Document Type : Full Lenght Research Article
Authors
^{1} Mechanical Engineering Dept., Shahrood university
^{2} University of Shahrood
Abstract
Keywords
Main Subjects


History: Submitted 25 March 2014 Revised 1April 2016 Accepted 27 September 2016
Keywords:
Natural convection Lattice Boltzmann method Open cavity Hydrodynamic boundary conditions Thermal boundary conditions

A B S T R A C T A lattice Boltzmann method is applied to compare the results of simulating natural convection in an openend cavity using different hydrodynamic and thermal boundary conditions. The Prandtl number in the present simulation is 0.71, Rayleigh numbers are 104,105, and 106, and selected viscosities are 0.02 and 0.05. The ongrid bounceback method with firstorder accuracy and the noslip method with secondorder accuracy are employed for implementation of hydrodynamic boundary conditions. Moreover, two different thermal boundary conditions (with first and secondorder accuracy) also are presented for thermal modelling. The results showed that first and secondorder boundary conditions (thermal/hydrodynamic) are the same for a twodimensional, singlephase, convectiveheat transfer problem, including geometry with straight walls. The obtained results for different hydrodynamic and thermal boundary conditions are useful for researchers in the field of the lattice Boltzmann method in order to implement accurate conditions on the boundaries with different physics. © 2016 Published by Semnan University Press. All rights reserved. 
A Comparative Solution of Natural Convection in an Open Cavity using Different Boundary Conditions via the Lattice Boltzmann Method
Mohsen Nazari^{*} and Mohammad Hassan Kayhani
Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, IRAN. P.O. Box: 3619995161, nazari_me@yahoo.com 

Journal of Heat and Mass Transfer Research
Journal homepage: http://jhmtr.journals.semnan.ac.ir 
Natural convection in cavities has many engineering applications, such as solar collectors and heat exchangers. There have been some publications related to natural heat transfer in closed and openend cavities. Chan and Tien [1] performed a numerical study on natural convection in shallow open cavities. Bilgen and Muftuoglu [2] investigated natural heat transfer in a cavity with slots.
Corresponding Author: Mohsen Nazari Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, IRAN Email: nazari_me@yahoo.com

A thermal boundary condition also has been studied in many researches [1620]. D’Orazio et al. [10] proposed a novel thermal boundary condition, which can be applied in the case of either a constant wall temperature (Dirichlet boundary condition) or a constant heat flux on the wall (Newman boundary condition). D’Orazio and Succi [21] proposed a thermal boundary condition in which the unknown distribution functions are considered to be the equilibrium distribution functions with a counterslip internal energy and developed thermal boundary conditions for both Dirichlet and Newmann conditions. A secondorder accuracy thermal boundary condition was developed by Kuo and Chen [22] using a nonequilibrium mirrorreflection scheme. They also investigated properties of the temperature gradient that were calculated directly from the thermal lattice Boltzmann method.
Natural convection heat transfer in an openend cavity has been studied by applying the lattice Boltzman Method [23]. In that study, the D2Q9 model was used for flow while a D2Q4 model was applied for temperature. The bounceback boundary condition (firstorder accuracy) was used for flow on the solid walls, and since the north and south walls were insulated, the bounceback boundary condition (adiabatic) for temperature was applied on these walls. The boundary condition of the west wall was calculated using the wall temperature. Dixit and Babu [24] used the lattice Boltzmann method to simulate naturalconvection heat transfer with a high Rayleigh number in a square cavity. In that study, a standard D2Q9 model was used for both flow and temperature. As for the boundary conditioning, for both flow and temperature, a noslip boundary condition (secondorder accuracy) was applied on all walls. The north and south walls were adiabatic, while west and east walls were maintained at constant but different temperatures. Double diffusive natural convection has been studied in an openend cavity using the lattice Boltzmann method [25]. In this study, south and north walls were insulated, and the left wall was at a constant temperature (the right side was open). A D2Q9 model was applied for the flow while a D2Q4 model was used for temperature and species concentration. In this work, for the flow, the bounceback boundary condition was imposed on solid walls, and since north and south walls were adiabatic (or have no flux), this type of boundary condition also was used on these walls for temperature (or concentration). The boundary condition of the west wall was calculated using a constant wall temperature (or concentration).
The current study presents comparative results of simulated natural convection with different boundary conditions for flow in an openend cavity using the lattice Boltzmann Method. The results show accuracy of each boundary condition and their accordance to one another. Comparisons between different thermal and hydrodynamic boundary conditions, with different orders of accuracy, have not been presented completely in the literature, and this topic, to the best of our knowledge, is an open research topic.
2. Method of solution
In this work, a simple D2Q9 algorithm is used for both flow and temperature. Fig. 1 shows the cavity boundaries and the known and unknown distribution functions. South and north walls are adiabatic (zerotemperature gradient), while east and west walls have constant temperatures.
Flow streaming and collision are presented as the following [26, 27]:
(1) , 
(1) 
where is the streaming part and the righthand side of the equation is the collision term. is the equilibrium distribution function, and τ is the relaxation time. The equilibrium distribution function is given by
, 
(2) 
where and are density and microscopic velocity, respectively, and are the weight factors, which, for the D_{2}Q_{9} model, are defined as
. 
(3) 
The velocities are [23]
, 
(4) 
where c = ∆x/∆t, ∆x is the lattice space, and ∆t is the lattice time step size. In Eq. (1), is the force term in each of the directions and can be defined as
, 
(5) 
where is defined as
, 
(6) 
where , , and ∆T are the thermal expansion coefficient, gravity acceleration, and temperature difference, respectively. The macroscopic velocity and density can be obtained through the first and zeroth moment of the particle distribution f, i.e. [26, 27],

