Environ Fluid MechDOI 10.1007/s10652-013-9316-1

ORIGINAL ARTICLE

Large eddy simulation of dispersion around an isolatedcubic building: evaluation of localized dynamickSGS-equation sub-grid scale model

Farzad Bazdidi-Tehrani · Mohammad Jadidi

Received: 18 February 2013 / Accepted: 7 October 2013© Springer Science+Business Media Dordrecht 2013

Abstract In the present study, the prediction accuracy of a dynamic one-equation sub-gridscale model for the large eddy simulation of dispersion around an isolated cubic build-ing is investigated. For this purpose, the localized dynamic kSGS-equation model (LDKM)is employed and the results are compared with the available experimental data and twoother classic sub-grid scale models, namely, standard Smagorinsky–Lilly model (SSLM)and dynamic Smagorinsky–Lilly model (DSLM). It is shown that the three SGS modelsgive results in good agreement with experiment. However, near the ground level of the lee-ward wall, dimensionless time-averaged concentration, 〈K 〉, profile is not quite similar tothe experimental data. It is also demonstrated that the LDKM predicts the values of 〈K 〉 onthe roof, leeward and side walls more acceptably than the SSLM and DSLM. Whereas, thestreamwise elongation of time-averaged structures of the plume shape is more over-estimatedwith the LDKM than with the other two SGS models. In terms of numerical difficulty, theLDKM is found to be stable and computationally reasonable. In addition, it does not sufferfrom a flow dependent constant such as the Smagorinsky coefficient employed in the SSLMmodel.

Keywords Large eddy simulation · Sub-grid scale models · Localized dynamickSGS-equation model · Dispersion

List of symbols

〈 〉 Time-averaged valuec ConcentrationCD Dynamic coefficient in dynamic Smagorinsky–Lilly modelCμ = 0.09 ConstantCs Smagorinsky model constant

F. Bazdidi-Tehrani (B) · M. JadidiSchool of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846-13114, Irane-mail: bazdid@iust.ac.ir

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Cτ Dynamic coefficient in localized dynamic kSGS-equation modelCFL Courant–Friedrichs–Lewy numberDm Molecular diffusion coefficientDSGS Sub-grid scale turbulent diffusivityc Filtered ConcentrationI Turbulence intensityJ SGS

j SGS scalar fluxk Turbulence kinetic energyK Dimensionless concentrationkSGS Sub-grid scale turbulence kinetic energyLi j Leonard stress tensorQC

x = 〈ui 〉〈c〉 Convective flux in streamwise directionQT

x = 〈u′i c′〉 + J SGS

i Turbulent flux in streamwise directionSi j Rate of strain tensorScSGS Sub-grid scale turbulent Schmidt numbert∗ = t/T, T = Hb/Ub Dimensionless timeTi j Subtest scale stress tensoru′, v′ and w′ Velocity fluctuation componentsuτ Friction velocityUb = 3.3 m/s Velocity at the building heightUe = 0.63 m/s Velocity of effluentU∞ = 4.5 m/s Free stream velocity (mean velocity at y = δ)

y+ Dimensionless wall distancey+

p y+ At the first point from the wallκ = 0.41 Von Karman constant� Grid filter width� Test filter widthε Turbulence dissipation rateλuu−z Integral length scale based on streamwise velocity fluctuation in

z directionλvv−z Integral length scale based on normal velocity fluctuation in z

directionλww−z Integral length scale based on spanwise velocity fluctuation in z

directionλww−x Integral length scale based on spanwise velocity fluctuation in x

directionυSGS Sub-grid scale eddy viscosityτi j SGS stress tensorξ = x A − x B Separation distance between points A and B

Superscript

′ Fluctuation componentC Convective fluxT Turbulent flux

Subscript

rms Root-mean-squared

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SGS Sub-grid scaletest Test filterx Streamwise directiony Normal directionz Spanwise direction− Filtereda Test filtered

1 Introduction

In the lower level of atmospheric boundary layer, plume dispersion and its prediction areessential for the appropriate design of exhaust stacks and air intakes. However, existence ofcomplex vortical flow structures around buildings makes it difficult to accurately predict thepollutant dispersion. For the assessment of dispersion around an isolated model building,several experimental studies have been reported [1–3]. Li and Meroney [1] have investigatedthe dispersion of effluent plumes emitted on or in the near-wake region of a cubical modelbuilding. Time-averaged concentration measurements have been made on the model buildingfor three different roof vent locations and three different building orientations. It is found thatdue to the presence of sharp edges on the model building the concentration level on the leeface is greatly reduced. In addition, the optimum location for the intake vent on the buildingis recommended. This experimental work has been adopted numerously as a benchmark forthe computational fluid dynamics (CFD) predictions of dispersion around an isolated cubicbuilding [2,4,5].

At the present time, CFD is widely employed to study the flow field, pollutant transportand its prediction around buildings [6,7]. Since flow dynamics around buildings with asharp edge are highly unsteady and turbulent, turbulence modeling is very important andit has a significant influence on the accuracy of results. Turbulence models based on theReynolds averaged Navier–Stokes (RANS) equations may show poor prediction accuracyof concentration distribution compared to the experiment, as reported by Tominaga andStathopoulos [5] and Blocken et al. [8]. In addition, the unreliable results by some of thenon-linear k-ε models concerning the clear periodic vortex shedding behind an isolatedbuilding show that the unsteady RANS based models can not capture the entire turbulencespectrum [9]. In contrast, transient approaches such as the large eddy simulation (LES) seemto be further consistent with the complex vortical flow fields and time-dependant turbulentphysics of this kind of dispersion problem. Therefore, the LES approach may be required toachieve more accurate results [2,10,11]. It is expected that the utilization of LES for variouswind engineering applications will increase considerably in the near future [7,12].

