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On the performance and wake aerodynamics of the Savonius wind turbine Fernando, Mahamarakkalage Saman Udaya Kumar 1987

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O N T H E P E R F O R M A N C E A N D W A K E A E R O D Y N A M I C S O F T H E S A V O N I U S W I N D T U R B I N E by M A H A M A R A K K A L A G E S A M A N U D A Y A K U M A R F E R N A N D O B.Sc.Eng.(Hons) , University of Peradeniya, Sri Lanka, 1980 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A July 1987 © Mahamarakkalage Saman Udaya K u m a r Fernando, 1987 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at The University of Br i t i sh Columbia . I agree that the library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of Br i t i sh Columbia 2324 M a i n M a l l Vancouver, Br i t i sh Columbia Canada V 6 T 1W5 July 1987 c i i i A B S T R A C T The objective of the thesis is to establish methodology for development of a wind turbine, simple in design and easy to maintain, for possible application in devel-oping countries. To that end the Savonius configuration is analyzed in detail both experimentally and analytically to lay a sound foundation for its performance eval-uation. Following a brief review of relevant significant contributions in the field (Chapter I), an extensive wind tunnel test-program using scale models is described which assesses the relative influence of system parameters such as blade geometry, gap-size, overlap, aspect ratio, Reynolds number, blockage, etc., on the rotor output. The parametric study leads to an opt imum configuration wi th an increase in efficiency by around 100% compared to the reported efficiency of « 12 — 15%. O f particular interest is the blockage correction procedure which is v i ta l for application of the wind tunnel results to a prototype design, and facilitates comparison of data obtained by investigators using different models and test facilities. W i t h the design and performance results in hand, Chapters III — V I focus attention on analytical approaches to complement the test procedure. Using the concept of a central vortex, substantiated by a flow visualization study, Chapter III develops a semi-empirical approach to predict the rotor performance using measured stationary blade pressure data. The objective here is to provide a simple yet reliable design tool which can replace dynamical testing wi th a significant saving in time, effort, and cost. The simple approach promises to be quite effective in predicting the rotor performance, even in the presence of blockage, and should prove useful at least in the preliminary design stages. Chapter I V describes in detail a relatively more sophisticated and rigorous Boundary Element Approach using the Discrete Vortex Mode l . The method at-tempts to represent the complex unsteady flow field with separating shear layers in a realistic fashion consistent with the available computational tools. Important steps in the numerical analysis of this challenging problem are discussed at some length in Chapter V and a performance evaluation algorithm established. Of con-siderable importance is the effect of computational parameters such as number of elements representing the rotor blade, time-step size, location of the nascent vor-tices, etc., on the accuracy of results and the associated cost. iv Results obtained using the Discrete Vortex Model are presented and discussed in Chapter V I , for both stationary as well as rotating Savonius configurations. A detailed parametric study provides fundamental information concerning the starting and dynamic torque time histories, power coefficient, evolution of the wake, Strouhal number, etc. A comparison wi th the flow visualization and wind tunnel test data (Chapter II) shows remarkable correlation suggesting considerable promise for the approach. The thesis ends with concluding remarks and a few suggestions concerning pos-sible future research in the area. V T A B L E O F C O N T E N T S Abstract i i i Lis t of Tables v i i i Lis t of Figures ix Nomenclature xiv Acknowledgement xv i i Chapter I I N T R O D U C T I O N 1 1.1 Development of W i n d Turbines 4 1.1.1 Betz maximum efficiency 4 1.1.2 Types of wind turbines 5 1.1.3 Vert ical axis wind turbines 6 1.2 The Savonius Turbine 7 1.2.1 Experimental work 7 1.2.2 Theoretical investigations 14 1.3 Scope of the Present Investigation 16 Chapter II O P T I M U M C O N F I G U R A T I O N S T U D I E S 19 2.1 Definition of the Blade Shape 19 2.2 Experimental Methodology 21 2.3 Experimental Setup 24 2.3.1 W i n d tunnel 24 2 3.2 Mode l fabrication and mounting arrangement 24 2.3.3 Measuring devices 28 2.3.4 Bearing friction calibration 30 2.3.5 Pressure distribution measurements 33 2.3.6 Start ing torque 35 2.4 Experimental Results 36 2.4.1 Gap-size 36 2.4.2 Blade overlap 40 vi 2.4.3 Blade aspect ratio 40 2.4.4 Blade shape factor pjq 43 2.4.5 Blade arc angle 47 2.4.6 Reynolds number 47 2.4.7 Blockage ratio 52 2.5 P r e s s u r e D i s t r i b u t i o n 55 2.6 S t a r t i n g T o r q u e 63 2.7 E r r o r E s t i m a t i o n 65 C h a p t e r I I I A S E M I - E M P I R I C A L A P P R O A C H F O R P E R F O R M A N C E P R E D I C T I O N 67 3.1 I n a d e q u a c y o f the Q u a s i - S t e a d y A p p r o a c h 67 3.2 M o d i f i e d Q u a s i - S t e a d y A p p r o a c h 69 3.2.1 Flow visualization study 71 3.2.2 Central vortex approach 73 3.2.3 Emperical relations 73 3.3 R e s u l t s a n d G e n e r a l R e m a r k s 79 C h a p t e r I V A M A T H E M A T I C A L M O D E L F O R R O T O R A N A L Y S I S 84 4.1 B o u n d a r y E l e m e n t M e t h o d ( B E M ) 85 4.2 M a t h e m a t i c a l F o r m u l a t i o n 87 4.2.1 Basic assumptions 87 4.2.2 Governing equations 87 4.2.3 Uniqueness of the solution 90 4.2.4 Indirect Boundary Element formulation 91 4.3 D i s c r e t e V o r t e x M e t h o d ( D V M ) 94 4.3.1 Model geometry 94 4.3.2 Flow model 97 4.3.3 Vortex shedding model 99 4.3.4 Pressure, torque and power 102 4.4 M o d e l l i n g o f t h e W i n d T u n n e l B l o c k a g e 103 V I I Chapter V N U M E R I C A L I N V E S T I G A T I O N 106 5.1 Formation of the System Matrices 106 5.1.1 Unconfined flow model 106 5.1.2 W i n d tunnel blockage model 108 5.2 Calculation of Loading on the Blade 109 5.2.1 Velocity distribution along the blade 109 5.2.2 Calculat ion of | f term 115 5.2.3 Evaluat ion of torque and power coefficient 118 5.3 Calculation Procedure 119 5.3.1 Computat ional algorithm 119 5.3.2 Computat ional parameters 121 5.4 Effect of Computational Parameters 123 5.4.1 Number of elements m 123 5.4.2 Vortex core radius (l imiting distance o) 127 5.4.3 Position of the nascent vortex ts 130 5.4.4 Balancing parameter n 130 5.4.5 Time-step size At 130 Chapter V I R E S U L T S A N D D I S C U S S I O N 138 6.1 Stationary Blade 139 6.2 Rotat ing Blade 170 6.2.1 Effect of tip-speed ratio A 171 6.2.2 Effect of p/q variation 193 6.3 Effect of W i n d Tunnel Blockage 193 6.3.1 Stationary blade 200 6.3.2 Rota t ing blade 204 Chapter VII C O N C L U D I N G R E M A R K S 211 B I B L I O G R A P H Y 216 vii i L I S T O F T A B L E S T A B L E Page 5.1 Effect of number of elements on the computational cost 1 2 7 5.2 Effect of time-step size on the computational cost 1 3 7 6.1 Effect of k on the computational cost 1 5 5 6.2 Strouhal number at different blade angles 1 6 8 6.3 Effect of K on the computational cost 1 7 1 ix L I S T O F F I G U R E S F I G U R E Page 1- 1 A n illustrative schematic diagram of a wind energy operated irrigation system 3 l-2a The Savonius configuration 8 1- 2b Geometry of the Bach rotor 8 2- 1 Blade geometry and the associated parameters 20 2-2 Boundary layer wind tunnel used in the test program 25 2-3 Mode l mounting frame and associated subassemblies 27 2-4 Hydraulic dynamometer 29 2-5 Modified hydraulic dynamometer 29 2-6 Test arrangement for bearing friction calibration 31 2-7 Bearing friction characteristic 32 2-8 Pressure tap locations and numbering scheme 34 2-9 A schematic diagram showing model of the single-stage Savo-nius rotor used in the gap-size and blade overlap studies 37 2- 10 Plots showing the effect of gap-size on power coefficient^/** = 11.5%, i E n = 4 x 10 5) 38 2-11 Effect of gap-size on peak power coefficient {b/d = 11.5%, Rn = 4 x 10 5) 39 2-12 Variat ion of the maximum power coefficient with percentage overlap (a/d= 0,Rn = 4 x 10 5) 41 2-13 Effect of the blade aspect ratio on the peak power coefficient (a/d = 0, b/d = 0,Rn=4x 10 5) 42 2-14 A typical model of the two-stage Savonius rotor used to study the effect of blade geometry parameter p/q 44 2-15 P o w e r - R P M characteristic of the two-stage model (p/q = 1.0), showing the effect of bearing friction loss 45 2-16 Variat ion of the uncorrected power coefficient with wind speed for the two-stage rotor (p/q — 1.6) 46 X F I G U R E Page 2-17 Bearing loss corrected characteristic curves at three different wind speeds for the two-stage (p/q — 1.6) model 48 2-18 Effect of the blade geometry parameter p/q on the power coefficient at a constant blockage 49 2-19 Variat ion of the peak power coefficient wi th the blade geometry parameter p/q at a constant blockage of 16.4% 50 2-20 Effect of the blade circular arc angle 6 on the power coefficient for the single-stage model (B = 10%,p/q = 0.2) 51 2-21 Effect of blockage on power output of the Savonius rotor 53 2-22a Effect of blockage on peak power coefficient for two different blade geometries 54 2-2 2b Effect of blockage on the tip-speed ratio at peak power for two different geometries 54 2-23a-f Surface pressure distribution over the Savonius rotor as affected by the blade orientation: /? = 0° — 150° 56 2- 24 Variat ion of the starting torque with angular position of the Savonius rotor (p/q = 0.2, B = 10%) 64 3- 1 A typical velocity triangle at the blade element i combining the effect of free stream velocity and blade rotation 68 3-2 Results of the quasi-steady analysis 70 3-3 A schematic diagram showing the towing tank facility and associated equipment used in the flow visualization study 72 3-4 A typical set of flow visualization pictures showing the presence of the central vortex filament 74 3-5 Velocity distribution due to the potential vortex 75 3-6 Identification of the modified quasi-steady approach parameter S2 as a function of p/q when F2 and blockage are kept constant 77 xi F I G U R E Page 3-7 Identification of the modified quasi-steady approach parameter F2 as a function of blockage ratio when 5 2 and p/q are kept constant 78 3-8 Empir ica l relationship between S2 and the geometric parameter p/q 80 3- 9 Empir ica l relationship between F2 and the blockage ratio B 81 3-10 Predictions of the modified quasi-steady approach compared to the experimental results 82 4- 1 General description of the problem 89 4-2 Simpified geometry of the Savonius rotor under investigation 95 4-3 Description of the problem parameters — .96 4- 4 Geometry for blockage modelling 105 5- 1 Uniformly distributed vortex 110 5-2 Velocity distribution for a uniformly distributed vortex sheet compared to that for a point vortex 112 5-3a Velocity at the centre of the ith element 114 5-3b Points at a radial distance t from the blade 114 5-4 The block diagram of the computation process 120 5-5 Time history of the torque coefficient as affected by the number of elements 125 5-6 Flow patterns after 60 time-steps as affected by the number of elements 126 5-7 Effect of the l imit ing distance parameter o on the time history of CT 128 5-8 Flow patterns after 60 time-steps as affected by a 129 5-9 T ime history of the torque coefficient for different nascent vortex positions 131 5-10 F low patterns after 60 time-steps for different ts 132 5-11 Var ia t ion of the torque coefficient as affected by 77 133 5-12 Flow patterns after 60 time-steps for different 77 134 X I I F I G U R E Page 5-13 T ime history of CT as affected by the time-step size 135 5- 14 Flow patterns at t = 0.792 s for different time-step sizes 136 6- 1 Time history of the torque coefficient at 0 — 0° 141 6-2 Flow patterns at 0 = 0 ° : k - 30,60,90 142 6-3 Time history of CT at 0 - 30° 143 6-4 F low patterns at 0 = 30° : k = 30 - 150 145 6-5 T ime history of CT at• 0 - 60° 147 6-6 Flow patterns at 0 = 60° : ifc = 10 — 90 148 6-7 T ime history of CT at 0 = 90° 151 6-8 Flow patterns at 0 = 90° : k = 10 - 150 152 6-9 Time history of CT at 0 = 120° 157 6-10 Flow patterns at 0 = 120°: k = 40 - 200 158 6-11 Time history of CT at 0 = 150° 162 6-12 Flow geometry at the blade orientation of 0 = 150° 163 6-13 Flow patterns at 0 = 150°: k = 10 - 200 164 6-14 Variat ion of the average starting torque as affected by the blade orientation 169 6-15 T ime history of C? at A = 0.4 as affected by the resolution parameter K 172 6-16 T ime history of CP at A = 0.4 as affected by K 173 6-17 Flow patterns after k = 70,90,110, time-steps at A = 0.4 with K = 1 174 6-18 Flow patterns after k = 35,45,55, time-steps at A = 0.4 with K = 2 175 6-19 Evolut ion of the flow pattern for A = 0.4 and its compari-sion wi th the flow visualization study: k = 10 — 150 177 6-20 F low patterns after k = 20 - 100 time-steps: A '= 0.4,0.8,1.2; p/q = 0.2; K = 2 184 6-21 T ime history of CT at A = 0.8 189 6-22 T ime history of CT at A = 1.2 190 F I G U R E Page 6-23 Theoretically predicted variation of the average torque coefficient as affected by the tip-speed ratio A for p/q = 0.2 191 6-24 Theoretically predicted variation of the average power coefficient as affected by the tip-speed ratio A for p/q = 0.2 192 6-25 Evolut ion of the flow pattern as affected by p/q for A = 0.6 (Jk = 20 - 100) 194 6-26 Time history of CT as affected by p/q: A = 0.6; p/q = 0,1.0 . . . . 197 6-27 Theoretically predicted Cp vs. A curves for p/q = 0,0.2,1.0 199 6-28 Time history of the torque coefficient for a stationary rotor, wi th p/q - 0.2 and f3 = 120°, as affected by the blockage 201 6-29 Flow pattern after 100 time-steps for a stationary rotor, wi th p/q = 0.2 and 0 = 120°, as affected by the blockage 202 6-30 Variation of the average starting torque coefficient w i th blade angle (3 for a Savonius rotor: p/q = 0.2; B = 10% 203 6-31 Theoretically predicted characteristic curves for a rotor with p/q = 1.0 as affected by the blockage 205 6-32 Theoretically predicted characteristic curves for a rotor with p/q — 0.2 as affected by the blockage 206 6-33 Theoretically predicted variation of Cpmax with blockage for different p/q 207 6-34 Flow pattern after 100 time-steps for a rotor with p/q — 0.2 as affected by the blockage(A = 0.8,1.2) 209 X I V N O M E N C L A T U R E a blade gap-size A aspect ratio, h/d A square matrix of order (m + 2 ) x (m + 2 ) b blade overlap b column matrix of order (m + 2 ) B blockage ratio, (h.d)/(H.W) Cp pressure coefficient, (p — P o o ) / \ p U 2 base pressure coefficient cP power coefficient, Pj\pVzhd CT torque coefficient, Tj\pXJ2hdR d rotor diameter ddisc end plate diameter dA elemental area in 0 dS elemental length of the boundary S DVM discrete vortex method FuF2 empirical parameters (Equation 3 . 2 ) F normal derivative of the Green's function. 4 ^ ' a n G Green's function h blade height H height of the wind tunnel H space derivative of the Green's function, 4^-k number of time-steps K resolution parameter m number of elements representing the blade contour Mw number of elements representing the wind tunnel wal l n normal vector parameters defining blade geometry (Figure 2 - 1 ) Vii Poo pressure at the ith tap and infinity, respectively p power Pf frictional power loss X V (jy*y) polar coordinates of the blade contour wi th respect to the main radius R rotor radius, d/2 Rn Reynolds number A s length of the blade element Si, S2 empirical parameters (Figure 3-5) Sn,S<p boundary where | ^ and $ known, respectively (Figure 4-1) t time t* t ime elapsed since the generation of the vortex t" period over which the power is averaged (Equation 4.29) A t time-step for convection calculation AT time interval at which the boundary conditions are satisfied T torque u,v velocities in x and y directions, respectively U free stream velocity vn velocity normal to the surface V velocity vector w(z) complex velocity W(z) complex potential W width of the tunnel xs, £B points on the problem boundary za, Zb end points of the uniformly distributed vortex a angle between the element and the x-axis 0. angle of attack of the blade (Figure 2-8) 0] ini t ial blade angle 7 ( f s ) vorticity density at £ s r , s , Tw strengths of the bound vortices T(f) known vorticity density at point r es distance between the nascent vortex and the edge of the blade (Figure 4-3) r field point in fi 77 balancing parameter (Equation 4.26) A tip-speed ratio, du/2U /i viscosity xv i V kinematic viscosity p density of air o core radius of the free vortices 4> angle between the x-axis and the normal to the blade contour $ potential function w angle between the normal and radius vector at the blade contour stream function angular velocity of the rotor n problem domain (Figure 4-1) §c integration along an arbitrary path around the rotor excluding singular points h integration along the blade contour S u b s c r i p t s indices n nodal points s singular points wake max peak value xvii A C K N O W L E D G E M E N T The author wishes to thank D r . V . J . M o d i for his generous assistance during the course of this research work. His help and sharing of experience during the conduct of the experiments and the preparation of this thesis are very much appreciated. The author is grateful to the departments of Mechanical Engineering and Physics for the use of various facilities. Thanks are also due to the technicians of the machine shop for assistance during construction of the models. The project was supported by the Natural Sciences and Engineering Research council of Canada's grant (A - 2181) to D r . M o d i . I also want to recognize the K i l l a m Foundation's generosity in awarding the Predoctoral Fellowship. Final ly , the author would like to express his gratitude to his wife, Dhammika and children for their support and encouragement in completing this thesis. x v i i i To my parents. C H A P T E R I I N T R O D U C T I O N Since the beginning of time, man has worshipped the elements of nature, often out of fear, and has also strived to utilize them to advantage. In particular, light, water and wind represent the elements frequently exploited by the Indus-Valley Civ i l iza t ion of Ariyans and the Egyptian Dynasty of the Pharaohs. Windmil ls were in existence in Persia as early as in 134 B . C . (Ahamedi , 1978). In western Europe, windmills have been used for grinding grains and pumping water since the middle ages. In the United States, more than six mill ion windmil ls have been built since 1880, particularly in the midwest and southwest, and around 100,000 of them are sti l l in use (Savino and Eldridge, 1975). The attention towards ut i l izat ion of wind energy (besides other alternative en-ergy sources) may be attributed to increased consciousness about the depleting character of the fossil fuel supply. For the technologically advanced countries, a favourable balance of payments has always depended heavily on the exported prod-ucts of their technologies. Unfortunately, the heavy manufacturing industry, which supplies these products, uses an inordinate quantity of energy. Heavy consumption of fossil fuels with the accompanying degradation of the quality of air, excessive damming of the rivers for hydro-electric power production, thermal pollut ion, and possible risk of nuclear-electric plants; all have come under attack by environmental-ists and other concerned groups. Consequently, the " clean " natural energy sources 1 Chapter I: Introduction 2 such as solar . geothermal, tidal and wind have gained in importance (Blackwell et a l . , 1975). It has been demonstrated that the per capita energy consumption is directly related to the standard of l iving. Conversely, by improving the energy supply to a certain community the standard of l iving can be upgraded. A large percentage of the population in the developing countries live in rural areas. Being away from major cities and scattered over a vast region, economical distribution of the grid electric power is practically impossible for such communities. Al though the rural community subsists on farming, it does not produce enough for the village's au-tonomy whereas,in the Uni ted States, 1% of the population feed the nation and export the surplus. This disparity is largely due to old farming techniques and the lack of energy. Pumping water for irrigation is an important use of energy in rural communities. As pointed out earlier, needed power cannot be supplied from con-ventional energy sources due to remoteness of these areas and a small scale of the energy use. In this situation, windmills appear particularly promising. A schematic diagram of a wind energy operated irrigation system suitable for rural communities is shown in Figure 1-1. Obviously, the wind turbine for rural application should be simple in design, use locally available materials, and be easy to fabricate and maintain. Furthermore, preferably, it should be self-starting and have a reasonably high efficiency. Al though the wind has been used as an energy source for many centuries, the potential of wind energy for today's world as an alternative energy source needs to be emphasized. Consider the energy available in a wind stream. The kinetic energy per unit mass of a moving air parcel is | c 7 2 , and the mass flow rate through a stream tube of cross-sectional area A is pAU. The product of these two is the F I G U R E 1-1 A n illustrative schematic diagram of a wind energy operated irrigation system Chapter I: Introduction 4 power available, P = [\U2){PAU) = -PU3A. 2 Density of air (p), wind velocity (U), and stream tube area (A) are the variables that influence the power availability in the wind. Seasonal density variation may be about 20%. The stream tube area, which is directly related to the wind turbine frontal area, increases wi th the size of the turbine. W i n d velocity is the primary variable that affects the available power because of the cubic relationship. For example, a l O m / s wind contains 8 times more power than a 5m/s w ind . Since size of the turbine is a design parameter, it is convenient to use power per unit area (power flux) for the meteorological purposes. The power flux being proportional to the cube of the wind speed, mean wind speed data alone are not adequate for estimating power availability. Instead, the mean of U3, which is always greater than the cube of mean wind speed, should be used to estimate the power availability at a given site. 1.1 D e v e l o p m e n t o f W i n d T u r b i n e s 1.1.1 Betz maximum efficiency Design of ancient windmil ls was based on empiricism and engineering ski l l . A d -vent of the fluid mechanics as a science, and more specifically aerodynamics of wind turbines, is more recent. Perhaps the earliest significant development in this area came about after the first world war and may be attributed to Betz (1926). Betz power coefficient, CptBetz = 16/27, is considered the maximum possible efficiency for wind turbines. In particular, since the derivation of the Betz efficiency appears to be firmly based on the fundamental laws of conservation, it has been accorded the same status as the Carnot efficiency for heat engines. However, such an analogy Chapter 1: Introduction 5 is rather misleading because Betz derived his theory under restrictive assumptions -the velocity field of the wind must be stationary, non-viscous and incompressible. In addit ion, it must allow for a control volume which envelopes the rotor area and is bounded by streamlines. Glauert (1935) improved Betz efficiency prediction by taking wake rotation into account. As a result Glauert efficiency is dependent on the tip-speed ratio. According to Glauert , the efficiency is zero at zero tip-speed ratio and reaches the Betz efficiency as the tip-speed ratio tends to infinity. There exists in the literature remarks to the effect that the efficiency of the real windmills can exceed the Betz-Glauert l imit ing prediction (Hutter 1977, Inglis 1979). The derivation of the Betz efficiency is also critisized by Greet (1980), and by Rauh and Seelert (1984). They claim that the Betz optimum efficiency cannot be rigorously justified within Betz model itself and conclude that there exists no theoretical opti-mum for the power coefficient of windmills which is firmly based on the fundamental laws. 1.1.2 Types of wind turbines Many different types of wind turbines have been invented. A distinction can be made between turbines driven mainly by drag forces versus those powered mainly due to the lift. Further classification may be based on the orientation of the axis of rotation: turbines with the axis parallel to the ground (Horrizontal Axis Wind Turbines - H A W T ) ; and those wi th the axis perpendicular to the ground (Vertical Ax i s W i n d Turbines - V A W T ) . The efficiency of the wind turbines driven primarily by drag forces is usually low (De Vries, 1979) compared to the lift force driven turbines. This may be attributed to a large viscous dissipation of energy due to flow separation in the drag devices. One the other hand, lift based devices tend to be relatively more efficient. Most vertical axis wind turbines have the advantage of their performance being independent of the wind direction, whereas a horrizontal axis wind turbine has to be yawed into the wind direction to optimize performance. Chapter I: Introduction 6 1.1.3 Vertical axis wind turbines The first modern vertical axis wind rotor was developed by S. J . Savonius (Savo-nius, 1928), who published the results of its performance in 1931 (Savonius, 1931). Another vertical axis configuration was patented about the same time by G . J . M . Darrieus (Darrieus, 1931). Although both are vertical axis turbines, they work on completely different principles. The Savonius turbine is, in part at least, a drag driven machine whereas the Darrieus turbine is a lift force device. Over the years, attention has centred particularly on the Darrieus rotor because of its relatively high aerodynamic efficiency. Furthermore, its high rotational speed is favourable in the generation of electric power. It has received attention, particularly in techno-logically developed countries, for generation of electric power on a large scale. This type of turbine is also amenable to theoretical modelling due to its well-investigated aerodynamic shape. Consequently, a considerable body of theoretical and exper-imental data exists on the Darrieus turbine performance. Al though it has higher efficiency, it suffers from the disadvantage of a low starting torque. A n external prime mover is required to start the machine adding more sophistication to the system. Research on wind energy received a considerable boost due to the "energy crisis" of 1973-74. Aerodynamics of Darrieus rotors was investigated in depth by Lissaman et al . ( l974, 1976), Holme (1976), Base and Russel (1976), Templin (1974), Strick-land et a l . ( l975, 1976) and many others. Corresponding experimental results on the performance of the Darrieus rotor have also been presented by Lissaman et al.