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Prof.univ.Dr.ing. DUMITRU DINU S.L.drd.ing. STAN LIVIU
HYDRAULICS AND
HYDRAULIC MACHINES
2
CONTENTS
PART ONE
HYDRAULICS
1. BASIC MATHEMATICS 11
2. FLUID PROPRIETIES 17
2.1 Compressibility 18 2.2 Thermal dilatation 20 2.3 Mobility 22 2.4 Viscosity 22
3. EQUATIONS OF IDEAL FLUID MOTION 29
3.1 Euler’s equation 29 3.2 Equation of continuity 32 3.3 The equation of state 34 3.4 Bernoulli’s equation 35 3.5 Plotting and energetic interpretation
of Bernoulli’s equation for liquids 39 3.6 Bernoulli’s equations for the relative
movement of ideal non-compressible fluid 40
4. FLUID STATICS 43 4.1 The fundamental equation of
hydrostatics 43 4.2 Geometric and physical interpretation
of the fundamental equation of hydrostatics 45 4.3 Pascal’s principle 46
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4.4 The principle of communicating
vessels 47 4.5 Hydrostatic forces 48 4.6 Archimedes’ principle 50 4.7 The floating of bodies 51
5. POTENTIAL (IRROTATIONAL) MOTION 57 5.1 Plane potential motion 59 5.2 Rectilinear and uniform motion 63 5.3 The source 66 5.4 The whirl 69 5.5 The flow with and without circulation around a circular cylinder 71 5.6 Kutta – Jukovski’s theorem 75
6. IMPULSE AND MOMENT IMPULSE THEOREM 77
7. MOTION EQUATION OF THE REAL FLUID
81
7.1 Motion regimes of fluids 81 7.2 Navier – Stokes’ equation 83 7.3 Bernoulli’s equation under the permanent regime of a thread of real fluid 87 7.4 Laminar motion of fluids 90 7.4.1 Velocities distribution between two plane parallel boards of infinit length 90 7.4.2 Velocity distribution in circular conduits 93 7.5 Turbulent motion of fluids 97 7.5.1 Coefficient λ in turbulent motion 99 7.5.2 Nikuradze’s diagram 102
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8. FLOW THROUGH CIRCULAR CONDUITS 105 9. HYDRODYNAMIC PROFILES 113 9.1 Geometric characteristics of hydrodynamic profiles 113 9.2 The flow of fluids around wings116 9.3 Forces on the hydrodynamic profiles 119 9.4 Induced resistances in the case of finite span profiles 123 9.5 Networks profiles 125
5
PART ONE
HYDRAULICS
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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:
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 by those vectors, directed in such a manner that the trihedral a ,b and ba × should be rectangular.
zyx
zyx
bbbaaakji
ba =× . (1.4)
The modulus of the vectorial product is
given by the relation:
7
( )∧
=× 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
cccbbb
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 operator ∇is defined by:
zk
yj
xi
∂∂
+∂∂
+∂∂
=∇ . (1.9)
∇ applied to a scalar is called gradient.
.ϕϕ grad=∇
kz
jy
ix ∂
∂+
∂∂
+∂∂
=∇ϕϕϕϕ . (1.10)
∇ scalary applied to a vector is called divarication. .adiva =∇
8
za
ya
xaa zyx
∂∂
+∂∂
+∂∂
=∇ . (1.11)
∇ vectorially applied to a vector is called rotor. .arota =×∇
zyx aaazyx
kji
a∂∂
∂∂
∂∂
=×∇ . (1.12)
Operations with ∇: ( ) ψϕψϕ ∇+∇=+∇ . (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 doesn’t act must, in the end, be placed in front of the operator;
- it mustn’t be placed alone at the end.
( ) ( ) ( ) ϕψψϕψϕψϕψϕ ∇+∇=∇+∇=∇ cc . (1.16) ( ) ( ) ( ) ϕϕϕϕϕ ∇+∇=∇+∇=∇ aaaaa cc . (1.17)
( ) ( ) ( ) ϕϕϕϕϕ ∇×−×∇=×∇+×∇=×∇ aaaaa cc . (1.18)
9
( ) ( ) ( )cc bababa ∇+∇=∇ , (1.19)
( ) ( ) ( )bababa cc ∇−∇=×∇× , (1.20) ( ) ( )babrotabac ∇+×=∇ , (1.21)
( ) ( )abarotbba c ∇+×=∇ , (1.22)
( ) ( ) ( )abarotbbabrotaba ∇+×+∇+×=∇ . (1.23)
cϕ - the scalar ϕ considered constant,
cψ - the scalar ψ 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 of
its 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 to the vector velocity in its corresponding point:
10
zyx vdz
vdy
vdx
== . (1.26)
The whirl line is a curve tangent in each of its points to the whirl vector of the corresponding point ( )kji zyx ωωωω ++= .
vrot21
=ω . (1.27)
The equation of the whirl line is obtained by writing that the tangent to whirl line is parallel with the vector whirl in its corresponding point:
zyx
dzdydxωωω
== . (1.28)
Gauss-Ostrogradski’s relation:
τστσ
dadna ∫∫ ∇= , (1.29)
where τ - volume delimited by surface σ . The circulation of velocity on a curve (C) is defined by: =Γ ,rdv
C∫ (1.30)
in which
dsrd τ= (1.31) represents the orientated element of
the curve (τ - the versor of the tangent to the curve (C )).
11
Fig.1.1
( )∫ ++=ΓC
zyx dzvdyvdxv (1.32)
The sense of circulation depends on the
admitted sense in covering the curve.
ABMAAMBA Γ−=Γ . (1.33) Also:
BAAMBAMBA Γ+Γ=Γ . (1.34) Stokes’ relation:
( ) σσ
∫ ∫==ΓC
dnvrotrdv (1.35)
in which n represents the versor of the
normal to the arbitrary surface σ bordered by the curve (C).
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2. FLUID PROPRIETIES
As it is known, matter and therefore fluid bodies as well, has a discrete and discontinuous structure, being made up of micro-particles (molecules, atoms, etc) that are in reciprocal interaction.
The mechanics of fluids studies phenomena
that take place at a macroscopic scale, the scale at which fluids behave as if matter were continuously distributed.
At the same time, fluids don’t have their
own shape so are easily deformed. A continuous medium is homogenous if at a
constant temperature and pressure, its density has only one value in all its points.
Lastly, a continuous homogenous medium is
isotropic as well if it has the same proprieties in any direction around a certain point of its mass.
In what follows we shall consider the fluid
as a continuous, deforming, homogeneous and isotropic medium.
We shall analyse some of basic physical
proprieties of the fluids.
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2.1. Compressibility
Compressibility represents the property of fluids to modify their volume under the action of a variation of pressure. To evaluate quantitatively this property we use a physical value, called isothermal compressibility coefficient, β , that is defined by the relation:
,1 2
⎥⎦
⎤⎢⎣
⎡−=
Nm
dpdV
Vβ
(2.1)
in which dV represents the elementary variation of the initial volume, under the action of pressure variation dp.
The coefficient β 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, namely dv/ dp < 0.
The reverse of the isothermal compressibility coefficient is called the elasticity modulus K and is given by the relation:
.12 ⎥⎦⎤
⎢⎣⎡−==mN
dVdpVK
β (2.2)
Writing the relation (2.2) in the form:
,Kdp
VdV
−= (2.3)
we can underline its analogy with Hook’s law:
14
.El
dl σ= (2.4)
a) The compressibility of liquids
In the case of liquids, it has been
experimentally ascertained that the elasticity modulus K, and implicitly, the coefficient β , vary very little with respect to temperature (with approximately 10% in the interval C0600− ) and they are constant for variations of pressure within enough wide limits. In table (2.1) there are shown the values of these coefficients for various liquids at C00 and pressure 200≤p bars.
Table 2.1. Liquid
[ ]Nm /2
β [ ]2/ mN
K
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 β may be considered constant. Consequently, we can integrate the
differential equation (2.2) from an initial state, characterised by volume 0V , pressure 0p and
density 0ρ , to a certain final state, where the
state parameters will have the value pV ,1 and ρ respectively; we shall successively get:
15
∫ ∫−=V
V
p
p
dpVdV
0 0
,β (2.5)
or
( ) .00
ppeVV −−⋅= β (2.6) b) The compressibility of gases
For gases the isothermal compressibility
coefficient depends very much on pressure. In the case of a perfect gas, the following relation describes the isothermal compressibility:
pV = cons.,
which, by subtraction, will be:
.VdV
pdp
−= (2.8)
By comparing this relation to (2.3) we may
write:
.1 pK ==β
(2.9)
It follows that, in the case of a perfect
gas, the elasticity modulus is equal to pressure. 2.2 Thermal dilatation Thermal dilatation represents the fluid
property to modify its volume under the action of
16
a variation of temperature. Qualitatively, this property is characterised by the volumetric coefficient of isobaric dilatation, defined by the relation:
,1dTdV
V⋅=α (2.10)
where dV represents the elementary variation of the initial volume V under the action of variation of temperature dT. Coefficient α is positive for all fluids, except for water, which registers maximum density (minimum specific volume) at C04 ; therefore, for water that has
Ct 04≤ we shall have .0<α
Generally,α varies very little with respect to temperature, therefore it can be considered constant. Under these circumstances, integrating the equation (2.10) between the limits 0V and V,
and respectively 0T and T, we get:
( ),ln 00
TTVV
−=α (2.11)
or else
( ).00
TTeVV −= α (2.12)
By dividing the relation (2.12) to the mass of the fluid ,00VVm ρρ == we get the function of state for an incompressible fluid:
( ) ,00
TTe −−= αρρ (2.13)
In the case of a perfect gas the value of the coefficient is obtained by subtracting the
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equation of isobaric transformation ⎟⎠⎞
⎜⎝⎛ = .consTV
; we
get:
,. dTTVdTconsdV =⋅= (2.14)
which, replaced into (2.10) enables us to write:
.1T
=α (2.15)
Thus, for the perfect gas, coefficient α is the reverse of the thermodynamic temperature.
2.3. Mobility
In the case of fluids, the molecular cohesion forces have very low values, but they aren’t rigorously nil.
At a macroscopic scale, this propriety can be rendered by the fact that two particles of fluid that are in contact, can be separated under the action of some very small external forces. At the same time, fluid particles can slide one near the other and have to overcome some relatively small 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 constant volume, the compression efforts are rigorously nil and, as a result, the change in shape of the fluid requires the overcoming of the tangent efforts, which are very small. Therefore the mechanical work consumed from the exterior will be 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, because they don’t have their own volume, have a higher mobility than liquids (a gas inserted in a container takes both the shape and the volume of that container).