(7) 
. 
(8) 
The kinematic viscosity in the D2Q9 method is defined as [26, 27]
(2) , 
(9) 
where is the sound velocity, which is .
Temperature streaming and collision are presented as the following [23]:
(10)
where is the thermal equilibrium distribution function and is the relaxation time. The thermal equilibrium distribution function is defined as
. 
(11) 
for the D2Q9 model is given by
, 
(12) 
where is the thermal diffusion coefficient. The temperature, then, can be calculated at any point of the domain:
. 
(13) 
The average Nusselt number also is defined as
, 
(14) 
where M is the number of lattice nodes on the Y direction.
3. Boundary conditions
In this study, two different hydrodynamic boundary conditions in the presence of two types of thermal boundary conditions have been used. Here, the bounceback boundary condition is compared with a noslip boundary condition, and two different thermal boundary conditions—one with firstorder accuracy and the other with secondorder accuracy—are compared with each other. The term “bounceback” is used here to mean bounceback at the nodes (or “onnode bounce back”), which coincides with the actual physical boundaries. It should be noted that the mentioned “bounceback” rule is different from the “link bounceback.” In the link bounceback method, the boundary nodes lie midway between the solid and fluid nodes, and the fluid particles moving along the links between solid and fluid nodes interact at these boundary nodes.
3.1 Hydrodynamic boundary conditions
At the east wall, a zero gradient is realized through the following conditions:
, 
(15) 
where n is the number of lattice nodes on the open wall and n1 is number of the lattice nodes next to the boundary inside the cavity. For solid walls, two different boundary conditions are used.
3.1.1 Bounceback boundary condition
On the solid walls (west, south, and north boundaries), the bounceback boundary condition is used. For instance, at the south wall, the unknown distribution functions would be , which are
, 
(16) 
where n is the number of lattice nodes on the south wall.
3.1.2 A noslip boundary condition
The lattice Boltzmann method is a method with secondorder accuracy. The bounceback boundary condition for applying zero velocity at a wall is a method of firstorder accuracy. This shows that using the bounceback boundary condition is not always the best choice for boundary conditioning in the lattice Boltzmann method. Secondorder accuracy in the bounceback method is achieved if there are straight walls, and we correct for the shift in the effective position of the wall. If = 1 and there is a straight wall, the effective wall is always at half the distance between fluid and solid boundary nodes, so that is fine. For simple flows (like a Poiseuille flow), the shift of the wall even can be computed analytically for a given . However, in our case, that is not so straightforward; therefore, we return to a firstorder accuracy. Here, a noslip boundary condition is used to overcome this issue, which is a secondorder accuracy method. In this method, unknown distribution functions are assumed to be an equilibrium distribution function with a counterslip velocity, which is determined so that fluid velocity at the wall is equal to the wall velocity. The velocity normal to the wall is equal to the velocity of the wall, while the velocity along the wall is not equal to that of the wall. The difference between the wall velocity and the velocity along the wall is called slip velocity.
For the south wall, the unknown distribution functions are , which are defined as [3]
, 
(17) 
where and are the velocity of the wall in x and y directions, respectively, and and are unknown parameters, which can be defined by the condition that fluid velocity on the wall is equal to that of the wall. is the counterslip velocity. Now, we have two equations for and and one equation for (density of the virtual fluid in the wall), which can be calculated using Eq. (8). Thus, the unknown parameters are:
. 
(18) 
Since , the unknown distributions are
.

