Finite volume formulation ofthermal lattice Boltzmann method

Ahad ZarghamiDepartment of Engineering, Science and Research Branch,

Islamic Azad University, Fars, Iran

Stefano UbertiniDEIM – Industrial Engineering School, University of Tuscia,

Viterbo, Italy, and

Sauro SucciIstituto per le Applicazioni del Calcolo, CNR, Rome, Italy

Abstract

Purpose – The main purpose of this paper is to develop a novel thermal lattice Boltzmann method(LBM) based on finite volume (FV) formulation. Validation of the suggested formulation is performedby simulating plane Poiseuille, backward-facing step and flow over circular cylinder.

Design/methodology/approach – For this purpose, a cell-centered scheme is used to discretize theconvection operator and the double distribution function model is applied to describe the temperaturefield. To enhance stability, weighting factors are defined as flux correctors on a D2Q9 lattice.

Findings – The introduction of pressure-temperature-dependent flux-control coefficients in thestreaming operator, in conjunction with suitable boundary conditions, is shown to result in enhancednumerical stability of the scheme. In all cases, excellent agreement with the existing literature is foundand shows that the presented method is a promising scheme in simulating thermo-hydrodynamicphenomena.

Originality/value – A stable and accurate FV formulation of the thermal DDF-LBM is presented.

Keywords Backward-facing step, Double distribution function approach, Finite volume,Flow over a circular cylinder, Poiseuille flow, Thermal lattice Boltzmann models

Paper type Research paper

1. IntroductionThe lattice Boltzmann method (LBM) has attracted significant attention as analternative technique to solve complex fluid flow problems and has been successfullyapplied to several isothermal hydrodynamic problems (Qian et al., 1992; Benzi et al.,1992; Chen and Doolen, 1998; Yu et al., 2003; Tosi et al., 2007; Zu et al., 2008;Chiappini et al., 2010). The major advantages of the LBM are that it is explicit, easy toimplement and natural to parallelize (Huang et al., 2005; De et al., 2009). To model fluidflows with heat transfer, three classes of thermal lattice Boltzmann models (TLBM)have been proposed in the literature, i.e. the multi-speed approach, the passive-scalarmodel and the double distribution function (DDF) approach.

The multi-speed approach is an extension of the LBM for isothermal flows, in whichonly single-particle distribution functions are employed and a higher order velocitymoment of these distribution functions is used to describe the temperature field.The main disadvantage of this method is that it usually suffers from serious numerical

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Received 1 November 2011Revised 16 March 201222 May 201218 July 20127 August 2012Accepted 8 August 2012

International Journal of NumericalMethods for Heat & Fluid FlowVol. 24 No. 2, 2014pp. 270-289q Emerald Group Publishing Limited0961-5539DOI 10.1108/HFF-11-2011-0234

HFF24,2

270

instability and is only suitable for problems with a restricted temperature range(Shi et al., 2004). In addition, the multi-speed models using a single relaxation time arelimited to problems with a fixed Prandtl number, which has only few engineeringapplications. Although some methods have been proposed to cure these problems, thedrawbacks of the multi-speed models still limit their practical usage.

In the passive scalar TLBM (He et al., 1998; Benzi et al., 1994; Peng et al., 2003a, b;Shi et al., 2004); an additional set of distribution functions is used to solve thetemperature distribution. Based on this approach, the evolution scheme proposed byHe et al. (1998) shows a better numerical stability than the multi-speed approach, andthe viscous heat dissipation and the compression work can be solved implicitly(He et al., 1998). Peng et al. (2003a, b) proposed a simplified version which neglectscompression work and viscous heat dissipation. Introducing a forcing function,Guo et al. (2010) proposed a thermal LB equation with viscous heat dissipation in theincompressible limit and they successfully simulated natural convection in a lid drivencavity. The major advantage of the passive scalar model over the multi-speed approachis the higher numerical stability, which could be further enhanced by adopting thehybrid finite-difference approach proposed in Lallemand and Luo (2003a, b).

The DDF model (He et al., 1998) employs an additional distribution function todescribe the temperature field evolution and shows great improvement in the stabilityover the previous models (Van der Sman, 1997; Palmer and Rector, 2000). It is based onthe recent discovery that the lattice Boltzmann isothermal models can actually bederived directly by discretizing the continuous Boltzmann equation in temporal,spatial, and velocity spaces. Following the same procedure, the DDF model is derivedby discretizing the continuous evolution equation for the internal energy distribution.The model has been successfully used to solve some thermal problems in twodimensions (Onishi et al., 2001; Shu et al., 2002; Peng et al., 2003a, b; D’Orazio et al.,2004; Li et al., 2009; Nwatchok et al., 2010). On the other hand, since the pressure comesfrom the second moment of the density distribution, and is independent of the internalenergy distribution, the method actually solves the passive advection problem.

