Considering interaction forces (gravity and buoyancy force, drag force, interaction potential force, and Brownian force) between nanoparticles and a base fluid, a twophase 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 gravitybuoyancy force have positive effects on the enhancement of natural convective heat transfer, while drag force has a negative effect.
Keywords
Two phase Lattice Boltzmann modelRayleigh numberNanofluidNatural convection
Background
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 stimuliresponsive magnetic nanoparticles and investigated the effect of nanoparticle fraction on its cleavage efficiency. Bora and Deb [2] developed a novel bioconjugate of stearic acidcapped maghemite nanoparticle (γFe_{2}O_{3}) with bovine serum albumin. Guo et al. [3] produced magnetic nanofluids containing γFe_{2}O_{3} nanoparticles using a twostep 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 plasmaenhanced sputtering gas aggregationtype 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 finitevolume 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. AbuNada 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 finitevolume 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 liddriven flow filled with Cuwater, CuOwater, 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 fluidsolid 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 singlephase models, and the nanofluid was seen as a singlephase 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 twophase 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 twophase Lattice Boltzmann model to investigate suddenstart 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 singlephase 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 twophase 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
The density distribution function for a singlephase fluid is calculated as follows:
where ${\tau}_{f}^{\sigma}$ is the dimensionless collisionrelaxation 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.
A nanofluid is a twophase fluid constituted by nanoparticles and a base fluid, and there are interaction forces (gravity and buoyancy force, drag force, interaction potential force, and Brownian force) between nanoparticles and the base fluid. Thus, the macroscopic density and velocity fields are simulated using the density distribution function by adding the forces term.
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 NavierStokes equation, based on the mass and momentum conservation, we used multiscale 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 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.
When the internal forces and external forces are considered, energy between nanoparticles and base fluid is exchanged, and the macroscopic temperature for nanoparticles and base fluid is then given as:
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.
The corresponding kinematic viscosity and thermal diffusion coefficients are respectively defined as follows:
From Equations 18 and 19, the collisionrelaxation time for the flow field and the temperature field can be calculated. For water phase, the τ_{
f
} collisionrelaxation times are respectively 0.51433 and 0.501433 at Ra = 10^{3} and Ra = 10^{5}, and the collisionrelaxation time τ_{
T
} is 0.5. For nanoparticle phase, the τ_{
f
} collisionrelaxation times are respectively 0.50096 and 0.500096 at Ra = 10^{3} and Ra = 10^{5}, and the collisionrelaxation time τ_{
T
} is 0.500025.
Interaction forces between base fluid and nanoparticles
As noted before, a nanofluid is, in reality, a kind of twophase 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 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.
The total per unit volume forces acting on nanoparticles of a single lattice is:
where n is the number of the particles in the given lattice, and V is the lattice volume.
In a nanofluid, the forces acting on the base fluid are mainly drag force and Brownian force. Thus the force acting on the base fluid in a given lattice is:
${\mathit{F}}_{w}=n\left({F}_{D}+{F}_{B}\right)$
(28)
Results and discussion
The twophase Lattice Boltzmann model is applied to simulate the natural convection heat transfer in a square cavity which is shown in Figure 1. The square cavity is filled with the Al_{2}O_{3}water nanofluid. The thermophysical properties of water and Al_{2}O_{3} are given in Table 1. The height and the width of the enclosure are both H. The left wall is kept at a high constant temperature (T_{H}), and the top cold wall is kept at a low constant temperature (T_{C}). The boundary conditions of the other walls (right wall and bottom wall) are all adiabatic. The initial conditions for the four walls are given as follows:
Thermophysical 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
In the simulation, a nonequilibrium extrapolation scheme is adopted to deal with the boundary, and the criteria of the program convergence for the flow field and the temperature field are respectively given as follows:
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.
As shown in Table 2, the grid independence test is performed in a square enclosure using successively sized grids, 128 × 128, 192 × 192, 256 × 256, and 320 × 320 at Ra = 1 × 10^{5}, Pr = 0.7. It can be seen from Table 2 that there is a bigger difference between the result obtained with grid sizes 128 × 128 and 192 × 192 and the result available from the literature [30] than when compared with the result obtained with grids 256 × 256 and 320 × 320. In addition, the result with grid 256 × 256 and the result with grid 320 × 320 are very close. In order to accelerate the numerical simulation, a grid size of 256 × 256 was chosen as a suitable one which can guarantee a gridindependent solution.
Table 2
Comparison of the mean Nusselt numbers with different grids (Ra= 1 × 10^{
5
},Pr= 0.7)
In order to validate the Lattice Boltzmann model proposed in this work, the temperature distribution at midsections of the enclosure at Ra = 1 × 10^{5}, Pr = 0.7 is compared with the numerical results from Khanafer et al. [31] and experimental results from Krane et al. [32] in Figure 2. It can be seen that the results of this paper have a good agreement with those numerical [31] and experimental [32] results. They are closer to the experimental [32] than the numerical [31] results. In addition, the Nusselt number results at different Rayleigh numbers of this paper are compared with other published literature listed in Table 3, and it can be seen that the results are in good agreement.
