International Journal of Heat and Mass Transfer 52 (2009) 4063–4073

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt

Brian Spalding: CFD and reality – A personal recollection
Akshai K. Runchal
Analytic & Computational Research, Inc.1931 Stradella Road, Los Angeles, CA 90077, USAa r t i c l e

i n f oArticle history:
Received 18 December 2008
Accepted 23 March 2009
Available online 4 May 2009
Keywords:
CFD
Spalding
CFD history
Navier-Stokes equations
Recirculating ﬂow
Imperial college
Combustion
Turbulencea b s t r a c t
Brian Spalding did not invent CFD. He did not even coin the name. But more than anyone else, he createdthe practice of CFD – its application to problems of interest to engineers. The author was associated with,and was an integral part of the team led by, Prof. Spalding that developed the basic engineering practicethat came to be known as the Imperial College (IC) approach to ‘‘CFD”. Most of today’s commerciallyavailable CFD software tools trace their origin to the work done by the IC group in the decade spanningthe mid-60s and mid-70s.
This paper is a personal recollection of the key moments of the CFD developments at Imperial College
and the role played by Brian Spalding as a leader of, and as an active contributor to, the IC Group. His key
insights during this decade often made breakthroughs possible and re-directed the focus at critical
moments. The paper also explores the opportunities missed by the IC Group during this decade of breakneck progress in CFD.
Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction
I ﬁrst met D. Brian Spalding (popularly known as DBS) in 1965.
If you search the web for DBS today, other than the Aston Martin
DBS V12, one of the items that prominently pops up is Deep Brain
Stimulation – an innovative development in Neurology. How
appropriate! Of course I did not know that when I ﬁrst met him.
My association with him has certainly been Brain stimulatingand has truly changed the course of my life and the point of view
with which I view science and engineering.
D. Brian Spalding (Fig. 1) was born on 9th January 1923 in New
Malden in the now picturesque suburbia of London. The town ﬁnds
mention in the Domesday Book as Meldone and seems then to havebeen a typical rural English village. The fact that Brian grew up close
to a pigsty in a semi-rural environment will later lead to a remarkable professional observation. Spalding attended Kings College
School from the age of 9–18 and was then admitted to Oxford University where he obtained his B.A. in Engineering Science at the
Queens College in 1944. He then worked at Shell for a year. In
1945 he joined the newly established Rocket Propulsion Establishment (RPE) of the Ministry of Aircraft Production. Great Britain did
not yet have any rockets at that time; RPE was set up to develop
the technology in response to the success of the German V2 Missile.
Soon thereafter Brian was dispatched to Germany to learn the secrets and intricacies of rocket engines. During 1945–1946 he wasat the Luftfahrtforschungsanstalt Herman Goering (Herman Goering
Institute of Aeronautical Research) at Voelkenrode, near Braunschweig and its out-station at Trauen on the Lueneberge Heide.
E-mail addresses: runchal@ACRiCFD.com, runchal@gmail.com
URL: http://www.ACRiCFD.com
0017-9310/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijheatmasstransfer.2009.03.058

The V2 team, led by Werner von Braun, was already in the American
zone; but the British collected 10 members of a different groupwhich had developed the motor for the Messerchmidt 163 rocketpropelled airplane, the propellants of which were hydrazine hydrate
and hydrogen peroxide. They brought this team to Trauen and set
them to work converting their rocket motor to burn kerosine and liquid oxygen. The work continued until 1946, at which time the Allies
agreed that no further such work was to be done in Germany. The
Trauen team was then transported to England to continue its workat RPE, which was little more than a collection of huts on a disused
airﬁeld. Brian was the mentor of the German team led by Johann
Schmidt. Indeed he lived with the German team in one of the huts
until his marriage in 1947 to Eda Goericke, who, having formerly
worked at a hydrogen-peroxide-making plant in the Harz Mountains, had moved to Voelkenrode when the war ended.
Somewhat later, the reconstruction of the UK Scientiﬁc Civil
Service resulted in Brian’s being transferred, much to his disappointment, to the Metrology Department of the National PhysicalLaboratory (NPL). An unanswered question remains if this transfer
had anything to do with Brian’s membership in the communist
party during his student days. This was the beginning of the cold
war and the Burgess and McLean affair hit the news shortly thereafter in 1951. From the point of view of Brian’s career, this proved
to be a blessing in disguise because during this time Brian became
thoroughly familiar with instrumentation and the art and scienceof measurements. This stood him in good stead during the nextstage of his career. It also resulted in his not standing, as he otherwise would have done, by the side of Johann Schmidt when an
explosion of the kerosine-fuelled rocket motor, strong enough tobreak apart the bolts holding the window through which he waswatching, exploded and killed him instantly.

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A.K. Runchal / International Journal of Heat and Mass Transfer 52 (2009) 4063–4073

Fig. 1. Brian Spalding.

In 1948 Brian got an ICI (Imperial Chemical Industries, Ltd.) Fellowship to go to Cambridge University (Pembroke College) for a
Ph.D. With his RPE background, he knew that he wanted to do research on the combustion of liquid fuels. The Head of Department,
John Baker, appointed A.L.L. Bird as his supervisor since he hadsome interest in diesel engines. Bird and Spalding had very little
to do with one another. Brian early on demonstrated his independence and his tendency to go out on a limb. He had a fairly deﬁnitive idea of what he wanted to do – and it was not diesel engines.
When Bird tried to get Brian to use an existing experimental rig,
Brian protested to Baker. Baker advised Bird to ‘‘give him enoughrope to hang himself”. Thereafter Brian was on his own. Bird retired soon thereafter and since the regulations demanded thatevery Ph.D. must have a supervisor, a new recruit to the staff, Dudley Robinson was appointed his supervisor. Having worked at RPE,
Brian already knew more combustion than his supervisor andwound up advising Robinson on his research. Brian’s previous
experience at RPE and NPL came very handy for his Ph.D. researchwhich was remarkable in that it was entirely self-motivated. Brianobtained his Ph.D. in 1952 with Will Hawthorne and E.S. Sellers as
his examiners.
2. Early professional career: 1951–1964
The origin of Spalding’s later contribution to CFD goes back to
his days at Cambridge University and his Ph.D. Thesis [1]. It is aremarkable thesis in that it ‘‘uniﬁed” the key hydrodynamic concepts of von Karman [2], the heat transfer concepts of Kruzhilin[3] and the mass transfer concepts of Eckert [4]. He synthesizedthese to create a general theory of heat and mass transfer withand without combustion. In the process he made a then unforeseen
prediction that the chemical-reaction-rate constants had no inﬂuence on combustion until a critical rate of mass transfer was
reached. This was later borne out by experiments. Spalding deduced these critical rates by adapting the concepts of Zeldovichand Frank-Kamenetsky [5], and Semenov [6], who had been concerned with the quite-different phenomenon of steady laminar
ﬂame propagation. This led to a general theoretical framework
for the prediction of ﬂame-extinction which was a breakthroughfor combustion engineers [7]. His other notable contributions in
combustion include the ’centroid rule’ [8] which caused the predictions of a range of ﬂame-speed studies to fall on to a single curve,
the cooled-liquid-ﬁlm burner for measuring combustion rates andan innovative method for measuring extinction conditions [1], anda cooled porous burner for measuring ﬂame speeds [9]. He also
developed an electrical analogue of combustion [10]. To my knowledge this was a novel and unique concept and I am not aware ofother electrical analogues of combustion.

