Zehba A.S.Raizah and Abdelraheem M.Aly,2,＊

1Department of Mathematics,College of Science,King Khalid University,Abha,62529,Saudi Arabia

2Department of Mathematics,South Valley University,Qena,83523,Egypt

Abstract: This study simulates natural convection flow resulting from heat partitions in an H-shaped enclosure filled with a nanofluid using an incompressible smoothed particle hydrodynamics (ISPH) method.The right area of the H-shaped enclosure is saturated with non-Darcy porous media.The center variable partitions of the H-shaped enclosure walls are kept at a high-temperature Th.The left and right walls of the H-shaped enclosure are positioned at a low temperature Tc and the other walls are adiabatic.In ISPH method,the source term in pressure Poisson equation(PPE)is modified.The influences of the controlling parameters on the temperature distributions,the velocity field and average Nusselt number are discussed.The performed simulations proofed that the length of the heated partitions augments the velocity field and temperature distributions in an H-shaped enclosure.Rayleigh number rises the fluid velocity and heat transfer in an H-shaped enclosure.The porous layer on the right side of the H-shaped enclosure at a lower Darcy parameter causes a high resistance force for the fluid flow and heat transfer characteristic inside an H-shaped enclosure.Added nanoparticles reduces the velocity field and enhances the heat transfer inside an H-shaped enclosure.

Keywords: H-shaped enclosure; ISPH; natural convection; nanofluid;porous medium

1 Introduction

Fundamental numerical studies on heat transfer in porous media by using finite element method have been introduced by Lewis et al.[1,2].In addition, due to the wide applications of using various shapes of cavities and their effects in the flow formations and performance of the heat transfer.Then, there are many studies in the convection flow and heat transfer inside different geometries of the cavities.Esfe et al.[3] introduced numerical studies for natural convection inside a T-shaped cavity.By using the Boussinesq approximation, Li et al.[4] studied two-phase mixed convection over four rotating cylinders in a porous H-shaped cavity filled with a non-Newtonian nanofluid.Bhowmick et al.[5] investigated the transient natural convection flow in a valley-shaped triangular cavity filled by stratified water.Bhowmick et al.[6] studied the natural convection in a V-shaped cavity heated from below.Ma et al.[7] used lattice Boltzmann method (LBM) to analyze the natural convection from a hot obstacle inside a U-shaped cavity filled with Al2O3-water or TiO2-water nanofluid.Simulation of natural convection in H-shaped cavity filled with nanofluid using lattice Boltzmann was introduced by Rahimi et al.[8].In addition, Izadi et al.[9] used LBM to investigate the natural convection in a ⊥shaped enclosure filled with a hybrid nanofluid.Ahmed et al.[10] simulated the magnetohydrodynamics (MHD) ferroconvective in an inclined double-lid driven L-shaped enclosure.The nanofluid thermo-gravitational convection inside aΓ-shaped enclosure was evaluated by Mohebbi et al.[11].Liu et al.[12] performed a numerical work on the serpentine microchannel including fan-shaped reentrant cavities.Purusothamana et al.[13] studied MHD free convection flow in a tilted V-shaped electronic assembly.Ma et al.[14] adopted LBM method to analyze the natural convection in I-shaped heat exchanger filled with a nanofluid.Aly [15] applied the finite volume method with a SIMPLE algorithm to study the impacts of thermo-diffusion on the buoyancy flow over two circular cylinders inside a porous cavity filled with a nanofluid.

All of the previous studies are depending on the mesh numerical methods to solve the governing equations of the case study.But, there are some cases are requiring an effective method.One of these methods is the SPH method, which has been applied in the computational fluid dynamics.SPH method is a promising numerical method for simulating impact fluid flows, free surface flow and bouyancy forces [16-24].Shao [22] applied incompressible version of SPH method for simulating wave interactions with a porous medium.Kazemi et al.[23] introduced a novel SPH method for fluid flow in an open channel over natural porous beds.Kazemi et al.[24]developed SPH method to treat the interaction at an interface between free fluid flow with porous media.Ahmed et al.[25] used improved ISPH method to study the buoyancy-driven flow inside a nanofluid-filled enclosure including a cross shape.Aly et al.[26-34] modified the ISPH method to simulate several numerical problems concerning in heat and mass transfer inside different cavity shapes below different boundary conditions.

