Lattice Boltzmann simulation of turbulent natural convection in a ...

Journal of Mechanical Science and Technology 27 (8) (2013) 2341~2349 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-013-0618-5

Lattice Boltzmann simulation of turbulent natural convection in a square cavity using Cu/water nanofluid† H. Sajjadi1,*, M. Beigzadeh Abbassi2 and GH. R. Kefayati3 1

Department of Engineering, Shahid Bahonar University of Kerman, Kerman, Iran Department of Mechanical Engineering, Sirjan University of Technology, Sirjan, Iran 3 School of Computer Science, Engineering and Mathematics, Flinders University, Adelaide, Australia 2

(Manuscript Received December 27, 2011; Revised January 14, 2013; Accepted May 15, 2013) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract In this paper, lattice Boltzmann simulation of turbulent natural convection with large-eddy simulations (LES) in a square cavity which is filled by water/copper nanofluid has been investigated. The present results are validated by experimental data at Ra = 1.58×109 .This study is conducted for high Rayleigh numbers (Ra = 107-109) and volume fractions of nanoparticles (0 ≤ Φ ≤ 0.06). In this research, the effects of nanoparticles are displayed on streamlines and isotherms counters, local and average Nusselt numbers. The average Nusselt number is enhanced by the augmentation of nanoparticle volume fraction in the base fluid while the manner has an erratic trend toward different Rayleigh numbers. Keywords: Lattice Boltzmann method; Natural convection; Nanofluid; Turbulence; Heat transfer ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction Turbulence in fluids is ubiquitous in nature and technological systems and represents one of the most challenging aspects in fluid mechanics. The difficulty stems from the inherent presence of many scales that are generally inseparable among many other factors. Nevertheless, considerable progress has been made over the years towards more fundamental physical understanding of turbulence phenomena through measurements, statistical phenomenological theories, modeling and computation [1, 2]. Also a lot of experimental investigations have been done on it for instance Ampofo and Karayiannis [3] studied low-level turbulence natural convection in an air filled vertical square cavity while the hot and cold walls of the cavity were isothermal at 50 and 10°C respectively giving a Rayleigh number of Ra = 1.58×109. Applying a fluid with high heat transfer in systems of diverse industries such as cooling systems for electronic devices, chemical vapor deposition instruments (CVD), furnace engineering, solar energy collectors, phase change material and so forth was a permanent parameter. Whereas the fluids which were utilized in these industries had low thermal conductivity and heat transfer so millimeter and micrometer size solid particles with high thermal conductivity were used and improved *

Corresponding author. Tel.: +98 9153860633, Fax.: +98 5842231268 E-mail address: [email protected] † Recommended by Associate Editor Yang Na © KSME & Springer 2013

this problem. But these particles cause many troublesome problems such as poor suspension stability and clogging in different systems. At first Choi [4] solved the uncomplimentary phenomenon by producing of nanoparticles. So fluids with nanoparticles suspended in them are called nanofluids. Many numerical, experimental and theoretical investigations were performed about natural convection flow of nanofluid in different shapes. Xuan and Li [5] investigated experimentally the convective heat transfer and flow characteristics for a Cuwater nanofluid flowing through a straight tube with a constant heat flux under laminar and turbulent flow conditions. The results of the experiment showed that the suspended nanoparticles remarkably enhanced the heat transfer performance of the conventional base fluid and their friction factor coincided well with that of the water. Furthermore, they also proposed new convective heat transfer correlations for prediction of the heat transfer coefficients of the nanofluid for both laminar and turbulent flow conditions. Khanafer et al. [6] numerically investigated buoyancydriven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. They showed the Nusselt number for the natural convection of nanofluids is increased by enhancement of the volume fraction. Putra et al. [7] conducted the experiment for observation on the natural convective characteristics of water based on Al2O3. They reported that natural convective heat transfer in an enclosure is decreased with the increment of the volume fraction of nanoparticles. Kim et al.

