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 .

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

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.

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 dont 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.

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 Reynolds number, called critical value ().

For circular smooth conduits, the critical value of Reynolds number is .

For values of Reynolds number inferior to the critical value (), the motion of liquid will be laminar, while for , 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:

(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, (in fig. 7.2, efforts manifest on an elementary parallelipipedic volume of fluid with the sides ):

(7.2)

The tensor is symmetrical:

(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:

(7.4)

According to the theory of elasticity:

(7.5)

Thus:

But , according to the equation of continuity for liquids.

Then:

(7.7)

Similarly:

(7.8)

(7.9)

Hence:

(7.10)

(7.11)

Unlike the ideal fluids, in dAlemberts principle the viscosity force also appears.

(7.12)

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

(7.13)

or:

(7.14)

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

(7.15)

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 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 fluids energy is converted into thermal energy, is irreversibly spent to overcome the resistance brought about by its viscosity.

Denoting this specific energy (load) by , Bernoullis equation becomes:

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

(7.17)

By reporting the loss of load to the length l of a straight conduit, we get the hydraulic slope (fig.7.3):

Fig.7.3

. (7.18)

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

(7.19)

In the case of uniform motion ():

(7.20)

Experimental researches have revealed that irrespective of the regime under which the motion of fluid takes place, the losses of load can be written in the form:

(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.

for laminar regime;

for turbulent regime.

If we logarithm (7.21) we get:

(7.22)

In fig. 7.4 the load variation with respect to velocity is plotted in logarithmic co-ordinates.

Fig.7.4

For the laminar regime . The shift to the turbulent regime is made for a velocity corresponding to .

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:

(7.23)

from the equation of continuity , it results:

(7.24)

therefore velocity does not vary along the axis Ox.

b)  the movement is identically reproduced in planes parallel to xOz:

(7.25)

From (7.24) and (7.25) it results that .

c)  the motion is permanent:

(7.26)

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

e)  the fluid is incompressible.

The first equation (7.15) becomes:

(7.27)

Integrating twice (7.27):

(7.28)

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

(7.29)

Subsequently:

(7.30)

Then the law of velocity distribution will be:

(7.31)

It is noticed that the velocity distribution is parabolic, having a maximum for :

(7.32)

Computing the mean velocity in the section:

(7.33)

well notice that .

The flow that passes through a section of breadth b will be:

(7.34)

7.4.2 Velocity distribution in circular conduits

Lets consider a circular conduit, of radius and length l, through which an incompressible fluid of density and kinematic viscosity (fig.7.6) passes.

We report the conduit to a system of cylindrical co-ordinates (), 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:

(7.35)

The equation of continuity , written in cylindrical co-ordinates:

(7.36)

becomes:

(7.37)

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

On the other hand, taking into consideration the axial symmetrical character of the motion, velocity will neither depend on variable .

As a result, for a permanent motion, it will only depend on variable r, that is .

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

Noting by and the versors of the three directions of the adopted system of cylindrical co-ordinates, we can write vector velocity:

(7.38)

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

(7.39)

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

(7.40)

since, as we have seen, velocity only depends on variable r.

On the other hand, in cylindrical co-ordinates, the term may be rendered in the form:

(7.41)

Keeping in mind the permanent character of the motion, relation (7.40) and (7.41) the projection of equation (7.14) onto the axis Ox may be written in the form:

(7.42)

since, on the hypothesis of a horizontal conduit, .

Assuming that the gradient of pressure on the direction of axis Ox is constant (), and integrating the equation (7.42), we shall successively get:

(7.43)

(7.44)

The integrating constants and are determined using the limit conditions:

in the axis of conduit, at r = 0, velocity should be finite, thus constant should be nil;

on the wall of conduit, at , velocity of fluid should be nil; consequently:

(7.45)

and relation (7.44) becomes:

(7.46)

From (7.46) we notice that if the motion takes place in the positive sense of the axis , then ; 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:

(7.47)

where is the fall of pressure on the length l of the conduit. Subsequently, relation (7.41) becomes:

(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:

(7.49)

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:

(7.50)

and:

(7.51)

The mean velocity has the expression:

(7.52)

Further on we can write:

(7.53)

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

(7.54)

is the hydraulic resistance coefficient for laminar motion.

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 ;

b)          mean velocity

(7.55)

c)          pulsation velocity

(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)                Karamans 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 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 depends on 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:

(7.57)

or:

(7.58)

(7.59)

- relative rugosity

where:

(7.60)

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 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 the thickness of the laminar sub-layer, then we can classify conduits as follows:

conduits with smooth walls;  ;

conduits with rugous walls;  .

From (7.60) we notice that, unlike the laminar motion in turbulent motion is a complex function of and .

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

(7.61)

valid for .

Using Blasius relation in (7.59) we notice that under this motion regime the load losses are proportional to velocity to 1,75^th power.

Also for smooth conduits, but for higher Reynolds numbers we can use Konakovs relation:

(7.62)

In turbulent flow through conduits, coefficient no longer depends on Reynolds number, and it can be determined with the help of Prandtl Nikuradses relation:

(7.63)

Some of the most important formulae for the calculus of coefficient are given in table 7.1, the validity field of each relation being also shown [7].

Table 7.1

No.	a I	Relation	Regime	Field
I		III	IV	V
	Poisseuille		Laminar
	Prandtl		Smooth turbulent
	Blasius
	Konakov
	Nikuradze
	Lees
	Colebrook-White		Demi-rugous	Universal
	Prandtl-Nikurdze		Turbulent rugous
	Sifrinson

7.5.2 Nikuradzes 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 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 , distinctly differs.

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

(7.64)

corresponding to the laminar regime . On this line all the doted curves are superposed, which represents variation for different relative rugosities .

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

Area III corresponds to the smooth hydraulic conduits. In this area coefficient 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 , for higher values of Reynolds number, we use Kanakovs 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 for different relative rugosities becomes horizontal, which emphasises the independence of on number . Therefore this area corresponds to the rugous turbulent regime, where is determined by (7.63).

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

For this reason the rugous turbulent regime is also called square regime.

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 decreases with the decrease of relative rugosity.