The nanofluid flow over blockages has several engineering applications including nuclear fuel sub assembly with flow blockage, indoor building, storage and drying.The aim of this study is to perform numerical simulations of the natural convection flow in a nanofluid-filled H-shaped enclosure.A porous medium is filled the right area of the H-shaped enclosure and the heated partitions were positioned at the center of the H-shaped enclosure.The results showed that the length of the heated source can control the buoyancy force inside an H-shaped enclosure.An extra number of the cooled blockages reduces the fluid flow and heat transfer inside an H-shaped enclosure.Lower Darcy parameter strengths the porous resistance and consequently a decrease in Darcy parameter decreases the fluid intensity and temperature distributions inside the right side of the H-shaped enclosure.Rayleigh number plays an important role in enhancement heat transfer and strengths of fluid velocity inside an H-shaped enclosure.The average Nusselt number is affected by varying the length of the hot source, number of the cooled blockages and Darcy parameter.The average Nusselt number increases as Rayleigh number increases and it decreases as solid volume fraction increases.Adding nanoparticles until 5% reduces the velocity field in an H-shaped enclosure due to a higher viscosity.

2 Mathematical Formulation

Fig.1 presents the initial schematic diagram of the current physical models and their particle generation.The heated area with a high temperatureThis positioned at the center of the H-shaped enclosure with an equal variable lengthB.The other center walls and the horizontal walls are thermally insulated.The sidewalls of the H-shaped enclosure have a low-temperatureTc.H-cavity length isL= 1.4 and its height isW= 1.6.Center length and height of the H-shaped enclosure areLH=0.4 andWH=0.4, respectively.Model 2 shows the inner blockages with variable numbers inside an H-shaped enclosure.These blockages are maintained at a low temperatureTcwith an initial zero velocity.The blockages are taken as square shapes with a length 0.1 and their positions are settled in the center of the H-shaped enclosure.Tab.1 introduces the physical properties of the water as a base fluid and a copper as nanoparticles.

Figure 1: Initial physical models and their particles generations.(a) Initial physical model (1).(b) Particles (mesh) model (1).(c) Initial physical model (2).(d) Particles (mesh) model (2)

Table 1: Physical properties of the water and copper (Cu)

In this study, one phase model and Brinkman’s-extended non-Darcy model are used for a nanofluid and a porous medium, respectively.The Lagrangian form of governing equations are written as [35,36]:

where,

KandFare Forchheimer’s coefficient:

The properties of the nanofluid are defined as [37-40]:

The Lagrangian form of the dimensionless equations are defined as:

The dimensionless quantities are:

2.1 Boundary Conditions

In this study, the dimensionless boundary conditions are:

On the outer side-walls of H-cavity:U=0;V=0;θ=0,

On the center walls of H-cavity:U=0;V=0;θ=1,

On the horizontal and center walls of H-cavity:U=0;V=0;=0,

On the embedded square blockages:U=0;V=0;θ=0,

The average Nusselt number is defined as:

where n andLwis the normal vector and total length of the heated partition in the H-cavity.

3 Numerical Method

Here, the solving steps for the implicit scheme in the ISPH method are mainly depend in the projection method [41].Prediction step:

where, Pors=δ.

Pressure Poisson equations (PPE):

where,γ:(0 ≤γ≤1)is a relaxation parameter.Hyder et al.[42] used a new framework for solvingχ-stochastic Poisson equation.

The corrector step:

Thermal equation:

Update the particles positions:

In this study, the shifting technique according to [43] is applied to avoid particles disorders:

whereFis any hydrodynamic function andis a gradient of particle concentration.

