Hydraulics and Hydraulics Machines - D.dinu, S. Liviu

7/30/2019 Hydraulics and Hydraulics Machines - D.dinu, S. Liviu

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Prof.univ.Dr.ing. DUMITRUDINUS.L.drd.ing. STAN LIVIU

HYDRAULICSAND

HYDRAULIC MACHINES

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CONTENTS

PART ONE

HYDRAULICS

1. BASIC MATHEMATICS 11

2. FLUID PROPRIETIES 17

2.1 Compressibility 182.2 Thermal dilatation 202.3 Mobility 222.4 Viscosity 22

3. EQUATIONS OF IDEAL FLUID MOTION

29

3.1 Eulers equation 293.2 Equation of continuity 323.3 The equation of state 343.4 Bernoullis equation 353.5 Plotting and energetic interpretation of

Bernoullis equation for liquids 393.6 Bernoullis equations for the relative

movement of ideal non-compressible fluid40

4. FLUID STATICS 43

4.1 The fundamental equation ofhydrostatics 43

4.2 Geometric and physical interpretation

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of the fundamental equation ofhydrostatics 45

4.3 Pascals principle 46

4.4 The principle of communicatingvessels 47

4.5 Hydrostatic forces 484.6 Archimedes principle 504.7 The floating of bodies 51

5. POTENTIAL (IRROTATIONAL) MOTION57

5.1 Plane potential motion 595.2 Rectilinear and uniform motion 635.3 The source 665.4 The whirl 695.5 The flow with and without circulation

around a circular cylinder715.6 Kutta Jukovskis theorem 75

6. IMPULSE AND MOMENT IMPULSETHEOREM 77

7. MOTION EQUATION OF THE REALFLUID81

7.1 Motion regimes of fluids 817.2 Navier Stokes equation 837.3 Bernoullis equation under the

permanent regime of a thread of real fluid877.4 Laminar motion of fluids 90

7.4.1 Velocities distribution betweentwo plane parallel boards of infinit length

907.4.2 Velocity distribution in circular

conduits 93

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7.5 Turbulent motion of fluids 977.5.1 Coefficient in turbulent motion

997.5.2 Nikuradzes diagram 102

8. FLOW THROUGH CIRCULARCONDUITS 105

9. HYDRODYNAMIC PROFILES

113

9.1 Geometric characteristics ofhydrodynamic profiles 113

9.2 The flow of fluids around wings1169.3 Forces on the hydrodynamic profiles

1199.4 Induced resistances in the case of

finite span profiles 1239.5 Networks profiles 125

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PARTONE

Hydraulics

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1. Basic mathematics

The scalar product of two vectors

kajaiaa zyx ++= and kbjbibb yx 2++= is ascalar.

Its value is:

zzyyxx babababa ++= . (1.1)

a b a= b ( )

bacos . (1.2)

The scalar product is commutative:

a =b b a . (1.3)

The vectorial product of two vectors a and b is a

vector perpendicular on the plane determined bythose vectors, directed in such a manner that the

trihedral a ,b and ba should be rectangular.

zyx

zyx

bbb

aaa

kji

ba = . (1.4)

The modulus of the vectorial product is givenby the relation:

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( )

= bababa sin . (1.5)

The vectorial product is non-commutative:

abba = (1.6)

The mixed product of three vectors a ,b and c is

a scalar.

( )

zyx

zyx

zyx

ccc

bbb

aaa

cba = . (1.7)

The double vectorial productof three vectors a ,b

and c is a vector situated in the plane ( )cb, .

The formula of the double vectorial product:

( ) ( ) ( ) cbacabcba = . (1.8)

The operator is defined by:

zk

yj

xi

+

+

= . (1.9)

applied to a scalar is called gradient.. grad=

kz

jy

ix

+

+

=

. (1.10)

scalary applied to a vector is calleddivarication. .adiva =

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z

a

y

a

x

aa z

yx

+

+

= . (1.11)

vectorially applied to a vector is calledrotor. .arota =

zyx aaazyx

kji

a

= . (1.12)

Operations with :

( ) +=+ . (1.13)

( ) baba +=+ . (1.14)

( ) baba +=+ . (1.15)When acts upon a product:- in the first place has differential and only

then vectorial proprieties;- all the vectors or the scalars upon which it

doesnt act must, in the end, be placed infront of the operator;

- it mustnt be placed alone at the end.

( ) ( ) ( ) +=+= cc . (1.16)

( ) ( ) ( ) +=+= aaaaa cc . (1.17)

( ) ( ) ( ) =+= aaaaa cc . (1.18)

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( ) ( ) ( )cc bababa += , (1.19)

( ) ( ) ( ) bababa cc = , (1.20)

( ) ( ) babrotaba c += , (1.21)

( ) ( ) abarotbba c += , (1.22)

( ) ( ) ( ) abarotbbabrotaba +++= . (1.23)

c - the scalar considered constant,

c - the scalar considered constant,

ca - the vector a considered constant,

cb - the vector b considered constant.

If:

,vba == (1.24)

then:

( ) vrotvvvv +=

2

2

. (1.25)

The streamline is a curve tangent in each ofits points to the velocity vector of the

corresponding point ( )kvjvivv zyx ++= .The equation of the streamline is obtained by

writing that the tangent to streamline is parallel tothe vector velocity in its corresponding point:

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zyx v

dz

v

dy

v

dx== . (1.26)

The whirl line is a curve tangent in each of itspoints to the whirl vector of the corresponding

point ( )kji zyx ++= .

vrot2

1= . (1.27)

The equation of the whirl line is obtained by

writing that the tangent to whirl line is parallel withthe vector whirl in its corresponding point:

zyx

dzdydx

== . (1.28)

Gauss-Ostrogradskis relation:

dadna = , (1.29)

where

- volume delimited by surface .The circulation of velocity on a curve (C) isdefined by:

= ,rdvC

(1.30)in which

dsrd = (1.31)represents the orientated element of the

curve ( - the versor of the tangent to the curve(C )).

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Fig.1.1

( ) ++=C

zyx dzvdyvdxv (1.32)

The sense of circulation depends on theadmitted sense in covering the curve.

ABMAAMBA = . (1.33)

Also:

BAAMBAMBA += . (1.34)

Stokes relation:

( )

==C

dnvrotrdv (1.35)

in which n represents the versor of the normal

to the arbitrary surface bordered by the curve(C).

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2. FLUID PROPRIETIES

As it is known, matter and therefore fluidbodies as well, has a discrete and discontinuousstructure, being made up of micro-particles(molecules, atoms, etc) that are in reciprocalinteraction.

The mechanics of fluids studies phenomenathat take place at a macroscopic scale, the scale atwhich fluids behave as if matter were continuouslydistributed.

At the same time, fluids dont have their ownshape so are easily deformed.

A continuous medium is homogenous if at aconstant temperature and pressure, its density hasonly one value in all its points.

Lastly, a continuous homogenous medium isisotropic as well if it has the same proprieties inany direction around a certain point of its mass.

In what follows we shall consider the fluid as acontinuous, deforming, homogeneous and isotropicmedium.

We shall analyse some of basic physicalproprieties of the fluids.

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2.1. Compressibility

Compressibility represents the property offluids to modify their volume under the action of avariation of pressure. To evaluate quantitativelythis property we use a physical value, called

isothermal compressibility coefficient, , that is

defined by the relation:

,1 2

=

N

m

dp

dV

V

(2.1)

in which dV represents the elementary variation ofthe initial volume, under the action of pressurevariation dp.

