ABSTRACTThe modeling of convective flows with 3D latticeBoltzmann fluid dynamics is presented using alattice BGK approach for low Mach number flowswith variable density combined with a large eddyturbulence model. The ability to resolve non−Boussinesq density varying problems is depicted fortwo dimensional Rayleigh−Bénard−convection at aRayleigh numberRa= 800,000.

A complex three−dimensional example shows thestatus of our work presenting turbulent flow in andaround a building, so far without consideration ofthe energy equation in the 3D full scale case.Integrated within a CAD environment, the spatialgeometric model, based on an IFC building productdata model, is discretized using a hierarchic datastructure. Results are presented forRe = 75,000computed on a high−performance parallel vectorcomputer. State−of−the−art visualization techniquesintegrate the simulation results into the CAD modelin a virtual reality environment.

INTRODUCTIONBuilding energy simulation has been developedthroughout the past decades in the context of energyefficient building design, taking considerableadvantage of improvements in efficient numericalalgorithms. Moreover, the availability of powerfulhardware allows a detailed treatment of physicalprocesses, e.g. convective heat transfer, withinacceptable simulation time. A further significantincrease of performance is obtained when the com−putational code is implemented for parallel vectorarchitectures [1] or optimized for a parallel Linuxcluster environment using domain decompositionand message passing techniques. Considering costsvs. efficiency, the latter is expected to gainincreasing influence on engineering practice in thenear future.

One of the most commonly used methods within thedynamic building energy simulation is a multi−zonenetwork air flow and thermal model. Satisfying thefirst law of thermodynamics, a finite−difference(FD) or finite−volume based heat balance approachis used to represent the one−dimensional heat

transfer through the buildings’ fabric. Due to thecomplex geometry of buildings, the spatial andtemporal discretization usually constrains thequality of simulation. Hence, the consideration ofindoor airflow is often unsatisfactory, if macro−scopic models represent large air volumes by singlenodes only.

Since, on the other hand, detailed calculations arecomputationally very expensive, high−resolutionmodeling in the building simulation is currentlylimited to single zones and the consideration ofshort time intervals [2]. Common techniquesprovide the conflation of high−resolution indoor airflow modeling with the heat balance approach byestimating heat transfer coefficients from local CFDflow conditions, or by calculating the convectiveheat transfer directly via CFD. Additionally, short−and longwave radiation models are used inconjunction with a nodal network approach. On theother hand, some CFD tools consider more detailedradiation models for calculating the combinedconvective and radiative heat transfer. Comparingall mentioned approaches, each of these take intoaccount only one problem in particular whiledrastically simplifying or neglecting the otheraspects.

Considering this present situation, the long−termobjective of our group’s research is a fluid−structure−temperature coupling, targeting at thedirect simulation of at least a single room within thewhole building energy simulation. This is possibleby a partitioned solution approach for the analysis oftransient thermal storage and heat transfer, includingindoor air flow and the opaque and translucentfabric, the spatial geometric model being connectedto a building product model based on theIndustryFoundation Classes (IFC). Furthermore, thisdetailed model will be embedded into a multi−zonenetwork model in order to allow for a coupling toambient boundary conditions.

In this paper, emphasis is laid on a special aspect ofthis overall goal, namely on the modeling ofconvective flow and heat transfer using a new latticeBGK approach (LBGK) for flows with variabledensity [3] combined with a large eddy turbulence

DIRECT BUILDING ENERGY SIMULATION BASED ON LARGE EDDYTECHNIQUES AND LATTICE BOLTZMANN METHODS

C. van Treeck, M. Krafczyk, S. Kühner, E. Rank

Technische Universität München, Lehrstuhl für BauinformatikArcisstr. 21, D−80290 München, Germany

treeck@bv.tum.de, http://www.inf.bv.tum.de

Seventh International IBPSA ConferenceRio de Janeiro, Brazil

August 13-15, 2001

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model [1, 4].

The paper is organized as follows. After a briefintroduction into the basic lattice Boltzmannmethod, which provides a technique for solving thecompressible Navier−Stokes equations at low Machnumber limit, the LBGK approach is extended by afinite difference discretization for the solution of theenergy equation (non−Boussinesq−approximation).The feasibility of this approach is presentedconsidering an example of two−dimensionalRayleigh−Bénard−flow and compared to a standardfinite difference discretization of the Navier−Stokesequations for Boussinesq−incompressible flow [5].The lattice Boltzmann modifications for the LESapproach are discussed subsequently. Finally, acomplex three−dimensional example will show thestatus of our work presenting a complete sequencefrom pre− to postprocessing: Integrated within aCAD environment, the geometric model, based onan IFC building product data model, is discretizedusing a Cartesian grid based on a tree topology. Theoctree grid generation is followed by thecomputation of the 3D turbulent flow in and aroundthe building, so far without the consideration of theenergy equation in the 3D full scale case. State−of−the−art visualization techniques are presented tointegrate the simulation results into the spatial CADmodel.