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[8] analytically researched the convective instability driven by buoyancy and heat transfer characteristics of nanofluids with theoretical models. Wen and Ding [9] investigated convective heat transfer of nanofluids under laminar flow in the entry region. They found that the inclusion of nanoparticles could significantly enhance the convective heat transfer coefficient which increased with increasing Reynolds number and particle concentration. Tsai et al. [10] investigated gold-deionized water nanofluid flowing in a conventional heat pipe with a diameter of 6 mm and a length of 170 mm. Wen and Ding [11] investigated the heat transfer enhancement using waterTiO2 nanofluid filled in a rectangular enclosure heated from below. They reported that the natural convection heat transfer rate increasingly decreased with the increase of particle concentration, particularly at low Rayleigh number. Their data showed that the nanofluid causes a significant reduction in the thermal resistance of the heat pipe compared with deionized water at given concentrations. Natural convection heat transfer of nanofluids in a square cavity, heated isothermally from the vertical sides, has been investigated numerically by Ho et al. [12]. They investigated the effect of various formulas for the effective thermal conductivity and dynamic viscosity of Al2O3 nanofluid on the heat transfer characteristics and showed that the uncertainties associated with different models adopted to model the nanofluids have a great influence on the natural convection heat transfer characteristics in the enclosure. Santra et al. [13] demonstrated that heat transfer decline with increase of cupper volume fraction in water for any Rayleigh numbers. Jahanshahi et al. [14] numerically investigated the effects due to uncertainties in effective thermal conductivity according to experimental and theoretical formulations of the SiO2-water nanofluid on laminar natural convection heat transfer in a square enclosure. They observed heat transfer increases at various volume fractions for the whole range of Rayleigh numbers with usage of experimental thermal conductivity. lattice Boltzmann method (LBM) is an ideal mesoscopic approach to solve nonlinear macroscopic conservation equations because of its simplicity and ease for parallelization. For incompressible isothermal flows, the LBM is found to be at least as stable, accurate and computationally efficient, as traditional computational methods that it is a powerful method for simulation fluid flow and transport problems for single and multiphase flows [15-22]. Kefayati et al. [23] utilized this method (LBM) for simulating natural convection in tall enclosures using water/ SiO2 nanofluid. They obtained that the average Nusselt number increases with volume fraction for the whole range of Rayleigh numbers and aspect ratios and the effect of nanoparticles on heat transfer augments as the enclosure aspect ratio increases. Turbulent flows are modeled by various methods that the most tradition of method is large-eddy [24, 25]. This model is applied at various applications such as analysis of geophysical phenomena in the atmosphere, oceans and magnetosphere and provides a starting point for modeling these phenomena; the

Fig. 1. Geometry of the present study.

confinement of thermonuclear plasmas and in superfluid and superconductive behavior of thin films. S. Chen [26] proposed a novel and simple large-eddy-based lattice Boltzmann model to simulate two-dimensional turbulence. He showed that the model is efficient, stable and simple for two-dimensional turbulence simulation. Recently Sajjadi et al. [27] studied numerical analysis of turbulent natural convection in square cavity using large-eddy simulation in lattice Boltzmann method. They exhibited this method is in acceptable agreement with other verifications of such a flow. Also nanofluid are used in the other fields, the nanofibers are defined as continuous fibers with diameters less than 100 nm [28]. The nanofibers have many fascinating properties, such as unusual strength, high surface energy, high surface reactivity, high thermal and electric conductivity. Electrospinning is a simple method for producing nanofibers, and has attracted much recent interest. A detailed description of the electrospinning procedure is available on the monograph [28]. The aim of present paper is to study effects of turbulence on flow field and temperature distribution in nanofluid filled enclosure. Furthermore, the ability of lattice Boltzmann method (LBM) to solve nanofluid and various models of turbulent flows is demonstrated precisely.

2. Numerical method 2.1 Problem statement In this section, the proposed model is applied to simulate natural convection in a square cavity with side walls maintained at different temperatures. The left vertical wall is maintained at a high temperature TH while the right vertical wall is kept at a low temperature TC. The horizontal walls are assumed to be insulated, non conducting, and impermeable to mass transfer. The cavity is filled with a mixture of water and solid copper (Cu) (Fig. 1). The nanofluid in the cavity is Newtonian, incompressible, and laminar. Thermophysical properties of the nanofluid are assumed to be constant (Table 1). The density variation in the nanofluid is approximated by the standard Boussinesq model.

