Natural convection in a differentially heated cavity plays a major role in the understanding of flow physics and heat transfer aspects of various applications. Parameters such as Rayleigh number, Prandtl number, aspect ratio, inclination angle and surface emissivity are considered to have either individual or grouped effect on natural convection in an enclosed cavity. In spite of this, simultaneous study of these parameters over a wide range is rare. Development of correlation which helps to investigate the effect of the large number and wide range of parameters is challenging. The number of simulations required to generate correlations for even a small number of parameters is extremely large. Till date there is no streamlined procedure to optimize the number of simulations required for correlation development. Therefore, the present study aims to optimize the number of simulations by using Taguchi technique and later generate correlations by employing multiple variable regression analysis. It is observed that for a wide range of parameters, the proposed CFD-Taguchi-Regression approach drastically reduces the total number of simulations for correlation generation.

1. nd
th
Proceedings of the 22 National and 11 International
ISHMT-ASME Heat and Mass Transfer Conference
December 28-31, 2013, IIT Kharagpur, India
HMTC1300663
TAGUCHI BASED METHODOLOGY TO GENERATE CORRELATION VIA NUMERICAL ANALYSIS
FOR NATURAL CONVECTION IN A DIFFERENTIALLY HEATED CAVITY
D. Jaya Krishna
Assistant Professor
Department of Mechanical Engineering
BITS PILANI Hyderabad campus,
Hyderabad (INDIA)
djayakrishna.iitm@gmail.com
ABSTRACT
Natural convection in a differentially heated cavity plays a
major role in the understanding of flow physics and heat
transfer aspects of various applications. Parameters such as
Rayleigh number, Prandtl number, aspect ratio, inclination
angle and surface emissivity are considered to have either
individual or grouped effect on natural convection in an
enclosed cavity. In spite of this, simultaneous study of these
parameters over a wide range is rare. Development of
correlation which helps to investigate the effect of the large
number and wide range of parameters is challenging. The
number of simulations required to generate correlations for
even a small number of parameters is extremely large. Till
date there is no streamlined procedure to optimize the number
of simulations required for correlation development.
Therefore, the present study aims to optimize the number of
simulations by using Taguchi technique and later generate
correlations by employing multiple variable regression
analysis. It is observed that for a wide range of parameters,
the proposed CFD-Taguchi-Regression approach drastically
reduces the total number of simulations for correlation
generation.
NOMENCLATURE
AR
Avg.
ER
g
H
L
Nu
OA
P
Pr
Ra
T
u
aspect ratio
average
error
acceleration due to gravity, (m/s2)
cavity height, (m)
cavity length, (m)
Nusselt number
Orthogonal array
pressure, (N/m2)
Prandtl number
Rayleigh number
temperature, (K)
component of velocity in x-direction
Ruturaj R. Deshpande
M.E Thermal
Department of Mechanical Engg.
BITS PILANI Hyderabad
campus, Hyderabad (INDIA)
ruturaj1988@gmail.com
v
component of velocity in y-direction
X, Y coordinates axis
Greek symbols
Coefficient of thermal expansion, (1/K)
Density, (kg/m3)
Kinematic viscosity, (m2/s)
Subscripts
c
cold
cor
correlation
E
equally spaced levels
h
hot
R
randomly spaced levels
Ref
reference value
sim
simulation
Superscripts
i, i-1 present and previous grid size
1 INTRODUCTION
The phenomenon of Natural convection in enclosures filled
with fluid is an important study. It finds wide applications in
solar collectors, double walled insulations, air circulation in
buildings, electronic cooling systems, geophysics and nuclear
science applications [1]. The importance can be quickly
realized from the fact that, the amount of research carried in
the last 30-40 years related to this phenomenon is enormous.
A brief review of these studies is documented by Ostrach [2].
Natural convection in enclosures where the side walls are
maintained at different temperatures and remaining sides
being insulated has attracted many researchers to carry out
research using both experimental as well as numerical
methods. The contribution of de Val Davis [3], and Hortmann
et al. [4] using numerical methods for square cavity are worth
mentioning. These results are being used by several
researchers for benchmarking their numerical studies. Ramesh
[5] has considered the effect of surface radiation in his
experimental work related to natural convection and has
shown that natural convection in enclosures is not limited to a

2. particular set of parameters but may contain extra influential
parameters. Successful attempts were made by Berkovsky and
Poleviko [6], and Balaji and Venkateshan [7] to develop
correlations considering a fairly wide range of parameters to
have a better understanding of the phenomenon.
