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TRANSCRIPT
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PROFESORI TITULARI :
PROF.UNIV.DR.ING. DUMITRU DINUS.L.DRD.ING. STAN LIVIU
HIDRODINAMICASI
TEORIA VALURILOR
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CONTENTS
PART ONE
HYDRAULICS
1.BASIC MATHEMATICS 11
2.FLUID PROPRIETIES 17
2.1 Compressibility 182.2 Thermal dilatation 202.3Mobility 222.4Viscosity 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 relativemovement of ideal non-compressible fluid
40
4. FLUID STATICS 43
4.1 The fundamental equation ofhydrostatics 43
4.2Geometric and physical interpretationof the fundamental equation of
hydrostatics 45
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4.3 Pascals principle 46
4.4 The principle of communicatingvessels 47
4.5 Hydrostatic forces 484.6Archimedes principle 504.7 The floating of bodies 51
5. POTENTIAL (IRROTATIONAL) MOTION 57
5.1 Plane potential motion 595.2 Rectilinear and uniform motion 635.3 The source 665.4 The whirl 69
5.5 The flow with and withoutcirculation around a circular cylinder 71
5.6 Kutta Jukovskis theorem 75
6. IMPULSE AND MOMENT IMPULSE
THEOREM 77
7. MOTION EQUATION OF THE REAL FLUID81
7.1 Motion regimes of fluids 817.2 Navier Stokes equation 837.3 Bernoullis equation under the
permanent regime of a thread of real fluid 877.4 Laminar motion of fluids 90
7.4.1 Velocities distributionbetween two plane parallel boards of infinit
length 907.4.2 Velocity distribution in
circular conduits 937.5 Turbulent motion of fluids 97
7.5.1 Coefficient in turbulentmotion 99
7.5.2 Nikuradzes diagram 102
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8. FLOW THROUGH CIRCULAR CONDUITS 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 offinite span profiles 123
9.5 Networks profiles 125
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PARTONEHYDRAULICS
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1.Basic mathematics
The scalar product of two vectors
kajaiaa zyx ++= and kbjbibb yx 2++= is a
scalar.
Its value is:
zzyyxxbabababa ++= . (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 aandb is avector 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 isgiven by 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 cis a scalar.
( )zyx
zyx
zyx
ccc
bbb
aaa
cba = . (1.7)
The double vectorial product of 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 operatoris 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 called
divarication. .adiva =
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z
a
y
a
x
aa z
yx
+
+
= . (1.11)
vectorially applied to a vector is called
rotor. .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 andonly then vectorial proprieties;
- all the vectors or the scalars upon whichit doesnt act must, in the end, beplaced in front 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 ( )kvjvivvzyx
++= .The equation of the streamline is obtained
by writing that the tangent to streamline isparallel to the vector velocity in itscorresponding point:
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zyxv
dz
v
dy
v
dx== . (1.26)
The whirl line is a curve tangent in each of
its points 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 isparallel with the 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)is defined by:
= ,rdvC
(1.30)
in which
dsrd = (1.31)represents the orientated element of
the curve (- the versor of the tangent to thecurve (C )).
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Fig.1.1
( ) ++=C
zyx dzvdyvdxv (1.32)
The sense of circulation depends on the
admitted 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 thenormal to the arbitrary surface bordered bythe 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 scaleat which fluids behave as if matter werecontinuously distributed.
At the same time, fluids dont have theirown shape so are easily deformed.
A continuous medium is homogenous if at aconstant temperature and pressure, its densityhas only one value in all its points.
Lastly, a continuous homogenous medium isisotropic as well if it has the same proprietiesin any direction around a certain point of itsmass.
In what follows we shall consider the fluidas a continuous, deforming, homogeneous andisotropicmedium.
We shall analyse some of basic physicalproprieties of the fluids.
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2.1. Compressibility
Compressibility represents the property of
fluids to modify their volume under the action ofa variation of pressure. To evaluate
quantitatively this property we use a physicalvalue, 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 variationof the initial volume, under the action ofpressure variation dp.
The coefficient is intrinsic positive;
the minus sign that appears in relation (2.1)takes into consideration the fact that the volume
and the pressure have reverse variations, namelydv/ dp < 0.
The reverse of the isothermalcompressibility coefficient is called theelasticity modulus K and is given by therelation:
.
1
2
==
m
N
dV
dp
VK (2.2)
Writing the relation (2.2) in the form:
,K
dp
V
dV= (2.3)
we can underline its analogy with Hooks law:
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.El
dl = (2.4)
a)The compressibility of liquids
In the case of liquids, it has beenexperimentally ascertained that the elasticity
modulus K, and implicitly, the coefficient ,
vary very little with respect to temperature
(with approximately 10% in the interval C0
600 )and they are constant for variations of pressurewithin enough wide limits. In table (2.1) thereare shown the values of these coefficients for
various liquids at C00 and pressure 200p bars.Table 2.1.
Liquid
[ ]Nm /2
[ ]2/ mNK
Water 101012,5 91095,1
Petrol 101066,8 91015,1
Glycerine 101055,2 91092,3 Mercury 1010296,0 9107,33
Therefore, in the case of liquids,
coefficient may be considered constant.
Consequently, we can integrate thedifferential equation (2.2) from an initial
state, characterised by volume 0V , pressure 0p anddensity 0 , to a certain final state, where the
state parameters will have the value pV ,1 and
respectively; we shall successively get:
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=V
V
p
p
dpV
dV
0 0
, (2.5)
or
( ).00
ppeVV
= (2.6)
b)The compressibility of gases
For gases the isothermal compressibilitycoefficient depends very much on pressure. In thecase of a perfect gas, the following relationdescribes the isothermal compressibility:
pV = cons.,
which, by subtraction, will be:
.
V
dV
p
dp= (2.8)
By comparing this relation to (2.3) we maywrite:
.1
pK ==
(2.9)
It follows that, in the case of a perfectgas, the elasticity modulus is equal to pressure.
2.2 Thermal dilatation
Thermal dilatation represents the fluidproperty to modify its volume under the action of
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a variation of temperature. Qualitatively, this
property is characterised by the volumetriccoefficient of isobaric dilatation, defined bythe relation:
,1
dT
dV
V= (2.10)
where dV represents the elementary variation ofthe initial volume V under the action ofvariation of temperature dT. Coefficient ispositive for all fluids, except for water, whichregisters maximum density (minimum specific
volume) at C0
4 ; therefore, for water that has
Ct04 we shall have .0
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equation of isobaric transformation
= .cons
T
V; we
get:
,. dTT
VdTconsdV == (2.14)
which, replaced into (2.10) enables us to write:
.1
T= (2.15)
Thus, for the perfect gas, coefficient isthe reverse of the thermodynamic temperature.
