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Akshai K. Runchal
Analytic and Computational Research, Inc.,
1931 Stradella Road,
Los Angeles, CA 90077
e-mail: runchal@ACRiCFD.com;
runchal@gmail.com
Emergence of Computational
Fluid Dynamics at Imperial
College (1965–1975):
A Personal Recollection
This paper is a personal recollection of the development of computational fluid dynamics
(CFD) at Imperial College (IC) in the 1960s by a group founded by Brian Spalding who
was the charismatic leader of a dedicated team for a decade. I was a member of this team
during the development of the basic engineering practice that came to be known as the
IC approach to “CFD.” I hope to capture the essence and the significance of those devel-
opments. A version of this paper was delivered at the occasion of the seminar organized
at Villanova University to celebrate the award of the 2010 Franklin Prize to Prof. Spald-
ing. In a very strict sense the IC group invented neither the art and science of CFD nor
the name. However it did excel in adapting and developing the methodology and technol-
ogy of CFD, testing and verifying it against empirical data, and developing innovative
and practical computational tools that had widespread application and relevance to
problems of interest to engineers. In a nutshell the IC group heralded the CFD revolution
and was the pioneer of the practice and technology of CFD. Most of today’s commer-
cially available software tools trace their origin to the work done by the IC group in the
decade spanning the mid-60s to mid-70s. The group at IC benefitted tremendously and
borrowed liberally from the innovative and groundbreaking work being carried out
around this time at Los Alamos National Laboratory under the leadership of Frank
Harlow—this is described elsewhere in this volume. [DOI: 10.1115/1.4007655]
Keywords: CFD, computational fluid dynamics, finite volume methods, FVM, history of
CFD, heat transfer, combustion, two-phase flow, unified theory
Introduction
Growing up in an isolated, time-forgotten hamlet in the hills of
Himalayas, without electricity, antibiotics or television, I could
never have imagined that I would participate in the birth of a new
branch of engineering science—or that one day this remote hinter-
land would be known throughout the world. Yet that is exactly
what happened. I witnessed, and participated in, the birth of
CFD—a revolution in engineering design that, in turn, is contrib-
uting to the emergence of virtual reality as an engineering design
tool. And Dharamsala, the place where I grew up, became a sym-
bol of rejuvenation of Buddhism, the home of the Dalai Lama and
a major cause ce´le`bre for Human Rights.
The Navier–Stokes equations, that govern the flow of fluids in
continua, are recognized to pose one of the most intractable math-
ematical problems in physics. These equations were formulated
by Claude-Louis Navier in 1822 and subsequently refined by
George Gabriel Stokes in 1842 who postulated the stress–stain
relations. Other than limited subsets such as nonviscous, two-
dimensional or steady state flows, or one-dimensional viscous
flows, the solution of these equations has remained elusive for
most problems of practical interest that are three-dimensional and
transient in nature with strong influence of viscosity. Further,
most practical flows involve turbulence, a chaotic and stochastic
display of the seemingly simple and ordered Navier–Stokes equa-
tions that has not been understood to this day. The Nobel laureate
Richard Feynman famously described turbulence as “the most im-
portant unsolved problem of classical physics.” In fact a general
solution to the Navier–Stokes equations in three-space and time
dimension is still one of the six open Millennium Prize Problems
that carry a reward of 1 million US dollars from the Clay Mathe-
matics Institute. CFD now provides a tantalizing promise that we
may finally be able to tease out some useful information for a
wide range of practical problems from the Navier–Stokes
equations.
The emergence of CFD at Imperial College in 1960’s can be
traced to three major developments that took place in the UK, to-
gether with a key development at the Los Alamos National Labo-
ratory in the USA. The first of these was the Ph.D. thesis that
Spalding [1] submitted in 1948 to Cambridge University on
“Combustion of Liquid Fuels.” The most remarkable feature of
this thesis is that it laid the basis of a general theory of heat and
mass transfer [2] that for the first time in recorded literature
“unified” the key hydrodynamic, heat transfer and mass transfer
concepts of von Karman [3], Kruzhilin [4], and Eckert and Lie-
blein [5]. The second important development took place in 1964
with Spalding’s publication of “A Unified Theory (UT) of Fric-
tion, Heat Transfer and Mass Transfer in the Turbulent Boundary
Layer and Wall Jet” [6] that transformed the engineering practice
of turbulent shear flows. The third development resulted when
Spalding realized the importance of empirical data in the verifica-
tion and refinement of his UT and commissioned an extensive and
in-depth literature survey of the empirical data, conducted by a
number of his students, notably Ricou, Nicoll, Escudier, and Jaya-
tilleke [7–10], to determine the “universal” constants and empiri-
cal relations that would allow an engineer to determine the
entrainment rates and the growth and decay of boundary layers.
In parallel to these developments, computers were being used
increasingly for computational purposes. The focus of the devel-
opment of the modern computer was in the USA. The modern age
of electronic computers can be traced to the IBM701 that became
commercially available in 1952. The T-3 group led by Frank Har-
low at the Los Alamos National Laboratory immediately put it to
Manuscript received March 15, 2012; final manuscript received June 9, 2012;
published online December 6, 2012. Assoc. Editor: Gerard F. Jones.
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good use. They used the finite difference method (FDM) to solve
a number of complex fluid dynamic problems [11] of theoretical
importance as well as those of practical interest to the weapons
program.
When the first commercial computer, IBM 7090/7094, arrived
at Imperial College in the early 1960’s, the IC group led by Spald-
ing was well-placed to explore this new branch of science based
on reincarnation of FDM for fluids that had its recent origin in a
weapons laboratory in the USA, for engineering and industrial
applications related to fluid dynamics, heat and mass transfer, and
combustion. This technology eventually came to be called
“computational fluid dynamics”—or CFD for short. The stages
that occurred in this development are described below.
