Introduction to mechanical engineering part 2

Reynolds number Similar to pipe flows, the boundary layer over the surface of a body is initially laminar, but will soon become turbulent as the Reynolds number increases.. The boundary

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An Introduction to

Mechanical Engineering

Michael Clifford (editor)

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Michael Clifford (editor), Richard Brooks, Kwing-So Choi, Donald Giddings, Alan Howe, Thomas Hyde, Arthur Jones and Edward Williams

An Introduction to

Mechanical

Engineering

Part 2

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4.7 Modifying steady-state characteristics of a load using a transmission 265

4.8 Sources of mechanical power and their characteristics 266

4.9 Direct current motors and their characteristics 268

4.11 Inverter-fed induction motors and their characteristics 292

4.12 Other sources of power: pneumatics and hydraulics 305

4.13 Steady-state operating points and matching of loads to power sources 311

5.2 Feedback and the concept of control engineering 318

5.3 Illustrations of modelling and block diagram concepts 321

5.4 The s domain: a notation borrowed from mathematics 326

5.5 Block diagrams and the s notation: the heater controller and tensioning system 331 5.6 Working with transfer functions and the s domain 334

5.9 Conversion of the block diagram to the transfer function of the system 348

5.10 Handling block diagrams with overlapping control loops 350

5.11 The control algorithm and proportional-integral-derivative (PID) control 352

5.12 Response and stability of control systems 354

5.13 A framework for mapping the response of control systems: the root locus method 365

6.3 Response of damped single-degree-of-freedom systems 405

6.4 Response of damped multi-degree-of-freedom systems 424

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‘I think I should understand that better’, Alice said very politely, ‘if I had it written down: but I can’t quite

follow it as you say it.’

(Alice’s Adventures in Wonderland by Lewis Carroll)

This book builds on the experience and knowledge gained from An Introduction to Mechanical

Engineering Part 1 and is written for undergraduate engineers and those who teach them These

textbooks are not intended to be a replacement for traditional lectures, but like Alice, we see

the benefit of having things written down

In this book, we introduce material to supplement the foundational units in Part 1 on solid

mechanics, thermodynamics and fluid mechanics In addition, the reader will encounter units

on control, electromechanical drive systems and structural vibration This material has been

compiled from the authors’ experience of teaching undergraduate engineers, mostly, but not

exclusively, at the University of Nottingham The knowledge contained within this textbook

has been derived from lecture notes, research findings and personal experience from within the

lecture theatre and tutorial sessions

The material in this book is supported by an accompanying website: www.hodderplus.co.uk/

mechanicalengineering, which includes worked solutions for exam-style questions,

multiple-choice self-assessment and revision material

We gratefully acknowledge the support, encouragement and occasional urgent but gentle prod

from Stephen Halder and Gemma Parsons at Hodder Education, without whom this book

would still be a figment of our collective imaginations

Dedicated to past, present and future engineering students at the University of Nottingham

Mike Clifford, August 2010

Introduction

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The authors and publishers would like to thank the following for use of copyrighted material in this volume:

Figure 1.04 reproduced with the permission of The McGraw-Hill Companies; photo of

butterfly on page 10 © Imagestate Media; photo of crane on page 10 © DLILLC/Corbis; photo

of dolphin on page 10 © Stockbyte/Photolibrary Group Ltd; photo of whale on page 10 © Xavier MARCHANT – Fotolia.com; Figure 1.23 © Rick Sargeant – Fotolia.com; Figure 1.24b

McGraw-Hill Companies; Figure 1.27 D.S Miller, 1986, Internal Flow Systems, 2nd edn, Cranfield: BHRA,

p 215, reproduced with permission of the author; Figure 1.28 27 D.S Miller, 1986, Internal Flow

Systems, 2nd edn, Cranfield: BHRA, p 207, reproduced with permission of the author; Figure

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2.1 Drax Power Limited; Figure 2.14 reproduced by permission of the Chartered Institution of Building Services Engineers; Figure 2.22a © chukka_nc/released with a Creative Commons 2.0 licence; Figure 2.41 reprinted by permission of John Wiley & Sons Inc.; Figure 2.44

reprinted by permission of John Wiley & Sons Inc.; Figure 2.59 R.A Bowman, A.C Mueller

and W.M Nagle, 1930, ‘Mean temperature difference in design’, Transactions of the ASME, vol 12,

417–422; Figure 2.62 R.A Bowman, A.C Mueller and W.M Nagle, 1930, ‘Mean temperature

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Every effort has been made to trace and acknowledge the ownership of copyright The

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1.1 Introduction

Fluid dynamics is the study of the dynamics of fluid flow Here we learn how flows behave

under different external forces and conditions In a sense this is similar to rigid body dynamics

in physics, where Newton’s second law is used to describe the motion of rigid bodies Here,

we must apply Newton’s second law to fluid flows in a different way since fluids do not

behave exactly like rigid bodies This will be discussed in Section 1.2, where basic equations to

describe fluid motion are derived and explained Some discussions on laminar and turbulent

flows are also given there, paving the way for what will follow

The fluid that we deal with in this unit is a viscous fluid, so the velocity of fluid flow becomes

zero at the solid surface The consequence of this no-slip condition is that flow velocity

changes from zero at the wall to the free-stream value sufficiently far away from the wall

surface This thin layer is called the boundary layer, an important concept in fluid dynamics,

which explains how the fluid forces are generated So, in Section 1.3, we learn the basic

behaviour of boundary layers to be able to estimate the viscous drag acting on the solid surface

The boundary layers over solid bodies behave differently depending on their shape For

example, the drag force acting on sports cars is much less than that on pickup trucks, where the

boundary layer is separated from the body surface of vehicle creating a strong flow disturbance