3.1 SPH Approach

The concept of SPH method for calculating any fluid hydrodynamics properties is:

Wis a kernel function:

The divergence and gradient in SPH method can be approximated as:

According to [44], the divergence of the velocity is corrected by a kernel gradient normalization as:

The gradient of the pressure and the divergence of the velocity vector are:

For the second derivative, Laplacian operator can be approximated as:

Laplacian of velocity, pressure and temperature are approximated as:

4 Validation Tests

In order to prove the efficiency of the present ISPH method for simulating the natural convection flow in a partial layer porous cavity, a comparison of the temperature profiles along different horizontal lines with numerical and experimental data from Beckermann et al.[45] is performed.In Fig.2, ISPH results for the temperature profiles at three different positions in a partial layer porous cavity are agreeing well with the available numerical and experimental data from Beckermann et al.[45].For the mesh independence test, three different particle sizesd0=0.02, 0.01 and 0.005 were investigated.Tab.2 presents the average Nusselt number for three different particle sizesd0=0.02, 0.01 and 0.005.There are small variations in the value of average Nusselt number at three different particle sizes.Hence, the particle sized0=0.01 has been chosen for all the performed simulations in the current study.

Figure 2: Comparison of the natural convection in a partitioned porous cavity between numerical and experimental data from Beckermann et al.[45] and present ISPH results

Table 2: Mesh independence test for average Nusselt number when the hot source length B=0.2, solid volume fraction ?=0.01, Rayleigh number Ra=104, Darcy parameter Da=10-3,and porosity ε=0.6

Table 2: Mesh independence test for average Nusselt number when the hot source length B=0.2, solid volume fraction ?=0.01, Rayleigh number Ra=104, Darcy parameter Da=10-3,and porosity ε=0.6

Particle size d0=0.02 d0=0.01 d0=0.005 Nu 0.0237 0.0258 0.0261

5 Results and Discussion

In this work, the numerical simulations of the natural convection in an H-shaped enclosure under the impacts of the key physical parameters were presented in the temperature and velocity field distributions as well as average Nusselt number profiles.The partitions of the hot source in the center boundary of the H-shaped enclosure were varied from 0 to 0.6.Rayleigh number varies from 103to 105and the Darcy parameter varies from 10-2to 10-5.Different numbers of the cooled square blockages inside an H-shaped enclosure were considered.In addition, adding more concentration of nanoparticles inside an H-shaped enclosure was limited on 5% to avoid the solidification between the nanoparticles and a porous medium.This section is divided into four parts depending on the impacts of the physical parameters and thermal conditions.The first part will discuss the presence of the cooled square blockages inside an H-shaped enclosure with variations on the number of cooled blockages.The second part will check the impacts of the variable length of the hot source.Third part will investigate the effects of the augmented buoyancy force (greater values of the Rayleigh number) with increasing the porous resistance (lower values of the Darcy parameter) on the right side of the H-shaped enclosure.Finally, an enhancement of heat transfer by adding more nanoparticles concentration will be discussed.

5.1 Presence of Cooled Blockages

Fig.3 presents the temperature distributions under the variations on the number of the cooled square blockages at Rayleigh numberRa=104, Darcy parameterDa=10-3,?=0.01, porosityε=0.6 and hot source lengthB=0.2.It is found that the number of the cooled square blockages plays an important role in decreasing the temperature distributions inside an H-shaped enclosure.It is seen that the cooled square blockages are reducing the temperature distributions in the center of the H-shaped enclosure.An extra number of the cooled square blockages (around eight blockages) reduces the temperature distributions beside the hot source inside an H-shaped enclosure.Fig.4 shows the velocity field distributions under the variations on the number of the cooled square blockages.In general, due to the porous resistance on the right side of the H-shaped enclosure, the velocity of the fluid flows on the right porous layer is lower than the velocity of the fluid flows on the left side (nanofluid layer) of the H-shaped enclosure.It seems that the velocity of the fluid flows is decreasing according to an increase in the number of the cooled square blockages.The time transitions from the unsteady state to steady-state for the temperature distributions under the variations on the number of the cooled square blockages have been shown in Fig.5.An extra number of the cooled blockages reduces the temperature distributions.