I have previously reviewed the specific contributions of Profes-
sor D. B. Spalding on the occasion of his 85th birthday [12,13].
But the emphasis here is on the development of CFD at Imperial
College in the decade spanning the mid-60s to mid-70s and its
relation to the developments that took place elsewhere.
As a curious aside, in the 1960s the IC group and the Los Ala-
mos group had practically no direct contact with each other. This
is of course not unusual in scientific history. During this period,
the developments that took place at IC proceeded independently
of those at Los Alamos. For example, Los Alamos had already
moved to primitive variables and was also exploring Lagrangian
algorithms while the IC group was still using the stream-
function–vorticity approach in an Eulerian framework. The IC
focus had shifted to finite volume method (FVM) while the work-
horse at Los Alamos was still the FDM. Neither Spalding nor Har-
low paid any reciprocal visits to each other’s facilities. To my
knowledge, there were no other direct telephonic or written com-
munications. Email did not yet exist. The first tangible contact
was a visit by Tony Hirt, a key member of the T3 Group, in early
1970s to the IC group. This resulted in broad and comprehensive
exchange and spirited discussions of the merits of the various
alternate approaches that were being explored by each. Of course
by then the IC group was already aware of the work being done at
Los Alamos and had integrated some of the innovations first
explored there into the development of the semi-implicit method
for pressure-linked equations (SIMPLE) algorithm and the k–e
turbulence model.
The Boundary Layer and the Unified Theory
Up to the 1960s, the dominant theory of fluid dynamics for
practical applications, that involve viscosity, was the “Boundary
Layer Theory” developed by Prandtl in 1904. This had been the
first significant advance since the formulation of the Navier–
Stokes equations almost 100 years earlier. It had revolutionized
engineering practice of fluid dynamics and had led to a number of
breakthroughs in aerospace, naval, and industrial applications. For
practical applications, an essential tool of the boundary layer
theory was the classical profile method that had been perfected by
Pohlhausen [14] and others. The profile method relies on analyti-
cal profiles that are so devised that they represent the flow well,
and have the right number of free parameters to satisfy the initial
and boundary conditions specific to the problem at hand. When
successful, it allows one to approximately solve the intractable
Navier–Stokes equations. It had had considerable success with
flows that are broadly classified as “boundary layer” flows includ-
ing those on planar surfaces, in ducts and tubes, and in jets and
wakes. However the profile method is necessarily a problem-
specific and complex task. It often proves cumbersome or intracta-
ble for practical flows that involve separation, high pressure
gradients, or complex nonlinear physics with turbulence, gravity,
chemical reactions, and such like. For these, the boundary theory
is grossly unsatisfactory.
Inspired by Morton et al. [15], who had shown the way forward,
Spalding decided to go further. He decided that the need of
the practising engineer was not just fluid flow but also heat and
mass transfer, turbulence, combustion and other varied physical
processes. Borrowing from his work at Cambridge, he laid the
foundations of a comprehensive mathematical basis outlined in
his 1964 treatise on Unified Theory [6]. This document is remark-
able in laying down a common foundation for fluid flow, heat and
mass transfer, turbulence and combustion and has influenced all
subsequent work and revolutionized the engineering practice of
fluid dynamics.
To generalize the boundary layer theory and to make it more
tractable, Spalding came to the conclusion that the analytic pro-
files of the boundary layer theory can be replaced by simple alge-
braic profiles by dividing the complex profile into a number of
arbitrary segments, each represented by a simple linear or polyno-
mial curve. This simple strategy can approximate any profile that
might describe the flow to any given accuracy. This replaced the
problem of finding a suitable analytic profile by the need to deter-
mine the matching constants that linked the segments of the
piece-wise profile into a single “universal” profile. Spalding
derived additional relations to determine the profile constants
from the integrals of the moments of momentum and energy equa-
tions after multiplication by a weighting function (which could be
the dependent variable itself). The matching constants for these
“universal” profiles could then be derived from these equations
and from the initial and boundary conditions. In the search for
greater flexibility and wider applicability, many families of
weighting functions, including subdomain weighting functions,
were considered. The most prominent of these were the kinetic
energy equation and the “T-square” equation. During this time of
exploration, Spalding also suggested a variety of independent var-
iables for the cross-stream coordinate in the boundary layer. The
distance yor the dimensionless distance y/dwere the obvious can-
didates; but he also used the von-Mises stream function (and its
dimensionless counterpart), that simplified the representation of
the cross-stream velocity or flow rate.
The Unified Theory, in replacing the complex problem-specific
analytic profiles with generally applicable linear (or sometimes
polynomial) segmented profiles, was highly suited to computation
by the digital computers that were just beginning to become avail-
able for wide commercial use. One of his students, Patankar,
developed a general-purpose “integral-profile” computer code
based on simple profiles for piece-wise segments. This computer
code was used with considerable success for a variety of boundary
layer, jet and wake flows. Simultaneously Spalding and a few of
his students had undertaken an extensive literature survey to deter-
mine optimal entrainment functions, log-law constants, and heat
and mass transfer resistance required to describe a wide range of
flows. This work by Ricou, Escudier, Nicoll, and Jayatilleke
[7–10] led to “universal” constants and empirical relations that
could be used to define the interaction of flow with a wall or ambi-
ent external flow through “wall functions” or “entrainment
relations.” This combination of theory and empiricism enabled
practically useful predictions for a very broad range of turbulent
flows of interest to industry.