The coefficient is intrinsic positive; the

minus sign that appears in relation (2.1) takes intoconsideration the fact that the volume and thepressure have reverse variations, namely dv/ dp 0 the metacentre will be above the weight

centre, and the moment rM , given by the

relation (4.24) will also be positive. Fromfig.4.8.it can be noticed that, in this case, the

moment rM will tend to return the floating body

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to the initial floating 0L ; for this reason it is

called restoring moment. In this case the floatingof the body will be stable.

b) if h < 0, the metacentre is below the centre ofweight (fig.4.9 a). It can be noticed that, in this

case, the moment rM will be negative and will

slant the floating body even further. As a result,it will be called moment of force tending tocapsize, the floating of the body being unstable.

c) If h = 0, the metacentre and the centre of hullwill superpose (fig.4.9 b). Consequently, therestoring moment will be nil, and the body willfloat in equilibrium on the slanting floating.

Fig.4.9 a, b

In this case the floating is also unstable. Thus,the stability conditions of the floating are: themetacentre should be placed above the weightcentre, namely

.0>= arh (4.26)

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According to (4.24) and (4.23), we may write:

( ) gfr MMaDrDarDM +=== sinsinsin , (4.27)

where:

sinrDMf = , (4.28)

is called stability moment of form, and:

sinaDMg = , (4.29)is called stability moment of weight.

As a result, on the basis of (4.27) we canconsider the restoring moment as an algebraic sumof these two moments.

In the case of small longitudinal slantings, the

above stated considerations are also valid, therestoring moment being in this case:

( ) sinsin aRDHDMr == , (4.30)

where

aRH = . (4.31)

represents the longitudinal metacentric height, andR is the longitudinal metacentric radius.

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5. POTENTIAL (IRROTATIONAL)MOTION

The potential motion is characterised by the

fact that the whirl vector is nil.

02

1== vrot , (5.1)

hence its name: irrotational.

If is nil, its components on the three axes

will also be nil:

.02

1

,02

1

,02

1

=

=

=

=

=

=

y

v

x

v

x

v

z

v

z

v

y

v

xy

z

zx

y

yzx

(5.2)

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or:

.

,

,

y

v

x

v

x

v

z

v

z

v

y

v

xy

zx

yz

=

=

=

(5.3)

Relations (5.3) are satisfied only if velocity vderives from a function :

.,,z

vy

vx

v zyx

=

=

=

(5.4)

or vectorially:

=v . (5.5)Indeed:

( ) 0== gradrotvrot . (5.6)

Function ( )tzyx ,,, is called the potential ofvelocities.

If we apply the equation of continuity forliquids,

02

2

2

2

2

2

=

+

+

=

+

+

zyxz

v

y

v

x

v zyx , (5.7)

we shall notice that function verifies equation ofLaplace:

0= , (5.8)thus being a harmonic function.

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5.1 Plane potential motion

The motion of the fluid is called plane orbidimensional if all the particles that are found onthe same perpendicular at an immobile plane,called director plane, move parallel with this plane,with equal velocities.

If the director plane coincides with xOy, then0=zv .

A plane motion becomes unidimensional if

components xv and yv of the velocity of the fluid

depend only on a spatial co-ordinate.

For plane motion, the equation of thestreamline will be:

yx v

dy

v

dx= , (5.9)

or else:

0= dxvdyv yx , (5.10)

and the equation of continuity:

0

=

+

y

v

x

v yx

. (5.11)

The left term of the equation (5.10) is anexact total differential of function , called thestream function:

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xv

yv yx

=

=

, , (5.12)

0== dxvdyvd yx . (5.13)

Function verifies the equation of continuity(5.11):

0

22

=

=

+

xyyxy

v

x

v yx . (5.14)

Function is a harmonic one as well:

02

1

2

12

2

2

2

=

+

=

=

yxy

v

x

vxy

z

, (5.15)

0= . (5.16)

The total of the points, in which the potentialfunction is constant, define the equipotentialsurfaces.

In the case of a potential plane motion:

- constant, equipotential lines of velocity; - constant, stream lines.

Computing the circulation of velocity along acertain outline, in the mass of fluid, between pointsA and B (fig.5.1), we get:

====B

A

B

A

AB

B

A

drdrdv . (5.17)

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Thus, the circulation of velocity doesntdepend on the shape of the curve AB, but only onthe values of the function in A and B. Thecirculation of velocity is nil along an equipotential

line of velocity ( .constBA == ).If we compute the flow of liquid through the

curve AB in the plane motion (in fact through thecylindrical surface with an outline AB and unitarybreadth), we get (fig.5.1):

Fig.5.1

( ) ===B

A

B

A

AByx ddxvdyvQ 11 . (5.18)

Thus, the flow that crosses a curve does notdepend on its shape, but only on the values offunction in the extreme points. The flow through

a streamline is nil ( ).constBA == .

A streamline crosses orthogonal on anequipotential line of velocity. To demonstrate this

propriety we shall take into consideration that thegradient of a scalar function F is normal on the

level surface F = cons. As a result, vectors and are normal on the streamlines and on the

equipotential lines of velocity.

Computing their scalar product, we get:

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0=+=

+

= yxyx vvvvyyxx

. (5.19)

Since their scalar product is nil, it follows thatthey are perpendicular, therefore their streamlinesare perpendicular on the lines of velocity.

Going back to the expressions of xv and yv :

.

;

xyv

yxv

y

x

=

=

=

=

(5.20)

Relations (5.20) represent the Cauchy-Riemanns monogenic conditions for a function ofcomplex variable.

Any potential plane motion may always beplotted by means of an analytic function of complexvariable,

( )ireziyxz =+= .

The analytic function;

( ) ( ) ( )yxiyxzW ,, += , (5.21)

is called the complex potential of the planepotential motion.

Deriving (5.21) we get the complex velocity:

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yx vivy

iyx

ixdz

dW=

=

+

=

. (5.22)

Fig.5.2

( ) ievivdz

dW == sincos . (5.23)

Having found the complex potential, letsestablish a few types of plane potential motions.

5.2 Rectilinear and uniform motion

Lets consider the complex potential:

( ) zazW = , (5.24)

where a is a complex constant in the form of:

Kviva = 0 , (5.25)

with 0v and Kv real and constant positive.

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Relation (5.24) can be written in the form:

( ) ( ) ( )ixvyvyvxvizW KK ++=+= 00 , (5.26)

where from we can get the expressions of functions and :

( )

( ) .,

,,

0

0

xvyvyx

yvxvyx

K

K

=

+=

(5.27)

By equalling these relations with constants weobtain the equations of equipotential lines and ofstreamlines.

.

.

20

10

consCxvyv

consCyvxv

K

K

====+

(5.28)

From these equations we notice that the

streamlines and equipotential lines are straight,having constant slopes (fig.5.3).

Fig.5.3

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.0

,0

0

2

0

1

>=

=>=

Ky

x

vv

vv(5.32)

The vector velocity will have the modulus:

22

0 Kvvv += , (5.33)

and will have with axis Ox, the angle 2 , given by

the relation (5.29).

We can conclude that the potential vector(5.25) is a rectilinear and uniform flow on a

direction of angle 2 with the abscissa axis.

The components of velocity can be alsoobtained from relations (5.20):

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.

,0

Ky

x

vxy

v

vyx

v

=

=

=

=

=

=

(5.34)

If we particularise (5.25), by assuming 0=kv ,the potential (5.24) will take the form:

( ) zvzW 0= , (5.35)

that represents a rectilinear and uniform motion onthe direction of the axis Ox.