In some earlier published papers [2,4,11], RANS and LES predictions for plume disper-sion have been compared. It is concluded that LES appears to be more accurate than RANSin the prediction of the concentration field since it is capable of capturing the concentrationfluctuations. Tominaga and Stathopoulos [2] have assessed the accuracy of LES in modelingplume dispersion near and around a simple building model and reported that LES computationcan provide important information on instantaneous fluctuations of concentration, which cannot be obtained by RANS computations. They have also concluded that simple LES model-ing [i.e., standard Smagorinsky–Lilly model (SSLM)] gives better results than conventionalRANS computation modeling of the distribution of concentration. They have applied theSSLM with the empirical constant, Cs = 0.12, for the sub-grid scale eddy viscosity model.Near the wall, the length scale is modified by a van Driest damping function and the sub-grid

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scale turbulent Schmidt number is set to 0.5. Gousseau et al. [4] have carried out RANS andLES computations for two configurations of isolated buildings with distinctive features. Theyhave focused on the relative influence of convective and turbulent mass fluxes in the transportprocess and the role of these fluxes in the prediction accuracy of RANS and LES simulations.For this purpose, two cases with distinctive features in terms of the transport process havebeen selected. In case 1, the stack is relatively high and discharges the pollutants outside thebuilding wake, which decreases the influence of the building on the dispersion of the plume.In case 2, the source is located directly on the roof of the building and the pollutant gas isreleased with low momentum ratio into the rooftop separation bubble. It is demonstrated thatin order to predict accurately the concentration field, the correct simulation of the convectiveand turbulent fluxes is necessary. In addition, it is emphasized that when the influence of thebuilding on the dispersion process is higher (case 2), the accuracy of LES is clearly betterbecause this model computes more accurate convective mass fluxes, especially in separationregions on the roof and in the wake of the building. More recently, Tominaga and Stathopou-los [11] have investigated the spatial distribution of the turbulent scalar flux within a buildingarray with a point source for the simple street canyon model by both LES and RNG turbu-lence models. They have reported that the RNG model under-estimates turbulence diffusionin the building canyon as compared with LES and there are large differences between thedistributions of the estimated eddy viscosity and eddy diffusivity. It is also shown that thesuperiority of flow field prediction by LES has a notable influence on the concentration dif-fusion field. In addition, the effects of inlet boundary condition [13], SGS turbulent Schmidtnumber and turbulent Schmidt number [14,15], thermal stabilities on turbulent transports[16,17] and the effects of adjacent buildings on pollutant transport [18] in the field of windengineering have been investigated. However, very few studies have addressed the importantissue of the dependence of LES results on the sub-grid scale models [19–22]. Iizuka andKondo [19] have simulated the turbulent flow over a two-dimensional steep hill employingtwo types of traditional sub-grid scale models in LES. It is reported that the dynamic SGSresults are not in good agreement with the available experiment, due to its comparativelyinaccurate estimation of the model coefficient near the ground surface. Thus, in order toimprove the prediction accuracy of the dynamic SGS model, a SGS model as a combinationof the standard and dynamic Smagorinsky models, is introduced. Gousseau et al. [22] haveperformed LES of wind flow around a high-rise building more recently. Several cases havebeen run and analyzed on two different grids and with two different SGS models, namely, theSSLM and its dynamic version. It has been shown that generation of inflow turbulence basedon the vortex method provides accurate results. Moreover, the agreement between numericaland experimental results has been quantified by different validation metrics and systematicgrid and model variation technique has been used to provide estimates of the modeling andnumerical error contributions.

Although the effect of SGS models on the turbulent flow regarding the field of windengineering has been considered in a few numerical investigations [22], the influence ofSGS models on the dispersion of a passive scalar (concentration) on and around a three-dimensional sharp edge model building has not yet been adequately addressed [22].

In the present paper, the influence of the localized dynamic kSGS-equation model (LDKM)on the performance of LES concerning the dispersion around an isolated cubic model buildingwith a flush vent located on the roof is examined. In addition, the effects of two othertraditional SGS models, namely, the SSLM and dynamic Smagorinsky–Lilly model (DSLM),are considered and their results are compared with that of LDKM. Streamwise and spanwisedistributions of time-averaged concentration, time-averaged velocity and its fluctuation arealso compared with the available experimental data of Li and Meroney [1] and also Tominaga

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and Stathopoulos [2]. In addition, sub-grid scale mass fluxes of the three SGS models arecompared with each other. The geometry of the model building and flow parameters areadopted from the wind tunnel data by Li and Meroney [1].

2 Governing equations

The basic idea in LES is to resolve large grid scales and to model small sub-grid scales. Forthis reason, the equations are filtered and by applying filters to the governing equations of flowand concentration fields, the LES equations for the resolved fields are derived. In the presentstudy, the top hat filter is used since the finite volume method is implemented. Although thecontribution of instantaneous small scales is removed by the filtering procedure, their effectsremain in unclosed residual terms representing the influence of the sub-grid scales on theresolved scales [23]. For full derivations, the reader is referred to [24]. Applying the top hatfilter procedure to the governing equations, the LES filtered equations are acquired as in Eqs.(1)–(3).

∂ρ

∂t+ ∂ρui

∂xi= 0 (1)

∂

∂t(ρui ) + ∂

∂x j(ρui u j ) = − ∂ p

∂xi+ ∂σi j

∂x j− ∂τi j

∂x j(2)

∂ρc

∂t+ ∂(ρu j c)

∂x j= − ∂

∂x j(J SGS

j ) (3)

σi j is filtered molecular viscosity stress tensor. In the present work, the concentration istreated as a scalar transported by an advection-diffusion equation as Eq. (3). These equationsgovern the evolution of the large, energy-carrying, scales of motion. The effect of the smallscales appears in the flow field by means of SGS stress tensor through Eq. (4) and in thescalar field through Eq. (5) for the sub-grid scalar flux, which must be modeled.