By the end of 1965 further development of the Unified Theory
came to an abrupt halt. Wolfshtein and Runchal, had started with
the Unified Theory but had switched to FDM, an alternative
approach that had proved very promising and was demonstrably
successful for a class of problems that had failed to respond to
UT. This led Spalding to abandon the "piece-wise-profile" method
of UT and switch over to FDM for the parabolic form of the two-
dimensional Navier–Stokes equations for boundary layers.
Though the essence of the UT was lost, the spirit of the UT, in
providing a common framework for treatment of fluid flow, heat
and mass transfer, and the “universal” empirical constants for the
wall and entrainment relations, was incorporated in a general
FDM approach to two-dimensional boundary layer flows. Patan-
kar and Spalding used the innovation of replacing the distance
normal to the wall by a nondimensional stream-function as the
cross-stream coordinate. Suitable entrainment formulas were used
at free boundaries to determine the growth of the calculation do-
main into the surrounding free stream. This method had the speed,
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flexibility, accuracy, and simplicity needed for a general proce-
dure for two-dimensional boundary layers, free shear flows, wall
jets, and duct flows. This work culminated in the paper by Patan-
kar and Spalding [16] at the 1966 International Heat Transfer
Conference in Chicago.
A new and improved version of the parabolic boundary layer
method, embodied in a computer program called GENMIX, was
published by Patankar and Spalding [17] in 1967 and then in re-
vised form in 1970 [18]. Much later, in 1977, Spalding published
a Users’ Manual for the GENMIX computer program [19] and
described improved procedures for entrainment and the unknown
pressure gradient for confined flows. The revised GENMIX also
contained the hybrid scheme that had been developed earlier for
high-Reynolds number flows by Runchal [20]. The solution proce-
dure was based on a noniterative “marching” scheme. In GEN-
MIX the calculation of the entrainment at the free boundaries
remained problematic and caused oscillations and instability since
the governing equations are nonlinear. Later in the general-
purpose CFD code, PHOENICS, Spalding [21] addressed these
issues and employed an iterative procedure for the boundary layer
equations to remove this instability.
The Shortcoming of the Unified Theory
By 1965, a very broad range of flows, beyond the so-called
“boundary layer” flows, could be represented by the Unified
Theory but those with strong pressure gradients such as the im-
pinging wall jet (favorable pressure gradient) and flows with sepa-
ration such as that behind a step in a pipe or in a cavity (adverse
pressure gradient) remained outside its purview. If the flow does
not have a monotonic profile or the gradient is too steep, the alge-
braic equations for the UT profile weighting factors can develop
singularities. Examples of such flows are the transition of a wall
jet to a boundary layer or that in the vicinity of a reattachment
point or an embedded eddy.
The core of the problem is that UT relies on the boundary layer
version of the Navier–Stokes equations derived by Prandtl. These
lead to “self-similar” profiles in the downstream direction that can
be described in terms of a few nondimensional parameters. Spald-
ing thought that unlike the analytic profiles (with limited degrees
of freedom) his segmented profiles (with unlimited degrees of
freedom) could evolve and represent separated and impinging
flows that otherwise had no "self-similar" behavior. He therefore
embarked on a mission to bring these important industrial flows
within the scope of the Unified Theory. Spalding asked Wolf-
shtein to solve the problem of the favorable pressure gradient and
Runchal to concentrate on the adverse pressure gradient. Wolf-
shtein was assigned the problem of the impinging jet on a flat
plate and Runchal started with the separated flows behind a
backward-facing step and that in a driven-lid square cavity. Both
these types of flows show distinctly elliptical, rather than para-
bolic, behavior and cannot be described in terms of self-similar
profiles. If successful, these extensions would have taken the UT
well beyond Prandtl’s Boundary Layer Theory and significantly
broadened the scope to include both the parabolic and elliptic
flows. However that was not to be so. By the end of 1965, Runchal
and Wolfshtein came to the conclusion that the underlying profile
structure of UT with an inherent assumption of "self-similarity”
could not be extended to elliptic flows. The profile method had a
fatal flaw and could not account for the transition of a monotonic
profile to a multivalued or complex evolving profile. Physical
processes such as axial diffusion and buoyancy destroy the self-
similarity that is the lynchpin of the boundary layer theory. It
quickly became apparent that the only general method then avail-
able to tackle the elliptic form of the Navier–Stokes equations
was the method of finite differences. The FDM is in one way simi-
lar to the UT in that it also uses piece-wise profiles but there is a
crucial difference. While UT solves for the matching constants
that constitute a segmented profile, the FDM uses a simple universal
piece-wise profile to solve directly for the value of the governing
variable. In essence, in the FDM, there is no explicit profile
beyond the immediate neighborhood of the point of interest (or
node) in the computational domain. This may be thought of as a
locally evolving profile rather than a generic profile that is the
essence of the UT or the classical boundary layer profile method.
With the recognition of this shortcoming, the further develop-
ment of the UT was abandoned at Imperial College. From early
1966, all further research on boundary layers shifted to the use of
FDM methods as described above.
The Finite Volume Method
Finite difference methods for Navier–Stokes equations have
been extensively explored by a large number of researchers for
almost a century. Thom [22] used the FDM in 1928 and Southwell
[23] explored FDM with an innovative coordinate transformation
in 1946 well before the advent of electronic computers. With the
availability of the electronic digital computers, FDM became the
method of choice for complex flow problems. Early practitioners
included von Neumann [24] and Courant et al. [25],butitwasFrank
Harlow [11] who did the pioneering and ground breaking work in
showing that FDM can be successfully applied to wide range of fluid
dynamic problems that had been intractable for almost two centuries
since the formulation of the Navier–Stokes equations.