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 i=0 0  Ci = c cos[(i − 1) π ],sin[(i − 1) π ] i =1− 4 2 2  c 2 cos[(i − 5) π + π ],sin[(i − 5) π + π ] 2 4 2 4 

Table 1. Thermophysical properties of water and copper. Property

Water

Copper

µ (kg/ms)

8.9 × 10−4

_

cp (j/kg K)

4179

383

ρ (kg/m3)

997.1

-1

β (K )

2.1 × 10

k (w/m K)

(

(

)

i = 5−8

(5)

8954 −4

1.67 × 10−5

0.6

400

2.2 Lattice Boltzmann method For the incompressible flow, if the transport coefficients are independent of the temperature, the energy equation can be decoupled from the mass and momentum equations. For the incompressible thermal problem, f and g are two functions called as flow distribution function and temperature distribution function respectively. These functions are utilized to obtain macroscopic characteristics of the flow like velocity, pressure, temperature and etc. In this paper a square grid and D2Q9 model is used for both flow and temperature functions [29]. By detachment of Navier-Stocks equations, governing equations for flow and temperature functions are as follow: For the flow field: fi ( x + ci ∆t , t + ∆t ) − fi ( x, t ) = −

)

1

 fi ( x, t ) − fi τv 

eq

( x, t ) + ∆tF . i

(1) For the temperature field: g i ( x + ci ∆t , t + ∆t ) − g i ( x, t ) = −

where ω0 = 4/9, ω1-4 = 1/9, ω5-9 = 1/36 and c = 3RTm (to improve numerical stability, Tm is the mean value of temperature for the calculation of c). Using a Chapman-Enskog expansion, the Navier-Stokes equations can be recovered with the described model. The kinematic viscosity υ and the thermal diffusivity α are then related to the relaxation times by 

1



1









ϑ = τ v −  cs2 ∆t and α = τ c −  cs2 ∆t 2 2

(6)

where cs is speed of sound and equal to c / 3 . In the simulation, the Boussinessq approximation is applied to the buoyancy force term .In that case, the external force F appearing in Eq. (1) is given by: Fi = 3ωi g y β∆T

(7)

where gy, β and ∆T are gravitational acceleration, thermal expansion coefficient and temperature difference, respectively. Finally, the macroscopic variables ρ, u, and T can be calculated using as follows; Flow density: r = ∑ f i .

(8)

Momentum: ρ u j = ∑ f i ci .

(9)

i

1

 g i ( x, t ) − g τc 

eq i

( x, t )

(2)

i

Temperature: T = ∑ g i .

(10)

i

where the discrete particle velocity vectors defined ci (Fig. 2(a)), ∆t denotes lattice time step which is set to unity. τv, τc are the relaxation time for the flow and temperature fields, respectively. fieq, gieq are the local equilibrium distribution functions that have an appropriately prescribed functional dependence on the local hydrodynamic properties which are calculated with Eqs. (3) and (4) for flow and temperature fields respectively . Moreover, F is an external force term.  c .u 1 (ci ⋅ u ) 2 1 u ⋅ u  f i eq ( x, t ) = ωi ρ 1 + i 2 + −  cs 2 cs4 2 cs2  

(3)

 c ⋅u  g ieq = ωi T 1 + i 2  . cs  

(4)

For the 2-D case, applying third-order Gauss-Hermite quadrature leads to the D2Q9 model with the following discrete velocities ci,

2.3 Large Eddy simulation method In this model the main aim is obtaining ν t and α t = (

νt Prt

)

where Prt is turbulent Prandtle number which is assumed to be 4. In order to evaluate ν t we perform as follow:  2 Pr ν t = (C ∆ ) 2  S + ∇T ⋅  Prt 

ur g ur g

1/ 2

   

.

(11)

C is considered as Smagorinsky constant and in this paper it is assumed as 0.1. It is gained from ∆ = (∆x)2 + (∆y ) 2 , ∆x and ∆y are grid extents in X and Y directions. For S we have: S = 2S αβ S αβ

(12)

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(

)

S αβ = ∂α u β + ∂ β uα / 2 .