The primary motivation for the current work is to develop a stable and accuratefinite volume (FV) formulation of the thermal DDF-LBM. The introduction of upwindweighting factors allows overcoming instability and accelerating the convergenceprocess. The method is tested against typical benchmark thermal problems, i.e. thermalplane Poiseuille, thermal backward-facing step and laminar thermal flow over circularcylinder.

The outline of this paper is as follows. In Section 2 we give a brief description of thethermal lattice Boltzmann equation (LBE). Section 3 describes the details of theproposed finite-volume scheme. Thermal boundary conditions are presented in Section4. In Section 5, the method is validated against typical thermal flow problems and inSection 6 some conclusions are drawn.

2. Discrete thermal LBEUpon ignoring the viscous dissipation term, the thermal LBE with theBhatnagar-Gross-Krook (BGK) collision model can be expressed as follows:

›f

›tþ kvi ·7f ¼ 2

1

tf

ð f 2 f eqÞ ð1Þ

Finite volumeformulation

271

›g

›tþ kvi ·7g ¼ 2

1

tgðg 2 g eqÞ ð2Þ

where f and g are the particle density and energy distribution functions along theparticle velocity direction kvi , respectively. tf and tg are the density and thermalrelaxation times to local equilibrium, given by f eq and g eq. The equilibrium densitydistribution functions, f eq

i , for the D2Q9 lattice are given by:

f eqi ðkx; tÞ ¼ wir½c1 þ c2ðkvi · kuÞ þ c3ðkvi · kuÞ2 þ c4ðku · kuÞ�i¼1,9 ð3Þ

where c1 ¼ 1; c2 ¼ 1=2c2s ; c3 ¼ 1=2c4

s ; c4 ¼ 21=2c2s , being c2

s the lattice soundspeed, wi are the weighting factors and equals 4/9 for i ¼ 0, 1/9 for i ¼ 1 , 4 and 1/36for i ¼ 5 , 8 and ku is the fluid speed. The energy equilibrium distribution functionsare given by He et al. (1998) and Peng et al. (2003a, b):

geq0 ¼ 2

2r1

3

ku 2

c 2ð4Þ

geq1,4 ¼

r1

9

3

2þ

3

2c2 kvi · ku þ9

2c4ðkvi · kuÞ2 2

3

2c 2~u 2

� �ð5Þ

geq5,8 ¼

r1

363 þ

6

c2 kvi · ku þ9

2c 4ðkvi · kuÞ2 2

3

2c 2 ku2

� �ð6Þ

where 1 ¼ RT is the internal energy, being R the gas constant, and c 2 ¼ 3R �T. Thecorresponding kinematic viscosity and thermal diffusivity are calculated as follows:

n ¼ c2stf ð7Þ

a ¼ 2c2stg ð8Þ

where the lattice speed of sound reads as cs ¼ c=ffiffiffi3

p(D’Orazio and Succi, 2004). The discrete

velocities for a D2Q9 lattice are given by kv0 ¼ 0 and kvi ¼ liðcosui; sinuiÞ with li ¼ 1,ui ¼ (i 2 1)p/2 for i ¼ 1 , 4 and li ¼

ffiffiffi2

p; ui ¼ ði 2 5Þp=2 þ p=4 for i ¼ 5 , 8. The

macroscopic density r, velocity ku and internal energy of the fluid 1, are determinedby r ¼

Pif i , rku ¼

Pif ikvi and 1 ¼

Pigi , respectively, and the macroscopic pressure is

given by the equation of state of an ideal gas, p ¼ c2sr (Ubertini and Succi, 2005).

The simulation of a thermal flow through the LBM requires the definition of thelattice variables. Two constraints determine this choice: first, the simulation should beequivalent, in a well defined sense, to the physical system; second, the parametersshould be fine-tuned in order to reach the required accuracy (i.e. the grid should besufficiently resolved, the discrete time step sufficiently small).

In general, solving the LBE consists of two steps. A physical system is firstconverted into a dimensionless one, which is independent of the original physicalscales, but also independent of the simulation parameters. In a second step, thedimensionless system is converted into a discrete simulation. The correspondencebetween these three systems, the physical one (P), the dimensionless one (D), andthe discrete one (LB), is made through dimensionless, or scale-independent numbers.The dynamics of incompressible thermal flows, for example, depends only on twodimensionless parameters, which are the Reynolds number (Re) and Prandtl number (Pr).

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272

Thus, the three systems (P), (D), and (LB) are defined so as to have the same Reynoldsand Prandtl numbers. The transition from (P) to (D) is made through the choice of acharacteristic length scale l0 and time scale t0, and the transition from (D) to (LB)through the choice of discrete space (Dx) and time (Dt) steps. It is important to note thatthe discrete variables Dx and Dt directly determine the accuracy and the stability of asimulation. They do not depend on the scales of the considered physical system, anddefinitely not on the arbitrary choice of physical units.

