06. GATE - 13 - Mech - Fluid Mechanics - V.venkateswarlu

7/28/2019 06. GATE - 13 - Mech - Fluid Mechanics - V.venkateswarlu

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MECHANICAL ENGINEERING FLUID MECHANICS

2 VIGNAN UNIVERSITY

Table 1.1 commonly used Derived Terms in Fluid Mechanics

Derived term Dimension SI unit Abbreviation

Area (L2) m2Volume (L3) m3

Velocity (LT-1) m/s

Acceleration (LT-2) m/s2Force (MLT-2) NPressure or stress (ML-1T-2) N/m2 Pascal; Pa = N/m2

Energy or work (ML2T-2) N.m Joule ; J=N.M.

Power (ML2T-3) J/s Watt ; W=J/sDensity (ML-3) Kg/m3

Viscosity (ML-1T-1) Kg/m.s (N.s/m2) Pa.s

Surface tension (MT-2) N/m

DENSITY, SPECIFIC VOLUME AND SPECIFIC WEIGHT

DensityThe density () of a fluid is its mass per unit volume. The units are kg/rn3. In general, thedensity of a fluid depends upon the temperature and pressure. For incompressible fluids(liquids), the variation of density with pressure is however small.

Specific VolumeThe reciprocal of mass density is known as specific volume; it represents volume perunit mass of the fluid and has units of m3/kg.

Specific WeightThe specific weight of a fluid is its weight per unit volume, thus,=

g in units of N/m2

The standard value of acceleration due to gravity g is 9.086 m 2/s and is usuallytaken as 9.81 m2/s. At 20C temperature and one atmospheric pressure (760 mm ofmercury) the density of water is 998 kg/rn3. Thus, the specific weight of water at 20Ctemperature and 1 atmospheric pressure (known as NTP = normal temperature andpressure) is

= g =998 x 9.81 = 9790N/m3

= 9.79 kN/m3 Relative Density (or Specific Gravity)

Relative Density (RD) of a fluid is the ratio of its density to that of standard referencefluid, water (for liquids) and air (for gasses). In engineering practice, the term specificgravity (SG or S) is used synonymously with the term relative density. Thus

RDliquid = (SGliquid) =)/(998

)/(3

3

mkg

mkgliquidofDeisnty

RDgas = (SGgas)=)/(205.1

)/(3

3

mkg

mkggasofDeisnty

For example if the relative density of a liquid is 0.85, it means that its density is 0.850 x998 = 848.3kg/m3. Commonly used values of approximate specific gravities in fluid flowcalculations are 1.0 for water and 13.6 for mercury. When no other information is


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available, the following values corresponding to NTP (20C temperature and oneatmospheric pressure) are used:

Item Water Air

Density

Specific gravity (=Relative density)Specific weight

998 mg/m31.00

9790 N/m3

(=9.79kN/m3)

1.205 kg/m31.0

11.82 N/m3

Unless otherwise stated, the above values are used for p and y(for water and air) in thisbook. [Note: In approximate/quick calculations, for water [= 1000 kg/m

3 and, = 9.8 or

10.0 kN/m3 are used].

VISCOSITYShear Stress:-While the pressure, a normal stress, is encountered in both fluid staticand dynamic conditions the shear stress (r) is encountered only in real fluids and alsoonly when they are in motion. The unit of shear stress is N/m2 and is designated in Pa orkPa depending on the magnitude.Viscosity:-Dynamic Viscosity is the resisting property of a fluid to shearing force. The

shear stress is related to the deformation rate in most of the commonly occurring fluidsby the Newtons law of viscosity, as

dy

du

Wheredy

du= velocity gradient in the Y direction and = coefficient of viscosity, which is

a fluid property. The fluids which obey Newtons law of viscosity are known asNewtonian fluids. Most of the common liquids like water, kerosene, petrol, ethanol,benzene, Glycerin and mercury are Newtonian, Further, all gases are Newtonian.

The coefficient of viscosity, , is also known variously as the coefficient of dynamic

viscosity, absolute viscosity or simply as viscosity. It has the units

sPamsN

m

sm

mN

dy

du./.

/

/ 22

Sometimes, the coefficient of dynamic viscosity p is designated by a unitpoise(abbreviated as P) or as centipoises (abbreviated as CP) where

1 poise2

sec.1

sec.1

cm

onddyne

ondcm

gm

sPamsN .

10001.

)10(10

222

5

1 centipoise sPapoise .1000

1

100

1

The coefficient of viscosity depends upon the temperature. Generally, for liquids the

value of decreases with an increase in temperature, and for gases, the value of p

increases with an increase in the temperature.


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Kinematic Viscosity: The ratio of dynamic viscosity to the density of the fluid is knownas kinematic viscosity. This term is designated by the Greek letter v(nu) and has the

dimensions

T

L2as shown below:

smmmg

smkg

mkg

msNv /

.

.

/

/. 23

11

3

2

Sometimes, the kinematic viscosity vis designated by a unit stoke or as centistoke where

1 stoke = sms

m

ond

cm/10)10(1

sec1 24

2

22

2

1 centistoke = smstoke /10100

1 26

Table given below gives the dynamic and kinematic viscosities of some commonly usedfluids at 20C and 1 atm pressure.

Fluids Density

DynamicViscosity

(Ns/m2)

Kinematicviscosity

v(m2/s)

Surfacetension

(N/m)

Bulkmodulus

K(N/m2)

a) LiquidsWaterSea waterPetrolKeroseneGlycerineMercurySAE 10 oilSAE 30 oilCastor oil

9981025680804126013550917917960

1.00 x 10-31.07 x 10-3

2.92 x 10-41.92 x 10-31.491.56 x 10-31.04 x 10-1

2.90 x l0-1

9.80 x 10-1

1.00x106

1.04 x 106

4.29 x 1072.39 x 104

1.18 x 103

1.15 x 10-7

1.13 x 10-4

3.16 x 10-4

1.02 x 10-3

7.28x10-2

7.2 x l0-2

2.16x10-22.80 x 10-26.33 x 1024.84x 10-1

3.60x10-23.50x10-23.92x10-2

2.19 x 1092.28 x 1099.58 x 1081.43 x I09

4.34 x 109

2.55 x 1010

1.31 x 1091.38 x 109

1.44 x 109

(kg/m3) (Ns/m

2) V(m2/sSpecific

heat ratio,k=cp/cv

b) GasesAirCarbon dioxideHydrogenNitrogenMethaneOxygenWater Vapour

1.2051.8400.0841.1600.6681.3300.747

1.80 x10-51.48 x 10-50.90 x 10-5l.76 x 1051.34 x 105200 x 105101 x 105

1.494 x 10-50.804 x 10-5

10.714 x 10-51.5I7 x 10-52.000 x 10-51504 x 10-51352 x 10-5

1.401.281.401.401.30140133

Non-Newtonian Fluids

While most of the common fluids like water, air, petrol, ethanol and benzene followNewtons law of viscosity there exists a large number of fluids which do not follow

thislinear relationship between the shear stress and the rate of deformation,dy

du.

Such fluids which donot obey Newtons law of viscosity are known as Non-


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Newtonian fluids. Typical examples of non- Newtonian fluids are blood, suspensionof corn starch in water, paint, slurries, pastes and polymer solutions.In the non-Newtonian fluids, such as the ones mentioned above, the

relationshipbetween rate of deformation,dy

du, and the shear stress can in

general be expressed as a power law relation liken

dy

dum

In this, m is known as consistency index and the power n is the flow index.

When n < 1, the fluid is known as non-Newtonian pseudo plastic fluid. Gelatine,milk and blood are typical examples of pseudo plastic fluids.

When n > 1, the fluid is known as non- Newtonian dilatant fluid. Starch suspension,sugar solution and high-concentration sand suspension are typical examples ofdilatant fluids.

It may be noted that in above Eq, the case of n = 1 represents a Newtonian fluid,

with m = . The relationship between and

dy

duis known as rheological behavior and Fig. 1.1

is a schematic representation of rheological classification of fluids.

Fig. 1.1 Rheological Classification of Fluids

In Fig. 1.1, the x-axis also represents a Newtonian fluid with = 0, that is a fluid

with zero viscosity. Such fluid called an ideal fluid or inviscid fluid. Whendy

duzero

for all , the situation is represents an elastic solid. Some non-Newtonian fluidscan be modeled as

dydu

py

Such fluids which require a yield stress z,, for the flow to be established, areknown as Bingham plastic.While the above non-Newtonian fluids are time independent, there exist somenon-Newtonian fluids which are time dependent, that is the shear stress andcorresponding deformation rate are functions of time.


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SURFACE TENSION

The horizontal components of cohesive force of the molecules keep a fluid

particle on the surface under tension and this tensile force acting normal to a unit

length on the surface is called surface tension (sigma).