Figure 3: Temperature distributions under different numbers of the cooled square blockages at Rayleigh number Ra=104, Darcy parameter Da=10-3, ?=0.01, porosity ε=0.6 and hot source length B=0.2.(a) Two blockages.(b) Three blockages.(c) Four blockages.(d) Eight blockages

Figure 4: Velocity field distributions under different numbers of the cooled square blockages at Rayleigh number Ra=104, Darcy parameter Da=10-3, ?=0.01, porosity ε=0.6 and hot source length B=0.2.(a) Two blockages.(b) Three blockages.(c) Four blockages.(d) Eight blockages

Figure 5: Time histories for the temperature distributions under different numbers of the cooled square blockages at Rayleigh number Ra=104, Darcy parameter Da=10-3, ?=0.01, porosity ε=0.6 and hot source length B=0.2.(a) Two blockages.(b) Eight blockages

—Effects of the Hot Source Length

Figs.6 and 7 show the temperature distributions under the impacts of the hot lengthBat Rayleigh numberRa=104, Darcy parameterDa=10-3,?=0.01 and porosityε=0.6.An increase on the length of the hot source augments the buoyancy force and consequently it rises the temperature distributions inside an H-shaped enclosure.When the length of the hot source is equal to 0.6, then the temperature distributions are fully filled out the both sides of the H-shaped enclosure.In addition, an increase in the length of the hot source augments the buoyancy force and consequently the velocities of the fluid flows inside an H-shaped enclosure are increasing.This behavior appears in Fig.7, in which the velocity fields in both sides of the H-shaped enclosure are increasing according to an increase in the length of the hot source.

Figure 6: Temperature distributions under the impacts of hot source length B at Rayleigh number Ra=104, Darcy parameter Da=10-3, ? =0.01 and porosity ε =0.6.(a) B=0.(b) B=0.1.(c) B=0.3.(d) B=0.6

Figure 7: Velocity field distributions under the impacts of hot source length B at Rayleigh number Ra=104, Darcy parameter Da=10-3, ?=0.01 and a porosity ε=0.6.(a) B=0.(b) B=0.1.(c) B=0.3.(d) B=0.6

—Effects of Rayleigh and Darcy Parameter

Figs.8 and 9 introduce the influences of Rayleigh and Darcy parameter on the temperature and velocity field distributions at?=0.01, porosityε=0.6 and hot lengthB=0.2.Generally,an increase in the Rayleigh number augments the buoyancy force and consequently both of the temperature distributions and strength of the velocity fields are increase.From Fig.8, as the Darcy parameter decreases fromDa=10-2to 10-5, the temperature distributions on the right side (porous layer) of the H-shaped enclosure are decreasing due to high porous resistance in this side for any value of Rayleigh number.Moreover, an increase on the Rayleigh number augments the temperature distributions and the physical reason returns to the high buoyancy force.From Fig.9, there are clear differences in the velocity fields between the left and right area of the H-shaped enclosure due to the presence of the porous layer on the right side.Hence, as the Darcy parameter decreases (higher porous resistance) leads to a strong decrease on the velocity fields on the right side (porous layer) of the H-shaped enclosure.

Figure 8: Effects of Rayleigh and Darcy parameter on the temperature distributions at ?=0.01,porosity ?=0.6 and hot length B=0.2

Figure 9: Effects of Rayleigh and Darcy parameter on the velocity field distributions at ?=0.01,porosity ?=0.6 and hot length B=0.2

Higher buoyancy force at higher Rayleigh number augments the strength of the velocity fields and the distributions of the velocity fields inside an H-shaped enclosure.It is seen that the maximum values of the velocity fields are 1.82, 15.79 and 84.31 when the Rayleigh number equalsRa=103, 104and 105, respectively.From these results, the Rayleigh number is considering the main factor in the enhancement of heat transfer and fluid flows inside an H-shaped enclosure.Fig.10 presents the average Nusselt number under the impacts of Rayleigh and Darcy parameter at?=0.01, porosityε=0.6 and hot lengthB=0.2.It seems that the highest value of the average Nusselt number appears at Rayleigh numberRa=105.As Darcy parameter rises from 10-5to 10-2, the average Nusselt number decreases when Rayleigh number equals toRa=105.At lower values of the Rayleigh numberRa≤104, the average Nusselt number has slight changes under the impact of a Darcy parameter.