In Section 1.4 we study the streamlining strategy to reduce the drag force of immersed bodies

We also discuss how the drag of immersed bodies is affected by the Reynolds number as well as

the wall roughness

Pipes and ducts are important engineering components used in many fluid systems It is

important, therefore, that the flow resistance can be correctly estimated for different type of

ducts and pipes In general there are two types of flow resistance One is due to the friction drag,

while the other relates to the loss of energy due to boundary layer separation In Section 1.5,

■ Drag on immersed bodies

■ Flow through pipes and ducts

■ Dimensonal analysis in fluid dynamics

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01.01 An Introduction to Mechanical Engineering: Part 2

j

i

Stress τij acts on a surface

whose normal is j-direction

Stress τij acts

in i-direction

Figure 1.1 Stress tensor tij The first index i of the stress tensor t ij

indicates the direction of the stress that is acting on the surface,

whose normal direction is indicated by the second index j.

we study a method of minimizing the flow resistance similar to the streamlining strategy for

immersed bodies The discussion of pipes and ducts is extended to non-circular shapes by

introducing the concept of hydraulic diameter

The final section of this unit deals with the non-dimensional numbers of fluid dynamics We

are already familiar with the Reynolds number, but there are many other non-dimensional

numbers in fluid dynamics In Section 1.6, we learn how to identify relevant non-dimensional

numbers for different types of fluid flow We also study how these non-dimensional numbers

are used to carry out model tests Applications of the similarity principle to fluid machinery are

given, emphasizing the importance of non-dimensional numbers in fluid dynamics

1.2 Basic concept in fluid dynamics

Navier–Stokes equations

The main aim of fluid dynamics is to understand the dynamic behaviour of fluid flows Since

all fluids are continuous, we can determine the velocities and pressure of flows as a function of

space and time To achieve this, we require the governing equations to represent the fluid flows:

the Navier–Stokes equations

For simplicity, we consider only two-dimensional, isothermal (no thermal input or output)

and Newtonian (the shear stress is linearly proportional to the strain rate) flows with constant

density and viscosity Therefore, u, v and p (velocities and pressure in Cartesian coordinates) are

function of x, y and t If the fluid flows are steady, they are only functions of x and y.

The Navier–Stokes equations can be derived by applying Newton’s second law of motion to

fluid flow By considering a small control volume dxdy with a unit depth, the fluid mass times

acceleration is given in vector form by

Here, we shall consider only those forces acting on the surfaces of the control

volume, although a body force will be introduced later Surface forces include

hydrostatic pressure p, the normal stresses t xx, tyy and the shear stresses txy, tyx

→

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From Figure 1.2, the total surface forces in the x-direction can be obtained as follows.

[  ( txx 1 ∂txx

Here, the forces acting on the surface at (x 1 dx) can be obtained from the forces on the surface at

x by using Taylor’s expansion.

In a similar way, the total surface force in the y-direction can be obtained from Figure 1.3 as

So far, we have considered only the surface forces There are flows, such as water waves around a

ship, where the gravity force plays an important role Therefore, we should also consider such a body

force acting on the control volume together with the surface forces In such a situation, the vertical

force F y in equation (1.10) must be replaced by

x

0

y+dy dy y

Figure 1.2 The balance of surface forces in the x-direction

01.03 An Introduction to Mechanical Engineering: Part 2 Barking Dog Art

�y

Control volume

Figure 1.3 The balance of surface forces in the y-direction

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For the Newtonian flow, the stress is linearly proportional to the velocity gradient, we can write

this relationship in a matrix form as follows

where, n 5 m/r is called the kinematic viscosity with dimension [L2 T 21]

Now, the Navier–Stokes equations consist of four parts, which are:

(Inertia force) 5 (Pressure force) 1 (Viscous force) 1 (Gravity force) (1.17)

In others words, the left-hand side of the Navier–Stokes equations represents the inertia force (i.e mass times acceleration), while pressure force represented by the pressure gradient and viscous force led by the viscosity are on the right-hand side The gravity force can usually be omitted from the equations unless the surface wave or the natural convection is involved

By examining the magnitude in each term of the Navier–Stokes equations, we can see that the

left-hand side of the equations of is of the order of u 2 /L, while the right-hand side is of the order of nu/L 2 Here, u represents the velocity scale (either u or v), L represents the length scale (x or y) and n is the kinematic viscosity The ratio of these two will give a non-dimensional value

called the Reynolds number:

n   ∝ inertia force _ viscous force (1.18)

Figure 1.4 Effect of the Reynolds number on the vortex shedding from a circular cylinder

(H Schlichting, 1968, Boundary Layer Theory, 6th edn, New York: McGraw Hill, reproduced with

permission of the McGraw-Hill Companies)

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Similarly, the ratio between the magnitude of the inertia term ( of the order of _ u L 2) and that of

the gravity term (of the order of g) is called the Froude number:

Here, it is customary to take a square root of the ratio to define the Froude number

For example, a yacht with a length of L 5 20 m travelling at u 5 10 m/s would have

Re 5 1.3 3 108 and Fr 5 0.7 assuming that n 5 1.5 3 10–6 m2/s

Continuity equation

The continuity equation can be derived in a similar way as we have done to obtain the

Navier–Stokes equation Here, however, we should consider the mass balance in the control

volume instead of the force balance.

Mass flux into the control volume in the x-direction is given by

Figure 1.5 Mass balance within the control volume

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Here again, we have used Taylor’s expansion to evaluate the mass flux out from the surface at

(x 1 dx) and (y 1 dy) from the mass flux into the surface at x and y, respectively.

Since there must be no changes in mass within the control volume:

expression for the pressure gradient of this flow if the fluid density is constant and the viscosity and gravity forces are negligible.

