Lecture Notes on Wind Tunnel Testing

C0NTENTS

Page Introduction

1

Chapter - 1 1.1 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.6 1.3.7

Wind Tunnel Introduction Wind Tunnel Classification The Type of Test Section The Type of Return Circuit The Speed of Flow in the Test Section Types of Wind Tunnels Subsonic Wind Tunnels Transonic Tunnel Supersonic Tunnel Hypersonic Tunnel Full Scale Tunnel Compressed Air Tunnel Other Tunnels

10 10 10 10 11 12 13 13 15 16 17 18 19 19

Chapter -

Wind Tunnel Intrumentation Introduction Pick-up or Transducer Variable Resistance Transducer The Wheatstone Bridge Principle Summing Circuit Differencing Circuit Variable Capacitance Transducer Variable Reluctance Transducer Piezoelectric Transducer Signal Conditioner Signal conditioner for Variable Resistance Transducer Excitation Supply Bridge Balance Shunt Calibration Signal Amplification Signal Conditioner for Variable Capacitance Transducer Signal Conditioner for Variable Reluctance Transducer Signal Conditioner for Piezoelectric Transducer Data Acquisition System Analog System Digital System

20 20 21 21 25 25 28 29 30 31 32 33 33 33 34 35 36 37 38 39 40 41

2 2.1 2.2 2.2.1 2.2.1.1 2.2.1.2 2.2.1.3 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.1.1 2.3.1.2 2.3.1.3 2.3.1.4 2.3.2 2.3.3 2.3.4 2.4 2.4.1 2.4.2

1

Chapter -

3 3.1 3.2 3.3 3.4 3.5 3.5.1 3.5.2

Tunnel Characteristics Introduction Air Speed Calibration Determination of Velocity Variation in Test Section Determination of Angular Flow Variation in Test Section Turbulence Level Drag Sphere Pressure Sphere

43 43 43 47 49 50 50 51

Chapter -

Flow Visualisation Introduction Incompressible Flow Visualisation Techniques Smoke Method Tuft Method Oil Flow Method Evaporation Method Compressible Flow Visualisation Techniques Shadowgraph Method Schlieren Method Interferometer Method

54 54 54 54 56 57 57 58 58 58 59

Pressure Measurement by Mechanical Device Introduction Measurement of Cp Without Pre–Calibration of the Tunnel With Pre–Calibration of the Tunnel Pressure Distribution on Circular Cylinder Model Pressure Distribution on Elliptical Cylinder Model Pressure Distribution on Spherical Model

60 60 61 61 63 63 67 70

Chapter -

Force and Moment Measurement by Mechanical Balance Introduction Calibration Measurements of Forces and Moments Evaluation of the Tare and Interference Drag Evaluation of the Tare and Interference Drag Separately Evaluation of the Sum of the Tare and Interference Drag

72 72 72 78 80 81 82

Chapter -

Pressure Measurement by Transducer Introduction Time Response Pressure Scanning Measurement of Cp With Pre–Calibration of Tunnel Without Pre–Calibration of Tunnel

84 84 86 86 89 89 89

4 4.1 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.3.1 4.3.2 4.3.3

` Chapter -

5 5.1 5.2 5.2.1 5.2.2 5.3 5.4 5.5 6 6.1 6.2 6.3 6.4 6.4.1 6.4.2 7 7.1 7.2 7.3 7.4 7.4.1 7.4.2

2

Chapter -

8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.7.1 8.7.2 8.7.3 8.7.4

Force and Moment Measurement by Internal (Sting) Balance Introduction Measurement of Lift Measurement of Pitching Moment Simultaneous Measurement of Lift and Pitching Moment Other Forces and Moments Interactions Effect Factors Affecting the Accuracy of Measurement Surface Preparation and Bonding of Strain Gauges Noise Suppression Thermal Effect Optimising Excitation Level

91 91 92 95 98 100 103 104 104 106 108 111

9 9.1 9.2 9.3 9.4 9.5

Force and Moment Measurement by External Balance Introduction General Description Operation Calibration Wind Tunnel Testing

114 114 114 117 118 123

Chapter - 10 10.1 10.2 10.3 10.4 10.5 10.6

Wind Tunnel Boundary Corrections (2D Flow) Introduction Horizontal Buoyancy Solid Blocking Wake Blocking Streamline Curvature Effect Summary of Two–Dimensional Boundary Corrections

124 124 125 128 130 133 135

Chapter - 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7

Wind Tunnel Boundary Corrections (3D Flow) Introduction Horizontal Buoyancy Solid Blocking Wake Blocking Streamline Curvature Effect Downwash Effect Summary of Three-Dimensional Boundary Corrections

138 138 138 139 140 140 142 143

Chapter - 12 12.1 12.2 12.3 12.4

Drag Measurement on 2D Circular Cylindrical Body Introduction Drag by Pressure Distribution on the Cylindrical Surface Drag by Measuring Distribution in the wake of the Cylinder Drag by Direct Weighing

144 144 145 149 153

Chapter - 13 13.1 13.2

Flow about an Aerofoil Section Introduction Formulation of the Problem

156 156 157

Chapter -

3

13.3 13.3.1 13.3.2 13.3.3 13.4 13.4.1 13.4.2 13.5 13.6 13.6.1 13.6.2 13.6.3 13.7

Solution Exact analytical Solution Approximate Solution Exact Numerical Solution Lanearised Theory Thickness Effect Camber Effect Exact Numerical Method (Panel Method) Overall Aerodynamic Characteristics Lift, Drag and Pitching Moment Coefficient Location of Aerodynamic Centre Location of Centre of Pressure Wind Tunnel Testing

159 159 159 160 160 161 161 163 166 167 171 171 172

Chapter – 14 14.1 14.2 14.2.1 14.2.2 14.3 14.3.1 14.3.2 14.4 14.4.1 14.4.2 14.5

Measurement of Laminar Boundary Layer Introduction Boundary Layer Parameters Displacement Thickness (s*) Other Parameters Laminar Boundary Layer in Zero Pressure Gradient Theoretical Calculation Wind Tunnel Testing Laminar Boundary Layer in Favourable Pressure Gradient Theoretical Calculation Wind Tunnel Testing Laminar Boundary Layer in Adverse Pressure Gradient

177 177 178 179 180 183 183 184 188 189 190 192

Chapter - 15 15.1 15.2 15.3 15.4 15.5

Measurement of Turbulent Boundary Layer Introduction Structure of Turbulent Boundary Layer Log Law Relation Power Law Relations Wind Tunnel Testing

194 194 194 196 197 199

Chapter - 16 16.1 16.2 16.3 16.4 16.5 16.5.1 16.5.2

Flow about Rectangular and Swept Wings Introduction Theory Prandtl’s Lifting Line Theory Vortex Lattice Method Wind Tunnel Testing Measurement of Pressure Distribution Measurement of Overall Forces and Moments Using Balance

202 202 205 206 208 210 210 215

Chapter - 17 17.1 17.2

Flow about a Slender Delta Wing Introduction Slender Wings in Attached Flow

217 217 217

4

17.3 17.4 17.4.1 17.4.2

Slender Wings in Separated Flow Wind Tunnel Testing Measurement of Pressure Distribution Measurement of Overall Forces and Moments

219 221 221 223

Chapter - 18 18.1 18.2 18.3

Flow about Composite Wings Introduction Straked Configuration Canard Configuration

224 224 225 228

Chapter - 19 19.1

Drag Measurement of Sphere Introduction

231 231

Chapter - 20 20.1 20.2 20.3 20.4 20.4.1 20.4.2 20.4.3 20.4.4 20.5

Supersonic Aerodynamics Introduction Shock Visualisation Run Time of Tunnel Determination of Mach Number By using Area-Local Mach Number Relation By Static Pressure Measurement on the Wall of the Test Section By using Rayleigh-Pitot Formula By using θ-β-M Relation (Shock Wave over a Wedge) Variation of Mach Number along the Axis of Divergent Section of C-D Nozzle Variation of Mach Number along Diffuser Axis Determination of the Exit Velocity

235 235 237 238 241 241 242 242 243

Appendix – 1

Notations

252

Appendix – 2

Note on Units

253

Appendix – 3

List of Facilities

255

Appendix – 4

2100 System : Strain Gage Conditioner and Amplifier System

256

References

281

20.6 20.7

5

244 245 246

INTRODUCTION

The basic aim of aerodynamics is to obtain the flow quantities (especially, pressure distribution and skin friction) about a body immersed in fluid. Very often, the interest is limited only to obtain the overall forces and moments acting on the body. There are two main ways these quantities can be found; theoretically and experimentally. Both the procedures have their relative advantages and disadvantages and have acted and are going to act as supplementary to each other in foreseeable future The limitation of theoretical methods basically stems from the fact that the governing equation of real fluid about a body – the Navier-Stokes equation can not, in general, be solved theoretically. The theoretical methods are usually based on some simplified form of this equation. With the assumption of inviscid (infinite Reynolds Number) and incompressible (zero Mach number) flow, i.e., the ideal flow, the Navier Stokes equation can simplified to Laplace’s equation. The solution of this ideal flow, because of the above simplification, differs from the experimental results. Efforts are then made to employ some ‘corrections’ due to the effects of viscosity and compressibility. Even with simplification of inviscid incompressible flow, it is not easy to solve the problem. For a few simple configurations, exact analytic solutions exist (Chap. 5). Configurations of arbitrary shape are not amenable to analytic methods and demand numerical solution. In the early days, a variety of approximate numerical methods were developed. Examples are the different variants of linearised theory by Munk, Weber etc. for aerofoil problems, Prandtl’s lifting line theory, Multhopp’s lifting surface theory, Jone’s slender wing theory etc. for wing problems. With the advent of high speed digital computers, more sophisticated exact numerical methods (Panel method) have been developed. A variety of computer based theoretical schemes are also developed for effecting the corrections due to viscosity and compressibility to these solutions. Alternatively, attempts have been made to develop Euler as well as Navier-Stokes codes with or without turbulence modeling.

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It is almost certainly the case that – however sophisticated these theoretical methods may eventually become – the engineer will always wish to validate his design, prior to manufacture, by means of physical experiment. In this respect, in aircraft industry, the wind-tunnel experimentation will always play the superior role of the two. Wind-tunnel testing, like the theoretical calculations, has its own deficiencies and difficulties. Broadly speaking these are : the high capital and running cost associated with a wind tunnel, the expenses, elapsed time and skill needed in manufacturing accurate scale models, the difficulty in obtaining the adequate data (forces, pressure distribution etc.), the difficulty of interrogating this data. Students of aeronautical engineering are well aware of the fact that the forces and moments etc. experienced in flight on an aircraft depends primarily on two nondimensional parameters : Reynolds number and Mach number. Reynolds number expresses the relative contributions of inertia and friction forces in the motion of the fluid. The Mach number is the ratio of the flight speed and the speed of sound. In general it can be stated that only a full scale model operating at full scale speed can give a totally correct simulation of a real aircraft in flight. However, because of power conservation problem (specially for high-speed flow) the wind-tunnel model is generally constructed at a much smaller scale than the real aircraft. This in itself presents numerous difficulties associated with the acquisition of sufficiently detailed data on such a small model. However a more serious problem arises in simultaneously recreating the Mach number and Reynolds number experienced in flight. If the working medium and its temperature are the same in the wind-tunnel as in fullscale flight in the atmosphere, then proper matching of the Mach numbers requires the air speeds to be the same in both cases. If this is not achieved, then at Mach numbers of interest of most aircrafts, the effects of compressibility will be different between the wind-tunnel and flight. On the other hand, if the speeds are kept same for Mach number simulation, the Reynolds number in the wind-tunnel will be reduced (proportional to the geometric scale of model) relative to the real aircraft. Clearly, if the wind-tunnel speed is increased to approach full-scale Reynolds number then the Mach number will be incorrectly simulated.

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Now many vital phenomena depend strongly on the Reynolds number and these include : development of boundary layer, transition from laminar to turbulent boundary layer, separation of boundary layer, vortex formation at high angle of attack etc. If the Reynolds number is not matched properly, viscosity will be incorrectly simulated. Numerous technological approaches have been proposed to overcome such difficulties. One of these consists of modifying the properties of working medium and in particular working at very low temperature or at high pressure. These approaches, in turn, present other difficulties. However, since the present study is restricted to low speed regime where compressibility effects are negligible, matching of both parameters is not necessary and simulation of Reynolds number alone is sufficient. The other difficulties associated with wind tunnel testing arise from the fact that the flow conditions inside the tunnel are not exactly the same as those in the free air. Primarily, the air in the tunnel is considered to be more turbulent than the free air this turbulence being produced in the tunnel by propeller, vibrations of the tunnel walls etc. This consequently increases the effective Reynolds number of the tunnel (Section 3.5). Excessive turbulence makes the test data unreliable and difficult to interpret. Secondly, the wind-tunnel model experiences spurious ‘constraint’ effects due to windtunnel walls (chapter 10 and 11) which will be absent in free air. These extraneous forces must be calculated and subtracted out. These forces arise from two sources. Due to formation and growth of

boundary layer in the test section, the effective area is

progressively reduced resulting in increase of velocity and decrease of static pressure downstream. This variation of static pressure produces a drag force known as ‘horizontal buoyancy’. Again, the presence of a model in the test section reduces the area through which air flows. This ‘blockage’ caused by the model and its wake effectively increases the average air speed in the vicinity of the model than they would be in free air, thereby increasing all forces and moments at a given angle of attack. Thirdly, the model in a tunnel is usually installed by some supports which in turn affect the flow. The effect of this supports (the so-called ‘Tare’ and ‘Interference’ effects, section 6.4) need to be calculated carefully and eliminated from observed values.

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The procedure involved in wind-tunnel testing may now be summarized. The prerequisite of any experimental work is the calibration and evaluation of the tunnel (Chapter – 3) itself. The wind-tunnel must be pre-calibrated to give the velocity of air flow during any testing (since it is not practical to measure the velocity by pitot-static tube while the model is in tunnel). The flow characteristics of the tunnel must be ascertained by measuring the variation of velocity (static pressure) in the test section, flow angularity and the turbulence level of the tunnel. Wind-tunnel testing, then, involves model making, installation of model in the tunnel and measuring forces, moments, pressure distribution etc. the forces and moments may be obtained by any of the three methods :

a) Measuring the actual forces and moments with wind-tunnel balance b) Measuring the effects that the model has on the airstream by wake survey (profile drag, section 12.2) c) Measuring the pressure distribution over the model by means of orifices connected to manometer and integrating the pressure distribution over the model surface. The data acquired is then to be corrected for the tunnel boundary and support effects.

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Chapter 1

WIND-TUNNEL 1.1 Introduction : The wind-tunnel is one of the most important facilities available for experimental work in aerodynamics. Its purpose is to provide a region of controlled airflow into which models can be inserted. This region is termed the working section or test section. For aeronautical work, the flow in the test section should ideally be perfectly uniform in speed, direction and vorticity. Such perfection can never be achieved in practice and the quality of a windtunnel is related to the closeness to which the airflow in the test section approaches the ideal.

1.2 Wind Tunnel Classification : Wind-tunnels are usually classified according to the three main criteria : i)

the type of test section

ii)

the type of return circuit

iii)

the speed of flow in the test section

1.2.1 The type of test section: The cross sectional form of a test section may be square, rectangle, octagonal, circular or elliptic. Again, it can be closed or open. A closed test section is one which is completely enclosed within solid walls, the airflow therefore being constrained by these walls. An open test section is one which is not enclosed within solid walls (Fig. 1.1). Because the flow is not constrained, it usually tends to expand, partly due to pressure difference and partly due to mixing between the air in the test section and that outside. To allow for this expansion, the downstream part of the tunnel is bell-mouthed.

10

Figure 1.1

Open test section

Comparing these two types of test section, the closed type has the following advantages : a) greater efficiency (i.e. reduced power losses) b) better control of air flow c) no loss of air d) less noise On the other hand, the open type of test section allows easier access to the model and easier visual study of the flow.

1.2.2 The type of return circuit A wind tunnel may either be open-circuit or closed-circuit tunnel. The open circuit tunnel which is open at the both ends has no guided return of the air (Fig. 1.2). After the air leaves the tunnel it circulates by devious paths back to intake.

11

Figure 1.2

Open circuit tunnel

The closed circuit tunnel has, as the name implies, a continuous path for the air (Fig. 1.3). The whole circuit, except possibly the test section, is enclosed.

1.2.3 The speed of flow in the test section: Five categories of speed are usually recognized : a) low speed (up to about 60 or 70 m/s) b) high speed subsonic (but Mach number less than 0.9) c) transonic (Mach number between 0.9 and 1.2) d) supersonic (Mach number between 1.2 and 5) e) hypersonic (Mach number greater than 5)

12

Figure 1.3

Close circuit tunnel

The first two categories, low speed and high speed subsonic, are often taken together as subsonic tunnels.

1.3

Types of Wind Tunnel :

1.3.1 Subsonic Wind Tunnel : The simplest kind of subsonic tunnel consists of a tube, open at both ends, along which the air is propelled. The propulsion is usually provided by a fan downstream of the test section (a fan upstream would create excessive turbulence in the working section. Fig. 1.2 represents a tunnel of this type. The following description relates to Fig. 1.2. The mouth is shaped to guide the air smoothly into the tunnel; flow separation here would give excessive turbulence and nonuniformity in velocity in the test section. To make the flow parallel and more uniform in speed and top give a little time for turbulence to decay, the mouth is followed by a settling chamber. The settling chamber usually includes a honeycomb and wire-mesh screens.

13

A honeycomb is a coarse mesh made of thin, broad plates set edgewise to the flow. It has two purposes. First, it helps to guide the air to flow parallel to the tunnel axis. Second, if there are any large eddies in the incoming flow, the honeycomb ‘cuts’ them into smaller ones which can decay more rapidly than would the original larger ones. The mesh-screens are fitted to reduce non-uniformities in flow speeds. A typical installation might have one or two. The effects of screens on dynamic pressure variation in the test section is shown in Fig. 1.4. The screen also serves to reduce the turbulence level of the tunnel.

Figure 1.4

Effect of screen

The contraction followed by the settling chamber improves the quality of flow in the test section. The air flows from the mouth of the tunnel at low speed into a comparatively short settling chamber with a honeycomb and mesh screens. It is then accelerated rapidly in the contraction. The contraction reduces turbulence and also non-uniformities in flow speed and direction. The test section is followed by a divergent duct, the diffuser. The divergence results in a corresponding reduction in the flow speed, which has two principle effects. Firstly, it enables an increased fan efficiency to be achieved. Secondly, the reduction in dynamic pressure leads to reduced power losses at the exit from the tunnel in the laboratory.

14

Leaving the diffuser, the air enters the laboratory, along which it flows slowly back to the mouth of the tunnel. A typical tunnel will have a working section of about 1 meter square and an overall length of some 5 to 7 meters. The speed in the test section, will be controllable, upto about 30 m/s.

1.3.2

Transonic Tunnel:

The main special feature of a transonic wind-tunnel is its test section. In this, test section walls are neither open nor closed but a combination of both. The walls usually have perforation or streamwise slots. The reason is as follows : If, as an Fig. 1.5 an aerofoil is being tested in a transonic flow, shock waves occur. If the walls were solid these shockwaves would be reflected from them and would impinge on the model. The flow over the model would therefore be very different from that in free flight and the test would be invalid. If the test section were open, there would be a boundary between the jet and the surrounding atmosphere; the shock (compression) waves would be reflected from this boundary as expansion (rarefaction) waves. These would impinge on the model, so again the test would be invalidated.

Figure 1.5

Reflection of shock wave

15

If the walls are perforated or slotted (i.e., the test section is partly opened and partly closed), the reflections are mixtures of compression waves and rarefaction waves and so, depending on the degree of perforation, these tend to cancel each other out. The flow over the model therefore approximates more closely to that in free flight.

1.3.3 Supersonic Wind Tunnel: The simplest form of supersonic wind-tunnel is the blow-down type (Fig. 1.6). It consists of a convergent-divergent duct whose upstream end is connected to a tank filled with compressed air. The downstream end is usually open to the atmosphere. The air in the tank then discharges through the duct. This means, of course, that the pressure in the tank fall continuously, and therefore a reducing valve is fitted to maintain a constant pressure at the inlet of the duct. The duration of each test run is necessarily limited with this type of tunnel. The blow-down type of tunnel is relatively cheap. In particular, a relatively low-powered pump can be used to pressurize the tank taking, of course, a correspondingly long time to do so. The power expanded in driving the tunnel during a test is many times greater than the power of the pump. The test section of this type of tunnel is followed by a convergent-divergent duct. It can be shown that if the pressure ratio between the two ends of a convergent-divergent duct exceeds 1.892, the flow is sonic (M=1) at the throat and supersonic downstream. A plane downstream of the throat can therefore be used as a test section in which the flow is supersonic.

