A DYNAMICALLY ADAPTIVE LATTICE BOLTZMANN ...A dynamically adaptive lattice Boltzmann method for thermal convection problems 3 e 1 e 2 e 5 e 6 e 3 e 7 e 4 e 8 Fig. 1.Numerical stencil

Int. J. Appl. Math. Comput. Sci., , Vol. , No. , –DOI:

A DYNAMICALLY ADAPTIVE LATTICE BOLTZMANN METHOD FORTHERMAL CONVECTION PROBLEMS

KAI FELDHUSEN a,b , RALF DEITERDING c,∗, CLAUS WAGNER a,b

aInstitute of Aerodynamics and Flow TechnologyGerman Aerospace Center (DLR), 37073 Göttingen, Germany

bInstitute of Thermo- and FluiddynamicsTechnische Universität Ilmenau, 98693 Ilmenau, Germany

cAerodynamics and Flight Mechanics Research GroupUniversity of Southampton, Highfield Campus, Southampton SO17 1BJ, United Kingdom

email: [email protected]

Utilizing the Boussinesq approximation, a double-population incompressible thermal lattice Boltzmann method (LBM) forforced and natural convection in two and three space dimensions is developed and validated. A block-structured dynamicadaptive mesh refinement (AMR) procedure tailored for LBM is applied to enable computationally efficient simulationsof moderate to high Rayleigh number flows which are characterized by a large scale disparity in boundary layers and freestream flow. As test cases, the analytically accessible problem of a two-dimensional (2D) forced convection flow throughtwo porous plates and the non-Cartesian configuration of a heated rotating cylinder are considered. The objective of thelatter is to advance the boundary conditions for accurate treatment of curved boundaries and to demonstrate the effect onthe solution. The effectiveness of the overall approach is demonstrated for the natural convection benchmark of a 2D cavitywith differentially heated walls at Rayleigh numbers from 103 up to 108. To demonstrate the benefit of the used AMRprocedure for three-dimensional (3D) problems, results from the natural convection in a cubic cavity at Rayleigh numbersfrom 103 up to 105 are compared with benchmark results.

Keywords: Lattice Boltzmann method, adaptive mesh refinement, thermal convection, incompressible

1. Introduction

In recent years, the lattice Boltzmannmethod (LBM) has emerged as a powerfulalternative to traditional Navier-Stokes (NS)solvers (Chen and Doolen, 1998) to predictthermal fluid flow (Guo et al., 2002; Kuzniket al., 2007; Peng et al., 2003), turbulent fluidflow (Jonas et al., 2006), multiphase fluid

∗Corresponding author

flow (Lee and Lin, 2005; Yu and Fan, 2009)and magnetohydrodynamics (Deller, 2002).Instead of discretizing the NS equationsdirectly, the LBM is based on solving asimplified version of the Boltzmann equationin a specifically chosen discrete phase space.Using a Chapman-Enskog expansion, it hasbeen shown that the approach recovers the NSequations in the limit of a vanishing Knudsennumber (Hähnel, 2004). Originally proposed for

2 K. FELDHUSEN et al.

the isothermal weakly compressible case, severalmethod enhancements for incompressibility (Heand Luo, 1997; Qian et al., 1992) as well asincorporation of a buoyancy-driven temperaturefield for thermal convection flows are available(He et al., 1998; Qian, 1993). In general, thereare two different categories of thermal latticeBoltzmann models. For the multispeed approach,the number of discrete velocity directions willbe increased and the equilibrium distributionfunction is supplemented by higher order velocityterms to solve the internal energy equation,cf. (McNamara and Alder, 1993; Alexanderet al., 1993; Qian, 1993). However, this modelis reported to exhibit numerical instabilities,cf. (Chen and Teixeira, 2000). Here, we havechosen to pursue the strictly incompressibledouble distribution function (DDF) approachproposed by Guo et al. (2002) for 2D and thestraightforward expansion to 3D by He et al.(2004) and Azwadi Che Sidik and Syahrullail(2009).While the original LBM is formulated on auniform Cartesian grid, an increase of localresolution is particularly necessary in the thermalboundary layers close to heated objects andwalls. Kuznik et al. (2007) and Peng et al.(2003) demonstrated the computational benefitof a non-uniform grid for a thermal DDF LBMmethod in two and three spatial dimensionsfor simulating thermal convection in Cartesiancavities. In both works, a static geometrytransformation is applied to the discretizationin order to stretch the Cartesian lattice in thecavity center and reduce the spacing continuouslytowards the walls. Solution adaptive meshingis not used and on-the-fly mesh adaptationseems to have been applied so far to DDFLBM methods only in the context of isothermaltwo-phase flows, cf. (Yu and Fan, 2009). Ourobjective in this paper is to close this gap. Wesupplement a thermal DDF LBM method withsolution adaptive, dynamic mesh refinement.While adaptive lattice Boltzmann methods in thepast have used primarily isotropic refinementof individual cells, cf. (Chen et al., 2006), weapply in here a block-based approach, which ismore suitable for the regular transport step of theLBM and thereby computationally significantly

