t i n g a i r bearing system 4 2. Figure 2-1 shows a photograph of two t y p i c a l models, while the relevant physical parameters are summarized i n Table 2-1. Table 2-1 Physical Properties of the C i r c u l a r Cylinder Models and Regions of Wake Survey d, i n . Material Blockage Ratio, S/C Aspect Ratio l/d Limiting x/d for Wake Survey 1 Aluminum 3% 27 40 4 * Aluminum 4.5 18 30 2 Aluminum 6 13.5 20 3 A c r y l i c 9 9 14 5 A c r y l i c 14.8 5.4 8 7 A c r y l i c 20.5 3.85 5 9 A c r y l i c 26.5 3 4 12 A c r y l i c 35.5 2.25 3 Also microphone models The poor response c h a r a c t e r i s t i c of the f l u c t u a t i n g pressure measuring set-up, employing a Barocel, a t high Figure 2-1 A t y p i c a l c i r c u l a r cylinder model with notation for the model and wake geometries 15 frequency (Figure 2-6) necessitated the use of a microphone as an unsteady pressure sensor, particularly at the lower end of the bluffness spectrum. Hence three additional models of diameter 1, 1 1/2 and 2 inches were designed. tube with an acrylic center-piece carrying a cavity, 0.525 inch diameter and 3 inch long, to accommodate a 1/2 inch condenser microphone. Pressure on the surface of the cylinder was communicated to the cavity through a pin hole 0.025 inch i n diameter. A sealing ring fixed around the microphone cap effectively minimized the cavity volume, giving the Helmholtz resonance frequency of = 490 cps. The other end of the cavity served as an outlet for the microphone cable, which passed through the hollow aluminum tube, came out of the test section and was connected to the externally located amplifier, f i l t e r , and display instrumentation. The details of the microphone support cavity are shown schematically in Figure 2-2. ' 2.1.2 Flat Plate Models A typical microphone model consisted of an aluminum (2.1) Based on the experimental results of the circular cylinder, i t was considered adequate to construct fewer models for a similar investigation with the f l a t plate. Thus, 16 r D > i 0.525" D,= I.D. of aluminum tube D - s O . D . of aluminum tube Figure 2-2 Details of the microphone model 17 four sharp edge f l a t plates, 27 inch long, were designed to span the wind tunnel test section, giving the blockage ratio in the range 4.5 - 35.5%. To minimize the lateral deflection, the models were b u i l t of steel, aluminum or acrylic depending on the section strength and the force they were subjected to. The chamfered edge has a large angle of 20° to provide r i g i d i t y and allow installation of surface pressure taps. This limited the plate inclination to 30° due to flow reattachment at lower angles. An acrylic center section provided with pressure taps was cemented to the model with the tubes brought out from the lower end. The smallest model, of 1 1/2 inch chord, did not carry any pressure taps due to i t s thin section (5/32 inch maximum thickness). The models were mounted on the turntable of the wind tunnel balance thus permitting a change of orientation. Figure 2-3 presents cross-sections of the models showing distribution of the pressure taps, and Table 2-2 gives the relevant dimensions of the models. Constructional details are indicated in Figure 2-4. 20° Pressure taps i i — * -t 1 1 • « 1 j i — L i —I—i — v — i - J — i i — Figure 2-3 Sectional geometry of the f l a t plate models showing distribution of the pressure taps h K Pressure tap block Groove for the polyethyline tubes 002 Aluminum cover sheet Support bracket ZJ Figure 2-4 Constructional details of a typical f l a t plate model 20 Table 2-2 Physical Properties of the Flat Plate Models and Regions of Wake Survey h,in. Material Thickness, t, i n . Blockage Ratio, S/C Aspect Ratio, l/h Limiting x/h for Wake Survey 4 Steel 5/32 4 .5% 18 30 3 Aluminum 1/4 9 9 14 7 Aluminum 3/8 20.5 3.85 5 12 Acrylic 1/2 35.5 2.25 3 2.2 Wind Tunnel Both the circular cylinder and the f l a t plate models were tested in a low speed, low turbulence return type wind tunnel where the air speed can be varied from 4 - 150 ft./sec. with a turbulence level less than 0.1%. The pressure di f f e r e n t i a l across the contraction section of 7:1 ratio can be measured on a Betz micromanometer with an accuracy of 0.2 mm. of water. The test section velocity i s calibrated against the above pressure d i f f e r e n t i a l . The rectangular cross-section, 36 i n . wide x 27 i n . high, i s provided with 45° corner f i l l e t s which vary from 6 i n . x 6 i n . to 4.75 in x 4.75 i n . to partly compensate for the boundary layer growth. Figure 2-5 shows the tunnel outline. Figure 2-5 Schematic diagram of the low speed wind tunnel used in the test program 22 2.3 Instrumentation and C a l i b r a t i o n Most of the instrumentation used i n the experimental program, e.g. manometers, rms voltmeters, f i l t e r s , etc., constitute standard equipment i n any aerodynamics laboratory and hence needs no elaboration. However, a b r i e f account of the unsteady pressure and wake geometry measurement set-up would be appropriate. The measurement of the acoustic l e v e l pressure v a r i a -tions caused by shedding v o r t i c e s was accomplished using the Barocel Modular Pressure Transducing system developed by Datametrics Inc. of Waltham, Mass. Barocel i s a high-precision, stable capacitive voltage d i v i d e r , the v a r i a b l e element of which i s a t h i n prestressed s t e e l diaphragm d e f l e c t i n g proportional to the magnitude of the applied pressure. An a.c. c a r r i e r voltage at 10 kc i s applied to the stationary capacitor p l a t e s . The diaphragm attains a voltage l e v e l determined by i t s r e l a t i v e p o s i t i o n between the fi x e d capacitor p l a t e s . With the Barocel appropriately arranged i n a bridge c i r c u i t , the output voltage i s determined by the r a t i o of the capacitance of the diaphragm to each of the stationary electrodes. The c a r r i e r voltage i s thereby amplitude modulated i n accordance with the input pressure. The u n i t i s capable of measuring pressure down to 0.1 micrometer of mercury. A high degree of l i n e a r i t y and s t a b i l i t y of response over a wide range of temperature are ad d i t i o n a l a t t r a c t i v e features of 23 the system. The constructional details and working of the 41 system are described at length by Wiland The Barocel i s accurately calibrated for steady pressure. However, for the fluctuating pressure transmitted through relatively long, small diameter tubes, considerable attenuation occurred. Therefore, the output e l e c t r i c a l signal required calibration against the known input fluctuating pressure at the surface. This was achieved using the calibration system developed by Wiland. Figure 2-6 shows the response plots for the transducer i n terms of attenuation as a function of the signal frequency. It should be noted that the sensitivity of the Barocel diminishes rapidly with increase in frequency. Figure 2-7 schematically shows a typical arrangement for unsteady pressure measurement. Since noticeable attenuation occurred because of the presence of a constriction in the tube, a l l calibrations were performed with pressure tap in the c i r c u i t . Moreover, for work in the wind tunnel i t was found impossible to use atmos-pheric pressure as a reference. This i s because the difference between the static pressure at a tap and the atmospheric pressure was found to be so large as to throw the Barocel off scale at sensitive settings. Moreover, surges in the mean wake close to the model affected the pressure f i e l d around i t s surface and gave rise to the same effect. Hence the f i n a l calibration and measuring set-up incorporated a damping volume between the pressure ports of the Barocel, thus using the 24 Figure 2-6 Calibration plots for Barocel pressure transducer with damping bottle 25 _ Polyethylene tube ' t=3' d f=0.066* Barocel pressure transducer Signal conditioner Damping bottle < Filter ( w rms voltmeter with r_c clamping circuit Oscilloscope Figure 2-7 A schematic diagram of the fluctuating pressure measuring set-up using Barocel pressure transducer 26 mean static pressure at the tap in question as reference. 4 4 In the Reynolds number range of 10 - 12 x 10 of the experiment, the Strouhal frequency higher than 50 cps was associated with the smaller models of one to two inches in diameter. Due to a sharp increase in attenuation of the pressure signal at a higher frequency, i t was necessary to adopt an alternate procedure for measurements in this range. This was accomplished using a microphone system having better high frequency response. The modified experimental arrangement (Figure 2-8) used Bruel and Kjaer 1/2 i n . condenser microphone (type 4134) i n conjunction with type 2615 preamplifier. The microphone performance, having the sensitivity of 0.99 mV/y-bar and linear response in the range of 45 - 1000 cps, i s essentially independent of ambient temperature and pressure variations, thermal coefficient being less than 0.01 db/°C (-50°C to + 60°C), and pressure coeffic-ient - - 0.1 db for ± 10% pressure changes. In the absence of condensation, influence of relative humidity i s less than 0.1 db. The resonant frequency of the microphone i s 25 KHz while the Helmholtz resonator frequency of the cavity formed by the microphone diaphragm in position within the model is 490 cps. The microphone and the preamplifier were powered by Bruel and Kjaer type 2606 measuring amplifier, which has frequency response in the range 2 Hz - 200 KHz with adjustable 27 Measuring amplifier Microphone model • « Filter 1 w rms voltmeter with r -c damping circuit Oscilloscope Figure 2-8 The arrangement for measuring fluctuating pressure using condenser microphone 28 amplification i n 10 db steps from -50 to 100 db covering the sensitivity range of lOyV - 300 V f u l l scale. The amplified output was f i l t e r e d and measured on the DISA rms voltmeter as in the case of the Barocel arrangement. The wake survey was carried out using a disc probe 42 constructed by Ferguson in accordance with the design discuss 43 in detail by Bryer et a l . The mean pressure calibration results for the probe showed i t to be relatively insensitive to pitch of ± 5° and yaw of ± 20° (Figure 2-9). The unsteady pressure response of the probe i s included in Figure 2-6. The wake traversing gear designed by Ferguson was used to position the probe at a desired location in the wind tunnel test section The accuracy in positioning the probe was approximately 0.02 in 2.4 Test Procedure 2.4.1 Mean Static Pressure on the Model Surface The mean pressure distribution was obtained using a Lambrecht manometer with ethyl alcohol as the working f l u i d The o s c i l l a t i o n of the alcohol column caused by the fluctua-ting component of the static pressure was reduced using a damping bottle. To f a c i l i t a t e reduction of data, the pressure on the model surface was measured relative to the static head far upstream of the model. Figure 2-9 Disc probe dimensions with the mean pressure calibration plots showing M i t s relative insensitivity to pitch and yaw vo 30 2.4.2 Vortex Shedding Frequency The Strouhal frequency was determined by displaying a fluctuating pressure signal on a storage scope and comparing i t with a known frequency sine wave from a function generator. A band pass f i l t e r was introduced i n the c i r c u i t to eliminate extraneous noise. 2.4.3 Fluctuating Static Pressure on the Model Surface Fluctuating pressure about the mean at a tap in question was measured using the instrumentation set-up in Figures 2-7 and 2-8. Due to large, seemingly random, amplitude modulations of the fluctuating pressure, i t was necessary to present the results as time average values. A true rms voltmeter with a b u i l t - i n variable time constant r-c damping c i r c u i t was used to give an average of the fluctuating pressure signal over more than 500 cycles. The band-pass f i l t e r attenuation was determined, for each frequency and f i l t e r cut-off setting (1/3 octave), using a sinusoidal signal from a low frequency function generator. 2.4.4 Wake Survey The determination of the wake geometry was accomplished by examining the fluctuating pressure f i e l d associated with the vortices shed from the model using the instrumentation 31 layout shown in Figure 2-10. For symmetrical configurations of the models, i.e. circular cylinders and normal f l a t plates, the measurements were confined to one side of the wake except for occasional checks to confirm i t s symmetry. Traversing the disc probe la t e r a l l y in the wake at various downstream stations and recording the rms value of the pressure signal gave a set of curves each having a peak near the vortex centerline. The transverse distance between the peaks at each y-station represented, approximately, the spacing between the rows of vortices. It was convenient to present the results of lateral traverse as a ratio of the probe to reference signals in the wake, the latter taken to be the peak fluctuating pressure occurring, approximately, one diameter downstream. The longitudinal spacing between consecutive vortices i n a given row was obtained through recognition of the fact that the distance corresponds to 360° phase difference between the fluctuating pressure signals associated with them. Using a pressure tap on the model as reference, the disc probe was moved downstream near the centerline of a vortex row. The phase difference between the fluctuating signals was obtained using the function correlator, manufactured by Princeton Applied Research Corporation. For the cross correlation mode, i t i s given by a simple formula, $ = 3.6* N V fv (2.2) Signal conditioner x -y chart recorder Signal correlator rms voltmeter with r-c damping circuit Figure 2-10 Instrumentation lay-out for wake survey 33 The phase was recorded as a function of the downstream co-ordinate x. The process was continued u n t i l limited by the travel of the traversing gear (40 inch). The plot of phase vs. x-coordinate gave continuous variation of the longitudinal spacing through the relation a = °L* ( 360) (2 .3) The Strouhal frequency being known, the vortex velocity i n the wake was also determined. The wake geometry measurement were confined to the mid-span of the body. 3 RESULTS AND DISCUSSION The two models selected for the present investigation are representative of many two-dimensional geometries of engineering interest. Together, they cover a wide spectrum of aerodynamic characteristics encountered in practice, e.g., the sharp edge separation and a wide wake of a f l a t plate as against the boundary layer separation and a relatively narrow wake for a circular cylinder. Also a plate provides a p o s s i b i l i t y of studying assymmetric flow conditions. In the following presentation, the results for circular cylinders and f l a t plates are given together, in the same section, to f a c i l i t a t e comparison and help establish general trends of constraint effects on bluff bodies. 