Analogically, assuming in (5.25) 00 =v , we get:

( ) zvizW K= , (5.36)

that is the potential vector of a rectilinear anduniform flow, of velocity Kv , on the direction of the

axis Oy.

The motion described above will have areverse sense if the corresponding expressions ofthe potential vector are taken with a reverse sign.

5.3 The source

Lets consider the complex potential:

( ) zQ

zW ln2

= , (5.37)

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where Q is a real and positive constant.

Writing the variable ierz= , this complexpotential becomes:

( ) ( )

irQ

izW +=+= ln2

, (5.38)

where from we get function and :

.2

,ln2

Q

rQ

=

=(5.39)

which, equalled with constants, give us theequations of equipotential and stream lines, in theform:

..,.

consconsr

== (5.40)

It can be noticed that the equipotential linesare concentric circles with the centre in the originof the axes, and the streamlines are concurrentlines in this point (fig.5.4).

Fig.5.4

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Knowing that:

sincos ryandrx == , (5.41)

in a point ( ),rM , the components of velocitywill be:

.01

,2

==

=

=

rv

r

Q

rv

S

r

(5.42)

It can noticed that on the circle of radius r =cons., the fluid velocity has a constant modulus,being co-linear with the vector radius of theconsidered point.

Such a plane potential motion in which theflow takes place radially, in such a manner that

along a circle of given radius velocity is constant asa modulus, is called a plane source.

Constant Q, which appears in the above -written relations, is called the flow of the source.

The flow of the source through a circularsurface of radius r and unitary breadth will be:

12 rvrQ = . (5.43)

Analogically, the complex potential of theform:

( ) zQ

zW ln2

= , (5.44)

will represent a suction or a well because, in thiscase, the sense of the velocity is reversing, the

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fluid moving from the exterior to the origin (whereit is being sucked).

If the source isnt placed in the origin of the

axes, but in a point 1O , of the real axis, of abscissaa , then:

( ) ( )azQ

zW = ln2

. (5.45)

5.4. The whirl

Let the complex potential be:

( ) zi

zW ln2

= . (5.46)

where is a positive and real constant, equal tothe circulation of velocity along a closed outline,which surrounds the origin.

Proceeding in the same manner as for theprevious case, we shall get the functions and :

,ln2

,2

r

=

=

(5.47)

from which we can notice that the equipotentiallines, of equation .const= are concurrent lines, inthe origin of axes, and the streamlines, having theequation .constr = , are concentric circles with theircentre in the origin of the axes (fig.5.5).

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Fig.5.5

The components of velocity are:

02

10 >

=

==

=rr

vandr

v Sr

. (5.48)

Thus, on a circle of given radius r, the velocityis constant as a modulus, has the direction of thetangent to this circle in the considered point and isdirected in the sense of angle increase.

If the whirl is placed on the real axis, in apoint with abscissa a , the complex potential ofthe motion will be:

( ) ( )azi

zW

ln2

= . (5.49)

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5.5. The flow with and withoutcirculation around a circular cylinder

The flow with circulation around a circular cylinderis a plane potential motion that consists of an axialstream (directed along axis Ox), a dipole of

moment *2=M (with a source at the left of suction)

and a whirl (in direct trigonometric sense).

The complex potential of motion will be:

( ) zi

z

rzvzW ln

2

2

0

0

+= , (5.50)

where we have done the denotation:

0

2

0

1

vr =

. (5.51)By writing the complex variable ierz= , we

shall divide in (5.50) the real part from theimaginary one, thus obtaining functions and :

2

cos

2

0

0

+

+=

r

rrv , (5.52)

rr

rrv ln

2sin

2

0

0

=

. (5.53)

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* The dipole or the duplet is a plane potential motion that consists of two equalsources of opposite senses, placed at an infinite small distance , so that the product,, called the moment of the dipole should be finite and constant. .


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The stream and equipotential lines areobtained by taking in relations (5.52), (5.53),

CC == , respectively. We notice that if in (5.53)we assume 0rr = , function will become constant;therefore we can infer that the circle of radius 0r

with the centre in the origin of the axes is astreamline (fig.5.8).

Admitting that this streamline is a solid

border, well be able to consider this motiondescribed by the complex potential (5.50) as beingthe flow around a straight circular cylinder of

radius 0r , having the breadth normal on the motion

plane, infinite.

If we plot the otherstreamlines we shall getsome asymmetric curves

with respect to axis Ox(fig.5.6). On the inferiorside of the circle of

radius 0r , the velocity

due to the axial streamis summed up with thevelocity due to the whirl.

Fig.5.6

As a result, here we shall obtain smallervelocities, and the streamlines will be more rare.

In polar co-ordinates, the components of

velocity in a certain point ( ),rM , will be:

cos12

2

0

0

=

r

rvvr , (5.54)

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If the considered point is placed on the circle

of radius 0r , well have:

.2

sin2

,0

0

0r

vv

v

S

r

+=

=

(5.55)

The position of stagnant points can bedetermined provided that between these points thevelocity of the fluid should be nil.

The flow without circulation around a circularcylinder is the plane potential motion made up ofan axial stream (directed along axis Ox) and adipole of moment 2=M (whose source is at theleft of suction).

Thus, this motion can be obtainedparticularising the motion previously described bycancelling the whirl.

By making 0= , in relations (5.50), (5.52) and(5.53) we get the complex potential of the motion,the function potential of velocity and the functionof stream, in the form:

( ) ,2

0

0

+=

z

rzvzW (5.56)

,cos

2

0

0

+= rr

rv (5.57)

.sin

2

0

0

=

r

rrv (5.58)

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By writing the equation of streamlines =cons. in the form:

.22

2

00 constCy

yx

ryv ==

+ (5.59)

we notice that the nil streamline (C = 0) is made upof a part of the real axis (Ox) and the circle of

radius 0r (fig.5.7).

The otherstreamlines aresymmetric curves withrespect to axis Ox.Obviously, if weconsider the circle of

radius 0r , as a solid

border, the motioncan be seen as a flow

of an axial streamaround an infinitelylong cylinder, normalon the motion plane.

Fig.5.7

The components of velocity are:

.sin1

,cos1

2

20

0

2

2

0

0

+=

=

r

rvv

r

rvv

S

r

(5.60)

which, on the circle of radius 0r , become:

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.sin2

,0

0 vv

v

S

r

==

(5.61)

The position of stagnant points is obtained by

making 0== Svv , which implies 0sin = . Thus thestagnant points are found on the axis Ox in the

points ( ),0rA and ( )0,0rB .

5.6 Kutta Jukovskis theorem

Let us consider a cylindrical body normal onthe complex plane, the outline C being the crossingcurve between the cylinder and the complex plane.

Around this outline there flows a stream,

potential plane, having the complex potential ( )zW .The velocity in infinite of the stream, directed in

the negative sense of the axis Ox, is

v .

In this case the resultant of the pressureforces will have the components:

.1

,0

==

vR

R

y

x

(5.62)

The forces are given with respect to the unitof length of the body.

The second relation (5.62) is the mathematicexpression of Kutta-Jukovskis theorem, which willbe only stated below without demonstrating it:

If a fluid of density is draining around a

body of circulation and velocity in infinite v , it

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will act upon the unit of length of the body with a

force equal to the product v , normal on thedirection of velocity in infinite called lift force(lift).

The sense of the lift is obtained by rotating

the vector of velocity from infinite with 090 in thereverse sense of circulation.

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6. IMPULSE AND MOMENTIMPULSE THEOREM

We take into consideration a volume of fluid.This fluid is homogeneous, incompressible, ofdensity , bordered by surface . The elementary

volume d has the speed v .

