Peube J-L. Fundamentals of fluid mechanics and transport phenomena
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482 Fundamentals of Fluid Mechanics and Transport Phenomena
generalize the properties of harmonic signals. However, we note that the definition
of an analytic signal
tj
e
SQ
2
for a sinusoidal function
t
Q
S
2cos
consists of
eliminating the negative frequencies of the Fourier transform of this analytical
signal, by folding over the negative frequencies onto the positive frequency axis.
This operation can be applied to any signal x(t) and it allows the analytical signal
D
x
(t) to be defined. Let
Q
x
F be the Fourier transform of the signal x(t). The
Fourier transform of the analytical signal
t
x
D associated with x(t) is defined by:
¯
®
!{
{
0for2
0for0
QQQ
QQ
D
D
xx
x
FF
F
0 ( ) 1
i.e. () () (). () with:
0 ( ) 1
ax x x
xSignx
Fv Fv SignvFv
xSignx
£
¦
¦
¤
¦
¦
¥
[A2.1]
As the inverse Fourier transform
.
1
Q
SignF
of the sign function is equal to
tj
S
([BEE 03], [PRI 91]), the analytical signal
t
x
D is thus obtained by
performing the inverse Fourier transform of expression [A2.1]
>@
tjHtxFFt
xxx
QD
D
1
where H
x
(t) designates the Hilbert transform of the signal x(t)
:
³
f
f
ut
du
uxVPtx
t
FSignjFtH
x
.
1
*
1
.
1
SS
QQ
The Hilbert transform is limited to positive time, contrary to the Fourier
transform. The Hilbert “filter” is a linear filter whose impulse response is 1/
S
t; it is
thus not causal. The transfer function of this filter is
Q
Q
Q
S
signjFH
tH
.
1
. The Hilbert filter transforms a cosine to a sine; it
is a perfect quadrature filter. It is unfortunately not achievable as it is non-causal;
we can only create quadratures which function in a limited band of frequencies.
Appendix 3
Cepstral Analysis
A3.1. Introduction
The cepstrum is an integral transform that can be calculated from a spectrum and
contributes to its analysis. For example, in a vibration problem, it makes it possible
to separate the impulse response from the excitation forces. The analysis of vibration
signals from rotating machinery is important. The different operating conditions of a
machine, as well as defects, can be observed in the spectral domain ([STR 96]).
It is also used in speech processing, the vocal signal coming from the
convolution of the excitation (source) and the impulse response of the vocal passage.
Another application is the extraction of incident sound from a signal containing
both the incident sound and its reflection (echo suppression). We will quickly cover
this application, which enables the demonstration of the use of logarithms and thus
of the cepstrum.
A3.2. Definitions
Let x(t) be a time signal and F
x
(
Q
) its Fourier transform; by definition the
complex cepstrum C
x
(t) is the inverse Fourier transform, denoted F
-1
, of the
logarithm of function
)(
Q
x
F ([BOU 98], [NOR 03 ], [STR 96])
^`
))((ln)(
1
QW
xx
FFC
if
)(
)()(
QM
QQ
j
xx
eFF , then:
^`
^`
)()(ln)(ln
QMQQ
jFF
xx
484 Fundamentals of Fluid Mechanics and Transport Phenomena
The power cepstrum may be defined in several ways, for example, as the square
of the modulus of the complex cepstrum of the signal:
^`
2
1
))((ln
Q
xpx
FFC
.