3.1 Mean Pressure Distribution and Forces 3.1.1 Circular Cylinder The mean static pressure distribution on the surface of the circular cylinders was measured in the Reynolds number 4 4 range of 10 - 12 x 10 . The measurements, confined to one side of the models due to symmetry of the cross-section, were carried out at the angular spacing of 10°, with the finer 3° interval in the region 60° _< 9 <_ 120°, to obtain precise information concerning the maximum negative pressure, adverse 35 pressure gradient and the beginning of the wake. A detailed investigation showed mean pressure d i s t r i -bution to be substantially independent of the Reynolds number except for the largest model of 35.5% blockage. Typical results on the Reynolds number dependency are presented in Figure 3-1. It i s interesting to note that even at this high bluffness (35.5%), the potential flow region remained relatively unaffected, and so did the location of the minimum pressure. The increase in the base pressure coefficient with the Reynolds number (Figure 3-lc) may be attributed to the transition effect as the confinement correction would 4 4 probably s h i f t the Reynolds numbers 10 x 10 and 12 x 10 into that region. A comparison of the mean pressure distribution for different blockage ratios at the same Reynolds number i s given 44 in. Figure 3-2. The results of Fage and Falkner for blockage ratios of 6% and 12% at the Reynolds number of 10^ are also included in Figure 3-2c. The plots suggest larger gradient of the pressure curves in the potential flow region at higher bluffness, with upstream movement of the zero pressure coeffic-ient point. An increase in the constraint leads to a correspond-ing increase of the negative pressure in the separated flow region as well as i t s peak value. It i s of particular significance to recognize that position of the separation point, approximately indicated by the location of the adverse pressure gradient region, remains Figure 3-1 Effect of the Reynolds number on the mean static pressure distribution around circular cylinders of different blockages (a) S/C = 4.5%; (b) S/C =20.5% w Figure 3-1 Effect of the Reynolds number on the mean static pressure distribution around circular cylinders of different blockage: (c) S/C = 35.5% 38 unaffected by the constraint in the Reynolds number range i n -vestigated. Although no direct measurement of the separation point was undertaken, the evidence i s clear from the similarity of the pressure distribution plots. Besides, the almost constant drag coefficient for each model tends to substantiate independence of the separation from the Reynolds number, except for the largest model tested. From Figure 3-2 i t i s apparent that, in general, the pressure in the separated flow region i s not uniform but tends to diminish towards the 180° position. The constraint leads to an increase of i t s rate of reduction. Variation of the mean pressure at the 180° position with the model bluffness i s plotted i n Figure 3-3. A comparison 44 45 46 with the results of Fage and Falkner , Gerrard and Roshko i s also included (Figure 3-3b). As observed by Gerrard, in the Reynolds number range of the experiment, the pressure at 180° i s usually considered independent of R. With reference to the present set of results, this was observed to be essentially true at the smaller blockage ratios. Fage and Falkner reported a sharp increase in the pressure with the approach of the transition regime, however, no such rise was observed in the present experiments, at least up to R = 12 x 4 10 , except for the largest model of 35.5% blockage (Figure 3-lc). It should be pointed out that the results presented are not corrected for the blockage effect which, in general, would tend to increase the effective Reynolds number. i 1 r (a) R = 15 x lO S_ C 3% ' L "1513 180 30 60 90 120 Figure 3-2 Mean static pressure distribution on circular cylinders showing the effect of wall confine-ment: (a) R = 1.5 x 10 4 Figure 3-2 Mean static pressure distribution on circular cylinders showing the effect of wall confine-ment: (b) R = 5 x 10 4 41 e Figure 3-2 Mean static pressure distribution on circular cylinders showing the effect of wall confine-ment: (c) R = 10 x 10 4 42 1 r T 1 i 1 1 r CL5 1.0 1 3 -H A A O 43 c 6% 9 • 14.8 o CO a V 2.0 o n 205 2.5 3.0 (a) ?63 353 3-5 13 • L J L J i L 4 5 6 8 X) !2xK) 4 Figure 3-3 Effect of the Reynolds number on C at 180° for circular cylinders of different blockage ratios: (a) present results 43 i i i i 1 1 s_ c 2 % Roshko 0 5 46 Gerrad 45 \ Fage & Falkner — - i — 12 ' 44 4.5 6 9 present results a* S asl-0.9-1.0-I.I • 1.2-1.31-1.i (b) 2 o 1.5l 1.5 2 5 6 8 10 12 15 20 30x10 Figure 3-3 Effect of the Reynolds number on C p at 180° for circular cylinders of different blockage ratios: (b) comparison with the available data 44 As the pressure in the separated flow region i s not uniform, i t i s convenient to express i t as an average value. Figure 3-4a shows variation of the average base pressure coefficient, cp^t with the Reynolds number and cylinder bluff-ness. As before, for the blockage ratio up to 26.5% there does not appear to be any significant change in the base pressure coefficient with R. However, distinct rise in b for the large blockage of 35.5% i s quite evident. Effect of the blockage and the Reynolds number on the drag, obtained by integrating the mean surface pressure, i s shown in Figure 3-4b. The close similarity between Cp^ and C D plots can be expected. As the change i n the potential flow region i s minor, the base pressure governs variation of the drag coefficient. Average of C D, for blockage ratios up to 6%, was around the well known value of 1.15 for a circular cylinder in a nearly i n f i n i t e stream. 3.1.2 Flat Plate Mean static pressure distribtuion on the surface was measured at the mid-span section and for several plate orientations. The velocity was confined to the Reynolds number , A4 T_ lft4 , ... . , , . , , 47,48,49 et a l . range 10 - 12 x 10 , where i t i s established ' ' that the mean surface loading is independent of the Reynolds number. The limitation on the model thickness and the presence of sharp narrow corners restricted the detailed pressure measurements at the edges. 45 Figure 3-4 Effect of wall confinement and the Reynolds number on: (a) the average base pressure; (b) the drag coefficient 46 Figure 3-5 gives a set of representative plots show-ing the Reynolds number effect on pressure distribution at two blockage ratios and model orientations. Independence of the surface loading in the indicated range of the Reynolds number is clear and i s further substantiated by the force coefficients data in Figure 3-9. Variation of the mean pressure distribution with angle of incidence i s plotted i n Figure 3-6 for three blockage ratios: 9, 20.5 and 35.5%. As can be expected, the surface loading for a normal plate (a = 90°) i s symmetrical, with the stagnation point at the geometric centerline. The pressure on the front showed close agreement with the potential flow solution except close to the separation point and the base pressure i s essentially uniform. As the plate angle i s decreased from the normal position, the base pressure coeffic-ient decreases; however, i t continues to remain uniform for a given a. The stagnation point moves upstream approaching the leading edge with a rapid f a l l of pressure towards the t r a i l i n g edge. A comparison of the mean surface loading at different blockage ratios for the same angle of incidence i s given in 50 Figure 3-7. Data by Fage and Johansen , for S/C = 7.15%, are also included. It i s observed that on the front of the plate, where the character of the incoming flow i s not yet substantially affected by the presence of the body, pressure coefficient showed, in general, only a slight decrease with an 47 (a) — r 9 % , Q = 30° o 4.5x10 • 6.1 xlO4 4 J H 1 H o - 2 - 3 - 4 _L (b) — =35.5%, a = 90 c o 9.2x10 v 1 2 x l 0 4 •0.5 - 0 . 4 L.E. -0.2 a o 0.2 0.4 0.5 T.E. Figure 3-5 Independence of the pressure distribution on a f l a t plate from the Reynolds number: (a) S/C 9%, a = 30°; (b) S/C = 35.5%, a = 90° Figure 3-6 Variation of the mean pressure distribution with plate orientation for the blockage ratios of: (a) S/C = 9%; (b) S/C = 20.5% 49 1 i • i • i • 1 1 * * 0 * A' # # * * A (c) -| - = 353% * \ 30°'«.. • * • 1 = 1 — - . 30° - 2 -- 3 , - « . . • ^ 90° A t. 4 i I . I . I -0 .5 -0.4 -0.2 0.0 0.2 0.4 03 L.E. y/h T.E. Figure 3-6 Variation of the mean pressure distribution with plate orientation for the blockage ratios of: (c) S/C = 35.5% 50 increase in the constraint. On the other hand, the wall confinement effect i s mainly to reduce the base pressure coefficient, at a l l inclinations, due to the increase of flow velocity according to continuity. As the main effect of the wall confinement i s on the base pressure, Cp^ i s given i n Figure 3-8 as a function of both plate orientation and blockage ratio. Similar data from Fage and Johansen^, and Abernathy 4^ are included for comparison. The figure suggests a f a l l of the base pressure coefficient with an increase i n blockage, more at higher angles. It i s noticed that influence of the constraint increases, almost linearly, with the effective blockage ratio as shown i n Figure 3-8b. The same conclusion i s arrived at by the theoretical analysis later. Thus, as the angle of attack i s reduced from the normal position, the constraint i s effectively diminished, correspondingly, i t s influence on Cp^ i s less. Abernathy's results follow the same trend, however, they show a lower pressure coefficient in general. The reason for this discrepancy i s not clear. Abernathy conducted his experiment in a 7 i n . x 3 i n . wind tunnel. It i s not mentioned i f the test section was provided with f i l l e t s for boundary layer compensation. As he bases the blockage ratio on the model to tunnel width, the presence of f i l l e t s would lead to higher area blockage. Although the free stream turbulence character during his experiment i s not specified, i t i s anticipated that this i s not l i k e l y to explain the discrepancy. As 1 U L o — i • — Fage & Johansen 50 J O Z «A*-a = 30* o o o • o . V J L = o°/ C 9 / o ^ o 7.15 ' 20.5 • 35.5 — I — I — I — I — ' — f — I o O ^ A . 0 • J t * 0 o 4 o (b) a = 60 -1 A A A - 2 - 3 — = 97 c y / o 20.5 35.5 o o o 0.5 -0.4 L.E. -0.2 0.0 0.2 0.4 0-5 T.E. Figure 3-7 Surface loading over a f l a t plate as affected by wall constraint: (a) a = 30°; (b) a = 60° 52 1 1 i —I »'• 1 • 1 1 • 0 / o # (c) O \ % % • \~ : o t Q = 90° o • 1 1 V 1 1 1 - 1 • » ~ - = 7.15% — 7 c P i- v — / - ^ - v - - - - - - - " 7 ~ * " A " ^ " _ A ^ 9 - 2 --8 e 2Q5 • - 3 mm - 4 - o o ° 1 . O o o 0 o o o < > < > 35.5 50 rage & Jonansen l . l . l . o -1 0. 5 - 0 . 4 1. E. - 0 . 2 0.0 0.2 y/h 0.4 0.5 T.E. Figure 3 - 7 Surface loading over a f l a t plate as affected by wall constraint: (c) a = 90° 53 Figure 3-8 Base pressure coefficient for f l a t plates as a function of: (a) plate orientation; (b) blockage ratio 54 27 discussed by Vickery , based on the tests with square cylinder, the effect of turbulence i s primarily evident i n the condition close to reattachment, and i s to increase the base pressure. The uniformity of force coefficients for a f l a t plate in the approximately free flow condition and in the Reynolds 4 4 45 number range 0.6 x 10 - 60 x 10 as reported by Flachbart was found to hold even at higher blockage and at a l l angles. Figure 3-9 gives representative plots for the blockage ratios 9 and 35.5%. The scatter in the data at the lower Reynolds numbers may be attributed to the reduced accuracy in measuring small values of dynamic head and forces. As a direct result of the pressure distribution data, one would expect larger variations of the drag coefficient at higher angles of attack over a given range of S/C. This was substantiated by the balance measurements given i n Figure 3-10b. The drag coefficient attains i t s maximum at the 90° position where the wake is the widest and Cp^ the minimum. With a decrease in a, C D decreases and would reach the value of the skin f r i c t i o n drag at a = 0°. It should be mentioned here that the f r i c t i o n drag i s negligibly small compared to the profile drag when the flow i s separated (about two orders of magnitude lower). The effect of the constraint on C^ i s less at the lower angles of attack due to reduction in the effective blockage ratios which, as pointed out before, i s similar to that of the base pressure. 55 4 'D • r A I —T" 1 1 1 1 1 - A A 90 -° o ° 75° O U o 60° • • • • — 45° --A --30° -- -• 1 1 1 1 I I I 1 1 i 1 1 — i — i — 1 — i r cr-Ci::::ov:xc::Hi:::S^ft^fk?^ a 60° 45 75 30 C ( o ) _] L J I I i I i I I L 4 5 6 8 10 12 15x10" R Figure 3 - 9 Plots showing independence of the f l a t plate force coefficients from the Reynolds number: (a) S/C = 9 % 2.5 56 2.0 1.5 'D L T.O 0-5 -0.0 - A — A . — r • * i 1 r m + "> * A i — 1 — r a c 90 "O CT Q Q n o 0 n n u u •A l\ A A -•—• m 75 - i • 60* J i I I I I i l . l I 1.5 i 1 1 1 1 r T — • — i r a >- "A -A A A * A A 1.0 . . . . F L A . A A A A . A A . 45 60 (b) • o — O - " D - " 0 " 0 * ° — ° - O G - O O 75 ' l i i I • i • L 1 4 5 R 6 8 10 12x10 Figure 3-9 Plots showing independence of the f l a t plate force coefficients from the Reynolds number: (b) S/C = 3 5 . 5 % 57 - • 1 1 1 1 i O 355 - o -50 Fage & Johansen ^ • • 205 -o o • o o 7.15 • 4 9 ° 4,5%-S . c 1 1 1 i 1 30 45 60 75 90 (S/C) N Figure 3-10 Variation of the force coefficients with plate orientation and blockage ratio: (a) the drag coefficient as a function of a ; (b) the drag coefficient as a function of (S/C). N 58 The mean l i f t coefficient, which arises due to un-symmetric plate orientation, reaches a maximum value at about 50° (Figure 3-10c). The f a l l of from the maximum may be explained by the reduction of the surface area normal to the flow direction as the angle increases beyond 50°, while i t i s due to the decrease of the pressure difference between the two sides of the plate for a < 50°. This behavior was observed to be valid at a l l blockage ratios. The constraint effect i s to increase the l i f t coefficient at a l l angles, however, the increase i s larger where C L attains i t s maximum. 