Figure 10: Effects of Rayleigh and Darcy parameter on the average Nusselt number at φ=0.01,porosity ?=0.6 and hot length B=0.2

5.2 Effects of Solid Volume Fraction

Figs.11 and 12 introduce the distributions of the temperature inside H-enclusre under the effects of solid volume fraction?at Rayleigh numberRa=104, Darcy parameterDa=10-3,porosityε=0.6 and hot source lengthB=0.2.In Fig.11, an increase in the solid volume fraction?enhances the temperature distributions.In Fig.12, adding nanoparticles augments the viscosity of the fluid and consequently the velocity fields were declined.In the current simulations, adding nanoparticles is limited to 5% to avoid the solidification within the porous medium.

Figure 11: Temperature distributions under the effects of solid volume fraction ? at Rayleigh number Ra=104, Darcy parameter Da=10-3, porosity ε=0.6 and hot source length B=0.2.(a) φ=0.(b) φ=0.03.(c) φ=0.05

Figure 12: Velocity fields distributions under the effects of solid volume fraction ? at Rayleigh number Ra=104, Darcy parameter Da=10-3, porosity ε=0.6 and hot source length B=0.2.(a) φ=0, (b) φ=0.03, (c) φ=0.05

Tab.3 presents the average Nusselt numberfor different values of blockages numbers,hot source lengthBand solid volume fraction?at Rayleigh numberRa=104, Darcy parameterDa=10-3and porosityε=0.6.Here, the average Nusselt number is varying as the number of the square blockages is increasing from 2 to 8.There is one peak in the average Nusselt number when the number of the square blockages is equal to four.The description of the peak is appeared when the number of blockages is increasing from two to four and from four to eight blockages.Average Nusselt number is decreasing as the length of the hot sourceBis increasing from 0 to 0.1.While,asBis increasing from 0.1 to 0.3, the average Nusselt number rises.There is a slight decrease on the average Nusselt number as the hot lengthBincreases from 0.3 to 0.6.In addition, adding more concentration of the nanoparticles from 0 to 5% moderates the average Nusselt number.

Table 3: Average Nusselt number for different values of blockages numbers, hot source length B and solid volume fraction ? at Rayleigh number Ra=104, Darcy parameter Da=10-3, and porosity ε=0.6

Table 3: Average Nusselt number for different values of blockages numbers, hot source length B and solid volume fraction ? at Rayleigh number Ra=104, Darcy parameter Da=10-3, and porosity ε=0.6

Blockages numberB?Average Nusselt numberNu 2 0.2 0.01 0.0134 3 0.0832 4 0.0862 8 0.0379 0 0 0.0494 0.1 0.0065 0.3 0.0338 0.6 0.0233 0.2 0 0.0265 0.01 0.0258 0.03 0.0233 0.05 0.0195

6 Conclusion

The improved ISPH method presented in this paper is a simple approach to simulate the natural convection from the heated partitions in a nanofluid-filled H-shaped enclosure saturated with a partial layer porous medium.ISPH method is modified in terms of the solving pressure Poisson equation and corrected the first derivative of the velocity.The simulations showed that the length of the hot source augments the buoyancy force and consequently the distributions of the velocity field and temperature are increased.An extra number of the cooled square blockages reduces the distributions of the temperature and velocity field inside an H-shaped enclosure.The Rayleigh number plays an important factor in the augmentation of the velocity field and temperature distributions inside an H-shaped enclosure.Due to the porous resistance on the right side of the H-shaped enclosure, there is clear difference in the velocity field between the left and right sides of the H-shaped enclosure.As the Darcy parameter decreases, then there is almost no fluid flow inside the right side (porous layer) of the H-shaped enclosure.The average Nusselt number is increasing as the Rayleigh number increases and it declines according to an increase in Darcy parameter.Adding nanoparticles until 5% reduces the velocity field and enhances the heat transfer inside an H-shaped enclosure.

Acknowledgement:The authors would like to extend their appreciations to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (R.G.P 2/70/41).

Funding Statement:Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia funded this work through project number (R.G.P 2/70/41).

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.