Referring to above figure, the molecule p with diameter 2aexperiences equal

attraction from surrounding molecules at all direction. But the molecule q on the

surface experiences a resultant inward pull due to unbalanced cohesive force of

the molecules.

The dimensional formula for surface tension is MT2, as is considered as forceper unit length. . The most common interfaces and values of , for clean

surface at 20C, are = 0.073 N/m for air-water interface and = 0.480 N/m for air-mercury interface.

Note that the surface tension has the dimension of force/unit length (N/m).When a liquid interface interacts with a solid surface, a contact angle is

formed. For water-clean glass surface = 0oand for mercury-clean glass =

130.

Due to surface tension, pressure changes occur across a curved interface. The pressuredifference between inside and outside of a curved surface lip is related to the radius ofcurvature R and surface tension as

(i) For the interior of a liquid cylinder

Rp

(ii) For a spherical droplet

Rp

2

(iii) A soap bubble has two surfaces and the pressure difference is given by

R

p4

Thus, the pressure inside a droplet or a soap bubble will be higher than the surroundingatmosphere. The pressure inside will be higher, the smaller the size of the droplet orbubble:

CapillarityLiquids have both cohesion and adhesion, which are forms of molecular attraction.Capillarity, the rise (or fall) of liquid in small-diameter tubes is due to this attraction.Liquids, such as water, which wet a surface cause capillary rise.


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In non-wetting liquids (e.g. mercury) capillary depression is caused.

For a cylindrical glass tube the capillary rise (or depression) h is given by

Rh

cos2 Where = contact angle,

= unit weight of the liquid ( g),

R radius of curvature of the glass tube = coefficient of surface tension.

For clean glass and water can be assumed to be zero. For clean mercury-air-glass

interface, =130.

COMPRESSIBILITYBulk modulus, Evis defined as the ratio of the change in pressure to the rate of changeof volume due to the change in pressure. It can also be expressed in terms of change ofdensity.

K = dp/(dv/v) = dp/(d/)

where dp is the change in pressure causing a change in volume dv when the original

volume was v. The unit is the same as that of pressure, obviously. Note that dv/v= d/.

The negative sign indicates that if dp is positive then dv is negative and viceversa, so that the bulk modulus is always positive (N/m 2). The symbol used in this textfor bulk modulus is K.

This definition can be applied to liquids as such, without any modifications. In thecase of gases, the value of compressibility will depend on the process law for thechange of volume and will be different for different processes.

The bulk modulus for liquids depends on both pressure and temperature. Thevalue increases with pressure as dvwill be lower at higher pressures for the same valueof dp. With temperature the bulk modulus of liquids generally increases, reaches amaximum and then decreases. For water the maximum is at about 50C. The value is in

the range of 2000 MN/m2

or 2000 106 N/m2

or about 20,000 atm. Bulk modulusinfluences the velocity of sound in the medium, which equals (K/)0.5.

Velocity of Propagation of Sound (C)

Sound is propagated in fluid due to compressibility of the medium, and the speed of soundC is given by

C =

K


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Where K = bulk modulus of elasticity of the medium and = mass density of the fluid.

VAPOUR PRESSUREVapor pressure is defined as the pressure at which a liquid will boil (vaporize). Vapor

pressure rises as temperature rises. In many liquid flow situations such as in hydraulicmachines and in flow through constricted passages, a low pressure approaching vapourpressure of the liquid may occur. When this happens, the liquid flashes into vapour forminga rapidly expanding cavity, This phenomenon, known as cavitation, has serious implicationson the operating performance of hydraulic machines and passages of high-speed flowsVapour pressure of a liquid depends upon temperature and increases with it. At 20C, waterhas a vapour pressure (pv) of 2.34 kPa (i.e. vapour pressure head = Pv/ = 0 24 m)

2. Fluid StaticsPRESSURE

Definition and UnitsPressure is the compressive stress on the fluid and is given by

Pressure p = AArea

FForce

for uniform pressure.

Pressure p =dA

Fdfor variable pressure

The units of pressure are N/m2 = Pa. (Pa is the abbreviation for Pascal)1 Pa = 1 Pascal= I N/m21 kPa = 1 kilo Pascal = 1000 N/m2 Bar is a unit extensively used in meteorology and in calculations involving atmosphereand high pressures. Here, 1 bar = 105 Pa = 100 kN/m2

One bar is approximately equal to standard atmospheric pressure at sea level which is

101,325 kN/m2

.

Atmospheric Pressure

The pressure of 101,325 N/m2 = 101.325 kPa is called one atmosphere and is denoted by1 atm.

The standard Temperature and Pressure (STP) defined by IUPAC, is air pressure at 0C(273.16K = 32F) at 1 atmospheric pressure ( 1 atm = 101.325 N/rn 2 = 101.325 kPa = 760mm of mercury = 10.336 m of water).

Normal Temperature and Pressure (NTP) is a standard commonly used in engineering

practice and refers to 20C temperature and I atmospheric pressure (1 atm 101.325 kPa).

It is common to express the pressure in terms of the height of an equivalent column of afluid of density. Thus

=gh = h and

h (meters of fluid) =)/(

)/(3

2

mN

mNp

g

p

In such cases, h is called the pressure head.


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For example,i) A pressure head of 5.0 m of water is equivalent to a pressure of 5.0 x 9790 = 48950 Pa48.95 kPa.ii) Similarly, a pressure of 4.0 kPa is equivalent to a pressure head h of mercury where

h = mmm 04.3003004.097906.13

4000

of mercury.

Pressure in a Static FluidThe basic law relating to the pressure (normal stresses) in a static fluid is Pascals lawwhich states that (he pressure at a point in a fluid at rest is same in all directions. Forincompressible fluids (i.e., for liquids and such of the gas flow situations wherecompressibility effects can be ignored), the variation of pressure in vertical direction in astatic fluid is given by

dz

dp

)()(2112zzpp = constant

Where = where y = Specific weight of the fluid

and Z = Vertical distance measured from a datum (positive upward).

At a free surface the pressure is atmospheric. If h is the depth below the free surface of apoint M, the absolute pressure at M (Fig. 2.2) is

atmmphabsp )(

If the pressure in excess of atmosphere is recorded then

hppabspmatmm

)(

Fig.2.2

Note: That h is measured positive downwards from the liquid surface.

The pressure Pm is then called gauge pressure.


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The linear variation of pressure with depth below the free surface is known as hydrostaticpressure distribution.The variation of gauge pressure in a liquid below the free surface is shown in Fig givenbelow. From this, P1 = h1 and p2= h2, or

)()(1212hhpp

Note that in the above the atmospheric pressure was assumed as the datum, i.e.,reference with a zero value, Different references can be taken and depending upon thereference pressures we have the following:

Absolute pressure is the pressure measured above the absolute zero, Absolute pressurescannot be negative.

Gauge pressure is the pressure measured with respect to local atmospheric pressure.Gauge pressures are extensively used in engineering practice and as such are indicatedwith a symbol or a numeral without any other explanatory notation. e.g. 14.0 kPa, 3.2kPa, Pm are gauge pressures.

Gauge pressures can be po sit ive or negative.

Negative gauge pressures are also cal led vacuum p ressures.

It is seen thatAbso lute pressure = (Local atmosph eric pressure) + (gauge pressure)

Pressure has the dimension of [Force/Area] = [FL-2] and is usually expressed in pascalskPa (= N/m3); kilo pascals kPa (= 103 N/m2); height h of a colunm of a fluid of specificweight y, in bars (= 105 Pa) or atmospheres ( number of standard atmosphcric pressurevalue). The pressures are commonly indicated as gauge pressures and unless a pressureis specifically marked absolute the pressure is treated as gauge pressure. Theatmosphere, however, is an exception and is an absolute pressure unit.

Gauge pressures are commonly measured by a Bourdon gauge. Differences in pressuresare measured by manometers.

Local atmospheric pressure (i.e. the absolute pressure of the atmosphere at a place) ismeasured by a mercury barometer. The local atmospheric pressure varies with theelevation above mean sea level and local meteorological conditions. For engineeringapplication, a standard atmospheric pressure at mean sea level at 15C is often used. Thevalue of this standard atmospheric pressure (called 1 atmosphere)is

1 atm = 10.336 m of water

760 mm of mercury = 101.325 kPa

= 10132.5 mbar


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Aneroid barometer is another instrument commonly used to measure local atmosphericpressure.

Aerostatics

The variation of pressure in the earths atmosphere is of importance in many aspects ofengineering. The study of atmosphere in its state of static equilibrium is known asaerostatics. It is generally observed that from sea level up to an elevation of about 11,000m the temperature varies linearly with the elevation. This region is known as troposphere.Beyond 11,000 m up to 24,000 m the region is known as stratosphere and thetemperature is found to be approximately constant at 216.5K in this region. Threeapproaches used in aerostatics studies are given below.

