Air Flow Bench

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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.

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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

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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 kilogram (kg)

length metre (m) time second (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

1 N = 1 kg m/s2

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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

density ρ 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

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

pressure p kg/m s2

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/ms}2_{, 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 lb foot ft second s

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These are related to SI units by the conversion factors

1 lb = 0.453 592 37 kg exactly

or 0.4536 kg for most purposes 1 ft = 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 = 1.488 N/m2 1 lbf/ft2 = 47.88 N/m2 1 lbf/in2 = 6895 N/m2

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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.

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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 across the section, and since no fluid crosses the walls of the streamtube,

δ &m

δ &m = ρuδA = constant along the infinitesimal streamtube

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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

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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. For a streamtube of finite cross-sectional area A the volume flow rate is:

Q = udA

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.

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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 dp ds s

+ δ along the element, and the cross-sectional area increases from A to A dA . The forces due to pressure on the element are:

ds s

+ δ

On the section at s: pA in the s-direction

On the section at s + δs: − + p   +  dp ds s A dA ds s δ δ in the s-direction

On the wall of the streamtube: the pressure varies from p to p dp ds s + δ ,

and the projected area of the wall on a plane normal to the s-direction is A dA ds s A +   δ  − , viz. dA

ds δ . So the component of force in the s-direction lies s between: pdA ds δ and ps dp ds s dA ds s + δ _ δ  

To first order of infinitesimal quantities this is:

pdA ds δ s

The net force in the s-direction due to pressure is thus:

pA p dp ds s A dA ds s p dA ds s − +  δ + δ  + δ

which reduces simply to

−Adp dsδ s

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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 −ρ δgA s

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

−ρgAdzδ ds s

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

δu = du dsδ s

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 ds s + δ

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

δt = 1 uδ s

The s-component of acceleration as is therefore

as = δ δ δ δ u t du ds s u s = 1

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or

as = udu

ds

Equating the mass-acceleration of the fluid to net force, the equation of motion in the s-direction is therefore ρ δA s udu δ ρ δ ds A dp ds s gA dz ds s = − −

which leads to the result

udu ds dp ds g dz ds + 1 + = 0 ρ (1-7)

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

1 2 2 u + +p gz = E ρ (1-8)

where E is a constant. Multiplication by ρ gives

1

2ρu2 + +p ρgz = P

(1-9)

and division by g gives

u g p g z H 2 2 +ρ + = (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

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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

2

ρu +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 1

2 2

ρu +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 = u a

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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

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T = t + 273.15

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

ρ = p t

287 2. ( +27315. )kg/m

3

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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 (N/m2) Temperature t (°C) Density ρ (kg/m3) 0.95 10 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 = ρ gh

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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

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2. THE AF10 AIRFLOW BENCH

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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

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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.

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3. THE AF10A INCLINABLE MULTITUBE MANOMETER

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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:

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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

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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.

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4. BERNOULLI’S EQUATION APPLIED TO A CONVERGENT-DIVERGENT PASSAGE

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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

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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.

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

P = 1 u +

2ρ 2 p

A

(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 _t

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(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 u A A t t = (4-3)

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

u u B B t t = (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 P p

t t

= 2 ( − )

ρ

(4-6)

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

u u P p P p t t = − − 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).

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Results

Air temperature 22°C = 295 K

Barometric pressure 1028 mb = 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.

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x (mm) Po (N/m2) P (N/m2) p (N/m2) B B t P p Pt pt − − 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 −130 −315 −540 −805 −945 −980 −980 −845 −630 −480 −355 −255 −170 −95 25 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 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− × 29 = 70 62 7 ( ) . So B B t = 44 = 62 7. 0 701.

and in diverging section, when x = 204 mm

B = 76− 76 44− ×100 = 190 59 2 ( ) . So B B t = 44 = 59 2. 0 744.

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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

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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

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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 convergent-divergent 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 TR

(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 kg m 1028 10 287 2 295 1213 5 3 . . .

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Pt −pt = 790 980+ = 1770N m 2 1 2 2 1213 1770 × . ×ut = u_t = 54 0. m s Also at = 14 287 2 295. × . × a_t = 344m s Ma u a t t t = = 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?

