Physica A - دانشگاه آزاد اسلامی واحد...

Physica A 509 (2018) 210–233

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Physica A

journal homepage: www.elsevier.com/locate/physa

Develop the nano scale method of lattice Boltzmann topredict the fluid flow and heat transfer of air in the inclinedlid driven cavity with a large heat source inside, Two casestudies: Pure natural convection & mixed convection

Marjan Goodarzi a, Annunziata D’Orazio b, Ahmad Keshavarzi c,Sayedali Mousavi d, Arash Karimipour d,∗

a Sustainable Management of Natural Resources and Environment Research Group, Faculty of Environment and Labour Safety, Ton Duc

Thang University, Ho Chi Minh City, Vietnamb Dipartimento di Ingegneria Astronautica, Elettrica ed Energetica, Sapienza Università di Roma, Via Eudossiana 18, Roma 00184, Italyc Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Irand Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

h i g h l i g h t s

• Develop LBM performance to simulate the heat flux of heat source.• Natural convection and mixed convection of inclined driven cavity by LBM.• Modify collision operator and macroscopic velocities equations in LBM.

a r t i c l e i n f o

Article history:

Received 18 February 2018

Received in revised form 23 April 2018

Available online xxxx

Keywords:

Lattice Boltzmann method

Heat flux boundary condition

Heat source

a b s t r a c t

Nano scale method of lattice Boltzmann is developed to predict the fluid flow and heattransfer of air through the inclined lid driven 2-D cavity while a large heat source isconsidered inside it. Two case studies are supposed: first one is a pure natural convection atGrashof number from 400 to 4000000 and second one is a mixed convection at Richardsonnumber from 0.1 to 10 at various cavity inclination angles. Using LBM to simulate theconstant heat flux boundary condition along the obstacle, is presented for the first timewhile the buoyancy forces affect the velocity components at each inclination angle; hencethe collision operator of LBM and also a way to estimate the macroscopic velocities shouldbe modified. Results are shown in the terms of streamlines and isotherms, beside theprofiles of velocity, temperature and Nusselt number. It is observed that the present modelof LBM is appropriately able to simulate the supposed domain. Moreover, the effects ofinclination angle are more important at higher values of Richardson number.

© 2018 Elsevier B.V. All rights reserved.

1. Introduction

Various particle base methods like lattice Boltzmann method (LBM) have been introduced by now to simulate the fluidflow. In general, all particle base methods like molecular dynamic (MD), direct simulation of Monte Carlo (DSMC) or LBM

∗ Corresponding author.E-mail addresses: [email protected] (M. Goodarzi), [email protected], [email protected] (A. Karimipour).

https://doi.org/10.1016/j.physa.2018.06.013

0378-4371/© 2018 Elsevier B.V. All rights reserved.

M. Goodarzi et al. / Physica A 509 (2018) 210–233 211

Nomenclature

e Internal energy

f Hydrodynamic distribution function

g Thermal distribution function

g Vector of gravity

Gr Grashof number

H Cavity height

L Cavity length

Nu Nusselt number

q Heat flux

Re Reynolds number

Ri Richardson number

Tc Cold wall temperature

u Vector of macroscopic velocity

U0 Upper wall velocity

(U, V ) Velocities in dimensionless forms

(X, Y ) Coordinates in dimensionless forms

Greek symbols

ρ Density

γ Cavity inclination angle

υ Kinematic viscosity

θ Dimensionless temperature

can be implied at different flow regimes including MEMS and NEMS; however using MD and DSMC would be much timeconsuming at macro or even at micro scales. As a result, LBM usage has increased in recent decade; and a great number ofworks can be addressed to improve LBM ability at macro, micro or nanoscales [1–15].

In addition, LBM is based on first-order PDE Boltzmann equation, which makes its discretization and coding easier thanusing the well-known Naiver–Stokes approach. Moreover, the pressure field is directly achieved by LBMwith no more needto other equations. It means LBM would be more useful to simulate the more complex or multi-phase flows in comparisonwith SIMPLE algorithm approach [16–33].

LBMworks based on the collision and streaming between the virtual particles located on the nodes of a lattice at each timestep. To do this, the Boltzmann equation is defined according to the microscopic density–momentum distribution functionof ‘‘f ’’ so that all other macroscopic variables can be achieved simply through it [34–56].

Various models of LBM have been developed to simulate the thermal domains. Among them, double populationdistribution functions model, showedmore desired performance in both aspects of accuracy and convergence. In this way, aseparate distribution function of ‘‘g ’’, called internal energy distribution function, was introduced for the temperature field,which was defined according to density–momentum distribution function of ‘‘f ’’ [57–71]. However, there are still severaldrawbacks in different presented models of LBM such as weak convergence procedure or undesired accuracy at differentboundary condition models in the form of LBM approach. In addition, LBM is a compressible way for the ideal gases, whichmeans the incompressible flows can be modeled by this way at near incompressible limit (low values of Mach number).Therefore, LBMwould be encounter to compressibility error at higher values of Mach number. Thesementioned facts besidesome other minor ones, have made researchers to work on LBM and improve its performance in different aspects as likedeveloping the better boundary condition models or using a suitable collision operator like BGK model or even introducingappropriate models of relaxation times to increase convergence. The last object leads to apply LBM for the fluids with higherPrandtl number as like the nanofluids [72–89].

