Air Flow Bench

A First Course in Airflow by Emeritus Professor E. Markland (with minor additions by TecQuipment)

© TecQuipment Ltd 2009 Do not reproduce or transmit this document in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system without the express permission of TecQuipment Limited. TecQuipment has taken care to make the contents of this manual accurate and up to date. However, if you find any errors, please let us know so we can rectify the problem. TecQuipment supply a Packing Contents List (PCL) with the equipment. Carefully check the contents of the package(s) against the list. If any items are missing or damaged, contact TecQuipment or the local agent.

EM/PE/djb/0409

CONTENTS

AUTHOR’S PREFACE

i

NOTE ON UNITS

iii

CHAPTER 1.

A Brief Introduction to Airflow

1

CHAPTER 2.

The AF10 Airflow Bench

11

CHAPTER 3.

The AF10A Multitube Manometer

15

CHAPTER 4.

Bernoulli’s Equation applied to a Convergent-Divergent Passage (AF11)

19

CHAPTER 5.

Drag Measurement on Cylindrical Bodies (AF12)

31

CHAPTER 6.

The Round Turbulent Jet (AF13)

53

CHAPTER 7.

Boundary Layers (AF14)

69

CHAPTER 8.

Flow Around a Bend in a Duct (AF15)

89

CHAPTER 9.

Jet Attachment (AF16)

99

CHAPTER 10.

Flow Visualisation (AF17)

109

CHAPTER 11.

Aerofoil with Tappings (AF18)

117

APPENDIX A.

Determination of the Area Beneath a Curve

A-1

APPENDIX B.

Installation and Operating Instructions for the Flow Visualisation Apparatus

A-7

AUTHOR’S PREFACE It is now over forty years since the set of experiments which I devised for a first year hydraulics laboratory were put together with a laboratory manual to form an instructional package. The continuing demand for this equipment is gratifying evidence of the need for simple experiments to illustrate principles of fluid motion and to allow students to become familiar with elementary laboratory techniques in hydraulics. In many educational establishments the subject is taught as fluid mechanics, encompassing flow of gases as well as liquids, and a complementary set of experiments in air flow is now presented to meet laboratory needs which are generated by this more general approach. Again the equipment has been made simple, with the intention of facilitating experiments which emphasise the basic principles. In many cases there is nevertheless scope for extending the tests reported here into project work, if so desired. I have also found the experiments useful for classroom demonstration during lectures in a first course in fluid mechanics. Again I have adopted the style of the laboratory report, usually introduced by some remarks on the usefulness and significance of the subject, to describe the experiments. Some teachers will find the guidance far too detailed, and others will find alternative ways of using the equipment which will suit their purposes better. I hope that, whatever the mode of use, the equipment will meet the expectations of those who work with it. Emeritus Professor E. Markland D.Sc. C.Eng. F.I.C.E. F.I.Mech.E. Member ASME

i

ii

NOTE ON UNITS Throughout this book we use the International System of Units (SI) in which the units of mass, length and time are: mass length time

kilogram metre second

(kg) (m) (s)

In this system the unit of force, defined as that force which when applied to a unit mass of 1 kg produces an acceleration of 1 m/s2, is the newton

(N)

The newton may be expressed in terms of unit of mass, length and time by the equation 1N

1 kg m/s2

=

(1) which follows from the definition given above. The density ρ of a fluid at a point is defined as the ratio of mass to volume of an element surrounding the point, so the SI units of density are ρ

density

kg/m3

Pressure p at a point in a fluid is defined in terms of the normal force acting on an element of a plane surface through the point; the pressure is the ratio of force to area. The units are therefore pressure

p

N/m2

and, expressing N in terms of kg, m and s from Equation (1) we find 1 N/m2

=

iii

1 kg/m s2

Pressure may thus be written in terms if unit mass, length and time, as pressure

kg/m s2

p

The term ½ρV2, the so-called dynamic pressure, occurs frequently. If ρ is expressed in units of kg/m3 and V in units of m/s, then ½ρV2 appears in units of kg/ms2, i.e. in the same units as those of p. The SI system provides for the use of particular names for certain derived units, and the pascal, represented by the symbol Pa, is used to represent a pressure of 1 N/m2, i.e. 1 Pa

1 N/m2

=

For many purposes this unit is rather small, and another unit, the bar, is used. This is defined by: 1 bar

=

105 Pa

=

105 N/m2

It so happens that 1 bar is roughly equal to the pressure of the atmosphere at sea level, so for meteorological practice, the millibar, written mbar (or simply mb), is a further convenient unit. So we have: 1 mbar

=

10−3 bar

=

102 Pa

=

102 N/m2

It is recommended, that students become familiar with the derived units of pressure mentioned. However, in this book the N/m2 is adopted as the preferred unit, requiring no effort of memory to relate it to the fundamental units of mass, length and time. Some readers may be more familiar with the Imperial system of units in which the fundamental units of mass, length and time are: pound foot second

lb ft s

iv

v

These are related to SI units by the conversion factors 1 lb

=

1 ft

=

0.453 592 37 kg exactly or 0.4536 kg for most purposes 0.3048 m exactly

The unit of force, defined as that force which when applied to a unit of mass 1 lb produces an acceleration of 1 ft/s2 is the poundal, pdl. The conversion factor for poundal to newton may be calculated from the foregoing conversions for mass and length units as follows: 1 pdl

=

1 lb ft/s2

= = =

0.4536 kg × 0.3048 m/s2 0.1383 kg m/s2 0.1383 N

Frequently the alternative unit of force, the pound-force, abbreviated to lbf, is used; it is defined as that force which when applied to a unit mass of 1 lb produces the standard acceleration of gravity, viz. 32.174 ft/s2. The pound force is therefore larger than the poundal, the conversion factor being 1 lbf

= 32.174 0 pdl

and the conversion from lbf to N is 1 lbf

= = =

32.1740 pdl 32.1740 × 0.1383 N 4.448 N

Pressures in the Imperial system are frequently measured in units of pdl/ft2, lbf/ft2 or lbf/in2, and the reader should verify the following conversions 1 pdl/ft2 1 lbf/ft2 1 lbf/in2

= 1.488 N/m2 = 47.88 N/m2 = 6895 N/m2

vi

1.

A BRIEF INTRODUCTION TO AIRFLOW

The concept of continuity and Bernoulli’s equation are fundamental to an elementary understanding of airflow, and these matters are dealt with in every textbook on fluid mechanics, hydraulics or aerodynamics. For the sake of completeness and to introduce some of the terms used in subsequent chapters, a brief outline is given here. The fluid is considered as a continuous medium, so that if we fix attention at one point of the space in which a fluid is in motion, we can observe fluid streaming continuously through that point, and the fluid velocity at the point may conveniently be represented in magnitude and direction by a vector. This is conveniently called the velocity vector, and the whole motion is described by the set of velocity vectors at all points of that space by the velocity field. If we now draw a line in the fluid so that at every point of the line the tangent is in the direction of the velocity vector, we have a streamline of the fluid, as illustrated in Figure 1.1.

Figure 1.1 Definition of a Fluid Streamline Bernoulli’s equation deals with the case where the velocity field is steady, by which we mean that the velocity at each point of the motion does not vary with time. Consider now a bundle of streamlines as shown in Figure 1.2. Since the direction of the fluid velocity at each point in the surface defined by the streamlines lies along the surface, no fluid crosses it and the fluid contained inside the bundle may be thought of as flowing in a tube, a so-called streamtube. 1

Figure 1.2 A Streamtube

Consider a section across a streamtube of infinitesimal area δA. If the velocity along the tube at this section is u, then in unit time the volume of fluid which passes across & the section is uδA. The mass of this fluid is ρuδA. This is the rate of mass flow δm across the section, and since no fluid crosses the walls of the streamtube, & δm

ρuδA

=

=

constant along the infinitesimal streamtube (1-1)

This is one form of the equation of continuity of flow. Various other forms may be derived as shown below. If the streamtube has a finite cross-sectional area, then the rate of mass flow is: & m

∫ ρudA

=

=

constant along the streamtube

A

(1-2) where the integration is taken over the area A of the cross-section. For the particular case in which ρ and u are both constant over the sectional area, & m

=

ρuA

=

constant along the streamtube (1-3)

2

It is sometimes convenient to deal in terms of volume flow rate. For the infinitesimal tube the volume δQ crossing the elementary area δA per unit time is: δQ = uδA (1-4) and this is constant along the streamtube only if ρ is constant along it. streamtube of finite cross-sectional area A the volume flow rate is:

For a

∫ udA

Q =

A

(1-5) and for the particular case of constant velocity over the section Q = uA (1-6) Again, Q is constant along the streamtube only if ρ is constant along it.

Figure 1.3 Forces Acting on an Element of a Streamtube

3

Consider now steady motion of a fluid along an elementary streamtube. Figure 1.3 shows an element of length δs and the forces acting on it. The pressure rises from p to p +

increases from A to A +

dp δs along the element, and the cross-sectional area ds

dA δs . The forces due to pressure on the element are: ds

On the section at s:

pA in the s-direction

On the section at s + δs:

dp   δs − p +  ds 

On the wall of the streamtube:

the pressure varies from p to p +

dA   δs in the s-direction A +  ds  dp δs , ds

and the projected area of the wall on a plane normal to the s-direction is dA dA   δs . So the component of force in the s-direction lies δs − A , viz. A +  ds  ds between: p

dA dp  dA  δs δs δ s and  p +  ds  ds ds

To first order of infinitesimal quantities this is: p

dA δs ds

The net force in the s-direction due to pressure is thus: dA dA  dp    δs  A + δs + p δs pA −  p +  ds ds  ds   which reduces simply to −A

dp δs ds

4

to first order of infinitesimals. The force due to the weight of the fluid is: ρgAδs acting downwards, and if the z-direction is taken vertically upwards, this is

−ρgAδs in the z-direction. The component of this in the s-direction is −ρgA

dz δs ds

The mass of fluid within the element is ρAδs. The s-component of fluid acceleration along the streamline may be derived by considering the velocity change δu over the length δs of the element, which is du δs ds

δu =

The time δt in which this velocity change takes place is the time required for fluid to travel the distance from one end, where the speed is u, to the other, where the speed is u+

du δs ds

So, to first order of small quantities, the time δt is 1 δs u

δt =

The s-component of acceleration as is therefore

as

=

δu δt

5

=

du δs ds 1 δs u

or as

=

u

du ds

Equating the mass-acceleration of the fluid to net force, the equation of motion in the s-direction is therefore ρAδs u

du ds

= −A

dz dp δs − ρgA δs ds ds

which leads to the result u

dz du 1 dp + +g ds ds ρ ds

= 0 (1-7)

Provided that ρ is constant, this equation may be integrated to give 1

2

u2 +

p + gz = E ρ (1-8)

where E is a constant. Multiplication by ρ gives 1

2

ρu 2 + p + ρgz = P

(1-9)

and division by g gives u2 p + +z = H 2g ρg

(1-10)

both of which are forms of Bernoulli’s equation. The constant P is called the total pressure and H the total head. It should be noted that the result has been derived for steady motion of a fluid under pressure and gravity forces - shear force due to viscous action on the wall of the streamtube has been neglected and that the density has been assumed constant, so the fluid has been assumed incompressible. Note also that the

6

integration has been with respect to s, that is, in the s-direction along the streamline. The result may thus be stated in words: “The total pressure (or total head) is constant along a streamline in steady motion of an inviscid, incompressible fluid.”

The equation says nothing about the way that the total pressure changes from one streamline to another. The first term in Equation (1-9) is dependent on fluid velocity u, so we refer to: ½ρu2 as the velocity pressure or dynamic pressure. The remaining terms depend on pressure p and elevation z, and we refer to: p + ρgz as the piezometric pressure. When the working fluid is air, the static pressure p is much more important than ρgz. We shall see that changes in p in the experiments described later are typically about 1000 N/m2. For a change of elevation of 1 m, the corresponding change in ρgz is about 10 N/m2. So in practice, Bernoulli’s equation for air is frequently written as: 1

2

ρu 2 + p = P

(1-11)

since the contribution of ρgz is usually negligible. The equation is strictly valid of course if p is now taken to indicate piezometric pressure. We have seen that Bernoulli’s equation applies only to a fluid of constant density, and it has been mentioned in the previous paragraph that it may be applied to air, for which the density clearly changes with temperature and pressure. Is it possible to make the constant density assumption for air? By an analysis which is beyond our present scope, it may be shown that P generally exceeds 12 ρu 2 + p by an amount which increases as the velocity increases. The governing parameter in the Mach number Ma, defined as the ratio of velocity u to the local velocity of sound a, so that: Ma =

7

u a

Some numerical values, calculated for airflow at 15°C, for which the value of a is 340 m/s, are shown in Table 1.1. Now according to Equation (1-11), the value of the quantity tabulated in the last column of the table should be exactly 1, so the amount by which the numbers in this column exceed unity may be taken as an indication of the error in Equation (1-11) due to the compressibility of the air. In the experiments described in the following chapters, the air velocity rarely exceeds 50 m/s, so the compressibility error is no more than 0.5%, and the flow may justifiably be regarded as incompressible. u (m/s)

Ma = u/a

(P − p) /½ρu2

50 100 200 300 340

0.147 0.294 0.588 0.882 1

1.005 1.022 1.089 1.210 1.276

Table 1.1 Calculated Difference between Total Pressure and Static Pressure as a Function of Mach Number

The air density ρ may be calculated from the barometric pressure p and temperature T from the gas equation p ρ

= RT (1-13)

in which the value of the gas constant R for dry air is R = 287.2 J/kg K (1-14)

or R = 287.2 Nm/kg K (1-14a)

J indicates the SI unit of energy, the joule (which is identical with the newton metre or Nm) and K indicates the unit of temperature, the kelvin. If t represents temperature in °C, then

8

T = t + 273.15 (1-15)

and from Equations (1-13), (1-14) and (1-15) ρ

=

p kg/m3 287.2 ( t + 27315 . ) (1-16)

In this equation, p is expressed in N/m2 and t in °C. Some typical values of ρ are given in Table 1.2. Pressure p × 105 2

(N/m )

Temperature t

Density ρ

(°C)

(kg/m3)

10 0.95

15 20 25

1.168 1.148 1.128 1.109

1.00

10 15 20 25

1.230 1.208 1.188 1.168

1.01325

10 15 20 25

1.246 1.224 1.203 1.183

1.05

10 15 20 25

1.291 1.269 1.247 1.226

Table 1.2 Density of Dry Air

Note that the pressure 1.01325 × 105 N/m2 is the pressure of a standard atmosphere. In the event of the barometric pressure being given in mm of mercury, this may be converted to units of N/m2 by the hydrostatic relationship p = ρgh (1-17)

9

which gives the pressure p due to a column of height h of liquid of density ρ, acted on by gravity g. Using the value 13590 kg/m3 for the density of mercury, and the value 9.807 m/s2 for the standard acceleration of gravity, then 1 mm of mercury produces a pressure p given by p = 13590 × 9.807 × (1/1000) or p = 133.3 N/m2 So, 1 mm Hg = 133.3 N/m2 (1-18)

The experiments described in this book use a manometer that registers air pressures in terms of millimetres of water gauge. It is therefore useful to establish the relationship between the velocity pressure ½ρu2 of an airstream, and the corresponding water gauge reading h mm. For air of standard density ρ = 1.224 kg/m3, ½ × 1.224 × u2

= 1000 × 9.807 × (h/1000)

or u = 4.00 h m/s (1-19)

10

2.