(l974, 1976), Rangi et al . (1972, 1974), Feltz and Blackwell (1975), etc., to quote a few important contributions. Blade element theory, vortex theory and panel methods are the most commonly used tools in the analysis of the Darrieus turbine. Thus a considerable amount of literature aimed at development of the Darrieus turbine, both aerodynamically and structurally, already exists. However, due to its unde-sirable starting characteristics and complexity in design, the Darrieus rotor is not Chapter I: Introduction 7 suitable for small scale applications, especially in the rural environment. 1.2 T h e S a v o n i u s T u r b i n e 1.2.1 Experimental work The Savonius rotor concept never became popular, until recently, probably be-cause of its low efficiency. However, it has the following advantages over the other conventional wind turbines: - simple and cheap construction; - acceptance of wind from any direction thus eliminating the need for re-orientation; - high starting torque; - relatively low operating speed (rpm). The above advantages may outweigh its low efficiency and make it an ideal economical source to meet small scale power requirements, especially in the rural parts of developing countries. It has also been proposed as an auxil iary starting device for the Darrieus turbine (Templin and South, 1976) and as a t idal power generator (Manser and Jones, 1975). The concept of the Savonius rotor was based on the principle developed by Flettner. Savonius used a rotor which was formed by cutt ing the Flettner cylinder into two halves along the central plane and then moving the two semi-cylindrical surfaces sideways along the cutting plane so that the cross-section resembled the letter 'S '(Figure l-2a). A n 'opt imum' geometry was obtained by systematically testing more than 30 different models in a wind tunnel, and Savonius reported encouraging results. He conducted further tests in natural wind and observed that the rotor ran at a higher speed than that in the wind tunnel for the same wind velocity. According to Savonius the best of his rotors had a maximum efficiency of 31% while the maximum efficiency of the prototype Chapter I: Introduction 8 F I G U R E l-2a The Savonius configuration F I G U R E l - 2 b Geometry of the Bach rotor Chapter I: Introduction 9 was 37% (Savonius, 1931). However, other researchers who have conducted similar experiments wi th the Savonius rotor have not agreed wi th the claimed efficiencies (Shankar 1976, Sivasegaram 1978). Following Savonius, Bach (1931) made some investigations on the Savonius rotor. He altered the basic shape of the blade by connecting two circular arc portions wi th different radii (Figure l-2b) which resulted in a considerable increase in the power coefficient. Bach reported a maximum efficiency of 24% for his rotor without correcting for the blockage effect (B « 10%). For nearly 40 years the field remained essentially dormant except for some iso-lated contributions. Simonds and Bodek (1964) reported the possibility of using empty oil drums to fabricate Savonius turbines for water pumping in rural areas. They used a prototype machine to assess feasibility of the rotor in real life. The Savonius design was found to be relatively insensitive to the quality of its aerody-namic surfaces compared to the Darrieus turbine. Newman (1974) conducted experiments to assess the effect of gap-size on the performance. He used a two-bladed Savonius configuration in a closed-circuit wind tunnel at different Reynolds numbers and at a blockage ratio of about 20%. New-man's results indicate that while the optimum gap is of the order of 10% of the diameter, a rotor without a gap has almost the same peak efficiency. He also found the efficiency of the rotor to increase wi th the increasing Reynolds number in the operating range of his experiment. Newman applied corrections of the order of 50% to his results for C p . O n the basis of his model tests it was concluded that the full scale performance cannot be predicted with certainty from the model test results because of the wind tunnel interferrence, however, model tests can be used to assess relative merits of different rotor designs. Further experimental investigations on the effect of Reynolds number and blade geometry were carried out by several researchers in late 1970's and early 1980's. Chapter I: Introduction 10 Shankar (1976) studied the effect of the number of blades, blade gap and blade curvature on the rotor performance. His experiments were carried out in an open-circuit wind tunnel and wi th a considerably low blockage ratio ( « 3%) at Reynolds numbers between 1 x 10 5 and 2 x 10 5 . Shankar found the efficiency of the rotor to increase wi th the Reynolds number in this range although the increment was not very significant. He concluded that except for a slightly higher starting torque, the three-bladed rotors have up to 35% lower peak efficiency than the two-bladed rotors. The opt imum blade geometry suggested by Shankar had a maximum power coefficient of about 20%. Sivasegaram (1977, 1978a, 1978b, 1982) has done a remarkable amount of work on the development of the Savonius configuration. His experiments were conducted in an open test-section wind tunnel using fairly small models. In summarizing the results he concludes that the two-bladed configuration has more potential for improvement compared to the multibladed design; and an increase in the Reynolds number has a favourable effect on the maximum power coefficient. He indicates a possibility of further improvements in performance although it might be relatively small . Use of augmentors to concentrate the wind power is also recommended. K h a n (1978) has done a fair amount of model tests to improve performance of the Savonius rotor. His tests were carried out in a closed-circuit wind tunnel. He used several different configurations of the two-bladed design and achieved a considerable improvement in performance. O n the basis of the results obtained from the model tests, a prototype unit with the opt imum configuration was built and tested in an open site. K h a n concluded that depending on the basic geometry, the blade over-lap and the blade-gap may have different opt imum values. The prototype test results confirmed what Savonius referred to as " the S-rotor instantly utilizes any increase in the wind speed without having to lose time in getting into the right position." Chapter I: Introduction 1.1 Alder (1978) also tested the Savonius rotor in a closed-circuit wind tunnel. The mean and periodic components of torque, drag and side force were measured. He did not observe any significant effect of the Reynolds number on the power coefficient in the range 2 x 10 5 — 6 x 10 5 . However, Alder did notice a significant change in the nature of the flow around the rotor at a tip-speed ratio of 0.2. Gov inda Raju and Narashima (1979) investigated the use of sails in the fabri-cation of Savonius rotors. They built a prototype turbine for pumping water which showed encouraging results. Sundaram and Govinda Raju (1980) tested sail type Savonius wind turbines in an open-circuit wind tunnel. The opt imum configura-tion gave a maximum power coefficient of 0.18. Some flow visualization studies using smoke to better understand the flow around the Savonius rotor were also conducted. Unfortunately, the flow visualization results are insufficient to provide a clear understanding of the flow. The special aspects such as the effect of augmentors, shear flow (earth boundary layer), wind tunnel blockage, etc., have also received some attention. Sivasegaram (1979), Sabzevari (1978), and Morcos and Khalafallah (1981) investigated the effect of various types of augmentors. Sabzevari tested several ductings, concentrators and diffusers on a s p l i t ' S ' Savonius rotor in a wind tunnel. The results of these tests have led to the design of the circularly ducted Savonius rotor equipped wi th a number of identical wind concentrators and diffusers along the periphery of its cylindrical housing. The design proved to be effective independent of the wind direction and showed about 125% increase in the maximum power coefficient compared to the base value of 0.20 (without blockage correction). Sivasegaram (1979) has tested several straight walled ducted concentrator con-figurations in the wind tunnel and obtained 'op t imum' geometric parameters. A l -though this type of augmentors are not directionally independent, a change in wind direction by ss 10° d id not show a significant change in performance. Simplicity in Chapter I: Introduction 12 design and fabrication was the main objective. He was able to increase the power by a factor of about 1.5 through moderate size concentrators. Morcos et a l . ( l98 l ) introduced a simple flat plate shield as a power augmentor. A systematic series of model tests in a low speed wind tunnel gave an optimum configuration which increased the peak power coefficient from 0.22 to 0.34. However, this design is also not directionally independent. Sivasegaram et al.(l983) have also shown the possibility of an 80% increase in power coefficient by using two moderate size-stationary vanes in their opt imum position. •(•V -•Aldos (1984) developed a mechanism allowing the rotor blade to swing back and align with the wind when it is on the upwind stroke. This reduces the drag on the upwind blade and hence improves the performance. According to Aldos the system is simple to construct and independent of the wind direction. Ogawa et al.(1986a) tested in a wind tunnel a Savonius rotor with straight guide vanes installed axisymmetrically along the periphery. Through a systematic variation in geometric parameters they were able to improve the performance by a considerable amount. Use of guide vanes improved starting characteristics of the rotor. 'Majola (1986) has examined, under field conditions, performance characteristics of Savonius rotors wi th seven different overlap ratios. Rotational speed, torque and power results were obtained over a range of wind speeds. Design criteria based on the avilable data were recommended. Ogawa et al.(1986b) have also tested the effect of end plates on the performance of the Savonius rotor. The results suggested an opt imum size for the end plates. Furthermore, contribution of the lift force to the rotor power was found to be significant. Use of a deflecting plate in front of the rotor increased the power by about 24%. It was also suggested that the deflecting plate can serve as a speed control and safety device for Savonius turbines. Chapter I: Introduction 13 Bowen and McAleese (1984) carried out detailed measurements of the pulsating wind flow around a Savonius rotor using: i) tuffts and a stroboscope; ii) a hot-wire anemometer; and iii) a turbulence meter. Data suggested that "active coupling" between the rotors might be useful in "redirecting" the wind more efficiently. In particular, it was shown that if two counter-rotating rotors are placed side by side in a wind tunnel, a natural phase locking occurs. Majo la and Onsanya (1981) investigated the effect of shear flows on performance of the Savonius rotor. They used the conventional Savonius rotor in an open test-section wind tunnel simulating a shear flow. It was concluded that the major effect of wind shear is to reduce the power coefficient below the inviscid flow level. The strong correlation between the power coefficient and the tip-speed ratio was noticed even in the shear flow. The field testing of the Savonius rotor has indicated that unsteadiness of the wind appears to be a far more important parameter governing the power-loss than the wind shear. The flow visualization results and the comments made on the flow patterns by Jones et al.(l979) deserve some attention. It was observed that, in general, a Savonius rotor with zero gap describes the characteristic flow behaviour quite well, and the performance can be improved by a careful blade design to delay separation on the convex surfaces. For the purpose of clarifying the mechanism of rotation of the Savonius geom-etry, loading on the rotating and stationary blades was measured in a water filled towing tank (Sawada et al, 1985). The flow around the rotor was visualized using aluminum powder on the water surface. After a series of experiments it was con-cluded that the lift force produces the torque over a large range of angular positions. Alexander (1978), using a variation of Maskell 's theory (Maskell , 1965) for bluff bodies, predicted the wind tunnel correction for Savonius rotors. Several wind tunnel tests were carried out to substantiate the approach. Considering the Chapter I: Introduction 14 magnitude of corrections, the agreement was good, particularly at the point of max imum efficiency. Suzuki and Oki t su (1982) investigated the characteristics of a Savonius rotor connected to a synchronous generator. The effect of changes in wind speed and load resistance on the performance was discussed. Based on a careful evaluation of the experimental procedures, measuring tech-niques and analyses of the available data, following remarks can be made: ' (a) As can be expected, earlier tests did not have the advantage of reliable and sophisticated instrumentation. (b) .Almost all the tests were carried out using models in fairly small wind tunnels. As a result the corrections due to blockage were substantial and hence the accuracy of the measurements suffered. (c) In a number of cases blockage corrections were completely ignored. When applied, the procedure was mostly empirical and unreliable. (d) In all the test results, bearing friction losses were neglected. Unfortu-nately, due to small models, they may affect the performance substan-tially. A systematic optimizat ion program for the two-bladed Savonius configuration wi th a desirable blade geometry (easy to construct) representing a circular arc followed by a straight line remains unrecorded in the available literature. 1.2.2 Theoretical investigations Available theoretical work on the analysis of the Savonius turbines is indeed very scarce, perhaps due to the complex time-dependant separated nature of the high Reynolds number flow. Several researchers have commented that a reliable analysis of the flow around a Savonius turbine is extremely difficult if not impossible (Wilson Chapter I: Introduction 15 et a l . , 1976; Sivasegaram, 1977). O n the other hand, there are several well developed theories to analyse the Darrieus and propeller type wind turbines where lift is the main driving force. The blade element theory (Glauert, 1935; Wilson and Lissaman, 1974; Holme, 1981; and others), which is one of the simplest, can predict the performance of a Darrieus turbine reasonably well. It assumes that the different spanwise blade elements are independent of each other and that the forces on the elements can be determined from the local flow conditions. The flow field is estimated by either a momentum or a vortex consideration or both. Vortex and panel methods are also widely used in the analysis of Darrieus type turbines. Instead of estimating induced velocities from the momentum equation, a better model analogous to the theory for a wing wi th a finite span can now be formulated. These methods are based on the "lifting line" approach of Prandt l (1918), the "lifting surface" theory of Weissinger (1947), the "panel" methods of Hess (1972) and others. The flow around the wing is assumed to be irrotational everywhere (except in the thin layer of trail ing vortices), and the velocity is given in terms of a perturbation potential. A z u m a and K i m u r a (1983) have developed a computationally sound approach based on local circulation method to calculate the air loading on the Darrieus tur-bine. The method when checked experimentally showed good agreement (Kimura et a l . , 1984). Due to the completely different nature of the flow around the Savonius rotor the above mentioned methods cannot be applied directly in its analysis. Al though the classical momentum theory was used by Betz (1926), it fails to distinguish between different types of wind turbines, not to mention the effect of various parameters of the Savonius geometry. For the Savonius rotor an analytical model was developed by Wilson et al . Chapter I: Introduction 16 (1976). Another model using a vortex sheet was presented by Van Dusen and Kirchhoff (1978). These are the very first theories found in the literature for the prediction of the performance and flow field of a Savonius rotor. However, these models assume attached flow throughout, which is unrealistic. Available literature describing the modelling of separated flows reveals the considerable potential of the discrete vortex method. A theoretical analysis using the discrete vortex method and assuming flow separation at the tips of the blades has been developed by Ogawa (1984). Al though the performance predicted is not quantitatively accurate, the model provides fairly accurate qualitative description of the flow. Hatayama et al . (1984) analysed the problem using the discrete vortex method together with conformal mapping. A conventional Savonius rotor wi th zero gap between the blades was studied. The rotor was transformed into a unit circle using a rather complicated mapping function and the analysis carried out in the mapped plane. The unsteady lift and drag forces as well as the moment were evaluated for three different geometries and at two different tip-speed ratios. As a result of the conformal mapping, the prediction curves were found to be relatively smooth compared to the results of the discrete vortex method used in the physical plane (Ogawa, 1984). 1.3 S c o p e of the P r e s e n t I n v e s t i g a t i o n This thesis describes the development of a Savonius rotor configuration, simple in design, fabrication and maintenance, yet having a reasonably high efficiency. It is suitable for rural small scale applications in developing countries. Attention is also directed towards the aerodynamic problem of modelling the unsteady separated flow around a Savonius rotor in order to predict its performance. In Chapter II the experimental methodology and the corresponding results of the systematic optimization process undertaken, to develop an efficient Savonius Chapter I: Introduction .17 rotor configuration, are presented. Special care has been taken to overcome the problems which led to unreliable results found in the literature. Focus is on the two-bladed configuration with geometric parameters varied in an extensive wind tunnel test program. Effect of the blade geometry, wind tunnel blockage as well as frictional losses in the model mounting bearings are assessed. The pressure distr ibution and starting torques have been measured for the stationary blade at different angles. Chapter III presents a semi-empirical approach to predict performance of the Savonius rotor. The analysis is based on a quasi-steady method, similar to the blade element theory, wi th a central vortex at the core, and the measured pressure data as an input. The model is based on the flow visualization study carried out in a tow-tank. The experimental results of Chapter II are used to evaluate empirical parameters. In Chapter I V , the boundary element method and its application to the analysis of a Savonius rotor are discussed in detail. The flow model and the corresponding governing equations are developed using first principles and assumptions made in the analysis are discussed. Discretization of the governing equations and their numerical solution procedure is described at length in Chapter V . The effect of flow-model parameters as well as computational problems are discussed. In Chapter V I results are obtained for starting torque at different angular positions and compared wi th experimental data. The flow fields corresponding to different angular positions of the blade are also obtained. The vortex shedding behaviour of the stationary blade at several angles is also discussed wi th reference to the Strouhal number. For the rotating blade, flow patterns at different tip-speed ratios for several geometries are obtained. The wake characteristics are compared wi th the corresponding flow visualization results. The time history of the torque coefficient, and hence the power coefficient, has been Chapter J: Introduction 18 calculated over a range of tip-speed ratio for different geometries of the rotor and compared with the experimental results of Chapter II. The mathematical model is also used to account for the wind tunnel blockage. The thesis ends wi th concluding remarks and recommendations for further study. C H A P T E R I I O P T I M U M C O N F I G U R A T I O N S T U D I E S This chapter discusses a comprehensive wind tunnel test program aimed at opti-mization of a simple blade geometry to give the maximum possible power coefficient. The simplest of blade geometries would be the original Savonius configuration (Fig-ure l-2a), but several researchers (Khan , 1978, Sivasegaram, 1978) have concluded that the possible improvement in efficiency through modifications of the blade would be little. Bach(1931) suggested a rather different geometry (Figure l-2b) involving portions of two circular arcs. Performance of the Bach rotor is indeed better com-pared to the original Savonius design, however, the configuration is relatively more complicated. In 1978 Sivasegaram altered this configuration replacing the second circular arc by a straight line (arc of infinite radius, Figure 2-1). He conducted some experiments aimed at improving the performance, however, the extent of im-provement that can be realized proved to be inconclusive. In the present study, the circular arc-straightline blade geometry is systematically studied in a parametric fashion to arrive at an optimum configuration. 2.1 D e f i n i t i o n o f the B l a d e S h a p e The basic blade shape shown in Figure 2-1 can be denned by the following param-eters: 19 Chapter II: Optimum Configuration Studies 2 1 a blade gap-size ; b blade overlap ; D shaft diameter ; d rotor diameter ; ddisc end plate diameter ; H height of the wind tunnel ; h blade height ; p -+ b length of the straight line portion ; q radius of the circular arc portion ; W width of the wind tunnel ; 6 blade circular arc angle . To help establish the two dimensionality of the flow, end plates are used. In general, all length parameters (except p,q) are non-dimensionalized with respect to the rotor diameter d. The non-dimensionalized blade shape parameter p/q 'defines the basic geometry. 2.2 E x p e r i m e n t a l M e t h o d o l o g y The power P developed by a rotor, operating under unconfined conditions, would be a function of the blade parameters, density p, wind speed U, rotor angular velocity u and viscosity p,. W i t h the wind tunnel wall restriction the additional blockage parameter B has to be introduced, ^ g maximum projected area of the model (h d) ^ cross-sectional area of the tunnel (H W) ' Chapter II: Optimum Configuration Studies 22 i.e., P = P{a, b, rf, h, p, g, D, ddi3C,0, w, p, fi, U, B) (2.2) Using the Buckingham IT-theorem, this expression can be rewritten in terms of nondimensional parameters (assuming a geometric constraint given by Equation 2.10) as, Cp = Cp{l^^l^^WB^) ' (2-3) where is referred to as the average power coefficient. The parameter du> is known as the tip-speed ratio and A = 1 ( 2 - 6 ) is called the blade aspect ratio. Rn is the Reynolds number based on free stream velocity and rotor diameter. Similarly, the average torque coefficient CT can be expressed in non-dimensional terms as Cj — CT{-,, A,-, — , d t s c , 6 , A , B , R n ) , (2.7) a a q a a where C t = T ^ H l . . (2.8) Note, A is the only independent parameter that varies wi th the loading. There-fore, Cp vs A curve represents the fundamental characteristic of each rotor. Chapter 11; Optimum Configuration Studies 23 As pointed out in Chapter I, the flow past the turbine blade is so complex that no exact theoretical method exists to predict it. The flow is unsteady relative to the blades and, at any given instance, it is partly separated. Even if a theoretical description of the flow past the rotor may appear reasonable, predictions based on such conjecture cannot be accepted wi th certainty. Under these circumstances the logical approach is model testing. Experimental approach to performance prediction requires that the model sat-isfy certain fluid dynamical similarity wi th the prototype machine. Geometric sim-ilarity (similarity in ratio between the principal dimensions) is relatively easy to achieve in model fabrication. Kinematic similarity (similarity in ratios between the velocities at various corresponding locations) is achieved by maintaining similar geometry and operating conditions. In the present study, this is ensured by main-taining the same tip-speed ratio for models and the prototype. Dynamic similarity (similarity between the forces acting on the turbine) is basically achieved when the operating Reynolds numbers(J2„) for the model and the full scale turbine are equal. In general, the Reynolds number has relatively small influence on the model per-formance. In laboratory tests, a model frequently has a smaller Reynolds number compared to the prototype. This leads the performance predicted by the model to be marginally smaller than that for a full scale machine. However, it is advisable to perform the model tests at Reynolds numbers comparable to those of the prototype turbines under field conditions. In the present test program, relatively larger models and higher wind speeds are used to achieve comparable Reynolds numbers. This has an added advantage of better accuracy in the measurement of power and torque as well as the wind velocity. Chapter II: Optimum Configuration Studies 24 However, the use of larger models presents the problem of wind tunnel block-age, and the associated correction could become significant. A closed wind tunnel requires negative corrections whereas a wind tunnel wi th open test-section needs positive corrections. Obviously, the prediction would not be reliable unless a proper method for blockage correction is available. In this study, a set of experiments is carried out to establish the effect of blockage, thus providing an approach to obtain reliable data. The test program involves a systematic variation of variables affecting the per-formance curve C p vs. A, thus leading to an 'opt imum' combination of parameters. 