3.1.3 A Comparative Study It i s apparent that wall confinement affects the mean pressure distribution over circular cylinders and f l a t plates in essentially the same way: the potential flow region remains vi r t u a l l y unchanged while the base pressure shows marked reduction with increasing blockage. Accordingly the force coefficients increase. The variations of both the pressure and force coefficients increase with the constraint. On the other hand, the separation condition remains essentially unchanged. As geometries considered represent extremes of character-i s t i c a l l y different flow conditions one can say, with a degree of confidence, that the above observations are l i k e l y to be valid for a general two dimensional bluff body. However, a Figure 3-10 Variation of the force coefficients with plate orientation and blockage ratio: (c) the l i f t coefficient as a function of a 60 systematic study with other geometries i s required to confirm this point. It i s relevant to emphasize here that a circular cylinder always experiences a continuous decrease in base pressure towards the 180 degree position. The presence of wall confinement only accentuates this drop and makes i t more noticeable. The fact that this behavior i s independent of bluffness i s substantiated by the f l a t plate data (Figure 3-6) where the base pressure remains uniform for a given blockage. The behaviour cannot be attributed to the difference in the character of the separation either. The base pressure 2 51 distribution for a 90° angle section , D-section and e l l i p t i c 3,41 cylinders of several eccentricities suggest the reduction to be associated with the extension of the bodies into the wake, i.e., the decrease in Cp^ occurs when part of the body i s between the two separated shear layers (Figure 3-11). Apparently this i s related to the aerodynamics of the near wake associated with a bluff body. As reported by 49 Roshko (circular cylinder and f l a t plate), the pressure along the center line of the wake reaches a minimum approxi-mately at the formation of the f i r s t vortex. It stands to reason that the portion of the body i n the wake would reflect a similar reduction in the base pressure. The f l a t plate being always upstream of one of the separated shear layers at any orientation, i t s base pressure remains v i r t u a l l y un-affected by the suction f i e l d of the formation region. 61 Model contour sides Figure 3-11 Illustrations of bluff body extension into the wake affecting base pressure distribution 62 3.2 Strouhal Number The factors governing the vortex formation and shedding frequency for bluff bodies are not yet completely understood. Several length parameters have been proposed which are thought to have direct relation with the shedding frequency. One of these i s the projected width of a model which forms with the incident velocity and vortex frequency the Strouhal number, S N. The importance of S N l i e s in the fact that, for the given orientation of a body, i t i s relatively constant over a wide range of the Reynolds number i n the subcritical regime. Attempts have been made to define a more general form of the Strouhal number that does not depend on the body shape or 4 orientation. The most widely used has been that due to Roshko where the normal distance between the separated shear layers after they become pa r a l l e l , h', and the separation velocity V are used as characteristic parameters to give (3.1) This 'universal' Strouhal number was found to have the value of 0.181 for circular cylinders, 90° wedges ( -»• <3 ) and normal f l a t plates. One may thus conclude that the distance between the shear layers represents one of the important parameters associated with the vortex shedding. 3 It would be pertinent here to mention Dikshit's experiments with a family of e l l i p t i c cylinders. His effort at correlating results over a wide range of cylinder eccen-t r i c i t y and angle of attack using the normal distance between separation points as the characteristic length showed considerable promise except for very slender ellipses (e = 0.92, 0.9 8) at small angles of attack (a < 20°). 3.2.1 Circular Cylinder The variation of the Strouhal number S^ (based on cylinder diameter) with the Reynolds number and bluffness i s presented in Figure 3-12. Up to the blockage ratio of 14.8%, S^ i s within the known range of 0.19 - 0.21 for a circular cylinder in an approximately i n f i n i t e stream. As in a l l the reported experiments, scatter i n the Strouhal number data 52 always existed. Similar results by Etkin et a l . and 53 Gerrard , for blockage ratios in the range 1-15%, are also included. Average values (arithmetic as well as the root mean square) of the Strouhal number show a gradual increase with the blockage ratio up to 14.8%, but there after the rise i s quite pronounced. 3.2.2 Flat Plate The Strouhal number is known to be independent of the Reynolds number in the subcritical regime. This was confirmed for a l l blockage ratios tested. Figure (3-13) records a sample of the results. Variation of the average Strouhal 64 T r T 1 1 — i 1 r—i r 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 _S_ C 1% 2.8 5 7.5 15 Etkin et al. 53 Gerrad 52 • • J L 13 2 4 5 6 R 8 10 1. ins" 35.5% O 26.5 • 20.5 • 14.8 o 9 A 6 • 4.5 O 3 • 20 30 40x10 Figure 3-12 The Strouhal number for circular cylinders of different blockage as affected by the Reynolds number 65 0.40 f i r — i — i — 1 — i — r - | - = 35.5%, a = 3 0 ° ^ O ^ ^ $ 0.35 0.30 35.5%, 90 o O 0.25 0.20 « r-4.5% , 60 2tt5% , 45 o o "O—u o o £> _» f i t 20.5% , 60 0.15 -e—o—u e o °o o 4.5% , 90° o.ioi J I L J i L 5 6 8 10x10 Figure 3-13 Typical plots for inclined f l a t plates showing the Strouhal number's independence of the Reynolds number 66 number (based on plate chord) as a function of the blockage ratio and plate orientation i s plotted i n Figure (3-14a). It i s observed that decreases with increase in the angle of incidence for a l l blockage ratios. For a given a, i t increases with blockage, being based on the measured free stream velocity V Q. In general, the data by Fage and Johansen^, 48 and Abernathy follow the same trend as the present results. Explanation for deviations i n the magnitudes has to be l e f t to the same reasoning as that used in discussing the base pressure results. A more r e a l i s t i c representation of the Strouhal number would be by considering the projected width (h sin a) as the characteristic length. The data shown in Figure (3-14b) reflects an interesting trend. The constraint results in increasing S N with the larger increase occurring at higher angles, This leads to the divergence of the plots i n the figure: S N increases with decrease in angle of attack for small blockage ratios but reverses the trend at the larger bluffness. 3.2.3 Universal Strouhal Number It would be pertinent here to attempt to explain the effect of constraint on vortex formation and shedding through the use of Roshko's Strouhal number. As mentioned in the introduction to this section, the normal distance between the paralle l shear layers, h', represents an important parameter 1 1 Fage & Johansen" 48 Abernathy present results S_ c O 35.5% 19 *J 205 Figure 3-14 Effect of wall constraint and plate attitude on the Strouhal number: (a) based on plate chord? (b) based on projected width 68 in such a study. Moreover, the analytical investigation of 54 vortex street formation by Abernathy and Kronauer showed that the shedding frequency and vortex strength depend very strongly on the length scales h* and a. Abernathy4** directly measured h'/h for inclined f l a t plates and found i t to be unaffected by blockage ratios up to 28%. Also, i n the present experi-ments, the wake geometry parameters a/h and b/h remained independent of the constraint for both circular cylinders and f l a t plates, except for reduced b/h corresponding to the plate of 35.5% blockage at ct = 60°, 90°. With this background one may take liberty to surmise that character of the wake geometry, suitably non-dimensionalized, remains unchanged with blockage, i.e., h'/h i s independent of the wall constraint. Furthermore, one i s inclined to conclude that the mechanism of the vortex formation also remains unchanged. Using Roshko's definition of the Strouhal number S (eqn. 3.1), results for circular cylinders and f l a t plates under confined conditions are presented i n Tables 3-1 and 3-2, respectively. The length h', defined as the distance between centerlines of the shear layers, i s taken to be 1.41 h sin a 48 . 5 5 for f l a t plates and 1.15 d for circular cylinders . Average values of V and f are used i n the case of circular cylinders due to slight scatter in the base pressure (Figure 3.4a) and the Strouhal number (Figure 3-12) data. The separation velocity i s defined as 69 1/2 (3.2) where C D i s the pressure coefficient at the point of separation. The important conclusion from the tables i s the exis-tance of a universal Strouhal number, valid for the majority of blockage ratios and angles of attack, which again affirms a negligible effect of the constraint on the mechanism of * vortex formation. S for the cylinders i s quite close to that for the plates. Small deviations are probably due to d i f f e r -46 54 ences i n the accuracy of experimental measurements of h' ' used i n the evaluation of S . For the plate of S/C = 35.5%, * S i s observed to be higher than the average of the values for other blockage ratios. The lack of information on h'/h for this bluffness led to the use of Abernathy's experimental results. The smaller b/h for this case suggests a lower value of h'/h which may account for the observed difference. Also at a l l blockage ratios, S for the 30° position i s noticeably higher than the average. Similar deviation was also reported by Abernathy. This may be attributed to the f i n i t e dimension of the chamfered edge which becomes significant, particularly at smaller angles of attack. Here, proximity of the shear layer to the chamfered edge leads to, i n effect, what may be loosely referred to as "incomplete separation". 70 Table 3-1 S for Circular Cylinders S/C 3% 4.5 6 9 14.8 20.5 26.5 35.5 sd .195 .195 .2 .2 .205 .21 .22 .245 * s .156 .154 .156 .153 .152 .151 .147 .151 Table 3-2 * S for Inclined Flat Plates S/G 4 .5% 9 20.5 35.5 Angle a Sh * S S h * S S h * S S h * S 90° .147 .138 .16 .144 .193 .148 .275 .177 75 .157 .14 .169 .147 .2 .148 .277 .173 60 .175 .144 .188 .15 .21 .147 .271 .17 45 .225 .226 .151 .25 .152 .282 .154 30 .33 .344 .181 .34 .167 .366 .17 71 3.3 Unsteady Surface Loading Experimental investigation of the unsteady surface loading over a circular cylinder has received, relatively speaking, more attention compared to other geometries. In particular, there i s a complete absence of the corresponding information for a f l a t plate. Even where test results are available the phenomenon i s s t i l l far from being f u l l y understood. In a sense, this i s reflected i n a lack of an exact theoretical model. Several basic studies of circular cylinder conducted by G e r r a r d 4 5 ' 5 3 ' 5 6 - 5 9 and o t h e r s 4 2 ' 5 2 ' 60-66, et a l . h a v e primarily shown that, i n the subcritical Reynolds number regime, the fluctuating pressure signal has an appearance of a pure tone with random amplitude modulations and a much smaller f i r s t harmonic. The fluctuating drag coefficient at twice the fundamental frequency i s an order of magnitude smaller than the fluctuating l i f t and hence of l i t t l e practical significance. This section presents results on the unsteady pressure over the surface as a function of bluffness, orientation (in the case of a f l a t plate) and the Reynolds number. An attempt i s made to explain some of the phenomena observed. It appears that the three dimensional character of the flow represents an important parameter in such a study. Unfortunately this 56 68 aspect has received relatively l i t t l e attention ' 72 3.3.1 Circular Cylinder Fluctuating pressure on the surface of a circular cylinder was measured using the Barocel ( f v < 50 cps) or the microphone arrangement ( f v > 50 cps) described in section 2.4.3. Typical results are presented in Figure 3-15 for four of the models at several values of the Reynolds number. In a l l the cases, the distribution of the fluctuating pressure showed considerable similarity. In general, the pressure increases sharply from the front stagnation point reaching the peak value around 80° followed by an approximately uniform pressure region extending over the range 80° - 150°. A sharp drop ensues reaching, theoretically, zero value at 180° due to cancellation effect of the out of phase vortices. It appears that the effect of the Reynolds number i s more significant at smaller bluffness. Of particular interest i s the location of the peak fluctuating pressure which i s close to the separation point (- 82°, Ref. 67) and remains relatively unaffected by the constraint. 42 63 The results of Ferguson and Feng for the models of the blockage ratio 9%, tested i n the same tunnel as the present experiment, are also given in Figure 3-15. The pressure distribution shows the same trend as discussed above. Feng's result are in good agreement with the present, while the magnitude of the pressure amplitude i s somewhat lower i n Ferguson's case. A significant difference i n the experimental -0.4 u n |  u II i II II i i t i u II | t i I fa o or o -a6 -as -1.0 -1.2 -1.4 o ° o o " o n o o w o o a a A O A A A A A • O -1.6 . (a) -§- = 3% A 1X10 o 3 • 4 - 1 . 8 . ' ' 3 0 ' ' J O ' ' A ' '126' ' l i b 1—JL e i i I i 8 • A 180 0 9 OA y u | y II | A A I W V u A O I A O $ * 9 % R 4 A 1.5x10 o 3 • 6 A o (b) A 1.5 Ferguson O 2 Feng 6 3 42 i i JL i i 30 A ' ' A ' '1*0' '156' '190 e Figure 3-15 Dependence of the unsteady pressure distribution on the Reynolds number *»• for circular cylinders of blockage ratios: (a) S/C = 3%; (b) S/C «= 9% fix O -0.1 -0.3 -0.5 -0 .7h -0.9 i i | n | i I | M | i i | l i o or o -1.1 -1.3' -1.5 8 o • I- A a A . I (c) u • 8 • | B " B j A A o A " -| - = 20.5% 4 • 5x10 o 8 • 12 . . I J L 35 60 90 I e° 180 e Figure 3-15 Dependence of the Unsteady pressure distribution on the Reynolds number for circular cylinders of blockage ratios: (c) S/C = 20.5%; (d)S/C = 35.5% 75 set-up may explain this discrepancy. Ferguson carried out unsteady pressure measurements using the LDR (light dependent resistance) transducer of his design, which had limited sensitivity (Barocel i s approximately 250 times more sensitive in the frequency range of interest), and required frequent calibration to guard against thermal d r i f t s . While in the present case, as well as in Feng's, the more sensitive and stable Barocel pressure transducer was used. Figure 3-16 shows typical results on the fluctuating pressure distribution as affected by the constraint for the given Reynolds numbers. As before, a similarity in the pressure distribution i s apparent. This i s further emphasized by the fact that over the bluffness or the Reynolds number range of the experiment, the dimensionless pressure ratio p*/P m a x along the cylinder circumference varied by less than 5%. The similarity of the fluctuating pressure plots presents an interesting possibility of representing the entire curve by a single point. Here, the peak value, C-, , was P max selected to serve this purpose. Thus information on the effect of the constraint and the Reynolds number can be presented in a condensed form as shown in Figure 3-17a. The results of 53 Gerrard for models of blockage ratios 2.8%, 5%, 7.5%, and 15% are shown separately in Figure 3-17b to f a c i l i t a t e comparison and emphasize the scatter in results which appears to be inherent to the study. The results indicate, in general, I 1 1 I 1 1 I • I I I I I T Q • T T T T • T a « a • • • • • n a • • • • o • • • • R=4x10 _S_ C • 14.8% • 20.5 T 26.5 e • i i i i i ; i i | i i | i i O O o • _ 0 0 0 0 ° % 0 T a a • T • • T T • 0 • a • • a \- a (b) J6' ' lol ' J6' '126' '156' '180 A • I • 30 R: 10x10 4 s c • 2Q5% T 26,5 0 35.5 90 1 1 120 B "I T • 120' ' l f e ' e 180 Effect of wall confinement on the unsteady pressure distribution over circular cylinders: (a) R = 4 x 10 4; (b) R = 10 x 104 77 the unsteady pressure to increase with the bluffness. However, i t should be pointed out that as observed by Keefe^, the decrease in aspect ratio below 5 also gives rise to an increase in the fluctuating l i f t . In the present experiment, an increase i n the blockage ratio i s accompanied by a reduction in the aspect ratio (Table 2-1), which may be partly responsible for the increase i n fluctuating pressure for blockage ratios above 14.8% (aspect ratio < 5). The i n i t i a l pressure increase with the Reynolds number (up to around R = 3 - 5 x 10 ) agrees well with the results of Gerrard and Keefe. Following a short region of approximately uniform pressure, there i s a tendency towards a gradual drop. The wall confinement seems to reduce the dependency of fluctuating pressure on the Reynolds number. It should be noted that Gerrard chose to represent his data by the average curve overlooking the effect of blockage. Closer examination of his results i s given in Figure 3-17c together with data from the present experiment at approximately the same blockage ratios. The results in general indicate the scatter of his data i s partly due to the blockage effect. Integrating the pressure over the surface, knowing that i t i s approximately in phase on one side of the cylinder and 180° out of phase on the other, gives the fluctuating l i f t as shown in Figure 3-18a. Its similarity with the peak fluctuating pressure plots can be expected and the previous discussion concerning blockage and the Reynolds number effects 78 3-17 Variation of the maximum fluctuating pressure coefficient with the Reynolds number and blockage ratio: (a) present results; (b) Gerrard's results 79 0.3 0 . 