2.3. Mobility
In the case of fluids, the molecularcohesion forces have very low values, but theyarent rigorously nil.
At a macroscopic scale, this propriety canbe rendered by the fact that two particles offluid that are in contact, can be separated underthe action of some very small external forces. Atthe same time, fluid particles can slide one nearthe other and have to overcome some relativelysmall tangent efforts.
As a result, from a practical point of view,fluids can develop only compression efforts.
In the case of a deformation at a constantvolume, the compression efforts are rigorouslynil and, as a result, the change in shape of thefluid requires the overcoming of the tangentefforts, which are very small. Therefore themechanical work consumed from the exterior willbe very small, in fact negligible.
We say that fluids have a high mobility,meaning that they have the property to take the
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shape of the containers in which they are.
Consequently we should stress that gases, becausethey dont have their own volume, have a highermobility than liquids (a gas inserted in acontainer takes both the shape and the volume ofthat container).
2.4 Viscosity
Viscosity is the property of the fluid tooppose to the relative movement of its particles.
As it has been shown, overcoming some small
tangent efforts that arent yet rigorously nilmakes this movement.
To qualitatively stress these efforts, weconsider the unidimenssional flow of a liquid,
which takes place in superposed layers, along aboard situated in xOy plane (fig.2.1).
Fig.2.1.
Experimental measurements have shown thatvelocity increases as we move away from the board
in the direction of axis Oy, and it is nil in thenear vicinity of the board. Graphically, the
dependent ( )yfv = is represented by the curve .This simple experiment stresses on two aspects,
namely:
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- the fluid adheres on the surface of thesolid body with which it comes intocontact;
- inside the fluid and at its contact withthe solid surfaces, tangent effortsgenerate which determine variation invelocity. Thus, considering two layers offluid, parallel to the plane xOy and thatare at an elementary distance dy one fromthe other, we shall register a variation
in velocity dydy
dv, due to the frictions
that arise between the two layers.
To determine the friction stress, Newton
used the relation:
dy
dv = , (2.16)
that today bears his name. This relation thathas been experimentally verified by Coulomb,Poisseuille and Petrov shows that the frictionstress is proportional to the gradient ofvelocity. The proportionality factor is called
dynamic viscosity.
If we represent graphically the dependent
( )dydvf /= we shall get the line 1 (fig.2.2)where =ty .
The fluids that observethe friction law (2.16) arecalled Newtonian fluids
(water, air, etc). Thedependent of the tangenteffort to the gradient of
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velocity is not a straight
line (for example curve (2)in fig. 2.2), for a series ofother fluids, generally thoseof organic nature. Thesefluids are globally called
non-Newtonian fluids.Fig.2.2
The measures for the dynamic viscosity are:
- in the international standard (SI):[ ]
sm
Kg
m
sN
=
=
2 (2.17)
- in the CGS system:[ ]
scm
g
cm
sdyn
=
=
2 . (2.18)
The measure of dynamic viscosity in CGSsystem is called poise, and has the symbol P.We can notice the existence of relation:
Psm
Kg101 =
. (2.19)
We can determine the dynamic viscosity ofliquids with the help of Hpplers viscometer,whose working principle is based on theproportionality of dynamic viscosity to the timein which a ball falls inside a slanting tube thatcontains the analysed liquid.
The kinematic viscosity of a fluid is theratio of dynamic viscosity and its density:
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= . (2.20)
The measures for kinematic viscosity are:
- in the international system:[ ]
s
m2
= . (2.21)
- in CGS system:[ ]
s
cm2
= . (2.22)
the latter bearing the name stokes (symbol ST):
s
m
s
cmST
24
2
1011== . (2.23)
Irrespective of the type of viscometer used(Ubbelohde, Vogel-Ossag, etc) we can determinethe kinematic viscosity by multiplying the time(expressed in seconds) in which a fixed volume of
liquid flows through a calibrated capillary tube,under normal conditions, constant for that
device.
In actual practice, the conventionalviscosity of a fluid is often used; this value isdetermined by measuring the time in which a
certain volume of fluids flows through a specialdevice, the conditions being conventionally
chosen. The magnitude of this value thusdetermined is expressed in conventional units.
There are several conventional viscosities (i.e.Engler, Saybolt, Redwood etc) which differ from
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one another both in the measurement conditions
and in the measure units.Thus, Engler conventional viscosity,
expressed in Engler degrees E0
is the ratio
between the flow time of 200 cubic cm of theanalysed liquid at a given temperature and theflow time of a same volume of distilled water at
a temperature of C0
20 , through an Englerviscometer under standard conditions.
The viscosity of a fluid depends to a great
extent on its temperature. Generally, viscosityof liquids diminishes with the increase intemperature, while for gas the reverse phenomenontakes place.
The dependence of liquids viscosity ontemperature can be determined by using Gutman andSimons relation:
0
0
T
B
TC
B
e
+
= . (2.24)
where the constants B and C depend on the natureof the analysed liquid (for water we have B=511,6 K and C= -149,4 K).
For gases we can use Sutherlands formula
TS
TS
T
T
+
+
= 0
2/3
0
0 . (2.25)
where S depends on the nature of the gas (for airS=123,6 K).
In relations (2.24) and (2.25), and 0
are the dynamic viscosities of the fluid at the
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absolute temperature T, and at temperature
)0(15,273 00 CKT = respectively.
In table 2.2 there are shown the dynamic andkinematic viscosities of air and water at
different temperatures and under normalatmospheric pressures.
Table 2.2
Temperature
C0
-10 0 10 20 40 60 80 100
Ai r 0, 162
0, 172
0, 175
0, 181
0, 191
0, 20
0, 289
0, 218
sm
Kg910
Water
- 1, 79
1, 31 1, 01 0, 658
0, 478
0, 366
0, 295
Ai r 1, 26 13,3
14, 1 15, 1 16,9
18,9
20,9
23,1
s
m2
610
Water
- 1, 79
1, 31 1, 01 0, 658
0, 478
0, 366
0, 295
We must underline the fact that viscosity isa property that becomes manifest only during themovement of liquids.
A fluid for which viscosity is rigorouslynil is called a perfect or ideal fluid.
Fluids may be compressible ( )[ ]p = orincompressible ( is constant with respect to
pressure).
We should emphasise that the idealcompressible fluid is analogous to the ideal (orperfect) gas, as defined in thermodynamics.
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The movement of fluids may be uniform
(velocity is constant), permanent v = v (x,y,z)or varied v = v (x,y,z,t).
