Numerical simulation of natural convection in a square enclosure filled with nanofluid using the two-phase Lattice Boltzmann method
- Cong Qi^{1},
- Yurong He^{1}Email author,
- Shengnan Yan^{1},
- Fenglin Tian^{1} and
- Yanwei Hu^{1}
DOI: 10.1186/1556-276X-8-56
© Qi et al.; licensee Springer. 2013
Received: 23 October 2012
Accepted: 22 January 2013
Published: 4 February 2013
Abstract
Considering interaction forces (gravity and buoyancy force, drag force, interaction potential force, and Brownian force) between nanoparticles and a base fluid, a two-phase Lattice Boltzmann model for natural convection of nanofluid is developed in this work. It is applied to investigate the natural convection in a square enclosure (the left wall is kept at a high constant temperature (T_{H}), and the top wall is kept at a low constant temperature (T_{C})) filled with Al_{2}O_{3}/H_{2}O nanofluid. This model is validated by comparing numerical results with published results, and a satisfactory agreement is shown between them. The effects of different nanoparticle fractions and Rayleigh numbers on natural convection heat transfer of nanofluid are investigated. It is found that the average Nusselt number of the enclosure increases with increasing nanoparticle volume fraction and increases more rapidly at a high Rayleigh number. Also, the effects of forces on nanoparticle volume fraction distribution in the square enclosure are studied in this paper. It is found that the driving force of the temperature difference has the biggest effect on nanoparticle volume fraction distribution. In addition, the effects of interaction forces on flow and heat transfer are investigated. It is found that Brownian force, interaction potential force, and gravity-buoyancy force have positive effects on the enhancement of natural convective heat transfer, while drag force has a negative effect.
Keywords
Two phase Lattice Boltzmann model Rayleigh number Nanofluid Natural convectionBackground
Compared with common fluids such as water, nanofluid, using nanoscale particles dispersed in a base fluid, has an effect of enhancing the performance of natural convection heat transfer due to its high heat conductivity coefficient. Many researchers investigated nanoparticles and nanofluid in recent years. Wang et al. [1] synthesized stimuli-responsive magnetic nanoparticles and investigated the effect of nanoparticle fraction on its cleavage efficiency. Bora and Deb [2] developed a novel bioconjugate of stearic acid-capped maghemite nanoparticle (γ-Fe_{2}O_{3}) with bovine serum albumin. Guo et al. [3] produced magnetic nanofluids containing γ-Fe_{2}O_{3} nanoparticles using a two-step method, measured their thermal conductivities and viscosity, and tested their convective heat transfer coefficients. Pinilla et al. [4] investigated the growth of Cu nanoparticles in a plasma-enhanced sputtering gas aggregation-type growth region. Yang and Liu [5] produced a kind of stable nanofluid by surface functionalization of silica nanoparticles. Zhu et al. [6] developed a wet chemical method to produce stable CuO nanofluids. Nadeem and Lee [7] investigated the steady boundary layer flow of nanofluid over an exponential stretching surface. Wang and Fan [8] reviewed the nanofluid research in the last 10 years.
Natural convection is applied in many fields, and extensive researches have been performed. Oztop et al. [9] and Ho et al. [10] respectively investigated natural convection in partially heated rectangular enclosures and discussed the effects of viscosity and thermal conductivity of nanofluid on laminar natural convection heat transfer in a square enclosure by a finite-volume method. Saleh et al. [11] investigated heat transfer enhancement utilizing nanofluids in a trapezoidal enclosure by a finite difference approach. Ghasemi et al. [12], Santra et al. [13], and Aminossadati et al. [14] numerically simulated natural convection in a triangular enclosure and studied the behavior of natural convection heat transfer in a differentially heated square cavity, described a study on natural convection of a heat source embedded in the bottom wall of an enclosure, and used the SIMPLE algorithm to solve the governing equation. Kargar et al. [15] used computational fluid dynamics and an artificial neural network to investigate the cooling performance of two electronic components in an enclosure. Abu-Nada et al. [16] investigated the effect of variable properties on natural convection in enclosures filled with nanofluid, and the governing equations are solved by an efficient finite-volume method. Hwang et al. [17] investigated the thermal characteristics of natural convection in a rectangular cavity heated from below by Jang and Choi's model [18].