2. Problem statement and present work novelty

Fluid flow and heat transfer of air (Pr = υ/α = 0.7) through the inclined lid driven 2-D cavity is studied by thenano scale method of LBM while a large heat source is considered inside it (Fig. 1). Upper and lower walls of the obstacleare imposed by a constant heat flux of q′′0 . Moreover the aspect ratio of cavity equals to L/H = 3 and its sidewalls areadiabatic.

212 M. Goodarzi et al. / Physica A 509 (2018) 210–233

Fig. 1. The schematic view of the cavity with heat source inside.

Fig. 2. Lattice of D2Q 9 .

Two case studies are assumed: first one is a pure natural convection at Grashof number chosen as Gr = gβq′′0H4/(Kυ2)=400, Gr = 40 000 and Gr = 4000 000 while the cold upper and lower walls are fixed with no motion; and thesecond case is a mixed convection at Richardson number changes as Ri = Gr/Re2 = 0.1, Ri = 1 and Ri = 10 for the fixedvalue of Reynolds number (Re = U0H/υ = 200) while the cold upper wall moves with constant velocity of U0. Cavityinclination angle corresponds from horizontal to vertical positions as γ = 0◦, γ = 30◦, γ = 60◦ and γ = 90◦ through allstates of the second case.

As a result, present work aims to develop LBM performance to simulate the fluid flow and heat transfer in such supposedphysical configuration. Using LBM to simulate the constant heat flux boundary condition along the obstacle, is presentedfor the first time while the buoyancy forces affect the velocity components at each inclination angle; hence the collisionoperator of LBM and also a way to estimate the macroscopic velocities should be modified.


Fig. 3. Profiles of velocity and temperature versus Iwatsu et al. [3].

3. Equations

3.1. Lattice Boltzmann Method

Hydrodynamic and thermal Boltzmann equations are written as follows using D2Q9 lattice (Fig. 2),

∂ fi

∂t+ ciα

∂ fi

∂xα= −

fi − f eiτf

(1)

∂gi

∂t+ ciα

∂gi

∂xα= −

gi − geiτg

− fiZi (2)


Fig. 4. The comparison of dimensionless temperature against Habchi & Acharya [4].

Fig. 5. Pure natural convection streamlines and isotherms of horizontal cavity at Gr = 400.

c i=0 = (0, 0) , c i=1,2,3,4 =(cos

i− 12

π, sini− 12

π

)c

c i=5,6,7,8 =√2

(cos

[(i− 5)

2π +

π

4

], sin

[(i− 5)

2π +

π

4

])c

(3)

Zi = (ciα − uα)[

δuα

δt+ ciα

∂uα

∂xα

](4)


Fig. 6. Pure natural convection streamlines and isotherms of horizontal cavity at Gr = 40 000.

where c shows the microscopic velocity. i and α also represent the lattice velocity directions and Cartesian geometrycomponents, respectively. Moreover τf and τg imply the hydrodynamic and thermal relaxation times. The equilibriumdistribution functions:

f ei=0,1,...,8 = ωiρ[1+

3c i · uc2

+9(c i · u)2

2c4−

3u2

2c2

], ω0 = 4/9, ω1,2,3,4 = 1/9, ω5,6,7,8 = 1/36 (5)

ge0 = −2

3ρe

[u2

c2

], ge1,2,3,4 =

1

9ρe

[1.5+ 1.5

c1,2,3,4 · uc2

+ 4.5(c1,2,3,4 · u)2

c4− 1.5

u2

c2

]

ge5,6,7,8 =1

36ρe

[3+ 6

c5,6,7,8 · uc2

+ 4.5(c5,6,7,8 · u)2

c4− 1.5

u2

c2

] (6)

Now discretized figure of Boltzmann equation:

fi(x+ c i∆t, t +∆t)− fi(x, t) = −∆t

τf(fi(x, t)− f ei (x, t)) (7)

in which υ = τf RT , leads to:

−fi(x, t)− f ei (x, t)

τf= −

∆t

2τf

[fi(x+ c i∆t, t +∆t)− f ei (x+ c i∆t, t +∆t)

]−

∆t

2τf

[fi(x, t)− f ei (x, t)

](8)

fi(x+ c i∆t, t +∆t)− fi(x, t) = −∆t

2τf

[fi(x+ c i∆t, t +∆t)− f ei (x+ c i∆t, t +∆t)

]−

∆t

2τf

[fi(x, t)− f ei (x, t)

](9)

and through the same procedure for the thermal equation, using α = 2τgRT :

−gi(x, t)− gei (x, t)

τg− fi(x, t)Zi(x, t) = −

∆t

2τg

[gi(x+ c i∆t, t +∆t)− gei (x+ c i∆t, t +∆t)

]

−∆t

2fi(x+ c i∆t, t +∆t)Zi(x+ c i∆t, t +∆t)−

∆t

2τg

[gi(x, t)− gei (x, t)

]−

∆t

2fi(x, t)Zi(x, t)

(10)

gi(x+ c i∆t, t +∆t)− gi(x, t) = −∆t

2τg

[gi(x+ c i∆t, t +∆t)− gei (x+ c i∆t, t +∆t)

]

−∆t

2fi(x+ c i∆t, t +∆t)Zi(x+ c i∆t, t +∆t)−

∆t

2τg

[gi(x, t)− gei (x, t)

]−

∆t

2fi(x, t)Zi(x, t)

(11)


Fig. 7. Dimensionless vertical velocity and temperature profiles along the horizontal centerline of cavity in pure natural convection state.

