American Journal of Computational Mathematics, 2011, 1, 271-280
doi:10.4236/ajcm.2011.14033 Published Online December 2011 (
Copyright © 2011 SciRes. AJCM
Bringing out Fluids Experiments from Laboratory to
in Silico—A Journey of Hundred Years
Manickam Siva Kumar1, Pichai Philominathan2
1Departmentof Ph ysi cs, Indian School Muscat, Muscat, Sultanate of Oman
2PG and Research Department of Physics, A VVM Sri Pusp ha m C ol l ege , Thanjavur, India
Received August 19, 2011; revised Septembe r 20, 2011; accepted Se pt em b e r 28, 2011
By making use of the developments in the fields of numerical methods, computational technology and fluid
dynamics models, computational fluid dynamics (CFD) progress forward to play an active role today in
various industrial, academic and research activities. In many cases, CFD simulations replace expensive and
time consuming laboratory experiments successfully by allowing engineers and scientists to capture pressure,
velocity and force distributions. Researchers are now able to test various theoretical conditions unavailable in
the laboratory and CFD studies help them to get deeper insights on existing theories. The century-old history
started just to solve some stress analysis problems numerically and today CFD methodology is being applied
not only in fluid dynamics also in chemical engineering, mineral processing, fire engineering, sports, medical
imaging and even in acoustics. This paper reviews the growth of CFD as a discipline and discusses its con-
temporary methodology.
Keywords: Fluid Dynamics, Computational Fluid Dynamics, Numerical Methods and Algorithms
1. Introduction
Fluid dynamics saw a rapid growth during 18th and 19th
century through the contributions from Bernoulli (1700-
1782) who derived Bernoulli’s equation and Leonhard
Euler (1707-1783) who described conservation laws
through his famous Euler equations. The introduction of
viscous transport into the Euler equation by Claude
Louis Marie Henry Navier (1785-1836) and George Ga-
briel Stokes (1819-1903) changed the scenario of fluid
dynamics by forming Navier-Stokes equation in which
the modern day Computational Fluid Dynamics (CFD)
based on.
In the first part of 20th century, much work was done
on improving theories of boundary layers and turbulence.
In particular Ludw ig Prandtl (1875-1953) gave boundary
layer theory and investigated the mixing length concept,
compressible flows, and introduced Prandtl number.
Theodore von Kar man (1881-1963) investigated swirling
vortices produced by the unsteady flow separation of a
fluid over bluff bod ies. Geoffrey Ingram Taylor’s (1886-
1975) statistical theory of turbulence and George Keith
Batchelor’s (1920-2000) theory of homogeneous turbu-
lence gave remarkable improvements to our understand-
ing on flui d dynamic s.
The roots of modern computational fluid dynamics
dates back to 1910 when Richardson published his paper
on the computation of stress in masonry dam using hu-
man computers [1]. In 1947, Kopal compiled massive
tables of the supersonicflow over sharp cones by nu-
merically solving governing differential equations using
primitive computers [2]. Kawaguti (1953), by working
20 hours per week for 18 months (cited as: “a consider-
able amount of labour”) obtained a solution for flow
around a cylinder [3].
During 1960s, theoretical division at Los Alamos con-
tributed many numerical methods. In 1965, Horlow and
Fromm in their article in Scientific American stated very
first time the idea of in silico experiments-experiments
on computers [4]. Roache’s text book on CFD made sci-
entists and engineers to realize the need for keeping CFD
as a separate discipline from theoretical and experimental
fluid dynamics [5]. Till today CFD is heading with phe-
nomenal progress with the help of growing computing
power. Figure 1 gives the timeline history of important
milestones in CFD.
In 1980, Suhas V. Patankar’s contribution to SIMPLE
Semi-Implicit Method for Pressure-Linkage Equations) (
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Figure 1. Some important milestones in CFD.
algorithm and his ground breaking CFD book Numerical
Heat Transfer and Fluid Flow [6], made him one of the
most cited authors in science and engineering [7].1980’s
and 1990’s saw a revolutionary change in CFD—the
availability of commercial CFD codes removed the prac-
tice of writing program to perform CFD calculations.
Today, CFD is considered as a better alternative for
experimental investigation to test theoretical conditions
unavailable in the laboratory and to examine prototyp ing,
optimizing design and checking industry compliance etc.