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5. DRAG MEASUREMENT ON CYLINDRICAL BODIES

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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

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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 = p_a −p_o

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

2ρU2 = 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 non-dimensionalised, and the following dimensionless terms are defined:

Pressure coefficient, c p U

p = ₁

2ρ 2

(5-2)

Skin friction coefficient, c

U f = τ ρ 1 2 2 (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. A component in the direction of U, called the drag force, of intensity D per unit length of cylinder.

2. A component normal to the direction of U, called the lift force, of intensity L per unit length of cylinder.

3. A moment about the origin C, called the pitching moment, of intensity M per unit length of cylinder.

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

Drag coefficient, C D U d D = ₁ 2ρ 2 (5-4) Lift coefficient, C L U d L = ₁ 2ρ 2 (5-5)

Pitching moment coefficient, C M U d

M = ₁

2ρ 2 2

(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 = pcosθ τ+ sinθ δs

and integrating this round the whole perimeter yields

(

)

D =

∫

pcosθ τ+ sinθ ds This may now be cast in dimensionless form:

D U d d p U U ds 1 2 2 12 2 12 2 1 ρ ρ θ τ ρ θ =  +      ⌠ ⌡  cos sin

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or

(

)

∫

C d c c D = p + f 1 cosθ sinθ ds (5-7) Similarly

(

)

∫

C d c c L = − p + f 1 sinθ cosθ 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

δs = Rδθ = d 2δθ

(5-9)

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

C d c d d D = ⌠ p _ _ ⌡  1 2 cosθ θ ∴ C_D = 1₂

∫

c_p 0 2 cosθ θ π d d From consideration of symmetry, we may therefore write

C_D =

∫

cπ_pcosθ θ 0 2 (5-10) C_L = C_M = 0 (5-11)

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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 = 2Sinθ

(5-12)

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

P_o = p_o +1 U = p_a +

2ρ 2 12ρu 2

(5-13) The gauge pressure p at A is thus

p = pa −po = 12ρU2 −12ρu 2

From Equation (5-12)

(

)

p = 1 U − Si

2ρ 2 1 4 n2θ

so the pressure coefficient, cp, is

c p U Sin p = ₁ = − 2 2 2 1 4 ρ 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.

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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:

At the upstream section 2h po

At the downstream section −2h pe

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:

∫

ρu dy

h h

2 −

and over the upstream section is

∫

ρU dy

h h

2 −

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

∫

2h p_o 2h p_e D u dy2 U dy2 h h h h − − = − − − ρ ρ (5-15)

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Rearranging and making non-dimensional gives the result: C D U d h d p p U d u U dy D o e h h = = − +  −      ⌠ ⌡  − 1 2 2 12 2 2 2 2 2 1 ρ ρ (5-16) The integral may also be made non-dimensional by the substitution

y = η h (5-17) so that 1 1 2 2 2 2 1 1 −       ⌠ ⌡  = ⌠_ − _ ⌡  − − u U dy h u U d h h η

and the final result is

C h d p p U h d u U d D = o e − +  −      ⌠ ⌡  − 2 2 1 1 2 2 2 2 1 1 ρ η (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.

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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.

Test model AF12 Main unit

Pitot tube assembly Quick release coupling

Clear plastic tube Stainless steel tube

To manometer AF12 AF10 Contraction section (Outlet Duct) Thumb Nut Balance Arm Locking Screw

Cylinder

Quick release coupling

Protractor (located at back of main unit) To manometer AF12 Main unit AF12 Balance Arm AF10 Contraction section (Outlet Duct) AF12 Test model

AF12 Main unit Quick release coupling AF10 Contraction section (Outlet Duct) 10 g Weight 100 g Weight Retaining Screw

Test Model Clamp Screws (x2) Test Model Thumbnuts (x2)

Drag Force Pressure Distribution

Wake Traverse

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.

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Po

po

p

Pe

p = 0, atmospheric datum_e Pitot tube for traverse in exit plane

AIR FLOW BENCH AF10

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.

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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.

