Large-eddy simulation of turbulent flow and …wang44/Courses/MECH3492...Constrained large-eddy simulation of wall-bounded compressible turbulent flows Phys. Fluids 25, 106102 (2013);

Large-eddy simulation of turbulent flow and dispersion over a matrix of wall-mounted cubesMohammad Saeedi and Bing-Chen Wang Citation: Physics of Fluids 27, 115104 (2015); doi: 10.1063/1.4935112 View online: http://dx.doi.org/10.1063/1.4935112 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/27/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A stochastic perturbation method to generate inflow turbulence in large-eddy simulation models:Application to neutrally stratified atmospheric boundary layers Phys. Fluids 27, 035102 (2015); 10.1063/1.4913572 Constrained large-eddy simulation of wall-bounded compressible turbulent flows Phys. Fluids 25, 106102 (2013); 10.1063/1.4824393 A wall-layer model for large-eddy simulations of turbulent flows with/out pressure gradient Phys. Fluids 23, 015101 (2011); 10.1063/1.3529358 Designing large-eddy simulation of the turbulent boundary layer to capture law-of-the-wall scalinga) Phys. Fluids 22, 021303 (2010); 10.1063/1.3319073 High-pass filtered eddy-viscosity models for large-eddy simulations of transitional and turbulent flow Phys. Fluids 17, 065103 (2005); 10.1063/1.1923048

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PHYSICS OF FLUIDS 27, 115104 (2015)

Large-eddy simulation of turbulent flow and dispersionover a matrix of wall-mounted cubes

Mohammad Saeedi and Bing-Chen Wanga)

Department of Mechanical Engineering, University of Manitoba,Winnipeg, Manitoba R3T 5V6, Canada

(Received 9 April 2015; accepted 16 October 2015; published online 13 November 2015)

Turbulent flow over a matrix of wall-mounted cubic obstacles along with continuousrelease of a passive scalar from a ground-level point source has been investigatedusing wall-modeled large-eddy simulation (LES). The cubes are fully submergedin a modeled urban atmospheric boundary layer with high turbulence intensities.An inlet boundary condition has been proposed to reproduce the high turbulencelevel of the approaching flow based on generation of grid turbulence. Coherent flowstructures induced by the cubes and their influences on dispersion of the concen-tration plume in the context of the highly disturbed flow are also investigated. Thespatial evolution and temporal cascades of the kinetic and scalar energies havebeen examined in terms of their transport equations and resolved spectra. In or-der to validate the LES approach, numerical predictions of turbulence statistics forboth velocity and concentration fields have been thoroughly validated against a setof comprehensive water-channel measurement data. C 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4935112]

I. INTRODUCTION

The massively developing urban areas with different buildings in proximity of each other alongwith the safety requirements for areas with dense human habitation make it an important topic tostudy wind engineering and turbulent dispersion of contaminants in built-up (urban) environments.To this purpose, extensive experimental and numerical studies have been conducted to investigateturbulent flows1–3 and dispersion4,5 over a group of wall-mounted obstacles and buildings.

In order to investigate the flow and pollutant dispersion in urban environments, a number oflarge-scale field experiments have been performed, including URBAN 2000 meteorological andtracer field campaign conducted in Salt Lake City, Utah,6 Basel UrBan Boundary Layer Experiment(BUBBLE) which measured the wind flow through and above a homogeneous urban area,7 andMock Urban Setting Trial (MUST) conducted at US Army Dugway Proving Ground which studiedplume dispersing through an array of building-size obstacles.8 Besides the MUST field trials, Yeeet al.9 and Hilderman and Chong10 conducted several sets of scaled water-channel experiments toinvestigate flow and dispersion in modeled urban environments. They used fiber-optic laser Doppleranemometry (LDA) and laser induced fluorescence (LIF) to measure the velocity and concentrationfields, respectively. Brown et al.1 conducted a series of high-resolution measurements using pulsedwire anemometer (PWA) in a boundary-layer wind tunnel over different arrays of building-likeobstacles. They studied the first- and second-order turbulence statistics in the central vertical planeof a two-dimensional (2-D) array of wide obstacles and a three-dimensional (3-D) array of cubicobstacles immersed in a simulated atmospheric boundary layer (ABL).

Numerical simulation of the pollutant dispersion in an urban area has been a challenging sub-ject owing to the fact that the geometry of the domain is usually complex, the Reynolds numbers ofthe approaching flow are typically high, and the interaction of instantaneous concentration field with

a)Author to whom correspondence should be addressed. Electronic mail: [email protected]. Tel.: +1 (204)474-9305. Fax: +1 (204) 275-7507.

1070-6631/2015/27(11)/115104/33/$30.00 27, 115104-1 ©2015 AIP Publishing LLC


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115104-2 M. Saeedi and B.-C. Wang Phys. Fluids 27, 115104 (2015)

the dynamically evolving coherent flow structures occur over a wide range of spatial and temporalscales. In view of these challenges, there have been essentially two major approaches for numericalmodeling of urban flow and dispersion; namely, the Reynolds-averaged Navier-Stokes (RANS) andlarge-eddy simulation (LES) approaches.

In RANS approaches, use of numerical models has been primarily focused on predictionof the first- and second-order concentration statistics for urban pollutant plumes. Andronopouloset al.11 investigated the dispersion field at the cross-junction of two street canyons by explicitlysolving the transport equation for the concentration variance and used the so-called gradient transferassumption for closing the concentration variance equation. Milliez and Carissimo12 introduced analgebraic model for closure of the concentration variance equation and validated their model perfor-mance by comparing the obtained results against MUST measurement data. Wang et al.13 useda non-linear k-ϵ model for simulating instantaneous release of contaminant from a ground-levelpoint source. They proposed a so-called dissipation length-scale model required for closure of theconcentration variance equation. Santiago et al.3 used the standard k-ϵ model to study the turbulentflow over a regular array of cubes submerged in a simulated urban ABL. They observed that thecenter of the vortex formed in the canyon region was located at 3/4 of the cube height and thedownwash flow dominated the upwash flow inside the canyon region.

As reviewed above, traditionally, numerical studies have heavily relied on the RANS approach,which however, cannot provide detailed temporal and spatial information of the dynamically evolv-ing flow and concentration fields. Cheng et al.14 compared the effectiveness of LES and RANSapproaches in their numerical study of turbulent flow over a matrix of cubes at a relatively lowReynolds number. They observed a better performance of LES over RANS in terms of the predic-tion of the resolved Reynolds stresses and spanwise mean velocity. Salim et al.15 compared thepredictive performances of the RANS and LES approaches in their study of the concentration fieldreleased from two parallel line sources in a street canyon. They observed that the LES results tend tohave a better agreement with wind-tunnel data in terms of the predicted mean concentration profiles.Schmidt and Thiele16 studied flow structures over wall-mounted cubes using different modelingapproaches (including RANS, LES, and detached eddy simulation (DES)), and demonstrated theeffectiveness of DES in resolving the dominant flow patterns. Xie and Castro17 compared the perfor-mances of LES and RANS in numerical simulation of turbulent flow over staggered and randomarrays of wall-mounted cubes and reported that the RANS results were not as satisfactory as thoseof LES due to the inherent unsteadiness of the flow.

With the fast advancement of computational technology, high-resolution 3-D numerical simu-lations have become more and more accessible. Although there have been some studies based ondirect numerical simulation (DNS) of flow around a group of bluff bodies (e.g., Coceal et al.18

and Lee et al.19), conducting a DNS of urban flows at a high Reynolds number of practical in-terest can be prohibitively expensive due to the high demand for spatial and temporal resolutions.Furthermore, detailed flow and dispersion information at the finest Kolmogorov and Batchelorscales (for the velocity and scalar fields, respectively) obtained from DNS is not always necessaryin engineering practice. In view of this, LES can be considered as an optimum tool for transientsimulation of turbulent flow and dispersion over an idealized urban area. Hanna et al.2 performedLES of turbulent flows over different types of arrays of wall-mounted cubic obstacles immersedin a fully developed boundary layer. They compared regular and staggered arrays of obstacles andshowed that the flow quickly approached a quasi-equilibrium state in both configurations after thethird or fourth row. Liu et al.20 conducted LES to predict the flow field and pollutant dispersionfrom ground-level line-sources in 2-D modeled street canyons. They observed a higher pollutantconcentration inside street canyons when the aspect ratio of obstacles was increased. Niceno et al.21

studied the flow over a matrix of wall-mounted internally heated cubes using LES and showed thatthe total heat transfer from the cubes to the fluid is significantly influenced by the coherent vorticalstructures around the cubes.

It should be noted that for wall-resolved LES, the grid resolution requirement is as demandingas DNS in the near-wall regions, as in both approaches, the near-wall flow field must be fullyresolved (at viscous scales). In fact, conducting wall-resolved LES of turbulent flow and dispersionin a complex urban environment at Reynolds numbers of practical interests is still prohibitively


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expensive. Although LES has been used by a number of researchers to study urban flow and disper-sion, some of the studies are not rigorous in terms of the grid resolution required for resolvingnear-wall dynamics. For example, Shi et al.22 performed LES of a wind field over a group of stag-gered modeled building obstacles. In their study, only 7 grid points were used across an individualobstacle for a highly turbulent flow at Re = 22 400 (based on the mean velocity and obstacle height).In the LES study of Niceno et al.,21 the computational expenses have been significantly reducedby considering a portion of an array of obstacles submerged in the boundary layer. This is basedon the observation that flow quickly becomes self-similar after the first several rows of obstacles,such that use of the assumption of fully developed boundary-layer flow is justified and periodicboundary conditions can be applied to the streamwise and spanwise directions. In view of the highdemands involved in wall-resolved LES for either complex flows or high Reynolds number flows,the method of wall-modeled LES has been developed and has become more and more popularover the past two decades.23–26 Although wall modeling has been primarily used for LES of flowswith simple domain geometries such as plane channel flows and flat-plate boundary layer flows,26

a number of attempts have been also made towards applications to more complex geometries withflow separations. For instance, Mason and Callen23 proposed a wall model for rough surfaces, andXie and Castro17 utilized the log-law to infer the boundary condition on the shear stresses over anarray of wall-mounted cubes.