Density-Pressure Relationship in Compressible Fluids

For a compressible fluid, the density changes with pressure and temperature. For aperfect gas

p = pRT (2.4)Where p = absolute pressure

= mass density

T = absolute temperature (in Kelvin),

R = gas constant

Since = gdz

dp (2.5)

dzfdp

Depending upon the process involved, i.e., isothermal, constant temperature lapse rate or

adiabatic, the corresponding variation of pressure with Z can be determined.(1) Isothermal Process

In an isothermal process, T= T0 = constant.

SinceRT

p

RT

p

dZ

dp

2

1

2

1

dZRT

g

p

dp

o

0

12

1

2

)(exp RT

zzg

p

p

(2) Non-Isothermal Atmosphere

It is usual to consider that in troposphere the temperature decreases linearly with elevationas

ZTTo


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where To = Absolute temperature at sea level (that isat Z= 0)

T = Temperature at an elevation Z above sea level

= a constant known as lapse ratet

For standard atmosphere, = 6.5 K/km andat sea level, tempereture To = 285 K and

density

Po 101.325 kg/m3.

Variation of Pressure with Elevation

)( ZTR

p

RT

p

o

Substituting equation

)( ZTR

pgg

dz

dp

o

)( ZTR

gdZ

p

dp

o

on integration

o

o

oT

ZT

R

g

p

p

lnln

Rg

ooT

Z

p

p/

1

3) Adiabatic Process

For the case of adiabatic process (zero heat transfer), if there is no friction (isentropic)

k

p

pConstant = Cs

where k adiabatic constant for the gas. Combining with perfect gas law (Eq. 2.4) we get

1k

T

Constant

And by using (Eq. 2.5), on integration

k

k

T1

= constant

Substituting equation and on simplification

1

1

1

12

1

2 )()1(

1k

k

p

zzg

k

k

p

p

The variation of the temperature with Z in adiabatic process is given by

1

12

1

2)()1(

1RT

zzg

k

k

T

T

The rate of variation of the temperature with elevationdZ

dTis known as lapse rate (L) and

for the atmosphere having adiabatic process it is given by


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k

k

R

g

dZ

dTL

1

Measurement of pressureManometers

(i )Simple Manometer.

Simple manometers are those which measure pressure at a point in a fluidcontained in apipe orvessel.

Types.

(a) Piezometer:Measures gauge pressure only. Gas pressure cannot bemeasured as they do not form free atmospheric surface. Piezometers arealso used to measure pressure heads in pipes where the liquid is in motion. Such tubes should enter the pipe in a direction at right angles to the directionof flow.

(b) U-tube manometer: The tube contains a liquid of specific gravity greaterthan that of thefluid which the pressure is to be measured.

01

2

1

s

syz

ws

PA

1

2

1 s

syz

ws

PA

01

212

1

s

shh

wS

PA

2

1

2

1

1

hs

sh

wS

PA

AU-tube manometer can be used to measure negative or vacuum pressure.

- (c) Single column manometer:

A

asss

s

h

ws

PA )(

122

1

2

1

In case of inclined,

A

asss

s

h

ws

PA )(sin

122

1

1

1

Advantage: Only one reading is required. Negative gauge pressure canbe measured.


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(i i)Differential Manometers.

Differential manometers measure the difference of pressure between any two points in afluid contained in the pipe or vessel. These are used for measuring the pressure differencebetween any two points in a pipe or in two pipes or containers.

Types.(a) Two piezometer manometer.(b) Inverted U-tube manometer.

1hs

w

P

w

PBA

When U-tube is filled with a liquid of specific gravity S2, where, S2 < S1 then

)( 21 sshw

P

w

P BA

(i i i)U-tube differential manometer.

)( 12 ssxw

P

w

P BA

(iv)Micro manometer.These are used for the measurement of very small pressure difference.

(vii) MECHANICALGAUGES.

These are pressure measuring devices.Generally, these are used to measure highpressures and where high precision is not required.

Commonly used pressure gauges are

(i) Bourdon tube pressure gauge(ii) Diaphragm pressure gauge: low pressure intensities similar to

averoid parameter.(iii) Bellows pressure gauge.

(iv) Dead-cut pressure gauge: used to serve as a comparisondevice.

Corrections for Manometers and Gauges.

(a) At the gauge point hole should be drilled normal to the surface.

(b) Hole should be about 3 mm to 6 mm.

FORCES ON PLANE SURFACES

An important problem in the design of hydraulic structures and other structures whichinteract with fluids is the computation of hydrostatic forces on plane surfaces.Computations of magnitude and point of application of hydrostatic forces on planesurfaces are described.

Magnitude of Force on a Plane

When a plane area is immersed in a static liquid with its plane making an angle with thefree liquid surface the total hydrostatic force on one side of the area is

AhF

Where = specific weight of the liquid


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h = depth of the centre of gravity of the area below the free surface

A = area of the immersed plane.

It may be noted that the force F is independent of the angle of inclination so long as the

depth of the centroid h is unchanged.

Centre of PressureThe point of application of the force F on the submerged area is called the centre ofpressure. Considering the line of intersection of the plane area with the liquid surface(Line OX) as the reference axis, the centre of pressure is located along the planeat

yA

Iyy

gg

p Where

ggI = moment of inertia about an axis parallel to OX and passing through the centre of

gravity of the area

y = location of the centre of gravity with respect to the axis OX

A = area of the plane area

Note that the distances y are measured along the plane from the axis OX.

The lateral position of the centre of pressure with respect to any axis Perpendicular to OXand lying in the plane of the lamina is

yA

Ixx

xy

p

Where

xyI = product of inertia (= dAsy ) of the area about axis GY, passing through the centreof gravity of the area and parallel to OY and OX.

When either of the centroidal axes x = x or y = y = is an axis of symmetry,Ixy = 0 and x=

1.

Properties of some commonly encountered simple geometrical shapes are collated inTable 2.6

FORCES ON CURVED SURFACES

When the fluid static force on a curved submerged surface is desired, it is convenient toconsider the horizontal and vertical components of the force separately.


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Horizontal Component

The horizontal component of hydrostatic force in any chosen direction on any area (planeor curved) is equal to the projection of the area on a vertical plane normal to the chosendirection. The horizontal force acts through the centre of pressure of the vertical projection.

Vertical Component

The vertical component of the hydrostatic force on any surface (plane or curved) is equalto the weight of volume of liquid extending above the surface of the object to the level ofthe free surface. This vertical component passes through the centre of gravity of thevolume considered. The volume and the free surface can be real or imaginary.

Tensile Stress in a Pipe or Shell

In a circular pipe subjected to high pressure, the pressure centre can be taken to be at thecentre of the pipe. The tensile circumferential stress (hoop stress) in a pipe wall subjected-to an internal pressure of p (Fig. 2.6) is

I Moment of inertia about indicated axis

Ic = Moment of inertia about indicated axis passing through the centre of gravity of thearea

hoop stress

r

pDh

2

where D = diameter of the pipe

t = thickness of pipe.

This formula assumes t/D < 0.1 and hence is based on thin cylinder theory. If the ends of

a cylinder are closed and the cylinder has a fluid under pressure, a longitudinal stress L

is produced in the cylinder. This stress is given by

t

hL

pD

42

1

For thin spherical shells the tensile stress is

t

pD

s 4


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Sketch Area

Locationof

Centroid

I and Ie

Rectangle

bh

2

hy

c

12

3bh

Ic

Triangle 2

bh

3

hy

c

36

3bhI

c

Circle 4

2D

2

Dy

c 64

4DI

c

Semicircle 8

2D

3

4ryc

128

4DI

c

Ellipse 4

bh

2

hy

c

64

3bh

Ic

Semi-ellipse4

bh

3

4hyc

16

3bhI

c

Parabola3

2bh

8

3bx

c

8

3b

yc

7

2 3bhI

c

BUOYANCY

When a body is submerged or floating in a static fluid the resultant force exerted on it bythe fluid is called buoyancy force. This buoyancy force is always vertically upwards, andhas the following characteristics.

1. The buoyancy force is equal to the weight of the fluid displaced by the solid body.

2. The buoyancy force acts through the centre of gravity of the displaced volume, calledthe centre of buoyancy.

3. A floating body displaces a volume of fluid whose weight is equal to the weight of thebody.

Stability

A submerged body is stable if the centre of gravity of the body lies below the centre ofbuoyancy.