The Lattice Boltzmann method is a new way to investigate natural convection. Compared with the above traditional methods, the Lattice Boltzmann method has many merits including that boundary conditions can be conveniently dealt with, the transform between macroscopic and microscopic equations is easily achieved, the details of the fluid can be presented, and so on. In addition, nanofluid as the media can enhance heat transfer due to factors such as nanofluids having higher thermal conductivity and the nanoparticles in the fluid disturbing the laminar flow. Therefore, many researchers undertook investigations on the natural convection of nanofluids by the Lattice Boltzmann method. Barrios et al. [19] developed a Lattice Boltzmann model and applied it to investigate the natural convection of an enclosure with a partially heated left wall. Peng et al. [20] presented a simple a Lattice Boltzmann model without considering thermal diffusion, and this model is easily applied because it does not contain a gradient term. He et al. [21] proposed a new Lattice Boltzmann model which introduced an internal energy distribution function to simulate the temperature field, and the result has a good agreement with the benchmark solution. Nemati et al. [22] simulated the natural convection of a lid-driven flow filled with Cu-water, CuO-water, and Al_{2}O_{3}-water nanofluids and discussed the effects of nanoparticle volume fraction and Reynolds number on the heat transfer. Wang et al. [23] presented a Lattice Boltzmann algorithm to simulate the heat transfer of a fluid-solid fluid, and the result has a satisfactory agreement with the published data. Dixit et al. [24] applied the Lattice Boltzmann method to investigate the natural convection of a square cavity at high Rayleigh numbers. Peng et al. [25] developed a 3D incompressible thermal Lattice Boltzmann model for natural convection in a cubic cavity. The above Lattice Boltzmann methods are all single-phase models, and the nanofluid was seen as a single-phase fluid without considering the interaction forces between nanoparticles and water. In addition, the effects of these interaction forces on heat transfer were disregarded.
There are few two-phase lattice Boltzmann models that consider the interaction forces between nanoparticles and a base fluid for natural convection in an enclosure. Xuan et al. [26] proposed a two-phase Lattice Boltzmann model to investigate sudden-start Couette flow and convection in parallel plate channels without researching the effect of forces on volume fraction distribution of nanoparticles. Because these forces were not investigated before our work, the effects of forces between water and nanoparticles on the fluid flow patterns were unknown. In addition, as we know, the nanoparticles in the fluid easily gather together and deposit, especially at high volume fraction. Hence, the nanoparticle distribution in the fluid flow is important for nanofluid application, which is another objective in our paper. However, the single-phase model cannot be used to investigate nanoparticle distribution. Furthermore, natural convection of a square enclosure (left wall kept at a high constant temperature (T_{H}), and top wall kept at a low constant temperature (T_{C})) filled with nanofluid is not investigated in the published literatures. In this paper, a two-phase Lattice Boltzmann model is proposed and applied to investigate the natural convection of a square enclosure (left wall kept at a high constant temperature (T_{H}), and top wall kept at a low constant temperature (T_{C})) filled with Al_{2}O_{3}-water nanofluid and the inhomogeneous distribution of nanoparticles in the square enclosure.
Methods
Lattice Boltzmann method
where ${\tau}_{f}^{\sigma}$ is the dimensionless collision-relaxation time for the flow field, e_{ α } is the lattice velocity vector, the subscript α represents the lattice velocity direction, ${f}_{\alpha}^{\sigma}\left(r,t\right)$ is the distribution function of the nanofluid with velocity e_{ α } (along the direction α) at lattice position r and time t, ${f}_{\alpha}^{\mathit{\sigma eq}}\left(r,t\right)$ is the local equilibrium distribution function, δ_{ t } is the time step, δ_{ x } is the lattice step, the order numbers α = 1,…,4 and α = 5,…,8, respectively represent the rectangular directions and the diagonal directions of the lattice, ${F}_{\alpha}^{\sigma \text{'}}$ is the external force term in the direction of the lattice velocity without interparticle interaction, G = - β(T_{ nf } - T_{0})g is the effective external force, where g is the gravity acceleration, β is the thermal expansion coefficient, T_{ nf } is the temperature of the nanofluid, and T_{0} is the mean value of the high and low temperature of the walls.