τi j = ui u j − ui u j (4)

J SGSi = ui c − ui c (5)

2.1 SGS stress modeling

The turbulent scales in LES which cannot be resolved by the grid need to be modeledaccurately. Consequently, the unknown SGS stresses resulting from the top hat filteringoperation must be modeled. The SGS models in the present simulation employ the Boussinesqhypothesis [23,25], computing the SGS stresses from Eq. (6).

τi j − 1

3τkkδi j = −2υSGSSi j (6)

υSGS is the SGS eddy viscosity and Si j is the rate of strain tensor for the resolved scale, asdefined by Eq. (7). The isotropic part of the SGS stresses, τkk , is not modeled but it is addedto the filtered static pressure term.

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Si j = 1

2

(

∂ui

∂x j+ ∂u j

∂xi

)

(7)

υSGS is modeled on the basis of three approaches in this paper, namely, the SSLM, DSLMand LDKM. These approaches are described in the following sub-sections.

2.1.1 Standard Smagorinsky–Lilly model (SSLM)

The SSLM [26] is regarded as the simplest model whereby the SGS eddy viscosity is modeledby Eq. (8).

υSGS = (Cs�)2∣

∣S∣

∣ (8)

and the magnitude of the strain rate is computed as:

∣

∣S∣

∣ =√

(

2Si j Si j)

(9)

Cs is the Smagorinsky constant and the filter width of the computational cell [� =(�x�y�z)

1/3] is taken as the local grid size. The SGS viscosity becomes quite large near awall since the velocity gradient is very large at the wall. However, because the SGS turbulentfluctuations near a wall approach to zero the SGS viscosity should be modified in that region.A damping function is added to ensure this. A more convenient way to dampen the SGSviscosity near a wall is the upper limit length scale modification as follows:

� = min{

(�x�y�z)1/3, κn

}

(10)

n is the distance to a nearest wall and κ is von-Karman constant. In this paper, the filter-widthis modified near a wall as in Eq. (10).

It is important to note that Cs in Eq. (8) is not a universal constant and it is flow dependent.This is the most serious shortcoming of SSLM. It has been reported to vary in the range of0.065–0.25 [27]. Nonetheless, the Cs value of around 0.1 [28] has been found to yield thebest results for a wide range of flows. In the present study, Cs = 0.1 is used in SSLM.

2.1.2 Dynamic Smagorinsky–Lilly model (DSLM)

As mentioned above, Cs is not a universal constant and its value varies from one flow toanother. In the DSLM, proposed by Germano et al. [29] and Lilly [30] the model constant isdynamically computed based on the information provided by the resolved scales of motion.A test filter is applied to the filtered equations in order to remove the smallest scales of theresolved field. Application of the test filter to Eq. (2) leads to:

∂ρui

∂t+ ∂

(

ρui u j)

∂x j+ ∂p

∂xi= ∂

(

σi j)

∂x j− ∂Ti j

∂x j(11)

Ti j is the subtest scale stress tensor:

Ti j = ui u j − ui u j (12)

The fundamental concept of the dynamic modeling is the assumption that Ti j is similar toτi j so that Ti j can be modeled in the same way as τi j , as represented in Eq. (13).

T ai j = Ti j − 1

3δi j Tkk = −2υSGSSi j

where υSGS = CD�2∣

∣

∣

Si j

∣

∣

∣ with CD = C2s

(13)

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� is the test filter width. The same coefficient is used in the models of both stress tensors,τi j and Ti j . Test filtering τi j and subtracting it from Ti j results in:

Li j = ui u j − ui u j = Ti j − τi j (14)

The Leonard stresses, Li j , can be interpreted as the resolved scales between the test filterscale, �, and the grid filter scale,�. Using the assumption that CD�2 is only weakly varyingin space it is taken out of the test-filter operation, resulting in Eq. (15):

Li j − 13δi j Lkk = −2CD

(

�2∣

∣

∣

Si j

∣

∣

∣

Si j − �2

∣

∣Si j∣

∣ Si j

)

= CD Mi j

Mi j = −2(

�2∣

∣

∣

Si j

∣

∣

∣

Si j − �2

∣

∣Si j∣

∣ Si j

) (15)

The right hand side of Eq. (15) is written as the model factor, CD , which has to be determinedand a factor, Mi j , which is calculated from the filtered velocity data. CD is obtained byminimizing the difference between the left and right hand sides of Eq. (15) in a least squaressense, leading to:

Q ≡(

Li j − 1

3δi j Lkk − CD Mi j

)2

(16)

∂ Q

∂C= 0 ⇒ CD = Li j Mi j

Mi j Mi j(17)

To avoid any numerical instability, CD is clipped at 0.0. It is also clipped at the positive sideto 0.23 [28].

2.1.3 Localized dynamic kSGS-equation model (LDKM)

DSLM suffers from the drawback of numerical instability, associated with the negative val-ues and large variation of the CD coefficient. Usually, this problem is fixed by averagingthe coefficient in some homogeneous flow direction [23]. In real applications, local smooth-ing and clipping is used [28]. In addition to the above-mentioned two models, a dynamicone-equation model for the SGS turbulence kinetic energy (LDKM), suggested by [31], ispresented. The main objective of this model is to be applicable to real industrial flows. Fur-thermore, being a dynamic model, LDKM has the great advantage that the coefficients arecomputed rather than being prescribed. A transport equation is solved for the SGS turbulencekinetic energy, kSGS, as presented in Eq. (18).