For a floating body the stability depends upon the type of couple that is formed for smallangular displacements. For a body shown in Fig. 2.7(a) the centre of gravity is G and the


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centre of buoyancy is B. Initially it is stable with G above B. Figure 2.7(b) shows the samebody with a small displacement. If B is the new centre of buoyancy a vertical from Bintersects the line of symmetry through G at M. M is known as the meta centre. If M isabove G, then MG the metacentric height is positive and the equilibrium is stable. If M isbelow G, MG is negative and equilibrium is unstable. The metacentric height MG isindependent of magnitude of angular rotation (so long as it is small) and is given by

BGMG

1

Fig 2.7In this equation

I =Moment of inertia of the water line area about an axis through the centre of the areaand perpendicular to the axis of tilt (longitudinal axis).

BG = Vertical distance between the centre of gravity and centre of buoyancy.

= Volume of the fluid displaced by the body.

If M coincides with G, MG is zero, the body is said to be in neutral equilibrium.

RIGID BODY MOTION

When a fluid mass in a container is subjected to a motion such that there is no relative

motion between the particles, such a motion is known as rigid body motion. The motioncan be either translation or rotation at constant acceleration or a combination of both. Asthere is no relative motion there is no shear stress in such a motion and the pressuredistribution is similar to that in fluids at rest, of course modified by the combined action ofgravity and fluid acceleration.

Translation

If a container with a fluid is given a translation (a linear motion) with a uniform accelerationthe piezometric head will have a gradient in the direction of motion.

If the motion is in the x-direction with a constant acceleration a, then

g

a

dx

dhx tan :

Where h = (p/ + z) = piezometric head above datum

= Inclination of hydraulic grade line.

= Inclination of water surface, measured clockwise with respect to the x-direction.

Thus, if a vessel containing a liquid is given an acceleration a in x-direction (Fig. 2.8) thesurface will back up against the farthest side, i.e., it will have increasing depth in (-x)direction.


19/97



Fig.2.8

If a closed tank without a free surface is involved, an imaginary free surface equivalent tothe piezometric head line can be considered. This piezometric head line will be inclined tothe x-direction such that

gax /tan

It follows from the above that if acceleration is solely in the vertical direction (+ z direction)then ax = 0 and tan = 0. This means that the liquid surface will remain horizontal.

However, the pressure ph at any depth h below the free surface will now be

g

ahP zh 1

In this az = vertical acceleration in + z direction (if the acceleration is vertically downwards,az is taken as negative).

In vertical acceleration the liquid suffers an apparent gravity equal to (g + az).

If the acceleration is a in any direction s, then the components ax and az in x- and z-directions are considered. The fluid surface will now have an inclination tan given by

)(tan

z

x

ag

a

dx

dh

Rigid Body RotationWhen a vessel containing a liquid with a free surface is rotated about an axis, the freesurface will be a paraboloid of revolution given by

g

ry

2

22

where = angular velocity

y = height of the free surface above the vertex at a radial distance r from the

At any two points r1 and r2 from the axis

)(2

)( 21

2

2

2

12rr

gyy

Since r = V = tangential velocity.

yyy )(12 = difference in the liquid surface elevation between the points 2 and 1 (Fig.

2.9)


20/97



Fig.2.9

g

V

g

V

g

V

222

22

1

2

2

= difference in the velocity head at these two points

The pressure distribution in any vertical line at a radial distance r will, however, remainhydrostatic. At point 2, h2 = + Z for all values of A on this vertical line.

If the free surface does not exist, the piezometric head will follow the relation for y as:

g

rhh

2)(

22

0

where

h = piezometric head above a datum at any radial distance r from the axis

h0 = value ofh at r = 0, i.e. on the axis

= angular velocity.

The piezometric head h =

zp

will vary with r as a paraboloid of revolution and this

surface can be considered as an imaginary liquid surface. The volume of a paraboloid ofrevolution is one half the volume of the circumscribing cylinder.

3. Fluid Flow Kinematics

Classification of flow

A) Steady flow: Fluid flow conditions at any point do not change with time. For example

0

t

V, 0

t

p, 0

t

In a steady flow steam line, path line and streak line are identical.

Unsteady Flow: Flow parameters at any point change with time, e.g., 0

t

V.

B) Uniform flow: The velocity vector V is identically same at al points at a given instant.

Non-Uniform Flow: The velocity vector V at any instant varies from point to point.


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Streamline

In a fluid flow, a continuous line so drawn that it is tangential to the velocity vector at everypoint is known as a streamline. If the velocity vector V = iu + jv + kw then the differentialequation of a streamline is given by

w

dz

v

dy

u

dx

Stagnation Point:

A point of interest in the study of the kinematics of fluid is the occurrence of points wherethe fluid flow stops. When a stationary body is immersed in a fluid, the fluid is brought to astop. When a stationary body is immersed in a fluid, the fluid is brought to a stop at thenose of the body. Such a point where the fluid flow is brought to rest is known as thestagnation point. Thus, a stagnation point is defined as a point in the flow field where the

velocity is identically zero. This means that all the components of the velocity vector V ,

viz., u, v, and w are identically zero at the stagnation point. Pitot tube which is used tomeasure the velocity in a fluid flow is an example where the properties of the stagnation

point are made use.

Acceleration:

Accleration is a vector.

i) In the natural co-ordinate system, viz., along and across a streamline.

dt

dVa and 22 ns aaa

In the tangential direction:

s

VV

t

Va s

s

s

Fig 3.1

In the normal directionr

V

t

Va snn

2

Where r = radius of curvature of the streamline at the point, Vs = tangential component ofthe velocity V and Vn = normal component of velocity generated due to change in


22/97


23/97



Fig.3.3

(ii) When there is a variation of velocity across the cross section of a conduit, for anincompressible fluid discharge.

21 AA

vdAAvdA

In Differential Form

Cartesian co-ordinates:

0)()()(

z

w

y

v

x

u

t

For incompressible fluid (/t) = 0) and hence above Eq. is simplified as

0

z

w

y

v

x

u

ROTATIONALAND IRROTATIONAL MOTION

Consider a rectangular fluid element of sides dx and dy. Under the action of velocitiesacting on it let it undergo deformation as shown in fig given below in a time dt.

1 = angular velocity of element AB =

x

v

2 = angular velocity of element AD =

y

u

Considering the anticlockwise rotation as positive, the average of angular velocities oftwo mutually perpendicular elements is defined as the rate of rotation.

Thus rotation about z-axis

y

u

x

vx

2

1

For a three-dimensional fluid element, three rotational components as given in thefollowing are possible:


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Fig.3.5 (a) Fig.3.5 (b)

About z axis,

y

u

x

vz

2

1

About y axis,

x

w

z

uy

2

1

About x axis,

z

v

y

wx2

1

Fluid motion with one or more of the termsx

, y or z different from zero is termed

rotational motion. Twice the value of rotation about any axis is called as vorticity along that axis.Thus the equation

for vorticity along z-axis is =

y

u

x

vwzz 2

A flow is said to be irrotational if all the components of rotation are zero,

Viz. 0 zyx

i.e., 0

y

u

x

v, 0

y

w

x

u;

and 0

z

v

y

w

Thus for a two-dimensional irrotational flow

02

1

y

u

x

vz

Or 0

y

u

x

v

Circulation

In rotational fluid motion, circulation is very useful concept. Circulation is defined as theline integral of the tangential component of the velocity taken around a closed contour.The limiting value of circulation divided by the area of the closed contour, as the areatends to zero, is the vorticity along an axis normal to the area.


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Circulation is taken as positive in anticlockwise direction. Referring to Fig.

Fig.3.6

CC

wdzvdyudxSdV )(.

CcurveclosedofareaVorticity along the axis perpendicular to the plane containing the

closed curve C.

STREAM FUNCTION

In a two-dimensional flow consider two streamlines S1 and S2. The flow rate (per unit depth) ofan incompressible fluid across the two streamlines is constant and is independent of the path,(path a or path b from A to B in Fig. 3.7). A stream function is so defined that it is constant

along a streamline and the difference of s for the two streamlines is equal to the flow rate

between them. Thus AB = flow rate between S1 and S2. The flow from left to right is en as

positive, in the sign convention. The velocities u and v in x and s directions are given by

yu

And

xv

The stream function is defined as above for two dimensional flows only.

For an irrotational flow,yu

xv

= 0 and hance, 02

2

2

2

yx

That is, the Laplace equation 02

2

2

2

yx

is satisfied by the stream function in

irrotational flow. Conversely, if does not satisfy 2 =0, then the flow is rotational.

In polar coordinates vr=

r

1and

rv


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POTENTIAL FUNCTION

In irrotational flows, the velocity can be written as a gradient of a scalar function called

velocity otential.

xu

,

yv

and

zw

Considering the equation of continuity (Eq. 3.14) for an incompressible fluid.

0

z

w

y

v

x

u

and substituting the expressions for u, v and w in terms of ,

022

2

2

2

2

zyx

Thus the velocity potential satisfies the Laplace equation. Conversely, any functionwhich satisfies the Laplace equation is a possible irrotational fluid flow case.