LATTICE BOLTZMANN MET HODSDuring the last decade, lattice Boltzmann methodshave emerged as a new technique for the computa−tion of fluid flow phenomena [6, 7]. This approachhas proven to be an efficient tool, which isespecially well−suited to a parallel implementationand to complex geometries [1]. For a detailedreview, see e.g. [8, 9].

Common numerical methods for solving theNavier−Stokes equations are based on atop−downapproach with the discretization of nonlinear partialdifferential equations by e.g. FD− or FV−techniques. On the other hand, lattice−gas cellularautomata (LGCA) and its later derivative, latticeBoltzmann methods (LBM) represent abottom−upapproach by starting at a discrete microscopic modelwhich by construction conserves the desiredquantities, such as mass and momentum forobtaining the hydrodynamic properties described bythe Navier−Stokes equations.

The basic idea is to simulate the behaviour of amulti−particle system through artificial micro−worlds of particles residing on a computationallattice. Considering a sufficiently large ensemble ofparticles, hydrodynamic properties can be obtainedat a macroscopic scale. In the field of LGCA, a gaswas described as an ensemble of many ’molecules’

interacting locally at thenodes of a regular latticeby collisions. Due to someunphysical constraints,such as the non−isotropic advection term and thenoise particularly of the pressure field, a few yearslater McNamara and Zanetti [10] introduced latticeBoltzmann methods as a numerical method.

The main difference to the LG approach consists ina transition from explicit discrete particles to thepropagation and collision of an ensemble ofparticledistribution functions. Since a description of indi−vidual particle−particle interactions defined bynewtonian or quantum dynamics is computationallyunfeasible, remedy is found using methods of statis−tical physics. TheBoltzmann equation(1) describesthe probability to find a system of particles at agiven time t and state

xi ,

i , where

xi and

i

denote position and velocity of thei th particle. Forthe derivation of the Boltzmann equation(1),significant simplifications are additionally assumed:particles are statistically uncorrelated (molecularchaos hypothesis) and assumed to be similar.Furthermore only two−particle collisions areconsidered.

f

t

f

x

f (1)

Although the local collision operatorΩ(f) can be ofarbitrary complexity, it is sufficient to use the BGKapproximation, proposed by Bhatnagar, Gross andKrook [8], where Ω(f) is approximated by thesingle time relaxation approximation.The particledistribution functions are relaxed towards anequilibrium, the local Maxwell distributionf (0), on amicroscopic relaxation time scale given byτ. Thediscrete Boltzmann equation(2) is obtained by thediscretization of the Boltzmann equation with BGKapproximation with respect to its microscopicvelocity space.

f a

t

a

f a

x

1 f a

f a0 (2)

Considering small Knudsen numbersε, an ansatz ismade for the distribution functions with respect tothe microscopic velocity

. The expansion around

global equilibrium introducesas another constraintthe small Mach number(Ma) limit. In three−dimensional space, for the so−called D3Q19 model[11] the distribution functions are evaluated at thecollocation points depicted in fig. 1.

Macroscopic flow variables, as mass density andmomentum density, can be obtained as moments ofthe distribution functions

a

f a ,

u a

a f a . (3, 4)

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Fig. 1: Collocation points of the D3Q19 model [12].

The spatial and temporal discretization of thediscrete Boltzmann equation(2) by an explicit finitedifference scheme yields thelattice Boltzmannequation(5).

f a t

t ,x

a

t

f a t ,x

t

f a t ,x f a

0 t ,x (5)

The corresponding algorithm can be divided intotwo essential steps per time step:

The calculation of the new distribution functionswith respect to the right−hand side of (5), the socalledcollision, and

the streaming of the distribution functions to thenext neighbour nodes, usually referred to aspropagation.

The corresponding macroscopic equations, i.e. theNavier−Stokes equations, can be derived from (2) or(5) by a multiscale analysis, the Chapman−Enskogexpansion [13]. The errors introduced by thenumerical scheme are the discretization error oforderO(∆t2, ∆x2), the Knudsen−error of orderO(ε2)and the compressibility effect of orderO(Ma2). Thekinematic shear viscosity can be obtained from (6),where a unit time step and lattice spacing isassumed for simplification, i.e.∆t = ∆x = 1.

