2.7.1.2. The lattice Boltzmann method
The thermal LB model utilizes two distribution functions,
f and
g, for the flow and temperature fields, respectively. Thermal LBM employs modeling of movement of fluid
particles to capture macroscopic fluid quantities, such as velocity, pressure, and temperature. In this approach, the fluid domain is discretized to uniform Cartesian cells. The probability of finding particles within certain range of velocities at a certain range of locations replaces tagging each particle as in the computationally intensive molecular dynamics simulation approach. In LBM, each cell holds a fixed number of distribution functions, which represent the number of fluid particles moving in these discrete directions. The D2Q9 model has been implemented and values of
w0=4/9 for |c
0| = 0 (for the static particle),
w1−4=1/9 for |c
1–4| = 1 and
w5−9=1/36 for
|c5−9|=2–√ are assigned in this model. The density and distribution functions, that is,
f and
g, are calculated by solving the LBE, which is a special discretization of the kinetic Boltzmann equation. After introducing the BGK approximation, the general form of LBE with external force is
For the flow field:
fi(x+ciΔt,t+Δt)=fi(x,t)+Δtτv[feqi(x,t)−fi(x,t)]+ΔtciFk
(2.98)
For the temperature field:
gi(x+ciΔt,t+Δt)=gi(x,t)+ΔtτC[geqi(x,t)−gi(x,t)]
(2.99)
Here ∆
t denotes lattice time step,
ci is the discrete lattice velocity in direction
i,
Fk is the external force in direction of lattice velocity,
τv and
τC denote the lattice relaxation times for the flow and temperature fields. The kinetic viscosity
υ and the thermal diffusivity
α are defined in terms of their respective relaxation times, that is,
υ=c2s(τv−1/2) and
α=c2s(τC−1/2), respectively. Note that the limitation 0.5 <
τ should be satisfied for both relaxation times to ensure that viscosity and thermal diffusivity are positive.
Furthermore, the local equilibrium distribution function determines the type of problem being simulated. It also models the equilibrium distribution functions, which are calculated with Eqs.
(2.100) and
(2.101) for flow and temperature fields, respectively:
feqi=wiρ[1+ci.uc2s+12(ci.u)2c4s−12u2c2s]
(2.100)
geqi=wiT[1+ci.uc2s]
(2.101)
where
wi is a weighting factor and
ρ is the lattice fluid density.
To incorporate buoyancy force in the model, the force term in Eq.
(2.98) has to be evaluated in the vertical direction (
y):
F=3wiρ[gyβ(T−Tm)]
(2.102)
where
gy and
β are gravitational acceleration and thermal expansion coefficients. For natural convection, the Boussinesq approximation is applied. To ensure that the thermal LBM code works in the near-incompressible regime, the characteristic velocity of the flow for the natural convection regime
(Vnatural≡βgyΔTH−−−−−−−√) must be small compared with the fluid speed of sound. In the present study, the characteristic velocity is adopted as 0.1 that of sonic speed. Finally, the macroscopic variables are computed with the following formulae:
Flow density: ρ=∑ifi,Momentum: ρu=∑icifi,Temperature: T=∑igi.
(2.103)
2.7.1.2.1. Boundary conditions
Bounce-back boundary conditions were applied on all solid boundaries, which indicate that incoming boundary populations are equal to outgoing populations after the collision. Bounce back type boundary conditions are proven to provide more accurate numerical approximations for LBM simulations. For instance, for the east boundary, the following conditions are imposed:
f3,n=f1,n,f6,n=f9,n,f7,n=f5,n
(2.104)
where n is the node on the lattice.
