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

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Fluid Dynamics Equations 193



enthalpy

energy kinetic

difference pressure chydrostati

pressure dynamicU

termviscosity

on termaccelerati

0p0

www

xxu

dtdu

LUUL

jji

The Eckert number (or the Mach number) represents the influence of

compressibility on the properties of the flow whose kinetic energy is due to enthalpy

variations in the steady flow (conservation of the total enthalpy [4.33]).

The Reynolds number usually takes on very large values (from 10

to 10

which signifies the globally weak effects of viscosity and thermal conduction in

most flows. We will see in Chapter 6 that this property is not true in zones close to

solid boundaries (boundary layers).

Note that if the viscosity or other physical properties are not constant we must

introduce further non-dimensional functions.

4.6.1.3.3.

Non-dimensional boundary conditions

The treatment of boundary conditions is immediate: the conditions U and L are

transformed into a single condition equal to 1. The thermal problem requires, for

example, the temperature T

, given at the wall to be transformed into a non-

dimensional parameter

. If other velocity conditions or geometric dimensions

are given, they provide supplementary similarity parameters (velocity ratio, shape

factor, etc.).

The boundary conditions determine the order of magnitude of the similarity

parameters and the associated phenomena. Simplifications can result from this. For

example:

– if the Mach number is small (less than 0.3 in practice), it has to be shown (see

section 4.3.2.3.3) that the density variations can be neglected for the study of fluid

motion and that incompressible fluid relations can be used. Thermal transfer thus

occurs at constant pressure and the dissipation function is generally negligible. The

only unknown of the problem is therefore the temperature

TT  , and we take as a

reduced temperature



;

194 Fundamentals of Fluid Mechanics and Transport Phenomena

– for fully immersed flows of a liquid (section 4.4.2.4), we have seen that gravity

no longer has a direct effect on the flow; we therefore take the driving pressure

zgpp



to be the only variable, gravity disappearing from the equations

and from the boundary conditions: the Froude number is no longer a parameter.

4.6.1.3.4.

Non-dimensional expression of results

The non-dimensional quantities of a problem are functions of non-dimensional

coordinates and similarity parameters. They allow the calculation of dimensional

results which take on a specific form. For example, the pressure p can be written:

( , , Re, M, ...)

p  Upxt





Note that in place of

n dimensional data, we now have only n – 4 similarity

parameters (Re, Pr, Ec, Fr,

). For a purely dynamic problem (no thermodynamic

equation), we would have only

n – 3 similarity parameters. These results, which can

be obtained from a dimensional analysis (without writing the specific equations of

the problem), constitute the Vaschy-Buckingham theorem. Note that the

dimensional analysis does not allow us to see if certain similarity parameters

(Froude number) can be eliminated.

In practice, in place of the preceding non-dimensional parameters, we use other

quantities defined by custom. For example, we define the

local pressure coefficient

which is equal to

. The local viscous friction stress at a solid

boundary, equal to

can be expressed by means of the

friction coefficient

, which can also be written:

The

local thermal flux density at the wall



can be written in

non-dimensional form of the local

Nusselt number Nu:





Fluid Dynamics Equations 195

The global quantities (components of the force exerted on an obstacle, thermal

flux, etc.) can be obtained by integration of the local quantities over the wall surface.

We here define the non-dimensional coefficients, which can be immediately

deduced from the local coefficients. For example, the force

X exerted on an obstacle

in the

x-direction can be calculated by integration over the surface S of the obstacle:

³³



dsp

LUdsp

sin

cos

sincos

T-UT-P

Instead of the non-dimensional integral in the above formula, we can use the

drag coefficient C

in the x-direction, often defined by means of the surface S

projected frontal area (projection of

S on to the plane perpendicular to the x-axis):

4.6.1.3.5.

Case of unsteady flows

The procedure is identical to that outlined above; the temporal conditions

introduce further similarity parameters, for example the Strouhal number

St = ,

NL / U where N

is a characteristic frequency which appears among the data of

the problem.

However, we will see in section 6.6 that a problem posed with steady boundary

conditions will often have an unsteady or turbulent solution: the non-dimensional

solution is therefore a function of the variable

. For example, if the wake of the

cylinder discussed above comprises a preferred frequency

, the non-dimensional

frequency St =

will be a function of the similarity parameters St(Re,M,...).

