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

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472 Fundamentals of Fluid Mechanics and Transport Phenomena

of the solution (Figure 8.13). This more rational method of discretization has the

advantage of reducing the number of variables.

Once a discretization is chosen for our knowledge model of a time-invariant

linear system, we have to deal with a lot of modes that have no physical

significance, as we have seen at the beginning of this section. The best reduction

method thus consists of using the properties of the modal solution, where only first

modes, corresponding to the actual evolution of the system, are retained. Returning

to the definition of an exact reduced state representation (section 8.6.2.1), let us

assume that the reduced state vector

= R X with s components (s < n), derived

from

X by means of the reduction matrix R, satisfies reduced state representation

[8.98]:

UBXA

rrr

..  [8.98]

With condition [8.99] obtained by multiplying the left-hand side of equation

[8.93] by

R and comparing with [8.98]:

 

.00;; RXXRBBRARA

rrr

[8.99]

we verify that an eigenmode of [8.94] is also an eigenmode of complete state

representation [8.93]:

   

00 4/ 4/ 4/

iiriiii

RIARIRAIA

The

s eigenvalues

of the reduced model are thus eigenvalues of the complete

model, the corresponding eigenvectors being equal to

, the choice of which

modes to retain in the reduced model results from the definition of matrix

R. The

reduction matrix

R is thus such that the reduced state vector is constituted of the s

retained modes. Taking eigenvalues as a basis (modal basis), the matrix

A is the

diagonal eigenvalue matrix

. We assume that the s eigenvalues to be retained have

been placed in the first

s elements of the diagonal. In the modal basis, the reduction

matrix

R, of dimension s*n contains 1 on the diagonal, in correspondence with the

first diagonal element. The reader can verify that the reduced square matrix

contains the

s eigenvalues which were retained.

The same result can in principle be obtained in another manner. We have seen in

section 8.2.2.2 that the suppression of a mode amounts to requiring that its

coefficient be zero in the modal development at each instant: this results in an

instantaneous linear relationship between the state vector components (formula

Eigenvectors which have not been retained belong to the kernel of R.

Thermal Systems and Models 473

[8.19]). The suppression of n-s modes of the model comes down to writing n-s

relations between the variables, which allows us to eliminate

n-s state variables. We

thus obtain a representation by

s state variables of the system which is reduced. This

method does not require us to calculate the modal basis. While it is in principal

arbitrary in the preceding methods, the choice of physical variables which are not

eliminated should nonetheless be performed so as to be conducive to physical

interpretation. The reader will also note that the principle of modal reduction itself

leads to a situation where

discontinuous data leads to a discontinuity of the

variables of the reduced representation.

8.6.2.5. Reduction of input-output representations

8.6.2.5.1. Introduction

Simple input-output representations are useful:

– for obtaining a global knowledge of the behavior of a system in view of the

implementation of control;

– for the dimensioning of components during the realization of a system;

– for the representation of sub-systems in models of complex systems.

The properties of these reduced representations differ according to their

utilization: command or engineering formulae.

8.6.2.5.2.

Command models

The essential characteristics to know are thus relatively global and it is sufficient

to know one or several “response times” corresponding to the inputs on which we

act. The output variables allow us to define values of actions to be effected. We will

here consider a model of the form [8.59] of low order: for example, for a single

input single output system, we often consider a second order differential equation

which is satisfied by the output variable and whose input is the right-hand side. We

will here only consider an open-loop system (without any retroaction by the

command); the system operation in close-loop depends on the command structure

and is not the object of this work (see [OBI 00]).

Second order differential equations represent accumulation (of mass or heat for

example), damping and stiffness mechanisms, which are themselves represented,

respectively, by second order terms, first order terms and zero order terms (the

function). It is thus possible to represent aperiodic or stable oscillatory systems. The

coefficients of such differential equations can be determined by imposing an

impulse excitation as an input, and then by “best” identifying the response of the

system to an expression of the form







tae

cos . This is equivalent to

performing a spectral study of the system (section 8.5.2.4).