The elementary impulse will be:

dvId = . (6.1)

=

dvI . (6.2)

=

d

dt

vd

dt

Id. (6.3)

At the same time

iFdt

Id= . (6.4)

But: 0=++ ipm FFF (dAlembert principle).(6.5)

Therefore:

epm FFFdt

Id =+= . (6.6)

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The total derivative, of the impulse with

respect to time, is equal to the resultant eF of the

exterior forces, or

iieee vMvMF = , (6.7)

where ei MM , are the mass flows through entrance/

exit surfaces.

Under permanent flow conditions of idealfluid, the vectorial sum of the external forces whichact upon the fluid in the volume , is equal withthe impulse flow through the exit surfaces (fromthe volume ), less the impulse flow through theentrance surfaces (to the volume ) .

r- the position vector of the centre of volume

with respect to origin of the reference system.

The elementary inertia moment with respect

to point O (the origin) is:

( ) dvrdt

dd

dt

vdrMd i =

= , (6.8)

since

( ) .dt

vdr

dt

vdrvv

dt

vdrv

dt

rdvr

dt

d=+=+= (6.9)

then

( ) ==

dvrdt

dMdM ii . (6.10)

If:

dvId = the elementary impulse, (6.11)

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dvrkd = the moment of elementary impulse,(6.12)

=

,dvrk (6.13)

( ) iMdvrdt

d

dt

kd==

. (6.14)

The derivative of the resultant moment ofimpulse with respect to time is equal with theresultant moment of inertia forces with reversiblesign.

expm MMMdt

kd=+= , (6.15)

where

mM - the moment of mass forces,

pM - the moment pressure forces,

exM - the moment of external forces.

oioe rr , - the position vector of the centre of

gravity for the exit /entrance surfaces.

( ) ( )ioiieoeeex vrMvrMM = . (6.16)

Under permanent flow conditions of idealfluids, the vectorial addition of the moments ofexternal forces which act upon the fluid in thevolume , is equal to the moment of the impulseflow through the exit surfaces less the moment ofthe impulse flow through the entrance surfaces.

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7. MOTION EQUATION OF THEREAL FLUID

7.1 Motion regimes of fluids

The motion of real fluids can be carried outunder two regimes of different quality: laminar andturbulent.

These motion regimes were first emphasisedby the English physicist in mechanics OsborneReynolds in 1882, who made systematic

experimental studies concerning the flow of waterthrough glass conduits of diameter mmd 255 = .

The experimental installation, which was thenused, is schematically shown in fig.7.1.

79

Fig.7.1


80/132

The transparent conduit 1, with a veryaccurate processed inlet, is supplied by tank 2, fullof water, at a constant level.

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The flow that passes the transparent conduitcan be adjusted by means of tap 3, and measuredwith the help of graded pot 6.

In conduit 1, inside the water stream weinsert, by means of a thin tube 4, a coloured liquidof the same density as water. The flow of colouredliquid, supplied by tank 5 may be adjusted bymeans of tap 7.

But slightly turning on tap 3, through conduit1 a stream of water will pass at a certain flow andvelocity.

If we turn on tap 7 as well, the coloured liquidinserted through the thin tube 4, engages itself inthe flow in the shape of a rectilinear thread,parallel to the walls of conduit, leaving theimpression that a straight line has been drawninside the transparent conduit 1.

This regime of motion under which the fluidflows in threads that dont mix is called a laminarregime.

By slowly continuing to turn on tap 3, we cannotice that for a certain flow velocity of water, thethread of liquid begins to undulate, and for highervelocities it begins to pulsate, which shows thatvector velocity registers variations in time(pulsations).

For even higher velocities, the pulsations ofthe coloured thread of water increase theiramplitude and, at a certain moment, it will tear, theparticles of coloured liquid mixing with the mass ofwater that is flowing through conduit 1.

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The regime of motion in which, due topulsations of velocity, the particles of fluid mix iscalled a turbulent regime.

The shift from a laminar regime to theturbulent one, called a transition regime ischaracterised by a certain value of Reynolds

number * , called critical value ( crRe ).

82

* Number , is the number that defines thesimilarity criterion Reynolds.


83/132

For circular smooth conduits, the critical value

of Reynolds number is 2320Re =cr .

For values of Reynolds number inferior to the

critical value ( crReRe < ), the motion of liquid will belaminar, while for crReRe > , the flow regime will beturbulent.

7.2 Navier Stokes equation

Navier Stokes equation describes themotion of real (viscous) incompressible fluids in alaminar regime.

Unlike ideal fluids that are capable to develop

only unitary compression efforts that areexclusively due to their pressure, real (viscous)fluids can develop normal or tangentsupplementary viscosity efforts.

The expression of the tangent viscosity effort,defined by Newton (see chapter 2) is the following:

y

v

= . (7.1)

Newtonian liquids are capable to develop,under a laminar regime, viscosity efforts and ,that make-up the so-called tensor of the viscosity

efforts, vT (in fig. 7.2, efforts manifest on an

elementary parallelipipedic volume of fluid with the

sides dzanddydx, ):

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=

zzyzxz

zyyyxy

zxyxxx

vT

. (7.2)

The tensor vT is symmetrical:

yzzyxzzxxyyx === ;; . (7.3)

Fig.7.2

The elementary force of viscosity that isexerted upon the elementary volume of fluid in thedirection of axis Ox is:

( ) ( ) ( )

.dzdydxzyx

dydxdzz

dydxdyy

dzdydxx

dF

zxyxxx

zxyxxx

vx

+

+

=

=

+

+

=

(7.4)

According to the theory of elasticity:

84

z

x

y


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.

;

;2

+

=

+

=

=

z

v

x

v

y

v

x

v

x

v

xz

zx

xy

yx

x

xx

(7.5)

Thus:

.

2

2

2

2

2

2

2

2

2

2

2

22

2

2

dydzdxz

v

y

v

x

v

z

v

y

v

x

v

x

zx

v

z

v

y

v

yx

v

x

vdF

xxxzyx

zxxyx

vx

+

+

+

+

+

=

=

+

+

+

+

=

(7.6)

But 0=

+

+

z

v

y

v

x

v zyx, according to the equation

of continuity for liquids.

Then:

dzdydxvdF xx = . (7.7)

Similarly:

,dzdydxvdF yvy = (7.8).dydydxvdF zvz = (7.9)

Hence:

, dvFd v = (7.10)

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.=

dvFv (7.11)

Unlike the ideal fluids, in dAlembertsprinciple the viscosity force also appears.

.0=+++ ivpm FFFF (7.12)

Introducing relations (3.3), (3.5), (3.7) and(7.11) into (7.12), we get:

=

+

0ddt

vdvpF , (7.13)

or:

dt

vdvpF =+

1. (7.14)

Relation (7.14) is the vectorial form of Navier-Stokes equation. The scalar form of this equationis:

.

1

;1

;1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

zz

yz

xzzzzz

z

z

y

y

y

x

yyyyy

y

zx

yx

xxxxxx

x

vz

v

vy

v

vx

v

t

v

z

v

y

v

x

v

z

p

F

vz

vv

y

vv

x

v

t

v

z

v

y

v

x

v

y

pF

vz

vv

y

vv

x

v

t

v

z

v

y

v

x

v

x

pF

+

+

+

=

+

+

+

+

+

+

=

+

+

+

+

+

+

=

+

+

+

(7.15)

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7.3 Bernoullis equation under thepermanent regime of a thread of real fluid

Unlike the permanent motion of an ideal fluid,where its specific energy * remains constant along

the thread of fluid and where, from one section toanother, there takes place only the conversion of apart from the potential energy into kinetic energy,or the other way round, in permanent motion of thereal fluid, its specific energy is no longer constant.It always decreases in the sense of the movementof the fluid.