2.4 Viscosity
Viscosity is the property of the fluid to oppose to the relative movement of its particles.
As it has been shown, overcoming some small tangent efforts that aren’t yet rigorously nil makes this movement.
To qualitatively stress these efforts, we consider the unidimenssional flow of a liquid, which takes place in superposed layers, along a board situated in xOy plane (fig.2.1).
Fig.2.1.
Experimental measurements have shown that
velocity increases as we move away from the board in the direction of axis Oy, and it is nil in the near vicinity of the board. Graphically, the dependent ( )yfv = is represented by the curve Γ. This simple experiment stresses on two aspects, namely:
19
- the fluid adheres on the surface of the solid body with which it comes into contact;
- inside the fluid and at its contact with the solid surfaces, tangent efforts generate which determine variation in velocity. Thus, considering two layers of fluid, parallel to the plane xOy and that are at an elementary distance dy one from the other, we shall register a variation
in velocity dydydv
, due to the frictions
that arise between the two layers.
To determine the friction stress, Newton used the relation:
dydvητ = , (2.16)
that today bears his name. This relation that has been experimentally verified by Coulomb, Poisseuille and Petrov shows that the friction stress τ is proportional to the gradient of velocity. The proportionality factor η is called dynamic viscosity.
If we represent graphically the dependent ( )dydvf /=τ we shall get the line 1 (fig.2.2)
where ηθ =ty .
The fluids that observe the friction law (2.16) are called Newtonian fluids (water, air, etc). The dependent of the tangent effort to the gradient of
20
velocity is not a straight line (for example curve (2) in fig. 2.2), for a series of other fluids, generally those of organic nature. These fluids are globally called non-Newtonian fluids.
Fig.2.2
The measures for the dynamic viscosity are:
- in the international standard (SI):
[ ]sm
Kgm
sN⋅
=⋅
= 2η (2.17)
- in the CGS system:
[ ]scm
gcm
sdyn⋅
=⋅
= 2η . (2.18)
The measure of dynamic viscosity in CGS
system is called “poise”, and has the symbol P. We can notice the existence of relation:
Psm
Kg 101 =⋅
. (2.19)
We can determine the dynamic viscosity of liquids with the help of Höppler’s viscometer, whose working principle is based on the proportionality of dynamic viscosity to the time in which a ball falls inside a slanting tube that contains the analysed liquid.
The kinematic viscosity of a fluid is the
ratio of dynamic viscosity and its density:
21
ρην = . (2.20)
The measures for kinematic viscosity are: - in the international system:
[ ]s
m 2
=ν . (2.21)
- in CGS system:
[ ]s
cm 2
=ν . (2.22)
the latter bearing the name “stokes” (symbol ST):
sm
scmST
24
2
1011 −== . (2.23)
Irrespective of the type of viscometer used (Ubbelohde, Vogel-Ossag, etc) we can determine the 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 conventional viscosity of a fluid is often used; this value is determined by measuring the time in which a certain volume of fluids flows through a special device, the conditions being conventionally chosen. The magnitude of this value thus determined is expressed in conventional units. There are several conventional viscosities (i.e. Engler, Saybolt, Redwood etc) which differ from
22
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 the analysed liquid at a given temperature and the flow time of a same volume of distilled water at a temperature of C020 , through an Engler viscometer under standard conditions.
The viscosity of a fluid depends to a great extent on its temperature. Generally, viscosity of liquids diminishes with the increase in temperature, while for gas the reverse phenomenon takes place.
The dependence of liquids viscosity on temperature can be determined by using Gutman and Simons’ relation:
00
TB
TCB
e−
+=ηη . (2.24) where the constants B and C depend on the nature of the analysed liquid (for water we have B= 511,6 K and C= -149,4 K).
For gases we can use Sutherland’s formula”
TSTS
TT
++
⎟⎟⎠
⎞⎜⎜⎝
⎛= 0
2/3
00ηη . (2.25)
where S depends on the nature of the gas (for air S=123,6 K).
In relations (2.24) and (2.25), η and 0η are the dynamic viscosities of the fluid at the
23
absolute temperature T, and at temperature )0(15,273 0
0 CKT = respectively. In table 2.2 there are shown the dynamic and
kinematic viscosities of air and water at different temperatures and under normal atmospheric pressures.
Table 2.2
Temperature
[ ]C0
-10 0 10 20 40 60 80 100
Air
0,162
0,172
0,175
0,181
0,191
0,20
0,289
0,218 ⎥⎦
⎤⎢⎣⎡
⋅ smKg910
η
Water
-
1,79
1,31
1,01
0,658
0,478
0,366
0,295
Air
1,26
13,3
14,1
15,1
16,9
18,9
20,9
23,1 ⎥
⎦
⎤⎢⎣
⎡s
m 2610
ν
Water
-
1,79
1,31
1,01
0,658
0,478
0,366
0,295
We must underline the fact that viscosity is a property that becomes manifest only during the movement of liquids.
A fluid for which viscosity is rigorously
nil is called a perfect or ideal fluid. Fluids may be compressible ( )[ ]pρρ = or
incompressible ( ρ is constant with respect to pressure).
We should emphasise that the ideal
compressible fluid is analogous to the ideal (or perfect) gas, as defined in thermodynamics.
24
The movement of fluids may be uniform (velocity is constant), permanent v = v (x,y,z) or varied v = v (x,y,z,t).
25
3. EQUATIONS OF IDEAL FLUID MOTION
3.1 Euler’s equation We shall further study, for the most
general case, the movement state of a fluid through a volume τ that is situated in the fluid stream; we shall not take into consideration the interior frictions(i.e.viscosity), so we shall analyse the case of perfect (ideal) fluids that are on varied movement.
The volume τ is situated in an accelerated
system of axes, joint with this system. The equations, which describe the movement of the fluid, will be obtained by applying d’Alembert’s principle for the fluid that is moving through the volume τ .
The three categories of forces that act upon
the fluid that is moving through the volume τ ,bordered by the surface σ (fig.3.1), are:
Fig.3.1
- the mass forces mF ;
26
- the inertia forces iF ;
- the pressure forces pF (with an equivalent effect; these forces replace the action of the negligible fluid outside volume τ ).
According to d’Alembert’s principle, we shall get:
0=++ pim FFF . (3.1) Equation (3.1) represents in fact the
general vectorial form of Euler’s equations. Let’s establish the mathematical expressions
of those three categories of forces. If F is the mass unitary force
(acceleration) that acts upon the fluid in the volume τ , the mass elementary force that acts upon the mass τρ d , will be:
τρ dFFd m = , (3.2)
hence:
∫=τ
τρ dFFm . (3.3)
As the fluid velocity through the volume
τ is a vectorial function with respect to point and time: ( )trvv ,= , upon the mass τρd that is
moving with velocity v the elementary inertia will act:
27
τρ ddtvdFd i −= . (3.4)
So, the inertia will be:
∫−=τ
τρ ddtvdFi . (3.5)
If σd is a surface element upon which the
pressure p acts, and n - the versor of the exterior normal (Fig.3.1), the elementary force of pressure is:
σdnpFd p −= . (3.6)
Having in mind Gauss-Ostrogradski’s theorem,
the resultant of pressure forces will be:
τστσ
dpdnpFp ∫∫ ∇−=−= . (3.7)
By replacing equations (3.3), (3.5) and
(3.7) in the equation (3.1), we shall get:
0=⎟⎟⎠
⎞⎜⎜⎝
⎛−∇−∫
τ
τρρ ddtvdpF , (3.8)
Hence:
dtvdpF =∇−
ρ1
, (3.9)
Or
28
( )vvtvpF ∇+∂∂
=∇−ρ1
, (3.10)
The equation (3.10) – Euler’s equation in a
vectorial form for the ideal fluid in a non-permanent movement.
Projecting this equation on the three axes, we shall obtain:
zx
yx
xxx
x vzv
vyv
vxv
tv
xpF
∂∂
+∂∂
+∂∂
+∂∂
=∂∂
−ρ1
;
zy
yy
xyy
y vz
vv
yv
vxv
tv
ypF
∂
∂+
∂
∂+
∂
∂+
∂
∂=
∂∂
−ρ1
; (3.11)
zz
yz
xzz
z vzv
vyv
vxv
tv
zpF
∂∂
+∂∂
+∂∂
+∂∂
=∂∂
−ρ1
.
3.2 Equation of continuity
This equation can be obtained by writing in two ways the variation in the unity of time for the mass of fluid that is in the control volume τ , bordered by the surface σ (fig.3.1). By splitting from the volume τ one element τd , and taking into consideration that the density is a scalar function of point and time, ( )tr,ρρ = , we can write the total mass of the volume τ :
∫=τ
τρdm . (3.12)
29
The variation of the total mass in the unity of time will be:
∫ ∂∂
=∂∂
τ
τρ dtt
m . (3.13)
The second form of writing the variation of
mass is obtained by examining the flow of the mass through surface σ that borders volumeτ .
Denoting by n the versor of the exterior normal to the area element σd , and by v the vector of the fluid velocity, the elementary mass of fluid that passes in the unity of time through the area element σd is:
σρ dvdM n−= . (3.14)
In the unity of time through the whole
surface σ will pass, the mass:
∫−=σ
σρ dvM n (3.15)
that is the sum of the inlet and outlet mass in volume τ , by crossing surface σ .
By equalling equations (3.13) and (3.15), it will result:
∫ ∫ =+∂∂∂
τ σ
σρτρ 0dvt n . (3.16)
According to Gauss-Ostrogradski’s theorem:
( )∫ ∫∇=σ τ
τρσρ dvdvn . (3.17)
30
Taking into consideration (3.17), the equation (3.16) will take the form:
( ) 0=⎥⎦⎤
⎢⎣⎡ ∇+∂∂
∫ τρρ
τ
dvt
, (3.18)
hence, successively:
( ) ,0=∇+∂∂ v
tρ
ρ (3.19)
0=∇+∇+∂∂ vv
tρρ
ρ, (3.20)
0=∇+ vdtd
ρρ
. (3.21)
The equation (3.21) represents the equation
of continuity for compressible fluids. In the case of non-compressible fluids
( .cons=ρ , 0=dtdρ
), the equation of continuity
takes the form: 0=∇v , (3.22)
or
0=∂∂
+∂
∂+
∂∂
zv
yv
xv zyx . (3.23)
It follows that the inlet volume of non-
compressible liquid is equal to the outlet one in and from the volume τ .