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Laminar and turbulent flows

The flow through a pipe remains smooth and steady below the

critical Reynolds number, given by

Re  U · d n  5 13,000 (1.24)

However, the flow becomes fluctuating and random when the

Reynolds number exceeds this value Figure 1.6 shows this

process where the transition from laminar to turbulent flow is

demonstrated with dye injected into a pipe through a needle

We can see the change in flow behaviour as a function of the

Reynolds number Figure 1.7(a) shows a laminar flow at subcritical

Reynolds number, where the dye filament stays straight through until

the end of the pipe As the Reynolds number is increased in Figures 1.7(b) and (c) the flow

develops patterns, signifying the flow transition that is taking place Finally, turbulent flow is

reached in Figure 1.7(d) where the dye patterns seem to be random and chaotic

Although the Reynolds number is an important parameter affecting the transition process to

turbulence, there are other influential factors such as the wall roughness, initial disturbance and

external disturbance For example, a sharp inlet to the pipe will disturb the flow, and therefore

reduce the critical Reynolds number Vibration of the experimental setup from the floor, as

well as noise transmitted to the flow, will accelerate the process of transition

Transitional flow

Turbulent flow 01.07 An Introduction to Mechanical Engineering: Part 2 Barking Dog Art

Transitional flow

Turbulent flow

Barking Dog Art

Figure 1.6 The flow in a pipe changes from laminar flow (a) to turbulent flow (b)

Figure 1.7 Flow transition to turbulence in a pipe flow Laminar flow (a), where the injected dye

stays straight through the pipe; transitional flow (b), where the flow develops certain patterns;

turbulent flow (c, d), where the flow becomes random and chaotic.

Substituting these into the Navier–Stokes equations, we get

(2xy)(2y) 1 (2y2)(2x) 5 2 1

r   _ ∂p

∂x 2xy2 5 2 1

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There is, however, no precise definition of turbulence Indeed, it is very difficult to determine

whether a particular flow is turbulent or not For example, random waves that can be observed

over the surface of the water in a swimming pool are probably not turbulence Therefore, we

have to look at the symptoms of the flow to determine whether it is turbulent or not

Symptoms of turbulence include:

Every turbulent flow is different, yet they have many common characteristics as listed above

We must, therefore, check whether a particular flow satisfies all of these characteristics before

declaring that it is a turbulent flow

1.3 Boundary layers

The boundary layer is a thin layer created over the surface of a body

immersed in a fluid (Figure 1.8), where the viscosity plays a significant

role Due to the non-slip condition of viscous flows, the velocity at

a solid surface is always zero This means that the velocity gradually

increases from zero at the wall to the freestream velocity U o at the

edge of the boundary layer This creates a thin, highly sheared region

called the boundary layer, over a body surface in a moving fluid (or

over a moving body in a still fluid) Over a large commercial aircraft,

for example, the boundary layer a few millimetre thick near the

cockpit can grow to as much as half a metre thick towards the end of

the fuselage

Reynolds number

Similar to pipe flows, the boundary layer over the surface of a body is initially laminar, but

will soon become turbulent as the Reynolds number increases Here, the Reynolds number to

describe the state of the boundary layer flow can take any of the following forms

Re  U o x

n   , U od 

*

n   , U ou 

While the pipe diameter is always used to define the Reynolds number of pipe flows, the

length scale of boundary layer flows takes either the streamwise length x along the body surface

or one of the boundary layer thicknesses such as d, d * or u The boundary layer thickness d is

Learning summary

By the end of this section you should have learnt:

3 the Navier–Stokes equations are governing equations for fluid motion, which can be derived from

Newton’s second law of motion;

3 the continuity equation guarantees the conservation of mass;

3 the Reynolds number indicates a relative importance of inertial force in flow motion to viscous force;

3 the Froude number signifies the importance of inertial force in flow motion against the gravity force;

3 all flows become turbulent above the critical Reynolds number.

Figure 1.8 A boundary layer being developed over a wing

Edge of the boundary layer

U0

δ

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defined as the distance from the wall to the point where the boundary layer velocity reaches

the freestream velocity Discussions on other boundary layer thickness, such as d* or u, will be

given later in this section

The boundary layer thickness over a flat plate at the streamwise length x from the leading edge

can be obtained using the following formulae

for turbulent flow (Rex 3 3 106) (1.27)

The boundary layer growth depends on the flow condition, whether it is laminar or turbulent

The growth is much faster for turbulent boundary layers (Figure 1.9) as the diffusivity increases

as a result of transition to turbulence (Section 1.1) The critical Reynolds number for the

boundary layer over a flat plate is usually around 3 3 106 as indicated in equations (1.26) and

(1.27) It may take quite a different value, however, depending on the initial as well as boundary

conditions of the flow For the boundary layer over a rough surface, for example, the critical

Reynolds number is less than 106 This means that the boundary layer transition takes place

much earlier over a rough surface as compared to that over the smooth surface

Worked example

A laminar boundary layer is being developed over a flat plate with the

free-stream velocity of 1.5 m/s Assuming that the pressure gradient along the plate

is zero, obtain the distance x from the leading edge where the boundary layer

thickness d becomes 10 mm Using the critical Reynolds number R xc of 10 6 ,

determine the transition point where the boundary layer becomes turbulent

The kinematic viscosity and density of the fluid (air) are 1.5 3 10 –5 m 2 /s and

Figure 1.9 Development of the boundary layer over a flat plate parallel to the flow

Note that the boundary layer thickness grows faster when it is turbulent.