∂

∂t(ρkSGS) + ∂

∂xi(ρkSGSui ) = ∂

∂xi

[

μSGS

σk

∂kSGS

∂xi

]

− τi j∂ui

∂xi− Cε

k3/2SGS

�(18)

σk is equal to 1 and kSGS is defined by Eq. (19).

kSGS = 1

2(ukuk − ukuk ) (19)

τi j , quoted in Eq. (6), is modeled as follows:

τi j − 2

3δi j kSGS = −2υSGSSi j = −2Cτ� k1/2

SGSSi j (20)

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� is the grid filter width. Also, the model constants, Cε and Cτ , are determined dynamically[31]. The Leonard stresses are defined according to Eq. (21).

Li j ≡ ui u j − ui u j = Ti j − τi j (21)

and by the trace of the tensor, Li j , an expression for the kinetic energy resolved at the testlevel, ktest , is obtained:

ktest = 1

2

(

ui ui − ui ui

)

= Lii

2(22)

The production of ktest is given by:

Ptest = −Li j

(

∂ui

∂x j

)

(23)

and its dissipation by:

εtest = (υ + υSGS)

(

∂ui

∂x j

∂ui

∂x j− ∂ui

∂x j

∂ui

∂x j

)

(24)

The Leonard stress tensor is modeled as:

Li j − 1

3δi j Lkk = −2ρCτ

�k1/2test

Si j = Cτ Xi j (25)

Si j is computed from the data on the test level. The coefficient, Cτ , is determined by the leastsquares technique, minimizing the difference between left and right hand sides in Eq. (25),resulting in:

Cτ = Li j Xi j

Xi j Xi j(26)

A similarity assumption between εSGS and εtest is adopted to obtain an expression for Cτ :

εtest = Cεk3/2test /

�. (27)

The value of Cε is acquired by inserting the values of ktest and εtest from Eqs. (22) and (24).

2.2 SGS scalar flux, J SGSi , modeling

In line with the standard gradient diffusion hypothesis [23], eddy diffusivity models para-meterize the SGS fluxes which are proportional to the resolved scalar gradient, Eq. (28).Simplicity has turned eddy diffusivity models into the most popular since the eddy viscosityassumption is first introduced by [26].

J SGSi = − (Dm + DSGS)

∂c

∂xi= −

(

Dm + υSGS

ScSGS

)

∂c

∂xi(28)

Dm is the molecular diffusion coefficient and DSGS is the model coefficient (SGS turbulentdiffusivity). It is related to the SGS eddy viscosity through the SGS Schmidt number, ScSGS,as defined in Eq. (29).

DSGS = υSGS

ScSGS(29)

Since the SGS eddy viscosity is related to the velocity field, the value of ScSGS has to bedefined. Most of the prior studies have selected it to be a constant value [2,3]. In the presentpaper, ScSGS = 0.7 is considered for the sub-grid scalar flux modeling.

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Table 1 Distance (y/Hb) from adjacent walls of five initial cells for each grid resolution

Top wall Windward wall Leeward wall Side wall Floor

Fine grid (456327 cells)

First node 0.07586 0.0436 0.07702 0.07138 0.03704

Second node 0.15628 0.08854 0.15674 0.14704 0.07612

Third node 0.24154 0.13486 0.23924 0.22724 0.11734

Fourth node 0.3319 0.18264 0.32464 0.31224 0.16084

Fifth node 0.42768 0.23188 0.41302 0.40236 0.20674

Medium grid (148734 cells)

First node 0.0784 0.04698 0.07356 0.07212 0.03704

Second node 0.1803 0.10038 0.15998 0.15866 0.07612

Third node 0.3128 0.16106 0.26154 0.26252 0.11734

Fourth node 0.48502 0.23002 0.38086 0.38714 0.16084

Fifth node 0.70892 0.30838 0.52106 0.53668 0.20674

Coarse grid (80347 cells)

First node 0.07583 0.03978 0.07398 0.08274 0.03704

Second node 0.189574 0.09282 0.1694 0.1911 0.07612

Third node 0.36019 0.16356 0.29248 0.33308 0.11734

Fourth node 0.616114 0.25786 0.45128 0.51908 0.16084

Fifth node 1 0.38358 0.65612 0.76272 0.20674

3 Computational methodology

3.1 Computational domain and grid

The computational domain under investigation is adopted based on the experimental setupof Li and Meroney [1], as illustrated in Fig. 1. The dimensions of domain are 26Hb, 6Hb,and 13Hb (Hb = height of cube) in the x, y, and z directions, consecutively, in line withthe Architectural Institute of Japan (AIJ) guidelines [32] for practical applications of CFDto pedestrian wind environment around buildings. An upstream distance of 5.5Hb with theorigin of the coordinate system at the center of the cube’s bottom face is considered accordingto [32].

Fig. 1 Computational domain and boundary conditions of isolated cubic building: a front view, b top view

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Fig. 2 a Cross-section grid resolutions at different planes, b contour of y+p accompanied with grid resolutions

on different walls of cubic building

In order to find an appropriate grid resolution in the present study, three computationalgrids, namely fine, medium, and coarse grids with 456327, 148734 and 80347 hexahedralnon-uniform computational cells are employed and evaluated using the two-point correlationtechnique [28]. A different initial cell size using a geometric progression with a commonratio of 1.02 up to 1.06 is adopted for the grid (see Fig. 2a). Distance from the adjacent wallsfor five initial cells of the grid resolution is listed in Table 1 for each surface separately. Thepresent grid is refined near the wall to provide a first cell y+ height of between 14 and 35on different walls in the computational domain. As an illustration, the y+

p contour on thedifferent wall of cubic building is presented in Fig. 2b. The distribution of y+

p is non-uniformbecause the wall shear is not constant over the surfaces.

Two-point correlations are useful when describing some characteristics of the turbulence[33,34]. From the two-point correlations, it can be found that by how many cells the largestscale is resolved which is very informative about the grid resolution in LES. For calculatingthe two-point correlation in the x direction, two points along the x axis, say x A and x B attime t0 are picked and their fluctuating velocity (u′, v′, and w′) are sampled. The correlationof u′ at these two points is calculated as in Eq. (30).