31
3.3. The equation of state From a thermodynamically point of view, the
state of a system can be determined by the direct measurement of some characteristic physical values, 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 are link relationships, explained by the laws of physics.
In the case of homogenous systems, there is
only one implicit relationship, which carries out the link among the three state parameters, in the form of:
( ) 0,, =TpF ρ . (3.24)
Adding to vectorial equations (3.10) and
(3.21) the equation of state, we get three equations with three unknowns: ( ) ( ) ( )trptrtrv ,,,,, ρ , that enable us solve the problems of motion and repose for the ideal fluids.
3.4. Bernoulli ‘s equation Bernoulli’s equation is obtained by
integrating Euler’s equation written under a different form (Euler – Lamb), that stresses the rotational or non-rotational nature of the ideal fluid (see the relation (1.25)).
32
Euler – Lamb’s equation:
vrotvvtvpF ×−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛∇+
∂∂
=∇−2
12
ρ. (3.25)
Considering the case when the mass force
derives from a potential U, thus being a conservative force (the mechanical energy-kinetic and potential-will be constant):
UF −∇= . (3.26)
In the case of compressible fluids, when ( )pρρ = , we insert the function:
( )∫= pdpPρ
. (3.27)
Thus:
( ) pp
P ∇=∇ρ
1. (3.28)
The equation (3.25) takes the form:
vrotvtvvPU ×−∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛++∇−
2
2
. (3.29)
The equation (3.29) can be easily integrated
in certain particular cases.
In the case of permanent motion 0=∂∂
tv
, and:
- along a stream line:
33
zyx vdz
vdy
vdx
== , (3.30)
- along a whirl line:
zyx
dzdydxωωω
== , (3.31)
- in the case of potential motion 0=vrot :
0=== zyx ωωω , (3.32) -in the case of helicoid motion (the velocity vector v is parallel to the whirl vector):
z
z
y
y
x
x vvvωωω
== . (3.33)
Multiplying by rd the equation (3.29), we
shall get under the conditions of permanent
motion ( 0=∂∂
tv
):
( )vrotvrdvPUd ×−=⎟⎟⎠
⎞⎜⎜⎝
⎛++−
2
2
. (3.34)
Since ω2=vrot , we shall get:
zyx
zyx vvvdzdydx
vPUd
ωωω
22
2
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛++ . (3.35)
The determined is zero for one of the four
cases above. By integrating in these cases we shall get Bernoulli’s equation:
34
CvPU =++2
2
. (3.36)
If the fluid is a non-compressible one,
then ρpP = .
If the axis Oz of the system is vertical,
upwards directed, the potential U is:
CgzU += . (3.37)
It results the well known Bernoulli’s equation as the load equation:
Czpg
v=++
γ2
2
. (3.38)
The kinetic load g
v2
2
represents the height at
which it would rise in vacuum a material point, vertically and upwards thrown, with an initial velocity v, equal to the velocity of the particle of liquid considered.
The piezometric load γpis 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 an arbitrary chosen reference plane.
Bernoulli’s equation, as an equation of loads, may be expressed as follows: in the permanent regime of an ideal fluid, non-compressible, subjected to the action of some conservative forces, the sum of the kinetic,
35
piezometric and position loads remains constant along a streamline.
Multiplying (3.38) by γ we get the equation of pressures:
Czpv=++ γρ
2
2
, (3.39)
where:
2
2vρ dynamic pressure;
p piezometric (static) pressure;
zγ position pressure.
Multiplying (3.38) by the weight of the fluid G, we get the equation of energies:
CzGpGg
vG =++γ2
2
, (3.40)
where:
gvG2
2
- kinetic energy;
γpG - pressure energy;
Gz - position energy.
3.5. Plotting and energetic interpretation of Bernoulli’s equation for liquids
36
Going back to the relation (3.38) and considering C = H (fig.3.2):
Hzpg
v=++
γ2
2
. (3.41)
Fig.3.2 The sum of all the terms of Bernoulli’s
equation represents the total energy (potential and kinetic) with respect to the unit of weight of the moving liquid.
This energy measured to a horizontal reference plane N-N, arbitrarily chosen is called specific energy and it remains constant during the permanent movement of the ideal non-compressible fluid that is under the action of gravitational and pressure forces.
3.6. Bernoulli’s equation for the relative movement of ideal non-compressible fluid
Let’s consider the flow of an ideal non-
compressible fluid through the channel between two concentric pipes that revolve around an axis Oz with angular velocity ω (fig.3.3.).
37
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 the velocity ωru = .
Upon the liquid besides the gravitational
acceleration g, the acceleration r2ω acts as well.
The unitary mass forces decomposed on the
three axes will be:
.
;
;2
2
gF
yF
xF
z
y
x
−=
=
=
ω
ω
(3.42)
In this case, the potential U will be:
CrgzU +−=2
22ω. (3.43)
By adding (3.43) to Bernoulli’s equation, we
get:
38
Czpgr
gw
=++−γ
ω22
222
, (3.44)
or
Czpg
uw=++
−γ2
22
. (3.45)
In the theory of hydraulic machines we use
the following denotations: v – absolute velocity; w – relative velocity; u – peripheral velocity. The equation (3.45) written for two
particles on the same streamline is:
22
22
22
11
21
21
22z
pguw
zpuw
++−
=++−
γγ (3.46)
39
4. FLUID STATICS The fluid statics – hydrostatics – is that
part of the mechanics of fluid which studies the repose conditions of the fluid as well as their action, during the repose state, on solid bodies with whom they come into contact.
Hydrostatics is identical for real and ideal
fluids, as viscosity becomes manifest only during motion. In hydrostatics the notion of time does no longer exist.
4.1 The fundamental equation of
hydrostatics If in Euler’s equation (3.9) we assume that
0=v , we get:
01=∇− pF
ρ. (4.1)
We multiply everywhere by rd :
01=∇− rdprdF
ρ. (4.2)
40
or
ρdpdzFdyFdxF 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=+ρdpgdz . (4.4)
In the case of liquids ( ρ = cons.), by integrating equation (4.4) we get:
.constpgz =+ρ
(4.5)
or
.constpz =+γ
(4.6)
or
.constzp =+γ (4.7) Equation (4.7) is called the fundamental equation of hydrostatics.
41
If 0p is the pressure at the surface of water (in open tank the atmospheric pressure), pressure p, situated at a distance h from the surface, will be (fig.4.1):
Fig.4.1
102 zpzp γγ +=+ , (4.8)
hpp γ+= 0 . (4.9) p is called the absolute pressure in the
point 2, and hγ is the relative pressure. 4.2 Geometrical and physical
interpretation of the fundamental equation of hydrostatics (fig.4.2)
Fig.4.2
42
According to (4.6) we can write:
22
21
1
1 zp
zp
+=+γγ
. (4.10)
In fig.4.2 we have:
γp - piezometric height corresponding to
the absolute hydrostatic pressure;
2,1z - the quotes to an arbitrary plane (position heights).
4.3 Pascal’s principle We rewrite the fundamental equation of
hydrostatics between two points 1 and 2.
2211 zpzp γγ +=+ . (4.11) Supposing that in point 1, the pressure
registers a variation pΔ , it becomes pp Δ+1 . In order that the equilibrium state shouldn’t be altered, for point 2 the same variation of pressure has to be registered.
222111 zppzpp γγ +Δ+=+Δ+ . (4.12) Hence:
21 pp Δ=Δ . (4.13)
43
Pascal’s principle: “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 communicating
vessels Let us consider two communicating vessels
(fig.4.3) that contain two non-miscible liquids, which have specific weights 1γ and 2γ , respectively. Writing the equality of pressure in the points 1 and 2, situated in the same horizontal plane N – N that also contains the separation surface between the two liquids, we get:
220110 hphp γγ +=+ , (4.14) or else
1
2
2
1
γγ
=hh
, (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 the two liquids.
44
Fig.4.3
If ,21 γγ = 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 that limits a non-compressible fluid, with specific weight γ (fig.4.4).
45
Fig.4.4
46
Then the hydrostatic pressure force will be:
∫ ===A
yMAzzdAF γγγ 0 , (4.17)
where:
0z - the quote of the specific weight for surface A;
yM - the static moment of the surface A with respect to the axis Oy.
The application point of the pressure force
F is called pressure centre. It has the following co-ordinates:
y
yA
MI
zdA
dAz
F
zdF===
∫∫∫∫∫γ
γζ
2
, (4.18)
y
yzA
MI
zdA
yzdA
F
ydF===
∫∫∫∫∫γ
γξ .
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”.
47
The above statement represents the hydrostatic paradox and is illustrated in fig.4.5. The force that presses on the bottom of the three different shaped containers, is the same because the level of the liquid in the container is the same, and the surface of the bottom is the same.
Fig. 4.5
4.6 Archimedes’ principle
Let’s consider a solid body and further to simplify a cylinder, submerged in a liquid; we intend 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:
48
.
,
0''
0'
xx
xx
AzF
AzF
γ
γ
−=
= (4.19)
The vertical forces will have the value:
.
;
2''
1'
zz
zz
AzF
AzF
γ
γ
−=
= (4.20)
Thus their resultant will be:
( ) VhAzzAFFF zzzzz γγγ −=−=−−=+= 12
''' . (4.21)
This demonstration may easily be extended for 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 liquid
is called a floating body. The submerged part is called immerse part or
hull. The weight centre of the hull’s volume is
called the hull centre. The free surface of the liquid is called
floating plane. The crossing between the floating plane and
the floating body is called the floating surface.
49
Its weight centre is called floating centre, and its outline is called floating line or water line.
In order that the floating body be in
equilibrium, it is necessary that the sum of the forces that act upon it as well as the resultant moment should be nil.
Upon a floating body there can act two forces: 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 achieve
the equilibrium is:
VmgD γ== , (4.22) where m is the mass of the floating body, V is the volume of the hull, and γ is the specific weight of the liquid.
Furthermore, in order that the moment of the resultant should be nil these two forces must have the same straight line as support or, in other words, that the weight centre G should be found on the same vertical with the centre hull.
Equation (4.22) is called the equation of
flotability.
50
Stability is the ability of the floating body to return on the initial floating of equilibrium after the action of perturbatory forces that drew it out of that position has ceased.
