Numerical study of natural convection in an enclosed

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
__________________________________________________________________________________________
Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 18
NUMERICAL STUDY OF NATURAL CONVECTION IN AN ENCLOSED
SQUARE CAVITY USING CONSTRAINED INTERPOLATED PROFILE
(CIP) METHOD
Asish Mitra
Reviewer: Heat and Mass Transfer, WSEAS
Associate Professor& HOD of Basic Sciences & Humanities Department,
College of Engineering & Management, Kolaghat, East Midnapur, West Bengal -721171
mitra_asish@yahoo.com
Abstract
In the present study, Constrained Interpolated Profile (CIP) method was used to simulate the natural convection heat transfer and
fluid flow in an enclosed square cavity with differentially heated side walls. The fundamental idea of this method is to solve the
advection phase equation with CIP method and the non-advection phase equation is calculated with finite difference method.
CIPNSE is applied to predict the temperature and velocity profiles in a square cavity for various Rayleigh number: Ra=103
, 104
and
105
. The streamline and isotherms obtained under these conditions were then compared with those published in literature and found a
good agreement.
Keywords: Constrained Interpolated Profile (CIP), Finite Difference Method (FDM) and Lattice Boltzmann Method
(LBM), Natural Convection, Square Cavity, Stream-Function Vorticity.
----------------------------------------------------------------------***------------------------------------------------------------------------
LITERATURE SURVEY:
The natural or free convection is the phenomenon of heat
transfer between a surface and a fluid moving over it with the
fluid motion caused entirely by the buoyancy forces that arise
due to the density changes that result from the temperature
variations in the flow. Since the early works by researchers
([1], [2], [3], [4]) a great deal of theoretical and experimental
researches was dedicated to investigate this phenomenon. The
fundamental interest comes from the concern to understand
the heat transfer mechanism ([5], [6],[7]) and fluid flow
behavior around the surfaces ([8],[9]). On the other hand, a
similar interest was provoked by the wide range of
engineering applications utilizing this type of phenomenon
([10], [11]). Among the problems related to natural
convection, many researchers focused their investigation on
the heat transfer and fluid flow behavior from a differentially
heated side walls in a cavity ([12], [13], [14]). They frequently
considered adiabatic boundary condition for the top and
bottom walls. However, very few investigated the effect of
perfectly conducting top and bottom walls although it plays
important roles in real engineering applications ([15]).
In present study, numerical investigation of natural convection
in a square cavity is carried out by considering perfectly
conducting boundary condition for top and bottom walls. The
left and right walls were maintained at hot and cold
temperature respectively. The objective of this paper is to
extend the formulation of Constrained Interpolated Profile
(CIP) method for Navier-Stokes equations to predict
temperature and velocity profiles in a differentially heated
square enclosure.
List of symbols
Cp- Specific heat
g - Gravitational acceleration
H - Length of cavity
k - Thermal conductivity
L - Length
p - Pressure
t - Time
T - Temperature
Th - Surface wall with hot temperature
Tc - Surface wall with cold temperature
u - Velocity in x direction
U - Dimensionless velocity in x direction
v - Velocity in y direction
V - Dimensionless velocity in y direction
x - Axial distance
X - Dimensionless axial distance
y - Vertical distance
Y - Dimensionless vertical distance

__________________________________________________________________________________________
Dimensionless Parameters
AR - Aspect Ratio
Gr - Grashof Number
Pr - Prandtl Number
Ra - Rayleigh Number
Greek Symbols
ρ - Density
β - Volumetric thermal expansion
τ - Dimensionless time
θ - Dimensionless temperature
µ - Dynamic viscosity
ν - Kinematic shear viscosity
α - Thermal diffusivity
ω - Vorticity
Ω - Dimensionless vorticity
ψ - Stream function
Ψ - Dimensionless stream function
∇ - Nabla operator
Superscript
n - Current value
n + 1 - Next step value
* - Non advection phase value
Subscript
i - x direction node
j - y direction node
max i - x direction maximum node
max j - y direction maximum node
FORMULATION, GOVERNING EQUATIONS
AND NUMERICAL METHODS
The conservation equations for 2D incompressible square
cavity flow in Cartesian form (Fig 1) are:
Fig1. Schematic geometry of a square cavity
1
2
3
4
The pressure terms are eliminated by taking the y-
derivative of (3) and subtracting from it the x-derivative
of (2). This gives
5
Using the definitions of stream function (ψ) and vorticity (ω)

