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\dateposted{2024-11-05}
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%\makeatletter
%\def\TITREspecial{\relax}
%\def\cdr@specialtitle@english{More than a half century of Computational Fluid Dynamics}
%\def\cdr@specialtitle@french{Plus d'un demi-si\`ecle de m\'ecanique des fluides num\'erique}
%\makeatother

\CDRsetmeta{articletype}{foreword}

\title{Foreword to more than a half century of Computational Fluid
Dynamics (CFD)}

\alttitle{Plus d'un demi-si\`{e}cle de M\'{e}canique des Fluides Num\'{e}riques (MFN) : avant-propos}

\author{\firstname{Mohammed} \lastname{El Ganaoui}\CDRorcid{0000-0003-4910-6052}\IsCorresp}
\address{Professeur des Universit\'{e}s, Universit\'{e} de Lorraine,
Nancy-Metz, France}
\email[M. El Ganaoui]{mohammed.el-ganaoui@univ-lorraine.fr}

\author{\firstname{Patrick} \lastname{Bontoux}\CDRorcid{0000-0002-3750-8727}}
\address{Directeur de Recherche CNRS (\'{e}m\'{e}rite), Universit\'{e}
d'Aix-Marseille, France}

\maketitle

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\vspace*{2pc}

\section*{La version fran\c{c}aise suit la version anglaise}

The present issue entitled ``More than a Half Century of Computational
Fluid Dynamics (CFD)'' explores several aspects of a discipline that
has impacted both theoretical knowledge and applications in the field
of fluid physics.

Fluid mechanics, starting from its nascent 
stage~\cite{Benard1901,Eiffel1912,Rayleigh1916}, has managed to address
the fundamental aspects of knowledge, as well as practical
applications, through global international cooperative efforts, even
during periods of conflict, to arrive at its current state where its
contribution is of considerable benefits to human
society~\cite{Charru2023}. At the same time, Computational Fluid
Dynamics (CFD), a natural extension of theoretical and experimental
fluid mechanics, coupled with Machine Learning (ML) and Artificial
Intelligence (AI), is incubating a path to address practical problems
of complex flows, as discussed here~\cite{Runchal2023}.

CFD has gradually established itself as an indispensable means of
predicting the phenomena of fluid physics. It provides a means of
integrating the cost-effective optimization of objective functions for
system designs. It has benefited extensively from the impressive
evolution of computer capabilities, as hypothesized by Moore's law, and
from significant developments in novel algorithms for discretization of
differential terms and advanced methods for solving matrices of
algebraic equations. Furthermore, computer architecture developments in
vector, parallel, and cloud computing and the latest advances in ML and
AI have provided considerable momentum for the application of CFD. This
has led to the development of numerous software tools that enable
scientists to advance their understanding of the physics of phenomena
and meet the complex demands of a continually expanding industrial
sector. It is worth noting that CFD has been an excellent example of
collaborative efforts by both researchers and developers from academia
and industry.

CFD has also become firmly anchored in the field of modern engineering
practice as an essential component, allowing for in-depth understanding
and optimization of systems based on fluid behavior, such as
aeronautics or meteorology~\cite{Voldoireetal2023}. Not to mention the
coupling of the dynamic field with other fields~ensures physical and
applicative completeness to the problems addressed. Fluid physics in
microgravity is an example of the outcome of the complementarity
between CFD and experimental physics to leverage the benefits of the
space environment for terrestrial applications impacting materials,
energy, and health~\cite{Microgravity2004}.

This issue pertains to the evolution of CFD and its significant
contributions over the last half century, as captured by the
specialists and practitioners who were active during this rapidly
changing era. Their perspectives on this issue contribute to creating a
comprehensive yet fair and balanced image of a discipline that has
benefited from the rapid convergence of numerous international efforts.

CFD also presents itself as an important avenue to address the open
problem of turbulence, which is notably listed as one of the seven most
important unsolved Millennial Prize problems of mathematics by the Clay
Mathematical Institute~\cite{Clay2017}.