After the failure of UT to predict separated and impinging
flows, it became apparent to the IC group that it would be far sim-
pler to use the FDM than to modify the UT to accommodate such
flows. Burggraf [26] had recently applied the FDM to the sepa-
rated flow in a square cavity where he reported success for low Re
numbers but failure to obtain solutions beyond Re ¼400. In Janu-
ary 1966, Runchal and Wolfshtein combined their individual
research goals and started writing a general-purpose FDM com-
puter program. Their immediate interest was in solving two-
dimensional flows so the stream-function–vorticity (w–x) form of
the Navier–Stokes equation that eliminates pressure from the
equations and results in a set of two rather than three coupled par-
tial differential equations was the focus of their attention. It was
apparent that there were limitations of the standard, and then pre-
ferred, option of using central differences for FDM where it failed
to converge for high Reynolds number flows. Stability analysis
showed that this was due to the matrix coefficients that were no
longer positive-definite at high Reynolds numbers. Spalding
then made the analogy of how the wind from the pigsty always
stinks—or that in the Northern hemisphere northerlies bring the
cold. These discussions led to the development and use of the
“upwind” concept that provided a robust and versatile tool for
solving the two-dimensional Navier–Stokes equations. This led to
a number of papers on finite difference method that had been
developed at IC [27–32].
In fact, the IC team had reinvented upwind differences because,
before the days of the internet and air travel for the masses, it
was not unusual for researchers to be unaware of each other’s
work—often for years. Mathematicians had extensively explored
the properties of the central and one-sided difference methods for
a long time. In 1952, Courant et al. [25], in a classical paper, had
defined the concept of the upwind method that used one-sided dif-
ferences based on the direction of flow. Harlow and his colleagues
[33] had successfully exploited the upwind method in their work
and Barakat and Clark [34] had used the upwind method in a
more recent paper. By 1968, it began to be widely recognized that
upwind differences led to undue false (numerical) diffusion except
for a limited subset of flows. Wolfshtein [35] and Hirt [36] pre-
sented qualitative and quantitative dependence of numerical diffu-
sion on the speed of the flow, the angle of the stream-lines to the
grid, and the local gradients. Spalding [37] proposed an exponen-
tial method and in a companion paper Runchal [20] proposed the
“hybrid” method that automatically blended the central and
upwind difference methods based on the local Peclet number.
For quite some time the “hybrid” difference stayed the defacto
standard approach for solving high-Reynolds number flows.
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Subsequently, a number of higher order upwinding and flux-
splitting [38–41] schemes have been developed though none has
yet proved to be satisfactory and robust for general and arbitrary
flow conditions.
Perhaps the lasting and unique contribution of the IC group was
the development of the FVM as the core of mathematical tech-
nique for solving fluid flow problems. It is very distinct in its phi-
losophy from the traditional FDM, and the then emerging FEM.
The FVM treats the flow field of interest as a series of “tanks”
(that hold the nodes) that are connected by “tubes” (that convey
the fluid along the grid). The emphasis is on the fluxes that move
in and out of these “tanks” or “finite volumes.” This is distinct
from the FDM concept where the gradients are approximated by
the Taylor series or the FEM concept where the local variation of
a primitive variable is represented by simple polynomials with
local weighing functions. The FDM and FEM approaches stress
the mathematical concepts such as the gradients of variables while
the FMV stresses physical concepts such as the mass flux across a
boundary. There is also one more important difference, the FDM
and FEM require continuity of gradients to the second order while
the FVM requires continuity only for the first order.
By June of 1966, Wolfshtein reported results for the impinging
wall jet and Runchal for the separated flows behind a backward-
facing step and in a driven-lid square cavity for low and high
Reynolds number flows [27–32]. By late 1969, the FVM was
well-established [42,43] and began to be used widely by other
researchers all over the world. It was subsequently published in
the first book that described the application of a general FVM to
fluid dynamics problems [44]. It is noteworthy that Edwards [45]
had used the essentials of the FVM and unstructured grid
approach (both unique at that time) in a code called TRUMP at
Lawrence Livermore starting in the mid-1960s. But his work
never became widely known except to a small community of
researchers in National Laboratories in the USA. It was only in
the mid-1970’s when a group at the Lawrence Berkeley National
Laboratory of the University of California started using it for
applications related to geohydrology.
Around 1967, I realized that the Taylor series based FVM
approach could be replaced by that based on the Gauss theorem
followed by a strict integral to derive the algebraic analogue for
the Navier–Stokes equations. Gauss theorem leads to an elegant
formalism to derive the discrete set of equations in any arbitrary
shaped control volume. The Taylor series versus the Gauss theo-
rem is akin to the classic argument between the “differential”
approach of Newton and the “integral” approach of Leibnitz. One
advantage of the integral approach is that it does not require
smoothness or the existence of a second order derivative. 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 and he correctly
pointed out that the same set of algebraic equations can be derived
from either. Wolfshtein used the Taylor series approach for his
thesis [42], whereas Runchal [43] follows the integral approach
which is the first recorded use of the integral approach for CFD.
Most finite volume codes today use the integral approach.
In 1967, Spalding introduced the concept of a "General Trans-
port Equation" for a general variable called “A” that expresses all
second order convective–diffusive transport equations. This for-
mulation appears in the 1969 book on the IC method [44]. The
“A” represents velocity components, enthalpy, chemical species
or any other transported property of the fluid. Thus the Navier–
Stokes, energy and mass transport equations can all be expressed
by a single second order transport equation
@
@tqUðÞþdiv qUUðÞ¼div CgradUðÞþS(1)
The four components of this equation represent the four distinct
physical processes that are present to varying degree in all trans-
port problems: (1) storage, (2) convection, (3) diffusion, and (4)
sources or sinks. This unified approach led to the fast development
of CFD methodology since one can concentrate efforts at solving
this single equation rather than look at the specifics of each
variable.