2.3 E x p e r i m e n t a l S e t - u p 2.3.1 Wind tunnel Most of the model tests were carried out in the boundary layer wind tun-nel(Figure 2-2). The partially return type tunnel has a 2.44 m wide and 24.4 m long test-section consisting of eight 3.05 m long bays. The height of the test-section can be varied using rachets operating on the top panels. For this study the first bay wi th smooth uniform flow was used using a height setting of 1.6 m at the test-section. The estimated maximum turbulence level in this section is less than 0.4 %. W i n d is driven by a variable pitch blower running at a constant speed of 700 rpm. Power consumption of the blower is rated at 80 kW. The tunnel can provide a stable wind speed in the range 2.5 m/s — 25 m/s and is fitted with emergency stop switches located every 6m along the test-section. 2 3.2 Model fabrication and mounting arrangement Dimensions of a model are determined basically by the blockage ratio (B), Axivane Series 2000 Rotor. 2.44 m dia.. 16 Cast aluminum blades. 125 h.p. electric motor, 175,000 cfm at 700 rpm. Fisher 480-60 pneumatic variable pitch control 1 honeycomb and 4 screens in 4 x 4 i settling section Test-Section, 2.44 x 1.6 m F I G U R E 2-2 Boundary layer wind tunnel used in the test program. Chapter It: Optimum Configuration Studies 26 aspect ratio (A), shape factor (p/q), blade gap (a), blade overlap (b) and blade arc angle (0). The blockage ratio fixes the value of the projected area whereas the aspect ratio limits the diameter to height ratio. These two parameters determine the blade diameter (d), and height (h). For the convenience of using the same mounting structure, the diameter of the main shaft (D) was kept constant at 38mm. Depending on the p/q, a and 0 value, the dimension q can be evaluated using: S V - 2S2Cq3 - (d2 - C2) { — + 1 )q2 + ^ ~ ^ Cq + (fjl^l) * = 0 ; (2.10) 2 | 2 (a + 0 > n) or (J2 _ (-,2) q2{S2 + 2 S s i n 0 + 2 ( l -cos t? )} -qC(l-cos0)- '- - 0 ; (2.11) (a + 0 < 7 r ) where: 5 = ?; C = D + a; (2.12) - i t S a = tan 1 -2q Note, for C < 2q, a ^ t a n " 1 S. The models were fabricated using 16 gauge aluminum sheets. Blades were rolled to the desired shape using a sheet metal rolling machine, and pop-rivets were used for fastening. Both single and two stage models were built depending on the re-quirement. A frame made of angle iron (Figure 2-3) supported the models in the wind tun-nel. The model, mounted on a vertical steel shaft, was supported by two self aligning Chapter II: Optimum Configuration Studies Chapter II: Optimum Configuration Studies 28 ball bearings. The lower end of the shaft was flexibly coupled to a dynamometer. The frame was tied to the wind tunnel walls to avoid undesirable vibrations. 2.3.3 Measuring devices The wind velocity was measured using a pitot-static tube connected to an in-clined alcohol manometer. Since the model was mounted in the first bay, the wind velocity measurement was not possible upstream of the model. Therefore, the pitot-static tube was placed in the last bay (w 25 diameters downstream), at the centre line of the flow parallel to the tunnel, where the wake can be assumed to have reached steady state. The roof height was adjusted to compensate for the growing boundary layer in the tunnel. Velocity measurements at the first and the last bay showed no difference in the absence of the model. Even in the presence of a model wi th 10% blockage, a scan of the velocity field, at the centre line of the last bay showed the deviation to be less than 1%. The rotor rpm was measured using a strobotac. The emphasis was on measurement of the torque. Since the torque varied over a large range, different types of dynamometers were required in different ranges. However, the basic approach to the measurement was similar in each case. Es-sentially the force was measured using a cantilever beam with four strain gauges attached near its root, two on either side. The output signal from the strain gauges, forming a part of the Wheatstone bridge, is amplified using a Bridge Amplifier Me-ter ( B A M ) and measured by a digital voltmeter. The sensitivity of the system was calculated as 0.5 x 1 0 ~ 3 Nm. A t the beginning of each trial the force measuring system was calibrated using known loads. A higher accuracy is required when measuring smaller torques. The hydraulic dynamometer was designed and fabricated for this purpose(Figure 2-4). Load was applied by filling the outer cylinder wi th high viscosity oil ( S A E lOWfJO). The shaft of the turbine is connected to the central rotating vanes through a flexible Dynamomete support frame rn A Flexible couplings Bearings Stationary cylinder Bearings Vanes Strain gage torquemeter Oil drain F I G U R E 2-4 Hydraulic dynamometer. A Dynamometer if Flexible couplings support frame _l*fl Bearings Stationary cylinder Additional Vanes Vane Strain gage torquemeter Oil drain F I G U R E 2-5 Modif ied hydraulic dynamometer. Chapter fj: Optimum Configuration Studies 30 coupling. The motion of the outer cylinder is restricted by a string connecting the cylinder to the strain-gauged cantilever. The maximum torque that can be measured using this type of dynamometer depends on the rotational speed of the shaft. The present device is capable of measuring torques up to 2.2 Nm at a rotational speed of 180 rpm. The maximum measurable torque decreases rapidly wi th a decrease in shaft rotational speed. To measure a higher torque the dynamometer was modified by inserting verti-cal vanes on the inside wall of the outer cylinder (Figure 2-5). Though the concept seems promising, improvement in results was not as expected. In fact, performance of the modified dynamometer deteriorated at higher rotational speeds, due to up-ward creeping motion of the oil between the stationary vanes of the outer cylinder, leaving a clear gap between stationary and rotating vanes. The concept is indeed sound, however, design modifications are needed to make the device effective. To save time it was decided to leave this development as a part of a separate project. A prony brake type dynamometer was routinely used for larger torque measure-ments. One end of the rope-brake is connected to a known weight through a pulley while the other end is attached to the strain-gauged cantilever. The system was loaded through dead weights. The arrangement yields reasonably accurate measure-ments in the higher torque range. Since the weight imposes transverse loading on the main shaft, the bearing friction is slightly affected by this type of dynamometer. 2.3.4 Bearing friction calibration A t higher rotational speeds the frictional loss due to bearings cannot be ne-glected. Therefore a set of experiments was carried out to establish the frictional power loss characteristics of the bearings. The arrangement shown in Figure 2-6 was used for this purpose. The bearings support a shaft driven by a variable speed D . C . Bearings under test Bearings Strain gage transducer Belt drive Variable speed drive *"' T» I n Frame T M r F I G U R E 2-6 Test arrangement for bearing friction calibration. 10.0-1 8.0-7.0 00 ^ 6.0 8 ! o CL 3.0 2.0 1.0 0.0 . A " P f =0. 0308 f RPM-2 .1538 A ^ ^ A . A ' P f = 0 .0250 RPM- 1 I.O ^ A" 80 100 120 140 160 180 200 220 240 260 280 300 320 340 RPM F I G U R E 2-7 Bearing friction characteristic. GO 1x3 Chapter II: Optimum Configuration Studies 33 motor. The frictional torque of the bearing was measured using the strain-gauged cantilever over a range of speeds. It is evident from Figure 2-7 that, in general, the bearing loss characteristic is piece-wise linear and can be expressed as: f 0.025RPM - 1.0, 0 < RPM < 200; . . f ~ \ 0.0Z0SRPM - 2.1538, RPM > 200: ^ ' ' where Pf is the frictional power loss. 2.S.5 Pressure distribution measurements To better understand behaviour of the flow around the Savonius rotor, it was considered desirable to measure the pressure distribution along the blade. Mea-surements on the rotating blade are extremely difficult and need very sophisticated instrumentation. In this study the pressure measurement is l imited to the station-ary blade at different angles. This provided rather important information such as the-positions of the separation points, base pressure coefficient, etc.,which proved to be useful in theoretical modelling. A single-stage model wi th 46 pressure taps was used for this purpose. The pressure taps were mounted wi th the openings on the convex surface of one blade and the concave surface of the other blade (Figure 2-8), and were connected to a scanivalve (48J9-1331) v i a ' tygon' tubes (<£§")• Pressure at a tap was monitored using a transducer ( B A R O C E L 511J-10) attached to the scanivalve. Care was taken to minimize the interference due to the tubes by keeping them in the wake region. The wind speed was set at 6.76 m/s for all pressure measurements, which corresponds to & ,Rn of 3.2 x 10 5 (based on free stream velocity and rotor diameter). The model had a blockage ratio of 10%. The Savonius turbine model was mounted on two clamps wi th the facility to Front Side F I G U R E 2-8 Pressure tap locations and numbering scheme. Chapter If: Optimum Configuration Studies 35 lock it in a desired angular position. The angle was measured using a protracter scale attached to the lower clamp wi th 1° accuracy. The pressure was measured at 10° intervals from 0° to 360°. Measurements over one cycle were appropriately combined to give the pressure distribution on both sides of each blade. The pressure transducer was calibrated before each run and all the taps were checked for leakage. The pressure coefficient at the ith tap is defined as, where pi is the pressure measured at the ith tap; Ptx>, the static pressure at infinity; p. the density of air; and U, the free stream velocity. 2.3.6 Starting torque Start ing torque for the rotor with p/q = 0.2, 10% blockage ratio, single-stage model was measured over a range of angular positions. The model was mounted vertically on two bearings as explained before. A drum of 127 mm diameter,attached to the lower end of the shaft, had a string tangential to its surface connected to the strain gauge arrangement mentioned earlier to measure the static torque. The angular position of the turbine was recorded using a protractor scale mounted at the lower bearing. B y varying the length of the string, different angular positions were obtained. The arrangement permitted measurement of negative starting torque when rotated through 180°. The strain-gauged cantilever system was calibrated using known loads wi th sufficient time allowed for warm up. The torque measurements were carried out at two different wind speeds (Rn = Chapter II; Optimum Configuration Studies 2.1 x 10 5 , 2.7 x 10 5 ) . 36 2.4 E x p e r i m e n t a l R e s u l t s Basically two models were used in the test program. A smaller two blade model (Figure 2-9) with a maximum blade diameter of 330mm and a height of 301mm, resulting in the aspect ratio of 0.91 and projected area of 0 . 1 m 2 was primarily designed to study the effect of gap-size and overlap. The blades, rolled into the de-sired shape were supported by two plexiglass end plates, 6.35 mm thick and 381 m m in diameter. Rigidi ty of the plates was augmented by reinforcing them with alu-minum discs, 203 m m in diameter and 6.35 m m in thickness. A 25.4 m m diameter high prescision straight shaft supported the blade assembly in a pair of self-aligning bearings. For blade-gap and overlap studies a model with p/q = 1.9, 0 — 112° was used. The model was tested in a low speed low turbulence return type wind tunnel wi th a test-section of 0.91 x 0.68 (blockage 17% ) at a wind speed of 17.9 /s. The estimated turbulence level of the tunnel is less than 0.1%. 2.4-1 Gap-size Typica l results for the variation of power coefficient wi th tip-speed ratio and blade-gap are shown in Figure 2-10. More informative would be the effect of percent-age gap-size on the maximum power coefficient as presented in Figure 2-11. This clearly shows that as separation a is increased, the maximum power diminishes. The peak power coefficient of 0.158 occurred at zero gap-size. Chapter II: Optimum Configuration Studies End Plate Rotar Blades 381 mm 203 mm-330 mm 301 mm F I G U R E 2-9 A schematic diagram showing model of the single-stage Savonius rotor. Chapter II: Optimum Configuration Studies 38 0.16-1 0.14' 0.12-0.10-0.08-0.06-0.04 0.02 0.00 A a / d = 0.000 X a / d = 0.018 • a / d = 0.039 a / d = 0.058 S a / d = 0.106 I I I I I 0.5 0.6 0.7 0.8 0.9. 1.0 1.1 1.2 1.3 1.4 1.5 A F I G U R E 2-10 Plots showing the effect of gap-size on power coefficient [b/d = 11 .5%, i E n = 4 x, 1 0 5 ) . Chapter II: Optimum Configuration Studies 3 9 0.20 0.18 H 0.10 | i 1 j 1 ' 1 1 1 1 1 ' 1 0.0 3.0 6.0 9.0 12. a/d % F I G U R E 2-11 Effect of gap-size on peak power coefficient (b/d - 11.5%, Rn = 4 x 105). Chapter II: Optimum Configuration Studies 40 2.4.2 Blade overlap To investigate the effect of blade overlap, models similar to that described earlier but with zero blade gap and specified overlap ratios were used. B y combining power coefficient vs. tip-speed ratio curves for different blade overlaps, a more informative curve of maximum power coefficient vs. percentage overlap was established (Figure 2-12). It is apparent that an optimum value for blade overlap is around 10%. The plot also shows that the maximum power coefficient remains essentially unaffected for overlaps less than 10%. These results are uncorrected for blockage and bearing power loss. As the blockage ratio for models in this set of experiments is the same, it is not likely to be a significant parameter affecting the opt imum. On the other hand, the bearing friction correction may have a significant, effect on the final prediction since it is proportional to the rpm. Fortunately, since the measured power outputs are reasonably high, the error due to bearing friction is not likely to affect the opt imum blade setting by a significant amount. 2.4-8 Blade aspect ratio Of considerable interest is the effect of the aspect ratio (A) on the wind turbine performance. To that end, a set of models similar to the one discussed earlier but wi th different aspect ratios (keeping projected area constant) and fixed a/d, b/d (•- 0) were fabricated and tested in the wind tunnel in a similar manner. The results showing the variation of maximum power coefficient vs. aspect ratio (A) are presented in Figure 2-13. The results are corrected for bearing loss. This shows an opt imum aspect ratio of 0.77 giving a max imum power coefficient of 23.5%. Besides providing the useful information concerning the opt imum blade config-Chapter II: Optimum Configuration Studies 41 F I G U R E 2 -12 Variat ion of the maximum power coefficient wi th percentage overlap showing the peak around 10% (a/d = 0,Rn = 4 X 105). Chapter II: Optimum Configuration Studies 42 0.25 0.24 H A F I G U R E 2-13 Effect of blade aspect ratio on the peak power coefficient (a/d = 0,6/rf = 0 , ,R n = 4 x 10 5 ) . Chapter II: Optimum Configuration Studies 43 uration, the single-stage model study emphasized, as expected, the presence of dead spots when the blades are aligned with the wind and the rotor fails to start on its own. Thus for self-starting of a two blade rotor, it is necessary to have at least a two-stage system wi th blades in the individual stage oriented orthogonal to one another. Furthermore, results suggest that to generate even 100 W of power at a wind speed of 25 km/h it would require a projected area of 3.5 m 2 . In addition, the ease of fabrication being a guiding criterian, particularly in a rural environment, suggested a multi-stage construction. It was, therefore, decided to conduct tests wi th models of a two-stage rotor to assess interference effects due to staging. To establish the effect of blade shape, two-stage models with a projected area « 0.6 m 2 were used. 2.4-4 Blade shape factor p/q It is reasonable to assume that the basic blade shape has a significant effect on its performance. As shown in Figure 2-1 the parameter p/q governs the basic shape of the blade. A set of two-stage models, diameter 635 m m , stage height 489 m m (A = 0.77), and end-plate diameter 847mm was constructed with gauge 16 aluminum sheet to study the effect of shape factor p/q (Figure 2-14). For simplicity in fabrication 6 was taken to be 135° for all models. A typical power vs. rpm plot for p/q — 1, without friction correction, is presented in Figure 2-15 . It shows the maximum power of 39 W at 215 rpm. Note, the corrected curve indicates a peak power of 44 W at 222 rpm. Thus the bearing losses, if unaccounted for, would lead to an error of approximately 11% in peak power and 3% in the prediction of the corresponding rpm. Uncorrected Cp vs. A curves at three different wind speeds for p/q = 1.6 are Chapter II: Optimum Configuration Studies 44 F I G U R E 2-14 A typical model of the two-stage Savonius rotor used to study the effect of blade geometry parameter p/'g (projected areas: 0 .6m 2 ) . Chapter II: Optimum Configuration Studies 45 45.0 40.0-35.0-30.0 % 25.0 £ 20.0 O Q_ 15.0-10.0-5.0-0.0 -X X> X <x x x X X x x x x x X X A X A A A * l iA * A A ... .A. .. x X A " A A A X > • A X • L \ X • A -U = 6.76 m/s A Uncorrected x Corrected • • 150 170 190 210 230 250 270 290 310 330 350 RPM F I G U R E 2-15 P o w e r - R P M characteristic of the two-stage model (p/q = 1.0), showing the effect of bearing friction loss. Chapter II: Optimum Configuration Studies 4 6 0 . 3 5 ^ 0 .30 0.25H 0.20H 0.15 H 0.10 H 0 . 0 5 H 0.00 _ r n • n . • • X X • 1 X > < X u c X x > • < c X > • < X A A A * ^  A A A L A L X A A u X u • u 1 1 1 1 1 , 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 A F I G U R E 2 - 1 6 Variat ion of power coefficient with wind speed for the two-stage rotor (p/q = 1 .6) emphasizing the effect of bearing power loss. Chapter H: Optimum Configuration Studies 47 shown in Figure 2-16. In general, one would expect variation of the power coefficient with tip-speed ratio to be essentially independent of the wind speed. However, Figure 2-16 suggests a marked dependence due to the uncorrected character of the data. Accounting for the bearing dissipation led to near collapse of the results on a single curve as shown in Figure 2-17. The slight discrepancy between corrected curves may be attributed to the Reynolds number effect, which in this case is relatively insignificant. Corrected C p vs. A curves for different p/q values are shown in Figure 2-18. In this set of experiments the blockage ratio (B) was kept constant at 16.4%. Note, the geometric parameter p/q has a significant effect on the power coefficient. The more informative graph in Figure 2-19 shows the variation of maximum power coefficient with p/q. Under the given conditions, the maximum power coefficient was found to be as high as 0.5 (uncorrected for blockage) for p/q — 0.2. 2-4-5 Blade arc angle To investigate the effect of blade arc angle a set of single-stage rotors wi th p/q = 0.2 and 10% blockage were constructed. Each had a diameter of 704 m m and a height of 542 m m giving the aspect ratio of 0.77, wi th circular end plates measuring 939 m m diameter. The projected area was « 0.37 m 2 . The associated p and q were found using equation (2.10) or (2.11). The corrected Cp vs. A curves for 6 = 112°, 135° and 150°, presented in Figure 2-20, suggest the opt imum value for 6 to be around 135°. 2.4-6 Reynolds number The operating Reynolds number of the prototype turbine would be around Chapter II: Optimum Configuration Studies 48 0.40 0.35H 0.30 H 0.25 H C p 0.20 H 0.15H 0.10 0.05 A 0.00 1 fi-5 3 • n 8 s « i L \ ' S A R ..A • Corrected A u X u • u I I I I I I I 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 F I G U R E 2-17 Bearing loss corrected characteristic curves at three different wind speeds for the two-stage (p/q = 1.6) model. Chapter II: Optimum Configuration Studies 49 0.55-1 0.50 0.45 0.40 0.35 0.30 0.25 0.20-0.15-0.10-0.05-0.00 • • > >> <• ^ •> N -c j : c \ \ 3 • [ a"* x.--... » % • r . . 1 I / B = 1 6 . 4 % A p/q = 0.0 x p/q - 0.2 • p/cj = 0.4 B p/q = 1.0 H p/q = 1.6 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 A F I G U R E 2-18 Effect of the blade geometry parameter p/q on the power coefficient' at a constant blockage. Chapter II: Optimum Configuration Studies 50 0.55-1 C p/q F I G U R E 2-19 Variat ion of the peak power coefficient wi th the blade geometry parameter p/q at a constant blockage of 16.4%. Chapter II: Optimum Configuration Studies 51 > f-L \ L ^  > N < z L c ^ .[ y' : [ : c D-[ X 3 N \ -p/q = 0.2 A e -112° x 6 = 135° • e -150° 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 A FIGURE 2-20 Effect of the blade circular arc angle 6 on the power coefficient for single-stage model (p/q = 0.2, B = 10%). Chapter II: Optimum. Configuration Studies 52 3 x 10 5 -- 9 x 10 5 . On the other hand, most of the wind tunnel tests were carried out in the Reynolds number range 1.7 x 10 5 — 4 x 10 s . Fortunately, CP vs. A plots in this range suggest that the effect of the Reynolds number in the operating range is likely to be insignificant. 2.4-7 Blockage ratio W i n d tunnel results presented by different investigators often do not correlate because of different test conditions. One of the major parameters affecting the test data is the blockage. To have some appreciation of the wall confinement effects, four single-stage and one two-stage rotor models with an identical geometric shape but with different blockage ratios were tested in the boundary layer wind tunnel. The results presented in Figure 2-21 clearly show a dramatic increase in the maximum power coefficient, pr imari ly due to an increase in the local velocity, with blockage. Note, an increase in wall confinement from 5 to 20% can raise the C p m a i by around 70% thus leading to a highly optimistic performance if the blockage effect is not corrected. Figure 2-22a shows the variation of Cpmax wi th blockage (B) for p/q = 1. It should be noted that when operating in the unconfined environment the power co-efficient reduces to 0.2. To obtain similar information at other p/q values would involve an extensive test program. However, of particular interest here is the cor-rected power coefficient corresponding to the opt imum p/q of 0.2. To this end, two models wi th blockage ratios of 10% and 16.4% were constructed. Recognizing the fact that the associated wake aerodynamics remains essentially the same, the vari-ation of Cprnax wi th blockage is expected to have the same trend. This infers that the Savonius rotor wi th an opt imum combination of parameters has an efficiency of Chapter II: Optimum Configuration Studies 53 0.45 0.40-0.35 0.30 0.25 0.20-0.15 0.10-0.05-0.00 -• .. XI" •1— ~ x K p / q = 1.0 A B = 2.0% x B = 5.0% • B - 10.0% ia B = 16.4% E B = 20.0% 1 1 1 1 1 1 1 I 1 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 FIGURE 2-21 Effect of blockage on power output of the Savonius rotor. Chapter II: Optimum Configuration Studies 54 0.6 0.5-F I G U R E 2-22a Effect of blockage on peak power coefficient for two different blade geometries. F I G U R E 2-22b Effect of blockage on the tip-speed ratio at peak power coeffi-cient for two different blade geometries. Chapter II: Optimum Configuration Studies 55 about 32% in unconfined conditions. The plot of tip-speed ratio (A) at the peak power coefficient vs. B% (Figure 2-22b) shows a linear variation. This yields the zero blockage values of A as 0.71 and 0.79 for p/q = 1.0 and 0.2, respectively. The corresponding results of Maskell (1965) for the flat plate blockage correction also showed a similar linear variation for small B. However, the free air power coefficient predicted using Maskell 's procedure gives an error of about 25%. The considerably large side-force and the asymmetric wake due to blade rotation, which had been neglected in the Maskel l derivation may account for this discrepancy. The systematic optimization process has indeed proved to be quite rewarding. B y conducting a series of carefully planned experiments it has been possible to improve the rotor performance by a factor of two. The opt imum blade configuration (for the end plate parameter d/ddisc = 0.75) can be summarized as follows: a/d non-dimensional blade gap-size = 0 ; b/d non-dimensional blade overlap = 0 ; A blade aspect ratio = 0.77 ; p/q blade shape parameter = 0.2 ; 6 blade arc angle = 135°. 