The wall-modeled LES of Wang and Moin25 has been tested in the context of channel flowswith great success. In this research, we aim at extending test of this method to simulation of turbu-lent flow and dispersion over an array of 3-D wall-mounted obstacles. A new inlet boundary condi-tion is proposed to simulate the very high turbulence level encountered in the water-channel exper-iment. The obtained numerical results are validated against a set of comprehensive water-channelmeasurement data. The transport of flow kinetic energy (KE) and scalar energy (SE), evolution ofcoherent flow structures, and dispersion of the concentration field are also investigated.

The reminder of this paper is organized as follows: in Section II, the test case, boundary condi-tions and numerical algorithms will be described; in Section III, the results on turbulent coherentstructures will be analyzed, and flow and concentration statistics will be validated against availablewater-channel measurement data; and finally, in Section IV, major findings of this research will beconcluded and future research subjects will be discussed.

II. TEST CASE AND NUMERICAL ALGORITHM

A. Test case and computational domain

The test case is based on the water-channel experiments of Yee et al.9,27 and Hilderman andChong.10 In their experiments, a regular array of 16 × 16 wall-mounted cubes with side-length ofd = 31.75 mm was immersed in an emulated neutrally stratified urban ABL. Figure 1(a) shows theschematic of the array of wall-mounted cubes in the water-channel experiments. The cubes werestrictly aligned with a uniform spacing d in the streamwise and spanwise directions. The Reynoldsnumber based on the cube side-length and free-stream velocity (U∞ = 0.38 m/s) was Re = 12 005.The concentration field was generated by continuous release of sodium fluorescein dye from aground-level point source located in the central (specifically, the eighth) column and between thefirst and second rows of obstacles. The Schmidt number was 1920 based on the kinematic viscosityof water and molecular diffusivity of sodium fluorescein. The source location and coordinate systemare illustrated in Fig. 1(b). In this paper, we also use tensor notations, and the x, y , and z coordinatesare represented using xi (for i = 1, 2, and 3, respectively).

The velocity field was measured using a 4-beam 2-component TSI fiber-optic laser LDA pow-ered by an argon-ion laser. Titanium dioxide was used as seed particles for the LDA, and the datarate for the LDA measurements was typically 50–500 Hz. A one-dimensional (1-D) LIF linescansystem was used for measuring the instantaneous concentration field in the dispersing plume. So-dium fluorescein dye was continuously released from the point source at a rate of 12 ml min−1, andilluminated using a laser beam powered by an argon-ion laser. The dye source was released froma small vertical stainless steel tube (with an inner diameter d0 = 2.8 mm). A Dalsa monochrome


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FIG. 1. Schematic of the matrix of 16×16 wall-mounted cubes and the coordinate system. The side-length of the cube isd = 31.75 mm. The ground-level point source is located at the midpoint between rows 1 and 2 in the central (or eighth)column of the domain, with x/d = 1.5, y/d = 0, and z/d = 0. (a) Matrix of 16×16 cubes. (b) Ground-level point sourcelocated in the central (or eighth) column.

digital linescan CCD camera (1024 × 1 pixels) with 12-bit (4096 gray levels) amplitude resolutionwas used to measure the intensity of the dye fluorescence at a sampling rate of 300 Hz. Velocitymeasurements were made along vertical lines at different locations in two central cells in the firstand sixth rows. Here, a “cell” represents the repeating unit required for generating the regulararray of obstacles which occupies an area of 2d × 2d in the x-z plane with the cube placed in itsupper-left quadrant. Figure 2 schematically shows the first- and sixth-row cells and the selected fourmeasurement locations. In Fig. 2(a), point A1 represents the location on top of the obstacles, pointsB1 and C1 represent locations in the streamwise canyon street and point D1 is inside the canyon atthe midpoint between the first and second rows of obstacles. Similar measurement locations in cell 6are shown in Fig. 2(b).

FIG. 2. Locations in cell 1 and cell 6 where the comparison of the predicted and measured velocity profiles is made. Thenon-dimensional coordinate (x/d) in these two subfigures specifies the streamwise location with respect to the coordinatesystem defined in Fig. 1. (a) Cell 1. (b) Cell 6.


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In order to compare predicted concentration statistics against the experimental data,9,10,27

it is important to note that there are measurement uncertainties, which are attributed to base-line system biases and sampling errors. Yee et al.9 and Yee27 compared their wind-tunnel andwater-channel experiments of flow and dispersion in a similar modeled urban environment (at 1:50and 1:205 scales, respectively). They indicated that the uncertainty bounds on the mean concen-trations and fluctuation intensities in their wind-tunnel experiment were determined to be approx-imately 10% of the absolute value of the statistics near the plume centerline, and may increase toabout 15%-25% close to the plume edges (the mean concentration level is very low at the plumeedges). Also, because the sampling time was much longer in their water-channel experiment than intheir wind-channel experiment, the sampling errors in their water-channel experiment were furtherreduced.

Given the fact that the streamwise velocity is high and the concentration level is very low in thefar spanwise region, only the 5 central columns of cubes have been considered in the simulation.The range of the computational domain is x/d ∈ [−22.5,47], y/d ∈ [0,7], and z/d ∈ [−5,5]. Thisimplies that an inlet fetch of 22.5d is added before the first row of cubes, and an outlet fetch of 16dis considered after the last row of cubes. In total, 1232 × 80 × 304 grid points have been used to dis-cretize the domain in the streamwise, vertical, and spanwise directions, respectively. Figure 3 showsthe non-uniform grid distribution in the streamwise, vertical, and spanwise directions. As demon-strated in this figure, the grid resolution is refined near all solid surfaces in all three directions, andthe grid size is smoothly stretched (with a stretching rate kept less than 10%) to avoid numericalinstabilities arising from sudden change in grid sizes. According to Madabhushi and Vanka,28 a slowgrowth rate of the grid size is required in order to maintain a global second-order truncation errorin non-uniform grid systems. In order to perform parallel computing, the computational domain isdivided into 11 × 10 × 2 sub-domains in the x, y , and z directions, respectively.

FIG. 3. Grid distribution in streamwise, vertical, and spanwise directions. (a) x direction. (b) y direction. (c) z direction.


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B. Governing equations and subgrid-scale models

The filtered continuity, momentum, and scalar transport equations for LES of turbulent disper-sion of a passive scalar in the context of an incompressible fluid flow take the following form in aCartesian coordinate system:

∂ui

∂xi= 0, (1)

∂ui

∂t+

∂

∂x j(uiu j) = − 1

ρ

∂ p∂xi+ ν

∂2ui

∂x j∂x j−∂τi j

∂x j, (2)

∂c∂t+

∂

∂x j(u j c) = α

∂2c∂x j∂x j

−∂h j

∂x j, (3)

where ui, p, and c represent the filtered velocity, pressure, and scalar fields, respectively, ν isthe kinematic viscosity of the fluid, and α is the molecular diffusivity of the scalar. In the aboveequation, τi j and h j are the so-called subgrid-scale (SGS) stress tensor and SGS scalar-flux vector,respectively. The appearance of the SGS stress tensor and scalar flux is the result of the filteringprocess, which are defined as τi j

def= uiu j − uiu j and h j

def= u jc − u j c, respectively. In order to close

the above system of governing equations, the SGS stress tensor and SGS scalar-flux vector need tobe modeled.

The SGS stress model used for closure of the filtered momentum equation is the dynamicnon-linear model (DNM) proposed by Wang and Bergstrom,29 which expresses the SGS stresstensor as

τ∗i j = −CS βi j − CWγi j − CNηi j, (4)

where the base tensors are defined as βi jdef= 2∆2|S|Si j, γi j

def= 4∆2(SikΩk j + SjkΩki), and ηi j

def=

4∆2(Sik Sk j − SmnSnmδi j/3). Here, Si jdef= (∂ui/∂x j + ∂u j/∂xi)/2 is the resolved strain rate tensor,

Ωi jdef= (∂ui/∂x j − ∂u j/∂xi)/2 is the resolved rotation rate tensor, |S| = (2Si j Si j)1/2 is the norm of

Si j, δi j is the Kronecker delta, and an asterisk superscript denotes the trace free form of a tensor,i.e., (·)∗i j = (·)i j − 1

3 (·)kkδi j.Following the least squares procedure of Lilly,30 the three local dynamic model coefficients of

the DNM can be determined as

Mi jMi j Mi jWi j Mi jNi j

Wi jMi j Wi jWi j Wi jNi j

Ni jMi j Ni jWi j Ni jNi j

·

CS

CW

CN

= −

L∗i jMi j

L∗i jWi j

L∗i jNi j

, (5)

where Li j is the resolved Leonard type stress defined as Li jdef= uiu j − ˜ui ˜u j, and Mi j

def= αi j − βi j,

Wi jdef= λi j − γi j and Ni j

def= ζi j − ηi j are differential tensors, respectively; with αi j

def= 2 ˜∆

2| ˜S| ˜Si j,

λi jdef= 4 ˜∆

2( ˜Sik˜Ωk j +

˜S jk˜Ωki), and ζi j

def= 4 ˜∆

2( ˜Sik˜Sk j − ˜Smn

˜Snmδi j/3) being base tensors at the test-grid level. For the wall-modeled LES approach considered in this research, the resolved quantitiesat the grid level are denoted using an overbar, while quantities filtered at the test-grid level for thedynamic procedure are denoted using a tilde. In order to perform the dynamic modeling procedure,the filter size ratio between the grid and test-grid levels is set to 2, i.e., ˜∆/∆ = 2.