Figure 1.6

Supersonic wind tunnel 16

The Mach number at the test section will depend only on the cross-sectional areas at the throat and the test section. AT .S 1 5 M 2     M  6  A

3

(1.1)

This shows that the test section Mach number is determined solely by the shape of the tunnel (provided the pressure ratio is sufficient to maintain supersonic flow through the test section). Because of this supersonic tunnels frequently consist of a basic ‘frame’ to which various liners can be fitted. Each liner gives a unique area ratio and therefore a unique Mach number in the test section. The shapes of some different liners for various Mach number are illustrated in Fig. 1.7.

Figure 1.7

Shapes of liners

1.3.4 Hypersonic Wind Tunnel : The main special feature of hypersonic wind tunnel is that provision must be made for preheating the air before entering the tunnel.

17

By suitable design of lines i.e. providing the large area ratio AT*S/ A* for generating high Mach number, the Mach number in the test section of a supersonic wind-tunnel may be increased to hypersonic regime. But another consequence of expanding air to high speed, namely its change in temperature, becomes limiting criterion. The equation for the temperature ratio along a streamline originating in a region where the flow is at rest with temperature T0 and terminating where the temperature is T is given by

T0 M2  1 T 5

(1.2)

For M = 10, this equation gives T =T0/21. Now if T0 be the absolute temperature 228k then the wind temperature in the test section will be 13.5K. This is well below the temperature where air becomes liquid. Thus a limiting Mach number in the test section would be one at which air remains gaseous. The obvious choice for increasing this limiting Mach number is not preheat the air to be used in the tunnel to such an extent that the very low temperature in the test section is not realized. Another choice is to use a gas which has very much lower condensation temperature than air, e.g. helium. The majority of hypersonic tunnels, however use the preheating method. The preheating of air may be done by heating the reservoir air or alternatively to allow the air to pass through a heat exchanger as it leaves the reservoir to enter the working section. Apart from these wind tunnels, other types of wind tunnels are also designed and fabricated. The effort to simulate both Mach number and Reynolds number of free flight in wind-tunnel has resulted in development of two types of tunnels :

1. Full Scale Tunnel 2. Compressed Air Tunnel

1.3.5 Full Scale Tunnel : The Full Scale Tunnel is capable of testing actual aircrafts of moderate size under near flight condition. The wind tunnel, developed at Langley Field, U.S.A., attains wind velocities up to 53m/s with an open jet 18m wide and 9m high. Apart from providing a

18

total simulation of Mach number and Reynolds number, such wind tunnels also serve a useful purpose in giving a correlation between flight and small model tests.

1.3.6 Compressed Air Tunnel : The use of high pressure and therefore a high density in the test section can help to achieve full scale Reynolds number with relatively small and low speeds. Some tunnels are therefore completely enclosed in a large tank which can be pumped up to several times atmospheric pressures. Such tunnels are termed compressed air tunnels. It is worth mentioning that high pressure is no cure-all for getting a high Reynolds number since model strength may be a limiting factor.

1.3.7 Other Tunnels : There are also other types of tunnels built for various purposes. Some of these tunnels are:

Smoke tunnel

: For flow visualization

Spin Tunnel

: For studying spin recovery

Low Turbulence tunnel

: For testing at high Reynolds number

Stability Tunnel

: For studying dynamic stability

Gust Tunnel

: For studying effects of gust on models

V/STOL

: For studying V/STOL configurations

Ice Tunnel

: For studying formation and removal of ice on models subjected to icing condition.

Automobile Wind Tunnel : For testing full scale automobiles.

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Chapter 2

WIND TUNNEL INSTRUMENTATION 2.1 Introduction Instrumentation plays an important role in wind tunnel testing. The accuracy of experimental results depends not only on the quality of the tunnel but also on the performance of he measuring equipments. The quantities which are frequently measured in wind tunnel testing are generally pressure distribution and over all forces and moments acting on a model. Velocity, in general, can be calculated from the pressure and hence need not be measured. However, in some cases velocity itself (for example, fluctuating velocity components in turbulent flow) may be of importance and need to be measured. Also, measurement of skin friction may be necessary in some experiments. Measuring instruments may, broadly, be classified as two types: mechanical and electronic. Examples of mechanical type of instruments are the liquid-level manometers for pressure measurement and wind-tunnel mechanical balances for measurement of overall forces and moments. Such instruments lack the first response, capability of measuring high and low values and amenability to automation required for unsteady or short-duration high speed tunnel. All these limitations may be overcome in electronic instrumentation system. An electronic system usually consist of: a) pick-up or transducer b) signal conditioner c) data acquisition system The pick-up or transducer receives the physical quantity (pressure/force) under measurement and delivers a proportional electrical signal to the signal conditioner. Here the signal is amplified, filtered or otherwise modified to a format acceptable to the data acquisition system. The data acquisition system may be a simple indicating meter, an oscilloscope or a chart recorder for visual display. Alternatively, it may be a magnetic

20

tape recorder for temporary or permanent storage of data or a digital computer for data manipulation or process control.

2.2 Pick-up or Transducer: A transducer may be defined as a device which provides an electrical output signal for a physical quantity (pressure/force), whether or not auxiliary energy is required. Many other physical parameters (such as heat, light, intensity, humidity) may also be converted into electrical energy by means of transducers. Transducers used in wind tunnel testing may be classified according to the electrical principles involved, as follows: 1)

Variable resistance transducer (resistance strain gauge)

2)

Variable capacitance transducer

3)

Variable reluctance transducer

4)

Piezoelectric transducer

Of all these transducers, resistance strain gauge, because of its unique set of operational characteristics, has dominated in transducer field for the past twenty years.

2.2.1 Variable Resistance Transducer: The strain gauge is an example of variable resistance transducer that converts a physical quantity into a change of resistance. A strain gauge is a thin, wafer-like device that can be attached (bonded) to a variety of materials. Metallic strain gauges are manufactured from small diameter resistance wire such as constantan, or etched from thin foil sheets (Fig. 2.1). For simultaneous measurement of strain in more than one direction, two-element or three-element rosettes are used. The resistance of the wire or metal foil changes with length as the material to which the gauge is attached undergoes tension or compression. In a gauge diaphragm pressure transducer, strain gauges are directly bonded on the diaphragm while in a sting balance used for force measurement, strain gauges are bonded on he sting (Fig. 2.2). While the load is applied, the resistances of the strain gauges increase or decrease, depending on nature of stress (tensile or compression). The sensitivity of a strain gauge is described in terms of characteristics called the gauge factor, G, defined as the unit change in resistance per unit change in length

21

Or,

G = (RR)  (L/L)

where

G = Gauge factor

(2.1)

R = Gauge resistance R = change in gauge resistance L = normal length (unstressed condition) L = change in length. The term L/L is the strain , so that equation (2.1) may be written as G = (RR)   Where

(2.2)

 = strain in the lateral direction.

Figure 2.1 Strain gauges (a: wire, b: foil)

The resistance R of a wire of length L can be calculated by using the expression for the resistance of conductor of uniform cross-section.

R

Where

length L  area   2  d 4

(2.3)

 = specific resistance of conductor material L = length of the conductor d  diameter of the conductor

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Figure 2.2 Sting balance Tension on the conductor causes an increase L in its length and a decrease d in its diameter. The resistance of the conductor then changes to

R  R   .

L  L   4d  d 2

 .

L1  L L   4d 2 1  2d d 

(2.4)

Equation (2.4) may be simplified by using Poisson’s ratio, , defined as a ratio of strain in lateral direction to strain in axial direction. Therefore,

  d d  L L

(2.5)

Substituting equation (2.5) in equation (2.4) gives R  R  

1  L L L 2  4d (1  2 L L)

 R1  L L1  2 L L  R1  1  2 L L

[neglecting higher order term]

The gauge factor can now be obtained as

G  R / R L L  1  2 

(2.6)

23

Poisson’s ratio for most metals vary from 0.25 to 0.5 and the gauge factor is then of the order of 1.5 to 2.0. For strain-gauge application, a high sensitivity is very desirable. A large gauge factor means a relatively large resistance change which can be more easily measured than a small resistance change. Semi-conductor gauges are now developed, which have gauge factor of the order of 120. The semi-conductor strain gauges are however neither so practical nor so widely used as the conventional metallic gauges in general purpose, high accuracy transducers. It is worth nothing that semi-conductor gauges were originally considered advantageous because of their high output. This has less importance today because the same semiconductor technology which created the type of gauge has also created smaller and less expensive amplifiers high gain for use with conventional strain gauges. Conventional metallic strain gauges are generally of four types : Constantan, Karma, Isoelastic and platinum-tungsten. Constantan, a copper nickel alloy, of gauge factor 2.0 is the most popular alloy for transducer gauges. It possesses an exceptional linearity over a wide strain range and is readily manufactured. It is also easily solderable. Its primary limitation in precision applications is a slow irreversible drift in grid resistance when exposed to temperature above 75 C. Because the drift rate increases exponentially with temperature, Constantan is not recommended for transducers operating continuously at high temperature. Karma (gauge factor 2.1) is a nickel-chromium alloy used in a variety of modified forms for strain sensing. Like Constantan it displays extremely good linearity over a wide strain range. It has greater resistivity than Constantan making higher grid resistance feasible. A major advantage is its improved resistive stability, particularly at high temperature. Isoelastic alloy offers exceptionally good fatigue life and a gauge factor 3.1, about 50% higher than Constantan or Karma alloys. It has limited use in transducers because of its poor zero stability with temperature variation. Because of its good fatigue life, it is normally used for dynamic measurements. Platinum-tungsten alloys, like Isoelastic, find their primary use in dynamic transducer applications. With a gauge factor approximately two times greater than Constantan and Karma, and with very good fatigue life, platinum-tungsten gauges are used almost exclusively in ‘fatiguerated’ dynamic transducers.

24

2.2.1.1 The Wheatstone Bridge Principle : The change in resistance due to applied load can be converted into a change in voltage by the Wheatstone bridge circuit. Two types of Wheatstone bridge circuits are possible : ‘summing circuit’ and ‘differencing circuit’. Generally, in wind tunnel testing, differencing circuit is used for measuring moment.

2.2.1.2 Summing Circuit : In the summing circuit, resistance undergoing tension and compression are connected in opposite sides of the Wheatstone bridge. Four unstressed strain gauges R1, R2, R3, R4 are connected to form a Wheatstone bridge in summing circuit is shown in Fig. 2.3. The current passing through the resistance R1 and R3 is I13 where

I 13 

V R1  R3

(2.7)

Similarly, the current passing through resistances R4 and R2 is I42 where

I 42 

V R4  R2

(2.8)

Figure 2.3 Summing circuit

The voltage at A is therefore,

25

V A  V  I 13 R1  V 

V .R1 R1  R3

The voltage at B is, VB  V  I 42 R4  V 

V .R4 R4  R2

The voltage across A and B is,



 

V

V



V  V AB  V A  VB  V  R1   V  R4  R1  R3   R1  R2    R4 R1    V    R4  2 R1  R3  V

V V 

or,

R3 R 4  R1 R2 R1  R3 R2  R4 

R3 R4  R1 R2 R1  R3 R2  R4 

(2.9)

Now, the output voltage V will be exactly zero, if (1)

or, (2)

R3 R4  R1 R2  0

or,

R1  R2  R3  R4  R

R1 R4  R3 R2

(say)

no matter what the input voltage V may be. If any of the resistance changes due to applied load, the output voltage V will change. Provision may be made to change only one resistance (quarter active bridge) or two resistance (half active bridge) or three resistance (three-quarter bridge) or all four resistances (fully-active bridge). For the fully active bridge (Fig. 2.2), the output voltage due to applied load is calculated in a simple manner. The resistance R1 and R2 are subjected to compression and will therefore have a decrease in resistance value while resistance R4 and R3 will have a increase in resistance. The changed values of the resistances may be written as

26

R1  R  R  R2  R  R  R3  R  R 

R4  R  R

(2.10)

R, R are the changes in resistances due to changes in strain at positions 1 and 2 (Fig. 2.2). Substituting the values in equation (9) yields

V V



R  RR  R  ( R  R)( R  R) ( R  R  R  R )R  R   R  R 



2 RR   2 RR  2 4 R 2  R   R 



2 RR   R  4r



R   R 

(2.11)

2R

If the strain gauges are bounded very close to each other, it can be assumed

R  R  R and the equation (2.11) is reduced to

V V

or,

V V





4 RR 4R 2

R

(2.12)

R

The equation shows a linear relationship. However, for quarter-bridge and half bridge a non linearity appears in the expression for output voltage. For example, if only R4 is active (quarter-bridge) and the other three resistance are passive (not bonded on the sting), the expression for output voltage is

V V



R 4R  R 

(2.13)

For a half-bridge (taking only R4 and R3 active)

V V



R (neglecting higher order terms) 2R  R 

27

(2.14)

Similarly, for a three-quarter bridge (taking R4, R3 and R2 )

V V



3R 4R  R 

(2.15)

Because of the linearity in relationship, fully-active bridge is usually used in measurement techniques. It also has another advantage compared to others i.e. the temperature compensation effect. In a fully active bridge, all resistances have same temperature (neglecting the thermal e.m.f. effect) while in other bridges, the temperature of active gauges may be different from those of the passive gauges which will cause a change in resistance values resulting in further non-linearities.

2.2.1.3

Differencing Circuit :

The arrangement of resistance in the Wheatstone bridge in ‘differencing circuit’ is shown in Fig. 2.4. Using the similar procedure, the output voltage V in this circuit is obtained as

Figure 2.4

V V



Differencing circuit

R2 R4  R1 R3 R1  R2 R4  R3 

28

R  RR  R  R  R( R  R 2R  R  R. 2R  R  R 2 RR   R   2 4 R 2  R   R  2 RR   R  2 [neglecting R  R 

=

4R



2

with respect to 4R2]

R   R 

(2.16)

2R

If the strain gauges are pasted close to each other, the output voltage will be virtually zero since R will be almost equal to R.

2.2.2 Variable Capacitance Transducer : The capacitance of parallel-plate capacitor is given by

C

Where

k . A.  0 ( farads) d

A = area of each plate (m2) d  distance between the plates (m)

0 = 9.85  10 -12 (F/m) k  dielectric constant

Since the capacitance is inversely proportional to the spacing of the parallel plates, d, any variation in d causes a corresponding variation in the capacitance. This principle is applied in the variable capacitance pressure transducer (Fig. 2.5). A pressure, applied to a diaphragm that functions as one plate of a simple capacitor changes the distance between the diaphragm and the static plate. The resulting change in capacitance can be measured with an AC bridge but it is usually measured with an oscillator circuit. The transducer, as a part of the oscillatory circuit, causes a change in the frequency of the oscillator. This change in frequency is a measure of the magnitude of the pressure applied.

29

Figure 2.5

Variable capacitance transducer

2.2.3 Variable Reluctance Transducer : Such transducers employ magnetic diaphragms as sensing element (Fig. 2.6). When a differential pressure deflects the magnetic diaphragm, the air gaps (initially about 0.025 mm) also changes differentially and so does the reluctance. The two coils are connected on a two-active arm bridge so that an output proportional to pressure is obtained.

Figure 2.6 Variable reluctance transducer

30

Another type of variable reluctance transducer is based on linear variable differential transformer (LVDT). The LVDT is a three-coil device with a movable magnetic core (Fig. 2.7). Two outer coils are connected in ‘opposition’ so that induced voltages are 180 out of phase with each other. When the armature is centered, these voltages are equal in magnitude giving zero output. The pressure activates the diaphragm and when it moves the magnetic fluxes are unbalanced to produce an output proportional to the pressure applied.

Figure 2.7

Linear variable differential transducer

2.2.4 Piezoelectric Transducer: The Greek word piezo means ‘to squeeze’. The piezoelectric effect is appropriately described as generating electricity by squeezing crystals. This type of sensor is selfgenerating, that is, it does not require external electrical power as do the variable resistance or variable reluctance sensors. A piezoelectric transducer is illustrated schematically in Fig. 2.8. The sensitivity can be enhanced at the expense of resonant frequency by ‘stacking’ a series of elements together with the appropriate electrical connection.

31

Figure 2.8

Schematic diagram of piezoelectric transducer

A variety of piezoelectric materials are used, with quartz being most popular. Although piezo-electric transducers may be used for near static pressure measurements, they are more frequently employed for transient measurement.

2.3

Signal Conditioner:

Signal originating from the transducer is fed to the signal conditioner in which it is transformed into a form acceptable to the data acquisition system. Broadly speaking, the signal conditioner provides circuitry for amplification, noise suppression, filtering, excitation, zeroing, ranging, calibration and impedance matching. Because the operating principles of the different transducers are different, a variety of signal conditioners have been developed. The different types of signal conditioner for different transducers are outlined below.

2.3.1 Signal conditioner for Variable Resistance Transducer : The signal conditioner usually provides supporting circuitry for resistance strain gauge transducer. Usually, the equipment is able to accept quarter-bridge, half-bridge and fullbridge by providing appropriate dummy gauges. The circuitry usually provides excitation power, balancing circuits, calibration elements, signal amplification etc.

2.3.1.1 Excitation Supply : Normally DC excitation is used for resistance strain gauge transducer. Although AC excitation can be used, the disadvantages outweigh the advantages. The accuracy of an AC system is not as good as that of DC system. Also the noise rejection near the carrier frequency is poor. Earlier DC amplifier circuit was based on the ‘chopper’ principle in

32

which the DC is first converted to AC and then amplified and later converted to DC. Such a DC amplifier is fairly expensive. However, with the advent IC chips, DC amplifiers are no longer more costly than AC system. However, the DC power supplied must have high stability. To achieve this, the power supply should be isolated from all other ‘common lines’ and from the AC power line. In the other words, it should have a very low coupling to the power line and to the ground.

2.3.1.2 Bridge Balance : The Wheatstone bridge circuit should ideally have zero voltage output under no load condition, equation (2.9). However, because of normal gauge-to gauge resistance variations and additional resistance changes during gauge installation, the bridge circuit is usually in a resistively unbalanced state when first connected. It is advantageous to have a balancing network to nullify any residual signal.

Figure 2.9 Parallel balance network

The most common arrangement uses a shunt on one side of the bridge as shown in Fig. 2.9, the fixed resistor in the potentiometer wiper lead being used to omit the loading effect on the active arms of the bridge. If all the resistance strain gauges are of exact equal values, the voltage at A and B will be 0.5 V and the output V will be zero. In this hypothetical case, the potentiometer wiper

33

lead will be at the center (position C) and the voltage there will also be 0.5 V and therefore there will be no current through R4. However, if due to any of reasons mentioned above, the output V is not zero, the voltage at A is then either higher or lower than the voltage at B. in either case, bridge can be balanced by moving the wiper lead downward (C2) or upward (C1) respectively. The range of the balance network is given by

V V



R 4R4

if R4>>R

where V is the maximum out-of –balance (zero offset) that can be nullified. The range can be extended by decreasing the value of R4. However, R4 can not be decreased indefinitely because it will then have loading effect on the power supply. Usually, to limit the loading effect, R4 is many times higher than R (of the order of 75 k to 100 k ).

2.3.1.3 Shunt Calibration : Usually, in all signal conditioners, shunt resistors are provided across the arms connected to balance network. The shunt resistor, when connected, can usually accommodate a 0.4% change of resistance of the arm shunted. This actually amounts to simulating 2000 strain on the arm shunted as shown in Fig. 2.10. From equation (2.2),  = (R/R)/G. For R = 120 , G = 2.0, R = 0.48,  becomes 0.002 or 2000.

34

Figure 2.10

Shunt calibration

2.3.1.4 Signal Amplification : Signal amplification is the major function of a signal conditioner. Usually, the output voltage V (equations 2.12, 2.16) of a wheatstone bridge circuits is of the order of microvolts since the change in resistance is usually of the order of 10 –5 to 10 –6 ohms. Such a weak signal may not be accepted by the data indicator or recording system (although microvoltmeters are now available) and therefore the signal originating from transducer need to be amplified. Signal requirements for amplifier are quite stringent. These include impedance matching with the data indicator or recording device, high signal-to-noise ratio (SNR), low drift (change in output voltage with time is called drift) etc. With low impedance devices such as resistance strain gauges, no special problems arise in the operational mode. A fairly conventional voltage amplifier with an input impedance of 100k or greater in suitable for use with the data indicator system (such as DVM) or

35

C.R.O. For bridge circuits in which neither output terminal is grounded, a differential amplifier is needed. Such amplifiers offer good common mode rejection characteristics. The philosophy underlying noise cancellation is outlined in Fig. 2.11.