Table 3
Comparison of average Nusselt numbers with other published data (Pr= 0.7)
Due to the temperature balance between water and nanoparticles, the temperature nephogram of water and nanoparticles for each of the nanoparticle fractions are identical. The temperature nephograms of nanofluid at Ra = 1 × 10^{3} and Ra = 1 × 10^{5} are presented in Figure 3. It can be seen that isotherms are more crooked with the higher Rayleigh number, which denotes that the heat transfer characteristic transforms from conduction to convection.
Because there are fewer nanoparticles than water molecules, and the drag force of nanoparticles on water is small, the velocity vectors of nanofluid with different nanoparticle fractions have such small differences that it is difficult to distinguish them. However, the differences can be observed in the Nusselt number distribution. For this reason, only the velocity vectors of nanofluid components with φ = 0.03 at different Rayleigh numbers are given as an example in Figure 4. Separating the nanofluid into its two constitutive components, it can be seen that the velocity vectors of the water component are larger than those of the nanoparticle component due to the law of conservation of momentum. The velocity difference between the water component and the nanoparticle component gives rise to the drag force. In addition, it can be seen that velocity increases with Rayleigh number, which can also explain that the heat transfer characteristic transforms from conduction to convection.
Driving force and interaction forces have a big effect on nanoparticle volume fraction distribution and the flow and heat transfer characteristics of the nanofluid. The main driving force in this work is the temperature difference. Interaction forces between nanoparticles and base fluid include gravitybuoyancy force, drag force, interaction potential force, and Brownian force. In order to compare the effects of these forces, the ranges of them are presented in Table 4. We used doubleprecision variables in our code. From Table 4, we can find that the temperature difference driving force F_{S} is much bigger than the other forces (interaction forces between nanoparticles and base fluid). The driving force has the greatest effect on nanoparticle volume fraction distribution, and the effects of other forces on nanoparticle volume fraction distribution can be ignored in this case. However, these other forces play an important role in the flow and heat transfer of the nanofluid. Apart from the temperature difference driving force, the Brownian force is much larger than other forces, which is different from other twophase fluids. For this reason, the Brownian force can enhance the heat transfer of the nanofluid by disturbing the flow boundary layer and the thermal boundary layer. Drag force comes about due to the velocity difference between nanoparticles and water molecules, and the nanoparticles in the water decrease the velocity of nanofluid in the enclosure, which in turn attenuates the natural convection of nanofluid in the enclosure. The interaction potential force prevents the nanoparticles from gathering together and keeps the nanoparticles dispersed in the water. In addition to the above forces, there is the gravitybuoyancy force, that is, the sum of gravity of the nanoparticles themselves and the buoyancy force of the water. The gravitybuoyancy force and temperature difference driving force together give rise to the velocity vectors of the nanofluid within the enclosure. In summary, Brownian force, interaction potential force, and gravitybuoyancy force contribute to the enhanced natural convective heat transfer, while drag force contributes to the attenuation of heat transfer.
Table 4
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
6E06
3.2E19
5E13
2E14
9E19
8E16
1.6E15
Maximum
6E06
2E20
5E13
2E13
1E19
1.2E15
1.6E15
The temperature difference driving force distribution in the square at different Rayleigh numbers is given in Figure 5. From Figure 5, we can see that the temperature difference driving force along the left wall (high temperature) and the top wall (low temperature) is high. Its direction along the hightemperature wall is upward, and that along the lowtemperature wall is downward, while the temperature difference driving force in other regions far away from the two walls (left wall and top wall) is small. From Figure 3, it can be seen that the temperature gradient near the left wall and the top wall is higher than that in other regions, which causes a high temperature difference driving force near there. Similarly, the temperature gradient in other regions is small, causing only a low temperature difference driving force in that vicinity. In addition, it can be seen that the same driving force line at a high Rayleigh number becomes more crooked than that at a low Rayleigh number. This is because the driving force is caused by the temperature difference (temperature gradient); a bigger temperature gradient causes the same driving force line to become more crooked. It can be seen from Figure 3 that isotherms are more crooked at a higher Rayleigh number, and the isotherm changes correspond to the changes of temperature gradient. Thus, the conclusion that the same driving force line at a high Rayleigh number becomes more crooked than that that at a low Rayleigh number is obtained.