Buu−x (x A, x B , t0) = u′(x A, t0)u′(x B , t0) (30)

where, indices u and x represent the component of velocity and direction of sampling, respec-tively. Often, it is expressed as:

Buu−x (ξ, t0) = u′(x A, t0)u′(x A + ξ, t0) (31)

where, ξ = x A − x B is separation distance between points A and B. It is expected that thetwo-point correlation function be related to the largest eddies. It is convenient to normalizeBuu−x so that it varies between −1 and +1. The normalized two-point correlation is definedas:

Ruu−x (ξ, t0) = u′(x A, t0)u′(x A + ξ, t0)

urms(x A, t0)urms(x A + ξ, t0)(32)

The behavior of Ruu−x exhibits a correlation length which characterizes the size of largesteddies. This macro-scale is usually called the integral length scale, λuu−x , which is definedas the integral of Ruu−x in the direction of x over the separation distance. It is calculated

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as:

λuu−x =∞

∫

0

Ruu−x (ξ, t0)dξ (33)

Ratios of integral length scale (λ) to grid spacing is considered to be an appropriate methodfor the evaluation of grid resolution since these ratios demonstrate the number of cells in theresolved largest scale [33,34].

Figure 3 shows the two-point correlations in the lateral (z) direction for streamwise,normal and spanwise velocity components (Ruu−z, Rvv−z and Rww−z), on a line located atx/Hb = 1 and y/Hb = 0.5. The integral length scales based on the two-point correlationsare calculated and normalized by mean grid spacing in the lateral direction (˜�z). The meangrid spacing in the lateral direction is calculated as:

˜�z = Lz

Nz(34)

where, Lz and Nz are length of computational domain and number of cells in the lateraldirection, respectively. In Table 2, the ratio of integral length scale (λ) to the mean grid spacingin the lateral direction (˜�z) is tabulated. As expected, the ratios of λuu−z/�z, λvv−z/�z andλww−z/�z decrease when the mesh becomes coarser. From Table 2, it is found that 6 cellshave been included in the lateral integral length scale for the fine grid seeming to be sufficientbased on [33,34]. Although, it may reasonably be doubted as to whether that extra fine gridresolution would lead to a better prediction accuracy.

Figure 4 demonstrates the time-averaged and instantaneous plume shapes obtained byapplying the three grid sizes, at t∗ = 264. The time-averaged structures of the plume com-puted by all the grids are different in comparison with each other. It is uncommonly elongatedin the streamwise direction for the coarse grid as compared to the medium and fine grids. Inaddition, the instantaneous structures of the plume reveal that the coarse and medium gridsare more diffusive than the fine grid in the streamwise direction.

Figure 5 depicts the spanwise distributions of streamwise convective and turbulent massfluxes, respectively, at x/Hb = 3, y/Hb = 1. Inspection of the convective and turbulentmass fluxes demonstrates that dispersion is mostly under the influence of the convective fluxrather than the turbulent one in the streamwise direction. The convective flux is nearly 3 timeshigher than the turbulent flux in the core of the plume, at z/Hb = 0. It can be seen that the gridresolution has considerable effects on both of these fluxes such that they both decrease whenthe grid resolution is enhanced. In addition, the differences between the concentration fields(plume shapes) obtained by the three different grids can be investigated by the examinationof the convective and turbulent mass fluxes concerning each grid. As mentioned in Fig. 4,the plume shape for the coarse grid elongates in the streamwise direction more than thatfor the other two grids. This uncommon elongation of the plume shape for the coarse gridmay be because of the higher streamwise convective and turbulent flux values displayed inFig. 5. Results show that an increase of 40 % in the total flux (convective plus turbulent)leads to nearly 60 % more streamwise elongation for the coarse grid as compared with thefine grid.

3.2 Boundary conditions

The present boundary conditions are set as: at the inlet of computational domain, a velocityprofile acquired from the experimental data (Fig. 6a) is employed. At the exhaust outlet, a

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Fig. 3 Two-point correlations inlateral direction for a streamwise,b normal and c spanwise velocitycomponents, at x/Hb = 1 andy/Hb = 0.5

Table 2 Values of ratiosλuu−z/˜�z, λvv−z/˜�z andλww−z/˜�z

Grids λuu−z/˜�z λvv−z/˜�z λww−z/˜�z

Fine grid 4.63 2.42 6.1Medium grid 2.86 1.7 5.9Coarse grid 2.41 1.51 4.65

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Fig. 4 Time-averaged and instantaneous plume shapes computed by three different grids (at t∗ = 264)

Fig. 5 Spanwise distributions of a streamwise convective flux and b streamwise turbulent flux, at x/Hb = 3and y/Hb = 1

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Fig. 6 Inflow boundary condition based on experiment [1]: a velocity profile b turbulence intensity

Table 3 Flow parameters basedon the experiment [1]

Boundary layer thickness (δ) (cm) 30Exit velocity ratio (Ue/Ub) 0.19Velocity at the building height (Ub) (m/s) 3.3Velocity of effluent (Ue) (m/s) 0.63Contaminant release rate (Qe)(cm3/s) 12.5Height of cubic building (Hb) (cm) 5Free stream velocity (U∞) (m/s) 4.5

uniform velocity profile of pure helium release at a flow rate of 1.25×10−5 m3/s, is adopted.The top boundary is treated as a free stream boundary (slip boundary condition) and bottomplane satisfies the no-slip condition. At the outflow, a fixed value boundary condition for thepressure and a zero gradient boundary for the velocity (outlet boundary condition) is used.The symmetry condition is used at the side walls. The Reynolds number is equal to 1.1×104

on the basis of the cube height and time-averaged inlet velocity at cube height. A list of theflow parameters is represented in Table 3. Also, the thermal stratification is assumed to beabsent. Thickness of the boundary layer (δ) and free stream velocity (U∞) are equal to 0.3 mand 4.5 m/s, respectively. Moreover, the Werner and Wengle [35] wall boundary condition isimplemented for all wall surfaces in the simulation.