2 1

6(6)

LBGK WITH VARIABLE DENSITYFor the consideration of the energy equation withina lattice Boltzmann simulation several approacheshave been proposed recently. In [14], an internalenergy density distribution function is introduced forthe simulation of the temperature field. Otherauthors describe a passive scalar based thermal LBapproach and the modeling of internal energy as amoment of the distribution functions. To improvenumerical efficiency, we use a simplified approachbased on [3], where the LB method for low Machnumber flows is extended by a finite difference

discretization for the numerical solution of theenergy equation (non−Boussinesq−approximation).

The fluid flow is computed using an LBGK methodfor incompressible flows [15]. In [3], appropriatemodifications are applied to the LB approach to takeinto account the spatial and temporal variation ofthe mass density. The momentum and energyequations are considered to be decoupled during atimestep, and the density becomes a function oftemperature. The energy equation (7) is solvednumerically using an explicit FD scheme as descri−bed in [5].

T

t

u T T (7)

Eq. (7) describes the conservation of energy withconstant thermal conductivity, negligible viscousdissipation and neglecting only the substantialderivative of the pressure.α denotes the thermaldiffusivity coefficient.

For a coupled procedure of both numerical schemes,the following algorithm is obtained:

If the macroscopic variables are assumed to beknown at a time t, the lattice Boltzmannequation, adapted from (5), is solved by thecollision and propagation procedure.

At the new time stept+∆t, the energy equation(7) is numerically solved for the temperature e.g.by an explicit Euler scheme.

From the updated temperature field, new valuesfor the mass density can be obtained using theequation of state for an ideal gas (8), consideringthe thermodynamic part of the pressure beingconstant, assuming an open system.

T const. (8)

Additionally, the temperature dependence of thecollision timeτ has to be considered, because ofthe kinematic shear viscosity is a function of thedynamic viscosity and the mass density.

Finally, modified equilibrium distributionfunctions f (0) are calculated using the approachproposed by [3], which by definition takes intoaccount the variable density.

Compared to (3) and (4), the variation of theequilibrium distribution functions yields a change inthe meaning of the moments of the distributionfunctions. Hence, the zeroth moment results in aterm describing macroscopically the dynamic part ofthe pressure and acorrective term. The latter isdefinded in order to correct the stress tensor of theincompressible Navier−Stokes equations withrespect to its classical definition. A detailed deri−

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vation based on multiscale analysis is given in [3].

Results are presented for a class of two−dimensionalenclosure phenomena, see fig. 2 and 3, resultingfrom a temperature gradient parallel to gravity. Theupper and lower sides of the upright rectangular boxare cooled and heated, respectively. The com−putational grid size is 400x200 nodes. The observedRayleigh−Bénard−flow is theoretically initiated at aRayleigh number of approximatelyRa≈ 1708.

Fig. 2: Transient velocity field for Rayleigh−Bénard−flow (LBGK) atRa= 800,000

Fig. 3: Isotherms for Rayleigh−Bénard−flow(LBGK) at Ra= 800,000

A comparison of the LB simulations with resultsobtained by a standard FD discretization [5] are ingood agreement regardingthe critical Rayleigh−number behaviour [16] and for a wide range ofRayleigh numbers with respect to the velocity fieldand the isotherms.

MODIFICATIONS FOR LESFlows around buildings of technical relevance arecharacterized by high Reynolds numbers at the orderof O(106_108). As turbulent flows contain a widerange of length and time scales, a computationalgrid for a direct numerical simulation (DNS) has torepresent the largest and smallest dynamicalstructures in time and space, e.g. from the largest tothe smallest eddies. The expected number of gridpoints NG and time steps NT needed for acomputation can be estimated from (9) and (10).According to Kolmogorov [1], the minimum scalesof time, length and velocity are approximated by

NG ≈ Re9/4, (9)

NT ≈ Re3/4 . (10)

Even if supercomputing capabilities are available,the DNS of the Navier−Stokes equations of buildingaerodynamics phenomena still is and will persist tobe an unsolvable problem within the near future. Forindoor air flows, the Reynolds numer is of the orderO(105). If, additionally, a turbulence model is used,the simulation of the flow dynamics is possible.