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3.EQUATIONS OF IDEAL FLUIDMOTION
3.1 Eulers equation
We shall further study, for the most
general case, the movement state of a fluidthrough a volume that is situated in the fluidstream; we shall not take into consideration theinterior frictions(i.e.viscosity), so we shallanalyse the case of perfect (ideal) fluids thatare on varied movement.
The volume is situated in an acceleratedsystem of axes, joint with this system. Theequations, which describe the movement of thefluid, will be obtained by applying dAlemberts
principle for the fluid that is moving throughthe volume .
The three categories of forces that act uponthe fluid that is moving through the volume,bordered by the surface (fig.3.1), are:
Fig.3.1
- the mass forces mF ;
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- the inertia forces iF ;- the pressure forces pF (with an
equivalent effect; these forces replacethe action of the negligible fluidoutside volume ).
According to dAlemberts principle, weshall get:
0=++ pim FFF . (3.1)
Equation (3.1) represents in fact thegeneral vectorial form of Eulers equations.
Lets establish the mathematical expressionsof those three categories of forces.
If F is the mass unitary force(acceleration) that acts upon the fluid in the
volume , the mass elementary force that acts
upon the mass d , will be:
dFFd m = , (3.2)
hence:
=
dFFm . (3.3)
As the fluid velocity through the volume
is a vectorial function with respect to pointand time: trvv ,= , upon the mass d that is
moving with velocity v the elementary inertiawill act:
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ddt
vdFd i = . (3.4)
So, the inertia will be:
=
ddt
vdFi . (3.5)
If d is a surface element upon which the
pressure p acts, and n- the versor of theexterior normal (Fig.3.1), the elementary force
of pressure is:
dnpFd p = . (3.6)
Having in mind Gauss-Ostrogradskis theorem,the resultant of pressure forces will be:
dpdnpFp == . (3.7)
By replacing equations (3.3), (3.5) and(3.7) in the equation (3.1), we shall get:
0=
ddt
vdpF , (3.8)
Hence:
dt
vdpF =
1, (3.9)
Or
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( )vvt
vpF +
=
1, (3.10)
The equation (3.10) Eulers equation in avectorial form for the ideal fluid in a non-permanent movement.
Projecting this equation on the three axes,we shall obtain:
z
x
y
x
x
xx
xv
zvv
yvv
xv
tv
xpF
+
+
+
=
1 ;
z
y
y
y
x
yy
y vz
vv
y
vv
x
v
t
v
y
pF
+
+
+
=
1; (3.11)
z
z
y
z
x
zz
z vz
vv
y
vv
x
v
t
v
z
pF
+
+
+
=
1.
3.2 Equation of continuity
This equation can be obtained by writing intwo ways the variation in the unity of time forthe mass of fluid that is in the control volume, bordered by the surface (fig.3.1). By
splitting from the volume one element d , and
taking into consideration that the density is a
scalar function of point and time, tr, = , wecan write the total mass of the volume :
=
dm . (3.12)
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The variation of the total mass in the unity
of time will be:
=
dtt
m. (3.13)
The second form of writing the variation ofmass is obtained by examining the flow of themass through surface that borders volume.
Denoting by n the versor of the exterior
normal to the area element d , and by v thevector of the fluid velocity, the elementary massof fluid that passes in the unity of time through
the area element d is:
dvdM n= . (3.14)
In the unity of time through the whole
surface will pass, the mass:
=
dvM n (3.15)
that is the sum of the inlet and outlet mass involume , by crossing surface .
By equalling equations (3.13) and (3.15), itwill result:
=+
0dv
tn . (3.16)
According to Gauss-Ostrogradskis theorem:
=
dvdvn . (3.17)
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Taking into consideration (3.17), the
equation (3.16) will take the form:
( ) 0=
+
dvt
, (3.18)
hence, successively:
( ) ,0=+
v
t
(3.19)
0=++
vv
t
, (3.20)
0=+ vdt
d
. (3.21)
The equation (3.21) represents the equationof continuity for compressible fluids.
In the case of non-compressible fluids
( .cons= , 0=dtd ), the equation of continuity
takes the form:
0=v , (3.22)
or
0=
+
+
z
v
y
v
x
vzyx . (3.23)
It follows that the inlet volume of non-compressible liquid is equal to the outlet one inand from the volume .
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3.3. The equation of state
From a thermodynamically point of view, thestate of a system can be determined by the direct
measurement of some characteristic physicalvalues, that make up the group of state
parameters (e.g. pressure, volume, temperature,density etc.).
Among the state parameters of a
thermodynamically system generally there arelink relationships, explained by the laws ofphysics.
In the case of homogenous systems, there isonly one implicit relationship, which carries out
the link among the three state parameters, in theform of:
( ) 0,, =TpF . (3.24)
Adding to vectorial equations (3.10) and(3.21) the equation of state, we get three
equations with three unknowns: trptrtrv ,,,,, ,
that enable us solve the problems of motion andrepose for the ideal fluids.
3.4. Bernoulli s equation
Bernoullis equation is obtained byintegrating Eulers equation written under a
different form (Euler Lamb), that stresses therotational or non-rotational nature of the ideal
fluid (see the relation (1.25)).
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Euler Lambs equation:
vrotvv
t
vpF
+
=
2
12
. (3.25)
Considering the case when the mass forcederives from a potential U, thus being aconservative force (the mechanical energy-kineticand potential-will be constant):
UF = . (3.26)
In the case of compressible fluids, when
( )p = , we insert the function:
( )=
p
dpP
. (3.27)
Thus:
( )p
pP =
1. (3.28)
The equation (3.25) takes the form:
vrotvt
vvPU
=
++
2
2
. (3.29)
The equation (3.29) can be easily integrated
in certain particular cases.
In the case of permanent motion 0=
t
v, and:
- along a stream line:
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zyxv
dz
v
dy
v
dx== , (3.30)
- along a whirl line:
zyx
dzdydx
== , (3.31)
- in the case of potential motion 0=vrot :0=== zyx , (3.32)
-in the case of helicoid motion (the
velocity vector v is parallel to the whirlvector):
z
z
y
y
x
x vvv
== . (3.33)
Multiplying by rd the equation (3.29), weshall get under the conditions of permanent
motion ( 0=
t
v):
( )vrotvrdvPUd =
++
2
2
. (3.34)
Since 2=vrot , we shall get:
zyx
zyx vvv
dzdydxv
PUd
22
2
=
++ . (3.35)
The determined is zero for one of the four
cases above. By integrating in these cases weshall get Bernoullis equation:
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Cv
PU =++2
2
. (3.36)
If the fluid is a non-compressible one,
then
pP = .
If the axis Oz of the system is vertical,upwards directed, the potential U is:
CgzU += . (3.37)
It results the well known Bernoullisequation as the load equation:
Czp
g
v=++
2
2
. (3.38)
The kinetic loadg
v
2
2
represents the height at
which it would rise in vacuum a material point,vertically and upwards thrown, with an initialvelocity v, equal to the velocity of the particleof liquid considered.