Imperial College group also introduced the "compass" notation to
express the FDM or FVM stencil. This simplified the visualization
and expression of the fluxes and meshed in with the "tank-and-
tube" approach. The six points (or control volumes) surrounding a
central point (called “P”) are designated “East,” “West,” “North,”
“South,” “Up,” and “Down.” Thus, the standard form of the linear
algebraic equation that results from discretization of Eq. (1) is
written, in the so-called zero-residual form, as
aEUEUP
ðÞþaWUWUP
ðÞþaNUNUP
ðÞþaSUSUP
ðÞ
þaUUUUP
ðÞþaDUDUP
ðÞþS¼0(2)
The generalized FVM revolutionized the engineering practice
and gave rise to the new field that came to be known as CFD. The
actual use of the acronym CFD only came in vogue with the publi-
cation of the book “Computational Fluid Dynamics” by Roache
[46] in 1972. General and flexible computer codes could be writ-
ten to solve fluid flow problems in any complex geometry at any
Reynolds number. In an important way the FVM also changed the
basic thinking behind computational fluid dynamics. Unlike the
mathematical approach used by FDM and the FEM, an under-
standing of the FVM was enhanced by a physical approach to the
problem. The focus of interest was not the variable and its interre-
lationship parameters with other nodes; the focus now was on the
control volumes and the fluxes crossing their boundaries and the
sources or sinks contained within them. In a way FVM mimicked,
and reinforced, the control volume analysis that was the essence
of classical fluid mechanics and hydraulics.
By the end of 1968, the major developments of the FVM were
complete. The first international conference devoted entirely to
CFD as its theme was held at Monterey, California in 1968. A
conference (called a Post-Experience Course), was organized at
Imperial College in 1969 and representative from both academic
and industrial communities were invited to learn of the develop-
ments and the possibilities that had opened up. Academic Press
published the work done by the IC group for wider audience [44].
Both Wolfshtein and Runchal 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 assume custody of the computer code devel-
oped by Runchal and Wolfshtein. That code, called ANSWER,
made it to the first book on CFD [44] that dealt with the numerical
solution of the elliptical form of Navier–Stokes equations. The
authorship of this book illustrates the strict rule that Spalding fol-
lowed that 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 incorpo-
rated the first CFD based consulting company—Combustion Heat
And Mass Transfer Ltd. (CHAM) that was later superseded by
Concentration, Heat And Mass Transfer, Ltd.
1969 can be considered the year that ushered in the CFD as an
engineering tool. Both the elliptic FVM and the parabolic FDM
were firmly established. IC technology had gained a considerable
following and was widely used by researchers at a number of aca-
demic and commercial organizations around the world. The work
done at Imperial College on generalized transport equations, and
the computer codes to solve these equations, became widely avail-
able. 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 CHAM in 1969. The entire focus of the IC
group shifted to application and improvement of this technology.
The CFD codes had provided a convenient vehicle for testing of
turbulence, combustion, radiation and other models for physical
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processes. Development of mathematical models for turbulence
and combustion was in full swing. The two-flux and six-flux mod-
els for radiation [47] and formulations for particle concentrations
in different size ranges were also implemented in the framework
of these methods.
The Simple Algorithm
With the development of a general numerical procedure for
two-dimensional flows, it was natural that the interest of the Impe-
rial College group began to shift to three-dimensional flows. It
was immediately recognized that the delightful simplicity, and the
distinct advantages, of the stream-function–vorticity (w–x)
approach in the 2D simply disappear in its extension to 3D flows
where it leads to a set of six second-order differential equations.
In contrast, the primitive variable approach—based on the veloc-
ity components and pressure—consists only of four equations.
Therefore the primitive variable based methods dominate CFD
applications today.
First attempts at solving three-dimensional flows were focused
on developing a computational procedure for 3D boundary layers.
It was quickly realized that the development of a 3D boundary
layer procedure was not a simple or straightforward extension of
the two-dimensional boundary layer procedure. The three-
dimensional boundary layers show a parabolic behavior in the
stream-wise direction but an elliptic behavior in the cross-stream
plane. A number of alternatives, including extension of the 2D
stream-function based methodology, were tried. The first success
came with the coupled SIVA scheme of Caretto et al. [48]. How-
ever SIVA was deemed to be computationally expensive and
focus shifted to the search for an efficient sequential or segregated
method of solving the governing equations.
Harlow [11] had already developed a very successful algorithm
that employed a staggered grid and the pressure splitting method
to solve the continuity equation to determine pressure. Chorin
[49], borrowing from the classical Helmholtz–Hodge decomposi-
tion of the vector velocity field into its solenoidal (divergence-
free) and irrotational (curl-free) parts, had laid the foundations of
a general pressure projection method. It is then known that the so-
lenoidal component is related to a vector whereas the irrotational
component is related to a scalar field. Patankar and Spalding [50]
combined these ideas with the FVM technology and hybrid-
differencing developed earlier to arrive at a robust and general 3D
Navier–Stokes solver called SIMPLE. The parabolic 3D boundary
layer required a marching solution procedure that can be split into
cross-stream elliptic equations and parabolic stream-wise equa-
tion. For the cross-stream equations they used the continuity equa-
tion to derive the local pressure in the manner of Harlow and
co-workers. They then assumed that the stream-wise momentum
equation was driven by a single pressure gradient dp/dx across the
whole boundary layer at a given x. For unconfined flows the dp/dx
is imposed by the external free stream whereas for confined flows
it can be obtained from global mass conservation. The heart of the
SIMPLE algorithm, like that of the work by Harlow [11] and Cho-
rin [49], is the recognition that we could always solve the momen-
tum equations for any specified pressure field; but then the
resulting velocity field would not normally satisfy the continuity
equation. We could then propose “corrections” to the specified
pressure (scalar) field so that the corrected velocities satisfy the
continuity equation.