more efficient. The underlying data structuresincluding distributed memory parallelization areborrowed from the finite volume mesh refinementsystem AMROC (Deiterding, 2011). In orderto fit smoothly into AMROC, the DDF LBMis formulated on cell-centered data structuresand not node-based as it is mainly used forLBM in order to simplify the implementationof physical boundary conditions. In addition,complex geometry boundary condition treatmentfor possibly moving structures is incorporated.The update of the non-uniform lattice and thedynamic refinement procedure are orchestratedwith the recursive Berger-Collela algorithm(Berger and Colella, 1988). While the efficiencyof this algorithm is undisputed for time-explicitfinite volume schemes, its application to LBMis a novelty. In summary, our adaptive methodis uniquely designed for the efficient simulationof real-world thermal flow problems. In thispaper, the underlying computational techniquesare described and the required validation forwell-understood thermal convection problems isprovided.In Section 2, we discuss the details of thenumerical method, including the advancedthermal lattice Boltzmann approach, theblock-based AMR method and the treatmentof geometrically complex boundaries inthe originally Cartesian scheme. Section 3presents the computational results, where theanalytic solution of the 2D flow between twomoving porous plates, the 2D flow around arotating heated cylinder and the well-knownbenchmark case of a two-dimensional cavity withdifferentially heated walls are considered. Theresult section is closed presenting the solution ofthe flow in a 3D cubic cavity with differentiallyheated walls. The conclusions including a shortoutlook are given in Section 4.

2. Numerical method

2.1. Thermal lattice Boltzmann scheme.The incompressible two-dimensional LBMconstructed under Boussinesq approximationused in the present work has been proposed byGuo et al. (2002). For the three-dimensionalcase the incompressible LBM operator by

A dynamically adaptive lattice Boltzmann method for thermal convection problems 3

e1

e5e2e6

e3

e7 e4e8

Fig. 1. Numerical stencil of D2Q9 - Discrete velocitydirections in a computational cell.

He et al. (2004) is applied. By using theBhatnagar-Gross-Krook (BGK) collision model(Bhatnagar et al., 1954), the lattice Boltzmannequation for the partial probability distributionfunction fi with force field term Fi can beformulated as

fi (x + cei∆t, t+ ∆t) = fi (x, t)

− 1τν

(fi (x, t)− f (eq)i (x, t)

)+ ∆tFi. (1)

In the DDF approach, a set of correspondinglattice Boltzmann equations

gi (x + cei∆t, t+ ∆t) =

gi (x, t)−1

τD

(gi (x, t)− g(eq)i (x, t)

)(2)

is introduced based on distribution functions githat are used to convect the macroscopic scalarquantity, here temperature, with the flow field. Inthe latter, ei is the unit velocity vector in directionof the ith discrete velocity space direction, t and∆t denote the time and time step, x the position,∆x the spatial increment, and c = ∆x/∆t isthe particle speed. The relaxations times are τνfor the flow field and τD for the temperaturefield. The respective equilibrium distributionfunctions are denoted by f (eq)i and g

(eq)i . In

the two-dimensional case, a model with ninediscrete unit velocities is used to compute the flowfield (D2Q9) and an operator with four discretevelocities for the temperature field (D2Q4). Theorientation of the discrete unit length velocities eiused to compute the velocity fields are depicted inFig. 1. In the three-dimensional case, an operatorwith nineteen unit velocities is used for the flowfield (D3Q19) and a model with six discretevelocities for the temperature field (D3Q6). The

extended version of the orientation of the discreteunit length velocities ei are given in (3).

ei=

(0, 0) i=0,

(±1,0,0),(0,±1,0),(0,0,±1) i=1,...,6,(±1,±1,0),(±1,0,±1),(0,±1,±1) i=7,...,18

(3)The basic LBM algorithm is divided into the

steps of transport (or streaming) and collision,which are applied basically identically to (1)and (2). The following transport step representsthe advection of fluid particles along thecorresponding discrete velocities and is

T : f̃i (x + cei∆t, t+ ∆t) = fi (x, t) . (4)

Relaxation of the distribution functions towardsthe local equilibrium is performed on thetransported distribution functions in the collisionstep

C : fi (·, t+ ∆t) = f̃i (·, t+ ∆t)

− 1τν

(f̃i (·, t)− f̃ (eq)i (·, t)

). (5)

With the pressure p and the velocity vector uas independent variables, the specific equilibriumdistribution function f (eq)i for the D2Q9 model isdefined as (Guo et al., 2002)

f(eq)i =

−4σ pc2 − si(u), for i = 0,λ pc2 + si(u), for i = 1, ..., 4,γ pc2 + si(u), for i = 5, ..., 8,

(6)where the parameters σ, λ, and γ satisfy λ+ γ =σ and λ + 2γ = 1/2. The functions si(u)depend on the macroscopic velocity vector u andthe discrete velocity vector ei and obey

si (u)=ωi

[3ei ·uc

+4.5(ei ·u)2

c2−1.5 |u|

2

c2

],

(7)where the coefficients are given by ω0 =4/9, ω1,...,4 = 1/9, and ω5,...,8 = 1/36. Using(6) and (7), the macroscopic values for velocityand dynamic pressure are given as

u =∑i>0

ceifi, p =c2

4σ

[∑i>0

fi + s0(u)

].