(13)

2.4 Lattice Boltzmann method based on large eddy simulation model Large eddy model is easily applied in Lattice Boltzmann method the way ν t affects relaxation time [29-31]. ν total = cs 2 (τ m − 0.5) = ν 0 + ν t

(ν

0

+ν t ) 2 s

c

+ 0.5 =

ν0 2 s

c

+ 0.5 +

νt c

= τ0 +

2 s

νt

.

cs2

(15)

To obtain ν t in lattice Boltzmann method we have: S =

3 Q 2τ m

Fig. 2. (a) The discrete velocity vectors for D2Q9; (b) Domain boundaries.

Q = ∑ i =0 eiα eiβ ( f i − fi eq ) . 8

The thermo-physical properties of the nanofluid are assumed to be constant (Table 1) except for the density variation, which is approximated by the Boussinesq model. The effective properties of the nanofluid are defined as follows [11]: Density:

ρ nf = (1 − ϕ ) ρ f + ϕ ρ s . (16) (17)

in Eq. (12):

 9 2 Pr ν t = (C ∆ )  2 Q + ∇T .  4τ ν Prt  2

ur g ur g

(21)

Heat capacitance: Cnf =

If we put S

(b)

(14)

where ν total and ν 0 are total viscosity and initial viscosity respectively.

τν =

(a)

ϕρ s Cs + (1 − ϕ ) ρ f C f . ρ nf

(22)

Effective thermal conductivity:

1/ 2

   

(18) knf kf

=

( − ϕ (k

). −k )

k s + 2k f + 2ϕ k f − ks k s + 2k f

f

(23)

s

and if we substitute the above equation in Eq. (14): Viscosity:

τ total

ur  2 Pr g 2 9 (C ∆) Q + ∇T . ur  4τ ν2 Prt g  = τ0 + 2 cs

1/ 2

   

µnf = .

(19)

To obtain relaxation time in temperature function equation we have: τ c = τ D0 +

where τ D 0 =

αt cs

2

α0 cs 2

= τ D0 +

ν t / Prt cs

2

(20)

+ 0.5 .

Substituting new relaxation time in Eqs. (1) and (2) yields to lattice Boltzmann equations based on large eddy model. 2.5 Lattice Boltzmann method for nanofluid The dynamical similarity depends on two dimensionless parameters: the Prandtl number Pr and the Rayleigh number Ra, as it is assumed that nanofluid behaves like a pure fluid. Consequently, nanofluid qualities are obtained by Eqs. (21)-(23); thereafter; they are applied for Rayleigh and Prandtl numbers.

µf

(1 − ϕ )

2.5

(24)

.

where Eqs. (23) and (24) is appropriate for spherical and equal nanoparticles. 2.6 Boundary conditions: 2.6.1 flow Implementation of boundary conditions is very important for the simulation. The unknown distribution functions pointing to the fluid zone at the boundaries nodes must be specified (Fig. 2(b)). Concerning the no-slip boundary condition, bounce back boundary condition is used on the solid boundaries. For instance the unknown density distribution functions at the boundary east can be determined by the following conditions: f 6,n = f8,n ,

f 7,n = f 5,n ,

f 3,n = f1,n

where n is the lattice on the boundary.

(25)

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Table 2. Comparison of mean Nu with previous works. Ra number 7

10

Mean Nu Mean Nu [24] (this work)

Mesh

Mean Nu [25]

256×256

17.2

__

16.8

10

512×512

31.2

32.3

30.5

109

1024×1024

58.1

60.1

57.4

8

(a)

2.6.2 Temperature The north and south boundaries are adiabatic; therefore, the bounce back boundary condition is utilized on them. Temperature at the west and east walls are known, in the west wall TH = 1.0. Since the D2Q9 model is applied, the unknowns are g1, g5, g8 at the west wall which are evaluated as follows; g1 = TH (ω1 + ω3 ) − g 3

(b)

Fig. 3. Comparison of the velocity on the axial midline: (a) and local Nusselt number; (b) between the present results and experimental results by Ampofo and Karayiannis [3] (Ra = 1.58×109).