or by taking the inverse Fourier transform of logarithm of the modulus square:
^
`
¸
¹
·
¨
©
§
2
1
)(ln
Q
xpx
FFC
A3.3. Example of echo suppression
Let x(t) be a sound signal comprising the superposition of an original sound s(t)
and an unwanted echo s
r
(t):
)()()( tststx
r
The reflected sound is attenuated and dephased compared with the original signal
)()(
0
ttsats
r
The Fourier transform of x(t) can be written:
0
2
1)()(
tj
sx
eaFF
QS
QQ
and the square of the modulus of this transform is:
0
2
22
2cos21)()( taaFF
sx
QSQQ
If we take the logarithm of this quantity, the echo phenomenon is seen in
frequency space by the addition of a periodic term, of period 1/t
0
:
^
`
^
`
0
2
22
2cos21ln)(ln)(ln taaFF
sx
QSQQ
We now take the inverse Fourier transform of this expression ([ALL 04], ([MAD
98], [NOR 03]): the first two terms will be, respectively, the power cepstrums x(t)
and x(t); the third term will be the inverse Fourier transform of the periodic function
shown above, which will be comprised of Dirac functions of different amplitude,
separated by t
0
A
pp
endices 485
^
`
0
21
2cos21ln)()( taaFCC
pspx
QSWW
We thus find ourselves in a pseudo-temporal space, whose variable is known as
“quefrency”, in which we perform a “liftering” of the Dirac signals (the terms
“cepstrum”, “quefrency” and “liftering” are respectively anagrams of the words
spectrum, frequency and filtering). The echo signal is thus eliminated by the
suppression of the Dirac signals in this space. The initial signal is thus reconstructed
without its echo by an inverse process ([NOR 03]); however, we note that phase
information has been lost with this procedure of cepstrum power.
A3.4. General case
We consider a source signal x(t) going through a passive linear system whose
impulse response is h(t). The output signal y(t) is the convolution product of x by h:
)(*)()( txthty
The convolution product becomes a simple product after Fourier transform:
QQQ
xy
FHF
The cepstrum transforms the product to a sum:
)()()(
WWW
xhy
CCC
In order to perform the deconvolution, in other words the separation of the
source from that of the medium, some assumptions must be made regarding the class
of functions which comprise either the source or the medium:
– in the case of speech, we assume that the vocal excitation is comprised of a
periodic impulse t
0
(this property is of course only applied to the stable parts of
sounds) and that the cepstral contribution of the vocal passage is found in the low
quefrencies; we can thus obtain a smoothed spectrum which only contains
information on the vocal passage. The inverse Fourier transform of the spectrum
gives the estimated impulse response of the passage and is used for voice synthesis
([JUR 08], [MAD 98]).
– in mechanics, free oscillatory periods of a medium are generally small
compared to the periods of the excitations: the cepstrum of the impulse response of
the structure is in the left part, while that of the excitation is in the right part of the
cepstrum graph of structure response to excitation ([DES 00], [NOR 03], [ALL
04]).
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Appendix 4
Eigenfunctions of an Operator
A4.1. Eigenfunctions of an operator
Consider a linear operator L applied to a function
xf
which satisfies
homogenous boundary conditions at the extremities of the interval defined by [
a
,
b
].
The eigenvalues
O
i
of this operator and the associated eigenfunctions
f
i
satisfy the
relation:
ii
ffL
O
A4.2. Self-adjoint operator
A4.2.1. Eigenfunctions
Let us define the scalar product
! gf , for the foregoing functions f, by the
relation:
³
!
b
a
dxxgxfgf
.
The operator L is the self-adjoint if it satisfies the relation
³³
! !
b
a
b
a
dxxLgxfdxxgxLfgfLLgf
..
Under some general conditions:
488 Fundamentals of Fluid Mechanics and Transport Phenomena
the eigenvalues of the self-adjoint operator
L
are positive and form an infinite
denumerable set
1,2,..., ,...
i
in
O
. With each eigenvalue
O
i
we can associate an
eigenfunction
f
i
;
the eigenfunctions
f
i
form a basis for the class of functions
f
.
A4.2.2. Expression of a function f using an eigenfunction basis-set
Consider the following series development of
f
on the eigenfunction-basis:
xfcxf
j
j
j
¦
f
1
Let us calculate the scalar product
!
i
ff
:
! !
¦
f
xfxfcxfxf
ij
j
ii
1
[A4.1]
Now, the eigenfunctions are orthogonal for the scalar product. In effect, with the
assumption of a self-adjoint operator, we obtain for
ji
O
O
z
:
0
! ! !
jiijjiji
ffffLLff
OO
We also obtain:
2
iiiiiii
fffLff
OO
! !
Substituting previous results into [A4.1], we obtain the value of the coefficient
c
i
:
2
1
xfcxfxfcxfxf
iiij
j
ii
! !