As a promising method, the Large Eddy approach isbased on the time dependent, spatially averagedNavier−Stokes equations, in order to predict theturbulent flow. Since the smallest scales cannot beresolved, filter−algorithms (e.g. mean values in thesimplest case) are used to compute the influence ofthe unresolved small scales onto the large scales.This dissipative process is modeled by subtractingturbulent kinetic energy of the relevant part of thespectrum. The transient mean velocity field isobtained by subsequent averaging [17].

Accordingly, the LB approach is extended by aSmagorinsky subgrid scale model [4]. The turbulentdissipative effects are interpreted as an eddyviscosityνΤ. In general, within LBM the viscosity isdetermined by the relaxation timeτ. Thus, themolecular viscosityν0 and the turbulent virtualstresses give the total viscosityνtotal (11).

total

0

T

2 0

1

6

T

3(11)

Using Smagorinsky’s ansatz for the eddy viscosity,

T

T

3 CS

2 LB , (12)

the turbulent relaxation timeτT is obtained from(14). Here,εLBαβ denotes the local transient straintensor,

LB

3

2

1

0

T

SLB , (13)

which follows from the lattice Boltzmann stresstensor SLBαβ.

T

02 18C

S

2 Q1 2

1 2

0

2

, (14)

where Q LB . (15)

SIMULATIONIn the following, a complex three−dimensionalexample will show the present status of our work.For the turbulent flow in and around a building acomplete outline of the whole procedure ofmodeling, discretization, CFD simulation andvisualization is presented. Up to now, the energyequation is not considered at the 3D full scale case.

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The building model is generated using the IFC 1.51scheme [18], which can be obtained using manycommon CAD platforms supporting this format, e.g.AutoCAD Architectural Desktop, NemetschekAllplan FT or GraphiSoft ArchiCAD. With its IFCspecifications, the International Alliance forInteroperability (IAI) provides an object orienteddescription of building product model data to ensuresoftware interoperabilityin the building industry.

The given example basically consists of theinstances IFCWall, IFCDoor and IFCWindow,which are derived from the objectIFCBuilding−Element. These objects are located at the Inter−operability Layer of theIFCSharedBldgElementsscheme of the IFC implementation.

Fig. 4: IFC model in the Architectural Desktop [20]

For preprocessing, the generation of the three−dimensional computational lattice has beenintegrated into theArchitectural Desktop(ADT)environment [19]. The link to the CAD system isestablished in conjunction with theObjectARXandthe OMF1 interface, respectively. Fig. 4 shows theIFC model in the CAD system.

Fig. 5: Interfaces solid−fluid

Based on this IFC model, a surface mesh for each

1 The AEC Object Modeling Frameworkis aprogramming API based on ObjectARX.

mass element consisting of triangular and quadri−lateral facets is generated at the interfaces betweenfluid and solids first (see Fig. 5).

Subsequently, the computational domain is discre−tized using an octree data structure, i.e. a Cartesiangrid based on a hierarchical tree topology [20].

Fig. 6: Orthogonal projection of the octree

Fig. 7: Spatial representation of octree leaf nodeswith non−empty structure intersection

Fig. 8: Smoothed octree of the spatial CAD model,considering all leaf nodes

To initiate this procedure, the whole CAD object isencompassed by a cube, the so−called octree root−element. This bounding box represents the wholecomputational domain, including the expected wakearea of the flow. The root−element is then sub−divided into eight child elements, followed by the

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determination of the intersections with all faces ofeach building object. If a child element is comple−tely surrounded by an object, or none of the objectsreside within a child domain, this branch of theoctree needs no further refinement. The recursivealgorithm is continued until a given depth ofrefinement is achieved. Figures 6 to 8 illustrate atwo−dimensional projection, i.e. the correspondingquad−tree, and spatial representations of thesmoothed octree.

The nodes of all leafs of the octree with non−emptyintersections with the surface of the geometricmodel are subject to no−slip boundary conditions bydefault, alternative boundary conditions can beprescribed, e.g. in−/outflux velocity or pressure. Thesub−trees are then filled up to obtain theunstructured Cartesian computational grid. Theboundary of the resultingvoxel model is displayedin figure 9.

Fig. 9: Boundary of the building voxel model

The above presented, smoothed octree could evenbe used as a computational lattice directly. For theutilization of hierarchic data structures within anadaptive simulation, the interested reader is referredto [1], where this technique has been studied for atwo−dimensional model problem.

RESULTSThe results of the LB simulation are presented forthe 3D turbulent flow at a Reynolds number of Re =75,000. The grid size of the uniform lattice is400x100x200, inflow condition is a logarithmicboundary layer profile.