2.5 P r e s s u r e D i s t r i b u t i o n A few typical results showing pressure distributions over a single-stage Savonius rotor with opt imum geometric parameters and 10% blockage are presented in Figure 2-23. The tap numbering scheme was indicated in Figure 2-8. The side of the blade 1.5 1.0 X 0.5-0.0-A A'A A' A A A ' A -0.5--1.0-1 -1.5--2.0--2.5-^ A A A A A A A A A A A A A A A A '45 A A A A A A t -3.0 x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x .A : A A . . A . A /? = 0 A Front S ide x B a c k S ide T 5 10 15 30 35 20 25 Pressure Tap No. F I G U R E 2-23a Surface pressure distribution over the Savonius rotor as affected by the blade orientation: 0 = 0°. Note, two separation points (tap 0, 39) on the front (lower) surface and one on the back (upper) surface (tap 11). 40 45 1.5 1-0 " A A A ' A A ' A A A A A ' A A ' A ' A ' A " A A A A ' A A A A A 0.5 0.0 x -0.5 -1.0 -1.5 -2.0 -2.5-A A A A A A A A x X X X X X X X X X X X X X X X X X X 5^  X X X X V V' A X X X X X X X X A A X X •3.0 -r 0 (3 = 30 A Front Side x Back Side " T 5 10 15 20 25 30 35 40 Pressure Tap No. FIGURE 2-23b Surface pressure distribution over the Savonius rotor as affected by the blade orientation: 3 = 30° . Note, separation points at tap 11 and 38 on the back and front surfaces, respectively. 45 1.0 A A A A A A A A A A A A A ' A A A A A A A A A A A A 0.5 0.0 -0 .5 x x x x x x x x x x -1.0 -1.5 -2 .0 -2 .5 -3 .0 x x X X x x X X X. XX. X X X X X X A. XX. X X X X X X X X X x X X X A A A A A A p = 60 A Front Side x Back Side -1 45 5 10 15 T " 20 —r— 25 30 35 40 Pressure Tap No. F I G U R E 2-23c Surface pressure distribution over the Savonius rotor as affected by the blade orientation: 0 = 60° . Note, wake flow on the back surface and the separation at tap 41 on the front surface. 1.5-1.0 0.5 A A A A ' A A - A A - A A - A - A A A A A A - A A A A A ' A A A A A o.o-45 -0.5-1 X X X X x X x: X x xx x"x:"*"x X X x x x x x x x x x x x x x x Y x x x x x x -1.0 -1.5-x x x x x A A A A -2.0 -2.5 -3.0 p = 90 A Front S ide x B a c k S ide T 10 5 15 20 25 Pressure Tap No. 30 35 40 45 F I G U R E 2-23d Surface pressure distribution over the Savonius rotor as affected by the blade orientation: 0 = 9 0 c . Note near constant pressure along the back surface and smooth pressure variation on the front side. 1.5 1.0-0.5-0.0--0.5-•1.0 -1.5--2.0 -2.5 A A A A A A A A A A • A c A ^ A ' A x * : A X X X X > ( X X X X > < X X X X > < X X X X ) < X X X X > ' X x x X ( x x x x > > X x * X _ ... — o I (3 = 120° A Front Side x Back Side -3.0 10 15 20 25 Pressure Tap No. F I G U R E 2-23e Surface pressure distribution over the Savonius rotor as affected by the blade orientation: 3 = 120". Note the absence of surface flow separation. 1.5 1.0-0.5-0.0-A - A -A A A A A A A A A A A 45 A A A A A A A A A A A ^ A A A A -0.5 -x X X X X x X X X X X X X X X X X X X X X X X X X x X X X X x" x -1.0--1.5--2.0 -2.5 x x x x x x x x x x x x (3 = 150 A Front Side x Back Side -3.0 10 15 20 25 Pressure Tap No. 30 35 40 45 F I G U R E 2-23f Surface pressure distribution over the Savonius rotor as affected by the blade orientation: /3 = 150°. Note near constant back pressure and the separation at tap 15 on the front side. Chapter II: Optimum Configuration Studies 62 facing the free stream is referred to as the front side while the opposite is called the back side. Similarly, the upstream blade is called the leading blade and the downstream blade is referred to as the trai l ing blade(Figure 2-8). When the blades are aligned with the flow direction (/? = 0 ° , Figure 2-23a), stagnation point occurs at tap 0 on the front side of the leading blade. Pressure drops along the front surface reaching a minimum at tap 9 followed by a brief rise in pressure. Unable to negotiate the adverse pressure gradient, the flow separates at around the 11th tap. The flow remains separated and the pressure continues to be essentially uniform at the separation value. O n the back side, the flow separates at the t ip of the leading blade and reattaches in the vicini ty of the 2 3 r d tap resulting in a region of rise in pressure. Now the circular cylinder type pressure profile begins on the trail ing blade reaching a min imum in the vicinity of the 3 3 r d tap on the backward curvature, showing separation near the 39th tap, and maintaining a near constant value thereafter. Thus the flow separates twice on the back side and once on the front side. This pattern remains unchanged unti l about 0 = 30° . Obviously the local flow is rather complex with cause and effects often eluding detection. A t /? = 30° (Figure 2-23b), pressure on the front side remains at the stagnation value for the leading blade followed by a sharp drop. The reverse curva-ture causes an adverse pressure gradient and the flow separates around the 38th tap. The pressure on the back side decreases when the flow accelerates along the convex surface of the leading blade, goes through the familiar variation and separates at the \ \ t h tap. Thus there are two separation points, one on the back side of the leading blade and the other on the front side of the trailing blade. A t (3 — 60° , the front side pressure remains at the stagnation value for the leading blade followed by a drop and a part ial recovery on the trai l ing blade (Figure Chapter 11: Optimum Configuration Studies 63 2-23c). The back side pressure remains essentially uniform suggesting a wake flow. The same characteristics were observed for the range 40° < 0 < 80° . In this configuration there are only two separation points, one at the tip of the leading blade and the other at the front side of the trail ing blade. Separation points move to the tips at 0 — 90° (Figure 2-23d) as characterized by the constant back pressure. The smooth pressure drop along the front side of the trai l ing blade suggests accelerating flow and is followed by a partial pressure recovery before separation at the trail ing edge. Similar pressure variations were observed for 0 up to 140°(e.g. , refer to the pressure distribution at 0 = 120°. Note the stagnation point near the 10th tap on the front side). The configurations for 3 greater than 90° are similar to those wi th negative 0 values (e.g., 0 = 120° corresponds to 0 = —60°). To keep the direction of tap numbering the same as the flow direction, for 0 > 90° the equivalent negative angle has been considered. A t 0 — 150° (= - 3 0 ° ) , stagnation occurs in the vicinity of the 4 t h tap on the front side and separates at the leading edge of the blade. Proceeding in the direction of the flow, the pressure drops from the stagnation value, briefly rises, and becomes nearly uniform suggesting the wake condition. The back side pressure remains essentially constant indicating that this part of the blade is also in the wake (Figure 2-23f). 2.6 Starting Torque The starting torque coefficient for the 10% blockage model with opt imum blade geometry was measured at two wind speeds and is shown in Figure 2-24. Note, the effect of Reynolds number is minimal substantiating the earlier observations. The line represents the torque calculated through pressure integration assuming two k /'AXAxX * \x A . . A . . . .^x \ y V tX / V 1 1 t 1 1 > 1 X A ; / legend A R n = 2.1x105 x R n = 2.7x10s Calculated n 1 1 • 1 1 1 1 1 ' 1 1 1 1 1 , — i 1 1 0 20 40 60 80 100 120 140 160 180 (3 (Degrees) F I G U R E 2-24 Variation of the starting torque wi th angular position of the Savonius rotor (p/q = 0.2, B = 10%). Calculated results based on pressure data are also indicated. Chapter ././» Optimum Configuration Studies 65 dimensionality. The deviation between measured and calculated values is within an acceptable error margin (maximum error « 5%). This suggests that the assumption of two dimensional flow is reasonable for this class of models. Figure 2-24 shows a positive torque over the range of /? « 0° — 130° followed by the torque reversal. The peak positive torque occurs at around 30° suggesting that this rotor is not exactly a drag device. 2.7 Error Estimation The important parameters evaluated in this study are : CT Torque Coefficient ; Cp Power Coefficient ; Cp Pressure Coefficient ; A Tip-Speed Rat io . The quantities actually measured and the possible ranges of errors were: N ± 0.5 R P M of the rotor ; / ± 0.05 mm reading of the inclined alchol manometer ; V ± 0.005 v voltmeter reading of the load measuring system ; R ± 1 m m rotor radius ; h ± 1 m m blade height ; p ± 0.08 N/m2 reading of the pressure transducer. This gives the maximum possible error margins as \ , 0 .025 x , .0046" 6X = ± (0.0032 + — — )A + — 7 -Chapter IF: Optimum Configuration Studies 66 which amounts to 6X = ± Similarly = ± 0.0038 + 0.0135A 0.0025 + 0.0039A for wind speed = 4.43 m/s ; for wind speed = 6.76 m/s. bCr = ± ± 0.0016+ 0 .0751C T 0.0007 + 0 .0370C T at U = 4.43 m/s ; at U = 6.76 m/s. The maximum error margin for Cp is given by: SCP = ± = ± 0.0016 0.0038 A 0.0007 0.0025 CT A + 0.1052 + 0.0481 C p , at U = 4 . 4 3 m / s ; C p , at U = 6.76m/s. Final ly , the maximum error limits for pressure coefficient CP can be written as 6Cp = ± 0.0029 + 0.0286C 7, at U = 6.76 m/s. Thus the possible maximum errors in the measurements of tip-speed ratio, torque coefficient, power coefficient and pressure coefficient decrease wi th an in-crease in wind velocity. C H A P T E R I I I A S E M I - E M P I R I C A L A P P R O A C H F O R P E R F O R M A N C E P R E D I C T I O N This chapter discusses application of a semi-empirical approach to the Savonius rotor performance prediction using the stationary blade pressure distribution data presented in Chapter II. Such an approach, if successful, can replace dynamic testing of models wi th significant, reduction in cost, time and effort. 3.1 I n a d e q u a c y o f t h e Q u a s i - S t e a d y A p p r o a c h Consider the classical quasi-steady approach which has been successfully applied in the analysis of the Darrieus rotor as well as various other fluid dynamics problems. Here the blade is divided into a finite number of elements assuming two dimen-sional flow. For a given angular position of the blade, contributions of the free stream velocity and the.bla.de rotation at each element is determined using velocity triangles (Figure 3-1). As shown in Figure 3-1, ( r ; , ^ ) are the polar coordinates at the centre of the ith element wi th respect to the diameter of the rotor, ifci is the angle between the normal vector and the radius vector. ,/?t is the blade angle with respect to the relative velocity Vt- at the ith element, and OJ is the angular velocity of the turbine. The mean differential pressure coefficient obtained experimentally, at the ith element for the blade angle f3 = /?, was used in conjunction wi th the relative velocity Vj to evaluate the force acting on this element. In the present study, pres-67 Chapter III: Semi-Empirical Approach 68 F I G U R E 3-1 A typical velocity triangle at the blade element i combining the effect of free stream velocity and blade rotation. Chapter If]: Serni-Empirical Approach (59 sure distribution was measured at 10° intervals (section 2.5). Linear interpolation was used to estimate the pressure coefficients at intermediate locations. Using an appropriate moment arm for the blade element, integration over the blade gave the torque for a specified position /? of the rotor. Final ly , by evaluating the work done over a cycle, the mean power and power coefficient were established. Repeating the procedure for different angular velocities the effect of tip-speed ratio on the power coefficient was estimated. The results obtained using this procedure are compared wi th the corresponding experimental results (for p/q = 0.2, B — 10%, 6 = 135° model) in Figure 3-2. The large descrepancy between results suggests that the classical quasi-steady ap-proach is not successful in predicting the Savonius rotor performance. This may be attributed to disparity in the flow features associated wi th the stationary and rotational modes of the blade. The method, however, proved effective with the Darrieus rotor, where the flow character does show a degree of similarity. In the present case, since the relative direction of the flow varies significantly along the blade, interactions between elements of the rotor cannot be neglected. Unfortu-nately, the classical quasi-steady approach does not account for these interactions. Thus this approach cannot, be used effectively to analyse the Savonius rotor. 3.2 M o d i f i e d Q u a s i - S t e a d y A p p r o a c h Failure of the classical quasi-steady procedure emphasized the need for an alterna-tive. It was apparent that clear understanding of the flow character was a prerequi-site to approach the problem. To better appreciate the basic character of the flow around the Savonius rotor a flow visualization study was undertaken. Chapter III: Semi-Empirical Approach 70 X *x X legend Q-S M e t h o d x E x p e r i m e n t a l 0.0 0.3 0.6 0.9 A 1.2 1.5 F I G U R E 3-2 Results of the quasi-steady analysis. Note, the approach is woefully inadequate in predicting the actual performance. Chapter III: Serni-Empirical Approach 71 8.2.1 Flow visualization study The flow visualization study was carried out using the tow-tank facility located in the plasma physics laboratory of Professor B . Ahlborn . The tank is 5 m long and has a cross-section of 82 cm X 62 cm (Figure 3-3). A Savonius rotor model 20 cm in diameter and 30 cm high was used in the test. A 35 m m sti l l camera and a video camera moving wi th the model captured the flow patterns. The sti l l camera had a motor drive so that a series of pictures could be taken at a rate of two to four frames per second. The model was towed at a desired uniform speed using a step pulley combination driven by a half horse power constant speed motor. Towing speed could be varied from 2 to 200 cm/s. This gave a Reynolds number of the order 10 3 . Towing speed was estimated by electronically measuring the elapsed time between two points 10 cm apart. Safety switches were placed at either end of the cartway to guard against overshoot. The Savonius rotor model was mounted on an aluminum frame using two ball bearings. A pulley assembly and a flexible shaft connected to a variable speed electric motor was used to turn the model at a desired constant angular velocity. The rotor speed was measured using a " S H I M P O " noncontact type tachometer. For the surface flow investigations, aluminum filing tracers of « 0.5 m m size were applied by scraping an aluminum block wi th a file. The tracers were i l lumi-nated with four flood lamps placed at a low angle of incidence wi th respect to the fluid surface in order to avoid undesirable reflection of the surface waves. Special attention was paid to the fluid surface condition. Periodic skimming of the sur-face was needed to remove static electrical charges and contaminants, which altered properties of the surface significantly. A t higher tip-speed ratios, presence of large surface waves disturbed the flow sig-Still/Video Camera Flood Lights Camera Carriage Model Carriage Model T — R q i | — r Water Surface Seeded with Aluminum Filings • o a Speed Control FIGURE 3-3 A schematic diagram showing the towing tank facility and associated equipment used in the flow visualization study. Chapter III; Semi-Empirical Approach 73 nificantly thus affecting the simulation. In order to obtain better flow-visualization pictures the motor drive was removed and self-driven blades were used. A typical set of pictures obtained are shown in Figure 3-4. Perhaps the most significant feature is the central vortex filament in the time averaged flow. The existence of this vortex has also been reported by Jones, Li t ter and Manser (1979). Several other relevant features are discussed later in Chapter V I . 8.2.2 Central vortex approach The flow visualization study provided a rational modification of the classical quasi-steady approach to the problem by introduction of a central vortex filament. The flow is simulated by a potential vortex wi th a core radius S\R, a core angular velocity S2OJ and a uniform free stream velocity U. Here R and ui represent radius and angular speed of the rotor, respectively and S\, S2 are two empirical parameters. The velocity distribution of the vortex filament is shown in Figure 3-5. The relative velocity at each element is evaluated using the modified flow field and the angular speed of the rotor. Force at the individual element is evaluated using the differential pressure coefficient and the velocity of the resultant flow as discussed in section 3.1. The procedure leads to evaluation of power coefficient Cp in terms of tip-speed ratio A, Si and S2 . 8.2.3 Empirical relations Si and 5 2 were systematically varied and the corresponding Cp estimated over a range of tip-speed ratio A = 0.6 — 1.5. The results obtained, when compared to the experimental results presented in section 2.4, suggested the functional relations: F I G U R E 3-4 A typical set of flow visualization pictures showing the presence of the central vortex filament. Chapter III: Semi-Empirical Approach FIGURE 3-5 Velocity distribution due to the potential vortex. Chapter 111: Semi-Empirical Approach Cp -;- CP(Si, S2) (3.1) 5, • FAX) +F2{B,Rn) (3.2) S2 •• S2{p/q,Rn) (3.3) as logical. The effect of tip-speed ratio could be absorbed in Si while 52 showed a direct dependence on the variation of the geometric parameter p/q.The remainder of Si , when varied, appeared to model the blockage effect. Influence of the Reynolds number (Rn) is included here in both S\ and however, as seen before it is insignificant in the operating range. Using the experimental results for p/q — 0, B — 16.4% (Figure 2-18), F\(X) can be expressed as, wi th corresponding values for F2 and 52 as 1.7 and 1.25, respectively. According to the suggested functional forms, F2 = 1.7 combined wi th different 52 should produce a set of results corresponding to performance of constant blockage (l6.4%)models with different p/q values. A s shown in Figure 3-6, the calculated results for 52 = 1.25, 1.3, '1.37. agree with experimental curves of p/q — 0, 1.0, 0.4 at B = 16.4%, respectively when F2 is kept constant at 1.7. Similarly, when 52 is kept constant at 1.3 (corresponding to p;q -- l ) and F2 is varied, this should simulate results for p/q = 1 models with different blockage ratios. The results shown in Figure 3-7 verifies this prediction. Now it is evident that the above mentioned functional forms can be used to predict, the performance of Savonius rotors, if the relationship between F2 and blockage ratio, and 52 and p/q can be established for the operating Rn range. As shown in Figures 3-6 and 3-7, there is a slight descrepancy between the experimental and calculated results i at lower tip-speed ratios. This may be due to the fact that the pressure data for p/q — 0.2 were used to predict the performance Fi(X) = 0.27778A 2 - 1.08333A (3.4) Chapter III: Semi-Empirical Approach 77 0.5 0.4-F 2 = 1.7, B = 16.4% MQSM Sx = 1.30 MQSM . Sa= 1-37 Exper. .PA=.°. Exper. _PA=A Exper. p/o t= 0.4 "1 1.5 X F I G U R E 3-6 Identification of the modified quasi-steady approach parameter 5 2 as a function of p/q when i<2 and blockage are kept constant. Chapter III: Semi-Empirical Approach 78 0.5-1 0.4 0 . 3 -0 . 2 -0 .1-0.0 S 2 = 1.3, p/q = 1.0 MQSM . 2 - i v v F a = 1.65 MQSM r 2 = i.7o Exper. 6 =5% Exper. B = 10% Exper. B = 16.4% 0.5 0.7 . 0.9 1.1 X 1.3 1.5 F I G U R E 3-7 Identification of the modified quasi-steady approach parameter F2 as a function of blockage ratio when 5 2 and p/q are kept constant. Chapter ill: Semi-Empirical Approach 79 of other geometries. However, at moderate tip-speed ratios the effect of the central vortex is more dominant and the results may not be affected significantly by a slight change in pressure data. The agreement between calculated and experimental results shown in Figures 3-6 and 3-7 is used to establish 52 and F2 in terms of p/q and B, respectively. The empirical curves established for 52 and F2 for the operating Reynolds number range are shown in Figure 3-8 and Figure 3-9. The variation of S2 wi th p/q shown in Figure 3-8 resembles the Cp>max vs. p/q curve (Figure 2-19). Similarly, F2 vs. B curve (Figure 3-9) has the same basic shape as the C p , m a i vs. B curve shown in Figure 2-22. This indicates that 52 and F2 are directly related to the maximum power coefficient Cpimax. 3.3 R e s u l t s a n d G e n e r a l R e m a r k s Cp vs X curve for p/q = 0.2 and p/q = 1.6 at B = 16.4% were obtained experimen-tally in section 2.4.4 and were not used to establish the empirical relations in the modified quasi-steady approach. Similarly, the experimental results for p/q = 0.2, B = 10% were not involved in the development of the method. Therefore, these three cases can be used to verify the validity of the above approach. The com-parison of the predictions given by the modified quasi-steady approach wi th the experimental results for these three cases are shown in Figure 3-10. From engineering design considerations, the agreement may be considered quite acceptable although the results are under-predicted at lower tip-speed ratios. Superposition of the central vortex filament assumes a potential flow field. But the time-dependent shear layers, separated flow and high turbulence levels suggest that the.flow field is far from being potential. Furthermore, the wake is not con-Chapter III: Semi-Empirical Approach 81 F I G U R E 3-9 Empi r ica l relationship between F2 and the blockage ratio B. Chapter III: Semi-Empirical Approach 82 0.6 0.5-0.4-0.3-0.2 0.1 0.0 legend M Q S M F a = 1.70, S a = 1.45 M Q S M F 2 = 1.65, = JL45_ M Q S M F a = 1.70, S a = 1.26 E x p e r . B = 16.4% p/q_ = 0.2 E x p e r . B = 10% p / q : = 0.2 E x p e r . B = 16.4% p/q_ = 1.6 0.5 0.7 0.9 T 1.1 1.3 1.5 A F I G U R E 3-10 Predictions of the modified quasi-steady approach compared to the experimental results. Chapter III: Semi-Empirical Approach 83 sidered at all in this analysis. However, wi thin its limitations, the method permits prediction of model performance in the wind tunnel as well as that of the prototype in unconfined condition with a reasonable accuracy. A more sophisticated mathematical model considering the unsteadiness of the flow, shear layers, vortex shedding and the wake is discussed later in Chapters I V , V and V I . C H A P T E R I V A M A T H E M A T I C A L M O D E L F O R R O T O R A N A L Y S I S W i t h some appreciation as to the character of the complex flow associated with a Savonius rotor and-significant parameters affecting it, a more sophisticated ap-proach to its analysis is attempted here. The problem is indeed formidable, being characterized by viscous, turbulent, separated and time-dependent behaviour of the flow, and wi l l demand application of the complete Navier-Stokes equations. Even if a suitable algori thm is developed to account for the above mentioned complexi-ties, appropriate boundary conditions and mathematical consistency, the numerical solution w i l l be enormously time consuming wi th the available computer (Amdahl 5850) and equally costly. One is, therefore, forced to turn to an approximate ap-proach which makes the problem amenable to the numerical tools at hand without sacrificing its essential features. A n approximate method of solution of the Navier-Stokes equations in the oper-ational range of the Reynolds number (3 x 105 — 9 x 105) would involve use of an appropriate turbulent flow model and an integration scheme. Since the turbulent scales are relatively small around this Reynolds number the grid required in finite element and difference integration schemes becomes extremely small thus adding to the computational cost. Several such approaches, applicable to specific geometries have been reported in 84 Chapter IV: Mathematical Analysis 85 literature. They involve time stepping, finite element or difference solutions of the Navier-Stokes equations through Eulerian (Jordan and Fromm, 1972), Lagrangian (Chor in , 1973; Stansby and Dixon , 1983) and hybrid (Eulerian approach in the boundary layer region with Lagrangian procedure in the potential flow regime, Spalart et al . , 1983) schemes for simple geometries like circular cylinders. Since the boundary layer is simulated, the flow separates on its own and the point of separation need not be specified. Unfortunately, these methods are not applicable to bluff bodies wi th geometrical irregularities and sharp edges. Furthermore, they often suffer from computational instability or require finite element or difference grid of impracticably small size to obtain reliable results at high Reynolds numbers. This chapter briefly describes the basis of an approach which, through judicious simplifications, arrives at a mathematical model that can predict the performance of the Savonius rotor wi th an acceptable accuracy.. Since the rotor possesses sharp edges and hence fixed ini t ial separation condit ion, the difficulty of having a very fine grid for stable results is overcome through the use of a Boundary Element Method ( B E M ) as against the Fini te Element Me thod or Fini te Difference Method ( F E M or F D M ) . 4.1 B o u n d a r y E l e m e n t M e t h o d ( B E M ) The essence of this technique is the transformation of the governing differential equations into an equivalent set of integral equations as a first step to.their solution. This would result in a set of equations which would involve only values of the variables at the extremes of the range of integration (i.e., boundaries of the region). Thus any discretization scheme needed subsequently would only involve subdivisions of the boundary surfaces of the problem (Banerjee, 1981). In contrast to the F E M , Chapter IV; Mathematical Analysis 86 which involves subdivisions of the whole flow field, the B E M proves to be less expensive in terms of computer time. A l l boundary integral methods utilize the principle of superposition and there-fore are applicable to linear systems or those which can be approximated as incre-mentally linear (Hess et al . , 1962-76; Zienkiewicz, 1978; Banerjee et al . , 1979; M a r -jar ia et a l . , 1980). In problems involving elliptic differential equations the solutions are direct, whereas for parabolic and hyperbolic systems of equations, marching processes in time have to be introduced (Banerjee, 1981). In general, there are only a few problems solvable by a F E M which cannot be solved at least as efficiently by a B E M . These are the problems where: i) the material properties of almost every individual element are different; ii) those in which the general geometry of the problem is such that one or more dimensions are smaller compared to others but not sufficiently so as to genuinely reduce its effective dimensionality (e.g., moderately thick plates and shells, narrow thin strips, etc.). A B E M reduces the dimensionality of the basic problem by one, i.e., for two dimensional problems the analysis generates a one dimensional boundary integral equation. Similarly, for three dimensional problems only two dimensional surface integral equations arise. The method proves to be more advantageous compared to the F E M for problems with some boundaries at infinity. Since the B E M solu-tion procedure automatically satisfies admissible boundary conditions at infinity, no subdivisions of these boundaries arise. On the other hand wi th the F E M infinite boundaries have to be approximated by an appreciable number of distant elements. The indirect formulation of the B E M (Banerjee, 1981) can be easily applied to Chapter IV: MatharmUtdl Analysis 87 the Savonius rotor analysis. In this formulation, the integral equations are expressed in terms of a unit singular solution of the original differential equation distributed at a specific density over the boundaries of the region of interest. 