After completing his Ph.D. Spalding stayed at Cambridge for ashort time and was then recruited by Prof. Owen Saunders in
1954 to join as Reader in Applied Heat, in the Mechanical Engineering Department at the Imperial College, London. Spalding went on
to do seminal work in combustion and made key and innovative
contributions in evaporation burning of droplets. This work eventually led to the now universally adopted ‘‘B” factor and the Spalding Number. Spalding’s efforts at uniﬁcation led to his
remarkable book on Convective Heat and Mass Transfer [11] thathas greatly inﬂuenced subsequent work in this ﬁeld.
In late 1950 s Spalding turned his attention to the important
issue of the role that wall shear plays in most engineering ﬂows.
The turbulent velocity proﬁle for walls was conventionally represented by a three part proﬁle, a ‘‘viscous” sub layer, a ‘‘transitional” layer and a ‘‘fully turbulent” layer. Spalding found an
unconventional, elegant, and simple solution: express Y+ in termsof U+ rather than U+ as a function of Y+. This key insight enabled
him to develop a continuous ’wall law’, covering viscous, transitional and logarithmic regions [12]. He was also not comfortablewith the conventional method of treating wall boundary layers,
jets and wakes as distinct ﬂows – each with its own physics,
mathematics and terminology. Since all these ﬂows are primarily
governed by shear, he argued that the underlying physics andmathematics must be represented in a uniform manner. This led
to his Uniﬁed Theory of Turbulent Boundary Layers, Jets andWakes [13]. This was his ‘‘grand” design built upon the insights
of Taylor [14] to have a single theory that covered Boundary Layers, Wakes and Jets. This was based on the remarkable insight thatwith a ‘‘universal” entrainment law and a suitable two-part proﬁleto represent the wall and wake regions, all such ﬂows can be universally represented. A number of his students worked on deriving the entrainment formulae and other input needed for theUniﬁed Theory (e.g. [15–17]). Soon thereafter, Spalding came tothe conclusion that instead of searching for an optimum proﬁle,one can ‘‘universalize” a proﬁle by simply representing it as aset of piece-wise polynomial – or linear – segments and derive
the weighting functions from the governing initial and boundary
conditions. This freed one from having to ﬁnd an ‘‘ideal” proﬁleto ﬁt a given ﬂow. However it later became apparent that Spalding’s search for a uniﬁed theory was not yet over since this approach was found to generate solutions that were occasionallyspurious or even singular.
3. Convergence of our paths: 1965–1975
In 1965 Spalding occupied the Chair, Professor of Heat Transfer,
at Imperial College. He was appointed to this chair in 1958 when it
was created. He also headed the ‘‘Thermoﬂuids” Section which waslater renamed Computational Fluid Dynamics Unit. Though digitalcomputers had been around for a couple of decades, early 1960’s
coincided with the ‘‘advent” of the computer as a widely available
tool and led to the developments that eventually gave rise to what
is today known as CFD.
I graduated with a B.Sc in Engineering in 1964 (Fig. 2) and in1965 won an ICI scholarship in India that gave me the choice to
go to any college in the UK for my Ph.D. I wanted to work on dryingof sprays – a subject of much interest to ICI and other companies –that involved both heat and mass transfer. Since Spalding was oneof the most respected researchers in heat and mass transfer, I
wrote to him to accept me as his Ph.D. Student. The essence of
his reply was: ‘‘I am not interested in working on drying of sprays,
but I am happy with last year’s ICI scholar – Suhas Patankar (Fig. 3)– so I will accept you as my student and we will ﬁgure out what to
do once you get here”. I suspect another reason may have been his
soft corner for an ICI scholarship since he himself had completed
his Ph.D. under an ICI Fellowship.

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Fig. 4. Micha Wolfshtien.
Fig. 2. Akshai Runchal.

Fig. 3. Suhas Patankar.

Brian was then busy with his ‘‘Uniﬁed Theory”. At that time his
approach was to use piece-wise proﬁle methods that could approximate – to a given accuracy – any proﬁle that might describe theﬂow. His previous students had already determined optimal
entrainment functions, log-law constants and heat and mass transfer resistance required to describe a wide range of ﬂows. Patankar
had had a fair amount of success on the theoretical side in building
a general purpose ‘‘integral-proﬁle” computer code based on piecewise linear segments.
Spalding was at that time conﬁdent that most ﬂows of engineering interest can be represented by his Uniﬁed Theory andpiece-wise proﬁles. So far he had not been able to extend his Uniﬁed Theory to separated ﬂows or ﬂows with strong pressure gradients. Shear plays a key role in separated ﬂows, including thosewhere the boundary layer is destroyed by, say, an adverse pressure
gradient, or a geometry that induces separation. Brian therefore
asked me to extend his Uniﬁed Theory to such ﬂows. Flow behinda Backward Facing Step (BFS) in a channel and that in a Driven-Lid
Square Cavity (DSC) were to be the focus of my attention. Micha
Wolfshtein (Fig. 4) had already joined the group in October 1964
and Brian had asked him to tackle the problem of the Impinging
Jet on a Flat Plate – a ﬂow with strong favorable pressure gradients.
These extensions would have ﬁrmly established the Uniﬁed Theory
not only for ‘‘parabolic” ﬂows such as the boundary layers but alsofor ‘‘elliptic” ﬂows with strong pressure gradients, recirculationand impingement.