Lines of constant are called equipotential lines and it can be shown that these lines will

form orthogonal grids with = constant lines. This fact is used in the construction of flow

nets for fluid flow analysis.

RELATION BETWEEN AND FOR2-DIMENSIONAL FLOW

exists for irrotational flow only.

yx

u

xy

v

By continuity equation 02

2

2

2

yx

By irrotational flow condition, 02

2

2

2

yx

= constant along a streamline.

= constant along an equipotential line which is normal to streamlines.

Some common Formulae in Cylindrical Co-Ordinates

1. Equation of continuity:

0)(1)()(1

V

rV

rV

rrr

For incompressible fluid flow:

01

V

rr

V

r

V rr

2. Stream function :

r

Vr1


27/97



V

3. Potential Function :

r

Vr

rrV

1

4. Laplace equation

011

2

2

22

rrrr

ELEMENTARY INVISCID PLANE FLOWS

Since the Laplace equation is linear, several interesting potential flow sitwitions can beconstructed by using elementary solutions and method of superposition. The basic flowtypes are Uniform flow, Source, Sink and Vortex. These are briefly described below.

Uniform Flow

A stream of constant velocity U in x-direction is shown in Fig. 3.8 and has

Uy and Ux

In polar coordinates

sinUr and cosUr

Fig.3.8

Line Source and Sink

A two-dimensional flow emanating from a point in the x-y plane and imagined toflow uniformly in all directions is called a source. Since the two-dimensional sourceis a line in the z-direction, it is known as a line source.

The total flow per unit time per unit length of the line source is called the strength m

of the source. The velocity at a radial distance r from the source is

r

mvr

The stream function and the potential function for such line source is given

by m and m ln r

Fig. 3.9 n 3.10


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Line Vortex

Suppose we reverse the role of and in Fig. yielding

Fig.3.11

rKln and K (3.35)

from which we get 0rv and rKv /0 representing a circulating flow. Such a flow is known as

line vortex and K in Eq. 3.35 is known as Vortex strength. The centre of the vortex is a singularpoint and the circulation r of the vortex around a circular path about the centre is given by

K2

Two-Dimensional Doublet

The limiting case of a line source approaching a line sink of equal strength while keepingconstant the product of their strength and the distance between them ( ) is known as a two

dimensional doublet, For a doublet.

ryx

y

doublet

sin

)(

)(22

ryx

xdoublet

cos

)( 22

Figure 3.12 shows the streamlines and equipotential lines in a doublet.

Fig.3.12

Other Inviscid Flows

Using the basic flow elements described above various flow situations can be createdby the method of superposition. A few examples are given below in Table 3.1.


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Table 3.1 Some Ideal Fluid Flow Simulations

Sl.No. NameCombination (and flow

description)Equation of Stream

function

1Rankine Half

Body

Source + uniform flow

[curved, roughly ellipticalhalf body]

mUr sin

2 Rankine OvalSource - sink + Uniformflow [cylindrical ovalshaped body]

)(sin21

mUr

3 Circular CylinderUniform flow + doublet[circular cy1inder r

Ur

sin

sin

4Rotating CircularCylinder

Uniform flow + doublet +vortex

[rotating circular cylinder]

rKr

Ur lnsin

sin

4. Energy Equation and Its Applications:

BERNOULLI EQUATION

Euler equation: For the frictionless flow along a streamline of an incompressible fluidthe relationship among the pressure, elevation and velocity is given by the Euler equation.

01

s

VV

s

zg

s

p

t

V

Berloulli Equation: Integration of the Euler equation for steady, incompressible fluidflow, without friction, yields the Bernoulli equation

zg

Vp

2

2

= constant = H (4.2)

It can be shown that the Bernoulli equation is applicable across the streamlines also if theflow is irrotational.

In above Eq. the term V2/2 g represents kinetic energy of the flow per unit weight of thefluid. Similarly, Z represents potential energy per unit weight. The term p/y represents flow work,i.e. the work done by the fluid on the surroundings. All the terms in above Eq. have unit of [U =(N.m/N) of fluid. The constant H is called the total energy. For any two points in a steadyirrotational flow field of an ideal fluid,

0)(22

)(2121

2

2

2

121

HHZz

g

V

g

Vpp

PRACTICAL APPLICATIONS OF BERNOULLI EQUATION

In practical applications of Bernoulli equation the restriction of frictionless flow isaccommodated by introducing a loss of energy term and the restriction of irrotational flowis waived in most of the cases. Equation 4.2 is used as a special case of the generalenergy equation. The general energy equation dealing with the conservation of energy iswritten for steady, incompressible fluid flow between two sections 1 and 2 as

HI + HE HL = H2


30/97



Where

H1 = total energy at section 1

HE = energy input to the system between sections 1 and 2

HL = energy loss due to friction, etc. between sections 1 and 2

H2 = total energy at section 2.

Energy is transferred to the system as mechanical work done on the fluid by a pump.Similarly, energy is extracted from the system by a turbine. For incompressible fluid flowall non-recoverable energy such as change of internal energy and heat transfer are usuallyclubbed under a common term energy loss.

Thus for a fluid flow system shown in Fig. 4.1 the Bernoulli equation is

H1 + HE HL = H2

Where1

2

11

12

Zg

VH

Fig.4.1

2

2

22

22

Zg

VH

HE HP = energy input per unit weight of fluid per second by the pump

HL =energy loss between points 1 and 2

ENERGY EQUATION

The general equation for conservation of energy fcr an incompressible fluid flow can bewritten as

)(22

122

2

22

1

2

11 eeZg

VPHqZ

g

VpEw

(4.4)

where

qw = heat added per unit weight of fluide1, e2 = internal energy per unit weight of fluid at the respective states

HE= external work done (i.e. shaft work added) on the fluid per unit weight of fluid from adevice such as a pump.

If the total head H= Zg

Vp

2

2

then Equation4.4 is written as

H1 = HE = [(e2 e1)- qw] = H2


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The term:

(e2 - e1) - qw = (reversible + irreversible) head

In incompressible fluid flow irreversible head is called head loss HL and represents energyloss per unit weight of fluid due to friction and other causes. Thus for an incompressible fluid

2sec1sec tionatheadTotallossHead

pumpa

assuchamachine

todueadedHead

tionatTotalhead

or H1 = HE HL = H2 (4.3 a)

When a pump is used HE = HP (a positive quantity), and when a turbine is used HE = HP (anegative quantity).

Hydraulic Grade Line

A line joining the piezometric heads at various points ma flow is known as the hydraulicgrade line (HGL).

As the piezometric head zrph the HGL represents the variation of

zrph

measured above a datum.

Energy Line

The total energyg

vh

g

Vz

pH

22

22

A line joining the elevation of total energy of a flow measured above a datum is knownas energy line. The energy line lies above the HGL by an amount of V 2/2g.

Kinetic Energy Correction Factor,

In one-dimensional method of analysis, the average velocity V is used to represent thevelocity at a cross section. The actual velocity distribution in the cross section may be non-uniform. Hence, the kinetic energy calculated by using V must be multiplied by a correction


32/97



factor to obtain proper kinetic energy at the cross section due to non-uniform velocitydistribution.

Thus the velocity head in the Bernoulli equation will beg

V

2

2

where

dAV

v

A

3

1

The term is called the kinetic energy correction factor. For uniform velocity distribution = 1.0 and in all other cases it will be greater than 1.0. Greater the non-uniformity in velocity

distribution larger will be the value of . For laminar flow through a pipe, = 2.0 and for

turbulent flow through a pipe its value varies from 1.01 to 1.20. In the absence of specificinformation about the value of a, it is usual practice to assume its value as unity.

POWER

In the case of work done over a fluid the power input into the flow is

mQHP

Where = unit weight of fluid in N/rn

3,

Q = discharge in m3/s and

Hm= head added to the flow, in m

In a pump Hm = HP is positive. In a turbine Hm = Ht is negative and power is extracted from

the flow. Ifp

= efficiency of the pump, the power input required at the pump is

p

m

in

QHP

In the case of a turbine, in r is the efficiency of the turbine, power delivered by the turbineis tmou t QHP

5. Momentum Equation and Its Applications:

LINEAR MOMENTUM EQUATION

This equation states that the vector sum of all external forces acting on a control volume ina fluid flow equals the time rate of change of linear momentum vector of the fluid mass in thecontrol volume.

The external forces are of two kinds, viz, boundary (surface) forces and body forces.

Boundary forces consist of1. Pressure intensities acting normal to a boundar Fp, and

2. Shear stresses acting tangential to a boundary Fs.

Body forces are those that depend upon the mass of the fluid in the control volume, forexample weight, Fb.

The linear momentum equation in a general flow can be written for any direction x as


33/97



xxcvxbxpxx MMM

tFFF

ou t)(

where Mx momentum flux in x-direction xQV Suffixes out represent the flux going out of

the control volume and in represent the flux coming into the control volume.