THE AF10 AIRFLOW BENCH

AF10 Airflow Bench (shown with the AF10A)

11

Many readers will know that much of the research and development work in aerodynamics is done in wind tunnels, which provide the controlled flow of air required for tests. Some of these tunnels are large and complex, representing major engineering achievements in their own right, and require considerable power to operate. The AF10 Airflow Bench is in the nature of a simple miniature wind tunnel; it provides a controlled airstream for experiments which use matching test equipment. The bench and associated experiments are intended for use in conjunction with lecture courses in fluid mechanics or aerodynamics, particularly in the early stages. The experiments have generally been devised so that they may be set up for classroom demonstration to illustrate a lecture topic. In many cases a simple qualitative demonstration of an effect may be all that is needed; in other cases the lecturer may take one or two specimen results and use the data as the basis for a worked classroom example. Alternatively the experiments may be built into a formal laboratory programme of prescribed work in the subject. The expositions presented in this book are in the style of formal reports, as these provide a convenient means of developing the relevant theory and of describing the test apparatus, its range and its capabilities. Those lecturers who wish to use the equipment for project work should have little difficulty in formulating suitable projects; certain questions and suggestions are included which will be helpful in this regard. The airflow bench and its associated equipment is in many ways complementary to the hydraulics benches which have been manufactured for many years by TecQuipment. As the basic concepts of hydraulics and of incompressible airflow are identical, some of the experiments on the airflow bench bear marked similarities to their counterparts on the hydraulics bench, although the emphasis might be somewhat different. For example, the Bernoulli theorem experiment in airflow corresponds to the Venturi meter in the range of hydraulics equipment. On the other hand, an experiment on boundary layers is probably more relevant in a first course on aerodynamics than in a first course in hydraulics. The AF10 Airflow Bench comprises a fan which draws air from the atmosphere and delivers it along a pipe to an airbox which is above the test area. In the pipe is a valve which may be used to regulate the discharge from the fan. There is a rectangular slot in the underside of the airbox to which various contraction sections may be fitted. The air accelerates as it flows from the box along the contracting passage, and any unsteadiness or unevenness of the flow at the entry

12

becomes proportionately reduced as the streaming velocity increases towards the test section, which is fitted at the exit of the contraction. Discharge from the test section is in most cases directed towards the bench top, in which a circular hole is provided to collect the air so that it may be led through a duct to the rear of the bench. If necessary, the exhaust can be taken right out of the laboratory (for example, if, use is to be made of smoke traces) with the exhaust duct extended as necessary and an extractor fan fitted at the downstream end if required. The bench is mounted on wheels with jacking screws so that it may be moved without difficulty. It requires an earthed, AC single-phase electrical supply.

13

14

3.

THE AF10A INCLINABLE MULTITUBE MANOMETER

AF10A Inclinable Multitube Manometer

15

The AF10A is a 14 limb multitude manometer, as shown in Figure 3.1.

Figure 3.1 AF10A Inclinable Multitube Manometer The reservoir for the manometer liquid is mounted on a vertical rod so that it may be set to a convenient height. It is recommended that the manometer tubes at the two sides, marked A in Figure 3.1, and the reservoir connection, be normally left open to atmospheric pressure. Pressures p1, p2, p3 ... in tubes 1, 2, 3 are then gauge pressures, measured relative to an atmospheric datum. (Pressures relative to some other datum may be obtained by connecting the reservoir and the manometer tubes marked A to the required datum). The usual manometer liquid is water, although in some instances a paraffin-based liquid of low specific gravity is used. To aid visibility, the water may be coloured by a dye which is supplied with the equipment. The specific gravity of the water is not significantly altered by addition of the dye. To fill, the reservoir is positioned about halfway up the bar, and the fitting at the top is unscrewed. Using the funnel provided, manometer liquid is poured in until the level is halfway up the scale. Any air bubbles from the manometer tubes are then removed by tapping the inlet pipe, or by blowing into the tops of the tubes. The manometer scale is usually graduated in mm. Pressure readings taken in terms of mm of water may be converted to units of millibar (mb) from the relationship: 1 mm water = 0.0981 mb

16

The manometer may be used vertically, or, for increased sensitivity, inclined at some suitable angle to the vertical. Two predetermined settings are provided: 0°, giving a scale magnification of 1.0 (reading × 1); 60°, giving a scale magnification of 2.00 (reading × 0.5); 78°, giving a scale magnification of 5.00 (reading × 0.2). Scale readings are therefore to be divided by factors of 2.00 and 5.00 respectively to obtain equivalent readings on a vertical scale. The manometer is sufficiently accurate and sensitive for all of the experiments described in this book. The manometer must be levelled before taking readings. This can be done by using the adjustable feet, while observing the spirit level and the manometer liquid levels across all of the tubes under static conditions. It is possible that, as the air speed is increased, liquid may be driven out of the tops of the manometer tubes, or drawn down into the manifold at the base. The connection between tapping points and the manometer would then have to be cleared, or the reservoir may need to be refilled. It is therefore advisable, before starting a test, to guard against these eventualities by adopting the following setting-up procedure. With the fan at rest and the bench valve closed, the manometer should be set to the vertical position, with the liquid level at about mid-height. The fan should then be started, and the air speed raised gradually by carefully opening the bench valve, while observing the levels in the manometer tubes. As the pressures in the various tubes change, the reservoir level should be moved up or down, as found to be necessary to keep all the liquid levels within the bounds of the scale. A good setting would use most of the scale at full airspeed. If, however, only a small proportion of the scale is used, the procedure should be repeated with the manometer inclined to the vertical. To fill the manometer position the reservoir approximately halfway up the side bar. Unscrew the fitting on top of the reservoir and, using the funnel provided, pour in a quantity of water (and dye if required). Continue until the water level is halfway up the manometer scale. Check the system for air bubbles, and remove by tapping the inlet pipe, or by gently blowing into the manometer tube at the top. Having decided on a suitable manometer setting, a final height adjustment of the reservoir should then be made to bring the datum reading at tubes A to some convenient scale graduation - such as, for example, 120 mm. This is the value which

17

has to be subtracted from the scale readings of the pressures p1, p2... shown in Figure 3.1 to obtain gauge pressures. It is much easier to perform the subtraction with a datum that has been conveniently chosen.

18

4.

BERNOULLI’S EQUATION DIVERGENT PASSAGE

APPLIED

TO

AF11 Bernoulli’s Equation Apparatus

19

A

CONVERGENT-

Introduction This experiment demonstrates the use of a Pitot-static tube, and investigates the application of Bernoulli’s theorem to flow along a convergent-divergent passage.

Description of Apparatus

Figure 4.1 Arrangement of the Apparatus for an Experiment on Bernoulli’s Equation

A duct of rectangular section is fitted to the exit of the contraction which leads from the airbox, and liners placed along the inside wall of the duct produce a passage which contracts to a parallel throat and then expands to the original width. The shape

20

of this convergent-divergent passage is indicated in Figure 4.1, from which it may be noted that the convergent portion is shorter than the divergent portion. Air is blown through the passage, and a probe may be traversed along the centre line to measure the distribution of total pressure P and static pressure p. This probe is a Pitot-static probe. Pressure tappings are connected from the airbox and from the Pitot-static probe to a multitube manometer.

Theory The aim of the experiment is to measure the distribution of total pressure P and static pressure p along the duct and to compare these with the predictions of Bernoulli’s equation. Consider how the equation is applied to the present case. Figure 4.2 shows the duct as a stream tube.

Figure 4.2 Measurement of Total and Static Pressure

21

According to Bernoulli’s equation the total pressure P, defined by

P =

1

2 ρu

2

+p (4-1)

should be constant along this tube, provided the flow is steady and that the air is incompressible and inviscid. If Po denotes the total pressure in the airbox, then we should expect the measured value of P along the passage to be the same everywhere as Po, if Bernoulli’s theorem is valid for this motion. Now the total pressure P is measured with comparative ease by an open ended tube facing the flow. Figure 4.2 shows a streamline starting from the airbox, passing the duct, and arriving at the mouth of the Pitot tube. The motion is arrested at this point, so that in Equation (4-1) the local value of u is zero. The pressure recorded by the Pitot tube is therefore the local value of total pressure P. If Bernoulli’s equation applies along the whole length of the streamline from the airbox, then P should be the same everywhere as the initial total pressure Po. The value of Po may be found easily from a pressure tapping in the wall, since the air velocity in the box is so slight as to make the difference between total pressure and static pressure quite negligible. The variation of static pressure p may be measured by the static pressure tube. Figure 4.2 shows a further streamline emanating from the airbox and flowing close to the surface of the probe. Provided that the holes in the surface of the probe are placed far enough from the tip of the tube as to be unaffected by the disturbance in this locality (which means in practice about 6 tube diameters away from the tip) then the flow is undisturbed by the holes, which measure the undisturbed pressure, which is the static pressure, p. To compare the measured values of p with the results of calculations we must use the continuity equation as well as the Bernoulli equation. Taking the flow as one-dimensional, that is assuming the velocity over any chosen cross-section to be uniform over that section, then the continuity equation for incompressible flow gives the volume flow rate as:

Q = uA = u t A t (4-2)

22

(The suffix t indicates conditions at the throat). The velocity distribution along the duct may thus be written in the form of the ratio: u ut

At A

=

(4-3)

and since the depth of the duct is constant, and the cross-sectional area is proportional to the width, then u ut

Bt B

=

(4-4)

The velocity ratio following from continuity may therefore be calculated simply from the dimensions of the convergent-divergent passage. This now may be compared with the velocity ratio inferred from pressure distribution using Bernoulli’s theorem. For Equation (4-1) this gives the local velocity as: u =

2 ( P − p) ρ

(4-5)

and in particular the velocity ut at the throat is ut

2 ( Pt − p t ) ρ

=

(4-6)

so from Equations (4-5) and (4-6) u ut

P−p Pt − p t

=

(4-7)

The right-hand side of this equation may be evaluated from the measured pressure distribution and compared with the values from Equation (4-4).

23

Results

Air temperature Barometric pressure

22°C 1028 mb

= =

295 K 1.028 × 105 N/m2

The profile of the convergent-divergent passage is shown in Figure 4.3.

Figure 4.3 Dimensions of a Convergent-Divergent Passage

In Table 4.1, measurements of Po, P and p are recorded as the probe traverses along the duct. These pressures are ‘gauge pressures’, which are measured relative to atmospheric pressure. Note that the readings of P and p in a single line of the table do not represent the same physical position of the probe, because the static pressure holes lie 25 mm downwind of the tip. By measuring pressures at longitudinal spacings of 12.5 mm and 25 mm, P and p are obtained at identical stations but at different probe settings. The initial value, x = 4 mm, was a convenient starting point with the particular equipment under test and may vary somewhat from one test rig to another.

24

x (mm)

Po (N/m2)

P (N/m2)

p (N/m2)

Bt B

P−p Pt − p t

4 16.5 29 41.5 54 66.5 79 91.5 104 129 154 179 204 229 254 279 304

800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800 800

795 790 785 790 790 785 790 790 790 790 785 785 780 775 780 780 780

195 35

0.593 0.643 0.701 0.772 0.857 0.965 1.000 1.000 1.000 0.946 0.867 0.801 0.744 0.694 0.651 0.613 0.440

0.582 0.653 0.719 0.790 0.867 0.948 0.990 1.000 1.000 0.961 0.894 0.845 0.801 0.763 0.733 0.703 0.653

−130 −315 −540 −805 −945 −980 −980 −845 −630 −480 −355 −255 −170 −95 25

Table 4.1 Total and Static Pressure Distributions

Any convenient starting value may be chosen, the subsequent calculations being changed accordingly. The values of Bt/B are calculated from the known dimensions of the contraction. For example, in the converging section, when x = 29 mm

B = 76 − (76 − 44) ×

So

Bt B

=

44 62.7

29 70

= 62.7

= 0.701

and in diverging section, when x = 204 mm B = 76 − (76 − 44) ×

So

Bt B

=

44 59.2

25

100 190

= 59.2

= 0.744

Discussion of Results

Figure 4.4 and Figure 4.5 show the results in graphical form. The total pressure P is seen to remain very close to the airbox pressure Po over the whole length of the duct, despite the considerable fluctuation of static pressure p. Bernoulli’s equation has therefore been verified for the streamline along the centre of the duct, along which significant velocity changes take place. The distribution of velocity, measured by the Pitot-static probe, is compared in Figure 4.5 with the velocity distribution inferred from the continuity equation. In the converging section the results are almost identical, but in the diverging section downstream of the throat a steadily increasing discrepancy arises. The airstream is apparently decelerating less quickly than the geometrical shape of the passage would indicate.

Figure 4.4

It will be seen in a later experiment that a boundary layer forms adjacent to any fixed surface along which air flows, and in this layer the velocity reduces from the free stream value down to zero at the surface. The thickness of the layer increases in the direction of flow, and it is found experimentally that the growth in thickness is more 26

rapid in regions of rising pressure (i.e. where the main stream is decelerating) than in regions where the pressure is constant. The converse is true; where the pressure falls in the direction of flow the growth of boundary layer thickness is retarded.

Figure 4.5

The results presented in Figure 4.5 are consistent with this concept. In the converging section and the throat, the measured pressures agree closely with those calculated from the variation in duct width so the boundary layer has scarcely any effect. In the diverging section, however, thickening of the boundary layer would give the

27

appearance of the cross-section of the duct enlarging less rapidly than it actually does, the retarded air in the thickening boundary layer presenting a partial blockage to the flow. We may therefore conclude that the experiment as a whole has demonstrated that Bernoulli’s equation is sensibly valid along the central streamline of the convergentdivergent duct, since the total pressure has been shown to be virtually constant along its length. The calculated pressure distribution, which depends on the concept of continuity as well as constant total pressure, shows a significant discrepancy from the measured results in the divergent portion, and this may be explained by the growth of boundary layers on the walls of this portion.

Questions for Further Discussion

1.

What boundary layer thickness do your results lead you to expect. Can you infer this from the graph of Figure 4.5?

2.

What is the Mach number at the throat of the duct? For approximate calculation, you may assume that the static pressure and temperature there are approximately the same as in the airbox. The air velocity at the throat may be found from the Pitot-static reading, and the acoustic velocity may be estimated from the equation a =

γ RT

(4-7)

in which, γ is the ratio of specific heats = 1.4 for air R is the gas constant = 287.2 J/kg K T is the absolute temperature in K For the results given here: ρ =

p RT

=

1028 × 105 . kg m3 = 1213 . 287.2 × 295

28

Pt − p t

= 790 + 980 = 1770 N m2 1

2

× 1213 . × u 2t

ut

= 1770

= 54.0 m s

Also =

at

14 . × 287.2 × 295

at Ma t

=

= 344 m s

ut at

=

54.0 344

. = 0157

3.

What difference to the results would you expect if the flow direction were reversed? You may check your prediction by reversing the liners.

4.

What suggestions have you for improving the experiment?

5.

How might you check whether there is in fact a boundary layer of significant thickness at exit from the duct? A possible project would be to devise and construct a suitable simple traversing gear from a Pitot tube which would measure the velocity distribution. Would it be necessary to traverse along more than one axis?

29

30

5.

DRAG MEASUREMENT ON CYLINDRICAL BODIES

AF12 Drag Force Apparatus

31

Introduction The resistance of a body as it moves through a fluid is of great technical importance in hydrodynamics and aerodynamics. In this experiment we place a circular cylinder in an airstream and measure its resistance, or drag, by three methods. We start by introducing the ideas which underline these methods. Consider the cylindrical body shown in cross-section in Figure 5.1. The reader may be unfamiliar with the idea of a non-circular cylinder. In the present context the word ‘cylinder’ is used to describe a body which is generated by a straight line moving round a plane closed curve, its direction being always normal to the plane of the curve. For example, a pencil of hexagonal cross-section is by this definition a cylinder.

Figure 5.1 The curve shown in Figure 5.1 represents a section of an oval cylinder. An essential property of a cylinder is that its geometry is two-dimensional; each cross-section is exactly the same as every other cross-section, so that its shape may be described without reference to the dimension along the cylinder axis. We shall use the term circular cylinder to denote the particular and important case of the cylinder of circular cross section. Motion of the cylinder through stationary fluid produces actions on its surface which give rise to a resultant force. It is usually convenient to analyse these actions from the point of view of an observer moving with the cylinder, to whom the fluid appears to be approaching as a uniform stream. At any chosen point 32

A of the surface of the cylinder, the effect of the fluid may conveniently be resolved into two components, pressure p normal to the surface and shear stress τ along the surface. It is convenient to refer absolute pressure pa to the datum of static pressure po in the oncoming stream; p is then a gauge pressure, that is:

p = pa − po Let U denote the uniform speed of the motion and ρ the density of the fluid, then the dynamic pressure in the undisturbed stream, ½ρU2, is 1

2 ρU

2

= Po − p o (5-1)

where Po is the total pressure in the oncoming stream. This pressure is a useful quantity by which the gauge pressure p and shear stress τ may be nondimensionalised, and the following dimensionless terms are defined: Pressure coefficient, c p

p 1 ρ U2 2

=

(5-2)

Skin friction coefficient, c f

=

1

τ 2 2ρU (5-3)

The combined effect of pressure and shear stress (sometimes called skin friction) gives rise to resultant force on the cylinder. This resultant may conveniently be resolved into the following components acting at any chosen origin C of the section as shown in Figure 5.1: 1. 2. 3.