A personal example from this time illustrates Spalding’s practical and single-minded unconventional approach. When I asked
him what classes I should enroll in he told me there was no needto take any classes; he wanted me to concentrate on research
and asked me to start with a thorough literature study of analytic
and approximate methods for boundary layers, jets and wakes.
This is remarkable in the context of a student who had freshly
completed only his undergraduate studies from a university in another country and was enrolled in a Ph.D. program without a Master’s degree or any other professional experience. I wonder how
many of today’s professors would wish that they had that freedom
with their graduate students!
After a thorough review of the published literature I started
using the piece-wise proﬁle method to solve the BFS problem toextend the Uniﬁed Theory to elliptical form of Navier–Stokes equations. It is notable that the dominant thinking in the 1960’s wasthat there was no possibility of solving the Navier–Stokes equations with turbulence except for highly simpliﬁed one or twodimensional ﬂows. So for Spalding to even attempt to extend the
Uniﬁed Theory to separated ﬂows is remarkable in itself. During
the course of this work, I began to come across papers that used ﬁnite-difference methods which bore a similarity to the Uniﬁed Theory in terms of piece-wise proﬁles but had a distinctly different
ﬂavor in terms of implementation. ﬁnite-difference methods forNavier–Stokes equations had been around for a long time: Thom
[18] had used them well before the advent of the electronic
computers.
After about 4 months, I had failed to obtain an acceptable solution to the BFS problem. I started to have my doubts and asked for ameeting with Brian (Fig. 5) to brief him on my progress with theliterature review and my failure to make any headway with theUniﬁed Theory for separated ﬂows. I also mentioned the papers

Fig. 5. Brian Spalding 1977.

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A.K. Runchal / International Journal of Heat and Mass Transfer 52 (2009) 4063–4073

that had used ﬁnite-difference methods that reported success withseparated ﬂows at low-Reynolds numbers. Brian was quick to arrive at the key conclusion that there was no easy way in the Uniﬁed
Theory to represent the key role that axial diffusion played in separating and reversing the boundary layer. The proﬁle method
therefore needed to be modiﬁed. It quickly became apparent that
it will be simpler and more general to use the ﬁnite-difference(FD) method rather than modify the Uniﬁed Theory. I started working on writing a ﬁnite-difference computer program and was able
to report success in solving the BFS and the DSC problems.
At one of our subsequent meetings, Brian mentioned that
Wolfshtein was reporting success along similar lines and asked
us to get together. We soon realized that we were essentially using
the same approach on different problems and that we had approached them from different view points – one from a high-Reynolds number and the other from a low-Reynolds number
viewpoint. When we (Fig. 6) compared our notes, we realized thatthe main impact of the Reynolds number was on the nature of thematrix coefﬁcients. With the central difference scheme that wewere using, for any given grid, the matrix coefﬁcients changed
from positive-deﬁnite to non positive-deﬁnite at some critical Reynolds number and numerical instability ensued. Spalding thenmade the now famous analogy of how wind from the pigsty always
stinks – or the wind from the north always brings cold. He knew
this because he had grown up close to a pigsty. This led Spalding
to propose the ‘‘upwind” concept. Brian also made an important
physical analogy of likening the upwind and FD method to a seriesof tanks (control volumes) and tubes (grid). With these twochanges we were soon ‘‘free” of the Reynolds number constraint
and the tank-and-tube analogy changed our approach to thinking
in terms of ﬂuxes rather than the state variables. This led to the formalization of the ‘‘Finite Volume” concept with its emphasis on
volumes and ﬂuxes rather than on the mathematical descriptionin terms of distributions (basis-functions) and gradients. Once formalized, this eventually frees one from the limitations of the Taylor’s Series and equating ‘‘order” with ‘accuracy”. With thesechanges we started assembling a general-purpose Navier–Stokes
computer program to solve a diverse range of problems. Like manyother codes of that era, this was based on the stream-function andvorticity (w–x) variables.
One can see the beginnings of the FD in what Brian was doingwith his uniﬁed method. Instead of using piece-wise polynomials
to construct a local value to convert the differential equations directly to algebraic equations (as in FD), he was using piece-wisepolynomial to represent a set of values (proﬁle) and then integrat-

Fig. 6. Micha Wolfshtein, Akshai Runchal and Irit Wolfshtein.

ing the differential equations to obtain the algebraic equations
that will give the values of the constants of the proﬁle. But inhis characteristic fashion he used his insight to invent a ‘‘physical”
rather than a ‘‘mathematical” approach to the problem. It waseasy for him to see that the focus of interest should not be ‘‘variables” but their ‘‘ﬂuxes”. With his engineering background andextensive work on the usefulness of the ‘‘control volumes”, hequickly came to view each ‘‘node” of a ﬁnite-difference grid as
an independent ‘‘tank” which exchanges ‘‘ﬂuxes” with other tanksby ‘‘tubes”. Brian’s re-invention of the upwind scheme similarly
had a ‘‘physical” insight into the mathematical approach. Oncethe focus is ﬂuxes, upwinding is straightforward: ﬂuxes comefrom somewhere; they have a distinct speed and ﬂow in a certain
direction.
Soon thereafter we came across a paper by Barakat and Clark[19] and we could see that the upwind approach had already been‘‘discovered”. Little did we know that upwind and one-sided differences had been around far longer. A paper by Courant et al. [20]had used upwind concept more than a decade earlier and mathematicians had extensively explored the properties of one-sided
and central difference methods for far longer. However in thosedays the interaction between mathematicians and engineers was
somewhat limited. Also Burggraf [21] published his now classic paper on square cavity where he reported success at low-Re numbers
but failure to obtain solutions beyond Re = 400. We thought weshould publish our work before we were trumped up by anotherclaim. In our new-found enthusiasm, we were blissfully ignorantof the pitfalls of upwind. This led to our ﬁrst papers on ﬁnite-difference methods with the IC approach [22,23]. The second paper isalso a good example of why not to publish a paper in a hurry since
it contains results for Re = 1000 for driven square cavity whichwere proven to be wrong due to high numerical diffusion.
Later on we started becoming wise to the pitfalls of upstream
differences and this led to some work on numerical diffusion.
Wolfshtein [24] published a technical note where he showed thatnumerical diffusion is related both to the speed of the ﬂow andthe angle of the stream-lines to the grid. Spalding [25] proposed
an exponential method and in a companion paper Runchal [26]proposed the ‘‘hybrid” method [26] that automatically blended
the central and upwind difference methods based on the local Peclet number. The hybrid method eventually became the establishedpractice for all Imperial College codes.
Around 1967, I became fascinated with the Gauss Theorem andthe integral approach to derive the algebraic analogue for the Navier–Stokes equations. This had to do with the classic ‘‘ﬁght” between the ‘‘differential” approach of Newton and the ‘‘integral”
approach of Leibnitz. One advantage of the integral approach isthat it does not require smoothness or the existence of a 2nd orderderivative. This is an important consideration for ﬂows with sharp
discontinuities such as shocks and contact surfaces. Wolfshtein
and I had many discussions over the competing approaches andhe correctly pointed out that the same set of algebraic equations
can be derived from either. He eventually went on to write his thesis [27] in terms of Taylor Series whereas I submitted mine with
the integral approach [28]. Most Finite Volume codes today use
the integral approach due to the simplicity that results from reducing the volume integral to a surface integral.
By mid 1968 both Wolfshtein and I had completed our thesis
work. I took a hiatus and went to spend a long and productive summer in Cambridge (Massachusetts) to consult with Northern Research on the application of CFD to aircraft compressors. I cameback from Cambridge in September 1968 and Brian sprang a surprise on me. He informed me that London University did not granta Ph.D. in Engineering solely on the basis of theoretical work! I
took over the experimental rig of David Gosman (Fig. 7) who had
just ﬁnished and modiﬁed it to measure ﬂow behind a BFS at very