(8)


For the D3Q19 model the parameters changeto σ = 1/2, λ = 1/18, and γ = 1/36.Furthermore the weight coefficients are given byω0 = 1/3, ω1,...,6 = 1/18, and ω7,...,18 = 1/36.For the D2Q4 model used to compute thetemperature field, the equilibrium function g(eq)iis

g(eq)i =

T

4

[1 + 2

ei · uc

], for i = 1, . . . , 4 (9)

and the macroscopic temperature is T =4∑i=1

gi.

Analogously, in the D3Q6 model of thetemperature field, the equilibrium function reads

g(eq)i =

T

6

[1 + 3

ei · uc

], for i = 1, . . . , 6

(10)

and the macroscopic temperature T =6∑i=1

gi.

Since the fluid is assumed to be incompressible,a linear dependency between temperaturedifferences and gravitational forces is applied(Boussinesq approximation), cf. (Mohamad andKuzmin, 2010), which leads to the force termFi. The force in (11) acts only in the two directvertical directions. For 2D, this can be expressedaccording to Fig. 1 (Guo et al., 2002) as

Fi =1

2(δi2 + δi4) ei · F (11)

with

F = gβ (T − Tref ) , (12)

where g and β are the acceleration vector ofgravity and the coefficient of thermal expansion,respectively; Tref is the average temperature. Theforce term establishes the coupling between thelattice Boltzmann equations for the flow field (1)and the temperature field (2).

Note that through a multiscaleChapman-Enskoq expansion, the incompressibleNavier-Stokes equations can be derived fromthe discussed incompressible LBGK model.After neglecting the viscous heat dissipation andcompression work carried out by the pressure, thetemperature field obeys a passive scalar equation.In sum, the approximated incompressible

equations in this work are, cf. (Guo et al., 2002),

∇ · u = 0, (13)∂u

∂t+∇ · (uu) = −∇p+ ν∇2u + F, (14)

∂T

∂t+∇ · (uT ) = D∇2T. (15)

The kinematic viscosity ν and the thermaldiffusivity D are related to the dimensionlesscollision times by ν = 16 (2τν − 1) c∆x and D =14 (2τD − 1) c∆x. Introducing the physical speedof sound as cs = c/

√3 these expressions yield

the relations

τν =ν + c2s∆t/2

c2s∆t, τD =

D + 32c2s∆t/2

32c

2s∆t

,

(16)which can be used to evaluate the dimensionlesscollision times in (1) and (2) for givenmacroscopic gas properties ν, D and time step∆t.

2.2. Adaptive mesh refinement. For localdynamic mesh adaptation we have adoptedthe block-structured AMR method proposedby Berger and Colella (1988). This methodwas originally designed for time-explicit finitevolume schemes for hyperbolic conservationlaws, however, its recursive execution procedureand natural consideration of time step refinementmake it equally applicable to lattice Boltzmannschemes, which is is not surprising as ahyperbolic constant velocity advection equationis the theoretical underpinning of the transportstep (4). In order to fit smoothly into ourexisting, fully parallelized finite volume AMRsoftware system AMROC (Deiterding, 2011),we have implemented the LBM cell-based. Inthe block-based AMR approach, finite volumecells are clustered with a special algorithminto non-overlapping rectangular grids. Thegrids have a suitable layer of halo cells forsynchronization and applying inter-level andphysical boundary conditions. Refinement levelsare integrated recursively starting from thecoarsest level. With index l denoting the AMRlevel, the spatial mesh width ∆xl and the timestep ∆tl are refined by the same factor rl,where we assume rl ≥ 2 for l > 0 and


(a)

fC,ni,in

(c)

T −1(f̃C,ni,out)

(b)

fF,ni,in

f̃F,n+1i,out

Fig. 2. Visualization of distributions involved in dataexchange at a coarse (C) - fine (F ) bound-ary. The thick black lines indicate a physi-cal boundary. (a) Coarse distributions goinginto fine grid; (b) incoming interpolated finedistributions in halos (top), outgoing distribu-tions in halos after two fine-level transport steps(bottom); (c) averaged distributions replacingcoarse values before update is repeated in cellsnext to boundary.

r0 = 1. In the adaptive thermal LBM, it isof foremost importance that the dimensionlesscollision times of the DDF LBM are adjusted ona level basis according to (16) as the time stepis recursively refined. In addition, the interfaceregion requires a specialized treatment to ensureconsistent transport of coarse-grid distributionsinto refined cells and of fine-grid distributionsinto the coarse cells adjacent to the boundariesof refined regions. Since the D2Q4 stencil isjust a simplified version of the D2Q9 method, werestrict our description of the interface algorithmto the latter. Distinguishing between the transportand collision operators, T and C, respectively (cf.(4) and (5)), our method proceeds in the followingsteps if a refinement factor of 2 is considered:

1. Complete update on coarse grid: fC,n+1i :=CT (fC,ni )

2. Use coarse grid distributions fC,ni,in thatpropagate into the fine grid, cf. Fig. 2(a), toconstruct initial fine grid halo values fF,ni,in ,

cf. Fig. 2(b).

3. Complete transport f̃F,ni := T (fF,ni ) on

whole fine mesh. Collision fF,n+1/2i :=C(f̃F,ni ) is applied only in the interior cells(yellow in Fig. 2(b)).