[8]. In particular, the reasons for growing importance to
CFD in the field of engineering can be attributed to:
1) Quick forecast of performance;
2) Better alternative to costly and impossible experi-
3) Better insights unlike experiments with expensive
4) Availability of faster computational speed and lar-
ge r memory size.
This paper reviews CFD in the historical perspective
pertaining to laminar flow. As the body of literature is
very vast in this field, the application of CFD to thermal,
electromagnetic, acoustic and other fields was not taken
into account in this work.
2. The Overview of Present CFD
The present approach to CFD methodology is to consider
it as a numerical experiment which is modeled using go-
verning equations and observed through running a cho-
sen algorithm [8]. The final step, similar to any conven-
tional fluids experiment, is to interpret results and ana-
lyze (Figure 2).
In continuum mechanics non-conservation form of
equations governing fluid’s kinetic energy (K), internal
energy (U), mechanical power (M) and heat energy (Q)
can be written as[9],
+ M + Q
DtDt (1)
The conservation form of equations, so-called Na-
vier-Stokes equations, is given in terms of conservation
flow variables (U), convection flux variables (Fi), diffu-
sion flux variables (Gi) and source terms (B) is given by
tx x
 
 
UB (2)
CFD involves converting these partial differential
equations into discretized algebraic equations. These set
of equa tions are numerically solved at a given poin t/time
to get flow parameters. Any complex geometry of the
flow can be converted into grid or mesh with discrete
points where the flow variables are solved. Using appro-
priate interpolation schemes, flow variables are obtained
at non-grid point locatio ns also. Th e fo llowing flow chart
(Figure 3) details the steps involved in the contemporary
CFD methodology [10].
3. Discretization
The literature on discretization methods is huge and ex-
tensive and therefore the most common discretization
methods that are important for CFD,are discussed briefly
in this section.
The governing partial differential equations are con-
verted into a set of algebraic equations using the discre-
Figure 2. Fluids expe riment versus CFD studies.
Figure 3. The contemporary CFD methodology.
tization process (Figure 4) which describes continues
flow field in terms of discrete values at prescribed loca-
tions. Discretization offers the following advantages over
infinite unknowns become finite,
analytical approach is converted into numerical,
applicability is widened and
approximation becomes possible.
3.1. Finite Difference Method (FDM)
Finite Difference Method is based upon the differential
form of the PDE to be solved and it employs global
mapping of geometry. It is one of the oldest discretiza-
tion schemes [1]. Thom (1933) gave the first ever nu-
merical solution for flow over a circular cylinder [11].
This methodology involves the following steps (Figure
1) The infinite set of points is replaced by finite set of
points called nodes.
2) Navier-Stokes equations are enforced at these
3) Differential equation converted into stencils at mesh
4) Stencils relate velocity and pressure values
FDM is easy to implement in CFD analysis and pro-
vides discrete solution . But it is restricted to simple grids
and does not conserve momentum, energy, and mass on
coarse grids. Higher order FDM is hard to be locally
conservative.Modern FDM codes make use of overlap-
ping grids, where the solution is interpolated across each
3.2. Finite Element Method (FEM)
Finite Element Method is based upon an integral form of
the PDE to be solved and it uses local geometric map-
ping. Though Courant (1943) applied this method to
solve torsion problem, the name was given by Clough in
1960 [12]. For analyzing structural mechanics problems,
this method was refined greatly in 60’s and 70’s and ap-
plied for fluid flow in late 70’s. This methodology pro-
vides a continuous solution up to a point with local ap-
proa ch. The steps involv ed are (Figure 5(b)):
Figure 4. The process of discretization.
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1) The flow field is represented as large number of
known functions ;
2) By using Navier-Stokes equations, the one with the
best approximation properties is selected;
3) The domain is divided into elements in which the
candidate functions are constructed from interpolation
4) The value of the function at the nodes determines
the value of the function everywhere inside the element;
5) The mesh is formed by combining all the elements.
The main advantage of this method in CFD point view
is high adaptability an d accuracy on co arse grids. FEM is
excellent for diffusion dominated problems, viscous and
free surface problems. But FEM cannot handle fluid me-
chanics equations effectively and therefore this method
has limited accuracy, slow for large problems and not
well suited for turbulent flow.