where ${F}_{\alpha}^{\sigma}$ represents the total interparticle interaction forces, and B_{ α } is one of the weight coefficients. $\frac{2{\tau}_{f}^{\sigma}-1}{2{\tau}_{f}^{\sigma}}$ is a coefficient. Because the total interparticle interaction forces cannot be optionally added in the lattice Boltzmann equation, we introduce an unknown coefficient in the total interparticle interaction forces. In order to enable the lattice Boltzmann equation including the total interparticle interaction forces to recover to the Navier-Stokes equation, based on the mass and momentum conservation, we used multi-scale technique to deduce the unknown coefficient which is equal to $\frac{2{\tau}_{f}^{\sigma}-1}{2{\tau}_{f}^{\sigma}}$. Due to the very long derivation process, we directly gave the final result in the paper.
where ${c}_{s}^{2}=\frac{{c}^{2}}{3}$ is the lattice's sound velocity, and w_{ α } is the weight coefficient.
where τ_{ T } is the dimensionless collision-relaxation time for the temperature field.
where F_{p} represents the total forces acting on the nanoparticles, F_{w} represents the total forces acting on the base fluid, and L_{ x }L_{ y } represents the total number of lattices.
where Φ_{ αβ } is the energy exchange between nanoparticles and base fluid, ${\Phi}_{\mathit{\alpha \beta}}=\frac{{h}_{\mathit{\alpha \beta}}\left[{T}_{\beta}\left(x,t-{\delta}_{t}\right)-{T}_{\alpha}\left(x,t-{\delta}_{t}\right)\right]}{{\rho}_{\alpha}{c}_{\mathit{p\alpha}}{a}_{\alpha}}$, and h_{ αβ } is the convective heat transfer coefficient of the nanofluid.
From Equations 18 and 19, the collision-relaxation time for the flow field and the temperature field can be calculated. For water phase, the τ_{ f } collision-relaxation times are respectively 0.51433 and 0.501433 at Ra = 10^{3} and Ra = 10^{5}, and the collision-relaxation time τ_{ T } is 0.5. For nanoparticle phase, the τ_{ f } collision-relaxation times are respectively 0.50096 and 0.500096 at Ra = 10^{3} and Ra = 10^{5}, and the collision-relaxation time τ_{ T } is 0.500025.
Interaction forces between base fluid and nanoparticles
As noted before, a nanofluid is, in reality, a kind of two-phase fluid. There are interaction forces between liquid and nanoparticles which affect the behavior of the nanofluid. The external forces include gravity and buoyancy forces F_{H}, and the interparticle interaction forces include drag force (Stokes force) F_{D}, interaction potential F_{A}, and Brownian force F_{B}. We introduce them as follows.
where a is the radius of a nanoparticle, and Δρ^{'} is the mass density difference between the suspended nanoparticle and the base fluid.
where μ is the viscosity of the fluid, and ∆u is the velocity difference between the nanoparticle and the base fluid.
where A is the Hamaker constant, and L_{ cc } is the center-to-center distance between particles.
where n_{ i } is the number of the particles within the adjacent lattice i, n_{ i } = ρ^{ σ }V/m^{ σ }, m^{ σ } is the mass of a single nanoparticle, and V is the volume of a single lattice.
where G_{ i } is a Gaussian random number with zero mean and unit variance, which is obtained from a program written by us, and C = 2γk_{ B }T = 2 × (6πηa)k_{ B }T, γ is the surface tension, k_{ B } is the Boltzmann constant, T is the absolute temperature, and η is the dynamic viscosity.
where n is the number of the particles in the given lattice, and V is the lattice volume.