It is important to realize that approximations have played a
significant role in advancing CFD modeling to understand the essential
aspects of physics and benefit, in retrospect, from the analysis of
phenomena that led to major breakthroughs (Soret effect, Dufour effect,
etc.). One of the most famous approximations, that of Boussinesq,
receives detailed attention in this thematic issue~\cite{Lappa2022}. 
An example of coupled stratified radiative transfer with fluid
mechanics is described in this thematic issue~\cite{GolsePironneau2022}, 
and the development of multiphase formulations for
modeling forest fires~\cite{Morvanetal2022}.

Another synthesis concerning experimental and numerical approaches and
bridging the notions of continuous and discrete meshing is the visible
manifestation of the contribution of numerical analysis to the
discipline, which is included in this issue~\cite{Bonnet2022}. A
contribution ranging from fundamental aspects to geophysics and
astrophysics~\cite{Cambonetal2023} is also presented. Indeed,
continuous and discrete aspects have marked the forefront of CFD
advancement and continue to do so. Furthermore, a recent review
about the evolution of CFD toward an integrated discrete approach based
on the original mathematical modeling of generic equations is
also~presented in this issue~\cite{Caltagironeetal2023}.

Meshing, as a significant manifestation of the discrete aspect, has
always been the subject of extensive scientific exchanges. An excellent
paper translates a debate almost as old as CFD itself to address the
compressible Navier--Stokes equations. Although theoretically
unresolved, software developers have provided some evidence in favor of
unstructured meshes~\cite{Dervieux2022}. The meshing topic continues to
address new problems amenable to practical and theoretical
developments, bringing mathematical and physical communities closer.
One contribution discusses recent advances in domain decomposition
methods for large-scale saddle point
problems~\cite{NatafandTournier2022}.  The results suggest that this
combined approach (meshing, numerical algorithms modeling efficient
computational implementation) is highly effective for practical
applications in aeronautical and maritime design~\cite{Visonneau2022}.
Significant attention is given to the numerical methods that have
accompanied the development of CFD. Classical approaches, often linked
to curvilinear meshing, are instinctively associated with the finite
element method (FEM), which has enjoyed practical use by engineers
since the 1940s. This method, first described by~(Leibnitz 1646--1716),
has benefited from a methodological formalism and rigorous mathematical
framework that found its foundations in the variational 
approach~(Galerk in 1915, Trefftz in 1926).

The Finite Volume Method (FVM), which had its origins at Imperial
College under the guidance of  Professor 
Spalding~\cite{RunchalandWolfshtein1966,Runchaletal1969,PatankarandSpalding1966} 
is an alternative formalism to address the problems of fluid flow based
on flux and mass balance in discrete volumes. It has benefited from
further developments and practical  
applications~\cite{Patankar1980,Hanjalicetal1980}, and the formalism of
a rigorous mathematical  framework~\cite{Eymardetal1991}. In this
issue, a contribution concerning the recent developments of the method
is also presented~\cite{Eymardetal1991}.

In the 1980s, lattice Boltzmann methods (LBM) emerged, which are
finding increasing interest and offer advantages in representing
certain families of flows. LBM relies on theoretical tools imported
from the kinetic approach of Boltzmann from the French mathematical
school. An excellent review of the LBM methods is included in this
issue~\cite{Succi2022}.

Some of these developments in CFD are described in the series of
conferences ``Discrete Simulation of Fluid Dynamics (DSFD)'' that began
in 1986 at Los Alamos at the initiative of Gary D. Doolen and had its
23rd edition in Paris in 2014. Among the many topics addressed in this
conference series are LBM, dissipative particle dynamics, smooth
particle hydrodynamics, Monte Carlo, and other methods.

It should be noted that the contributions derived from high-precision
numerical approximation, such as Hermitian or compact finite
differences of the 4th and 6th order, spectral methods of tau type,
decomposition collocation based on Fourier developments, Chebyshev
polynomials, and others, saw their initial rise, particularly with
vector supercomputers (such as Cray), for various applications in CFD.
However, these developments do not appear directly in this thematic
issue of the Proceedings of the French Academy of Sciences.