¦
f
or:
2
i
i
i
fxf x
c
fx
Bibliography
[ABR 65] ABRAMOWITZ M.S., and STEGUN I.A., Handbook of Mathematical Functions,
Dover, New York, 1965.
[ALL 04] ALLEN R.L., MILLS D., Signal Analysis: Time, Frequency, Scale and Structure,
Inc NetLibrary, Wiley-IEEE, 2004.
[AME 58] AMES J.S., MURNAGHAN F.D., Theoretical Mechanics, An Introduction to
Theoretical Physics, Dover, 1958.
[AND 99] ANDREWS L. C., SHIVAMOGGI B. K., Integral Transforms for Engineers,
Society of Photo-optical Instrumentation Engineers, SPIE Press, 1999.
[ASH 05] American Society of Heating, Refrigerating and Air-Conditioning Engineers,
ASHRAE Handbook of Fundamentals, 2005.
[BAH 01] BAHER H., Analog & Digital Signal Processing, John Wiley & Sons, 2001.
[BAL 77] BALIAN R., PEUBE J.-L., Fluid Dynamics, Theoretical physics summer school
course, Les Houches 1973, Gordon and Breach Science Publishers, 1977.
[BAR 01] BARRASS B., YOUNG P., BARRASS C.V., Ship Stability, Notes and Examples,
Butterworth-Heinemann, 3rd edition, 2001.
[BAS 98] BASS H.E., SHIELDS F.D., “Ultrasonic relaxation processes” in Handbook of
Acoustics, Malcolm J. Crocker (ed.), John Wiley & Sons, Inc., 1998.
[BEE 03] BEERENDS R. J.,TER MORSCHE H. G.,VAN DEN BERG J. C., VAN DE VRIE
E. M,. Fourier and Laplace Transforms, Cambridge University Press, 2003.
[BEL 00] BELLANGER M., Digital Processing of Signals: Theory and Practice, Wiley,
2000.
[BEN 74] BENNETT C.O., MYERS J.E, Momemtum, Heat and mass Transfer, McGraw-
Hill, 2nd edition, 1974.
[BEN 99] BENDER C.M., ORSZAG S.A., Advanced Mathematical Methods for Scientists
and Engineers, Springer-Verlag, New York, 1999.
490 Fundamentals of Fluid Mechanics and Transport Phenomena
[BEN 06] BENESON W., HARRIS J.W. STOCKER H., LUTZ H., Handbook of Physics,
Springer, 2006.
[BER 84] BERGE P., POMEAU Y., VIDAL C., Order Within Chaos; Towards a
Deterministic Approach to Turbulence, John Wiley & Sons, 1984.
[BER 90] BERANEK L.L., Acoustics, American Institute of Physics, 1990.
[BIR 01] BIRD R.B., STEWART W.E., LIGHTFOOT E.N., Transport Phenomena, John
Wiley & Sons, 2nd edition, 2001.
[BLA 98] BLATTER C., Wavelets: A Primer, A K Peters Ltd, 1998.
[BOC 92] BOCCARA N., Les principes de la thermodynamique classique, Presses
Universitaires de France, 1992.
[BOR 00] BORGHI R., CHAMPION M., Modélisation et théorie des flammes, Editions
Technip, 2000.
[BOU 98] BOULENGER A., PACHAUD C., Diagnostic vibratoire et maintenance
préventive, Dunod, 1998.
[BRE 91] BREZINSKI C. and M. REDIVO-ZAGLIA M., Extrapolation Methods. Theory
and Practice, North-Holland, 1991.
[BRU 95] BRUUN H. H. Hot-wire Anemometry Principles and Signal Analysis, 1995,
Oxford University Press.
[BYU 02] BYUNG CHAN EU, Generalized Thermodynamics: Thermodynamics of
Irreversible Processes and Generalized Hydrodynamics, Springer, 1st edition, 2002.
[CAS 06] CASTANIÉ F., Spectral Analysis, ISTE Ltd, 2006.