With respect to a Cartesian system of axes Oxyz, having the plane xOy in the floating plane and axis Oz upwards directed (fig.4.8), the floating body has six degrees of freedom: three translations and three rotations. The rotation around Ox and Oy is most important.
These slantings are due to the actions of
waves or wind.
By definition, the rotation of the floating body thus produced as the volume of the hull to remain unchanged as a value – but which can vary in shape – is called isohull slanting.
Let 00 LL − be the plane of the initial floating. After the slanting of the isohull around 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 α , the centre of hull would move
51
further, in the sense of slanting, to a point
1C . This movement takes place due to the
alteration of the shape of the hull volume. The locus of the successive positions of the
centre of the hull for different isohull slantings around the same axis is called the curve of the centre of hull (trajectory C).
The curvature centre of the curve of the hull centres is called metacentre and its curvature radius is called metacentric radius.
For transversal slantings around the longitudinal axis Ox – we shall talk about a transversal metacentre M and about a transversal metacentric radius r (fig.4.8 a).
Fig.4.8 a, b
For longitudinal slantings – around the transversal axis Oy – the longitudinal metacentre will be denoted by μ , and the corresponding metacentric radius will be R (fig. 4.8 b).
52
Causing a transversal slanting to the floating body, isohull, with a small angle, α , the centre of hull will move to point 1C (fig.4.8 a). In this case, the force of flotability Vγ , normal on the slanting flotability 11 LL − , having as application point the point 1C won’t have the same support as the weight (displacement) of the floating body.
As a result, the two forces will make up a couple whose moment, rM , will be given by the relation:
αsinhDM r = , (4.23)
where
arh −= . (4.24)
is called metacentric height, and a is the distance on the vertical between the weight centre and centre of hull; denoting by Gz and Cz the quotes of these points to a horizontal reference plane, we shall have:
CG zza −= . (4.25)
The metacentric height, expressed by the relation (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
53
body to the initial floating 0L ; for this reason it is called restoring moment. In this case the floating of the body will be stable.
b) if h < 0, the metacentre is below the centre of 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 of hull will superpose (fig.4.9 b). Consequently, the restoring moment will be nil, and the body will float 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: the metacentre should be placed above the weight centre, namely
54
.0>−= arh (4.26) According to (4.24) and (4.23), we may
write: ( ) gfr MMaDrDarDM +=−=−= ααα sinsinsin , (4.27)
where:
αsinrDM f = , (4.28)
is called stability moment of form, and:
αsinaDM g −= , (4.29) is called stability moment of weight.
As a result, on the basis of (4.27) we can consider the restoring moment as an algebraic sum of these two moments.
In the case of small longitudinal slantings, the above stated considerations are also valid, the restoring moment being in this case:
( ) αα sinsin aRDHDM r −== , (4.30) where
aRH −= . (4.31) represents the longitudinal metacentric height, and R is the longitudinal metacentric radius.
55
5. POTENTIAL (IRROTATIONAL) MOTION
The potential motion is characterised by the
fact that the whirl vector is nil.
021
== vrotω , (5.1)
hence its name: irrotational.
If ω is nil, its components on the three axes will also be nil:
.021
,021
,021
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂
∂=
=⎟⎠
⎞⎜⎝
⎛∂∂
−∂∂
=
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−
∂∂
=
yv
xv
xv
zv
zv
yv
xyz
zxy
yzx
ω
ω
ω
(5.2)
56
or:
.
,
,
yv
xv
xv
zv
zv
yv
xy
zx
yz
∂∂
=∂
∂∂∂
=∂∂
∂
∂=
∂∂
(5.3)
Relations (5.3) are satisfied only if
velocity v derives from a function ϕ :
.,,z
vy
vx
v zyx ∂∂
=∂∂
=∂∂
=ϕϕϕ
(5.4)
or vectorially:
ϕ∇=v . (5.5) Indeed:
( ) 0== ϕgradrotvrot . (5.6)
Function ( )tzyx ,,,ϕ is called the potential of velocities.
If we apply the equation of continuity for liquids,
02
2
2
2
2
2
=∂∂
+∂∂
+∂∂
=∂∂
+∂
∂+
∂∂
zyxzv
yv
xv zyx ϕϕϕ
, (5.7)
we shall notice that function ϕ verifies equation of Laplace:
0=Δϕ , (5.8) thus being a harmonic function.
57
5.1 Plane potential motion
The motion of the fluid is called plane or bidimensional if all the particles that are found on the same perpendicular at an immobile plane, called director plane, move parallel with this plane, 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 the streamline will be:
yx vdy
vdx
= , (5.9)
or else:
0=− dxvdyv yx , (5.10) and the equation of continuity:
0=∂
∂+
∂∂
yv
xv yx . (5.11)
The left term of the equation (5.10) is an
exact total differential of function ψ , called the stream function:
58
xv
yv yx ∂
∂−=
∂∂
=ψψ , , (5.12)
0=−= dxvdyvd yxψ . (5.13)
Function ψ verifies the equation of
continuity (5.11):
022
=∂∂
∂−
∂∂∂
=∂
∂+
∂∂
xyyxyv
xv yx ψψ
. (5.14)
Function ψ is a harmonic one as well:
021
21
2
2
2
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂
∂=
yxyv
xv xy
zψψω , (5.15)
0=Δψ . (5.16)
The total of the points, in which the
potential function ϕ is constant, define the equipotential surfaces.
In the case of a potential plane motion:
ϕ - constant, equipotential lines of
velocity; ψ - constant, stream lines. Computing the circulation of velocity along
a certain outline, in the mass of fluid, between points A and B (fig.5.1), we get:
∫ ∫ ∫ −==∇==ΓB
A
B
AAB
B
A
drdrdv ϕϕϕϕ . (5.17)
59
Thus, the circulation of velocity doesn’t depend on the shape of the curve AB, but only on the values of the function ϕ in A and B. The circulation of velocity is nil along an equipotential line of velocity ( .constBA ==ϕϕ ).
If we compute the flow of liquid through the curve AB in the plane motion (in fact through the cylindrical surface with an outline AB and unitary breadth), we get (fig.5.1):
Fig.5.1
( )∫ ∫ −==−=B
A
B
AAByx ddxvdyvQ ψψψ11 . (5.18)
Thus, the flow that crosses a curve does not
depend on its shape, but only on the values of function ψ in the extreme points. The flow through a streamline is nil ( ).constBA ==ψψ .
A streamline crosses orthogonal on an
equipotential line of velocity. To demonstrate this propriety we shall take into consideration that the gradient of a scalar function F is normal on the level surface F = cons. As a result, vectors ψ∇ and ϕ∇ are normal on the streamlines and on the equipotential lines of velocity.
60
Computing their scalar product, we get:
0=+−=∂∂
∂∂
+∂∂
∂∂
=∇∇ yxyx vvvvyyxxψϕψϕ
ψϕ . (5.19)
Since their scalar product is nil, it
follows that they are perpendicular, therefore their streamlines are perpendicular on the lines of velocity.
Going back to the expressions of xv and yv :
.
;
xyv
yxv
y
x
∂∂
−=∂∂
=
∂∂
=∂∂
=
ψϕ
ψϕ
(5.20)
Relations (5.20) represent the Cauchy-
Riemann’s monogenic conditions for a function of complex variable.
Any potential plane motion may always be
plotted by means of an analytic function of complex variable,
( )θireziyxz =+= .
The analytic function; ( ) ( ) ( )yxiyxzW ,, ψϕ += , (5.21)
is called the complex potential of the plane potential motion.
Deriving (5.21) we get the complex velocity:
61
yx vivy
iyx
ixdz
dW−=
∂∂
−∂∂
=∂∂
+∂∂
=ϕψψϕ
. (5.22)
Fig.5.2
( ) θθθ ievivdzdW −=−= sincos . (5.23)
Having found the complex potential, let’s
establish a few types of plane potential motions.
5.2 Rectilinear and uniform motion
Let’s 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.
62
Relation (5.24) can be written in the form:
( ) ( ) ( )ixvyvyvxvizW KK −++=+= 00ψϕ , (5.26)
where from we can get the expressions of functions ϕ and ψ :
( )( ) .,
,,
0
0
xvyvyxyvxvyx
K
K
−=+=
ψϕ
(5.27)
By equalling these relations with constants
we obtain the equations of equipotential lines and of streamlines.
..
20
10
consCxvyvconsCyvxv
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
63
.0
,0
02
01
>=
<−=
vv
tg
vv
tg
K
K
θ
θ (5.29)
We can easily check the orthogonality of the
stream and equipotential lines by writing:
121 −=θθ tgtg . (5.30) Deriving the complex potential we get the
complex velocity:
KvivadzdW
−== 0 , (5.31)
that enables us to determine the components of velocity in a certain point:
.0,00
>=>=
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 2θ , 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 2θ with the abscissa axis.
The components of velocity can be also obtained from relations (5.20):
64
.
,0
Ky
x
vxy
v
vyx
v
=∂∂
−=∂∂
=
=∂∂
=∂∂
=
ψϕ
ψϕ
(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 on 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 a reverse sense if the corresponding expressions of the potential vector are taken with a reverse sign.
5.3 The source
Let’s consider the complex potential:
( ) zQzW ln2π
= , (5.37)
65
where Q is a real and positive constant.
Writing the variable θierz = , this complex potential becomes:
( ) ( )θπ
ψϕ irQizW +=+= ln2
, (5.38)
where from we get function ϕ and ψ :
.2
,ln2
θπ
ψ
πϕ
Q
rQ
=
= (5.39)
which, equalled with constants, give us the equations of equipotential and stream lines, in the form:
..,.
consconsr
==
θ (5.40)
It can be noticed that the equipotential lines are concentric circles with the centre in the origin of the axes, and the streamlines are concurrent lines in this point (fig.5.4).
Fig.5.4
66
Knowing that:
θθ sincos ryandrx == , (5.41) in a point ( )θ,rM , the components of
velocity will be:
.01
,2
=∂∂
=
=∂∂
=
θϕ
πϕ
rv
rQ
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 the considered point.
Such a plane potential motion in which the
flow takes place radially, in such a manner that along 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 the source.
The flow of the source through a circular
surface of radius r and unitary breadth will be: 12 rvrQ π= . (5.43)
Analogically, the complex potential of the
form:
( ) zQzW ln2π
−= , (5.44)
67
will represent a suction or a well because, in this case, the sense of the velocity is reversing, the fluid moving from the exterior to the origin (where it is being sucked).