Figure 2.11 Noise cancellation by amplifier common-mode rejection If the common mode rejection ratio is of the order of 105, the noise that appears at the output terminal is largely eliminated. Such transducers have the ability to handle direct coupled signals, the D.C. drift being less than 10Vhour after allowing one hour warmup. Low drift rates are fairly difficult to achieve and the cost of D.C. amplifier with this sort of performance is comparatively high.

2.3.2 Signal Conditioner for Variable Capacitance Transducer : A number of signal conditioner is available based on the following schemes i)

D.C. polarization as the input circuit for an amplifier.

ii)

An A.C. bridge circuit for use with and amplitude modulation system.

iii)

A frequency modulating oscillator circuit.

iv)

A pulse modulating circuit.

The D.C. polarization circuit, the simplest of these, is described here. It is effected by the circuit shown in Fig. 2.12. in which C represents the capacitance of the transducer together with that of the connecting cable and any stray parallel capacitance. The polarizing voltage V is usually a few hundred volts. If it is assumed that the capacitance C can be represented by a constant portion C0 plus a sinusoidally varying part C1 sinwt, then 36

C = C0 + C1 sinwt If C1 50000), the output becomes V 

Q CF

Thus a voltage proportional to charge Q is produced.

Figure 2.15

2.4

Charge amplifier

Data Acquisition System :

Data acquisition systems are used to measure, indicate and/or record signals originating from transducers and signal conditioning process. Such systems can be categorized into

39

two major classes : analog system and digital system. The type of data acquisition system, whether analog or digital, depends largely on the intended use of the recorded input data. In general, analog systems are used when wide bandwidth is required or when lower accuracy can be tolerated. Digital systems are used when the physical process being monitored is slowly varying (narrow bandwidth) and when high accuracy and low pre-channel cost is required. Digital data acquisition systems are in general more complex than analog systems both in terms of instrumentation involved and the volume and complexity of input data they can handle.

2.4.1 Analog System An analog system may be defined as continuous function such as a plot of voltage versus load (Fig. 2.16) or displacement versus pressure. Examples of the analog systems are the analog panel meter, CRO, strip-chart recorder, X-y plotter etc.

Figure 2.16 Analog system

A complete analog instrumentation system used in wind tunnel testing may consist of some or all of the following elements :

a) Transducers: for translating physical parameters into electric signal. b) Signal Conditioners: for amplifying, modifying etc. of these signals.

40

c) Visual Display Devices: for continuous monitoring of the input signals. These devices may include single or multi-channel CRO, storage CRO, panel meter, numerical display and so on. d) Graphic Recordings Instruments: for obtaining permanent records. These instruments include strip chart recorder to provide continuous records on paper charts, X-y plotter, ultraviolet recorders etc. e) Magnetic Tape Instruments:

for acquiring data, preserving their original

electrical form and reproducing them at a later data for more detailed analysis.

2.4.2 Digital System : Digital systems handle information in digital form. A digital quantity may consist of a number of discrete and discontinuous pulses (Fig. 2.17) which contains information about the magnitude or nature of quantity. Digital system may consist of digital panel meter, data-logger, computer etc. It is worth noting that if a digital system is employed, an analog-to-digital (A/D) converter must be used before since the output signal from the signal conditioner is in analog form.

Figure 2.17 Digital system

A complete digital instrumentation system may include some or all of the following elements (Fig. 2.18). a) Transducers: for translating physical parameters into electrical signals. b) Signals Conditioners: for amplifying, modifying, etc. of these signal.

41

c) Scanner or Multiplexer: for sequentially connecting multiple analog signals to one measuring/recording system. d) Signal Converter: translates the analog signal to a form acceptable by analog-todigital converter. An example of signal converter is an amplifier for amplifying log-level voltages generating by strain gauges. e) Analog to Digital (A/D) converter: converts the analog voltage to its equivalent digital form. f) Digital Recorder:

records digital information on punched cards, perforated

paper tape, magnetic tape, or a combination of these systems. g) Auxiliary Equipment:

this section contains instruments for system

programming functions and digital data processing. These functions may be performed by individual instruments or by a digital computer.

Transducer

Signal Conditioner

Scanner/ Multiplexer

Signal Conver -ter

A/D Conver -ter

Auxiliary Equipment and System Programming

Figure 2.18 Complete digital instrumentation system

42

Digital Recorder

Chapter 3

TUNNEL CHARACTERISTICS 3.1 Introduction : Once a wind tunnel is designed and constructed, the primary task is to calibrate and evaluate the tunnel characteristics in terms of uniformity in wind speed and direction, and also level of turbulence. A wind tunnel can be considered to have good characteristics if the flow in the test section has uniform speed, no angular variation in direction and low level of turbulence. Four tests are generally necessary for calibrating and evaluating a tunnel. These are: 1. Air speed calibration. 2. Determination of velocity variation in the test section. 3. Determination of angular flow variation in the test section. 4. Determination of turbulence level.

3.2 Air Speed Calibration : In any experiment, the wind tunnel flow speed (or dynamic pressure) must be known for calculation of flow quantities. However, it is not desirable top insert a pitot-static tube in the tunnel in the presence of a model. This is because of two reasons; firstly, the tube will interfere with the model and secondly the tube will not read true owing to the effect of model on it. It is therefore necessary to determine the airflow speed during an experiment without using the pitot-static tube. This is possible by a prior calibration of a wind tunnel manometer with respect to air speed. The pitot-static tube (Fig.3.1) at station J is considered. If P0 be the total pressure, pj be the static pressure and UJ be the oncoming flow speed at the test section, then from Bernoulli’s equation

P0  p J  or,

1 U J2 2

U J  2P0  PJ  

(3.1)

43

Figure 3.1 Calibration of wind tunnel manometer The pitot-static tube is connected to manometer M1 which shows a difference in waterlevel of hJ , then P0  PJ   water  hJ  g

The manometer M1 is inclined at an angle of 600,

P0  PJ  Water  hJ Sin60 0  g

(3.2)

From equation (3.1) and (3.2)

U J  2  Water  hJ Sin60 0  g 

(3.3)

The air flow speed at test section can now be calculated from equation (3.3)

44

 1  The calibration of flow speed UJ or dynamic pressure q J   U J2  can now be  2 

calibrated with the help of another manometer M2 . Applying Bernoulli’s equation at L and S stations gives

pL  or,

1 1 U L2  p S  U S2 2 2

p L  qL  pS  qS

where q is the dynamic pressure.

If the pressure drop between S and L stations due to friction is considered, total head at L will be slightly smaller by an amount (say qSK1 where K1is he loss coefficient), then p L  q L  pS  qS  qS k1

or,

p L  pS  qS 1  k1   q L

Applying equations of continuity between stations L and S

AS U S  ALU L ; Therefore,

U L   AS AL U S



p L  pS  q S 1  k1   AS AL 

2



(3.4)

Applying equation of continuity between S and J

AS U S  AJ U J ; or,

U S   AJ AS U J

q S   AJ AS  q J 2

Putting in equation (3.4)



p L  pS   AJ AS  q J 1  k1   AS AL  2

2



 k2q j

or,

q J   pL  pS  k 2

where k2 is a constant.

Now, if another manometer M2 is connected to stations L and S, then p L  pS   water  hLS  g

or,

q J   water  hLS  g  k 2

 khLS

(3.5)

where k is a constant. Equation (3.5) shows that the free stream dynamic pressure is linearly proportional to the pressure difference in terms of manometer water level difference h LS. Free stream speed

45

(U) at station J Is also therefore directly related to pressure difference (in terms of h LS ) between two points L and S. The lows peed wind tunnel (LSWT) in the department can be run at 11 different speed setting. For 11 different speeds a table can be made concerning free stream speed C at station J and hLS, as shown in Table 3.1.

Table 3.1. : Calibration of tunnel speed No. of runs

hJ (cm)

qJ (N/m2)

U at J (m/s)

hLS(cm)

1. 2. 3. 11.

Calibration graphs (Fig. 3.2) can now be made in terms of q vs hLS and U vs hLS. Using these graphs velocity or dynamic pressure in any subsequent experiment can be obtained simply from hLS (without using pitot-static tube).

q

U

(N/m2)

(m/s)

hLS (cm)

hLS (cm) Figure 3.2 calibration graphs

46

3.3 Determination of Velocity variation in test section : Velocity in the test section, even in the absence of model, is not uniform either in horizontal or vertical direction. Owing to the effects of viscosity, the velocity near the tunnel wall will be slower than the velocity on the centerline and velocity at downstream will be greater than at upstream. To achieve uniformity of speed various means like using guide vanes, breathers or screens are used. To check uniformity of speed in vertical direction velocity at different vertical positions (for example, points 1, 2, 3, 4, 5, in Fig. 3.3) can be measured by pitot-static tube. Velocity at these points for a particular tunnel speed setting can be obtained from

U  2   water  h sin 60 0  g 

(3.6)

Tunnel Roof

Exit 0

0

5

4

0

5

Test Section

0

4

Entrance

0

3

3

0 2

0

2

0

1

0 1

Tunnel Floor

Figure 3.3 Velocity measurement at five vertical and five horizontal positions

Velocity in the wind tunnel varies in longitudinal directions (i.e. along the axis of the test section) because of viscous effects. As the flow progresses towards the exit, the boundary layer is thickened resulting in an effective reduction of area, increase in velocity and decrease in static pressure. Because of the decrease of static pressure there is tendency of the model to be drawn downstream. This creates a drag force acting on the body, termed horizontal buoyancy (chapter 10, 11), which is to be calculated and subtracted in any drag measurement experiment. Velocities (dynamic pressure) at different points along the tunnel center line (1, 2, 3, 4, 5 in Fig. 3.3) can be measured using the pitot-static tube as before. Subtraction of

47

dynamic pressure from total pressure (atmospheric pressure) will give static pressure at these points. A table can now made for calculation of velocity variation in vertical and horizontal directions as shown below.

Table 3.2: Calculation of velocity at 9 points

Stations y cm

h cm

U m/s

Stations

1.

1.

2.

2.

3.

3.

4.

4.

5.

5.

x cm

U

U (m/s)

(m/s)

p (N/m2)

Height from floor,

h cm

U m/s

p (N/m2)

Distance along tunnel

y (cm)

Centerline, x (cm)

Figure 3.4 Velocity variation in vertical and horizontal direction

48

Velocity variation with tunnel height (y) and velocity and static pressure variation with distance along tunnel center line (x) can now be plotted (Fig. 3.4). Static pressure gradient (p/x) should be calculated and noted.

3.4 Determination of Angular Flow Variation in the Test Section : Due to defectiveness in design and construction, the flow in the test section may not be horizontal. It is therefore necessary to know whether such angularity in flow exists and if it exists then to measure it so as to allow compensations due to this angularity of flow. The angular variation in the flow can be checked by using a spherical yawhead as shown in Fig. 3.5. The yawhead has two smooth orifices usually 900 apart on the forward face of a sphere. Obviously, if they are exactly placed, they will read equal pressure when the flow is directed along the axis of the yawhead. If the pressure at the two points a and b are not equal then it will indicate that the flow is inclined at an angle. This angle of yaw may then be determined by simply rotating the yawhead till the pressures at these points become equal. The angle of rotation of yawhead is then the angle of yaw of the flow. A similar procedure can be adopted for measuring yaw in the horizontal plane by measuring pressure at two other points a and b in the horizontal plane again 900 apart.

Figure 3.5 Spherical yawhead

An alternative way of measuring yaw angle is to fix yawhead in tunnel and to determine the flow angularity by reading the pressure difference between two orifices and comparing with a previous calibration of the yawhead. It is believed that accurate testing can not be done if the variation in angle is greater than

49

 0.5 degree. The larger angles of yaw distorts the span load excessively.

Unfortunately, the variation of flow angle across the jet may change with the tunnel speed. If such a change is noted, a testing speed must be selected and the guide and antitwist vanes should be adjusted to give smooth flow at that speed.

3.5 Turbulence Level : The flow conditions inside the wind tunnel are not exactly same as those in free air. The flow inside the tunnel is more turbulent than the free air because of the effects of the propeller, the guide vanes and the vibrations of tunnel walls. This discrepancy in the turbulence level results in disagreement of tests made in the wind-tunnel and in the free air at the same Reynolds number. By the same reasoning, tests made in different tunnels at the same Reynolds number may not agree. A correction factor is therefore necessary for compensating the turbulence created in the tunnel. It is found that the flow pattern in the tunnel at a given Reynolds number corresponds closely to the flow pattern in the free air at a higher Reynolds number. The increase ratio is called the ‘turbulence factor’ and the effective Reynolds number RNe of the tunnel can be obtained from the calculated Reynolds number using the turbulence factor of the tunnel from RN e  TF  RN

(3.7)

The turbulence may be found with a sphere in two ways : a) Drag sphere b) Pressure sphere

3.5.1 Drag Sphere : The drag coefficient of sphere is affected greatly by change in velocity. Contrary to the layman’s guess, CD for a sphere decrease with increasing airspeed since the result of earlier transition to turbulent flow is that the air sticks longer to the surface of the sphere. This action decreases form or pressure drag, yielding a lower total drag coefficient. Obviously, the Reynolds number at which the transition occurs at a given point on the sphere is a function of the turbulence already present in the air and hence the drag coefficient of a sphere can be used to measure turbulence . The method is to measure the

50

drag, D, for a small sphere 15 or 20 cm in diameter, at many tunnel speeds. After subtracting the ‘horizontal buoyancy’ drag DB the drag coefficient may be computed from

CD 

D  DB



(3.8)



1  d 2 4 U 2 2

Figure 3.6

Variation of CD with Reynolds Number

The sphere drag coefficient is then plotted against the calculated Reynolds number, RN (Fig.3.6). The Reynolds number at which the drag coefficient equals 0.30 is noted and termed the critical Reynolds number, RNC. The above particular value of the drag coefficient occurs in free air at RN = 385000, so it follows that the turbulence factor may be given by TF = 385000/RNC

(3.9)

Once the turbulence factor (TF) is obtained from equation (3.9), the effective Reynolds number, RNe, can now be calculated from equation (3.7).

51

3.5.2 Pressure Sphere : An alternative method (which will be used) of measuring turbulence makes use of ‘pressure sphere’. No force tests are necessary and the difficulty of finding the support drag is eliminated. The pressure sphere has an orifice at the front stagnation point and 0

1 four more interconnected and equally spaced orifices at 22 from the theoretical rear 2

stagnation point (Fig.3.7).

Figure 3.7 Pressure Sphere A lead from the front orifices is connected across a manometer to the lead from the four rear orifices. After the pressure difference due to the static longitudinal pressure gradient is subtracted, the resultant pressure difference, p for each Reynolds number is divided by the dynamic pressure for the appropriate Reynolds number, and the quotient is plotted against Reynolds number (Fig. 3.8). It has been found that the pressure difference p/q is 1.22 when the sphere drag coefficient is 0.30 and hence this value of p/q determines the critical Reynolds number RNC. Once the turbulence factor is determined, the turbulence factor may then be determined, as before, from equation (3.9).

52

Figure 3.8 Variation of p/q with Reynolds number

This experiment is carried out on a sphere of diameter 20 cm. The following table may be made for plotting p/q vs Reynolds number.

Table 3.3 : Experimental measurement of turbulence factor

No.of

hLS

U from Fig.

q from

hj

Runs

(cm)

1.2 b (m/sec)

Fig.1.2 a (cm) (N/m2)

p = hjwg.sin600

p/q

RN = UD/

(N/m2)

1. 2. 3. 11.

Turbulence factor usually varies from 1.0 to 3.0. Values above 1.4 indicate that the tunnel has too much turbulence for reliable testing. Low turbulence factor is necessary for the test data to be reliable.

53

Chapter 4

FLOW VISUALISATION 4.1 Introduction : Flow visualization techniques are a means of obtaining the qualitative pattern of the flow about a body. Flows encountered in engineering application are often complex in nature. Such techniques of flow visualization helps in obtaining a better understanding of the flow characteristics. Many a times suitable mathematical methods have been developed for a particular flow problem based on such qualitative studies. Flow visualisation techniques can be classified as follows :

Flow visualisation techniques

Incompressible flow

Entire flow field

Compressible flow

Only on model

Flow pattern

Shock visualisation

1. Oil flow

1. Shadowgraph

1. Smoke

1. Tuft

2. Tuft on wire mesh

2. Oil flow

2. Interferometer

3. Evaporation

3. Schlieren

4.2 Incompressible Flow Visualisation Techniques :

4.2.1 Smoke Method : Flow visualisation with smoke is achieved in a smoke tunnel with a facility to emit cleaned smoke in streamer form (Fig. 4.1). Smoke is generated by burning kerosene or paraffin. Particular care is needed in introducing the smoke in the tunnel by a blower without disturbing the flow in the tunnel. This smoke follows the air flow and makes the

54

flow pattern visible. Smoke tunnels are usually low-velocity tunnels and most of them have two dimensional test sections. Such tunnels are usually open circuit type to prevent accumulation of smoke in the tunnel. The walls of test section are made of glass so that the flow can observed (Fig. 4.2) and/or photographed.

Figure 4.1 Smoke Tunnel

Figure 4.2 Flow separation at high angle of incidence

55

4.2.2 Tuft Method : Tufts are simplest and most often used. A large number of silk tuft are pasted at one end on the surface of the wing. The length of each tuft is taken about 2 cm. The most rapid method of installing the tufts is to attach them about every one inch to the tape and pasting the tape on the model (Fig. 4.3). To obtain clear photography the model is usually painted black while the tufts used are white. Since the open ended tufts align with the flow the general direction of he tufts indicate the direction of the flow on the surface of the body. Motion of tufts usually means that the flow in the boundary layer has become turbulent. Violent motion or tendency a tendency to lift from the surface and point upstream indicates separation. If the tufts are to be used to examine the entire flow field they may be supported on wires on a mesh installed inside the tunnel. Complete grids of wires normal to the flow can be fixed in the tunnel behind a wing model. Tufts attached on one end on the mesh junctions will align with the flow direction and show up trailing vortices.

Figure 4.3 Visualisation of flow over a straked wing by tuft method

56

4.2.3 Oil Flow Method : In this method the model is pasted with a semi-liquid mixture of mobil oil and grease and a dye. The dye taken for this purpose is a chemical known as Rhodamin B. When the model is installed in the tunnel, the air flow spreads the mixture along the streamlines so that after the tunnel has been stopped the flow pattern remains. The process requires about 30 minutes of continuous air flow in the tunnel. The model is thereafter removed from the tunnel and the flow pattern (Fig. 4.4) can be examined afterwards under ultraviolet light. An alternative approach is to mix mobil oil and titanium dioxide (dye) and paste on the model. In this case the mixture gets dried up in a few minutes and the flow pattern can be observed without using ultraviolet light. Care must be taken so that the oil does not follow machining marks on the surface.

Figure 4.4 Visualisation of flow over a straked wing by oil flow method

4.2.4 Evaporation Method : Napthaline may be dissolved in acetone and pasted on a model. When the tunnel runs naphthalene evaporated quickly from the turbulent portion making that portion white. If the model is painted black, transition from laminar to turbulent flow can be observed easily. Among the incompressible flow visualization techniques it may to be noted that tuft, oil flow and evaporation method gives pattern of flow on the surface of the model only while the smoke method (and tuft on mesh screen) gives the picture of the entire flow field.

57

Among the compressible flow visualization techniques, only the oil flow method, described in section 4.2.3, can be used. Other methods are not suitable because of the high speed involved.

4.3 Compressible Flow Visualisation Techniques :

4.3.1 Shadowgraph Method : A parallel beam of light is produced by a point source. It is passed through a converging lens and then through the working section. Since the flow in the working section is compressible, refraction of light rays through the compressible medium will be different. The screen will be illuminated where rays have converged. Shock waves then appear on the screen as two adjacent bands, one dark and one light, corresponding to the sudden increase in density gradient at the front of the shock and the sudden decrease in gradient at the rear.

Figure 4.5 Shadowgraph picture of flow about a sphere

4.3.2 Schlieren Method : Schlieren method is most widely used. It is sensitive to density changes whereas shadowgraph method is sensitive to change in density gradient. The light rays passing

58

through the varying density area (test section) will be deflected. The screen will be illuminated or darkened depending on the deflection of the light beam. This method is described in details in chapter 20.

4.3.3 Interferometer Method : A direct response to density changes is given by the interferometer which depends on the interference fringes formed on the recombination of two light rays from the same source which have taken different times to make the journey. If the two path lengths are same, interference fringes may be produced. The light paths are adjusted with no airflow disturbance to produce a uniform and parallel set of interference fringes on screen giving uniform illumination. When the tunnel is run with model installed, fringe spacing will change by an amount proportional to the phase change by the disturbance at any point which is in turn proportional to the change of fluid density integrated along the light path. If the interferometer is pre-calibrated, it will give absolute values of density.