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Reynolds number of living things

Reynolds number of living things depends on their size, flight or swim speed and the medium

they live in The following are the Reynolds numbers of some of familiar living things Unless

the body size is very small or the flight or swim speed is very low, as with butterflies, the flow

around the body of living things is most likely turbulent

Butterflies: Re  _ (0.3 m/s) 3 (0.08 m)(1.5 3 10–5 m2/s) 5 1600

Cranes: Re  (15 m/s) 3 (1.0 m)(1.5 3 10–5 m2/s) 5 1 000 000

Dolphins: Re  _ (1.0 3 10(10 m/s) 3 (2 m)–6 m2/s) 5 20 000 000

Whales: Re  _ (15 m/s) 3 (25 m)(1.0 3 10–6 m2/s) 5 375 000 000

Velocity profiles of boundary layers

The Blasius profile is the theoretical velocity profile of the laminar

boundary layer over a flat plate where the static pressure does not

change along the plate Here, the Blasius profile is independent of the

Reynolds number of the boundary layer, as shown in Figure 1.10

The Blasius profile can be approximated by the parabolic velocity

profile given by equation (1.28)

For the turbulent boundary layer, however, there is no theoretical

solution to represent the velocity profile The turbulent boundary

layer profile can be approximated by the one-seventh law given by

The atmospheric boundary layer is often simulated by the

one-seventh law in a wind tunnel, where building or bridge models are

tested Indeed, this empirical law has a reasonable agreement

with the actual profile of the turbulent boundary layer However,

the velocity gradient of the one-seventh law at the wall is always

0

1.0 0.8 0.6 0.4 0.2 0

Turbulent flows

Blasius profile Parabolic profile

U u0

Figure 1.10 Velocity profiles of laminar and turbulent boundary layers

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Therefore, it cannot be used to investigate the turbulent boundary

layer close to the wall surface Here, we should use the logarithmic

velocity profile instead (Figure 1.11) With the logarithmic velocity

profile (the log law, for short) a large part of the turbulent boundary

layer can be represented by

u

u* 5 5.75 log10 u

*y _

except for a very thin region near the wall (the viscous sublayer)

where the velocity is given by the following linear profile

be noted that the turbulent velocity profile is dependent on the

Reynolds number (Figure 1.10)

Worked example

An atmospheric boundary layer (d 5 80 m) with the free-stream velocity of

U0 5 10 m/s has the wall-shear stress of tw 5 0.077 Pa The kinematic viscosity

and density of air are n 5 1.5 3 10 –5 m 2 /s and r 5 1.2 kg/m 3 , respectively.

(1) Estimate the thickness of the viscous sublayer.

(2) Obtain the velocity at y 5 1 m from the ground.

(1) The thickness of the viscous sublayer is given by

y1 5 yu

* _

10 4

10 3

10 2 10

1

y+

Logatithmic region Equation (1.31)

Equation (1.32)

Viscous sublayer

u+

Figure 1.11 Logarithmic velocity profile of the turbulent boundary layer

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Friction velocity in turbulent boundary layers

Logarithmic velocity profile of turbulent boundary layer is given by

u u* 5 5.75 log10 u

turbulent pipe flows except in the viscous sublayer (y1 , 10), where we should use

We see that the velocity gradient within the viscous sublayer is constant

Effect of wall roughness

The wall roughness can affect the laminar boundary layer by promoting an early transition to turbulence For the turbulent boundary layer, the wall roughness enhances the fluid mixing to increase the near-wall velocity gradient, which leads to an increase in skin-friction drag Here,

it is important to know how rough the wall should be before it starts affecting the skin-friction drag of the turbulent boundary layer

The wall surface of the turbulent boundary layer is hydraulically smooth when 1   _ u*

n    , 5

Here,  is the roughness height, u* 5 √ _ tw

r     is the friction velocity and n is the kinematic viscosity of the fluid In other words, the wall surface is smooth as far as fluid dynamics of turbulent boundary layer is concerned if the Reynolds number 1, based on the roughness

height  and the friction velocity u*, is less than 5 This suggests that the skin-friction drag of the turbulent boundary layer will be increased only when 1 5 Recalling that the thickness

of viscous sublayer is given by y1 5 10, we can say that the wall surface is hydraulically smooth

if the roughness is completely submerged in the viscous sublayer

If we know the roughness height , the Moody chart for pipe flows (and the analogous chart

for the boundary layers) will give us the effects of roughness on the skin-friction drag in the turbulent boundary layer Wall roughness should be considered always relative to the boundary layer thickness, so we must use the non-dimensional roughness height in studying its effect on the skin-friction drag

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Momentum integral equation

It is possible to estimate the skin-friction drag of the boundary

layer using the momentum integral equation To derive the

equation, we consider a balance of forces on a control volume

within the boundary layer (Figure 1.13), which is being developed

over a flat plate Here, the line 1 and line 3 indicate the entry

and exit of the control volume, while the streamline just above

the boundary layer (line 2) and the flat plate (line 4) indicate the

upper and bottom surface, respectively

Since there is no flow across the streamline or the plate wall, the

mass flow rate to the control volume through line 1 must be equal

to the mass flow rate out of line 3 This gives

r Uo h 5 r ∫  0 d(x) u dy (1.33)

The only force acting on the control volume is the skin-friction drag over the flat plate since

the pressure gradient over a flat plate is zero Accordingly, the skin-friction drag must be equal

to the change in the momentum flux within the control volume Therefore, the skin-friction

drag over the flat plate is given by

By substituting equation (1.33) into equation (1.34), we obtain

5 r ∫  0 d(x) u(U o 2 u)dy Introducing the momentum thickness u, a measure of the momentum loss as a result of the

boundary layer growth, which is is given by:

Figure 1.12 Schematic view of the boundary layer The wall surface is

considered smooth (a); it should be considered rough (b) despite the

identical physical size of roughness.