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4067

with Fluent software in late 1970’s which eventually became Fluent Corporation in 1983. Creare had acquired the software fromProf. Jim Swithenbank of Shefﬁeld who in turn had acquired it from
Imperial College.
4. The CFD decade at the imperial college: 1965–1975

Fig. 7. David Gosman 1974.

high Schmidt numbers. Though I did not like having to do experiments at that time, I wish today that that rule was in force universally. It taught me the respect for experimental data, its inherent
uncertainties and the vagaries of the instrumentation. It alsotaught me that the real world of ﬂuids is inherently unpredictable,
never two-dimensional and never steady. I completed my experimental project in late 1968 and submitted my thesis [28] to London University.
By the end of 1968, Brian had realized the potential of thedevelopments that had taken place. He decided to organize a
Post-Experience Course at Imperial College in 1969 targeted atboth academic and industrial communities. Academic Press became interested in publishing the work done by our group. Both
Wolfshtein and I were leaving Imperial College and Spalding asked
David Gosman who had just completed his thesis on experimental
work to edit the book, and Sam Pun – another of his recent Ph.Ds –to take over the computer code from me and Wolfshtein. Thatcode, called ANSWER, made it to the ﬁrst book on CFD [29] that
dealt with Elliptical form of Navier–Stokes equations. The authorship of this book illustrates the strict rule that Spalding followedthat all joint publications carry the names in alphabetic order.
Though Spalding was the prime contributor, his name appears near
the end. It was also in 1969 that Spalding incorporated the ﬁrst
CFD based consulting company – CHAM (Combustion Heat andMass Transfer) Ltd.
If a date for the ‘‘birth” of CFD is to be picked, then 1969 was theyear that ushered in the CFD as an engineering tool. The work doneat Imperial College on Navier–Stokes equations, and the computercodes to solve these equations together with generalized transport
equations for any two-dimensional ﬂow, became widely available
in 1969 through the publication of the book [29]. The Post-Experience Course at IC in 1969 reached a large number of researchers in
the UK and later abroad through a series of courses and seminars at
various universities in the US and Europe. At the same time commercial services in CFD became available through CHAM1 in
1969. It should also be noted that the ﬁrst conference with CFD at
its theme was held at Monterey in 1968.
In late 1969, I accepted a teaching position at IIT Kanpur in India. I brieﬂy returned to Imperial College in the summer of 1970
and then came back to join as Senior Research Fellow at Imperial
College in 1972 and worked as Technical Director of CHAM till
1974. CHAM was the only consulting company providing commercial services in CFD and it stayed that way till Creare, Inc. started

1
Combustion, Heat And Mass Transfer, Ltd.; later superseded by Concentration,
Heat And Mass Transfer, Ltd.

The decade between 1965 and 1975 was a very fertile period for
CFD. These were the heydays of ‘‘CFD” at Imperial College. In retrospect it was a unique and amazingly productive period. The group
under Spalding included two young and dynamic faculty members:
Jim Whitelaw and Brian Launder (Fig. 8). This group of three, working symbiotically, transformed the theory and practice of ﬂuid
mechanics. Spalding working with his students and associates
transformed the emerging ﬁeld of computational ﬂuid dynamics
from an esoteric and mathematical branch of science to a fully
developed tool for practicing engineers. Whitelaw and his students
turned the emerging Laser Doppler Anemometry into a proven andpreferred experimental method for measuring ﬂows. Whitelaw
worked on the experimental side and many of his students used
the CFD to verify their experimental results. Brian Launder andhis students were active in the ﬁeld of Turbulence. They went on
to make signiﬁcant contributions in the theory and experimentsof turbulent ﬂows. All three sub-groups used CFD and experiments
in a highly synergistic manner to advance the theoretical andexperimental knowledge base of Fluid Dynamics.
By 1969 the CFD group consisted of more than 30 researchers
and there were weekly seminars mostly given by a member ofthe group. To my knowledge it was the largest CFD group in theworld at that time. I wonder if even today there is a larger groupof researchers focused on CFD under the guidance of a single person. Though signiﬁcant CFD work was going on at various locations
around the world, the only other large group at that time was the
T3 at Los Alamos National Laboratory (LANL) under Frank Harlow(Fig. 9). The developments that occurred at Imperial College – after
the emphasis shifted to three-dimensional ﬂow and primitive variables in 1970 – owe heavily to the work done by Harlow. Harlow
had experimented with a number of alternatives for solving theNavier–Stokes equations in primitive form. One of these was apressure projection algorithm with a staggered grid and a pressure
correction based on the continuity equation. However the pressure
correction step required solution of the equations in transientmode. It was this development that provided the basis for the ICgroup to devise the SIMPLE algorithm that could be also be applied
to steady ﬂows.
Harlow’s group worked on a wide variety of problems in ﬂuid
dynamics. His focus was mostly on transient ﬂows with steadyFig. 8. Jim Whitelaw and Brian Launder.