3. FV formulationThe LBM as fluid solvers have revealed to be a very efficient alternative to Navier-Stokesto solve complex fluid problems. In the original LBM the representative particles evolveon a regular Cartesian grid according to a simple streaming-and-collide resolution, whichmakes the method very efficient from a computational point of view. However, if thestructured form of the lattice represents the main advantage of the LBM, it is also itsmain limitation in practical engineering problems, especially when there is a need forhigh resolutions near the body or the walls. Consequently, much research has beendevoted to remove or alleviate this weakness, as curved boundary treatments (Mei et al.,1999) and grid refinement techniques (Filippova and Hanel, 1998) have been proposedsince the beginning. A particularly interesting option is represented by finite-volumeformulations, which loose the simplicity of the streaming-and-collide procedure of thestandard LBM, but allow retaining the other typical advantages with respect toNavier-Stokes (Ubertini et al., 2003, 2006; Rossi et al., 2005). There is no need of solvingany (expensive) Poisson problem and the streaming operator, even if no longer exact, isnevertheless still linear, which is to say that it does not depend on the solution itself.

In this paper a cell-centered FV formulation (Figure 1) of the thermal LBE on arectangular mesh system is employed (Zarghami et al., 2011). The integration of thefirst term of the discrete LBE is approximated as follows:

Zabcd

›wi

›tdA <

›wi

›t

� �I ; J

· AI ; J ð9Þ

where wi ¼ fi, gi and AI,J is the area of abcd. The integration of the collision term (righthand side of equations (1) and (2)) is performed through the following formulation(Xi et al., 1999):

2

Zabcd

1

twi2w

eqi

� �dA¼2

AI ; J

t

1

4wne

i

� �I ; J

�

þ1

8wne

i

� �Iþ1; J

þ wnei

� �I ; Jþ1

þ wnei

� �I21; J

þ wnei

� �I ; J21

n o

þ1

16wne

i

� �Iþ1; J21

þ wnei

� �Iþ1; Jþ1

þ wnei

� �I21; Jþ1

þ wnei

� �I21; J21

n o�

ð10Þ

where wnei ¼ wi 2 w

eqi is the non-equilibrium component of the distribution function.

Using a generic parameter, u, to represents pressure, p, for the density distributionfunction and temperature, T, for the energy distribution functions, the followingweighting factors are defined:

Finite volumeformulation

273

zab ¼DuabPuhorizontal

; zbc ¼DubcPuvertical

; zcd ¼DucdPuhorizontal

; zda ¼DudaPuvertical

ð11Þ

where Xuhorizontal ¼

XðuIþ1; J þ 2uI ; J þ uI21; J Þ;

Xuvertical ¼

XðuI ; Jþ1 þ 2uI ; J þ uI ; J21Þ

and

Duab ¼ uIþ1; J 2 uI ; J ;Dubc ¼ uI ; Jþ1 2 uI ; J ;Ducd ¼ uI ; J 2 uI21; J ;Duda

¼ uI ; J 2 uI ; J21:

As a result, after applying an upwind scheme, the convective fluxes may be writtenas follows (Stiebler et al., 2006):

Si ¼

Zabcd

vi ·7wi dA ¼

Zabcd

vix›wi

›xþ viy

›wi

›y

� dA

< ~vi · Nabðzab½wi�I ; J þ ð1 2 zabÞ½wi�Iþ1; J Þ þ ~vi · Nbcðzbc½wi�I ; J þ ð1 2 zbcÞ½wi�I ; Jþ1Þ

þ ~vi · Ncdðzcd½wi�I ; J þ ð1 2 zcdÞ½wi�I21; J Þ þ ~vi · Ndaðzda½wi�I ; J þ ð1 2 zdaÞ½wi�I ; J21Þ

ð12Þ

Figure 1.Schematic of the FVdiscretization withcell-centered lattice

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274

where Nk ¼ ðDyki 2 DxkjÞk is the outward unit vector normal to the edge and k ¼ ab, bc,cd and da. These coefficients are introduced to enhance transport downhill thepressure/temperature gradient and reduce it uphill, thus improving stability andaccuracy and reducing the iteration steps (Zarghami et al., 2011).

In flux modeling, especially at high Reynolds numbers or in the presence of stronggradients, the addition of artificial dissipation is inevitable to perform a stablesimulation. Therefore, in order to damp out spurious oscillations, a fourth-orderartificial dissipation is employed here and a modified, fifth order Runge-Kuttatime differencing scheme is used to advance the computations in time (Zarghami et al.,2011, 2012).

4. Boundary conditionsHydrodynamic boundary conditions are performed by introducing D2Q9 lattices atthe edge of each boundary cell. Then, boundary nodes are treated like internalnodes, except that the fluxes over the boundary edges have to be evaluated. Theimplementation of velocity boundary conditions are presented in a previous work(Zarghami et al., 2011).