The computation was performed on a high−perfor−mance parallel vector computer, theHitachiSR8000−F1[21]. Its current configuration consistsof 112 SMP nodes containing 9 physical CPUs (8COMPAS), each node delivering a peak perfor−

mance of 12 GFlop/s2 resulting in 1.3 TFlop/s forthe whole system. Our vectorized code isparallelized on an intra−node basis using COMPASdirectives (autoparallelizing compiler) and for inter−node communication by explicit message passingusing the MPI library. The present implementationgives a performance of 3 GFlop/s per node. Thecode is also implemented on a parallelCompaqAlphaplatform.

For visualization purposes, the applicationVirtualFluids is used, which was developed in ourgroup recently [22]. This application is designed fora combined visualization of simulation data and acorresponding spatial CAD model. Figure 10 showsstreamlines of the averaged velocity field in andaround the building. Since some of the windows areopen, indoor air flow is induced due to the outerflow field. In order to allow for a real−timevisualization of a system of approximately 8 milliongrid points, the simulation data are filtered to reducethe level of detail using hierarchic data structures.

Fig. 10: Turbulent flow, streamlines

Two orthogonal slice planes of the averagedvelocity field are plotted in figure 11.

Fig. 11: Turbulent flow, orthogonal slice planes ofthe averaged velocity field

2 Giga Floatingpoint operations/second

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Fig. 12: Particle tracing based onVirtualFluids.

Figure 12 shows particles injected at a line whichcan be scaled, transformed and positionedinteractively. A major part of the particles beinginjected at the line streams through the openwindows, while some of them are carried away bythe so−called horse−shoe vortex. Figure 13 showstwo orthogonal slice planes of the averaged velocityfield inside the building.

Fig. 13: Walk through the building, averagedvelocity field.

The graphical representation of the CAD model isprovided − independent of specific data formats −by reading these informations from a VRML file,which is an ISO certified standard for thedescription of three−dimensional scenes.

The geometry is visualized by its surfacerepresentation using textured surfaces. Virtualreality features are obtained using high levelgraphical libraries, such asOpenGL Optimizerwiththe scene graph APICosmo3D [22]. Hence, theapplication runs on high end graphic workstations,e.g. a holobench, as well as on low−cost PChardware. Furthermore, this technique supports theprocess of communication during the buildingdesign by providing architects with a descriptive,reduced and platform−independent dataset of thepostprocessing simulation data.

CONCLUSIONSAlthough being a relatively new simulationtechnique, the lattice Boltzmann method has provento be an efficient tool for solving various transportphenomena. Due to its algorithmic structure, thisapproach is especially well−suited to an efficientparallel implementation and to complex geometries.Further investigations will include the extension of

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the two−dimensional LBGK approach for flowswith variable density to the third dimension and theexamination of consequences for the turbulencemodel with respect to convective heat transfer.

The partitioned solution approach to conflate high−resolution air flow modeling with a multi−zonenetwork for a coupling to ambient boundaryconditions will be subject of future investigations.

ACKNOWLEDGEMENTSThe first author would like to thank the Dr.−Ing.Leonhard−Lorenz−Foundation for financiallysupporting the participation at the conference. Theauthors also want to express their gratitude to theircolleagues B. Crouse and J. Tölke for theircontributions, as well as to the HPC group at theLeibniz−Rechenzentrum Munich for valuabletechnical support.

REFERENCES[1] M. Krafczyk, Gitter−Boltzmann Methoden:

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[3] O. Filippova, D. Hänel,A Novel Lattice BGKApproach for Low Mach Number Combustion,J. Comp. Phys., 158: 139−160, 2000.

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[8] D.A. Wolf−Gladrow, Lattice−Gas CellularAutomata and Lattice Boltzmann Models,Springer Verlag, 2000.

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[16] A. Bejan, Convection Heat Transfer, 2nd Ed.,J. Wiley, 1995.

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[18] International Alliance for Interoperability,IFCRelease 2.0, Member CD, 1999.

[19] Autodesk,Architectural Desktop Release 2.0[20] B. Crouse,Automatische Gittergenerierung für

zweidimensionale Gebiete unter Verwendungvon Quadtree−Datenstrukturen, Diplomathesis, LS Bauinformatik, TU München, 1999.

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[22] S. Kühner, M. Krafczyk,VirtualFluids − Anenvironment for integral visualization andanalysis of CAD and simulation data, Proc. 5thInt. Fall Workshop ’Vision, Modeling andVisualization 2000’, Saarbrücken, 2000.

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