In order to close the filtered scalar equation, the dynamic eddy diffusivity model (EDM)proposed by Moin et al.31 is used for modeling the SGS scalar-flux vector, viz.,

h j = −Cθ∆2|S| ∂c

∂x j. (6)

Following the dynamic procedure (see Wang et al.32), the model coefficient can be obtained as

Cθ = −L jMj

MiMi, (7)


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where L jdef= u j c − ˜u j ˜c is a resolved scalar-flux vector which can be directly computed from the

filtered scalar field and Mjdef= a j − bj is a differential vector. Here, bj and a j are base vectors at the

grid and test-grid levels, respectively, defined as bjdef= ∆2|S| ∂c

∂x jand a j

def= ˜∆

2| ˜S| ∂ ˜c∂x j

.The Schmidt number is set to one (Sc = ν/α = 1) in the current simulations, which is often

seen in the literature on numerical simulation of passive scalar dispersion.33–35 This method ofsimplification is necessary in order to make it appropriate for calculating both velocity and scalarfields using only one computational grid system. In order to elaborate this common numericalpractice, we use the current test case as the example. If the Schmidt number of the current simu-lation is kept the same as the experiment (i.e., 1920), the finest turbulence scale of the scalar field(Batchelor scale) would be 43 times smaller than the Kolmogorov scale. For DNS, two sets of gridsystems would be needed: a coarse grid for the velocity field to capture the Kolmogorov scale anda much finer grid for the scalar field to capture the Batchelor scale. In a LES approach, the taskbecomes less demanding because of the involvement of SGS models. In LES, if the size of an eddyis smaller than the subgrid scale, its effects on the transport processes of momentum and scalarare approximated by SGS stress and scalar-flux models. Based on an analysis of their wind-tunnelmeasurement data on concentration plumes of a wide range of Schmidt numbers released froma point source, Vanderwel and Tavoularis36 showed that although the Schmidt number influenceshigher-order statistics of concentration fluctuations, the widely used counter-gradient diffusion(or eddy diffusivity) assumption is not subjected to strong Schmidt number effects. This confirmsthat in LES, if the goal is to focus on prediction of lower-order (first- and second-order) concentra-tion statistics, it is adequate to use the EDM of Moin et al.31 (which is based on the counter-gradientassumption) to perform the simulation, and use the EDM to account for all SGS transport processesof the scalar.

C. Numerical algorithm

The numerical simulation has been performed based on an in-house code developed usingthe FORTRAN 90/95 programming language, and fully parallelized using the message passinginterface (MPI) library. The governing equations have been discretized using a fully conservativefinite difference discretization scheme of Ham et al.37 based on a staggered grid arrangement. Asecond-order total variation diminishing (TVD) scheme is used for discretizing the scalar transportequation. A fully implicit four-level fractional step method of Choi and Moin38 coupled with asecond-order Crank-Nicolson scheme is used to advance the velocity field over a single time step.In the fractional step method, an intermediate velocity is calculated in the first step based on thepressure gradient of the previous time step using an alternative directional implicit (ADI) solver,and then in the second step, it is further modified to the second intermediate velocity by removinghalf of the old pressure gradient. In the third step, the Poisson equation is solved using a four-levelV-cycle multi-grid solver to obtain the new pressure field. Finally, half of the new pressure is used toupdate the velocity field.

For collecting statistics of the flow and concentration fields, it is important to ensure thatboth fields have reached a statistically stationary state. In our simulation, the code was first runfor approximately 2 flow-through times (the time required for the mean flow to travel throughoutthe domain) to allow the flow field to fully evolve to a statistically stationary state. At thisstage, the concentration release from the point source was activated and an additional period ofapproximately 2 flow-through times was spent to ensure the scalar field to become fully devel-oped. Then, flow and concentration statistics were collected through a course of approximately 10flow-through times. During the simulation, the time step was set to 3 × 10−4 s and the maximumCourant-Friedrichs-Lewy (CFL) number was approximately 0.3. The convergence criterion for thesolution of both flow and scalar transport equations was set to 10−6 for the maximum differencebetween two time steps in each quantity (i.e., u, v , w, and c) normalized by its spatially averagedvalue obtained in the previous time step. The simulation was conducted on a local 252-core com-puter cluster. In total, more than 150 000 CPU-hours have been spent to perform the simulation andcollect the statistics.


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1. Boundary condition for wall shear stresses

In order to estimate the wall shear stress, the wall model of Wang and Moin25 has been appliedto solid surfaces. This model considers the influence of pressure gradients on the wall shear stress,and calculates the wall-parallel component of the wall shear stress (τwi) as

τwi = µ∂ui

∂xn

xn=0=

ρ δ

0dxnν+νt

(uiδ −

1ρ

∂ p∂xi

δ

0

xndxn

ν + νt

), (8)

where δ is the height of the first grid point off the wall, n denotes the wall-normal direction, uiδ isthe wall-parallel velocity component at height δ taken from the outer LES solution, and νt is theeddy viscosity that can be determined from the mixing length theory as

νt/ν = κδ+D2, (9)

where κ is the Kármán constant, D = 1 − exp(−δ+/A+) is the wall damping function, δ+ = uτδ/νis the non-dimensional normal distance from the wall, and A+ = 19 (recommended by Wang andMoin25) is the van Driest constant for wall-modeled LES.

2. Inlet and outlet flow conditions

Since the purpose of the original water-channel experiment was to simulate a modeled rough-wall ABL, the approaching flow had an exceptionally high turbulence level. In fact, the lowestturbulence intensity at the inlet and above the cubes was approximately 10% of the mean flowvelocity. Indeed, reproducing such high turbulence level typical of a developing urban ABL of prac-tical interests using numerical method is a very challenging task. Many research works have beendevoted to developing effective inflow conditions for DNS and LES. Recent reported approaches oninflow conditions include synthetic turbulence and deterministic computations based on precursorsimulations and rescaling methods.39–43

A simple treatment is to assume that the flow is fully developed such that a periodical boundarycondition can be used in the streamwise direction (e.g., Cheng et al.,14 Xie and Castro,17 Cocealet al.,18 and Niceno et al.21). However, we will demonstrate in the following context that thissimplest treatment method significantly underestimates the turbulence kinetic energy (TKE) levelfor an urban ABL flow of practical interests, which typically features high turbulence levels (10%and above) and is typically developing (rather than fully developed) in the streamwise direction.In order to mimic such high turbulence level of a realistic developing urban ABL flow, it is oftenseen in the literature that some unphysical perturbations are added to the flow field. For instance, inthe studies of Hanna et al.2 and Shi et al.,22 at the inlet of the boundary-layer flow, time-correlatedrandom fluctuations are prescribed. In contrast to the previous methods based on either periodicboundary conditions or unphysical perturbations, in the present numerical study, we aim at gener-ating a highly turbulent urban ABL flow by using a physical inlet condition. In order to refine theresearch, a comparative study is conducted, which includes tests of four different inlet boundaryconditions: (1) prescribing the mean turbulent velocity profile with no fluctuations following theapproach of Schlüter et al.;43 (2) applying periodic boundary condition to the streamwise directionfollowing the approach of Cheng et al.;14 (3) using a turbulent plane channel flow simulation as aprecursor following the approach of Lund et al.;41 and (4) placing a solid grid in the inlet planeto accelerate flow transition and facilitate turbulent fluctuations. The solid grid placed at the inletconsists of a set of horizontal and vertical fine finite-width lines fitted on the computational grid. Inthis research, the blockage ratio for the grid is approximately 19%.

In order to evaluate the performances of these four inlet flow conditions, the profiles of theresolved streamwise root-mean-square (RMS) velocity predicted by LES at position D1 (at x/d= 1.5, in cell 1) and D6 (at x/d = 11.5, in cell 6) are compared against the experimental data. Asis evident in Figs. 4(a) and 4(b), classical methods 1 and 2 totally failed in the sense that they werenot able to produce more than 0.5% of turbulence in regions above the cubes. Also as a classicalapproach, method 3 performed slightly better than methods 1 and 2, and was able to generate up to3% of turbulence level in regions above the cubes. In contrast to the unsatisfactory performances of


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FIG. 4. Effects of four different inlet conditions on the predicted streamwise RMS velocity level at two measurementlocations D1 and D6. (a) Location D1. (b) Location D6.

classical methods 1, 2, and 3, the proposed method (i.e., method 4) was able to produce the highestturbulence intensity level (up to 7%). For method 4, we need to extend the inlet fetch (between theinlet and the first row of cubes) long enough to avoid any direct interference of the immediate wakesproduced by the solid grid at the inlet with the downstream cubes, and also to allow the flow to reachto a sustainable turbulence level. Once a sustaining turbulence level is achieved, the time-averagedvelocity profile obtained is expected to match the measured one, with the presence of a constantlevel of fluctuations. Based on several numerical tests, we observed that the required inlet fetchshould be at least 20d in length. However, since the grid resolution in the approaching region is notrequired to be considerably high, the additional computational cost arising from the extended inletfetch is only approximately 10%. Figures 5(a) and 5(b) show the instantaneous streamwise velocitycontours in the central x-y plane (at z/d = 0) in the approaching and cube regions, respectively.As is evident in Fig. 5(a), significant turbulence is triggered right behind the solid grid located atx/d ≈ −22. As the distance from the inlet increases, the turbulent patches quickly become uniform,reaching a stable sustainable level. Then, at x/d = 0, the turbulent flow starts to strike the first rowof cubic obstacles.

The outlet plane is kept 16d away from last row of the cubes and the Neumann boundarycondition has been applied to the outlet boundary. Slip condition has been considered for the upperfree-surface boundary.

FIG. 5. Contours of the resolved instantaneous streamwise velocity in the central x-y plane (located at z/d = 0).(a) Approaching region prior to obstacles (for −22.5 < x/d < 0). (b) Obstacle region (for 0 < x/d < 22.5).


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3. Boundary conditions for the concentration field

For the concentration field, its value has been set to zero at the inlet boundary. Zero-gradientcondition has been applied to the spanwise, upper, and outlet boundaries of the domain. Zero-fluxNeumann condition has been assumed at all solid surfaces to warrant the no-penetration condition.

In order to use LES to simulate the instantaneous release of the concentration from theground-level point source in the water-channel experiment, the point source is represented using onesingle control volume at the same position, and the size of the control volume is ∆x × ∆y × ∆z =1.283 × 0.829 × 1.283 mm. Note that the point source is located within the recirculation bubblebehind the obstacle. Once the scalar is released from the ground-level point source, it quickly mixeswith the scalar field within the bubble (due to the dominant recirculating flow pattern). Then, thescalar within the bubble disperses out through the bubble interface due to intense interactions withthe shear flows issued from the rooftop and two vertical sides of the cube. The scalar either quicklychannels downstream through the canyons below the canopy or spreads into the highly turbulentfree-stream flow above the canopy. Owing to this special mechanism of dispersion in the context ofthe recirculation bubble, the exact size and location of the point source are not sensitive parametersin either RANS or LES based on our many tests in this and previous studies.13,44 On the other hand,the source strength is a very sensitive parameter, which must be kept identical to the experimentalvalue. In order to compare the numerical and experimental results, the concentration field has beennon-dimensionalized using the source strength value cs.