Results and discussion
Thermo-physical properties of water and Al_{ 2 }O_{ 3 }[29]
Physical properties | Fluid phase (H_{2}O) | Nanoparticles (Al_{2}O_{3}) |
---|---|---|
ρ (kg/m^{3}) | 997.1 | 3970 |
c_{p} (J/kg k) | 4179 | 765 |
v (m^{2}/s) | 0.001004 | - |
k (W/m/K) | 0.613 | 25 |
where ε is a small number, for example, for Ra = 1 × 10^{3}, ε_{1} = 10^{-6}, and ε_{2} = 10^{-6}. About 2 weeks is needed to achieve the equilibrium state for the low Rayleigh number (Ra = 1 × 10^{3}), and about 1 month for the high Rayleigh number (Ra = 1 × 10^{5}).
In order to perform a grid independence test and validate the Lattice Boltzmann model proposed in this work, we used another square enclosure, because there are exact solutions for this square enclosure. The left wall is kept at a high constant temperature (T_{H}), and the right wall is kept at a low constant temperature (T_{C}). The boundary conditions of the other walls (top wall and bottom wall) are all adiabatic, and the other conditions are the same as those in Figure 1.
Comparison of the mean Nusselt numbers with different grids ( Ra = 1 × 10 ^{ 5 } , Pr = 0.7)
Physical properties | 128 × 128 | 192 × 192 | 256 × 256 | 320 × 320 | Literature[30] |
---|---|---|---|---|---|
Nu _{avg} | 4.5466 | 4.5251 | 4.5220 | 4.5218 | 4.5216 |
Comparison of average Nusselt numbers with other published data ( Pr = 0.7)
Comparison of different forces ( Ra = 10 ^{ 5 } , φ = 0.03)
Forces | |||||||
---|---|---|---|---|---|---|---|
F _{S} | F _{A} | F _{Bx} | F _{By} | F _{H} | F _{Dx} | F _{Dy} | |
Minimum | -6E-06 | -3.2E-19 | -5E-13 | 2E-14 | -9E-19 | -8E-16 | -1.6E-15 |
Maximum | 6E-06 | -2E-20 | 5E-13 | 2E-13 | -1E-19 | 1.2E-15 | 1.6E-15 |
It is also found that almost all the isolines behave with oscillations in Figures 6, 7, 8, 9, but smooth isolines are given in Figures 3 and 5. Due to the ruleless Brownian movement of nanoparticles, it is difficult for nanofluid to achieve a complete equilibrium state, which is the difference compared with other common two-phase fluids. In order to expediently judge the equilibrium state and save time, we choose the temperature equilibrium states of water phase and nanoparticle phase as the whole nanofluid equilibrium state in the computation. When the water-phase and nanoparticle-phase temperatures all achieve equilibrium state, the whole nanofluid (temperature distribution, velocity vectors, density distribution, and nanoparticle volume fraction distribution) is considered as being in an equilibrium state. Hence, the temperature isolines in Figures 3 and 5 look smooth due to a complete equilibrium state, and the density distribution in Figures 6 and 7 and nanoparticle volume fraction distribution in Figures 8 and 9 behave with oscillations due to an approximate equilibrium state. Although the interparticle interaction forces have little effect on heat transfer, they play an important role on the nanoparticle distribution.
Conclusion
A 2D two phase Lattice Boltzmann model has been developed for nanofluids and the simulation results of this two-phase Lattice Boltzmann model are in good agreement with published experimental results. This model is applied to investigate the natural convection of a square enclosure filled with Al_{2}O_{3} nanofluid. The effects of different nanoparticle fractions and Rayleigh numbers on natural convection heat transfer of nanofluid are investigated. In addition the effects of forces on the nanoparticles volume fraction distribution and the heat transfer are also investigated.