4.2 M a t h e m a t i c a l F o r m u l a t i o n 4.2.1 Basic assumptions The physical problem involves a time-dependent separated flow at high Reynolds number. Bu t for the purpose of analysis a simplified model is considered which partly accounts for the effect of viscosity. It is assumed that the influence of viscos-ity is confined to an infinitely thin layer of fluid adjacent to the solid surface of the rotor (i.e., boundary layer of zero thickness which does not affect the external invis-cid flow), and vorticity is contained within "wake sheets" issuing from sharp edges. These "wake sheets" correspond to physically identifiable regions of rotational fluid which transport the vorticity generated by viscous effect wi th in the boundary layer on the blade. The vorticity generated is carried away downstream wi th in a sheet of fluid whose shape is such that the pressure difference across it, and hence the difference in velocity magnitude, is zero at all points on its surface. The velocity is discontinuous across the shear layer. These shear layers roll up to form strong vortices which shed downstream with the flow. In practise, the effect of viscosity and diffusion lead to finite cores of rotational fluid. 4.2.2 Governing equations The problem in hand is to estimate performance of the Savonius rotor under various conditions. W i t h the above mentioned assumptions, torque and power gen-Chapter IV: Mathematical Analysis 88 erated can be determined uniquely from the velocity field which can be considered as the only unknown of the problem. The governing equations for the idealized incompressible, inviscid, irrotational two-dimensional flow outside the wake surface can be written as: £ + H<.'+.'>=/<«); <«> where <& is the velocity potential; p, the pressure; p, the density; / ( £ ) , an arbitrary function of time t; and u, v are the velocity components in the x and y directions, respectively. The velocity components (u,v) can be obtained from the potential V = V$, or ay It is necessarj^ to specify conditions on the boundary 5 of the domain of interest fl (Figure 4-1) in order to solve for <fr, which is the only unknown. This falls into the general class of boundary value problems. The boundary condition at the blade surface specifies that the flow normal to the surface be zero, ^ - - " „ = 0, • (4.4) on where n is the vector normal to the blade surface and vn is the normal velocity of the blade. Chapter IV: Mathematical Analysis 89 F I G U R E 4-1 General description of the problem. Chapter IV: Mathematical Analysis 90 Theoretically, the wake extends indefinitely downstream of the body. However, due to viscous dissipation, it effectively ceases to exist beyond some characteristic distance downstream. In the present analysis, the wake is artificially terminated at a large but finite distance downstream such that this truncation produces negligible effect on the body and the near wake. By virtue of the definition of S, the potential in the idealized physical problem is single valued and continuous throughout fi and S (although it is possible to include point singularities outside the domain O). 4-2.3 Uniqueness of the solution The uniqueness theorem for a function $ satisfying Equation (4.1) throughout fi can be expressed as: If at each point of the boundary of a given region either $ or its normal derivative d$jdn is known (Dirichlet, Neumann problem) and if § is single valued, then there exists at most one function $ satisfying these conditions, unless the value of $ is nowhere known, in which case $ is indeterminate to the extent of an additive constant (Hunt, 1980). Suppose a mathematical domain fim wi th boundary Sm geometrically identical to that of the physical problem (Figure 4-1) is constructed, then the potential $ m wi l l be identical to $ at all corresponding points of fim and fi (at least to within an additive constant) provided the following condition is satisfied: the potential 5>m and $ are identical at all corresponding points of selected regions S$> (where <& is known) of the surface Sm and S, and (d<& jdn)m and d<&/dn are identical at all corresponding points of the Chapter TV: Mathematical Analysis 91 remainder Sn (where d$jdn is known ) of these boundaries (Hunt, 1980). It is important to note that in principle any mathematical model which satis-fies the appropriate conditions may be used, even though the model may bear no physical resemblance to the flow. In the so called panel methods, the exact mathematical expressions derived in the following section are approximated. Note that it is not necessary to consider singularity distribution on the boundary surface in order to simulate the external flow field. For example, a point doublet of strength 27r, set wi th its axis parallel to a uniform flow of unit magnitude, produces a velocity field identical to the potential flow about a sphere of unit radius, in the region outside of the sphere. It follows that use of the geometric surface for the purpose of singularity dis-tr ibut ion does not act as a source of error in the mathematical formulation of the problem(Hunt, 1980). 4-2.4 Indirect Boundary Element formulation The fundamental solution of Equation (4.1) relates the potential $(x) generated at a field point x to a singularity of strength 7 ( f ) applied at a load point £. Note, here x and £ are coordinate vectors. Although the origin of coordinates x and £ are identical, it is absolutely necessary to reserve each set for a specific purpose. The classical singular solution can be written as, <D(x) = G ( x , £ h ( £ ) , (4-5) where G(x, £) is the ' two-point ' function (generally known as the Green's function) involving the coordinates (x, £) of the two points. Chapter IV: Mathematical Analysis 92 By differentiating $ with respect to x the velocity vector V ( x ) is obtained, Let dG/dx be H ( x , f ) , then V ( x ) = H(x ,eh(0 . (4-6) The Green's function G ( x , £) and the principle of superposition can be used to construct the solution to the physical problem in terms of an integral equation. The field points on S are designated by x s while £ s locate load points on S. Now singularities of unknown intensity 7 ( £ 8 ) per unit length of S are introduced. To complete the problem definition any specified singularity distribution T(c) per unit area in Q is also included. The net result of the 7 ( £ s ) and T(c) distributions at any field point x can be determined by integrating the fundamental solution, * ( x ) = / G ( x , £ s h ( £ s ) r f S ( £ s ) + / G(x,f)r(f)df7(c) + C . (4.7) Js Jn For existence of the solution it should be insured that the net flux across the infinite boundary is zero. This is achieved when / i(L)dS(es)+ [ r ( f ) r fn (c) = o. (4.8) J s Jn Similar operations using H ( x , £ s ) would yield the velocity at any field point x , v ( x ) = / H ( x , £ s b ( £ s H S ( £ s ) + / H(x,f)r(<r)dn(f) . (4.9) J s Jn Chapter TV: Mathematical Analysis 93 In principle, the only remaining formal step to arrive at a solution to the problem is to bring the field point x onto the boundary S. W i t h this, Equation (4.7) becomes ( x B ) = / G ( x B , £ B b ( £ B ) d S ( £ 8 ) + / G{xB,t)T{t)dn[$) + C. (4.10) Js Jn Similarly, from Equation (4.9), velocity normal to the boundary S is given by vnM= I F ( x 8 , £ 8 b ( £ s ) < 2 S ( £ s ) + / ^ ( x s , f ) r ( f ) d f i ( f ) , (4.11) Js Jn where Fix £ ) = ' ( x , ^ s j ^ , and fs represents an improper integral due to the singularity in F as £ s —• xs. In a well-posed problem, either $ ( x s ) or u(xs) is specified at every point of the boundary and Equations (4.10) and (4.11) are simultaneous integral equations which can be solved for the only unknown 7 ( £ s ) . Once 7 ( £ s ) is determined, Equations (4.7) and'(4.9) yield 3>(x) and V ( x ) , respectively, at any field point x of interest. It is important to notice that Equations (4.10) and (4.11) are scalar integral equations since the kernel functions (G,F), the singularities ( I \ 7 ) , and therefore are scalar quantities. If it is possible to integrate Equations (4.10) and (4.11) in closed forms and solve for 7 then the solution would be exact; however this is practically impossible and approximations have to be introduced. In the B E M inaccuracies arise from, and only from, numerical discretization and integration procedures. Thus by refining the approximations,, theoretically, any degree of precision is achievable. In practise, there is a trade off between computing time, effort and solution accuracy. Several different algorithms can be used to obtain worthwhile practical results. Chapter IV: Matherriatical Analysis 94 4.3 D i s c r e t e V o r t e x M e t h o d ( D V M ) In essence, this method is a version of the B E M where a point vortex is used as the distributed fundamental solution of Equat ion (4.1). The approach is physically more meaningful as a vortex shedding model is incorporated in the analysis. Equations (4.7) to (4.11) are discretized so that a numerical computational scheme can be ut i l ized. In general, in the discrete vortex method, the physical plane is mapped con-formally into a circle and analysed in the transformed plane. However, a complex physical configuration makes determination of the mapping function difficult. Fur-thermore, even if a complex mapping function is obtained, the rotating character of the system would require derivatives of the function thus reducing the inherent advantages of the mapping technique. 4-3.1 Model geometry In the present study, a Savonius rotor with th in blades is considered. As shown in Figure 4-2 the basic geometry is defined by the parameters: p/q - shape param-eter; 0 - size of the arc portion; R - radius of the rotor; and A - the blade aspect ratio. The blade contour is divided into m equal straight line segments (elements, Figure 4-3). A l l dividing points including the blade edges are considered as control points ( m - f 1 nodal points). Unknown discrete bound vortices Tsi(i — l , . . . , m ) , are located at the mid-points of each segment (singular points). The origin of the coordinate system coincides with the centre of rotation. The blade angular position is given by angle f3, whereas r s t , 7S{ (i = 1,... , m) and rni, 7 m - (i = 1,..., ra+ l ) represent the polar coordinates of singular and nodal points, respectively, when /? is zero (Figure 4-3). These coordinates are obtained in Chapter IV: Mathematical Analysis 95 F I G U R E 4-2 Simpified geometry of the Savonius rotor under investigation. Chapter IV: Mathematical Analysis F I G U R E 4-3 Description of the problem parameters. Chapter IV: Mathematical Analysis 97 terms of the number of elements and geometric parameters. In addit ion, the angle between the radius vector and the outward normal to the blade contour is calculated at all nodal and singular points, i.e., [i — 1,. • • ,m + l) andipsi (i = 1,. . . , m ) . For simplicity, the model reference frame is treated as complex Z-plane. The blade rotates in the positive j3 direction (counterclockwise) and the uniform flow is in the negative x-direction. 4.3.2 Flow model As discussed earlier, the flow past a bluff body separates at the sharp edges and the shear layers are shed downstream from the separation points. To represent starting shear layers an unknown free vortex is placed close to each separation point. These points are known as nascent points (Figure 4-3). Introduction of the first free vortices can be achieved in several ways. One approach would be to fix the positions of the first nascent vortices and make the strengths of the vortices unknown (Kuwahara, 1973; K i y a and Ar ie , 1977). Alternatively, one may specify the strengths of the vortices and to make their positions unknown (Sarpkaya, 1975). In this study the former approach is used. In the discrete vortex method, it is easier to carry out the analysis in the complex plane. A fundamental solution to Equat ion (4.1) can be given as a bound vortex Wi{z) = ^-Teiln{z- zst), (4.12) where Wi{z) = (4.13) and clockwise T is considered'positive. W^(z) is the complex potential at z due to a bound vortex of strength Tsi at Zg%. 5>i and ^ , are the potential and stream Chapter IV: Mathematical Analysis 98 functions, respectively. Thus the Green's function in Equation (4.5) has the form G{z,zsl) = J - l n ( * - * „ • ) . (4.14) Now Equation (4.7) can be written for the flow after k time-steps in the discretized form as follows, m 2k W{z) = ] T r s i l n ( z - zsi) + ^~Y1 TWJ \n{z - zwj) + C, (4.15) 1=1 y=i where C is a constant at a given time instant. Note, the $ in Equat ion (4.7) corresponds to the real part of W(.z). Similarly, Equation (4.9) can be written as, m r . 2fc _ w{z) = ~yT-^—, + — yl Twi (4.i6) V ^ 2 7 T {Z - 2 s i ) 2 7 T ^ ( 2 - 2 u ; y ) V ' Here w(z) is the wellknown complex velocity in the potential flow theory, w(z) = u — iv, (4-17) where u and v are the velocity components in x and y directions, respectively. B y looking at Equation (4.16) it is evident that w{z) —» 0 as z —» oo. But the model requires a uniform velocity — U in ^-direction at infinity. To satisfy this condition, Equat ion (4.16) and hence Equat ion (4.15) were modified in the following way: m „ . 2 k -p w{z) = — y- s i , +—y - 1 / ; ( 4 . 1 8 ) .V ; 2TT^(Z-ZSI) 2TT^(Z-ZWJ) V ; m . 2fc ^ ( ^ = R « L N ( 2 - *«) + T~ YlT^ ln(2 - - Uz + c- (4-19) . t ' = l • . j \ Chapter IV: Mathematical Analysis 99 Now the only remaining step is to apply Equat ion (4.18) at the (m + 1) nodal points of the blade contour. A t the begining of the kth time-step this equation contains m - f 2 unknowns r s i (i --- 1.. . . , ro) and Twj (j — 2k — 1,2A;). B y satisfying the boundary condition that the velocity normal to the blade at the nodal points is zero: <$>ni — tpni + Ini + 0 • angle between the x-axis and the normal vector at the ith node; u! — angular velocity of the rotor; (m -+ 1) simultaneous linear algebraic equations are obtained. Equat ion (4.8) yields the remaining relation which is also known as Kelv in ' s theorem: the sum of strengths of all bound and free vortices reduces to zero. B y solving Equations (4.20) and (4.21), the strengths of m bound vortices and two nascent vortices are determined, which, in turn, through Equations (4.18) and (4.19) establish the flow field. 4-3.3 Vortex shedding model A t any instant t. the strengths of m bound vortices and two nascent vortices are found by solving Equations (4.20) and (4.21). During the incremental time interval At, the shed free and the nascent vortices, move with the velocity induced by other vortices and the uniform flow at their respective locations, and the rotor advances (4.20) (4.21) Chapter IV: Mathematical Analysis 100 by an angle to At. A t the subsequent instant another set of unknown vortices are placed at the singular and the nascent points, and their strengths determined by solving the (m + 2) linear simultaneous equations. The position of the Ith vortex shed from the separation point is advanced in a small time interval At by the first order scheme: zwl{t + At) = zwi(t) + w{zwi)At- (4.22) • m p . 2k „ 27T ^ ( 2 u ; / - Zsl) 27T ^ ( z w / - Zwj) where w(zwi) is the complex conjugate of w(zwi), the complex velocity at the Ith free vortex due to all vortices except the Ith vortex itself and the uniform flow. A n y interception of a vortex by the blade is avoided as discussed later. The unit inviscid vortex expressed in Equation (4.14) induces a velocity whose magnitude is inversely proportional to the distance from its centre. Therefore, the vortices which approach each other too closely aquire extremely large velocities. To avoid this problem a rotational core of radius o is imparted to the free vortices while calculating velocities. In the core region the velocity is given by: w(z) = { 2 ^ f - i ^ (4.24) t r 27TC7 When a shedding vortex gets too close to the blade surface, to satisfy the zero normal flow condition, an equal and opposite image vortex wil l be generated on the boundary surface, thus cancelling the effect of its presence. Physically this cancellation is due to the viscous dissipation (Fage and Johansen, 1927; Ogawa, 1984). In the present study the l imi t ing spacing between a vortex and the blade surface for this cancellation is considered as a. Chapter IV: Mathematical Analysis 101 In real flow the core of a vortex grows with time. Giesing (1969) and Nagano et al . (1981), in their studies wi th flat plates and cylinders, suggested that o be of the form a = 2.24v/^i7, (4.25) where v is the kinematic viscosity and V is the time elapsed since the vortex was shed from the separation point. To reduce the computational cost, several investigators in their studies wi th flat plates and cylinders have suggested to replace the free vortices in a given cluster by a single equivalent vortex when located more than two diameters downstream (Sarpkaya, 1975). However, in this singular point vorticity model, it is important to preserve the mechanism of vorticity dissipation which in the above procedure is affected. Thus the cancellation effect of vorticity when two vortices of opposite circulation come close to each other is minimized. Hence, in the present study, representation of a cluster by a single vortex is avoided. In real flows only a fraction (=s 60%) of the circulation fed into the shear layers is found in the vortex clusters or in the concentrated vortices of the Karman vortex street (Mair and M a u l l , 1971). The loss of vorticity as a result of coming too close to the blade surface, and a small amount of cancellation between elementary vortices of opposite sign which come close to each other (Clements, 1973) are the only two mechanisms that could bring about some loss in vorticity, in this approach. Evidently these two mechanisms are not sufficient to account for an approximately 40% loss of circulation. Therefore Clements (1973), through his study wi th flat plates, concluded that the mechanism by which a large amount of vorticity is lost must be due to viscosity. Chapter IV: .Mathematical Ana-lysis 102 To account for this Stansby (1985) suggested a vorticity dissipation model T{t + At) = T{t)e-r>At, (4.26) where rj is a balancing parameter. Increasing rj increases the rate of shedding of vorticity into the wake by reducing the vorticity in the wake. In the present study the above model is used and the effect of different values of rj evaluated (Stansby suggested n = 0.03). 4.3.4 Pressure, torque and power Once the flow field is known, the corresponding pressure distribution and hence the torque and power produced by the rotor can be evaluated quite readily. Pressure distribution along the blade can be obtained by applying the unsteady flow Bernoull i Equat ion (4.2). The pressure coefficient defined in Equat ion (2.13) can be written as n P ~ Poo l \ 2 o ^ ( 4 - 2 ? ) w\\ 2 5 $ 1 ' U J U2 dt where |tu| is the magnitude of complex velocity. M u l t i p l y i n g the pressure by elemental area the force exerted on each element and hence the elemental torque, can be found. Integrating over the blade the total torque at a given instant can be calculated. Alternatively, by applying the generalized unsteady flow Blasius equation (Imai, 1974), Chapter fV: Mathematical Analysis 103 where: C represents an arbitrary path around the rotor excluding singular points; B is a closed loop along the blade contour; and W is as given by Equat ion (4.19). Average power generated over a period V* is now given as Torque and power coefficients are calculated as defined by Equations (2.8) and (2.4), respectively. 4.4 Modelling of the Wind Tunnel Blockage The mathematical model discussed above can be extended quite readily to account for the blockage imposed by the wind tunnel walls as shown in Figure 4-4. The two walls are placed symmetrically, at a distance H from the centre of the rotor. The blockage ratio in a two-dimensional case can be defined as where d is the diameter of the rotor. Each wall was divided into a finite number of elements (Mw) and unknown concentrated singularities placed at the centre of each element. This adds 2Mw additional unknowns to the problem. To satisfy the K u t t a condition at the downstream end of the wall , an unknown fixed nascent vortex is placed at a distance e£ from the end of each wall . Upstream ends of the walls being nodal points, automatically satisfy the tangential flow condition. The formulation results in 2(Mw + l ) additional unknowns. The boundary condition of no flow through the wal l , applied at 2(Myy + l ) nodal points, yields 2(Mw + 1) additional equations (similar to Equat ion 4.20). Thus there are 2(Mw + l ) + m + 2 (4.29) B = d 2E" Chapter IV: Mathematical Analysis 104 linear algebraic simultaneous equations and the same number of unknowns. The solution to this set of equations establishes the flow field and hence the required loading. Details of the numerical representation and computer application of the above discussed mathematical model are presented in Chapter V . Singularities i « 1 • i I 4 1-Tunnel Wall M w 1^1 U Rotor 2H i • 2 Nodes 3 • i M w —t • Nascent Vortex FIGURE 4-4 Geometry for blockage modelling. C H A P T E R V N U M E R I C A L I N V E S T I G A T I O N 5.1 F o r m a t i o n o f the S y s t e m M a t r i c e s 5.1.1 Unconfined flow model The set of linear simultaneous algebraic Equations (4.20) and (4.21) derived in Chapter IV can be represented in the matr ix form where A is a (m + 2) x (m + 2) square matrix and x , b are order (m + 2) x 1 column matrices, x represents the unknown strength of the distributed and nascent vortices, and b is a known matrix. The matrix A can be partitioned as where A j is known as the influence coefficient matrix and A 2 is a unit row matrix. Similarly, matrix b can be partitioned as A x = b , (5.1) (5.2) (5.3) where vector b i defines normal veloci ty at the nodal points due to known inputs 106 Chapter V.: .Numerical Investigation .107 (free stream, rotation, wake vortices, etc.,) and 2k b2 = - Y , T 3 = 1 after k time-steps. Twj is the strength of the j t h shedded vortex. Note, the last row of matr ix equation (5.1) represents Equation (4.21). Let one of the nascent vortices be specified as (Ti,zy) where T j is the strength and 2 l 5 the position. Similarly, let the distributed vortices be designated as (Tj,Zj) (j' — 2 , . . . , ro - f - .1.) and the nascent vortex at the other end as ( T m + 2 : - The unknown matrix x can then be written as x = r 2 V r m + 2 J (5.4) The element a,j of the influence coefficient matrix represents the velocity normal to the blade surface at the ith node due to a unit vortex at the j t h singular point. Using Equations (4.16) and (4.20), a,j can be written as: Real f _ L _ ^ ! _ l f,-=l,...,m+l; , \2-K{zni - Z j ) \ \j — 1 , . . . , m + 2; v " } where i and j are two positions fixed with respect to the blade. Now (zl — Zj) can be expressed as Zni Zj — TiyC where (r{j,&ij) are constants for given i, j and blade geometry. 0 is the blade angular position. Recalling that <j)ni — yjni + Ini + 0 (Equation 4.20), it is evident that aty is a constant irrespective of the blade rotation, i.e., the matrix A j need be calculated only once for a given geometry. Chapter V: Numerical Investigation 108 The element bn of the matr ix b i is the velocity at the ith node due to free stream, free vortices and blade rotation. Using Equations (4.16) and (4.20), after k time-steps, bu can be writ ten as bit = TniU! sin ipni — Real 2k _ »• }_wl 2 7T ^ {zni - Zwl) _ el4>-' \ - Ucos<j>ni, (5.6) where T^/ (/ = 1 , . . . , 2k) are the free vortices shedded in A; time-steps. The matrix equation (5.1) is solved using the Gaussian El imina t ion Procedure wi th partial pivoting, forward and backward substitution, followed by iterative im-provement. Once the unknown strengths of the vortices are found, the nascent vortices are renamed as shedding free vortices, i.e., zw(2k-\) ~ z l i Tw2k = + (5.7) zw2k = zm + 2i . T8i = Ti+i i = 1 , . . . , m . 