If the source isn’t placed in the origin of the axes, but in a point 1O , of the real axis, of abscissa a± , then:
( ) ( )azQzW ±= ln2π
. (5.45)
5.4. The whirl
Let the complex potential be:
( ) zi
zW ln2πΓ
−= . (5.46)
where Γ is a positive and real constant, equal to the circulation of velocity along a closed outline, which surrounds the origin.
Proceeding in the same manner as for the previous case, we shall get the functions ϕ and ψ :
,ln2
,2
rπ
ψ
θπ
ϕ
Γ−=
Γ=
(5.47)
from which we can notice that the equipotential lines, of equation .const=θ are concurrent lines,
68
in the origin of axes, and the streamlines, having the equation .constr = , are concentric circles with their centre in the origin of the axes (fig.5.5).
Fig.5.5
The components of velocity are:
02
10 >Γ
=∂∂
==∂∂
=rr
vandr
v Sr πθϕϕ
. (5.48)
Thus, on a circle of given radius r, the
velocity is constant as a modulus, has the direction of the tangent to this circle in the considered point and is directed in the sense of angle increase.
If the whirl is placed on the real axis, in
a point with abscissa a± , the complex potential of the motion will be:
( ) ( )azizW mln2πΓ
−= . (5.49)
69
5.5. The flow with and without
circulation around a circular cylinder The flow with circulation around a circular cylinder is a plane potential motion that consists of an axial stream (directed along axis Ox), a dipole of moment *2π=M (with a source at the left of suction) and a whirl (in direct trigonometric sense).
The complex potential of motion will be:
( ) zi
zr
zvzW ln2
20
0 πΓ
−⎟⎟⎠
⎞⎜⎜⎝
⎛+= , (5.50)
where we have done the denotation:
0
20
1v
r = . (5.51)
By writing the complex variable θierz = , we shall divide in (5.50) the real part from the imaginary one, thus obtaining functions ϕ and ψ :
θπ
θϕ2
cos2
00
Γ+⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
rr
rv , (5.52)
rrr
rv ln2
sin2
00 π
θψ Γ−⎟⎟
⎠
⎞⎜⎜⎝
⎛−= . (5.53)
* The dipole or the duplet is a plane potential motion that consists of two equal sources of opposite senses, placed at an infinite small
distance ε2 , so that the product QM ε2= , called the moment of the
dipole should be finite and constant. ( )az
MzWm
12π
= .
70
The stream and equipotential lines are obtained by taking in relations (5.52), (5.53), CC == ψϕ , respectively. We notice that if in (5.53) we assume 0rr = , function ψ 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, we’ll be able to consider this motion described by the complex potential (5.50) as being the flow around a straight circular cylinder of radius 0r , having the breadth normal on the motion plane, infinite.
If we plot the other streamlines we shall get some asymmetric curves with respect to axis Ox (fig.5.6). On the inferior side of the circle of radius
0r , the velocity due to the axial stream is summed up with the velocity due to the whirl.
Fig.5.6
As a result, here we shall obtain smaller velocities, and the streamlines will be more rare.
In polar co-ordinates, the components of
velocity in a certain point ( )θ,rM , will be:
71
θcos1 2
20
0 ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
rr
vvr , (5.54)
If the considered point is placed on the
circle of radius 0r , we’ll have:
.2
sin2
,0
00 r
vv
v
S
r
πθ Γ+=
= (5.55)
The position of stagnant points can be determined provided that between these points the velocity of the fluid should be nil.
The flow without circulation around a
circular cylinder is the plane potential motion made up of an axial stream (directed along axis Ox) and a dipole of moment π2=M (whose source is at the left of suction).
Thus, this motion can be obtained
particularising the motion previously described by cancelling the whirl.
By making 0=Γ , in relations (5.50), (5.52)
and (5.53) we get the complex potential of the motion, the function potential of velocity and the function of stream, in the form:
( ) ,2
00 ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
zr
zvzW (5.56)
,cos2
00 θϕ ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
rrrv (5.57)
.sin2
00 θψ ⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
rr
rv (5.58)
72
By writing the equation of streamlines ψ =
cons. in the form:
.22
20
0 constCyyx
ryv ==
+− (5.59)
we notice that the nil streamline (C = 0) is made up of a part of the real axis (Ox) and the circle of radius 0r (fig.5.7).
The other streamlines are symmetric curves with respect to axis Ox. Obviously, if we consider the circle of radius 0r , as a solid border, the motion can be seen as a flow of an axial stream around an infinitely long cylinder, normal on the motion plane.
Fig.5.7 The components of velocity are:
.sin1
,cos1
2
20
0
2
20
0
θ
θ
⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
rr
vv
rr
vv
S
r
(5.60)
which, on the circle of radius 0r , become:
73
.sin2,0
0 θvvv
S
r
−==
(5.61)
The position of stagnant points is obtained
by making 0== Svv , which implies 0sin =θ . Thus the stagnant points are found on the axis Ox in the points ( )π,0rA and ( )0,0rB .
5.6 Kutta – Jukovski’s theorem Let us consider a cylindrical body normal on
the complex plane, the outline C being the crossing curve 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 pressure
forces will have the components:
.1
,0
∞Γ==
vRR
y
x
ρ (5.62)
The forces are given with respect to the
unit of length of the body. The second relation (5.62) is the mathematic
expression of Kutta-Jukovski’s theorem, which will be only stated below without demonstrating it:
74
“ If a fluid of density ρ is draining around a body of circulation Γ and velocity in infinite ∞v , it will act upon the unit of length of the body with a force equal to the product
∞Γ vρ , normal on the direction 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 the reverse sense of circulation.
75
6. IMPULSE AND MOMENT IMPULSE THEOREM
We take into consideration a volume τ of fluid. This fluid is homogeneous, incompressible, of density ρ , bordered by surface σ . The
elementary volume τd has the speed v.
The elementary impulse will be:
τρ dvId = . (6.1)
∫=τ
τρ dvI . (6.2)
∫=τ
τρ ddtvd
dtId
. (6.3)
At the same time
iFdtId
−= . (6.4)
But: 0=++ ipm FFF (d’Alembert principle). (6.5)
Therefore:
epm FFFdtId
=+= . (6.6)
76
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 τ , is equal with the impulse flow through the exit surfaces (from the volume τ ), less the impulse flow through the entrance surfaces (to the volume τ ) “.
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:
( ) τρτρ dvrdtdd
dtvdrMd i ×−=⎟
⎠⎞
⎜⎝⎛−×= , (6.8)
since
( ) .dtvdr
dtvdrvv
dtvdrv
dtrdvr
dtd
×=×+×=×+×=× (6.9)
then
( )∫∫ ×−==ττ
τρ dvrdtdMdM ii . (6.10)
If:
77
τρ dvId = the elementary impulse, (6.11)
τρdvrkd ×= the moment of elementary impulse, (6.12)
∫ ×=τ
τρ ,dvrk (6.13)
( ) iMdvrdtd
dtkd
−=×= ∫τ
τρ . (6.14)
The derivative of the resultant moment of
impulse with respect to time is equal with the resultant moment of inertia forces with reversible sign.
expm MMMdtkd
=+= , (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 ideal
fluids, the vectorial addition of the moments of external forces which act upon the fluid in the volume τ , is equal to the moment of the impulse
78
flow through the exit surfaces less the moment of the impulse flow through the entrance surfaces”.
79
7. MOTION EQUATION OF THE REAL FLUID
7.1 Motion regimes of fluids
The motion of real fluids can be carried out under two regimes of different quality: laminar and turbulent.
These motion regimes were first emphasised by the English physicist in mechanics Osborne Reynolds in 1882, who made systematic experimental 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 very
accurate processed inlet, is supplied by tank 2, full of water, at a constant level.
Fig.71
80
The flow that passes the transparent conduit can be adjusted by means of tap 3, and measured with the help of graded pot 6.
In conduit 1, inside the water stream we insert, by means of a thin tube 4, a coloured liquid of the same density as water. The flow of coloured 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 a certain flow and velocity.
If we turn on tap 7 as well, the coloured liquid inserted through the thin tube 4, engages itself in the flow in the shape of a rectilinear thread, parallel to the walls of conduit, leaving the impression that a straight line has been drawn inside the transparent conduit 1.
This regime of motion under which the fluid
flows in threads that don’t mix is called a laminar regime.
By slowly continuing to turn on tap 3, we can 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 registers variations in time (pulsations).
For even higher velocities, the pulsations of the coloured thread of water increase their amplitude and, at a certain moment, it will tear, the particles of coloured liquid mixing with the mass of water that is flowing through conduit 1.
81
The regime of motion in which, due to pulsations of velocity, the particles of fluid mix is called a turbulent regime.
The shift from a laminar regime to the turbulent one, called a transition regime is characterised by a certain value of Reynold’s number *, called critical value ( crRe ).
* Number υvl
=Re , is the number that
defines the similarity criterion Reynolds.
82
For circular smooth conduits, the critical
value of Reynold’s number is 2320Re =cr . For values of Reynold’s 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 the motion of real (viscous) incompressible fluids in a laminar regime.
Unlike ideal fluids that are capable to develop only unitary compression efforts that are exclusively due to their pressure, real (viscous) fluids can develop normal or tangent supplementary viscosity efforts.
The expression of the tangent viscosity effort, defined by Newton (see chapter 2) is the following:
yv∂∂
=ητ . (7.1)
Newtonian liquids are capable to develop,
under a laminar regime, viscosity efforts σ and τ , 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, ):
83
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
zzyzxz
zyyyxy
zxyxxx
vT
σττ
τστ
ττσ
. (7.2)
The tensor vT is symmetrical:
yzzyxzzxxyyx ττττττ === ;; . (7.3)
Fig.7.2
The elementary force of viscosity that is
exerted upon the elementary volume of fluid in the direction of axis Ox is:
( ) ( ) ( )
.dzdydxzyx
dydxdzz
dydxdyy
dzdydxx
dF
zxyxxx
zxyxxxvx
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂
∂+
∂∂
=
=∂∂
+∂
∂+
∂∂
=
ττσ
ττσ
(7.4)
According to the theory of elasticity:
z
x
y
84
.
;
;2
⎟⎠
⎞⎜⎝
⎛∂∂
+∂∂
=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂
∂=
∂∂
=
zv
xv
yv
xv
xv
xzzx
xyyx
xxx
ητ
ητ
ησ
(7.5)
Thus:
.