Fpx, Fsx and Fbx represent x-component of pressure force, shear force and body force

respectively acting on the control volume surface.

cvxM

t)(

= rate of change of x-momentum within the control volume. This component is

zero in a steady flow.

Thus for a steady flow, in the x-direction.

inxou txbxsxpxMMFFF )()(

inxou tx QVQv )()(

Similar momentum equations are applicable to other coordinate directions, y and z also.

Application to One-dimensional Flow

Momentum Correction Factor In one-dimensional analysis the flow characteristic in onemajor direction, say longitudinal axis direction, is considered and the variation in otherdirections neglected. Thus, for example, in the two-dimensional transition shown in Fig.5.1, the velocity distribution of u with y is accounted for by taking average velocity V=

udyB1

and Vis used in the analysis.

The discharge Q = VA.

A momentum correction factor

dAuAV2

2

1 (5.3)

is used to account for the variation of the velocity moss the area in the calculation of themomentum flux. Thus the momentum flux at section 1 is

111QVM (5.4a)

and the momentum flux at section 2 is

222QVM (5.4b)

For uniform velocity distribution = 1 and for all ber cases, > 1.0. In laminar flow through a

circular tube, = 1.33 and for turbulent flow through pipes = 1.05. By definition /3 depends

upon the nature of the velocity distribution; larger the non-uniformity greater will be the value of6. If no other information is given, it is usual practice to assume = 1.0:


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Fig.5.1

Control Volume In the application of the linear momentum equation the control volume can beassumed arbitrarily. It is usual practice to draw a control volume in such a way that (Fig. 5.2):

i) Its boundaries are normal to the direction of flow at inlets and outlets.

ii) It is inside the flow boundary and has the same alignment as the flow boundary.

iii) Wherever the magnitudes of the boundary forces (due to pressure and shear stresses)are notknown, their resultant is taken as a reaction force R (with components, R, Rand R) on the control volume. This reaction R is the Force acting on thefluid in thecontrol volume due to reaction from the boundary. The Force F of the fluid on theboundary will be equal and opposite to the reaction R.

Fig.5.2

Reaction of the Boundary, R As indicated above, the reaction of the boundary R, withcomponent Rx and Ry is the force exerted by the boundary on the fluid. In most of theapplications, R is an unknown to be determined. As such, Rx and Ry are assumed to act inchosen directions and the momentum equation written. Upon solving for Rx and Ry dependingupon the sign of the answer, the assumption is corrected, if need be. Thus, Rx and Ry can beassumed to be in positive or negative direction of x and y respectively and upon solving, thefinal answer will emerge out with the proper direction of the reaction force, R. Also,

22

yxRRR (5.5)

And its inclination to x-axis is

x

y

R

Rtan (5.6)

When at a section is given, the momentum flux past the section in the chosen x-

direction is given by

xx QVM (5.7)


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In Fig. 5.2, the momentum flux in various directions are:

at 1, in x-direction: 111 QVMx

at 1, in y-direction: 1yM = 0

at 2, in x-direction: cos222 QVMx

at 2, in y-direction: sin222 QVMy

Discharge 2211 VAVAAVQ

Forces on Moving Blades

A major application of the momentum equation relates to impact of liquid jets on blades.Figure 5.3 shows a liquid jet of velocity V impacting on a curved blade moving at a velocity u.

The static pressure is atmospheric everywhere. Relative velocity of water entering the

blade = VrV1- u, where V1 = absolute velocity of the jet.If there is no friction, the relative velocity will remain constant all over the blade. At the

exist of the blade, the relative velocity V r2= Vr = V1-u. The absolute velocity V2 is obtained asvector sum of Vrand u as in fig.

uvV r 2

The relative velocity is always assumed to leave the blade tangentially. Hence, the momentumequation can be applied to the relative velocities.

If Px is the reaction of the blade on the fluid in the control volume.

)cos(0rrrxvvQP

)1(cos0 2 rx AVP

)cos1()( 21 uVAPx

Force on the blade || xx PF in the positive x-direction

Power developed Fxu (5.10)

If a series of vanes are so arranged on a wheel that the entire jet is intercepted by oneblade or other, the discharge to be used in Eq. (5.8) is the actual discharge of the jet Q insteadof Qr.

This principle is used in Pelton turbines. In reaction turbines, the pressure on the blade isnot atmospheric and the velocity triangles have to be written for both inlet and outlet of theblades.


36/97



Fig.5.3

Momentum Equation for Steady Flow

For a control volume lying in a horizontal plane, shown in Fig. 5.2, the linear momentumequation for steady flow is written as outlined below.

Let Rx along positive x-direction and Ry in negative y-direction be the reaction of theboundary on the fluid of the control volume (cv). Then in x-direction:

cvintogoing

fluxMomentum-x

cvofoutgoing

fluxMomentum-x

direction-in xcvon

forcesallofresultantthe

Thus

)cos(cos1122122211

QVQVMMRApAp xxx (5.11)

Similarly my-direction,

sinsin0221221

QVMMRAp yyy (5.12)

For any direction, that does not lie in a horizontal plane, the component of the body force(weight of fluid in cv) should be suitably included among the forces on cv.

In the solution of Eqs 5.11 and 5.12 often, depending upon the data, the continuityequation.

A1V1 = A2V2 (5.13)

And the Bernoulli equation

2

2

222

1

2

111

22Z

g

V

g

pZ

g

V

g

p

(5.14)

Will have to be used.

THE MOMENT OF MOMENTUM EQUATION

The moment of momentum equation is based on Newtons second law applied to arotating fluid mass system. Moment of momentum about an axis is known as angularmomentum. The moment of a force about a point is torque. The moment of momentum principlestates that in a rotating system the torque exerted by the resultant force on the body withrespect to an axis is equal to the time rate of change of angular momentum.

In a steady flow rotating system, i.e. when the rotating speed is constant,


37/97



cvthe

enteringfluidof

momentumangular

cvofout

leavingfluidof

momentumAngular

elementrotating

by thefluidon the

exertedTorque

])()[( inuou tu rVrVQT (5.15)

where Q = discharge, Vu = tangential component of absolute velocity, r = moment arm ofVu out and in denote items leaving or entering a control volume (cv) respectively.

Equation (5.15) fInds considerable application in the analysis of rotc) dynamic machines,viz,, turbines, pumps, propellors, etc. In the following section, the details of reaction with rotationwith a typical application to a lawn sprinkler is given.

6. LAMINAR FLOW BASIC EQUATIONS

The basic equations which govern the motion of incompressible viscous fluid in laminar motionare called as NavkrStokes equations. In Cartesian coordinates, for two-dimensional flow,

these are:

2

2

2

2

y

u

x

uu

x

pX

y

uv

x

uu

t

u ----- (1)

2

2

2

2

y

v

x

vu

y

pY

y

vv

x

vu

t

v ------- (2)

The continuity equation isy

v

x

u

-------- (3)

These equations can be solved exactly for only a few simple flow situations.

An important result that can be obtained from the above for the two-dimensional, steady,

uniform laws in the X-direction isyx

p

-------- (4)

Which stares that in steady uniform flow the pressure gradient depends upon the existence ofviscous shear losses and its variation across the flow.

Flow in Circular Conduits

Consider a horizontal circular pipe carrying an incompressible fluid in laminar motion, asillustrated in Fig. 7.1. The following relationships for the velocity distribution, shear stress and itsdistribution and for the head loss have been established analytically.

Fig.7.1


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Velocity distribution: )(4

1 22 rRdx

dpu

------ (5)

Maximum Velocity: 2

4

1R

dx

dpum

Hence

2

1R

ruum -------- (6)

Mean velocity: 2

8

1

2R

dx

dpuV m

----- (7)

Shear stress at the boundary:

dx

dpRo

2

D

Vo

8 ------- (8)

Variation of the shear stress:R

ro -------- (9)

Pressure gradient

dx

dp:

For a horizontal pipe, for two sections 1 and 2 distance L apart,L

p

L

pp

dx

dp

21

For inclined pipes, replace

Zp

dx

dby

dx

dp(

i.e., by

ds

dh where h = p/ +Z = Piezometric head

HereL

h

L

Zp

zp

L

hh

ds

dh

2

2

1

1

21

Head Lo ss hf

Designating hf=- h = head loss in a length L

L

h

ds

dh f

Note that for a uniform flow the velocity is same all along the length and hence the energy loss

= head, loss = drop in Piezometric head.