A part of the fluids energy is converted intothermal energy, is irreversibly spent to overcomethe resistance brought about by its viscosity.

Denoting this specific energy (load) by fh ,

Bernoullis equation becomes:

fhzp

g

vz

p

g

v+++=++ 2

2

2

2

1

1

2

1

22 . (7.16)

In different points of the same section, onlythe potential energy remains constant, the kineticone is different since the velocity differs in the

section, ( )zyxvv ,,= . In this case the term of thekinetic energy should be corrected by a coefficient, that considers the distribution of velocities in

the section ( )1,105,1 = .

fhzp

g

vz

p

g

v+++=++ 2

2

2

22

1

1

2

11

22

. (7.17)

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By reporting the loss of load fh to the length l

of a straight conduit, we get the hydraulic slope(fig.7.3):

Fig.7.3

l

h

l

zp

g

vz

p

g

v

If=

++

++

=2

2

2

22

1

1

2

11

22

. (7.18)

If we refer only to the potential specificenergy, we get the piezometric slope:

l

zp

zp

Ip

+

+

=2

2

1

1

. (7.19)

In the case of uniform motion ( ctv = ):

l

h

tgIIf

p === . (7.20)

Experimental researches have revealed thatirrespective of the regime under which the motionof fluid takes place, the losses of load can bewritten in the form:

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m

f vbh = , (7.21)

where b is a coefficient that considers the nature ofthe fluid, the dimensions of the conduit and thestate of its wall.

1=m for laminar regime;

275,1 =m for turbulent regime.

If we logarithm (7.21) we get:

vmbhf lglglg += . (7.22)

In fig. 7.4 the load variation fh with respect to

velocity is plotted in logarithmic co-ordinates.

Fig.7.4

For the laminar regime 045= . The shift to theturbulent regime is made for a velocity

corresponding to 2320Re =cr .

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7.4 Laminar motion of fluids

7.4.1 Velocities distribution between twoplane parallel boards of infinite length (fig.7.5).

To determine the velocity distribution

between two plane parallel boards of infinitelength, we shall integrate the equation (7.15)under the following conditions:

Fig.7.5

a) velocity has only the direction of the axisOx:

;0,0 == zyx vvv (7.23)from the equation of continuity 0=v , it results:

,0=x

vx (7.24)

therefore velocity does not vary along the axis Ox.

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b) the movement is identically reproduced inplanes parallel to xOz:

0=

y

vx. (7.25)

From (7.24) and (7.25) it results that ( )zvv xx = .

c) the motion is permanent:

0=t

vx . (7.26)

d) we leave out the massic forces (thehorizontal conduit).

e) the fluid is incompressible.

The first equation (7.15) becomes:

01

2

2

=+

dz

vd

x

p x

, (7.27)

Integrating twice (7.27):

( ) 212

2

1CzCz

x

pzvx ++

=

. (7.28)

For the case of fixed boards, we have theconditions at limit:

.0,

;0,0

====

x

x

vhz

vz(7.29)

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Subsequently:

.0

;2

1

2

1

=

=

C

hx

pC

(7.30)

Then the law of velocity distribution will be:

( ) ( )zhzx

pzvx

=2

1. (7.31)

It is noticed that the velocity distribution is

parabolic, having a maximum for2

hz= :

x

phvx

=

8

2

max* . (7.32)

Computing the mean velocity in the section:

( )

==h

xx

phdzzv

hu

0

2

12

1

, (7.33)

well notice that max3

2vu = .

The flow that passes through a section of

breadth b will be:

x

phbhbvQ

==12

3

. (7.34)

92

* is positive, since (the sense of the flow, the positivesense of axis Ox, corresponds to a decrease in pressure).


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7.4.2 Velocity distribution in circular conduits

Lets consider a circular conduit, of radius 0r

and length l, through which an incompressible fluidof density and kinematic viscosity (fig.7.6)passes.

We report the conduit to a system of

cylindrical co-ordinates ( andrx, ), the axis Ox,being the axis of the conduit. The movement beingcarried out on the direction of the axis, the velocitycomponents will be:

0,0 == vvv rx . (7.35)

The equation of continuity 0=v , written incylindrical co-ordinates:

( ) ( ) 01 =

+

+

=

xrvv

rvr

rv xr

, (7.36)

becomes:

0=x

vx , (7.37)

where from we infer that the velocity of the fluiddoesnt vary on the length of the conduit.

On the other hand, taking into consideration

the axial symmetrical character of the motion,velocity will neither depend on variable .

As a result, for a permanent motion, it will

only depend on variable r, that is ( )rvv = .

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The distribution of velocities in the section offlow can be obtained by integrating the Navier-Stokes equations (7.14).

Noting byrii, and i the versors of the three

directions of the adopted system of cylindrical co-ordinates, we can write vector velocity:

( ) irvv x= . (7.38)

Bearing in mind that in cylindrical co-ordinates, operator "" has the expression:

+

+

=r

i

ri

xi r . (7.39)

On the basis of (7.38), we can write:

( ) ( ) 0=

= xx vix

vvv , (7.40)

since, as we have seen, velocity xv only depends on

variable r.

On the other hand, in cylindrical co-ordinates, the

term v may be rendered in the form:

.

1

=

=

+

+

==

rr

v

rr

i

rx

v

xr

vr

r

v

rr

iviv

x

xxx

x

(7.41)

Keeping in mind the permanent character ofthe motion, relation (7.40) and (7.41) the projectionof equation (7.14) onto the axis Ox may be writtenin the form:

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x

pr

r

v

rr

x

=

1, (7.42)

since, on the hypothesis of a horizontal conduit,0== xx gF .

Assuming that the gradient of pressure on the

direction of axis Ox is constant ( ./ consxp = ), andintegrating the equation (7.42), we shallsuccessively get:

,2

1 1

r

Cr

x

p

r

vx +

=

(7.43)

,ln4

121

2 CrCrx

pvx ++

=

(7.44)

The integrating constants 1C and 2C are

determined using the limit conditions:

- in the axis of conduit, at r = 0, velocity

should be finite, thus constant 1C should be

nil;

- on the wall of conduit, at 0rr = , velocity offluid should be nil; consequently:

2

024

1r

x

pC

=

, (7.45)

and relation (7.44) becomes:

( )2204

1rr

x

pvx

=

. (7.46)

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From (7.46) we notice that if the motion takes

place in the positive sense of the axis ( )0>xvOx , then0/


97/132

breadth d r (fig.7.6 b). The elementary flow thatcrosses surface d A is:

rdrvdAvdQ xx 2== , (7.50)and:

( ) ==0

0

4

0

22

082

r

rI

drrrrI

Q

. (7.51)

The mean velocity has the expression:

28

max,2

0

xvr

I

A

Qu ===

. (7.52)

Further on we can write:

g

v

ddg

dv

dg

v

r

v

l

hI

f

2

1

Re

64Re32

328 2

2

2

22

0

=====

. (7.53)

Relation (7.53) is Hagen-Ppiseuilles law,which gives us the value of load linear losses in theconduits for the laminar motion:

g

v

d

l

g

v

d

lhf

22Re

64 22== , (7.54)

Re

64= is the hydraulic resistance coefficient for

laminar motion.

7.5 Turbulent motion of fluids

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In a point of the turbulent stream, the fluidvelocity registered rapid variation, in one sense orthe other, with respect to the mean velocity insection. The field of velocities has a complexstructure, still unknown, being the object ofnumerous studies.

