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Flow pattern and pollutant dispersion over three dimensional building arrays
Atmospheric Environment 116 (2015) 202e215
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Atmospheric Environment journal homepage: www.elsevier.com/locate/atmosenv
Flow pattern and pollutant dispersion over three dimensional building arrays Zhi Shen, Bobin Wang, Guixiang Cui*, Zhaoshun Zhang Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
h i g h l i g h t s Accomplishment of LES investigation of flow and pollutant dispersion in building arrays. Discovery of five identifiable flow patterns which is closely related to building density. Exploration of the relationship between flow patterns and pollutant dispersion.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 October 2014 Received in revised form 29 April 2015 Accepted 12 June 2015 Available online 15 June 2015
The flow pattern and pollutant dispersion in urban canopies is investigated by large eddy simulation of flow over an array of cubes. It had been found that the pattern of flow over an isolated cubical obstacle can be characterized by an external wake of horseshoe vortex around the lower part of the windward face and an internal wake of recirculation cavity leeward. The width of the external wake Wex and the size of the internal wake Lin in the isolated roughness flow are used as key parameters to determine the wake effects on the flow and dispersion in the canopy of the same roughness height h in the isolated roughness flow. Flow patterns are categorized into five types based on the packing density as a result. The five types of urban canopy flow are introduced as (1) Isolated roughness flow when the lateral building interval WL is much greater than Wex and the streamwise building interval WS is much greater than Lin; (2) External wake interference flow when WL is less than Wex while WS is greater than Lin; (3) Internal wake interference flow when WS is in the same order of the size of Lin; (4) Skimming flow when WL is less than Wex and WS is less than Lin (5) Street network flow when WL and WS are much less than the Wex and Lin respectively. Results of time-averaging velocity field and pollutant concentration contours are demonstrated for each type of flow patterns. It is concluded that the behavior of flow pattern and pollutant dispersion is governed by the packing density from a very low packing density case, approximated as the flow around an isolated roughness element, to a high packing density case, resembled as the network flow. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Large-eddy-simulation Urban canopy Pollutant dispersion Flow pattern
1. Introduction The pollutant dispersion in urban atmosphere is an important topic in urban environment and has been paid great attention since late last century. It is well known that the pollutant dispersion is governed by flow patterns that consist of complex turbulence in the urban canopy, of which the geometric configuration is complicated and different from case to case. For two dimensional (2D) urban canopies Oke (1988) proposed a street canyon model that the flow
* Corresponding author. E-mail address: [email protected] (G. Cui). http://dx.doi.org/10.1016/j.atmosenv.2015.06.022 1352-2310/© 2015 Elsevier Ltd. All rights reserved.
pattern over 2D street canyon is dependent on the ratio of the building interval of l to its height h. The flow pattern is denoted as isolated roughness if l/h > 2 (Fig. 1a), wake interference if l/h~1 (Fig. 1b) and skimming when l/h < 1/2 (Fig. 1c). The pollutant dispersion is closely related to the flow pattern in flow over two dimensional canyons. If the pollutant source is located in the isolated roughness canyon it is spreading far behind roughness element and high concentration exists in the recirculation zone behind the building. When the pollutant source is situated in the skimming canyon the pollutant concentration is great in the leeward surface of the canyon and the pollutant is transported to the next canyon with considerable concentration (Walton and Cheng, 2002). The classification of flow regimes for 2D street
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Fig. 1. The flow regimes in 2D street canopy (Oke, 1988).