5.1.2 Wind tunnel blockage model In the modelling of the wind tunnel wall confinement, matrix A in Equation (5.1) is of order (m -f Myy -f 4) x (m -r Mw + 4) and is a constant for the station-ary blade. However, for the rotating blade certain parts of the matr ix A become functions of time, i.e., the influence coefficients relating the blade to the tunnel boundaries vary with orientation of the rotating blade. In contrast, the unconfined model had a constant A matrix for both rotating and stationary blade. This implies Chapter V: Numerical Investigation 109 an increase in the computational effort not only due to an increase in the number of boundary elements but also as a result of the variable influence coefficient matrix. T ime marching process and the rest of the computational procedure remain essentially the same. 5.2 Calculation of Loading on the Blade This is achieved by applying Equation (4.27) along the blade. Several problems were encountered during application of Equations (4.18) and (4.19) along the blade, due to distributed concentrated vorticies. The improper integral which has a singularity in its kernel function in the domain of integration (Equation 4.11) was the reason for most of these problems. It is advised to use the Cauchy Pr incipal Value Integral with an added 'free term' from the singularity (Banerjee, 1981) to overcome this difficulty in the B E M . Definition of the 'free term' is rather complicated when concentrated, instead of distributed, vortices are used. 5.2.1 Velocity distribution along the blade For the purpose of calculating the velocity on either side of the ith element, the concentrated vortex in that element is treated as uniformly distributed. Thus the density of this uniformly distributed vortex is TSI/AS. where AS is the length of the element. The complex potential for a vortex sheet, with uniformly distributed vorticity of density 7 , which makes an angle 0 . with the positive real axis (Figure 5-1) can be written as where za and z\, are the coordinates of the end points. On evaluating the integral (5.8) Chapter V: Numerical Investigation 110 FIGURE 5-1 Uniformly distributed vortex. Chapter V : Numerical Investigation 111 it takes the form, t"ye ~ia \ ( z — za \ F(z) - — z In - r zi l n ( 2 - 2 b ) - za \r\(z - z a ) + ( z a - 25 ) 27T L V 2 - 2 b / • (5-9) The associated complex velocity is given by dF(z) i^e where l n ( f ^ ) is the key term. Let 2 - za = r a e i e " and 2 - zi, — ^^e^^f• (Figure 5-1). Now \ z - z b J n The velocity normal to the vortex sheet has a magnitude vn= — In — , 27T r\ whereas the velocity tangential to the sheet is Vt = ^-{6a ~ Ob)-Depending on the side of the sheet considered, (6a - 61) would be ZL7I. Figure 5-2 shows the velocity distribution due to a vortex sheet, compared to that of a concentrated vortex. The normal velocity tends to infinity at both ends of the element and is zero at the middle of the element. The tangential velocity is constant at -i- ^ and - ^ on either side of the element and is zero along the line of the element everywhere else. In the case of a concentrated vortex at the middle of the element, the normal velocity tends to infinity at the centre when approaching from either direction and is finite everywhere else. The tangential velocity is zero along the Chapter V: Numerical Investigation 112 F I G U R E 5-2 Velocity distribution for a uniformly distributed vortex sheet com-pared to that for a point vortex. Chapter V: Numerical Investigation 113 line of the element and becomes infinite at the centre. Both distributions are fairly similar away from the element. In the analysis, boundary conditions were satisfied only at the nodes. Looking at the velocity distribution for a concentrated vortex it is obvious that the 'no flow through the blade' condition is not satisfied along the element. This also suggests that calculation of the velocity at the centre of the ith element may cause some error. To avoid this situation calculations were made at the nodes i and i - r 1 (where the boundary conditions are satisfied) using Equation (4.18), and a linear variation in magnitude of the velocity along the element assumed. Now the velocity at the centre of the ith element can be given as w(zsi) \w{zni)\ + |u>(2NI+1)| exp(-i(j)si) ± —~- exp(-i<psl), (5.11) where w{zni) and w{zni+i) are the complex velocities at z n i and respectively, and <t>si is the inclination of the ith element to the positive real axis. -I- and - signs correspond to top and bot tom sides of the blade, respectively. Several other methods to determine the velocity distribution along the blade were also tested. In one approach calculations were made at the nodes and the velocity at the centre of the ith element obtained by averaging the velocities at ith and (i + l)th nodes, and adding the free term ± ( r s t / 2 A s ) exp(-idsl) to it, i.e., w{zsl) = ± ^ e x p ( - ? . 0 S ! ) . This yields the velocity at the centre, different from that in the tangential direction, on either side of the element (Figure 5-3a). Another method was to evaluate velocities at the nodes, and add averaged "free Chapter V: Numerical Investigation 114 FIGURE 5-3b Points at a radial distance t from the blade. Chapter V: Numerical Investigation 115 term" to obtain the total velocity at the node, i.e., w{zni) = w(zni) ± ( r ^' ' J 3 ' - 1 ) e x p ( - ^ n i ) , 4As where w(zni) is the velocity found by using Equation (4.18) at the ith node. The procedure is rational and gives logical results. Evaluat ion of Equat ion (4.18) was also attempted along points which were at a distance t away from the blade . This was tried at the nodes as well as at the singularities given by: ZA,Bni ~ Zm ± t e x p ( i > „ i ) ; ZA,Bsi = zsi ± t exp(i<f>si): where — and — signs denote the two sides of the blade (Figure 5-3b). As expected, the results were very sensitive to t and the method failed due to the singular char-acter of the solution at the blade. Alternatively, the velocity can be calculated at the ith singularity, using Equa-tion (4.18) neglecting the ith singularity, and adding the "free term" ±(T si/2As)e~t<t' This method also yields a non-tangential velocity on either side of the blade. How-ever, the results showed considerable improvement compared to those of the above method. None of these methods were as successful as the method used in the present analysis (Equation 5.11). 5.2.2 Calculation of 4jy term To find the pressure distribution using Equation (4.27), it is necessary to eval-Chapter V: Numerical Investigation 116 uate the term ^ where $ is the real part of W(z) in Equat ion (4.19), W(z) = * + t * , <9$ „ f dW — = Rea <^  — dt | df (5.12) Since the vorticity distr ibution is assumed to be uniform along the element in the calculation of velocity, it is necessary to use the same assumption in the calculation of Al though Equation (5.9) appears to have singularities, contributed by the first term, it can be rewritten as F(z) i~fe 2TI (z - za)z (z - zh)zi , M * > + ( « . - * ) Thus in the l imi t ing case as z —> za or z —» Z j , the associated terms cancel leaving the relation free of singularity. Therefore, the complex potential at the centre of the ith element after k time-steps can be written using Equation (5.9) together with contributions from the free stream, bound and free vortices as za \n(zsl - za) + (za - zb) Zwl) + Uzai + C[t), (5.13) where za and z^, are the end points of the ith element. By considering the time dependency of each term and simplifying it using the chain rule the following ex-le F(zsl) = ——Tei 2wAs zi, ln(zsl- - zt,) 2k - r — E T s n l n ( 2 s * ~ zsn) - — ^2 T w l ln(s*' II = 1 2TT Chapter V: Numerical Investigation 117 pression for ^ is obtained dF(zsi) i dTf + + + ± 2k i v - - r, {^si zwl) i uTSi(zni — zni+i) Zni\n(zsi — Zn{) — 2ru+l l n ( 2 s t ~~ zni+\) + + l — zni) zBi + i uUz£i + C(t ) , ie~l(p" 2TTAS t e - » > . . - dTsi 2TTAS ~dT e-1*'' dTei 2-nAs dt (5.14) where OJ is the angular velocity of the rotor and C is a constant at a given instant of t ime. Evaluat ion of the general term. 2.71 in Equation (5.13) using a computer leads to some error in the estimation of Recognizing that z — z,- = r,-c t f l ' , A,- = — 2^ In r, + i $i 6, i 2 7 T ' 2 7 T In r , . Here, - ^ corresponds to $ where the datum for 0, is somewhat ambiguous. Assum-ing 0, in the range 0 < 0, < 27r, and with a series of z,-'s such that the corresponding 0t's are in the 4th and 1st quadrants, the real part of AT w i l l have a discontinuity of 2TT w-hen crossing the positive real axis. It could be avoided by making the range — Ti < 8i < 7T. This would then result in a similar discontinuity when crossing the Chapter V: Numerical Investigation 118 negative real axis. Thus a discontinuity can be avoided by appropriately selecting the datum corresponding to the situation. In the computer algorithm, this problem was carefully studied and treated accordingly so that no discontinuity in 6t would occur along the blade. The results were found to be quite sensitive to the value of dTsi/dt. Therefore, it was evaluated with a second order accuracy by using the Gaussian Three Point Differential Formula at the middle point. The experimental data available in this study are not sufficient to evaluate the time-dependent constantC, which contributes to the value of ^ y , as unsteady pressure measurements were not carried out. To make the matter worse, variations in the range of 0, to obtain a smooth $ along the blade adds yet another time-dependent constant to Qjj term. It is indeed demanding, from the computational point of view, to keep track of these constants. Fortunately, the final results obtained for the average pressure distributions showed proper trends and none of these unknown constants affect prediction of the rotor performance (torque, power) since it depends only on the difference in pressure on the two sides of the blade. 5.2.3 Evaluation of torque and power coefficient. The torque coefficient can be found either by integrating the elemental torques due to the differential pressure over the blade or by applying Equation (4.28). Both methods were tested in the present study and led to similar answers. Equat ion (4.28) was simplified by using the 'residue theorem' and the chain rule for differentiation. The discretized, simplified form of Equat ion (4.28) can be Chapter V: Numerical Investigation 119 writ ten as . m 2k l_yy 71 izsi - Zwl) 2i -»-{: ££ +2U Sr"2"+1 *™)- (5'15) where rSJ is the magnitude of the radius vector at the ith singularity. Now the torque coefficient is estimated using the equation c - = WvW • < 5 ' 1 6 ) The power produced is calculated by discretizing the Equation (4.29) ^ * = —y^w (5.i7) p k A i h ^ where is the average power over k time-steps, T\ is the torque generated at the Ith time-step and UJ is the angular velocity of the rotor. The average power coefficient after k time-steps is given by 5.3 C a l c u l a t i o n P r o c e d u r e 5.3.1 Computational algorithm A block diagram for the computational process is shown in Figure 5-4. A t t = 0 the unknown distributed and nascent vortices T ; (j = l , . . . , m -j-2) are calculated by satisfying the boundary conditions and Kelv in ' s theorem (Equation 5.1). Next , the nascent vortices r^.j and r„,2 are shed downstream and all the vortices in the wake are convected according to Equations (4.22) and (4.23), in time-steps of interval At up to an instant tn^i = tn -f AT [AT = KAt, K = Chapter V: Numerical Investigation 120 Establish Problem Geometry t = 0 k = 1 Set Initial Conditions I E Calculate Solve Ax = fe Find the Flow Field G No k = k+1 FIGURE 5-4 The block diagram of the computation process. Chapter V: Numerical Investigation 121 an integer; Inamuro et al . , 1983). During the interval At vortices are convected without satisfying the boundary conditions at the blade. A t the end of the Kih interval, i.e., at t = tn+i-, strength and position of the distributed and nascent vortices are again evaluated by satisfying the boundary conditions and the same process is repeated. Here, the time interval At is chosen smaller than AT inorder to reduce the computational cost and to improve the accuracy of the convection calculation. To evaluate using the Gauss Three Point Formula at the middle point, the computational process is advanced by three time-steps and then returned to the second step to calculate the loading. Thus the earliest loading information is available only at the second step. 5.3.2 Computational parameters There are several important parameters which must be established before the calculation process begins. They are: ts distance between the nascent vortex and the edge of the blade; m number of elements representing the blade; At time-step size; o core radius of the vortices; n balancing parameter (Equation 4.26); K number of small time-steps At per large time-step AT (Resolution Parameter). Obviously, an increase in the number of elements would lead to a better accu-racy, however, it would also escalate the computational cost. Thus a compromise between these two competing parameters is necessary. The effect of the number of Chapter V: Numerical Investigation 122 elements on the final results was systematically tested inorder to arrive at a cost effective value. The size of the viscous core (o) is an important parameter which indirectly determines the amount of vorticity dissipation in the wake. K i y a and Arie (1977) suggest o to be 0.05 x R where R is the characteristic radius. W i t h this as a guideline, three different values for o were tested. Considering the fact that es should be greater than o, Ogawa (1984) selected es as 1.1 x o. It can be presumed that there is an opt imum value for ts and location of the nascent point, which may depend on the position and the tip-speed ratio of the blade. In the present study, however, the nascent vortex position is fixed on the extension line of the blade contour. Stansby (1985) has suggested a value for the balancing parameter rj to be 0.03. In the present study, several different values of n were used to assess their effect. Based on accuracy and computational cost considerations, the resolution pa-rameter K was taken to be 2. A change of K from 1 to 2 reduces the computational cost by at least a factor of 4, without substantially affecting the final results. The most important parameter in the computational process is the time-step size At. This should be small enough for the convecting vortices to follow stream lines, and large enough to reduce the computational cost. A smaller value of At was necessary so that the free vortex may not intercept the blade after one time-step. This is particularly necessary at a large tip-speed ratios; the blade rotation angle uAt, in one time-step may become large raising the possibility of vortex interception. Based on the above consideration together wi th values suggested by Clements (1973), K i y a and Ar ie (1977), and K a t z (1981), At was selected in the range 0.03 - 0.1 x (R/U), depending on the tip-speed ratio A. Chapter V: Numerical Investigation 123 Taking a constant angular displacement of the rotor per time-step of 3° the following relation can be obtained, A t larger values of A, At gets quite small resulting in a slow rate of convection downstream. To avoid this the following model was used at higher tip-speed ratios, where 6 is determined according to the range of A of interest. 5.4 Effect of Computational Parameters In order to determine the computational parameters which may lead to an im-proved calculation efficiency, several test runs were carried out and the effect of each parameter on the final result established. The calculations were performed, in double precision, using the Amdah l 5850 main frame computer. Al though the increasing 0 direction was taken as counter-clockwise in the modelling procedure, and the free stream flow along the negative x-direction, the results are presented wi th the flow from left to right with clock-wise rotation of the blade. The computer C P U time required for the calculations increases exponentially with the number of time-steps. Therefore, a restart facil-ity was incorporated in the program so that the computations can be stopped and restarted as desired. 0.04363372 (5.19) At = XU (5.20) 5.4-1 Number of elements m A word concerning the choice of number of elements would be appropriate. The Chapter V: Numerical Investigation 124 min imum length of the element is l imited by the size of the vortex core o to satisfy the boundary condition at the blade. This gives the maximum value for m as 33. W i t h this as a background three sets of results were obtained wi th the number of elements being 20, 25, and 30 with all the other parameters held constant: p/q Geometry parameter = 0.2; Q Circular arc radius = 0.191 m; A Tip-speed ratio = 0; 0 Blade angle = 120°; At Time-step size = 0.004 s; K Resolution parameter = 2; U W i n d speed = — 7 m/s; V Balancing parameter = 0; k Number of time-steps = 100; 6 Blade arc angle = 135°. The results obtained for torque coefficient (CV) as affected by the number of elements is shown in Figure 5-5. Note, the plot extends over the time interval of 0.8 5 during which the first vortex cluster moves approximately two rotor diameters downstream. The corresponding flow patterns in the wake after 60 time-steps are also presented in Figure 5-6. It is apparent that the results for m = 25 and 30 are essentially the same with the corresponding flow patterns remaining vir tual ly unaffected. The C P U time involved in the calculation process (a measure of cost) is compared in Table 5.1. Chapter V: Numerical Investigation 125 1.2-1 T T ime (Sec) F I G U R E 5-5 T ime history of the torque coefficient as affected by the number of elements. Chapter V: Numerical Investigation 1.5-1 126 Y 1.0-0.5-0.0 -0.5H -1.0 -1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 1.0 0.5 H 0.0 -0.5 -1.0 -1.5 H x x X X x x x x x T x x X x * x X x x X * * * ^ x x x * x x x < x x x M = 30 M = 25 M = 20 ^ i i i 1 1 l 1 i 1 . -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 X (meters) F I G U R E 5-6 Flow patterns after 60 time-steps as affected by the number of elements! Chapter V: Numerical Investigation 127 Number of C P U Time Elements (m) (s) 20 173 25 217 30 260 TABLE 5.1 Effect of number of elements on the computational cost Based on this information m — 25 was selected in the subsequent analysis. 5.4-2 Vortex core radius (limiting distance o) Another important parameter in the flow model is a , the core radius of the vortex or the l imit ing distance between a vortex and a solid surface. As mentioned before, it indirectly determines the dissipation of vorticity in the flow. To establish the effect of o on the final results, several calculations were carried out with a — 0.017?, 0.05i? and 0.1 R, keeping all the other parameters fixed. The results for time variation of the torque coefficient are shown in Figure 5-7. The plots for o = 0.05/? and O.l i? are similar in trend except for a phase shift suggesting the process of vorticity dissipation to be dependent on o. However, the character of torque variation wi th o = O.Oli? is fundamentally different. This disparity is also reflected in the corresponding flow fields (Figure 5-8) where the flow pattern for o = O.Oli? at the upstream blade is rather unrealistic. The vortex core radius o is directly related to the position of the nascent vortex (es — 1.1a). When o is small , the nascent vortex being too close to the singularity at the t ip of the blade causes this unrealistic behaviour. In the present study, the value of o was fixed at 0.05/?. Chapter V: Numerical Investigation 128 Time (Sec) FIGURE 5-7 Effect of the l imit ing distance parameter o on the time history of CT-Chapter V: Numerical Investigation 1.5 1 0.5H 0.0 -0.5-J -1.0-1 Y 5 1.0 0.5 0.0 -0 .5 -1.0 -1 .5 -1.0--1.0 x x x x x v X x X x* x x x x X X X X X x T 1 1 r XX X x * x x x X x ^ x x X x x X > < I I i i r x x X X x^x x a = 0.10 R T r ff = 0.05 R T r a = 0.01 R -1.5- i \ 1 1 1 T 1 1 1 1 p -0 .5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 X (meters) F I G U R E 5-8 Flow patterns after 60 time-steps as affected by o. Chapter V: Numerical Investigation 130 5.4-8 Position of the nascent vortex es Three different values, es = l.lo, 1.5a and 2.0a, of the distance between the nascent vortex and the tip of the blade were tested to evaluate its effect on the final result. Of course, es should be greater than o to satisfy K u t t a condition at the t ip. The results for the torque coefficient variation for different e3 are presented in Figure 5-9. The curves show basically similar time histories with a slight difference in the phase and the average magnitude. The corresponding flow patterns obtained at 60 time-steps are shown in Figure 5-10. They are also quite similar suggesting the effect of ee on the results to be relatively insignificant. However, as ts = 1.1a predicts the torque coefficient closest to the experimental value shown in Figure 2-30 it was selected for the model. 5-4-4 Balancing parameter n A similar procedure was adopted to evaluate the effect of the balancing param-eter (n) on the final predictions. The results are shown in Figure 5-11. Except for a small change in amplitude, the rotor response is essentially the same. The similarity of the wake is also apparent (Figure 5-12). However, as the computing cost involved is slightly higher for none-zero r/ it was set at zero in the present model. 5-4-5 Time-step size At In a time marching computational process, the size of the time-step is an impor-tant parameter. Conceptually, the smaller the time-step the better the prediction. However, this may not be true for the flow field. Calculation of the blade loading using smaller time-steps may give misleading results due to a large number of small vortices close to the blade causing an unrealistically large velocity. This effect is Chapter V: Numerical Investigation 131 1.2 0.9H 0.6H 0.3 H 0.0 H 7— / \ DVM - Solu. — 1 1 / , / ^ —^ is X \ V . C-S i . i v es = 1.5 o cs = 2.0 a \^  / / \ V 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 T ime (Sec) FIGURE 5-9 T ime history of the torque coefficient for different nascent vortex positions. Chapter V: Numerical Investigation 1.5 1.0-0.5 0.0 - 0 . 5 H -1.0 Y -1.5 1.0 0.5 0.0 - 0 . 5 -1.0 -1.5 1.0-0 .5 -0 .0 -- 0 . 5 -- 1 . 0 --1.5 X x x x x X x x X v x x x x X X v X x ^ x rn • • n • • ® -Xx x x xx v x Xx x x XX X x ^ x x x X x ^ x x X x x X > < c s = 2.0 a c, = 1.5 a i i r c s = 1.1 a 1 """""T™* i i i i i 1^ -""T™ -"•""T"™"" -T~"" i I - 0 . 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 X (meters) FIGURE 5-10 Flow patterns after 60 time-steps for different es. Chapter V: Numerical Investigation 133 T ime (Sec) FIGURE 5-11 Var ia t ion of the torque coefficient as affected by n. Chapter V: Numerical Investigation 1.5 1.0-0.5-0.0--0.5--1.0--1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 1 0.5-1 0.0 -0.5 -1.0 >x x X ^ x x x  xx X x x X X X x x x xx X x X x ^ x x X x x X > < fid* i r X X X / x x x x x £ > x X x x x X to* •n = 0.06 •n = 0.03 r) - 0.00 -5 f i l l 1 1 I 1 1 1 | -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 X (meters) FIGURE 5-12 Flow patterns after 60 time-steps for different n. Chapter V: Numerical Investigation 135 F I G U R E 5-13 Time history of Cj as affected by the time-step size. Chapter V: Numerical Investigation 1.5-1.0 0.5H 0.0 - 0 . 5 --1 .0--1.5 1.0 0.5 0.0 - 0 . 5 -1.0 -1.5 xxx n X X ^ O n or • • • • XX * x X x X x *xxxx* At = 0.008 At = 0.004 "I P At = 0.002 1 1 I I 1 1 I I 1 I ' - 0 . 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 X (meters) F I G U R E 5-14 Flow patterns at t = 0.792 s for different time-step sizes Chapter V: Numerical Investigation 137 evident in the predicted torque coefficient shown in Figure 5-13. A larger value of At leads to a relatively smaller amplitude for CT as the vortices are convected downstream to a longer distance. This is v ividly shown by the flow patterns in Figure 5-14 (at t = 0.792 s). Several interesting features are apparent. Although the wake vortices are similar in the basic shape, it is evident that there is an in-crease in the radius of curvature of the separating shear layers with an increase in At. Note, a larger At causes the vortices to come closer to the blade and hence be destroyed whereas at a smaller At they continue to persist. Since dissipation of vorticity in the wake plays an important role in the analysis a smaller At may not always give better results. In fact, in the present case, At = 0.008 s predicted the torque coefficient closest to the experimental results. There is yet another aspect to the selection of At: the computational cost involved in the execution of the program. The C P U time for the results shown in Figures 5-13 and 5-14 are given in Table 5.2. Time-Step C P U Time Size (s) (s) 0.002 1275 0.004 217 0.008 75 T A B L E 5.2 Effect of time-step size on the computational cost A n enormous increase in cost with a decreasing At is apparent. Based on these considerations, the stationary rotor analysis was carried out using At — 0.004 s. C H A P T E R V I R E S U L T S A N D D I S C U S S I O N From the foregoing discussion it is apparent that the mathematical model and its numerical evaluation, as developed in the last two chapters, are indeed quite chal-lenging both in terms of fluid dynamical phenomena and computational complexity. The difficulty is further accentuated by a large number of variables, some rather i l l defined, e.g., the l imi t ing distance parameter a and A r , which indirectly govern vorticity dissipation; the balancing parameter 77, which directly controls the vis-cous effects; and the nascent vortex position cs. Obviously by a systematic change of wind speed, rotor geometric variables, wall confinement, init ial conditions, and computational parameters an enormous body of data can be generated. However, the objective here is to develop methodology for approaching such a complex situa-tion and assess its effectiveness in the light of experimental data presented earlier. Hence, for conciseness, only a typical set of results useful in establishing trends are recorded here. To begin wi th , the static Savonius rotor is analyzed with different orientation of the blade; and the starting torque, wake geometry and frequency characteristics discussed. This is followed by the results for the rotating case accounting for the effect of tip-speed ratio (A), and blade geometry parameter [p/q). F inal ly , influence of the wind tunnel wall confinement for both stationary and rotating blades is assessed. 