2 -0.15 0.10 0.08 0.6 0.5 0.4 0.3 X o E ""lb? U 0.2 0.15 0.6 0.5 0.4 0.3 0.2 0.15 0.10 ( c ) J L 0.8 1 J L c 3% 2.8 Gerrad 53 ' ' • I I L J L 13 2 I I J I 4 5 6 R 14.8 15 Gerrad 8 X) 12 15 20 30x10 Figure 3-17 Variation of the maximum fluctuating pressure coefficient with the Reynolds number and blockage ratio: (c) comparison between the present results and Gerrard's data 80 applies to the l i f t c o e f f i c i e n t as w e l l . A comparison with 53 61 62 the r e s u l t s of Gerrard , Keefe and McGregor i s given i n Figure 3-18b. Gerrand's average p l o t shows substantial v a r i a t i o n of C=-, with the Reynolds number. However, t h i s Li dependency on R i s somewhat reduced i f the data f o r each blockage r a t i o are presented separately as i n the case of C-, (Figure 3-17c). Keefe's and McGregor's r e s u l t s agree Pmax with the general trend of the present experiment of increasing f l u c t u a t i n g l i f t c o e f f i c i e n t with the constraint. 3.3.2 F l a t Plate The f l u c t u a t i n g pressure on the surface of the plate was measured using the Barocel arrangement. Due to the s i m i l a r i t y of the pressure d i s t r i b u t i o n curves, d e t a i l e d measurements were confined to two speeds only, the Reynolds number e f f e c t being studied through the maximum f l u c t u a t i n g pressure. The unsymmetrical o r i e n t a t i o n of the plate introduced an extra v a r i a b l e over the c i r c u l a r cylinder case, however, the nature of the pressure sig n a l remained un-changed --for a l l angles of incidence i t was a pure tone with random amplitude modulations and a small f i r s t harmonic. The unsteady pressure d i s t r i b u t i o n f o r a plate model symmetrical with respect to the incident flow was found to be si m i l a r to that of the c i r c u l a r c y l i n d e r (a = 90°, Figure 3-19) . The pressure i n t e n s i t y reaches a maximum at the edge 81 3.0 T r i—i—i i i » i—r c T 35.5% 26.5 20.5 14.8 J ' • i • l » L Gerrad 53 H 61 Keefe 62 2.3 McGregor __| 5 6 8 10 12 15 "20 3QX104 0.6 0.8 1 Figure 3-18 Fluctuating l i f t coefficient for circular cylin-ders as affected by the Reynolds number and wall confinement: (a) present results (b) com-parison with the reported data 82 of the plate and decreases to a minimum at the geometrical centerline, along the free stream direction, due to cancellation of the pressure signals 180° out of phase. The intensity of the pressure i s higher at the base and the cancellation process less complete, probably due to irregular character of the separated flow. Figure 3~19 also shows variation of the pressure i n -tensity and the phase angle for several plate orientations and blockage ratios. It i s observed that the pressure dis-tributions are similar for a l l bluffness at a given angle of attack. The measured pressure intensity at a given location on the surface i s primarily a resultant influence of the complex flow in the near wake. However, looking from a rather simplified point of view (and thus risking a distortion of the true picture) i t appears that the following factors have substantial bearing on the phenomenon: (i) The unsteady pressure f i e l d appears to be divided into regions where associated vortices have pronounced influence as indicated i n Figure 3-20. Thus there are two zones where the individual vor-tex plays a dominant role while the out of phase character of the other vortex has only secondary effects. (ii) The minimums of the fluctuating pressure on the surface of a body close to the region representing transition zone of the influence of the vortices. 83 ( i i i ) The f l u c t u a t i n g pressure attains peak values close to separation, which f o r a f l a t plate would corres-pond to the sharp edges. The absolute maximum i s associated with the downstream separation point which can be expected i n the l i g h t of the wake data (section 3.4.2). E s s e n t i a l l y the same conclusion can be derived from the r e s u l t 2 4 1 3 of Slater ( s t r u c t u r a l angle s e c t i o n ) , Wiland and D i k s h i t 6 9 (family of e l l i p t i c cylinders) and Heine (rectangular cylinders of d i f f e r e n t aspect r a t i o s ) . Starting from the peak f l u c t u a t i n g pressure at the leading edge (point A, Figure 3-20), C-, gradually drops due to the c a n c e l l a t i o n e f f e c t of the vortex of opposit strength from the t r a i l i n g edge reaching a minimum near the t r a n s i t i o n zone. Progressing along the front face, the pressure gradually r i s e s under the influence of the t r a i l i n g edge vortex reach-ing a maximum at point C. On the base, the c a n c e l l a t i o n e f f e c t due to the phase difference, i s repeated followed by a gradual r i s e to the maximum at A. Figure 3-21 shows v a r i a t i o n of the maximum f l u c t u a t i n g pressure, C-, , with the Reynolds number and angle of Pmax incidence. A degree of randomness inherent to the phenomenon contributed to the scatter of the data. However, a general trend can be discerned. As i n the case of a c i r c u l a r cylinder C-, increases with the Reynolds number reaching a peak a f t e r Pmax which i t reduces again. V a r i a t i o n of C-, with R i s r e l -Mnax 84 Figure 3-19 Unsteady pressure and phase angle distributions over the surface of f l a t plates as functions of the blockage ratio and plate orientation: (a) S/C = 9% 85 B C D A y/h 3-19 Unsteady pressure and phase angle distributions over the surface of f l a t plates as functions of the blockage ratio and plate orientation: (b) S/C = 20.5% • H u I i I i i 1 1 -0 .5 -0 .25 0 6 7 2 5 5 3 6 7 2 5 0 _0.25 -075 A B C D A y/h Figure 3-19 Unsteady pressure and phase angle distributions over the surface of f l a t plates as functions of the blockage ratio and plate orientation: (c) S/C = 35.5% 87 LE. Region of influence of L.E. vortex x min. unsteady pressure point Separated shear layer W y Region of influence Figure 3-20 Schematic representation of stagnation stream-lines and separated shear layers near an i n -clined f l a t plate 88 tively smaller at the larger blockage ratios and at a l l incidences as further indicated in Figure 3-22. In general, the intensity of the pressure increases with a, which could be related to the increase in vortex strength, being directly proportional to the base pressure parameter K (Figure 3-7). The scatter in the results i s probably due to many factors influencing the pressure intensity. An attempt to explain even a few of them i s , at best, baffling. Apart from the magnitude of the vo r t i c i t y in the separated shear layers, factors like details of the formation and growth of the 59 60 f i r s t vortex ' , transition to turbulence in the separated 58 shear layer , percentage of the shed vorticity in the f u l l y formed vortex^, correlation length^"* etc., are important. However, their direct role in the formation of the vortex leading to unsteady pressure i s hardly understood and needs further investigation. The effect of the constraint i s to increase the i n -tensity of fluctuating pressure at a l l angles (Figure 3-22). Associated with an increase in the blockage i s a corresponding decrease in the aspect ratio, which leads to higher C-, and reduced dependency on the Reynolds number, as mentioned in section 3.3.1. Figure 3-21 Variation of the maximum fluctuating pressure with the Reynolds number for several plate orientations and blockage ratios 90 1 2 3 4 6 8 10 12x10' R Figure 3-22 Dependence of the maximum fluctuating pressure on the wall constraint and the Reynolds number for several plate orientations: (a) a = 90°, 75°, 60° 91 24) Hmax 1.0 0.8 0-6 0.4 0.6 0>4 0.3 0.2 • 1 • 1 • 1 ' 1 i i • i i - S c O = 353%^e<"~c> • - a = o 45 2CX5 — — 9 — - A --o 30 353 - B — b a—cy* 0 A A •' — — 9 AA-^f IS. — 2 0 . 5 ^ . > ' ' « .. i— 1 . i . i . I . I . I I 1 3 4 Figure 3-22 Dependence of the maximum fluctuating pressure on the wall constraint and the Reynolds number for several plate orientations: (b) a = 45°. 30° 92 3.3.3 Observations on the Influence of Three Dimensionality of the Flow and Vortex Formation on the Unsteady Surface Loading The three dimensionality of the flow and the mechanism of vortex formation are believed to a f f e c t the v a r i a t i o n of the f l u c t u a t i n g pressure i n t e n s i t y with the parameters of the present experiment, i . e . , the Reynolds number, blockage r a t i o , aspect r a t i o and the model o r i e n t a t i o n . The objective here i s to summarize the availa b l e information that would help explain the r e s u l t s of the present i n v e s t i g a t i o n . The flow i n the wind tunnel i s three-dimensional, to a c e r t a i n extent, at most wind speeds as the tapered f i l l e t e d corners compensate for the growth of the boundary layer at only one optimum speed. Moreover, the leakage of the high pressure a i r from the boundary layer at the ends of a model to i t s base, where the mean pressure i s negative, leads to addi t i o n a l spanwise flow. This three-dimensionality of the flow does not seem to a f f e c t the mean aerodynamic ch a r a c t e r i s -t i c s s u b s t a n t i a l l y , however, the unsteady character appears to be quite s e n s i t i v e to the l a t e r a l v a r i a t i o n s . A review of the three-dimensional structure of the wake was given by Gerrard together with his own in v e s t i g a t i o n , from which i t can be concluded that: (i) the vortex l i n e i s approximately s t r a i g h t and appears to t i l t backwards and forwards between the l i m i t s of i n c l i n a t i o n of ± 15° for a c i r c u l a r c y l i n d e r . (Similar findings are reported by 93 3 Dikshit for e l l i p t i c cylinders.) This t i l t i n g of the vortex line takes place at a frequency an order of magnitude lower than the fundamental frequency. (ii) Vortices do not shed at the same frequency over the span of the model due to variation of the effective speed caused by the three-dimensionality of the flow. ( i i i ) Vortices may be formed in loops due to this d i f f e r -ence i n frequency. The distance over which a loop forms i s directly related to the correlation length 63—65 reported by other investigators (iv) The wake flaps from side to side similar to a flag resulting in amplitude modulations of the fluctuating velocity, (v) Turbulent mixing in the wake, in the Reynolds number range of the present experiment, would lead to fluctuations of the vortex strength. Relevant to the current discussion i s the experimental investigation, by Drescher^, of the unsteady surface pressure and the wake of a circular cylinder i n the subcritical Reynolds number regime. It indicates that the maximum l i f t coefficient (and thus pressure) occurs when the wake swings out maximally to the opposite side. In other words, this correlates the amplitude modulations of the pressure signals to the flapping of the wake. 94 The above considerations only emphasize the complex character of the phenomenon. They merely suggest some of the important factors that may contribute to the modulations of the pressure signals, but i n no way they can explain, as yet, the details of the modulation spectrum. For both the circular cylinder and the f l a t plate, the intensity of C-, showed less dependency on the Reynolds number at the higher blockage ratios. As pointed out before, associated with an increase in S/C i s a corresponding reduction of the aspect ratio (Tables 2 - 1 , 2-2). With this reduction one would expect a better correlation of the fluctuating pressure signals over the span of the model^^". Thus i t may be concluded that with the reduction of the la t e r a l effects, variations of the pressure with the Reynolds number become less pronounced. It should be recognized that the vortex strength and position represent important factors governing the intensity of the fluctuating pressure. Measurements of the position of the f i r s t vortex, for the circular cylinder, are published by Bloor and G e r r a r d ^ ' ^ . The results suggest a forward s h i f t of the f i r s t vortex with an increase of the Reynolds 4 4 number in the range 10 - 2 x 10 . This would dxrectly lead to a corresponding increase in the fluctuating surface pressure. On the other hand, there does not appear to be any direct method for measuring the strength of turbulent vortices i n the wake i n the Reynolds number range of the present 59 59 experiment . Nevertheless, Bloor and Gerrard have presented an empirical relation--based on the model of t r a i l i n g v o r t i c e s 70 by Hoffmann and Jouber — t o determine the vortex strength by matching the measured unsteady v e l o c i t y i n the wake. They found the vortex strength to increase with the Reynolds 4 4 number i n the range 10 - 2 x 10 . I t i s i n t e r e s t i n g to note that t h i s agrees with the increase of C-, , for both the p max c i r c u l a r c y l i n d e r and the plate, as recorded i n the present experiment. 