It is found that the Nusselt number distribution along the heated surface firstly increases, and then decreases with Y at both low and high Rayleigh numbers. Average Nusselt numbers of the whole square enclosure both increase with nanoparticles volume fraction at a low and a high Rayleigh number. In addition, the enhancement of the average Nusselt numbers is much more pronounced at a high Rayleigh number than at a low Rayleigh number.
It is found that the temperature difference driving force is the biggest force and has the greatest effect on nanoparticle volume fraction distribution. For a low Rayleigh number, the nanoparticle volume fraction is low in the lower right corner and high in the top right corner and lower left corner. For a high Rayleigh number, the nanoparticle volume fraction is low at the bottom and high at the top.
Apart from the temperature difference driving force, Brownian force, interaction potential force, and gravity-buoyancy force contribute to the enhanced natural convective heat transfer, while the drag force contributes to the attenuation of heat transfer.
Nomenclature
a radius of nanoparticle (m)
A Hamaker constant
B_{ a } weight coefficient
c reference lattice velocity
c_{s} lattice sound velocity
c_{p} specific heat capacity (J/kg K)
F_{S} dimensionless temperature difference driving forces
F_{B} dimensionless Brownian force
F_{H} dimensionless gravity and buoyancy force
F_{D} dimensionless drag force
F_{A} dimensionless interaction potential force
g dimensionless gravitational acceleration
G dimensionless effective external force
G_{ i } Gaussian random number
h_{ a β } convective heat transfer coefficient (W/(m^{2} K))
H dimensionless characteristic length of the square cavity
k thermal conductivity coefficient (W/m/K)
k_{ B } Boltzmann constant
L_{cc} center-to-center distance between particles (m)
Ma Mach number
m^{ σ } mass of a single nanoparticle (kg)
n_{ i } number of the particles within the adjacent lattice i
Nu Nusselt number
Pr Prandtl number
r position vector
Ra Rayleigh number
T dimensionless temperature
T_{0} dimensionless average temperature (T_{0} = (T_{H} + T_{C})/2)
T_{H} dimensionless hot temperature
T_{C} dimensionless cold temperature
u^{ σ } dimensionless macro-velocity
u_{c} dimensionless characteristic velocity of natural convection
V_{ A } dimensionless interaction potential
V volume of a single lattice (m^{3})
w_{α} weight coefficient
x, y dimensionless coordinates
Greek symbols
β^{ σ } thermal expansion coefficient (K^{-1})
ρ^{ σ } density (kg/m^{3})
v kinematic viscosity (m^{2}/s)
η dynamic viscosity (Pa s)
χ thermal diffusion coefficient (m^{2}/s)
γ surface tension (N/m)
φ nanoparticle volume fraction
δ_{ x } lattice step
δ_{ t } time step
σ components (σ = 1, 2, water and nanoparticles)
τ_{ f } dimensionless collision-relaxation time for the flow field
τ_{ T } dimensionless collision-relaxation time for the temperature field
∆T dimensionless temperature difference (∆T = T_{ H } – T_{ C })
Δρ^{’} dimensionless mass density difference between nanoparticles and base fluid
∆u dimensionless velocity difference between nanoparticles and base fluid
Φ_{ αβ } dimensionless energy exchange between nanoparticles and base fluid
Error_{1} maximal relative error of velocities between two adjacent time layers
Error_{2} maximal relative error of temperatures between two adjacent time layers
Subscripts
α lattice velocity direction
avg average
C cold
nf nanofluid
H hot
w base fluid
p nanoparticle
Declarations
Acknowledgments
This work is financially supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (grant no. 51121004).