474 Fundamentals of Fluid Mechanics and Transport Phenomena

8.6.2.5.3. Engineering formulae

We will designate under this category the input-output relations derived from

knowledge models and which are simple enough to be immediately useful for

example for quick dimensional assessment of components of a system in view of a

realization or for the representation of a sub-system in a more complex system.

These formulae are the result of reduced state representations whose pertinence has

been previously verified. They can also be obtained by simplification of an

analytical solution when one exists, for example by truncating a series (example of

section 8.6.2.4.2) or when using composite representations (section 8.3.2.2.3).

8.6.2.5.4.

Established regimes

The parametric expression of established regimes, obtained in section 8.5.2 in

the form of a series, can be simplified by truncation of this. Let us recall that the

representation in the form of a series is only of interest if it is possible to limit this to

a small number of terms. If this is not the case, this indicates that thermal boundary

layers exist in the domain, and it is therefore preferable to find a direct expression

which represents the boundary layer and to describe a composite solution by

matched asymptotic expansion (see section 8.3.2.2.3).

8.7. Application in fluid mechanics and transfer in flows

The evolution equations of a discrete or continuous system as a time function are

parabolic (or irreversible). In the presence of flow, the evolution variable is the time

only when Lagrangian variables are used. In Eulerian variables, the evolution speed

of a quantity

g is no longer represented by its temporal derivative tg ww but by the

material derivative

dtdg . This representation does not change the parabolic

character of the balance equations for extensive quantities with Euler variables

expressed along curvilinear abscissa of trajectories (or characteristic curves), which

is a parabolic variable equivalent to the time with Lagrange variables (see

interpretation of section 5.2.1) upstream then becoming the equivalent of the past. In

steady flow, the time variable disappears and the evolution variable becomes the

coordinate of the particle trajectories: systems studied thus appear as dynamic

systems along the trajectories. The same is true of boundary layer equations, or more

generally of the evolution of fluid properties along trajectories. These preceding

ideas are thus applicable to problems encountered in flows.

The methods evoked in this chapter are encountered when dealing with the

solution of flow and transfer problems in boundary layers ([SCH 99], [YIH 77]).

However, these are generally non-linear and computations cannot be effected in as

complete a manner as in this chapter. Writing balance equations, ultimately with

approximations which may be more or less global, leads to state representations

Thermal Systems and Models 475

where time is replaced by a boundary layer coordinate. The inputs are therefore

often evolution laws for the velocity of a perfect fluid as a function of the spatial

coordinate, the initial conditions having been provided in the initial section of a

boundary layer or a pipe. It is thus possible to obtain approximate external

representations in forms which are more or less explicit.

In principle, the procedure is of the same nature as for linear problems: an

elementary analytical solution makes it possible to study situations which are more

or less similar, and to identify the parameters associated with the category of

solutions studied. However, non-linearity does not enable the easy use of series

developments of eigenfunctions or of integral transformations. On the other hand,

singular or regular perturbation methods remain useful for writing the equations as a

cascade of successive approximations (section 6.4.2.3). The differential equations

obtained must often be solved, either by numerical means or by some global

methods (section 6.3.1.2). In other words, the general methodology of the present

chapter is applicable to problems of flow and associated transfer by means of an

adaptation of the calculation methods. The preceding discussions of linear dynamic

systems often allow us to organize knowledge gained from these non-linear systems

in a more rational way.

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Appendix 1

Laplace Transform

A1.1. Definition

The Laplace transform of a real variable function f(t), considered for

0tt

(0)( tf for t < 0), is defined by the relation

 



dtetfpL

where p is a complex valued variable. It is particularly well adapted to causal

signals.

A1.2. Properties

– The Laplace transform is a linear application on integrable functions.