2
2
2
2
2
2
2
2
2
2
2
22
2
2
dydzdxzv
yv
xv
zv
yv
xv
x
zxv
zv
yv
yxv
xv
dF
xxxzyx
zxxyxvx
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
+∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂
∂+
∂∂
∂∂
=
=⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂+
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
∂+
∂∂
=
ηη
ηηη
(7.6)
But 0=∂∂
+∂
∂+
∂∂
zv
yv
xv zyx , according to the
equation of continuity for liquids.
Then:
dzdydxvdF xx Δ=ην . (7.7) Similarly:
,dzdydxvdF yvy Δ=η (7.8)
.dydydxvdF zvz Δ=η (7.9) Hence:
,τη dvFd v Δ= (7.10)
85
.∫ Δ=τ
τη dvF v (7.11)
Unlike the ideal fluids, in d’Alembert’s
principle 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:
∫ =⎟⎟⎠
⎞⎜⎜⎝
⎛−Δ+∇−
τ
τρηρ 0ddtvdvpF , (7.13)
or:
dtvdvpF =Δ+∇− υ
ρ1
. (7.14)
Relation (7.14) is the vectorial form of
Navier-Stokes’ equation. The scalar form of this equation 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
zy
yy
xyyyyy
y
zx
yx
xxxxxx
x
vzvv
yvv
xv
tv
zv
yv
xv
zpF
vzv
vyv
vxv
tv
zv
yv
xv
ypF
vzvv
yvv
xv
tv
zv
yv
xv
xpF
∂∂
+∂∂
+∂∂
+∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
+∂∂
−
∂
∂+
∂
∂+
∂
∂+
∂
∂=⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂+
∂
∂+
∂∂
−
∂∂
+∂∂
+∂∂
+∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
+∂∂
−
υρ
υρ
υρ
(7.15)
86
7.3 Bernoulli’s 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 potential energy into kinetic energy, or the other way round, in permanent motion of the real fluid, its specific energy is no longer constant. It always decreases in the sense of the movement of the fluid.
A part of the fluid’s energy is converted into thermal energy, is irreversibly spent to overcome the resistance brought about by its viscosity.
Denoting this specific energy (load) by fh , Bernoulli’s equation becomes:
fhzp
gv
zp
gv
+++=++ 22
22
11
21
22 γγ. (7.16)
In different points of the same section,
only the potential energy remains constant, the kinetic one is different since the velocity differs in the section, ( )zyxvv ,,= . In this case the term of the kinetic energy should be corrected by a coefficient α , that considers the distribution of velocities in the section ( )1,105,1 ÷=α .
* the weight unit energy
87
fhzp
gv
zp
gv
+++=++ 22
222
11
211
22 γα
γα
. (7.17)
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
hl
zp
gv
zp
gv
I f=⎟⎟⎠
⎞⎜⎜⎝
⎛++−⎟⎟
⎠
⎞⎜⎜⎝
⎛++
=2
2222
11
211
22 γα
γα
. (7.18)
If we refer only to the potential specific
energy, we get the piezometric slope:
l
zp
zp
I p
⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛+
=2
21
1
γγ. (7.19)
In the case of uniform motion ( ctv = ):
lh
tgII fp === θ . (7.20)
Experimental researches have revealed that
irrespective of the regime under which the motion
88
of fluid takes place, the losses of load can be written in the form:
mf vbh = , (7.21)
where b is a coefficient that considers the nature of the fluid, the dimensions of the conduit and the state of its wall.
1=m for laminar regime;
275,1 ÷=m for turbulent regime.
If we logarithm (7.21) we get:
vmbh f 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=θ . The shift to the turbulent regime is made for a velocity corresponding to 2320Re =cr .
89
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 infinite length, we shall integrate the equation (7.15) under the following conditions:
Fig.7.5
a) velocity has only the direction of the
axis Ox:
;0,0 ==≠ zyx vvv (7.23)
from the equation of continuity 0=∇v , it results:
,0=∂∂
xvx (7.24)
therefore velocity does not vary along the axis Ox.
90
b) the movement is identically reproduced in
planes parallel to xOz:
0=∂∂
yvx . (7.25)
From (7.24) and (7.25) it results that ( )zvv xx = . c) the motion is permanent:
0=∂∂
tvx . (7.26)
d) we leave out the massic forces (the
horizontal conduit).
e) the fluid is incompressible.
The first equation (7.15) becomes:
012
2
=+∂∂
−dz
vdxp xυ
ρ, (7.27)
Integrating twice (7.27):
( ) 212
21 CzCz
xpzvx ++∂∂
=η
. (7.28)
For the case of fixed boards, we have the conditions at limit:
.0,;0,0
====
x
x
vhzvz
(7.29)
91
Subsequently:
.0
;21
2
1
=∂∂
=
C
hxpC
η (7.30)
Then the law of velocity distribution will be:
( ) ( )zhzxpzvx −∂∂
−=η21
. (7.31)
It is noticed that the velocity distribution
is parabolic, having a maximum for 2hz = :
xphvx ∂∂
−=η8
2
max*. (7.32)
Computing the mean velocity in the section:
( )∫ ∂∂
−==h
x xphdzzv
hu
0
2
121
η, (7.33)
we’ll notice that max32 vu = .
The flow that passes through a section of
breadth b will be:
xphb
hbvQ∂∂
−==η12
3
. (7.34)
* maxxv is positive, since 0<∂∂xp
(the sense of
the flow, the positive sense of axis Ox, corresponds to a decrease in pressure).
92
7.4.2 Velocity distribution in circular conduits
Let’s consider a circular conduit, of radius
0r and length l, through which an incompressible fluid of density ρ and kinematic viscosity υ (fig.7.6) passes.
We report the conduit to a system of
cylindrical co-ordinates ( θandrx, ), the axis Ox, being the axis of the conduit. The movement being carried out on the direction of the axis, the velocity components will be:
0,0 ==≠ θvvv rx . (7.35)
The equation of continuity 0=∇v , written in
cylindrical co-ordinates:
( ) ( )01
=⎥⎦
⎤⎢⎣
⎡∂
∂+
∂∂
+∂
∂=∇
xrvv
rvr
rv xr
θθ , (7.36)
becomes:
0=∂∂
xvx , (7.37)
where from we infer that the velocity of the fluid doesn’t 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 θ .
As a result, for a permanent motion, it will only depend on variable r, that is ( )rvv = .
93
The distribution of velocities in the section of flow can be obtained by integrating the Navier-Stokes’ equations (7.14).
Noting by rii, and θi 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:
θθ
∂∂
+∂∂
+∂∂
=∇ri
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 vΔ may be rendered in the form:
.
1
⎟⎠
⎞⎜⎝
⎛∂∂
∂∂
=
=⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛∂∂
∂∂
+⎟⎠
⎞⎢⎣
⎡∂∂
∂∂
+⎟⎠
⎞⎜⎝
⎛∂∂
∂∂
=Δ=Δ
rrv
rri
rxv
xrv
rrv
rriviv
x
xxxx θθ
(7.41)
Keeping in mind the permanent character of
the motion, relation (7.40) and (7.41) the
94
projection of equation (7.14) onto the axis Ox may be written in the form:
xpr
rv
rrx
∂∂
=⎟⎠⎞
⎜⎝⎛∂∂
∂∂
ρυ 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 =∂∂ ), and integrating the equation (7.42), we shall successively get:
,21 1
rC
rxp
rvx +
∂∂
=∂∂
η (7.43)
,ln41
212 CrCr
xpvx ++∂∂
=η
(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
of fluid should be nil; consequently:
202 4
1 rxpC∂∂
−=η
, (7.45)
and relation (7.44) becomes:
( )2204
1 rrxpvx −∂∂
−=η
. (7.46)
95
From (7.46) we notice that if the motion
takes place in the positive sense of the axis ( )0>xvOx , then 0/ <∂∂ xp ; therefore pressure
decreases on the direction of motion if I is the piezometric slope (equal in this case to the hydraulic slope), we can write:
Ilp
xp
γ=Δ
=∂∂
− , (7.47)
where pΔ is the fall of pressure on the length l of the conduit. Subsequently, relation (7.41) becomes:
( )2204
rrIvx −=ηγ
. (7.48)
Fig.7.6
It can be noticed that the distribution of
velocities in the section of flow is parabolic (fig.7.6 a), the maximum velocity being registered in the axis of conduit (r = 0), therefore we get:
20max 41
rI
vx ηγ
= . (7.49)
96
Let us now consider an elementary surface d A in the shape of a circular crown of radius r and breadth d r (fig.7.6 b). The elementary flow that crosses surface d A is:
rdrvdAvdQ xx π2== , (7.50)
and:
( )∫ =−=0
0
40
220 82
r
rI
drrrrI
Q πηγ
ηγ
π . (7.51)
The mean velocity has the expression:
28max,2
0xv
rI
AQu ===
ηγ
. (7.52)
Further on we can write:
gv
ddg
dv
dgv
rv
lh
I f
21
Re64Re
32328 2
2
2
220
=====υ
γη
. (7.53)
Relation (7.53) is Hagen-Ppiseuille’s law,
which gives us the value of load linear losses in the conduits for the laminar motion:
gv
dl
gv
dlh f 22Re
64 22
λ== , (7.54)
Re64
=λ is the hydraulic resistance coefficient
for laminar motion.
97
7.5 Turbulent motion of fluids
In a point of the turbulent stream, the fluid velocity registered rapid variation, in one sense or the other, with respect to the mean velocity in section. The field of velocities has a complex structure, still unknown, being the object of numerous studies.
The variation of velocity with the time may be plotted as in fig.7.7.
Fig.7.7
A particular case of turbulent motion is the
quasipermanent motion (stationary on average). In this case, velocity, although varies in time, remains a constant means value.
In the turbulent motion we define the
following velocities:
a) instantaneous velocity ( )tzyxu ,,, ;
b) mean velocity
( ) ( )∫=T
dttzyxuT
zyxu0
,,,1,, ; (7.55)
c) pulsation velocity
98
( ) ( ) ( )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 kept constant.
b) Theory of whirl transports (Taylor)
where the rotor of velocity is presumed constant.
c) Karaman’s theory of turbulence, which
states that, except for the immediate vicinity of a wall, the mechanism of turbulence is independent from viscosity.
7.5.1 Coefficient λ in turbulent motion
Determination of load losses in the turbulent motion is an important problem in practice.