An original illustration of 35 years of aerodynamics and insect flight,
described in a contribution by~\cite{Engelsetal2022}, paves the way for
future topical connections. Now, a parallel can be drawn between the
fluidity of flows and the presence of CFD in all springs of scientific
activity, often implicitly benefiting recent domains where automation
and digitalization combine and manifest in progress made in hardware,
software, big data, AI, IoT, and virtual and augmented reality. The
range of tools offered to an engineer will enable them to access a
level of visualization and perception of interaction and dynamic object
manipulation far beyond what is currently available.

In brief, this testimony of half a century of coupling between the
physics of fluids, the physical world of applications, and the digital
universe serves as a true bridge between the extraordinary development
that fluid understanding has experienced and the potential that
artificial intelligence brings to the dynamics of tomorrow.

\selectlanguage{french}
\section*{{Version fran\c{c}aise}}

Le pr\'{e}sent num\'{e}ro sp\'{e}cial, intitul\'{e} \textit{Plus d'un
demi-si\`{e}cle de M\'{e}canique des Fluides Num\'{e}rique (MFN)},
explore des ann\'{e}es 1950 \`{a} nos jours plusieurs facettes d'une
discipline qui a marqu\'{e} autant les savoirs abstraits que les champs
applicatifs, \`{a} savoir la physique des fluides.

En effet, la m\'{e}canique des
fluides~\cite{Benard1901,Eiffel1912,Rayleigh1916} a transcend\'{e} et
invit\'{e} \`{a} une coop\'{e}ration active les th\'{e}matiques
fondamentales des savoirs, et \'{e}volu\'{e} sur le plan international
au travers des conflits mondiaux pour perfectionner aujourd'hui des
r\'{e}alisations majeures, pour le plus grand bien de la
soci\'{e}t\'{e} humaine~\cite{Charru2023}. La MFN est \`{a} la fois le
prolongement naturel de la m\'{e}canique des fluides th\'{e}orique et
exp\'{e}rimentale et une voie d'incubation de l'intelligence
artificielle pour les \'{e}coulements complexes, largement
\'{e}voqu\'{e}e de nos jours~\cite{Runchal2023}.

La MFN s'est impos\'{e}e progressivement comme une voie incontournable
de pr\'{e}vision des ph\'{e}nom\`{e}nes de la physique des fluides,
notamment en int\'{e}grant l'optimisation \`{a} moindre co\^{u}t de la
conception des objets et des syst\`{e}mes. Elle a b\'{e}n\'{e}fici\'{e}
conjointement de la loi de Moore pour l'\'{e}volution impressionnante
des capacit\'{e}s des ordinateurs, mais aussi du d\'{e}veloppement
algorithmique, et du calcul vectoriel, parall\`{e}le et intensif qui
prolonge les \'{e}volutions du mat\'{e}riel informatique. Ceci a permis
le d\'{e}veloppement de nombreux logiciels, permettant \`{a} des
scientifiques de faire avancer la compr\'{e}hension des
ph\'{e}nom\`{e}nes et de r\'{e}pondre aux attentes complexes d'un
secteur industriel en continuelle expansion. Notons d'ailleurs que ces
scientifiques n'\'{e}taient pas toujours form\'{e}s aux
math\'{e}matiques appliqu\'{e}es ou \`{a} l'analyse num\'{e}rique.

La MFN est \'{e}galement devenue une composante essentielle de
l'ing\'{e}nierie moderne, et a permis une compr\'{e}hension approfondie
et une optimisation des syst\`{e}mes bas\'{e}s sur le comportement des
fluides, comme en a\'{e}ronautique ou en m\'{e}t\'{e}orologie~\cite{Voldoireetal2023},
ou encore le couplage du champ dynamique avec d'autres
champs, assurant la compl\'{e}tude physique et applicative aux
probl\`{e}mes abord\'{e}s. La physique des fluides en microgravit\'{e}
fournit un exemple de retomb\'{e}e de la compl\'{e}mentarit\'{e} entre
MFN et m\'{e}canique des fluides exp\'{e}rimentale, associ\'{e}es pour
tirer profit des avantages de l'environnement spatial pour les
applications terrestres impactant mat\'{e}riaux, \'{e}nergie,
sant\'{e}, etc.~\cite{Microgravity2004}.