III. RESULTS AND DISCUSSIONS

In this section, the LES results are analyzed to investigate the characteristics of the flow andconcentration fields. Turbulence statistics of the flow and concentration fields obtained from thenumerical simulation will be also compared against the available water-channel measurement dataof Yee et al.9 and Hilderman and Chong.10

A. General description of the flow field

The characteristics of the highly disturbed flow within and above the cubic obstacles havebeen demonstrated vividly using the contours of the resolved instantaneous velocity in Fig. 5(b),which has been briefly discussed in Sec. II. From the viewpoint of boundary-layer theory, the flowfield studied here can be classified as a “d”-type rough-wall flow (or, skimming flow), since theratio of the downstream pitch (or, length of a cell) to the height of cubes arrays is only 2d/d = 2.Above the canopy, the high-speed flow is highly turbulent and quickly skims over the cubic obstaclearray. Within the canopy, however, the speed of the flow is much reduced and the flow exhibitsrecirculation patterns (indicated by negative streamwise velocities). These features can be betterviewed in Fig. 6, which shows the time-averaged streamlines in the central x-y plane around thefirst seven rows of obstacles. As shown in Fig. 6(a), a large vortex is observed in the stagnant region(impinging zone) in front of the first row of obstacles. This is the signature of the horseshoe vortex,which will be discussed separately in the next paragraph. Also, a small recirculation region has beenformed only above the rooftop of the first obstacle which is due to the strike of the flow at thesharp front edge of the obstacle. At the rooftop of the first obstacle, a strong shear layer with largemean velocity gradients forms, which further triggers flow instability and induces boundary layerseparations. It is also clearly observed in Figs. 6(a) and 6(b) that the vortical structures (in terms ofthe location and size of recirculating vortices between the cubes) in the canyon regions graduallyevolve towards a self-similar state as the distance from the first row increases. The size and patternof recirculating vortices between two adjacent cubic obstacles become increasingly similar after thefifth row.

In order to analyze the structure of the horseshoe vortex formed around the first row of cubes,a top view of contours of the time-averaged resolved streamwise velocity ⟨u⟩ in the central column(for −1 < x/d < 2 and −1 < z/d < 1) at elevation y/d = 0.25 is displayed in Fig. 7(a). In Fig. 7(b),contours of the time-averaged pressure coefficient (defined as Cp = (p − p∞)/q∞, where p∞ is the


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FIG. 6. Time-averaged streamlines around obstacles in different rows, demonstrated in the central x-y plane (located atz/d = 0). (a) Rows 1-3. (b) Rows 5-7.

resolved free-stream pressure and q∞ = 12 ρU2

∞ is the free-stream dynamic pressure) at the samelocation are demonstrated, superimposed with time-averaged streamlines. As is evident in Fig. 7(a),there are five distinct flow zones around the first-row obstacles: (a) a stagnant zone formed in theimmediate adjacency of the windward face of the first-row obstacles due to the impingement of theflow onto the obstacle, (b) an arch-shaped negative velocity zone located in front of the stagnantregion, (c) a pair of acceleration zones formed symmetrically on the two sides of the obstacle, wherethe streamwise velocity drastically increases due to the local blockage effects, (d) a pair of smallrecirculation zones formed immediately adjacent to the side walls (under the acceleration zones)due to boundary layer separation, and (e) a large recirculation zone formed behind the obstacle. Thefive flow regions mentioned above and the characteristic horseshoe vortex pattern around a first-rowobstacle are also clearly shown using time-averaged streamlines in Fig. 7(b). Corresponding to thefive flow regions shown in Fig. 7(a), the pressure distribution shown in Fig. 7(b) exhibits an oppositetrend in terms of its magnitude. Zone a centers around the stagnation point, where the pressurereaches the highest value. The high stagnation pressure then causes the reverse flow towards zone b.It shows intuitively in Fig. 7(b) that the horseshoe streamline pattern in the front of the obstacle isa result of this reverse flow and the approaching flow from the upstream. Also from Fig. 7(b), it isclear that due to the acceleration of the flow on both sides of the obstacle, the pressure value drops

FIG. 7. Horseshoe vortex in front of the first-row obstacle in the central column, visualized using time-averaged resolvedstreamwise velocity, pressure coefficient, and streamlines in the x-z plane at elevation y/d = 0.25. (a) Contours oftime-averaged u. (b) Contours of time-averaged Cp and streamlines.


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significantly in zone c. Further, according to Prandtl’s classical boundary-layer theory, the pressuregradient in the z-direction (vertical to the mean flow direction) is trivial within the boundary layer.As such, the mean pressure in zone d is close to that in zone c, and therefore, also maintains at alow level. In comparison with the high pressure level in zone a, the pressure level in zone e is muchlower. This pressure difference between the front and the rear of the obstacle is the exact cause ofthe form drag in building aerodynamics and wind engineering.

Figure 8 shows the time-averaged resolved streamwise velocity contours and streamlines in thehorizontal plane at half-cube height in three streamwise regions (inlet, self-similarity, and outletregions). In Fig. 8(a), the streamlines show that the free-stream flow strikes at the first-row obsta-cles, and vortical structures behind the cubes gradually evolve towards a self-similar state. Thelocal acceleration of the flow caused by the blockage of the first-row obstacles in their side regionsis well captured by the mean streamwise velocity contours. From the figure, it is also observedthat the boundary layer separation on the two sides of the first-row obstacles induces two smallrecirculation bubbles. In Fig. 8(b), it is evident that the flow has reached a self-similar state and arepeating pattern in the recirculation flows behind the cubes is clearly observed. Figure 8(c) showsthe streamlines around the last two rows of obstacles. Due to a lack of a downstream restriction,the counter-rotating vortices behind the last-row obstacles are apparently different than those in theupstream region between two rows of cubes. The streamlines passing through the streets diverge

FIG. 8. Contours of the time-averaged resolved streamwise velocity and time-averaged streamlines in x-z planes at elevationy/d = 0.5 in different streamwise regions. (a) Inlet region. (b) Similarity region. (c) Outlet region.


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towards the recirculation region behind the last row of obstacles, exhibiting a clustered and parallelpattern. Meanwhile, this local divergent pattern of streamlines results in a local deceleration alongthe street canyons immediately downstream of the last row of obstacles.

In order to obtain more physical insights into the canyon flow in the self-similar region, theinstantaneous and time-averaged vector plots and streamlines of the resolved velocity field betweenrows 7 and 8 in the central plane (z/d = 0) are displayed in Fig. 9. As shown in Fig. 9(a), thehigh-speed flow skimming over the obstacles interacts intensively with the windward face of thecube (especially around the top edge), creating turbulent eddies and a downwash flow pattern. Thisdownwash flow along the windward face further drives the flow to recirculate within the canyon. InFig. 9(b), four stable vortices are observed, with one large vortex (or primary vortex) marked as “a”in Fig. 9(b), and three small vortices (or secondary vortices) at three corners marked with “b,” “c,”and “d.” The four eddies observed in Fig. 9(b) can be better visualized using streamlines shown inFig. 9(c). The formation of these four vortices between two cubes resembles the classical lid-drivencavity flow pattern which has been extensively studied (see the comprehensive review by Shankarand Deshpande45).

Figures 10(a)-10(c) show the temporal energy spectra for the streamwise, vertical, and span-wise resolved velocity fluctuating components (i.e., for u′′1 , u′′2 , and u′′3 , respectively), at positionx/d = 13.5, y/d = 1, and z/d = 0 (located in the central plane of the domain between rows 7 and8 at the rooftop elevation). At this position, flow has reached the self-similar state and is dominatedby a very strong shear layer formed from the rooftop of the upstream (seventh row) obstacle. Inthe figure, the energy spectra have been non-dimensionalized as Eii/(U∞d). The frequency has alsobeen non-dimensionalized and is represented using the Strouhal number defined as Sr = f d/U∞.

FIG. 9. Lateral view of the instantaneous and time-averaged vector plots and streamlines of the resolved velocity field insidethe canyon in the self-similar region (between rows 7 and 8). The plots are made in the central x-y plane located at z/d = 0.(a) Instantaneous velocity vector plot. (b) Time-averaged velocity vector plot. (c) Streamlines of time-averaged velocity field.


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FIG. 10. Non-dimensionalized temporal energy spectra for the streamwise, vertical, and spanwise resolved velocity fluctua-tions at position x/d = 13.5, y/d = 1, and z/d = 0 (located within the self-similar region between rows 7 and 8 at the rooftopelevation in the central plane of the domain). (a) Streamwise. (b) Vertical. (c) Spanwise.

Three distinct subranges are expected to appear in the energy spectra: the first subrange containsthe flow TKE at very small time frequencies (spatially, corresponding to energetic large-scaleeddy motions), the second part is the so-called inertial subrange where the energy cascade con-serves across each frequency and the spectrum slope features a constant value, −5/3, followingthe well-known “K41 theory”, and the third part is the viscous dissipation subrange correspondingto high-frequency turbulent motions (spatially, corresponding to small-scale eddy motions). Bycomparing Figs. 10(a)-10(c), it is observed that the pattern of the spectra of three resolved velocityfluctuation components is similar to each other; however, the resolved TKE level of the streamwisecomponent is slightly higher than those of the vertical and spanwise components, especially at lowfrequencies. As is evident in Figs. 10(a)-10(c), the spectra of all three resolved TKE componentsshow that the energy cascades from large- to small-scale eddies, going through the inertial subrange.This indicates that the simulation has been able to resolve most of the energy of turbulent eddies(i.e., energy containing eddies), implying that the resolution in all directions is fine enough tocapture the dominant flow features at this selected position.