Authors’ Affiliations
References
- Wang SX, Zhou Y, Guan W, Ding B: Preparation and characterization of stimuli-responsive magnetic nanoparticles. Nanoscale Res Lett 2008, 3: 289–294. 10.1007/s11671-008-9151-3View ArticleGoogle Scholar
- Bora DK, Deb P: Fatty acid binding domain mediated conjugation of ultrafine magnetic nanoparticles with albumin protein. Nanoscale Res Lett 2009, 4: 138–143. 10.1007/s11671-008-9213-6View ArticleGoogle Scholar
- Guo SZ, Li Y, Jiang JS, Xie HQ: Nanofluids containing γ-Fe_{2}O_{3} nanoparticles and their heat transfer enhancements. Nanoscale Res Lett 2010, 5: 1222–1227. 10.1007/s11671-010-9630-1View ArticleGoogle Scholar
- Pinilla MG, Martínez E, Vidaurri GS, Tijerina EP: Deposition of size-selected Cu nanoparticles by inert gas condensation. Nanoscale Res Lett 2010, 5: 180–188. 10.1007/s11671-009-9462-zView ArticleGoogle Scholar
- Yang X, Liu Z: A kind of nanofluid consisting of surface-functionalized nanoparticles. Nanoscale Res Lett 2010, 5: 1324–1328. 10.1007/s11671-010-9646-6View ArticleGoogle Scholar
- Zhu H, Han D, Meng Z, Wu D, Zhang C: Preparation and thermal conductivity of CuO nanofluid via a wet chemical method. Nanoscale Res Lett 2011, 6: 1–6.Google Scholar
- Nadeem S, Lee C: Boundary layer flow of nanofluid over an exponentially stretching surface. Nanoscale Res Lett 2012, 7: 1–6. 10.1186/1556-276X-7-1View ArticleGoogle Scholar
- Wang L, Fan J: Nanofluids research: key issues. Nanoscale Res Lett 2010, 5: 1241–1252. 10.1007/s11671-010-9638-6View ArticleGoogle Scholar
- Oztop HF, Abu-Nada E: Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int J Heat Fluid Flow 2008, 29: 1326–1336. 10.1016/j.ijheatfluidflow.2008.04.009View ArticleGoogle Scholar
- Ho CJ, Chen MW, Li ZW: Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity. Int J Heat Mass Transfer 2008, 51: 4506–4516. 10.1016/j.ijheatmasstransfer.2007.12.019View ArticleGoogle Scholar
- Saleh H, Roslan R, Hashim I: Natural convection heat transfer in a nanofluid-filled trapezoidal enclosure. Int J Heat Mass Transfer 2011, 54: 194–201. 10.1016/j.ijheatmasstransfer.2010.09.053View ArticleGoogle Scholar
- Ghasemi B, Aminossadati SM: Brownian motion of nanoparticles in a triangular enclosure with natural convection. Int J Therm Sci 2010, 49: 931–940. 10.1016/j.ijthermalsci.2009.12.017View ArticleGoogle Scholar
- Santra AK, Sen S, Chakraborty N: Study of heat transfer augmentation in a differentially heated square cavity using copper–water nanofluid. Int J Therm Sci 2008, 47: 1113–1122. 10.1016/j.ijthermalsci.2007.10.005View ArticleGoogle Scholar
- Aminossadati SM, Ghasemi B: Natural convection cooling of a localised heat source at the bottom of a nanofluid filled enclosure. Eur J Mech B/Fluid 2009, 28: 630–640. 10.1016/j.euromechflu.2009.05.006View ArticleGoogle Scholar
- Kargar A, Ghasemi B, Aminossadati SM: An artificial neural network approach to cooling analysis of electronic components in enclosures filled with nanofluids. J Electron Packaging 2011, 133: 1–9.View ArticleGoogle Scholar
- Abu-Nada E, Chamkha AJ: Effect of nanofluid variable properties on natural convection in enclosures filled with a CuO-EG-water nanofluid. Int J Therm Sci 2010, 49: 2339–2352. 10.1016/j.ijthermalsci.2010.07.006View ArticleGoogle Scholar
- Hwang KS, Lee JH, Jang SP: Buoyancy-driven heat transfer of water-based Al_{2}O_{3} nanofluids in a rectangular cavity. Int J Heat Mass Transfer 2007, 50: 4003–4010. 10.1016/j.ijheatmasstransfer.2007.01.037View ArticleGoogle Scholar