The variation of velocity with the time may beplotted as in fig.7.7.

Fig.7.7

A particular case of turbulent motion is thequasipermanent motion (stationary on average). Inthis case, velocity, although varies in time, remainsa constant means value.

In the turbulent motion we define thefollowing velocities:

a) instantaneous velocity ( )tzyxu ,,, ;

b) mean velocity

( ) ( )=T

dttzyxuT

zyxu0

,,,1

,, ; (7.55)

c) pulsation velocity

( ) ( ) ( )zyxutzyxutzyxu ,,,,,,,,' = . (7.56)

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There are several theories that by simplifyingdescribe the turbulent motion:

a) Theory of mixing length (Prandtl), whichadmits that the impulse is kept constant.

b) Theory of whirl transports (Taylor) wherethe rotor of velocity is presumed constant.

c) Karamans theory of turbulence, whichstates that, except for the immediate vicinity of awall, the mechanism of turbulence is independentfrom viscosity.

7.5.1 Coefficient in turbulent motion

Determination of load losses in the turbulentmotion is an important problem in practice.

It had been experimentally established that in

turbulent motion the pressure loss p depends onthe following factors: mean velocity on section, v ,diameter of conduit, d , density of the fluid andits kinematic viscosity , length l of the conduitand the absolute rugosity * of its interior walls;therefore:

( )= ,,,,, ldvfp , (7.57)or:

d

lvp

2

2= , (7.58)

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d

l

g

vphf

2

2

=

= , (7.59)

ror

d

- relative rugosity

where:

=d

Re,2 1 . (7.60)

100

*mean height of the conduit prominence ; -relative rogosity.


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As it can be seen from relation (7.60), inturbulent motion the coefficient of load loss maydepend either on Reynolds number or on therelative rugosity of the conduit walls.

In its turbulent flow through the conduit, thefluid has a turbulent core, in which the process ofmixing is decisive in report to the influence ofviscosity and a laminar sub-layer, situated near thewall, in which the viscosity forces have a decisiverole.

If we note by l the thickness of the laminar

sub-layer, then we can classify conduits as follows:

- conduits with smooth walls; l .

From (7.60) we notice that, unlike the laminarmotion in turbulent motion is a complex function

ofRe andd

.

It has been experimentally established that inthe case of hydraulic smooth conduits, coefficient depends only on Reynolds number. Thus, Blasius,by processing the existent experimental material(in 1911), established for the smooth hydraulicconduits of circular section, the following empiricalformula:

25,0

4/1

Re

3164,03164,0 =

=

dv

, (7.61)

valid for 510Re000,4


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Also for smooth conduits, but for higher

Reynolds numbers ( )710Re000,3

3 Blasius 25,0Re3164,0 =5

10Re

000,4Re

4 Konakov

( ) 25,1Relg8,1 =710Re

000,3Re

5 Nikuradze

237,0Re221,00032,0 +=6

5

102Re

10Re

6 Lees35,03 Re61,010714,0 +=

6

3

103Re

10Re

102

II

Author


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7 Colebrook-

White Re

51,2

72,3lg2

1+

=

d

Demi-rugous

Universal

8 Prandtl-

Nikurdze

2

0 74,1lg2

+

=

r

Turbulent rugous

5 10Re10

d

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7.5.2 Nikuradzes diagram

On the basis of experiments made withconduits of homogeneous different rugosity, whichwas achieved by sticking on the interior wall somegrains of sand of the same diameter, Nikuradze hasmade up a diagram that represents the waycoefficient varies, both for laminar and turbulentfields (fig.7.8).

Fig.7.8

We can notice that in the diagram appear five

areas in which variation of coefficient , distinctlydiffers.

Area I is a straight line which represents inlogarithmic co-ordinates the variation:

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Re

64= , (7.64)

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corresponding to the laminar regime ( )2320Re < . Onthis line all the doted curves are superposed, which

represents variation ( )Ref= for different relativerugosities 0/ r .

Area II is the shift from laminar regime to theturbulent one which takes place for

( )2300Re4,3Relg .

Area III corresponds to the smooth hydraulicconduits. In this area coefficient can bedetermined with the help of Blasius relation (7.61),to which the straight line III a corresponds, calledBlasius straight. Since the validity field of relation(7.61) is limited by 510Re = , for higher values ofReynolds number, we use Kanakovs formula, towhich curve III b corresponds. It is noticed that thesmaller the relative rugosity is, the greater thevariation field of Reynolds number, in which the

smooth turbulent regime is maintained.

In area IV each discontinuous curve, which

represents dependent ( )Ref= for different relativerugosities becomes horizontal, which emphasisesthe independence of on number Re . Thereforethis area corresponds to the rugous turbulentregime, where is determined by (7.63).

It is noticed that in this case the losses of load(7.59) are proportional to square velocity.

For this reason the rugous turbulent regime isalso called square regime.

Area V is characterised by the dependence ofthe coefficient both on Reynolds number and onthe relative rugosity of the conduit.

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It can be noticed that for areas IV and V,coefficient decreases with the decrease ofrelative rugosity.

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8. FLOW THROUGH CIRCULARCONDUITS

In this chapter we shall present the hydrauliccalculus of conduits under pressure in a permanent

regime.

Conduits under pressure are in fact ahydraulic system designed to transport fluidsbetween two points with different energetic loads.

Conduits can be simple (made up of one orseveral sections of the same diameter or differentdiameters), or with branches, in this case, settingup networks of distribution.

By the manner in which the outcoming of thefluid from the conduit is made, we distinguishbetween conduits with a free outcome, whichdischarge the fluid in the atmosphere (fig.8.1 a)and conduits with chocked outcoming (fig. 8.1 b).

Fig.8.1a, b

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If we write Bernoullis equation for a streamof real liquid, between the free side of the liquidfrom the tank A and the end of the conduit, takingas a reference plane the horizontal plane N N, weget:

fhzp

g

vz

p

g

v+++=++ 2

2

2

22

1

1

2

11

22

, (8.1)

which, for the case presented in fig.8.1 a, when

01 v , 021 ppp == , 121 == , hzz += 21 , becomes:

fhg

vh +=

2

2

, (8.2)

where 2vv = is the mean velocity in the section ofthe conduit , and h is the load of the conduit.

In the analysed case shown in fig. 8.1 b, byintroducing in equation (8.1) the relations

1022112011 ,,,,0 hppzhhzvvppv +=++=== and121 == , we shall get the expression (8.2).

From an energetic point of view, this relationshows that from the available specific potentialenergy (h), a part is transformed into specific

kinetic energy ( gv 2/2 ) of the stream of fluid, which

for the given conduit is lost at the outcoming in theatmosphere or in another volume. The other part

fh is used to overcome the hydraulic resistances

(that arise due to the tangent efforts developed bythe fluid in motion) and is lost because it isirreversibly transformed into heat.

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Analysing the losses of load from the conduitwe shall divide them into two categories, writingthe relation:

'''

fff hhh += . (8.3)

The losses of load, denoted by fh ' are brought

about by the tangent efforts that are developedduring the motion of the fluid along the length of

the conduit ( l) and, for this reason, they are calledlosses of load distributed. These losses of loadhave been determined in paragraph 7.4.2, gettingthe relation (7.54) which we may write in the form:

d

l

g

vh f

2

2' = , (8.4)

where the coefficient of losses of load, , calledDarcy coefficient is determined by the relations

shown in table 7.1 ; the manner of calculus beingalso shown in that paragraph. Generally, inpractical cases, the values of coefficient vary in adomain that ranges between 04,002,0 .