B. Statistics of the velocity field

Figures 11 and 12 compare the numerical predictions of the time-averaged resolved streamwisevelocity against the measurement data in cells 1 and 6, respectively. As is evident in both figures,the LES predictions match very well with the experimental data. Figure 11(a) shows the profileof ⟨u⟩/U∞ at location A1 (which is on the rooftop of the first-row obstacle). From the figure, itis observed that in the vicinity of the top surface of the obstacle (y/d ≈ 1), a small region with


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FIG. 11. Vertical profiles of the non-dimensionalized mean streamwise velocity at different locations in cell 1. (a) LocationA1. (b) Location B1. (c) Location C1. (d) Location D1.

negative mean velocities is predicted by LES, which is consistent with the previous analysis of theboundary layer separation and the concomitant recirculation pattern near the rooftop of the first-rowobstacle in Fig. 6. Immediately above this small recirculation region, a very strong shear layer(corresponding to a sharp vertical velocity gradient d⟨u⟩/dy) appears, which has been well capturedby LES. Figures 11(b) and 11(c) show the velocity profiles in the street canyon at locations B1 andC1, respectively. Due to the no-slip boundary condition, the velocity predicted is zero at the wall,and approaches free-stream velocity U∞ as the elevation increases. Figure 11(d) shows the meanvelocity profile at location D1. As is evident in this figure, the reverse flow (for ⟨u⟩ < 0) under thecanopy (for y/d < 1) has been correctly predicted by LES, which is consistent with the analysisof the recirculation pattern shown previously in Figs. 6(a), 7(a), and 8(a). Figures 12(a)-12(d)compare the predicted and measured velocity profiles at four locations (A-D) in cell 6 (locatedin the self-similar region). Although Figs. 11 and 12 share some general flow features, a carefulcomparison of these two figures indicates that some differences are present in the mean velocityprofiles between cells 1 and 6. Specifically, as shown in Fig. 12(a), the small recirculation regionnear the rooftop of the first-row obstacle (at location A1 in cell 1) observed in Fig. 11(a) is no longerpresent at location A6 in cell 6. This indicates that the boundary-layer separation which leads toflow recirculation on the rooftop of the first-row obstacle is a result of direct strike of the approach-ing free-stream flow at the sharp-edged cubic obstacle. This striking and blockage effect is muchattenuated in the follow-up downstream rows, and the mean flow gradually becomes self-similar toreflect the characteristics of a fully developed d-type rough-wall boundary layer. This observation isconsistent with the streamwise velocity contours presented in Fig. 5 and streamline pattern shown inFig. 6.


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FIG. 12. Vertical profiles of the non-dimensionalized mean streamwise velocity at different locations in cell 6. (a) LocationA6. (b) Location B6. (c) Location C6. (d) Location D6.

Figures 13 and 14 compare the profiles of the non-dimensionalized resolved streamwise RMSvelocity urms/U∞ at four different locations in cells 1 and 6, respectively. As is evident in Fig. 13(a),the RMS velocity reaches its maximum around the top of the canopy (y/d = 1). The large meanvelocity gradient at the rooftop (shown previously in Fig. 11(a)) results in a strong shear productionrate for TKE. As such, the magnitude of the RMS velocity peaks at this special elevation at bothlocation A1 (cf. Fig. 13(a)) and its downstream location D1 (cf. Fig. 13(d)). Because of the verticalspreading and dissipation of TKE, the elevation of the peak increases while the magnitude of thepeak decreases as the fluid flows from A1 to D1. By comparing the numerical results with themeasurement data, it is seen that this physical feature has been reproduced by LES. It is especiallysatisfying to see in Fig. 13(a) that both the elevation (near the top of the canopy) and the magnitudeof the maximum RMS velocity have been correctly reproduced in the simulation. However, inregions above the cubes (y/d > 1), the magnitude of the RMS velocity has been underpredictedby the simulation. This is due to the exceptionally high turbulence level in this region discussed inSection II C 2. As shown in Figs. 13(b) and 13(c), the peak value of the RMS velocity has beenshifted towards the ground at locations B1 and C1. Owing to vertical turbulent dispersion of TKE,the elevation of the peak at location C1 is higher than that at its upstream location B1.

In Figs. 14(a)-14(d), the profiles of the streamwise RMS velocity at four measurement locationsin cell 6 are presented. In general, the profiles of the streamwise RMS velocity in cell 6 are similarto those at the corresponding measurement locations in cell 1 presented in Fig. 13, except that themagnitude of the maximum RMS velocity peak in location A6 is about 40% lower than that inlocation A1, and shape of peaks at B6 and C6 is wider than that at B1 and C1. This is consistentwith the previous discussion that the free-stream flow strikes the windward side of the obstacles


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FIG. 13. Vertical profiles of the non-dimensionalized RMS streamwise velocity at different locations in cell 1. (a) LocationA1. (b) Location B1. (c) Location C1. (d) Location D1.

of the first row much more intensely than does the sixth row, resulting in a stronger shear layer(see Figs. 11(a) and 12(a)) and boundary-layer separation at the rooftop of the first-row obstacles.As the flow passes from cell 1 to cell 6, spatial transport of TKE takes place due to convection,diffusion, and dissipation, and inevitably, the profiles of the RMS velocity evolve spatially due tothese mechanisms.

C. Budget of the resolved kinetic energy

In Subsection III B, the resolved turbulent fluctuations in terms of the streamwise RMS veloc-ity have been analyzed. In this subsection, we further extend this discussion by investigating thetransport processes of the resolved KE and TKE of the flow. The filtered KE of the flow (defined asE def= 1

2 uiui) can be decomposed as

E = kr + ksgs, (10)

where krdef= 1

2 uiui is the resolved KE and ksgsdef= 1

2 (uiui − uiui) is the SGS KE. It can be shown thatthe transport equation for the time-averaged resolved KE of the flow takes the following form:32

u j∂kr∂x j

advection

= − 1ρ

u j

∂p∂x j

pressurediffusion

+2ν∂Si jui

∂x j

viscousdiffusion

−2ν⟨Si jSi j⟩ viscous

dissipation

−∂τ∗i jui

∂x j

SGSdiffusion

+⟨τ∗i jSi j⟩ SGS

dissipation

. (11)


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FIG. 14. Vertical profiles of the non-dimensionalized RMS streamwise velocity at different locations in cell 6. (a) LocationA6. (b) Location B6. (c) Location C6. (d) Location D6.

Figures 15(a)-15(c) show the budget balance of kr across the stream in the self-similarityregion at mid-point (x/d = 13.5) between rows 7 and 8 and behind the central column (−1 < z/d <1) at three different elevations for y/d = 0.5, 1, and 1.5 (at the half-cube height, rooftop, and abovethe canopy, respectively). All the terms shown in these figures have been non-dimensionalized usingU3∞/d. Also, all the budget terms on the right-hand side (RHS) of Eq. (11) shown in Figs. 15(a)-

15(c) have included the addition/subtraction signs in Eq. (11). As such, the pressure diffusion,viscous dissipation, and SGS diffusion terms shown in the budget balance in these figures are

− 1ρ

u j

∂p∂x j

, −2ν⟨Si jSi j⟩ and −

∂τ∗i jui

∂x j

, respectively. As shown in Fig. 15(a), at elevation y/d =

0.5, the advection, pressure diffusion and SGS dissipation rate are the dominant terms in the trans-port equation of kr , and furthermore, the advection term is primarily balanced by the pressurediffusion and SGS dissipation terms. The peak values of advection and pressure diffusion occur atz/d ≈ ±0.5, directly downstream of the two sides of the obstacle. At these locations, the flow inthe canyons strikes the two vertical side edges of the cube, forming a strong shear layer on eachside which further triggers the flow instability and entrains the recirculating region behind the cube.As a consequence of the enhanced turbulence level due to the pressure difference between side andrear regions and the strong shear layers formed on both sides of the obstacle, the magnitudes of thepressure diffusion and advection terms peak around z/d ≈ ±0.5. The highest value in the magni-tude of the SGS dissipation rate occurs at z/d ≈ ±0.35, which is slightly displaced from the locationof the advection and pressure diffusion peaks towards the recirculation bubble behind the obstacle.As shown previously in Fig. 8, at these special locations, strong shear layers intensely interactwith a pair of counter-rotating recirculating vortices. At z/d = 0 (inside the recirculation region),the level of the advection and pressure diffusion is generally low while the SGS dissipation rate


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FIG. 15. Budget of the time-averaged resolved kinetic energy (kr) at x/d = 13.5 within −1 < z/d < 1, at three differentelevations. All the quantities shown in the figures have been non-dimensionalized usingU3

∞/d. (a) y/d = 0.5. (b) y/d = 1.0.(c) y/d = 1.5.

reaches it absolute minimum. This is because all these three dominant terms respond positivelyto high turbulence levels caused by pressure difference, convection, shear instability, and intenseinteraction of eddies of different scales, and all these effects are relatively weak at the center of therecirculation bubble trapped between two cubes (see, Fig. 9). The overall influence from viscousdiffusion and viscous dissipation is not significant in comparison with other terms, because this flow(with exceptionally high free-stream turbulence level and strong disturbances from cubes) is domi-nated by inertial forces rather than viscous forces. LES is an excellent tool for simulating this flowwhich features high TKE levels at large resolved scales. As shown in Fig. 15(a), the level of SGSdiffusion is much smaller than that of SGS dissipation. This is because the SGS dissipation term⟨τ∗i j Si j⟩ represents KE transfer between the large resolved and small subgrid scales. The cascadeof KE can be positive or negative, representing backward or forward scatter of energy betweenthese two scales. The effect of SGS dissipation becomes stronger as the Reynolds number increases.The SGS dissipation rate reflects a major feature of SGS dynamics, as it represents a key physicalquantity that determines the cascade of energy in LES. In contrast, the SGS diffusion representsre-distribution of KE due to SGS shear stresses (which are much smaller than resolved turbulentshear stresses), and its value is expected to be smaller than that of the SGS dissipation rate. The SGSdissipation rate will be further thoroughly investigated later in this subsection.