Figure 4.6 Schematic diagram of the interferometer system

59

Chapter 5

PRESSURE MEASUREMENT BY MECHANICAL DEVICE 5.1 Introduction : Pressure, at different points on the surface of model, can be obtained by drilling holes on the surface and connecting tubes from these points to a mechanical device like a multitube liquid level manometer (Fig. 5.1). liquid levels, which are initially in the same level, undergo changes in height proportional to the pressure applied and pressure at different points in the surface can be calculated from the heights of the columns.

Figure 5.1 Liquid level manometer

Multi-tube, indicated schematically in Fig. 5.1 may be used in vertical position. For increased sensitivity the manometer may be inclined at various angles in which readings are multiplied by appropriate factors. Also, in stead of water, liquid of specific gravity less than 1.0 may be used.

60

The reservoir for manometer liquid is usually mounted on a vertical rod at a height which is adjustable. It is recommended that the reservoir be normally left open to atmospheric pressures. Pressures p1, p2, p3,….are then gauge pressures i.e., pressures relative to atmospheric datum. Pressure relative to some other chosen datum may be obtained by connecting the reservoir and one manometer tube to the required datum. Manometers are generally graduated so that height of liquid level may be read in cm and the pressure is calculated from the height of the liquid column in the relevant tube. Some manometers are graduated directly in N/m2 or in millibar (1mb = 100 N/m2 ).

5.2 Measurement of Cp : Pressure is usually expressed in non-dimensional form as pressure coefficient Cp . by definition Cp is given by

Cp 

p  p 1 U 2 2

(5.1)

Using a liquid-level manometer as shown in Fig. 5.1, pressure coefficient Cp can be obtained in two ways depending on whether the tunnel is precalibrated or not.

5.2.1 Without Pre-Calibration of the Tunnel : If the tunnel is not pre-calibrated to give U, two holes are to be drilled on the walls of the settling chamber and the test section and directly connected to the manometer in addition to connecting pressure port of the configuration. Now, by Bernoulli’s theorem, P0  p 

1 U 2  PS 2

where PS is the settling chamber pressure. Or,

1 U 2  p S  p 2

If the manometer is graduated in N/m2 ,(p - p) and (PS - p) can be obtained directly in units of N/m2 and Cp can be obtained as the ratio of the two given by

Cp 

p  p PS  p

(5.2)

61

Non-dimensional pressure coefficient is thus obtained simply as a ratio of pressure differences and value of U is not needed. If U is needed (e.g., to calculate Reynolds number) U can be obtained in a simple manner by assuming no frictional loss between settling chamber and test section. Under this assumption, U can be obtained as

U   2  PS  p  

(5.3)

If the manometer is graduated to give height of liquid column, Cp can be obtained as ratio of column heights as shown below. p  p   liquid  h  h   g

1 U 2  p S  p   liquid  hS  h  2

and Where,

hS = height of column in the tube connected to settling chamber. h = height of the water column in the tube connected to the pressure port on the configuration where pressure is being measured. h = height of the column in the tube connected to test section This gives ,

Cp 

p  p PS  p



 liquid  g  h  h   liquid  g  hS  h 



h  h hS  h

(5.4)

Cp is then obtained as ratio of height difference of liquid columns. By assuming zero frictional loss between settling chamber and test section U can be obtained as

U   2  PS  p    2   liquid  hS  h   g 

62

(5.5)

5.2.2 With Pre-Calibration of the Tunnel : If the tunnel is pre-calibrated to give U, pressure coefficient can be derived in terms of U. Cp 

p  p h  h    liquid  g  1 1 2 U  U 2 2 2

If the manometer is inclined at 600, then Cp 

h  h    liquid  g  sin 60 0 1 U 2 2

If the liquid is water, height is graduated in cm and density of air is taken as 1.225 kg/m 3, then

Cp 

h  h   1000  9.81 0.866 1  1.225  U 2 2

 138.70  h  h  U 2

(5.6)

Experimental measurement of pressure distribution on a few simple models are described in the following sections. In all models several holes are drilled on the surface and connected to the multi-tube manometer. Pressure distribution can then be obtained from equation (5.4) or (5.6) depending on whether the tunnel is pre-calibrated or not. These models include : a) Circular cylinder model b) Elliptical cylinder model c) Spherical model

5.3 Pressure Distribution on Circular Cylinder Model : Exact analytical solutions are available for limited cases of direct potential flow problems. The problem of two dimensional flow about a cylindrical body is one of such problems. For steady, inviscid, incompressible irrotational flow, for which the governing equation is Laplace’s equation, the non lifting two dimensional flow about a cylindrical

63

body can be simulated by placing a doublet in uniform flow. The total velocity at any point P (Fig. 5.2) is obtained as qt  2U  sin 

(5.7)

Figure 5.2 Circular cylinder in uniform flow The pressure distribution can be obtained from Bernoulli’s equation,

2

 q  p  p CP   1   t   1  4 sin 2  1 U  U 2 2

(5.8)

It may be noted that the expression for total velocity or pressure is independent of the diameter of the cylinder. The ‘ideal’ pressure distribution, given by equation (5.8), over the surface of the cylinder will be symmetrical about the axis in the direction of the flow and about the plane normal to it. Consequently, the net forces, lift and drag, are zero. An experimental study can be undertaken to check how far the ‘real’ solution deviates from the ‘ideal’ solution. For the case of uniform flow of real fluid, both the effects due to compressibility and viscosity are to be taken into account. For the low speed test case (0.1 Mach number) the effect due to compressibility may be justifiably ignored. However, effect of viscosity alone will change the flow pattern considerably. Primarily, the flow will be asymmetric about the axis normal to the uniform stream and hence pressure distribution will also be asymmetric resulting in a net force (drag) acting

64

on the cylinder along the flow direction. However, the flow is still symmetrical about the axis in the direction of the flow and hence no lift force acts on the cylinder. Secondly, while the ‘ideal’ flow is always attached to the body surface, in real fluid, the flow may separate under adverse pressure gradient. In the forward face of the cylinder ( between 00 to 900), the flow speed increases and pressure decreases, hence the flow is not likely to separate in this region. In the backward face, ( between 900 to 1800), the speed decreases and pressure increases. Under the action of this increasing pressure (i.e. adverse pressure gradient), the flow is likely to separate. This separation is the so-called ‘boundary layer separation’. Since the flow velocity is less in the boundary layer than in the free stream outside the boundary layer, the flow separates in the boundary layer. The exact process of separation is yet little understood. Generally speaking, at low speed the flow in the boundary layer is laminar and will be attached to the body. Since the flow speed is less, kinetic energy associated with the flow is also less, and the laminar flow is more susceptible to separation. As the flow speed is increased, the boundary layer becomes turbulent. Transition for laminar to turbulent flow is governed primarily by the Reynolds number of the flow. The model chosen for experimental work is a circular cylinder of diameter 10.8 cm and span 60.8 cm which extends from wall to wall (so that the flow is two dimensional). 0

1 Sixteen pressure holes are equally spaced at 22 apart (Fig. 5.3) on the surface of the 2

cylinder and are connected to a multi-tube manometer. Advantage, however, can be taken for this circular cylinder model. Only one hole can be drilled and pressure at different points on the circular section can be obtained by simply rotating the model (chapter –12).

65

Figure 5.3 Pressure holes on cylinder surface

Both the theoretical and experimental Cp distribution can now be obtained from equation (5.8) and equation (5.4) or (5.6) and plotted against . The difference is due to viscous effects. The following table may be made for plotting Cp vs.  (Fig. 5.4).

Table 5.1 : Pressure distribution on circular surface

Tap



hLS

U

h

h

points

1.

0

2.

22.50

3.

450

16.

337.50

66

Cp

Cp

(Theoretical) eq. (5.7)

(Experimental) eq. (5.6)

Cp -Ve

0

90

180

270

300

330

360



Figure 5.4 Pressure distribution on cylinder surface

5.4 Pressure Measurement on Elliptical Cylinder Model : Exact analytical solution exists also for the case of potential flow about elliptical; sections. Using conformal transformation, flow around a circular section can be conformed into a flow around an elliptical section in such a way that the condition at infinity is unaltered. The flow at any point (r, ) on the surface of a circular section is given by equation (5.7) qt  2u sin 

(5.7)

The flow past a circular section can be transformed into the flow past an elliptical section by a conformal transformation (Fig. 5.4)

 Z

b2 Z

(5.9)

67

Figure 5.5 Conformal Transformation The velocities for corresponding points can be related by

qellipse qcircle Now,



dz d

d dz  1   1

(5.10)

b2 b 2 2i  1  e z2 a2

[since z = aei]

b2 cos 2  i sin 2  a2 1

2 2  b 2   b2   2 d Therefore, dz  1  a 2 cos 2    a 2 sin 2       

1

 2b 2 b4 2  1  2 cos 2  4  a a  

or,

dz  2b 2 b4   1  2 cos 2  4  d a a  



1 2

Therefore,

68

qcircle

qellipse 

 2b 2 b4  1  2 cos 2  4  a a  

[from eq. (5.9)]

1 2

2U  sin 



 2b 2 b4  1  2 cos 2  4  a a  

(5.11)

1 2

The pressure distribution of the surface of the ellipse may be obtained from Bernoulli’s equation, cp  1

qellipse

2

(5.12)

U

The pressure distribution on the ellipse can be experimentally determined by a elliptical model extending from tunnel wall to wall so that two dimensional flow is obtained. The major and minor axis of the elliptical model are 15.75 cm and 10.9 cm respectively. Static pressure holes are made at sixteen points on the surface for measurement of pressure. Cp at these sixteen points can be obtained from water level in the manometer. A table can be made, as shown, for plotting of theoretical and experimental pressure distribution vs  . The comparison of theoretical and experimental pressure distribution may be shown in a similar manner as for circular cylinder.

Table 5.2 : Pressure distribution on elliptical cylinder surface Tapping



hLS

U

h

cp

cp

(Theoretical) (Experimental) eq. (5.12) eq. (5.6)

points 1.

0

2.

22.50

3.

450

16.

h

337.50

69

5.5 Pressure Measurement on Spherical Model : The potential flow about a spherical body can be mathematically simulated by placing a three-dimensional point doublet in a uniform stream. The flow about a sphere of radius ‘a’ can be shown to be generated by placing a doublet of strength  (=2a3U) in uniform stream U (Fig. 5.5). The perturbation velocity components due to a doublet of strength , placed at origin, at any point (x, y, z) are

 (3 x 2  r 2 ) 5 4r 3xy v 4r 5 3xz w 4r 5

u

(5.13)

where,

r  x2  y2  z 2 and

 = 2a3U

Taking doublet strength  = 2a3U the perturbation velocity components on the surface of the sphere at the center section are obtained as 1 3 u   cos 2   U  [putting x = acos, y=0 and z = asin] 2 2

v=0 w

3 cos  sin U  2

(5.14)

The total velocity components due to the combined flow is

1  3 U   cos 2    1U  2  2 3 W  U  cos  sin  2 The total velocity is given by

70

3 qt  U 2  W 2  U sin  2 The pressure distribution can be obtained from Bernoulli’s equation

(5.15)

2

 q  c p  1   t  U  9 (5.16)  1  sin 2  4 The spherical model undertaken for experimental work is drilled at 16 equally spaced

points for pressure measurement (Fig. 5.6)

Figure 5.6 Pressure holes on spherical surface

A table similar to that used for previous two experiments can be made and theoretical (equation 5.16) and experimental (equation 5.6) pressure distribution can be plotted. The theoretical pressure distribution is symmetrical over the surface of the sphere and hence no force or moment acts on the sphere. The discrepancy with experimental results is due to viscous effects.

71

Chapter 6

FORCE MEASUREMENT BY MECHANICAL BALANCE 6.1 Introduction : A three component mechanical balance (Fig. 6.1) is basically designed to measure two force components along mutually perpendicular axes (lift and drag) and a single moment about an axis perpendicular to those of the forces (pitching moment). This type of balance is usually a roof-top balance to be installed on the top wall of the tunnel. The model is suspended from three vertical struts – two forward and one at the rear. Only these vertical struts emerge in the tunnel. The main lift beam, in conjunction with the pitching unit beam, gives the total lift. The two front struts are connected to the main lift beam through hinges. The main lift beam has two scale-pans for placing weights and two riders moving along a graduated scale. The two front struts are to be attached to the main lifting surface of the model and they transfer the lift force to the main lift beam. The weight placed in the beam scale pan together with the rider displacements required to balance the torque imposed on the main lift beam by the lift force gives a measure of the lift force provided the beam is properly pre-calibrated. The rear strut also gives a part of the lift, which is measured in the same way as in the main lift beam. The pitching wheel, when rotated, guides a block along a threaded rod. As the block travels up or down the attitude of the model in pitch is changed. The drag beam comes under a torque due to the horizontal force on the front struts. This horizontal force or drag is transferred from the front struts to the drag beam through appropriate linkages. The moment on the drag beam is balanced by beam pan weights and rider movement which gives a measure of the drag force encountered by the model.

72

Figure 6.1 Wind tunnel mechanical balance

6.2 Calibration : The calibration of a balance require certain equipments and the idea is to make these equipment as permanent as possible since calibration checks are needed many times during the life of a balance. The first equipment needed for calibration is a loading ‘TEE’ (Fig. 6.2). The tee facilitates the application of static loads in order to simulate the lift and drag forces as they arise from model tests in the tunnel. The tee is fitted to the struts of the balance, its head to the two front struts and its tail to the rear strut.

73

Figure 6.2 Loading TEE for calibration

To simulate drag, a horizontal force is applied to the tee in the direction of the drag force through a string attached to the tail of the tee. The string passes over a pulley and carries drag weights in a scale pan attached to the free end of the string. Static lift forces are simulated by dead weights placed in a weight pan hanging from the middle of the tee’s head. Riders are provided for minute adjustment in balancing. For the lift and pitching unit two riders (left rider and right rider) are provided while for the drag beam one rider is provided. Rider movements need to be calibrated. This is done for the lift unit in the following way. The step is initially balanced with no weight in either dead weight pan (which is hanging from the tee via the pulley) or in the main lift beam pan (Fig. 6.3) and the rider positions are recorded. Next, 50 gms of static lift is provided by placing this weight in the dead weight pan. The set-up is again balanced first by moving the right rider (keeping left rider stationary) and then by moving the left rider (keeping the right rider stationary) and these displacements are then recorded.

74

Figure 6.3 Simulation of lift and drag by dead weight

Table 6.1 : Rider calibration

Load in dead weight pan

Left Rider (mm)

Right Rider (mm)

0

80

80

50gms

80

90

50 gms

70

80

The table shows that each rider movement of 1 mm is equivalent of 5 gms of load. It may be noted that the riders move from left to right in balancing the static load. The combined load from dead weight pan and rider movement is termed equivalent lift and this is plotted against the load in beam pan.

Table 6.2 : Calibration of lift beam Initial Positions : Left Rider =

mm; Right Rider = mm

1 mm of rider displacement = 5 gms of static loa

75

Load (lift)

LR

RR

dLR

dRR

Load from

Load

Equivalent

weight pan

mm

mm

mm

mm

rider

In beam

lift (gms)

(gms)

weight pan

(1)+(6)

5

6

7

1+6

(gms) 1

2

3

4

169.62

60

100

0

0

0

16.81

169.62

250

60

65

0

35

165

34.64

415

450

60

65

0

0

0

50

450

750.8

60

85

0

-20

-100

78.23

650.8

Similarly, tables can be prepared for drag beam and pitching unit.

Table 6.3 : Calibration of drag beam Initial positions : Rider = mm 1 mm of rider displacement = gms of static load

Load in dead

Rm

dR

Drag load

Load in

Equivalent

weight pan

mm

mm

from rider

beam weight

drag (gms)

(gms)

pan

(4)

(5)

(gms) (1)

(2)

(3)

Table 6.4 : Calibration of pitching unit Initial position : LR = mm ; RR = mm 1 mm of left rider displacement = gm of static lift load. 1 mm of right rider displacement = gm of static lift load.

76

(1+4)

Load (lift)

LR

RR

dLR

dRR

Load from

Load in beam

Equivalent

in dead

Mm

Mm

mm

mm

rider (gms)

weight pan

lift (gms)

weight pan (1)

(gms) (2)

(3)

(4)

(5)

(6)

(7)

(1)+(6)

Calibration charts may now be made for lift, drag and pitching unit as shown below.

Equivalent lift

Load in beam weight pan

Figure 6.4 Calibration of main lift beam

Equivalent drag

Load in beam weight pan

Figure 6.5 Calibration of drag beam

77

Equivalent lift

Load in beam weight pan

Figure 6.6 Calibration of pitching unit 6.3 Measurement of Forces and Moments : Once the balance is calibrated, forces and moment acting on a model installed in the tunnel can be computed easily from the beam weight pan load and calibration chart and rider displacements. Tests are carried out on two models 1)

a flat plate of dimensions 45.75 cm  30.5 cm

2)

A rectangular wing of dimensions 61 cm  30.5 cm (which spans the jet width) with NACA 0012 as its aerofoil section.

The interest is in finding the variations of CL, CD and CM with angle of incidence . The tests with these models are carried out at the same speed for different angles of incidence determined from the pitching screw settings.

Table 6.5 : Flat plate lift from main lift beam Initial position: LR = mm, RR = mm Incidence

(1)

LR

(2)

RR

(3)

dLR

(4)

dRR

(5)

Lift from

Load in

Lift

Total lift

rider

beam

from

L2 (gms)

weight pan

Fig. 6.4

(7)

(8)

(6)

78

(6) + (8)

Table 6.6 : Flat plate lift from pitching unit Initial position : LR = mm, RR = mm

Incidence

(1)

LR

(2)

RR

(3)

dLR

dRR

Lift from

Load in

Lift

Total lift

rider

beam

from

L2 (gms)

weight pan

Fig. 6.6

(7)

(8)

(5)

(4)

(6)

(6) + (8)

Table 6.7 : Flat plate lift coefficient Incidence

Total lift

Total lift (L)

(degrees)

(L1 + L2 )

(L1 + L2 ) + 9.81

gms

1000 Newton

½ ? U 2 (N/m2)

CL 

L 1 U 2 S 2

Table 6.8 : Flat plate drag coefficient Initial position : R = mm

Incid-

R mm

ence

dR

Drag

Load in

Drag

Total

Total

mm

load

drag

from

drag D

drag =

from

beam

Fig.

gms

(D9.8)/

rider

pan gms

6.5

1000

gm

Newton

gms (1)

(2)

(3)

(4)

(5)

(6)

79

(4)+(6)

CD 

D 1 2 U 2

Table 6.9 : Flat plate pitching moment coefficient

Incid-

LR

dLR

dRR

Lift

ence

mm

mm

mm

degree (4)

(5)

Load

Lift

Total

due to in

from

pitch-

rider

beam

Fig. 11.6 ing

gms

pan

gms

lift

M0.25C

N.m

Results can now be presented in graphical form. On the graph, the value of CLmax should be noted. Similar table can be created for the wing model.

6.4 Evaluation of the effects of the Support (Tare and Interference Drag) : In any wind tunnel the model needs to be supported in some manner and the supports, in turn, affects the drag measurement. Any strut connecting the model will add three quantities to the forces read. The first is obvious drag of the exposed strut (tare) the second is the effect of the strut’s presence on the free air flow about the model and the third is the effect of the model on the free air flow about the strut. The last two items are usually lumped together under the term “interference” and their existence should make clear the impossibility of evaluation the total drag of the struts with the model out. This procedure will primarily expose parts of the model support not ordinarily in the air stream (although the extra length may be made removable) and will fail to record

the

interference drag and will record only the tare drag. The tare drag can be reduced by shielding a large part of the strut by fairings not attached to the balance; only a minimum of strut

bayonet is exposed to the air stream.

Theoretically the tare drag can be eliminated entirely by shielding the supports all the way into the model (with adequate clearances inside), however the added size of shield (and the presence of shield so close to model) will probably increase the interference drag so much that no net gain will be achieved.

80

The tare and interference drag can be evaluated separately or jointly using the ‘mirror’ or ‘image’ technique. The separate evaluation of drag items is long and also unnecessary. This separate evaluation approach is therefore rarely used. However, both the methods are outlined below.

6.4.1 Evaluation of tare and Interference Drag Separately : The model is first tested in the normal manner. Symbolically, Dmeasured = DN = IUB/M + IM/UB + IUSW + TU where DN

(13.1)

= drag of model in normal position.