4. Repeat 3. to obtain f̃F,n+1/2i :=T (fF,n+1/2i ) and f

F,n+1i := C(f̃

F,n+1/2i ).

5. Average outgoing distributions from finegrid halos (Fig. 2(c)), that is f̃F,n+1/2i,out in theinner halo layer and f̃F,ni,out (outer halo layer)to obtain f̃C,ni,out.

6. Revert transport for averaged outgoingdistributions, f̄C,ni,out := T −1(f̃

C,ni,out), and

overwrite those in the previous coarse gridtime step.

7. Synchronization of fC,ni , f̄C,ni,out on entire

level.

8. Repeat complete update on coarse gridcells next to coarse-fine boundary only:fC,n+1i := CT (f

C,ni , f̄

C,ni,out)

In this description and in Fig. 2, the timesteps on the coarse level C are indexed by thesuperscript n, index F denotes the fine level andthe subscripts in and out indicate distributionswhich are convected in- and outwards of thefine grid along the coarse-fine boundary. Theoverall algorithm is computationally equivalent tothe method by Chen et al. (2006) but explicitlytailored to the Berger-Collela recursion thatupdates coarse grids in their entirety before finegrids are computed. The complete update ofthe entire respective coarse mesh and subsequentcorrection is the basis of the computationalefficiency of the Berger-Collela method; however,this approach has so far hardly been applied tolattice Boltzmann methods. Previous adaptiveLBM, cf. (Chen et al., 2006), update thefine grid before the respective coarse level andprovide no apparent avenue for implementingtime-interpolated fine level interface conditions.While not being used above, the benefitof interpolating in time the non-equilibriumportion of coarse-grid distributions crossing


the coarse-fine interface in Step 4 has beendemonstrated by Dupuis and Chopard (2003) andwill be considered in our implementation in thefuture.

2.3. Wall boundary treatment. The correctimplementation of boundary condition isvery important for numerical stability. Forthe considered test cases we need differentimplementations of boundary conditions forthe velocity and temperature partial distributionfunctions. No-slip or adiabatic boundaryconditions are realized via a bounce-backapproach for the unknown partial distributionfunctions as described in (Succi, 2001). Toprescribe fixed macroscopic values on the wallin form of Dirichlet boundary conditions we usea second order extrapolation scheme from Guoet al. (2002). The outflow boundary conditionsare implemented via a linear propagation asprescribed in (Mohamad, 2011). We use a setof halo cells around the computational domainto manipulate the unknown partial probabilitydistribution functions in the transport step.

2.4. Curved boundary treatment. Werepresent non-Cartesian boundaries implicitly onthe adaptive Cartesian grid by utilizing a scalarlevel set function ϕ that stores the distance tothe boundary surface. The boundary surface islocated exactly at ϕ = 0 and the boundary outernormal in every mesh point can be evaluatedas n = −∇ϕ/|∇ϕ|, (Deiterding, 2011). Wetreat a fluid cell as an embedded ghost cellif its midpoint satisfies ϕ < 0. In orderto implement non-Cartesian boundary conditionswith the LBM, we have chosen to pursue fornow a 1st order accurate ghost fluid approach.In our technique, the density distributions inembedded ghost cells are adjusted to modelthe boundary conditions of a non-Cartesianreflective wall moving with velocity vector wbefore applying the unaltered LBM. The laststep involves interpolation and mirroring of p,T , u, across the boundary to p′, T ′ and ū andmodification of the macroscopic velocity vectorin the immersed boundary cells to u′ = 2w −ū, cf. (Deiterding, 2011). From the newlyconstructed macroscopic values the distributions

in the embedded ghost cells are simply set tofeqi (p

′,u′) and geqi (T′).

3. ResultsFor the setup of physical configurations itis useful to recall the definitions of thedimensionless Rayleigh and Prandtl numberwhich is

Ra =gβ∆TH3

νD, Pr =

ν

D. (17)

The characteristic velocity U for thermalconvection flows is generally set to the buoyancyvelocity U =

√gβ∆TH , where H denotes a

problem-dependent geometric height. A cell(j, k) is flagged for refinement if any of thescaled gradient relations

|φj+1,k−φj,k|>�φ, |φj,k+1−φj,k|>�φ,|φj+1,k+1−φj,k|>�φ

(18)

is satisfied for a particular macroscopiccomponent φj,k and a prescribed limit �φ.If not stated otherwise, �T is set to 1% ofmaximum temperature and �u, �v , �w are set to5% of characteristic velocity.

3.1. Porous Plate. In order to validate thebasic numerical method, we selected the problemof forced thermal convection between two porousplates also employed by Guo et al. (2002). Thisproblem is set up as a Couette flow between twoporous plates of which the upper is in motion. Aconstant flow is injected normal to the lower plateand leaves the domain through the top plate withthe same rate. The bottom plate is cooled, whilethe upper plate is heated. The analytic solutionsfor the horizontal velocity and the temperatureprofile in steady state are

u∗(y) = U0

(eRe·y/H − 1eRe − 1

), (19)

T ∗(y) = TC + ∆T

(eRePr·y/H − 1eRePr − 1

), (20)

where U0 is the velocity of the upper plate. TheReynolds number Re is based on the injection


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

y/H

u/U0

Ana: Re=5

LBM: Re=5

Ana: Re=10

LBM: Re=10

Ana: Re=20

LBM: Re=20

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

y/H

(T-TC)/(TH-TC)

Ana: Re=5

LBM: Re=5

Ana: Re=10

LBM: Re=10

Ana: Re=20

LBM: Re=20

Fig. 3. Comparison of velocity and temperature distri-bution predicted for different Re in comparisonwith analytic solution.