Two distribution functions of f̃i and g̃i are used to overcome the difficulty of the implicitness of two last equations.

f̃i = fi +∆t

2τf(fi − f ei )−

∆t

2F and g̃i = gi +

∆t

2τg(gi − gei )+

∆t

2fiZi (12)

Collision and streaming steps:

f̃i(x+ c i∆t, t +∆t)− f̃i(x, t) = −∆t

τf + 0.5∆t[̃fi(x, t)− f ei (x, t)

](13)

g̃i(x+ c i∆t, t +∆t)− g̃i(x, t) = −∆t

τg + 0.5∆t[g̃i(x, t)− gei (x, t)

]−

τg∆t

τg + 0.5∆tfiZi (14)

Finally determine the macroscopic variables as follows,

ρ =∑

i

f̃i, ρe = ρRT =∑

i

g̃i −∆t

2

∑

i

fiZi, ρu =∑

i

c ĩfi (15)

q =(∑

i c ĩgi − ρeu− 0.5∆t∑

i c ifiZi)τg

τg + 0.5∆t(16)


Fig. 8. Streamlines at Ri = 0.1 for γ = 0 (top), γ = 30, γ = 60 and γ = 90◦ (bottom).

3.2. Gravity

Buoyancy force is introduced as G = βg(T − T ) and F = G.(c− u)f e/RT applying the Boussinesq approximation,

∂t f + (c.∇)f = −f − f e

τf+ F (17)

and the equations discretized forms, considering the gravity effects:

∂t fi + (c i.∇)fi = −fi − f ei

τf+

G.(c i − u)RT

f ei (18)


Fig. 9. Isotherms at Ri = 0.1 for γ = 0 (top), γ = 30, γ = 60 and γ = 90◦ (bottom).

f̃i(x+ c i∆t, t +∆t)− f̃i(x, t) = −∆t

τf + 0.5∆t[̃fi − f ei

]

+(

∆tτf

τf + 0.5∆t3G(cix − u)

c2f ei

)sin γ +

(∆tτf

τf + 0.5∆t3G(ciy − v)

c2f ei

)cos γ

(19)

f̃i = fi +∆t

2τf(fi − f ei )−

∆t

2F ⇒ fi =

τf f̃i + 0.5∆tf eiτf + 0.5∆t

+0.5∆tτf

τf + 0.5∆tF (20)


Fig. 10. Streamlines at Ri = 1 for γ = 0 (top), γ = 30, γ = 60 and γ = 90◦ (bottom).

fi =τf f̃i + 0.5∆tf eiτf + 0.5∆t

+0.5∆tτf

τf + 0.5∆tG.(ci − u)

RTf ei (21)

Through a simple math procedure on Eq. (21):

fi =τf f̃i + 0.5∆tf eiτf + 0.5∆t

+(

0.5∆tτf

τf + 0.5∆tG(cix − u)

RTf ei

)sin γ +

(0.5∆tτf

τf + 0.5∆tG(ciy − v)

RTf ei

)cos γ (22)


Fig. 11. Isotherms at Ri = 1 for γ = 0 (top), γ = 30, γ = 60 and γ = 90◦ (bottom).

Finally the macroscopic variables, including the influences of gravity and inclination angle,

ρ =∑

i

f̃i and u = (1/ρ)∑

i

f̃icix +∆t

2G sin γ and v = (1/ρ)

∑

i

f̃iciy +∆t

2G cos γ (23)

3.3. Hydrodynamic boundary conditions

No-slip boundary condition is simulated using the non-equilibrium bounce back model of LBM.


Fig. 12. Streamlines at Ri = 10 for γ = 0 (top), γ = 30, γ = 60 and γ = 90◦ (bottom).

For example for the west wall:

ρ =∑

i

f̃i ⇒ f̃1 + f̃5 + f̃8 = ρw − (̃f0 + f̃2 + f̃3 + f̃4 + f̃6 + f̃7) (24)

u = (1/ρ)∑

i

f̃icix +∆t

2G sin γ ⇒ f̃1 + f̃5 + f̃8 = ρwUw + (̃f3 + f̃6 + f̃7)−∆t/2ρwG sin γ (25)


Fig. 13. Isotherms at Ri = 10 for γ = 0 (top), γ = 30, γ = 60 and γ = 90◦ (bottom).

v = (1/ρ)∑

i

f̃iciy +∆t

2G cos γ ⇒ f̃5 − f̃8 = ρwVw + (−̃f2 + f̃4 − f̃6 + f̃7)−∆t/2ρwG cos γ (26)

f̃1 − f̃ e1 = f̃3 − f̃e3 ⇒ f̃1 = f̃3 +

2

3ρwUw. (27)


Fig. 14. Dimensionless horizontal velocity profiles at vertical centerline of cavity.

Applying Eq. (27) in Eqs. (25) and (26),

f̃8 = f̃6 −f̃4 − f̃2

2+

1

6ρwUw −

1

2ρwVw +

∆t

4ρwG(cos γ − sin γ )

f̃5 = f̃7 +f̃4 − f̃2

2+

1

6ρwUw +

1

2ρwVw −

∆t

4ρwG(cos γ + sin γ )

(28)

3.4. Thermal boundary conditions

GPTBC model of D’Orazio, according to non-equilibrium bounce back model, is used to represent the cold wall tempera-ture. For example, for the cavity top moving lid [20,21]:

g̃4,7,8 = ρ(e+ e′

)×

(ge4,7,8/ρe

)and ρe′ = 2ρe+ 1.5∆t

∑

i

fiZi − 3K (29)

g̃7 = (3ρe+ 1.5∆t∑

i

fiZi − 3(̃g0 + g̃1 + g̃2 + g̃3 + g̃5 + g̃6))[3.0− 6U0 + 3.0U20 ]1

36

g̃4 = (3ρe+ 1.5∆t∑

i

fiZi − 3(̃g0 + g̃1 + g̃2 + g̃3 + g̃5 + g̃6))[1.5− 1.5U20 ]1

9

g̃8 = (3ρe+ 1.5∆t∑

i

fiZi − 3(̃g0 + g̃1 + g̃2 + g̃3 + g̃5 + g̃6))[3.0− 6U0 + 3.0U20 ]1

36

(30)


Fig. 15. Dimensionless horizontal velocity profiles along the vertical centerline of cavity at various Ri.