Being proportional to the length of theconduit, the distributed losses of load are alsocalled linear losses.

The second category of losses of load is

represented by the local losses of load that arebrought about by: local perturbation of the normalflow, the detachment of the stream from the wall,whirl setting up, intensifying of the turbulentmixture, etc; and arise in the area where theconduit configuration is modified or at the meetingan obstacle detouring (inlet of the fluid in the

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conduit, flaring, contraction, bending andderivation of the stream, etc.).

The local losses of load are calculated with thehelp of a general formula, given by Weissbach:

g

vhf

2

2'' = , (8.5)

where is the local loss of load coefficient that is

determined for each local resistance (bends,valves, narrowing or enlargements of the flowsection etc.).

Generally, coefficient depends mainly onthe geometric parameters of the consideredelement, as well as on some factors thatcharacterise the motion, such as: the velocitiesdistribution at the inlet of the fluid in the examinedelement, the flow regime, Reynolds number etc.

In practice, coefficient is determined withrespect to the type of the respective localresistance, using tables, monograms or empiricalrelations that are found in hydraulic books.Therefore, for curved bends of angle 090 ,coefficient can be determined by using the

relation:

0

0

5,3

5,3

90

16,013,0

+=

d, (8.6)

where andd are the diameter and curvature

radius of the bend, respectively.

Coefficient , corresponding to the loss ofload at the inlet in their conduit, depends mainly on

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the wall thickness of the conduit with respect to itsdiameter and on the way the conduit is attached tothe tank. If the conduit is embedded at the level ofthe inferior wall of the tank, the losses of load thatarise at the inlet in the conduit are equivalent withthe losses of load in an exterior cylindrical nipple.

For this case, 5.0 .

If on the route of the conduit there are several

local resistances, the total loss of fluid will be givenby the arithmetic sum of the losses of loadcorresponding to each local resistance in turn,namely:

=g

vhf

2

2'' , (8.7)

Using relations (8.4) and (8.7), we get thetotal loss of load of the conduit:

gv

dlhf

2

2

+= , (8.8)

that allows us to write relation (8.2) in the form:

g

v

d

lh

21

2

++= , (8.9)

where from the mean velocity in the flow sectionwill result:

++=

d

l

hgv

1

2

. (8.10)

The flow of the conduit is determined by:

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++==

d

l

hgdv

dQ

1

2

44

22

, (8.11)

which allows us to express the load of the conduit,h, and diameter, d, with respect to flow Q; we get:

++= dl

d

Q

gh 1

84

2

2 , (8.12)

and respectively:

( )++=

dldh

Q

gd

2

2

5 8. (8.13)

Sometimes in the calculus of enough long

conduits, the kinetic term ( )gv 2/2 and the locallosses of load are negligible with respect to the

linear losses of load.

In the case of such conduits, called longconduits, relation (8.2) takes the form:

d

l

g

vhh f

2

2' == , (8.14)

and relations (8.10), (8.11), (8.12) and (8.13)become:

l

gdhv

2= , (8.15)

l

gdhdQ

2

4

2

= , (8.16)

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ld

Q

gh

5

2

2

8= , (8.17)

and, respectively:

lh

Q

gd

2

2

5 8= . (8.18)

With the help of the above written relations

all problems concerning the computation ofconduits under pressure can be solved. Generally,these problems are divided into three categories:

a)to determine the load of the conduit, whenlength, rugosity, flow and rugosity of interiorwalls of the conduit are known;b)to determine the optimal diameters whenflow, length, rugosity of the walls of conduitas well as the admitted load are known;c)to determine the flow of liquid conveyed

through the conduit when diameter, length,nature of the wall of conduit and its load areknown.

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9. HYDRODYNAMIC PROFILES

9.1 Geometric characteristics ofhydrodynamic profiles

A hydrodynamic profile is a contour with anelongated shape with respect to the direction ofstream, rounded at the front edge-called leadingedge-and having a peak at the back edge, calledtrailing edge.

In what follows we shallstress on some of theelements, whichcharacterise the profile.

a) The chord of theprofile is defined as thestraight line which joinsthe trailing edge A, withthe point B, in which thecircle

Fig.9.1

with the centre in A is tangent to the leading edge;the length of the chord will be noted by c (fig.9.1).

b) The thickness of the profile is measured on thenormal to the chord and is noted by e. Thisthickness varies along the chord and reaches amaximum in a section which is called section of

maximum thickness, situated at the distance ml

to the leading edge.

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c) Relative thickness, , and maximum relative

thickness, m , are defined by the relations:

c

eand

c

e mm == . (9.1)

d) The framework of a profile, or the line of meancurvature, is the curve that joins the meanthickness points. The shape of the framework is

an important geometric parameter and is linkedto the curvature motion of the profile.From this point of view, profiles can be withsimple curvature (fig.9.1) or with doublecurvature (9.2).

e) The arrow of the profile, f, is the maximumdistance, measured on the normal to the chord,between the framework and the chord of theprofile.

f) The extrados and intradosof the profile represent theupper and lower part of theprofile, respectively.

By the geometric shape ofthe trailing edge, whichplays an important part inthe theory of profiles, wemay distinguish amongthree categories of profiles:

Fig.9.2

- Jukovski profiles, profiles with a sharp edge,for which the tangents to the trailing edgeat extrados and intrados superpose (fig.9.3a)

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- Karman-Trefftz profiles, or profiles with adihedral tip, for which the

tangents to the extrados and the intradosmake an angle in the

trailing edge (fig.9.3 b),

- Carafoli profiles, or profiles with therounded tip, for which the trailing

edge ends in a rounded contour, with asmall curvature radius.

(fig.9.3c).It is generally studied the

plane potential motionaround the hydrodynamicprofile, considered as theintersection of the complexplane of motion with acylindrical object (calledwing), normal on this planeand having an infinite length

(called span).

In reality, wings have afinite span and, from ageometrical point of view,they are characterised bythe section of the wing,which, generally, alters

Fig.9.3 a, b, cthe length of the wing and the shape of the wing inplane.

By the shape of the wing in plane, there are:rectangular wings (fig.9.4), trapezoidal wings (9.4b), elliptical (9.4 c), and triangular wings (9.4 d).

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Fig.9.4 a, b, c, d

An important parameter of the wing is therelative elongation defined by the relation:

S

l2= , (9.2)

where l and S represent the span and the surface ofthe wing, respectively.

In the particular case of rectangular wing, the

length of the chord is constant 0cc = and relation(9.2) becomes:

0/ cl= ,since:

0clS = .We can classify wings by their elongation ;

into:

- wings of infinite span, when 6> ;- wings of finite span, when 6


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9.2 The flow of fluids around wings

Kutta-Jukovskis relation (5.62) can be appliedto any solid body in relative displacement withrespect to a fluid.

It indicates that whenever there is acirculation around a body, there arises a lift force

yR , whose value is determined, under the same

circumstances of environment ( vand ), by the

intensity of circulation.To get a higher circulation around bodies, we

can act in two ways:

- for geometrical symmetric bodies: they are

asymmetrically placed with respect to v

direction or a rotational motion is induced(an infinitely long cylinder, sphere-Magnuseffect).

- for asymmetrical bodies: study of shapesmore proper to circulation.

On the basis of many theoretical andexperimental studies, we have come to designingwings with a high lift, called hydrodynamic profiles.