A component in the direction of U, called the drag force, of intensity D per unit length of cylinder. A component normal to the direction of U, called the lift force, of intensity L per unit length of cylinder. A moment about the origin C, called the pitching moment, of intensity M per unit length of cylinder.

33

These components may be expressed in dimensionless terms by definition of drag, lift, and pitching moment coefficients as follows: Drag coefficient, C D

=

1

D 2 2ρU d (5-4)

Lift coefficient, C L

=

L 1 ρ U2d 2 (5-5)

Pitching moment coefficient, C M

=

1

M 2 2 2ρU d (5-6)

in which d denotes a suitable dimension which characterises the size of the cylinder. In Figure 5.1 this is shown as the width measured across the cylinder, normal to U, which is the usual convention. (An important exception is the aerofoil, where the length in the direction of flow or ‘chord’ of the section is used instead). The coefficients CD, CL and CM are of prime importance since they are invariably used for correlating aerodynamic force measurements. We may see how pressure and skin friction coefficients are related to lift and drag coefficients. Consider an element of length δs of the surface, at a point where the normal is inclined at angle θ to the direction of U, as shown in Figure 5.1. The element of drag δD per unit cylinder length due to p and τ is δD =

( p cos θ + τ sin θ) δs

and integrating this round the whole perimeter yields D =

∫ (p cos θ + τ sin θ) ds

This may now be cast in dimensionless form: D 1 ρU 2 d 2

=

1⌠ p cos θ +  2 1 d ⌡  2 ρU

34

 τ sin θ ds 2 1 ρU  2

or

CD

1 d

=

∫ (c p cos θ + c f sin θ) ds (5-7)

Similarly CL

=

1 d

∫ (−c p sin θ + c f cos θ) ds (5-8)

These results show that the drag of a cylinder may be found by measuring p and τ over the surface and calculating the drag coefficient by Equation (5-7). Now it is easy to measure the distribution of p over a cylinder merely by drilling fine holes into its surface, but measurement of τ is a much more difficult task. For the case of the circular cylinder, however, the contribution to drag from shear stress (the ‘skin friction drag’) is found to be very much smaller than from pressure (the ‘pressure drag’) and may be neglected. Making this assumption and writing d δθ 2

δs = Rδθ =

(5-9)

for the circular cylinder of Figure 5.2 simplifies Equation (5-7) to: CD

=

∴ CD

1⌠  d c p cosθ   dθ   2 d⌡ =

1 2

2π

∫c 0

p

cosθ dθ

From consideration of symmetry, we may therefore write CD

=

2π

∫c 0

p

cosθ dθ (5-10)

CL

= CM

= 0 (5-11)

35

for the circular cylinder. Equation (5-10) allows us to calculate CD from the measured pressure distribution over the cylinder surface.

Figure 5.2 The Circular Cylinder

At the point marked S in Figure 5.2, the oncoming airstream is brought to rest. S is called the stagnation point, and the streamline arriving at S is the dividing streamline. Moving around the cylinder from S, we expect the velocity over the surface to increase from zero at S, and so according to Bernoulli’s equation, we might expect the pressure and therefore the pressure coefficient to fall. By an analysis which is beyond our scope, the velocity u over the surface is given in terms of θ by the simple equation u U

= 2 Sinθ (5-12)

provided that the fluid is incompressible and non-viscous.

36

Writing pa as the absolute static pressure at A, Bernoulli’s equation is: = po +

Po

1

2 ρU

2

= pa +

1

2 ρu

2

(5-13)

The gauge pressure p at A is thus p = pa − po

=

p =

2

1

2 ρU

2

−

1

2 ρu

2

From Equation (5-12) 1

2 ρU

(1 − 4 Sin θ) 2

so the pressure coefficient, cp, is cp

=

p 1 ρ U2 2

= 1 − 4 Sin 2 θ (5-14)

This is the theoretical result for an incompressible, inviscid fluid, and forms the basis of comparison with experimental results.

Figure 5.3 Application of the Momentum Equation to Flow along a Duct past a Cylindrical Body

There is an entirely different way of finding the drag on a cylinder which depends on the application of the momentum equation to the airflow. This equation is derived in textbooks on the subject, but again for completeness a brief exposition is given here.

37

Consider the flow of a fluid along a duct of width 2h past a cylindrical body which spans the duct, so that the motion is two-dimensional as indicated in Figure 5.3. The velocity is U and the pressure is po in the oncoming flow. Downstream of the cylinder the velocity is no longer uniform; let the velocity be u at distance y from the duct centreline. 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. The forces in the x-direction acting on the fluid in the control volume are, per unit length of cylinder: 2h po −2h pe

At the upstream section At the downstream section At the cylinder

−D

Note the minus sign for the force exerted by the cylinder on the fluid, which is equal and opposite to the force exerted by the fluid on the cylinder. Forces due to shear stress on the wall of the duct and due to the fluid weight are neglected. The momentum flux per unit width over the downstream section is: h

2 ∫ ρu dy

−h

and over the upstream section is h

2 ∫ ρU dy

−h

Equating the net force in the x-direction to the momentum flux out of the control volume 2 h po − 2 h pe − D =

h

h

−h

−h

2 2 ∫ ρu dy − ∫ ρU dy

(5-15)

38

Rearranging and making non-dimensional gives the result: h

CD

=

D 1 ρU 2 d 2

=

u2  2h po − pe 2⌠ + − 1   dy  d 12 ρU 2 d⌡ U2  −h

(5-16)

The integral may also be made non-dimensional by the substitution y = ηh (5-17)

so that h

1

−h

−1

⌠ ⌠ u2  u2    1 − 2  dy = h  1 − 2  dη U  U  ⌡ ⌡

and the final result is 1

CD

=

u2  2h po − pe 2h ⌠  + − 1   dη  d 12 ρU 2 d ⌡ U2  −1

(5-18)

Equation (5-18) provides a means to calculate CD from the pressure drop along the duct and the velocity distribution in the wake. Note 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 D which comes into the momentum equation. The skin friction drag on the walls also contributes to the momentum change and is also included in D. It is also worth mentioning that Equation (5-18) applies only to the case of flow along a duct where the flow is confined between parallel walls. The foregoing analysis shows how drag force may be found from the pressure distribution over the surface of the cylinder and by measurements in the wake. The results obtained from both of these methods may be compared with the drag measured by direct weighing, and this is described in the next section.

39

Description of Apparatus and Test Procedure

Figure 5.4 shows the three main experiment arrangements for the AF12. Figure 5.4a shows the main pressure tapping points and their notation. These arrangements allow study of the drag of various bodies, the pressure distribution around a cylinder and the wake of different bodies. The balance arm is attached to the side of the main unit and holds the models in place inside the main unit. It is used with adjustable weights in the drag force experiments and as a fixed support in the wake traverse experiments. It is not used in the pressure distribution experiment. For the pressure distribution experiment, a protractor and cylinder model is fitted in the main body. The cylinder is connected to the separate AF10a manometer to display the pressure distribution as the cylinder is rotated. For the wake traverse experiments, a pitot tube assembly is attached to the main unit. Drag Force

Pressure Distribution

AF10 Contraction section (Outlet Duct)

AF10 Contraction section (Outlet Duct)

Quick release coupling

Quick release coupling

10 g Weight AF12

AF12

AF12 Main unit

AF12 Main unit

Test Model Thumbnuts (x2)

Protractor (located at back of main unit) Retaining Screw

100 g Weight

Test Model Clamp Screws (x2)

To manometer

Test model

Cylinder Balance Arm

Wake Traverse AF10 Contraction section (Outlet Duct) Quick release coupling AF12 Main unit

AF12

Balance Arm Locking Screw Thumb Nut Pitot tube assembly Stainless steel tube

Test model

To manometer

Clear plastic tube

Figure 5.4 Diagram of Apparatus

The apparatus is supplied with the cylinder and protractor model for pressure distribution experiment and three other models for the wake traverse and drag force experiments. The models include: a circular cylinder, a flat plate (inverted prism) and a symmetrical aerofoil section. 40

AIR FLOW BENCH AF10

Po

po

p

Pe Pitot tube for traverse in exit plane pe = 0, atmospheric datum

Figure 5.4a Pressure Tappings The Drag Force Experiments - Procedure Fit the circular cylinder model in position and adjust the weights to achieve

equilibrium. Record the value of the weights. Every 1 mm on the front scale (100 g weight) represents 1 g and every 1 mm on the back scale (10 g weight) represents 0.1 g (use for fine adjustment). Switch on the fan and adjust the wind speed to a low level and readjust the weights to achieve equilibrium. Record the value of the weights and the total pressure (Po) and the static pressure (po) at the inlet. Increase the wind speed in increments up to its maximum level, each time readjust the weights to achieve equilibrium and record the pressures and the value of the weights. Subtract the initial value of the weights (with zero wind speed) from all the readings to give an actual value of equilibrium at each wind speed. Repeat the procedure with the other two models.

41

The Pressure Distribution Experiments - Procedure Fit the circular cylinder and protractor model into the AF12. Connect the model to the

manometer. Set the wind speed to maximum and the protractor to zero angle. Record the surface pressure (p) and the static pressure po. Rotate the protractor in 5° intervals for readings over the front half (0 to 90°) and 10° intervals for readings over the rear half (90 to 180°). Monitor the total pressure (Po) and static pressure (po) at the inlet to ensure that the wind speed is kept constant throughout the experiment. The Wake Traverse Experiments - Procedure Fit the circular cylinder model into the AF12. Fit the pitot traverse assembly. Set up a constant wind speed (a pressure difference of approximately 450 N/m2 is

recommended). Traverse the Pitot tube across the section, in increments of 2 mm near the centre, increasing to 5 mm or 10 mm in regions where the total pressure is seen to be substantially constant. Record the values of Pe and pe The readings in the wake are extremely unsteady because of the high level of turbulence. Some users may wish to damp the oscillations by placing a tubing clip on the flexible connection to the manometer. If this method is used, take care to avoid excessive damping that can cause error. It is better to use too little than too much. Repeat the experiment with the other two models.

42

Results

Table 5.1 gives the drag force measured in units of gram-force by direct weighing at various air speeds. The drag force is written as the product of the drag D per unit length and the length l of the cylinder.

Drag Force

Po

po

Po−po = ½ρU2

Dl (gmf)

(mm H2O)

(mm H2O)

2 (N/m )

39.2 38 34 27.6 19.3 13.5 4.9

190 188 181 170 155 145 128

128 128 126 125 124 123 121

608.22 588.6 539.55 441.45 304.11 215.82 68.67

Table 5.1 Drag Force on the cylinder measured by Direct Weighing

Air temperature Barometric pressure

24°C 1040 mb

Air density

ρ =

= =

p RT

=

Diameter of cylinder d Length of cylinder l Half width of working section h h d

297 K 1.04 × 105 N/m2 1.04 × 10 5 = 1.219 kg/m3 287.2 × 297

= = =

12.5 mm = 0.0125 m 48 mm = 0.048 m 50 mm = 0.050 m

=

4

In Figure 5.5 the force is plotted against dynamic pressure and a good linear relationship is established with a slope

Dl 38 = 2 1 ρU 600 2 38 × 9.81 × 10− 3 m2 = 6.213 × 10 −4 m2 600 (Note that 1 gmf = 981 dyn = 9.81 × 10−3 N)

43

CD may now be found by substitution in Equation (5-4) =

CD

=

D 1 ρU 2 d 2

=

D1 1 ρU 2 dl 2

6.213 × 10 −4 = 1.03 0.0125 × 0.048

Value of drag coefficient by direct weighing CD = 1.03 40

35

Drag Force D1 (gmf)

30

25

20

15

10

5

0 0

100

200

300

400

500 2

600

700

2

Dynamic Pressure 1/2ρU (N/m )

Figure 5.5 Measured Drag Force on a Circular Cylinder The measured pressure distribution round the cylinder is given in Table 5.2, and graphs of cp and cp cosθ as functions of angle θ from the front are presented in Figures 5.6 and 5.7. The velocity pressure at inlet during this test was: Po − po = 588.6 (x2), 598.41, 598.4 (x7) : Mean 596.448 N/m2 ∴½ρU2 = 596.448 N/m2

44

The pressure is seen to be relatively symmetrical about the line θ = 0. The pressure coefficient falls over the front portion to a minimum at θ = 70°, and thereafter rises a little. Over the rear half of the cylinder the pressure is reasonably uniform. The distribution of cp for a cylinder in inviscid, incompressible fluid is shown for comparison, calculated from Equation (5-14). θ

p − po 2

Cp =

p − po 1 ρU 2 2

(degree)

(N/m )

0

559.17

0.94

5

549.36

10

Cp Cosθ

θ

p − po 2

Cp =

p − po 1 ρU 2 2

Cp Cosθ

(degree)

(N/m )

0.94

0

549.36

0.92

0.92

0.92

0.92

−5

549.36

0.92

0.92

529.74

0.89

0.87

−10

519.93

0.87

0.86

15

461.07

0.77

0.75

−15

470.88

0.79

0.76

20

372.78

0.63

0.59

−20

392.4

0.66

0.62

25

245.25

0.41

0.37

−25

284.49

0.48

0.43

30

117.72

0.20

0.17

−30

147.15

0.25

0.21

35

-19.62

-0.03

-0.03

−35

19.62

0.03

0.03

40

-186.39

-0.31

-0.24

−40

-137.34

-0.23

-0.18

45

-353.16

-0.59

-0.42

−45

-313.92

-0.53

-0.37

50

-480.69

-0.81

-0.52

−50

-451.26

-0.76

-0.49

55

-598.41

-1.00

-0.58

−55

-559.17

-0.94

-0.54

60

-696.51

-1.17

-0.58

−60

-676.89

-1.13

-0.57

65

-755.37

-1.27

-0.54

−65

-765.18

-1.28

-0.54

70

-765.18

-1.28

-0.44

−70

-794.61

-1.33

-0.46

75

-716.13

-1.20

-0.31

−75

-774.99

-1.30

-0.34

80

-657.27

-1.10

-0.19

−80

-735.75

-1.23

-0.21

85

-627.84

-1.05

-0.09

−85

-667.08

-1.12

-0.10

90

-608.22

-1.02

0.00

−90

-637.65

-1.07

0.00

100

-618.03

-1.04

0.18

−100

-637.65

-1.07

0.19

110

-608.22

-1.02

0.35

−110

-627.84

-1.05

0.36

120

-608.22

-1.02

0.51

−120

-627.84

-1.05

0.53

130

-618.03

-1.04

0.67

−130

-627.84

-1.05

0.68

140

-618.03

-1.04

0.79

−140

-627.84

-1.05

0.81

150

-618.03

-1.04

0.90

−150

-637.65

-1.07

0.93

160

-618.03

-1.04

0.97

−160

-627.84

-1.05

0.99

170

-618.03

-1.04

1.02

−170

-608.22

-1.02

1.00

180

-608.22

-1.02

1.02

−180

-608.22

-1.02

1.02

Table 5.2 Pressure Distribution around a Cylinder

45

1.00

0 to 180

0.00

0 to -180

Cp

Ideal Fluid Theory

-1.00

-2.00 Calculated from equation 5-14

-3.00 0

20

40

60

80

100

120

140

160

180

θ (Degrees)

Figure 5.6 Pressure Distribution around a Circular Cylinder

1.2 1 0.8

Cp cos θ

0.6 0.4 0.2

0 to180 0 to -180

0 -0.2 -0.4 -0.6 0

20

40

60

80

100

120

140

θ (Degrees)