(19) 
3.2 Thermal boundary conditions
At the east wall (open wall), the boundary condition depends on whether the flow penetrates the cavity or leaves it. If the flow penetrates the cavity, then the temperature should be ambient , and if the flow is leaving the cavity, there is no diffusion [23]. At the open wall, unknown distribution functions are
(20)
For solid walls two different boundary conditions with first and secondorder accuracy have been used.
3.2.1 Firstorder accuracy thermal boundary condition
For north and south walls (adiabatic walls), bounceback boundary conditions are used. At the west wall ( ), the boundary condition can be calculated using the wall temperature:
. 
(21) 
3.2.2 Secondorder accuracy thermal boundary condition
Here, a thermal boundary condition (D’Orazio’s approach [10]) is used, which is a secondorder accuracy boundary condition. In this method, unknown distribution functions are assumed to be an equilibrium distribution function at a temperature . The temperature is calculated considering whether the wall is insulated (Newman boundary condition) or has a constant temperature (Dirichlet boundary condition). Since we have a zero velocity on the walls, from Eq. (11) for unknown distributions, we have
. 
(22) 
For the west wall (the Dirichlet boundary condition), the unknown distribution functions are , , and . According to Eq. (22), we have
(23) 
The wall temperature is calculated using Eq. (13):

(24) 
where is the sum of the known distribution functions:
. 
(25) 
From Eq. (24), can be calculated as

(26) 
and unknown distribution functions are
(27) 
For insulated walls, there is no temperature gradient. For instance, on the north wall, the unknown distribution functions are , , and . Using Eq. (22), we have
(28)


(29) 
where is defined by
. 
(30) 
Therefore, the unknown distribution functions can be calculated by
(31)

4. Results and discussion
This study compared two hydrodynamic and two thermal boundary conditions in order to find the most convenient one. For this purpose, some features, such as the Nusselt number, velocity, and temperature at the center of the cavity, were compared with each other in different cases using all methods. From these figures, we can understand the differences and similarities of these methods’ influences on the mentioned features, and as can be seen, one can simply use the bounceback boundary condition for flow and the firstorder accuracy thermal boundary condition instead of the more complicated noslip boundary condition and secondorder accuracy thermal boundary condition.
Table 1. Nusselt number comparison resulting from different hydrodynamic methods in the presence of the firstorder accuracy thermal boundary condition in the case of viscosity = 0.02
Ra 

LBM/Error(%) grid=64×64 
LBM /Error(%) grid=128×128 
LBM/Error(%) grid=256×256 
Mohamad [23] LBM 
Mohamad [30] FVM 
Hinojosa et. al [31] FVM 
10^{4} 
Bounce Back 
3.297/1.016 
3.283/0.596 
3.361/2.977 
3.377 
3.264 
3.57 
No Slip 
3.299/1.082 
3.284/0.612 
3.361/2.981 

10^{5} 
Bounce Back 
7.254/0.095 
7.250/0.017 
7.256/0.064 
7.323 
7.261 
7.75 
No Slip 
7.257/0.054 
7.261/0.003 
7.257/0.059 

10^{6} 
Bounce Back 
14.19/0.843 
14.33/1.831 
14.33/1.839 
14.38 
14.076 
15.11 
No Slip 
 