Barking Dog Art

δ

δ ε

Flat plate

Boundary layer

L

Figure 1.13 Control volume used in the derivation the momentum integral equation

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Therefore, the momentum thickness u represents the skin-friction drag D over a flat plate A

differential form of this equation can be given by

This is the Kármán’s momentum integral equation, which is valid for both laminar and

turbulent boundary layers as long as the pressure gradient is zero The Kármán’s momentum integral equation indicates that the local skin-friction coefficient is exactly twice the

streamwise change of the momentum thickness

Another important parameter in boundary-layer theory is the displacement thickness d*

displacement thickness d * (Figure 1.14) Therefore, we can estimate the change in flow rate though a duct, for example, without considering the change in velocity profile

The ratio of the displacement thickness d* to the momentum thickness u is called the shape

of the laminar boundary layer is H  2.6, while the shape factor of the turbulent boundary

layer is H  1.4 For the boundary layer under transition, the shape factor takes a value between

h h

u(y) y

δ *

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The shape factor can also be used to find out if it is close to flow separation (to be discussed

next in this section) Since the velocity profile will become tall and thin as the flow separation

is approached, the shape factor of the boundary layer will increase in value whether it is

laminar or turbulent Usually this increase in the H value is quite rapid, giving warning that the

boundary layer flow is about to detach from the wall

Formulae for the boundary layer development

Obtain the displacement thickness,

momentum thickness and the shape

factor of the boundary layer when

the velocity profile is given

δ

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Boundary-layer equations

Assuming that the boundary-layer thickness is small as compared to the streamwise length of

development, we can derive a special form of the Navier–Stokes equations, called Prandtl’s

boundary-layer equations In order to do that, we must make a number of assumptions.

(1) The length scale in the vertical (normal) direction of the boundary layer is much smaller

than that of the longitudinal (streamwise) scale In other words,

Dy ≪ Dx or ∂  _

∂y ≫ ∂  _

∂x (2) The velocity scale in the vertical direction of the boundary layer is much smaller than that in

the longitudinal direction:

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Then, the Navier–Stokes equations (1.15) and (1.16) will take a very simple form

Equation (1.46) suggests that p 5 p(x), therefore the pressure is constant across the boundary

layer Since the pressure gradient of the boundary layer over a flat plate is always zero, i.e

∂p/∂x 5 0, this means that the pressure is constant everywhere in the boundary layer over a flat

plate

The derivation of the boundary-layer equations can be done using the order of magnitude

analysis Here, we set

x  L (x is of the same order of magnitude as the plate length L)

y  d (y is of the same order of magnitude as the boundary layer thickness d)

p  ru 2 ( p is of the same order of magnitude as the dynamic pressure ru2)

v  (d/L) · u (using the continuity equation, we find that v is of the same order of magnitude as

(d/L) · u)

After replacing x, y, p, u and v with L, d and u, we find that the first viscous term in the

Navier–Stokes equations is much smaller than the second viscous term It should also be noted

that all terms in the y-equation become zero, except for the pressure gradient term.

Effect of pressure gradient

So far, we have studied only the boundary layers with zero pressure gradient However, we

can extend the boundary-layer theory to cover situations with non-zero pressure gradient

Figure 1.16 shows a typical development of the boundary layer over a curved surface, where a

dramatic change in the velocity profile is taking place Where the pressure gradient is negative,

or favourable, i.e ∂p/∂x , 0 (from point A to C in Figure 1.16), the lost energy of the

boundary layer due to skin-friction drag can easily be replenished by the pressure force acting

in the flow direction Therefore, flow separation does not generally take place easily in such a

pressure gradient condition However, when the pressure gradient is positive, or adverse,

∂p/∂x 0 (from point C onward in Figure 1.16), the pressure force cannot easily replenish

the lost momentum This is because the pressure force acts against the boundary layer under

an adverse pressure gradient This leads to a separation of the boundary layer, or simply a flow

separation, creating a region of flow reversal (Figure 1.16).

The flow separation point is defined as a location where the wall-shear stress becomes zero, i.e

tw  (  ∂u _

∂y ) y 5 0 5 0 (1.47)

Therefore, the velocity gradient at y 5 0 (at the wall) also becomes zero.

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Worked example

A sports equipment company is required to design a new swimming

costume for the next Olympic Games to help swimmers break the world record in the 200 m freestyle Answer the following questions assuming that the typical swim speed is 2 m/s.

(1) Suggest how you might design a new swim cap In doing so, you need to explain the design concept based on fluid mechanical principles.

(2) A new swimming costume must also cover the entire arms and legs How should the design concept for these parts of the costume differ from that of a cap? Again, your answer must be based on fluid mechanical principles.

(3) What considerations should be given to the choice of fabric for the swim cap and the swimming costume? Your answer must be accompanied by clear and sound reasons.

(1) The Reynolds number of the boundary layer over a swim cap is estimated as

skin-(b) Promote turbulent flow by tripping the boundary layer, which can reduce the

pressure drag of the swimmer’s head by moving the separation point further downstream

(2) To increase the thrust, the drag on arms and legs must be increased by

(a) making the surface of the swimming costume covering the arms and legs rough, thereby increasing the skin friction drag, or

(b) making the surface smooth as possible, thereby allowing the laminar flow to separate early

(3) We must choose the fabric of the swimming costume carefully to achieve these objectives: a smooth fabric where we want laminar flow, and a rough fabric in certain parts of the costume to promote turbulence

3 viscous fluid does not slip at a solid wall surface This is called the non-slip condition of flow motion;

3 the boundary layer is a thin fluid layer near a solid wall surface, where the velocity is less than the freestream velocity;

3 the momentum thickness signifies the loss of momentum in the boundary layer due to skin-friction drag;

3 the displacement thickness is a measure of mass flow deficit in the boundary layer;

3 the boundary layer equations are a simplified form of the Navier–Stokes equations;

3 flow separation occurs over a curved surface when the static pressure increases in the flow direction.