The actual dynamics of thermal flows in the LBM is highly dependent on the thermalboundary conditions. He et al. (1998) extended the bounce-back rule of non-equilibriumdistributions proposed by Zou and He (1997) to impose thermo-hydrodynamicboundary conditions. Similarly to the counter-slip approach of Inamuro et al. (1995) andD’Orazio and Succi (2004) proposed that the incoming unknown thermal populationscan be assumed to be equilibrium distributions with a counter-slip thermal energy. Thelatter method is easy to implement and allows imposing exactly prescribed velocity andtemperature or heat flux at the wall.

Similarly to Tang (Tang et al., 2005), we decompose the unknown energydistribution at a boundary node into its equilibrium and non-equilibrium parts. Thenon-equilibrium part is extrapolated from the non-equilibrium distribution at theneighboring fluid node and the first-order extrapolation is proved to be of second-orderaccuracy for flat boundaries. So, the gi at the node of the boundary cell can bedetermined as follows:

giðxbÞ ¼ geqi ðxb; rb; 1bÞ þ giðxnÞ2 geq

i ðxn; rn; 1nÞ� �

ð13Þ

where subscripts b and n denote the boundary cell and its nearest neighboring cell,respectively. The unknown rb in boundary cells is assumed to be equal to rn. For theDirichlet type condition, the given temperature by energy distribution function isapplied directly on the boundary. Meanwhile, the Neumann type condition istransferred to the Dirichlet type condition through the conventional second-order finitedifference approximation in order to obtain the temperature at the boundary (Shu et al.,2002). When the temperature gradient is given, the temperature at the boundary can becalculated as follows (Mohammad et al., 2009):

›T

›y

x;1

¼23Tx;1 2 4Tx;2 2 Tx;3

2Dyð14Þ

The above equation yields the corresponding Dirichlet type boundary condition forboth the adiabatic and constant heat flux boundary conditions.

Finite volumeformulation

275

5. Results and discussionTo validate the present scheme, we have applied it to a plane Poiseuille flow, a thermalbackward-facing step thermal flow and a thermal flow over circular cylinder.Quantitative comparisons between the simulation results and literature data are alsopresented. Computations were carried out to investigate convergence and accuracy ofthe method.

5.1 Thermal Poiseuille flowAs a first test case for open flows, we consider a two-dimensional plane Poiseuille flowwith different thermal boundary conditions. The channel aspect ratio is defined asL/H ¼ 10 where L and H are the channel height and length, respectively. Uniformvelocity, Uin, and temperature, Tin, profiles are given at the inlet boundary. Therelaxation parameters are determined by fixing the Prandtl number, Pr ¼ n/a ¼ 0.7.Numerical simulations have been performed in two different test cases that differ fromeach other for the boundary conditions at the walls:

. Case 1. The temperature at the lower and upper walls is uniform and constantlyequal to Tlow ¼ 1 and Tup ¼ 3, respectively.

. Case 2. The same uniform heat flux is applied to the lower and the upper walls,qlow ¼ qup ¼ 0.01.

The inlet temperature has been selected Tin ¼ 5 for case 1 and Tin ¼ 1 for case 2.Figure 2 shows the local Nusselt number distribution at the lower wall as a function

of the Graetz number for the different boundary conditions. The Graetz and the Nusseltnumbers are defined as follows:

Gz ¼ ðx=H ÞRePr ð15Þ

Nux ¼ Dhqw;x=½kðTw;x 2 Tb;xÞ� ð16Þ

Figure 2.Local Nusselt numberdistribution at lower planefor cases 1 and 2

0

35

70

0.001 0.01 0.1

Notes: Logarithmic scale has been used for the horizontal axes;numerical results are compared to those of Tang

1

Gz

Nu

Case 1, FV-LBM

Case 1, Tang et al

Case 2, FV-LBM

Case 2, Tang et al

Source: Tang et al. (2005)

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276

where

Dh is hydraulic diameter,k is thermal conductivity coefficient; qw, x ¼ k(›T/›y)w is the wall local heat

flux; and

Tb ¼

Z H

0

Tr · udy=

Z H

0

r · udy

is the bulk temperature.In Figure 2 present results are compared with the numerical data produced by

Tang et al. (2005), who used a standard LBM scheme to simulate the thermal Poiseuilleflow with a 1,001 £ 101 lattice corresponding to an aspect ratio of L/H ¼ 10, and agood agreement is observed.