Figure 5.7 Distribution of cp cosθ around a Cylinder 46

160

180

CD may be obtained from Figure 5.7 from the area beneath the curve. The area A beneath the mean curve is A =

π

∫ c p cosθ dθ

o

which, from Equation (5-10), we recognise as the drag coefficient CD. This area can be evaluated in various ways; Appendix A describes how it is done by use of Simpson’s rule. The result is A = 1.07 The value of drag coefficient obtained by plotting CD = 1.07

y

y d

Pe − pe

U2

(mm

 U2  U   U  × 1 −  U2   1  1

H2O)

(mm)

y

y d

(mm)

Pe − pe

U2

 U2  U   U  × 1 −  U2   1  1

(mm H2O)

0

0

19

5.53

0.24

0

0

19

5.53

0.24

1

0.08

20

5.67

0.24

-1

-0.08

20

5.67

0.24

3

0.24

21

5.81

0.24

-3

-0.24

21

5.81

0.24

5

0.4

22

5.95

0.24

-5

-0.4

23

6.08

0.23

7

0.56

27

6.59

0.21

-7

-0.56

25

6.34

0.22

9

0.72

33

7.29

0.18

-9

-0.72

31

7.06

0.19

11

0.88

37

7.72

0.16

-11

-0.88

35

7.50

0.17

13

1.04

41

8.12

0.13

-13

-1.04

40

8.02

0.14

15

1.2

47

8.70

0.08

-15

-1.2

47

8.70

0.08

17

1.36

50

8.97

0.06

-17

-1.36

50

8.97

0.06

19

1.52

53

9.24

0.03

-19

-1.52

53

9.24

0.03

22

1.76

55

9.41

0.02

-22

-1.76

55

9.41

0.02

25

2

57

9.58

0.00

-25

-2

55

9.41

0.02

35

2.8

58

9.66

-0.01

-35

-2.8

56

9.49

0.01

40

3.2

58

9.66

-0.01

-40

-3.2

57

9.58

0.00

45

3.6

48

8.79

0.07

-45

-3.6

57

9.58

0.00

-49

-3.92

43

8.32

0.11

Table 5.3 Results of Velocity Traverse in the Wake

47

In table 5.3 the value of U2 is calculated from the experiment results: 2 g ( Pe − p e ) U2 =

ρ

To obtain a value of drag coefficient from the Wake Traverse method, the effects of the duct must be taken into consideration. Because of this the value of free stream velocity (U1) is calculated using the free stream flow of the duct (the area outside the wake of the cylinder. Looking at the results of Pe & pe in table 5.3, taken during the experiment, the free stream flow region can be easily identified as the stable region. In this case the values obtained between 25 mm & 40 mm. The average values of Pe & pe are: Pe = 16.7 Pe = 11 Using these values, U1 is calculated using; U1 =

U1 =

2 g ( Pe − pe )

ρ

2 × 9.81× (16.7 − 11) 1.22

U = 9.57 m/s 0.3 0.25

0 to 45 mm (U2/U1)*(1-(U2/U1))

0.2

0 to -45 mm

0.15 0.1 0.05 0 -0.05 -4

-3

-2

-1

0

1

2

y/d

Figure 5.8 Velocity Traverse in the Wake of a Cylinder

48

3

4

The drag coefficient may be obtained by use of the curve of

U2 U1

 U2 1 −  U1

  in Figure 

5.8. The area beneath the curve is found using the trapezoidal rule (see Appendix A) to be: y/d

⌠ U2   ⌡  U1

−y / d

 U2 1 −  U1

 y   d = 0.525   d

Using this Equation gives y/d

⌠ U C D = 2   2 ⌡  U1 −y / d

 U2 1 −  U1

 y   d   d

CD = 2 x 0.525 = 1.05 Value of drag coefficient from wake traverse CD = 1.05

Discussion

The values obtained for drag coefficient of the circular cylinder are as follows By direct weighing By pressure plotting By wake traverse

CD CD CD

= 1.03 = 1.07 = 1.05

Taking the first of these as being the most reliable, we see that pressure plotting yields a result within the probable limits of accuracy. There have been very many measurements of drag of circular cylinders, and the variation of CD with Reynolds number Re, where Re =

Ud ν

is well established. In this experiment, the absolute viscosity is, at 24°C,

49

µ = 2.40 × 10-5 kg/m s so the kinematic viscosity is 2.40 × 10 −5 µ ν = = 1.97 × 10−5 m2/s = ρ 1.216 The velocity U at the typical value of velocity pressure ½ρU2 = 500 N/m2 is 1

 2 × 500  2 U=   = 26.6 m/s  1.22  So a typical value of Re for these tests is: Re =

Ud

ν

=

28.6 × 0.0125 1.97 × 10 −5

Re = 2.4 × 104 It is well established that in the range of Re from 104 to 105, CD is almost constant, the value usually quoted for a cylinder being CD = 1.20 l is so great that the three d dimensional flows at the ends have no significant effect on the result. Our l = 3.9 for which end effects must be experiments are made for a cylinder with d significant. Moreover, the cylinder presents an appreciable blockage to the cross section of the flow - its projected area is about 1/8th of the cross sectional area of the working section, so the results cannot be compared directly with those obtained from cylinders in unconfined flows.

This is for the case of a long cylinder, for which

50

Questions for Further Discussion

1.

A yawmeter is an instrument for finding the direction of a fluid stream. One type of yawmeter consists of a circular cylinder with two surface holes at different positions in the same diametral plane. It is placed in the stream and rotated slowly about its axis until a balance is obtained between the observed pressures at the holes. Suggest a suitable angular spacing between the holes to give good sensitivity.

2.

Measure the drag coefficients of a flat plate and an aerofoil section by direct weighing, and comment on the results obtained. Could pressure plotting be used to establish the drag coefficients of these sections? (CD = 2.0 for plate, CD = 0.04 for aerofoil)

3.

The circular cylinder presents a substantial blockage to the flow along the working section. Suppose instead of using the approach velocity U, the velocity U1 past the cylinder was used as the velocity on which the reference value of velocity head is based. Show that Ul =

8 U 7

for the results presented here, and find the corresponding value CD1 from C D1 =

D 1 ρU 2 d 2 1

(0.76)

51

52

6.

THE ROUND TURBULENT JET

AF13 Round Turbulent Jet Apparatus

53

Introduction The behaviour of a jet as it mixes into the fluid which surrounds it has importance in many engineering applications. The exhaust from a gas turbine is an obvious example. In this experiment we establish the shape of an air jet as it mixes in a turbulent manner with the surrounding air. It is convenient to refer to such a jet as a ‘submerged’ jet to distinguish it from the case of the ‘free’ jet where no mixing with the surrounding medium takes place, as is the case when a smooth water jet passes through the atmosphere. If the Reynolds number of a submerged jet (based on the initial velocity and diameter of the jet) is sufficiently small, the jet remains laminar for some length - perhaps 100 diameters or more. In this case the mixing with the surrounding fluid is very slight, and the jet retains its identity. Laminar jets are important in certain fluidic applications, where a typical diameter may be 1 mm, but the vast majority of engineering applications occur in the range of Re where turbulent jets are produced.

Figure 6.1 Schematic Representation of a Round Turbulent Jet

54

The essential features of a round turbulent jet are illustrated in Figure 6.1. The jet starts where fluid emerges uniformly at speed U from the end of a thin-walled tube, of cross-sectional radius R, placed in the body of a large volume of surrounding fluid. The sharp velocity discontinuity at the edge of the tube gives rise to an annular shear layer which almost immediately becomes turbulent. The width of the layer increases in the downstream direction as shown in the diagram. For a short distance from the end of the tube the layer does not extend right across the jet, so that at section 1 there is a core of fluid moving with the undisturbed velocity U, the velocity in the shear layer rising from zero at the outside to U at the inside. Further downstream the shear layer extends right across the jet and the velocity uo on the jet axis starts to fall as the mixing continues until ultimately the motion is completely dissipated. There is entrainment from the fluid surrounding the jet by the turbulent mixing process so that the mass flux in the jet increases in the downstream direction. The static pressure is assumed to be constant throughout, so there is no force in the direction of the jet. The momentum of the jet is therefore conserved. The kinetic energy of the jet decreases in the downstream direction due to the turbulent dissipation. It should be emphasised that the velocity profiles indicated in Figure 6.1 are mean velocity distributions, and that the very severe turbulence in the jet will cause instantaneous velocity profiles to vary considerably from these mean ones.

Velocity Distribution and Momentum Flux Consider the jet of Figure 6.1. If we assume that the flow pattern is independent of Reynolds number, then we might expect the velocity on the jet axis to depend on position in the dimensionless form

 x = f   R

uo U

(6-1)

In the core of the jet, we have already observed that

uo U

= 1

55

Far downstream, when the length of the core ceases to have influence, there is some theoretical justification (supported by experiment) for expecting the centreline velocity to decay inversely as x, i.e. uo U

=

c x (6-2)

where c is a constant. The velocity u at any position (r, x) in the jet may also be written in the dimensionless form u uo

 x r = g ,   R x

(6-3)

x is R large. We might reasonably expect that the velocity distribution across the section would not depend appreciably on the precise detail of the flow near the tube exit, so x we might ignore the dependence upon and simply write R

Consider now the velocity distribution over a section far downstream, i.e. where

u uo

 r = g   x

(6-4)

far downstream. Velocity profiles of this type, in which the velocity ratio depends on a parameter, are frequently called ‘similar’, in the sense that a single expression is used to characterise the velocity distribution at any number of chosen sections. Using certain assumptions about the nature of the turbulent processes, it is possible to show that Equation (6-4) should take the form u uo

=

1 2   λ r   1 + 0.25    x  

2

(6-5)

where λ is a constant which is to be determined by experiment.

56

Values of u/uo computed from this expression are presented in Table 6.1. The value λr/x = 1.287 is included, as this makes u/uo = 0.5. When comparing with experimental results it is useful to have this value, since the radius at which u/uo = 0.5 is easily identified on the velocity profile. λr/x

u/uo

0 0.2 0.4 0.6 0.8 1.0 1.287 1.4 1.6 1.8 2.0 2.25 2.5 3.0 4.0

1.000 0.980 0.925 0.842 0.743 0.640 0.500 0.450 0.372 0.305 0.250 0.195 0.152 0.095 0.040

Table 6.1 Calculated Velocity Profile of Round Jet

Figure 6.2 Annular Element of a Round Jet

57

Coming now to mass, momentum and energy flux, we see in Figure 6.2 an annular element of the jet through which fluid of density ρ is flowing with velocity u. The area of the element is δA = 2πr δr & through it is So the mass flux δm

= 2πρur δr

& δm

& through the section of the jet is The total mass flux m ∞

& = 2 πρ ∫ ur dr m 0

(6-6)

The momentum flux J through the section is similarly found to be ∞

= 2 πρ ∫ u 2 r dr

J

o

(6-7)

and the kinetic energy flux E to be ∞

E = 2 πρ ∫ 12 u 3 r dr o

(6-8)

It is convenient in many instances to relate these to the corresponding fluxes at the tube exit, as follows:

&o m Jo Eo

= πρ UR 2 = πρU 2 R 2 = π 12ρU 3 R 2

with the results

58

& m &o m

∞

⌠ u  r   r  = 2  d  ⌡ U  R   R  0

(6-9)

J Jo

2

∞

⌠ u   r   r  = 2   d  ⌡ U   R   R  0

(6-10) ∞

E Eo

⌠ u 3 r   r  = 2   d  ⌡ U   R   R  0

(6-11)

Description of Apparatus and Procedure

The round jet is produced by discharging air from the airbox through a short tube as indicated in Figure 6.3. The inlet of the tube is rounded to prevent separation so that a substantially uniform velocity distribution is produced at the tube exit. A traversing mechanism is supported on the tube so that a Pitot tube may be brought to any desired position in the jet. Measurements are normally made in one plane, but if it is necessary to check on the symmetry of the jet about the axis, the traversing mechanism may be rotated as a whole to any position. The Pitot tube is first brought into the plane of the exit of the jet tube and the scale readings are noted for which the axial position x and the radial position r are zero. The latter may be obtained by taking the average of the readings when the tube is set in line with one side and then the other side of the tube. The pressure Po in the airbox is then brought to a convenient value and traverses are made at various axial stations along the length of the jet. The readings of total pressure P fluctuate violently because of the turbulence and some damping is required; however, excessive damping must not be used. It is recommended that graphs of total pressure P against radius r be plotted as the experiment proceeds to ensure that the profile is well-established by a sufficient number of readings in the critical regions.

59

Po in air box

Traversing gear mounted on tube

U

R Scale for x

x

Scale for r r

Figure 6.3 Arrangement of Jet Apparatus Results and Calculations

Diameter D of jet tube Radius R Pressure Po in airbox

51.6 mm 25.8 mm 900 N/m2

Air temperature

22oC

= 295 K

Barometric pressure

1025 mb

=

Air density ρ

1025 . × 105 287.2 × 295

= 1.210 kg/m3

1.025 × 105 N/m2

= 1.82 × 10−5 kg/ms

Coefficient of viscosity, µ

60

P

Coefficient of kinematic viscosity, ν

Velocity U at tube exit, ∴

=

µ ρ

= 150 . × 10−5 m2 s

½ρU2

= 870 N/m2

U

=

Reynolds number Re at tube exit Re

=

2 × 870 . 1210

UD ν

=

=

37.9 m/s

37.9 × 0.0516 × 105 150 .

= 1.30 × 105

Re

The velocity along the axis of the jet was first found by traversing axially, the results being presented in Table 6.2 and Figure 6.4.

x (mm)

P (N/m2)

uo U

0 50 75 100 125 150 175 200 225 250 300 350 400

870 860 845 835 830 810 775 730 675 620 505 430 340

1.00 0.99 0.99 0.98 0.98 0.96 0.94 0.92 0.88 0.84 0.76 0.70 0.63

Table 6.2 Velocity Distribution along Jet Axis

61

1.0 0.8

uo/U

0.6 0.4 0.2 0

5

x/R 100

10 200

x (mm)

15 300

400

500

Figure 6.4 Centreline Velocity along Jet

For the initial portion the centreline velocity uo is seen to be almost constant, and further downstream it starts to fall more rapidly as the shear layer extends to the centre. Extrapolating the falling curve backwards to the line uo/U = 1 shows the length of the core to be xc = 175 mm

or

xc/R = 6.8

The results of radial traverses made at various values of x are shown in Table 6.3 and in Figures 6.5(a) to 6.5(d). It may be noted that for x = 300 mm a check was made to find whether the velocity distribution was symmetrical about the axis, and this established that there was no appreciable departure from roundness. The profile at x = 75 mm shows a distinct region of constant velocity in the core, and at x = 150 mm there is still some evidence of a flat top to the profile. Further downstream, however, this has disappeared. In Figure 6.6 a dimensionless comparison of the profiles is made by dividing the radius by the radius at which the velocity ratio is 0.5. If you take a set of readings further downstream than x = 300 mm and plot the velocity profile, it will be very similar to the x/R = 11.6 curve. This shows a similar profile. The transition from the square-topped profile at the tube exit to the similarity profile is clearly demonstrated. The curve calculated from Equation (6-5) as shown in Table 6.1 is also plotted. There is good agreement with the similarity profile near the centre of the jet, but Equation (6-5) over estimates u/uo at the outer edge.