In general, the variation of the head loss h, due to uniform laminar flow in a length L of a pipe ofdiameter D is given by,

2

32

D

YLhf

---------- (10)

This equation is known as Hagen- Poiseuille equation. Since the mean velocity


39/97



2

4D

QV

Where Q = discharge

4

128

D

QLhf

------ (11)

Power, P

Power required to overcome a head H is

QHP

Hence, in laminar flow the power required to overcome frictional resistance in a pipe oflength L and diameter D, carrying a discharge Q of a fluid of specific weight and viscosity p is

4

2128

D

LQQhP f

---- (13)

Frict ion Factor,f

It is usual to designate the frictional resistance to flow in a pipe by Darcy- Weisbach equation as

gDfLVhf2

2

------- (14)

where f= friction factor. For laminar flow

gD

fLV

D

VLhf

2

32 2

2

HenceRe

64642.

3222

VDLV

gD

D

VLf

------- (15)

orRe

64f

whereVVDRe = Reynolds number

Flow Between Two Stationary Parallel Plates

For uniform laminar flow between two stationary parallel plates separated by a distance B, anexact solution of the NavierStokes equations yields:

Fig.7.2Velocity distribution

)(2

1 2yBydx

dpv

--------(16)

2

2/2/2

B

y

B

yvm --------- (16 a)


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Maximum velocity 2

8

1B

dx

dpvm

------ (17)

123

2 2B

dx

dpvVm

-----(18)

Shear stress at the boundaryBVB

dxdp

o 6

2

--- (19)

Variation of the shear stress

2/1

20

B

yy

B

dx

dp for y < B/2

and

1

2/B

yo

for y > B/2

The head loss h in a length L is2

12

B

VLhf

Viscous Flow with a Free Surface

When a viscous uniform how takes place in laminar regime down an inclined plane with a freesurface (Fig. 7.3), the flow is similar to flow between two parallel plates. Here the depth of flow d= B12 = half the spacing between the plates.

Fig.7.3

On this basis the various parameters of the flow, viz, the velocity distribution and shear stressdistribution can be estimated.For the head loss equation,

2

3sin

d

VS

L

ho

f

---- (22)

where S0 = slope of the inclined plane

= slope of the liquid surface

= slope of the hydraulic grade line

Coutte Flow

The flow between two stationary parallel plates is a special case of a general flowsituation representing flow under pressure gradient in the gap between two parallel plates, withone of the plates moving relative to the other. This general flow, schematically represented inFig. 7.4 is called Genera! Coutte flow. In Fig. 7.4,

U = Velocity or the plane

u = velocity at a distance y from, the bottom

B = gap between the two plates.


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The solution of two-dimensional Navier-Stokes equation for the boundary conditionsrepresented in Fig. yields

By

dxdpBy

BUyu 1

2----- (23)

(dp

In this equation

dx

dp1 = pressure gradient in the direction of flow. Using the non-

dimensional pressure gradient

dx

dp

U

yBP

2

2

the velocity distribution of Equation (23) can be

represented as

B

y

B

yp

B

y

U

u1 ----- (24)

Fig.7.4

The variation of U

u

with

PB

y

fn , is shown in fig. as the variation of U

u

with B

y

for various

values of P

For non-horizontal Coutte flow, the pressure p is to be replaced by piezometic head h as

hZp

yp

Thus for inclined Coutte flow

B

y

dx

Zg

pd

gBy

B

Uyu 1

2----(25)

Fig.7.5 velocity distribution in coutte flowLimts of General Coutte flow

If U= 0,11 is easy to see that we get the case of flow between two fixed parallel plates (knownas 2-D Poisuille tIow) discussed in Sec. 7.1.2.


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Plain Coutte Flow The particular case of Coutte flow with 0

dx

dpis known as Simple

or Plain Coutte Flow.

In plain Coutte flow,

B

y

U

u or

B

yUu i.e., the velocity varies linearly from zero at the fixed

boundary to U at the moving boundary.

The velocity gradient isB

U

dy

du is constant all across the gap.

Fig.7.6velocity and shear stressdistribution in plain coutte flow

Fig. 7.6 shows the variation of velocity and shear stress across the gap between the platesin a simple Coutte flow.

CREEPING MOTION

Very slow motion of an object in an infinite expanse of a viscous fluid is known as creepingmotion. For the case of a sphere of diameter D moving with a velocity V0 in a viscous fluid, thecreeping motion occurs at the Reynolds number

0.1Re v

DVo - -----(26)

Through an analytical procedure Stokes has shown that the net longitudinal force F exertedupon the sphere is

oVDF 3 ----- (27)

This equation, known as Stokes Equation, finds application in the determination of the fallvelocity of small particles.

LUBRICATION

Whenever there is relative motion of two surfaces ui contact there exists friction and consequentloss of energy. In machine elements having moving pans. the friction is considerably reducedthrough application of lubrication and use of bearings. There are a wide variety of bearings inuse and the mechanics of commonly used bearings can be modeled through laminar flow inpassages of simple geometries. Examples of common hearings that can be analyzed by simplelaminar flow concepts include journal bearing, conical bearing, collar bearing, pedestal hearingand slipper bearing & few examples to illustrate the analysis procedure are given in the exampleset that follows. In mechanics of flow related to lubrication, it is always assumed that the flow islaminar.

VISCOMETERS

A viscometer is a device for determining the viscosity of a liquid. Many of these instruments uselaminar flow situations to estimate the viscosity of the liquid. The capillary tube viscometerutilizes the Hagen-Poiseuille equation to estimate the coefficient of viscosity M of the liquid.


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INTERNAL AND EXTERNAL FLOWS

It may be realized that the examples considered in this chapter are flows bounded by walls.Such flows are known as internal flows. If the flows are not bounded by walls such flows areknown as external flows. Both laminar flows and turbulent flows exist as internal or external

flows.8. Boundary Layer Concepts:

Fig. Boundary Layer Growth Fig. Boundary Layer Thickness

As u reaches the free stream velocity U asymptotically, the boundary Layer thickness is

defined as the value of y at which u = 0.99 U.

Rex =v

Ux is called the local Reynolds number, where v = kinematic viscosity of the fluid.

Initially the boundary layer wilt be Laminar. But around a value of Re x= 5 x I05 the flow in

the boundary layer will undergo a transition phase and soon becomes turbulent. A boundaryLayer in which the flow is turbulent is called turbulent boundary layer. In a laminar boundarylayer the velocity distribution (i.e variation of u with y) is parabolic, while it is logarithmic in aturbulent boundary Layer.

In boundary layer theory the following thicknesses of the boundary layer are defined andused:

Nominal thickness, 8: ft is the thickness, measured from the boundary at which the x-component velocity attains 99% of the free stream velocity U, i.e. the value of y at which u =O.99U.

Displacement thickness, *: It is defined as

0

* 1 dyUu

Momentum thickness, : It is defined as

0

1 dyU

u

Energy thickness, ** : it is defined as


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dyU

u

U

u

0

2

**1

Shape factor, H: The ratio /*H is said to be the shape factor of the boundary Layer.

BOUNDARY CONDITIONS

For a laminar boundary layer, the boundary conditions are:

1. At the wall y=0,u=0 andv=0.

2. At the outer edge y= ,u=U.

3. Shear stress at the wall,0

y

oy

u

The flow over a flat plate which is described in this section is a particular case in which

U= constant or the pressure gradient

dx

dpis zero. This case is also known as zero pressure

gradient flow

There are situations in which the pressure gradient can be favorable

veis

dx

dpor

adverse

veis

dx

dpin which U = f(x). These are beyond the scope of this book. The boundary

layer thickness and the local shear stress i are functions of x.

LAMINARBOUNDARYLAYER OVER A FLAT PLATE

For laminar flow over a flat plate, Blasius solved the basic boundary layer equations and

obtained analytical solution which have been verified experimentally to be remarkably accurate.The classic Blasius solution for laminar boundary layer are:

xx Re

0.5

Wherev

Uxx Re

by defining2

2UC

fo

(8.2)

where Cf= local shear stress coefficient.

We havex

fCRe

664.0 (8.3)

If the total drag force on one side of a plate of length L and width B is defined as,

2).(

2

0

UBLCdxBF

Df

L

oD


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ThenL

DfCRe

328.1 (8.4)

Wherev

ULLRe and DfC = total frictional drag coefficient.

KARMAN MOMENTUM INTEGRAL FORMULATION

This is an approximate but simple method of solving boundary layer equations. By theapplication of momentum principle to a steady boundary layer over a flat plate it can be shownthat,

xdy

U

u

U

u

xC

Uf

o

0

21

2

1(8.5)

Putting )(fU

u where /y

dffxUo )(1()(

1

0

2

Let dff )(1()(1

0

Thenx

Uo

2

(8.6)

From the boundary conditions for a laminar boundary layer,

00

)(

d

dfU

y

u

y

o

Putting

0

)(d

df

U0

(8.7)

Equating the two expressions for0

U

dx

dU 2

dxU

ud

1

Integrating with the boundary condition (x = 0, =0)

x

x

Re)/(2

xx Re

)/(2

(8.8)

Substituting the value of Sin Eq. (8.7) foro

, and simplifying,


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x

o

fU

CRe

12

2/2

(8.9)

The drag force on one side of the plate, for a plate of unit width is

L

DdxF

0

0

LDfD CLUF Re/2)5.0/(2

(8.10)

In Table 8.1, some of the commonly adopted forms of u/U= f() and the corresponding

boundary layer parameters , Cf, CDf, obtained by using the Karman momentum integral

equations, are given. After studying the Example 8.5, the reader is advised to derive all theelements listed in Table 8.1 as a good exercise.