GPTBC model can also be used for the adiabatic sidewalls. For example for the cavity west wall, including qx = 0 inEq. (15):

∑

i

cix̃gi = 0.5∆t∑

i

cixfiZi (31)

ρe′ = 3(̃g6 + g̃3 + g̃7)+ 1.5∆t∑

i

cix

cfiZi − ρe (32)

The hydrodynamic and thermal boundary conditions of the obstacle in the LBM form, can be written through the sameway. Moreover the following equations are used in order to present the results in dimensionless forms,

X = x/H, Y = y/H, θ = (T − Tc)/∆T , Nux =H

∆T

∂T

∂y, ∆T = q′′0H/K

Natural convection: U = u/√H2βgq′′0/K , V = v/

√H2βgq′′0/K

Mixed convection: U = u/U0, V = v/U0

(33)

In the case of pure natural convection, the first pair of U & V in the above equation is used; Obviously for the state ofmixed convection, the velocities would be presented in dimensionless forms based on the last objects of Eq. (33).


Fig. 16. Dimensionless temperature profiles at vertical centerline of cavity.

Table 1

Grid independency study for γ = 0, Ri = 0.1 at X = 1.5 and Y = 0.9.Mesh

Parameters 180× 60 210× 70 240× 80U 0.366 0.372 0.375

θ 0.0728 0.0735 0.0739

4. Results and discussion

4.1. Grid study and validation

Table 1 shows the grid independency study for the state of γ = 0 and Ri = 0.1 at the point of X = 1.5 and Y = 0.9. It isobserved that the differences between the results corresponded to 210× 70 and 240× 80 are negligible so that the latticenodes of 210× 70 are found suitable for the following computations.

The first validation is presented in Fig. 3 with those of Iwatsu et al. [3] concerned an enclosure which is heated fromupper lid and cooled from the lower wall while its adiabatic sidewalls are kept adiabatic at Gr = 102 and Re = 400. Thesecond case study for the validation is chosen in a way to investigate the simulation of obstacle existence through the fluidflow as well as a work reported by Habchi & Acharya [4] involved mixed convection in a vertical channel (x-direction) witha attached hot block to its right wall. Fig. 4 illustrates the comparison of dimensionless temperature from present work by


Fig. 17. Dimensionless temperature profiles along the vertical centerline of cavity at various Ri.

LBM versus Ref. [4] along the channel cross-section of x/L = 0.77 at Ra = 105 for Ri = 0.1 and Ri = 5. The good agreementsare seen in both figures of 3 and 4.

4.2. Pure natural convection

Figs. 5 and 6 represent the pure natural convection streamlines and isotherms of horizontal cavity at Gr = 400 andGr = 40 000, respectively. Two strong cells around the cavity heat source and their corresponded symmetric isothermscan be seen in these figures. Perpendicular isotherms to the adiabatic walls and completely symmetric position of them,than to the vertical cavity centerline, are other visual achievements. Moreover isotherms are dense and compact around thehorizontal walls of obstacle due to imposing heat flux at these areas.

Dimensionless vertical velocity (V ) and temperature profiles (θ ) along the horizontal centerline of cavity in pure naturalconvection state are presented in Fig. 7. More Gr corresponds to more V which means generating two stronger rotationalcells at higher values of Grashof number in the left and right hand sides of the heat source.

4.3. Mixed convection

Figs. 8, 9, 10, 11, 12 and 13 show the streamlines and isotherms for γ = 0, γ = 30, γ = 60 and γ = 90◦ at Ri = 0.1,Ri = 1 and Ri = 10 respectively; while the cavity upper lid moves with constant velocity of U0 which represent the mixedconvection state. A large and long cell dominates the whole domain of cavity which no changes are seen through it at higher


Fig. 18. Dimensionless temperature profiles at horizontal centerline of cavity.

inclination angle (γ ) at Ri = 0.1 (which implies the force convection domination state). However it will change and divideto several smaller cells at higher γ at Ri = 10. In the all states, the center of the main cell is seen in the right hand side ofobstacle. Moreover dense isotherms are generated adjacent to the heat source upper wall and less dense ones can be seenaround the obstacle lower wall. Isotherms are normal to the cavity and heat source adiabatic sidewalls. More γ correspondsto more change in isotherms at Ri = 10 which represents the free convection domination.

Dimensionless horizontal velocity profiles (U) along the vertical centerline of cavity are presented in Fig. 14 for Ri = 1(mixed convection domination state) and Ri = 10. It is observed that U = 0 at Y = 0, Y = 1 (cavity upper and lowerwalls) and at Y = 1/3 , Y = 2/3 (heat source upper and lower walls) according to no-slip boundary condition. At the stateof Ri = 1, the flow direction is in the opposite of X inside the bottom domain of obstacle while the flow direction is in thesame way with cavity upper lid (the same direction with X) inside the top domain of obstacle. At the state of Ri = 10, morechanges in U profiles are seen with γ so that the absolute value of U equals to 0.74 at Y = 0.1 which is significantly higherthan that of Ri = 1. These trends can also be seen in Fig. 15 which represents the effects of Richardson number on U profilesat both horizontal and vertical positions of cavity.