For a thermally fully developed flow (i.e. for large inverse Graetz numbers) thepredicted Nusselt numbers at the lower plane compared with the analytical solutions(Shah and London, 1978) are shown in Table I. The relative error defined as follows isless than 2.5 percent for both performed test cases:

�Nuanalytical 2 Nunumerical

�=Nuanalytical

£ 100 ð17Þ

In order to analyze the effect of grid resolution on the scheme accuracy, the simulationof the test case 2 have been performed with four different grids, 501 £ 51, 751 £ 51,1,001 £ 101 and 1,251 £ 126. The results summarized in Table II clearly show that themaximum Nusselt relative error tends to zero by increasing grid resolution and isconsistent with standard LBM-DDF simulations available in literature (D’Orazio et al.,2004; Tang et al., 2005; Shi et al., 2004; Li et al., 2009; Nwatchok et al., 2010; Guo et al.,2010). Given the same lattice (i.e. 1,001 £ 101) and aspect ratio, our numerical schemewith enhanced stability gives a maximum Nusselt relative error of 1.72 percent, close tothe 1.82 percent error measured by Tang et al. (2005).

5.2 Thermal backward facing step flowIn this section the proposed numerical scheme is tested against a laminar thermal flowover a backward facing step, which is one of the most fundamental forms of flowseparation in fluid mechanics and has been extensively studied both numerically andexperimentally (Armaly et al., 1983; Kim and Moin, 1985; Barton, 1997; Erturk, 2008).

Boundary conditions Nuanalytical Nunumerical Relative error (%)

Case 1 4.0 3.94 1.50Case 2 8.24 8.41 2.06

Table I.Comparison of the

Nusselt numbers in thethermal fully developedregion for different BCs

Mesh size 501 £ 51 751 £ 75 1,001 £ 101 1,251 £ 126

Relative error (%) 5.12 2.06 1.75 1.59

Table II.Relative Nusselt error vs

mesh size for case 2 ofthermal Poiseuille flow

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277

The channel configuration is shown in Figure 3(a). The main geometrical parametersare the channel height H and the step height, h. The expansion ratio ER is given byH/h. The entrance channel is 10h, thus preventing downstream effects of the inletboundary conditions. The outlet channel is truncated at 50h in order to avoid anyalteration of the flow region behind the step.

The flow has been numerically simulated for a Reynolds number ranging between100 and 600, being the Reynolds number defined as Re ¼ 4Umax(H 2 h)/3n, whereUmax is the maximum speed at inlet. The 1,800 £ 101 non-uniform rectangular meshshown in Figure 3(b) has been employed for the simulations.

The heat transfer in a backward-facing step flow is mainly dependent on Re, ER andPr. The influence of these parameters on the heat transfer has been investigated andhas been tested in the two cases featuring thermal boundary condition:

. Case 1. The step-side lower wall downstream of the step is maintained at aconstant temperature higher than the inlet temperature; the other walls are setadiabatic.

. Case 2. All the walls are at a constant temperature lower than the inlettemperature.

Figure 4(a) shows the influence of the Reynolds number on the local Nusselt numberdistribution on the bottom heat transfer surface with ER ¼ 1.5 and Pr ¼ 0.7 for case 1.Here, the abscissa is the distance from the step normalized by the step height.

Results depict that, as the x/h ratio increases the maximum local Nusselt numbermoves towards the end of the channel and increases in value with increasing Reynoldsnumber. The downstream shift of this peak value is most likely related to themovement of the main recirculation length, X1. To better examine this point, the localNusselt number is re-plotted in Figure 4(b) in function of x/X1 for different Reynoldsnumbers.

This result reveals that the peak value of Nu is always located around, but notexactly on the flow reattachment point and the relationship between the Nusseltpeak value and the reattachment length, X1, considerably depends on the Reynolds

Figure 3.(a) Schematic and (b)non-uniform mesh ofbackward-facing step flow

(a)

(b)

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278

number. Moreover, Figure 4(a) shows a good agreement between our numericalresults and those by Kondoh et al. (1993), obtained through a standard CFDsimulation. In the latter, a regular grid Nx £ Ny ¼ 120 £ 60, with non-uniform linespacing for main channel, is employed, the convection terms are discretized througha third-order upwind scheme and the other spatial derivatives are evaluated bycentral differences. Finally, the Crank-Nicolson scheme has been used for the timeintegration.

Figure 5(a) shows the effect of the channel expansion ratio on the local Nusseltnumber distribution for case 1 and Re ¼ 105 and Pr ¼ 0.7. As the expansion ratio

Figure 4.Case 1: (a) comparison of

the Nusselt numberbetween Kondoh

(Kondoh et al., 1993) andthe present scheme for

ER ¼ 1.5 and Pr ¼ 0.7 and(b) local Nusselt

distribution vs x/X1

0

0.5

1

1.5

2

2.5

0 10 20

(a)

(b)

30

x/h

Nu

FV-LBM (Re = 50) Kondoh et al (Re = 50)FV-LBM (Re = 100) Kondoh et al (Re = 100)FV-LBM (Re = 200) Kondoh et al (Re = 200)FV-LBM (Re = 500) Kondoh et al (Re = 500)

