In the present work, we numerically study the laminar natural convection of a nanofluid confined in a square cavity. The vertical walls are assumed to be insulated, non-conducting, and impermeable to mass transfer. The horizontal walls are differentially heated, and the low is maintained at hot condition (sinusoidal) when the high one is cold. The objective of this work is to develop a new height accurate method for solving heat transfer equations. The new method is a Fourth Order Compact (F.O.C). This work aims to show the interest of the method and understand the effect of the presence of nanofluids in closed square systems on the natural convection mechanism. The numerical simulations are performed for Prandtl number ( ), the Rayleigh numbers varying between and for different volume fractions varies between 0% and 10% for the nanofluid (water + Cu).
In a better description, nanofluids are engineered colloidal suspensions of nanoparticles (1 - 100 nm) in a base fluid. Common base fluids include water, oil, and Ethylene Glycol while nanoparticles are typically made of chemically stable metals, metal oxides or carbon in various forms. The use of particles of nanometer dimension was first continuously studied by a research group at the Argonne National Laboratory a decade ago. S. Choi [
A numerical study of natural convection of copper-water nanofluide in a two dimensional enclosure was conducted by Khanafer et al. [
Sandeep Naramgari and C. Sulochana [
The effects of Rayleigh number, volume fraction and partitions location on the average Nusselt number are studied. I.El Bouihi and R. Sehaqui [
In order to optimize and improve heat transfer by natural convection in closed square cavity. Although extensive research has been given to cases of rectangular cavity filled nanofluid, few studies have focused on the study, theoretical or numerical discretizations high order (≥2). The numerical study of systems of equations in the heat transfer area is usually treated by various methods, sometimes numérical classics like Finite Elements (FE), Volume Finite (VF) and Finite Differences (DF). or using some software adapted as “FLUENT”. To solve fluid mechanics problems such as conductive heat transfer, convective or mixed into regular geometries.
More specific order four schemes have been used to solve the Navier-Stokes in enclosures without considering the energy equation by Ecran Erturk et C. Gokcol [
Consider a square cavity filled with a nanofluid. The vertical walls are assumed to be insulated, non-conducting, and impermeable to mass transfer. The horizontal walls are differentially heated, the low is maintained at hot condition (sinusoidal) when the high one is cold (
We have considered the continuity, momentum and energy equations for a Newtonian, Fourier constant property fluid governing an unsteady, two-dimensional flow. It is further assumed that radiation heat transfer among sides is negligible with respect to other modes of heat transfer. Under the assumption of constant thermal properties, the Navier-Stokes equations for an unsteady, incompressible, two-dimensional flow are:
Continuity equation:
x-momentum equation:
y-momentum equation:
Energy equation:
where
The viscosity of the nanofluid can be estimated with the existing relations for the two-phase mixture. The equation given by Brinkman [
The effective density of the nanofluid at reference temperature is
The heat capacitance of the nanofluid is expressed as Abu-Nada [
The effective thermal conductivity of the nanofluid is approximated by the Maxwell-Garnetts model [
Equations (1)-(4) can be converted to the dimensionless forms by definition of the following parameters as:
Hence, the governing equations of continuity, linear momentum and energy for unsteady laminar flow in Cartesian coordinates take the following dimensionless form:
Physical properties | Pure water | Cu |
---|---|---|
4179 | 383 | |
997.7 | 8933 | |
0.613 | 400 | |
2.1 × 10−4 | 1.67 × 10−5 |
The enclosure boundary conditions consist of no-slip and no penetration walls, U = V = 0 on all four walls. The thermal boundary conditions on the bottom wall is such that
The governing equations for the present study in
Kinematics equation:
Vorticity equation:
where:
Energy equation:
Before turning to the application of the method of fourth order in the equations governing our problem we will combine Equations (12) and (13) in a condensed form by introducing a dummy variable
All these terms are listed in
Dimensionless boundary conditions for
For:
For:
For vorcicite Störtkuh et al. [
Quantities transported | |||
---|---|---|---|
Vorticity Equation | |||
Equation of Energy | 0 |
For corner points, we again follow Störtkuh et al. [
where V is the speed of the wall in our case which is equal to 0 for the four stationary walls.