Dans ce num\'{e}ro, on trouvera des contributions importantes sur
l'\'{e}volution de la MFN, r\'{e}dig\'{e}es par des sp\'{e}cialistes et
des t\'{e}moins d'une \'{e}poque en pleine mutation. Leurs regards
crois\'{e}s dans ce num\'{e}ro contribuent \`{a} la formation d'une
image non exhaustive, mais juste et \'{e}quilibr\'{e}e, d'une
discipline qui a b\'{e}n\'{e}fici\'{e} de la convergence rapide de
l'effort international.

En premier lieu, les approximations ont jou\'{e} un r\^{o}le important
pour faire continuellement \'{e}voluer la mod\'{e}lisation, dans le but
de mieux comprendre certains aspects essentiels de la physique et
d'apporter un \'{e}clairage, \textit{a posteriori} et via leur prise en
compte de l'analyse, \`{a} des ph\'{e}nom\`{e}nes mineurs qui ont
ensuite conduit \`{a} des avanc\'{e}es majeures (effet Soret, effet
Dufour,\,\ldots). Une des plus c\'{e}l\`{e}bres approximations, celle de
Boussinesq, b\'{e}n\'{e}ficie d'une note d\'{e}taill\'{e}e dans ce
num\'{e}ro th\'{e}matique~\cite{Lappa2022}. Un exemple de transfert
radiatif stratifi\'{e} coupl\'{e} aux \'{e}quations de la m\'{e}canique
des fluides est d\'{e}crit dans ce num\'{e}ro 
th\'{e}matique~\cite{GolsePironneau2022} 
de m\^{e}me que le d\'{e}veloppement de
formulations multiphasiques appliqu\'{e}es \`{a} la mod\'{e}lisation
des incendies de for\^{e}t~\cite{Morvanetal2022}. 

La MFN se pr\'{e}sente comme une des pistes de r\'{e}ponse au
probl\`{e}me de la turbulence, qui reste encore ouvert pour ce
si\`{e}cle, au point de compter parmi les sept probl\`{e}mes retenus
par la fondation Clay~\cite{Clay2017}. Une autre synth\`{e}se concernant
les approches exp\'{e}rimentales et num\'{e}riques et faisant le lien
entre les notions de continu et de discret (le maillage \'{e}tant la
manifestation visible de l'apport de l'analyse num\'{e}rique \`{a} la
discipline) figure dans ce num\'{e}ro~\cite{Bonnet2022}, qui comprend
\'{e}galement une contribution retra\c{c}ant l'histoire de la
turbulence, depuis son aspect fondamental jusqu'\`{a} la
g\'{e}ophysique et l'astrophysique~\cite{Cambonetal2023}.

De fait, les aspects continu et discret ont marqu\'{e} le front
d'avancement de la MFN et continuent \`{a} le faire. Une revue
r\'{e}cente sugg\'{e}rant l'\'{e}volution de la MFN vers une approche
discr\`{e}te int\'{e}gr\'{e}e, bas\'{e}e sur une mod\'{e}lisation
math\'{e}matique originale des \'{e}quations g\'{e}n\'{e}riques, est
d'ailleurs apport\'{e}e dans ce num\'{e}ro~\cite{Caltagironeetal2023}.

Le maillage, comme importante manifestation du discret, a toujours fait
l'objet de grands \'{e}changes scientifiques~: un excellent papier
revient sur ce d\'{e}bat presque aussi ancien que la MFN, pour aborder
le mod\`{e}le de Navier--Stokes compressible~\cite{Dervieux2022}. Il
montre que bien que cela ne puisse \^{e}tre tranch\'{e}
th\'{e}oriquement, les d\'{e}veloppeurs des logiciels ont apport\'{e}
un arbitrage (de fait) en faveur des maillages non structur\'{e}s. 

La th\'{e}matique du maillage continue \`{a} poser de nouveaux
probl\`{e}mes, susceptibles d'engendrer des d\'{e}veloppements
pratiques et th\'{e}oriques et de rapprocher encore davantage les
communaut\'{e}s math\'{e}matique et physique. Une contribution aborde
le progr\`{e}s r\'{e}cent dans les m\'{e}thodes de d\'{e}composition de
domaine, pour le probl\`{e}me du point de selle \`{a} grande
\'{e}chelle~\cite{NatafandTournier2022}.