Figure 15(b) shows the cross-stream budget balance of the resolved kinetic energy kr at thesame streamwise location but at the rooftop elevation for y/d = 1. At this elevation, the threeseparated boundary layers from the rooftop and two sides of cube interact with the reverse flowbetween the cubes and significantly enhance turbulent mixing. As shown in Fig. 15(b), the advec-tion, pressure diffusion, and SGS dissipation rates dominate the transport process of kr , which is


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similar to that observed in Fig. 15(a) (under the canopy for y/d = 0.5). As mentioned above, thesethree terms reflect enhanced turbulence activities caused by pressure gradients, convection, shearinstability, and intense mixing of eddies of different scales, and all these effects are strong at thisspecial position. The cascade of the resolved TKE as a result of eddy motions and interactionsat this special position has been analyzed using the energy spectra shown previously in Fig. 10.Although in both Figs. 15(a) and 15(b), the transport of kr is dominated by advection, pressurediffusion, and SGS dissipation terms, the profiles of these three dominant terms exhibit differentpatterns at these two elevations (for y/d = 0.5 and 1). The magnitude of the advection term isthe largest among all terms. By comparing Figs. 15(b) and 15(a), it is observed that although twolocal maxima are still preserved at z/d ≈ ±0.5 to reflect strong instable shear layers from both cubesides, the largest peak of the advection term appears around the midspan (at z/d = 0) for y/d = 1,which is in sharp contrast to the pattern shown in Fig. 15(a) (in which the advection term reachesits local minimum around the midspan). This largest peak in the magnitude of the advection termis a consequence of the strong unstable shear layer from the rooftop. Also by comparing Fig. 15(b)with 15(a), it is interesting to observe that the value of the advection term has generally increasedby a factor of approximately 2.5, indicating that as the elevation increases from y/d = 0.5 to 1,the convection and its effects on the transport of kr have increased drastically, especially aroundthe midspan. Also as a response to the enhanced convection level at the top of the canopy, themagnitude of the SGS and viscous dissipation rates have also increased in comparison with theircounterparts at the half-cube height shown in Fig. 15(a).

In Fig. 15(c), the cross-stream budget balance of kr is shown at an elevation above the canopy(for y/d = 1.5). As is evident in Fig. 15(c), the advection term is primarily balanced by the pressurediffusion term. The impact of the SGS dissipation rate on the transport of kr still ranks the third,however, its magnitude has reduced as the elevation increases from y/d = 1 to 1.5. In this figure,the effects of the shear layers from the cube sides are much reduced in comparison with those shownin Figs. 15(a) and 15(b), and no local maxima are observed for the advection and pressure diffusionterms around the cube sides located at z/d ≈ ±0.5. However, there is an increasing trend in thecross-stream distribution of the advection and pressure diffusion terms that their magnitudes reachthe maximum in the midplane (z/d = 0). This indicates that at this elevation above the canopy, interms of the transport of kr , the flow is still significantly influenced by the strong unstable shearlayer formed on the rooftop (through vertical spreading), but no longer directly sees the two strongunstable shear layers formed by the two cube side faces (as they are buried deeply under the can-opy). It is interesting to observe that the profile of the SGS dissipation term is evolving towards ahorizontal line, implying that the eddy motions and interactions represented by this SGS term arebecoming increasingly homogeneous in the spanwise direction above the canopy. The flow is highlydisturbed by the cubes and is highly heterogeneous under the canopy; however, the influence ofcubes is expected to reduce as the elevation above the canopy increases, and eventually vanishesallowing the flow to reach a spanwise homogeneity state in regions far above the canopy.

The local KE transfer between the resolved and unresolved (subgrid) scales can be quan-tified using the SGS KE dissipation rate, which appears in both the resolved KE (kr) equation(i.e., Eq. (11)) as τ∗i j Si j and the SGS KE (ksgs) equation as −τ∗i j Si j. As such, it functions as a KE sink

term to the large resolved motions and a KE source term to the SGS motions. Define εsgsdef= −τ∗i j Si j.

The value of εsgs can be either positive or negative, representing forward and backward scatterof local KE between the resolved and subgrid scale motions, respectively. In Figs. 16(a)-16(c),the time-averaged cross-stream profiles of ⟨εsgs⟩ are displaced at streamwise location x/d = 13.5,across the spanwise range of −1 < z/d < 1 at three elevations for y/d = 0.5, 1, and 1.5. In orderto gain some deeper insights into the relative strength of the SGS and viscous dissipation rates,the value of the SGS dissipation rate has been non-dimensionalized with the value of the localviscous dissipation rate ⟨εr⟩ def

= 2ν⟨Si j Si j⟩ (see, Eq. (11)). In the context of the conventional SGSeddy viscosity (Smagorinsky) modeling approach, it can be shown that this ratio is equivalent tothe ratio between the SGS and kinetic molecular viscosities, i.e., ⟨εsgs⟩/⟨εr⟩ = νsgs/ν. Although theconcept of SGS eddy viscosity is not directly applicable to the SGS stress model (DNM) used inthis research, this ratio reflects the effective SGS dissipation rate over the viscous dissipation rate.32


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FIG. 16. Profiles of the time-averaged SGS KE dissipation rates (⟨εsgs⟩) at x/d = 13.5 within −1 < z/d < 1, at threeelevations. All the quantities shown in the figures have been non-dimensionalized using the averaged viscous dissipationrate ⟨εr⟩. (a) y/d = 0.5. (b) y/d = 1.0. (c) y/d = 1.5.

In Fig. 16, the forward scatter (⟨ε+sgs⟩) and backscatter (⟨ε−sgs⟩) of local KE flux have been separated.Naturally, these two quantities must verify ⟨εsgs⟩ = ⟨ε+sgs⟩ + ⟨ε−sgs⟩. As shown in Figs. 16(a)-16(c),the magnitude of the forward scatter is approximately 4 times larger than that of the backwardscatter, resulting in a net KE transfer from the resolved to unresolved scales. At the half-cubeheight (y/d = 0.5), as displayed in Fig. 16(a), the maximum value of the SGS dissipation rate andforward scatter occurs at z/d ≈ ±0.3. As shown in Figs. 16(b) and 16(c), as the elevation increasesto y/d = 1.0 and 1.5, the location for the maximum values moves closer towards the center ofthe domain (at z/d ≈ 0). Under the canopy, as demonstrated in Fig. 16(a), the local KE flux issignificantly influenced by the two strong shear layers formed on two vertical sides of the cube (atz/d = ±0.5) and their interactions with the recirculating zone behind the cube (see Fig. 8). This isthe reason that the KE flux peaks exhibit a symmetrical pattern in the spanwise direction behind thecube. However, these two vertical shear layers are immersed under the canopy and their influenceon flow decreases significantly in regions above the canopy (when y/d > 1). In the region at orimmediately above the canopy, the flow is influenced by the shear layer formed at the rooftop ofthe cube and the concomitant intense eddy motions associated with it. As shown in Fig. 16(b), thismechanism tends to elevate the KE flux magnitudes (in comparison with the KE flux magnitudesunder the canopy shown in Fig. 16(a)).

In order to complement the above analysis of the resolved KE transport processes, we canfurther study the transport of the resolved TKE. Conceptually different from the resolved KE kr andSGS KE ksgs, the resolved TKE of the flow is defined as k def

= 12 ⟨u′′i u′′i ⟩, where u′′i is the resolved

fluctuating velocity obtained based on the following decomposition method: u′′ = u − ⟨u⟩. The


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time-averaged resolved KE ⟨kr⟩ relates to the resolved TKE k as

⟨kr⟩ = 12⟨ui⟩⟨ui⟩ + k . (12)

It can be shown that the transport equation of k takes the following form:

⟨u j⟩ ∂k∂x j

advection

= −⟨u′′i u′′j ⟩∂⟨ui⟩∂x j

production

−12

∂

∂x j

u′′i u′′i u′′j

turbulent diffusion

− 1ρ

u′′j

∂p′′

∂x j

pressure diffusion

+2ν∂

∂x j⟨S′′i ju′′i ⟩

viscous diffusion

−2ν⟨S′′i jS′′i j⟩ viscous

dissipation

− ∂

∂x j

u′′i τi j

SGS diffusion

+⟨τ∗i jS′′i j⟩ SGS

dissipation

. (13)

Figures 17(a)-17(c) show the budget balance of k at the streamwise location x/d = 13.5 acrossthe spanwise range −1 < z/d < 1 at three elevations for y/d = 0.5, 1, and 1.5. All the budgetterms on the RHS of Eq. (13) shown in Figs. 17(a)-17(c) have included the addition/subtractionsigns in Eq. (13). For example, the production and viscous dissipation terms shown in Fig. 17 are−⟨u′′i u′′j ⟩ ∂⟨ui⟩∂x j

and −2ν⟨S′′i jS′′i j⟩, respectively. At y/d = 0.5, as shown in Fig. 17(a), the advectionterm is primarily balanced by the production, turbulent diffusion, pressure diffusion, and SGS dissi-pation terms. The maximum TKE production rate occurs at z/d ≈ ±0.5, directly downstream ofthe two vertical sides of the cube. At this special location, a strong shear layer with sharp velocitygradients is formed on each side of the cube which enhances the TKE production rate. As shownin Figs. 17(a)-17(c), at all three elevations, there are three local minima in the cross-stream profileof the TKE production rate, which are located at z/d = 0 (corresponding to the central x-y plane

FIG. 17. Budget of the time-averaged resolved TKE (k) at x/d = 13.5 within −1 < z/d < 1, at three elevations. All thequantities shown in the figures have been non-dimensionalized using U3

∞/d. (a) y/d = 0.5. (b) y/d = 1.0. (c) y/d = 1.5.


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of the domain) and z/d ≈ ±1 (corresponding to the central x-y plane of the neighboring streetcanyons). At these three special spanwise locations, the cross-stream mean velocity profile is locallysymmetrical, and therefore, the spanwise mean velocity gradient is trivial. This further results ina significant reduction in the TKE production rate. As shown in Fig. 17(b), the magnitudes ofthe turbulent advection, TKE production, and SGS dissipation rates have increased by a factor ofapproximately 3 at the rooftop level (y/d = 1) in comparison with their values at half-cube height(y/d = 0.5). This is an indication of increased turbulent activities as the elevation increases fromy/d = 0.5 to 1.0 (consistent with the vertical profiles of urms/U∞ in the self-similar region shown inFig. 14). At this elevation (y/d = 1), the separated shear layers from the side edges and top surfaceof the cube all contribute considerably to transport of resolved TKE, and all these three shear layersinteract directly with the recirculation zone behind the cube. Above the canopy at y/d = 1.5, asshown in Fig. 17(c), the magnitudes of all budget terms have decreased by a factor of approximately6 in comparison with their counterparts at y/d = 1. This is an indication of decayed turbulenceactivities above the cube rooftop, which is in agreement with the previous observation of decreasedstreamwise turbulence intensity levels in the vertical profiles of urms/U∞. As the elevation increasesabove the canopy, the disturbances from the cubes are much reduced and flow becomes increasinglyhomogeneous in the spanwise direction.