In explicit notation, for the wall points shown in
Similarly, for the corner points also shown in
The reader is referred to Störtkuh et al. [
The local and averaged heat transfer rates at the bottom hot wall of the cavity are presented by means of the local and averaged Nusselt numbers, Nu and
High-Order Compact (HOC) formulations are becoming more popular in computational fluid dynamics (CFD) field of study. Compact formulations provide more accurate solutions in a compact stencil. In finite differences, a standard three-point discretization provides second-order spatial accuracy and this type of discretization is very widely used. When a high-order spatial discretization is desired, i.e. fourth-order accuracy, then a five- point discretization has to be used. However, in a five point discretization there is a complexity in handling the points near the boundaries. High-order compact schemes provide fourth-order spatial accuracy in a 3 × 3 stencil
and this type of compact formulations does not have the complexity near the boundaries that a standard wide (five- point) fourth-order formulation would have. Dennis and Hudson [
We will use the equations of streamlines
Stream function:
General equation of conservation:
For first and second-order derivatives the following discretizations are fourth-order accurate:
where
If we apply the discretizations in Equations (24) and (25) to Equations (22) and (23), we obtain the following equation
In these equations we have third and fourth derivatives
For example, when we take the first and second x-derivative of the stream function Equation (22) we obtain
And also, by taking the first and second y-derivative of the stream function Equation (22) we obtain
Using standard second-order central discretizations given in
When we substitute Equations (35) and (37) into Equation (28) we obtain the following finite difference equation.
Derivations | Discretizations |
---|---|
We note that the solution of Equation (38) is also a solution to stream function Equation (22) with fourth-or- der spatial accuracy. Therefore, if we numerically solve Equation (38), the solution we obtain will satisfy the stream function equation up to fourth order accuracy.
In order to obtain a fourth-order approximation for the vorticity equation and energy (23), we follow the same procedure. When we take the first and second derivatives of the general equation of conservation (23) with respect to x- and y-coordinates we obtain:
If we substitute Equations (39) and (41) for the third derivatives of the general equation of conservation and into Equations (29), (40) and (42) and also if we substitute Equations (34) and (36), for the third derivatives of stream function into Equations (29), (40) and (42) and finally, if we substitute Equations (40) and (42) for the fourth derivatives of the general equation of conservation into Equation (29), then we obtain the following:
where:
and
Again we note that the solution of Equation (43) satisfy the vorticity and energy Equation (23) with fourth- order accuracy.
As the final form of our FOC scheme, we prefer to write Equations (38) and (43) as
where
We note that the finite difference Equations (44) and (45) are fourth-order accurate
Equations (47) and (48) are the standard second-order accurate
function and the general equation of conservation (22) and (23). When we use Equations (44) and (45) for the numerical solution of the stream function and general equation of conservation, we can easily switch between second and fourth-order accuracy just by using homogeneous values for the coefficients A, B, C, D, E and F or by using the expressions defined in Equation (46) in the code.
We note that the numerical solutions of Equations (44) and (45), strictly provided that second-order discretizations in
As a measure of convergence to the steady state, during the iterations we monitored three residual parameters. The first residual parameter, RES1, is defined as the maximum absolute residual of the finite difference equations of steady stream function and general Equations (44) and (45). These are, respectively, given as
The magnitude of RES1 is an indication of the degree to which the solution has converged to steady state. In the limit RES1 would be zero. The second residual parameter, RES2, is defined as the maximum absolute difference between two iteration steps in the stream function, vorticity and energy variables. These are, respectively, given as