Although Patankar and Spalding presented the SIMPLE algo-
rithm in the context of three-dimensional parabolic flows, the
method contained within it a general procedure for elliptic flows.
It became widely used all over the world and went on to achieve a
classic status and led to the commercialization of the new CFD
technology. Improved variants of the method, such as SIMPLEC
[51], SIMPLER [52], and SIMPLEST [53] were later developed
by different researchers, to strengthen the coupling between ve-
locity and pressure and to provide more robust algorithms with
faster convergence. Ironically, Vanka [54] later showed that the
SIVA procedure that was abandoned in favor of SIMPLE proved
superior for strongly coupled three-dimensional flows.
Both the SIMPLE and the method originally developed by Har-
low required a staggered grid where the velocity components
were defined at locations that were staggered from those of pres-
sure and density. This has some unintended consequences. For
example, the decomposition cannot adequately deal with systems
where total mechanical energy or the total pressure must be con-
served since the pressure and velocities are defined at different
locations and any interpolation to one or the other location can only
be arbitrary and inaccurate. With the popularity of unstructured
grids in later years, it became convenient to look for colocated vari-
able alternatives to SIMPLE. Runchal had proposed [55]acolo-
cated grid pressure projection method but at the time it was not
judged to offer any distinctive advantage over staggered grids and
required additional memory that was scarce with the computer sys-
tems then available. Subsequently Rhie and Chow [56]perfected
the colocated grid that is today a preferred option for unstructured
grids and offers distinct advantage for complex geometries.
Turbulence Modeling
The core interest of the Imperial College group was understand-
ing and prediction of Industrial flows. Turbulent flows form an
important, indeed prominent, part of these flows. Hence prediction
of turbulence, right from the days of the Unified Theory, was at
the heart of their endeavor. Till the early 1960 extensions of
Prandtl’s mixing-length theory were at the core of predicting tur-
bulent flows. Spalding and his students undertook an extensive lit-
erature survey that resulted in a number of advancements in this
field. Spalding and his students [7–10] published extensively in
this field. With successful development of predictive tools for
elliptic flows, the IC group started to look for turbulence methods
that could be applied to such flows. There was no easy way to
extend Prandtl’s mixing-length theory to such flows since a single
dominant flow direction—and a prominent boundary layer—does
not exist in elliptic flows.
In the 1940s and 1950s Kolmogorov [57], Prandtl [58], Chou
[59], and Rotta [60] had already developed the basic mathematical
framework for turbulence modeling that included the concepts of
turbulence energy (k), and turbulence length (l) and time (x)
scales. Davidov [61] had formally derived an equation for the rate
of dissipation of turbulent energy, called epsilon (e). However the
resulting equations were so complex and the empirical evidence
to determine the attendant constants so meager, that no attempt
had been made to solve these equations.
Spalding, Launder and Whitelaw, started to focus their attention
on improving the prediction of turbulent flows. Whitelaw and his
students started an extensive set of measurements of turbulent
flows with the then nascent laser-Doppler methodology for flow
visualization. Spalding and Launder started looking at theoretical
alternatives to mixing-length. The equation for the kinetic energy
of turbulence (k) can be easily solved with the CFD Tools. How-
ever one or more equations were needed to replace the Prandtl’s
mixing length (l). The first attempts of the IC College group, fol-
lowing Rotta’s lead, were therefore focused at models that solve
combinations of kand and a length scale, l. A number of options,
including k-l,k-kl, and k-kl
1/2
and x(mean-square fluctuation of
vorticity) were explored. Launder and his students also explored
direct solution of the turbulent (Reynolds) stress equations.
Though each of these led to successful prediction of a limited set
of flows, none of these efforts proved versatile, robust and suc-
cessful for a wide range of turbulent flows.
Harlow and Nakayama [62] had used the turbulence dissipation,
e, rather than the length scale, l, in a two-equation k–eeddy vis-
cosity model that had shown promise for a few flows. Hanjalic
[63], a Ph.D. student of Launder, tested it in both a k–eand a (k–e-
<uv>) model and applied it to both wall and free flows. Most tur-
bulence models apply in the high turbulence number region of the
boundary layer. They cannot be extended to the so-called viscous
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sublayer or the transition (low turbulence Reynolds number)
regions of the boundary layer. Jones and Launder [64] developed
an extension of the k–emodel to apply it across the boundary layer
right up to the wall. From theoretical analysis one can show that
the diffusion coefficient was more likely to be a constant for the
“e” than for the “l”or“x” formulations that were being investi-
gated. Also “e” has an easy interpretation: it is simply the energy
dissipation. This led Launder and Spalding [65,66] to adopt the
k–eas the model of choice that led to the formalization,
“standardization” and acceptance of the k–emodel as a general
predictive tool for engineering practice.
There is an interesting footnote to the search for a versatile tur-
bulence model. It is easier to envisage “vorticity” being trans-
ported and diffused. The “dissipation,” as inherent loss of energy,
is hard to visualize as being “diffused.” Therefore “vorticity”
seems the more intuitive and elegant representation of a cascade
of turbulence eddies. After early and extensive exploration of the
k–xmodel, Spalding had abandoned it in favor of the k–emodel.
It was Saffman [67] and Wilcox [68] who eventually went on to
establish it as a viable and preferable tool for certain classes of
flows. It should be noted that Rotta had already shown that all
two-equation models are essentially identical in that the same dif-
ferential equation governs them all but they differ in the source
terms and may confer distinct numerical properties.