62

r (mm) 0 5 10 15 17.5 20 22.5 25 27.5 30 32.5 35 40 45 50 55 60 65 70 75 80 85 90

x = 75 mm P u/uo (N/m2) 840 1.00 840 1.00 840 1.00 835 1.00 810 0.98 755 0.95 630 0.87 465 0.74 275 0.57 160 0.44 50 0.24 5 0.08 0 0

x = 150 mm P (N/m2) u/uo 810 1.00 800 0.99 770 0.97 720 0.94

x = 300 mm P (N/m2) u/uo 520 1.00 520 1.00 495 0.98 450 0.93

590

0.85

375

0.85

430

0.73

330

0.80

250

0.56

245

0.69

135 70 25 10 5 0

0.41 0.29 0.18 0.11 0.08 0

185 155 110 75 45 30 20 15 5 0

0.60 0.55 0.46 0.38 0.29 0.24 0.20 0.17 0.10 0

Table 6.3 Velocity Distribution at Various Sections of the Jet

63

Figure 6.5(a)-(c) Velocity Profiles in Jet at Various Distances Downstream 1.0 x/R = 2.91 x/R = 5.81 x/R = 11.6

0.8

u/uo

0.6 0.4

Equation (5-5)

0.2 0

0.2

0.4

0.6

0.8

1.0

1.2 r/r0.5

1.4

1.6

1.8

Figure 6.6 Dimensionless Velocity Profiles in the Jet

64

2.0

2.2

2.4

0.6 x = 75 mm

x = 150 mm

(u/U) r/R

0.4 2

x = 300 mm

0.2

0

0

2

r/R

4

Figure 6.7 Momentum Flux in the Jet A check on momentum conservation may be made by application of Equation (6-10). 2

r  u r are drawn as functions of for each of the sets On Figure 6.7 the curves of    U R R of radial traverses. The areas under these curves represent the integrals ∞

2

⌠ u   r   r     d  ⌡ U   R   R  0

and so are a measure of momentum flux. The measured areas lead to the results of Table 6.4. The values do not remain constant at 1.0 as expected, but rise significantly as the jet develops. There can be no doubt that the momentum flux does not increase since there is no force acting in the direction of the jet, so the apparent rise must be due to 65

experimental error. The most likely source is turbulence which could have the effect of giving an excessive mean velocity pressure.

x (mm)

x/R

⌠ u 2  r  d  r  2       ⌡ U   R   R 

75 150 300

2.91 5.81 11.6

1.01 1.10 1.15

Table 6.4 Momentum Flux in Jet

Conclusion

The diffusion of a turbulent air jet into the surrounding atmosphere has been measured by velocity traverses along the centreline and along several radii. The first part of the jet is found to have a central core of almost constant velocity which extends for a length xc = 6.8R along the axis. Thereafter the centreline velocity reduces and the velocity profile rapidly tends to similarity, that is to a profile which may be characterised by the single parameter r/x. The momentum flux in the jet, which must be constant in a constant-pressure atmosphere, appears to rise by approximately 15% along its length. The discrepancy is attributed to measurement error due to turbulence.

66

Suggestions for Further Experiments

1.

Obtain the angle at which the jet spreads by establishing the trajectory along which u/uo = 0.5.

2.

Compare the variation of centreline velocity with Equation (6-2).

3.

Investigate the effect of initial turbulence in the jet by placing a wire gauze over the exit of the tube and comparing the results with those obtained with a plain exit.

67

68

7.

BOUNDARY LAYERS

AF14 Boundary Layer Apparatus

69

Introduction It is a fact well-established by experiment that when a fluid flows over a solid surface there is no slip at the surface. The fluid in immediate contact with a surface moves with it, and the relative velocity increases from zero at the surface to the velocity in the free stream through a layer of fluid which is called the boundary layer. Consider steady flow over a flat smooth plate as shown in Figure 7.1, where the streaming velocity U is constant over the length of the plate. It is found that the thickness of the boundary layer grows along the length of the plate as indicated in the diagram. The motion in the boundary layer is laminar at the beginning, but if the plate is sufficiently long, a transition to turbulence is observed. This transition is produced by small disturbances which, beyond a certain distance, grow rapidly and merge to produce the apparently random fluctuations of velocity which are characteristic of turbulent motion. The parameter which characterises the position of the transition is the Reynolds number Rex based on distance x from the leading edge:

Re x

=

Ux ν (7-1)

Figure 7.1 General Characteristics of the Boundary Layer over a Flat Plate

70

The nature of the process of transition renders it prone to factors such as turbulence in the free stream and surface roughness of the boundary, so it is impossible to give a single value of Rex at which transition will occur, but it is usually found in the range 1 × 105 to 5 × 105.

Definitions of Thickness

A little consideration will show that the boundary layer thickness δ, shown in Figure 7.1 as the thickness where the velocity reaches the free stream value, is not an entirely satisfactory concept. The velocity in the boundary layer increases towards U in an asymptotic manner, so the distance y at which we might consider the velocity to have reached U will depend on the accuracy of measurement.

Figure 7.2 Velocity Distribution and Displacement Thickness of Boundary Layer

A much more useful concept of thickness by which fluid outside the layer is displaced away from the boundary by the existence of the layer, as indicated schematically in Figure 7.2, by the approaching streamline. In Figure 7.2 the curve 0A shows the distribution of velocity u within the layer as a function of distance y from the boundary. If there were no boundary layer, the free stream velocity U would persist right down to the boundary as shown by the line CA. The reduction in volume flow rate (per unit width normal to the diagram) due to the reduction of velocity in the layer is therefore h

∆Q = ∫ ( U − u) dy 0

(7-2)

71

which corresponds to the shaded area 0AC in the diagram, the dimension h being chosen so that u = U for any value of y greater than h. If the volume flow rate is now considered to be restored by displacement of the streamline at A’A away from the surface to a position B’B through a distance δ*, the volume flow rate between A’A and B’B is also ∆Q, and this is seen to be

∆Q = Uδ * (7-3)

Equating the results of Equation (7-2) and (7-3) gives h

∫ ( U − u) dy

1 U

δ* =

0

or h

u ⌠ δ * =   1 −  dy  U ⌡ 0

Now h is any arbitrary value which satisfies the condition u = U or 1−

u U

= 0

for all values of y greater than h. The value of h may therefore be increased indefinitely without affecting the value of the integral, so we allow h to increase towards infinity: h → ∞ and obtain the result ∞

⌠ u δ * = 1− dy ⌡ U 0

(7-4)

We shall see that in the practical measurement of δ* from a measured velocity distribution the infinite upper limit presents no difficulty. 72

A further definition is required when momentum effects within the boundary layer are considered. Consider a control volume of length δx, height h (greater than the boundary layer thickness δ) and unit thickness normal to the plane of the diagrams & at the left-hand end, and the rate of shown in Figure 7.3. The rate of mass inflow is m & dm & + mass outflow at the right-hand end is m δx . Consideration of continuity then dx & −dm δx . The momentum equation shows the outflow through the upper surface to be dx

may now be derived as follows.

Figure 7.3 Mass and Momentum Flux in Boundary Layer & from the control volume is The net rate of efflux of x-component of momentum M the sum of & & + dM δx at the right-hand end M dx

& at the left-hand end −M and −U

& dm δx at the upper surface dx

Note: Over the upper surface, the x-component of velocity is U, and the mass & −dm outflow rate is δx . dx

If the surface shear stress is τw acting in the direction shown in the diagram, the momentum equation is then

73

& & & + dM δx − M & − U dm − τ w δx = M δx dx dx which simplifies to τw

= U

& & dm dM − dx dx

or τw =

d (Um& − M& ) dx (7-5)

Now h

& = ρ ∫ u dy m 0

(7-6)

and h

& = ρ ∫ u 2 dy M 0

(7-7)

so substituting these results into Equation (7-5) gives

τw

d = ρ dx

h  2  ∫ Uu − u dy  0 

(

)

or τw

d = ρU 2 dx

  h  ⌠ u  1 − u  dy  U  ⌡U   0

Since u = U for all values of y greater than h, the arbitrary upper limit may be replaced by infinity, giving  ∞ u  2 d ⌠ u  τ w = ρU  1 −  dy dx   U  ⌡U   0 (7-8) It is convenient to express τw in the dimensionless form of a local skin friction coefficient

74

cf

τw 1 ρU 2 2

=

(7-9)

and if this is done, Equation (7-8) becomes ∞  d ⌠ u  u  cf = 2  1 − dy  dx ⌡ U  U   0  (7-10)

The writing of this result is simplified if we now define ∞

u ⌠ u Θ =  1 − dy ⌡ U U 0

(7-11)

where Θ is known as the momentum thickness of the boundary layer, and Equation (7-10) becomes cf

= 2

dΘ dx (7-12)

The total skin friction force per unit width on a plate of length L is L

Df =

∫τ

w

dx

0

(7-13)

Writing τw in terms of cf from Equation (7-9) L

Df

=

1

2 ρU

2

∫c o

and from Equation (7-12)

75

f

dx

L

Df

=

1

⌠ dΘdx 2 ρU × 2  ⌡ dx 2

0

When x = 0, Θ = 0, and writing ΘL for the momentum thickness at distance L from the leading edge,

Df =

1

2 ρU

2

× 2Θ L (7-14)

The skin friction force Df is now written in terms of a dimensionless overall skin friction coefficient CF where Df 1 ρ U2 L 2

CF =

and substituting Df from Equation (7-14) gives CF

=

2Θ L L (7-15)

This equation gives the overall skin friction coefficient on a flat plate very simply in terms of the momentum thickness at the trailing edge and the length of the plate. It is frequently useful to refer to the ratio of displacement thickness δ* to momentum thickness Θ, and this is called the shape factor H: H =

δ* Θ (7-16)

The calculation of the velocity profiles and thickness of boundary layers is beyond the scope of this manual, but for reference and for comparison with results of experiments a few results are presented here. For a laminar boundary layer along a flat plate with uniform free stream velocity, the velocity profile has been calculated and some numerical results are presented in Table 7.1

76

y Re x x

u/U

0

0.5

1.0

1.5

2.0

3.0

0 0.1659 0.3298 0.4868 0.6298

0.8461

4.0

5.0

6.0

0.9555 0.9916 0.9990

Table 7.1 Velocity distribution in laminar boundary layer along flat plate

Note the dimensionless parameter y Re x x used in the table, which generalises the results to any value of distance x along the plate. For this laminar layer, the displacement thickness δ* and the momentum thickness Θ are given by 1721x . Re x

δ* =

(7-17)

and Θ =

0.664 x Re x

(7-18)

from which it may be noted that the thickness along the plate grows in proportion to x . The shape factor is H = 2.59 (7-19)

For a turbulent boundary layer along a smooth flat plate there are no corresponding calculated results. Frequently the velocity distribution is expressed in the form u U

1

 y n =    δ (7-20)

where n is an index which varies from about 5 to 8 as the value of Rex increases in the range of 105 to 109, although there are many alternative expressions. The displacement and momentum thickness are frequently quoted as δ* =

0.046 x Re x 0.2

(7-21) 77

Θ =

0.036 x Re x 0.2

(7-22)

with the shape factor H = 1.29 (7-23)

The Effect of Pressure Gradient

The preceding discussion has related to boundary layer development along a smooth plate with uniform flow in the free stream - in conditions of zero pressure gradient along the plate. If the stream is accelerating or decelerating, substantial changes take place in the boundary layer development. For an accelerating free stream, the pressure falls in the direction of flow, the pressure gradient being given by differentiating Bernoulli’s equation in the free stream as dp dx

= − ρU

dU dx (7-24)

The boundary layer grows less rapidly than in zero pressure gradient and transition to turbulence is inhibited. For a decelerating free stream, the reverse effects are observed. The boundary layer grows more rapidly and the shape factor increases in the downstream direction. The pressure rises in the direction of flow, and this pressure rise tends to retard the fluid in the boundary layer more severely than that in the main stream since it is moving slower. Energy diffuses from the free stream through the outer part of the boundary layer down towards the surface to maintain the forward movement against the rising pressure. However, if the pressure gradient is sufficiently steep, this diffusion will be insufficient to sustain the forward movement, and the flow along the surface will reverse, forcing the main stream to separate. It is this separation, or stall as it is sometimes called, which leads to the main component of drag on bluff bodies and to the collapse of the lift force on an aerofoil when the angle of incidence is excessive.

78

Description of Apparatus

Figure 7.4 shows the arrangement of the test section attached to the outlet of the contraction of the airflow bench. A flat plate is placed at mid height in the section, with a sharpened edge facing the oncoming flow. One side of the plate is smooth and the other is rough so that by turning the plate over, results may be obtained on both types of surface.

Figure 7.4 Arrangement of Test Section

A fine Pitot tube may be traversed through the boundary layer at a section near the downstream edge of the plate. This tube is a delicate instrument which must be

79

handled with extreme care to avoid damage. The end of the tube is flattened so that it presents a narrow slit opening to the flow. The traversing mechanism is spring loaded to prevent backlash and a micrometer reading is used to indicate the displacement of the Pitot tube. Liners may be placed on the walls of the working section so that either a generally accelerating or generally decelerating free stream may be produced along the length of the plate, depending on which way round they are fitted. With the liners removed, uniform free-stream flow conditions are obtained over the plate length. To obtain a boundary layer velocity profile, the Pitot tube is set approximately 10 mm from the surface and the desired wind speed is established by bringing the pressure Po in the airbox to the required value. Readings of total pressure P measured by the Pitot tube are then recorded over a range of settings of the micrometer as the tube is traversed towards the plate. At first the readings should be constant, indicating that the traverse has started in the free stream; if this is not the case, go back and start with an initial setting further from the plate. As the Pitot tube reading begins to fall, the step length of the traverse should be reduced so that at least 10 readings are obtained over the range of reducing readings. The reading does not fall to zero as the tube touches the wall because of its finite thickness, so the traverse is stopped as soon as contact is indicated either by the electrical circuit or by the readings becoming constant as the micrometer is advanced towards the surface. Readings obtained in turbulent boundary layers are subject to unsteadiness which leads to difficulty in obtaining average readings on the manometer. Damping may be provided by squeezing the connecting plastic tube, but care should be taken that the restriction is not too severe, which can lead to false readings.

Results and Calculations (a)

Turbulent Boundary Layers on Smooth and Rough Surfaces

The plate was installed in the test section without the liners fitted, and measurements were made in the boundary layer formed on the smooth surface and then on the rough surface. Air temperature

19°C

=

80

292 K

Barometric pressure Air density

ρ

1010 mb

1.010 × 105 N/m2

=

1010 . × 105 = 287.2 × 292

=

Coefficient of viscosity µ Coefficient of kinematic viscosity ν =

µ ρ

1.204 kg/m3

=

1.80 × 10−5 kg/m s

=

1.49 × 10−5 m2/s

Length of plate from leading edge to traverse section, L = 0.265 m Thickness of Pitot tube at tip, 2t = 0.40 mm ∴Displacement of tube centre from surface when in contact, t = 0.20 mm Pressure in airbox: 640 N/m2 Readings of Pitot pressure P are tabulated in Table 7.2 and Table 7.3. Values of y shown in the tables are obtained from the micrometer reading at which the tube just touched the surface, making allowance for the initial displacement t due to the thickness of the Pitot tube. Values of u/U are found from u U

=

P Po

where Po is the Pitot tube reading in the free stream. The free stream velocity U is obtained from: 1

2 ρU

∴U =

Re =

UL ν

=

2

= 550 N m2 2 × 550 1204 .