The Laplace transform

of the derivative of a function is written:

         

00'

fppLfdtetfpdtetfpL

ptpt

 

³³



The derivative is here taken in the sense of distributions: any discontinuity of the

function leads to a Dirac distribution at that point for the derivative.

478 Fundamentals of Fluid Mechanics and Transport Phenomena

– The Laplace transform of a primitive is written:

  

)(

pLduuftg



– The Laplace transform of a delayed temporal signal can be written:

if:



W WtW ttgttftg 0)(,)(

     



xpptpt

epLdxexfdtetfdtetgpL







³³³

)(

Conversely, if:

etftg



)()(

  



)(

apLdtetfdteetfpL

tapptta



³³





– The Laplace transform of a convolution product of two functions f and g is

equal to the product of the Laplace transforms of these functions:

)()()(

pLpLpL

gfgf

A1.3. Some Laplace transforms

– The Laplace transform of a Dirac function is equal to 1:

  

³³

f



dtetdtetpL

ptpt

being zero for negative values of t.

– The unit step H(t), equal to 1 for t positive, is the primitive of the Dirac

distribution:



dtt ; its transform is:



ppL

1 .

– The unit ramp t is the primitive



duuH

of the unit step H(f); its transform

is:

1)( ppL

In general, we obtain:





ppL

–

  

foo



ppLtff

limlim0

Appendices 479

–

  

limlim

ofo

ppLtff

– Consider the limited development P

(t) to order m for small values of time t.

We obtain:

  

¦¦





pLtatP

The lowest order terms in p

-1

represent the Laplace transform which corresponds

to small values of time t.

For a limited development, and for large values of t, (t o f) we obtain:

  

¦¦









mni

pLtatQ

The highest order terms in p

-1

represent the Laplace transform which corresponds

to large values of t.

– The Laplace transforms of exponential and harmonic functions are:



sin

cos

exp

)(

exp

;





dtepL

tpa

A1.4. Application to the solution of constant coefficient differential equations

Consider the system:



0. XXBUXA

 .

The Laplace transform L

(p) of vector X(t) satisfies the equation:

 

XpLpLApI

BUX

 

From this we deduce:



)()( XpLApIpL

BUX



-1

480 Fundamentals of Fluid Mechanics and Transport Phenomena

The matrix



pI A

is comprised of elements which are rational fractions of the

frequency p which can be developed in a complete series in p. We can write this in

the form:





DpApI

The transformed linear system can thus be written:



)()()( XApIpLDXApIpLDppL

BUi

1-1-

 

¦¦

The second term represents the transient response to the initial conditions. The

steady response is thus:



)(

tUBDX

Appendix 2

Hilbert Transform

The amplitude a and the frequency

of an oscillatory harmonic motion appear

naturally if the signal is written as the real part of a complex number in the

trigonometric form



, in which the time derivative of the instantaneous phase



 tt 2 is equal to 2

. In the same way, we can associate a complex-

valued function



D (called an analytical signal) with any real signal x(t) by

adding to it an imaginary-valued function of time jy(t). This complex signal can be

written in the trigonometric form:



])(Re[)(:with.)()()(

)()( tjtj

etatxetatjytxt



The modulus



ta and the argument



tM of this complex number can be defined

as the amplitude and the instantaneous phase of the signal x(t); we thus define the

instantaneous frequency as the derivative



2' t .

These definitions are only meaningful if the point whose affix is the complex

function turns nearly regularly in the positive direction around the origin, in a way

quite similar to a complex exponential function. This is the case for amplitude- or

phase-modulated harmonic signals, or for the sounds of musical instruments. A

narrow-band signal x(t) appears on an oscilloscope as a carrier of the neighbouring

frequency

, whose amplitude [a(t)] varies slowly and whose phase [

(t)] is

fluctuating.

Given the real part of a complex number, it cannot be determined uniquely,

because we only know the product



cos . As the signals to be represented are

quite close to harmonic signals, it is natural to apply the Fourier transform and to