In the following, dimensionless temperature profiles (θ ) at vertical centerline of cavity are presented in Fig. 16. At thestate of Ri = 1, it is seen that θ = 0 on both cavity cold walls and θ = 0.18 on heat source lower wall and θ = 0.31 atheat source upper wall which they both illustrate the domain under the constant heat flux. At Ri = 10, the temperature onthe heat source lower wall increases with γ while the temperature of heat source upper wall decreases with γ . These factscan also be traced in Figs. 17 and 18 concerned θ profiles along the vertical and horizontal centerlines of cavity at variousamounts of Ri, respectively.

More focus on heat transfer rate, Figs. 19, 20 and 21 present the local Nusselt number along the cavity upper and lowerwalls at various inclination angles and Richardson number. More severed oscillations are observed through those of along


Fig. 19. Local Nusselt number at various cavity inclination angles.

the cavity upper wall; Moreover NuX values are in general greater along that wall rather than the cavity lower wall. To havea better vision, Fig. 22 illustrates the averaged Nusselt number (Num) along the cavity horizontal walls. It is seen that Numis significantly higher on cavity upper wall in comparison with its amount along the cavity lower wall, which implies themain heart transfer rate is achieved along the top lid. At Ri = 0.1 and Ri = 1, Nusselt number does not change significantlywith γ however this event is completely different from the state of Ri = 10; so that more γ corresponds to larger Num onthe cavity lower wall. It should be mentioned that fact occurs in inverse on the cavity upper wall.

5. Conclusion

Fluid flow and heat transfer of air in the inclined lid driven cavity was studied by the nano scale method of latticeBoltzmann while a large heat source was considered inside it. Upper and lower walls of the obstacle were imposed by aconstant heat flux of q′′0 . Two case studies were assumed: first one was a pure natural convection and the second case was amixed convection at Richardson number changes as Ri = 0.1, Ri = 1 and Ri = 10.

Present work developed LBM performance to simulate the fluid flow and heat transfer in such supposed physicalconfiguration. Using LBM to simulate the constant heat flux boundary condition along the obstacle, was presented for thefirst time while the buoyancy forces affected the velocity components at each inclination angle; hence the collision operatorof LBM and also a way to estimate the macroscopic velocities were modified.


Fig. 20. Local Nusselt number at various Ri for γ = 0.

Moreover the following physical points can be addressed in brief:

Pure natural convection state: Two strong cells around the heat source and their corresponded symmetric isotherms canbe seen. Perpendicular isotherms to the adiabatic walls and completely symmetric position of them, than to the verticalcavity centerline, are other visual achievements. Moreover isotherms are dense and compact around the horizontal walls ofobstacle due to imposing heat flux at these areas. Higher Gr corresponds to larger V , which means generating two strongercells at higher values of Grashof number in the left and right hand sides of obstacle.

Mixed convection state: A large and long cell dominates the whole domain of cavity which no changes are seen throughit at higher inclination angle at Ri = 0.1. However it will change and divide to several smaller cells at higher γ at Ri = 10. Inthe all states, the center of the main cell is seen in the right hand side of obstacle. Moreover dense isotherms are generatedadjacent to the heat source upper wall and less dense ones can be seen around the obstacle lower wall. Isotherms are normalto the cavity and heat source adiabatic sidewalls. More γ corresponds to more change in isotherms at Ri = 10. Num issignificantly higher on cavity upper wall in comparison with its amount along the cavity lower wall, which implies the mainheart transfer rate is achieved along the top lid. At Ri = 0.1 and Ri = 1, Nusselt number does not change significantly with γhowever this event is completely different from the state of Ri = 10; so that more γ corresponds to larger Num on the cavitylower wall. It should be mentioned that fact occurs in inverse on the cavity upper wall.


Fig. 21. Local Nusselt number at various Ri for γ = 90.

Fig. 22. Averaged Nusselt number along the cavity horizontal walls.


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transfer in a microchannel using lattice boltzmann method, Int. J. Therm. Sci. 54 (2012) 142–152.

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applied to mixed convection in inclined lid driven cavity, Meccanica 50 (4) (2015) 945–962.

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methods for solving natural convection heat transfer problems inside cavities and enclosures, in: Abstract and Applied Analysis, vol. 2014, Hindawi

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convection in the inclined lid driven cavity using lattice Boltzmann method, J. Theoret. Appl. Mech. 51 (2) (2013) 447–462.

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water nanofluid in a shallow inclined lid driven cavity using the lattice Boltzmann method, Physica A 402 (2014) 150–168.

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copper–water nanofluid in a microchannel in slip flow regime using the lattice Boltzmann method, Eur. J. Mech. B Fluids 49 (2015) 89–99.

[17] Arash Karimipour, New correlation for Nusselt number of nanofluid with Ag/Al 2 O 3/Cu nanoparticles in a microchannel considering slip velocity

and temperature jump by using lattice Boltzmann method, Int. J. Therm. Sci. 91 (2015) 146–156.

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water/copper nanofluid, Int. J. Therm. Sci. 52 (2012) 91–101.

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Therm. Sci. 52 (2012) 137–144.