= 30.2 m s

30.2 × 0.265 × 105 . 149

81

= 5.37 × 105

Micrometer Reading (mm)

y (mm)

P (N/m2)

u/U

21.0 20.0 19.0 18.0 17.0 16.5 16.0 15.8 15.6 15.4 15.2 15.14

6.06 5.06 4.06 3.06 2.06 1.56 1.06 0.86 0.66 0.46 0.26 0.20

550 555 550 530 495 460 415 380 360 320 250 180

1.00 1.00 1.00 0.98 0.95 0.91 0.87 0.83 0.81 0.76 0.67 0.57

Table 7.2 Velocity Distribution in Boundary Layer on Smooth Flat Plate, Re = 5.37 × 105

Micrometer Reading (mm)

y (mm)

P (N/m2)

u/U

25.0 24.0 23.0 22.0 21.0 20.0 19.0 18.5 18.0 17.5 17.0 16.5 16.3 16.10

9.10 8.10 7.10 6.10 5.10 4.10 3.10 2.60 2.10 1.60 1.10 0.60 0.40 0.20

540 540 525 515 500 470 420 375 335 275 215 150 125 100

1.00 1.00 0.99 0.98 0.96 0.93 0.88 0.83 0.79 0.71 0.63 0.53 0.48 0.43

Table 7.3 Velocity Distribution in Boundary Layer on Rough Flat Plate, Re = 5.37 × 105

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Figure 7.5 shows the velocity distributions plotted for both smooth and flat plates. Also shown are the curves of u/U(1 − u/U), which are easily deduced from the curves of u/U by reference to Table 7.4:

Figure 7.5 Velocity Distribution in Turbulent Boundary Layer on Smooth Plate Re = 5.37 × 105 83

u/U

0.5

0.6

0.7

0.8

0.9

1.0

u u 1 −  U U

0.25

0.24

0.21

0.16

0.09

0

Table 7.4 Values of u/U (1 - u/U)

The appropriate areas under the curves, evaluated by use of Simpson’s rule, as described in Appendix A, are: Smooth plate:

Rough plate:

δ*

=

u ⌠   1 − U  dy ⌡

= 0.53 mm

Θ

=

⌠ u  1 − u  dy    ⌡ U U

= 0.40 mm

∴H

=

δ* Θ

= 1.32

δ*

=

u ⌠   1 − U  dy ⌡

= 1.50 mm

Θ

=

⌠ u  1 − u  dy    ⌡ U U

= 0.98 mm

∴H

=

δ* Θ

= 1.53

In Figure 7.5 a 1/7th power law is shown, corresponding to u U

1

 y 7 =    3

and this is seen to compare reasonably well with the experimental results on the smooth plate. For the rough plate, however, the velocity distribution does not fall towards zero at y = 0. This is because the origin of the traverse has been taken from the highest points of the rough surface. An examination of the structure of the roughness would be required to establish the position of the mean surface from which y should be measured.

84

The values of δ* and Θ calculated for a turbulent boundary layer along a smooth surface from Equations (7-21) and (7-22) are, with the length L = 0.265 m inserted. δ* =

and Θ =

0.046 × 0.265

(5.37 × 10 )

5 0.2

0.036 × 0.265

(5.37 × 10 )

5 0.2

= 0.85 × 10−3 m = 0.87 mm

= 0.68 × 10−3 m = 0.68 mm

The experimental results for the smooth plate are noticeably lower than these values, indicating that over part of the length of the surface the boundary layer is laminar, yielding an overall skin friction less than if the whole length of the layer was turbulent. For the rough plate the boundary layer thickness is more than twice that of the smooth plate. Also it is more than the calculated values for a smooth surface, showing that the roughness has produced a significant increase in skin friction drag to a value higher than could be obtained on a smooth plate even if the whole length of the boundary layer was turbulent.

(b) Effect of Pressure Gradient

The test was repeated with the liners fitted to give a generally decelerating flow over the plate length. The Reynolds number based on the main stream velocity at the exit and the length of the plate was Re = 4.90 × 105 which is not sufficiently different from the previous value to affect the results. The procedure is the same as before, so full details of the working are not presented. Figure 7.6 shows the measured velocity profile in comparison with the previous results, from which it is clear that the layer has grown appreciably thicker in the rising pressure which is produced by the decelerating flow.

85

Figure 7.6 Effect of Pressure Gradient on Boundary Layer on Smooth Plate Re = 4.90 × 105

86

The thickness and shape factor are: u ⌠ δ * =   1 −  dy = 0.86  U ⌡ u ⌠ u Θ =   1 −  dy = 0.60 mm  U ⌡U H = 1.43

Summary and Conclusions

Velocity traverses in turbulent boundary layers have been made with a specially shaped fine Pitot tube traversed close to the surface. The resulting velocity profiles have shown that roughness of the plate surface, and a rising pressure gradient, both serve to increase the rate of growth of the boundary layer.

Suggestions for Further Experiments

1.

Investigate the effect of a falling pressure gradient on the boundary layer by repeating the tests with the liners reversed.

2.

It has been suggested that the velocity distribution in a turbulent boundary layer may be approximated by a logarithmic profile u U

 y = A + B log    δ

where A and B are constants. Check whether your results fit this expression. 3.

Observe the growth of boundary layer along a plate with a constant main stream velocity by making traverses at successive stations along the plate length. (The plate may be withdrawn from the working sections to a variety of positions allowing traverses to be made at different stations along it). Plot the growth of Θ along the plate and consider what information this gives about the skin friction coefficient cf.

87

4.

Consider the possibility of making measurements of laminar boundary layers in this apparatus. If the minimum main stream velocity for which reasonably accurate velocity traverses may be obtained is 10 m/s, and the laminar layer along the plate persists up to Re = 1 × 105, show that the layer will extend about 0.15 m from the leading edge. Show also that the displacement thickness at this section would be about 0.8 mm according to Equation (7-17).

88

8.

FLOW AROUND A BEND IN A DUCT

AF15 Flow Around a Bend Apparatus

89

Introduction The engineer is frequently presented with problems of flow contained within tubes and ducts. Such flows may be classified as internal flows to distinguish them from flows over bodies such as aerofoils, called external flows. It is sometimes necessary to shape a duct in such a way that particular requirements are met. For example, it may be necessary to change the shape of cross-section from square to rectangular with a small loss of total pressure, or it may be required to form a bend in such a way that the distribution of velocity at the exit is as nearly uniform as it can be made. Due to the presence of boundary layers along the duct walls, the fluid mechanics of such flows are sometimes extremely complicated. Separation may be produced where the pressure rises in the direction of flow, as illustrated in Figure 8.1(a).

Figure 8.1(a) Schematic Representation of Separating and Reattaching Flow in a Duct This shows a duct of increasing cross-sectional area in which the flow decelerates with an accompanying rise of pressure. Separation of flow from one wall is shown, followed by a region of severe turbulence in which there is mixing between the main flow and the region of recirculating flow (often called the separation bubble). The turbulent mixing leads to loss of total pressure, the size of this loss depending on the

90

extent of the separation. It should be emphasised that the flow shown in the figure is schematic only. The separation line is rarely steady. The size of the separated zone often fluctuates violently, and in some cases the separation is intermittent. Separation might occur over more than one surface and would not normally take place uniformly over one side as shown for illustrative purposes in the diagram. A further complication arises from secondary flow which is again due to boundary layer effects.

Figure 8.1(b) Formation of Secondary Flow in a Bend of a Duct Figure 8.1(b) shows one example of the formation of a secondary flow in a gentlycurving duct of rectangular cross-section. The curvature of the flow is accompanied by a pressure gradient which rises across the section from the inner to the outer wall. The pressure gradient extends over the whole section, so that the boundary layers on the upper and lower walls are subjected to the same pressure gradient as the main flow. But because the streaming velocity in the boundary layer is less than in the main part of the flow, the curvature of the streamlines in the boundary layer is more severe, as indicated. This gives rise to a net inward-directed flow adjacent to the upper and lower walls, which sets up a secondary flow in the form of a double rotation, superimposed on the main stream. The motion emerging from the curve in the duct is therefore a pair of contra-rotating spirals, the strength of which depends on the amount of curvature and on the thickness of the boundary layer. 91

Simple Theory of Flow in a Bend

Figure 8.2 Assumed Velocity Distribution in a Bend In this experiment we investigate the flow around a 90° bend in a duct of rectangular section, using pressure tappings along the walls to establish the pressure distributions. Figure 8.2 indicates flow approaching a bend with a uniform velocity U. Within the bend we shall assume a free vortex distribution of velocity, given by

u =

C r (8-1)

where u is the streaming velocity at radius r from the centre of curvature of the bend. Separation and secondary flow will be neglected. The constant C may be found by applying the equation of continuity as follows: r2

∫

Q = Ub ( r2 − r1 ) = b u dr r1

(8-2)

where b is the width of the section of the duct. Substituting for u from Equation (8-1) and performing the integration leads to the result 92

C = U

r2 − r1 ln ( r2 / r1 ) (8-3)

so the velocity distribution is, in dimensionless form, u U

r2 − r1 r ln ( r2 r1 )

=

(8-4)

The corresponding pressure distribution may be found by assuming that Bernoulli’s equation may be applied between the upstream section and a section within the bend as follows:

po +

1

2 ρU

2

= p +

1

2 ρu

2

(8-5)

where po is the static pressure upstream and p is the pressure at radius r in the bend. It is convenient to express p in the form of a dimensionless pressure coefficient cp where cp

=

p − po 1 ρU 2 2 (8-6)

From Equation (8-5) this may be written cp

= 1−

u2 U2 (8-7)

which may be evaluated for any radius r by substituting the appropriate value of u/U obtained from Equation (8-4). A comparison with measured values of cp may be made as shown in Table 8.2.

Description of Apparatus

Figure 8.3 shows the dimensions of the bend and the positions of the pressure tappings. There is a reference pressure tapping 0 on the side face near the entry, and three sets of tappings; one set of 10 along the outer curved wall, one set of 10 along 93

the inner curved wall and a set of 9 along a radius of the bend. Air from the contraction section is blown along the duct and is exhausted to atmosphere.

Figure 8.3 Dimensions of the Bend and the Positions of Pressure Tappings

Experimental Procedure and Results

The pressure tappings along the outer wall, the reference tapping 0 and the pressure tapping in the airbox are all connected to the manometer. The air speed is adjusted to a value slightly below the maximum, as indicated by the airbox pressure, and the pressures are recorded. (The setting of air speed slightly below the maximum is to ensure that the same setting may be repeated in later tests). The tappings on the inner wall are then connected in place of the ones on the outer wall. The airbox pressure is adjusted to the previous value and a further set of readings are recorded. Finally the procedure is repeated with the third set of pressure tappings. In Table 8.1 the pressures p are recorded relative to an atmospheric datum and the pressure coefficients cp are calculated from Equation (8-6).

94

Airbox pressure

= P

= 630 N/m2

Reference tapping pressure

= po

= 80 N/m2

Tapping

Outer Wall

= 550 N/m2

1 ρU 2 2

Velocity pressure of uniform flow along duct P − po =

Inner Wall

Radial Section

No.

p (N/m2)

cp

p (N/m2)

cp

p (N/m2)

cp

1 2 3 4 5 6 7 8 9 10

90 145 205 325 330 320 240 85 35 0

0.02 0.12 0.23 0.45 0.45 0.44 0.29 0.01 –0.08 –0.15

70 40 –240 –415 –440 –395 –255 –25 10 0

–0.02 –0.07 –0.58 –0.90 –0.95 –0.86 –0.61 –0.19 –0.13 –0.15

–265 –150 –40 50 115 170 215 260 295

–0.63 –0.42 –0.22 –0.05 0.06 0.16 0.25 0.33 0.39

Table 8.1 Measured Pressures and Pressure Coefficients

From Figure 8.3 the inner and outer surfaces of the bend have radii r1 = 50 mm r2 = 100 mm From Equation (8-4) the velocity distribution across the section according to the free vortex assumption is therefore u U

=

50 r ln(2)

=

72.1 r

where r is expressed in mm. In Table 8.2 we calculate this ratio and the corresponding value of cp from Equation (8-7) for a number of values of r.

95

r (mm)

u U

cp

50 55 60 65 70 75 80 85 90 95 100

1.443 1.312 1.202 1.110 1.030 0.962 0.902 0.849 0.801 0.759 0.721

–1.081 –0.720 –0.445 –0.232 –0.062 0.075 0.187 0.280 0.358 0.423 0.480

Table 8.2 Calculated Pressure Coefficients

Figure 8.4 Distribution of Pressure Coefficient cp over Walls

Figure 8.4 shows the distribution of measured pressure coefficient over the curved walls and compares the measured and calculated values across the radial section. It may be seen that the pressure across the inlet section is nearly uniform. As the flow approaches the bend, the pressure on the inner wall falls rapidly and on the outer wall 96

rises rapidly to values which remain substantially constant round most of the curve. This indicates that the curvature of the flow is also likely to be reasonably constant. The distribution of cp over the radial section follows the calculated curve quite closely, indicating that the assumption of a free vortex velocity distribution made in Equation (8-1), together with the assumption that Bernoulli’s equation applies to the flow, give a fairly accurate distribution of the pressure field. The measured pressure distribution varies rather less steeply than calculated, indicating a vortex strength C somewhat less than that given by Equation (8-3). Downstream of the bend, the wall pressures readjust until at the duct exit the pressure is constant across the section. It is, however, a little lower than the reference pressure at the inlet, and this difference represents a pressure loss round the bend. It is convenient to express this loss ∆p in terms of the velocity pressure ½ρU2 in the uniform approaching flow by the expression K =

1

∆p 2 2 ρU (8-8)

where K is the dimensionless loss coefficient. In this case we find, from the change in cp from the inlet to the outlet sections, the value: K = 0.15 (8-9)

Conclusion

The distribution of pressure over the curved walls of a 90° bend of rectangular section has been established by pressure plotting. The pressure coefficient is negative and almost constant round the inner wall, and positive and almost constant round the outer wall. Across the 45° cross-section the pressure distribution may be predicted with reasonable accuracy by assuming free-vortex velocity distribution over the section. The value of loss coefficient K is 0.15 for this bend.

97

Questions for Further Discussion

1.

Do you consider that there is likely to be any separation of flow anywhere in the bend, and can you suggest any way by which this might be investigated?

2.

Do you consider that there might be any secondary flow in the stream, downstream of the bend, and can you suggest how this might be investigated?

3.

It has been proposed to measure flow rate Q in a duct system by placing pressure tappings on the inner and outer walls at the 45° section of any convenient 90° bend which occurs in the line of the duct, and measuring the differential pressure ∆p between the tappings. Using Equations (8-2) to (8-5) show that Q is given by

4. Q =

(r

b r1 r2

2

2

− r12

)

r  ln  2   r1 

2 ∆p ρ

where b, r1 and r2 are defined in Figure 8.2 Noting that the measured pressures do not quite agree with the theoretical values, this equation may be modified to Q = Cd

(r

b r1 r2

2

2

−

r12

)

r  ln  2   r1 

2 ∆p ρ

in which Cd is a discharge coefficient. Show that Cd is given by

Cd

=

[(r2 r1) − 1] [(r2 r1)2 − 1] ( r2 r1 ) ln ( r2 r1 )

1

2

1

2

ρU 2 ∆p

and hence find Cd from the experimental results (Cd = 1.06)

98

9.

JET ATTACHMENT

AF16 Jet Attachment Apparatus

99

Introduction In a previous experiment on the flow round a circular cylinder, the phenomenon of separation of flow from a surface had been observed. The present experiment deals with an effect which in some respects is the reverse, namely, the tendency for a plane jet to attach itself to an adjacent wall and to flow along it. Figure 9.1(a) shows a typical configuration. A plane jet emerges from the slit which discharges into the atmosphere alongside a wall. It is found to deflect sideways and to attach to the wall. If the wall curves as shown, it will follow the curve and so may suffer a considerable change of direction. Exploitation of this phenomenon was proposed in the 1930’s by Henri Coanda, who made several inventions which used this form of jet deflection; the phenomenon is therefore sometimes referred to as the “Coanda Effect”.