Table 8.1 Laminar Boundary Layer Solutions for Various Velocity Profiles

u/U = f()x

x Re)/( xx Re)/(*

xfC Re

LDfC Re

Exact(Blasius)

5.00 1.729 0.664 1.328

4322 5.84 1.752 0.686 1.372

3

2

1

2

3

4.64 1.740 0.626 1.292

22 5.48 1.826 0.730 1.460

2sin

4.80 1.741 0.654 1.308

3.46 1.730 0.577 1.154

BOUNDARY CONDITIONS FOR A PROPER f()

A proper function u/U = f() must satisfy the following essential and desirable boundary

conditions:

Essential Desirable

At the wall, y =0; u = 0 02

2

y

u

At y = ; u = U 02

2

y

u

y

u


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TRANSITION FROM LAMINAR BOUNDARY LAYER

As the flow passes down the plate, i.e. as Rex increases the boundary layer thicknessincreases and soon it becomes unstable. Turbulence persists and grows in the boundary layerat higher values of Rex. It is generally believed that the transition from laminar to turbulentboundary layer takes place between Rex = 1.3 x 10

5 and 4 x 106, with the mean value of Rex =(Re

x) = 5 x 105 taken as the commonly accepted critical Reynolds number.

In a flow past a long plate, the initial part in the boundary layer up to Xcrit will be laminarand the onwards the flow of the boundary layer will be turbulent.

TURBULENT BOUNDARY LAYER

The turbulent boundary layer will have much steeper velocity gradients at the boundarythan the laminar boundary layer. The velocity distribution is logarithmic and could be

conveniently expressed i the form of a power law,nyUu /1)/(/ over a range of Reynolds

number. The power n can be 5 to 10 depending on the Reynolds number range.

Next to the boundary, in a turbulent boundary layer over a smooth bed, there exists a thinlayer called as laminar sub layer.

For Rex between 5 x 106 and 2 x 107 the velocity distribution can be expressed by the 1/7

power law,7/1)/(/ yUu . The turbulent boundary layer characteristics found by experiments

and analytical calculations, to be valid for 5 x 105


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If correction for initial laminar boundary layer is applied then,

Lx

DfC

Re

1700

)Re(log

455.058.2

The term 1700/ReL is so small that omitting it does not cause any appreciable error. In Fig. 8.3,the values of CDf are plotted against the Reynolds number ReL as obtained from variousequations for various regimes of flow.

LAMINAR SUBLAYER

The laminar sub-layer is usually very thin and its thickness 3 is found by experiments to

be*/6.11' uv

where

ou * shear velocity.

If the roughness magnitude of a surface is very small compared to ' i.e., ' , thensuch a surface is said to be hydrodynamically smooth. Roughness does not have any influencein such flows while the viscous effects predominate. Usually 25.0'/ is taken as the criterionfor hydrodynamically smooth surface (Fig. 8.4).

Fig 8.4

If the laminar sublayer thickness ' is very small compared to roughness height e,

)'.,.(, ei in such flows viscous effects are not important and the boundary is said to be

hydrodynainical rough. Usually 6'/ is taken as the criterion for hydrodynamically roughboundaries.

In the region 6'/25.0 , the boundary is in the transition regime and both viscosity androughness control the flow.


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Rough flat plate: For flow on a completely rough flat plate the local friction coefficient Cfand total drag coefficient Dfare given by

5.2

log58.187.2

1

xC

f

5.2

log62.189.1

1

LC

Df

ESTABLISHMENT OF FLOW IN A PIPE

When a flow enters a pipe from a reservoir a bounds layer forms in the pipe at theentrance. The thickness of the boundary layer in the radial direction grows along the length ofthe pipe till it merges at the centre line at a distance Le known as entrance length. The flow isuniform beyond Le. The establishment length in the laminar and turbulent flow is given by thefollowing formulae:

In laminar flow: Re07.0D

Le

In turbulent flow: 50D

Le

BOUNDARY LAYER SEPARATION

Separation Phenomenon

The flow past a flat plate held parallel to the flow is a case of boundary layer with zeropressure gradients. Flows in converging boundaries are examples of favourable pressuregradient and flows in diverging conduits or diverging boundaries are examples of adversepressure gradient flows.

In adverse pressure gradient boundary layer flow the boundary layer may at some section

leave the boundary. This is called as separation and downstream of the separation sectionturbulent eddis exist and this disturbed region is called as a wake (Fig. 8.5). Separation can takeplace in both laminar and turbulent boundary layers. The location of the separation section onthe surface of a body and the size of the wake have important bearing on the total drag forceexperienced by the body.

Fig 8.5At the separation point, the shear stress is zero and the velocity gradient 0

y

u.

Control of Separation

Separation of flow from the boundary leads to inefficiency of the flow unit. In the liftingsurfaces such as aerofoils, it may cause reduction of lift and even stalling. Diffusers, conduittransitions, pump and turbine blades and aerofoils are some common flow units whereseparation may impair the performance.

Common procedures to control separation are based on the following methodologies:

Streamlining of blunt body shapes


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Fluid injection into the boundary layer

Suction of fluid from the boundary layer

Creating a motion of the boundary wall.

Fig.8.69. TURBLENT PIPE FLOW

CHARACTERISTICS OF TURBULENCE FLOWS

Turbulence is the breakdown of orderly laminar flow in to a state of random fluctuations ofvelocity. The source of turbulence is the formation of eddies at the shear layer formed either atthe boundary or at the layer of separation at the surfaces of discontinuity in the flow. If theturbulence is generated at the wall as in internal flows it is known as wall turbulence and thosedeveloped in external flows, away from any boundary, such as in free jets, is known as freeturbulence.

For purposes of modeling, a turbulence property such as a velocity is considered to bemade up of a mean value and a fluctuating component. Thus the velocity components are

'uuu , 'vvv , 'www

Where dtuT

u

T

0

1etc. for v and w. It is obvious that 0''' wvu

The rms value (root mean square value) of the fluctuations is an important statisticalproperty of turbulence. Thus for x-component

2

1

0

22 '1

'

dtuT

urms

T

Similarly 2'v and 2'w are defined. These rms values are measures of average values

of turbulence intensities in x, y and z-directions.

The intensity of turbulence of the flow is expressed as

222 '''3

11wvu

VI

where V is the mean velocity of flow given by

)(3

1 222 wvuV

The average kinetic energy of turbulence per unit of mass is defined as

KE per unit mass = )'''(2

1 222 wvu

A correlation exists between the various turbulent fluctuations u, v and w. These are

represented as, for example, ''vu T

dtvuT

0

''1

. Similarly for ''wv , uw' and so on. These

correlations of fluctuations of velocities cause additional tangential stresses andnormal stresses due tomomentuth exchange and could be represented in a compactform as


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2

2

2

'''''

'''''

'''''

wvwuw

wvvuv

wuvuu

In this 2'u ), 2'v and 2'w are normal stresses on planes normal to x, y and z

directions respectively. The remaining are tangential (shear) Stresses Oil appropriate

planes. For example, ''vu is the turbulent shear stress on xy plane. Obviously ''vu

= ''uv and so on. These turbulent shear stresses play a very important role in the flow

mechanism and energy losses of turbulent flows.

The continuity equation for turbulent flow is written for the mean motion as

0

x

w

x

v

x

uand it should satisfy the continuity condition for the fluctuations as

0'''

x

w

x

v

x

u

Shear Stress

In turbulent flow the shear stress r is expressed as

dy

du

dy

du

dy

duturblamt

)(

where = dynamic viscosity and = eddy viscosity which is not a fluid property but

depends upon turbulence conditions of the flow. Different models are proposed for theestimation of the turbulent shear

stressdy

dut

Prandtls model assumes

22

dy

dul

t

dy

dul2

Where mixing length l = ky

In which k is the Karmans coefficient = 0.4

Karmans model assumes the mixing length to be

22 /

/

dyud

dydukl

2

2

dy

dul

t

Turbulent Flow Near a Wall

For turbulent flow near a wall typical shear stress and the velocity distributions are shownin Fig. 10.1. Three important regions are to be noted:


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06. GATE - 13 - Mech - Fluid Mechanics - V.venkateswarlu