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A.K. Runchal / International Journal of Heat and Mass Transfer 52 (2009) 4063–4073

Fig. 9. Frank Harlow.

state as an asymptotic state of the ﬂow. Many applications involved compressible ﬂow or free surface. They often involved moving boundaries and multiple phases. Primary focus of the T3 group
was on the physics and science of problems related to the weapons
program at LANL. The primary focus of Spalding’s group was onengineering ﬂows of interest to the industry. Most of these ﬂows
could be treated as steady and incompressible – at least to a ﬁrst
approximation. Moving boundaries were not of much interest to
the IC group. Multiple phases were approached by the IC groupin an ad-hoc manner or as equivalent single-phase with approximations such as a void-fraction. Compressible and transient ﬂows
were treated as ‘‘extensions” of the steady, incompressible ﬂow.
These philosophical and practical differences had a profound effecton the ‘‘world-view” of the two groups and their approach to CFD.
Harlow’s approach was by far the more rigorous and often stayed
closer to the physics of the problem. Los Alamos at that time hadsome of the most sophisticated computational resources in theworld. The computer resources generally affordable by the industry were signiﬁcantly limited. It was computationally expensive
to use the Los Alamos methods developed for transient ﬂow tocompute the steady state ﬂows. The computer programs developedat Los Alamos were available as listings in technical reports or on
personal request. Harlow made little attempt to distribute themto outside researchers. He was more interested in innovative research than in disseminating his technology or spending his timeteaching others how to use it. The excellent and path-breaking
work done at Los Alamos was not widely used outside a select research community. It was not till Tony Hirt became leader of the T3
Group, around 1973, that computer programs developed at Los
Alamos became generally available to outside researchers though
the US Department of Energy distribution sites.
With Spalding’s focus on engineering application, he looked foralternatives and tools that will allow his methods to work efﬁciently with limited computer resources. Computational economy
was a major concern and a driving force. He often made boldassumptions and used his keen insight to separate the essentialfrom the inconvenient. The technology developed by his group
was made widely available through personal contacts, a post-experience course, distribution of the computer programs, and publication of books. It is important to note that Spalding has always
emphasized that a poor solution is better than no solution. It iscountered by some that no solution is better since it will not lull
one to the dangers inherent in a poor solution. However Brian
has shown that with insight, some caution, and testing against
empirical data, one can obtain useful engineering information froman approximate solution even though one is aware of the shortcomings inherent in it.
Around 1970 Brian became convinced that the w–x approach
had no distinct advantages for 3D ﬂows. He was quick to abandon

it and turned to primitive variable form of the Navier–Stokes equations. Characteristically again, he used the available tools and technology to create something entirely new and useful for engineers.
Just as he ‘‘uniﬁed” the boundary later, heat and mass transfer concepts for his Ph.D. work, he set about to change CFD by combining
existing concepts with key insight and bold assumptions. By thistime Harlow [30] had introduced a staggered grid and a decoupledpressure based on continuity equation for transient computations,and Chorin [31] had pointed out that any scalar can be used in lieu
of pressure. The key advancement of splitting the pressure contribution into two stages was obvious since Patankar and Spalding[32,33] had already used this approach successfully for parabolic
ﬂows where the axial pressure can be decoupled from the component that governs the cross-axial velocities. Patankar and Spalding[34] combined these insights and arrived at the SIMPLE algorithm
that revolutionized the CFD practice. The depth of that insight andachievement can be gauged by the simple fact that many of thesuccessful commercial CFD codes even today employ SIMPLE or
its variations at least as one of the available options.
Brian knew that from the point of view of practicing engineers,
CFD will not be a really useful tool unless it dealt with the intractable problems of turbulence and chemical reactions. With his
deep background in the physics and theory of ﬂows (and his ﬂuency in German and Russian) he built upon the work of Kolmogorov [35], Prandtl [36], Chou [37] and Rotta [38]. However theequations derived by these researchers were so complex, and solittle information was available about the attendant constants thatno attempt had been made to solve these equations of turbulence.
Spalding was one of the ﬁrst to realize that, with the availability ofthe digital computers, the set of equations developed by them
could form the basis of practical predictive tools if one could derive
the constants needed to quantize these equations. He turned togetting these constants from experimental data with bold assumptions about the ‘‘universality” (or better – usefulness) of these constants. This led to breakthroughs such as one of the ﬁrst k, k–l andk–x methods and the eddy break-up method for turbulence-kinetics interaction. About the same time Harlow and coworkers [39]independently had come to the same conclusion and publishedtheir ﬁrst paper on a 2-equaion (k–e) model of turbulence. Working
with Launder and others, Spalding adapted the k–e method as apreferable tool primarily due to the computational advantages related to the fact that the so-called diffusion coefﬁcient was more
likely to be a constant for the ‘‘e” than for the ‘‘l” or ‘‘x” formulations that were being investigated. Another reason was the easeof interpretation of ‘‘e” compared to other variables; it was simply
related to the energy dissipation near the wall. It was during thisperiod that turbulence modeling became established as a practical
tool. Spalding worked extensively on turbulent ﬂows for a while
but then moved away to concentrate on other ﬁelds of more imme-

Fig. 10. Frank Schmidt.

A.K. Runchal / International Journal of Heat and Mass Transfer 52 (2009) 4063–4073

Fig. 11. William Kays.

diate interest to him. In fact later on he started espousing MultiFluid rather than the 2-equaion models for turbulence.
During this decade, the Imperial College group had distinguished visitors. These included J.C. Rotta (Goettingen), Frank
Schmidt (Penn State; Fig. 10), William Kays (Stanford; Fig. 11),
Chang-Lin Tien (Berkeley), Joseph Kestin (Brown), P.D. Richardson(Brown), Bill Reynolds (Stanford; Fig. 12), Philip Saffman (Caltech),Peter Bradshaw (NPL), David Pratt (Washington), Larry Caretto(Berkeley; Fig. 13), Graham de Vahl Davis (New South Wales;
Fig. 14), Tony Hirt (Los Alamos; Fig. 15), Harry McDonald (United
Aircraft Res. Lab.), David Dyers (Alabama), and many others. These
visits and interactions were very valuable in dissemination of the
IC CFD technology and its wide acceptance. Following a conversation with Jim Whitelaw at the 1970 International Heat Transfer
Conference in Paris, Frank Schmidt organized a series of short

4069

Fig. 14. Graham de Vahl Davis.