3.3. Finite Volume Method (FVM)
Finite Volume Method is based upon a piecewise repre-
Figure 5. Discretization method s (a) FDM; (b) FEM; and (c)
sentation of the solution in terms of specified basis func-
tions. Evans & Harlow (1957) well documented the first
use [13]. During late 70’s body fitted gr ids an d early 90’s
unstructured grid methods had appeared. FVM is con-
sidered as integral version of FDM and can be derived
also from FEM [8]. This methodology involves the fol-
lowing steps to be followed (Figure 5(c)):
1) The volume of the fluid is divided into a finite num-
ber of volumes, or cells;
2) Navier-Stokes equations are converted into equiva-
lent integral form and applied to each cell;
3) The local form of the governing equations balances
mass and momentum fluxes across the faces of each in-
dividual cell.
There are efficient and well developed solvers are
available using FEM, though it is applicable on coarse
grids, there are issues like false diffusion, difficult stabil-
ity and convergence analysis.
3.4. Other Methods
Boundary Element Method (BEM) is used for potential
flows where the integrals over the whole domain are
transformed over the boundaries [14,15]. This method is
also called as panel method since each element plays the
role of a panel on the surface of airfoil. BEM requires
less data, less time and since discretization takes place on
the surface, system of equations are smaller. BEM is
effective for external flows such as potential and stokes
flows. At the same time, this method is not good for
non-linear flow problems and involves complex mathe-
matics different from other CFD schemes.
Coupled Eulerian-Lagrangian (CEL) method is an-
other method used widely in CFD. In the case of moving
mesh points along with the fluid particles, Lagrangian
coordinates have to be used in computing variables.
Therefore it is convenient to have both Eulerian and La-
grangian coordinates coupled, called as the CEL method
[8]. This method combines advantages of both Eulerian
and Lagrangian methods and useful in highly distorted
and multiphase flows. But heat transfer analysis is lim-
ited and mass scaling is not supported. Due to approxi-
mations, the corners of solids are rounded.
The modern Spectral Methods (SM) was first pro-
posed by Gottlieb and Orszag [16 ]. This meth od invo lv es
multi-dimensional discretization which is formulated as
tensor products of one-dimensional constructs in or-
thogonal simply-connected domains. Such methods are
broadly classified as collocation methods and Galerkin
methods. Sun et al. (2006) proposed another improvised
method combining with finite volume, called spectral
finite volume method for 3D space [17]. It takes a global
approach and has exponential convergence which leads
Copyright © 2011 SciRes. AJCM
to high accuracy CFD analysis. But it can handle only
simple geometries and applicable for limited boundary
conditions. Huynh (2007) has extended spectral method
using flux reconstruction approach and applied to high-
order schemes successfully [18].
Unlike other methods, Lattice Boltzmann Methods
(LBM) models the fluid consisting of fictive particles
[18]. It makes use of particles on hexagonal grids where
particles move according to discrete rules. The macro-
scopic motion of the particles resembles the Navier-
Stokes equation. This method is efficient in handling
with complex boundaries, microscopic interaction and
parallelization of the algorithm [19]. But its main disad-
vantage is rrequirement of higher order terms in solving
compressible flows and coupling density to temperature
The smoothed particle hydrodynamics (SPH) makes
use of mesh-free method and involves modeling fluid as
particles and smoothening them using kernel function
[19]. This method is capable of simulating flow in real
time, but its limitation includes requirement of large
number of pa rticles for better resolution.
For fine grids each type of method gives the same so-
lution. But some methods are more suitable to specific
cases than others and the preference is determined by the
attitude of the develop er [10].
4. Meshing
Grid or mesh is a discrete representation of the geometric
domain where problem is to be solved. Mesh divides the
solution domain into subdomains like nodes, elements
and control volumes.Since from the first attempt of ob-
taining numerical solutions to partial differential equa-
tions, the concept of mesh generation has been ass ociate d
with computational methods [20]. Slowly mesh genera-
tion steadily evolved as a separate discipline drawing on
ideas from mathematics and computer science. In 1950’s
FDM was put into use through two dimensional simple
boundary shapes. By using coordinate transformations
mesh that is aligne d wi th b ou ndaries was in use [21].
Due to the development of FEM which involved com-
plex boundaries, manual mesh creation was used during
late fifties and early sixties [22]. In order to apply FDM
for their calculations, CFD community started using me-
shes with simple generic shapes like a rectangle or circle
to represent complex geometry [23]. Since then CFD
became key drivers in simulating the development of
various mesh generating techniques. Though FEM is
powerful and versatile, the quality of mesh can greatly
affect the accuracy of result. Manual generation of high
quality mesh is time consuming and error-prone. It is
important to note that the appropriate type of mesh and
its resolution is problem dependent.