To develop a correlation, large number of
experimentssimulations is to be carried out and each
experimentsimulation follows a plan. Each plan has
predetermined values of parameters, and a method in which all
possible experiments are considered is known as full factorial
design. For a full factorial design, the number of
experimentalsimulation runs to be carried out is extremely
large. There is an additional toll on the resources to carry out
experimentssimulations and to handle large quantity of data a
highly evolved statistical tools are required. With the above
mentioned disadvantages conventional full factorial design in
any type of engineering analysis is almost impossible [8].
Various attempts were made to come up with a fractional
factorial design to overcome the disadvantages of full factorial
design. The most successful and widely used methods are
Taguchi method, Central Composite Design (CCD), Optimal
Space Filling Design (OSFD), and Box-Behnken Design
(BBD). These methods were developed for a specific problem.
Over a period of time these are evolved into more robust, userfriendly and accurate methods to study multiple parameters
affecting the dependent parameter [9].
Taguchi method for design of experiments was developed
after the World War II in Electrical Communication
Laboratories (ECL-JAPAN) by Dr. Taguchi. The motivation
behind the development of Taguchi method was to optimize
the process of engineering experimentation. Originally the
Taguchi method was used for quality control. This method
was later used by companies like Ford Motor for quality
control and product improvement [8].
Taguchi method along with Analysis of Variance
(ANOVA) is successfully used by researchers to cut down on
the number of experimental runs with the help of orthogonal
arrays [10-12]. Till date a full theory that includes all the
possible number of plans is not yet developed [13]. A set of
orthogonal arrays which serve as a plan for experiments to be
carried out can be prepared by the use of computer codes.
However, one can also use orthogonal arrays available at
various resource libraries [14]. Even though selected and less
number of experiments are carried out, conclusions drawn
from Taguchi-ANOVA analysis is valid over an entire
experimental region spanned by the independent factors.
The first attempt to apply the Taguchi method to
investigate the statistical performance of CFD solver and its
interaction was done by Stephen [15]. Taguchi method was
used by Follett et al. [16] to optimize different nozzle
configurations for bell annular tri-propellant engine. This
approach was adopted to overcome difficulties such as little or
no previous experience, time constraints, and nearly infinite
number of parameters. Juan and Piniella [17] used CFDTaguchi method to study flow over four digit NACA airfoil
using signal to noise ratio (S/N) concept. S/N concept was
used by modifying the Reynolds number reported by NACA.
Mousavi et al. [18] used Taguchi method to effectively
optimize biomass production. Zeta Dynamics Ltd. used
Taguchi method in conjunction with the CFD package
PHOENICS-VR 3.4 to find the optimum design of a piece of
hardware known as the Zeta Linear Coanda Unit (ZLCU) [19].
Jafari et al. [20] used CFD-Taguchi method for evaluation of
the controllable parameters effect on thermomagnetic
convection in ferro-fluids. Recently, Ravinder [21]
successfully optimized cold spraying of Hydroxyapatite using
CFD-Taguchi-ANOVA approach where Taguchi orthogonal
array helped to plan the simulations.
Once all the experimentalsimulation outcomes are noted,
a mathematical tool is required for the analysis and
development of correlation. Regression analysis serves as a
tool to find out the relationship between dependent and
independent parameters. It also helps to predict and explain
the behavior of the dependent variable. However, the
computation required for multivariable regression analysis is
highly complicated, and is performed using computer codes.
Various models are used for regression analysis such as linear,
exponential, weighted averages, parabolic, hyperbolic etc. An
incorrectly specified model may cause biased estimates of
parameters. An initial model may be based on a combination
of educated guesses, availability of data, and observation from
the scatter plot. The model selected for regression analysis
may not be optimal for several reasons some of them are: (i)
Model may contain too many variables and is said to be over
specified (ii) Model may not contain right variables (iii) The
relation between parameters differ from the one used in the
model. If the model is inadequate for either or both of the last
two reasons then the model exhibits specification error i.e. the
inputs for regression analysis are not correct. It is difficult to
judge the error due to omission of variables and can hardly be
avoided. Parity plots are useful to find out errors due to
mathematical relationship. A general thumb rule is that the
spectrum of Parity plot must be as small as possible [22].
Normally 4(independent parameters) are the number of data points
required to come up with a fairly accurate correlation [23].