IUB/M

= interference of upper surface bayonets on model.

IM/UB

= interference of on model on upper surface bayonets.

IUSW

= Interference of upper support windshield.

TU

= free air tare drag of upper bayonet.

Next, the model is supported from the tunnel floor by the ‘image’ or ‘mirror’ system. The supports extend into the model but a small clearance is provided (Fig. 6.7) so that the balance record only the drag of the exposed portion of the support (in the presence of the model).

Figure 6.7 Arrangement for determining tare and interference drag separately

81

That is, Dmeasured = IM/UB + TU

(6.1)

For the interference run, the model is inverted and run with the mirror supports just clearing the attachment points. This gives

Dmeasured = Dinverted + ILB/M + IUB/M + IM/UB + IUSW + ILSW + TL

(6.2)

where Dinverted = drag of the model inverted (should equal that the drag of the model normal, except for misalignment). The upper supports are removed and a second inverted run is made giving

Dmeasured = Dinverted + ILB/M + IM/LB + ILSW + T1

(6.3)

The difference between the two inverted runs is the interference of the supports of the upper surface. That is eq. (6.2) minus eq. (6.3) yields IUB/M + IUSW

(6.4)

By subtracting eq. (6.1) and (6.4) from the first run, the actual model drag is determined if the balance is aligned.

6.4.2 Evaluation of the Sum of Tare and Interference Drag : In this procedure, the sum of tare and interference drag can be found in three runs instead of four in the previous method. In this case, the normal run is made, yielding

Dmeasured = DN + IU + TU

(6.5)

where IU = IUB/M + IM/UB + IUSW Next model is inverted and this gives

Dmeasured = Dinverted + IL + TL

(6.6)

Then the dummy supports are installed. Instead of clearance being between the dummy supports and the model, the exposed length of the support strut is attached to the model and the clearance is in the dummy supports. This configuration yields (Fig. 6.8). Dmeasured = Dinverted + IL + IU + TL + TU

82

(6.7)

Figure 6.8 Arrangement for determining the sum of tare and interference drag

The differenc between eq. (6.7) and eq. (6.6) yields the sum of tare and interference T L and IL . A third method which is crude but simple is to minimize the interference drag by using strut of aerofoil shape (and not using the windshield). The drag of the supports is than essentially the tare drag which can be measured easily by the balance with the model out. The drag of the suspension system can be measured this way and subtracted from the drag measured of the two dimensional wing.

83

Chapter 7

PRESSURE MEASUREMENT BY TRANSDUCER 7.1 Introduction : Pressure measurement in wind tunnel is of interest not only for determining pressure distribution on aerodynamic shapes but also for determining test conditions in the wind tunnel test sections. Up to about 15 to 20 years ago, the majority of wind tunnel pressures were

measured with liquid manometers and readings were taken manually. These

manometers , however, are not suitable for measuring very high or very low values of pressure or for measurement of pressure in unsteady or short duration tunnel.

To overcome this limitations, these manometers have been replaced to a large extent by pressure transducers in a scanning system with automated data recording system. Pressures from 2  10-7 psia to approximately 1000,000 psia are successfully measured in wind-tunnel with the aid of transducers whose electric output signal emanates from a deflection or deformation caused by a pressure activated elastic sensing element. The most common type of elastic sensing elements are the diaphragms. In order to produce an electric signal, the elastic elements operates in conjunction with electrical sensing element which provides an electrical change in response to the deflection or deformation of the sensing element. The most frequently used electrical sensing elements include metallic or semi-conductor strain gauges, variable capacitance device, variable reluctance device and piezoelectric elements.

A summary of general performance of different pressure transducers as given in Table 7.1.

84

Table 7.1. Summary of performance of pressure transducers : Type of transducers

Range of pressure Operating temperature Resonant measurement (psi)

( 0F)

frequency (KHz)

Variable resistance

10-4 to 100,000

- 430 to 300 0F

Upto 1

Variable

2  10-7 to 10,000

- 55 to 225 0F

Upto 200

Variable reluctance

3  10-5 to 10,000

- 63 to 250 0F

Upto 25

Piezoelectric

5  10-4 to 100,000

- 400 to 500 0F

Upto 500

capacitance

In variable resistance transducer, a pressure change is converted into a change in resistance caused by the strain in a strain gauge or gauges. Most strain gauge pressure transducer incorporate four active strain gauge elements in a Wheatstone bridge circuit. Variable reluctance transducer employs diaphragm as the sensing element. This diaphragm is supported between two inductance core assemblies. A magnetic circuit with core is completed. As the pressure is applied, reluctance changes. In variable capacitance transducer pressure is applied on one plate of the capacitor. Since the capacitance varies inversely proportional to the plate distance capacitance changes due to applied pressure. In piezoelectric transducer, voltage is generated when pressure is applied due to the squeezing effect of the crystal. This type of transducer is self generating and does not require any external power supply. Variable resistance transducers are most widely used. In such transducers strain gauges are bonded directly on diaphragm (Fig. 7.1). A diaphragm is essentially a thin circular plate fastened around its periphery to a support shell. Stainless steel or beryllium copper is generally used as the diaphragm material. If the strain gauges are located as shown in Fig. 7.1, elements 1 and 2 will be in tension and 3 and 4 will be compression. By electrically connecting these gauges a fully active Wheatstone bridge is realized. Recent advances in the semi-conductor field have led to the development of an integrated Wheatstone bridge consisting of four strain sensitive resistive arms formed directly on the diaphragm.

85

Figure 7.1 Variable resistance pressure transducer

7.2 Time Response : Response time of a transducer is not critical for a continuously running tunnel. When measurements are required in short duration facility or when unsteady or transient data are required, time response becomes primary in importance. In such cases transducer must be flush mounted or be connected by very short tube lengths. The response time, t, for a flush mounted transducer is

t

1 2 f 1 h2

where f = transducer resonant frequency h = damping ratio

7.3 Pressure Scanning : Pressure distribution on models described in chapter 5 can also be obtained by use of pressure transducer. However, measuring pressure at multiple numbers of ports on a body poses a problem. Either it will require an equal number of pressure transducers, signal

86

conditioner and readout systems ( which will be expensive) or a single transducer will be connected to all pressure holes one by one (which will be time consuming). This problem can be overcome by a pressure scanning system. A number of tubes from various pressure ports are routed to a common point and then applied individually to a single transducer and readout system as shown on Fig. 7.2.

Figure 7.2 Pressure scanning system The most important part of the pressure scanning system is the scanivalve (Fig 7.3). in a scanivalve, the transducer is sequentially connected to the various pressure ports via a radial hole in the rotor which terminates at the collector hole. As the rotor rotates, this collector hole passes under the ports in the stator. Referring to the cutaway drawing (Fig. 7.3) the rotor is seen to be rigidly supported by a ball thrust bearing. The stator is elastically connected to the block in a manner which allows the stator to follow the surface of the rotor. Thus the pneumatic forces (pressure  area) at each port which tends

87

to blow the rotor away from the stator are withstood by the ball thrust bearing. The stator is epoxied into the block to prevent rotating.

Figure 7.3 Scanivalve

A scanivalve may be hand driven or solenoid driven. For a solenoid driven scanivalve, further instruments are required. A solenoid controller is necessary for controlling the stepping speed of the scanivalve. Also, interface controller is necessary for controlling scanivalve port location.

88

7.4 Measurement of Cp : The excitation voltage applied to a pressure transducer is generally 12 volts. A typical value of full scale output of a PDCR23 transducer at this excitation is 17.5 mV for 1 psi of pressures. If the reference side of diaphragm is connected to wall of test section (i.e. datum pressure is free stream static pressure p ), the pressure coefficient at any measurement location can be calculated from the output in two ways depending on whether the tunnel is pre-calibrated or not.

7.4.1 With Pre-calibration of tunnel : By definition, pressure coefficient Cp is given by

Cp 

p  p 1 U 2 2

If the tunnel is pre-calibrated, i.e., the free-stream speed U is known, Cp can be derived in terms of U. Cp 

readout.in..mV   6894.6 / 17.5 1 U 2 2

(1 psi = 6894.6 N/m2)

Since the transducer output is usually small, an amplifier is usually used. Taking the amplifier gain into account Cp can be written as Cp 

readout..in..mV   6894.6 1  1.225  U 2  amplifier ..gain   17.5 2

 643.228 

(readout..in..mV ) U  (amplifier ..gain ) 2 

(7.1)

7.4.2 Without pre-calibration of tunnel : Cp can be derived in an alternative way if the tunnel is not pre-calibrated to give U. Cp can be expressed as

Cp 

p  p 1 U 2 2 89



p  p P0  p



p  p PS  p

(PS = settling chamber pressure)

There are uaually 48 pressure port locations in a scanivalve. If the reference side is connected to wall of test section to sense p and 48th port location is connected to wall of settling chamber then for pressure upto 47 points can be measured one by one by a scanivalve as a simple ratio of digital outputs, given by equation (7.2).

90

Chapter 8

FORCE AND MOMENT MEASUREMENT BY ELECTRONIC INTERNAL (STING) BALANCE

8.1 Introduction : Aerodynamic forces and moments acting on a model in wind tunnel can be accurately measured by variable resistance strain gauges. Usually a sting balance, also known as internal balance or strain gauge balance, is used where strain gauges are bonded on the sting (Fig. 2.2). These strain gauges are connected in Wheatstone bridge arrangement (differencing circuits for forces and summing circuits for moments). When the tunnel is started, the forces acting on the model change the resistance of strain gauges. The voltage of the ‘unbalanced’ Wheatstone bridge is then a measure of the forces acting on the model. An aircraft model is subjected to three aerodynamic forces along three axes (lift, drag and side force) and three moments (yawing, rolling and pitching) about the three axes. In general, Wheatstone bridge circuit needs to be constructed for each of the forces and moments. A general six component balance will then require six Wheatstone bridge circuits (consisting of 24 strain gauges), a six channel signal conditioner, separate power supply for each channel and appropriate data acquisition system. Philosophy underlying bonding of strain gauges and Wheatstone bridge circuits needs to be studied for each component separately. It is worth nothing here that six components of forces and moments are measured in a sting balance about the body axes (since the sting is attached to the body and moves with the body) and not about wind axes. Hence body axes need to be converted to wind axes which are more familiar.

91

8.2 Measurement of lift : Generally, the symmetrically placed aircraft model will experience a lift L, drag force D and pitching moment M at the aerodynamic center. The sting is like a cantilever beam on which lift and pitching moment act. The sting is also subjected to axial stress due to drag in addition to the bending stress due to lift (Fig. 8.1). it can be shown easily that the output voltage will be proportional to lift force only if the ‘differencing’ circuit is used.

Figure 8.1 Differencing and summing circuits

The output voltage of the Wheatstone bridge circuit, from equation (2.16), is

V V

where



R'R' '

(8.1)

2R

R'  Rb'  Ra R' '  Rb''  Ra Rb' , Rb'' = changes in resistance due to bending stress. Ra = changes in resistance due to axial stress.

Therefore,

V V

 

R

' b

 

 Ra  Rb''  Ra 2R



Rb'  Rb''

(8.2)

2R

92

The above expression shows that the output voltage is independent of axial stress (i.e., drag forces). Now the gauge factor G is defined as G

R R R R  L L 

and the change in resistance is expressed in terms of change in strain as R = R G 

(8.3)

The longitudinal strains on the four strain gauges can be written as LZ  Z 1  h EI LZ  Z 2  2   h EI LZ  Z 2  3  h   2 EI

1  

4 

(8.4)

LZ  Z1  h   1 EI

The changed resistance value of four strain gauges will be R1 = R + RG 1 R2 = R + RG 2

(8.5)

R3 = R + RG 3 R4 = R + RG 4 The output voltage is then

V V



R2 R4  R1 R3 R1  R2 R3  R4 



R  RG 2 R  RG 4   R  RG 1 R  RG 3  RG 1  RG 3   RG 3  RG 4 



1  G 2 1  G 4   1  G 1 1  G 3  2  G 1  G 2 2  G 3  G 4 

 

2G 2   1 



4  G 2  1   2  2 1 2 2

2

(since 3 = - 2 & 4 = - 1 )



G  2   1  (Neglecting higher order term) 2

93



G Z 2  Z1 hL 2 EI

 K1 L where

K1 

(8.6)

Gh( Z 2  Z1 ) = Constant 2 EI

The output voltage is seen to be linearly proportional to lift force, L. It is to be noted that the circuit can be made more sensitive by increasing the distance (Z2 – Z1) between the strain gauges. It is worth noting here that the relationship will not remain linear except for a fully active bridge. The calibration constant K1 can be obtained by putting appropriate values of G, h, E, I and (Z2 – Z1) in the expression K1 = Gh (Z2 – Z1)2EI. Alternatively, it can be obtained by a calibration procedure as shown in Fig. 8.2. In this procedure, K1 is obtained by putting known weight (W) at position (AA) on the string through a pulley and noting the voltmeter readings. For an excitation voltage of 4.0 volts and amplifier gain 1000, typical value of the constant K1 is of order of 0.028 mV/gm.

Sting 1

2

A

3

4

A

W

Figure 8.2 Calibration procedure for obtaining K1 The lift coefficient CL in any subsequent experiment can be directly related to the millivolt output as C L  C N cos  

V 1 K1 U 2 S  1000 2

 9.81  cos 

94

(8.7)

8.3 Measurement of pitching moment : The pitching moment can be obtained from four strain gauges by constructing the ‘summing’ circuit (Fig. 8.1), which is constructed simply by interchanging R 2 and R3 of ‘differencing’ circuit. The output of the summing circuit can be shown to be independent of the drag force. The output voltage can be written as

V V



R3 R4  R1 R2 R1  R3 R2  R4 

The changed values of the resistances may be written as

R1  R  R ' b  Ra R 2  R   R '' b   R a R3  R  R '' b  Ra R 4  R  R ' b  R a Substituting in previous equation yields

V V



R  Rb  Ra R  Rb  Ra   R  Rb R  Rb  Ra  2R  RbRb  2Ra 2R  Rb  Rb  2Ra 



2RRb  RRb  Ra Rb  RaRb  2R  2Ra 2  Rb  Rb 2



2RRb  RRb  4R 2



(neglecting higher order term)

Rb  Rb

(8.8)

2R

Equation 8.8, analogous to equation (2.11), shows the output voltage V to be independent of the axial stress due to the drag force. The output of the summing circuit can now be shown to be proportional to pitching moment. The output voltage is

V V



R4 R3  R1 R2 R1  R3 R4  R2 

95

Using equation (8.5),

V V

 

R  RG 4 R  RG 3   R  RG 1 R  RG 2  2R  RG 1  RG 3 2R  RG 4  RG 2   2G 1  2G 2

(since 3 = - 2 & 4 = - 1 )

4  G  1  G 2 2  2G 1 2



2

2

2

G 1   2  2

(neglecting higher order term)

G  2   1   G 1 2 Gh   K1 L  LZ  Z 1  2 EI 

  K1 L  K 2 M

(8.9)

where K2 = Gh / 2EI is a constant, M is the moment L (Z – Z1) due to L and K1 is the constant defined earlier for lift. The output voltage from equation (8.9) is seen to be dependent on both lift L and pitching moment M. However, it is to be noted that K1, for this summing circuit, is made very small to be negligible. This is because (Z2 – Z1) is made very small in summing circuit in comparison to differencing circuit by fixing strain gauges very close to each other. Equation (8.9) can be written as

V V

 K2M

(8.10)

This equation shows that the output voltage in ‘summing’ circuit is proportional to pitching moment only. The value of the constant K2 can, in principle, be obtained theoretically by putting appropriate values of G, h, E, I and (Z2 – Z1). However, it is desirable to determine its value through static calibration. This can easily be done by using dummy weight (W) at position AA, as before, to simulate lift and noting the change in output with increasing load (Fig. 8.3).

96

1

2

A

A 3

4

W

Figure 8.3 Calibration procedure for obtaining K2 However, if an accurate estimate of both K1 and K2 are needed weights can be placed at two positions, first at AA and then at BB, as shown in (Fig. 8.4) and the two values of voltages are to be noted. f

1

2

B

A

3

4

B

A

Figure 8.4 Calibration procedure for obtaining K1 and K2 From equation (8.9) the output voltages will be

V V

V V

  K1 L  K 2 M

(8.11)

  K1 L  K 2 M  Lf 

(8.12)

BB

AA

From equations (8.11) and (8.12) K2 is obtained as

97

V K2 

V

 AA

V V

BB

(8.13)

Lf

Once K2 is obtained from equation (8.11), K1 can be calculated from equation (8.11) or (8.12). The pitching moment M can be easily obtained from K2M 

or,

V V

 K1 L

 V  M   K1 L  K 2 V 

(8.14)

8.4 Simultaneous Measurement of Lift and Pitching Moment : Lift and pitching moment cam be measured simultaneously by using eight strain gauges to form two Wheatstone bridge circuits (both circuits as summing circuit) as shown in Fig. 8.5. The output voltages of the two circuits will vary because of the variation in moment with the distance f. the output voltages can be written from equation (8.9), as

V V

V V

 K1 L  K 2 M 1

(8.15)

 K3L  K4 M 2

(8.16)

AA

BB

A

B

1

2

3

4

6

5

7

A

8

B

W

Figure 8.5 Calibration procedure for obtaining K2 Now, M2 = M1 + Lf

(8.17)

98

Using equations (8.15), (8.16) and (8.17)

K1 L  K 2 M 1 

K 4 f

V V

(8.18) AA

 K 3 L  K 4 M 1 

V V

(8.19) BB

In matrix form,

K1

K2

V

L

V

AA

= K3 + fK4

K4

V

M

V

BB

On inverting,

L

K1

K2

V

-1

V

AA

= M

(8.20) K3 + fK4

V

K4

V

BB

Equation (8.20) gives lift and pitching moment directly from the outputs of Wheatstone bridge circuits provided, of course, four coefficient are measured from calibration. It is however, to be noted that K1 and K3 are very small to be almost negligible. This is because both Wheatstone bridge circuits are summing circuits where strain gauges are pasted very near to each other. If K1 and K3 are assumed to be negligible, lift and moment can be readily obtained as

99

V M1 

V AA K2

V M2  and L 

V BB K4

M 2  M1 f

(8.21)

8.5 Other Forces and Moments : The mathematics underlying strain gauge instrumentation is described in details for lift and pitching moment measurement in the previous sections. The same principles are easily extended to measuring other forces and moments. In general, differencing circuit (for forces) and summing circuits (for moments) are to be used. However, to measure other components, cantilever or the sting needs to be specially machined for suitable positioning of strain gauges for particular component. Arrangement for measuring drag in a 3 component balance is shown in Fig. 8.6.

Figure 8.6 Three component balance

100

The normal forces (CN ) and axial force (CX ) obtained by the sting balance are converted to lift (CL ) and drag (CD ) force by a simple conversion of axes (Fig. 8.7).

Figure 8.7 Axes system

CL = CN cos - CX sin

(8.22)

CD = CN sin + CX cos

(8.23)

The side force and yawing moment can be obtained using the same principle for measuring lift and pitching moment. However, unlike in the previous case, the strain gauges are to be bonded on the side surface of the sting as shown in Fig. 8.8. To measure rolling moment, the sting to be machined such that the cross section is as shown in Fig. 8.9 and strain gauges bonded are connected as a ‘summing’ circuit.

101

Figure 8.8 Measurement of side force and yawing moment

Figure 8.9 Measurement of rolling moment

102

8.6 Interactions Effect : While it is desirable to design a strain gauge balance to make each bridge sensitive to only one load component, it is not truly possible to eliminate completely interactions due to other components. It may be therefore be necessary to take into account the presence of non-linear interactions as well as the linear interactions. As the most general case, a six-component strain gauge balance is considered. Such a balance would measure six load components, i.e., three pure forces components (L, D, Y) and three moments (M, R, N). Each bridge indicator reading, as a consequences of interactions, is function of all six components. If only linear interactions assumed it can be written in general case R  K L L  K D D  K Y Y  K M M  K R R  K N N

(8.24)

Where R is the indicator reading corresponding to ,  being any one of L, D, Y, M, R, N. There will be six equations of this type, one for each reading where K L ,KD, etc. are constant coefficients ( the so-called calibration coefficients). A total of 36 coefficient ( first order coefficients) are to be calculated. All these coefficients can be calculated by loading the balance with each component independently. Repeating this procedure 36 time will give the 36 balance coefficients. If single load component  is applied to the balance, where , like , is one of the component is present, from Eq. (8.24), R is given by R  K  

(8.25)

By plotting R against  for several values of , K can be obtained as the slope of the curve. If the plot is linear, the slope at once gives K = R . If plot is non-linear, the effect of non-linear interactions is also to be taken into account. If non-linear interaction takes place, equation can be written in a polynomial form as R  K L L  K D D  K Y Y  K M M  K R R  K N N  K  ( LD) LD  K  ( LY ) LY  ....  K  ( RN ) RN

(8.26)

103

To obtain the second order calibration coefficients, it will be necessary to load the balance in combination of various pairs of components. If two load components,  and  (say) are applied to the balance simultaneously, then the six readings are determined by relations, each of the form R  K    K     K  ( )

(8.27)

If one of the applied loads, , is maintained constant and the other, i.e.,  is varied then a plot may be made of R against . Measurement of the slope at  = 0 gives

R  K    K  ( )    0

(8.28)

Comparing this with the values of (R  ) = 0 when  = 0 will show whether or not K( ) is negligible. If K( ) is not negligible, then R should be determined for a few different values of . The slope of a graph of R against  gives K( ). In this way, all the calibration coefficients in the six equations (8.24) may be determined.