To generate a time-dependent inlet boundary condition, the vortex method (VM) technique[36] is employed to produce the instantaneous velocity profiles from the given turbulenceintensity data (Fig. 6b) at the inlet. With this approach, a perturbation is added on a specifiedtime-averaged velocity profile (Fig. 6a) via a fluctuating two-dimensional vorticity field(two-dimensional in the plane normal to the streamwise direction). For this purpose, the time-averaged streamwise velocity component is set to obey the expressed power law, as depictedin Fig. 6a. This expression represents the flow of atmospheric boundary layer conditions whileother velocity components are assumed to be zero. Also, the value of n = 0.19 corresponds tothe wind tunnel experiment performed by Li and Meroney [1]. The implementation of VMrequires the turbulence kinetic energy, k, and turbulence dissipation rate, ε, profiles at theinlet of computational domain. In the present work, k and ε are estimated based on the AIJguidelines for practical applications of CFD to pedestrian wind environment around buildings

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Table 4 Numerical schemes Pressure-velocity coupling algorithm SIMPLE algorithm

Spatial discretization Bounded centraldifferencing scheme

Time discretization Second order implicit schemeTime step size, �t 5 × 10−5 (s)Sampling time 4 (s) (≈80,000 time steps)

[32] as follows:

k(y) (I (y) × u(y))2

ε(y) Pk(y) C0.5μ k(y) ∂u

∂y(35)

Pk is the production of turbulence kinetic energy. It should be mentioned that the value of ε

is calculated by the local equilibrium assumption [23].

3.3 Numerical technique

An unsteady algorithm on the basis of the finite volume technique is employed for hexahe-dral non-uniform computational cells using OpenFOAM [37]. The convection term in themomentum equation is discretized using a bounded central differencing scheme. The SIM-PLE algorithm [38] is employed for the pressure-velocity coupling. The resulting algebraicequation for all the flow variables except pressure is solved iteratively using a preconditionedbi-conjugate gradient method at each time step. Time marching is performed using a secondorder implicit scheme.

In the present work, since the effects of initial conditions are typically remembered bythe flow for a considerable time period, initial values from a separate RANS simulation areused for all variables. Initialization based on a separate RANS simulation not only reducesthe time needed before the data sampling for LES, but also decreases any instability problemthat may happen due to an unrealistic initial condition. The initial transient condition needsaround 20,000 time steps in order to be washed out and a statistically steady state operatingcondition is reached.

Afterwards, time averaging is begun and first and second order statistics for all relevantquantities (velocity, concentration) are collected. Averaging continued until acceptable sta-tistical convergence is achieved. This usually required 80,000 time steps equivalent to 10–15domain flow-through times traveled by the mainstream. Since the source of dispersion islocated in a zone of high turbulence intensity, a long averaging period is required to get sta-tistically steady results. In the present work, the required time periods obtaining the statisticalvalues for the DSLM and LDKM are 1.19 and 1.37 times, consecutively, more than that forthe SSLM.

The physical time step is chosen such that the Courant–Friedrichs–Lewy number, CFL,is restricted to 0.5 in the most refined portions of each grid enforcing the constant timestep expressed as a non-dimensional quantity (�t × Ub/Hb) of 3.3 × 10−3. The details ofnumerical method regarding the dispersion around an isolated cubic building are summarizedin Table 4.

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Fig. 7 a Streamwise distribution of 〈K 〉 on centerline of roof and leeward wall, b lateral distribution of 〈K 〉on centerline of roof and side wall

4 Discussion of results

4.1 Concentration field

Figure 7a demonstrates the distributions of dimensionless time-averaged concentration, 〈K 〉,on the roof centerline and leeward wall in the streamwise direction. Eq. (36) represents thedefinition of 〈K 〉:

〈K 〉 = 〈c〉Qe

HbUb

(36)

c is concentration and the rest including Qe, Hb and Ub are provided in Table 3. A sig-nificant characteristic of the plume dispersion around an isolated cubic building is thatthe maximum concentration on the roof takes place somewhat upwind of the stack. Thisis clearly demonstrated in Fig. 7a by both the present LES and LES of Tominaga andStathopoulos [2]. As stated in the Introduction section, they have employed the SSLMwith a damping function whereby the sub-grid scale turbulent Schmidt number is con-sidered as 0.5. The distributions of 〈K 〉 on centerline of roof and leeward wall show thatthe present three sub-grid scale models are capable of capturing the concentration on thedifferent wall surfaces reasonably well. However, the results of the LDKM and DSLMmodels provide slightly better agreement with the available experimental data of Li andMeroney [1] than the SSLM model. In the streamwise direction and on the second half ofroof centerline, the values of 〈K 〉 predicted by the SSLM gives a small under-predictionwhile on the first half the SSLM provides an over-estimation in comparison with theDSLM and LDKM models. In addition, on the centerline of leeward wall all the threemodels slightly over-predict the value of 〈K 〉. Furthermore, at the ground level of the lee-ward wall surface the trend of 〈K 〉 profile is not quite similar to the experimental dataand an over-prediction of 〈K 〉 profile is observed with a relative error of 42.8 % on aver-age.

As illustrated by Fig. 7b, the lateral profiles of 〈K 〉 on the roof centerline are relativelyclose to each other regarding the three sub-grid scale models. Whereas, on the side wall, the

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Fig. 8 Comparison of experimental [1] and present contours of 〈K 〉 on roof, leeward and side walls

LDKM shows moderately better consistency with the experimental data as compared withthe other two models. However, for all the present models, an over-estimation occurs relativeto the experiment which is also indicated by the LES results of Tominaga and Stathopoulos[2]. The SSLM and DSLM models show more tendency towards an over-estimation of lateral〈K 〉, particularly on the side wall.