138 Chapter VI: Results and Discussion 139 6.1 Stationary Blade The flow around a stationary blade at different angular positions and the cor-responding average strating torque characteristics for the Savonius rotor were es-tablished using the mathematical model discussed in Chapters IV and V with the computational parameters: m = 25 ; o =0.05# ; is = 1.1a ; n = 0 ; At = 0.004 s. To minimize the computational cost the resolution parameter K (section 5.3.1) was fixed at 2. q (Figure 2-1) was taken as 0.191 m and p/q fixed at 0.2 to facilitate comparison with the experimental results presented in Chapter II (section 2.6). The wind speed was maintained at 7 m/s simulating the experimental conditions. As discussed in section 2.5 the pressure distribution over the blade varies with the blade angle of attack. Depending on the angle the flow may have different separation points. However, the mathematical model developed assumes the flow to separate at the tips of the blade, which is valid only for angles (/?) between 90° and 120°. Al though the model agrees with the separation points in this range it cannot be expected to give a constant base pressure on the backside of the blade as observed experimentally due to its potential character ( D V M ) . At other angles, the flow does not separate at the tips. However, due to high Reynolds number the separated shear layer remains close to the tip of the blade making the flow geometry not much different from the separation at the tips. But in the calculation of loading it makes a significant difference due to a considerable change in the pressure distribution. Chapter VI: Results and Discussion 140 T w o distinct procedures were used to account for these effects. The results (section 2.5) show that separation occurs due to an adverse pressure gradient and the pressure remains at the separation value in the wake. To minimize the effect of this discrepancy, the experimentally established separation points were used in the analysis and the pressure calculated at the separation point was held constant in the wake. The results obtained using these assumptions are named as " D V M with surface separation". In the second approach, the shear layers separate at the blade tips as implicit in the D V M model, however, the base pressure is assumed constant. This is referred to as the " D V M with Cpb = constant". It is of interest to point out that actual value of Cpb is not required as it does not contribute to the torque. The time variations of the torque coefficient predicted at 0 = 0° using the above three approaches are compared in Figure 6-1. Constant base pressure cases give almost the same result whereas the potential flow solution yields relatively higher values for Cj. High frequency irregularities present in all the three solutions may be attributed to the free vortices shed from the front tip passing close to the blade. There is no evidence of low frequency vortex shedding (Karman vortex shedding) at this angle. The flow patterns at k =30, 60 and 90 time-steps are presented in Figure 6-2 substantiate the above observation. Figure 6-3 shows variation of the torque coefficient with time at 0 = 30°. Per-haps the most str iking feature in these curves is their smoothness. Plots representing the constant base pressure assumption are fairly close to each other. However, the solution using the potential flow assumption gives a torque coefficient well above the other two as before. Again there is no evidence of the Ka rman vortex shedding at this angle, i.e., the rotor continues to behave as a fiat plate. Chapter VI: Results and Discussion 141 1.2 1.0-0.8-0.6-0.4-0.2-0.0-[3 = 0 DVM Seporatton C p b  tip recovery tip consfT surface const. 0.0 0.1 0.2 0.3 0.4 0.5 T ime (Sec . ) FIGURE 6-1 T ime history of the torque coefficient at 0 = 0°. Chapter VI: Results and Discussion 142 1.5-r 1.0-0.5--1.0--1.5-j 1 1 1 1 1 1 1 : ! 1 1 1 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-2 Flow patterns at (3 = 0°: k = 30,60,90. Chapter VI: Results and Discussion 143 (3 = 3 0 ° PVM Separation C p b tip recovery tip const. surface const. 1 1 r n 1 i 1 1 1 f 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time (Sec . ) F I G U R E 6-3 Time history of CT at 0 = 30° . Chapter VI: Results and Discussion 144 The corresponding flow patterns obtained at time-steps 30, 60, 90, 110, 130, and 150 are shown in Figure 6-4 . These patterns are quite similar to that for a flow past an aerofoil. A l l the vortices shed from the front edge of the blade are destroyed and the flow separates at the rear t ip. Thus validity of the results predicted by the constant base pressure assumption are in doubt. The flow patterns suggest attached flow over the blade simulating an aerofoil at a high angle of attack without stalling. However, this is rather unrealistic. Therefore the results obtained at this angle using the potential flow solution would be highly optimistic and unrealistic. When the blade is oriented at 60° to the free stream, a trace of a low frequency vortex shedding becomes apparent (Figure 6-5) which is also evident in the flow patterns (Figure 6-6). As can be expected, the time needed to reach a steady state depends on the init ial conditions and the blade orientation. Note, the solution corresponding to t ip separation with a constant base pressure shows a periodic character. On the other hand, the case of the surface separation leads to a periodic solution with a low frequency mean associated wi th a high frequency modulation (Figure 6-5). The potential flow solution was observed to reach a steady state at around 0.35 seconds, with several peaks, corresponding to formation of vortex clusters, having a periodicity of K 0.41 s. The periodic nature of the flow is evident in the flow patterns shown in Figure 6-6. Evolut ion of the vortex clusters and their drift downstream is apparent. The first vortex cluster from the bottom edge separates at around 30 time-steps fol-lowed by a similar separation from the top at around k =60. Around 50 time-steps, the first vortex cluster separated from the blade passes two diameters downstream where the solution can be approximated as having reached a steady state. Immedi-ately afterwards, at around 80 time-steps, the second cluster from the bottom edge Chapter VI: Results and Discussion 1.5 145 1.0H 0.5 0.0--0.5--1.0--1.5 1.0-0.5-Y 0.0 -0.5 -1.0-1 -1.5 1.0 0.5 0.0 -0.5--1.0-nuiiEiiirrBiiiMirwrm-T i r X^\s>_^jnrrjm: n L- II • li I- II !• !• I*P H !• H • 11. H t. <. ii .t i M , |. n H H H f7Trn-n-pTTr TTrF--r-t—I1-T r T r j f f i i i i n i i i r n i . 1 • i . i H ' M i , l i M . | | | | | T i r m : k = 30 T r k = 6 0 TOW win ; n-i i t.ri.|..m.»:.itirC33-k = 9 0 -1.5 i 1 1 1 i i 1 1 r -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-4 Flow patterns at f3 = 30°: fc = 30,60,90. Chapter VI: Results and Discussion 146 2.0-1.5-1.0-0.5-0.0--0.5--1.0--1.5 -2.0 1.5-1.0-0.5-Y 0.0 -0.5 -1.0--1.5--2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 y n||lll—III '"»"•"'"" "• "•iiiniiiiiiiiiii"'»n»"iMiiiiiiiiiiimrmT k = 110 T— i r T P T P m i H i m i i m m • millliTTTTTTTTT k = 130 T P i r T 1 1 1 1 P T 1 P muHUBHiHHnimiHmminimi mm ••••.•• I l l| l l, lm„„„„ n „ m m t T ^ k = 150 -2.0 H 1 1 1 1 — I — I — i — i — i — i — i — i — i — i — i — • ' 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 X (meters) F I G U R E 6-4(cont.) F low patterns at 0 = 30°: k = 110,130,150. Chapter VI: Results and Discussion 147 1.8 1.54 124 0.9H 13 = 60 DVM Seporatfon 9pb flp recovery t ip" const. surface const. A T ime (Sec . ) F I G U R E 6-5 Time history of CT at 0 = 60° . Chapter VI: Results and Discussion 148 1.5 1.0H 0.5 0.0--0.5 -i.oH -1.5 1.0 0.5-Y o.o--0.5-- 1 . 0 --1.5 1.0 0.5-1 o.o-l -0.5 -1.0 - 1 . 5 k = 10 k = 20 k = 30 x * x k = 40 k = 50 rftb X X x X x* k = 60 -0.5 0.0 0.5 • 1.0 1.5 2.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 X (meters) F I G U R E 6-6 Flow patterns at 0 = 60°: k = 10,20,30,40,50,60. Chapter VI: Results and Discussion 1.5-149 1.0-0.5-0.0--0.5--1.0--1.5 1.0 H 0.5 Y o.o -0.5 -1.0-1 -1.5 1.0-0,5-0.0 -0.5-j -1.0-I -1.5-* x x So x x x x x X x x x x X x x xxxxxx X v XxX x x^ xxxxxx X X x >$< x x x x x x x k = 70 k = 80 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-6(cont . ) Flow patterns at /? = 60° : k = 70,80,90. Chapter VI: Results and Discussion 150 separates which clearly shows mixing of vortices with opposite rotation (k =90) resulting in a partial cancellation effect among the vortices. Note, evolution of the flow closer to the blade follows the same pattern and the process continues in a similar way shedding vortex clusters of opposite signs intermittently. The effect of the vortices far downstream becomes progressively negligible on the blade. The periodic variation of CT (Figure 6-5) is primari ly governed by the periodic nature of the flow described above. A t 90 time-steps (Figure 6-6) the distance between the two vortex clusters shed from the bottom edge was found to be approximately 2.5 m . W i t h the free stream velocity of 7 m / s , it gives a vortex shedding frequency of fa 2.8 Hz resulting in a period of fa 0.38 s which is comparable to the period of the CT fluctuation as shown in Figure 6-5. The nature of the flow at 0 = 90° is quite similar to that at 6 0 ° . The flow separates at the tips as in the real case. Therefore only two sets of results are presented for the variation of the torque coefficient (Figure 6-7). B o t h numerical solutions show periodic nature once the steady state is established. Compared to the case of 0 = 60° the time taken to reach the steady state is significantly larger (ss 0.65 S for 0 = 90° compared to fa 0.35 s for 0 = 60°) . Obviously, this is due to a higher bluffness at 0 = 90° , which also results in a lower Strouhal frequency as reflected in the time history of CT-The corresponding flow patterns are shown in Figure 6-8. Unlike the earlier case, here vortex clusters from both top and bottom tips grow almost at the same rate init ial ly. A s a result shedding of the first vortex cluster is delayed. As the time increases, the top cluster advances further and ultimately sheds around 0.7s (k fa 90). The cluster from the bottom edge also separates soon after(k = 110). Consequently attainment of the steady state is delayed. A t 150 time-steps, the two Chapter VI: Results and Discussion 151 1.2 0.8-0.4-0.0--0.2 _ 1 1 1 1 1 .1 [\ i » i • i » 1 1 1 1 \ • • f i 1 \ • > i • / i • < 1 \ \ \ A. ! • • '• -\ ] V Ws / • J t V. « • i i i • ( \> r t » * li A i / + f3 = 9 0 ° PVM Separation C p b tip recovery tip const. t 0.0 0.2 0.4 0.6 0.8 T ime (Sec . ) 1.0 1.2 FIGURE 6-7 T ime history of CT at 0 = 90° . Chapter VI: Results and Discussion 1.5-152 1.0 H 0.5 0.0 -0.5--1.0--1.5 0.5 -0.5H -1.0 -1.5 .0.5 -1.0 -1.5 ^^^^^^ ^ x x x * X * x * k = 10 k = 40 i i l I \ X x3< ) x x x x k = 20 i i i i i * i -6 xx X x k = 50 i i i i C x X X x x x x* k = 30 k = 60 i r -0.5 0.0 0.5 1.0 1.5 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 X (meters) FIGURE 6-8 Flow patterns at (3 = 90°: k= 10,20,30,40,50,60. Chapter VI: Results and Discussion 1.5-, 153 1.0H -1.0-_1-5 -1 1 1 1 1 1 ; ! : 1 : 1 1 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-8(cont . ) Flow patterns at f3 = 90° : k = 70,80,90. Chapter VI: Results and Discussion 1.5 154 1.0 0.5 0.0 -0.5-1 -1.0H -1.5H 1.0 0.5-Y o.o--0.5 -1.0H - 1 . 5 1.0-1 0 . 5 , 0 . 0 • 0 . 5 -- 1 . 0 -- 1 . 5 XX KXXXXX, X* X r I Do D F k = 110 x * x > x *>S<>K k = 130 X T * * # „ x JL>& x X X x X x E xxx ^ x \ x X  x X x *>*xxxx *xx xx*x>< x x x £ x x < * £ x x xxx X • r k = 150 i . I I I I T i i i . i -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-8(cont<) Flow patterns at 0 = 90°: k = 110,130,150. Chapter VI: Results and Discussion 155 vortex clusters shedded from the top edge were observed to be approximately 3 meters apart corresponding to a vortex shedding frequency of % 2.3 Hz or a period of 0.43 s, which is comparable to the period of CT variation (Figure 6-7). It was necessary to carry out the computational process long enough to establish periodic nature of the flow. In general, it took around 0.35 — 0.7 s for the flow to attain a steady state (k 40 — 60). The true periodicity of the flow can be assessed only after this point. For k = 100, usually it was possible to observe only one cycle of variation. To help establish the periodic character of the flow more conclusively, the computational process was extended to 200 time-steps for 0 — 120° and 150°. As can be expected, the number of time-steps has an enormous effect on the computational cost. Since every time-step adds two vortices, the number of free vortices and hence the amount of calculations needed to evaluate the flow field, increases with the number of time-steps. Table 6.1 shows variation of the computational cost ( C P U time) with number of time-steps. Number of C P U Time Time-Steps (k) 00 50 75 100 217 150 550 200 1278 250 2811 TABLE 6.1 Effect of k on the computational cost Thus the cost increases exponentially wi th the number of time-steps. Hence, having established periodic behaviour of the flow, the computation was limited to Chapter VI: Results and Discussion 156 the first cycle. The prediction of CT variation wi th time for 0 = 120° is presented in Figure 6-9. This clearly shows a regular periodic behaviour specially with the constant base pressure case. The potential flow solution also shows a periodic pattern with essentially the same period. The corresponding flow patterns are shown in Figure 6-10 up to A" = 200 (note the scales on different pages are not the same). Shedding of the first vortex cluster from the top edge is apparent at k — 40. Formation of the vortex cluster at the lower edge and its shedding follows. The first cluster travels two diameters downstream around k = 50. A t k = 100 shedding of the second vortex cluster from the top edge is noticeable. During the period k = 100 — 200 (0.8 — 1.6 s), it is obvious that the distance between the vortex clusters shed from the top edge remains constant. At 200 time-steps shedding of the third cluster from the top edge can be seen. The distance between the second and the third cluster is approximately equal to that between the first and the second cluster (=: 3.5 m) . This represents a vortex shedding frequency of about 2 Hz which corresponds to a period of 0.5 s, the same as the CT variation (Figure 6-9). However, the vortices shed from the bottom edge do not seem to have as regular a behaviour as the vortices shed from the top edge. It is of interest to note that distance between the two clusters shed from the bottom edge gets shorter with time. A t k — 200 mixing of the second clusters from the top and bottom edges, leading to a cancellation effect, is apparent. This results in a spacing between vortices in the top row to be different from that in the bot tom row in the near wake. However, for a distance greater than around ten rotor diameters downstream an evenly spaced street of vortex clusters wi th alternate signs is established. Chapter VI: Results and Discussion 157 1.4 f 1.1-P = 120 DVM Seporatfon 9pb tip _ recovery tip consfT 0.84 C T 0.5 0.2 H - o . H -0.4 0.0 0.2 0.4 0.6 0.8 T ime (Sec . ) 1.0 1.2 1.4 FIGURE 6-9 T ime history of CT at .0 = 120c Chapter VI: Results and Discussion 158 1.5 1.0 0.5 0.0 - 0 . 5 -1.0 -1.5 1.0 0.5 Y o.o - 0 . 5 -1.0 -1.5 1.0 0 .5 -0 .0 -- 0 . 5 -1.0 * x x \ k = 40 T r xx X x X x x X x k = 60 T 1 xVx X Y XX xx x V * x fe •5U 5Bfe%^  k = 80 i i i i i i l I | i 1 | | - 0 . 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-10 Flow patterns at /? = 120°: k = 40,60,80. Chapter VI: Results and Discussion 159 Chapter VI: Results and Discussion 160 2.5-1 -2.5 1.5 0.5 -0.5 -1.5 -2.5 1.5 0.5 -0.5 -1.5--2.5 x V x ^ k = 160 T — 1 — i — • — r " i — 1 —r • k = 180 ' I ' I—1—I—1 I 1 T — 1 — r 4* k = 200 -> 1 — . — | ,—p—.—p—,—p i 1 i • i — • — r -0.5 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 X (meters) F I G U R E 6-10(cont.) Flow patterns at 0 = 120°: k = 160,180,200. Chapter VI: Results and Discussion 161 Note, the cancellation of vortices is identified through following development of the wake over a relatively long distance (around ten rotor diameters) thus adding substantially to the computing cost. In general, the distance required for the wake to attain steady state would also depend on the blade inclination. In the earlier results for /5 = 6 0 c , 9 0 ° , as the numerical integration of the equation extended only over 4 — 5 diameters to minimize cost, the vorticity cancellation process won't be complete, and hence the calculated frequency must be corrected to account for this effect. As in the case of /3 = 120°, the integration process was extended to k = 200 to firmly establish periodic character of the flow. The computed transient character-istics of CT for f3 — 150° is shown in Figure 6-11. A l l the three cases show fairly regular periodic behaviour, however, as can be expected, the local details differ. For the potential flow solution, time as well as spatial variation of the base pres-sure leads to high frequency modulation of the torque coefficient and the results cannot be expected to represent the real situation. T i p separation with constant base pressure dramatically improves the prediction and the frequency of the torque coefficient reflects that of the fluctuating pressure on the upstream face of the blade. The surface separation case models the real situation sti l l better. The attached flow is only over a small portion of the blade (Figure 6-12) with the vortex separating from the upper tip having little influence on the pressure in that region. Only the vortex shedded from the lower t ip, being closer, affects the pressure thus reducing the frequency of the transient torque by around a factor of two. Figure 6-13 shows the corresponding flow patterns. Basic features governing evolution, separation, clustering, downstream convection and periodic cancellation of vortices as discussed before are all quite apparent here. For example, for k « Chapter VI: Results and Discussion 162 p = 150° DVM I ' 1 0.2 0.4 FIGURE 6-11 = 150°. Chapter VI: Results and Discussion 163 FIGURE 6-12 Flow geometry at the blade orientation of 0 = 150°. Note, pressure in the attached flow region is primari ly affected by the vortex cluster formed near the lower t ip due to its proximity. Chapter VI: Results and Discussion 1.5 164 0.5 -0.5 Y 0.0 -0.5 -1.0 1.0 -1.5 * * * x * x x * x v x k = 10 k = 40 ' I I I I 1 1 1 1 1 1 * x x x ^ x x k = 20 k = 50 ' I I I I I I I 1 1 1 * * * x X x x x x k = 30 k = 60 -0.5 0.0 0.5 1.0 1.5 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 X (meters) FIGURE 6-13 Flow patterns at 0 = 150°: k = 10,20,30,40,50,60. Chapter VI: Results and Discussion 1.5-1 -1.5-1.0-0.5-Y 0.0--0.5--1.0--1.5 1.0 ,0.5-0.0--0.5--1.0--1.5 x# x x x X X X v x x X k = 70 X x B y x * x x x x x x x x x X X k = 80 T 1 1 r i r x x A x x X y c x \ x x x x< x ^ k = 90 • • 165 . . , ! ! ! ! , ! ^ ! 1  -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) FIGURE 6-13(cont.) Flow patterns at (3 = 150°: k = 70,80,90. Chapter VI: Results and Discussion 166 2.0-1 X (meters) FIGURE 6-13(cont.) Flow patterns at (3 = 150°: k = 100,120,140. Chapter VI: Results and Discussion 167 2 .5 n X (meters) F I G U R E 6-13(cont . ) Flow patterns at 0 = 150°: k = 160,180,200. Chapter VI: Results and Discussion 168 180 — 200, the third vortex cluster is clearly visible. A s in the case of 0 = 120°, the distance between the first and second vortex clusters ( « 4 ro) is substantially larger than that between the second and third vortex clusters ( « 2.75m). Similar behaviour can be observed for the vortices shed from the lower edge. This gives an approximate steady state vortex shedding frequency of 2.54 Hz leading to a period of s= 0.4 s, which compares with the steady state period of CT (Figure 6-11). The Strouhal number data for various blade orientations are summerized in Table 6.2. Blade Strouhal Angle (0°) Number 60 0.125 90 0.12 120 0.09 150 0.13 T A B L E 6.2 Strouhal number at different blade angles Numerical ly obtained variation of the average torque coefficient as affected by the blade orientation is compared wi th the test data presented earlier (Figure 2-29, uncorrected for blockage) in Figure 6-14. This clearly brings to light strengths and l imitations of the different approaches developed in the thesis. In particular, it provides designers with a tool to predict static performance of the Savonius rotors using several different methods, depending upon the available data, and the associated measure of accuracy. As can be expected, although the potential flow model qualitatively predicts correct trends, the actual magnitudes are rather off. Of course, this is primarily due to the method's inability to predict constant pressure in the wake. On the positive 1.8 1.4 1.2 1.0 CjO.8-0.6 0.4 0.2 0.0--0.2 -• Experimental PVM Sepgrgtion C p b • tip recovery x tip const. • surface const. ---- 1 1 X p • > < • • X / • / c : —> 1 • 1 > • C / X 1 1 • E 1 ^ / i 1 ' 0.0 20.0 40.0 F I G U R E 6-14 60.0 80.0 100.0 120.0 140.0 P Variation of the average starting torque as affected by the blade orientation. Note, the experimental results are uncorrected for the blockage of 10%. 160.0 180.0 CO Chapter VI: Results and Discussion 170 side, the method is self contained, i.e., it does not require any input information. Hence, it may be used to advantage during the preliminary design requiring trends in torque variation. On the other hand, the approach involving tip separation with constant base pressure appears quite attractive. Note, it assumes constant base pressure but does not require its explicit value. Thus, as before, no input information is required, however, the correlation between experimental and numerical results is vastly im-proved. Of course, in practice, the flow does not always separate from the tip. If the surface separation information, as obtained from the pressure data or by simple flow visualization study, is used in conjunction with the constant base pressure, the correlation is, as expected, better. As discussed later, blockage corrections wil l further improve the numerical predictions. 6.2 R o t a t i n g B l a d e The main goal of the mathematical model developed earlier is to predict perfor-mance of the rotating blade. In this case further complications arise in modelling of the flow. Since the shear layer separation positions are not known for rotating blades, the assumption of t ip separation has to be contended wi th . There is no ev-idence to assume a constant base pressure for a rotating blade. Therefore, at least in the present case, one is forced to resort to the potential flow approach discussed earlier. This may prove to be favourable in the case of pressure recovery through reattachment of the flow, as the mathematical model is able to partially account for it. To confirm that satisfying the boundary conditions at every other time-step Chapter VI: Results and Discussion 171 (A' = 2), as in the stationary blade case, continues to yield satisfactory results, trial runs were conducted using K =1 and 2, wi th: p/q X Pi (initial blade angle) At = 0.2 ; = 0.4 ; = 30° ; = 0.0057 s The torque and power coefficients as well as flow fields are compared in F ig -ures 6-15 to 6-18. Surprisingly the correlation is excellent even in the transient state (Figures 6-15, 6-16). Furthermore, the vortex patterns in the wake compare favourably (Figures 6-17, 6-18). Note, a saving in the computational cost with K — 2 is significant as indicated in the following table. K C P U (s) 1 350 2 95 T A B L E 6.3 Effect of K on the computational cost. In this analysis, the tip-speed ratio X was taken to be 0.4. As the angle of rotation of the blade per time-step is taken to be constant irrespective of A, the solution wi l l improve with an increase in tip-speed ratio due to an increase in the number of free vortices in the field. 6.2.1 Effect of tip-speed ratio X Effect of the tip-speed ratio for a rotor with p/q - 0.2 (optimum value given by the experimental data, Figure 2-19) was analysed using the numerical model rather thoroughly through a systematic variation of A in the range 0.4 — 1.8. Only three Chapter VI: Results and Discussion 172 Chapter VI: Results and Discussion 173 0.8 0.6 0.5 0.4 0.3 0.2 \ \\ \ \ \ » \\ \ \ \ \ \ \ \ \ \ - - ^ D V M K = 1 K = 2 U . U H 1 1 1 1 1 1 1 1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (Sec.) FIGURE 6-16 T ime history of CP at A = 0.4 as affected by K. Chapter VI: Results and Discussion 174 1.5-1.0-K = 1 0.5-0.0-- 0 . 5 -k = 70 x T, +ve v T, -ve -1 .0- o T 2 +ve + T 2 -ve -1.5-1.0-1 1 1 1 0.5-Y o.o-- 0 . 5 --1.0-*~+^ +# + + k = 90 x r, +ve v F, -ve o T 2 +ve + F 2 -ve -1.5-1.0-1 1 1 1 0.5-0.0-- 0 . 5 --1.0-k = 110 x T, +ve v r, -ve o T 2 +ve + F 2 -ve -1.5- I 1 1 1 1 1 1 1 1 1 1 1 1 -0 .5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-17 Flow patterns after k = 70,90,110, time-steps at A = 0.4 with K = 1. Chapter VI: Results and Discussion 175 1.5-1.0-0.5H 0.0 -0.5--1.0--1.5H 1.0 0.5H Y o.o -0.5 -1.0H -1.5 1.0 H 0.5 0.0 -0.5 -1.0 -1.5 V x x xx 0 x x x x ^ +++ ^ + + + + + ++ + + + K = 2 X r, +ve v r, -ve o r 2 +v« + T2 -ve k = 35 x 3* vx„ ^xx X^xxv .++ + + k = 45 X x X X y ^ 4 X X X X V + + v X + t + + + A + + + + + + ++ + k = 55 1 1 1 1 1 1 1 1 1 1 1 1 T -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-18 Flow patterns after k = 35,45,55, time-steps at A = 0.4 with K = 2. Chapter VI: Results and Discussion 176 typical sets of results are presented here to help establish trends. A sample of flow visualization photographs are also included for the case of A = 0.4. It should be emphasized that there are three fundamental aspects of interest here: a) effect of blade rotation on evolution of the wake compared to the station-ary case studied earlier; b) flow visualization study of the wake to assess validity of the numerical model; and c) effect of the tip-speed ratio on the wake and the rotor performance. Figure 6-19 shows evolution of the wake for rotating blades wi th A = 0.4. Al though, the basic process of generation of the wake is the same as in the stationary case there are some interesting differences. The rotation is initiated wi th a disturbance /?, = 30° at t = 0, and evolution of the wake, which is now sensitive to the instantaneous position of the blade, remains essentially the same unti l k » 40. For the stationary case, vortices associated with the two tips were of opposite sign as expected. However, wi th the rotating blades, the situation can be different as seen at A- = 50 where there is shedding of counter-clockwise (—ve) vortices from both the tips. The process continues until k =s 70, beyond which the wake pattern bears no resemblance to the stationary case. Note interaction between vortices shed from the two tips which was not present in the stationary case. A t k > 100, the wake attains a steady state and the important features of the flow field, also substantiated by the flow visualization, can be summarized as follows: a) a clockwise (+ve) vortex is captured near the centre of the rotor; Chapter VI: Results and Discussion 1.5-17' 1.0 0.5 0.0 -0.5H -1.0 -1.5 0.5 Y o.o -0.5 -1.0 -1.5 0.5 -1.0 -1.5 : k = 10 x F,+ve v r, -ve o r 2 +ve k = 40 i i i i i k = 20 c — +t+ k = 50 i t i i i k = 30 k = 60 -0.5 0.0 0.5 1.0 1.5 2.