14.33/1.839 
14.33/1.843 
The errors are calculated with respect to the F.V. solution [30]. The numbers of lattices in [23] are 64*64, 128*128 and 256*256 for the Ra. Numbers of 10^{4}, 10^{5}, and 10^{6}, respectively.
Table 2. Nusselt number comparison resulting from different types of boundary conditions in the case of viscosity = 0.05 and Ra = 10^{5}
Type of boundary condition 
Nu 

grid=64×64 
grid=128×128 
grid=256×256 

flow=B.B & Temperature=1^{st} order accuracy 
7.244257 
7.257607 
7.246472 
flow=Noslip & Temperature=1^{st} order accuracy 
7.242551 
7.257314 
7.246411 
flow=B.B & Temperature=2^{nd} order accuracy 
7.253804 
7.263481 
7.255821 
flow=Noslip & Temperature=2^{nd} order accuracy 
7.252012 
7.263239 
7.255728 
West wall ( ) 
1 
4 
3 
2 
8 
5 
6 
7 
1 
4 
3 
2 
8 
5 
6 
7 
1 
4 
3 
2 
8 
5 
6 
7 
1 
4 
3 
2 
8 
5 
6 
7 
East (open) wall ( ) 
North wall (Adiabatic) 
South wall (Adiabatic) 
Fig.1 Cavity boundaries and velocity directions.
Table 1 illustrates the comparison of the Nusselt number in both types of hydrodynamic boundary conditions in the presence of the firstorder accuracy thermal boundary condition in the case of different Rayleigh numbers and the number of lattice nodes. This table also includes the Nusselt numbers reported by other authors using the lattice Boltzmann method and finitevolume method. One can see that the Nusselt number resulting from this study is in good agreement with the Nusselt number resulting from the mentioned works. Table 2 shows the Nusselt number using all of the mentioned thermal and hydrodynamic boundary conditions. As can be seen in this table, in a certain viscosity, the Nusselt numbers calculated using different combinations of thermal and hydrodynamic boundary conditions are almost equal in all cases of grid numbers (number of lattice nodes).
Figs. 2 and 3 illustrate the streamlines and isotherms, respectively, for the cases of bounceback and firstorder accuracy thermal boundary conditions for and. These visualizations of flow and temperature fields will help to understand the physics and the importance of the choice of boundary conditions.
The results for each type of boundary condition are presented in Figs. 4–14. In the figures, “visco,” “B.B.,” and “Temp” stand for “viscosity,” “Bounceback boundary condition,” and “temperature,” respectively. In Figs. 4–12, the applied thermal boundary condition is a firstorder accuracy one. Figs. 4–6 show that Nusselt numbers at a fixed viscosity are almost the same for both bounceback and noslip boundary conditions.
Fig.2 Streamlines in the case of bounceback boundary conditions for Ra = 10^{5} and = 0.05.
Fig.3 Isotherms in the case of bounceback boundary conditions for Ra = 10^{5} and = 0.05.
Moreover, the value of viscosity has little influence on the Nusselt number in both types of boundary conditions. In other words, the Nusselt number does not change with viscosity in either of the methods, and when it does, this mostly is seen along with high Rayleigh numbers. As it has been illustrated, the number of lattice nodes has little effect on the Nusselt number, and the Nusselt number is pretty much the same in all cases (a–c). Here the Nusselt number of the last iteration is used, and it does not change with iteration after this equilibrium time (number of iterations).
Figs. 7–9 illustrate the temperature at the center of the cavity. It can be observed that both bounceback and noslip boundary conditions have almost a similar effect on the temperature in a certain viscosity. Regarding (a), (b) and (c) in all figures, it can be seen that each case has a different behavior in the path to converging. One can see that, in the case of higher numbers of lattices, there is a greater difference between the temperature results in the two different viscosities. This difference decreases as the time steps (number of iterations) increase. The same result is observed when the Rayleigh number increases. It can be observed that, in the case of higher Rayleigh numbers in a certain number of lattices, there is more variance between the results of two different viscosities. But this variance is negligible.
