Peube J-L. Fundamentals of fluid mechanics and transport phenomena

Physics of Energetic Systems in Flow 123

In particular, for components of the acceleration, we have:

VVgrad

t

V

dt

Vd

x

u

t

u

dt

du

j

i

j

ii

i

GG

G

¸

¹

·

¨

©

§



w



w

JJ

or

[3.27]

Acceleration [3.27] can also be written in the form:

V

grad

t

V

dt

Vd



w

ZJ

G

2

[3.28]

where

Vtor

G

GG

2

1

Z

is the rotation vector. The vorticity vector is twice the rotation

vector.

NOTE –

The material derivative

dt

dg

, defined in a Galilean reference frame, is a

physical quantity which does not depend on the reference frame chosen for its

evaluation; on the contrary, the temporal derivative

t

g

w

of the spatial representation

corresponds to an observation of the quantity

g at a fixed point and it does depend

on the reference frame chosen to represent the field. Consider two Cartesian

reference frames of coordinates





tx

j

,

and





tx

j

~

,

~

:

.

~

;

~

tttUxx

jjj



The reader can easily verify the relation:

j

x

g

U

t

g

t

g

~

w



w

where we let



.

~

,

~

, txgtxg

ii

The time derivative calculated in a reference frame depends on the choice of

reference frame. Its value is notably associated with spatial inhomogenities of the

quantity considered and which is displaced with respect to this reference frame.

124 Fundamentals of Fluid Mechanics and Transport Phenomena

For example, the reader can verify that the field of the quantity g, steady in the

reference frame



tx ,

1

and with spatial period O

¸

¹

·

¨

©

§

O

S

1

2

cos),(

x

Atxg

is seen by a fixed observer in the reference frame

>@

ttUtxx 

~

;

~

11

as an

oscillating field with temporal period

UT

O

~

. We will come back to this

phenomenon, which is known as the

Doppler-Fizeau effect in acoustics.

3.3.3.2.

Flux of the quantity G

We saw (section 2.1.3.1) that the transfer of an extensive quantity G is

characterized by a

flux density vector

G

q

G

whose flux across a surface is equal to the

amount of the quantity

G which crosses this surface per unit time. The

demonstration of the existence of this vector is independent of the transfer

mechanisms considered. In particular, the movement of matter with respect to a

surface leads to a flux of the quantities associated with the matter across this surface,

which we often refer to as a convective flux of the quantity

G.

ds

n

G

V

G

G

t



V

G

Figure 3.4. Balance in a flow through ds

for an elementary displacement

When a quantity G is transported by moving matter, the quantity G which

crosses a surface

ds of unit normal n

G

in time

G

t occupies an oblique cylinder of

length

tV

G

(Figure 3.4). We thus derive an expression for the flux density vector of

G:

(g: quantity of per volume unit)

g

qgV G



G

[3.29]

The convective flux of the quantity

G crossing a surface S per unit time is thus

equal to:





dsnugdsnVgdsnq

jjG

SS

gG

³³³

6

MM

G

.. [3.30]

Physics of Energetic Systems in Flow 125

The volume and mass transported by the matter’s motion are respectively

characterized by the volume flux density vector

V

G

and the mass flux density vector

V

G

U

which correspond, respectively, to

1

g

and

U

g

.

The elementary volume flux

dq

v

and the elementary mass flux dq

m

are

1

:

dsnVdqdsnVdq

mv

G

.and.

U

The

volume flux q

v

(or volume flow rate) and the mass flux q

m

(or mass flow

rate) across a surface

S can be written:

³³

S

m

S

v

dsnVqdsnVq

G

.and.

U

When the quantity

G

is a vector (for example the momentum

Vm

G

), the

preceding results can simply be applied to each of the components

i

g . We thus

define a

flux density tensor

G

q

of the quantity G

G

by the relation:

Vgqugq

GjiGij

G

 or [3.31]

The flux of the quantity

G

crossing the surface S is thus the vector

GS

M

G

:

³

S

GGS

dsnq

G

M

3.3.3.3.