0

2.5

0 1 2

Note: The values of X1/h for Re = 50, 100, 200 and 500 are equal to1.38, 1.54, 1.75 and 2.18 respectively

x/X1

Nu

Re = 50Re = 100Re = 200Re = 500peak value of Nu (Re = 50)peak value of Nu (Re = 100)peak value of Nu (Re = 200)peak value of Nu (Re = 500)

Finite volumeformulation

279

increases, the Nu peak value moves upstream and grows consistently. This movementof the Nu peak location is considered again to be due to the movement of the flowreattachment point, since the latter moves upstream with increasing expansion ratios. Ifthe abscissa is normalized by the main recirculation length, X1, as shown in Figure 5(b),it is understood that the Nu peak value occurs slightly downstream the flowreattachment point and by increasing the expansion ratio, value moves closer to it.

Figure 6 shows the influence of the Re number on the local Nusselt distribution forCase 2 and Pr ¼ 0.7. A pattern similar to Figure 4(a) is observed and the differencesnear the step are obviously due to the different thermal boundary conditions.

Figure 5.Case 1: (a) effect of channelER on local Nusseltdistribution for Re ¼ 105and Pr ¼ 0.7 and (b) localNusselt distribution vsx/X1

0

0.5

1

1.5

2

2.5

0 10 20

(a)

(b)

Note: The values of X1/h for ER = 1.25, 1.5 and 2 at Re = 105 are equal to0.95, 1.39 and 2.05 respectively

30

x/h

Nu

ER = 2

ER = 1.5

ER = 1.25

0

2.5

0 1 2x/X1

Nu

ER = 2ER = 1.5ER = 1.25Peak value of Nu (ER = 2)Peak value of Nu (ER = 1.5)Peak value of Nu (ER =1.25)

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280

Figure 7 shows the effect of the expansion ratio on the local Nusselt distribution atRe ¼ 105 and Pr ¼ 0.7. By increasing the expansion ratio, the Nu peak value movesupstream and grows consistently, as a consequence of the movement of the flowreattachment point. It is naturally understood that the compression of the thermalboundary layer by the recirculation flow becomes stronger as the expansion ratiogets higher. Again, a good agreement with literature data is observed (Chen et al.,2006).

The effect of the Prandtl number on the heat transfer in a separating andreattaching flow is shown in Figure 8, which reports the local Nusselt distribution on

Figure 6.Case 2: comparison of theNusselt number between

Chen (Chen et al., 2006)and the present scheme for

ER ¼ 2 and Pr ¼ 0.7

0

1

2

0 4 8 12 16

x/h

Nu

FV-LBM (Re = 65) Chen et al (Re = 65)FV-LBM (Re = 135) Chen at al (Re = 135)

FV-LBM (Re = 205) FV-LBM (Re = 275)

Figure 7.Case 2: effect of channel

ER on local Nusseltdistribution for Re ¼ 105

and Pr ¼ 0.7

0

0.4

0.8

0 5 10 15 20

x/h

Nu

FV-LBM (ER = 2) Chen et al (ER = 2)

FV-LBM (ER = 1.5) Chen et al (ER = 1.5)

FV-LBM (ER = 1.25) Chen et al (ER = 1.25)

Note: Results are compared with Chen

Finite volumeformulation

281

the lower wall at different Prandtl numbers. The Re and the ER are fixed at 100 and 2,respectively. The notable heat transfer enhancement by the recirculation flow isidentified around the flow attachment location. Local Nusselt continues to increasewith the Prandtl number in the whole region and the heat transfer enhancement aroundthe flow reattachment location becomes more and more evident.

In the end of this section we investigate the effect of temperature-based weightingfactors in simulating the thermal backward facing step flow. For this reason, thetemperature convergence criterion is set to:

Re sth ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPI ; J Tnþ1

I ; J 2 TnI ; J

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPI ; J Tn

I ; J

nþ1r ð18Þ

Figure 9 compares the convergence to steady-state solution of the present method withthat of a flux averaging scheme in log-log scale, where power laws appear as straightlines. The residual decay slope of proposed scheme line is clearly higher than the flux

Figure 8.Case 2: influence of thePrandtl number on localNusselt distribution forRe ¼ 100 and ER ¼ 2

0

0.25

0.5

0.75

1

0 5 10 15 20

x/h

Nu

Pr = 1

Pr = 0.7

Pr = 0.4

Figure 9.Case 2: comparing theconvergence of the presentscheme with fluxaveraging schemes in thelog-log scale at(a) Re ¼ 100 and(b) Re ¼ 200

Re = 100

(a) (b)