Fig. 15. Tony Hirt.

courses at Penn State that were delivered by IC faculty and research staff. The courses started in 1972 and continued till 1994
and covered a number of subjects including computational methods for boundary layers, re-circulating ﬂows, combustion and turbulence. These courses were along the line of the 1969 PostExperience Course at Imperial College and were directed at bothacademic and industrial communities. Frank Schmidt, Jim Whitelaw and Brian Launder also arranged a series of very successful
conferences on Turbulent Shear Flows starting in 1977 that in a
modiﬁed form survive to this day. Starting around 1970, Steve
Cline with support from Bill Reynolds, William Kays and others
was instrumental in arranging a number of ‘‘Olympiads” wherecompeting researchers presented the results from their computational methods for boundary layer ﬂows and turbulence. The methods were then formally ‘‘judged” in terms of agreement of theFig. 12. Bill Reynolds.

Fig. 13. Larry Caretto.

Fig. 16. John Argyris.

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A.K. Runchal / International Journal of Heat and Mass Transfer 52 (2009) 4063–4073

Fig. 17. Olgierd Zienkiewicz.

Fig. 18. George Raithby.

predictions with experimental data. These activities led to the IC
CFD technology being widely known around the world.
The IC group also interacted with Argyris (Fig. 16) and Zienkiewicz (Fig. 17) about the merits of different approaches to
CFD and their espousal of the ﬁnite-element (FE) method. Though
IC group acknowledged some advantages of FE method, the overwhelming feeling was that the method was unsuitable for highReynolds number ﬂows and lacked clear theoretical basis (sincethe Hamiltonian does not exist for non-linear systems). This, atleast to some extent, has been vindicated by subsequent developments since most of the current commercial technology has
adopted FV approach. It must be stated, of course, that over theyears, both approaches have borrowed ideas from each other. Most
visibly, the FE has moved away from minimizing a Hamiltonianand has implemented upwind methodology whereas the FV has
adopted the unstructured and boundary-ﬁtted grids. With today’s
technology, the differences are more of semantics than of substance. It can be shown that FD, FE and FV, with appropriate
assumptions, can all lead to identical algebraic equations. However
profound conceptual difference remain; FD and FE both use amathematical approach based on differential form of equations
whereas FV uses a physical approach based on integral form ofequations that stresses ﬂuxes of variables.

and Chow [44] perfected the co-located grid which is today a preferred option for unstructured grids and offers distinct advantage
for complex geometries. We also failed to fully explore the impactof numerical diffusion and truncation. Wolfshtein [24] did sometentative work on the subject but it was left to Hirt [45] to produce
a formal and heuristic method to deﬁne these effects.
Another important example of a missed opportunity was the‘‘vorticity-ﬂuctuation” (k–x) method for turbulent ﬂows. Vorticity
ﬂuctuation seems a more intuitive and elegant representation of acascade of turbulence eddies than dissipation. It is easier to envisage ‘‘vorticity” rather than ‘‘dissipation” being transported and diffused. After early and extensive exploration of it, the focus shifted
and stayed with the k–e model because diffusion coefﬁcient for thedissipation equation was easier to deﬁne. It was Saffman [46] andWilcox [47] who eventually went on to establish it as a viable andpreferable tool for certain class of ﬂows. Of course, as is clear fromthe work of Rotta [38], it can be shown that all two equations models are identical in that the same differential equation governs all
such models except that they differ in the source terms and may
have distinct numerical properties.

5. The missed opportunities

6. The post: 1975 period

The single-minded and focused approach followed by Spalding
had some drawbacks. Spalding has a distinctive trait that once he isconvinced of the usefulness of an approach, he is able to completely focus on the path that lies ahead and completely ignore
any idea that might sidetrack him from that path. This is a common
trait of genius and of high achievers. However the drawback of thisapproach is that sometimes it is the off-the-path ideas that lead togreener pastures. Of course one may also waste a lot of effort inlooking for greener pastures.
During this period Spalding’s group explored and discarded
many ideas that in hindsight would have proved fruitful. Late in1960s we experimented with and abandoned what later became
Vector-Differencing because it did not conserve ‘‘extrema”. Since‘‘hybrid” scheme was working reasonably well no attempt wasmade to ﬁnd a ‘‘limiter” for this scheme. Raithby [40] (Fig. 18)
found a way to make Vector Differencing a practical option. The
single-mindedness of the group was also responsible for the premature abandonment of the coupled primitive variables SIVA algorithm [41] because of the focus on the segregated SIMPLE
algorithm. It was subsequently shown by Vanka [42] and others
to be a superior method for a class of strongly coupled ﬂows. Another example is the early abandonment of the co-located grid,
Runchal [43] because it was felt that co-located grids offered no
distinctive advantage over staggered grids. Subsequently Rhie

Spalding stayed at Imperial College till 1988 when he retired to
devote his full attention to the development of the PHOENICS codeand the consulting activities of CHAM. PHOENICS debuted in 1978
and was the ﬁrst commercially available software tool in CFD. At
this time the only other widely available CFD tool was the TEACH
code from Imperial College which was severely limited in its scopeand capabilities. PHOENICS provided a general framework for solving any problem within its scope and allowed users to extend thecapabilities of the code through a formal framework that was included in its design.
Spalding subsequently went on to invent highly useful and simple tools and techniques such as the IPSA Inter-Phase Slip Algorithm, IPSA, [48] for predicting multi-phase ﬂuid ﬂow, a simple
algorithm to determine the wall distance (needed for many turbulence and radiation computations) for complex geometries [49], amulti-ﬂuid approach to turbulence [50], a multi-ﬂuid approachfor turbulent combustion [51], a novel approach for integrated
radiation computations [52], and a methodology to unify ﬂuid
and solid mechanics [53]. He integrated all these diverse phenomena and brought them within the scope of CFD. All these are examples of Spalding’s ‘‘physical” approach to develop practical tools.
These produce approximate and plausible results even when one
knows that the actual physics is more complex but also more
intractable.