[31] Arash Karimipour, Habibollah Alipour, Omid Ali Akbari, Davood Toghraie Semiromi, Mohammad Hemmat Esfe, Studying the effect of indentation on

flow parameters and slow heat transfer of water-silver nano-fluid with varying volume fraction in a rectangular two-dimensional micro channel,

Indian J. Sci. Technol. 8 (15) (2015) 51707.

[32] OmidAli Akbari, Davood Toghraie, Arash Karimipour, Impact of ribs on flowparameters and laminar heat transfer ofwater–aluminumoxide nanofluid

with different nanoparticle volume fractions in a three-dimensional rectangular microchannel, Adv. Mech. Eng. 7 (11) (2015) 1687814015618155.

[33] Omid Ali Akbari, Davood Toghraie, Arash Karimipour, Numerical simulation of heat transfer and turbulent flow of water nanofluids copper oxide in

rectangular microchannel with semi-attached rib, Adv. Mech. Eng. 8 (4) (2016) 1687814016641016.

[34] A. Amrollahi, A.M. Rashidi, Convection heat transfer of functionalized MWNT in aqueous fluids in laminar and turbulent flow at the entrance region,

Int. Commun. Heat Mass Transf. 37 (2010) 717–723.

[35] Zahra Nikkhah, Arash Karimipour, Mohammad Reza Safaei, Pezhman Forghani-Tehrani, Marjan Goodarzi, Mahidzal Dahari, Somcha Wongwises,

Forced convective heat transfer of water/functionalized multi-walled carbon nanotube nanofluids in a microchannel with oscillating heat flux and

slip boundary condition, Int. Commun. Heat Mass Transfer 68 (2015) 69–77.

[36] Habibollah Alipour, Arash Karimipour, Mohammad Reza Safaei, Davood Toghraie Semiromi, Omid Ali Akbari, Influence of T-semi attached rib

on turbulent flow and heat transfer parameters of a silver-water nanofluid with different volume fractions in a three-dimensional trapezoidal

microchannel, Physica E 88 (2017) 60–76.

[37] Seyed Ali Sajadifar, Arash Karimipour, Davood Toghraie, Fluid flow and heat transfer of non-newtonian nanofluid in a microtube considering slip

velocity and temperature jump boundary conditions, Eur. J. Mech. B Fluids 61 (2017) 25–32.

[38] Omid Ali Akbari, et al., The effect of velocity and dimension of solid nanoparticles on heat transfer in non-newtonian nanofluid, Physica E 86 (2017)

68–75.

[39] Mohammad Hemmat Esfe, Wei-Mon Yan, Mohammad Akbari, Arash Karimipour, Mohsen Hassani, Experimental study on thermal conductivity of

DWCNT-ZnO/water-EG nanofluids, Int. Commun. Heat Mass Transfer 68 (2015) 248–251.


[40] MohammadHemmat Esfe, Ali AkbarAbbasianArani, ArashKarimiopour, Seyed SadeghMirtalebi Esforjani, Numerical simulation of natural convection

around an obstacle placed in an enclosure filled with different types of nanofluids, Heat Transfer Res. 45 (3) (2014) 279–292.

[41] Meissam Esfandiary, Babak Mehmandoust, Arash Karimipour, Hossein Ali Pakravan, Natural convection of Al 2 O 3–water nanofluid in an inclined

enclosure with the effects of slip velocity mechanisms: Brownian motion and thermophoresis phenomenon, Int. J. Therm. Sci. 105 (2016) 137–158.

[42] W. Zhao, L. Wang, W.A. Yong, On a two-relaxation-time D2Q9 lattice Boltzmann model for the Navier–Stokes equations, Physica A 492 (2018) 1570–

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[43] H. Otomo, B.M. Boghosian, F. Dubois, Two complementary lattice-Boltzmann-based analyses for nonlinear systems, Physica A 486 (2017) 1000–1011.

[44] M. Nemati, A.R.S.N. Abady, D. Toghraie, A. Karimipour, Numerical investigation of the pseudopotential lattice Boltzmann modeling of liquid–vapor

for multi-phase flows, Physica A 489 (2018) 65–77.

[45] H. Noorian, D. Toghraie, A.R. Azimian, Molecular dynamics simulation of Poiseuille flow in a rough nano channel with checker surface roughnesses

geometry, Heat Mass Transf. 50 (1) (2014) 105–113.

[46] D.T. Semiromi, A.R. Azimian, Nanoscale Poiseuille flow and effects of modified Lennard–Jones potential function, Heat Mass Transf. 46 (7) (2010)

791–801.

[47] D.T. Semiromi, A.R. Azimian, Molecular dynamics simulation of annular flow boiling with the modified Lennard-Jones potential function, Heat Mass

Transf. 48 (1) (2012) 141–152.

[48] O.A. Akbari, D. Toghraie, A. Karimipour, M.R. Safaei, M. Goodarzi, H. Alipour, M. Dahari, Investigation of rib’s height effect on heat transfer and flow

parameters of laminar water–Al2O3 nanofluid in a rib-microchannel, Appl. Math. Comput. 290 (2016) 135–153.

[49] D. Toghraie, M. Mokhtari, M. Afrand, Molecular dynamic simulation of copper and platinum nanoparticles Poiseuille flow in a nanochannels, Physica

E 84 (2016) 152–161.

[50] M. Rezaei, A.R. Azimian, D. Toghraie, Molecular dynamics study of an electro-kinetic fluid transport in a charged nanochannel based on the role of

the stern layer, Physica A 426 (2015) 25–34.

[51] M. Rezaei, A.R. Azimian, D.T. Semiromi, The surface charge density effect on the electro-osmotic flow in a nanochannel: a molecular dynamics study,

Heat Mass Transf. 51 (5) (2015) 661–670.