3.4 Wake Geometry Three regions could be i d e n t i f i e d i n the wake of a b l u f f body: (i) the near wake, where the separated shear layers undergo t r a n s i t i o n to turbulence before r o l l i n g into turbulent v o r t i c e s ; ( i i ) the intermediate region of the organized Karman vortex s t r e e t ; ( i i i ) the far wake, where the vortices d i f f u s e and d i s s i -pate forming a turbulent f i e l d . Much of the ava i l a b l e l i t e r a t u r e i s concerned with the l a s t 71 region . In the current i n v e s t i g a t i o n the i n t e r e s t i s con-fined to (i) and ( i i ) . In the near wake, the length a f t e r which transion to turbulence occurs and the appearance of the f i r s t vortex (the formation length) are given as functions of the Reynolds number in references 60 and 59, respectively. The transition length has i t s importance in the determination of the vortex strength while the formation length i s necessary for predicting the unsteady pressure intensity. Investigations related to the v o r t i c i t y content of the separated shear layer, the mechanism of vortex formation and vortex strength were under-55 54 taken by Fage and Johansen , Abernathy and Kronauer , 58 59 72 Gerrard , Bloor and Gerrard , Wood , et a l . It was found that the strengths of the two separated shear layers are equal for even unsymmetrical bodies and proportional to the base pressure parameter, K. For a symmetrical body, there i s a loss of about 35% of the available v o r t i c i t y during formation of a vortex from the separating shear layer. Several length parameters associated with the near wake have been found important in predicting the shedding frequency. The normal distance between the separated shear layers before they r o l l into vortices, h', was proposed by 49 Roshko and was examined at length in section 3.2.3. The others have been the formation length and transverse distance 58 between the separation points suggested by Gerrard and 3 Dikshit , respectively. In the intermediate region, the vortices are organized in a constant longitudinal spacing, a, and the increasing lateral spacing, b, as they travel downstream. The geometry of a turbulent vortex as well as i t s velocity f i e l d are not yet clearly defined, however, occurrence of the maximum 97 fluctuating pressure near i t s center has been established with 73 74 a f a i r degree of accuracy ' . The geometry of the vortex street in this intermediate region has i t s effect on the 54 75 76 formation of the turbulent vortex , the strouhal number ' 19 75 and the mean drag coefficient ' . The lateral spacing being varying downstream, 'a/h' represents a better wake characteris-77 t i c parameter compared to the cla s s i c a l ratio b/a. 3.4.1 Circular Cylinder Using the procedure described in section 2.4.4/wake geometry data were obtained at the Reynolds number of: 4 (i) 3 x 10 for blockage ratios 3 and 4.5%; (i i ) 5 x 10 4 i n the blockage ratio range 6-20.5%; 4 ( i i i ) 10 x 10 for blockage ratios 26.5 and 35.5%. The choice of the specific Reynolds number was dictated by the poor sensitivity of the Barocel at high frequency and the pressure decay in the wake. Figure 3-23 shows the lateral variation of the fluctu-ating pressure amplitudes at several downstream stations. Due to the symmetry of the models, the unsteady pressure plots are symmetrical about the wake centerline. In the downstream direction, the pressure intensity diminishes due to diffusion and dissipation of vorti c i t y , and the pressure plots tend to become f l a t . The figure indicates no significant effect of blockage. From the lateral pressure distribution results similar to those in Figure 3-23, but obtained at finer interval in 98 (a) Figure 3-23 Lateral variation of the fluctuating pressure amplitude i n the wake of circular cylinders of blockage ratios: (a) S/C = 3, 4.5, 6 and 9% Figure 3-23 Lateral variation of the fluctuating pressure amplitude in the wake of circular cylinders of blockage ratios: (b) S/C = 14.8, 20.5, 26.5 and 35.5% 100 the downstream direction, the decay of the peak pressure amplitude was obtained as shown i n Figure 3-24. It i s of i n -terest to note that the results are substantially independent of the blockage ratio. The scatter of data i n the near wake may be attributed to the inherently unsteady character of the formation region. The presence of the probe would only tend to enhance the irregularities through interference. The two peaks i n the near wake may be associated with the formation and shedding of the f i r s t vortex. The fluctuating pressure there after decays exponentially. The amplitude of the fluctuating pressure as represented i n Figures 3-23 and 3-24 i s relative to that of the absolute peak pressure i n the wake, p* c c ' rw,max The ratio of p* to *w ,max the maximum fluctuating pressure on the model surface, p m m a x# are l i s t e d i n Table 3-3. The table indicates no significant dependence on the blockage. Thus one would anticipate the fluctuating pressure in the wake to follow the same trend as that of the surface pressure. Table 3-3 Maximum Fluctuating Pressure in the Wake of Circular Cylinders S/C 3 4.5 6 9 14.8 20.5 26.5 35.5% RxlO - 4 3 3 5 5 5 5 10 10 rw,max — i %,max 0.77 0.78 0.745 0.73 0.76 0.74 0.78 0.75 101 T 1 1 1 1 J 1 1 1 1 1 1 • 1 « 1 1 1 r _ «6 on a Q o o X o n (a) o v o A A 9 9 j I . I . I . I i I i L i 1 1 I i I. . 0.2 0.4 0.6 0.8 1.0 U M H3 0 5 2 . 0 x/d f- , 1 i • i • i —i 1 i — r —1— 1 ' -1— s c V A 3% _ Q O 6 • 14.8 8 • 20.5 A 8 A V 35.5 H A (b) a A A O t A A -O o A . i . 1 i 1 • 1 . 1 . 1 i 1 1 v . q 8 10 12 14 16 18 20 X/d Figure 3-24 Decay of the peak fluctuating pressure i n the wake of the confined circular cylinders: (a) near wake; (b) intermediate wake 102 Figure 3-25 shows the variation of the late r a l spacing obtained by considering location of the peak fluctuating pressure as an approximate position of the vortex core. 73 However, as pointed out by Hooker , the maximum velocity fluctuations (and hence pressure variations) do not occur along the path of vortex centers as some experimenters have asserted but rather develop in the neighborhood of the core farthest from the street centerline. Thus, the distance between the peaks as presented here would overestimate the wake width by an amount equal to the diameter of the core. 74 Application of Schafer and Eskinazi's mathematical model for vortex street i n a viscous f l u i d showed correction i n the wake width to be negligible (< 1%) at the model and less than 5% twenty diameters downstream. It may be pointed out, how-ever, that the correction f a i l s far downstream as i t does not account for turbulence or wake i n s t a b i l i t y which would influence the position and growth of the vortex core. From Figure 3-25 i t i s apparent that, within the experimental accuracy, there i s no variation of the dimension-less l a t e r a l spacing b/d with the blockage ratio up to 35.5%, the largest model tested. The length of the test-section downstream of the model support was about 40 inches which limited the wake traverse to x/d as given in Table 2-1 and indicated on the plot. The results show that the vortices always move away from the wake centerline with increasing x. This observation agrees with the experimental measurements 103 50 2 of Fage and Johansen for a f l a t plate and of Slater for an angle section. It may be pointed out that such detailed measurements of wake geometry are not reported for a circular cylinder in the subcritical range investigated. The results obtained by Wille and Timme80 (R = 200) and Tyler 8(R = 4900) are included i n Figure 3-25 for comparison. Both results show a f a i r agreement with the present experiment. However/ i t should be noted that Wille and Timme's results are for the stable laminar vortex street, while Tyler's and the present data are for the periodic turbulent wakes. Reference should 79 also be made here to Kovasznay's measurement of the lateral spacing i n the wake of a circular cylinder at the low Reynolds number of 56. Although the characteristic of the wake was the same as that during Wille and Timme's experiment the deviation in the magnitude of b/d i s substantial. Neverthe-less, both the results follow the same trend. The results on phase distribution i n the wake of the circular cylinders are presented in Figure 3-26. The variations remained the same for a l l models indicating the phase to be independent of the constraint. The maximum scatter of data about the plotted average value was observed to be less than 20° for the entire range of bluffness covered. The phase variation increased rapidly up to 5 diameters downstream then became linear indicating that the vortices to have reached a uniform streaming velocity, and thereby a constant longitudin-al spacing. Using the fact that a phase difference of 360° Figure 3-25 Growth of the wake in the downstream direction and i t s independence of the cylinder blockage 3TT w i 1 | — r ~ — i r 2TT TT -i \ 1 s r Position of the blockage ratio indicates extent of the traverse for the given cylinder J i I i I I I I T — r 10 12 x/d 14 ' » i • ' 16 18 Figure 3-26 Longitudinal variation of phase angle in the wake of circular cylinders under constraint o 106 e x i s t s between the f l u c t u a t i n g pressure signals associated with consecutive v o r t i c e s i n one row, the l o n g i t u d i n a l spacing was obtained as shown i n Figure 3-27a. In the uniform v e l o c i t y region,a/d was 4.37 which compares favorably with Fage and 55 Johansen's value of 4.27 for the c i r c u l a r c y l i n d e r of blockage r a t i o 3.5% at R = 3 x 10 4. The r e s u l t s of T y l e r 7 8 , and Wille 80 and Timme also show a good agreement. Since a/d i s independent of the constraint the v a r i a t i o n of the dimensionless vortex v e l o c i t y , i s i d e n t i c a l to that of the Strouhal number. Combining the r e s u l t s of the l a t e r a l and the l o n g i t u d i n -a l spacings, gives the v a r i a t i o n of the spacing r a t i o b/a, Figure 3-27b. I n i t i a l l y the r a t i o decreases with the downstream distance reaching a minimum at about 4 diameters, followed by the monotonic increase exceeding Kantian's value of 0.281. Of p a r t i c u l a r i n t e r e s t i s the close agreement between the current r e s u l t s and those by Wille and Timme i n s p i t e of a wide difference i n t h e i r Reynolds number regimes. This may suggest r e l a t i v e i n s e n s i t i v i t y of the parameter to the Reynolds number, although a systematic study i s e s s e n t i a l to confirm t h i s observation. o (3.3) 107 Figure 3-27 Streamwise variation of: (a) longitudinal vortex spacing; (b) wake geometry ratio for circular cylinders 108 3.4.2 Flat Plate Similar to the circular cylinder case, the fluctuating pressure traverses at several downstream stations were carried out, for each model, at a = 30°, 60°, 90°. Generally, due to the diffusion and dissipation of vo r t i c i t y i n the downstream direction, the pressure intensity diminishes exponentially. This leads to the flattening of the lateral variation of the fluctuating pressure plots (Figure 3-28). Due to symmetry of the model at the 90° attitude, pressure traverses are symmetrical about the wake centerline, which coincides with the geometrical centerline of the tunnel. For an unsymmetrical model orientation, corresponding asymmetry i n the wake i s expected. However, at a given station the resulting difference in the unsteady pressures i s unusually high, particularly at the lower angles of attack (Figure 3-28, a = 60°, 30°). The wake pressure was observed to be higher on the t r a i l i n g edge side as was the surface pressure (section 3.3.2). This difference in the pressure may be attributed to unequal vortex strengths. I t i s known that the vorticity content of the two separated shear layers i s the same at any 55 model orientation . However, i t appears that the layer separated from the leading edge undergoes a greater loss in vor t i c i t y before r o l l i n g into a vortex. Increased width of 48 this layer and the longer distance i t travels during the formation results in larger area for convection of vorticity to the outer flow. The resulting weaker vortex accordingly Figure 3-28 Fluctuating pressure traverses in the wake of inclined f l a t plates of blockage ratios: (a) S/C =4.5% 113 leads to a lower fluctuating pressure f i e l d . A direct measure-ment of the vortex strength i s necessary to support this argument. The downstream decay of the wake fluctuating pressure i s given as a function of the model attitude and blockage ratio i n Figure 3-29. The pressure rises to a peak about one chord length downstream/ close to the location of the f i r s t vortex, followed by a progressive decay due to dissipation of vorticity. The narrow wake at lower angles of incidence promotes greater interaction between shedding vortices leading to a rapid development of a turbulent f i e l d as indicated by a sharp drop i n the fluctuating wake pressure. The pressure spectrum becomes relatively f l a t and the fundamental frequency i s not as well defined as that i n the case of a higher a. As against a circular cylinder, for which v i r t u a l l y no constraint effect was observed for S/C as large as 35.5%, the plate results show a slight dependence of the pressure decay on blockage, particularly at higher S/C, due to a comparatively wider wake. Position of the vortex rows as affected by plate incidence i s presented i n Figure 3-30. The plots indicate a continuous movement of vortices away from the wake center-line defined by the position of the minimum unsteady pressure. As can be anticipated, the centerline of the unsymmetrical wake moves i n the direction of the weaker vortex. The actual movement i s probably governed by, besides other factors, the ™ 1 1 "T T 1 1 1 —I •s ~1 1.0 a 9 0 ° c -0.8 o 4.5% o a 9 A 9 0.6 2 o o • 205 -o • 35.5 0.4 ° 6 6 0.2 1.0 _ o 60 -o • o 0.8 o o « 0.6 o . o A O -$ o A 0.4 — • 6 6-0.2 6 -1.0 o • _ o 30 • A 0.8 — • O o A 0.6 — 0.4 - * o 2 o Q e fl o -0.2 - o • • • A 1 1 1 1 « I • 1 0 1 2 3 4 5 6 7 8 9 10 x/h Figure 3-29 Downstream decay of the unsteady pressure in the wake of inclined f l a t plates under con-straint 115 general skewed character of the flow together with the unequal strength of the vortices on the two sides. Considering the amount of uncertainty inherent in the measurement of the later a l spacing and the scatter of data (Figure 3-31)/ the effect of constraint appears to be negligible for blockage ratios as high as 20.5%. This concurs with Abernathy's observation concerning the spacing between shear layers as affected by the constraint. However, the wall confinement effect begins to be noticeable for the largest plate tested (S/C = 35.5%), i n the near wake where the width appears to be di s t i n c t l y smaller for a = 60°, 90°. It i s anticipated that the proximity of the walls would affect the expansion of the vortex rows. In the present experiment with circular cylinders, blockage up to 35.5% showed v i r t u a l l y no effect on lateral spacing. On the other 20 hand, investigation by Rosenhead and Schwabe , in the Reynolds number range of 50-800, indicated a substantial reduction i n the wake width