Various mesh strategies were in use during the last
few decades. For example, in 70’s algebraic methods,
80’s multiblock types and 90’s composite and overset
methods were used for generating mesh. Thereare large
number of types of mesh available as of today (Figure
In order to generate various types of mesh, there are
large number of methods available. Some of them in-
clude: mesh topology first (mesh smoothing [24]), nodes
first (topology decomposition, node connection [25]),
adapted mesh template (grid-based [26], mapped element
[27], conformal mapping [28]) and simultaneous nodes
and elements approach (geometry decomposition [29]).
The mesh generation itself is becoming a separate dis-
cipline and more detailed methods of mesh generation
can be found in various dedicated texts [30-32]. It is
also important to note now-a-days CFD users have a
choice of number of mesh generating software such as
Gambit from Ansys (, Gridgen from
Pointwise ( and Gridpro from PDC
( At the same time, many commercial
CFD applications have built-in and user-friendly mesh
generators. (e.g.: Solidworks flow simulation from DSS
inc. (, Flow-3D from Flowscience
( etc).
5. Boundary Conditions
While solving the Navier-Stokes and continuity equa-
tions, boundary conditions need to be applied. Boundary
conditions for fluid flow are generally complicated due
to coupling of velocity fields with pressure distribution.
Incorrect, over or under boundary condition will lead to
wrong result s .
In CFD analysis there are two types of requirements
regarding boundary conditions. Some variables will take
Figure 6. Some available mesh types.
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a constant value at the boundary (Dirichlet condition)
while some variable may have constant gradients (Neu-
mann condition s). In one dimensions, as FEM, FDM and
FVM provide identical final algebraic equations, there-
fore Dirichlet boundary conditions will give same results
for all these methods [8]. Neumann boundary conditions
are approximated before applying in FDM, but applied
exactly in FEM and FVM.We can also apply mixture of
Dirichlet and Neumann as a boundary condition. It im-
portant to note that at a given boundary, different variables
may be prescribed with different boundary conditions.
The general boundary conditions used in CFD studies
include pressure inlet and outlet, velocity inlet and out-
flow conditions. Compressible flows boundary condi-
tions include mass flow inlet and pressure far-field. In
addition with these there are special boundary conditions
like inlet/outlet vent, intake/exhaust fan etc. Many com-
mercial solvers allow users to set boundary conditions
easily (Figure 7).
Wall boundary conditions are applied while using
bound fluid and solid regions. In viscous flows, no slip
(tangential velocity is equal to wall velocity and normal
velocity is zero) conditions are usually applied. For tur-
bulent flows, wall shear stress and wall roughness can be
defined. For moving wall boundary conditions sliding or
moving mesh techniques are used [10].
In the case of symmetric flow field and geometry, us-
age of symmetry boundary conditions reduces computa-
tional effort during CFD simulation. No inputs for these
boundary conditions are required but one has to define
boundary locationscorrectly. Such boundary conditions
are used in modelling slip walls in viscous flow. In case
of periodically repeating flow pattern, periodic bounda-
ries reduce computational load in CFD simulation s.
6. CFD Solvers
The process of discretization finally brings out set of
Figure 7. Boundary conditions available in Flow 3D © com-
mercial flow solver v 9.3.
coupled algebraic equations which may be linear or
non-linear. Irrespective of the method of discretization
these equations should be solved for a discrete solution
using direct or iterative methodologies. Direct methods
make use of standard methods of linear algebra. Matrix
methods are employed in order to devise efficient solu-
tion techniques. Iterative methods involve guess-and-
correct methodology till solution is converged. One of
the simplest iterative methods is Jacobi method involving
matrix diagonlization. Gauss-Seidel method or Liebmann
method is twice as fast as the Jacobi method, makes use
of successive displacements in solving set of linear equa-
tions. But owing to slow convergence, they are not used
in practice.
In any typical present day CFD simulations, there are
multi -dimens ions with larger nu mber of g rid poin ts. There-
fore both Jacobi and Gauss-Seidel becomes ineffective
and expensive. Peaceman and Rachford (1955) proposed
Alternating Direct Implicit (ADI) method which consid-
ers multidimensional problem as a set of low dimen-
sional problems [33]. Stone (1968) proposed Strongly
Implicit Procedure (SIP) which involves matrix appro-
ximation and Lower-Upper factorization [34]. This is used
in some commercial CFD codes to solve non-linear
equations in the case of multigrid methods.