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Figure 8 displays comparisons of the contours of 〈K 〉 on the roof and wall surfaces (leewardand side walls) acquired from the present three sub-grid scale models and the experiment byLi and Meroney [1]. The overall distributions of 〈K 〉 given by the present LES are similar tothe experiment, although the LES results tend to be slightly diffusive especially at the sideand leeward wall surfaces. Also, the lateral spreading of 〈K 〉 is a little over-predicted on theroof by the three sub-grid scale models, where the SSLM tends to intensify the spreadingout of 〈K 〉 more than the DSLM and LDKM. The LDKM shows a good consistency withthe experimental results for the lateral and streamwise expansions of 〈K 〉. For instance,experiment shows that the outer extend of 〈K 〉 = 0.5 is within the first quarter of the side wallwhereas it is predicted inside the third quarter by the SSLM and within the second quarter bythe LDKM. As stated in Fig. 7, on the leeward and side walls, the three models over-estimatethe value of 〈K 〉 where the DSSL and LDKM do relatively better than the SSLM.

In addition, at the outer extend on the side wall and near the ground level, all the modelsover-predict 〈K 〉 and the LDKM shows the best agreement with the experimental data.In general, the distributions of 〈K 〉 obtained by three sub-grid scale models show goodagreement with the experiment which is attributed to an appropriate convection and turbulencediffusion obtained by the LES. However, the LES approach indicates discrepancies in thewake of the isolated cube near the ground level on the walls, as reported by the earlier study[2] and also the present work. An over-estimation in 〈K 〉 is distinctive in the stated region.The prediction accuracy of LDKM appears to be appropriate and in good consistency withthe experimental data.

The vertical and horizontal (streamwise) distributions of 〈K 〉 and their corresponding iso-surface of 〈K 〉 = 1 in the wake region of the isolated cube at z/Hb = 0 plane, are depicted inFig. 9. It is evident that the three SGS models behave nearly in the same way in the verticaldirection, while in the streamwise direction the LDKM provides a slightly longer predictionthan the other two models. Especially at level 〈K 〉 = 1 the over-estimation is more apparent.Elongation of iso-surface of 〈K 〉 = 1 by the SSLM is estimated to be 2.7 % more than thatof experiment, whilst for the LDKM it reaches 27.8 %. Generally, all the SGS models over-estimate the time-averaged concentration in the streamwise direction, as compared with theexisting experimental data [1]. However, the vertical distribution of 〈K 〉 is more acceptableexcept near the ground and close to the leeward wall of the isolated cubic building.

Figure 10 shows the lateral and vertical contours of 〈K 〉 in the near wake region atx/Hb = 1 and 3. It is observed that all the three SGS models are not able to predict thelateral diffusion very accurately and they under-estimate it to some extent for the lower val-ues of 〈K 〉 and over-estimate it for the higher values, in comparison with the experiment[1]. For example at x/Hb = 3 plane, the experiment shows that the scalar at the low levelof 〈K 〉 = 0.1 diffuse up to the z/Hb = 2, whereas in the LES results by the three sub-gridscale models, it reaches only up to the z/Hb = 1.1 indicating a relative error of 45 % onaverage in comparison with the experimental measurement. In addition, in the vertical direc-tion, distributions of 〈K 〉 are slightly over-predicted by all the three models. For instance,experimnet shows that the lowest extend of 〈K 〉 = 1 at x/Hb = 3 is at y/Hb = 0.7 whilein the LES results it comes down to y/Hb = 0.4 representing a relative error of 43 % onaverage.

Figure 11 depicts the lateral profiles of 〈K 〉 in the near wake region (x/Hb = 3) onthe ground level. As mentioned before (Fig. 9), the LES approach shows discrepancies inthe prediction of 〈K 〉 in the wake region of the isolated cube near the ground level. It isunambiguous that in the wake region of the isolated building, particularly at the centerplane (z/Hb = 0), the three SGS models over-estimate the time-averaged concentration. Forinstance, the LDKM gives an over-prediction of nearly 20.8 % on average in comparison

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Fig. 9 Comparison of a, b, d, f contours of 〈K 〉 and c, e, g iso-surface of 〈K 〉 = 1 in wake region withexperiment, at z/Hb = 0

with the experimental data, whereas it is reduced reasonably on the ground level in the lateraldirection to less than 15 % on average.

Figure 12 demonstrates the presently simulated time-averaged and instantaneous plumeshapes, at t∗ = 264, employing the three SGS models. The time-averaged structures ofthe plumes computed by all the models are nearly similar to each other. However, for theLDKM model, it is to some extent longer in the streamwise direction, as compared with theSSLM and DSLM. Also, the instantaneous structures of the plumes reveal that the LDKMand DSLM are more diffusive in streamwise than the SSLM as explained before in Fig. 9.Additionally, it can be observed that (see also Figs. 7, 8) only little dispersion reaches theupper edge of side walls when applying the LDKM and DSLM, while it is not the case forthe SSLM.

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Fig. 10 Comparison of contours of 〈K 〉 in near wake region with experiment, at x/Hb = 1 and 3

4.2 Sub-grid scale mass flux

Mass transport is considered to be a combination of three transport mechanisms. Those aremolecular, convective and turbulent mass transport. The molecular mass transport is smallrelative to the two others in the turbulent flows. The convective flux is calculated basedon the resolved variable as in Eq. (37) and the turbulent flux is computed as outlined byEq. (38).