The piezometric load
pis the height of the
column of liquid corresponding to the pressure p
of the particle of liquid.
The position load z represents the height at
which the particle is with respect to anarbitrary chosen reference plane.
Bernoullis equation, as an equation ofloads, may be expressed as follows: in the
permanent regime of an ideal fluid, non-compressible, subjected to the action of someconservative forces, the sum of the kinetic,
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piezometric and position loads remains constantalong a streamline.
Multiplying (3.38) by we get the
equation of pressures:
Czpv
=++ 2
2
, (3.39)
where:
2
2v
dynamic pressure;
p piezometric (static) pressure;
z position pressure.
Multiplying (3.38) by the weight of thefluid G, we get the equation of energies:
CzGpGg
vG =++ 2
2
, (3.40)
where:
g
vG
2
2
- kinetic energy;
pG - pressure energy;
Gz - position energy.
3.5. Plotting and energetic
interpretation of Bernoullis equation for
liquids
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Going back to the relation (3.38) and
considering C = H (fig.3.2):
Hzp
g
v=++
2
2
. (3.41)
Fig.3.2The sum of all the terms of Bernoullis
equation represents the total energy (potentialand kinetic) with respect to the unit of weightof the moving liquid.
This energy measured to a horizontalreference plane N-N, arbitrarily chosen is calledspecific energy and it remains constant duringthe permanent movement of the ideal non-compressible fluid that is under the action ofgravitational and pressure forces.
3.6. Bernoullis equation for the
relative movement of ideal non-compressible fluid
Lets consider the flow of an ideal non-
compressible fluid through the channel betweentwo concentric pipes that revolve around an axisOz with angular velocity (fig.3.3.).
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Fig.3.3
In the equation (3.38) v is replaced by w,which represents the relative velocity of the
liquid to the channel that is revolving with thevelocity ru = .
Upon the liquid besides the gravitational
acceleration g, the acceleration r2
acts as
well.
The unitary mass forces decomposed on thethree axes will be:
.
;
;
2
2
gF
yF
xF
z
y
x
=
=
=
(3.42)
In this case, the potential U will be:
Cr
gzU +=2
22
. (3.43)
By adding (3.43) to Bernoullis equation, weget:
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Czp
g
r
g
w=++
22
222
, (3.44)
or
Czp
g
uw=++
2
22
. (3.45)
In the theory of hydraulic machines we usethe following denotations:
v absolute velocity;w relative velocity;u peripheral velocity.
The equation (3.45) written for twoparticles on the same streamline is:
2
2
2
2
2
2
1
1
2
1
2
1
22
zp
g
uwz
puw++
=++
(3.46)
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4.FLUID STATICS
The fluid statics hydrostatics is thatpart of the mechanics of fluid which studies therepose conditions of the fluid as well as theiraction, during the repose state, on solid bodieswith whom they come into contact.
Hydrostatics is identical for real and idealfluids, as viscosity becomes manifest only duringmotion. In hydrostatics the notion of time doesno longer exist.
4.1 The fundamental equation of
hydrostatics
If in Eulers equation (3.9) we assume that
0=v , we get:
01
= pF
. (4.1)
We multiply everywhere by rd :
01
= rdprdF
. (4.2)
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or
dpdzFdyFdxF
zyx=++ . (4.3)
If the axis Oz of the system xOyx is
vertical, upwards directed, then:
0== yx FF , ,gFz =
and equation (4.3) becomes:
0=+
dpgdz . (4.4)
In the case of liquids (= cons.), by
integrating equation (4.4) we get:
.constp
gz =+
(4.5)
or
.constp
z =+
(4.6)
or
.constzp =+ (4.7)
Equation (4.7) is called the fundamentalequation of hydrostatics.
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If 0p is the pressure at the surface of
water (in open tank the atmospheric pressure),pressure p, situated at a distance h from thesurface, will be (fig.4.1):
Fig.4.1
102 zpzp +=+ , (4.8)
hpp += 0 . (4.9)
p is called the absolute pressure in the
point 2, and h is the relative pressure.
4.2 Geometrical and physicalinterpretation of the fundamental equation
of hydrostatics (fig.4.2)
Fig.4.2
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According to (4.6) we can write:
2
2
2
1
1
1 zp
zp
+=+
. (4.10)
In fig.4.2 we have:
p- piezometric height corresponding to
the absolute hydrostatic pressure;
2,1z - the quotes to an arbitrary plane
(position heights).
4.3 Pascals principle
We rewrite the fundamental equation of
hydrostatics between two points 1 and 2.
2211 zpzp +=+ . (4.11)
Supposing that in point 1, the pressure
registers a variation p , it becomes pp +1 . Inorder that the equilibrium state shouldnt bealtered, for point 2 the same variation ofpressure has to be registered.
222111 zppzpp ++=++ . (4.12)
Hence:
21 pp = . (4.13)
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Pascal s pr i nci pl e:
Any pressure variation created in a certain
point in a non-compressible liquid in
equilibrium, is transmitted with the same
intensity to each point in the mass of this
liquid.
4.4 The principle of communicatingvessels
Let us consider two communicating vessels(fig.4.3) that contain two non-miscible liquids,
which have specific weights 1 and 2 ,
respectively. Writing the equality of pressure inthe points 1 and 2, situated in the samehorizontal plane N N that also contains the
separation surface between the two liquids, weget:
220110 hphp +=+ , (4.14)
or else
1
2
2
1
=
h
h, (4.15)
where 1h and 2h are the heights of the two liquid
columns that, according to this relation, are in
reverse proportion to the specific weights of thetwo liquids.
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Fig.4.3
If ,21 = then 21 hh = .
In two or more communicating vessels, that
contain the same liquid (homogenous and non-
compressible), their free surfaces are on the
same horizontal plane.
4.5 Hydrostatic forces
The pressure force that acts upon a solid
wall is determined by means of the relation:
=A
dAnpF , (4.16)
where dA is a surface element having the versor
n, and p is the relative pressure of the fluid.
Let A be a vertical plane surface thatlimits a non-compressible fluid, with specific
weight (fig.4.4).
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Fig.4.4
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Then the hydrostatic pressure force will be:
===A
yMAzzdAF 0 , (4.17)
where:
0z - the quote of the specific weight for
surface A;
yM - the static moment of the surface A withrespect to the axis Oy.
The application point of the pressure forceF is called pressure centre. It has the following
co-ordinates:
y
yA
M
I
zdA
dAz
F
zdF
===
2
, (4.18)
y
yzA
M
I
zdA
yzdA
F
ydF
===
.
yI - the inertia moment of surface A with
respect to the axis Oy;
yzI - the centrifugal moment of surface A
with respect to axes Oy and Oz.