A.K. Runchal / International Journal of Heat and Mass Transfer 52 (2009) 4063–4073

Uniﬁcation of ﬂow, heat and mass transfer has stayed the primary goal of Spalding from his Cambridge days in early 1950s to
the present. He has now added structural dynamics, turbulence
and multiple phases to his plate. Spalding’s approach to researchis always intuitive and physical as distinct from theoretical ormathematical. He strongly believes in the ‘‘art of the possible”
and is not encumbered by theoretical limitations that may exist.
The question that Spalding always asks is: ‘‘Can this be useful toa practicing engineer?”
7. Synopsis
Brian’s impact on engineering practice is incredibly broad andtruly amazing. He has worked in diverse ﬁelds of engineeringand science over his career. Most people today know him for his
CFD work. Yet the fact is that before he started with CFD in
1964, he had already made his mark in combustion [7], heat andmass transfer [11] and boundary layer theory [14]. He then madeseminal and enduring contributions in CFD, turbulence and multi-phase ﬂows. He was the ﬁrst to formally propose that, in thecontext of numerical solution, all 2nd order transport equations
can be expressed and solved as a single generalized transport
equation. Thus to some extent, he has achieved his life-long goalof ‘‘unifying” the treatment of ﬂuid ﬂow, heat and mass transport,
and mechanical stresses by a single ‘‘universal” method in a ‘‘uniﬁed” manner. His post-1980 work on turbulence, multi-phase, solid-ﬂuid interaction and wall distance computation has not yet
seen the same popular adoption as these but he has pointed toimportant path-breaking research that may bear fruit in the future.
Throughout Spalding’s career a recurring theme – and prime
objective – has been to invent predictive tools that are useful to,and easily used by, practicing engineers. He abhors piece-mealsolutions to problems. So uniﬁcation is an important goal; whether
that is the uniﬁcation of ﬂow, heat and mass transfer concepts orthat of seemingly different shear ﬂows. Another recurring theme
is a readiness to challenge the prevailing wisdom and explore
unorthodox ideas. His simple solution of obtaining the adiabatic
ﬂame speed (which is unobtainable from any practical experiment)
as the limiting case of vanishing heat transfer and obtaining Y+ interms of U+ are good examples of his unconventional out-of-thebox thinking. He has an intuitive feel for the importance of theexisting ideas to his goals and he is able to boldly adapt and build
upon the work of others. He also has a tremendous knack of
expressing his ideas in clear and cogent terms to reach a wide audience of different backgrounds. He developed a clear methodology
to express heat and mass transfer concepts and he can be credited
to some extent for unifying the terminology and language used bychemical and mechanical engineers which was different before hearrived on the scene.
This pattern of ‘‘innovation” by ‘‘uniﬁcation” and ‘‘adoption” ofother’s work with ‘‘bold insights” are the common and recurrent
themes of his professional career. If I were to characterize one
common distinctive thread in Spalding’s thinking, it is that hethinks ‘‘physics” rather than ‘‘mathematics”. Though his key contributions involve complex mathematical concepts, his breakthroughs have come because he looked at the physics behindthose concepts. He was able to see the ‘‘trees” because he refused
to get lost in the forest of mathematical equations and looked at
the processes these equations were trying to express. He thinks
like an engineer. He repeatedly asks the question: ‘‘What can I dothat will help solve a given problem?” To me this has been his
key contribution above all. He labored on with ‘‘gaps” in our
knowledge about multi-phase, turbulence, boundary conditions
and such like. Each time he came across an ‘‘obstacle” he foundan engineers’ solution. If it worked – but was theoretically a bit
shaky – it was good enough to get the job done. He proposed things

4071

like eddy break-up for combustion when there were so many gaps
in our knowledge that no one thought you could attempt combustion with CFD!
So did Brian Spalding invent CFD? The answer is obviously not
an unqualiﬁed yes. Origins of CFD can be traced as far back as 1928
or perhaps even earlier. Harlow at Los Alamos, and others in academia and research organizations, had already contributed immensely to what would later become CFD. They had already exploredmany of the key concepts that would be used or re-invented by
the IC group. Spalding did not even coin the name. Formally thename was ﬁrst used by Pat Roache [54] in his famous book. But
let us ask another question. Would there be CFD as we know it today without Spalding? And the answer is an unqualiﬁed No. Morethan anyone else, he created CFD as an Engineering Tool – theapplication of CFD to problems of interest to engineers. Most of today’s commercially available CFD software tools trace their origin
to the work done by Spalding and his group in the decade spanning
1965–1975.
He was an active leader and the key contributor of novel ideasthat led to the development of CFD methodology that was efﬁcientand ‘‘engineer-friendly”. His key insights during this decade often
made breakthroughs possible and re-directed the focus at critical
moments. He has had over 100 graduate students. If you consider
the period between 1965 and 1972 when both Launder and Whitelaw worked closely with Spalding, the number of students whocarried the torch of ‘‘IC” methodology is truly amazing for a group
under the close supervision of a single person. Many of these students have gone on to make signiﬁcant professional contributions
and a distinctive mark on the world stage. Many well-known
names in CFD, turbulence or combustion today have a 1st or 2nd
generation IC connection.
The fact that any vector can be expressed as a rotational (vector)
and irrotational (scalar) components has been known for ages. That
pressure is related to the scalar component and vorticity to thevector is part of classical hydrodynamics. Harlow and Chorin independently suggested that we decompose velocity correction into avector and an arbitrary scalar stage. Harlow’s group used it to turnthe continuity equation into an equation for pressure. But it was an
inefﬁcient mechanism for steady ﬂows – of routine interest to
engineers – since it required transient solution. It took the workof Patankar and Spalding [34] to weave it into an economical andpractical tool that could also be used for steady ﬂows – SIMPLE.
So we can not credit Spalding for pressure projection methods
but we should recognize the manner in which he used and turned
it into a practical and viable alternative for steady state ﬂows.
Take another example. ﬁnite-differences have been around forages. Mathematicians have played with and described discrete
spaces, Banach, Hilbert, etc., for ages. Long line of researchers –Thom [18], von Neumann [55], Southwell [56], Courant et al.
[20], and others – applied it to ﬂuids. Spalding started with thesame tools, quickly realized the limitations of FD – and gradients-based Taylor Series – in dealing with non-linear equations
that span from elliptical to hyperbolic. He could see the connections between the mathematics as it was written and the real
physics where ‘‘quantities” (ﬂuxes, heat, etc.) move in a conserved
manner. This eventually led to the FV approach. It turned out the ICgroup was not alone in using the FV approach; Harlow’s group had
essentially thought in FV terms and expressed their equations interms of ﬂuxes and conserved quantities. However they never expressed it by a simple visual analogy such as a ‘‘Tank and Tube”.
Neither did they resort to formal integration around a control volume for their published work. Edwards [57] had used the essentials of the FV and (unique at that time) unstructured grid
approach in a code called TRUMP at Lawrence Livermore startingin the mid-1960s. But I bet most CFD researchers, even today, do
not even know of Edwards work. I certainly had not heard of him