It had been experimentally established that in turbulent motion the pressure loss pΔ depends on the following factors: mean velocity on section, v , diameter of conduit, d , density ρ of the fluid and its kinematic viscosity υ , length l of the conduit and the absolute rugosity
*Δ of its interior walls; therefore:
( )Δ∂=Δ ,,,,, ldvfp ρ , (7.57) or:
99
dlvp
2
2ρλ=Δ , (7.58)
dl
gvph f 2
2
λγ
=Δ
= , (7.59)
ror
dΔΔ - relative rugosity
where:
⎟⎠⎞
⎜⎝⎛ Δ
=d
Re,2 1ϕλ . (7.60)
*mean height of the conduit prominence ;r
or ΔΔd
-
relative rogosity
100
As it can be seen from relation (7.60), in turbulent motion the coefficient of load loss λ may depend either on Reynolds number or on the relative rugosity of the conduit walls.
In its turbulent flow through the conduit, the fluid has a turbulent core, in which the process of mixing is decisive in report to the influence of viscosity and a laminar sub-layer, situated near the wall, in which the viscosity forces have a decisive role.
If we note by lδ the thickness of the laminar sub-layer, then we can classify conduits as follows:
- conduits with smooth walls; lδ<Δ ;
- conduits with rugous walls; lδ>Δ .
From (7.60) we notice that, unlike the laminar motion in turbulent motion λ is a
complex function of Re and dΔ.
It has been experimentally established that in the case of hydraulic smooth conduits, coefficient λ depends only on Reynolds’ number. Thus, Blasius, by processing the existent experimental material (in 1911), established for the smooth hydraulic conduits of circular section, the following empirical formula:
25,0
4/1
Re3164,03164,0 =⎟
⎠
⎞⎜⎝
⎛=υ
λdv
, (7.61)
valid for 510Re000,4 << . Using Blasius’ relation in (7.59) we notice
that under this motion regime the load losses are proportional to velocity to 1,75th power.
101
Also for smooth conduits, but for higher Reynolds’ numbers ( )710Re000,3 << we can use Konakov’s relation:
( ) 25,1Relg8,1 −−=λ . (7.62)
In turbulent flow through conduits,
coefficient λ no longer depends on Reynolds number, and it can be determined with the help of Prandtl – Nikuradse’s relation:
2
0 74,1lg2−
⎟⎠
⎞⎜⎝
⎛+
Δ=
rλ . (7.63)
Some of the most important formulae for the
calculus of coefficient λ are given in table 7.1, the validity field of each relation being also shown [7].
Table 7.1
No.
Relation
Regime
Field
I III IV V
1 Poisseuille Re
64=λ Laminar 2320Re <
2 Prandtl ( ) 28,0Relg2
−−= λλ
710Re000,3Re
<
>
3 Blasius 25,0Re3164,0 −=λ 510Re000,4Re
<
>
4 Konakov ( ) 25,1Relg8,1 −−=λ 710Re
000,3Re<
>
5 Nikuradze
237,0Re221,00032,0 −+=λ 6
5
102Re10Re
<
>
6 Lees 35,03 Re61,010714,0 −− +=λ
Smooth turbulen
t
6
3
103Re10Re
<
>
II
Auth
102
7 Colebrook-White λλ Re
51,272,3
lg21+
Δ−=
d Demi-
rugous Universal
8 Prandtl-
Nikurdze
20 74,1lg2
−
⎟⎠
⎞⎜⎝
⎛+
Δ=
rλ 5 10Re10 <<
9 Sifrinson
25,0
11,0 ⎟⎠⎞
⎜⎝⎛ Δ=
dλ
Turbulent rugous
500Re >Δd
103
7.5.2 Nikuradze’s diagram On the basis of experiments made with
conduits of homogeneous different rugosity, which was achieved by sticking on the interior wall some grains of sand of the same diameter, Nikuradze has made up a diagram that represents the way coefficient λ varies, both for laminar and turbulent fields (fig.7.8).
Fig.7.8 We can notice that in the diagram appear
five areas in which variation of coefficient λ , distinctly differs.
Area I is a straight line which represents in logarithmic co-ordinates the variation:
Re64
=λ , (7.64)
104
corresponding to the laminar regime ( )2320Re < . On this line all the doted curves are superposed, which represents variation ( )Ref=λ for different relative rugosities 0/ rΔ .
Area II is the shift from laminar regime to the turbulent one which takes place for
( )2300Re4,3Relg ≅≅ .
Area III corresponds to the smooth hydraulic conduits. In this area coefficient λ can be determined with the help of Blasius relation (7.61), to which the straight line III a corresponds, called Blasius’ straight. Since the validity field of relation (7.61) is limited by
510Re = , for higher values of Reynolds’ number, we use Kanakov’s formula, to which curve III b corresponds. It is noticed that the smaller the relative rugosity is, the greater the variation field of Reynolds number, in which the smooth turbulent regime is maintained.
In area IV each discontinuous curve, which represents dependent ( )Ref=λ for different relative rugosities becomes horizontal, which emphasises the independence of λ on number Re. Therefore this area corresponds to the rugous turbulent regime, where λ 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
is also called square regime.
105
Area V is characterised by the dependence of the coefficient both on Reynolds’ number and on the relative rugosity of the conduit.
It can be noticed that for areas IV and V, coefficient λ decreases with the decrease of relative rugosity.
106
8. FLOW THROUGH CIRCULAR CONDUITS
In this chapter we shall present the hydraulic calculus of conduits under pressure in a permanent regime. Conduits under pressure are in fact a hydraulic system designed to transport fluids between two points with different energetic loads. Conduits can be simple (made up of one or several sections of the same diameter or different diameters), or with branches, in this case, setting up networks of distribution. By the manner in which the outcoming of the fluid from the conduit is made, we distinguish between conduits with a free outcome, which discharge the fluid in the atmosphere (fig.8.1 a) and conduits with chocked outcoming (fig. 8.1 b).
Fig.8.1a, b
107
If we write Bernoulli’s equation for a stream of real liquid, between the free side of the liquid from the tank A and the end of the conduit, taking as a reference plane the horizontal plane N – N, we get:
fhzp
gv
zp
gv
+++=++ 22
222
11
211
22 γα
γα
, (8.1)
which, for the case presented in fig.8.1 a, when
01 ≅v , 021 ppp == , 121 ==αα , hzz += 21 , becomes:
fhg
vh +=2
2
, (8.2)
where 2vv = is the mean velocity in the section of the conduit , and h is the load of the conduit.
In the analysed case shown in fig. 8.1 b, by introducing in equation (8.1) the relations
1022112011 ,,,,0 hppzhhzvvppv γ+=++===≅ and
121 ==αα , we shall get the expression (8.2).
From an energetic point of view, this relation 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 the outcoming in the atmosphere or in another volume. The other part ( )fh is used to overcome the hydraulic resistances (that arise due to the tangent efforts developed by the fluid in motion) and is lost because it is irreversibly transformed into heat.
108
Analysing the losses of load from the
conduit we shall divide them into two categories, writing the relation:
'''fff hhh += . (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 along the length of the conduit ( l) and, for this reason, they are called losses of load distributed. These losses of load have been determined in paragraph 7.4.2, getting the relation (7.54) which we may write in the form:
dl
gvh f2
2' λ= , (8.4)
where the coefficient of losses of load, λ , called Darcy coefficient is determined by the relations shown in table 7.1 ; the manner of calculus being also shown in that paragraph. Generally, in practical cases, the values of coefficient λ vary in a domain that ranges between 04,002,0 ÷ .
Being proportional to the length of the conduit, the distributed losses of load are also called linear losses.
The second category of losses of load is represented by the local losses of load that are brought about by: local perturbation of the normal flow, the detachment of the stream from the wall, whirl setting up, intensifying of the turbulent mixture, etc; and arise in the area
109
where the conduit configuration is modified or at the meeting an obstacle detouring (inlet of the fluid 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 by Weissbach:
gvh f 2
2'' ζ= , (8.5)
where ζ is the local loss of load coefficient that is determined for each local resistance (bends, valves, narrowing or enlargements of the flow section etc.).
Generally, coefficient ζ depends mainly on the geometric parameters of the considered element, as well as on some factors that characterise the motion, such as: the velocities distribution at the inlet of the fluid in the examined element, the flow regime, Reynolds’ number etc.
In practice, coefficient ζ is determined with respect to the type of the respective local resistance, using tables, monograms or empirical relations that are found in hydraulic books. Therefore, for curved bends of angle 090≤δ , coefficient ζ can be determined by using the relation:
0
0
5,3
5,3
9016,013,0 δ
ρζ ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
d, (8.6)
where ρandd are the diameter and curvature radius of the bend, respectively.
110
Coefficient ζ , corresponding to the loss of
load at the inlet in their conduit, depends mainly on the wall thickness of the conduit with respect to its diameter and on the way the conduit is attached to the tank. If the conduit is embedded at the level of the inferior wall of the tank, the losses of load that arise at the inlet in the conduit are equivalent with the losses of load in an exterior cylindrical nipple. For this case, 5.0≅ζ .
If on the route of the conduit there are
several local resistances, the total loss of fluid will be given by the arithmetic sum of the losses of load corresponding to each local resistance in turn, namely:
∑= gvh f 2
2'' ζ , (8.7)
Using relations (8.4) and (8.7), we get the
total loss of load of the conduit:
gv
dlh f 2
2
⎟⎠⎞
⎜⎝⎛ += ∑ζλ , (8.8)
that allows us to write relation (8.2) in the form:
gv
dlh
21
2
⎟⎠⎞
⎜⎝⎛ ++= ∑ζλ , (8.9)
where from the mean velocity in the flow section will result:
111
∑++=
ζλdl
hgv
1
2. (8.10)
The flow of the conduit is determined by:
∑++==
ζλ
ππ
dl
hgdvdQ1
244
22
, (8.11)
which allows us to express the load of the conduit, h, and diameter, d, with respect to flow Q; we get:
⎟⎠⎞
⎜⎝⎛ ++= ∑ζλ
π dl
dQ
gh 18
4
2
2 , (8.12)
and respectively:
( )∑++= ζλπ
dldh
Qg
d2
25 8
. (8.13)
Sometimes in the calculus of enough long
conduits, the kinetic term ( )gv 2/2 and the local losses of load are negligible with respect to the linear losses of load.