Material derivative of a volume integral

Let ),(),( txgtxg

ii

U

be the volume density of the quantity

G (g designates

the corresponding mass density). The amount of the quantity

G contained in a

domain

D of fluid inside the closed surface 6 is:

³

D

dvtxgG

i

),(

The laws of physics imply the balance of the extensive quantities (mass,

momentum, etc.) associated with the matter which, in a flow, is in motion. It is

therefore necessary to calculate the variation of the quantity

G associated with a

material domain

D in motion using the usual Eulerian representation.

n

G

designates

the unit normal to the surface

6 directed towards the outside of the fluid domain D.

1 We shall use the usual scalar notation

q

v

and

q

m

for volume or mass fluxes which evidently

are fluxes and not flux densities; confusion is not possible, flux densities being at least

vectors.

126 Fundamentals of Fluid Mechanics and Transport Phenomena

The extensive quantities are associated with the matter and the balance of the

quantity

G must be performed on a material domain. Consider (Figure 3.5a) the

domain



tD occupied by the matter at instant t, of external surface

6

, and then the

domain



tt

G

D occupied by the matter at time tt

G



. Between these two

instants, the material domain is displaced such that it leaves

6

over the section

6



whereas it is displaced towards the interior of



tD over the surface

6



(Figure

3.5b); let

D

0

be the volume common to



tD and



tt

G



D . Let

1

D

G

and

2

D

G

be

the additional parts of

D

0

in D(t) and



tt

G



D generated by the surfaces

6

1

and

6

2

and the vector displacement

tV

G

. The (positive) amounts of the quantity G

contained in each of these domains are, respectively,

³

6

1

. tdsnVg

G

and

³

6



2

. tdsnVg

G

.

(a) (b)

n

G

x

1

x

2

x

3

O

D

6

ds

D

0

6

1



6

2



n

G

n

G

V

G

D

1

G

D

2

Figure 3.5. Balance of an extensive quantity in a flow:

(a) domain at one instant; (b) displacement of the material domain

The variation

G

G of the quantity G contained in

D

between the instants t and

tt

G



is thus:

>@





³³³

66





21

..)()(

)()(

tdsnVgtdsnVgdvtgttg

tGttGG

GGG

G

0

D

By regrouping the surface integrals, taking the limit as

0

ot

G

, and applying

Ostrogradski’s theorem, we obtain:



³³³

qq

¸

¹

·

¨

©

§



w



w

6 DD

dvVgdiv

t

g

dsnVgdv

t

g

dt

dG

G

. [3.32]

Physics of Energetic Systems in Flow 127

or, using notation with indices:



³³³

qq

¸

¹

·

¨

©

§

w



w



w

6 DD

dv

x

ug

t

g

dsnugdv

t

g

dt

dG

i

ii

[3.33]

The notation

d

t

dG

in equation [3.32] or [3.33] indicates a Lagrangian derivative.

Formula [3.33] shows that the flux of the quantity

G

across the surface

6

can be

interpreted as due to the integration over the volume of the

local volume source





Vgdiv

G

.

Material derivatives can also be expressed as a function of the mass density

g

:



³³³³

66



w



w

qq

mv

gdqdvg

t

dqgdv

t

g

dt

dG

DD

U

Let us take the elementary case where

1

g

. The quantity

G

is the volume of

the domain

D. For a small domain of volume

v

G

, we have:



vVdiv

dt

vd

G

.

G

The quantity

Vdiv

G

is thus the local velocity of the volume expansion



dt

vd

v

G

1

.

If the quantity

G

is a vector (for example the volume momentum

V

G

U

), the

preceding steps can simply be applied to each of the components

g

i

. We thereby

define a

tensor flux of the quantity

G

by the relation:

or , (with : Cartesian product). 

G

Gij i j G

qgu qgV

NOTE

–

A more mathematical demonstration of expression [3.33] could be

performed by using the fact that the domain



tt

G



D can be derived from the

domain



t

D by means of a one-to-one geometric transformation which is precisely

defined by the displacement of fluid particles between these instants ([ZIL 06]).

128 Fundamentals of Fluid Mechanics and Transport Phenomena

3.3.3.4.

Material derivative of a flux integral

Consider now the integral

³

S

dsnB

G

.

M

, in which the vector

B

G

and the surface

S

are

attached to the moving matter

. The surface

S(t)

moves to position



ttS

G



,

which we will call

S

1

at time

tt

G



, having thus swept the surface

S

A

(Figure 3.6).