Fig.9.5

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In fig.9.5, the arising of circulation around thehydrodynamic profile, alters the spectre of lines of

rectilinear stream, of velocity v as follows: on the

extrados the sense of circulations coincides withthat of motion and is seen as a supplement ofvelocity v , and on the intrados velocity isdecreased with v .

According to Bernoullis law, the velocitiesasymmetry brings about the static pressuresasymmetry (high pressure on the intrados, lowpressure on the extrados) as well as the arising oflift force.

Applying Bernoullis relation between a pointat and a point on the profile, we get:

22

22

S

S

vp

vp

+=+ . (9.3)

The pressure coefficient is defined by therelation:

2

2

21

2

=

=v

v

v

ppC SSp

. (9.4)

In fig. 9.6 it is shown the distribution ofpressure and of the pressure coefficient on ahydrodynamic profile at a certain angle of

incidence, * .

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Fig.9.6

The alteration of the incidence angle leads tothe shift in the pressures distribution.

9.3 Forces on the hydrodynamicprofiles

The forces which act upon hydrodynamic oraerodynamic profiles: lift, shape resistance, frictionforce or the force due to the detachment of thelimit layer give a resultant R which decomposes by

the direction of velocity in infinite and by adirection which is perpendicular on it (fig.9.7).

121

The angle between and the chord of the p

rofile.


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Component xR is called resistance at advancement,

and component yR , lift force.

They are usually written in the form:

.

2

;22

2

Sv

CR

Sv

CR

yy

xx

=

=

(9.5)

where xC is called the coefficient of resistance at

advancement, and yC the lift coefficient ( lcS = forprofiles of constant chord).

Fig.9.7

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Force R can also decompose by the direction

of chord (component tR ) and by a direction

perpendicular on the chord (component nR ).

These components may also be expressedwith the help of coefficients:

tC - the coefficient of tangent force and nC - the

coefficient of normal force.

For a certain angle s, is the distancebetween the leading edge and the pressure centre(the application point of hydrodynamic force).

The relation expresses the moment of theforce R with respect to the leading edge:

sincos sRsRsRM xyn +== . (9.7)

Also, moment M can be expressed by ananalytic form similar to that used for thecomponents of hydrodynamic force:

Sv

cCM m2

2

= . (9.8)

Using (9.5), (9.7), and (9.8), we get:

sincos xy

m

CC

C

c

s

+= . (9.9)

In the case of small incidence angles:

y

m

C

C

c

s . (9.10)

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The usage of coefficients xC , yC and nC is

often met in actual practice. Their variation isstudied in different conditions and given in theform of tables and graphics of great importance forthe calculus and design of systems, which deal withprofiles.

Coefficients xC , yC and nC depend on the

following main elements:- the shape of the profile;- the span of the profile (finite or infinite,

finite of small span or great span);- the type of the flow (Reynolds number);- rugosity of surfaces;- the angle of incidence.

For each shape of profile, at certain differentrelative elongation, , (see paragraph 9.1), in the

case of certain flow velocities (numbers Revariable), there are diagrams experimentally

established ( ) ( ) ( ) myx CandCC , .

Fig.9.8

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In fig. 9.8 there are plotted the diagrams ofcoefficients for resistance at advancement and forlift force for a NACA 6412 profile, of relativeelongation 3, at a number Re = 85,000.

Another type of diagram often used is the

polar profile, namely the function ( )xy CC at differentslanting angles (fig.9.9). The polar allows us todefine two characteristics of the profile:

- the floating or gliding coefficient:

y

x

C

Ctg == , (9.11)

- aerodynamic accuracy:

x

y

C

Cf ==

1. (9.12)

Fig.9.9

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9.4 Induced resistance in the case offinite span profiles

For wings of great span, considered infinite=l , the motion around the profile is plane.

Circulation may be replaced by a whirl.

In reality, atthe tips of thewing, because ofthe difference inpressure, therearises a motionof fluid fromintrados toextrados (9.10).The greater theweight of this

motion, thesmaller the wingspan is.

Fig. 9.10

As aconsequence,circulation isno longerconstant; at the

tips there is aminimum.(fig.9.11).This leads to analteration ofhydrodynamicparameters,through the

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arising of the so-called inducedresistance.

Fig.9.11

In fig.9.12 the scheme of hydrodynamic forcesfor the wing of finite span is plotted.

Due to the arising of an induced velocity iv ,

created by the free whirl, perpendicular on thevelocity in infinite v , the resultant velocity

becomes:

ivvv += . (9.13)

Fig.9.12

As a consequence there will appear an

induced incidence angle i , which thus decreases

the incidence angle .

The alteration of direction and value ofvelocity bring about the corresponding alteration oflift, which, as we have already shown, isperpendicular on the direction of stream velocity.

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If yR is the lift of the infinite profile and F is

the lift under the circumstances of an inducedvelocity (perpendicular on the direction of velocity

v ), then:

.cos

;sin

iy

ii

FR

FR

==

(9.14)

In the conditions of very small values of i , wemay assume that FRy , namely lift does not alter.

Component iR acting on the direction Ox is

called induced resistance and may be written in theform:

Sv

CR xii2

2

=

. (9.15)

The total resistance of the wing of infinitespan is the sum between the resistance of wing of

infinite span xR and the induced resistance iR .

9.5 Network profiles

Several profiles that are in the stream of fluidare in reciprocal influence, behaving in a differentmanner within the assembly, rather than solitary.Networks of profiles are often met in practice in the

hydraulic or pneumatic units, propellers, etc.To study the behaviour of profiles in network,

let us consider a system made up of severalidentical profiles, of span l and control contourABCD (fig.9.13). The pitch of the network is t.

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Fig.9.13

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Velocities v in points 1 and 2 have the

components xv and yv , according to the system of

axes shown in the figure. Assuming that thedensity of fluid doesnt alter in a significant way

when passing through the network, 21 = , then

21 xxvv = .

Indeed, applying the equation of continuity:

ltvltvm xx 21

.

== , (9.16)

it results xxx vvv == 21 .

We have denoted by m the massic flow.Applying the theorem of impulse, we get

component yR of the lift force in the network:

( ) ( )2121.

yyxyyy vvltvvvmR == . (9.17)

The circulation of velocity on the controlcontour will be:

+++==ABCD

C

B

D

C

A

D

y

B

A

y dsvdsvdsvdsvdsv 21 . (9.18)

The integrals on the segments of contour BCand AD cancel reciprocally. There only remains:

( ) tvvdsvdsvD

C

yyy

B

A

y == 2121 . (9.19)

Therefore:

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tvv yy

=

21. (9.20)

Replacing (9.20) into (9.17), we get:

= lvR xy . (9.21)

The axial component xR is due to the

difference of pressure:

( ) tlppRx 21 = . (9.22)

Applying Bernoullis equation between the points 1and 2, we get:

22

2

2

2

2

1

1

vp

vp

+=+ , (9.23)

or else:

( ) ( )222

122

1

2

2

2

1

2

221

yy

yy

vv

t

vvvvpp+

===

. (9.24)

Replacing (9.24) into (9.22). we get:

+

= lvv

Ryy

x2

12 . (9.25)

The resultant force will be:

( )tl

vC

vvvlRRR

r

yy

xyxr 24

22

222 21

=

+

+=+= . (9.26)

In relation (9.26) we have denoted by rC the

coefficient of the network and by v the mean

velocity in the network (fig.9.14).

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( )4

2

2 21 yy

x

vvvv

++= . (9.24)

Fig.9.14

The lift force is perpendicular on v .

Coefficient rC is different from the hydro-

aerodynamic coefficient corresponding to aseparate profile.

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Hydraulics and Hydraulics Machines - D.dinu, S. Liviu