Table 1. Spatial averaged error: porous plate problem.Re Eave(u) [%] Eave(T ) [%]5 1.08 1.1410 0.64 0.9820 0.19 0.38

velocity V0 and is given by Re = V0·Hν . Westudy three different configurations with varyingReynolds number. The Prandtl number is fixedand set to Pr = 0.71, which corresponds to airand the Rayleigh number is set to Ra = 100.The velocity of the upper plate is also fixed andset to U0 = 0.1. Finally, the dimensionlessrelaxation time τν on the coarsest level isprescribed as τν = 1/1.25. The simulationsare performed for the Reynolds numbers Re =5, 10 and 20 using a base grid of 64 × 32 cells.Successive embedded static refinement with fouradditional levels with refinement factors r1,...,4 =4 is realized in the complete computationaldomain [0, 64] × [0, 32]. In detail, we havethe finest resolution r4 near the top and bottomboundaries [0, 64] × ([0, 4] & [28, 32]), then r2in [0, 64] × ([4, 8] & [24, 28]) and r3 in [0, 64] ×([8, 12] & [20, 24]). The coarsest refinement level

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Ve

locity E

rro

r ave [

%]

Iteration (*1000)

Re=5

Re=10

Re=20

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Te

mp

era

ture

Err

or a

ve [

%]

Iteration (*1000)

Re=5

Re=10

Re=20

Fig. 4. Averaged L2-norm error for computed macro-scopic velocity and temperature over iterationsteps for different Re.

r1 is in the center region [0, 64] × [12, 20]. Theentire velocity field is initialized at rest as (0, 0)T

and the temperature field to the constant valueTC . We compare the numerical predictionsof the velocity and temperature distributionswith the analytic solution. Figure 3 plots thenormalized numerical results vs. the analyticsolutions. From the point of validation, themacroscopic values for the horizontal velocityand scalar temperature are being calculated ineach cell midpoint along each vertical line. Themacroscopic values in the cells are averaged alongthe horizontal lines. The L2-norm error of theaveraged macroscopic quantities Φ are calculatedwith (21) and displayed for the last iteration stepin Table 1.

Eave(Φ) =

√∑i

|Φave(xi)− Φ∗(xi)|2√∑i

|Φ∗(xi)|2(21)

The agreement is obviously excellent and below2% for all three cases. It is noteworthy thatthe error for the velocity is smaller than theone for the temperature. When increasing the


v = 0, ∂u∂y = 0,∂T∂y = 0

∂u∂x = 0∂v∂x = 0∂T∂x = 0

v = 0, ∂u∂y = 0,∂T∂y = 0

u = U∞

v = 0

T = TC

TH

u = 0, v = 0

ω

Fig. 5. Setup for the flow past the heated rotating cylin-der.

discrete velocity directions for the temperaturedistribution functions from 6 to 9, this errorshould decrease. Figure 4 plots the averagederror for the computed macroscopic velocityand temperature over the computational iterationsteps. The convergence to a fixed value isobvious.

3.2. Fluid flow past a heated rotat-ing cylinder. In order to test the dynamicadaptation capabilities and boundary conditionsfor embedded complex geometries, we studythe setup of a two-dimensional fluid flow pasta heated isothermal rotating cylinder. Theorigin of the coordinate system is located inthe center of the cylinder. As shown inFigure 5, the left boundary is an inlet withconstant temperature TC , zero vertical velocityand constant inflow velocity U∞. On the righthand side of the domain, an outlet is modeledby imposing zero horizontal gradient boundaryconditions for velocity and temperature. Slipadiabatic wall boundary conditions are appliedat the upper and lower boundary. The cylinderboundary is modeled as a no-slip wall, which isisothermally heated to the constant temperatureTH and has the constant prescribed angularvelocity Ω. In terms of the cylinder radius R =15, the computational domain has the extensions[−6R, 16R] × [−8R, 8R], which is sufficientlylarge to eliminate boundary influences on thesolution (Yan and Zu, 2008). A base grid of288× 240 cells is used and three additional levelsrefined by the factors r1 = 2 for level 1 and r2,3 =4 for the other levels are applied. The dynamicrefinement is based on scaled gradients of the

velocity components as well as the temperature.The entire velocity field is initialized as (U∞, 0)T