Les r\'{e}sultats sugg\`{e}rent que cette approche combin\'{e}e
(maillage, algorithmes num\'{e}riques, mod\'{e}lisation, mise en
\oe{}uvre informatique efficace) est d'une grande efficacit\'{e}
industrielle dans la conception a\'{e}ronautique et 
maritime~\cite{Visonneau2022}. Une place importante est accord\'{e}e
aux m\'{e}thodes num\'{e}riques qui ont accompagn\'{e} le
d\'{e}veloppement la MFN. Des approches classiques, souvent li\'{e}es
au maillage curviligne, sont associ\'{e}es \`{a} la m\'{e}thode des
\'{e}l\'{e}ments finis. Cette technique a b\'{e}n\'{e}fici\'{e} d'un
usage pratique par des ing\'{e}nieurs d\`{e}s les ann\'{e}es 40, avant
de profiter d'un formalisme m\'{e}thodologique et d'un cadre
math\'{e}matique rigoureux, fond\'{e} sur l'approche 
variationnelle~(Galerkin en 1915 et Trefftz en 1926), et m\^{e}me, sur les
th\'{e}ories d'auteurs encore plus anciens~(Leibnitz 1646--1716).

La M\'ethode des Volumes Finis (MVF), qui a vu le jour \`{a}
l'Imperial College sous la direction du professeur   
Spalding~\cite{RunchalandWolfshtein1966,Runchaletal1969,PatankarandSpalding1966}, 
est un formalisme alternatif permettant d'aborder les probl\`{e}mes
d'\'{e}coulement des fluides, bas\'{e} sur l'\'{e}quilibre des flux et
de la masse dans des volumes discrets. Il a b\'{e}n\'{e}fici\'{e} de
d\'{e}veloppements ult\'{e}rieurs et d'applications 
pratiques~\cite{Patankar1980,Hanjalicetal1980},  ainsi
que du formalisme d'un cadre math\'{e}matique rigoureux et
f\'{e}cond~\cite{Eymardetal1991}.  Dans ce num\'{e}ro, une contribution
concernant les d\'{e}veloppements r\'{e}cents de la m\'{e}thode est
\'{e}galement pr\'{e}sent\'{e}e~\cite{Eymardetal1991}.

Les ann\'{e}es 80 virent appara\^{i}tre les m\'{e}thodes dites de gaz
sur r\'{e}seaux (LBM), qui connaissent \`{a} pr\'{e}sent un
int\'{e}r\^{e}t croissant et pr\'{e}sentent des avantages de
repr\'{e}sentation pour les \'{e}coulements. Sur ce sujet, la s\'{e}rie
de conf\'{e}rences~\guillemotleft{}~Discrete Simulation of Fluid Dynamics (DSFD)~\guillemotright{}~a
commenc\'{e} en 1986 \`{a} Los Alamos \`{a} l'initiative de Gary D.
Doolen, et a connu 23$^{\mathrm{\`{e}me}}$ \'{e}dition \`{a} Paris
en 2014. Parmi les nombreux sujets trait\'{e}s dans cette s\'{e}rie de
conf\'{e}rences~: les sch\'{e}mas de Boltzmann sur r\'{e}seau,
particulaires dissipatives, l'hydrodynamique particulaire, les
m\'{e}thodes de Monte Carlo, etc. Les sch\'{e}mas de Boltzmann sur
r\'{e}seau s'appuient sur des outils th\'{e}oriques import\'{e}s de
l'approche cin\'{e}tique de Boltzmann, un th\`{e}me de pr\'{e}dilection
pour l'\'{e}cole math\'{e}matique fran\c{c}aise. Une excellente revue
des m\'{e}thodes LBM figure dans ce num\'{e}ro~\cite{Succi2022}.