__________________________________________________________________________________________
6
7
8
Eq (5) can be written as
9
In terms of the stream function, the equation defining the
vorticity (8) becomes
10
The following dimensionless variables are now introduced:
11
12
13
14
15
Where
16
In terms of these variables, (5), (10), (4) become
17
18
19
CONSTRAINED INTERPOLATED PROFILE
NAVIER STOKES EQUATION (CIPNSE)
In the CIPNSE method, the equation is divided into two parts:
advection phase and non advection phase. The non advection
phase will be solved independently through finite
difference method while the solutions of advection phase will
be obtained using two-dimensional (2D) CIP method.
Solution of Vorticity Transport Equation
The vorticity transport equation (17) can be written as
20
Where
D
X Y
Ra
X
= + +Pr[ ] .Pr( )
∂
∂
∂
∂
∂θ
∂
2
2
2
2
Ω Ω
21
The X- and Y-derivatives of (20) are
22
23
Putting the X- and Y-derivatives of D from (21) into (22) and
(23) we get

__________________________________________________________________________________________
∂
∂
∂
∂τ
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ∂
∂ θ
∂
∂
∂
∂
∂
∂
∂
∂
∂X
U
X X
V
X Y X X Y
Ra
X X
U
X Y
V
X
( ) ( ) ( ) Pr( ) .Pr( )
Ω Ω Ω Ω Ω Ω Ω
+ + = + + − −
3
3
3
2
2
2
24
∂
∂
∂
∂τ
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ∂
∂
∂
∂ θ
∂ ∂
∂
∂
∂
∂
∂
∂
∂
∂Y
U
Y X
V
Y Y X Y Y
Ra
X Y X
U
Y Y
V
Y
( ) ( ) ( ) Pr( ) .Pr( )
Ω Ω Ω Ω Ω Ω Ω
+ + = + + − −
3
2
3
3
2
25
Non-advection phase
∂
∂τ
∂
∂
∂
∂
∂θ
∂
Ω Ω Ω
= + +Pr( ) .Pr( )
2
2
2
2
X Y
Ra
X 26
∂
∂
∂
∂τ
∂
∂
∂
∂ ∂
∂ θ
∂
∂
∂
∂
∂
∂
∂
∂
∂X X X Y
Ra
X X
U
X Y
V
X
( ) Pr( ) .Pr( )
Ω Ω Ω Ω Ω
= + + − −
3
3
3
2
2
2
27
∂
∂
∂
∂τ
∂
∂ ∂
∂
∂
∂ θ
∂ ∂
∂
∂
∂
∂
∂
∂
∂
∂Y X Y Y
Ra
X Y X
U
Y Y
V
Y
( ) Pr( ) .Pr( )
Ω Ω Ω Ω Ω
= + + − −
3
2
3
3
2
28
Eqs (26) - (28) are solved by using central finite difference
method (FDM), which finally gives
29
30
31
Advection phase
32
33
34
In CIP method, the profile between the lattice points is
interpolated using cubic polynomial as in eq (35)
35
Where
, and
36