(26a) (26b) (26c)

g 5 = TH (ω5 + ω7 ) − g 7 g8 = TH (ω8 + ω6 ) − g 6 .

Nusselt number Nu is one of the most important dimensionless parameters in the description of the convective heat transport. The local and average Nusselt numbers at the hot wall are calculated as: H ∂T ∆T ∂x 1 H = ∫ NU y dy . H 0

NU y = −

(27)

NU avg

(28)

Because of the convenience, a normalized average Nusselt number is defined as the ratio of Nusselt number at any volume fraction of nanoparticles to that of pure water that is as follows: NU avg (db)(ϕ ) =

NU avg (ϕ ) NU avg (ϕ = 0)

.

Fig. 4. Comparison of the temperature on axial midline between the present results and numerical results by Khanafer et al. [6] and Jahanshahi et al. [14] (pr = 6.2, φ = 0.1, Gr = 104).

be utilized for different Rayleigh numbers. For the second part, the solution method for nanofluid by Lattice Boltzmann Method is validated by the results of Khanafer et al. [6] and Jahanshahi et al. [14]. As Fig. 4 shows a comparison with temperature at the mid section of the cavity for Cu-water nanofluid where volume fraction is equal to φ = 0.1.

(29)

3. Code validation and mesh results The nanofluid in the cavity with different heated vertical sides is chosen as Cu-water mixtures. The research is carried out for various values of nanoparticles volume fractions (0 < φ < 0.06) and Rayleigh numbers (107 < Ra < 109). The present numerical method is proved for two topics which have formed the study. The two topics are effect of nanoparticle and turbulence on natural convection in a cavity. At the first part, Table 2 shows the comparison of average Nusselt number for different Rayleigh numbers between present results and the finds of Barakos et al. [24], Dixit [25] as the cavity was filled by air with Pr = 0.71. A comparison with temperature and velocity at the middle section of the cavity is considered by experimental results of Ampofo and Karayiannis [3] in Fig. 3. Evidently, it is observed that the present results match previous studies. Furthermore, this table demonstrates various meshes should

4. Result and discussion Fig. 5 provides a comparison between pure fluid (φ = 0) and nanofluid (φ = 0.05) for various Rayleigh numbers in the isotherms and the streamlines contours. When Rayleigh number increases, the symmetry state diminishes and the centralization of the streamline in the core of the cavity tends to shift to the hot wall. The heat transfer process ameliorates as Rayleigh number enhances. The process is obvious where two isotherms of T = 0.1 and 0.9 move to the cold and hot walls respectively when Rayleigh number rises. The effect of nanoparticle on the streamlines is apparent where the streamlines of nanofluids expand more than the pure fluid. The phenomenon demonstrates the increment of heat transfer process in the presence of nanoparticle. In addition, the augmentation of the maximum streamline value in the presence of nanoparticle declines with the increment of Rayleigh numbers as at Ra = 109, the value for nanofluid is less than the pure fluid. Fig. 6 examines the local Nusselt number on the hot wall

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ψ min

f

ψ min

= 7.9 × 10−4

nf

= 8.1× 10−4

(a)

(b)

(a)

(c)

ψ min

f

= 7.8 × 10

−4

ψ min

nf

ψ min

nf

= 7.96 × 10

−3

(b)

ψ min

f

= 6.9 × 10−4

Fig. 6. Nusselt number distributions on the hot wall at different volume fractions and Rayleigh numbers: (a) Ra = 107; (b) Ra = 108; (c) Ra = 109.

= 6.1× 10−4

(c)

(a)

(b)

Fig. 5. Comparison of the isotherms and streamlines between nanofluids (---) (φ = 0.05) and base fluid (—) (φ = 0) at various Rayleigh numbers: (a) Ra = 107; (b) Ra = 108; (c) Ra = 109.