Figure 9.1 Mechanics of Jet Attachment to a Wall A descriptive explanation of why a jet should exhibit this behaviour can be made along the following lines. It is known that a jet emerging from a tube or slit will entrain fluid from the surroundings as it mixes into the ambient fluid; in most cases of engineering importance the mixing is turbulent and entrainment is much more intense than if the process were laminar. Consider now entrainment into the jet of Figure 9.1(a). In the first moment after starting, the jet is straight and entrainment takes place equally on both sides. The inflow from the surroundings into the jet is, however, restricted on one side due to the

100

presence of the wall and this restriction results in a reduction of pressure which bends the jet towards the wall. This in turn further restricts the supply of fluid to this side of the jet, causing further reduction of pressure and further jet curvature until very soon the jet moves over to the wall as shown in the diagram. In the final condition shown, there is a zone of separated flow in which recirculation takes place. The rate at which fluid is entrained into the jet is balanced by the return flow into the separation zone from the region of jet attachment. The pressure in the separation zone is approximately constant, and is lower than in the atmosphere on the opposite side of the jet. Figure 9.1(b) shows the effect of opening a hole from the atmosphere into the separation zone. Fluid flowing through the hole is entrained into the jet; the entrainment rate now balances the sum of recirculation and inflow through the hole. The pressure in the separation zone rises somewhat, the jet curvature is reduced and the separation zone lengthens. If the inflow rate from the atmosphere is sufficiently high, the entrainment rate may be insufficient to maintain the balance, and the jet will detach from the wall. Returning to Figure 9.1(a), we see that having attached to the wall, the flow tends to stick to it because separation would require a supply of fluid to the space between the surface and the separating jet. Separation is therefore unlikely to occur unless the curvature of the surface is unduly severe or if an adverse pressure gradient is encountered. The previous explanations have assumed two-dimensional flow. This assumption will be valid if the jet is very wide in the direction normal to the plane of the diagram in comparison to its width. If this is not so, however, flow from the atmosphere round the ends of the jet into the separation zone will have considerable effect. Because of the importance of these end effects, the ratio of jet width (in the direction normal to the diagram) to jet thickness has acquired a specific description, known as the ‘aspect ratio’. The principle of wall jet attachment has recently found an application in the technology of fluidics. Figure 9.2 shows a typical fluidic switch in which the supply jet S is directed to either outlet 01 or 02 depending on to which of the two walls the jet attaches. Suppose the outlet 01 is active. By introduction of fluid at control hole C1, the jet may be switched across to outlet 02, and it will remain there after the inflow at C1 has ceased. Further action at C1 will not have any effect; to switch the jet back to 01 requires a signal at C2.

101

The switch is a fluidic counterpart of an electronic ‘flip-flop’ and is called a fluidic flip-flop. Other switches or gates, such as AND and OR gates, may be constructed, and fluidic logic circuits may be developed by interconnection.

Figure 9.2 A Fluidic Flip-Flop In the experiments which follow, the attachment of a jet to a single adjacent wall is first studied, and the behaviour of a flip-flop is then observed.

Description of Apparatus and Procedure The essential features of the arrangement are shown in Figure 9.3. The equipment, which fixes to the outlet flange of the contraction section of the airflow bench, consists of a nozzle plate which houses a rectangular supply nozzle. The jet which emerges from this nozzle is contained between side plates that may be moved laterally so that the offset between the nozzle and the attachment wall be varied. The aspect ratio of the nozzle may be altered by removing one of the nozzle blocks and fitting a different sized block.

102

Figure 9.3 Details of Apparatus and Notation For the single-wall tests, the left-hand attachment wall is fitted between the side plates. It is mounted on a spindle which terminates at a control used to rotate the attachment to any desired angle. For tests on the flip-flop, a further attachment wall and a splitter block are added. The space between the nozzle block and the attachment wall may be left open to atmosphere or may be sealed by closing a flexible seal as indicated. The left-hand attachment wall is first fitted and the flexible seal attached. The second, larger control is used to lock the wall at any desired angle. The right-hand seal should be in its open position. The offset, dimensioned y in Figure 9.3, and the wall angle β are both set to zero. The wind speed is then brought up to a convenient value close to the maximum and is then held constant by maintaining a constant airbox pressure, throughout the tests. The wall angle β is now slowly increased until separation is observed at angle βs. This is most easily detected simply by holding the hand in the jet, some 150 mm downstream of the trailing edge of the wall so as not to interfere with the flow along it, and noting when the flow pattern suddenly changes. The change is usually audible. The process should be repeated once or twice to ensure that the value of βs is established to within about 1°. Then, starting with a detached jet and reducing the wall angle, the value at which reattachment occurs βr, is established. The same procedure is repeated at several different values of offset y in the range from approximately −4 mm to 20 mm. Intervals of 2 mm are recommended, but smaller steps may be required where large changes in βs or βr are seen to take place. The whole test may then be repeated with the flexible seal removed.

103

Proceed now to construct a flip-flop by inserting the second attachment wall, and fixing the flexible seals to both walls. Centralise the assembly on the centreline of the nozzle. Starting with parallel walls, both set at zero wall angle, slowly increase the wall angles until the jet is clearly attached to one or the other of the walls. Then lock the walls in position. Try to switch the jet by prising open the seal on the ‘attached’ side. If this does not produce the desired switching, increase the angles in small steps until it does. It should then be possible to demonstrate flip-flop action, switching the jet back and forth at will by briefly prising open the seals. Further increase the angles to discover the upper limit at which a satisfactory flip-flop action is possible. The central splitter may now be added. It will be observed that this enhances the bi-stability, as the range of wall angles which give satisfactory switching is considerably increased.

Results Nozzle width w Nozzle breadth b Aspect ratio Attachment wall length L Attachment wall radius R Length from nozzle to attachment wall x

= = = = = =

Flexible Seal Closed y mm βs (°) βr (°) 37 53 −4 39 58 −2 44 63 0 47 68 2 52 72 4 58 76 6 65 82 8 44 86 10 37 90 12 35 96 14 32 100 16 29 110 18 22 73 20

10 mm 100 mm 10 100 mm 9.5 mm 12.5 mm

Flexible Seal Open y mm βs (°) βr (°) 28 45 −4 36 52 −2 44 63 0 36 51 2 32 46 4 29 38 6 26 34 8 25 31 10 23 28 12 22 26 14 22 25 16 21 24 18 21 23 20

Table 9.1 Separation and Reattachment Angles at a Single Wall

104

Table 9.1 gives results obtained for separation and reattachment of the jet at a single wall, and the results are plotted on Figure 9.4. The following observations were made when two walls were used. Offset y to each wall = 13.5 mm Without splitter fitted: β = 15° β = 20° - 30° β = 35°

Jet attaches to one wall and will not separate when seal is lifted. Jet attaches to one wall and may be switched back and forth as a flip-flop. Jet attaches to one wall but when seal is fitted it moves to centre position instead of attaching to opposite wall.

With splitter fitted β = 15° - 80°

Jet attaches to one wall and emerges from the passage between the wall and splitter without spilling into the opposite passage. Switches as a flip-flop.

Discussion Figure 9.4 exhibits the way in which the jet may be deflected through very considerable angles, exceeding 90°, by the Coanda effect. When the wall projects into the jet, i.e. when y is negative, the behaviour is much the same with a seal fitted as when it is removed. There is some 17° hysteresis between detachment and reattachment in this range of y. As soon as the wall is moved out of the jet, however, the two conditions behave entirely differently. With the seal fitted, the separation angle and the reattachment angle both continue to grow until, at a value of y about 8 mm, the hysteresis range is suddenly increased by a sharp drop in the reattachment angle. There is some 50° of hysteresis at this condition. The separation angle continues to grow as y increases to 20 mm, reaching a maximum of about 110°. When the wall is unsealed, as y increases from zero, both separation and reattachment angles decrease steadily and the hysteresis also decreases, diminishing to approximately 3° when y reaches 20 mm.

105

Figure 9.4 Detachment and Reattachment for a Single Wall The behaviour with two attachment walls is in fair agreement with predictions that may be made from Figure 9.4. Consider the separation condition. At y = 13.5 mm (which is the value used in the experiment with two walls), Figure 9.4 shows that the jet will separate from the unsealed wall at βs = 26°. The experimental result observed when two walls were used is that for βs greater than 20°, the jet detaches when the seal is lifted. So separation occurs at a somewhat smaller angle when there is a further sealed wall on the other side of the jet. Again, at y = 13.5 mm, Figure 9.4 shows that the jet will reattach to a sealed wall at βr = 35°. This agrees exactly with the observed behaviour of flip-flop action up to 30°, but for β = 35° or more, the jet moves to the centre position when the seal is lifted instead of attaching to the opposite wall.

106

The presence of a splitter improves the bi-stability. It generates a recirculation, as indicated in Figure 9.2, which strengthens the attachment very considerably, thereby increasing the range of successful flip-flop action.

Conclusion Tests on a single-wall configuration have shown that the wall attachment effect, or Coanda effect, may be used to divert a plane jet along a plane wall through angles up to 90° or more. Considerable hysteresis is found between the condition for jet separation and that for jet attachment. The effect of offset between the jet and the wall, and venting the space between the nozzle and the wall, have been investigated. A fluidic flip-flop which exploits the phenomenon of wall attachment has been constructed and satisfactory switching has been observed.

Suggestions for Further Experiments 1.

Investigate the effect of changing the aspect ratio of the jet on the single-wall detachment and reattachment characteristics. Change the nozzle width to, say, 8 mm and 5 mm in turn.

2.

Determine the significance of the side plates on the single-wall results. Blank off, 20 mm at each end of the breadth of the nozzle with adhesive tape, leaving 40 mm clear in the middle, so that the jet is not in contact with the side plates when it emerges from the nozzle, and repeat the tests.

3.

Consider the possibility of an asymmetric fluidic switch. Suppose it were desired to produce a switch which stayed ‘on’ until a control signal switched it ‘off’. Could this be done and if so, can such a switch be made from the parts described here?

107

108

10.

FLOW VISUALISATION

AF17 Flow Visualisation Apparatus

109

Introduction In experiments with fluid mechanics, it is often necessary to study the nature of the motion by direct observation of part or of the whole of the flow pattern. Important features may be observed, such as regions of steady or unsteady flow, thickening boundary layers and separation, secondary flows and so on. Flow visualisation can show characteristics of the motion, which might previously have been totally unexpected. It frequently explains phenomena which may otherwise defy explanation. It sometimes provides the starting point for new theoretical or analytical studies. A good historical example is provided by Osborne Reynolds’ visualisation of laminar and turbulent flows in glass tubes, which was the starting point for a rational understanding of the resistance to flow experienced by a fluid as it moves along a pipe. There are many techniques for visualisation. In water, dye filaments as used by Osborne Reynolds are still used. A more recent innovation is the use of tiny bubbles of hydrogen produced by an electrode which sheds a sheet of bubbles, or which may be arranged to shed discrete streams of bubbles which behave like dye filaments. In air, the most common practice is to use smoke injected through a tube or a row of tubes, usually called a rake. The smoke must be of nearly neutral density so that it does not rise or fall through the flow due to the effect of gravity and needs to be quite dense if the traces are to be observed for more than a short length downstream of the injector tube.

Description of Apparatus Figure 10.1 shows the main parts of the flow visualisation module. Smoke from the generator passes through flexible tubing into a streamlined manifold that spans the duct at the inlet to the working section. This duct contracts from a settling chamber, which contains honeycomb and gauzes to reduce turbulence, to the working section. Smoke emitted from the rake of tubes (smoke comb) accelerates with the airflow along the inlet duct into the working section and the filaments may be observed through the clear plastic front surface against a matt black background. A strong, diffused light from either side can help to give good contrast. The apparatus exhausts through the outlet in the bench top; a flexible air hose should be fitted to conduct the exhaust to a suitable ventilator, otherwise the smoke will easily fill a laboratory in minutes. The smoke is harmless, but you must ventilate it away for safety and common sense. 110

Figure 10.1 Flow Visualisation Apparatus The individual filaments retain their separate laminar identities for the whole length of the working section for speeds up to approximately 1 m/s. It is important to adjust the injection rate so that the velocity of injection matches the air velocity past the injection tubes, otherwise premature turbulence in the smoke filaments is likely to occur. It is recommended that students gain a little experience in setting the air velocity and smoke injection velocity with the working section clear of any models, before proceeding to visualisation of flow around various bodies.

Operation Important – make sure that you use this equipment in a well-ventilated area and the wind tunnel outlet is directed to a suitable air extractor. Make sure the wind tunnel fan is off. Fit your chosen model into the working section. Put the smoke generator onto the wind tunnel bench top as shown in the picture at the start of this section. Connect the outlet of the smoke generator to the smoke comb connector. Refer to Appendix B for full details on operation of the equipment.

111

Typical Results Various objects may be placed in the working section and the flow pattern observed. Three examples of the models supplied for use with the apparatus are described briefly here. Figure 10.2 shows the flow around a circular cylinder. The motion over the front part of the cylinder is steady as indicated by the almost unwavering smoke filaments. Separation occurs at around 80° from the front of the cylinder. A wake forms, which is shown to be unsteady by the mixing of the smoke. The unsteadiness is transmitted to the flow outside the wake. It is interesting to note that the pressure in the separated flow, as indicated by the surface pressure readings recorded in Figure 5.6, is almost constant.

Figure 10.2 Flow around a Circular Cylinder

112

Figure 10.3a Flow over an Aerofoil Small Incidence

Figure 10.3b Flow over an Aerofoil Large Incidence

Figure 10.3a shows flow over an aerofoil at a small incidence. The flow remains attached to the surface over almost the whole chord; this represents the normal or ‘unstalled’ condition, at which useful lift is generated, while the drag is comparatively small. The lift is due to the difference in pressures on the upper and lower surfaces of the aerofoil. Over the upper surface, convergence of the smoke traces indicate an acceleration of the flow, particularly over the first quarter of the chord, and this is accompanied by a fall of pressure which contributes much of the total lift on the aerofoil.

113

Figure 10.3b shows what happens if the angle of incidence is increased too much. The flow no longer sticks to the upper surface but separates, causing stall. The lift is reduced considerably, and the drag increases as a consequence of the wider wake which results from the separation.

Figure 10.4 Flow through a Sharp-Edged Slit

Figure 10.4. shows an example of flow through a sharp-edged slit. The contraction of a vena contracta at approximately one half-slit width downstream of the edges can readily be seen.

114

Further Observations 1.

Extend the observations to flow around a flat plate placed across the stream, flow along a long straight, flow through a round-edged slit, flow through a convergent-divergent pipe, and other cases which come to mind. Sketch the flow patterns and identify unsteady zones, separation points and so on.

2.

The photographs of Figures 10.2 to 10.4 were taken with the smoke filaments running in the mid plane between the front and rear walls of the working section. Repeat these tests and observe what happens when the smoke is directed close to the front wall. If different patterns are now observed, what inferences may be drawn? Do you consider that secondary flows are present, and if so, how do they arise?

115

116

11.

AEROFOIL WITH TAPPINGS (AF18)

Note: Professor Markland did not create this section of the manual. TecQuipment created this section to complement Professor Markland’s work.

The Aerofoil with Tappings (AF18)

117

Introduction The Aerofoil with Tappings fits onto the AF10 and shows the pressure distribution around a symmetrical NACA0020 aerofoil and the characteristics of lift. It is a small-scale model wing, but the results from its tests can be scaled up to compare with larger aerofoils. Description A clear-sided duct contains the aerofoil in ‘closed ends’ arrangement, so airflow is only across the curved surfaces of the aerofoil, and not around its ends (or wing-tips). This stops any wing-tip vortices or drag caused by the wing tips. Therefore, the aerofoil shows two-dimensional flow. This arrangement also gives the aerofoil a theoretical unlimited span or ‘infinite span’. The duct holds the aerofoil in a vertical position, relative to the ground. Pressure tappings are along the top and bottom surfaces of the aerofoil. They connect to a set of numbered, small pipe connectors on a plate next to the aerofoil. A set of larger bore pipes (supplied) connects the numbered pipe connectors to the AF10A Inclinable Multi-tube Manometer or other suitable manometer. Just above the aerofoil at the inlet to the duct is an extra pressure tapping for measurement of the static pressure upstream of the aerofoil, and for use with the tapping on the AF10 to calculate air velocity upstream of the aerofoil. A control on the front of the unit allows the user to adjust the aerofoil’s ‘angle of incidence’ (also known as the ‘angle of attack’). A scale shows the angle, relative to the airflow from the AF10.

Figure 11a. Angle of Incidence

118

Tapping Number

Tapping Position From the leading edge (mm)

1

2.2

2

3.9

3

6.1

4

8.7

5

11.8

6

14.8

7

20.0

8

25.6

9

31.4

10

37.3

11

43.4

12

49.5 Table 11a. Tapping Positions

Table 11a shows the positions of the tappings along the surface of the aerofoil. Note that tappings 1, 3, 5, 7, 9 and 11 are all on one side of the aerofoil. Tappings 2, 4, 6, 8, 10 and 12 are all on the other. Table 11b shows the details of the aerofoil.