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The coefficients of a1 ,a2,…a7 are determined so that the
interpolation function and its first derivatives are continuous at
both ends [16]. With this restriction, the numerical diffusion
can be greatly reduced when the interpolated profile is
constructed. The spatial derivatives are then calculated as
37
38
In two-dimensional case, the adverted profile is approximated
as follow
39
40
41
Where
and
42
The newly calculated spatial quantities are then be used to
solve non-advection phase of Eqns. (26) to (28) and vorticity
formulation of Eqn. (18). In present study, the explicit central
finite different discretisation method is applied with second
order accuracy in time and space. For example, the treatment
for eqn. (18) is
43
In summary, the evolution of the proposed scheme consists of
three steps.
1. The initial value of
Ωi j
n
, ,
Ωx i j
n
, , and
Ωy i j
n
, , are specified at
each grid point.
2. Solve for
Ωi j
n
,
*
Ωx i j
n
, ,
*
and
Ωy i j
n
, ,
*
using the constrained
interpolation process [from eqs (39), (40) and (41) ].
3. The values of
Ωi j
n
,
+1
,
Ωx i j
n
, ,
+1
and
Ωy i j
n
, ,
+1
are then computed
from the newly advected values in step 2 by solving the
nonadvection phase of the governing equations [from eqs (29)
and (31) ].
Then the interpolation and the advection processes are
repeated.
Solution of Energy Equation
The energy equation (19) can be written as
∂θ
∂τ
∂θ
∂
∂θ
∂
+ + =U
X
V
Y
M
44
Where
M
X Y
= +
∂ θ
∂
∂ θ
∂
2
2
2
2
45
The X- and Y-derivatives of (44) are
∂
∂
∂θ
∂τ
∂
∂
∂θ
∂
∂
∂
∂θ
∂
∂
∂
∂θ
∂
∂
∂
∂θ
∂
∂
∂X
U
X X
V
X Y X
M
X
U
X Y
V
X
[ ] [ ] [ ] [ ]+ + = − −
46
∂
∂
∂θ
∂τ
∂
∂
∂θ
∂
∂
∂
∂θ
∂
∂
∂
∂θ
∂
∂
∂
∂θ
∂
∂
∂Y
U
Y X
V
Y Y Y
M
X
U
Y Y
V
Y
[ ] [ ] [ ] [ ]+ + = − −
47
Putting the X- and Y-derivatives of M from (45) into (46) and
(47) we get
∂
∂
∂θ
∂τ
∂
∂
∂θ
∂
∂
∂
∂θ
∂
∂θ
∂
∂θ
∂ ∂
∂θ
∂
∂
∂
∂θ
∂
∂
∂X
U
X X
V
X Y X XY X
U
X Y
V
X
[ ] [ ] [ ]+ + = + − −
3
3
3
2
48
∂
∂
∂θ
∂τ
∂
∂
∂θ
∂
∂
∂
∂θ
∂
∂θ
∂ ∂
∂θ
∂
∂θ
∂
∂
∂
∂θ
∂
∂
∂Y
U
Y X
V
Y Y X X Y X
U
Y Y
V
Y
[ ] [ ] [ ]+ + = + − −
3
2
3
3
49

__________________________________________________________________________________________
Non-advection phase
∂θ
∂τ
∂ θ
∂
∂ θ
∂
= +
2
2
2
2
X Y 50
∂
∂
∂θ
∂τ
∂ θ
∂
∂ θ
∂ ∂
∂θ
∂
∂
∂
∂θ
∂
∂
∂X X X Y X
U
X Y
V
X
[ ] = + − −
3
3
3
2
51
∂
∂
∂θ
∂τ
∂ θ
∂ ∂
∂ θ
∂
∂θ
∂
∂
∂
∂θ
∂
∂
∂Y X X Y X
U
Y Y
V
Y
[ ] = + − −
3
2
3
3
52
Eqs (50) - (52) are solved by using central finite difference
method (FDM), which finally gives
53
54
55
Advection phase
56
57
58
The same procedure is applied for getting the advection phase
of energy equation.
RESULTS AND DISCUSSION
In this numerical research work, the natural convection
phenomena in an enclosed square cavity have been studies
using CIP method. Applying the numerical procedure
mentioned in the last section, the streamlines and isotherms
plots are obtained for various values for Rayleigh Numbers of
Ra=103, 104 and 105 at a fixed Prandtl Number of 0.71. An
in-house code (in Matlab) has been developed for the whole
simulation. These are compared with those obtained from
Lattice Boltzmann method (LBM) [17]. Figure 2 illustrates the
comparison of the streamline plots for various Rayleigh
number between the present and LBM methods, while Figure
3 demonstrates the comparison of the isotherm plots.
Ra = 103

__________________________________________________________________________________________
Ra = 104
Ra = 105
Fig 2 Comparison of the streamline plots for various number
between the present CIP (left) and LBM (right)
Fig 2 Comparison of the isotherm plots for various number
between the present CIP (left) and LBM (right)
CONCLUSIONS
Natural convection in a square cavity was studied using
CIPNSE method. The velocity and temperature profiles for
Rayleigh numbers 103, 104 and 105 obtained in this approach
were compared with those obtained from lattice-Boltzmann
formulations. A good agreement between the present results
and the past indicates that like finite difference and lattice-
Boltzmann formulations, the CIPNSE method may be an
efficient and stable numerical scheme in natural convection.

__________________________________________________________________________________________
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Numerical study of natural convection in an enclosed

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