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1,000 10,000 100,000

Iteration

Res

idu

alth

Flux Averaging Scheme

Scheme with Weighting Factors

Re = 200

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1,000 10,000 100,000

Iteration

Res

idu

alth

Flux Averaging SchemeScheme with Weighting Factors

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282

averaging scheme. Note that the flux averaging scheme is obtained after applying theGreen’s theorem to the second term of equation (1) or (2) as follows:Z

abcdkvi ·7wi dA ¼

Zabcd

›ðkvix ·wiÞ

›xþ

›ðwi · kviyÞ

›y

� dxdy ¼

Iaround I ; J

ðkvix wi dy 2 kviy wi dxÞ

<½wi�I ; J þ ½wi�Iþ1; J

2kvi · Nab þ

½wi�I21; J þ ½wi�I ; J

2kvi · Nbc

�

þ½wi�I ; J þ ½wi�I ; Jþ1

2kvi · Ncd þ

½wi�I ; J21 þ ½wi�I ; J

2kvi · Nda

ð19Þ

As it is seen, the use of temperature based factors improves accuracy and stability,thus reducing the number of iterations to reaches convergence. Furthermore, it isimportant to note that the flux averaging scheme becomes unstable for Reynoldshigher than 300.

5.3 Laminar thermal flow over circular cylinderIn this section the numerical method is tested against a confined flow past a heatedcircular cylinder at Re ¼ 40, where Re ¼ Ud/n, d is cylinder diameter and U is theaverage velocity at inlet. Figure 10 shows a schematic of the geometry and the relevantdimensions considered in this test case. A 256 £ 64 non-uniform rectangular mesh hasbeen employed for the simulations. On either side of the cylinder, two adiabatic channelwalls are placed at a distance H/2 from the cylinder axis. Tests have been performedwith two different blockage ratios, BR ¼ d/H ¼ 0.2 and 0.5. Fluid enters the duct witha fully developed laminar velocity profile at a temperature T1. The cylinder ismaintained at a constant temperature Tc . T1, being DT ¼ Tc 2 T1 ¼ 40.

The dimensionless velocity profiles upstream of the cylinder at different distancesXus are shown in Figure 11 for BR ¼ 0.2 and 0.5. This figure shows that the role of thecylinder on the hydrodynamic behavior of the fluid becomes apparent at closer sectionsat the front side of the cylinder especially for BR ¼ 0.5. This indicates that, due to the latehydrodynamic response of the flow domain at higher BR, the fluid decelerates morerapidly to zero toward the stagnation point (Xus ¼ 0 ! Xus ¼ 0.125d ) resulting in anaugmented momentum velocity in the neighborhood of the cylinder. Furthermore, onecan see that the order of the upstream velocity profiles reverses at the intersection points.

Figure 12 shows the dimensionless downstream velocity profiles at differentdistances Xds for BR ¼ 0.2 and 0.5. Note that, the recirculation length becomeshorter with higher BRs (He and Doolen, 1997; Buyruk et al., 1998; Sen et al., 2009).

Figure 10.Flow configuration for

simulation of flow past acylinder placed

symmetrically in a planarchannel

dy

x

Adiabatic

Adiabatic

H

24d12d

xdsxus

q

Finite volumeformulation

283

Regardless of the level of BR, a backflow (u/U , 0) is clearly observable downstreamthe cylinder and the backflow velocity increases with the blockage ratio. Regarding themomentum transfer around the separation point, it can be said that the growth of thebackflow values is highly characterized by the comprehensively augmented throatvelocity values of Figure 12 at higher BR. In other words, a higher BR not onlyaugments the throat momentum transfer but also moves the separation pointdownstream (Buyruk et al., 1998; Sen et al., 2009).

Figure 13 shows the dimensionless temperature profiles, u ¼ (T 2 Tc)/(T1 2 Tc),upstream of the cylinder for BR ¼ 0.2 and 0.5. It can be seen that significant decreasesin temperature are appraised while approaching the stagnation point (Xus ¼ y/d ¼ 0).Furthermore, it can be observed that similarly to the hydrodynamic boundary layer

Figure 13.Dimensionlesstemperature profiles at theupstream of the cylinderfor the (a) BR ¼ 0.2 and (b)BR ¼ 0.5

BR = 0.2

–0.6

–0.3

0

0.3

0.6

0 0.25 0.5 0.75 1

Teta

y/d

X_us = 0X_us = 0.025 dX_us = 0.125 d

BR = 0.5

–0.6

–0.3

0

0.3

0.6

(a) (b)

0 0.25 0.5 0.75 1

Teta

y/d

X_us = 0X_us = 0.025 dX_us = 0.125 d

Figure 12.Dimensionless velocityprofiles downstream thecylinder for (a) BR ¼ 0.2and (b) BR ¼ 0.5

BR = 0.2

–0.5

–0.25

0

0.25

0.5

–0.05 –0.025 0 0.025

(a) (b)