[CHA 91] CHAPMAN S, COWLING T. G., The Mathematical Theory of Non-uniform
Gases: An Account of the Kinetic Theory of viscosity, Thermal Conduction and Diffusion in
Gases, Cambridge University Press, 3rd edition, 1991
[CAN 01] CANDEL S., Mécanique des fluides: Cours, Dunod, 2001.
[CAR 59] CARSLAW H.S., JAEGER J.C., Conduction of Heat in Solids, Clarendon Press,
(2nd edition), 1959.
[CHA 70] CHANG P. K., Separation of Flow, Pergamon Press, 1970.
[COI 97] COIRIER J., Mécanique des milieux continus, Dunod, 1997.
[COL 68] COLE J.D., Perturbation Methods in Applied Mathematics, Blaisdell Pub. Co.,
1968.
[COM 02] COMOLET R., Mécanique expérimentale des fluides, volume I, Dunod, 2002.
[COM 94] COMOLET R., Mécanique expérimentale des fluides, volume II, Masson, 1994
[COU 89] COURANT R., HILBERT D., Methods of Mathematical Physics, vol. II, Partial
Differential Equations, Wiley Interscience, 1989.
[COU 89] COUSTEIX J., Aérodynamique, Turbulence et couche limite, Cepadues-éditions,
1989.
Bibliography 491
[CRO 05] CROWS C.T., ELGER D.F., ROBERSON J.A., Engineering Fluid Mechanics,
Wiley, 2005.
[DEB 06] DEBNATH L., BHATTA D., Integral Transforms and their Applications, CRC
Press, 2006.
[DEG 62] DE GROOT S.R., MAZUR P., Non-equilibrium Thermodynamics, North-Holland,
Amsterdam, 1962: Dover Editions, 1984.
[DEM 97] DEMMEL J. W., Applied Numerical Linear Algebra, SIAM, 1997.
[DES 00] DE SILVA W., DE CLARENCE W., Vibration Fundamentals and Practice, CRC
Press, 2000.
[DOU 95] DOUGHERTY T., Systems & Control, An Introduction to Linear, Sampled &
Non-linear Systems, World Scientific, 1995.
[DUR 81] DURST F., MELLING A., WHITELAW J.H., Principles and Practice of laser-
Doppler Anemometry, Academic Press, 1981 (2nd edition)
[EU 92] EU B. C., Kinetic Theory and Irreversible Thermodynamics, John Wiley, 1992.
[EVA 98] EVANS L. C., Partial Differential Equations, American Mathematical Society,
Providence, 1998.
[EVE 89] EVETT J.B., LIU C., 2,500 Solved Problems in Fluid Mechanics and Hydraulics,
McGraw-Hill, 1989
[PRE 64] FREDRICKSON A.O., Principles and Applications of Rheology, Prenctice Hall,
1964.
[GAB 98] GABARD S., PEUBE J., “The influence of a narrow tube on the measurement of
pressure fluctuations”, ACTA ACUSTICA, 1998, vol. 84(6), p.1151.
[GAR 06] GARDE R J., Fluid Mechanics Through Problems, New Age Publishers, 2006.
[GER 94] GERARD D,. KUIKEN C., Thermodynamics of Irreversible Processes:
Applications to Diffusion and Rheology, John Wiley, 1994
[GIL 64] GILES R., Mathematical Foundations of Thermodynamics, Pergamon, Oxford,
1964.
[GRI 71] GRIMSON J., Advanced Fluid Dynamics and Heat Transfer, McGraw-Hill, 1971.
[GUP 83] GUPTA S., HASDORFF L., Fundamentals of Automatic Control, Krieger Pub Co,
1983.
[GUY 01] GUYON E., HULIN J.P., PETIT L., MITESCU C.D., Physical Hydrodynamics,
Oxford University Press, 2001.
[HAN 05] HANSEN C. D., JOHNSON C. R., The Visualization Handbook, Academic Press,
2005.
[HAR 98] HARRIS J., STOCKER H., HARRIS J.W., Handbook of Mathematics and
Computational Science, Springer, 1998.
[HIG 93] HIGGINS G.R., Sampling Theory in Fourier and Signal Analysis Foundations,
Oxford University Press, 1996.