- Jang SP, Choi SUS: Role of Brownian motion in the enhanced thermal conductivity of nanofluids. Appl Phys Lett 2004, 84: 4316–4318. 10.1063/1.1756684View ArticleGoogle Scholar
- Barrios G, Rechtman R, Rojas J, Tovar R: The lattice Boltzmann equation for natural convection in a two-dimensional cavity with a partially heated wall. J Fluid Mech 2005, 522: 91–100.View ArticleGoogle Scholar
- Peng Y, Shu C, Chew YT: Simplified thermal lattice Boltzmann model for incompressible thermal flows. Phys Rev E 2003, 68: 026701.View ArticleGoogle Scholar
- He X, Chen S, Doolen GD: A novel thermal model for the lattice Boltzmann method in incompressible limit. J Comput Phys 1998, 146: 282–300. 10.1006/jcph.1998.6057View ArticleGoogle Scholar
- Nemati H, Farhadi M, Sedighi K, Fattahi E, Darzi AAR: Lattice boltzmann simulation of nanofluid in lid-driven cavity. Int Commun Heat Mass Transfer 2010, 37: 1528–1534. 10.1016/j.icheatmasstransfer.2010.08.004View ArticleGoogle Scholar
- Wang J, Wang M, Li Z: A lattice Boltzmann algorithm for fluid–solid conjugate heat transfer. Int J Therm Sci 2007, 46: 228–234. 10.1016/j.ijthermalsci.2006.04.012View ArticleGoogle Scholar
- Dixit HN, Babu V: Simulation of high Rayleigh number natural convection in a square cavity using the lattice Boltzmann method. Int J Heat Mass Transfer 2006, 49: 727–739. 10.1016/j.ijheatmasstransfer.2005.07.046View ArticleGoogle Scholar
- Peng Y, Shu C, Chew YT: A 3D incompressible thermal lattice Boltzmann model and its application to simulate natural convection in a cubic cavity. J Comput Phys 2003, 193: 260–274.View ArticleGoogle Scholar
- Xuan Y, Yao Z: Lattice Boltzmann model for nanofluids. Heat Mass Transfer 2005, 41: 199–205.Google Scholar
- Russel WB, Saville DA, Schowalter WR: Colloidal Dispersion. Cambridge: Cambridge University Press; 1989.View ArticleGoogle Scholar
- He C, Ahmadi G: Particle deposition in a nearly developed turbulent duct flow with electrophoresis. J Aerosol Sci 1999, 30: 739–758. 10.1016/S0021-8502(98)00760-5View ArticleGoogle Scholar
- Abu-Nada E: Effects of variable viscosity and thermal conductivity of Al_{2}O_{3}-water nanofluid on heat transfer enhancement in natural convection. Int J Heat Fluid Flow 2009, 30: 679–690. 10.1016/j.ijheatfluidflow.2009.02.003View ArticleGoogle Scholar
- Hortmann M, Peric M, Scheuerer G: Finite volume multigrid prediction of laminar natural convection: benchmark solutions. Int J Numer Methods Fluid 1990, 11: 189–207. 10.1002/fld.1650110206View ArticleGoogle Scholar
- Khanafer K, Vafai K, Lightstone M: Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transfer 2003, 46: 3639–3653. 10.1016/S0017-9310(03)00156-XView ArticleGoogle Scholar
- Krane RJ, Jessee J: Some detailed field measurements for a natural convection flow in a vertical square enclosure. Proc 1st ASME-JSME Thermal Eng Joint Conf 1983, 1: 323–329.Google Scholar
- D'Orazio A, Corcione M, Celata GP: Application to natural convection enclosed flows of a lattice Boltzmann BGK model coupled with a general purpose thermal boundary condition. Int J Therm Sci 2004, 43: 575–586. 10.1016/j.ijthermalsci.2003.11.002View ArticleGoogle Scholar
- De Vahl DG: Natural convection of air in a square cavity: a bench mark numerical solution. Int J Numer Meth Fluids 1983, 3: 249–264. 10.1002/fld.1650030305View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.