This shows that even for a small number of parameters the
number of required data points is huge. Traditionally Taguchi
method is used for the optimization of parameters by
performing very few experimental runs. Application of
Taguchi method to understand the phenomenon itself is rarely
explored. Use of CFD-Taguchi-Regression analysis will not
only help to optimize the whole phenomenon but also to
understand and predict the behavior of dependent parameters.
Perhaps for the first time, an attempt is being made, to use
CFD-Taguchi-Regression analysis to come up with a
systematic approach for correlation development. The success
of this approach lies in the realization of a fairly accurate
correlation with less number of simulations than the number
which is conventionally suggested. In this study the numerical
analysis for a differentially heated cavity has been carried out
using a commercial software Ansys-Fluent. The independent
parameters considered for the study are Rayleigh number
(103≤Ra≤106), Prandtl number (0.71≤Pr≤10), and aspect ratio
(1≤AR≤10). The dependent parameter under study is Nusselt
number.
2 MATHEMATICAL MODEL
A schematic representation of a differential heated cavity is
shown in Fig. 1, where L and H are the length and height of
the cavity in X and Y directions respectively .The standard
gravitation field is acting in the negative Y-direction.
2.1 Governing Equations
The flow and heat transfer solutions are obtained by
solving a simultaneous system of Eqs. (1- 4) describing the
conservation of mass, momentum and energy.
Continuity equation
(1)

3. than 0.5%. Table 1 provides details pertaining to mesh sizes
employed in the present study.
X-Momentum equation
(2)
Y-Momentum equation
Energy equation
(
) (3)
(4)
2.2 Boundary Conditions
The left and right walls of the cavity shown in Fig. 1 are
maintained at Tc and Th respectively. Lower and upper walls
of the cavity are insulated. No-slip condition is applied at the
walls.
Velocity Boundary conditions.
(5)
(7)
To check the consistency of the present numerical
methodology, comparison of the simulation results is carried
out with the results available in the literature [3, 24-26]. Table
2 shows an extensive comparison of the present results with
benchmark data for an average Nusselt number. From Table 2
it may be noted that the present numerical methodology is
rigorously validated and the results agree very well with the
results available in the literature. Figure 2 provides isotherms
(top) and stream function (bottom) for Prandtl number 1(left)
and 10 (right) for a fixed Rayleigh number (106) and aspect
ratio (5). It may be noted that with the increase in Prandtl
number a marginal increase in vortex strength may be
observed which in turn lead to the marginal increase of
Nusselt number. To study the influence of aspect ratio on
thermal hydraulics Fig. 3 is plotted for AR=1 and 10 for a
fixed Rayleigh number (106) and Prandtl number (5.35). It
may be noted that with the increase in aspect ratio in spite of
increase in vortex strength the Nusselt number is observed to
reduce due to the increase of the boundary layer.
Table 1. MESH SIZES CONSIDERED TO GENERATE
RESULTS
Temperature Boundary conditions
(6)
AR
Mesh
1
90×90
2
90×180
4
8
90×280
90×320
10
90×320
For lower and upper walls
Table 2. COMPARISON OF PRESENT RESULTS WITH
LITERATURE
S.
No
1
103
0.71
1
1.114
1.116
[3]
2
Figure 1. SCHEMATIC REPRESENTATION OF
DIFFERENTIAL HEATED CAVITY
Ra
10
4
0.71
1
2.198
2.234
[3]
10
5
0.71
1
4.494
4.510
[3]
10
6
0.71
1
8.805
8.798
[3]
5
10
3
0.71
4
1.110
1.090
[24]
6
105
0.71
8
3.373
3.390
[24]
7
10
3
10
1
1.113
1.141
[25]
8
10
6
10
1
9.236
9.232
[25]
9
104
7
8
1.843
1.844
[25]
0.71
10
1.68
1.706
[26]
3
4
The governing equations are solved by using finite volume
method. SIMPLE algorithm is employed for pressure-velocity
coupling. The convective terms of the momentum and energy
equations are discretized using the second order upwinding
scheme. Convergence is assumed to be achieved when all the
residuals are reduced to 10-6.
2.3 Grid Independence and Validation
Careful study is carried out to establish grid independence
using Eq. (7), where represents the average Nusselt number,
i and i-1 represents present and previous grid size respectively.
A non-uniform grid with suitable growth factor is considered.
