The numerical solution of compressible flows has become more prevalent than that of incompressible flows. With the help of the artificial compressibility approach, incompressible flows can be solved numerically using the same methods as compressible ones. The artificial compressibility scheme is thus widely used to numerically solve incompressible Navier-Stokes equations. Any numerical method highly depends on its accuracy and speed of convergence. Although the artificial compressibility approach is utilized in several numerical simulations, the effect of the compressibility factor on the accuracy of results and convergence speed has not been investigated for nanofluid flows in previous studies. Therefore, this paper assesses the effect of this factor on the convergence speed and accuracy of results for various types of thermo-flow. To improve the stability and convergence speed of time discretizations, the fifth-order Runge-Kutta method is applied. A computer program has been written in FORTRAN to solve the discretized equations in different Reynolds and Grashof numbers for various grids. The results demonstrate that the artificial compressibility factor has a noticeable effect on the accuracy and convergence rate of the simulation. The optimum artificial compressibility is found to be between 1 and 5. These findings can be utilized to enhance the performance of commercial numerical simulation tools, including ANSYS and COMSOL.

Heat transfer problems commonly occur in many industries, most notably in the strategic oil and gas and petrochemical industries. In these problems, heat is mainly transferred via convection, conduction, and radiation [

Artificial compressibility methods have widely been used by several researchers. Qin et al. [

Mousa et al. [

Zhang et al. [

The literature survey demonstrates the importance of artificial compressibility methods for researchers in different numerical fields. Different researchers also worked on the accuracy of available schemes to improve their accuracy. Marti et al. [

The second section of this work describes the governing equations and the novel numerical technique. In the third section, the problem-solving process and several code components are discussed. In the subsequent stage, the results of numerical simulations are obtained and compared to find suitable answers to the research questions.

The following equations are applied to two-dimensional and incompressible fluid flow with heat transfer. Density and viscosity are assumed constant for Newtonian fluids with laminar flows. The first equation represents continuity, and the second and third ones signify momentum equations. The fourth equation expresses the energy equation [

A virtual compressibility sentence

For many thermos-flows, a faster convergence can be achieved using the Boussinesq model instead of defining density as a function of temperature.

In the above relations,

In

The following equation is produced by applying the Green theorem and converting the double integral on the cell surface to the integral on the cell boundaries.

The discretization of the resulting equation in a two-dimensional space with quadrilateral cells leads to the following relation for cells

The convective terms are obtained by the second-order averaging scheme.

The fourth-order averaging scheme is used to calculate viscous fluxes on the boundary of secondary cells.

The fifth-order Range-Kutta has been used for time discretization

In the case of the upper index,

A square rectangular cavity with a movable lid is chosen in this study as a benchmark, illustrated in _{2} nanofluid). Upper and lower walls move at a constant speed of U, while the remaining walls are fixed and insulated. The no-slip law is used to determine the velocity and temperature boundary conditions whereas a second-order extrapolation is used to calculate the pressure.

The current work is performed in several steps. First, the dimensionless governing equations are discretized in a finite volume framework. Then viscous fluxes and convective fluxes are obtained by a fourth-order and second-order averaging scheme, respectively. Time marching is done by a fifth-order Runge-Kutta method and the velocity, pressure, and temperature fields are obtained. The grid independence and validation of the results are done. In the next step, simulations are repeated for different artificial compressibility factors. Finally, the accuracy and convergence speed are compared for different artificial compressibility factors to find its optimum value. A code has been written in FORTRAN software to perform the above steps. This code contains different subroutines and the main part. The subroutines are then called in the main part. Gird generation is done in the first subroutine. Initial and boundary conditions are given in the second subroutine. Convective fluxes and viscous fluxes are obtained using the third and fourth subroutines. Charts are obtained using these files’ data via TECPLOT software. The flowchart of this work is shown in

Simulations were run in four distinct grids in order to ensure the accuracy of the numerical solution. The simulations are carried out for

To find the optimum grid, high accuracy and less computational time are considered. As seen in

One of the well-known problems of viscous flow is the flow inside a square cavity. Depending on the Reynolds number, one or more vortices have been formed inside the cavity. As the Reynolds number increases, the number of vortices increases. The incompressible and viscous steady flow inside the cavity is a well-known example of a case test used to validate a novel numerical scheme. The simple and the clustered quadrilateral grid have been used in this work. The natural and mixed convection nanofluid flows are considered in this paper. First, natural convection (Gr = 0) is considered and then mixed convections with non-zero Grashof numbers are simulated. In the first case, the continuity and momentum equations are solved simultaneously and the velocity and pressure fields are determined. By placing the velocity and pressure fields in the energy equation, the temperature field is determined. Nevertheless, in mixed convection, the momentum equation is a function of Grashof number and temperature. Therefore, all the equations should be solved simultaneously to determine the pressure, velocity and temperature fields. In numerical techniques, the Courant number is taken into account and is defined as follows:

Courant number is a factor used to determine the convergence of the numerical method. Convergence speed improves as the Courant number is increased. However, the numerical method diverges with higher Courant numbers. Mathematical approaches can be used to determine the range of courant numbers required for convergence in simpler cases with simple governing equations. Nevertheless, in more complex equations, this range is determined by numerical experiments. To find this range, the written code for the numerical scheme is run for different Courant numbers until the code output diverges and the highest achievable value for Courant numbers is determined. Convergence history for various Courant numbers with

To assess the effect of the artificial compressibility coefficient on the results, the friction coefficients at the bottom plate are calculated and compared for various artificial compressibility coefficients, as illustrated in

It is noted that the deviation from the results occurs when the artificial compressibility coefficient is changed abruptly. With 200 times the artificial compressibility coefficient, the resulting change is observed (

The governing equations are numerically solved, and velocity, pressure and temperature fields are obtained. The resulting temperature field is used to display the isotherms (

In this study, a numerical simulation of a two-dimensional incompressible flow with heat transfer inside a square cavity was investigated using an artificial compressibility method. The effect of the artificial compressibility coefficient on convergence speed and accuracy of results was also assessed. It is found that the accuracy of the results is affected if the artificial compression ratio is either too small or too large. In other words, the artificial compressibility term may not tend to zero at the final steps of time marching. The results also demonstrated that the optimal values of the artificial compressibility factor lie between 1 and 5 for viscous incompressible flows. Using the artificial compressibility method necessitates a measure comparable to network independence, as indicated by the results. In other words, simulations should initially be conducted with varying values of the synthetic compressibility factor, and if the results do not change, simulations should continue. By determining the optimal number for the artificial compressibility factor, researchers in the field of numerical techniques can ensure the speed of convergence as well as the accuracy of the results. This study employed the fifth-order Rung-Kutta method in conjunction with the artificial compressibility method. Simulations could utilize more numerical approaches. The convective fluxes are obtained using the method of averaging. The characteristics-based method could be used.

_{p}

Specific heat capacity (J/kg.K)

Eckert number

Gravitational acceleration (m/s^{2})

Grashof number

Thermal conductivity (J/m.K)

Local Nusselt number

Pressure (Pa)

Prandtl number

Reynolds number

Temperature (K)

Time (s)

Velocity components (m/s)

Coordinates (m)

Artificial compressibility coefficient

_{ex}

Thermal expansion coefficient

Coefficient of viscosity (Pa.s)

Kinematic viscosity (m^{2}/s)

Density (kg/m^{3})

_{2}O

_{3}-H

_{2}O and γAl

_{2}O

_{3}-C

_{2}H

_{6}O

_{2}nanofluids spray along a stretching cylinder