A bounce-back boundary condition (adiabatic) is used on the north and south of the boundaries. For instance at the north boundary, the following conditions are imposed:
g8,n=g8,n−1,g4,n=g4,n−1,g7,n=g7,n−1
(2.105)
Temperatures at the rectangular body and active side walls are known. For instance at the west boundary of rectangular body Th = 1, as we are using D2Q9, the unknowns are g3, g6, and g7 which are evaluated as
g3=Th(w(3)+w(1))−g1g6=Th(w(6)+w(8))−g8g7=Th(w(7)+w(5))−g5
(2.106)
2.7.1.2.2. Second law analysis
According to Bejan
[18], one can find the volumetric entropy generation rate as
S∙gen=HTI+FFI
(2.107)
where HTI is the irreversibility due to heat transfer in the direction of finite temperature gradients and FFI is the contribution of fluid friction irreversibility to the total generated entropy.
In terms of the primitive variables, HTI and FFI become
HTI=k(∇T.∇T)T2FFI=μφT
(2.108)
One can also define the Bejan number, Be, as
Be=HTIHTI+FFI
(2.109)
Note that a Be value more/less than 0.5 shows that the contribution of HTI to the total entropy generation is higher/lower than that of FFI. The limiting value of Be = 1 shows that the only active entropy generation mechanism is HTI while Be = 0 represents no HTI contribution.
The dimensionless form of entropy generation rate, Ns, is defined as
Ns=(HΩ)2S∙genk
(2.110)
one finds that
Ns=(∂θ∂x)2+(∂θ∂y)2Ω2(Ω−1+θ)2+Ge ϕRaΩ2(Ω−1+θ)
(2.111)
where the dimensionless temperature difference is defined as
Ω=Th−TcTc
(2.112)
The dimensionless viscous dissipation function, addressed in Eq.
(2.111), takes the following form:
ϕ=2(∂u∂x)2+2(∂v∂x)2+(∂u∂x+∂v∂x)2
(2.113)
One easily verifies that, as both Ω and
θ can put on values smaller than unity, one cannot neglect any of them in favor of the other as noted by Hooman and Ejlali
[19] for a forced convection problem. Keep in mind that Ω can be
O(
1) for special cases. Hence, one should be very careful when one simply neglects either Ω
−1 or
θ in the denominator of Eq.
(2.111). Here, Ge is the Gebhart number which is defined as
Ge=gβHCp
(2.114)
and throughout this work Ge = 10
−5 to be a real choice for most of engineering applications. In the light of Nield
[20] one knows that the proper dimensionless number to show the effects of viscous dissipation in a free convection problem in a cavity, filled with or without a porous insert, is the Gebhart number which, interestingly, does not contain viscosity; see also Hooman et al.
[21]. Unlike the previous findings of Dagtekin et al.
[22], proper scaling shows that with a fixed Ge value FFI decreases with Ra. Interestingly, both HTI and Ns increase with Ra and this will be elaborated on in the forthcoming discussion. It should also be noted that there are certain cases where viscous dissipation effects are not important in the thermal energy equation but can be significant when it comes to study second law aspects of a convection problem as outlined by Bejan
[18]. The local and average values of Be are found to convey little information as they are very close to unity; hence, we did not present any graph or contour for Be. Average Ns is denoted by [Ns], where the angle brackets show an average taken over the area, as
[Ns]=∫ANsAdA
(2.115)
Selecting the fluid, trapped between the heated plate and the cavity, as the thermodynamic system, one observes that the amount of heat entered through the heated plate is equal to the one transferred to the surroundings via the isothermal walls. Moreover, one notes that the total volumetric entropy generation rate is obtainable as
[S∙gen]=q′′H(1Tc−1TH)
(2.116)
where, in terms of Nu, it reads
[S∙gen]=4NukH2Ω21+Ω
(2.117)
Applying perturbation techniques for small values of Ω, say Ω<<1, one has
[S∙gen]≈4 Nu Ω2(1−Ω) kH2
(2.118)
The dimensionless entropy generation number can be obtained as
[Ns]=4 Nu1+Ω
(2.119)
The average Nusselt numbers are defined as follows:
Nuave=kn fkf∫0114(∂T∂x∣∣X=0+∂T∂x∣∣X=1)dy.
(2.120)