Fig. 7. Values of the horizontal velocity on the axial midline (X = 0.5): (a) and the vertical velocity on the axial midline (Y = 0.5); (b) for Ra = 107.

for different volume fractions and Rayleigh numbers. Generally, the local Nusselt numbers of nanofluids have an incremental pattern with the augmentation of volume fraction. It demonstrates the effect of nanoparticle is decreased by the move to upward partition of the hot wall where the local Nusselt number value for the nanofluid with φ = 0.04 sets lower rate than the pure fluid. Moreover, the greatest effect of nanpaticle on the local Nusselt number is observed in the middle of the plot that it is exhibited by the circles. Fig. 7 displays the values of the vertical and horizontal velocities on the axial midline at Ra = 107 for different nanoparticles. The plot has erratic manners but at the minimum and the maximum values, it obtains regular behavior towards the growth of the volume fraction. As the pictures are zoomed, it can be observed the gradual trend at the plot in the presence of naoparticles. The plot proves that the effects of the nanoparticles on the horizontal and vertical velocities are ameliorated by the velocity enhancement for the both directions. Since the

velocity of the fluid is increased by the nanoparticles, the heat transfer process improves. Fig. 8 illustrates the influence of the nanoparticle volume fraction φ on the average Nusselt number (NUavg) and the normalized average Nusselt number (NUavg(db)) along the heated surface for different volume fractions and Rayleigh numbers. It shows the values of the average Nusselt number and normalized average Nuseelt number for different volume fractions and Raleigh numbers. The average Nusselt number increases with the augmentation of nanoparticles. The increment is different for various Rayleigh numbers and the volume fractions. The average Nusselt number indicates a linearly manner toward the increase in the volume fractions at Ra = 107. However, there is a jump at φ = 0.01 for Ra = 108 as the plot has the same trend with Ra = 107. It is noticeable that the effects of nanoparticles at Ra = 109 are weak on the average Nusselt number in comparison with the lower Rayleigh numbers.


(a)

(b)

(c) Fig. 8. Values of the average Nusselt number (the left side) and normalized average Nuseelt number (the right side) for different volume fractions and Raleigh numbers: (a) Ra = 107; (b) Ra = 108; (c) Ra = 109.

5. Conclusions Natural convection in a cavity which is filled with a water/Cu nanofluid has been conducted numerically by lattice Boltmann method (LBM). This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra = 107-109, the volume fractions 0-5% and some conclusions are summarized as follows: (a) A proper validation with previous numerical investigations demonstrates that lattice Boltzmann method is an appropriate method for turbulent and multiphase flows problems. (b) Generally, the increase in the volume fractions and the Rayleigh numbers result in the augmentation of heat transfer. (c) The effect of nanoparticle on the average Nusselt number at Ra = 109 is less than other considered Rayleigh numbers. (d) The nanoprticles increases the velocity of the base fluid with a regular manner toward the increment of the volume fractions for different Rayleigh numbers.