at S/C = 66%. The picture i s expected to be somewhat different with f l a t plates where the bluffness effect would be f e l t even at a lower blockage due to a wider wake (b/h for a plate i s approximately 1.6 times b/d for a circular cylinder). Thus although details of the wake measurements for S/C = 35.5% are limited, the observed trend i s indeed consistent with the anticipated constraint effect. Values of the universal Strouhal number given i n Tables 3-1 and 3-2 further substantiate this observation. It i s of interest to Figure 3-30 Streamwise position of the vortex rows in the wake of inclined f l a t plates of different blockage ratios 1 1 T 1 .... | I 1 1 2.4 a = 90° o 2.0 A A O 1.6 _ e • AO _ 1.2 S o • O A o oc i & <* <» ° • • Fage & Johansen^ -0.8 -• o J L = 7.15%. R = 3.4x10* C -t 2.0 60° 1.6 o s — 1.2 -A O O * * 8 o % c o 4.5 A 9 • 205 -0.8 O 35.5 0.8 3 0 ° 6 ft © O — 0.4 1 1 1 1 1 I i i i 0 1 2 3 4 5 6 7 8 9 1( x/h Figure 3-31 Lateral spacing of vortices in the wake as affected by plate orientation and blockage ratio 118 recognize that the l a t e r a l spacing becomes considerably indepen-dent of the plate i n c l i n a t i o n i f i t i s nondimensionalized with respect to the projected width, h s i n a. As i n the case of the c i r c u l a r c y l i n d e r , the phase angle measurements were found to be e s s e n t i a l l y independent of the bluffness (Figure 3-32). The r e s u l t s showed considerably less scatter (maximum of about 20°) compared to that observed during the l a t e r a l vortex spacing study. The l o n g i t u d i n a l spacing of v o r t i c e s , Figure 3-33a, i s thus unaffected by the blockage r a t i o . The spacing increased r a p i d l y i n the f i r s t few chord lengths downstream reaching a constant value. In the uniform spacing region, a/h equal 5.15, 4.5 and 2.55 at angles of 90°, 60° and 30°, r e s p e c t i v e l y . If based on the projected width, a/h s i n a w i l l have the corresponding values of 5.15, 5.2 and 5.1. Hence, i n the l i g h t of the previous observation concerning the l a t e r a l spacing, the wake geometry r a t i o , b/a, can be expected to remain e s s e n t i a l l y uniform with plate incidence a f t e r few chords downstream (Figure 3-33b). The figure shows the same trend as that observed during the c i r c u l a r c y l i n d e r study. T 1 1 1 1 1 1 1 1 1 1 1 r 1 i i i i i i i i i i » i i I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x/h Figure 3-32 Phase angle distribution in the wake of inclined f l a t plates ID Figure 3-33 Downstream variation of: (a) longitudinal vortex spacing; (b) wake geometry ratio for inclined f l a t plates 121 4 ANALYTICAL PROCEDURE FOR WALL CONFINEMENT CORRECTION This chapter aims at establishing and extending the range of v a l i d i t y of the several well known theories for blockage correction. Moreover, with the better understanding of the effect of wall confinement on bluff body aerodynamics gained through the experimental program, a potential flow model i s developed to predict mean surface loading under the con-straint. The va l i d i t y of the approach i s assessed through comparison with the experimental results. 4.1 Evaluation of the Reported Theories There are two clas s i c a l correction methods, due to 21 22 Glauert and Maskell , for the mean surface loading over bluff bodies. Both the methods are widely used and considered satisfactory for small blockage. However, at the higher blockage ratios, one has to depend on empirical formulae 24 developed for specific configurations such as rectangular 25 26 and square plates, and solid prisms . Due to a lack of any analytical correction procedure for the Strouhal number, 27 2 unsteady l i f t force and wake geometry, Vickery and Slater have suggested the use of an extension to Maskell's model. The approach was relatively successful in the low blockage ratio range of their experiments. 122 4.1.1 Glauert's Correction Method Glauert developed a cor r e c t i o n formula f o r the mean drag given by [ /• 0.822 A (S/C)2] [ /-7(S/CJ] -2 (4.1) where measured mean drag c o e f f i c i e n t corrected mean drag c o e f f i c i e n t shape f a c t o r , 1.0 f o r c i r c u l a r c y l i n d e r s and 0.5 f o r f l a t p l a t e s n. = empirical fa c t o r determined from experiment, 0.3 f o r c i r c u l a r c y l i n d e r s and 1.0 f o r normal f l a t p l a t e s . The f i r s t bracket i n the formula gives the s o l i d blockage g c o r r e c t i o n as evaluated by Lock using the image method, while the second bracket was introduced by Glauert to correct f o r the wake blockage e f f e c t . On applying t h i s formula to the experimental data for the c i r c u l a r cylinders, the r e s u l t s for blockage r a t i o up to 20.5% collapsed around an average value of 1.195 with the maximum deviation of ± 7% (Figure 4 - l a ) . To f i n d a value of n that would improve the agreement, equation (4-1) was f i t t e d to the experimental data. The value of n = 0.6 was found to r e s u l t i n a better collapse over the e n t i r e range of blockage 123 ratio. However, this was at the expense of decreasing the average drag coefficient to 1.1, as shown in Figure 4-lb. It should be noted that the expression giving the wake blockage effect was introduced by Glauert a r b i t r a r i l y and n was chosen to match Fage's** experimental results. Now an attempt was made to change Glauert's formula by including higher order terms in S/C with a hope to extend i t s valid i t y to the larger blockage. To this end, equation (4.1) was modified as follows, C 0 C0 [ I + 0.822 i(S/cff { I. , f | (S/CTlf . . . .(4.2) By varying the value of n and determining n through curve f i t t i n g of the experimental data, the best collapse was achieved with n = 3 and = 0.39. This results i n an average correct drag coefficient of 1.14 with maximum deviation of about ±5% for blockage ratios as high as 35.5%, as shown in Figure 4-lc. Application of Glauert's correction procedure to the normal f l a t plate showed a more favorable trend compared to the circular cylinder case. With the recommended value of n = 1.0, the experimental data showed a good collapse around the mean corrected drag coefficient of 1.76 with the maximum deviation of ± 5% for blockage ratios as high as 35.5%. 1 . 6 , — , — , — | — | — r u 1.2 1.0 1.4 12 1.0 1.4 1.2 1.0 n = 1 Tl = 0.3 _S_ ^ /s C O O ^ T • O 35.5% w v T • • 26.5 • D 2 C X 5 a D 0 8 o. • . l 4 > 3 J I 1 * « • o • • 6 o o 4.5 (a) A 3 n = 1 T\ - 0.6 cb B D a a a • • o (b) n = 3 Y] = 0.39 8 " • g (c) i ' ' L 6 8 10 12xl04 R Figure 4 -1 Application of Glauert's correction method to circular cylinders: (a) direct application of Glauert's formula; (b) use of experimentally determined n ; (c) modified expression for the drag 125 Further improvement was achieved with the value of n = 0.85 derived by curve f i t t i n g the present data r e s u l t i n g i n the average drag c o e f f i c i e n t of 1.88 (as against the true value of 1.96, Figure 4-11) with the reduced deviation of only ± 2.5%. 4.1.2 Ap p l i c a t i o n of Maskell's Theory 22 Maskell developed a theory f o r blockage c o r r e c t i o n based on momentum balance between the undisturbed flow upstream of the body and that downstream where the e f f e c t i v e wake reaches i t s maximum width, B. For a square plate normal to the flow, and assuming the pressure i n the wake to be uniform and equal to the base pressure, p^, the drag c o e f f i c i e n t was given by where m = B/S. By hypothesis, he derived an expression for the e f f e c t of blockage on the wake width as c (4.3) Co -C{ Dc S (4.4) C where the subscript c stands for corrected values. This gives the c o r r e c t i o n formula 126 2.2 -I n = i 1.0 1 -2.0 mm _ C D c 1.8 e • C D c = 1.76 -1.6 (a) • -n = 0.85 2.0 • C ° c = 1.88 -1.8 — • • -1.6 (b) -i i • 1 i ( ) 0.1 0.2 0.3 0.4 S/C Figure 4 -2 Correction of the mean drag for normal f l a t plates using Glauert's method: (a) n = 1, as determined by Glauert; (b) n = 0.94, as evaluated from the present data 127 2 where terms of order (S/C) were considered negligibly small. Now the correction for drag and pressure coefficients are directly given by i- C" = Ss. = j&L = % (4.6) For non-uniform base pressure, Maskell suggested the use of a mean value defined by ft = " f l - j p ( 4 - 7 With the integration performed over the base of the body and the surface of the effective wake. It should be mentioned here that the theory considers invariance of the separation point under constraint, thus Maskell doubted i t s applicability to well rounded bodies. The experimental results for the circular cylinders indicated some differences from Maskell's assumptions, the 2 main discrepancy being the variation of the ratio CD/K with the constraint. Also the pressure over the base was 128 not uniform, especially at the higher blockage ratios. On the other hand, the point of separation was found to remain unchanged under constraint thus suggesting i t s possible va l i d -i t y . Applying the conventional form of Maskell's theory with the base pressure taken as: (i) average over the separated flow region; or (ii) average over ± 30° around the 180° position; or ( i i i ) pressure at the 180° position showed the f i r s t and second choices to underestimate the mean drag coefficient with the third alternative giving closer results. Hence a l l the calculations were carried out using pressure at the 180° position as the base pressure (Figure 4-3a). It i s apparent that the theory i s able to account for the bluff-ness of around 5-10%. To obtain a correction procedure applicable to bodies with larger bluffness, i t was decided to modify Maskell's f i n a l correction relation, equation (4.5), by including the higher order terms i n S/C. The corrected results using this approach are shown in Figure 4-3b. The procedure leads to a distinct improvement in the correlation of C D for different blockage ratios. However, the corrected drag coefficient i s slightly underestimated value for the lower blockage ratios and i s overestimated at the higher bluffness. The theory being developed for the square plate con-figuration with uniform base pressure and fixed separation points, i t was anticipated to show better agreement in correct-129 12x10 Figure 4-3 Maskell's correction for circular cylinders: (a) direct application of the formula; (b) use of higher order terms 130 2 ing the measured f l a t plate data. However, the ratio Cp/K s t i l l showed variation over the blockage ratio range of the experiment. In the application of the theory the effective blockage ratio (S/C) N based on the projected area of inclined plate i s used i n accordance with Maskell's analysis. Applying directly the modified form of Maskell's expression, i.e. including the higher order terms i n (S/C) N, the corrected results obtained are given i n Figure (4-4). It i s observed that v a l i d i t y of the theory i s limited to the lower blockage ratio range, as the discrepancy tends to increase with the constraint, particularly at higher angles. 4.2 Empirical Correction Formulae Based on Maskell's correction formula, equation (4.5), curve f i t t i n g of the experimental results i n a polynomial of the form was thought to represent a satisfactory correction relation. The least square f i t approach was used in conjunction with a d i g i t a l computer to determine the order and coefficients of the polynomial that would result in the best collapse of the data. This approach was applied to correct the mean drag and fluctuating l i f t coefficients and the Strouhal number for 131 Figure 4-4 Corrected mean drag coefficient of inclined f l a t plates using the modified Maskell's formula 132 the circular cylinders (Figure 4-5). A straight line f i t was found to be the best for the three measured parameters. The correction relations are: (4.9) with the average C D , C-, and S, as 1.058, 0.672 and 0.191, c c c respectively. In the absence of a precise theory, these correction formulae may be used, with some confidence, to account for the bluffness effect. For the f l a t plate, the mean drag coefficient was also expressed by a polynomial f i t of the form (4.8). A straight line relation represented the variation of C D over the blockage ratio range of the experiment for a l l angles, where the effective blockage ratio (S/C) N was used in the polynomial. The correction formulae are given by Figure 4*5 Empirical correction formulae for circular cylinder aerodynamics (a) mean drag coefficient Figure 4-5 Empirical correction formulae for circular cylinder aerodynamics: * (b) unsteady l i f t coefficient 136 where the coefficient Aft and the average C D have the values as l i s t e d below: o a 30 45 60 75 90 A a 1.49 1 .6 1 .76 1 . 8 3 1 .69 0 . 5 1 .06 1 .42 1 .56 1 . 9 The possibility of arriving at a universal Strouhal number independent of the body shape and blockage ratio was discussed in section 3 . 2 . 3 . This provides an alternate approach to the correction of the Strouhal number data. The knowledge of the normal distance between the separated shear layers, h 1, i s essential here, nevertheless i t was found to be relatively unchanged over a wide range of the blockage ratio. Equation 3 . 1 leads to the relation where f v i s the vortex shedding frequency and V g represents the separation velocity. On evaluating the correct values of V , and h* i f i t i s changing, the correct frequency and thus the Strouhal number follow directly. 137 4.3 Free-Streamline Model A free-streamline approach was f i r s t proposed by Kelm-81 holz i n his c l a s s i c a l model i n which the wake of a b l u f f body was represented by a region of stagnant flow at the mean s t a t i c pressure pg, bounded by two free-streamlines represent-ing the separated shear l a y e r s . Using the model and applying the hodograph technique, K i r c h o f f analyzed the flow past a normal f l a t p l a t e . The predicted drag c o e f f i c i e n t was found 8 2 to be l e s s than i t s measured value. Roshko presented a more r e a l i s t i c model where the wake pressure was taken to be equal to that measured on the base. The predicted surface loading and drag f o r a normal f l a t p l a t e , a c i r c u l a r c y l i n d e r and a 90° wedge were found to be i n good agreement with the experi-48 mental data. Abernathy , successfully extended Roshko's approach to the case of an i n c l i n e d f l a t p l a t e . A d i f f e r e n t procedure was proposed by Parkinson and 83 Jandali i n which the flow was treated i n the complex plane r e s u l t i n g i n a simpler expression for the pressure d i s t r i b u t i o n on the body. In the transform plane, two wake sources create stagnation condition at the separation points which now become the c r i t i c a l points of the transformation. The p o s i t i o n and strength of the sources are determined by the condition of separation and the base pressure c o e f f i c i e n t . This method i s extended here to predict the mean surface loading on two dimensional b l u f f bodies under wall confinement. An exponential function maps the confining wall into the 138 axis of symmetry of the body. Using the value of the base pressure at a given blockage, i t i s now possible to evaluate the pressure distribution over a wide range of constrained conditions. The analytical results are compared with the experimental measurement up to the blockage ratio of 35.5%. 