D. General description of the concentration field

Following the above discussion of the velocity field, here the concentration field is analyzedbased on the statistics of the instantaneous scalar values, temporal spectrum, and budget balance ofthe transport equation of the scalar energy.

Figure 18 shows the contours of the time-averaged concentration field (non-dimensionalizedusing the source strength cs) in x-z planes at two different elevations under and above the canopy(for y/d = 0.5 and 1.5, respectively). At both elevations, the plume is symmetrical with respectto the central line of the domain (z/d = 0). Above the canopy at y/d = 1.5, no direct interferencebetween the obstacles and the concentration field occurs and the plume exhibits a typical Gaussiandistribution across the stream.

Figure 19 shows the streamwise evolution of the vertical profiles of the mean and standarddeviation of the concentration field (i.e., ⟨c⟩ and crms =

⟨(c − ⟨c⟩)2⟩, respectively). The profiles areextracted from midpoints of 8 spanwise canyons starting from the second row (x/d = 3.5) alongthe central column (z/d = 0). In order to make the profiles at different locations comparable, thevalue of ⟨c⟩ and crms has been non-dimensionalized using their local maximum values along eachvertical line (that goes through the midpoint between two adjacent cubes). The vertical position ofthe maximum concentration (corresponding to ⟨c⟩/⟨c⟩max = 1) represents the vertical centroid ofthe plume, which, as shown in Fig. 19(a), elevates as the streamwise distance from the point source

FIG. 18. Contours of the time-averaged resolved concentration field (⟨c⟩) in x-z planes at two different elevations below andabove the canopy. The concentration field has been non-dimensionalized using the source strength cs. (a) Under the canopy(at y/d = 0.5). (b) Above the canopy (at y/d = 1.5).


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FIG. 19. Streamwise evolution of vertical profiles of the mean and standard deviation of the concentration normalized bytheir local maxima along the vertical lines in the central plane of the domain (z/d = 0). (a) Mean concentration. (b) Standarddeviation.

increases. In fact, clean fluid packets (upstream or above the plume) entrain the plume and contam-inated fluid packets engulf into the background clean fluid at the plume edges. This process resultsin the growth of the plume size. Also because of entraining and engulfing effects, the plume edgesbecome highly intermittent, which significantly increases the level of concentration fluctuations. Asimilar trend is observed in the crms profiles shown in Fig. 19(b). However, by comparing Figs. 19(a)and 19(b), it is evident that the peak values of the standard deviation are above those of the meanconcentration at all streamwise locations. Similar observations have been made by Fackrell and

FIG. 20. Cross-stream profiles of the non-dimensionalized mean concentration at elevation y/d = 0.25 and differentstreamwise locations. (a) x/d = 3.5. (b) x/d = 5.5. (c) x/d = 7.5. (d) x/d = 11.5.


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Robins46 for a point source in a turbulent boundary layer, and by Lavertu and Mydlarski47 for aline source in a turbulent plane channel flow. As is evident from Fig. 19(a), the maximum verticalgradient of the mean concentration (d⟨c⟩/dy) occurs at elevations immediately above that of themaximum mean concentration. This is related to the fact that the mean concentration gradientis typically the highest at plume edges. As a consequence, the production rate of the concentra-tion variance (i.e., −2⟨u′′j c′′⟩ ∂⟨c⟩

∂x j) is enhanced, further leading to an increase in the concentration

variance at plume edges.

E. Concentration statistics

In the water-channel experiments of Yee et al.9 and Hilderman and Chong,10 concentrationstatistics at three elevations are available, two under the canopy (for y/d = 0.25 and 0.5) andone above the canopy (for y/d = 1.25). In this section, the predicted and measured cross-streamprofiles of the mean concentration at these three elevations are thoroughly compared throughFigs. 20-22, which show a good agreement between the simulations and experiments. Figures 20and 21 compare the mean concentration profiles at four streamwise locations under the canopy (fory/d = 0.25 and 0.5, respectively). From both Figs. 20 and 21, it is clear that as the distance fromthe source increases, the plume width increases in the cross-stream direction, however, its peakvalue decreases as a result of plume dispersion. At x/d = 3.5, the characteristic dual-peak patternhas been well captured by the numerical simulation. This dual-peak pattern is a direct result ofthe presence of the cube immediately behind the point source (which is located at x/d = 1.5 inthe central column). The cube introduces a local blockage to the concentration field and the plumegoes downstream by sweeping around the cube. This results in a symmetric dual-peak pattern in thecross-stream concentration profile over a long sampling time. By comparing Fig. 22 with Fig. 21,

FIG. 21. Cross-stream profiles of the non-dimensionalized mean concentration at elevation y/d = 0.5 and different stream-wise locations. (a) x/d = 3.5. (b) x/d = 5.5. (c) x/d = 7.5. (d) x/d = 11.5.


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FIG. 22. Cross-stream profiles of the non-dimensionalized mean concentration at elevation y/d = 1.25 and differentstreamwise locations. (a) x/d = 3.5. (b) x/d = 5.5. (c) x/d = 7.5. (d) x/d = 11.5.

it is clear that above the canopy (at y/d = 1.25), the plume width decreases apparently at the samestreamwise locations, indicating a reduced amount of concentration at a higher elevation. Also, dueto the absence of obstacles, no apparent dual-peak pattern is observed at this elevation and theprofiles tend to exhibit a Gaussian distribution across the stream.

Figures 23-25 compare the predicted and measured cross-stream profiles of the standard devi-ation (or RMS value) of the resolved concentration field at the same measurement locations asfor the mean concentration profiles presented previously in Figs. 20-22. By comparing the profilespresented in Figs. 23-25, a general trend is observed in the streamwise evolution of the crms profiles:in regions close to the source, the magnitude of crms is high and the width of its profile is narrower,reflecting an intense turbulent fluctuation level; however, as the distance from the source increases,the peak value of crms decays rapidly and its profile becomes wider. As shown in Figs. 23 and 24,under the canopy at y/d = 0.25 and 0.5, a distinctive dual-peak pattern with a local minimum alongthe plume centerline (at z/d = 0) has been well captured by the simulation. However, the simulationhas under-predicted the peak values of crms. The dual-peak pattern is due to the large spanwiseconcentration gradient around two plume edges (see Figs. 20-22), which significantly increases theproduction rate for the concentration variance (i.e., −2⟨u′′j c′′⟩ ∂⟨c⟩

∂x j). As shown in Figs. 25(a)-25(d),

above the canopy at elevation y/d = 1.25, the dual-peak pattern is much less apparent in compar-ison with that at lower elevations, which is a direct consequence of the absence of cubic obstacles.

F. Spectra and scalar energy of the concentration field

The spectrum of the resolved concentration fluctuations (Ecc) relates to the resolved turbulentscalar energy (TSE) as


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FIG. 23. Cross-stream profiles of the non-dimensionalized standard deviation of concentration at elevation y/d = 0.25 anddifferent streamwise locations. (a) x/d = 3.5. (b) x/d = 5.5. (c) x/d = 7.5. (d) x/d = 11.5.

ks = ⟨c′′c′′⟩ = fc

0Eccdf , (14)

where ksdef= ⟨c′′c′′⟩ is the resolved TSE (or scalar variance). Figure 26 demonstrates Ecc of the

concentration field at the same position (x/d = 13.5, y/d = 1, and z/d = 0, located in the centralplane of the domain between rows 7 and 8 at the rooftop elevation) as for the energy spectra of theresolved velocity fluctuations plotted in Fig. 10. At this particular position, the flow has reached aself-similar state. As indicated by Fackrell and Robins,46 similar to temporal energy spectra for theresolved velocity field, the inertial subrange for the resolved TSE spectra also features the slopeof −5/3. As shown in Fig. 26, the inertial subrange characteristic of the cascade of the resolvedTSE and associated energetic motions for mixing of the plume have been well captured by thecurrent LES, with the mixing effects at higher (unresolved) frequencies being reflected in the SGSscalar-flux model.

Following Jiménez et al.,48 the so-called resolved SE is defined as kr sdef= c2, which is analogous

to the definition of the resolved KE kr of the flow field discussed previously in Section III C. Thetime-averaged transport equation for the resolved SE takes the following form:

u j∂kr s∂x j

advection

= α

∂2kr s∂x j∂x j

moleculardiffusion

−2α∂c∂x j

∂c∂x j

moleculardissipation

−2∂ch j

∂x j

SGSdiffusion

+2h j

∂c∂x j

SGSdissipation

. (15)

Figures 27(a)-27(c) present the budget balance of the resolved SE kr s at midpoint (x/d = 13.5)between rows 7 and 8 across the stream (within −1 < z/d < 1) at three different elevations (atthe half-cube height, rooftop, and above the canopy, for y/d = 0.5, 1, and 1.5, respectively). All


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FIG. 24. Cross-stream profiles of the non-dimensionalized standard deviation of the concentration at elevation y/d = 0.5and different streamwise locations. (a) x/d = 3.5. (b) x/d = 5.5. (c) x/d = 7.5. (d) x/d = 11.5.

the budget terms on the RHS of Eq. (15) shown in Figs. 27(a)-27(c) have included the addi-tion/subtraction signs in Eq. (15). For instance, the molecular dissipation term and SGS diffusionterm shown in the budget balance in Fig. 27 are −2α

∂c∂x j

∂c∂x j

and −2

∂ch j

∂x j

, respectively. In

Eq. (15), the molecular and SGS dissipation act as sink terms for the resolved SE. As shown inFig. 27(a), under the canopy at elevation y/d = 0.5, the advection term is negative within the rangefor |z/d | < 0.7 and is positive for |z/d | > 0.7. This indicates that at the half-cube height, the effectof the mean flow is to carry the resolved SE out of the rear region of the cube (where concentrationrecirculates with the flow) towards the two canyons beside the cube. At the rooftop level (y/d = 1),however, the advection term is constantly negative, which indicates that the effect of the mean flowis to carry the resolved SE away. The minimum of the advection term occurs at z/d = ±0.65, wherethe contaminated fluid is dispersed away at the highest rate through convection mechanisms. Asshown in Fig. 27(c), above the canopy, the advection term is always positive, indicating that theresolved SE is carried to this elevation from the ground level. In Figs. 27(a)-27(c), the advectionterm exhibits extrema for z/d ∈ ±[0.5–1.0], which is due to the following two factors: (1) there aretwo strong shear layers issued by the two vertical sides of the cube (located at z/d = ±0.5), whichhave a significant impact on convention immediately downstream of them; and (2) the center of thestreet canyon is located at z/d = ±1.0, where the streamwise convection (streamwise velocity) isthe highest under the canopy.