QCi = 〈ui 〉〈c〉 (37)

QTi = 〈u′

i c′〉 + J SGSi (38)

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Fig. 11 Lateral profiles of 〈K 〉 in near wake region on ground level, at x/Hb = 3

Fig. 12 Time-averaged and instantaneous plume shapes at t∗ = 264

Sub-grid scale turbulent flux of a scalar, J SGSi , is mostly modeled using the SGS turbulent

Schmidt number, ScSGS, where, the J SGSi of a scalar, c, is related to the scalar gradient by the

generalized gradient diffusion hypothesis (GGDH) assumption [23]. Where, J SGSi is being

proportional to the resolved scalar gradient of a scalar as follows.

J SGSi = −DSGS

∂c

∂xi(39)

DSGS is the SGS turbulent diffusivity and it is related to the SGS eddy viscosity, υSGS, byScSGS as:

DSGS = υSGS

ScSGS(40)

υSGS is computed in different ways based on the choice of the SGS models.

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Fig. 13 Time history of ratio of υSGS/ScSGS at two different locations: a, c, e z/Hb = 0, y/Hb = 0.5 andx/Hb = 1 and b, d, f z/Hb = 0, y/Hb = 0.5 and x/Hb = 2

Table 5 Time-averaged valuesof νSGS/ScSGS

Sub-grid scale νSGS νSGSmodel /ScSGS(x/Hb = 1) /ScSGS(x/Hb = 2))

SSLM 3.3 × 10−5 6.7 × 10−5

DSLM 6.7 × 10−5 8.8 × 10−5

LDKM 7.2 × 10−5 9.4 × 10−5

The time history of the ratio of υSGS/ScSGS for each of the three present SGS models, attwo positions of x/Hb = 1 and x/Hb = 2, is depicted in Fig. 13. The ratio of υSGS/ScSGS

estimated by the SSLM is not similar to the DSLM and LDKM in terms of fluctuation andmagnitude. It is clear that for the DSLM and LDKM, the value of υSGS/ScSGS is deviatedfrom its mean value dramatically and it fluctuates sharply about the mean value whilst it is notthe case for the SSLM. Table 5 provides the computed time-averaged value of υSGS/ScSGS

for each SGS model. This ratio for the LDKM and DSLM is shown to be 118 and 103 %higher than that of the SSLM. According to Fig. 13, the instantaneous value of υSGS/ScSGS

is totally different for each SGS model.

4.3 Numerical stability

The time history of dynamic coefficients, Cτ and CD , according to LDKM and DSLM sub-grid scale models is shown in Fig. 14 at two positions of x/Hb = 1 and x/Hb = 2. It canbe observed that the time history of dynamic coefficient is much smoother for the LDKMthan the DSLM model. The reason is that in the LDKM model, the similarity assumption ismade between the grid level and the intermediate level. Therefore, the denominator, Xi j Xi j ,in Eq. (26) does not tend to zero as much as the denominator, Mi j Mi j , in Eq. (17). Due tothe small variation of the dynamic coefficient in LDKM, no sign of numerical problems is

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Fig. 14 Time history of dynamic coefficients Cτ and CD at four positions of a x/Hb = 1, y/Hb =0.5, z/Hb = 0, b x/Hb = 2, y/Hb = 0.5, z/Hb = 0, c x/Hb = 1, y/Hb = 0.75, z/Hb = 0, dx/Hb = 2, y/Hb = 0.75, z/Hb = 0

Table 6 Time-averaged and standard deviations (sdev) of dynamic coefficient

Sub-grid scale model (x/Hb = 1, y/Hb = 0.5) (x/Hb = 2, y/Hb = 0.5)

〈C〉 sdev(C) =√

1N (C − 〈C〉)0.5 〈C〉 sdev(C) =

√

1N (C − 〈C〉)0.5

LDKM 0.015 0.008 0.016 0.010DSLM 0.038 0.023 0.034 0.021

observed. Moreover, the relatively small variation of the dynamic coefficient in the LDKMmodel leads to the reduction of number of iterations per time step. For example, the numberof iterations per time step is about 5 in LDKM model while it is 7 for DSLM confirmingthe numerical stability of LDKM. However, since an extra transport equation is solved forthe sub-grid scale turbulence kinetic energy in the LDKM model, it is computationally moreexpensive (i.e., it requires more CPU time) than the DSLM. Correspondingly, DSLM isnearly 24 % faster than LDKM in each time step. In Table 6, the time-averaged and standarddeviations [23] of dynamic coefficients for both the LDKM and DSLM sub-grid scale modelsare also presented at two positions of x/Hb = 1 and x/Hb = 2.

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5 Conclusion

In the present study, an evaluation of the influence of the LDKM on the performance ofthe LES of dispersion around an isolated cubic model building is carried out. The results ofLDKM are compared with those of two other traditional SGS models, SSLM and DSLM,and the available experimental data. All the present SGS models show fairly good agreementwith the available experiment. However, on the centerline of leeward wall and in the wakeregion, all models over-predict the values of 〈K 〉. It is also demonstrated that the LDKMmodel predicts the values of 〈K 〉 more acceptable in comparison with the SSLM and DSLMon the roof, leeward and side walls of the isolated cubic building.

Additionally, the time-averaged structure of the plume computed by all the models islonger than expected and it is slightly more elongated in the streamwise direction for theLDKM and DSLM. It is also demonstrated that all the SGS models under-predict the lateraldiffusion in the wake region of model building for low value of 〈K 〉. Furthermore, the timehistory of SGS turbulent diffusivity shows that it is deviated from its mean value dramaticallyfor the LDKM and DSLM, fluctuating sharply about the mean value whilst it is not the casefor the SSLM. Generally, the LDKM represents a number of advantages, as compared withthe SSLM and DSLM. Contrary to the SSLM, its model factor in the definition of the SGSeddy viscosity is not a constant value needing to be used prior to a simulation. Moreover,for the LDKM the local values of the dynamic coefficients could be used without seriousnumerical problems whereas the DSLM suffers from a numerical instability.

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