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A.K. Runchal / International Journal of Heat and Mass Transfer 52 (2009) 4063–4073

till 1978. So just like ‘‘upwinding”, the FV was perhaps re-invented
and Spalding is the one who should be credited for the wide-spreaduse of this technology.
More or less the same story repeats with the two-equation
models of turbulence. The basic equations were written down by
Kolmogorov [35] and Prandtl [36]. Rotta [38] had added to theknowledge in the 1950s and elaborated on the concept of lengthscales and frequencies. Davidov [58] seems to have been the ﬁrst
to formally derive and propose a closed form equation for e as partof a very complex 3rd-moment closure. Harlow and Nakayama[39] developed an epsilon equation and demonstrated its use in a2-equation eddy viscosity scheme for a few ﬂows. Hanjalic(Fig. 19), then a Ph.D. student with Launder, knew of the difﬁculties
being experienced by his IC colleagues in using k–l and k–kl based
variables to predict both wall and free ﬂows. He adapted the coefﬁcients in Davidov’s epsilon equation, tested it ﬁrst in a k–e eddyviscosity formulation and found the same form could mimic bothwall and free ﬂows. He then embedded it in a 3-equation model(k–e – huvi) to predict ﬂows where shear stress and strain vanishedin different locations [59]. Bill Jones [60] extended the k–e modelso that it could be applied across the sublayer right to the wall (enabling situations where the sub-layer was not universal to be predicted) while others made further contributions. Spalding till thenexploring, k, k–l, k–kl, and k–x models could see the advantages ofa ‘‘standard” model and the order that it will bring to future research. He dropped further work on other models, and the IC group
adopted k–e as the model of choice. This resulted in the Launderand Spalding [61] book on the subject and the seminal review ofLaunder and Spalding [62] that led to the formalization, ‘‘standardization” and acceptance of the k–e model as a general tool for engineering practice.
It was due to Spalding’s insight and single-minded focus onpractical tools that these techniques became established and usefulto engineers. He focused the attention away from the mathematics
and to the physics of the phenomena and application to practicalproblems. In my view it will be true to say that Brian did not inventthe ‘‘science” of CFD but he is still the person who honed the ‘‘art”
and ‘‘technology” of CFD for engineering design and practice.
In Newton’s words: Spalding stood on the shoulder of giants
and saw farther than his peers. He foresaw that unifying ﬂow,
heat and mass transport will lead to practical tools for engineers.
He foresaw that looking at the physics rather than the mathematics will lead to wider acceptance of CFD tools. He foresaw that
CFD – once turned into to a design tool – would revolutionize
engineering. He foresaw that there were commercial opportunities in CFD.
Brian to me is the best example of Richard Feynman’s famous
saying: ‘‘What Do You Care What Other People Think?” He says that

Fig. 19. Kemo Hanjalic.

Fig. 20. Brian and Colleen Spalding – 2008.

his most distinguishing trait is that ‘‘he did what he did because hedidn’t know any better”. He quotes Virgil: ‘‘Possunt quia possunt
videntur” (They can because they think they can). He would liketo change it to: ‘‘They can because they don’t know they can’t.”
At many stages in his life he simply did not know (or more probably, did not care) that others had already declared a problem to be
impossible or too ill-posed to be solved. And they had gone on toprove it by elaborate mathematical and physical arguments. He
simply found a way to devise a practical solution to deal withthe essence of the problem.
Brian has immense intellectual capability to grasp the essenceof a problem and suggest a resolution. More than once when I
brought a problem to him that I was struggling to grasp, I foundthat he could see the underlying essence while I was having trouble enunciating it! Brian (Fig. 20) has a tremendous capability to
not only generate new ideas but to bring out creativity in others
who work with him. Other than his towering intellect and his keen
insight, Brian has a single-mindedness that has allowed him to focus on the problem of immediate interest and achieve so much inhis professional career. When Brian in focused on a problem, he isable to ignore everything else that is not immediately relevant to
his purpose. I would dare to say that Brian looks at the world as divided in to two groups: those he can work with and those not relevant to his purpose.
References[1] D.B. Spalding, The Combustion of Liquid Fuels, Ph.D. Thesis, Cambridge
University, 1951.
[2] T. von Karman, Z. Angew, Math. Mech. 1 (1921) 233.
[3] G. Kruzhilin, J. Tekh. Phys. (USSR) Col. 3 (1936) 183.
[4] E.R.G. Eckert, V. Lieblein, Forschung 16 (1949) 33.
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[7] D.B. Spalding, Some Fundamentals of Combustion, Butterworths, London,
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[8] D B Spalding, Predicting the laminar ﬂame speed in gases with temperatureexplicit reaction rates, Combust. Flame 1 (3) (1957) 296–307.
[9] J.P. Botha, D.B. Spalding, The laminar ﬂame speed of propane–air mixtures with
He at-extraction from the ﬂame, Proc. Roy. Soc. A 225 (1954) 71–96.
[10] D.B. Spalding, Analogue for high-intensity steady-ﬂow combustion
phenomena, in: Proc. I. Mech. E. London, [171], vol. 10, 1957, pp. 383–411.
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[19] H.Z. Barakat, J.A. Clark, Transient Natural Convection in Flows in Closed
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[30] F.H. Harlow, J.E. Welch, Numerical calculation of time-dependent viscousincompressible ﬂow with free-surface, Phys. Fluids 8 (12) (1965) 2182–2189.
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Thesis, Imperial College, London University, UK, 1967.
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by W. Rodi as Imperial College Mech. Eng. Dept. Tech. Notes TWF/TN/38 andTWF/TN/39].
[39] F.H. Harlow, P.I. Nakayama, Transport of Turbulence Energy Decay Rate, LA3854, Los Alamos Scientiﬁc Laboratory, University of California, 1968.
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involving ﬂuid ﬂow, Comput. Methods Appl. Mech. Eng. 9 (1976) 153–164.
[41] L.S. Caretto, R.M. Curr, D.B. Spalding, Two numerical methods for threedimensional boundary layers, Comput. Methods Appl. Mech. Eng. 1 (1) (1972)
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Variable Navier–Stokes Equations and a Concise Notation for Difference
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Dept., Imperial College, London, 1971.
[44] C.M. Rhie, W.L. Chow, Numerical study of the turbulent ﬂow past an airfoil
with trailing edge separation, AIAA J. 21 (11) (1983) 1525–1532.
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Phys. 2 (4) (1968) 339–355.
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Turbulent Shear Flows, Symposium on Turbulence and Dynamical Systems,
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