Figure 11b. The Aerofoil Aerofoil Type

NACA0020 (symmetrical)

Aerofoil Chord

63 mm

Aerofoil Wing Span

49 mm

Effective surface area

0.0031 m2

Table 11b. Aerofoil Details

119

Theory

Figure 11c. Pressure around a symmetrical Aerofoil

120

In aerodynamics and fluid mechanics, an object in a flow produces lift when the pressure on one side of the object is greater than the pressure on the other. A symmetrical aerofoil in an airflow produces no lift when its angle of incidence is zero, as the pressures above and below the aerofoil are equal. As the angle of incidence increases positively from 0, the aerofoil starts to produce lift. Pressure decreases above the main part of the aerofoil as the incidence angle increases. The aerofoil shape and the airflow velocity determine the pressure distribution above and below the aerofoil. When the incidence angle reaches a certain point (the stall angle), the airflow over the upper surfaces separate from the wing. The pressure distribution changes and the centre of lift moves its position. The aerodynamic centre of the aerofoil moves backwards. These factors ’stall’ the aerofoil, so the lift reduces and drag increases. Engineers use lift curves to show the performance of an aerofoil (see Figure 11d). They test the aerofoil’s lift at a range of incidence angles for any given air velocity, until it stalls. This shows them the operating area for the aerofoil at that velocity. The angle (slope) of an ideal lift curve from theoretical calculation is 2π CL/radian, but real results are never be greater than the absolute maximum ideal value of 5.7 CL/radian.

Figure 11d. Lift against Incidence Angle for a Symmetrical Aerofoil You can measure the performance of a wing in two ways: 1. By direct measurement - with force sensors on the aerofoil. 2. By measurement of the pressures on the surface of the aerofoil, relative to local air pressure. The AF18 allows students to measure the pressures on the aerofoil surface. The results can be shown as readings of pressures against and tapping positions. For comparison with other aerofoils of different sizes you must convert your readings into dimensionless values. These are the pressure coefficient and the chord ratio.

121

Cp =

p − p∞ 1 ρV∞2 2

Where: Cp = Pressure Coefficient (dimensionless) p = Pressure at a given point (tapping) (Pa or N.m-2) p∞ = Static Pressure (Pa or N.m-2) ρ = Density (kg.m-3) V∞ = Free Stream Velocity (m.s-1) The chord ratio (x/c) is simply the position (x) of the tapping with respect to the chord (c) of the aerofoil, measured from the front of the aerofoil and when both measurement units are the same (millimetres or metres). Therefore, x/c is dimensionless.

122

Experiment Procedure

Figure 11e. Pressure Pipe Connections Fit the AF18 to the AF10 as shown in the picture at the start of this section. It has locating holes, so you cannot make a mistake. Use the pipes (supplied) to connect the aerofoil tappings and the static pressure tapping to a multi-tube manometer. So that you will understand and clearly see what is happening, connect tappings 1, 3, 5, 7, 9 and 11 to the first set of manometer tubes, and tappings 2, 4, 6, 8, 10 and 12 to the next set of tubes. Leave one manometer tube and the common manometer connection open to atmosphere, so that all pressures are shown with respect to atmosphere.

123

A small locking screw at the back of the AF18 holds the aerofoil in position. Loosen the locking screw and set the aerofoil to 0 (zero) angle. Tighten the locking screw. a) Connect the static pressure tapping at the top of the wind tunnel to the last manometer tube and measure the pressure (with respect to the manometer tube that measures atmospheric pressure). The diagram shows this as a dotted line. b) Move this manometer connection down to the inlet static pressure tapping just above the aerofoil and measure the pressure (with respect to the manometer tube that measures atmospheric pressure). c) Record all the manometer readings for the tappings of the aerofoil (with respect to atmosphere). Increase the angle of incidence to be more positive in steps of 5 degrees (clockwise), up to 25 degrees. At every five degree angle, repeat steps a, b and c. Also, take extra measurements at 17.5 and 22.5 degrees. Convert all your pressure readings into Pascals (Pa). Results Analysis The pressure distribution around an aerofoil helps you to understand its lift

characteristics. As shown in the theory, it is normal to show the pressure distribution in a non-dimensional way, so that you can compare it with other aerofoils of different size and at different air velocity. So you must convert all your pressure readings into pressure coefficient (CP) values as shown in the theory. However, you need the free stream velocity to find the pressure coefficient. The wind tunnel airbox pressure (pairbox) and duct inlet pressure will not give you the free stream velocity at the aerofoil. Because the aerofoil is in a duct, and the boundary layer thickness increases, this gives a smaller effective area around the aerofoil. The free stream velocity at the aerofoil is higher than it is at the inlet. To allow for this you must find the ‘effective static pressure’ (peff), around the aerofoil, and then use this to find the correct free stream velocity around the aerofoil. To find the effective static pressure, you must interpolate between the duct inlet pressure (before the aerofoil) and atmospheric pressure (after the aerofoil). The duct inlet tapping is 135 mm upstream of the exit of the duct. The centre of the aerofoil is 85 mm downstream from this tapping. So: peff = po + 85/135 x (pa – po) Where po = pressure at the duct inlet and pa = atmospheric pressure

124

For each angle, use peff to find the correct free stream velocity with the equation: V∞ =

2 × ( pairbox − peff )

ρ

For each tapping point on the aerofoil, find the pressure coefficient with the equation: Cp =

p n − p eff 1 ρV 2 2 ∞

Where pn = pressure at the tapping. The aerofoil is in a vertical position, but to understand the results more easily, you must consider pressure tappings 1, 3, 5, 7, 9 and 11 to be on the lower surface of the aerofoil with respect to the airflow, and tappings 2, 4, 6, 8, 10 and 12 to be on the upper surface of the aerofoil. For each angle, plot the pressure coefficient against the chord ratio (mentioned in the theory). Draw best-fit lines for the pressures on the upper and lower surfaces. To compare your results with theory, extend your curves to the zero pressure coefficient line at a chord ratio of zero and 1. You should find that the pressure coefficient near to the leading edge is zero, because the aerofoil forces the air to stop moving at this (stagnation) point. The exact position of this stagnation point changes with incident angle. The area between your curves is the coefficient of lift. Use the trapezium rule or other suitable method to find this area for each angle. Plot the coefficient of lift and compare it to the lift curve shown in the theory. From your curve, find: The gradient of the lift curve The maximum lift coefficient The stall angle What do you notice about the pressure distribution at angles greater then the stall angle?

125

Typical Results Pressure Profile at 0 degrees

Coefficient of Pressure CP

-4 -3 -2 -1 0 1 0.0

0.2

0.4

X/C

0.6

0.8

1.0

0.8

1.0

0.8

1.0

Pressure Profile at 10 degrees

Coefficient of Pressure CP

-4 -3 -2 -1 0 1 0.0

0.2

0.4

X/C

0.6

Pressure Profile at 17.5 degrees

Coefficient of Pressure CP

-4 -3 -2 -1 0 1 0.0

0.2

0.4

X/C

126

0.6

Pressure Profile at 20 degrees

Coefficient of Pressure CP

-4 -3 -2 -1 0 1 0.0

0.2

0.4

X/C

0.6

0.8

1.0

0.8

1.0

0.8

1.0

Pressure Profile at 22.5 degrees

Coefficient of Pressure CP

-4 -3 -2 -1 0 1 0.0

0.2

0.4

X/C

0.6

Pressure Profile at 25 degrees

Coefficient of Pressure CP

-4 -3 -2 -1 0 1 0.0

0.2

0.4

X/C

127

0.6

Lift against Incidence Angle

Coefficient of lift ( CL )

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

5

10

15

20

25

Incidence Angle (Degrees)

Conclusions The results for the aerofoil show a stall angle of approximately 20 degrees and a

maximum lift coefficient of approximately 1.25. TecQuipment used the trapezium rule to measure the area between the curves, which gave a dimensionless measurement of lift coefficient. At angles greater than stall, the pressure ‘peak’ near the front of the aerofoil drops, as the air flow separates from the upper surface. The slope of the curve is approximately 0.08 CL/degree or 4.56 CL/radian.

128

APPENDIX A: DETERMINATION OF THE AREA BENEATH A CURVE We sometimes need to evaluate the area enclosed below a curve that has been established by experiment. Suppose that such a curve, as shown in Figure A.1, has ordinates y0, y1, y2….yn, spaced at equal intervals h over the range from zero to L in the x-direction. The true area under the curve is L

A =

∫ y dx o

We aim to approximate to this result in terms of the ordinates y0, y1, y2, ... yn, as measured from the curve.

Figure A.1

The Trapezoidal Rule provides the simplest approximation. Imagine the curve to be replaced by the straight lines shown dashed in Figure A.1. The area of the set of trapeziums produced by this replacement is At =

1 2

( y o + y1 ) h

+

1 2

( y1 + y 2 ) h

+ ..... 2 ( y n −1 + y n ) h 1

which reduces to At =

[

]

1 y 0 + 2 ( y1 + y 2 + ......+ y n −1 ) + y n L 2n

A-1

The trapezoidal rule introduces error, which is obviously the sum of the differences between the areas enclosed by the curve and by the trapezoids. A much smaller error is given by Simpson’s Rule. This gives the area enclosed under the curve between three successive points such as y0, y1, and y2 as* As =

1 ( y o + 4 y1 + y 2 ) h 3

Applying this rule repeatedly over the whole set of ordinates from yo to yn gives the result As =

1 1 1 ( y o + 4 y1 + y 2 ) h + ( y 2 + 4 y 3 + y 4 ) h +.... ( y n −2 + 4 y n −1 + y n ) h 3 3 3

which reduces to As =

[

]

1 y o + 4( y1 + y 3 + .... y n −1 ) + 2( y 2 + y 4 +.... y n −2 ) + y η L 3n

This repeated application of Simpson’s rule obviously involves an even number of intervals, i.e. n will be an even number. If, for some reason, it is necessary to divide the length L into an odd number of intervals, then Simpson’s rule may be used up to the penultimate interval, and the trapezoidal rule then used for the remaining last step. For example, when analysing Figure 5.7, we need the area π

A =

∫c

p

cos θ dθ

o

The mean of the two curves of Figure 5.7 is reproduced in Figure A.2, with the required area shown shaded. Note that part of the area, from about 35° to 90°, is to be reckoned negative. The following ordinates have been measured from the curve:

*

Simpson’s rule is exact for any cubic curve of the form y = a + bx + cx2 + dx3

So error arises only from term of 4th and higher orders of x. This is very much better than the trapezoidal rule, which is exact only to 1st order of x. A-2

θ (°)

0

20

cpcosθ

0.98

0.62

40

60

80

−0.18 –0.52 –0.17

100

120

140

160

180

0.16

0.47

0.72

0.88

0.92

Figure A.2

Since the number of steps is odd, Simpson’s rule is used in the range 0° to 160°, and the trapezoidal rule for the last step from 160° to 180°. For 0° to 160°: Simpson’s rule with n = 8, L = 160 = 160 (π/180) radians: As =

1  π  0.98 + 4(0.62 − 0.52 + 0.16 + 0.72) + 2( −0.18 − 0.17 + 0.47) + 0.88] × 160  [  180  3×8

As = 0.70 (Note the negative ordinates, and the conversion factor (π/180) from degrees to radians) For 160o to 180o: Trapezoidal rule with n = 1, L = 20 = 20(π/180) radians:

A-3

At

=

1  π  0.88 + 0.92] × 20   [  180  2 ×1

= 0.31

Total area A from 0 to 180 : A = As + At = 1.01 So

π

∫ c p cosθ dθ = 1.01

o

Figure A.3

As a further example, consider Figure 7.5, reproduced as Figure A.3, from which it is required to evaluate the integral: ∞

δ* =

∫ (1 − u U) dy o

A-4

This is represented by the shaded area. Note that (1 − u/U) is dimensionless, and y has dimensions of length, being measured in units of mm. So the shaded area will also appear in units of mm. Values of (1 – u/U) read from the curve are as follows: y (mm)

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

>5.0

(1−u/U) 1.00 0.25 0.15 0.10 0.06 0.04 0.03 0.02 0.01 0.01 0.00 0.00

Simpson’s rule with n = 10, L = 5.0 mm:

As =

1 1.00 + 4(0.25 + 0.10 + 0.04 + 0.02 + 0.01)    × 5.0 mm 3 × 10  + 2(0.15 + 0.06 + 0.03 + 0.01) + 0.00

A = 0.53 mm So ∞

δ* =

∫ (1 − u / U) dy o

5

=

∫ (1 − u / U) dy o

A-5

= 0.53 mm

A-6

APPENDIX B: INSTALLATION AND OPERATING INSTRUCTIONS FOR THE AF17 FLOW VISUALISATION APPARATUS

Installation

NOTE: For safety and local transport regulations, the gas cylinder is supplied empty. You must fill it with compressed carbon dioxide before use. 1.

Use the Airflow Bench in a well-ventilated area.

2.

Remove the standard contraction piece from the AF10 Airflow Bench

3.

Put the bottom (outlet ducting) of the smoke tunnel into the hole in the Airflow Bench.

4.

Use the four clips of the airbox to fix the smoke tunnel contraction piece and visualisation section in place.

5.

Attach the outlet ducting to the bottom of the visualisation section using the two clips.

6.

Attach the exhaust pipe to the bench outlet and connect it to a suitable laboratory ventilation system, or out to atmosphere. Do not vent directly to atmosphere in windy conditions, as this can seriously affect the airflow.

7.

Put the smoke generator in position on the bench top. Put the gas bottle in its holder to the back of the Airflow bench. Connect the regulated output of the gas bottle outlet to the smoke generator. Fit the metal rake adaptor to the flexible tube (supplied). Fit the other end of the flexible tube to the outlet of the smoke generator. Connect the rake adaptor to the smoke rake connection.

8.

Make sure that the smoke generator switch is off, and connect its mains lead to one of the power sockets on the Airflow Bench.

A-7

The Smoke Generator Principle of Operation

The smoke generator works by heating non-toxic food-grade oil and forcing it out of the smoke generator by means of compressed carbon dioxide gas. The oil vapour condenses and becomes ‘smoke’ when it meets normal cold air at its outlet. The ‘smoke’ is actually a suspension of oil drops in the air, rather than a product of combustion.

Operating the Smoke Generator

Read the smoke generator manufacturer’s instructions and make sure that the smoke generator has enough oil in its reservoir.

Isolator Valve

Delivery Pressure

Control Valve

Figure B.1

There are two pressure gauges on the compressed gas bottle. One shows the pressure available from the bottle, the other shows the delivery pressure after the control valve. There are two valves at the bottle. The isolator valve is on the top of the bottle to completely isolate the gas supply. The pressure regulator control valve is between the gauges and sets the outlet (delivery) pressure. Shut the control valve (turn clockwise). Fully open the isolator valve. Switch on the smoke generator. Its display will show numbers, one is the heater temperature and the other is the temperature that it needs to reach so that it works (approximately 314 degrees Celsius). Switch on the smoke button. Slowly open the control valve until the delivery pressure is at 1 bar. Switch off the smoke button. When the numbers in the display become the same, the ready lamp will go on. A-8

Operating the AF17

The experiment is supplied with four models: Aerofoil Flat plate Cylinder Sharp-edged orifice 1. 2. 3.

4.

Remove the back panel of the visualisation section and attach a model. Replace the back panel. Start the smoke generator as described earlier and press its smoke button. Open the valve on the airflow bench to approximately one quarter of fully open. If the air speed is too high, the smoke trails will be poor and the smoke may be forced back out of the smoke generator and into the room. After a few minutes smoke will begin to flow into the visualisation section.

Controlling the Smoke

1.

The airflow bench control sets the quality of the smoke trails. You cannot control the amount of smoke from the smoke generator. A low air flow rate gives the best results, but you must experiment to find the best setting for your model.

2.

The sharp-edged orifice obstructs the flow the most, so you must take more care to adjust the flow correctly for this model. Too much or too little air flow will force smoke to come out of the smoke generator outlet.

3.

If condensation happens inside the smoke rake, its holes may become blocked. To unblock the holes, switch off the smoke generator, remove the back of the visualisation section and use a clean cloth or tissue to wipe the smoke rake and absorb the oil.

A-9