Grid independence is assumed to be achieved when the
percentage deviation for the average Nusselt number is less
10
10
4
Pr
AR
Average Nusselt
number
Present Literature
Study
Reference

4. Figure 3. COMPARISON OF TEMPERATURE CONTOURS
(LEFT) AND STREAM FUNCTION (RIGHT) FOR (a) AR=1; (b)
6
AR=10 WITH FIXED Ra(10 ) AND Pr (5.35).
3 RESULTS AND DISCUSSION
3.1 Taguchi Method
Figure 2. COMPARISON OF (a) TEMPERATURE
CONTOURS ; (b) STREAM FUNCTION CONTOURS FOR
6
Pr=1(LEFT) AND Pr=10 (RIGHT) WITH FIXED Ra(10 ) AND
AR (5).
The smallest Orthogonal Array (OA) suggested by the
Taguchi method for 3 independent parameters is L-4 OA. The
number of data points available from the L-4 OA is 4 but
considering the complex nature of the problem these are very
few in number. Formulation of correlation based on the L-4
OA array is mathematically impossible for 3 independent
parameters and one dependent parameter which are related in
a non-linear fashion. The next smallest OA suggested by
Taguchi is L-9 OA. The L-9 OA has 9 data points i.e. it
suggests 9 simulations to be carried out which are less than the
number of simulations suggested by conventional analysis
procedures (4(independent parameters) = 64) [23]. In the present study
selection of values for different levels for each independent
parameter is carried out by following two rules: (i) Simulation
plan must have combinations of extreme levels of considered
independent factors. If not, extra simulations are to be planned
with all levels set to extreme values (ii) Selection of levels
other than extreme levels should be unbiased. This is achieved
by choosing equally spaced intervals or randomly generated
levels. Table 3 gives different levels for independent
parameters using equal interval and also by using random
level. Since the original L-9 OA does not suggest a simulation
when all the levels are set to 3 an additional simulation i.e.
number 10 is carried out.
Table 3. EQUAL AND RANDOM INTERVAL LEVELS
Ra
Pr
3
3.16×10
10
6
6×10
10
6
0.71
5
Random
2
4
Equidistant
Random
103
Random
Equidistant
103
Equidistant
Levels
1
AR
0.71
1
1
5.35
7
5.5
5
10
10
10
10

5. 3.2 Regression Analysis
As mentioned earlier multivariable regression analysis is
considered to be more complicated and is performed using
computer codes. Hence, Regression analysis is carried out
using commercial code-LAB Fit. The input given to Lab Fit
are (i) Tabular data containing the values of dependent and
independent variables and (ii) The form of the equation to be
fitted to this data [27]. The general selection of the particular
form of an equation is based on either pure guess or literature
or on the trend of dependent variables. It may be noted that the
present study focuses to provide a streamlined methodology to
optimize the number of simulations/experiments to generate
correction. Therefore to avoid the influence due to the
form/type, the following equations i.e. Eqs. (8): power law,
(9): Berkovsky and Poleviko [6] and (10): modification of Eq.
(9) have been employed for the data provided in Table 4.
Even though three different forms of equations are used to
carry out regression analysis, Fig. 5 shows the inability of
correlation suggested by equation 10R to accurately predict the
Nusselt number over the whole range of parameters. This
helps to eliminate the form of equation suggested by Eq. (10).
The correlation developed from Eqs. (8 and 9) seem to be
valid for the entire range of independent parameters, the
analysis of parity plot shows Eq. (9) is the best equation
available after regression analysis. It is to be noted that the
correlation suggested as Eqs. (8E, 9E, 8R, and 9R) are valid for
the entire range of the independent parameter. In order to
further test the validity of obtaining correlations (Eqs. 8 E, 9E,
8R, and 9R) 10 more simulations have been considered with
randomly generated parameters.
(8)
(9)
(10)
Where a, b, c, d and e are constants. The constants for Eqs. (8
–10) are obtained by employing equidistant values
(represented by suffix E) and also by randomly spaced values
(represented by suffix R) for independent variables. The
values pertaining to the generation of correlation may be
referred from Table 3. The constants for Eqs. (8-10) are
obtained through regression analysis by carrying out
simulations based on L-9 OA (Table 4).
