In this paper, natural heat convection inside square and equilateral triangular cavities was studied using a meshless method based on collocation local radial basis function (RBF). The nanofluids used were

In heat transfer, the heat convection occurs when heat is transferred from one part of the fluid to another. Natural convection has become an active research topic due to its lower cost and wide applications in engineering. Several models were developed for calculating the efficiency of thermal conductivity in solid and fluid systems [

Several numerical methods, such as finite elements, finite volumes, and finite differences have been used to solve the thermal convection equations and simulate convective heat flow in various geometries. These methods are mesh based methods and are widely used. Several researchers have used these methods to model natural convection in nanofluids. Oztop et al. [

Finite elements have also been widely used. For example, Bhowmick et al. [

It is worth noting that, over the last two decades, the lattice Boltzmann method has seen significant advances and is now commonly used in heat transfer problems, particularly for nanofluids. In Izadi et al. [

Recently, Petrov-Galerkin local meshless method was used to study natural convection in a porous medium [

Other lines of investigation relevant to our study were explored. Kavusi et al. [

The present work aims to develop an accurate method using local radial basis function combined with the artificial compressibility model to solve the natural convective heat transfer problems in complex geometries. These methods have rarely been used in the case of nanofluids. A square cavity and an equilateral triangular cavity filled with a

This paper is structured as follows: The problem and the mathematical implementation of the governing equations are described in

To demonstrate the simplicity and efficiency of the proposed method to deal with complex geometries, we choose two domains. The proposed configurations are illustrated in

A two-dimensional square cavity. The left and right walls of the square cavity are the hot wall (

An equilateral triangular cavity where the bottom wall is the hot wall while the other two upper walls are kept adiabatic.

The cavities considered are filled with a nanofluid composed of a nanoparticle-water mixture. The nanofluid is considered Newtonian, and the flow is laminar. The thermo-physical properties of the base fluid and the nanoparticle are taken as [

Properties | |||
---|---|---|---|

997.1 | 8933 | 3970 | |

4179 | 385 | 765 | |

0.613 | 401 | 40 | |

21 | 1.67 | 0.85 |

The equations of the continuity, momentum, and energy for buoyancy-driven laminar fluid flow and heat transfer of the nanofluid inside a cavity are

L is the characteristic length as well as the length of the cavity edge. For this work, we set it to 1. The above equations can be written in the dimensionless form:

For the cases studied, we have used three types of walls; hot, cold or adiabatic wall. The corresponding boundary conditions are

For calculating the local and average Nusselt numbers, the following formulas were used, respectively

In this work, the values of the Nusselt number were calculated on horizontal or vertical walls, where for a horizontal wall

The thermophysical parameters of the base fluid, such as density, viscosity, and heat conductivity, are affected by the addition of nanoparticles. In this study, the models that describe the thermophysical properties were obtained from the literature and are as follows:

The effective density of the nanofluid is given as

Here

The thermal expansion coefficient of the nanofluid is given by

Finally, the effective viscosity of the nanofluid, based on Brinkman model [

We employ the radial basis functions method, which is a meshless method originally proposed by Kansa [

Radial basis function interpolation approximates the solution function by an expansion. According to the Kansa method, the solution is approximated on a set of

Many radial basis functions were proposed In the literature. In the current study, we use the infinitely smooth multiquadric radial basis function defined as

The numerical solution of an equation involving partial derivatives can be approximated by using linear approximation of the partial derivatives at the

This can be rewritten in a more concise form as

Note that in the governing equations the first and second derivatives are calculated for each node by using several matrix operations on

Using this approach, the first and second spatial derivatives of a function

In attempting a numerical solution of the

Here

For simplicity, the

On a set of

The problem in

It should be noted that radial basis function approaches for time-dependent partial differential equations with diffusive terms can be stably advanced in time with a suitable choice of time step size [

Here

Several preliminary tests were performed to evaluate the sensitivity of the results to the number of nodes. These tests were used to determine the optimal distribution of nodes that ensures good accuracy with reasonable computational cost, Three distributions are used for this investigation and are shown in

The meshless method proposed in this work was then validated in the case of natural convection on two problems. The first was the natural convection in a cavity filled with pure water. In the second, we considered the

In the first investigation, we ran a simulation of the effects of differentially heated boundary conditions on natural convection in a square cavity filled with a nanofluid. We used the cavity previously for the validation of meshless method, but extend the range of the Raylieh number to

^{6}. Finally, as Rayleigh number and volume fraction grow, the intensity of heat transfer increases, and the behavior of isotherms for

In the second investigation, we looked at the natural convection in an equilateral triangular enclosure. The two nanofluids employed in the previous case were used. The two inclined walls of the cavity were kept cold, while the bottom wall is heated. Pranowo et al. [

In order to quantify the heat exchange within the cavity filled with

A two-dimensional numerical investigation of natural convection in enclose cavities was carried out in this study. Two geometries were used, namely those of a square and a triangular cavity. The laminar heat transfer equations of nanofluid were solved using the radial basis function method coupled with an artificial compressibility, which is classified as a meshless method.

The implemented meshless method offers high flexibility in dealing with complex geometries due to the simplicity of the numerical evaluations of space derivative. This method provides a valuable and efficient way to analyze natural convection in nonrectangular geometries.

To overcome the pressure velocity coupling problem that occurs in equation systems like the ones examined, the artificial compressibility model was implemented. The quality of the results was found to be of the same accuracy order as the other classical methods.

Results clearly indicate that the addition of nanoparticles has produced a substantial enhancement of heat transfer as compared to that of the pure fluid. As mentioned above, convective transfer dominates conduction when Rayleigh numbers are increased. As a consequence, heat transfer from the hot walls to the cool walls is improved. This remains true for both of the cases investigated. Furthermore, we conclude that the heat transfer enhancement is not clear in low Rayleigh numbers or when the enclosures have geometric singularities, as was the case in our study, for the case of the triangular cavity.

Number of nodes

Specific heat,

Acceleration due to gravity,

Thermal conductivity,

Nusselt number

Prandtl number

Rayleigh number

Pressure,

Dimensionless pressure

Temperature,

Dimensionless temperature

Dimensional velocity components in x and y,

Dimensionless velocity components in X and Y

Dimensional Cartesian coordinates

Dimensionless Cartesian coordinates

First-order differentiation matrix

Second-order differentiation matrix

thermal diffusivity,

thermal expansion coefficient,

effective viscosity,

density,

artificial compressibility parameter

shape parameter

pseudo time

base fluid

nanofluid

solid particles

cold

hot

The authors would like to acknowledge the financial support from King Faisal University, Saudi Arabia, Project No. AN000675.

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_{4}/water hybrid nanofluids using LBM