and the temperature field to the constant valueTC . The Reynolds number is given by Re =2U∞R/ν and is set to Re = 200, whereU∞ = 0.01 is used. The peripheral velocityV of the rotating cylinder is given by V =ΩR. With the parameter k = V/U∞ = 0.5prescribed, we can determine V and the angularvelocity Ω. To allow the direct comparisonto the experimental results by Coutanceau andMenard (1985) the Prandtl number is set to Pr =0.5 and all variables are normalized with thereference length R and U∞ as velocity. Further,T−TCTH−TC defines the reference temperature andthe time normalization factor follows as R/U∞.Figure 6 shows the dynamic adaption duringthe computation at four different time pointsby displaying streamlines and the domains ofdifferent mesh refinement levels. The onset ofvortex shedding can be inferred. The finestrefinement level (red) is located directly aroundthe cylinder. Namely, where the boundary layersare located and detach from the cylinders surface.The unrefined regions, colored in blue, are in theouter regions of the domain. The refined levelsmove downstream with the shedding vorticesand the cylinder wake increases over time.Figure 7 compares the temporal evolution of thevelocity components along representative pointson the x-axis obtained in the simulation and withdata from the experiment, while Fig. 8 displaysthe time evolution of the scalar temperatureversus numerical results reported by Lai andYan (2001). The latter adopted a finite volumemethod with non-orthogonal grids. Again, oursimulation results are in good agreement withsome differences in the u-velocity componentat t∗ = 8 when the vortex is shed (see Fig.6). A possible explanation is our rather simpletemperature operator with only four discrete unitydirections and with the used boundary conditionsfor the curved boundary explained in Section 2.4.However, by using the bounce back scheme forcurved moving boundaries from Bouzidi et al.(2001) and Li et al. (2013) with a global uniformmesh the differences are considerably reduced,cf. Fig. 9. Therefore, the next step is toimplement the curved boundary treatment in the


t∗ = 3

t∗ = 6

t∗ = 8

t∗=12

Fig. 6. Evolution of the velocity field and the adaptivemesh refinement regions for Re = 200 andk = 0.5.

-0.5

0

0.5

1

1 2 3 4 5 6

u*

x*

t*=3 (LBM)

t*=3 (Exp.)

t*=8 (LBM)

t*=8 (Exp.)

-0.5

0

0.5

1

1 2 3 4 5 6

v*

x*

t*=3 (LBM)

t*=3 (Exp.)

t*=8 (LBM)

t*=8 (Exp.)

Fig. 7. Time evolution of the velocity componentsalong the x-axis for Re = 200 and k = 0.5.

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6

T*

x*

t* = 3 (LBM)

t* = 3 (FVM)

t* = 6 (LBM)

t* = 6 (FVM)

t* = 12 (LBM)

t* = 12 (FVM)

Fig. 8. Time evolution of the temperature along the x-axis for Re = 200, Pr = 0.5 and k = 0.5.


0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6

T*

x*

t*=12 (AMR - OldBC)

t*=12 (UNI - NewBC)

t* = 12 (FVM)

t*=6 (AMR - OldBC)

t*=6 (UNI - NewBC)

t* = 6 (FVM)

Fig. 9. Comparison of simulation results with differentused curved boundary conditions: Time evo-lution of the Temperature along the x-axis forRe = 200, Pr = 0.5 and k = 0.5.

∂T/∂y = 0, u = 0, v = 0

T = TC

u = 0

v = 0

∂T/∂y = 0, u = 0, v = 0

T = TH

u = 0

v = 0g

H

H

yx

Fig. 10. Configuration of the two dimensional cavity.

AMR method.

3.3. Natural convection in a square 2D-cavity.In order to benchmark the overall method weemploy a two-dimensional square cavity withdifferentially heated walls. At the verticalwalls isothermal temperatures TH and TC areprescribed and adiabatic boundary conditionsare applied at top and bottom. Further, atall four walls we prescribe no-slip boundaryconditions for the velocity field. Figure 10depicts this setup. The flow is characterized bythe Prandtl number Pr = 0.71 (air) and theRayleigh numbers Ra = 10j with j = 3, . . . , 8with accordingly increasing velocities U . Thereference temperature is given by Tref = (TH +

TC)/2. The simulations were terminated afterreaching steady state. Two additional levels ofrefinement with r1,2 = 2 are used and the basemesh has (H∆x0)2 cells, whereby ∆x0 = 1and H is given in the left column of Table 2.For simulations with Ra = 103, · · · , 106 we usethe defined refinement thresholds for horizontaland vertical velocity �u, �v with 2.5% of thecharacteristic velocity and 1% of the maximumtemperature. The thresholds for Ra = 107 and108 remain as previously stated. We compare ouradaptive simulation results to published referencedata by De Vahl Davis (1983), who solved theNS equations on a uniform square mesh witha second order finite difference method, and byGuo et al. (2002), who used the incompressiblethermal LBGK approach presented above with auniform mesh. Further results by Kuznik et al.(2007), who used a D2Q9 DDF LBM approachwith non-uniform mesh resolution, are listed inTable 2. Table 2 contains the obtained maximalhorizontal velocity umax along the vertical centerline at x = H/2 and the location ymax of itsoccurrence and similarly for the horizontal centerline at y = H/2, the maximal vertical velocityvmax and its location xmax. Furthermore, theaverage Nusselt number

Nuave = −H∫0

1

∆T

∂T

∂x

∣∣∣∣x=0

dy (22)

is compared. Velocity values in Table 2 arenormalized by the reference diffusion velocityD/H . As expected, umax, vmax and Nuaveincrease with increasing Rayleigh number Ra.Comparing the Nu numbers predicted by ouradaptive method to the literature data, anagreement within 2 % is found for all Ranumbers. Figure 11 shows the vertical velocitycomponent in the horizontal mid-plane for alldiscussed Rayleigh numbers. The velocityprofiles plotted in Fig. 11 reveal the developmentof a boundary layer close to the heated/cooledwalls with velocity maxima/minima whose valuesincrease/decrease with increasing/decreasing Ra.This increase of the magnitude of the verticalvelocity with increasing Ra is also reflected inTable 2. To give an impression of the flowsolution, contours of the temperature fields and


Table 2. Comparison of the simulation results: naturalconvection in the square cavity.