4.3.1 Analytical Development Consider a two-dimensional body (in the complex physical plane z) with i t s axis of symmetry parallel to the uniform flow V Q and located midway between the two parallel i n f i n i t e walls, distance H apart, as shown in Figure 4-6. The x-axis is taken on the centerline while the y-axis passes through the separation points S^, S^. The flow over the body i s considered potential upstream of S^, S 2 and outside the separating shear layers which become par a l l e l , distance h^ apart, at i n f i n i t y . Consistent with the free-streamline approach, the pres-sure in the wake, p^, i s taken to be constant and the velocity at the separation i s thus given by K V Q , where K represents 1/2 the base pressure parameter, (1-Cr, ) ' . The transformation (4.12) maps the two walls on the u-axis in the intermediate w-plane and reduces the uniform flow to the source M, of strength V „ H , at the o r i g i n (Figure 4 - 6 ) . Due to the p o t e n t i a l flow consider-ation, portion S ^ D S 2 of the body downstream of the separation i s ignored. The upstream section, i n general, would be transformed into a symmetrical arc with free-streamlines remaining tangential. The y-axis i s mapped into the u n i t c i r c l e and the separation points make an angle ±u = TTh/H with the negative u-axis. The condition of zero normal v e l o c i t y at the walls i s s a t i s f i e d . I d e a l l y one would l i k e to transform the arc A S 2 B S ^ A into a c i r c l e keeping the source and the walls along the axis of symmetry. Although i n general t h i s may not be possible, there are several s i t u a t i o n s i n which i t can be achieved exactly (normal f l a t p l a t e ) , or with a f a i r degree of accuracy ( c i r c u l a r c y l i n d e r ) . Consider the transformation = 3 C C O ) (4.13) which can accomplish t h i s . Following the procedure of Parkinson 83 and Jandali the arc can now be mapped into a c i r c l e y, of radius r ^ , i n the £ - plane (Figure 4 - 7 ) , where and S 2 are made the c r i t i c a l points of the transformation of which f'(?) has simple zeros. Accordingly, angles of i n t e r s e c t i o n are halved at these points i n £ - plane and the free-streamlines become normal to the surface of the c i r c l e . The source M appears at a distance £ from the o r i g i n . The o v e r a l l trans-formation function between z and £ planes i s given by Figure 4 - 6 Representation of a two-dimensional symmetrical body i n : (a) the physical plane; (b) the intermediate transform plane 141 g = (H/2 7T) L f(<;) . « ( (4.14) In £ - plane, the flow past the c i r c l e y in the presence of the source M i s satisfied by adding an equal source M* at the image point ^ and a corresponding sink at the origin. A pair of double sources of strength 2Q, symmetrically located on the c i r c l e at angle ± 6, and their image sinks at the origin are combined with the above singularities. The complex potential of the resulting flow thus becomes . . . .(4.15) and the corresponding complex velocity i s given by Figure 4-7 Mapping of the body and the singularities walls in the £ - plane, and the position of 143 where q = ( 2 Q/HV Q) and r, = r e i o . To make the free-stream-lines normal to the surface at the points of separation the source strength Q and the angle 6 are adjusted i n such a way as to make S^, S 2 stagnation points. By setting W ( CJ H = 0 (4.17) at these points, a relation between Q and <5 i s obtained. As the velocity in the physical plane i s W(3) = ( 4 . 1 8 ) a measured value of C or the separation velocity Pb M3)/s s ft = f l . C , f for a given blockage ratio (S/C)^ can now be used to evaluate Q and 6. The surface loading for this ratio can then be determined using the routine procedure. For extending the method to other blockage ratios i n -cluding the body in an i n f i n i t e stream, an additional relation i s required to evaluate the unknown cp^. Geometric similarity between wakes under constraint as found during the experimental program, section 4 . 4 , suggested the choice of a constant length parameter for this purpose. The analysis was conducted by fixing: 144 (i) hm/h representing the relative separation at i n f i n i t y between the free streamlines; or (ii) the source angular position 6 in ? - plane; or ( i i i ) the relative positions of the sources, y (6)/h, in the physical plane. The third condition seemed to give the best agreement with the experimental results for both the cases of a normal f l a t plate and a circular cylinder. With this, the source position <5 can be derived for any blockage ratio once i t i s known for (S/C^. Now the strength Q i s obtained from equation (4.17) and the base pressure coefficient follows using (4.18). The pressure distribution on the surface of the body upstream of separation i s given by Bernoulli's equation with the base pressure coefficient considered constant. Inte-grating the surface loading yields the drag coefficient as (4.20) h/2 C, 0 (4.21) with no contribution to the drag from parts of the body above S~ or below S,. The shape of the free-streamlines i s given by the solution of (4.22) with the reference streamline = 0 fixed to the walls, Figure (4-6). Their position in the physical plane i s obtained using (4.14) and the pressure distribution along these streamlines i s found from (4.20). The downstream velocity at i n f i n i t y , i s obtained through continuity as and the spacing h between the free-streamlines i s given by two-dimensional bluff body once the transformation (4.13) i s defined. Here i t i s applied to a normal f l a t plate and a circular cylinder representing extreme cases of sharp edge and boundary layer separations. The main limitation of the model i s the concept of a f i n i t e wake extending to i n f i n i t y . In r e a l i t y , i n s t a b i l i t y of the shear layers, represented here by free-streamlines, lead to the formation of vortices. V = V J. 2 Q H (4.23) (4.24) In principle, the model i s valid for any symmetrical 146 4.3.2 Application of the Theory 4.3.2.1 Normal Flat Plate Consider a f l a t plate of blockage ratio h/H placed normal to the flow between two parallel walls as shown in Figure (4-8a) . Using the transformation (4 .12), the plate i s represented by an arc of the unit c i r c l e (Figure 4-8b) in the co-plane. Next, the plate and the walls are mapped into the c i r c l e y and 5-axis (Figure 4-7) respectively, with the overall transformation as . (4.25) where, on the plate, (4.26) The separation points are symmetrically located at ±3 = (IT/2) (1-h/H) in £ - plane, with the sources M and M' at , = cot p 'm m and the radius of the c i r c l e as r^ = cosec 3 . The complex velocity W(C) on the c i r c l e thus becomes Figure 4-8 (a) Flat plate geometry; (b) Mapping of the f l a t plate into a circular arc 148 W f « ? ) r i J£ sinp smo- [" V0 2Costs', Cos p-sec p 1 ] (4.27) 2 cos <r - ? c*os S Equating (4.27) to zero at a = ±3 to make S^, S 2 stagnation points gives <j 2 C*S (4.28) Sec p - Cos p leading to the velocity in the physical plane as W(z) r 4 s , „ <r CosA -2Co$%-> seep V0 (Cos C-Cos <&)(<5ecJ3 -Cosp) (4.29) Applying equation (4.19) gives If = 2 cot p Sin £ Cos p - CoS 5 (4.30) The measured value of at (S/C)^ i s used in equation (4.30) to obtain 6 ^ and the position of the source in the physical plane, y ( < 5 ^ ) , i s thus obtained from (4.26). The angular position 6 for any arbitrary blockage ratio i s then determined from condition ( i i i ) . The relative source strength q and the base pressure parameter K follow from (4.28) and (4.30). The pressure distribution on the front of the plate thus becomes 149 CK ( <r) r I _ ( K P-Co^ )2 ( 4 > 3 1 ) with y ( a ) given by (4.26). The drag coefficient as obtained from (4.21) leads to a rather lengthy expression and hence not presented here. The free-streamlines are obtained by solving (4.22) through iteration, and the relative distance h^/h between them at i n f i n i t y i s given by i hluJJ u-32) 4.3.2.1 Circular Cylinder For a circular cylinder with blockage ratio d/H (Figure 4-9a), the separation angle 9 , assumed known, is s considered unaffected by the constraint. This i s in f a i r agreement with the experimental results described in section 3.1.1. The surface upstream of the separation points is mapped into a pseudo-circular arc in the w - plane, by transformation (4.12), as shown in Figure 4-9b. One can 84 transform this into a circular arc using Theoderson's approximate approach, however, here i t i s represented by 150 the c i r c u l a r arc passing through the points A, Si and S 2 with error < 0.8% i n r a d i a l distance f o r blockage r a t i o s up to 40%. This e f f e c t i v e l y avoids the complication of numerical evalu-a t i o n inherent i n Theodersen's analysis without a f f e c t i n g the accuracy. The radius of the c i r c u l a r arc i s given by r / = * » * ( ^ ) * - J ? ( 4 . 3 3 ) H I J/ where and the angle <(> i s obtained from Sin <fi = ± <5in (J£L \ (4.34) Using the procedure s i m i l a r to that i n the case of the pl a t e , the arc i s mapped into the c i r c l e y« The o v e r a l l transformation function has the form ^ = J L L \± Sin ( mi) f c -f c°l P - 1 1 . Cos 7£h.} . ±E 0 27T ( 2 P ^ + C o t p J H ] 8 (4.35) 0 0 0 0 where B = ( T T - < { > ) / 2 , with sources M and M1 at Figure 4-9 (a) Circular cylinder in confined flow; (b) Trans-formation of the cylinder into a pseudo-circular arc 152 <~ = cot (2£k \ - e*l yB , • cr = G>S«: A5 •2 Angles 6 and a i n the two planes are related by 0 , flrclan J -where 3= & ' * A » W) -i 2 7 T p _ cjp (7Th/H) Cosec P sin ( / / Cos 6* Cos ? ) ' (cos ^ + Cos p ) z + sin2 & Cot(?±) - Coif 6*7 V 9 ^ (TTfl/r/J Cot P {Cos e1 -f Cos p ) z * <zinz ^ (4.36) The complex velocity i n C - plane i s given by 153 - < JZ e <5tn B S i n & 7T ' 4 Cos fl"- Cos <o) Cot (h/4)- Cotp 200* [Cot (hif4) -Cot pj + 2 cot(h/4) Cos? - CosecfS-(4.37) Cbt*(hl4)°>inP - Cosp C0t p with the relative source strength as H = 2 (Cos P + CosS) [tot {.h/4) -Cot P] Cos p Cot P - Co sec p - Cot Z(hl4) 3f>J3 (4.38) The velocity at the surface of the cylinder in the physical plane thus becomes v0 l -Zi6J C Sin 6" [ Cot p . Cot (h/4)]j ^ j (e% Cos P) [( e!a,+ Cos?) (Cot P -cot ty) y ie'^cosecp *n^~\ t (Cos - CosS) Cosec2 ( h/4) t 2(CoSCb+CosP)[cetLhl4)-G,tp]- Sinp Cosecz(n/4) •, * I 2 (Costs'* Cos?) [cot (/>/*}- Cotf>] - <5inp Cosec* (.h/4) . . . . .(4.39) 154 and the boundary condition (4.17) leads to ff r 2 [ f ^ *'» ( Wl) - Cot yS S/V)2 ( Cosec(hk) Siri'p j . . . .(4.40) The angular position 6 of the source for (S/C)^ and the arbitrary blockage ratios are evaluated as before using (4.40), (4.36) and condition ( i i i ) , following which the base pressure parameter i s determined readily. The pressure distribution over the wetted surface of the cylinder i s obtained from (4.20) and (4.39) with the angular position 9 given by (4.36). The expression for C^(o) being rather long, the drag coefficient i s evaluated by numerical integration of (4.21). The free-streamlines were obtained using iteration procedure as previously indicated. 4.3.3 Discussion of Results 4.3.31 Flat Plate The required specified value of the base pressure co-e f f i c i e n t appearing in the analysis i s taken to be -1.38 at (S/C)^ = 7.15% as measured by Fage and Johansen^. This f a c i l i t a t e s comparison of results as obtained by this model 155 with those of Roshko and Parkinson et a l . (Figure 4-10a). The present model being generalization of Parkinson and Jandali's approach to account for blockage, equation (4.31) degenerates to their i n f i n i t e stream case for $ = IT/2. I t may be pointed out that the results of Roshko, and Parkinson et a l . correspond to the blockage ratio of 7.15% and hence d i f f e r , though slightly, from the free flow condition. The pressure distribution results as given by the theory and the experiment are shown in Figure (4.10b). The agreement appears to be satisfactory (maximum error of 7% in Cp^ for the blockage ratio of 20.5%). The experimental results show the base pressure to be essentially uniform thus agreeing with the assumption used in the analysis. The constraint appears to have l i t t l e effect on the potential flow region, however, the base pressure i s substantially affected. Figure 4-11 gives a comparison between the calculated values of Cp^ and C D for blockage ratios up to 50% and the experimental data from the present investigation as well as those obtained by others. The good agreement as shown permits the prediction of parameter values for a plate in i n f i n i t e stream. The figure suggests the effect of wall confinement to substantially increase with the blockage ratio, which explains why several approximate formulae could account for this effect at smaller blockage but f a i l in the higher range. 1.0 0.8 0.4 0.0 C P 0.4 0.8 .1.2 .1 .6 0.0 T — 1 1 1 r (a) 0.1 Present theory 82 83 Roshko & Parkinson et al. Fage et al. , experiment 0.2 y/h 0.3 0.4 156 0.5 Figure 4-10 Pressure distribution over a f l a t plate: (a) comparison of the present theory with the reported analyses and experimental data for S/C = 7.15% 157 Figure 4-10 Pressure distribution over a f l a t plate: (b) comparison between results for different blockage ratios as given by the present theory and the experiment 158 Figure 4-11 Variation with blockage ratio of C and C n for a f l a t plate u 159 4.3.3.2 Circular Cylinder In general, application of a free-streamline model to the flow past a cricular cylinder presents several problems, primarily associated with separation and wake conditions. The base pressure, although may be assumed constant over the surface downstream of the separation points for low blockage ratios, shows a considerable decrease towards the 180° position at higher constraint, Figure 4-13. It continues to rise beyond separation reaching a peak at around 90° and decreases gradually thereafter. The discussion of the conditions at separation, given 85 by Woods , and Parkinson et a l . , suggests that positive i n f i n i t e a e i s { f i n i t e (4.41) negative i n f i n i t e depending on the magnitude of the base pressure coefficient C and the angle of separation. If (3C /38) 0 i s negative the solution is physically inadmissible because i t does not allow for separation to occur. Furthermore, i f i t i s negative i n f i n i t e , the free-streamline curvature at w i l l be i n f i n i t e and convex, as viewed from outside the wake, thus causing the separation-streamline to intersect the cylinder. 160 The separation angle 6g, approximately indicated by the position of the adverse pressure gradient region, was found to be essentially unaffected by the constraint, section 3.1.1. 