8.7 Factors Affecting the Accuracy of Measurement : To obtain high accuracy in strain gauge instrumentation, careful attention must be made on different aspects, in general, these include : i)

Surface preparation and bonding of strain gauges.

ii)

Noise suppression.

iii)

Thermal effect.

iv)

Optimizing excitation level.

8.7.1 Surface Preparation and Bonding of Strain Gauges : Strain gauges can be bonded satisfactorily to almost any solid material if the material surface is prepared properly. Concept of surface preparation is based on the understanding of cleanliness and contamination. Negligence to surface preparation may yield most unsatisfactorily gauge installation and hence erroneous result. The system of surface preparation includes five basic operations : 104

-

Solvent degreasing

-

Abrading

-

Application of gauge layout lines

-

Conditioning

-

Neutralising

Degreasing is performed to remove oils, greasing, organic contaminations and soluble chemical residues. Degreasing is done to avoid having subsequent abrading operation drive surface contaminants into the surface material. Degreasing can be accomplished using a hot vapour degreaser or aerosol spray cans of chlorothene SM or Freon. Spray applicators of cleaning solvent are always preferable because dissolved contaminants can not be carried back to the parent solvent. If possible, entire test piece should be degreased. Otherwise, for large objects, area sufficiently larger than the gauge area should be cleaned. Surface abrading is done to remove any loosely bonded adherents (rust, paint etc.) and to develop a surface texture suitable for bonding. The abrading operation can be performed in a variety of ways, depending upon the initial condition of the surface and the desired finish for gauge installation. For rough surface it may be necessary to start with a file. For moderately smooth surface, abrading can be done by silicon-carbide paper of appropriate grit. The normal method of accurately locating and orienting a strain gauge on the test surface is to first mark the surface with a pair of crossed reference line at the point where the strain measurement is to be made. The reference or layout lines should be made with a tool which burnishes rather than scribes. A scribed line usually raises a burr or creates a stress concentration. On aluminium and most other alloys a 4H drafting pencil is a satisfactory and convenient burnishing tool. After the gauge layout lines are marked, final surface preparation is accomplished by water based cleaners. To dislodge and remove oxides and mechanically bound contaminants, a mild phosphoric acid compound is used for conditioning the surface. This is immediately followed by the neutralizing any chemical reaction introduced by the acidic conditioner to produce optimum surface conditions for strain gauge bonding.

105

Once the surface material is prepared, strain gauges are to be bonded on the surface properly. Because a strain gauge can perform no better than the adhesive with which it is bonded to the test piece, the adhesive is a vitally important component in every strain gauge installation. Ideally, the adhesive would cause the strain gauge to act as an integral and inseparable part of the surface material – without adding influence of its own. One adhesive which is widely used for routine measurement in stress applications under environmental conditions the cyanoacrylate adhesive. This adhesive transforms from a liquid to solid when pressed into a thin film between the gauge and the mounting surface. This adhesive is very easy to handle and cures almost instantly to produce an essentially creep-free, fatigue-resistant bond. Other types of adhesive include mainly the epoxy adhesives. The epoxies form a large class of adhesives used for strain gauge bonding because of the wide range of characteristics available. Some of the epoxies are single-component and others are twocomponent. Epoxy-phenolic adhesive are used for higher operating temperature.

8.7.2 Noise Suppression : Strain measurements are often made in the presence of electric and/ or magnetic field which can superimpose electrical noise on the measurement signals. If not controlled, the noise can lead to inaccurate results and incorrect interpretation of the strain signals altogether. It is therefore necessary to apply noise-reduction measures top any strain gauge experimentation. Virtually every electrical device which generates, consumes or transmits power is potential source for causing noise in strain gauge circuits. In general, the higher the voltage or current level, and the closer the strain gauge circuit to the electrical device the greater will be the electrical noise. The common sources of electrical noise include : AC power lines, motor starters, transformers, relays, generators, rotating and reciprocating machinery, are welders, vibrators, fluorescent lamps, radio transmitters etc. Electrical noise from these sources can be categorized into two basic types: electrostatic and electromagnetic. The two types of noise are fundamentally different and thus require different noise-reduction measures. Unfortunately, most of the common noise sources

106

listed above produce combinations of the two noise types, which can complicate the noise-reduction problem. Electrostatic fields are generated by the presence of voltage – with or without current flow. Alternating electrical fields inject noise into strain gauge systems through the phenomenon of capacitive coupling. Fluorescent lighting is one of the more common sources of electrostatic noise. The simplest and most effective barrier against electrostatic noise is conductive shield. It functions by capturing the charges that would otherwise reach the signal wiring. Once collected, these charges must be drained off to a satisfactory ground (Fig. 8.10). If not provided with a low resistance drainage path, the charges can be coupled into signal conductors through the shield-to-cable capacitance.

Figure 8.10 Electrostatic shielding

Another source of electrostatic noise is leakage to ground through the strain gauges. This leakage, if excessive, can cause noise transfer from the test piece to the gauge circuit. Any strain gauge installation on a conductive specimen forms a classic capacitor which can couple noise from the test piece to the gauge. It is therefore essential to make certain that the test piece is properly grounded and the leakage between gauge circuit and test piece is well within bounds. Electromagnetic fields are ordinarily created either by the flow of electric current or by the presence of permanent magnet. In order for noise voltage to be developed in a conductor, magnetic lines of flux must be ‘cut’ by the conductor. Signal conductors in the vicinity of moving or rotating machinery are generally subjected to noise voltages from

107

this source since moving machine member (made of iron and steel which are ferromagnetic) redirect existing lines of flux. The most effective approach to minimizing electromagnetic noise is not to attempt magnetic shielding of the sensitive conductors but to ensure that noise voltages are induced equally in both side of the amplifier input. Achievement of noise cancellation by this approach is discussed in section 2.3.1.4 (Fig. 2.11). The noise, electrostatic or electromagnetic, can be effectively assessed by the signal conditioner by a simple but significant feature- a control for removing excitation from the Wheatstone bridge. With such a control, the instrument output can be easily checked for noise, independently of any strain signal. A simple but effective way of reducing noise is to reduce amplifier gain and compensate by increasing bridge excitation voltage.

8.7.3 Thermal Effect : Ideally, a strain gauge bonded to a test piece would respond only to the applied strain in the material and be unaffected by other variable in the environment. Unfortunately, the resistance strain gauge is somewhat less than perfect. The electrical resistance of the strain gauge varies not only with strain but with temperature as well. In addition, the relationship between strain and resistance change (i.e., the gauge factor C) itself varies with temperature. These deviations from ideal behavior can be important under certain circumstances and can cause significant error if not properly accounted for. Once an installed strain gauge is connected to a strain indicator and the instrument balanced, a subsequent change in the temperature of the gauge installation will generally produce a resistance change in the gauge. However, because this change in resistance due to the thermal effect will be registered by the strain indicator as strain, the indication is usually referred to as ‘temperature-induced apparent strain’ or ‘apparent strain’ in the test material. The net apparent strain is caused by two concurrent algebraically additive effects in the strain gauge installation. First, the electrical resistivity of the strain gauge is temperature dependent and any resistance change due to this effect appears as strain to a strain indicator. The second contribution to apparent strain is caused by the differential thermal

108

expansion between the strain gauge and the test material to which it is bonded. With temperature change, the test piece expands or contacts, and since the strain gauge is firmly bonded to the test material, the gauge grid is forced to undergo the same expansion or contraction. To the extent that the thermal expansion coefficient of the grid differs from that of the test material, the grid is mechanically strained in conforming to the free expansion or contraction of the test material. Since the grid is, be design, strain sensitive, the resultant resistance change appears to the strain indicator as strain in test material. The net ‘apparent strain’ can be expressed as the sum of resistivity and differential expansions effects:



G G

  T   G T

(8.29)

where BG

= Thermal coefficient of resistance of grid material.

G

= Gauge factor.

T - G = Difference in thermal expansion coefficients between test piece and grid respectively. T

= Temperature change.

It should not be assumed from the form of equation (8.27) that the apparent strain is linear with temperature because the coefficient within the bracket are themselves functions of temperature. The equation clearly demonstrates, however, that the apparent strain exhibited with temperature change depends not only upon the strain gauge but also on the material on which the gauge is bonded. The first part of apparent strain, i.e., the strain due to thermal expansion of grid can be eliminated by compensating gauges. For an quarter-arm bridge, an identical compensating or ‘dummy’ gauge connected on an adjacent arm of the Wheatstone bridge is mounted on an unstrained specimen for the identical material as the test piece and subjected always to the same temperature as the active gauge. Under these hypothetical conditions, the apparent strains in the active and dummy strain gauges should be identical. Since identical resistance change in adjacent arms of the Wheatstone bridge does not unbalance the circuit, the apparent strains in the active and dummy gauges should cancel exactly. This part of the apparent strain can be cancelled by

109

same philosophy by using half-bridge (for example, two active gauges on the two sides of a thin bending beam will have same temperature and cancel the apparent strain if connected in adjacent arms of the Wheatstone bridge) and full bridge. The second part of apparent strain which is due to difference in thermal expansion of strain gauge and test material can be eliminated by the concept of self-temperaturecompensation. The metallurgical properties of certain gauge alloys – in particular, constantan and modified Karma – are such that these alloys can be processed to minimize the apparent strain over a wide temperature range when bonded to test materials with thermal expansion coefficients for which they are intended. Strain gauges employing these specially processed alloys are referred to as ‘self-temperature-compensated’. Fig. 8.11 illustrates the apparent strain characteristics of the self-temperaturecompensated by this figure, the gauges are designed to minimize the apparent strain over the temperature range from about 00 F to 4000 F. When the self-temperature-compensated gauge is bonded to a test material having the thermal expansion coefficient for which the gauge is intended and when operated within the temperature range of effective compensation, strain measurements can usually be made without the necessity of correcting for apparent strain. Table 1 shows a number of common materials and gives the thermal expansion coefficients for each, along with the recommended S-T-C number. For apparent strain cancellation, strain gauges of appropriate S-T-C number should be bonded to the test material.

110

Figure 8.11 Variation of apparent strain with temperature

Table 8.1 : S-T-C number of different materials Test material

Thermal expansion coefficient

Recommended S-T-C

per degree Fahrenheit

number

Aluminium

12.9

13

Brass

11.1

13

Bronze phosphor

10.2

09

Copper

9.3

09

Molybdenum

2.2

03

Steel

6.0 (average)

06

Stainless steel

9.0 (average)

09

Tilonium

4.9

05

8.7.4 Optimising Excitation Level : The excitation voltage applied to a strain gauge bridge creates a power loss in each arm, all of which must be dissipated in the form of heat. Only a negligible fraction of the power input is available in the output circuit. This causes the sensing grid of every strain gauge to operate at a higher temperature than the test material to which it is bonded. It can be considered that heat generated within a strain gauge must be transferred by

111

conduction to the mounting surface. The heat flow through the specimen causes a temperature rise in the test material, which is a function of its heat-sink capacity and gauge power level. Consequently, both sensing grid and test material operate at temperatures higher than ambient. When the temperature rise is excessive, gauge performance will be affected in a number of ways. Firstly, a loss of self-temperature-compensation (S-T-C) occurs when the grid temperature is considerably above the specimen temperature. Secondly, hysteresis and creep effects are magnified since these are dependent on backing and glueline temperature. Thirdly, zero (no-load) stability is strongly affected by excessive excitation. One of the simple but effective way of determining the optimum excitation level is to gradually increase the bridge excitation under zero-load condition until a definite zero instability is observed. The excitation should then be reduced until the zero reading becomes stable again without a significant offset from the low-excitation zero reading. For most applications, this value of bridge voltage is the highest that can be safely used without significant performance degradation. Optimum strain gauge excitation level can also be determined on the basis of heat sink property of test material and gauge size and resistance. Heavy sections of high thermal conductivity metals such as copper or aluminium are excellent heat sinks. Thin section of low-thermal-conductivity metals such as stainless steel or titanium are poor heat sinks. Higher excitation level is permissible for test material having good heat-sink properties. Similarly, higher strain gauge resistances permit higher excitation level. Power dissipated in grid (watts) may be given by

PG 

V2 4R

(8.30)

while power density in grid (watts / m2) may be given by PG  PG / A

(8.31)

112

where R = Gauge resistance in ohms A = Grid area (Active gauges length  gauge area) V = Bridge excitation in volts

When grid area (A), gauge resistance ( R ) and grid power density ( PG ) are known :

V  2 R  PG  A

(8.32)

Table 2 provides the values of power density of various metals.

Table 8.2 : Heat sink conditions Accuracy

Excellent

Good

Fair

Poor

requirement

Aluminium or Copper

Thick Steel

Thin Steel

Plastic

High

2–5 3.1 – 7.8

1–2 1.6 – 3.1

0.5 – 1 0.78 – 1.6

0.1 - 0.2 0.16 - 0.31

Moderate

5 -10 7.8 - 16

2–5 3.1 – 7.8

1–2 1.6 – 3.1

0.2 – 0.5 0.31 – 0.78

Low

10 – 20 16 - 31

5 – 10 7.8 - 16

2–5 3.1 – 7.8

0.5 – 1 0.78 – 1.6

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Chapter 9

FORCE AND MOMENT MEASUREMENT BY ELECTRONIC EXTERNAL BALANCE 9.1 Introduction : Measurements of forces and moments on a model in wind tunnel are made either mechanically or electronically. The basic advantages of electronic measuring system, i.e., fast response, high and low values capability and amenability to automation are outlined in Chapter 2. In an electronic system pick up or transducer converts the physical quantity under measurement into electrical signal. Internal electronic balance or sting balance (where strain gauge is used as pick-up or transducer) for measurement of forces and moments are discussed in details in Chapter 8. in this chapter an external electronic balance (3-component) is described. The advantage of this system is that, unlike in sting balance, it is kept outside the tunnel and hence flow is not disturbed by it. In this balance, aft lift, fore lift and drag are measured by three load cells are obtained from three digital voltmeters. Pitching moment is obtained by simple manipulation.

9.2 General description : The general arrangement of the external balance is shown in Fig. 9.1. it is mounted on the side wall of the working section outside the tunnel and is designed for airflows from right to left when balance is viewed from front.

114

Figure 9.1 Three component external balance The balance is constructed mainly of aluminium alloy and its main frame work comprises a mounting plate which is secured to the tunnel test section and carries a triangular force

115

plate. The force plate and mounting plate are connected by three supporting legs, disposed at the corners of the force plate. Each leg is attached to the force plate and mounting plate by spherical universal joints. The effect of this is to constrain the force plate to move in a plane parallel to the mounting plate, while leaving it free to rotate about a horizontal axis. The necessary three degrees of freedom are thus provided. Models for use with the balance are provided with a 12 mm diameter mounting stem and this is inserted in the bore of model support and secured by a collet tightened by model clamp. The model support is graduated on the periphery and is free to rotate in the force plate for adjustment of the angle of attack of the model, while its position may be locked by an incidence clamp. The force plate may be locked in position by two centering clamps. It is to be noted that “ this plate should always be tightened when balance is not in use or when changing models”. The forces acting on the force plate are transmitted by way of flexible cables to strain gauge load cells which measure respectively the fore and aft lift forces and the drag force. The drag cable which lies horizontally, acts on a line through the center of model support while the two lift cables act vertically through points disposed equidistantly from the center of the model support and in the same horizontal plane as the support. The distance between the fore and aft lift tapes is 0.127 m (5.0 inch) and sum of the force in these tapes thus gives the lift on the model while the difference when multiplied by 0.127 gives the pitching moment in Newton meters. A drag balance spring acts on the force plate to apply preload to the drag load cell. The output from each load cell is taken to a strain gauge amplifier carried on the mounting plate and then via a flexible cable to a display unit comprising a set of three voltmeters shoeing the output from the respective load cell circuits. Lift and drag forces are then calculated directly from the load cell outputs by using the calibration factors. When calibrating the balance there is possibility of slight friction in the force plate supports. To overcome this, small vibration is provided. The motor which requires a 12 volt DC supply is carried on the mounting plate and controlled by the vibrator push button. It is not usually necessary to use the vibrator when using the balance for force

116

measurements on the model as their usually sufficient vibration present to overcome any friction in the mechanism.

9.3 Operation : To fit a model, centering clamps are tightened, model is set at zero incidence and incidence clamp is tightened. Model supporting stem is slided into model support and model clamp is tightened. Centering clamps are released to ensure that model moves freely without touching tunnel walls. It is to be noted that under “ no circumstances model clamp is tightened in the absence of model otherwise the collet will be damaged”. After switching on the supply it is desirable to allow a warm up time of 15 minutes for the load cells before taking any readings. Once centering clamps are released the display unit will indicate values corresponding to the zero readings of lift and drag. Vibrator is to be separated before recording the zero reading of fore and aft lift and drag. It will generally be found most convenient to set the incidence of the aerofoil models to give a lift force acting downwards, thus giving positive values of lift load cell read-out. To measure the aerodynamic forces tunnel speed is set to a desired value and hold display button is pressed on the display unit. Reading of digital voltmeter are then recorded. When it is desired to make a series of measurement of lift and drag over the range of model angles of incidence this angle may be set by releasing the incidence clamp, rotating the model support to the desired angle and retightening the clamp. “the centering clamps must be locked before releasing the incidence clamp or handling the force plate in any way. Otherwise there is risk of damaging the load cells”. At the end o the test sequence, zero readings of load cells are taken for recheck. Range of loading as per manufacturer’s specifications are as follows :

Lift Force

0 – 100 N

Drag Force

0 – 50 N

Pitching Moment

0 – 2.5 Nm

117

Initial calibration factors given by manufacturer are :

Fore Lift

7.570 N/volt

Aft Lift

7.418 N/volt

Drag

7.496 N/Volt

If  is model angle of incidence, a, f and d are aft load cell, fore load cell and drag load cell read out respectively, a0, f0 and d0 being the zero readings then Aft lift (A)

:

A = a1(a – a0)

Fore lift (F)

:

F = f1 (f – f0 )

Drag (D)

:

D = d1 (d – d0 )

Total Lift (L)

:

L=A+F

Moment (M)

:

M = 0.127 (F – A)

Where a1 , f1 and d1 are the aft lift, fore lift and drag calibration factors respectively. 9.3.1 Setting up load cells : At times it may be necessary to readjust the cables connecting the force plate to the load cells. It is essential that this is done correctly otherwise there is possibility of overloading the cells. To readjust, the centering clamps are first tightened. Forces are transmitted from lift and drag cables to the load cells by way of a conical nipple brazed to the cable and an adjusting screw secured by locknut which contacts the nipple and transmits the load to the cell. To adjust, locknut is loosened and adjusting screw is turned anti-clockwise until the cable is just tight. After that it is turned clockwise by one half of a revolution and locknut is tightened. There will then be a play of approximately 0.25 mm between cable and load cell when centering clamps are locked.

9.4

Calibration :

It is desirable to calibrate the balance periodically. Essentially the calibration procedure involves the application of known lift and drag forces using dead weights. Fig 9.2 shows

118

the set up for calibrating in the open circuit. The balance is usually mounted on a frame attached to the wind tunnel. However, a separate fixture also can be prepared for calibration where balance can be mounted.

Figure 9.2 Schematic arrangement off calibration rig

The balance is supplied with calibrating arm having a 12 mm diameter stem which could be secured in the model clamp. A pivoted link is fitted to the arm in one of the three positions, either on the axis of the arm or at the points displaced 63.5 mm on either side of the axis. The calibrating arm is locked in the model clamp with the arm cross member lying horizontally and the projection of the arm from the balance set so that arm loading point lies approximately on the axis of the wind tunnel. To calibrate lift load cells the pivoted link is fitted at the location point on axis of the calibrating arm and dead weights are applied to the loading link using a suitable hook.

