Numerical modelling of natural convection in a square cavity: effect of nanofluid volume fraction and inclination

In this paper, the phenomenon of natural convection in a square cavity filled with copper (Cu) based nanofluid (Nf) has been studied. The influences of the volume fraction (ɸ) on the convective flow, as well as the effect of the inclination (γ) of the cavity on thermal performances were examined. The dimensionless governing equations formulated using stream function, vorticity, and temperatures were solved by finite difference methods, where the Successive Over-Relaxation (SOR) iteration and the upwind scheme were adopted. The developed code served as a verification of results where it showed a good agreement with previous works. The calculation code was then used to visualize the influence of the inclination on the thermal enhancement and showed that the transfer becomes more efficient with an inclination of π/6.


Introduction
The phenomenon of convection remains a current and much diversified research axis. This is justified by a set of recently published research works. The mastery and performance management of this type of transfer has become a major issue in the industry, its study has been the subject of several research projects crowned by publications in the field [1][2][3][4][5]. At the laboratory scale, several works have been focused on the study of convective flows in several geometries, namely square cavities.
Karatas et al. [6] carried out an experimental study of the phenomenon of convection in a rectangular cavity effect of the nanoparticle volume fraction on the improvement of the heat transfer will allow the validation of the code, by comparing the present results with previous works. The code will then be used to analyze the effect of cavity inclination on thermal performance.

Mathematical formulation
The studied configuration is a square cavity with rigid walls, filled with a nanofluid. This configuration is subjected to a temperature gradient with the hot temperature (Th) imposed on the left wall while the cold the phenomenon of convection in a rectangular cavity subjected to a radiation considered as a source of thermal excitation. In the same context, convection within a ventilated cavity has been the subject of a numerical work by Ismael el al. [7]. Thus, several numerical methods have been implemented to solve the problem of convection in cavities. Latice Boltzman Methode (LBM) has been used by Haouat et al. [8] to analyze the same problem. Wang et al. [9] have opted for Chebyshev spectral method to analyze convection in a cavity with a porous media.
Other works concentrated on the study of the nanofluids in the square cavities [10]. Boualit et al. [11] and Khanafer et al. [12] used finite element as well as finite volume methods to solve the same problems. temperature (Th) imposed on the left wall while the cold temperature (Tc) is on the right wall (Dirichlet conditions). The other walls are adiabatic as it is shown in the schema presented in Fig.1. The objective of this work is the development of a numerical code to study the phenomenon of convection in a square cavity filled with nanofluids. Thus, the simulation of the natural convection of the fluid under the effect of a temperature gradient, and the study of the The used nanofluid is based on a suspension of copper (Cu) nanoparticles. The fluid flow in the cavity is assumed to be Newtonian and incompressible. The governing equations formulated using stream function, vorticity, and temperatures are expressed by the following system of equations: With the following initial and boundary conditions: And: In this work, the particle diameter used is =10 , and the characteristic dimension of the cavity "L" is taken equal to 1cm, Here, x and y are spatial coordinates, u and v are the velocities components, T is the temperature, α nf the nanofluid thermal diffusivity, μ eff is the effective dynamic viscosity, ρ nf , ρ s and ρ f are respectively the densities of nanofluid, nanoparticles and the density of fluid, β nf , β f and β s are respectively the thermal expansion of nanofluid fluid and nanoparticles, ɸ is the volume fraction, k d is the thermal conductivity of the dispersion, and (k ) is the effective stagnant conductivity [12], number (Pr) and the Grashof number (Gr) as well as the coefficient which appears in the term of thermal expansion are given by: 3 2 Pr , , In order to quantify the thermal transfer within the cavity, and to characterize the effect of increasing nanofluid volume fraction, the value of the local Nusselt and the Nusselt average must be determined. These parameters are calculated as follows: (11) and (k eff ) stagnant is the effective stagnant conductivity [12], C p the specific heat, the constant C is determined experimentally C=0.4 [11][12] for cooper-water nanofluid, ψ is the stream function and ω is the vorticity.
To obtain the dimensionless equations, we introduce the dimensionless spatial variables undermentioned: , , , The system of dimensionless equations will be written as follows:

Numerical method
The governing Eqs. (6)-(8) with corresponding initial and boundary conditions have been solved by finite difference method, where the Successive Over-Relaxation (SOR) iteration is used for the stream function equation (8) Vorticities at the boundaries are found from Taylor's developments of the stream functions, and are be given as follows:

Volume fraction effects
To validate the developed code, a series of numerical experiments carried out to describe the flow profiles into the cavity with γ=0°. These experiments are done for several mesh spacing from 41×41, to 91×91. We choose the spacing 81×81 for his better ratio between simulation time and accuracy of the results. These results are compared with those of Khanafer et al.   Fig. 6-a.
This figure shows a good agreement between the results. It can be also seen that the transfer intensifies by increasing the volume fraction that is explained by the Nu increasing.  In this section, after testing the calculation code, which gave a good agreement with result of Khanafer et al. [12], the influence of the inclination of the cavity on the nanofluid flow, as well as, on the thermal performances are investigated, where a series of simulations are performed. A cooper (Cu) nanoparticle volume fraction of 10% is used with different inclinations ranging from 0° to π/2 and for values of Grashof number ranging from 10 3 to 10 5 . effect on the intensification of the contours and on the profiles of temperature, where we can notice the migration of the temperature isolines from the hot source towards the cold source

Conclusion
In this paper, a code based on a finite difference method using the S.O.R iteration and the upwind system Fig. 6. a) The comparison of Nusselt between the present work Figure 6-b illustrates the evolution of the average number of Nusselt in the vicinity of the hot wall, thus allowing quantifying the rate of improvement of the heat transfer by convection in the cavity. As shown in Fig . 6, the heat transfer becomes more intense by increasing the number of Grashof (Gr). On the other hand, the inclination of the cavity has an influence on the number of Nusselt and marks a maximum for an inclination of π/6 and a minimum of performance at an inclination of π/3. method using the S.O.R iteration and the upwind system was developed. Several numerical simulations have been performed to validate the code, comparing the results with previous works. The good agreement of the results of the code was thus confirmed, which allowed using this code as a prediction tool in the study of the influence of the nanofluid concentration as well as the inclination of the cavity on the thermal performances. An analysis of the results concluded that the improvement in heat transfer is proportional to the increase in the volume fraction of the nanofluid, and that the variation of the inclination affects the thermal enhancement, where the transfer becomes more efficient with tilt of π/6. To analyze the effect of inclination on stream function and temperatures, the case of a nanofluid with a volume fraction of 10% was adopted. Fig.7 shows that increasing the angle of inclination produces a change in the stream function and temperature profiles. Indeed, we can notice that the increase of the inclination has a direct