4.6.1.3.6.

Validity of the similarity approach and comparison with experiments

The posing of a problem in non-dimensional form constitutes a

mathematical

model of the phenomena

studied, which is based on a choice of equations supposed

to be representative. The experimental verification of the model involves ensuring

that the non-dimensional results are correct, in other words that the non-dimensional

coefficients are only functions of similarity parameters defined with the model. For

example, in the fully immersed flow of an incompressible Newtonian fluid, the

results should depend on the Reynolds number alone. The problem has been badly

posed if this is not the case.

We can take an historic example: at the beginning of the 20

century, L. Prandtl

and G. Eiffel independently measured the drag of a sphere in a wind tunnel. They

found different drag coefficients for the same Reynolds number. This raised

196 Fundamentals of Fluid Mechanics and Transport Phenomena

questions over the validity of the similarity approach. An understanding of this

phenomenon was obtained 20 years later: one of the experiments comprised a

laminar flow in the zone of the boundary layer where the wake was generated,

whereas the turbulence had already been triggered in the other experiment. The

solution of the Navier-Stokes equations was not the same in the two experiments

because of instabilities generated by the residual present turbulence that is different

in the two wind tunnels.

Questioning the similarity methodology amounts to denying the validity of the

basic laws of classical physics

, which seems, at the very least, both erroneous and

presumptuous. It is clearly the specific model used which must be questioned when

agreement is not obtained by means of the non-dimensional representation of the

results (for example, the friction coefficient as a function of the Reynolds number).

4.6.1.3.7.

Similarity in diffusion problems

We will limit ourselves to the case of weak concentrations [4.60] by considering

the steady flow of a fluid of initial concentration c

which reacts with a reactive wall

according to a law of the form [2.70]. The concentration

satisfies equation:



cDuc

with the condition at the wall P:



This can be written in non-dimensional form by letting

ccc

. The non-

dimensional problem can be written:



with the absorption condition for a constituent, due to heterogeneous reaction at the

wall:



Lkc

[4.80]

The similarity parameters of the diffusion problem are:

Fluid Dynamics Equations 197

– the Péclet

diffusion number

Re.Sc

;

– the

Schmidt number:

Sc (

kinematic viscosity);

– the

Damköhler

number

Lkc

m 1



which characterizes the speed of the

chemical reaction.

Condition [4.80] shows that if the number

Lkc

m 1



is large compared to 1, the

wall concentration is zero, and the flux of the corresponding constituent is limited by

diffusion. If, on the other hand, we have





Lkc

, we have

and the

reaction speed is equal to

: diffusion is no longer important.

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Chapter 5

Transport and Propagation

The objective of this chapter is to present the general properties of the equations

which describe the flow of matter. The substantial derivative describes the physical

idea of the properties associated with matter which is in motion. It results in a

particular structure of the equations of fluid mechanics and the properties associated

with the displacement of material quantities. This leads to a specific means of posing

and numerically solving fluid mechanics problems. The corresponding mathematical

techniques which underlie the physical concepts are well known, but unfortunately

they are not widely taught; we will here recall them without providing detailed

demonstrations.

5.1. General considerations

5.1.1. Differential equations

We will first quickly recall the general properties of a differential equation which

is satisfied by a scalar function:

– the order n of the equation: the greater the maximum order n of the derivatives,

the greater the complexity of the solution (n time constants correspond to n

independent solutions for a linear differential equation with constant coefficients).

We know that, in general, the solutions of a differential equation form a family of

functions which depend on the n parameters. This means that the number of

boundary conditions which must be specified in order to determine a unique solution

is generally equal to the order n;

200 Fundamentals of Fluid Mechanics and Transport Phenomena

– the general structure of the differential equation: the solutions of a linear

differential equation belong to a vector space, which means that we can express a

solution by means of suitable linear combinations of other solutions. The dimension

of the vector space of the solutions is equal to the order of the differential equation;

– the properties of the coefficients of the differential equation, and in particular

their eventual singular properties: a coefficient which cancels itself out at some point

very often leads to particular properties of the solutions at that point. Such is the

case for solutions to problems of revolution about an axis for the point which

corresponds to the axis (generally denoted r = 0). Other particular points can play an

important role, as we will see with regard to stability problems;

– the nature of the boundary conditions imposed determines the kind of

differential problem which is posed:

- a Cauchy problem is determined by n boundary conditions given at a point

for the function and its n – 1 successive derivatives; such problems generally have a

unique solution

in the vicinity of this point. This kind of problem is often

encountered in mechanics for initial values of a motion. Regular behavior is only

ensured in the vicinity of the point, and the regular behavior can eventually extend

to the entire domain considered; however, in numerous cases we encounter accidents

in the behavior of the system far from the said point (divergence of the solution,

instabilities and random behavior, etc.),

- the n boundary conditions required may be specified at the two points of the

extremities of an interval. This case is common in physics, for field problems of

physical quantities (electromagnetic fields, velocity and displacement fields, etc.) for

which the solid boundaries impose particular conditions. The existence and

uniqueness of the mathematical solution can be obtained if the problems have been

well posed in physical terms for the entire domain.

In the preceding particular case where we impose zero conditions at two points,

the differential equation generally has a zero solution. However, there may exist

particular coefficient values for which a non-zero solution exists, and which

therefore depends on some parameter. These particular values correspond to the

eigenvalues of the differential operator. We will treat these problems in more detail

in Appendix 4.

The case of a system of coupled differential equations can amount to the study of

a higher order differential equation for one of the unknown functions after

elimination of the other unknown functions. For example, the system of two second

1 I.e. when we have mathematical properties of regularity, for the rest more or less satisfied in

application conditions of physics and mechanics.

2 We wish to say that any approximation with notable physical consequences has been made

when writing the equations for the problem.

Transport and Propagation 201

order equations with two unknown functions y(x) and z(x) can immediately be

transformed:

),,',,'(''),,',,'('' xyyzzgzxyyzzfy

Deriving z'' leads to the appearance in the expression for z''' the derivatives y'' and

z'' which we replace with their earlier expression f and g, giving for z''' an expression

of the form

),,',,'( xyyzzh . Doing the same for z'''', we get a similar expression

),,',,'( xyyzzm . The elimination of y, y' from the three equations:

),,',,'(''''),,,',,'('''),,,',,'('' xyyzzmzxyyzzhzxyyzzgz

gives a fourth order differential equation for the function z(x).

Conversely the differential equation '''' ( ''', '', , )zfzzzx for the unknown

function z(x) is immediately transformed into a system of four first-order differential

equations by letting

wzvzuz ','','''



wzvwuvxzwvufu '.;';';,,,,'

More generally, the system of first-order differential equations:



njitxftx

jii

,...,1,,

can be transformed into a differential equation of order n for one variable, for

example x

in effect, by differentiating



, the two other derivatives



appear, which we replace as a function of the quantities x

(t); we thus obtain:

 

txgtx

txtx

'''

Doing the same for the subsequent derivatives up to order n, we obtain n

equations which express the first n derivatives of the function x

as a function of all

the functions x



njitxgtx

,...,2,1,,

)(

Eliminating the n – 1 functions



(i = 2, 3,…, n) between the n preceding

equations leads to the n

order differential equation satisfied by the function x



t .

202 Fundamentals of Fluid Mechanics and Transport Phenomena







:,,..,,,

)1(

)(

txxxxFtx



The initial conditions



 

1,...,00

 nkx

can be obtained in the same way

as a function of the initial values



x .

The preceding procedure is only appropriate if the equations of the system are

suitably coupled. Otherwise, new variables X

must be chosen, such that the

equations are coupled. We will leave it to the reader to verify this for the elementary

linear system (let: X

= x

. + x

and X

.= x

– x

5.1.2. The Cauchy problem for differential equations

Consider the differential equation:

  

fxgxf ,' [5.1]

Solving the Cauchy problem involves calculating the value of the solution in the

vicinity of the value x

of the variable x, at which the value f

= f (x

) is given. It is

equivalent to saying that given the point (x

, f

) of the plane (x,f), we must find the

variations dx and df which it is possible to calculate from equation [5.1]:



dfdxxf ' [5.2]

It is clear that the system of equations [5.1] and [5.2] with the unknown f '(x)

should be of rank 1, hence the condition:

dfdx

The curve of the plane



fx,

on which the value of f extends (or is transmitted)

is therefore given by the relation:



fxg

dfdx

The preceding presentation may seem unnecessarily formal, but its interest is

that it can be extended to the study of partial differential equations.