Mentionnons que les apports d\'{e}riv\'{e}s de la pr\'{e}cision
sup\'{e}rieure de l'approximation num\'{e}rique (acquise avec les
M\'ethodes aux Diff\'erences Finies hermitiennes ou compactes du
4$^{\mathrm{\`{e}me}}$ et 6$^{\mathrm{\`{e}me}}$ ordre, les
m\'{e}thodes spectrales de type tau, ou la collocation en
d\'{e}composition sur la base des d\'{e}veloppements de Fourier, des
polyn\^{o}mes de Chebyshev et autres) ont d\'{e}but\'{e} leur essor en
particulier gr\^{a}ce aux supercalculateurs vectoriels (de type Cray),
m\^{e}me si leurs d\'{e}veloppements n'apparaissent pas directement
dans ce num\'{e}ro th\'{e}matique \textit{Comptes Rendus
M\'{e}canique}.

Enfin, une illustration originale sur trente-cinq ans
d'a\'{e}rodynamique et d'\'{e}tude du vol des insectes, d\'{e}crite
dans une contribution~\cite{Engelsetal2022}, tente d'\'{e}largir le
sujet \`{a} de futures nouvelles th\'{e}matiques. D\'{e}sormais, un
parall\`{e}le est possible entre la fluidit\'{e}~des \'{e}coulements et
la prise en compte de la MFN dans tous les ressorts de l'activit\'{e}
scientifique, souvent de mani\`{e}re implicite, profitant aux domaines
r\'{e}cents dans lequels l'automatique et le num\'{e}rique s'associent
et se manifestent dans les progr\`{e}s r\'{e}alis\'{e}s en
mat\'{e}riel, logiciels, grands volumes de donn\'{e}es, IA, IoT, et
r\'{e}alit\'{e}s virtuelle et augment\'{e}e. L'\'{e}tendu des outils
offerts \`{a} un ing\'{e}nieur lui permettra dor\'{e}navant
d'acc\'{e}der \`{a} un niveau de visualisation et de perception des
possibilit\'{e}s d'interaction et de manipulation des objets de
mani\`{e}re dynamique, ce qui compense l'exigence en abstraction en
donnant corps \`{a} cette derni\`{e}re.

Ainsi, ce t\'{e}moignage sur un demi-si\`{e}cle de couplage entre la
physique des fluides et l'univers num\'{e}rique souhaite \^{e}tre une
vraie passerelle entre l'essor extraordinaire qu'a connu la
compr\'{e}hension des fluides en \'{e}coulement et la promesse
qu'apporte l'intelligence artificielle pour la dynamique de demain.

\section*{Acknowledgments/Remerciements}

Thanks to the reviewers who worked on this issue: G. Accary, R.
Bennacer, H. Ben Hamed, O.~Bouloumou, PB, J.P. Bonnet, J.P.
Caltagirone, A. Dervieux, G. Destefano, F. Dubois, M. El Ganaoui, L.~El~Haj, 
J.P. Fontaine, T. Gallouet, M. Gander, S. Houat, F. Hubert, L. Halpern,
J.~Hristov, M.~Lappa, A. Mataoui, S. Meradji, D. Meiron, R. Moreau, D.
Morvan, B. S. Morsli, Morrone, M.C. Neel, J.~M.~Nunzi, G. Pezzella, O.
Pironneau, A. Saad, A. Runchal, R. Schiestel, S. Succi, J.L. Dufresne,
S.~Valcke, G. Krinner, B. Joseph, T. Truman Clark, K. Schneider, M.
Visonneau, C. Maliska, P.~Sparlat, F. Menter, M. Leschziner, B.
Launder, A. Filkov, E. Mueller, C.~Clements, Y.~Rogaume, A.~Ouahda.

To the editorial service and staff: J.-B. Leblond, A. Lopes, J. Fabre,
L. Pons.

This foreword is also an opportunity to recall that this issue was
partly concocted during the pandemic crisis. Very affected by the
support of colleagues and particularly Patrick Bontoux during the
period of hospitalization that I underwent. This project was a great
moral support to me (MEG). 

We have lost during this work very eminent colleagues from Marseille
and other places in the Fluid Mechanics field. Pierre Haldenwang (5
December 2021), Marcel Lesieur (22 March 2022), Abdelhak Ambari (26
April 2022) and Jean Pierre Guibergia (3 July 2023). 

May this topical issue of \textit{Comptes Rendus M\'ecanique} be a kind of
continuation of their dedication to fluid mechanics and the love they
communicated concerning this matter for their students.

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\bibliography{crmeca20240195}
\refinput{crmeca20240195-reference.tex}

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