(a)
(b)
(c)
Fig.4 Nusselt number per iteration in the presence of the firstorder accuracy thermal boundary condition at Ra = 10^{4} for the number of lattice nodes on horizontal and vertical walls: a = 64×64, b = 128×128, and c = 256×256.
(a)
(b)
(c)
Fig.5 Nusselt number per iteration in the presence of the firstorder accuracy thermal boundary condition at Ra = 10^{5} for the number of lattice nodes on horizontal and vertical walls: a = 64×64, b = 128×128, and c = 256×256.
It also can be seen that in higher Rayleigh numbers, the path to a steady solution is oscillatory. It should be noted that, in the case of higher Rayleigh numbers, less time steps are required to reach convergence. When viscosity increases, the temperature at the center of the cavity increases, and this is true for both mentioned hydrodynamic boundary conditions. However, by decreasing viscosity, the time steps necessary to reach the convergence solution increases. This result is expected since the time scale is a function of υ (relaxation time) [29].
(a)
(b)
Fig.6 Nusselt number per iteration in the presence of the firstorder accuracy thermal boundary condition at Ra = 10^{6} for the number of lattice nodes on horizontal and vertical walls: a = 128×128 and b = 256×256.
(a)
(b)
(c)
Fig.7 Temperature at the center of the cavity per iteration in the presence of the firstorder accuracy thermal boundary condition at Ra = 10^{4} for the number of lattice nodes on horizontal and vertical walls: a = 64×64, b = 128×128, and c = 256×256.
(a)
(b)
(c)
Fig.8 Temperature at the center of the cavity per iteration in the presence of the firstorder accuracy thermal boundary condition at Ra = 10^{5} for the number of lattice nodes on horizontal and vertical walls: a = 64×64, b = 128*128, and c = 256×256.
(a)
(b)
Fig.9 Temperature at the center of the cavity per iteration in the presence of the firstorder accuracy thermal boundary condition at Ra = 10^{6} for the number of lattice nodes on horizontal and vertical walls: a = 128×128 and b = 256×256.
Figs. 10–12 show changes in the velocity at the center of the cavity over the number of iterations for different Rayleigh numbers. On the one hand, it can be observed that, at higher Rayleigh numbers, it takes less time (less iterations) to reach a steadystate solution. On the other hand, by decreasing viscosity, the number of iterations for reaching the steadystate solution increases. The figures show that, in a certain viscosity, both mentioned boundary conditions have similar results. One can see that, in both types of hydrodynamic boundary conditions, the velocity at the center of the cavity is a negative value, and the absolute value of velocity in the case of viscosity = 0.02 is lower than of that in the case of viscosity = 0.05 (the ratio of velocity in the case of viscosity = 0.02 to that of viscosity = 0.05 is equal to 0.4).
(a)
(b)
(c)
Fig.10 Velocity at the center of the cavity per iteration in the presence of the firstorder accuracy thermal boundary condition at Ra = 10^{4} for the number of lattice nodes on horizontal and vertical walls: a = 64×64, b = 128×128, and c = 256×256.
This is expected because, in the scaling process, the viscosity’s conversion factor is a product of the conversion factors of velocity and length, and since M (the dimensionless value of length) is fixed in both cases, the conversion factor of length is constant, so the velocity ratio is equal to the viscosity ratio. Consider H as the height of the cavity, is the actual viscosity, and is the actual velocity. Then, the conversion factors of length, viscosity, and velocity are defined as