Patankar and Spalding (1972) proposed a class of it-
erative methods called Semi-Implicit Method for Pres-
sure-Linkage Equations (SIMPLE) for coupling pres-
sure-velocity for an incompressible flow [35]. This is
available today in almost all commercial codes (Figure
8). Followed by this, Van Doormal and Raithby (1984)
proposed SIMPLEC (SIMPLE-Consistent) which omits
less significant terms in velocity correction equation [36].
Issa (1986) proposed Pressure Implicit with Splitting of
Operators (PISO) algorithm, which involves an addi-
tional pressure correction equation for speedy conver-
gence [37]. These solvers are readily available for the
users in various commercial CFD software (Figure 9).
At the same time, multigrid methods offer fastest nu-
merical algorithms for solving systems of equations [38].
Geometric multigrid methods offer optimal scaling and
memory cost, but highly depending on geometric infor-
mation with trouble in plug-in black-box. Based on same
principle, algebraic methods have been proposed with is
robust and ideal without any other information. Unlike
other methods, this is abstract, complex and involving
repeat overhead costs.
Saad and Schultz have developed general minimum
residual method (GMRES) algorithm to solve nonsym-
metric linear system of equations [39]. This algorithm is
capable of non-linear scaling and preconditioning for
better performance. It is now considered as very robust,
memory intensive and easy to parallelize therefore used
widely in many commercial codes.
Copyright © 2011 SciRes. AJCM
Initial guess
Step1: Solving
momentum equations
Step2: Solving
pressure correcti on
Step3: Correcting
pressure velocity
Step4: Solving other
transport equations
Replace Initial
Figure 8. SIMPLE algorithm.
Figure 9. Solvers available in ANSYS © FLUENT V13.0.0.
Irrespective of the procedures used, the solutions ob-
tained should be accurate (free from modelling and trun-
cation errors), consistent (to ensure refines mesh/time
step yield more accurate results) and stable (single, mesh
independent solution).
7. Post-Processing
The output of computational results of CFD study is usu-
ally colourful and vivid. Usually almost all CFD study
results are presented graphically through one of the fol-
lowing categories:
1) XY plots (time/iterative history of residuals and for ces);
2) 2D/3D contour plots (dynamic-, absolute-, and to-
tal-pressure, velocity, vorticity, eddy viscosity etc.);
3) 2D/3D velocity vectors;
4) 3D Iso-surface plots (pressure, vorticity magnitude,
Q criterion etc.);
5) Streamlines, pathlines and streaklines etc;
6) Animations.
Various techniques have been used to analyse and
visualize the fluid parameters of the study. Early visuali-
zations made use of glyphs to represent vector fields in
the data [40]. Later, line integral convolution (LIC) and
texture based approaches [41] were used to depict realis-
tic flow. Feature based visualization were also used on
extracted features directly to analyse flow data. Accurate
core feature detection algorithms [42] were widely used
in many commercial codes. During post processing de-
rived variables (vorticity, shear stress etc.) integral vari-
ables (forces, lift/drag coefficients) and turbulent quanti-
ties (Reynolds stresses, energy spectra) are calculated.
Figure 10 shows a typical post processed results using
CFD methodology [43].
Almost all commercial CFD applications have their
built-in GUI visualization tools. In addition with them
there are some standalone CFD visualisation tools like
ENSIGHT ( are available today. GNUPLOT
( is another open-source popular plotting
package in CFD industry.
8. Conclusions
The application domain of CFD is widening with the
advancement with accurate numerical methods, compu-
tational models and visualisation techniques. CFD that
was just meant for fluid dynamics, now it is used as re-
search tool (to test theoretical advances unavailable in
the lab), educational tool (to teach thermal and fluid sci-
ences), design optimization tool (in aerospace and auto-
mobile industry). It has been also extended in studying
chemical, mineral processing and civil, environmental
engineering, power generation a nd even sport s .
With the exorbitant growth of computer simulation
technology, CFD is evolving rapidly with more tech-
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Figure 10. Top: flow trajectories past sphere at Reynolds number 150. Bottom: post processed results showing velocity con-
tours across a cross-section of a rotating solid helix in viscous fluid at zero Reynolds number [43].
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creasing hardware costs and increasing processor speeds
challenge CFD developers to help in what is called zero
prototype engineering.
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