Ref. umax ymax vmax xmax Nuave

Ra = 103 a 3.640 0.810 3.688 0.180 1.115U = 0.01 b 3.649 0.813 3.697 0.178 1.114H=100 c 3.655 0.813 3.699 0.180 1.115

d 3.636 0.809 3.686 0.174 1.117

Ra = 104 a 16.161 0.823 19.595 0.118 2.239U = 0.02 b 16.178 0.823 19.617 0.119 2.245H=150 c 16.076 0.820 19.637 0.117 2.248

d 16.167 0.821 19.597 0.120 2.246

Ra = 105 a 34.666 0.855 68.457 0.066 4.504U = 0.05 b 34.730 0.855 68.590 0.066 4.510H=200 c 34.834 0.859 68.267 0.062 4.535

d 34.962 0.854 68.578 0.067 4.518

Ra = 106 a 64.756 0.850 220.125 0.038 8.804U = 0.05 b 64.630 0.850 219.360 0.038 8.806H=200 c 65.361 0.852 216.415 0.039 8.778

d 64.133 0.860 220.537 0.038 8.792

Ra = 107 a 140.255 0.887 702.459 0.021 16.429U = 0.05 d 148.768 0.881 702.029 0.020 16.408H=256

Ra = 108 a 297.145 0.945 2228.4130.012 29.954U = 0.05 d 321.457 0.940 2243.36 0.012 29.819H=256

a = Present (LBM-AMROC), b = (De Vahl Davis, 1983)(FDM - uniform), c = (Guo et al., 2002) (LBM - uniform),

d = (Kuznik et al., 2007) (LBM - nonuniform).

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 0.2 0.4 0.6 0.8 1

v

x/H

Ra=103

Ra=104

Ra=105

Ra=106

Ra=107

Ra=108

Fig. 11. Vertical velocity in the horizontal mid-planeof the 2D cavity for different Rayleigh num-bers.

streamlines are presented in Fig. 12 for threeconsidered Ra numbers. For all three Ranumbers the streamlines reflect that fluid risesat the heated wall and descends at the cooledwall. This generates a circulation around thecenter where the velocity is zero. For the lowerRa numbers the computed flow field are ingood agreement with results reported in previousstudies (De Vahl Davis, 1983; Guo et al., 2002;Azwadi Che Sidik and Irwan, 2010; Kuznik et al.,2007; Abdelhadi et al., 2006). In the graphwith the contours predicted for Ra = 107 themesh refinement levels realized in the domainare additionally highlighted by colors. From thepredominantly vertical isotherms obtained for thelow Ra number case it can be concluded thatthe heat conduction dominates the heat transportbetween the heated walls. For larger Ra theisotherms are aligned more horizontally in thecavity’s center due to the thinner boundary layers.The denser isotherms near the hot and cold wallfurther reflect the lower thermal boundary layerthickness for higher Rayleigh number. It is inthis region where on-the-fly mesh resolution isparticularly beneficial.

3.4. Natural convection in a cubic cavity. Tobenchmark the three-dimensional implementationof the method, we employ a 3D cubic cavitywith differentially heated walls. As before, at thevertical walls the constant temperatures TH andTC are prescribed. At the bottom, top and front,back walls adiabatic boundary conditions areused for the temperature, while no-slip boundaryconditions at all six walls are realized for thevelocity fields. In summary, Fig. 13 representsthis numerical setup. Again, the Prandtl numberis Pr = 0.71 (air) and in the 3D simulationsthe Rayleigh numbers Ra = 10j is varied fromj = 3, . . . , 5. Here, we focus on the flowfor Ra ≤ 105, since for higher Ra the flowis expected to become unsteady and eventuallyturbulent. To benchmark our method for aturbulent flow is however beyond the scope ofthis paper. As discussed above, the buoyancy(reference) velocity U rises with increasing Raand the reference temperature is given by Tref =(TH+TC)/2. Two additional levels of refinementwith r1 = 2, r2 = 4 are used and the base


Ra = 103

Ra = 106

Ra = 107

Fig. 12. LBM results of natural convective flow in thesquare cavity for three Ra numbers. Left:contours of isotherms. Right: streamlines.