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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 Dl (gmf) Po (mm H2O) po (mm H2O) Po−po = ½ρU2 (N/m2) 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 24°C = 297 K Barometric pressure 1040 mb = 1.04 × 105_N/m2 Air density ρ = p RT = 287.2 297 1.219 10 04 . 1 5 = × × kg/m3 Diameter of cylinder d = 12.5 mm = 0.0125 m Length of cylinder l = 48 mm = 0.048 m Half width of working section h = 50 mm = 0.050 m

h

d = 4

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

600 38 2 2 1 _U = Dl ρ 3 10 81 . 9 600 38 _× _× − _m2₌₆_.₂₁₃_×₁₀−4 _m2

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CD may now be found by substitution in Equation (5-4) C D U d D U dl D = ₁ = 2 2 12 2 1 ρ ρ 03 . 1 048 . 0 0125 . 0 10 213 . 6 4 = × × = −

Value of drag coefficient by direct weighing CD = 1.03

0 5 10 15 20 25 30 35 40 0 100 200 300 400 500 600 700

Dynamic Pressure 1/2ρU2_(N/m2₎

D

rag Force D

1 (gmf)

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

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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 (degree) p − p_o (N/m2) Cp = p po− U2 1 2ρ Cp Cosθ θ (degree) p − p_o (N/m2) Cp = p po− U2 1 2ρ Cp Cosθ 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 100 110 120 130 140 150 160 170 180 559.17 549.36 529.74 461.07 372.78 245.25 117.72 -19.62 -186.39 -353.16 -480.69 -598.41 -696.51 -755.37 -765.18 -716.13 -657.27 -627.84 -608.22 -618.03 -608.22 -608.22 -618.03 -618.03 -618.03 -618.03 -618.03 -608.22 0.94 0.92 0.89 0.77 0.63 0.41 0.20 -0.03 -0.31 -0.59 -0.81 -1.00 -1.17 -1.27 -1.28 -1.20 -1.10 -1.05 -1.02 -1.04 -1.02 -1.02 -1.04 -1.04 -1.04 -1.04 -1.04 -1.02 0.94 0.92 0.87 0.75 0.59 0.37 0.17 -0.03 -0.24 -0.42 -0.52 -0.58 -0.58 -0.54 -0.44 -0.31 -0.19 -0.09 0.00 0.18 0.35 0.51 0.67 0.79 0.90 0.97 1.02 1.02 0 −5 −10 −15 −20 −25 −30 −35 −40 −45 −50 −55 −60 −65 −70 −75 −80 −85 −90 −100 −110 −120 −130 −140 −150 −160 −170 −180 549.36 549.36 519.93 470.88 392.4 284.49 147.15 19.62 -137.34 -313.92 -451.26 -559.17 -676.89 -765.18 -794.61 -774.99 -735.75 -667.08 -637.65 -637.65 -627.84 -627.84 -627.84 -627.84 -637.65 -627.84 -608.22 -608.22 0.92 0.92 0.87 0.79 0.66 0.48 0.25 0.03 -0.23 -0.53 -0.76 -0.94 -1.13 -1.28 -1.33 -1.30 -1.23 -1.12 -1.07 -1.07 -1.05 -1.05 -1.05 -1.05 -1.07 -1.05 -1.02 -1.02 0.92 0.92 0.86 0.76 0.62 0.43 0.21 0.03 -0.18 -0.37 -0.49 -0.54 -0.57 -0.54 -0.46 -0.34 -0.21 -0.10 0.00 0.19 0.36 0.53 0.68 0.81 0.93 0.99 1.00 1.02 Table 5.2 Pressure Distribution around a Cylinder

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-3.00 -2.00 -1.00 0.00 1.00 0 20 40 60 80 100 120 140 160 180 θ (Degrees) Cp 0 to 180 0 to -180 Ideal Fluid Theory

Calculated from equation 5-14

Figure 5.6 Pressure Distribution around a Circular Cylinder

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 20 40 60 80 100 120 140 160 180 θ (Degrees) Cp cos θ 0 to180 0 to -180