Combustion
Combustion, and turbulent combustion, was very much at the
heart of much of the work done at IC and Spalding was well
known for his work on combustion before he came to be known
for his work on CFD. Many of the key ideas in combustion today
can be traced back to Spalding’s pioneering developments. In his
Ph.D. thesis he had already concluded that the mixing of fuel and
oxidant often controlled combustion rate and also that multiscale
and surface phenomenon at the flame front played a crucial part in
the combustion process. What also emerged was that for many
turbulent, premixed, confined, bluff-body-stabilized flames, the
angle of spread of the flame was relatively independent of experi-
mental conditions (such as fuel type, equivalence ratio, mean
speed of the approaching flow or temperature and pressure levels).
He then went on to conclude that the chemical kinetics were sub-
sidiary to aerodynamic processes for a wide range of flame condi-
tions. His 1971 paper recognized the crucial role that turbulence
plays through “mixedness” that that came to be known as the
“eddy-breakup,” or EBU, model [69]. This paper is widely cited
and has achieved a classic status in turbulence combustion as hav-
ing foreshadowed much of the subsequent research that followed.
The process envisioned for EBU is akin to that of classical turbu-
lent mixing where large parcels of fuel gas break into smaller
ones that enhance the surface area available for the kinetics to act
at the molecular level. Therefore, at sufficiently large Damko¨ hler
numbers, where the fluid residence time is significantly larger
than the reaction time scale, the rate of combustion can be mod-
elled by recourse to the rate of eddy breakup.
Under equilibrium conditions, the rates of production and decay
of turbulence kinetic energy in turbulent flames are nearly equal.
Thus the reaction rate in this regime can be related to the local
mixture reactedness that depends solely on the turbulence mixing
time scale. For conditions where kinetics controls the reaction
rate—that is low Damko¨ hler numbers—the molecular Arrhenius
reaction rate was assumed to be controlling the reaction by a
strategy that preferentially picked the smaller of the turbulence
mixing and Arrhenius reaction rates. The development of the
EBU, in combination with the two-equation k–eturbulence model,
put combustion modelling firmly within the framework of the
CFD Tools that had been developed earlier. This combination was
soon widely adopted as the defacto standard for combustion
modelling.
Later, Khalil et al. [70] made one of the first attempts at a
comprehensive, fully elliptic multidimensional modelling of a
combustion chamber with a two-equation turbulence model and a
four-flux radiation model. This paper expressed the EBU model in
terms of the variance of a scalar mixture-fraction which later
became the preferred approach. Prior to this publication combus-
tion modelling was primarily parabolic simulations of laboratory
flames.
The EBU model had assumed that the rate of mixing was
related to the local mean-square fluctuations of the reactants
which, under equilibrium conditions, can be estimated from the
mean values of the reactants and the local turbulent time scale. It
was obvious that this model could be extended to determine the
mean-square fluctuations of species from their own governing
transport equations. Thus the EBU model, in turn, led to vigorous
research in developing multi-equation extensions where the single
“mixedness” variable was replaced with one or more field varia-
bles that represented the mean-square fluctuations of the species
or their interactions. A number of papers were subsequently pub-
lished with such extensions [70–73].
The EBU model assumed that “eddies” stretch as they transport
and mix and thus provide increased surface area for effective
chemical reaction of the species. Spalding regarded the EBU
model as a first crude attempt to account for the fact that the flame
aerodynamics often controls the combustion rate and the increased
surface area due to eddy stretching enhances the chemical reaction
rate. Further research by the IC group [70–76] led to considerable
refinement of the EBU model and, drawing from the ideas of
many previous authors, the ESCIMO model [77] was formulated.
ESCIMO assumes that turbulent combustion occurs when the
“coherent bodies of gas are squeezed and stretched as they travel
through the flame” in the shear regions of the flow. This can be
regarded as the first multiscale interpretation of turbulent combus-
tion suitable for embodiment in a CFD code. It provides a frame-
work to compute the local distribution of the scales of these
eddies and concentration of the fuel and oxidant within each fold
that accounts for molecular diffusion and chemical reaction as
they take place.
Although the ESCIMO did not gain the status of the EBU, its
key ideas were incorporated in later models. All flamelet models
use the same notion of distributed thin reaction layers; these use
the inverse of a time scale (the dissipation rate) rather than a
length scale (the fold width). Kerstein [78] developed a linear-
eddy model that employs an embedded one-dimensional calcula-
tion to account for mixing and chemical reaction at the scales that
cannot be resolved by the 3D mesh. Of course, unlike Spalding,
Kerstein computes the small scales in an Eulerian framework and
simulates turbulent mixing through stochastic one-dimensional
variable, rather than a local eddy distribution.
Multiphase Flow
As the interest of the IC group expanded to include multiphase
flows, the IPSA algorithm [79,80] was developed. It was essen-
tially the SIMPLE algorithm where the two fluids are fully mixed
yet retain some distinct identity. The interphase capillary pressure
is ignored so that there is a common fluid pressure but it is
assumed that the fluids can move independently. The two phases
can be simultaneously present in each control volume but the total
volume between the two phases, say gas and liquid, is preserved.
IPSA makes no distinction whether the phases are distributed as
droplets, films or plugs of fluid. Spalding later presented exten-
sions for removing the inherent weaknesses [81] of these simplify-
ing models.
The conservation equation for each phase is identical to the
single-phase equations but with additional terms to describe the
interphase exchange that can occur due to differential convection,
diffusion or sources and sinks. The velocity difference between
the two phases at a point can lead to a force term due to the shear
force exerted by one on the other. The temperature difference
between the phases can lead to interphase diffusive and convec-
tive heat transfer. The phase change can lead to source terms due
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to evaporation or condensation. These interphase terms strongly
depend on factors such as the surface area available, the size of
the droplets of the particulate phase, distribution of the phases
(droplets, films or plug flow) and a number of other parameters.