RES2 gives an indication of the significant digit on which the code is iterating. The third residual parameter, RES3, is similar to RES2, except that it is normalized by the representative value at the previous time step. This then provides an indication of the maximum percent change in
In our calculations, for all Rayleigh numbers we considered that convergence was achieved when both
In the present grid independence test, the Prandtl number is set to Pr = 6.2 (pure water). The nanoparticles are chosen to be copper (Cu) with a solid volume fraction
Our code has been tested for natural convection fluid flows in differentially heated cavities and in Rayleigh- Bénard configuration for Rayleigh numbers between 103 and 106 (
Grid | 32 × 32 | 42 × 42 | 62 × 62 | 82 × 82 | 102 × 102 |
---|---|---|---|---|---|
4.7357 | 4.6657 | 4.6072 | 4.5848 | 4.5741 | |
2.06977 | 2.0668 | 2.0621 | 2.0593 | 2.0593 |
G.de Val Davis [ | 1.118 | 2.243 | 4.519 | 8.799 |
Markatos and perikleous [ | 1.108 | 2.201 | 4.430 | 8.754 |
G.V.Hadjisophcleous et al. [ | 1.141 | 2.290 | 4.964 | 10.390 |
R.K. Tiwari, M.K. Das [ | 1.087 | 2.195 | 4.450 | 8.803 |
I. El Bouihi and R. Sehaqui [ | 1.042 | 2.024 | 4.520 | 8.978 |
Present work | 1.012 | 2.214 | 4.103 | 8.293 |
Difference with I. El Bouihi % | 2.87 | 9.38 | 9.22 | 7.62 |
In this section, the nanofluid-filled enclosure is studied for a range of solid volume fraction
The heat transfer distribution through the hot wall is displayed in
The variations of average Nusselt number (Nu) with Ra and χ are shown in
In this study the heat transfer enhancement in a two dimensional enclosure filled with nanofluids is studied
1.64 | 1.80 | 9.75 | 1.96 | 19.51 | 2.14 | 30.48 | 2.33 | 42.07 | |
1.73 | 1.88 | 8.67 | 2.04 | 17.91 | 2.21 | 27.74 | 2.39 | 38.15 | |
1.99 | 2.11 | 6.03 | 2.24 | 12.56 | 2.39 | 20.10 | 2.54 | 27.63 | |
4.42 | 4.75 | 7.46 | 5.09 | 15.15 | 5.46 | 23.52 | 5.86 | 32.57 | |
4.81 | 5.11 | 6.23 | 5.45 | 13.30 | 5.83 | 21.20 | 6.26 | 30.14 | |
6.42 | 6.73 | 4.82 | 7.16 | 11.52 | 7.67 | 19.47 | 8.22 | 28.03 | |
7.61 | 7.89 | 3.67 | 8.29 | 8.93 | 8.87 | 16.55 | 9.54 | 25.36 |
numerically. This study presented a new fourth-order compact formulation and investigated the effect of a sinusoidal thermal boundary condition, for different Rayleigh number Ra and volume fractions of nanoparticles. The flow and temperature fields are symmetric near the middle plane of the enclosure due to the imposed symmetry condition on the bottom wall boundary. From the results of this work, the following main conclusions may be drawn:
・ The fourth-order accurate compact formulation was developed and was in agree with previous studies.
・ Our numerical code has been validated for different Rayleigh number.
・ A comparative study illustrates that the suspended nanoparticles substantially increase the heat transfer rate with an increase in the nanoparticles volume fraction for different Rayleigh number Ra Rayleigh number. Moreover, the nanofluid flows as well as the cooper nanoparticles increase.
In the near future, this study will be extended for different geometry studies and other types of base fluids and nanoparticles.
Mostafa Zaydan,Naoufal Yadil,Zoubair Boulahia,Abderrahim Wakif,Rachid Sehaqui, (2016) Fourth-Order Compact Formulation for the Resolution of Heat Transfer in Natural Convection of Water-Cu Nanofluid in a Square Cavity with a Sinusoidal Boundary Thermal Condition. World Journal of Nano Science and Engineering,06,70-89. doi: 10.4236/wjnse.2016.62009
i x-direction grid point
j y-direction grid point
Cp Specific heat capacity (J・K−1)
g Gravitational cceleration (m・s−2)
h ocal heat transfer coefficient (m−2・K−1)
H Height of cavity (m)
qw Heat flux (W・m−2)
t Dimensional time (s)
τ Non-dimensional time
T Temperature (K)
p pressure (pa)
(x, y) Dimensionless Cartesian coordinates (m)
u, v velocity components in x, y directions (m・s−1)
α Fluid thermal diffusivity (m2・s−1)
β Thermal expansion coefficient (K−1)
ν Kinematic viscosity (m2・s−1)
ρ Density (kg・m−3)
μ Dynamic viscosity (N・s・m−2)
κ Thermal conductivity (W・m−1・K−1)
ψ dimensional stream function
θ non-dimensional temperature
ω dimensionalvorticity
χ nanoparticle volume fraction
c cold wall
eff effective
h hot wall
s solid
f pure fluid
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