The flux variation

S

M

between these two instants is:

³³

 

SS

SSS

dsntBdsnttBttt

G

).().()()(

1

GMGMGM

[3.34]

Consider the domain

D bounded by the surface

A

SSS  6

1

;

W

G

d

is the

elementary arc of the curve

C

which encloses the surface

S;

we take the normal in

the direction

W

G

dV 

, on the lateral surface element

WG

G

dtV 

swept out by

W

G

d

during time

t

G

.

ds

S

1

S

A

Ĳd

G

tV

G

n

G

B

G

V

G

C

Figure 3.6. Material derivative of a flux integral

The flux of

B

G

leaving the closed surface

6

at time

t

can be written:

   





³³³³³



6 CSS

dVBtdsntBdsntBdvtBdivdsntB

WG

G

....

1

D

Taking account of this expression, flux variation [3.34] can be written:



³³





C

SS

S

dVBtdvtBdiv

dsntBdsnttB

WG

GGM

G

GGG

G

.

).().(

11

D

Applying Ostrogradski’s theorem, and letting t

G

tend to zero, we obtain the

material derivative of the flux

S

M

crossing S:

Physics of Energetic Systems in Flow 129



³³³



w

SCS

S

dsBdivnVdVBdsn

t

B

d

t

d

G

...

W

M

which is, after the application of Stokes’ theorem:



³

¸

¹

·

¨

©

§



w

S

dsnBdivVVBtor

t

B

td

d

G

.

M

[3.35]

If the

field B

G

is conservative

(0

Bdiv

G

), the total derivative of the flux

S

M

can

be written:



³

¸

¹

·

¨

©

§



w

S

dsnVBtor

t

B

td

d

G

.

M

[3.36]

3.3.3.5.

Unsteady and quasi-steady flows

In Eulerian variables, a flow (or the transfer of a quantity

G

) is considered

unsteady if the time

t

appears as a variable in the description of the flow:

/0or /0

zz

ss ss

i

ut gt

In Lagrangian variables, the idea of a flow’s unsteadiness does not have any

direct meaning.

If the term

tu

i

ww

/ (respectively

tg ww

/ ) is very small and negligible compared

with the other terms involving the corresponding material derivative, the flow

(respectively the transfer of quantity

G

) is quasi-steady and the differential character

with respect to time no longer features in the balances. The time

t

thus becomes a

parameter, and the

unsteady properties of the problem studied are those of a steady

flow

(or of the transfer of the quantity

G

). The

physical causes of variations of the

quantity G in the matter are thus solely balanced by convective fluxes

.

Following the note made in section 3.3.3.1, the flow can only be steady in a

given, particular reference frame.

3.3.4. Deformation rate tensors

The determination of the stresses undergone by the matter in a flow is related to

our understanding of the manner in which the matter is deformed. Any deformation

of a solid requires the action of a stress. A fluid which does not have any particular

130 Fundamentals of Fluid Mechanics and Transport Phenomena

shape or form can be deformed easily without stresses, on condition that the

corresponding action occurs slowly.

In a fluid stresses are thus associated with the structure of the velocity field.

However, velocity fields corresponding to changes of reference frame, or to

displacements, will not be associated with stresses.

The study of the local structure of a velocity field at any time is made by

considering two neighboring points

M

G

and MdM

G



at the respective coordinates

x

i

and



ii

dxx 

(Figure 3.7) whose velocities are





ji

xu

and





jji

dxxu 

.

x

2

x

3

x

1



MV

GG





MdMV

G

MdM

G



M

G

Figure 3.7. Velocity field at two neighboring points

The study of the velocities at these two points will allow the nature of the motion

of the matter to be determined. We perform a Taylor expansion about the point

M,

G

at coordinates

x

j

:

)()()(

hoh

x

u

xuhxu

j

i

jijji



w

 

We can separate the symmetric and anti-symmetric components of the

tensor

ji

xu ww

:

11

with: ;

22

jj

iii

ij ij ij ij

jjiji

uu

uuu

%%

xxxxx

::

¬¬

ss

sss









    









sssss





®®

[3.37]

The anti-symmetric part

jij

h

:

of the velocity field

du

i

can be written:

¸

¹

·

¨

©

§



¸

¹

·

¨

©

§

¸

¹

·

¨

©

§

::

::

::

:

1221

3113

2332

3

2

1

3213

3221

1321

0

hh

h

jij

ZZ

Physics of Energetic Systems in Flow 131

with:

213132321

: : :

Z

The preceding expression

jij

h

:

, written in vector form h

G



Z

, expresses the

velocity field due to the instantaneous rotational velocity vector

Z

K

:

Vrot

G

2

1

Z

[3.38]

The remaining symmetric tensor

ij

H

characterizes the deformation rates (strain

rates), as the displacement velocities have already been accounted for.

We can easily verify that the diagonal terms of the tensor

ij

H

are expansion (or

compression) rates, the other terms corresponding to shear (or angular) expansion

rates. By considering for example the velocity field



>@

0,

3221

uuxu we see

that the deformation generated comprises a transverse motion of the segment MM'

(Figure 3.8a) with respect to the velocity.

If the segment MM' is parallel to the velocity (



0,

3211

uuxu ), the

segment MM' is expanded or compressed according to the sign of

1

du (Figure

3.8b).

x

1

x

1

+dx

1

x

1

u

1

u

1

+du

1

x

2

x

1

x

2

+dx

2

x

2

u

1

u

1

+du

1

M

M'

M

M'

(a):

(b):

0

11

zww xu0

21

zww xu

Figure 3.8. Deformation rates: (a) shear

12

/0;zssux

(b) expansion

0

11

zww xu

The volume expansion rate Vdiv

G

(section 3.3.3.3) is furthermore always the

trace of the matrix

H

ij

, i.e. the sum of the linear expansion rates following the three

axes. We will leave it to the reader to verify this.

132 Fundamentals of Fluid Mechanics and Transport Phenomena

3.4. Phenomenological laws of viscosity

3.4.1. Definition of a fluid

3.4.1.1. Introduction

We saw in Chapter 2 that the stresses in a continuous medium can be represented

by a tensor

ij

V

. A fluid particle undergoing a bulk movement (translation and

rotation) is at rest in a moving Cartesian reference frame. It is not subject to stresses

other than those induced by pressure forces. On the other hand, this is no longer the

case in the presence of strain. The internal stresses in this continuous medium do not

depend on the relative position of the fluid particles, but on their relative velocities:

an infinitely slow fluid movement will not generate any stresses, contrary to what

occurs in solid bodies. This distinction between fluids and solids does not always

exist: certain bodies can have the properties of elastic solids for motions which

occur at the scale of seconds or fractions of seconds, and their shape may be

changed by a flow at the scale of many hours or many days (e.g. viscoelasticity,

creep in solids).

3.4.1.2.

Viscous fluids

A viscous fluid is one in which the stresses at a given instant are a function only

of the deformation rates at that instant. We will separate pressure stresses and the

viscous stress tensor

W

ij

using the relation:

(Kronecker symbol : 0 1)

ij ij ij ij ii

 p  si i j,    v 

[3.39]

This definition corresponds with the idea of a fluid as defined in fluid statics

(section 2.2.1.1). In keeping with what has already been said, the viscous stress

tensor depends only on the strain rates, and not the deformations. There exist “visco-

elastic” bodies in which these two kinds of stress generation can co-exist. We will

not cover the more complex cases ([FRE 64], [GER 94], [TAN 00], [COI 97]),

limiting our attention to the study of viscous fluids.

As in other domains of physics, the “relationship” between viscous stresses and

strain rates become complex when the structure is complex at the molecular level. In

particular, the flow may provoke changes in the “molecular cohesion”, or preferred

orientations in the presence of macromolecules. The establishment of a flow in a

fluid (gas or liquid) containing solid particles which are more or less dispersed may

lead to their becoming suspended. For example, the wind can carry sand in the

desert or on beaches, in many industrial applications powders are often transported

using fluidized beds (airflows which are sufficiently energetic to raise and transport

solid particles, these particles being deposed when the fluid velocity drops

sufficiently), etc. A snow avalanche is a heavy fluid which flows, whereas the snow

Peube J-L. Fundamentals of fluid mechanics and transport phenomena

Подождите немного. Документ загружается.