119

Since the lift load cells are disposed symmetrically on each side of the balance axis, it may be assumed that dead loads so applied are divided equally between the two cells. To calibrate the drag load cell, a horizontal force is applied to the calibrating arm by way of loading link using dead weights, a nylon chord and a pulley (Fig. 9.2). If desired, the individual fore and aft lift load cells may be calibrated by applying dead weights using the loading link, at the locating points at each end of the transverse member of the calibrating arm. The calibrating procedure is as follows :

(a)

The power supply to the balance is switched on and left for twenty minutes for warm up.

(b)

The centering clamps are released and the zero readings of load cells are recorded.

(c)

The dead weights are applied and the load cell outputs are recorded.

(d)

The dead weights are removed and the zero readings are noted again.

(e)

The procedure is repeated ten times thus collecting ten set of readings.

(f)

Average values of load cell outputs are calculated.

Suitable loads for calibration are 100 N and 50 N for lift load cell calibration and 50 N and 25 N for drag cell. It is usually desirable to carry out calibration at two set of weights : one at rated load and the other at half rated load to confirm linearity of the relationship between load cell output and the load. It will be observed that in amplifier box there are three holes labeled ‘set zero’ and three labeled ‘set bridge volts’. In each case the hole gives access to an adjusting screw that may be reached by a small screwdriver. It should not normally be necessary to make any adjustment to these settings, if they are changed they will need recalibration. The set zero adjustment is made with the force plate clamped in which condition none of the load cells is subjected to any loading. The output from each cell as shown on the display unit should then be approximately zero, although this setting is not critical.

120

To check the bridge voltage use is made of the calibration cable provided with the balance. The cable has a male and female termination and may be inserted between each load cell in turn and the input plug to the amplifier. The calibration cable has two free leads which can be connected to a high grade digital voltmeter. After warming up the bridge supply voltage should be set to 10.000 volts 0.005 volts on all three circuits.

Table 9.1 : Calibration of drag cell

(a) Weight = 0.5 kg Serial No.

Reading with load

Reading with no load

(x)

(y)

1.

2.060

1.416

2.

2.081

1.430

3.

2.078

1.456

4.

2.078

1.443

5.

2.074

1.445

6.

2.081

1.453

7.

2.067

1.460

8.

2.081

1.459

9.

2.076

1.455

10.

2.066

1.460

Mean

2.0741

1.4511

d = 2.0741

d0 = 1.4511

Calibration factor = (0.5  9.81)  (2.0741 – 1.4511)

121

= 7.865 Nv

Table 9.2 : Calibration of lift load cells (a)

Weight = 1000gms

Serial No.

Fore lift reading With load

Aft lift reading

Without load

With load

Without load

1.

3.777

3.123

3.201

2.703

2.

3.802

3.132

3.279

2.701

3.

3.804

3.140

3.283

2.704

4.

3.805

3.142

3.287

2.710

5.

3.800

3.140

3.283

2.708

6.

3.816

3.135

3.207

2.713

7.

3.806

3.139

3.292

2.711

8.

3.814

3.146

3.290

2.706

9.

3.813

3.142

3.288

2.715

10.

3.814

3.144

3.289

2.712

Mean

3.8051

3.1383

3.2859

2.7083

Calibration factor: (b)

8.483 N/v

Weight = 2035 gms

Serial No.

7.349 N/v

Fore lift reading With load

Aft lift reading

Without load

With load

Without load

1.

4.473

3.124

3.937

2.692

2.

4.493

3.131

3.920

2.692

3.

4.522

3.126

3.924

2.697

4.

4.495

3.116

3.933

2.699

5.

4.494

3.104

3.925

2.706

6.

4.466

3.105

3.925

2.702

7.

4.475

3.112

3.914

2.706

8.

4.486

3.104

3.936

2.697

9.

4.473

3.102

3.920

2.701

10.

4.498

3.094

3.946

2.713

Mean

4.4875

3.1118

3.928

2.7005

Calibration Factor:

7.248 N/v

8.123 N/v

122

9.5 Wind Tunnel Testing : Wind tunnel testing is carried out on two-dimensional wings : one with NACA 0012 section and the other a supercritical aerofoil. Both the models are 30.48 cm (1ft) in chord and 61 cm (2 ft) in span. After the models are installed in tunnel zero reading are recorded before starting the tunnel. After the tunnel is started readings may be taken from the three digital voltmeters (if necessary, by pressing the push button to hold the display). Fore and aft lift and drag may now be determined by subtracting the respective zero readings and using the calibration charts. Total lift is obtained by summing the fore and aft lift and pitching moment at the holding point can be determined by multiplying 0.127 with the difference of the two lifts. Pitching at any other point (e.g. leading edge or ¼ c) can be derived by moment transfer theorem.

123

Chapter 10

WIND TUNNEL BOUNDARY CORRECTIONS (2D FLOW) 10.1 Introduction : The conditions under which a model is tested in a wind tunnel are not the same as those in free air. The closed (or open) boundaries of test section in most cases produce extraneous forces. This must be subtracted out in order for the results to be comparable with those in free air. The presence of test section boundary walls produces : i)

A variation in static pressure along the test section due to formation and subsequent thickening of boundary layer downstream. The effective area is reduced progressively downstream resulting in an increase of velocity and decrease of pressure downstream. The change in pressure upstream and downstream of the model produce a drag force known as ‘horizontal buoyancy’.

ii)

A lateral constraint to the flow pattern about a body, known as ‘solid blocking’. In a closed wind tunnel, solid blocking is the same as an increase of speed, increasing all forces and moments at a given angle of attack. It is usually negligible with an open test section, since the airstream is then free to expand in a normal manner.

iii)

A lateral constraint to the flow pattern about the wake known as ‘wake blocking’. The effect increases with an increase of wake size and in a closed test section increases the drag of the model. Wake blocking is usually negligible with an open test section since the airstream is then free to expand in normal manner.

iv)

An alternation to the local angle of attack along the span. In a closed test section the angle of attack near the wingtip of a model with large span is increased excessively, making the tip stall early. The effect of an open jet is just the opposite (tips unstalled). In both cases the effect is diminished to the point of negligibility by keeping model span less than 0.8 times the tunnel test section span.

124

v)

An alternation to the normal curvature of the flow about a wing. The wing moment coefficient, wing lift and angle of attack are increased in a closed wind tunnel and are decreased with an open jet.

vi)

An alternation to the normal downwash so that the measured lift and drag are in error. The closed jet makes the lift too large and the drag too small. An open jet has just the opposite effect.

It is to be noted that the additional effects resulting from the customary failings of wind tunnels – local variations in velocity, angularity of flow, tare and interference etc. – are extraneous to the basic wall corrections and it is assumed that the errors due to these effects have already been removed before wall effects are considered. Methods governing their removal are discussed in Chapter 6. Since the manner in which the two and three dimensional walls affect the model and are simulated is quite different they will be considered individually. Wall corrections for two-dimensional testing are given here. Wall corrections for three-dimensional testing are discussed in next chapter, Chapter 11. In order to study effects primarily concerned with two dimensional flow, it is customary to build models of constant chord which completely span the test section from wall to wall. The trailing vortices are then practically eliminated. Consequently, corrections due to downwash and spanwise variation of local angle of attack are not needed. These corrections, needed for three dimensional testing, are given in Chapter 11. Corrections for tests under two-dimensional flow conditions include : i)

horizontal buoyancy.

ii)

Solid Blocking

iii)

Wake blocking

iv)

Streamline curvature effect

10.2 Horizontal Buoyancy : Almost all wind tunnels with closed throats have a variation in static pressure along the axis of the test section resulting from the thickening of the boundary layer as it progresses toward the exit and to the resultant effective decrease of the jet area. It follows that the pressure is usually progressively reduced as the exit is approached and there is a tendency

125

of the model to be ‘drawn’ downstream. The static pressure variation along a jet is usually as shown in Fig, 10.1.

Static pressure (N/m2 )

distance along tunnel center line Figure 10.1 Variation of static pressure along tunnel center line The variation of cross-sectional area of the model (NACA 0012 model of Chapter 9) is shown in Fig. 10.2. it is seen that the variation of static pressure from, say station 2 to station 3, is (p2 - p3). This pressure difference acts on the average area (S2 + S3)/2. The resulting force for that segment of the model is therefore

 S 2  S3    2 

DB   p 2  p3 

Figure 10.2 Variation of cross-sectional area of aerofoil model This equation is simply solved by plotting local static pressure against body section area, the buoyancy then becoming the area under the curve. However, for the case where the

126

longitudinal static pressure gradient is a straight line (as shown in Fig. 10.1), the equation gives horizontal buoyancy as

DB  S X dp ds ds where

SX = model cross-section area at station x S = distance from model nose dp/ds = slope of longitudinal static pressure gradient

Now,  SXds = body volume, horizontal buoyancy can be obtained as DB = -(dp/ds) (body volume)

(10.1)

Aerofoil body volume can be obtained approximately by the expression Model volume = 0.7  model thickness  model span. In deriving eq.(10.1) only the pressure gradient effect is taken into account. But the existence of failing static pressure gradient not only implies that the test section is getting effectively smaller but also that the streamlines are getting squeezed. This squeezing effect should also be incorporated in the calculation of horizontal buoyancy. More accurate formulae incorporating this squeezing effect have been derived by Glauert and also by Allen and Vincenti. The expression for horizontal buoyancy by Glauert is

DB   where



1t 2 dp ds 

2

(10.2)

t = body thickness 1 = body-shape factor (about 4.2 for NACA 0012 aerofoil)

The expression for horizontal buoyancy derived by Allen and Vincenti is

DB   where

 8

2 c dp ds   6 2

h2



2 dp ds 

(10.3)

h = tunnel height c = model chord

 

2 

2 c

2

  48  h 

16



1

(10.4) 1 2

 z c1  p 1  dz dx  d x c  0

127

(10.5)

= 0.24 for NACA 0012 aerofoil (x, z are the aerofoil coordinates), c its chord and p its no chamber (basic) pressure distribution. The amount of ‘horizontal buoyancy’ (DB) is then subtracted from the observed values of drag in order for the result to be comparable with free air condition. This is usually small for wings, but large for fuselages and nacelles.

10.3 Solid Blocking : The presence of a model in the test section reduces the area through which air must flow, and hence by Bernoulli’s principle increases the velocity of air as it flows over the model. The increase of velocity, which may be considered constant over the model

for

customary model sizes, is called ‘solid blocking’ (Fig. 10.3). its effect is a function of model thickness, thickness distribution, and model size, and is independent of the camber. The velocity increment at the model due to solid blocking can not simply be obtained by direct area reduction. It is much less than the increment one obtains from the direct area reduction since it is the streamlines far away from the model that are most displaced. To understand the mathematical approach, solid blockage for a circular cylinder in a twodimensional tunnel is considered. The cylinder in an open free stream may be mathematically simulated by placing a doublet of strength  = 2a2U in a uniform stream where ‘a’ is the radius of the cylinder and U is the free stream speed. Next, the presence of tunnel roof and floor is to be mathematically simulated. It is well known that any boundary wall near a source, sink, doublet or vortex can be simulated easily by the addition of a second source, sink, doublet or vortex above the boundary wall. A solid boundary is formed by the addition of such an image system which produces a zero streamline matching the solid boundary. An open boundary, on the other hand, requires an image system that produces a zero velocity potential line which matches the open boundary. After the image system, as shown in Fig. 10.3, is established, its effect on the model is the same as that of boundary it represents. The image system is developed in the following way. A single doublet of strength  at A is considered, which is to be contained within the solid walls 1 and 2. To simulate wall 1, a doublet is needed at B and for wall 2,another doublet is needed at C. Now, doublet B

128

needs a doublet B’ to balance it from wall 2 and doublet C needs a doublet C’ to balance it from wall 1 and so on out to infinity. The image system for a closed rectangular test section thus becomes a doubly infinite system of doublets.

Figure 10.3 Mathematical simulation of solid body between tunnel roof and floor The axial velocity in the tunnel centerline due to first doublet at B is ,

U    2h 2  2a 2U  2h 2 or,

U  U

a2  2 h

Since velocity by a doublet varies inversely with the square of the distance from the doublet, the doubly infinite doublet series may be summed as

129

 Sb  

U  U



 2 1

total

1 a2 n2 h2

 2 a2

(10.6)

3 h2

It is seen that a 0.25 m radius cylinder in a tunnel 2.5 m high act as though the clear jet speed (U) were increased by 3.3 percent. Now, the blockage due to a given aerofoil of thickness t may be represented on that due to “ equivalent ” cylinder of diameter t(1)1/2 . with this approach the solid blocking for a two-dimensional aerofoil may be found from simple doublet summation. Glauert wrote this solid blocking velocity increment as

 Sb 

 2 1 t 2 3 4 h2

 0.8221

t2 h2

(10.7)

where 1 = 4.2 for NACA 0012 aerofoil. Allen and Vincenti obtained their expression by rewriting eq.(10.7). introducing  as in eq.(10.4) and using 2 = 4 1 t2/ c2, solid blocking correction is obtained as Sb = 2  (2 = .24 for NACA 0012 aerofoil)

(10.8)

A simple form of solid blocking correction is given by Thom as

 Sb 

K1 (mod el ..volume) C3 2

(10.9)

where K1 = 0.74 for a wing spanning the tunnel breadth C = tunnel test section area

10.4 Wake Blocking : Any real body without suction-type boundary layer control will have a wake behind it , and this wake will have a mean velocity lower than the free stream (Fig. 10.4). According to the law of continuity, the velocity outside the wake in a closed tunnel must be higher than free stream in order that a constant volume of fluid may pass through the test section. The higher velocity in the mainstream has, by Bernoulli’s principle, a lowered pressure and this lowered pressure, arising on the boundary layer (which later becomes the wake) grows on the model and puts the model in a pressure gradient resulting in a velocity increment at the model.

130

Figure 10.4 Velocity characteristics of wake To compute this wake effect, the wake and tunnel boundaries are to be mathematically simulated. The wake simulation is fairly simple. In the two-dimensional case a line source at the wing trailing edge emitting, say ‘blue’, fluid will result in a ‘blue’ region simulating the wake. In order to preserve continuity a sink of same strength should be added far downstream. The simulated wake may be ‘contained’ within the floor and ceiling by an infinite vertical row of source sink combination (Fig. 10.5).

131

Figure 10.5 Mathematical simulation of wake of a body contained between tunnel roof and floor. The axial velocity induced at the model is

U   where

 h

(10.10)

2

 = strength of source-sink. h = tunnel height.

The increment in horizontal velocity due to wake blocking can be written as

 wb 

U  U



ch CD 4

(10.11)

Maskell suggests that the correction be

 wb 

U  U



ch CD 2

(10.12)

132

10.5 Streamline Curvature Effect : The presence of ceiling and floor prevents the normal curvature of the free air that occurs about any lifting body and – relative to the straightened flow – the body appears to have more camber (around 1% for customary sizes) than it actually has. Accordingly, the aerofoil in a closed wind tunnel has too much lift (and moment about quarter-chord) at a given angle of attack and, indeed, the angle of attack is too large as well. This effect is not limited to cambered aerofoils only, since, using the vortex analogy, any lifting body produces curvature in the airstream. Streamline curvature effect may be estimated by assuming that the aerofoil in question is small and may be approximated by a single vortex at its quarter-chord point. The image system necessary to contain this vortex between floor and ceiling consists of vertical row of vortices above and below the real vortex (Fig. 10.6). the image system extends to infinity both above and below and has alternating signs.

Figure 10.6 Mathematical simulation of streamline curvature

133

The first image pair may be considered first. It is apparent they induce no horizontal velocity since the horizontal components cancel, but as will be seen, the vertical components add. From simple vortex theory, the vertical velocity at a distance x from the lifting line will be w

 x 2 2 h  x 2

(10.13)

Substitution of reasonable values for x and h in the above equation reveals that the boundary induced upwash angle varies almost linearly along the chord, and hence the stream curvature is essentially circular. The chordwise load for an aerofoil with circular camber may be considered to be a flat plate loading plus an elliptically shaped loading. Considering the flat plate loading first, the upwash induced at half chord by the two images closest to the aerofoil, by eq. (10.13), may be given by

w2 Since  

c4  2 2 h  c 42

1 cC LU  2

 

the angular correction needed for the nearest image becomes

w 1 c2  CL U  8 h 2  c 42

Assuming that (c/4)2 is smaller to h2 and again using

 

2 c

2

  48  h 

the angular correction is obtained as  6 3 

  

 C L 

The second pair of vortices induces a upwash velocity

w  2

c4  2 2 2h   c 42

and an angular correction ,

  

1 6 CL 43

134

Adding for all the infinite pairs of images, angular correction may be obtained as

 SC 

6  1 1 1  1     ........C L 3    4 9 16 

6 2  .C L  3 12 1   .C L ......radian 2 



57.3  .C L 2

degrees

(10.14)

The additive lift correction is

C Lsc  2 . . sc  2 .

1 . .C L 2

  .C L

(10.15)

and the additive moment correction is

C

1 M c 4



 4

C Lsc

(10.16)

10.6 Summary of Two-dimensional Boundary Correction : The data concerned for the NACA 0012 aerofoil model in Chapter 9 are the following : Free stream speed

= U m/s

Free stream dynamic pressure = q N/m2 Reynolds number

= Re

Angle of incidence

= 

Drag

= D

Lift

= L

Applying the wind tunnel boundary corrections the corrected values can be obtained as summarized below. Sb is given by eq.(10.6) or (10.8). To get wb from eq.(10.11) or (10.12), CD needs to be corrected first. Considering horizontal buoyancy into account, CD may be corrected by using

135

CD 

D  DB 1 U 2 S 2

where DB is given by eq.(10.1), (10.2) or (10.3). wb can then be obtained from eq. (10.11) or (10.12). Corrected value of free stream speed U may be obtained from U C  U  (1   Sb   wb )  U  (1   )

Corrected value of dynamic pressure q may be obtained from

qC  q 1     q 1  2  2

and the Reynolds number from

RC  R1    Lift coefficient CL is found from CL 

L 1 U 2 S 2

The corrected lift coefficient taking blockage effect can be obtained from C Lc 

or,

L 1 U 2 1   2 S 2

C Lc  C L 1 2 

Taking both blockage effect and streamline curvature effect into account C Lc  C L 1  2    [from eq.(10.15)]

Where  can be obtained from eq.(10.4). Corrected value for incidence  is

C   

57.3  .C L 2

[from eq. (10.14)]

Corrected drag coefficient CD (taking both solid and wake blocking into account) may be obtained as C DC  C D 1  3 Sb  2 wb 

136

Corrected moment coefficient CM(1/4)C may be given as

C

1 M C 4

C

1 M C 4

1  2    .C L 4

Both the uncorrected and corrected values can be put tabular form as shown in Table 10.1.

Table 10.1 : Uncorrected and corrected values of different parameters

Uncorrected U

R



CL

Corrected CD

CM(1/4)C

137

U

R



CL

CD

CM(1/4)C

Chapter 11

WIND TUNNEL BOUNDARY CORRECTIONS (3D FLOW) 11.1 Introduction : Wind tunnel boundary corrections for three dimensional testing follow the same reasoning used for two dimensional testing (Chapter 10). The correction factors are, however, different since both vertical and horizontal wall effects are now taken into account. Also, an additional correction is needed for the wall effects on downwash by the trailing vortices issuing from the trailing edges of the wing models. The corrections for three dimensional testing include : i) horizontal buoyancy ii) solid blocking iii) wake blocking iv) streamline curvature effect v) downwash effect

11.2 Horizontal Buoyancy : The philosophy behind the buoyancy correction has been described in Chapter 10. for the three dimensional case, the correction for pressure gradient effect only may be written, as before, as

 dp  DB    (body volume)  ds 

(11.1)

The total correction for both pressure gradient and streamline squeezing effect has been given by Glauert as DB  

 4

3t 3

dp ds

(11.2)

where 3 = body shape factor for three-dimensional bodies = 4.2 for NACA 0012 wing and

t

= maximum body thickness = .05856m for the case of 1.6 aspect ratio rectangular wing

138

11.3 Solid Blocking : The solid blocking correction for three dimensional flow follows the same philosophy described earlier for two dimensions. The body can be represented by a source-sink distribution is now contained by the walls of the tunnel. The simulation of the tunnel walls for the three-dimensional case requires image system for horizontal boundary walls (floor and ceiling ) as well as for side vertical walls as shown in Fig. 11.1. The image system as before extends to infinity on all sides.