Figures 6 and 7 give the density distribution of the water phase at Ra = 1 × 10^{3} and Ra = 1 × 10^{5}. For a low Rayleigh number (Ra = 1 × 10^{3}), when the water near the left wall is heated, its density decreases and flows upward, so the density of water near the top right corner also becomes smaller. Then when the water is cooled by the top wall, the density of the water becomes larger. Then the denser water flows downward to the lower right corner, and so, the density of water in the lower right corner is larger than that in other regions. Because the temperature gradient (corresponding to the temperature difference driving force) is small and the temperature is high in the lower left corner, the density of water in the lower left corner is thus low. For a high Rayleigh number (Ra = 1 × 10^{5}), the temperature gradient and the corresponding driving force become larger, then the lowerdensity water, including that in the lower left corner, rises to the top right corner. The denser water is cooled by the top wall and flows downward to the lower right corner, and the area where the denser water in the lower right corner becomes larger.
Figures 8 and 9 respectively present the nanoparticle distribution of nanofluid with volume fractions at Ra = 1 × 10^{3} and Ra = 1 × 10^{5}. For a low Rayleigh number (Ra = 1 × 10^{3}), the driving force along the left wall is upward, and many nanoparticles are driven to the top right corner, which contributes to the high nanoparticle volume fraction in the top right corner. However, the temperature gradient in the lower left corner is small and causes a correspondingly small temperature difference driving force. Thus, many nanoparticles are left in the lower left corner, which contributes to the high nanoparticle volume fraction in the lower left corner. There is a large temperature gradient in the lower right corner, and the large driving force displaces the nanoparticles off the lower right corner, which contributes to the low nanoparticle volume fraction in the lower right corner. For a high Rayleigh number (Ra = 1 × 10^{5}), the convection heat transfer is enhanced and the velocity of the nanofluid becomes larger, and the temperature gradient and the corresponding driving force become bigger. Thus, many nanoparticles from the bottom are driven to the top by the driving force, which contributes to the low nanoparticle volume fraction at the bottom and a high nanoparticle volume fraction at the top. In addition, we can see that the nanoparticle volume fraction distribution is opposite to that of the waterphase density distribution. From Table 4, we can see that the temperature difference driving force is the biggest one, and the changes of the waterphase density and the inhomogeneous nanoparticle distribution are mainly due to the driving force. Through the above analysis, it is found that the nanoparticles migrate to locations where the water density is small, and thus, the conclusion that the nanoparticle volume fraction distribution is opposite to that of the waterphase density distribution is obtained.
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 twophase 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 waterphase and nanoparticlephase 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.
Figure 10 shows the Nusselt number distribution along the heated surface using Al_{2}O_{3}water nanofluid at Ra = 10^{3}. It can be seen that the Nusselt number along the heated surface increases with nanoparticle volume fraction at low Y (0 < Y < 0.58) and decreases with nanoparticle volume fraction at high Y (0.58 < Y < 1). Because the heat transfer is more sensitive to thermal conductivity than viscosity at low Y, while it is more sensitive to viscosity than thermal conductivity at high Y.
Figure 11 shows Nusselt number distribution along the heated surface using Al_{2}O_{3}water nanofluid at Ra = 10^{5}. It can be seen that the Nusselt number along the heated surface increases with nanoparticle volume fraction at low Y (0 < Y < 0.875) and decreases with nanoparticle volume fraction at high Y (0.875 < Y < 1). Compared with Figure 7, the Nusselt number becomes larger, and the enhanced heat transfer section also gets longer. The high Rayleigh number increases the velocity and then enhances the heat transfer.
Figure 12 presents the average Nusselt numbers at different Rayleigh numbers. Although the Nusselt number distribution along the heated surface increases with nanoparticle volume fraction in one section and decreases in the other section, the average Nusselt numbers at Ra = 10^{3} and Ra = 10^{5} both increase with nanoparticle volume fraction. For this square enclosure (left wall is kept at a high constant temperature (T_{H}), and top cold wall is kept at a low constant temperature (T_{C})), adding nanoparticles can enhance the average heat transfer at both 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.
Conclusion
A 2D two phase Lattice Boltzmann model has been developed for nanofluids and the simulation results of this twophase 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 gravitybuoyancy 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)
e_{
α
} lattice velocity vector
${f}_{\alpha}^{\sigma}$
density distribution function
${f}_{\alpha}^{\mathit{\sigma eq}}$
local equilibrium density distribution function
${F}_{\alpha}^{\sigma \text{'}}$
dimensionless external force in direction of lattice velocity
${F}_{\alpha}^{\sigma}$
dimensionless total interparticle interaction forces
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} centertocenter 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 time (s)
${T}_{\alpha}^{\sigma}$
temperature distribution function
${T}_{\alpha}^{\mathit{\sigma eq}}$
local equilibrium temperature distribution function
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 macrovelocity
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 collisionrelaxation time for the flow field
τ_{
T
} dimensionless collisionrelaxation 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
(1)
School of Energy Science and Engineering, Harbin Institute of Technology
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