In the case of such conduits, called long conduits, relation (8.2) takes the form:
dl
gvhh f 2
2' λ== , (8.14)
and relations (8.10), (8.11), (8.12) and (8.13) become:
112
lgdhvλ
2= , (8.15)
lgdhdQλ
π 24
2
= , (8.16)
ldQ
gh λ
π 5
2
2
8= , (8.17)
and, respectively:
lh
Qg
d λπ
2
25 8= . (8.18)
With the help of the above written relations all problems concerning the computation of conduits under pressure can be solved. Generally, these problems are divided into three categories:
a) to determine the load of the conduit, when length, rugosity, flow and rugosity of interior walls of the conduit are known; b) to determine the optimal diameters when flow, length, rugosity of the walls of conduit as 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 are known.
113
9. HYDRODYNAMIC PROFILES
9.1 Geometric characteristics of hydrodynamic profiles
A hydrodynamic profile is a contour with an elongated shape with respect to the direction of stream, rounded at the front edge-called leading edge-and having a peak at the back edge, called trailing edge.
In what follows we
shall stress on some of the elements, which characterise the profile. a) The chord of the profile is defined as the straight line which joins the trailing edge A, with the point B, in which the circle
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
the normal to the chord and is noted by e. This thickness varies along the chord and reaches a maximum in a section which is called
114
section of maximum thickness, situated at the distance ml to the leading edge.
c) Relative thickness, ε , and maximum relative thickness, mε , are defined by the relations:
ce
andce m
m == εε . (9.1)
d) The framework of a profile, or the line of
mean curvature, is the curve that joins the mean thickness points. The shape of the framework is an important geometric parameter and is linked to the curvature motion of the profile. From this point of view, profiles can be with simple curvature (fig.9.1) or with double curvature (9.2).
e) The arrow of the profile, f, is the maximum distance, measured on the normal to the chord, between the framework and the chord of the profile.
f) The extrados and intrados of the profile represent the upper and lower part of the profile, respectively.
By the geometric shape of the trailing edge, which plays an important part in the theory of profiles, we may distinguish among three categories of profiles:
Fig.9.2
- Jukovski profiles, profiles with a sharp edge, for which the tangents to the
115
trailing edge at extrados and intrados superpose (fig.9.3 a)
- Karman-Trefftz profiles, or profiles with a dihedral tip, for which the tangents to the extrados and the intrados make an angle δ in the trailing edge (fig.9.3 b), - Carafoli profiles, or profiles with the rounded tip, for which the trailing edge ends in a rounded contour, with a small curvature radius. (fig.9.3c).
It is generally studied the plane potential motion around the hydrodynamic profile, considered as the intersection of the complex plane of motion with a cylindrical object (called wing), normal on this plane and having an infinite length (called span).
In reality, wings have a
finite span and, from a geometrical point of view, they are characterised by the section of the wing, which, generally, alters
Fig.9.3 a, b, c the length of the wing and the shape of the wing in plane.
By the shape of the wing in plane, there are: rectangular wings (fig.9.4), trapezoidal wings (9.4 b), elliptical (9.4 c), and triangular wings (9.4 d).
116
Fig.9.4 a, b, c, d
An important parameter of the wing is the relative elongation defined by the relation:
Sl 2
=λ , (9.2)
where l and S represent the span and the surface of the wing, respectively.
In the particular case of rectangular wing, the length of the chord is constant 0cc = and relation (9.2) becomes:
0/ cl=λ , since:
0clS = . We can classify wings by their elongation
λ ; into:
- wings of infinite span, when 6>λ ; - wings of finite span, when 6<λ .
117
9.2 The flow of fluids around wings Kutta-Jukovski’s relation (5.62) can be
applied to any solid body in relative displacement with respect to a fluid.
It indicates that whenever there is a
circulation Γaround a body, there arises a lift force yR , whose value is determined, under the
same circumstances of environment ( ∞vandρ ), 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-Magnus effect).
- for asymmetrical bodies: study of shapes more proper to circulation.
On the basis of many theoretical and
experimental studies, we have come to designing wings with a high lift, called hydrodynamic profiles.
118
Fig.9.5
In fig.9.5, the arising of circulation around the hydrodynamic profile, alters the spectre of lines of rectilinear stream, of velocity ∞v as follows: on the extrados the sense of circulations coincides with that of motion and is seen as a supplement of velocity vΔ , and on the intrados velocity is decreased with vΔ .
According to Bernoulli’s law, the velocities asymmetry brings about the static pressures asymmetry (high pressure on the intrados, low pressure on the extrados) as well as the arising of lift force.
Applying Bernoulli’s relation between a
point at ∞ and a point on the profile, we get:
22
22S
Sv
pv
p ρρ
+=+ ∞∞ . (9.3)
The pressure coefficient is defined by the
relation:
2
2
2 1
2∞∞
∞ −=−
=vv
vpp
C SSp
ρ. (9.4)
In fig. 9.6 it is shown the distribution of
pressure and of the pressure coefficient on a hydrodynamic profile at a certain angle of incidence, *α .
119
Fig.9.6
The alteration of the incidence angle leads
to the shift in the pressures distribution. 9.3 Forces on the hydrodynamic
profiles
The forces which act upon hydrodynamic or aerodynamic profiles: lift, shape resistance, friction force or the force due to the detachment of the limit layer give a resultant R which decomposes by the direction of velocity in infinite and by a direction which is
• The angle between ∞v and the chord of the p
120
perpendicular on it (fig.9.7). Component xR is called resistance at advancement, and component
yR , lift force. They are usually written in the form:
.2
;2
2
2
Sv
CR
Sv
CR
yy
xx
∞
∞
=
=
ρ
ρ (9.5)
where xC is called the coefficient of resistance
at advancement, and yC the lift coefficient
( lcS = for profiles of constant chord).
Fig.9.7
121
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 expressed with
the help of coefficients:
tC - the coefficient of tangent force and nC - the coefficient of normal force.
For a certain angle s,α is the distance between the leading edge and the pressure centre (the application point of hydrodynamic force).
The relation expresses the moment of the force R with respect to the leading edge:
αα sincos sRsRsRM xyn +== . (9.7)
Also, moment M can be expressed by an analytic form similar to that used for the components of hydrodynamic force:
Sv
cCM m 2
2∞= ρ . (9.8)
Using (9.5), (9.7), and (9.8), we get:
αα sincos xy
m
CCC
cs
+= . (9.9)
In the case of small incidence angles:
y
m
CC
cs≅ . (9.10)
122
The usage of coefficients xC , yC and nC is
often met in actual practice. Their variation is studied in different conditions and given in the form of tables and graphics of great importance for the calculus and design of systems, which deal with profiles.
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
different relative elongation, λ , (see paragraph 9.1), in the case of certain flow velocities (numbers Re variable), there are diagrams experimentally established ( ) ( ) ( )ααα myx CandCC , .
123
Fig.9.8 In fig. 9.8 there are plotted the diagrams
of coefficients for resistance at advancement and for lift force for a NACA 6412 profile, of relative elongation 3, at a number Re = 85,000.
Another type of diagram often used is the
polar profile, namely the function ( )xy CC at different slanting angles (fig.9.9). The polar allows us to define two characteristics of the profile:
- the floating or gliding coefficient:
y
x
CC
tg == γε , (9.11)
- aerodynamic accuracy:
x
y
CC
f ==ε1
. (9.12)
124
Fig.9.9
9.4 Induced resistance in the case of
finite span profiles For wings of great span, considered infinite
∞=l , the motion around the profile is plane. Circulation Γ may be replaced by a whirl.
In reality, at
the tips of the wing, because of the difference in pressure, there arises a motion of fluid from intrados to extrados (9.10). The greater the weight of this motion, the
125
smaller the wing span is.
Fig. 9.10
As a consequence, circulation Γ is no longer constant; at the tips there is a minimum. (fig.9.11). This leads to an alteration of hydrodynamic parameters, through the arising of the so-called induced resistance.
Fig.9.11 In fig.9.12 the scheme of hydrodynamic
forces for the wing of finite span is plotted. Due to the arising of an induced velocity
iv , created by the free whirl, perpendicular on
the velocity in infinite ∞v , the resultant velocity becomes:
ivvv += ∞ . (9.13)
126
Fig.9.12
As a consequence there will appear an
induced incidence angle iα , which thus decreases the incidence angle α .
The alteration of direction and value of
velocity bring about the corresponding alteration of lift, which, as we have already shown, is perpendicular on the direction of stream velocity.
If yR is the lift of the infinite profile
and F is the lift under the circumstances of an induced velocity (perpendicular on the direction of velocity v), then:
.cos;sin
iy
ii
FRFR
αα
==
(9.14)
In the conditions of very small values of
iα , we may assume that FRy ≅ , namely lift does not alter.
Component iR acting on the direction Ox is called induced resistance and may be written in the form:
127
Sv
CR xii 2
2∞=
ρ. (9.15)
The total resistance of the wing of infinite span 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
fluid are in reciprocal influence, behaving in a different manner 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 several identical profiles, of span l and control contour ABCD (fig.9.13). The pitch of the network is t.
Fig.9.13
128
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 the density of fluid doesn’t alter in a significant way when passing through the network, 21 ρρ = ,
then 21 xx vv = .
Indeed, applying the equation of continuity:
ltvltvm xx 21
.ρρ == , (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 −=−= ρ . (9.17)
The circulation of velocity on the control contour will be:
∫ ∫ ∫ ∫∫ +++==ΓABCD
C
B
D
C
A
D
y
B
A
y dsvdsvdsvdsvdsv τττττ 21 . (9.18)
The integrals on the segments of contour BC
and AD cancel reciprocally. There only remains:
( )tvvdsvdsvD
Cyyy
B
Ay ∫∫ −=−=Γ
2121. (9.19)
Therefore:
129
tvv yy
Γ=−
21. (9.20)
Replacing (9.20) into (9.17), we get:
Γ= lvR xy ρ . (9.21)
The axial component xR is due to the difference of pressure:
( ) tlppRx 21 −= . (9.22) Applying Bernoulli’s equation between the points 1 and 2, we get:
22
22
2
21
1v
pv
pρρ
+=+ , (9.23)
or else:
( ) ( )222
1221
22
21
2221
yyyy
vvt
vvvvpp+Γ
−=−=−=−ρρρ
. (9.24)
Replacing (9.24) into (9.22). we get:
Γ+
−= lvv
R yyx 2
12ρ . (9.25)
The resultant force will be:
( )tlvC
vvvlRRR r
yyxyxr 24
22222 21 ∞=
++Γ=+= ρρ . (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).
130
( )4
22 21 yyx
vvvv
++=∞ . (9.24)
Fig.9.14
The lift force is perpendicular on ∞v .
Coefficient rC is different from the hydro-aerodynamic coefficient corresponding to a separate profile.