Figure 11.1 Mathematical simulation of solid body between horizontal as well as lateral boundaries of the tunnel. Summing the effect of images, velocity increment due to solid blocking for a wing may be given by

 Sb 

U  U



K1 1 .( wing ..volume) C

(11.3)

32

where, K1 = body shape factor (1.008 for NACA 0012 wing) 1

= factor depending on the tunnel test section shape and model span to tunnel width ratio

C

= tunnel test section area (61 cm  61 cm for low speed tunnel)

Thom’s short-form equation for solid blocking for a three dimensional body is

 Sb 

U  U



K .(mod el ..volume) C3 2

(11.4)

where K = 0.9 for a three-dimensional wing.

139

11.4 Wake Blocking : The correction for wake blocking follows the logic of the two dimensional case in that the wake is simulated by a source of strength Q at the trailing edge which is matched for continuity by adding a downstream sink of same strength Q. The image system consists of a doubly infinite source-sink system spaced at a tunnel height (h) apart vertically and a tunnel width (B) apart horizontally as shown in Fig. 11.1. The axial velocity induced by the image system is

U   Q 2BH The incremental velocity is

 wb 

U  U



S CD 4C

(11.5)

where, S = model wing area C = tunnel test section CD = drag coefficient of the wing The increase of drag due to pressure gradient may be subtracted by removing the wing wake pressure drag CD 

K1 1 .( wing ..volume) CD C3 2

(11.6)

where K1, C and 1 are as defined for eq. (11.3).

11.5 Streamline Curvature Effect : The correction for streamline curvature for three dimensional testing follow the same philosophy as those for the two dimensional case in that they are concerned with the variation of the boundary induced upwash along the chord. But for the three dimensional system is shown in Fig. 11.2. Basically it consists of the real wing with its bound vortex CD and trailing vortices C and D. The vertical boundaries are simulated by the infinite system of horse-shoe vortices and the horizontal boundaries are simulated by the infinite lateral system . Linking the two systems is the infinite diagonal system.

140

Figure 11.2 Mathematical simulation of streamline curvature effect

The effect of the image system can be summed up and without going into the details of the formulation the correction factors can be written as

 SC   2 1 S C .C L .57.3

(11.7)

where, 2 = factor representing the increase of boundary induced upwash at a point p behind the wing quarter-chord in terms of the amount at the quarter-chord. = 0.195 for the present model and tunnel. 

= a factor which is function of the span load distribution, ratio of model span to tunnel width, the shape of the test section and whether or not the model is on the tunnel centerline. = 0.137 for the present case.

S

= model surface area.

C = tunnel test section area. The additive lift correction is

141

C LS   SC a where

(11.8)

a = wing lift-curve slope = 0.088 per degree for a 3D wing

The additive correction for the moment coefficient is

C MSC  0.25.C LSC

(11.9)

11.6 Downwash Effect : The downwash induced by the trailing vortex system needs to be corrected for the tunnel wall effects. Through elementary vortex theory the correction factor for the tunnel boundary induced downwash can be developed. The only mathematical tools needed are the expression for the induced velocity w due to a vortex of strength  at a distance r w

 4r

(11.10)

and the relation between lift and circulation for a uniformly loaded wing of span b

  SU  2bC L

(11.11)

combining the two gives

 SU   w C L  8rb 

(11.12)

Now r represents the vortex spacing in the image system which may be expressed as some constant times a tunnel dimension, e.g. the tunnel height h and the model wing span may be expressed in terms of the tunnel width B. The induced angle at the centerline of the test section is then

  1  

w s  CL U  8K b B hb 

for any one image. Summing the whole field and setting B/8Kb =  and noting that hB is the test section area C,  for the complete system is obtained as

 i   1 S / C .C L 57.3)

(11.13)

Now, the induced drag coefficient may be written as

C Di   i C L

where i = induced angle

142

Therefore, the change in induced drag caused by the boundary induced downwash becomes

C Di   i C L   1 S C C L2

(11.14)

11.7 Summary of Three-Dimensional Boundary Corrections : Data (, CL, CD, U) obtained from testing of a wing model in a closed three dimensional tunnel may be corrected to free air conditions according to the following relations : The corrected value of wind speed is

where

U C  U  1 2 

(11.15)

   Sb   wb

(11.16)

The dynamic pressure is qC 

1 U 2 1   2 2



1 U 2 1  2  2

 q 1 2 

(11.17)

The Reynolds number is

RC  R1   

(11.18)

The lift coefficient is (from eq.{(11.7) & (11.8)) C LC  C L 1  2    2 1 (S C ).(57.3).a

(11.19)

The angle of attack is (from eq. (11.7) & (11.3))

 C     1 S C C L 57.31   2 

(11.20)

The drag coefficient is (from eq.(11.6) & (11.14)) C D.C  C D 1  2    1 S C .C L2

(11.21)

143

Chapter 12

DRAG MEASUREMENT ON CYLINDRICAL BODY 12.1 Introduction : The resistance experienced by a body as it moves through a fluid is what is commonly known as drag. Total drag of a body may be separated into a number of items each contributing to the total. As a first step it may be divided into “pressure drag” and “friction drag”. The pressure drag may itself be considered as the sum of three items : 1) boundary layer pressure drag 2) trailing vortex drag or induced drag 3) wave drag. Some of these items depend on viscosity, others may exist in inviscid fluid. Schematically, Total drag

Friction drag (depends on viscosity)

Boundary layer normal pressure drag (depends on viscosity)

Pressure drag

Trailing vortex drag (does not depend on viscosity)

Wave drag (does not depend on viscosity)

Trailing vortex drag can exist only in the case of flow about a three dimensional lifting body and depends on the lift generated. The wave drag is associated with the formation of shock waves in high speed flight. For the particular case of low speed two-dimensional flow about a circular cylinder, both the items can be eliminated. The drag components acting on the body are the friction drag and boundary layer normal pressure drag. The summation of these two components is the profile drag. Flow around a circular cylinder (Fig. 12.1) is considered. To analyze the drag force it is convenient to assume the cylinder t be moving in a stationary fluid. To an observer moving with the cylinder the fluid will appear to be approaching as a uniform stream. At 144

any point A on the surface of the cylinder, the effect of fluid may conveniently be resolved into two components, pressure (p) normal to the surface and shear stress( )along the surface. The combined effect of pressure and shear stress (skin friction) in the direction of oncoming fluid gives rise to the drag force (profile drag).

Figure 12.1 Uniform flow past a circular cylinder It is worth noting that in the case of ‘ideal’ fluid, shear stress is zero and the pressure distribution (given by cp = 1 – 4sin2 ) is symmetrical over the forward and backward face of the cylinder which cancel out exactly giving zero drag force. For a real fluid shear stress exists and the pressure distribution is no longer symmetrical resulting in an overall rearward force. This force is so called boundary layer normal pressure drag. There are three methods of measuring the drag force : 1) by measuring pressure distribution on the surface on the cylinder, 2) by measuring pressure distribution in the wake of the cylinder, 3) by direct weighing.

12.2 Drag by Measuring Pressure Distribution on the Cylinder Surface : Consider an element s on the surface of the cylinder at a point where the normal is inclined at an angle  to the direction of U as shown in Fig. 11.2. The element of drag d per unit cylinder length due to p and  is

D   p cos    sin  .ds

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and integrating round the whole parameter yields D    p cos    sin  .ds

Figure 12.2 Elemental drag due to p and 

Drag may be expressed in non-dimension form as CD 

D 1 U 2 S 2

where S is the area. For a bluff body like circular cylinder S represents the frontal projected area normal to U. For a cylinder of diameter d and length unity S becomes (d x l) d. the characteristics dimension is the diameter d i.e. the width measured across the cylinder normal to the flow. This is in contrast to the concept of wetted surface area used for streamlined body like aerofoil section where the characteristics dimension used is the length of the body along the direction of flow or the ‘chord’ of the aerofoil.

   1  p  CD    cos   sin  .ds 1 1 1  U 2 d d  U 2 U 2  2 2 2  D

or, C D 

1 C p cos  C f sin  .ds d

(12.1)

where, Cf = skin friction coefficient CD = drag coefficient 146

Cp = pressure coefficient This equation shows that the drag of cylinder may be found by measuring p and  over the surface. Now it is easy to measure the distribution of p over a cylinder merely by drilling fine holes into its surface, but measurement of  is a much more difficult task. For the case of a circular cylinder, however, the contribution on drag from shear stress (the skin friction drag) is found to be very much smaller than from pressure (boundary layer pressure drag) and may safely be neglected. Making this assumption and writing

 .s  R.  d / 2.d simplifies equation (12.1) to 2

CD 

1 C p cos  d 2.d d 0 2



1 C p cos  .d 2 0

(12.2)

Using equation (12.2) CD can be calculated from the measured distribution over the cylinder surface. The circular cylinder model is provided with a fine pressure tapping at one point on its surface. A protractor is attached to the cylinder and the pressure taping is connected to the manometer. By rotating the cylinder about its axis to successive angular positions (0 0 , 50, 100, …, 3600 ) the complete pressure distribution round the whole surface may be recorded. Pressure taping at three points are connected to the manometer for measuring inlet total pressure Po(i.e. the pressure in the settling chamber), inlet static pressure p and static pressure on the cylinder P (Fig. 12.3). The dynamic pressure of the oncoming flow q is q 

1 U 2  PO  p 2

(12.3)

Manometer reads directly in terms of millibar, written mbar (1 mbar =10-3 bar = 100 N/m2 ). A table may now be prepared.

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Figure 12.3: Pressure distribution on the surface of the cylinder

Table 12.1 : Pressure distribution on the surface of the cylinder  degrees 0

h0

h

h

(P0 - p) = h0 - h

p - p = h - h

Cp

Cp Cos

5 10 360

Cp can be obtained simply as the ratio of (h - h ) and (h0 - h). Two graphs of Cp and Cp cos as function of  can now be plotted [Fig. 12.4(a), 12.4(b)]. CD May be obtained from Fig. 12.4(b) by use of planimeter to measure the area beneath the curve. Usually 2

c

p

cos   2.02

0

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2

From equation (12.2),

CD 

1 c p . cos  .d  1.01 2 0

Figure 12.4 Variation of Cp and Cp cos with  To obtain Reynolds number of the flow, value of U is needed. This can be obtained from equation (12.3) written in the form

U   2 P0  p   Reynolds number is obtained from

R  U  d 

12.3 Drag by Measuring Distribution in the Wake of the Cylinder : The second method of determining the drag is based on the application of momentum equation to the air flow. The flow of a fluid along a duct of width 2h past a cylindrical body (Fig. 12.5) is considered. The velocity is U and the pressure is p at upstream. Downstream of the cylinder the velocity the velocity is no longer uniform; let the velocity be u at distance y from the duct center line. The pressure across the downstream section is assumed to be uniform and has the value pe. It is convenient to refer to the space bounded by the upstream section, downstream section and duct walls as the control volume and the surface formed by these boundaries as the control surface.

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Figure 12.5 Application of the Momentum equation

The forces in the direction acting on the fluid in the control, volume are, per unit length of cylinder :-

at the upstream section

2h p

at the downstream section

-2h pe

at the cylinder

-D

It is to be noted that the force exerted by the cylinder on the fluid (which has a minus sign) is equal and opposite to the force exerted by the fluid on the cylinder. Forces due to shear stress on the walls of the duct and due to the fluid weight are neglected. h

The momentum flux per unit width over the downstream section =

 .u

2

dy

h h

The momentum flux per unit width over the upstream section =

 .U

2 

dy

h

Equating the net force in the x-direction to the momentum flux out of the control volume

2hp  2hpe  D 

h

h

h

h

2 2  .u dy   U  dy

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Rearranging and making non-dimensional gives the result

CD 

D 1 U 2 d 2



h 2h p   p e 2  u2    1  2 dy d 1 U 2 d h U   2

The integral may also be made non-dimensional by the substitution y = h h

So that

1   u2  u2     1  . dy  h 1    U 2    U 2 .d  h 1

And then final result is 1 2h p   p e 2h  u2   .d CD   1 d 1 d 1 U 2  2 U  2

(12.4)

Equation (12.4) provides a means to calculate CD from the pressure drop along the duct and the velocity distribution in the wake. It is to be noted that the derivation does not restrict the result to pressure drag only; the contributions of both pressure and skin friction forces are contained in the force which comes into the momentum equation. The ski friction drag on the wall also contributes to the momentum change and is therefore included in D. it is also worth mentioning that equation (12.4) applies only to the case of flow along a duct where the flow is confined between parallel walls. diameter of the cylinder

d = 48 mm = .048 m

half width of working section h = 50 mm = .05 m

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Table 12.2 : Velocity traverse in wake. y



(mm)

(= y/h)

0

0.0

2

0.04

4

0.08

50

1.0

0

0.0

-2

-0.04

-4

-0.08

-50

-1.0

P0 - p 

1 U 2 2 N/m2

U m/s

Pe - pe 

1  .u 2 2 N/m2

u

u / U

1  u 2 U 2

m/s

Readings are recorded at successive values of the distance y from the center line, made dimensionless by dividing by h in the next column. The third column indicates the Pitot pressure Pe ( pe =0, atmospheric datum) and hence represents the local dynamic pressure 1  .u 2 at a point in the exit section. This is also made dimensionless by dividing by 2 1 U 2 . Next two columns show u / U and 1  u 2 U 2 which can be plotted as shown 2

in Fig. 12.6. In stead of determining u and U individually to calculate

1  u 2 U 2 ,an alternative

approach is to obtain u / U directly from the formula

u U 

Pe  pe  P0  p 

The drag coefficient may now be obtained by use of the curve 1  u 2 U 2 in Fig. 12.6. The area beneath the curve is usually found to be –0.074. CD may now be calculated from equation (12.4).

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Figure 12.6 Velocity traverse in the wake of the cylinder

12.4 Drag by Direct Weighing Fig. 12.7 shows the essential components of the working section in which drag may be measured by direct weighing. The body is mounted on an arm which extends through a hole in one wall of the working section and which is supported on a flexible link so as to form a balance. Now the drag experienced by the body in the air flow may be directly measured by balancing the setup with weights in the scale pan. It is recommended that exact balance is found by suitably trimming the wind speed rather than making small adjustments to weights in the scale pan. At each wind speed the total pressure P 0 and static pressure p at inlet are recorded.

Diameter of the cylinder d = 12.5 mm = .0125 m Frontal projected area of the cylinder S = .0125  .048 sq. m

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Figure 12.7 Drag measurement by a mechanical balance

Table 12.3 : Drag by direct weighing No. of runs

P0

p

P0 - p 1  . ?U 2 2

D gm

D

N

CD D

1  .U 2 S 2

N/ m2 1.

Three methods of drag measurement should yield almost identical values of CD. However, it is to be noted that both wake traverse and direct weighing include pressure as

154

well as skin friction component whereas surface pressure measurement method takes only pressure drag into account. This particular procedure can be repeated for 1) flat plate 2) aerofoil section 3) square cylinder section.

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Chapter 13

FLOW ABOUT AN AEROFOIL SECTION 13.1 Introduction : The problem is to obtain pressure distribution on the surface of an aerofoil in twodimensional steady incompressible flow and derive the overall aerodynamic characteristics of the aerofoil by integrating the pressure distribution. Cartesian coordinates Oxz are taken with the origin coinciding the leading edge of the aerofoil. The free stream velocity U is inclined at the angle of incidence  to the Ox axis as shown in Fig. 13.1.

Figure 13.1 Cartesian coordinate system

The equation of the aerofoil profile relative to the axis system is denoted by z = fu(x) on upper surface = f 1(x) on lower surface In the case of symmetrical profile

(13.1) fu(x) = - f 1(x)

(13.2)

The perturbation velocity induced due to the presence of the body in the flow may be assumed to be u and w in x and z directions respectively. The total velocity components at any point on the surface of the aerofoil are

156

U  U  cos   u W  U  sin   w and the total velocity and pressure are qt  U 2  W 2

(13.3)

C p  1  qt2

13.2 Formulation of the Problem : In theoretical analysis, the unknowns that are to be computed are the two components of perturbation velocity (u, w) and pressure (p). These unknowns can, in principle, be calculated from principles of conservation of mass and momentum. For steady inviscid, incompressible flow, these equations are : Eq. of continuity

:

U W  0 x z

Euler’s eq. of momentum : U

U U 1 p W  . x z  x

: U

U W 1 p W  x z  z

The situation

(13.4)

(13.5)

is simplified if the flow is considered to be potential (irrotational).

Condition of irrotationality is

U W  0 z x

(13.6)

The velocity field, under the assumption of irrotational flow, can now be expressed as the gradient of a scalar potential  such that U

  ,.....W  x z

(13.7)

Using (13.4) and (13.70, Laplace’s equation is obtained.

2  0

(13.8)

The simplicity of potential flow derives from the fact that the velocity field is determined from Laplace’s equation, eq. (13.8), which contains equation of continuity, eq. (13.4), and condition of irrotationality, eq. (13.6). The equation of momentum, eq. (13.5), is not used and the velocity is determined independent of pressure. Once the velocity field is

157

obtained, pressure can be known by integrating equation (13.5). Equation (13.5) can be integrated to give one of the forms of Bernoulli’s equation. For steady incompressible flow, Bernoulli’s equation becomes (in the simplest form) p

1  .qt2  Constant 2

Using this expression, pressure distribution Cp can be obtained as

q p  p Cp   1 t 1 U U 2 2

2

(13.9)

where p and U are the pressure and velocity at infinity. Since onset flow U always satisfies the Laplace’s equation, eq. (13.8) can be further simplified by assuming

     where

 = total potential  = perturbation potential  = potential due to onset flow U

Since

 2   0 , equation (13.8) can be written as  2  0

(13.10)

Laplace’s equation in perturbation potential is a second order linear differential equation and requires two boundary conditions for solution, one on the body surface and other at infinity. The boundary conditions are : i)

flow at the body surface must be tangential, i.e., the normal component must be zero, qn = 0 on body surface

ii)

(13.11)

the perturbation velocities must tend to zero at infinity i.e,

u, w  0 at infinity (13.12) The problem of calculating inviscid, incompressible, irrotational flow about a body finally reduces to solving equation (13.10) subject to the boundary conditions, equations (13.11) and (13.12).

158

13.3 Solutions : Several methods have been developed for solution of the problem formulated above. These methods may broadly be classified as Solution of Laplace’s equation

Approximate solution

Exact solution

Analytic

Numerical

13.3.1 Exact Analytic Solution : Exact analytic solution of Laplace’s equation can be obtained only for an extremely limited class of simple body surface (e.g. flow past a half-body, Rankine oval, circular body etc.). However, in two-dimensional flow problems, advantage can be taken of the fact that in two dimensions the problem of solving Laplace’s equation can be replaced by the problem of finding a suitable conformal transformation of the boundary. The use of this technique has resulted in a number of useful potential flow solutions. For example, using Joukowski’s transformation flow past a circular section can be mapped onto flow past a flat plate, elliptical section or Joukowski aerofoil. Nevertheless, these solutions comprise a restricted class.

13.3.2 Approximate Solution : The solution of Laplace’s equation presents difficulty because of the nonlinear boundary condition, equation (13.11). The governing equation, the Laplace’s equation is linear and requires no simplification. The non-linearity enters in the problem through the boundary condition. Because exact analytic solutions are limited and because exact numerical methods are beyond the capability of hand calculation, approximate methods were developed in the past. The earliest theory developed for solving this problem, the socalled linearised theory, is based on a through –going linearisation of the boundary condition. The boundary condition is linearised under the assumption of small

159

perturbations. These simplifying assumptions obviously place a limit on the accuracy of the solution. Usual assumption are : a) the body is slender with small local surface slope b) the perturbation velocity components due to body are small with respect to onset flow.

13.3.3 Exact Numerical Solution : With the advent of high speed digital computer, exact numerical methods have become feasible. These methods do not use any simplifying assumptions in the formulation and are applicable to a variety of body surfaces. However, since the solution is achieved numerically, numerical inaccuracies enter into the solution. A distinction must be made between approximate solution and numerically exact solution. In the later, the analytic formulation, including all equations, is exact and numerical approximations are introduced for the purpose of numerical calculation. Exact numerical methods have the property that the errors in the calculated solutions can be made as small as desired, by sufficiently refining the numerical calculations. In contrast, approximate solutions introduce analytic approximations into the formulation itself and thus place a limit on the accuracy that can be obtained in a given case regardless of the numerical procedure used.

13.4 Linearised Theory : Boundary condition of flow tangency, eq. (13.11) can be written as

dz U  sin   w  dx U  cos   u

(13.13)

In linearised theory (or thin aerofoil theory) the perturbation velocities u, w and angle of incidence  are assumed to be small, so that cos = 1 ,

sin = 

u,w