TH TC

H

H

H

g

x

y

z

Fig. 13. Configuration of the three dimensional cavity.

mesh has (H∆x0)3 cells, whereby ∆x0 = 1and H is given in the left column of Table 3.The adaptive mesh refinement obeys the scaledgradient criteria given above in (18). The usedthresholds for all three velocity components are1%, 2% and 5% of the reference velocity U forRa = 103, 104 and 105, respectively. As before,1% of TH is used as the temperature refinementthreshold. The computed results are comparedto published literature results after reachingsteady state. Azwadi Che Sidik and Syahrullail(2009) use a D3Q19 DDF LBM approach withD3Q6 operator for the temperature field and auniform cubic mesh to get excellent numericalstability and accuracy. Peng et al. (2003) usea three-dimensional incompressible LBM withDDF approach and two D3Q19 operators for thetwo fields and a non-uniform mesh resolution.Finally, Fusegi et al. (1991) use a high-resolution,finite difference NS solver with a uniform meshresolution result and obtain results which agreereasonably well with experimental measurements.Figure 14 visualizes the temperature isosurfacesin the cubic enclosure and the different meshrefinement levels in the symmetry plane for Ra =104, 105. Near the heated walls, the isosurfacesare predominantly vertical. Notice, that theisosurfaces in the center of the cavity becomemore horizontally with increasing Ra. The reasonis that the thermal boundary layer is becomingthinner. This observation is similar to that in theprevious chapter. Note that the shaping of themesh refinement levels for Ra = 104 is muchmore pronounced than for Ra = 105. As in theprevious chapter, we compare the results in thesymmetry plane z = H/2 in terms of maximalhorizontal velocity umax along the vertical centerline at x = H/2 and at the correspondinglocation ymax of its occurrence and similarlyfor the horizontal center line at y = H/2, themaximal vertical velocity vmax and its locationxmax. Furthermore, we use the average Nusseltnumber (22) for comparison. Our results arelisted in Table 3. The velocity values in Table3 are normalized with the reference velocity U .The Nusselt number increases with increasing Ranumber, which means that the convective partof the heat transfer predominates the conduction.Comparing the Nu numbers predicted with our


Table 3. Comparison of the simulation results: naturalconvection in the cubic cavity.

Ref. umax ymax vmax xmax Nuave

Ra = 103 a 0.132 0.195 0.132 0.829 1.099U = 0.01 e 0.132 0.186 0.132 0.841 1.096H=81 f 0.132 0.188 0.133 0.826 1.097

g 0.131 0.200 0.132 0.833 1.105

Ra = 104 a 0.197 0.194 0.220 0.887 2.270U = 0.02 e 0.200 0.182 0.224 0.883 2.301H=81 f 0.206 0.163 0.221 0.887 2.304

g 0.201 0.183 0.225 0.883 2.302

Ra = 105 a 0.141 0.152 0.242 0.935 4.583U = 0.1 e 0.151 0.142 0.248 0.930 4.670H=91 f 0.149 0.136 0.240 0.935 4.658

g 0.147 0.145 0.247 0.935 4.646

a = Present (LBM-AMROC), e = Azwadi et al. (Azwadi Che Sidikand Syahrullail, 2009) (LBM - uniform), f = Peng et al. (Peng

et al., 2003) (LBM - nonuniform), g = Fusegi et al. (Fusegiet al., 1991) (NS - uniform)

method to the literature, an agreement within 2%is found for all three Ra numbers, although thecomparison of the horizontal velocity componentshows larger differences. The reason for thismight be a lack of dynamic mesh refinement nearthe upper and bottom walls. The mesh refinementis more pronounced near the heated and cooledwalls, where the thinner thermal boundary layersare located.

4. ConclusionsA novel two and three dimensionalincompressible dynamically adaptive thermallattice Boltzmann method on block-basedhierarchical finite volume meshes withembedded complex geometric structures hasbeen developed and validated. The agreementfor a two-dimensional porous plate problem ona Cartesian grid is nearly perfect. Successfulvalidation against analytic solutions of theNavier-Stokes equations, e.g., for a heatedrotating cylinder for Pr = 0.5 has been achieved.While for this particular example the deviationsin velocity and temperature were found toincrease over time, a possible improvementcould be the implementation of a bounce-backboundary condition for curved boundaries. Forthe benchmark of a two-dimensional heatedcavity with Rayleigh numbers from Ra = 103

to 108, the predictions are in good agreement

Ra = 104

Ra = 105

Fig. 14. Simulation results of natural convective flowin the cubic cavity. Left: isosurfaces of tem-perature. Color indicates temperature fromhot (red) to cold (blue). Right: mesh refine-ment levels.

with published results. Our results in form of thecomputed Nusselt number reach an agreementwithin 2%. For higher Rayleigh numbers, thedeviations in the considered quantities are greaterin regions without refinement. The comparisonfor a three-dimensional heated cubic cavity withRayleigh numbers from Ra = 103 to 105 againstliterature results delivers a good agreementas well. In terms of the Nusselt number, theagreement with literature results is again under2%. A comprehensive analysis of CPU-timeand memory savings by employing our uniqueblock-based adaptive LBM will be conductedin the future. We will also take a closer look athow the results are influenced by the refinementcriteria. Finally, extension and validation of the3D approach to turbulent flows at higher Ra orRe numbers is planned.

5. BiographiesThe author biographies will be provided uponacceptance.


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A DYNAMICALLY ADAPTIVE LATTICE BOLTZMANN ...A dynamically adaptive lattice Boltzmann method for thermal convection problems 3 e 1 e 2 e 5 e 6 e 3 e 7 e 4 e 8 Fig. 1.Numerical stencil