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1.4 Drag on immersed bodies

Pressure drag

While the friction drag Dfric results from the viscous action of fluids on the body surface, the

pressure drag Dpres comes from the static pressure distribution around the body, mainly due to

boundary-layer separation The total drag acting on immersed bodies in incompressible flows,

therefore, consists of the friction drag and the pressure drag We can write

Dtot (total drag) 5 Dfric (friction drag) 1 Dpres (pressure drag) (1.48)

The relative importance of Dpres to Dfric depends on the body shape as well as the Reynolds

number When the immersed bodies are streamlined, the friction drag dominates the total drag

When the non-streamlined bodies (bluff bodies) are placed in a fluid flow, however, the total

drag is dominated by the pressure drag, and the contribution of the friction drag is usually

bodies, such as motor cars) or the plan-form area (for long bodies, such as aircraft wings)

In terms of drag coefficient, equation (1.49) can be written as:

Flow around a circular cylinder

For a circular cylinder with radius a and length b, the pressure drag is given by

whose drag coefficient is given by

CDpres 5 ∫  0 2p ab( p 2 p ∞ ) cos u du 

C p 5 _ p 2 p∞

1

_ 2 rV 2 (1.53)

It should be noted that the frontal area of circular cylinder

(2ab) is used to non-dimensionalize the drag to give the

pressure drag coefficient This equation suggests that CDpress

can be obtained by integrating the streamwise component

of C p over the circular cylinder The cylindrical coordinate

system being used in the computation is shown in Figure

1.17, where the angle u is measured clockwise from the

frontal stagnation point

U0

p V∞

a dθ

θ

Separation

(p–p∞) adθb

Wake

Figure 1.17 The coordinate system used for the integral

of static pressure around a circular cylinder to give the

pressure drag Here, p is the static pressure over the cylinder surface and p∞ is the freestream pressure.

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Figure 1.18 compares the distribution of pressure coefficient C p over a circular cylinder

between the laminar flow and the turbulent flow It should be noted that both C p curves

are asymmetric with respect to u 5 90°, indicating that the static pressure over the front of

the circular cylinder is much higher than that in the rear The integrated pressure difference

between the front and rear surfaces gives the pressure drag acting on the circular cylinder

Figure 1.18 also shows a significant difference in the static

pressure distribution between the turbulent flow and the

laminar flow While the static pressure for the laminar flow

stays near the minimum value of C p 5 21.0 in the rear of

the circular cylinder, the turbulent flow recovers to a much

greater value of C p 5 20.4 after reaching the minimum

value of C p 5 22.1 at around u 5 75° This reflects a

small CD value of 0.3 for the turbulent flow as compared

reason for the smaller CD value for turbulent flow is that

the flow separation takes place much further downstream

due to greater mixing capability of the turbulent flow As a

result, the wake region in the downstream of turbulent flow

separation is narrower than for laminar flow (Figure 1.21)

However, inviscid theory gives a symmetric C p curve

(Figure 1.18), suggesting that the drag on a circular

cylinder is zero for zero viscosity fluids Certainly this is

not a realistic assumption in calculating the drag force on

immersed bodies Indeed, inviscid theory cannot impose

the non-slip condition on the wall, so there will be no

boundary layer development or flow separation over the

immersed bodies

Drag of bluff bodies

As has been previously suggested, the drag coefficient CD of

immersed bodies is a function of the Reynolds number The drag

coefficient is gradually reduced with an increase in the Reynolds

number as seen in Figure 1.19 Once the Reynolds number

reaches the critical value (Re  5 3 105 for circular cylinders

and spheres with smooth surface) the drag coefficient will drop

suddenly This is the transition point where the flow around the

immersed bodies will become turbulent from laminar flow

Turbulent

Inviscid flow Laminar

Figure 1.18 Non-dimensional pressure distribution over

a circular cylinder, where the C p curve for the laminar and turbulent flow are compared with the solution of inviscid flow.

5 4 3 2

0 1

10 10 2 10 3 10 4 10 5 10 6 10 7

Circular cylinder

C D

Transitional Reynolds Number

Sphere

Re d

Figure 1.19 Drag coefficient of circular cylinder and sphere as a function of the Reynolds number, showing that the C D value reduces as the transition takes place.

Figure 1.20 Drag coefficient of a circular cylinder, showing that the

transition takes place at lower Reynolds number with an increase in

the surface roughness.

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If the wall surface is rough, the transition to turbulences over a circular cylinder takes place at

lower Reynolds number (Figure 1.20), but the drag coefficient CD for the turbulent flow (after

transition) is greater than that with a smooth surface Figure 1.20 also shows that the critical

Reynolds number reduces with an increase in the roughness ratio /d The flow over a sphere

is qualitatively similar to that over a circular cylinder

The dimples on a golf ball can reduce the pressure drag by making the ball surface rough This

reduces the transition Reynolds number by artificially forcing (tripping) the boundary layer

to turbulent flow at low Reynolds numbers As a result, the wake becomes narrower as can be

seen in Figure 1.21 Although the friction drag is increased in this case, the total drag of the golf

ball is reduced This is because the golf balls are bluff (non-streamlined) bodies, where Dpres is

much greater than Dfric

Worked example

Obtain the drag force on a baseball of 73 mm diameter at the critical Reynolds

number assuming that the flow around the ball is turbulent The density and

kinematic viscosity of air are 1.2 kg/m 3 and 1.5 3 10 –5 m 2 /s, respectively.