8 was taken to be the accepted value of 82.5° i n the present analysis. With 9 g constant, there i s one c r i t i c a l value of C~. below which the solution becomes inadmissable pb according to (4.41). Using the Cp^ value of -1.0 corresponding to the blockage ratio of 4.5% as given by the present experimental data, the base pressure predictions were in close agreement with the measured data i n the low blockage ratio range, where Cp^ i s essentially uniform (Figure 4-12a). Even for the higher values of wall confinement, the analytical results corresponded well with the average pressure in the wake. This i s reflected in the favorable drag estimate up to as high as 35.5% blockage (Figure 4-12b). In general, the model gives a good representation of the surface loading at low blockage ratios (Figure 4.13a). However, i t shows several discrepancies, particularly, at the higher blockage. This i s anticipated i n lig h t of condition (4.41), which when examined numerically, gives the c r i t i c a l base pressure value of -2.271 corresponding to the blockage of 30% in this case. The pressure distribution shows a f a i r agreement in the potential flow region. The trends suggesting forward movement of the zero pressure coefficient points and increase in the favourable pressure gradient with blockage, Figure 4-12 Variation of Cp^ and C D with blockage ratio for a circular cylinder 162 are confirmed by the test data. However, the theory over-estimates the extent of the forward movement of the zero pressure coefficient points and underestimates the changes in the pressure gradient. The theory erroneously suggests some downstream movement of the minimum pressure point 6 m (=11° over the range of the blockage ratio 6 - 30%) thus shrinking the adverse pressure gradient region. In the limit, the minimum pressure point coincides with the separation point and results in a negative pressure gradient at separation leading to a physically inadmisable solution for S/C > 30%. The descrepancy appears to be associated with the potential character of the model. It i s of interest to mention here the effect of decrease of Cpk on the potential flow past a convex body (viewed from 85 the flow field) as predicted by Woods' model. It was concluded that a decrease of C 0 results in an increase of pb the positive pressure gradient in the wake leading to a downstream movement of the minimum pressure point, 9 m . This i s i n agreement with the above observation. Of course, the conclusions of both the models were contradicted by the experimental results where the minimum pressure point remained unaffected by the constraint at approximately 70°. 163 Figure 4-13 Comparison of the theoretical and experimental pressure distributions over circular cylinders: (a) S/C = 6 and 14.8% Figure 4-13 Comparison of the theoretical and experimental pressure distributions over circular cylinders: (b) S/C = 20.5, 26.5 and 35.5% 165 4.3.4 Concluding Comments The model treats the problem of wall confinement using the well developed free-streamline representation of the separated flow past two-dimensional b l u f f bodies. The know-ledge of the base pressure at any one blockage r a t i o i s a l l that i s needed for evaluation of the surface loading at d i f f e r e n t c onstraint. By making 3 = IT/2, the method i s able to provide a more r e a l i s t i c representation of the free flow condition, a mechanism not inherent i n other free-streamline theories. In p r i n c i p l e , the model could be applied to any two-dimensional symmetrical b l u f f body i f the transformation (4.13) i s determined. V a l i d i t y of the approach does not seem to be r e s t r i c t e d by the Reynolds number, however, i t was not possible to substantiate t h i s observation due to lack of recorded experimental data on constraint e f f e c t s i n d i f f e r e n t flow regimes. Application of the model to f l a t plates and c i r c u l a r cylinders points out i t s considerable p o t e n t i a l i n p r e d i c t i n g average base pressure and drag c o e f f i c i e n t , for the blockage r a t i o as large as 35.5%. However, d e t a i l s of the pressure d i s t r i b u t i o n show considerable discrepancy for c y l i n d e r s i n highly confined condition. 166 5 CLOSING COMMENTS 5.1 Concluding Remarks Based on the present investigation, conclusions concern-ing the aerodynamics of bluff bodies i n general and as affected by wall confinement may be summarized as follows: (i) For a bluff body under constraint, mean pressure coefficient over the surface upstream of the separation i s reduced only slightly compared to a substantial drop of the base pressure coefficient. As a result, variation of the mean drag coefficient i s essentially proportional to the changes i n the base pressure parameter. In general, i t appears that the constraint does not introduce any changes in the characteristics of the flow other than increasing the local velocity. However, when the corrected speed results in shifting the Reynolds number into the c r i t i c a l regime, the corresponding effect on separation becomes apparent. When the base of a bluff body extends into the wake, the base pressure exhibits a drop in the downstream direction. Under highly confined condition, this decay becomes quite significant. 167 (ii ) As i s well known, interference introduced between the shear layers (splitter plate) promotes their s t a b i l i t y . However, outside interference in the form of wall confinement does not appear to have such a stabilizing influence. The mechanism of the vortex formation i s hardly affected by the constraint. Thus the Strouhal number, when based on local length and velocity parameters associated with the formation region, showed insensitivity to either blockage or model orientation. The shear layers behave i n a l i k e manner in forming the vortices independent of the geometry of the gener-ators . The existence of a universal Strouhal number offers a possibility of correcting the measured vortex frequency under constraint, ( i i i ) The unsteady flow characteristic i s strongly a function of the Reynolds number and the three-dimen-sionality of the flow. Under wall confinement, the intensity of the fluctuating pressure over the surface of a body and in i t s wake increases. This i s partly due to an increase in the vortex strength which i s directly proportional to the base pressure parameter. On the other hand, the relative intensity and the phase angle between pressure signals at different 168 points do not show significant dependence on the blockage or the Reynolds number. This would imply unchanged geometrical details of the shear layers and vortex spacing and was substantiated by the direct measurements. It seems that for the unsymmetrical model orientations the vortex shed from the t r a i l i n g edge is stronger than the other due to reduced v o r t i c i t y convection to the outer flow. As a result, the fluctuating pressure on the model surface and in the wake near the t r a i l i n g edge i s higher. At low angles of attack the wake develops rapidly into a turbulent f i e l d leading to very low surface fluctu-ating pressure. On the surface of a body, the f i e l d of each vortex approximately extends over a region bounded by the stagnation points of the potential flow solution. The pressure signal, which i s in phase with the vortex closer to i t , attains a maximum value near the separation region, while the minimum occurs at the stagnation where the signal changes phase. With a decrease in the aspect ratio of a body, the three-dimensionality effects on the unsteady flow characteristics reduces and the correlation of the pressure signal improves. The dependency of the unsteady pressure on the Reynolds number also reduces. 169 (iv) The longitudinal spacing between vortices i n the wake remains unaffected by the constraint over the blockage ratio range of the experiment. It appears to represent a better wake parameter compared to the spacing ratio. In general, the lateral spacing increases i n the downstream direction as the vortices diffuse. It i s v i r t u a l l y independent of the constraint for a relatively narrow wake. However, as can be a n t i c i -pated, the spacing reduces due to the proximity of the walls when the wake becomes wide, (v) The two classical correction methods due to Glauert and Maskell have only a limited range of applicability confined to low blockage ratios. Modifying the value of the empirical parameter, n, appearing i n Glauert's procedure improves collapse of the data, particularly at the higher blockage. Also, inclusion of the higher order terms in S/C i n Maskell's theory leads to better correlation of the results; however, i t s t i l l remains far from being satisfactory. Polynomial representation of the experimental results leads to empirical correction formulae, which should prove useful i n rapid evaluation of the wall constraint effects. (vi) The potential flow model developed for predicting the mean surface loading accounting for wall confinement effects appears to be quite promising. 170 It can also predict the pressure distribution for a body i n an i n f i n i t e stream. Although applied here to the particular cases of normal f l a t plates and circular cylinders, i n principle, the approach i s valid for any two-dimensional bluff body and over a wide range of the Reynolds number. 5.2 Recommendation for Future Work The present investigation quite v i v i d l y brought home to the author our inadequate understanding concerning the fundamental character of the bluff body aerodynamics. The f i e l d offers almost inexhaustable p o s s i b i l i t i e s for further exploration. The parameters l i k e l y to have substantial effects on the aerodynamics of a bluff body are essentially i t s geometry and attitude, the aspect ratio for a nominally two-dimensional body, the Reynolds number and turbulence character of the free stream. For better insight into the phenomenon, i t i s important to plan investigations that w i l l involve systematic variations of these parameters. Broadly speaking the associated problems are classif i e d under the categories of: (a) basic research; (b) extension of the present work. Some of the more important problems in each category are l i s t e d below. 171 (a) Basic Research: (i) Logical f i r s t step would be an attempt at determining a theoretical model that would predict the mean flow past a circular cylinder including the decay of the base pressure towards 180° position, (ii) It i s believed that a key to better understanding of this complex phenomenon l i e s in clearer appreciation of the near wake. Experimental exploration leading to information concerning periodic movement of separation points, mean and unsteady velocity profiles of the separating shear layer and i t s transition to turbulence, mechanism of the f i r s t vortex formation together with the generation, dissipation, diffusion and retention of vo r t i c i t y , mass transfer, etc. would go a long way in attaining this objective. The flow visualization may prove to be a useful tool in such a study. ( i i i ) Any attempt at rational mathematical representation of a f u l l y formed vortex (to evaluate i t s strength) would require detailed measurements of the associated mean and unsteady velocity profiles. 37 (iv) As pointed out by Mair and Maull , a l l available theoretical developments have been for two-dimensional flows. On the other hand, most experimental investi-gations have revealed a degree of three-dimensionality. Furthermore, much of the current interest of bluff 172 body flows i s associated with the three-dimension-a l i t y of a vortex shedding from a body such as a circular cylinder, nominally two-dimensional. Hence the three-dimensional character of the flow past a bluff body and in the near wake as affected by the system parameters should receive more attention, (v) As emphasized by Rosenhead^, what i s badly needed i s a quantitative theoretical treatment of the problem which w i l l consolidate measured data. Although quite challenging, the insight gained through ( i i ) - ( i v ) should make the process relatively less painful. Any attempt at analytical modeling of the unsteady flow f i e l d associated with a bluff body should include spatial and time dependent character of the vortex strength together with the Reynolds number of the flow as a parameter, (vi) Serious efforts should be made at correlating the avail-able information for different bluff body geometries over ranges of R (and planned experimentation to f i l l the gaps) with a view to establish discernable trends concerning wake-body interaction. (b) Extension of the Present Work (i) In view of the increasing attention received by the bluff body behaviour in turbulent flows and i t s obvious relevance to situations of practical impor-173 tance, blockage corrections for turbulent streams should constitute a logical extension to the present study. Of considerable interest would be the confine-ment corrections associated with dynamical condition of the model. Extension of the present and the projected analyses to wider range of the Reynolds number i s also of significance, (ii) Some attention should also be directed towards wall confinement effects for a cluster of bodies, and in multiphase flows. The latter has far reaching implications i n , beside other areas, chemical and nuclear industries, ( i i i ) Any effort at improving the theoretical representation of the flow past a circular cylinder under highly confined condition to accurately predict the condition at separation and the decay of the base pressure i s l i k e l y to be quite rewarding. This should provide us with better appreciation as to the fundamental character of the flow and would help us evolve a r e a l i s t i c model for predicting time dependent characteristics. (iv) Accessment of the applicability of the present analytical model to different geometries should prove to be of some interest as i t would extend the scope of i t s usefulness. The problem essentially involves determination of suitable transformation functions which would map various geometries into c i r c l e s or pseudo-circles. More challenging problem would be an extension of the method to unsymmetrical geometries such as a f l a t plate at an arbitrary inclination to the free stream. BIBLIOGRAPHY 1. Pankhurst, R.C., and Holder, D.W., Wind Tunnel Technique, 1st ed., Pitman and Sons Limited, London, 1952, Chapter 8. 2. S l a t e r , J.E., "Aeroelastic I n s t a b i l i t y of a St r u c t u r a l Angle Section," Univ. of B r i t i s h Columbia, Ph.D. Thesis, March 1969. 3. Dikshit, A.K., "On the Unsteady Aerodynamics of Stationary E l l i p t i c Cylinders During Organised Wake Condition," Univ. of B r i t i s h Columbia, M.A. Sc. Thesis, July 1970. 4. A l l e n , H.J., and Vin c e n t i , W.G., "Wall Interference i n a Two-Dimensional-Flow Wind Tunnel, with Consideration of the E f f e c t of Compressibility," NACA, Report 782, 1944, pp. 155-184. 5. 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