[52] M. Tohidi, D. Toghraie, The effect of geometrical parameters, roughness and the number of nanoparticles on the self-diffusion coefficient in couette

flow in a nanochannel by using of molecular dynamics simulation, Physica B 518 (2017) 20–32.

[53] M. Afrand, D. Toghraie, A. Karimipour, S. Wongwises, A numerical study of natural convection in a vertical annulus filled with gallium in the presence

of magnetic field, J. Magn. Magn. Mater. 430 (2017) 22–28.

[54] O.A. Akbari, D. Toghraie, A. Karimipour, Numerical simulation of heat transfer and turbulent flow of water nanofluids copper oxide in rectangular

microchannel with semi-attached rib, Adv. Mech. Eng. 8 (4) (2016) 1687814016641016.

[55] M.H. Esfe, S.S.M. Esforjani, M. Akbari, A. Karimiopour, Mixed-convection flow in a lid-driven square cavity filled with a nanofluid with variable

properties: effect of the nanoparticle diameter and of the position of a hot obstacle, Heat Transfer Res. 45 (6) (2014) 563–578.

[56] M. Afrand, D. Toghraie, B. Ruhani, Effects of temperature and nanoparticles concentration on rheological behavior of Fe3O4–Ag/EG hybrid nanofluid:

an experimental study, Exp. Therm Fluid Sci. 77 (2016) 38–44.

[57] M. Afrand, Experimental study on thermal conductivity of ethylene glycol containing hybrid nano-additives and development of a new correlation,

Appl. Therm. Eng. 110 (2017) 1111–1119.

[58] M. Mahmoodi, M.H. Esfe, M. Akbari, A. Karimipour, M. Afrand, Magneto-natural convection in square cavities with a source–sink pair on different

walls, Int. J. Appl. Electromagn. Mech. 47 (1) (2015) 21–32.

[59] M. Bahrami, M. Akbari, A. Karimipour, M. Afrand, An experimental study on rheological behavior of hybrid nanofluids made of iron and copper oxide

in a binary mixture of water and ethylene glycol: non-newtonian behavior, Exp. Therm Fluid Sci. 79 (2016) 231–237.

[60] M. Afrand, S. Rostami, M. Akbari, S. Wongwises, M.H. Esfe, A. Karimipour, Effect of induced electric field on magneto-natural convection in a vertical

cylindrical annulus filled with liquid potassium, Int. J. Heat Mass Transfer 90 (2015) 418–426.

[61] M. Afrand, A.A. Nadooshan, M. Hassani, H. Yarmand, M. Dahari, Predicting the viscosity of multi-walled carbon nanotubes/water nanofluid by

developing an optimal artificial neural network based on experimental data, Int. Commun. Heat Mass Transfer 77 (2016) 49–53.

[62] M. Goodarzi, A.S. Kherbeet, M. Afrand, E. Sadeghinezhad, M. Mehrali, P. Zahedi, S. Wongwises, M. Dahari, Investigation of heat transfer performance

and friction factor of a counter-flow double-pipe heat exchanger using nitrogen-doped, graphene-based nanofluids, Int. Commun. HeatMass Transfer

76 (2016) 16–23.

[63] M. Vafaei, M. Afrand, N. Sina, R. Kalbasi, F. Sourani, H. Teimouri, Evaluation of thermal conductivity of MgO-MWCNTs/EG hybrid nanofluids based on

experimental data by selecting optimal artificial neural networks, Physica E 85 (2017) 90–96.

[64] M. Afrand, S. Farahat, A.H. Nezhad, G.A. Sheikhzadeh, F. Sarhaddi, Numerical simulation of electrically conducting fluid flow and free convective heat

transfer in an annulus on applying a magnetic field, Heat Transfer Res. 45 (8) (2014) 749–766.

[65] N. Motamedifar, A. Shirneshan, An experimental study of emission characteristics from cylindrical furnace: effects of using diesel-ethanol-biodiesel

blends and air swirl, Fuel 221 (2018) 233–239.

[66] F. Mianzarasvand, A. Shirneshan, M. Afrand, Effect of electrically heated catalytic converter on emission characteristic of a motorcycle engine in

cold-start conditions: CFD simulation and kinetic study, Appl. Therm. Eng. 127 (2017) 453–464.

[67] M.J. Saheban Alahadi, A. Shirneshan,M. Kolahdoozan, Experimental investigation of the effect of grooves cut over the piston surface on the volumetric

efficiency of a radial hydraulic piston pump, Int. J. Fluid Power 18 (3) (2017) 181–187.

[68] R. Ranjbarzadeh, A. Karimipour, M. Afrand, A.H.M. Isfahani, A. Shirneshan, Empirical analysis of heat transfer and friction factor of water/graphene

oxide nanofluid flow in turbulent regime through an isothermal pipe, Appl. Therm. Eng. 126 (2017) 538–547.

[69] A. Shirneshan, B.H. Samani, B. Ghobadian, Optimization of biodiesel percentage in fuel mixture and engine operating conditions for diesel engine

performance and emission characteristics by Artificial Bees Colony Algorithm, Fuel 184 (2016) 518–526.

[70] A. Nedayali, A. Shirneshan, Experimental study of the effects of biodiesel on the performance of a diesel power generator, Energy Environ. 27 (5)

(2016) 553–565.

[71] A. Shirneshan, A. Nedayali, Investigation of the effects of biodiesel-diesel fuel blends on the performance and emission characteristics of a diesel

engine, J. Teknol. 78 (6) (2016) 169–177.