From Table 1.2, we find that CD  0.2 for a sphere when the flow is turbulent, where the

critical Reynolds number is Re 5 3 3 105 Therefore, the drag force on the ball can be

Figure 1.21 Comparison of laminar separation with turbulent

separation, showing that the separation point moves further

downstream when the boundary layer becomes turbulent This

reduces the pressure drag by making the wake narrower.

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Tables 1.1 and 1.2 give the drag coefficient of two-dimensional and three-dimensional bodies, respectively It should be noted here that the drag coefficient of sharp-edged bodies, such as squares and cubes, is insensitive to the Reynolds number since the flow is always separated at

the sharp edges In other words, the CD value of sharp-edged bodies remains constant whether the flow is laminar or turbulent as long as the Reynolds number is greater than 104

2.1

1.6 1.2

d L

1.15 0.90 0.85 0.87 0.99

Ellipsoid:

0.75 1 2 4 8

0.5 0.47 0.27 0.25 0.2

0.2 0.2 0.13 0.1 0.08

d L

Laminar Turbulent

Table 1.2 Drag coefficient of three-dimensional bodies at Re 104

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Worked example

The fork ball is a baseball pitch thrown like a straight ball but with little or

no rotation, where the ball initially travels straight but falls sharply as it gets

closer to the batter who is standing about 18 m away from the pitcher.

(1) Draw a figure showing the drag coefficient of the baseball as a function of

the Reynolds number by considering the baseball as a smooth sphere.

(2) Obtain the drag force on a baseball with 73 mm diameter at the critical

Reynolds number assuming that the flow around the ball is turbulent The

density and kinematic viscosity of air are 1.2 kg/m 3 and 1.5 3 10 –5 m 2 /s,

respectively.

(3) By what percentage does the drag force change if the flow becomes

laminar rather than turbulent?

(4) Explain the behaviour of the fork ball as described above using the

principles of fluid mechanics.

(5) Discuss how the pitcher should adjust the delivery of the fork ball for it to

remain effective if the ball surface becomes rough during a game.

(1)

Figure 1.22

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(2) CD  0.2 for a sphere when the flow is turbulent, where

CD 5 D

r 

increased to 2.5 times (150% increase)

(4) The fork ball is the result of reverse transition (from turbulent to laminar rather than

usual laminar to turbulent route) of flow around the ball During this transition, the

ball will experience a 150% increase in drag, resulting in a sharp drop near the batter

(5) As shown in Figure 1.20 for a circular cylinder (similar for a sphere), the drag

increase will be smaller when the surface is rough Therefore, the amount of drop

will be reduced as a result of ball roughness The critical Reynolds number for flow

transition will be lowered so that the pitcher must throw a fork ball with a lower

initial speed for it to be effective

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Streamlining strategy

An important strategy in reducing pressure drag Dpres of immersed bodies is to streamline them,

by shaping the bodies in such a way as to move the flow separation point further downstream

This will effectively reduce the width of wake (the area in the downstream of flow separation),

leading to a reduction of the low pressure region in the rear of the immersed bodies

Figure 1.23 shows a procedure for streamlining a rectangular cylinder that has sharp corners

The drag can be easily reduced to nearly a half by rounding the front corners of a cylinder,

which reduces the drag coefficient CD from 2.0 to 1.0 A more dramatic reduction in drag can

be obtained by tapering the rear corners, resulting in a reduction of drag to nearly one seventh

of its original It is surprising to observe that a fully streamlined cylindrical body is equivalent

in terms of the total drag with a circular cylinder one tenth of its width This shows the

effectiveness of streamlining strategy in reducing drag by tapering the rear of immersed bodies

Worked example

A man jumped from an airplane with a parachute of 7.3 m in diameter

Assuming that the total mass of the man and parachute is 80 kg, calculate the

speed of descent when he reaches terminal velocity.

The drag coefficient of parachute is CD 5 1.2 regardless of the Reynolds number, as it is

a “sharp-edged” body

The terminal velocity will be reached when the drag of the parachute is balanced by the

weight of the parachute and the man, i.e D 5 W.

no wonder a biplane (e) cannot fly very fast.

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The width of the wake region can be reduced if the flow separation is moved back towards the

rear of the body In practice, however, it is often difficult to do this as it may reduce the capacity

(volume) of the vehicle Instead, trucks can benefit much by attaching a deflector on top of the

cab (Figure 1.24), which reduces a large separation region in front of the trailer, leading to

is a non-dimensional form of the frictional head loss h f The friction factor is a function of the

Reynolds number R d 5 Vd _ n    as well as the relative surface roughness d , where V is the bulk  

velocity, d is the pipe diameter, n is the kinematic viscosity of fluid and  is the typical surface

roughness height

Figure 1.24 A deflector can reduce the pressure drag of a truck by 20%, by steering the

streamlines away from the frontal surface of the trailer If there is no trailer attached to the cab,

however, there will be a large increase in drag

3 pressure drag is a result of the boundary layer separation, where the static pressure difference is

created between the front and rear of the bodies;

3 drag coefficient of immersed bodies is reduced with an increase in the Reynolds number when the

flow is laminar;

3 drag coefficient of immersed bodies is suddenly reduced at the critical Reynolds number when the

flow becomes turbulent;

3 surface roughness will reduce the critical Reynolds number of immersed bodies, thereby reducing

their drag at lower Reynolds number;

3 streamlining is an effective strategy for reducing drag, where the immersed bodies are rounded at

the front and tapered at the rear.