The hydrostatic pressure force that acts
upon the bottom of a container does not depend on
the quantity of liquid, but on the height of the
liquid and on the section of the bottom of this
container.
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The above statement represents the
hydrostatic paradox and is illustrated infig.4.5. The force that presses on the bottom ofthe three different shaped containers, is thesame because the level of the liquid in thecontainer is the same, and the surface of the
bottom is the same.
Fig. 4.5
4.6 Archimedes principle
Lets consider a solid body and further tosimplify a cylinder, submerged in a liquid; weintend to compute the resultant of the pressure
forces that act upon it (fig.4.6).
Fig.4.6
The resultant of the horizontal forces'
xF
and ''xF is obviously nil:
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.
,
0
''
0
'
xx
xx
AzF
AzF
=
=(4.19)
The vertical forces will have the value:
.
;
2
''
1
'
zz
zz
AzF
AzF
=
=(4.20)
Thus their resultant will be:
( ) VhAzzAFFF zzzzz ===+= 12''' . (4.21)
This demonstration may easily be extendedfor a body of any shape.
An object submerged in a liquid is up
thrust with an equal force with the weight of the
displaced liquid.
4.7. The floating of bodies
A free body, partially submerged in a liquidis called a floating body.
The submerged part is called immerse part orhull.
The weight centre of the hulls volume iscalled the hull centre.
The free surface of the liquid is calledfloating plane.
The crossing between the floating plane andthe floating body is called the floating surface.
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Its weight centre is called floating centre,
and its outline is called floating line or waterline.
In order that the floating body be inequilibrium, it is necessary that the sum of the
forces that act upon it as well as the resultantmoment should be nil.
Upon a floating body there can act twoforces: the archimedean force and the weight
force also called displacement (D = mg)(fig.4.7)
Fig.4.7
As a result, a first condition to achievethe equilibrium is:
VmgD == , (4.22)
where m is the mass of the floating body, V isthe volume of the hull, and is the specific
weight of the liquid.
Furthermore, in order that the moment of theresultant should be nil these two forces must
have the same straight line as support or, inother words, that the weight centre G should befound on the same vertical with the centre hull.
Equation (4.22) is called the equation offlotability.
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Stability is the ability of the floating
body to return on the initial floating ofequilibrium after the action of perturbatoryforces that drew it out of that position hasceased.
With respect to a Cartesian system of axesOxyz, having the plane xOy in the floating plane
and axis Oz upwards directed (fig.4.8), thefloating body has six degrees of freedom: three
translations and three rotations. The rotationaround Ox and Oy is most important.
These slantings are due to the actions ofwaves or wind.
By definition, the rotation of the floating
body thus produced as the volume of the hull toremain unchanged as a value but which can varyin shape is called isohull slanting.
Let 00 LL be the plane of the initialfloating. After the slanting of the isohullaround a certain axis, the floating body will be
on a floating 11 LL .
If initially the centre of hull were
situated in the point 0C after the isohull
slating with an angle , the centre of hullwould move
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further, in the sense of slanting, to a point
1C .
This movement takes place due to thealteration of the shape of the hull volume.
The locus of the successive positions of thecentre of the hull for different isohullslantings around the same axis is called the
curve of the centre of hull (trajectory C).
The curvature centre of the curve of thehull centres is called metacentre and itscurvature radius is called metacentric radius.
For transversal slantings around thelongitudinal axis Ox we shall talk about atransversal metacentre M and about a transversalmetacentric radius r (fig.4.8 a).
Fig.4.8 a, b
For longitudinal slantings around thetransversal axis Oy the longitudinal metacentrewill be denoted by , and the corresponding
metacentric radius will be R (fig. 4.8 b).
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Causing a transversal slanting to the
floating body, isohull, with a small angle, ,
the centre of hull will move to point 1C
(fig.4.8 a). In this case, the force of
flotability V , normal on the slanting
flotability 11 LL , having as application point
the point 1C wont have the same support as the
weight (displacement) of the floating body.
As a result, the two forces will make up acouple whose moment, rM , will be given by the
relation:
sinhDMr = , (4.23)
where
arh = . (4.24)
is called metacentric height, and a is thedistance on the vertical between the weight
centre and centre of hull; denoting by Gz and Cz
the quotes of these points to a horizontalreference plane, we shall have:
CG zza = . (4.25)
The metacentric height, expressed by therelation (4.24) may be positive, negative or nil.We shall in turn analyse each of these cases.
a)if h > 0 the metacentre will be above the
weight centre, and the moment rM , given by the
relation (4.24) will also be positive. From
fig.4.8.it can be noticed that, in this case,
the moment rM will tend to return the floating
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body to the initial floating 0L ; for this
reason it is called restoring moment. In thiscase the floating of the body will be stable.
b)if h < 0, the metacentre is below the centreof weight (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 to capsize, the floating of the body
being unstable.c)If h = 0, the metacentre and the centre ofhull will superpose (fig.4.9 b). Consequently,the restoring moment will be nil, and the bodywill float in equilibrium on the slantingfloating.
Fig.4.9 a, b
In this case the floating is also unstable.Thus, the stability conditions of the floatingare: the metacentre should be placed above theweight centre, namely
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.0>= arh (4.26)
According to (4.24) and (4.23), we maywrite:
( )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,the restoring moment being in this case:
( ) sinsin aRDHDMr
== , (4.30)
where
aRH = . (4.31)
represents the longitudinal metacentric height,and R 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 threeaxes 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
yz
x
(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 ifvelocity v derives from a function :
.,,z
vy
vx
vzyx
=
=
=
(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 for
liquids,
02
2
2
2
2
2
=
+
+
=
+
+
zyxz
v
y
v
x
vzyx , (5.7)
we shall notice that function verifies
equation of Laplace:
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 or
bidimensional if all the particles that are foundon the same perpendicular at an immobile plane,
called director plane, move parallel with thisplane, with equal velocities.
If the director plane coincides with xOy,
then 0=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 vdy
vdx = , (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
the stream 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 thepotential function is constant, define the
equipotential surfaces.
In the case of a potential plane motion:
- constant, equipotential lines of
velocity;- constant, stream lines.
Computing the circulation of velocity along
a certain outline, in the mass of fluid, betweenpoints A 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 doesnt
depend on the shape of the curve AB, but only onthe values of the function in A and B. The
circulation 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 andunitary breadth), 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 considerationthat the gradient of a scalar function F isnormal on the level surface F = cons. As a
result, vectors and are normal on thestreamlines and on the equipotential lines ofvelocity.