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 X (meters) FIGURE 6-19 Evolut ion of the flow pattern for A = 0.4 and its comparison with the flow visualization study: k = 10,20,30,40,50,60. Chapter VI: Results and Discussion 178 1.5-1.0 H 0.5 -1.5 0.5 -1.0 -1.5 0.5 0.0 -1.5 x r,+ve v P, -ve o F 2 +ve •» T 2 -ve k = 70 i i i i i i i i +++ k = 80 i i i i i i i i i i i i k = 90 i r -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-19(con t . ) Evolut ion of the flow pattern for A =- 0.4 and its compar-ison with the flow visualization study: k — 70,80,90. Chapter VI: Results and Discussion 1.5-179 1 0.5-1 0.0 -0.5-1 -1.0-1 5-1.0-0.5 Y o.o -0.5 -1.0 -1.5 H 1.0 - 1 . 5 H * r , 4 v e v P , - v e o T 2 4 v e •* T 2 - v e 4 4 4 + + 4 + + + 4 4 + k = 100 +4-4 4+ 4+ 4 4 4 k = 110 I I + + 4 ^ X X ^ X X „ 4 v £ : + , ^ X ' , 4 + 4 4 A k = 120 i i : 1 : 1 1 1 1 : 1 1 : — . -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-19(cont.) Evolut ion of the flow pattern for A = 0.4 and its compar-ison wi th the flow visualization study: k = 100,110,120. Chapter VI: Results and Discussion 180 F I G U R E 6-19(con t . ) Evolution of the flow pattern for A = 0.4 and its compar-ison with the flow visualization study (k s= 100,110.120, flow visualization pictures). Chapter VI: Results and Discussion 181 1.5 1.0 0.5 o.oH -0.5 -1.0H -1.5 1.0 0.5-1 Y o.o-l -0.5 • -i.oH -1.5 1.0-0.5-0.0--0.5--1.0--1.5 XX X o v X . #1 L + + V ~ ^ x + ++++ " + x x „ + - H -o. v + ++ + X X xxxx V++ + + +* *x x r,+ve ^ T, -ve o f 2 +ve 4 f 2 -ve k = 130 k = 140 V k = 150 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-19(con t . ) Evolut ion of the flow pattern for A = 0.4 and its compar-ison with the flow visualization study (k = 130,140,150). Chapter VI: Results and Discussion 182 F I G U R E 6-19(con t . ) Evolut ion of the flow pattern for A = 0.4 and its compar-ison wi th the flow visualization study (k = 130.140.150. flow visualization pictures). Chapter VI: Results and Discussion 183 b) a shear layer wi th negative vorticity, separating from the upstream blade, has a small part of it captured at the centre. The captured vortex slowly advances towards the tip of the upper blade and is periodically shed; c) vortices shed from the tips form shear layers close to the blade; d) separating shear layers produce large vortex structures which are even-tually shed downstream. A video of the flow visualization clearly showed the existence of the central vortex, shear layers eminating from the tips and large vortex structures moving down-stream. The typical flow visualization photographs corresponding to k = 100 — 150 reveal the same features although they are a bit obscure. It is of interest to point out that an independent flow visualization study, using a smaller model of the ro-tor, by Yokomizo et al . (1987) also revealed the same basic flow pattern. Note, the assumption of the central vortex made in the semi-empirical approach (Chapter III) is supported by both the theoretical analysis and the flow visualization thus giving it a firm foundation. In practice, one would like to establish the opt imum geometry of the rotor leading to a peak power coefficient which, in turn, depends on the tip-speed ratio A. Figure 6-20 attempts to assess the effect of A on evolution of the wake for several values of k (k = 20,40,60,80,100). For k in the range of 20 — 40, variation of the tip-speed ratio has no significant effect on the character of the wake. However, for k = 60 the central vortex, which is distinctly present at A = 0.4and0.8, disappears at A = 1.2. This suggests that entrainment of vorticity at the centre is reduced or even eliminated as the rotor tip-speed ratio increases. The same trend persists at k — 80,100. A s mentioned earlier, now the central vortex is shed and a new one formed from the vorticity Chapter VI: Results and Discussion 184 1.5 1.0-0.5-0.0--0.5--1.0--1.5-1.0 0.5-Y 0.0 -0.5 -1.0 -1.5-1.0-0.5 o.oH -1.0-1 -1.5 k = 20 X = 0.4 X = 0.8 i i r X = 1.2 T 1 P k = 40 x T , +ve o T 2 +ve + T 2 - v e X = 0.4 X = 0.8 X = 1.2 + T 1 I P T I P -0.5 0.0 0.5 1.0 1.5 2.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 X (meters) FIGURE 6-20 F low patterns after k = 20 and 40 time-steps: A = 0.4,0.8,1.2; p/q = 0.2; K = 2. Chapter VI: Results and Discussion 185 1.5 "i I I 1 1 1 1 1 1 1 1 1 1 1 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6 -20(con t . ) Flow patterns after k = 60, time-steps: A = 0.4,0.8,1.2; p/q = 0.2; K = 2. Chapter VI: Results and Discussion 186 1.5-1.0-0.5-0.0--0.5--1.0--1.5 1.0H 0.5 Y o.o -0.5-1 -1.0-1 -1.5 9 o V O O o o X V V x x x x ^ + 4- n y S 7 ?w w V + + ++v + ++ + + + + + y x X k = 80 v r, -ve o T 2 +ve + T 2 -ve + + + A = 0.4 X - 0.8 -I T" A = 1.2 1-5 j | i | | | | i i i | | i -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-20(con t . ) F low patterns after k = 80, time-steps: A = 0.4,0.8,1.2; p/q = 0.2; K = 2. Chapter VI: Results and Discussion 187 2.0 1.0H 0.0H - i .oH - 2 . 0 H 1.0 H Y o.o -1.0 -2.0 1.0 o.o H - 1 . 0 H -2.0 po .++++ V V k = 100 x r, +ve v T, -ve o T 2 +ve + T 2 -ve ++ X = 0.4 I ' I ' I ' I ' I ' I 1 I 1 I 1 I 1 I I it' x = 0.8 i r—1 r - 1 i — 1 i 1 i — r i I 1 I o X = 1.2 "i 1 r -0.5 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 X (meters) FIGURE 6-20(cont.) Flow patterns after k = 100, time-steps: A = 0.4,0.8,1.2; p/q = 0.2; K = 2. Chapter VI: Results and Discussion 188 shed from the upper blade. Jones et al . (1979) observed similar phenomenon of the central vortex formation and shedding through a flow visualization study. Corresponding time variation of the torque coefficient for A = 0.8 and 1.2 is presented in Figures 6-21 and 6-22, respectively. In general, the torque coefficient approaches a periodic profile, around an average value, asymptotically. The results for average torque coefficient as a function of the tip-speed ratio are presented in Figure 6-23. The plot shows the maximum value of the average torque coefficient as 0.42 occuring at A = 0.4. The experimental data presented earlier showed the same trend with Cj^max (corrected for blockage) as 0.40 for A ss 0.45. The average power coefficient obtained by integrating the steady state CT VS. time curves, at different tip-speed ratios, is given in Figure 6-24. This shows a max-imum power coefficient of about 0.38 at a tip-speed ratio 1.2. The corresponding experimentally obtained Cp vs. A curve was presented earlier (Figure 2-17). It shows a peak power coefficient of about 0.50 at a tip-speed ratio of about 1.2. It is important to notice that the theoretical prediction is for zero blockage whereas the experimental results were obtained at a 16.4% blockage. The experimentally obtained maximum power coefficient wi th blockage correction for the p/q = 0.2 model is around 0.32 compared to the theoretical prediction of 0.38. Considering the highly complex, transient and separated character of the flow, the numerical pre-dictions should be considered rather good and of sufficient accuracy for engineering applications. The mathematical mode) forecasts the peak power coefficient at a tip-speed ratio greater than one, suggesting that the Savonius configuration is not exactly a drag type wind turbine. The experimental as well as the theoretical starting torque characteristics also suggest the same. Ogawa et al . (1986 b) and Sawada et al . Chapter VI: Results and Discussion 1 8 9 Chapter VI: Results and Discussion 190 Time (Sec. ) FIGURE 6-22 Time history of CT at A = 1.2. Chapter VI: Results and Discussion 191 0.45-1 A FIGURE 6-23 Theoretically predicted variation of the average torque coefficient as affected by the tip-speed ratio A for p/q — 0.2. Chapter VI: Results and Discussion 192 0.40-1 n 1 i 1 i 1 i 1 i 1 1 1 1 1 1 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 F I G U R E 6-24 Theoretically predicted variation of the average power coefficient as affected by the tip-speed ratio A for p/q = 0.2. Chapter VI: Results and Discussion (1985) have also come to the same conclusion. 193 6.2.2 Effect of p/q variation To assess the model's ability to predict the effects of geometric changes on performance of the Savonius rotor, several test runs were executed with different P/Q. A set of typical results obtained at a tip-speed ratio of 0.6 for p/q — 1.0, 0.2 and 0 is presented in Figure 6-25. The flow patterns are essentially similar in character to the ones seen earlier (Figure 6-19). The presence of the central vortex is evident at time-steps 60, 80, and 100 irrespective of the value of p/q. The corresponding typical variation of the torque coefficient is presented in Figure 6-26 for two representative values of p/q. The average power coefficient variation wi th the tip-speed ratio (Figure 6-27) shows the same trends as the experimental results presented earlier (Figure 2-27). Note, the theory predicts the peak Cp for p/q = 0.2, which is the same as that given by the test results. A n error of around 16% in prediction of CpjTnax for p/q — 0.2 is indeed remarkably good considering the challenging character of the problem and the quasi-potential nature of the model. 6.3 E f fec t o f W i n d T u n n e l B l o c k a g e The numerical computational scheme has the capability of changing the number of tunnel elements as well as the length of the tunnel. In the present analysis the tunnel wall was divided into 32 boundary elements. The finite tunnel walls extended three diameters upstream and five diameters downstream. The size of the rotor was kept constant while the width of the tunnel was altered with a consequent change Chapter VI: Results and Discussion k = 20 k = 40 x T, +ve k = 60 * r, -ve o T 2 +ve + T 2 -ve p/q = o . P/q = 0 p/q = 0 -P/q = 0.2 . P/q = 0.2 P/q = 0.2 . | . | i | i | i | i i I i I i I i I i I i p/q = 1-0 . P/q = 1-0 p/q = 1.0 I 1 I 1 I 1 I 1 I 1 I ' i • I 1 -0.5 0.5 -0.5 0.5 1.5 -0.5 0.5 1.5 2.5 X (meters) F I G U R E 6-25 Evolut ion of the flow pattern as affected by p/q for A = (k = 20,40,60). Chapter VI: Results and Discussion 195 v + c D f e 0 +++ + V * x * k = 80 x r, +ve v T, -ve o r 2 +ve + T 2 -ve p/q = 0 O V p , xxxxy p/q = 0.2 £ 2 X *X p/q = 1-0 •1.5-1 1 1 1 1 1 1 1 1 1 1 r -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-25(con t . ) Evolut ion of the flow pattern as affected by p/q for A = 0.6 (A; = 80). Chapter VI: Results and Discussion 196 1.5-1 1.0 0.5 0.0 -0.5 -1.0 -1.5 1.0 0.5 Y 0.0 -0.5 -1.0 -1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 + CPQ ^ 4 4 + & 4 5'0 o 4 O 4 X 4 X X x^ X X x x x v x —I H 4+ 4 4 + + 4 +++ f+ 4 + k = 100 x T, 4ve v T, -ve O T 2 4ve + T 2 -ve p/q = 0 + 4 + e^o V \ V V *8o° X X X X 4 X 4 4 X w 4+ 4 4 ++1 p/q = 0.2 P/q = 1-0 1 1 1 1 1 1 1 1 1 1 1 1 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) F I G U R E 6-25(con t . ) Evolut ion of the flow pattern as affected by p/q for A = 0.6 (k = 100). Chapter VI: Results and Discussion 197 1.8-1 i I i i i i i | i 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (Sec . ) FIGURE 6-26 T ime history of CT as affected by p/q: A = 0.6; p/q = 1.0. Chapter VI: Results and Discussion 198 1.8-1 : : : : • : ! 1.5H -0.3-f j i i i 1 i 1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 T ime (Sec . ) F I G U R E 6-26(con t . ) T ime history of CT as affected by p/q: A = 0.6; p/q = 0. Chapter VI: Results and Discussion 199 0.40-1 n , , , ! , ! , ( , ! , ! , , 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 A F I G U R E 6-27 Theoretically predicted CP vs. A curves for p/q = 0 , 0 . 2 , 1 . 0 . Chapter VI: Results and Discussion in blockage. 200 6.8.1 Stationary blade Theoretically obtained transient torque coefficient for a stationary rotor, with p/q = 0.2 and (3 = 120°, as affected by the blockage is presented in Figure 6-28. The procedure assumed was that of the surface separation with a constant base pressure. Corresponding results for unconfined condition are also included to facilitate comparison. Three interesting features become apparent : a) Time to attain steady state is governed by the wall confinement and increases wi th the blockage. b) Period of the torque fluctuations increases wi th the blockage. This is related to the vortex formation and shedding. As shown in Figure 6-29, distance between the successive vortices increases as the blockage ratio increases suggesting a delay in the vortex formation. c) As can be expected due to an increase in the local velocity wi th blockage, amplitude of the torque coefficient as well as the velocity with which the vortices are convected increases. Similar results were obtained for different angular positions of the stationary blade with identical trends. Theoretically obtained average starting torque coeffi-cient for a rotor with 10% blockage is compared with the corresponding experimental data over a range of blade orientation (/?) in Figure 6-30. The correlation between the results is indeed good. As expected, the effect of blockage is not significant for angles in the range 0 < j3 < 30° and 150° < p < 180°. Chapter VI: Results and Discussion 201 0.5 0.4-0.3 0.2 0.1-0.0--0.1--0.2--0.3 -0.4--0.5 * * / DVM - C p b = Const. B= 0 % B= 10 7o B = 20 % * / * t t / t * / * y . 3^ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (Sec . ) 0.8 FIGURE 6-28 Time history of the torque coefficient for a stationary rotor, with p/q = 0.2 at /? = 120°, as affected by the blockage. Chapter VI: Results and Discussion 202 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 1.0 0.5 Y 0.0 -0.5 -1.0 -1.5 1.0 0.5-0.0--0.5--1.0--1.5-X ^ X * * xXxx?x x ^ x x X x "^XsTxx 4 - H X X X :^<xxx5< + xx&xx^xx Nxx 4 + 4 4 +±. 4 + 4+4^ + " +4 4 x x x X x X X xx: X X X* k = 100 B = 0 % X ^ xxxx y x x x Xxxx x X X x B = 10 % x "x x x x xx x 4+ 4+44 x X X ***** x X B = 20 % I I 1 1 1 1 1 1 1 1 1 1 , 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) FIGURE 6-29 Flow pattern after 100 time-steps for a stationary rotor, with p/q — 0.2 and /? = 120°, as affected by the blockage. Chapter VI: Results and Discussion 203 Chapter VI: Results and Discussion 204 6.S.2 Rotating blade Effectiveness of the analytical model for predicting wall confinement effects for a moving rotor was also assessed through a comparison with the experimental data presented earlier. Four blockage ratios of 5%, 10%, 15% and 20% were used with the rotor configurations defined by p/q — 0.2 and 1. Variat ion of the average power coefficient with the tip-speed ratio is shown in Figures 6-31 and 6-32 for p/q = 1 and 0.2, respectively. A t the outset it is apparent that the theory is able to predict the trends quite accurately. Even the magnitudes correlated rather well as shown by the representative experimental results (dots). In general, the effect of blockage to increase the peak power coefficient which now occurs at a higher tip-speed ratio. Note, the theory consistently predicts the peak power coefficient, at a higher tip-speed ratio. This may be attributed to the incapability of the potential flow theory to model the surface separation and the associated constant base pressure in the wake. The numerical predictions can be improved further but at a computational cost. In the present study, walls were modelled using fairly large boundary elements, and the boundary conditions were satisfied only at the nodal points. Thus in essence, the model treats the rigid wall as permeable. Even a small increase in the number of elements (from 32 to 40) showed an improvement in the correlation between the theoretical and experimental results. However, the computation cost increased by about 40%. Figure 6-33 summarizes the variation of Cpmax with blockage for two values of p/q. Note, the effect of the blade geometry parameter is to shift the plot without altering its shape. Thus the functional variation remains essentially the same to a constant (this concept was used earlier in Figure 2-22 to obtain the C P r n a x variation for p/q = 0.2 using the experimental data for p/q = 1). Experimental data for the Chapter VI: Results and Discussion 205 0.50 0.45-0.40 0.35-0.30-0.25 0.20-0.15 0.4 p / q = 1 . 0 B = 0 %  B = 5 % B = 10 % B = 15 % B = 20 % Exper. B = 20 % 0.6 0.8 —I 1 1— 1.0 1.2 1.4 1.6 1.8 X F I G U R E 6-31 Theoretically predicted characteristic curves for a rotor with p/q = 1.0 as affected by the blockage. Chapter VI: Results and Discussion 206 0.50 0 . 4 5 -0.40 0 . 3 5 -0 . 3 0 -0 . 2 5 -0.20 0.15 0.4 p/q = 0.2 B = 0 %  B = 5 %  B = 10 % B = 15 % B = 201 % Exper . B = 10 % \ \ X Y \ 0.6 0.8 - 1 1.0 1.2 1.4 1.6 1.8 A F I G U R E 6-32 Theoretically predicted characteristic curves for a rotor with p/q — 0.2 as affected by the blockage. Chapter VI: Results and Discussion 207 0.55 0.50-0.45-Theoretical A p/q = 0.2 x p/q - 1.0 Exper. p/q = 0.2 — A FIGURE 6-33 Theoretically predicted variation of Cpmax w i th blockage for dif-ferent p/q. Note, the agreement with the experimental data is quite good. Chapter VI: Results and Discussion 208 opt imum configuration of p/q = 0.2 gave the maximum error of less than « 15%. The theoretical model was also used to predict the evolution of the wake for a rotating Savonius turbine in the presence of blockage as would be the case during wind tunnel tests. Only a small sample of data for the opt imum blade configuration (p/q = 0.2) are presented in Figure 6-34 for A =• 0.8 and 1.2. The basic flow features are the same as those in the unconfined case. For example, existence of the central vortex is noticeable, even in the presence of blockage, at A = 0.8, however, it is absent at A = 1.2 as in the unconfined case. As discussed for the stationary case, position of the vortex cluster continues to be farther downstream wi th blockage during rotation. Thus the vortex convective speed increases wi th blockage even during the blade rotation. Chapter VI: Results and Discussion 209 1.5-1 1.0 0.5 0.0 -0.5 -1.0H -1.5 1.0-0.5-Y 0.0--0.5--1.0 -1.5 1.0 0.5 0.0 -0.5--1.0--1.5 -to + + #x +4 + + 4 +_+ 4 r 4 X = 0.8 x P. 4ve v T, -ve o r2 4 v e + T 2 -ve B = 0 % B = 10 % B = 20 % 1 1 1 1 1 1 1 1 I 1 1 1 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 X (meters) FIGURE 6-34 Flow pattern after 100 time-steps for a rotor wi th p/q - 0.2 as affected by the blockage(A = 0.8). Chapter VI: Results and Discussion 210 1.5 -1-5 T 1 I 1 1 1 1 1 l 1 1 1 1 1 - 0 . 5 0 . 0 0 . 5 1.0 1.5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5 6 .0 X (meters) FIGURE 6-34 Flow pattern after 100 time-steps for a rotor wi th p/q = 0.2 as affected by the blockage(A = 1.2). C H A P T E R VII C O N C L U D I N G R E M A R K S The thesis has attempted to lay a sound foundation for the design, development and performance evaluation of a wind turbine, economical to fabricate and easy to maintain. Using the classical Savonius configuration as reference, it has tried to approach the objective through an organized experimental program, complemented by theoretical approaches and flow visualization, with innovative contributions at all the stages. The extensive experimental program, through a carefully planned parametric study, has led to an opt imum configuration for the Savonius rotor wi th a twofold increase in efficiency. Equally important are the results of blockage study which are helpful in the design of a prototype. They also provide a basis for comparison of results obtained by researchers using different sizes of models and test facilities. O n the analytical side attention was focussed on two aspects of practical signif-icance: i) To develop a simple and quick approach of engineering accuracy that can be used wi th confidence, at least during the preliminary design stage. The semi-empirical model, using the concept of a central vortex in con-junction with the measured pressure data on stationary blades, can ad-mirably play this role. It can account for the effect of blade geometry 211 Chapter VII: Concluding Remarks 212 as well as blockage, and can effectively replace dynamical testing with a considerable saving in cost. ii) To develop a sophisticated mathematical model that can effectively sim-ulate the transient, separated viscous flow associated wi th the rotor, yet maintaining simplicity of the potential flow. Furthermore, it was neces-sary to achieve a desired accuracy wi th an acceptable cost using available computational tools. The numerical, time-marching approach based on the Boundary Element Method is developed in the thesis which meets all of the criteria mentioned above. This elegant tool is able to predict performance of the rotor and the wake dynamics wi th an accuracy that is l imited, to an extent, by the computational cost. Its effectiveness can be further improved by judiciously incorporating information that can be measured quite readily. Furthermore, the approach is rather general and can be applied quite readily to a large class of transient aerodynamics problems involving a variety of slender and/or bluff geometries. Based on the wind tunnel test program, flow visualization study and mathe-matical modelling of the Savonius rotor, following general conclusions can be made: a) The opt imum configuration of the Savonius rotor corresponds to the blade geometry as follows: gap-size (a/d) blade overlap (b/d) aspect ratio (^ 4) 0 ; 0 ; 0.77 ; blade shape parameter [p/q) blade arc angle (0) = 0.2 ; = 135°. It is inferred that this has a peak power coefficient of around 0.32 at a tip-Chapter VIJ: Concluding Remarks 213 speed ratio of 0.79 (with a possible error of ±5 — 10%) in the unconfined condition representing a twofold improvement in performance compared to the reported values in literature ( « 0.12 — 0.15). This brings it close to more sophisticated and costly configurations like the Darrieus design in terms of performance, yet retaining its simplicity and self starting character. b) Characteristic plots of Cp vs. A suggest that the Savonius rotor is not a pure drag device. This is logical as at small 0 the rotor behaves like a slender body wi th lift contributing to the power. c) The effect of blockage is to increase the peak power coefficient and the corresponding tip-speed ratio. A variation in the blade shape parameter (p/q) shifts the C p > m a x vs. blockage plot along the ordinate without affecting its general shape. d) The semi-empirical approach using experimentally established parame-ters can predict performance of the rotor with a reasonable accuracy. e) Presence of a vortex filament at the centre of the rotor and its shedding which is predicted by the analytical model is also confirmed through the flow visualization study. f) A mathematical model for viscous separated flow wi th an unsteady wake, developed in the thesis, represents a versatile tool applicable to a large class of transient fluid dynamics problems. When applied to a Savonius rotor it is able to predict performance and evolution of the wake geometry wi th a reasonable accuracy. g) The wake of the stationary blade shows periodic behaviour over the range Chapter VII: Concluding Remarks 214 /? « 60° —150°. The vortex shedding frequency corresponds to a Strouhal number of « 0.12. The theoretically predicted peak starting torque coef-ficient occurs at (3 ss 30° which is confirmed by the experimental results. Starting torque becomes negative over the range of f3 « 130° — 180°. h) For the rotating blade, existence of the central vortex and its shedding is evident for tip-speed ratio less than one. However, for A > 1 theoretical results do not show presence of the central vortex. Periodicity of the wake geometry is reflected in the time history of CT for both stationary and rotating cases. i) For a given blockage ratio, the effect of p/q is to shift the C p vs. A characteristic curve along the ordinate. In general, p/q does not have a significant effect on the wake geometry. There are several avenues for future studies which are likely to be profitable and satisfying : a) Measurement of torque and pressure distribution during blade rotation should provide valuable information for comparison wi th and further re-finement of the mathematical model. b) W i t h an opt imum blade configuration in hand, the next logical step would be to bui ld a prototype model for field tests to investigate its operational performance, structural and mechanical integrity, and maintenance re-quirements. W i n d tunnel tests of the optimum configuration simulating the earth boundary layer and the turbulent character as encountered in field operation should provide valuable information. Its performance cor-relation wi th the field data has not been reported in the literature. Chapter VII: Concluding Remarks 215 c) The use of augmentors to further improve the performance presents an exciting possibility. d) Improvement in the semi-empirical model through the use of more sophis-ticated functional relations is likely to make the method more attractive. e) There are several possible avenues to further improve the Boundary E l -ement M o d e l . Perhaps the most obvious one is to incorporate a suitable boundary layer model to predict separation and base pressure with added accuracy. f) Appl ica t ion of the Boundary Element Method in the solution of transient fluid dynamics problems of aircraft and industrial aerodynamics as well as ocean engineering presents a challenging possibility. 216 B I B L I O G R A P H Y A h m e d i , G . , 1978,"Some Pr i l iminary Results on the Performance of a Small Vertical Axis Cyl indr ica l W i n d Turbine", Wind Engineering, V o l . 2, No . 2, pp. 65-74. Alder , G . 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