Barking Dog Art

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For laminar flows (R d , 2 3 103) it can be shown that the friction factor is the only function

of the Reynolds number, where

This is called the Darcy–Weisbach equation for laminar pipe flows It should be noted that the surface roughness does not affect the friction factor for laminar pipe flows

For turbulent pipe flows, Colebrook gave the following formula for f, covering a wide range of

Reynolds number and surface roughness

1

R d √_f ) (1.56)

Although accurate in presenting the friction factor for both transitional and fully turbulent pipe

flows (R d 4 3 103), this formula is difficult to use in practice since the friction factor is not given in a closed form In other words, iteration is required to obtain the friction factor from this equation for a given Reynolds number and roughness ratio It is for this reason Moody has

presented a chart where the friction factor can be easily read This is called the Moody chart,

where the friction factor is given as a function of the Reynolds number R d

and the roughness ratio d (Figure 1.25) Following a curve of constant   d value (as shown on  the right-hand side of the chart) to meet a constant Reynolds number line, one can read off the friction factor on the left-hand side of the chart Typical surface roughness  for pipes and ducts, from iron to concrete, can be found in Table 1.3

0.05 0.04 0.03 0.02 0.15 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0.0002 0.0001 0.000,05 0.000,01

Critical zoneTransition zone

Figure 1.25 Moody chart (taken from F M White, 2008, Fluid Mechanics, New York: McGraw Hill

and reproduced with permission of the McGraw-Hill Companies)

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Material Condition  mm Uncertainty, %

Stainless, new 0.002 650 Commercial, new 0.046 630

Galvanized, new 0.15 640

Table 1.3 Typical surface roughness in pipe and channel flows

There are two different definitions for the friction factor The Darcy friction factor is used

throughout this textbook, which is defined by

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Minor losses

Whenever there are changes in velocity magnitude or direction in a pipe or duct system, there will be associated pressure drops, called minor losses The minor losses are typically found at

• the entrance to the pipe or exit

• sudden expansion or contraction

• bends

• valves

The minor losses are caused by the internal flow separation as a result of changes in the

magnitude or direction of the flow through the pipes or ducts These are similar to the pressure reductions along the immersed bodies as a result of boundary layer separation Although they are called minor losses, the pressure drops can be a significant part of the total pressure drop when the pipe or duct has a short straight section

The minor head loss h m in a duct or pipe system is expressed by

is much greater in a flow through a square bend than through a circular bend This explains why the minor loss is much greater for the flow around a square bend Therefore, the use of square bends should be avoided in a pipe or duct system

The minor loss coefficient for mitre bends including that of the square bend is given in

Figure 1.27 There is no Reynolds number dependency on this value since the flow is always separated at the sharp corner Figure 1.28 shows the minor loss coefficient of circular bends

at Re 5 106 as a function of the bend-radius-to-pipe-diameter ratio r d and the bend angle u b Here, the minor loss coefficient depends on the Reynolds number, so a correction factor given

in Figure 1.29 should be applied to this value

Other examples for large minor loss in a pipe and duct system include sudden contraction and expansion, where the flow separation takes place at the junctions (insets in Figure 1.30) The

K-factors are a function of the rate of contraction or expansion.

Figure 1.26 Flow through a circular (a) and square (b) bends

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Figure 1.31 illustrates some commercial valve geometries Typical K-factor for the gate and disk

valves is K  0.2 when they are fully open, while it is K  4 for the globe valve Valves are the

main source of minor losses in a pipe system as one can see in Figure 1.32, showing typical

values of the K-factor when valves are partially open It is shown that the head loss in the pipe

or duct system will be increased by more than 100 times when a gate valve is closed by 75%

We must be careful, therefore, in the selection and use of the valves in a pipe and duct system

There are further sources for minor losses in a pipe system Figure 1.33 shows the entry losses

for different entry geometry As we would expect, the K-factor for the pipe entry is a function

of the relative radius and length of the entry Note that the K-factor is always unity for a sharp

exit from the pipe

Barking Dog Art

Figure 1.27 Mitre bend loss coefficient

180

90 60 50

10 20

150 120

40 30

0.16 0.14 0.12 0.10

θb

r d

Figure 1.28 The loss coefficient of circular bends

at Re 5 10 6 For other Reynolds numbers this coefficient must be multiplied by the correction factor given in Figure 1.29

Barking Dog Art

Figure 1.29 The Reynolds number correction factor for

circular bend loss coefficient

1.0 0.8 0.6 0.4

Sudden expansion

Sudden contraction 0.2

D d

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Barking Dog Art

D

h

Figure 1.31 Commercial valve geometries (a) gate valve; (b) globe valve; (c) angle valve; (d)

swing-check valve; (e) disk-type gate valve (F M White, 2008, Fluid Mechanics, New York:

McGraw Hill Reproduced with permission of The McGraw-Hill Companies)

Barking Dog Art

Figure 1.32 Typical minor losses of valves when they

are partially open (F M White, 2008, Fluid Mechanics,

New York: McGraw Hill Reproduced with permission

of The McGraw-Hill Companies)

0.6 Sharp- edged

v

l t

0.5

1.0

b a

Figure 1.33 Entry losses to re-entrant inlets (a), rounded inlets (b) and bevelled inlets (c) Note that

the exit losses are K 5 1 for all exit shapes (F M White, 2008, Fluid Mechanics, New York: McGraw

Hill Reproduced with permission of The McGraw-Hill Companies)

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Hydraulic diameter

When the pipes and ducts are not circular, we can use the hydraulic diameter D h in place for the

diameter of the circular pipe to calculate pipe losses The hydraulic diameter is defined by

D h 5 4 3 cross-sectional areas wetted perimeter (1.64)

With this concept, we can obtain the friction factor of non-circular pipes and ducts using the

Moody chart just as we have obtained the friction factor for a circular pipe from it Here, the

Reynolds number and the relative roughness can be defined by VD h

d D

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