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Computing their scalar product, we get:
0=+=
+
=
yxyxvvvv
yyxx . (5.19)
Since their scalar product is nil, itfollows that they are perpendicular, thereforetheir streamlines are perpendicular on the linesof 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 of
complex variable.
Any potential plane motion may always be
plotted by means of an analytic function ofcomplex variable,
( )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, lets
establish 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 offunctions and :
( )
( ) .,
,,
0
0
xvyvyx
yvxvyx
K
K
=
+=
(5.27)
By equalling these relations with constantswe obtain the equations of equipotential linesand of streamlines.
.
.
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
vvvv (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 motionon the 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 and
uniform flow, of velocity Kv , on the direction of
the axis Oy.
The motion described above will have areverse sense if the corresponding expressions of
the potential vector are taken with a reversesign.
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, inthe form:
..
,.
cons
consr
=
=
(5.40)
It can be noticed that the equipotentiallines are concentric circles with the centre inthe origin of the axes, and the streamlines areconcurrent lines 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 ofvelocity will be:
.01
,2
==
=
=
rv
r
Q
rv
S
r
(5.42)
It can noticed that on the circle of radiusr = cons., the fluid velocity has a constantmodulus, being co-linear with the vector radiusof the considered point.
Such a plane potential motion in which the
flow takes place radially, in such a manner thatalong a circle of given radius velocity is
constant as a modulus, is called a plane source.
Constant Q, which appears in the above -written relations, is called the flow of thesource.
The flow of the source through a circular
surface of radius r and unitary breadth will be:
12 rvrQ = . (5.43)
Analogically, the complex potential of theform:
( ) zQ
zW ln2
= , (5.44)
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will represent a suction or a well because, in
this case, the sense of the velocity isreversing, the fluid moving from the exterior tothe origin (where it is being sucked).
If the source isnt placed in the origin of
the axes, but in a point 1O , of the real axis, of
abscissa a , then:
( ) ( )azQ
zW = ln
2
. (5.45)
5.4. The whirl
Let the complex potential be:
( ) z
i
zW ln2
=
. (5.46)
where is a positive and real constant, equalto the circulation of velocity along a closedoutline, 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 equipotential
lines, of equation .const= are concurrent lines,
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in the origin of axes, and the streamlines,
having the equation .constr = , are concentriccircles with their centre in the origin of theaxes (fig.5.5).
Fig.5.5
The components of velocity are:
02
10 >
=
==
=
rrvand
rv
Sr
. (5.48)
Thus, on a circle of given radius r, thevelocity is constant as a modulus, has thedirection of the tangent to this circle in theconsidered point and is directed in the sense ofangle increase.
If the whirl is placed on the real axis, in
a point with abscissa a , the complex potentialof the motion will be:
( ) ( )azi
zW mln2
= . (5.49)
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5.5. The flow with and without
circulation around a circular cylinder
The flow with circulation around a circularcylinder is a plane potential motion thatconsists of an axial stream (directed along axis
Ox), a dipole of moment *2=M (with a source at
the left of suction) and a whirl (in directtrigonometric 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
20
1
vr = . (5.51)
By writing the complex variable ierz = , weshall divide in (5.50) the real part from theimaginary one, thus obtaining functions and :
2
cos2
0
0
+
+=
r
rrv , (5.52)
rr
rrv ln2sin
2
00
= . (5.53)
* The di pol e or t he dupl et i s a pl ane potent i al moti on t hat consi st sof t wo equal sour ces of opposi t e senses, pl aced at an i nf i ni t e smal l
di st ance 2 , so t hat t he pr oduct QM 2= , cal l ed t he moment of t he
di pol e shoul d be f i ni t e and const ant . ( )az
MzW
m
1
2= .
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The stream and equipotential lines are obtained
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 a
streamline (fig.5.8).
Admitting that this streamline is a solid
border, well be able to consider this motiondescribed by the complex potential (5.50) asbeing the flow around a straight circular
cylinder of radius 0r , having the breadth normal
on the motion plane, infinite.
If we plot the otherstreamlines we shallget some asymmetriccurves with respect to
axis Ox (fig.5.6). Onthe inferior side ofthe circle of radius
0r , the velocity due to
the axial stream issummed up with thevelocity due to thewhirl.
Fig.5.6
As a result, here we shall obtain smallervelocities, and the streamlines will be morerare.
In polar co-ordinates, the components of
velocity in a certain point ( ),rM , will be:
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cos12
2
0
0
=
r
rvvr , (5.54)
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 acircular cylinder is the plane potential motionmade up of an axial stream (directed along axis
Ox) and a dipole of moment 2=M (whose sourceis at the left of suction).
Thus, this motion can be obtained
particularising the motion previously describedby cancelling the whirl.
By making 0= , in relations (5.50), (5.52)and (5.53) we get the complex potential of themotion, the function potential of velocity andthe function of stream, in the form:
( ) ,2
0
0
+=
z
rzvzW (5.56)
,cos2
00
+=
r
rrv (5.57)
.sin2
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 madeup of 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
motion can be seen asa flow of an axial
stream around aninfinitely longcylinder, normal on
the motion plane.Fig.5.7
The components of velocity are:
.sin1
,cos1
2
2
0
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
the stagnant 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 thecrossing curve between the cylinder and thecomplex 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, isv .
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
unit of length of the body.
The second relation (5.62) is the mathematicexpression of Kutta-Jukovskis theorem, which
will be only stated below without demonstratingit:
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If a fluid of density is draining
around a body of circulation and velocity ininfinite v , it will act upon the unit of length
of the body with a force equal to the product
v , normal on the direction of velocity ininfinite called lift force (lift).
The sense of the lift is obtained byrotating the vector of velocity from infinite
with0
90 in the reverse sense of circulation.
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6.IMPULSE AND MOMENT IMPULSETHEOREM
We take into consideration a volume offluid. This fluid is homogeneous, incompressible,of density , bordered by surface . The
elementary volume d has the speed v.
The elementary impulse will be:
dvId = . (6.1)
=
dvI . (6.2)
=
ddt
vd
dt
Id. (6.3)
At the same time
iFdt
Id= . (6.4)
But: 0=++ ipm FFF (dAlembert principle).
(6.5)
Therefore:
epm FFFdtId =+= . (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 ideal
fluid, the vectorial sum of the external forces
which act upon the fluid in the volume , isequal with the impulse flow through the exit
surfaces (from the volume ), less the impulseflow through the entrance surfaces (to the volume
) .
r- the position vector of the centre ofvolume with respect to origin of the referencesystem.