As shown in Fig. 27(a), at the half canopy height (y/d = 0.5), the molecular diffusion is posi-tively valued across the spanwise direction. The concentration is released from a ground-level pointsource located at x/d = 1.5, which then channels through street canyons under the canopy. A posi-tively valued molecular diffusion term is expected at this low elevation, which shows that the effectof molecular activities is to diffuse the resolved SE to other regions (i.e., transporting the concen-tration to a higher elevation). However, as shown in Fig. 27(b), at the rooftop level (y/d = 1), the


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FIG. 25. Cross-stream profiles of the non-dimensionalized standard deviation of the concentration at elevation y/d = 1.25and different streamwise locations. (a) x/d = 3.5. (b) x/d = 5.5. (c) x/d = 7.5. (d) x/d = 11.5.

molecular diffusion keeps negative across the channel, showing that the net effect of molecularactivities is to carry the resolved SE c2 towards this region. Negative molecular diffusion maybe counterintuitive at first, which however, can be well-explained through the advection-diffusionequation for c2 (i.e., Eq. (15)). As discussed in the previous paragraph, at the rooftop level, the

FIG. 26. Non-dimensionalized temporal spectrum for the resolved turbulent scalar energy at position x/d = 13.5, y/d = 1,and z/d = 0.


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FIG. 27. Budget of the time-averaged resolved scalar energy (kr s) at x/d = 13.5 within −1 < z/d < 1 at different elevations.All the terms are non-dimensionalized using U∞c

2s/d. (a) y/d = 0.5. (b) y/d = 1.0. (c) y/d = 1.5.

flow and dispersion are much complicated by the three shear layers (one created by the top surfaceand two by the vertical sides of the cube) and their interactions with turbulent eddy motions. Anegatively valued diffusion term is a direct consequence of a negatively valued advection term atthis special position. Furthermore, as shown in Fig. 27(c), above the canopy, both the advectionand molecular diffusion terms remain positive across the channel. This shows that in the regionabove the canopy, the effect of obstacles on plume dispersion is much reduced. Positively valuedadvection and molecular diffusion indicate that the contaminated fluid packets engulf into the cleanbackground fluid in the region above the canopy, which facilitates the growth of the plume size andelevation of its vertical centroid in the streamwise direction. This is consistent with the previousobservation in Fig. 19. Another interesting observation is that at this elevation (y/d = 1.5), thecubes are deeply immersed under the canopy and their disturbances to the flow and dispersion areattenuated. As such, the profile budget terms shown in Fig. 27(c) tend to recover spanwise homoge-neity (as in an open channel), and this tendency is the most strongly expressed with the molecularand SGS dissipation terms.

As is evident in Figs. 27(a)-27(c), the SGS diffusion and dissipation terms play a significantrole in balancing the advection term, and their magnitudes are much larger than that of the molec-ular dissipation term. Similar to the function of the SGS dissipation term τ∗i j Si j in Eq. (11) for thetransport of the resolved KE, the SGS dissipation term 2h j

∂c∂x j

reflects the local SE flux between

the resolved and subgrid scale motions. As shown in Figs. 27(a)-27(c), the value of 2h j

∂c∂x j

re-

mains negative across the stream at all three elevations, indicating a net forward scatter of SE fromresolved to subgrid scales.


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IV. CONCLUSIONS

Turbulent flow and dispersion over a matrix of wall-mounted cubic obstacles at Reynolds num-ber 12 005 have been studied using wall-modeled LES. The statistics and spectra of the turbulentfields, flow structures and their interactions with the dispersing plume, and transport mechanismsof the kinetic and scalar energies have been thoroughly analyzed. One of the major challengesinvolved is to reproduce numerically the highly turbulent (with turbulence intensity of at least 10%)approaching boundary layer of the water-channel experiment. Based on a comparative study of fourinlet boundary conditions, we selected a method that utilizes a solid grid at the inlet of the domain totrigger the flow instability and produce physical perturbations. This approach enabled us to generatea highly turbulent boundary layer similar to that in the water-channel experiment with a stablesustainable turbulence level up to approximately 7% of the mean free-stream speed in regions abovethe wall-mounted cubes.

Due to the intense interaction of the approaching highly turbulent boundary layer with thecubic obstacles, the flow exhibits complex patterns, which dynamically evolve within and above thecubic obstacle array and have a significant impact on the transport of the momentum and scalar.The spatial evolution of the flow field has been carefully analyzed using the time-averaged velocitycontours, streamlines, and vector plots. A horseshoe vortex is observed in front of the first-rowobstacles at low elevations (e.g., y/d ≈ 0.25) as a result of the adverse pressure gradient in thewindward face of obstacles. It is observed that after the fifth row of obstacles, the flow quicklyreaches a self-similar state featuring a self-repeating pattern in time-averaged vortical structuresaround an obstacle.

A distinctive dual-peak pattern is observed in the cross-stream profiles of the mean concen-tration and its standard deviation under the canopy. The cube introduces a local blockage intothe concentration field such that the plume exhibits a distinct bimodal form as it sweeps aroundthe cube. However, above the canopy, a typical Gaussian pattern is observed in the cross-streamprofiles of the mean concentration. The simulation predictions of the mean concentration field arein excellent agreement with the water-channel measurement data. The shapes of the cross-streamprofiles of the concentration variance at different locations are also well captured by the simulation;however, their peak values are slightly under-predicted by the simulation under the canopy.

At plume edges, clean fluid packets (upstream or above the plume) entrain the plume andcontaminated fluid packets engulf into the background clean fluid. This process makes plume edgeshighly intermittent, which significantly increases the level of local concentration fluctuations. At allstreamwise locations, the peak values of the concentration standard deviation crms are above those ofthe mean concentration ⟨c⟩. This is due to the fact that the mean concentration gradient is typicallythe highest at plume edges. As a consequence, the production rate of the concentration variance issignificantly enhanced at plume edges.

The budget balances of the resolved KE and TKE transport equations have been analyzedbehind a typical obstacle located in the self-similar region at three different elevations. A studyof the local KE transfer between the resolved and subgrid scales indicates that the magnitude offorward scatter is approximately four times that of backward scatter, resulting in a net KE transferfrom the resolved to unresolved scales. By analyzing the budget balance of TKE, it is found thatthe TKE advection is primarily balanced with the turbulent diffusion, TKE production, and SGSdissipation rate. Under the canopy, the peak values of the TKE production term occur at z/d ≈ ±0.5directly downstream of the two vertical sides of the cube. Above the canopy at y/d = 1.5, themagnitudes of all TKE budgetary terms decrease by a factor of approximately 6 in comparison withtheir counterparts at the cube rooftop. As the elevation increases above the canopy, the disturbancesfrom the cubes are much decayed and flow becomes increasingly homogeneous in the spanwisedirection.

Through a careful analysis of the transport equation of the resolved SE c2, it is observedthat under the canopy at the half-cube height, the effect of the mean velocity field is to carry theconcentration out of the rear region of the cube towards the two canyons besides the cube. At therooftop level (y/d = 1), however, the advection term is constantly negative, which indicates that theeffect of the mean velocity field is to carry the resolved SE away. A positively valued molecular


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diffusion is observed at half-cube height indicating that the effect of molecular activities is to diffusethe resolved SE to higher elevations. Above the canopy, both the advection and molecular diffusionterms remain positive across the entire spanwise direction. This indicates that the contaminated fluidpackets engulf into the clean background fluid in the region above the canopy, which results in thegrowth of the plume size and elevation of its vertical centroid in the streamwise direction.

In an urban environment, accidental or deliberate release of toxic passive materials may incurserious threats to the safety of citizens. The major challenge associated with this subject involvesdeveloping a deeper understanding of the interactions of dynamically evolving flow structureswith the complex boundaries, the micro-mixing processes of the scalar, as well as the couplingof the transport processes of the momentum and scalar. The current research intends to pres-ent a comprehensive LES study of turbulent dispersion of a passive scalar in a modeled urbanenvironment, which, together with the experimental data, may serve as a benchmark test casefor future development of operational models for the prediction of urban pollution crises. Alsoin future studies, in order to establish a reliable LES approach for improved prediction of theconcentration field (in terms of its mean and RMS values), it may be beneficial to consider usinghigher-order (third-order or above) discretization schemes for the scalar transport equation, differenttypes of wall-functions (other than the current approach of Wang and Moin25) for wall-modeledLES, or even the wall-resolved LES method if the computing expenses are affordable. It wouldalso be interesting to study the predictive performances of advanced (non-counter-gradient type)SGS scalar-flux models32,36 in comparison with that of the EDM of Moin et al.31 based on thecurrent or other test cases of urban flow and dispersion. This suggested comparative study can bemeaningful, especially when the Schmidt number is high. This is because at high Schmidt numbers,the Batchelor scale is smaller than the Kolmogorov scale, and as a consequence, the “burden” onSGS scalar-flux modelling becomes heavier.

ACKNOWLEDGMENTS

The authors would like to thank Western Canada Research Grid (WestGrid) for an access tohigh-performance computing facilities. A research funding from Natural Sciences and EngineeringResearch Council (NSERC) of Canada to Bing-Chen Wang is gratefully acknowledged.

1 M. J. Brown, R. E. Lawson, D. S. DeCroix, and R. L. Lee, “Comparison of centerline velocity measurements obtainedaround 2D and 3D building arrays in a wind tunnel,” Technical Report LA-UR-01-4138, Los Alamos National Laboratory,2001.

2 S. R. Hanna, S. Tehranian, B. Carissimo, R. W. MacDonald, and R. Lohner, “Comparisons of model simulations withobservations of mean flow and turbulence within simple obstacle arrays,” Atmos. Environ. 36, 5067–5079 (2002).

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