0.05

u/U

y/d

X_ds = 0x_ds = 0.025 dX_ds = 0.125 d

BR = 0.5

–0.4

–0.2

0

0.2

0.4

–0.1 –0.05 0 0.05

u/U

y/d

X_ds = 0X_ds = 0.025 dX_ds = 0.125 d

Figure 11.Dimensionless velocityprofiles upstream of thecylinder for (a) BR ¼ 0.2and (b) BR ¼ 0.5

BR = 0.2

–0.625

0

0.625

0 0.5 1 1.5

u/U

y/d

X_us = 0X_us = 0.025 dX_us = 0.125 d

BR = 0.5

–0.625

0

0.625

(a) (b)

0 0.5 1 1.5 2

u/U

y/d

X_us = 0X_us = 0.025 dX_us = 0.125 d

HFF24,2

284

development, the thermal boundary layer also attains lower thickness values at higherblockage. The narrowing nature of the dimensionless temperature range highlightsthat the impact of blockage on heat transfer rates reduces in the neighborhood of thecylinder, especially at the stagnation point.

The dimensionless downstream temperature variations are shown in Figure 14,which demonstrate that the higher is the BR, the hotter becomes the fluid in thecylinder neighborhood. This determination puts forward that the momentum in thevicinity of the stagnation point has a higher potential to increase heat transfer rates, ifcompared to the downstream vortex system.

Figure 15 shows the variations of the Nusselt number, defined as Nu ¼ hud/k, on thecylinder surface, being hu ¼ 2 (k(›T/›n))/(Tc 2 Tbulk) the local convective heattransfer coefficient. The graph clearly shows that the Nusselt number increases withthe BR as also reported in Khan et al. (2004) and Dhiman et al. (2005) and decreases inthe angular direction approaching the stagnation point.

As per the computational efficiency, a serial computation on an XEON Quad-Core at2.00 GHz, takes approximately 5 microseconds to update a single volume, and thewhole simulation takes about 2 hours to reach steady state conditions after 70,000 timesteps, on a grid with 256 £ 64 volumes, which is roughly one order of magnitudecomputationally more expensive than standard LB. This is mostly due to the need ofkeeping the time-step at about one tenth of the standard LB counterpart, on account ofstability reasons. This is not surprising, since any generic grid is bound to surrender

Figure 15.Nusselt number

distribution on thecylinder surface for

BR ¼ 0.2 and 0.5

0

2

4

6

8

0 45 90θ

135 180

Nu

BR = 0.2

BR = 0.5

Figure 14.Dimensionless

temperature profiles at thedownstream of thecylinder for the (a)

BR ¼ 0.2 and (b) BR ¼ 0.5

BR = 0.2

–0.6

–0.3

0

0.3

0.6

0 0.1 0.2 0.3 0.4 0.5

θ θ

y/d

X_ds = 0X_ds = 0.025 dX_ds = 0.125 d

BR = 0.5

(a) (b)

–0.6

–0.3

0

0.3

0.6

0 0.1 0.2 0.3 0.4 0.5

y/d

X_ds = 0X_ds = 0.025 dX_ds = 0.125 d

Finite volumeformulation

285

the exact nature of the streaming operator, which is peculiar to LB and allows it tomarch in time at a unit kinetic CFL (Courant-Friedrichs-Lewy) numbers, vdt/dx ¼ 1.

6. ConclusionIn this paper, a cell-centered finite-volume formulation of a TLBM with DDF approachis presented. The introduction of pressure-temperature dependent flux-controlcoefficients in the streaming operator, in conjunction with suitable boundaryconditions, is shown to result in enhanced numerical stability with respect to standardschemes. Furthermore, we added an artificial dissipation to damp out spuriousoscillations, which is found to increase stability and accuracy of the method. Also, thedecomposition of energy distribution population at the boundary cells into equilibriumand non-equilibrium parts was applied. The method is numerically demonstrated forthe case of thermal Poiseuille flow, backward facing step and flow over a circularcylinder. In all cases, the results agree well with the available numerical results andshow that the present method represents a promising scheme in the simulation ofthermo-hydrodynamics problems, also in conjunction with multiscale applications(Succi et al., 2001).

The computational times clearly show that the present method is about one order ofmagnitude computationally more expensive than the traditional LBM. Even grantingthat this gap would shrink as the geometrical complexity increases, non-uniform gridsrequiring fewer nodes for a given spatial resolution, it appears unlikely that any LBmethod decoupling the numerical mesh from the lattice structure, could outperform thestandard LB on a full-domain basis.

However, similarly to local grid-refinement strategies, the main interest of thepresent method rests with prospective multiscale applications, combining the standardLB method in the bulk, with the FV version around critical regions only.

In conclusion, the proposed method is not meant to be a replacement of the standardLBE, but rather as a potential complement for future multiscale applications. In thisrespect, further investigations and consolidation of the present method, are surelywarranted.

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Corresponding authorAhad Zarghami can be contacted at: ahad.zarghami@gmail.com

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