The elementary inertia moment with respectto 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:
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dvId = the elementary impulse,(6.11)
dvrkd = the moment of elementary impulse,(6.12)
=
,dvrk (6.13)
( ) iMdvrdtd
dt
kd
== . (6.14)
The derivative of the resultant moment ofimpulse with respect to time is equal with theresultant moment of inertia forces withreversible sign.
expm MMMdt
kd=+= , (6.15)
where
mM - the moment of mass forces,
pM - the moment pressure forces,
exM - the moment of external forces.
oioerr , - the position vector of the centre of
gravity for the exit /entrance surfaces.
( ) ( )ioiieoeeex vrMvrMM = . (6.16)
Under permanent flow conditions of ideal
fluids, the vectorial addition of the moments of
external forces which act upon the fluid in the
volume , is equal to the moment of the impulse
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flow through the exit surfaces less the moment of
the impulse flow through the entrance surfaces.
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7.MOTION EQUATION OF THE REALFLUID
7.1 Motion regimes of fluids
The motion of real fluids can be carried outunder two regimes of different quality: laminarand turbulent.
These motion regimes were first emphasisedby the English physicist in mechanics OsborneReynolds in 1882, who made systematicexperimental studies concerning the flow of water
through glass conduits of diameter mmd 255 = .
The experimental installation, which was
then used, is schematically shown in fig.7.1.
The transparent conduit 1, with a veryaccurate processed inlet, is supplied by tank 2,full of water, at a constant level.
Fig.7
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The flow that passes the transparent conduit
can 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
liquid of the same density as water. The flow ofcoloured liquid, supplied by tank 5 may be
adjusted by means of tap 7.But slightly turning on tap 3, through
conduit 1 a stream of water will pass at acertain flow and velocity.
If we turn on tap 7 as well, the colouredliquid inserted through the thin tube 4, engages
itself in the flow in the shape of a rectilinearthread, parallel to the walls of conduit, leaving
the impression that a straight line has beendrawn inside the transparent conduit 1.
This regime of motion under which the fluid
flows in threads that dont mix is called alaminar regime.
By slowly continuing to turn on tap 3, wecan notice that for a certain flow velocity of
water, the thread of liquid begins to undulate,and for higher velocities it begins to pulsate,which shows that vector velocity registersvariations in time (pulsations).
For even higher velocities, the pulsations
of the coloured thread of water increase theiramplitude and, at a certain moment, it will tear,the particles of coloured liquid mixing with themass of water that is flowing through conduit 1.
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The regime of motion in which, due to
pulsations of velocity, the particles of fluidmix is called a turbulent regime.
The shift from a laminar regime to theturbulent one, called a transition regime is
characterised by a certain value of Reynolds
number *, called critical value ( crRe ).
* Number
vl
=Re , i s t he number t hat
def i nes t he si mi l ar i t y cr i t er i on Reynol ds.
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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 be laminar, while for crReRe > , the
flow regime will be turbulent.
7.2 Navier Stokes equation
Navier Stokes equation describes themotion of real (viscous) incompressible fluids ina laminar regime.
Unlike ideal fluids that are capable todevelop only unitary compression efforts that are
exclusively due to their pressure, real (viscous)fluids can develop normal or tangentsupplementary viscosity efforts.
The expression of the tangent viscosityeffort, defined by Newton (see chapter 2) is thefollowing:
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 inthe direction of axis Ox is:
( ) ( ) ( )
.dzdydxzyx
dydxdzz
dydxdyy
dzdydxx
dF
zxyxxx
zxyxxx
vx
+
+
=
=
+
+
=
(7.4)
According to the theory of elasticity:
z
xy
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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
vzyx , according to the
equation of continuity for liquids.
Then:
dzdydxvdFxx = . (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 ofNavier-Stokes equation. The scalar form of thisequation is:
.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
vv
y
vv
x
v
t
v
z
v
y
v
x
v
z
pF
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 the
permanent 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 to another, there takes place
only the conversion of a part from the potentialenergy into kinetic energy, or the other wayround, in permanent motion of the real fluid, its
specific energy is no longer constant. It alwaysdecreases in the sense of the movement of the
fluid.
A part of the fluids energy is convertedinto thermal energy, is irreversibly spent to
overcome the resistance brought about by itsviscosity.
Denoting this specific energy (load) by fh ,
Bernoullis equation becomes:
fhz
p
g
vz
p
g
v+++=++ 2
2
2
2
1
1
2
1
22 . (7.16)
In different points of the same section,only the potential energy remains constant, the
kinetic one is different since the velocitydiffers in the section, ( )zyxvv ,,= . In this casethe term of the kinetic energy should becorrected by a coefficient , that considers thedistribution of velocities in the section
( )1,105,1 = .
* t he wei ght uni tenergy
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fhz
p
g
vz
p
g
v+++=++ 2
2
2
22
1
1
2
11
22
. (7.17)
By reporting the loss of load fh to the
length l of a straight conduit, we get thehydraulic slope (fig.7.3):
Fig.7.3
l
h
l
zp
g
vz
p
g
v
I
f
=
++
++
=
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
htgII
f
p === . (7.20)
Experimental researches have revealed thatirrespective of the regime under which the motion
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of fluid takes place, the losses of load can be
written in the form:
m
fvbh = , (7.21)
where b is a coefficient that considers thenature of the fluid, the dimensions of theconduit and the state 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 tothe turbulent 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 two
plane 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 theaxis Ox:
;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
( )zvvxx = .
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
pzv
x++
=
. (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
pzv
x
=
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 ofbreadth b will be:
x
phbhbvQ
==
12
3
. (7.34)
* maxxv i s pos i t i ve, s i nce 0xvOx , then 0/
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Let us now consider an elementary surface d
A in the shape of a circular crown of radius rand breadth d r (fig.7.6 b). The elementary flowthat crosses surface d A is:
rdrvdAvdQxx 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
xv
rI
A
Qu ===
. (7.52)
Further on we can write:
g
v
ddg
dv
dg
v
r
v
l
h
I
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 inthe conduits 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.
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7.5 Turbulent motion of fluids
In a point of the turbulent stream, thefluid velocity registered rapid variation, in onesense or the other, with respect to the meanvelocity in section. The field of velocities hasa complex structure, still unknown, being theobject of numerous studies.
The variation of velocity with the time maybe plotted 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,remains a 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
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( ) ( ) ( )zyxutzyxutzyxu ,,,,,,,,' = . (7.56)
There are several theories that by
simplifying describe the turbulent motion:
a)Theory of mixing length (Prandtl),which admits that the impulse is keptconstant.
b)Theory of whirl transports (Taylor)where the rotor of velocity is presumedconstant.
c)Karamans theory of turbulence, whichstates that, except for the immediate vicinityof a wall, the mechanism of turbulence isindependent from viscosity.