[72] A. Shirneshan, M. Almassi, B. Ghobadian, A.M. Borghei, G. Najafi, Response surface methodology (RSM) based optimization of biodiesel-diesel blends

and investigation of their effects on diesel engine operating conditions and emission characteristics, Environ. Eng.Manag. J. 15 (12) (2016) 2771–2780.

[73] M. Goodarzi, M.R. Safaei, K. Vafai, G. Ahmadi, M. Dahari, S.N. Kazi, N. Jomhari, Investigation of nanofluid mixed convection in a shallow cavity using a

two-phase mixture model, Int. J. Therm. Sci. 75 (2014) 204–220.

[74] M.H. Esfe, A. Karimipour,W.M. Yan, M. Akbari, M.R. Safaei, M. Dahari, Experimental study on thermal conductivity of ethylene glycol based nanofluids

containing Al2O3 nanoparticles, Int. J. Heat Mass Transfer 88 (2015) 728–734.

[75] H. Togun, G. Ahmadi, T. Abdulrazzaq, A.J. Shkarah, S.N. Kazi, A. Badarudin,M.R. Safaei, Thermal performance of nanofluid in ductswith double forward-

facing steps, J. Taiwan Inst. Chem. Eng. 47 (2015) 28–42.

[76] M.R. Safaei, B. Rahmanian, M. Goodarzi, Numerical study of laminar mixed convection heat transfer of power-law non-newtonian fluids in square

enclosures by finite volume method, Int. J. Phys. Sci. 6 (33) (2011) 7456–7470.


[77] H. Yarmand, S. Gharehkhani, S.N. Kazi, E. Sadeghinezhad, M.R. Safaei, Numerical investigation of heat transfer enhancement in a rectangular heated

pipe for turbulent nanofluid, Sci. World J. 2014 (2014).

[78] M.R. Safaei, H.R. Goshayeshi, B.S. Razavi, M. Goodarzi, Numerical investigation of laminar and turbulent mixed convection in a shallow water-filled

enclosure by various turbulence methods, Sci. Res. Essays 6 (22) (2011) 4826–4838.

[79] H. Yarmand, G. Ahmadi, S. Gharehkhani, S.N. Kazi,M.R. Safaei,M.S. Alehashem, A.B.Mahat, Entropy generation during turbulent flow of zirconia-water

and other nanofluids in a square cross section tube with a constant heat flux, Entropy 16 (11) (2014) 6116–6132.

[80] M.R. Safaei, M. Safdari Shadloo, M.S. Goodarzi, A. Hadjadj, H.R. Goshayeshi, M. Afrand, S.N. Kazi, A survey on experimental and numerical studies of

convection heat transfer of nanofluids inside closed conduits, Adv. Mech. Eng. 8 (10) (2016) 1687814016673569.

[81] M. Goodarzi, M.R. Safaei, H.F. Oztop, A. Karimipour, E. Sadeghinezhad, M. Dahari, S.N. Kazi, N. Jomhari, Numerical study of entropy generation due to

coupled laminar and turbulent mixed convection and thermal radiation in an enclosure filled with a semitransparent medium, Sci. World J. (2014)

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[82] A. Karimipour, A. Taghipour, A. Malvandi, Developing the laminarMHD forced convection flow of water/FMWNT carbon nanotubes in amicrochannel

imposed the uniform heat flux, J. Magn. Magn. Mater. 419 (2016) 420–428.

[83] A. Karimipour, A. D’Orazio, M.S. Shadloo, The effects of different nano particles of Al2O3 and Ag on the MHD nano fluid flow and heat transfer in a

microchannel including slip velocity and temperature jump, Physica E 86 (2017) 146–153.

[84] M. Zadkhast, D. Toghraie, A. Karimipour, Developing a new correlation to estimate the thermal conductivity of MWCNT-CuO/water hybrid nanofluid

via an experimental investigation, J. Therm. Anal. Calorim. 129 (2) (2017) 859–867.

[85] M.H. Esfe, A.A.A. Arani, A.H. Niroumand, W.M. Yan, A. Karimipour, Mixed convection heat transfer from surface-mounted block heat sources in a

horizontal channel with nanofluids, Int. J. Heat Mass Transfer 89 (2015) 783–791.

[86] A. Karimipour, A.H. Nezhad, A. Behzadmehr, S. Alikhani, E. Abedini, Periodicmixed convection of a nanofluid in a cavitywith top lid sinusoidalmotion,

Proc. Inst. Mech. Eng. C 225 (9) (2011) 2149–2160.

[87] M. Goodarzi, A. Amiri, M.S. Goodarzi, M.R. Safaei, A. Karimipour, E.M. Languri, M. Dahari, Investigation of heat transfer and pressure drop of a counter

flow corrugated plate heat exchanger using mwcnt based nanofluids, Int. Commun. Heat Mass Transfer 66 (2015) 172–179.

[88] A.R. Shirneshan, M. Almassi, B. Ghobadian, G.H. Najafi, Investigating the effects of biodiesel from waste cooking oil and engine operating conditions

on the diesel engine performance by response surface methodology, Iran. J. Sci. Technol. Trans. Mech. Eng. 38 (M2) (2014) 289–301.

[89] M. Akbari, M. Afrand, A. Arshi, A. Karimipour, An experimental study on rheological behavior of ethylene glycol based nanofluid: Proposing a new

correlation as a function of silica concentration and temperature, J. Molecular Liquids 233 (2017) 352–357.

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