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

beneath the mean curve is

∫

A c_p

o

= π cosθ θd

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 (mm) d y Pe−pe (mm H2O) U2             − × 1 2 1 1 2 U U U U y (mm) d y Pe−pe (mm H2O) U2             − × 1 2 1 1 2 U U U U 0 1 3 5 7 9 11 13 15 17 19 22 25 35 40 45 0 0.08 0.24 0.4 0.56 0.72 0.88 1.04 1.2 1.36 1.52 1.76 2 2.8 3.2 3.6 19 20 21 22 27 33 37 41 47 50 53 55 57 58 58 48 5.53 5.67 5.81 5.95 6.59 7.29 7.72 8.12 8.70 8.97 9.24 9.41 9.58 9.66 9.66 8.79 0.24 0.24 0.24 0.24 0.21 0.18 0.16 0.13 0.08 0.06 0.03 0.02 0.00 -0.01 -0.01 0.07 0 -1 -3 -5 -7 -9 -11 -13 -15 -17 -19 -22 -25 -35 -40 -45 -49 0 -0.08 -0.24 -0.4 -0.56 -0.72 -0.88 -1.04 -1.2 -1.36 -1.52 -1.76 -2 -2.8 -3.2 -3.6 -3.92 19 20 21 23 25 31 35 40 47 50 53 55 55 56 57 57 43 5.53 5.67 5.81 6.08 6.34 7.06 7.50 8.02 8.70 8.97 9.24 9.41 9.41 9.49 9.58 9.58 8.32 0.24 0.24 0.24 0.23 0.22 0.19 0.17 0.14 0.08 0.06 0.03 0.02 0.02 0.01 0.00 0.00 0.11 Table 5.3 Results of Velocity Traverse in the Wake

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

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;

ρ ) ( 2 1 e e p P g U = − 22 . 1 ) 11 7 . 16 ( 81 . 9 2 1 − × × = U U = 9.57 m/s -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 -4 -3 -2 -1 0 1 2 3 4 y/d (U2 /U1 )*(1-(U 2 /U1 )) 0 to 45 mm 0 to -45 mm

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The drag coefficient may be obtained by use of the curve of _      − 1 2 1 2 ₁ U U U U in Figure 5.8. The area beneath the curve is found using the trapezoidal rule (see Appendix A) to be: 525 . 0 1 / / 1 2 1 2 ₌  ⌡ ⌠               − − d y d U U U U d y d y

Using this Equation gives

d y d U U U U C d y d y D  ⌡ ⌠               − = − / / 1 2 1 2 ₁ 2 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 CD = 1.03

By pressure plotting CD = 1.07

By wake traverse CD = 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

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µ = 2.40 × 10-5_{kg/m s}

so the kinematic viscosity is

ν = µ ρ = 1.216 10 40 . 2 _× −5 = 1.97 × 10−5 m2/s

The velocity U at the typical value of velocity pressure ½ρU2 = 500 N/m2 is 6 . 26 22 . 1 500 2 12 =       × = U m/s

So a typical value of Re for these tests is:

5 10 97 . 1 0125 . 0 6 . 28 Re ₋ × × = = ν Ud 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

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

d is so great that the three dimensional flows at the ends have no significant effect on the result. Our experiments are made for a cylinder with l

d = 3.9 for which end effects must be 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.

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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 = 8U

7

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

C D U d D1 ₁ 2 12 = ρ (0.76)

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6. THE ROUND TURBULENT JET

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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.

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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

u U f x R o ₌    (6-1) In the core of the jet, we have already observed that

u U

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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.

u U c x o ₌ (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 u g x R r x o =   ,  (6-3)

Consider now the velocity distribution over a section far downstream, i.e. where x R is 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 we might ignore the dependence upon x

R and simply write u u g r x o =    (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 u _r x o = +            1 1 0 25 2 2 . λ (6-5) where λ is a constant which is to be determined by experiment.

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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

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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

So the mass flux _{δ &m} through it is

δ &m = 2πρur δr

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

∫ &m = 2 ∞u 0 πρ r dr r dr (6-6)

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

∫

J u

o

= 2πρ∞ 2

(6-7) and the kinetic energy flux E to be

∫

E u o = 2 ∞1 2 3 πρ r dr R (6-8)

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

&mo = πρU 2

J_o = πρU R 2 2

Eo = π ρ12 U R 3 2

Air Flow Bench

A First Course in

Airflow

by

Emeritus Professor E. Markland

CONTENTS

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