7.5.1 Coefficient in turbulent motion
Determination of load losses in theturbulent motion is an important problem inpractice.
It had been experimentally established that
in turbulent motion the pressure loss p dependson the following factors: mean velocity on
section, v , diameter of conduit, d , density of the fluid and its kinematic viscosity ,length l of the conduit and the absolute rugosity
* of its interior walls; therefore:
( )= ,,,,, ldvfp , (7.57)or:
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d
lvp
2
2
= , (7.58)
d
l
g
vph
f2
2
=
= , (7.59)
ror
d
- relative rugosity
where:
=
dRe,2 1 . (7.60)
*mean hei ght of t he condui t promi nence ;r
or
d-
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As it can be seen from relation (7.60), in
turbulent motion the coefficient of load loss may depend either on Reynolds number or on therelative rugosity of the conduit walls.
In its turbulent flow through the conduit,
the fluid has a turbulent core, in which theprocess of mixing is decisive in report to theinfluence of viscosity and a laminar sub-layer,situated near the wall, in which the viscosityforces have a decisive role.
If we note by l the thickness of the
laminar sub-layer, then we can classify conduitsas follows:
- conduits with smooth walls; l .From (7.60) we notice that, unlike the
laminar motion in turbulent motion is a
complex function of Re and d
.
It has been experimentally established thatin the case of hydraulic smooth conduits,
coefficient depends only on Reynolds number.Thus, Blasius, by processing the existentexperimental material (in 1911), established forthe smooth hydraulic conduits of circularsection, the following empirical formula:
25,0
4/1
Re
3164,0
3164,0=
=
dv
, (7.61)
valid for510Re000,4
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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
6Lees 35,03 Re61,010714,0 +=
Smooth
turbulent
6
3
103Re
10Re
II
Auth
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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
5 10Re10
d
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7.5.2 Nikuradzes diagram
On the basis of experiments made withconduits of homogeneous different rugosity, which
was achieved by sticking on the interior wallsome grains of sand of the same diameter,
Nikuradze has made up a diagram that representsthe way coefficient varies, both for laminarand turbulent fields (fig.7.8).
Fig.7.8
We can notice that in the diagram appear
five areas in which variation of coefficient ,
distinctly differs.
Area I is a straight line which representsin logarithmic co-ordinates the variation:
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 differentrelative rugosities 0/ r .
Area II is the shift from laminar regime to
the turbulent one which takes place for
( )2300Re4,3Relg .
Area III corresponds to the smoothhydraulic conduits. In this area coefficient can be determined with the help of Blasius
relation (7.61), to which the straight line III acorresponds, called Blasius straight. Since the
validity field of relation (7.61) is limited by510Re = , for higher values of Reynolds number,
we use Kanakovs formula, to which curve III bcorresponds. It is noticed that the smaller therelative rugosity is, the greater the variationfield of Reynolds number, in which the smoothturbulent regime is maintained.
In area IV each discontinuous curve, which
represents dependent ( )Ref= for differentrelative rugosities becomes horizontal, which
emphasises the independence of on number Re.Therefore this area corresponds to the rugous
turbulent regime, where is determined by(7.63).
It is noticed that in this case the lossesof load (7.59) are proportional to squarevelocity.
For this reason the rugous turbulent regimeis also called square regime.
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Area V is characterised by the dependence
of the coefficient both on Reynolds number andon the relative rugosity of the conduit.
It can be noticed that for areas IV and V,
coefficient decreases with the decrease of
relative rugosity.
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8.FLOW THROUGH CIRCULAR CONDUITS
In this chapter we shall present thehydraulic calculus of conduits under pressure ina permanent regime.
Conduits under pressure are in fact ahydraulic system designed to transport fluidsbetween two points with different energeticloads.
Conduits can be simple (made up of one orseveral sections of the same diameter ordifferent diameters), or with branches, in thiscase, setting up 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 astream of real liquid, between the free side ofthe liquid from the tank A and the end of theconduit, taking as a reference plane thehorizontal plane N N, we get:
fhz
p
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:
fh
g
vh +=
2
2
, (8.2)
where 2vv = is the mean velocity in the sectionof the conduit , and h is the load of theconduit.
In the analysed case shown in fig. 8.1 b,by introducing in equation (8.1) the relations
1022112011 ,,,,0 hppzhhzvvppv +=++=== and
121 == , we shall get the expression (8.2).
From an energetic point of view, thisrelation shows that from the available specific
potential energy (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 theoutcoming in the atmosphere or in another volume.
The other part fh is used to overcome the
hydraulic resistances (that arise due to thetangent efforts developed by the fluid in motion)and is lost because it is irreversiblytransformed into heat.
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Analysing the losses of load from theconduit we shall divide them into two categories,writing the relation:
'''
fffhhh += . (8.3)
The losses of load, denoted by fh' are
brought about by the tangent efforts that are
developed during the motion of the fluid alongthe length of the conduit ( l) and, for this
reason, they are called losses of loaddistributed. These losses of load have beendetermined in paragraph 7.4.2, getting therelation (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, ,called Darcy coefficient is determined by therelations shown in table 7.1 ; the manner of
calculus being also shown in that paragraph.Generally, in practical cases, the values of
coefficient vary in a domain 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 isrepresented by the local losses of load that arebrought about by: local perturbation of thenormal flow, the detachment of the stream fromthe wall, whirl setting up, intensifying of theturbulent mixture, etc; and arise in the area
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where the conduit configuration is modified or at
the meeting an obstacle detouring (inlet of thefluid in the conduit, flaring, contraction,bending and derivation of the stream, etc.).
The local losses of load are calculated with
the help of a general formula, given byWeissbach:
g
v
hf 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 theflow section etc.).
Generally, coefficient depends mainly on
the geometric parameters of the consideredelement, as well as on some factors that
characterise the motion, such as: the velocities
distribution at the inlet of the fluid in theexamined element, the flow regime, Reynoldsnumber etc.
In practice, coefficient is determined
with respect to the type of the respective localresistance, using tables, monograms or empiricalrelations that are found in hydraulic books.
Therefore, for curved bends of angle090 ,
coefficient can be determined by using the
relation:
0
0
5,3
5,3
9016,013,0
+=
d, (8.6)
where andd are the diameter and curvature
radius of the bend, respectively.
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Coefficient , corresponding to the loss of
load at the inlet in their conduit, depends
mainly on the wall thickness of the conduit withrespect to its diameter and on the way theconduit is attached to the tank. If the conduitis embedded at the level of the inferior wall ofthe tank, the losses of load that arise at theinlet in the conduit are equivalent with thelosses of load in an exterior cylindrical nipple.
For this case, 5.0 .
If on the route of the conduit there areseveral local resistances, the total loss offluid will be given by