These interphase terms must be obtained from empirical data and
knowledge of the actual type of flow that might result for a given
multiphase situation.
The solution of the coupled conservation equations for the two
phases by the segregated method employed in the SIMPLE algo-
rithm results in numerical instabilities. To get around this problem
Spalding employed the Partial Elimination Algorithm that he had
previously developed for heat exchangers [80]. If the continuity
equation is solved in its mass-conservation form, a small error in
the mass of the denser fluid will leads to a comparatively magni-
fied error in the mass of the lighter fluid since often the density of
one fluid may be as much as a 1000 times the density of the other
fluid. To address this problem IPSA formulates the problem in
terms of a “volume continuity equation” [82] rather than the
mass-conservation equation. As in SIMPLE, the overall continuity
equation is solved by transforming it into a pressure-correction
equation for two phases. Though a single ‘shared’ pressure is the
default assumption, IPSA permits a distinct pressure for each
phase provided an algebraic relationship between individual phase
pressures is known.
IPSA was widely adopted and almost all multiphase CFD codes
today use an essentially similar methodology. In subsequent imple-
mentation of IPSA, the IC team introduced a number of innovations
including, the “algebraic-slip” between phases, drift-flux model for
multiphase, a diffusion term in the phase continuity equations to
account for the turbulent flux associated with fluctuating velocity and
volume fraction correlations. Later, Spalding used the two-phase and
the phase diffusion concepts to devise advanced turbulence and
combustion models [83–87] where the turbulent flow is assumed to
consist of a mixture of laminar and turbulent fluid parcels.
The CFD Decade—1965–1975: Impact
of the IC Technology
It was the decade spanning 1965–1975 that led to the emer-
gence of CFD as a practical tool. The developments that took
place at Los Alamos and Imperial College had an impact on engi-
neering practice that is incredibly broad and truly amazing. These
developments transformed CFD from an esoteric and mathemati-
cal branch of science to a practical, indeed essential, tool for engi-
neers. The confluence of factors that made it possible included:
(1) The electronic computer had become easily and widely
accessible.
(2) The basic methods and methodology of CFD had been
tested and honed at both Los Alamos National Laboratory
and at Imperial College.
(3) The Los Alamos group had shown that very complex prob-
lems (many related to weapons systems) could be success-
fully tackled.
(4) The Imperial College group had made unique contributions
to solve problems of practical interest to the engineering
industry that involved heat and mass transfer, turbulence
and combustion in addition to flow.
(5) The work on turbulence, by both groups, led to the first
major advancement in computation of turbulent flows since
Prandtl’s mixing-length theory.
(6) Important theoretical and computational advances were
made that made it possible to bring combustion, radiation,
multiphase flow and many other complex nonlinear physi-
cal processes within the gambit of CFD.
(7) Dissemination of the CFD know-how to the scientists and
engineers occurred through focused international conferen-
ces and training seminars.
(8) Books on CFD practice and CFD Tools began to be
published.
A general solution of the Navier–Stokes equations still remains
a dream but CFD has provided tantalizing glimpses of what is hid-
den in these equations. It certainly has provided incredibly useful
insight to tackle a vast range of practical problems. In some cases,
it has even minimized, if not eliminated, the need for the tradi-
tional wind tunnel testing that was the norm for engineering
design of yesteryears. CFD today is a engineering design tool that
is used in diverse fields of engineering and science. The technology
has matured to a point where it is employed even for nonscientific
fields such as games, entertainment, medicine, biology, ecology
and a number of other so-called soft-sciences. A companion paper
in this volume by Frank Harlow explores novel and unusual
application of CFD for sociopsychological sciences.
The IC group, under Spalding, developed a clear methodology
that unified the treatment of fluid flow, heat and mass transport,
by a single “universal” method. It was shown that in the context
of numerical solution, all second order transport equations can be
expressed and solved in the unified format of a generalized trans-
port equation. With the FVM, the IC group shifted the focus to the
“physical” rather than “mathematical” concepts of physics of flu-
ids. The former, such as fluxes, can be easily understood by engi-
neers as opposed to abstract concepts such as gradients. The IC
group also made bold attempts to go beyond the then “solvable”
problems to explore “unsolvable” problems such as turbulence,
combustion and multiphase flows. 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.
The IC group worked in a highly synergistic manner. The group
consisted of more than 30 researchers led by Spalding. It is per-
haps the largest CFD group that has ever focused on CFD under
the guidance of a single person. Spalding and his colleagues, Jim
Whitelaw and Brian Launder, together, with their students and
associates, used the emerging technology of CFD with targeted
experiments to advance the theoretical and experimental knowl-
edge base of fluid dynamics. Recently developed laser Doppler
anemometry (LDA) became a proven experimental technique for
measuring flows with the work of Whitelaw and his students. Sig-
nificant contributions to the theatrical and experimental data base
of turbulence were made by Launder, Spalding and their students.
Many from the CFD group during that decade have carried the
torch of “IC” methodology and gone on to make significant pro-
fessional contributions and a distinctive mark on the world stage.
Many well-known names in CFD, turbulence or combustion today
have a first or second generation IC connection.
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[71] Spalding, D. B., 1976, “Mathematical-Models of Turbulent Flames,” Combust.
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in the Properties of Free, Round-Jet, Turbulent, Diffusion Flames,” Combust.
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[79] Spalding, D. B., 1981, “IPSA New Developments and Computed Results,” Im-
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