(32)

where M is the number of lattice nodes (dimensionless value of length), is either 0.02 or 0.05, and is the dimensionless velocity. Since dimensionless numbers (in this case, the Reynolds number) should be equal in both physical and dimensionless systems, the following applies:
. 
(33) 
Then from equations (32) and (33):
, 
(34a) 
or

(34b) 
From equation (32):
. 
(35) 
where and are the velocities in the case of viscosity = 0.02 and viscosity = 0.05, respectively.
Fig. 13 shows the effect of each type of boundary condition on variations of the Nusselt number with the number of iterations. One can see that Nusselt numbers at a fixed viscosity are almost the same for all boundary conditions. Therewith, the value of viscosity does not have much effect on the Nusselt number. In other words, the Nusselt number does not change with viscosity in any of the methods. As it has been illustrated, the number of lattice nodes has little effect on the Nusselt number, and the Nusselt number is pretty much the same in all cases (a–c).
(a)
(b)
(c)
Fig.11 Velocity at the center of the cavity per iteration in the presence of the firstorder accuracy thermal boundary condition at Ra = 10^{5} for the number of lattice nodes on horizontal and vertical walls: a = 64×64, b = 128×128, and c = 256×256.
(a)
(b)
Fig.12 Velocity at the center of the cavity per iteration in the presence of the firstorder accuracy thermal boundary condition at Ra = 10^{6} for number of lattice nodes on horizontal and vertical walls: a = 128×128 and b = 256×256.
Fig. 14 illustrates the variation of the temperature at the center of the cavity with the number of iterations, using all mentioned thermal and hydrodynamic boundary conditions. It can be seen that all of the thermal and hydrodynamic boundary conditions have a nearly similar influence on the temperature in a certain value of viscosity. Moreover, the path to convergence changes with the number of lattice nodes. When there are more lattice nodes, there is a greater difference between the temperature results in the two different viscosities, with this difference decreasing as time steps (number of iterations) increase.
(a)
(b)
(c)
Fig.13 Nusselt number per iteration using all thermal and hydrodynamic boundary conditions at Ra = 10^{5} for the number of lattice nodes on horizontal and vertical walls: a = 64×64, b = 128×128, and c = 256×256.
(a)
(b)
(c)
Fig.14 Temperature at the center of the cavity per iteration using all thermal and hydrodynamic boundary conditions at Ra = 10^{5} for the number of lattice nodes on horizontal and vertical walls: a = 128×128 and b = 256×256.
From the results illustrated above, one can see that, in the case of the hydrodynamic boundary condition, both bounceback and noslip boundary conditions have similar results. This means that, in a problem with geometry and boundary conditions like this problem, it is convenient to use the simple and less complicated bounceback boundary condition because, with less cost, the desirable result will be achieved. The same argument can be made for the thermal boundary condition. As it was illustrated in the results, both first and secondorder accuracy thermal boundary conditions have similar outcomes. From these results, one can presume that, in a problem like this one, it is appropriate to use the firstorder accuracy boundary condition instead of the more complicated timeconsuming secondorder one. Finally, it can be claimed that, in such problems as this one (including straight walls), it is convenient to use the firstorder accuracy thermal or hydrodynamic boundary conditions and have acceptable outcomes with proper exactness.
5. Conclusion
Comparison results of simulated natural convection in an openend cavity were studied by applying two different hydrodynamic and two different thermal boundary conditions on solid walls. The results show that the Nusselt number is almost the same in all the different boundary conditions. The temperature at the center of the cavity is also analogous using either of the tested boundary conditions. The temperature also increases as viscosity increases in all cases, and the time steps required to achieve a steadystate solution have a decreasing behavior with increasing viscosity. The decrease in equilibration time with viscosity corresponds to what we would expect, since viscosity is a relaxation parameter. Velocity at the center of the cavity is also almost equivalent in both types of hydrodynamic boundary conditions.
The bounceback boundary condition is easier to apply because of its simple numerical implementation compared to the noslip boundary condition. In the noslip boundary condition, a system of equations needs to be solved on each wall before finding the unknown distribution functions, while in the bounceback boundary condition, the distribution functions can be found directly. It should be noted that the bounceback boundary condition is a firstorder accuracy method, and it is not desirable to be used with the lattice Boltzmann method, which is a secondorder accuracy method. However, as it was observed in the results in Section 4, the difference in the flow velocity and temperature profile between the two methods is not significant, and we can use the bounceback method instead of the more timeconsuming noslip boundary condition.
The same argument can be made for thermal boundary conditions. As it was observed in the results, both thermal boundary conditions with first and secondorder accuracy have the same effect on the outcomes. Thus, it is appropriate to use the firstorder accuracy thermal boundary condition instead of the more complicated and timeconsuming one with secondorder accuracy. Finally, it can be claimed that, in the geometry with straight walls, it is convenient to use firstorder accuracy thermal or hydrodynamic boundary conditions and have acceptable outcomes with proper exactness.
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