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Lattice Boltzmann simulation of turbulent natural convection in tall enclosures using Cu/water nanofluid, Numerical Heat Transfer Part A, 62 (2012) 512-530. [17] GH. R. Kefayati, S. F. Hosseinizadeh, M. Gorji and H. Sajjadi, Lattice Boltzmann simulation of natural convection in an open enclosure subjugated to Water/copper nanofluid, International Journal of Thermal Science, 52 (2012) 91-101. [18] GH. R. Kefayati, M. Gorji, H. Sajjadi and D. D. Ganji, Investigation of Prandtl number effect on natural convection MHD in an open cavity by Lattice Boltzmann Method, Engineering Computations, 30 (2013) 97-116. [19] GH. R. Kefayati, M. Gorji, H. Sajjadi and D. D. Ganji, Lattice Boltzmann simulation of MHD mixed convection in a lid-driven square cavity with linearly heated wall, Scientia Iranica Transactions B - Mechanical Engineering, 19 (2012) 1053-1065. [20] H. Sajjadi, M. Gorji, GH. R. Kefayati and D. D. Ganji; Lattice Boltzmann simulation of natural convection in an inclined heated cavity partially using Cu/water nanofluid, International Journal of Fluid Mechanics Research, 39 (2012) 348-372. [21] H. Sajjadi, M. Gorji, GH. R. Kefayati, D. D. Ganji and M. Shayannia, Numerical analysis of turbulent natural convection in a square cavity using large-Eddy simulation in lattice Boltzmann method, International Conference on Aerospace, Mechanical, Automotive and Materials Engineering, Dubai, January, 29-31, (2012). [22] H. Sajjadi, S. F. Hosseinizadeh, M. Gorji, Gh. R. Kefayati and M. Shayannia, Simulation of natural convection flow in an inclined open cavity using Lattice Boltzmann Method, International Conference on Mechanical Engineering, Paris, July 27-29, (2011). [23] G. H. R. Kefayati, S. F. Hosseinizadeh, M. Gorji and H. Sajjadi, Lattice Boltzmann simulation of natural convection in tall enclosures using water/ SiO2 nanofluid, International Communications in Heat and Mass Transfer, 38 (2011) 798-805. [24] G. Barakos, E. Mitsoulis and D. Assimacopoulos, Natural convection flow in a square cavity revisited: laminar and turbulent models with wall function, International Journal of Numerical Method in Fluids, 18 (1994) 695-719. [25] H. N. Dixit and V. Babu, Simulation of high Rayleigh number natural convection in a square cavity using the Lattice Boltzmann method, International Journal of Heat and Mass Transfer, 46 (2006) 727-739. [26] S. Chen, A large-eddy-based lattice Boltzmann model for turbulent flow simulation, Applied Mathematics and Computation, 215 (2009) 591-598. [27] H. Sajjadi, M. Gorji, S. F. Hosseinizadeh, GH. R. Kefayati and D. D. Ganji, Numerical analysis of turbulent natural convection in square cavity using Large-Eddy Simulation in Lattice Boltzmann Method, Iranian Journal of Science and Technology Transition B/Engineering, 35 (2011) 133-142. [28] JH He, Y Liu, LF Mo, YQ Wan and L Xu, Electrospun nanofibres and their applications, Shawbury: Smithers Rapra Technology (2008). [29] A. A. Mohamad and A. Kuzmin, A critical evaluation of

force term in lattice Boltzmann method, natural convection problem, International Journal of Heat and Mass Transfer, 53 (2010) 990-996. [30] S. Hou, J. Sterling, S. Chen and G. D. Doolen, A Lattice Boltzmann subgrid model for high Reynolds number flows, Fields Inst., Comm., 6 (1996) 151-166. [31] C. M. Teixeira, Incorporation turbulence models into the Lattice-Boltzmann method, International Journal of Modern Physic C, 9 (1998) 1159-1175.

Nomenclature-----------------------------------------------------------------------C ci cp F f feq g geq gy H k Nu Pr R Ra S αβ

T x,y Gr

: Lattice speed : Discrete particle speeds : Specific heat at constant pressure : External forces : Density distribution functions : Equilibrium density distribution functions : Internal energy distribution functions : Equilibrium internal energy distribution functions : Gravity : Cavity height : Thermal conductivity : Nusselt number : Prandtl number : Constant of the gases : Rayleigh number : Strain rate tensor : Temperature : Cartesian coordinates : Grashof number

Greek letters ωi β φ τc τv ρ µ ν ∆x ∆t

: Weighted factor indirection i : Thermal expansion coefficient : Volume fraction : Relaxation time for temperature : Relaxation time for flow : Density : Dynamic viscosity : Kinematic viscosity : Lattice spacing : Time increment

Subscripts avg C f H nf s db t

: Average : Cold : Fluid : Hot : Nanofluid : Solid : Normalized : Turbulence


H. Sajjadi, Ph.D, student of Mechanical Engineering at Shahid Bahonar University, received M.S. degree from School of Mechanical Engineering at Babol University of Technology. He received his B.S. degree from School of Mechanical Engineering at Isfahan University of Technology. His main research interests are CFD, LBM, nanofluids, MHD, turbulent flow. Gholamreza Kefayati, Ph.D, student of Mechanical Engineering at Flinders University. He received his M.S degree from School of Mechanical Engineering at Babol University of Technology. His main research interests are CFD, LBM, nanofluids, MHD, multiphase flow and non-Newtonian flow.

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M. Beigzadeh Abbassi, received Diploma degree (previous continuous German Master progam) from Technische Universität Clausthal-Zellerfeld, Germany. He received Ph.D. degree from Technische Universität Berlin. Associate professor in Sirjan University of Technology. His main research interests are internal combustion engines and construction.