It may be noted that the curve fitting using multiple linear
regression analysis would approximately require 4 n solutions,
where ‘n’ is the number of independent parameters [23]. As
the present study has three independent parameters based on
multiple linear regression analysis we would require a
minimum of 64 solutions. But by making use of Taguchi’s L9OA we could make use of only 10 solutions to generate
correlations. The correlations thus obtained are provided in
Table 5. Figs. 4 and 5 show the parity plot for Eqs. (8 – 10)
developed based on equidistant levels (8E, 9E and 10 E) and
random levels (8R, 9R and 10 R).
Figure 4. PARITY PLOT FOR EQUATIONS DEVELOPED
FROM EQUIDISTANT LEVELS
Table 4. L-9 OA FOR EQUALLY SPACED AND RANDOMLY
GENERATED LEVELS
Simulation
plan no
Ra
Pr
AR
1
1
1
1
2
1
2
2
3
1
3
3
4
2
1
2
5
2
2
3
Eq. no
6
2
3
1
8E
7
3
1
3
9E
8
3
2
1
9
3
3
2
10
3
3
3
Figure 5. PARITY PLOT FOR EQUATIONS DEVELOPED
FROM RANDOM LEVELS
Table 5. DIFFERENT FORMS OF CORRELATIONS
OBTAINED THROUGH REGRESSION ANALYSIS
10E
Equation/correlation
(
(
)
)

6. 8R
(
9R
10R
(
)
traditional multiple linear regression analysis. The
independent parameters have been considered based on both
equidistant and random values for Taguchi’s orthogonal array.
Three different forms of equations (Eqs. (8 – 10)) have been
considered. Based on the study the following conclusions may
be drawn.
)

The values for average Nusselt number for 10 randomly
considered parameters based on simulations and Eqs. (8 E, 9E,
8R, and 9R) are given in Table 6. The percentage error between
the simulated result and respective correlation is given by Eq.
(11). From Table 6 it may be observed that the maximum
percentage error based on Eq. (9E) is -5.41%. Based on the
study it may be noted that the present methodology can be an
attractive option in the generation of correlation with few
number of simulations.
(11)
Conclusions:





A systematic approach for generating correlation
with reduced number of simulations is developed
based on Taguchi’s orthogonal arrays.
Regression analysis has to be considered for multiple
proposed forms of equations.
An optimum form of the equation is to be found by
comparing the correlation generated from equally
spaced and randomly generated levels.
The final form of the equation is to be obtained by
comparing the correlations generated from equally
spaced and randomly generated levels.
Parity plots can be used to select the best form of
equation.
The correlation thus developed by making use of 10
solutions is observed to have a maximum percentage
error of -5.41% for average Nusselt number.
In the current work a problem of natural convection in a
differentially heated cavity has been considered for generating
correlation with fewer number of solutions. The parameters
considered for the study are 103≤ Ra≤106, 0.71≤ Pr≤10 and 1≤
AR≤10. The numerical methodology has been rigorously
validated with literature. The study deals with the application
of Taguchi method for correlation development instead of
Table 6. AVERAGE NUSSELT NUMBER FOR RANDOMLY CONSIDERED PARAMETERS BASED ON SIMULATIONS AND
EQS. 8E, 8R, 9E AND 9R
Average Nusselt number
Ra
Pr
AR
Simulation
104
104
105
106
103
105
103
106
104
104
7
7
0.71
0.71
0.71
0.71
10
10
7
0.71
4
2
1
1
4
8
1
1
8
10
Equation
8E
%
Error
Equation
8R
%
Error
Equation
9E
%
Error
Equation
9R
%
Error
2.179
2.39
4.494
8.805
1.11
3.373
1.113
9.236
1.843
1.68
2.04
2.308
4.787
8.692
1.115
3.221
1.466
8.77
1.788
1.7
6.38
3.43
-6.52
1.28
-0.45
4.51
-31.72
5.05
2.98
-1.19
1.945
2.223
4.657
8.833
0.991
3.118
1.347
9.193
1.702
1.575
10.74
6.99
-3.63
-0.32
10.72
7.56
-21.02
0.47
7.65
6.25
2.121
2.415
4.737
8.591
1.111
3.208
1.111
9.063
1.863
1.697
2.66
-1.05
-5.41
2.43
-0.09
4.89
0.18
1.87
-1.09
-1.01
1.953
2.232
4.608
8.742
0.98
3.089
1.347
9.197
1.71
1.56
10.37
6.61
-2.54
0.72
11.71
8.42
-21.02
0.42
7.22
7.14
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