canyon gives great help for understanding pollutant dispersion in building canopy and casting engineering models (Walton et al., 2002). However in real urban canopies the geometrical configuration is three dimensional (3D) and flow patterns are different from those in 2D street canyon. For instance the flow pattern of 3D isolated roughness, shown in Fig. 2 (Hanna et al., 1982), is completely different from the flow pattern over 2D isolated roughness elements (Fig. 1a). Results of flow visualization (Martinuzzi and Tropea, 1993) indicate that there exist a horseshoe vortex around the roughness element and a pair of circulating vortex behind the roughness element. In this case the pollutant will be entrained into the horseshoe vortex with high concentration if there is a pollutant source at the front of the roughness element. Obviously the flow pattern around 3D roughness element is completely different from that in 2D street canyon for the pollutant dispersion is limited in the cavity in 2D isolated roughness flow without any lateral dispersion which is the dominant dispersion mechanism in 3D
203
roughness case. Building wake effects on dispersion of pollutant and particles have been taken account of in urban and regional dispersion models. For instance, Holmes and Morawska (2006) and Schulman et al. (2000) developed a model considering plume captured by recirculation cavity in near wake and re-emitted to downstream far wake bounded by horseshoe vortex envelop. The 3D flow pattern and pollutant dispersion will be presented in detail in Section 3. The purpose of this paper is to explore whether flow patterns can be classified into some identifiable regimes in 3D urban building canopy and the essential feature of pollutant dispersion in the classifications. In 2D street canyon the idea of the classification of flow regimes is based on the geometrical parameters of street canyon, i.e. the interval and height of the building (Oke, 1988). The geometrical parameters are also important criteria for identifying flow patterns in 3D building canopy but they are not the unique set of parameters. The characteristic length scales of average flow structures around each individual roughness elements are important as well. In concrete the spacing of horseshoe vortex tails and the streamwise and lateral size of the recirculation cavity behind roughness elements (see Fig. 2) are also key parameters in classification of 3D building canopy flow patterns. The criteria of the classification are established based on the comparison between geometrical parameters of urban building canopy and the characteristic length scales in isolated roughness flow. The flow pattern over 3D building canopies can be classified into five types and will be demonstrated in Section 3. The tool of this investigation is modern computational fluid dynamics (CFD). The computational fluid dynamic method has been developed quickly for investigating complex turbulent flows, including atmospheric environmental flows, since late last century. It has been convinced that Large Eddy Simulation (LES) (one of modern CFD methods) is an appropriate method for investigating flow and pollutant dispersion in complex atmospheric environment flow (Britter and Hanna, 2003). In particular a number of authors applied LES to investigate the flow and dispersion in 2D
Fig. 2. Flow pattern around 3D isolated roughness element (Hanna et al., 1982).
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street canyons (Walton and Cheng, 2002; Walton et al., 2002; Li et al., 2008; Inagaki et al., 2012). Numerous studies on the flow and dispersion in 3D building canopy have been bloomed since late last century and various numerical models have been proposed. For instance a number of authors applied LES to investigate flows in simplified building canopy, e.g. cubical roughness arrays (Coceal et al., 2006; Tseng et al., 2006; Xie and Castro, 2006; Coceal et al., 2007; Shi et al., 2008). LES has also been applied to investigating atmospheric environment flow in real cities, for instance Liu et al. (2011, 2012) investigated the traffic pollutant dispersion in City Macao and a residence area of City Beijing, Tseng et al. (2006), BouZeid et al. (2009) applied LES to investigate the urban atmospheric flow of City Baltimore and Lausanne respectively. The LES is used in this paper for investigation of the flow pattern and pollutant dispersion, for easier computation and analyses the building canopy is simplified as a cubical roughness array while the cubes are fully resolved in computation. The paper is organized as follows. Section 2 presents the numerical method of LES and verification of the code. Section 3 will illustrate the five identifiable flow patterns in exemplified 3D urban canopies and the pollutant dispersion formulation. Analyses and discussions of the numerical results are given in Section 4 with brief concluding remarks.
Flow within building arrays is heterogeneous in horizontal plane and the eddy viscosity coefficient Cs is determined by a dynamic procedure which is based on the scale similarity between two filter scale D1 and D2 with D1 ¼ D and D2 ¼ 2D respectively. Unlike Germano's method (Germano, 1992) which is valid for turbulent flows with homogeneous directions, Meneveau et al. (1996) proposed a Lagrangian dynamic model which takes averages along trajectories of fluid particles and results in two time-integral values
Z0
ILM ¼
Lij Mij ðzðt 0 Þ; t 0 ÞWðt t 0 Þdt 0 ;
(4)
∞
IMM ¼
Z0
Mij Mij ðzðt 0 Þ; t 0 ÞWðt t 0 Þdt 0 :
(5)
∞ 2 e e eu e ug In equations (4) and (5)Lij ¼ u u , M ¼ 2½D i j i j ij 2 SSij g D21 SSij , and W(tet’) is a weight function. If the weight function takes the exponential form, i.e.Wðt t 0 Þ ¼ expððt t 0 Þ=TÞ=T, then ILM and IMM can be determined by following equations:
vILM vI 1 b j LM ¼ þu L M ILM T ij ij vt vxj
(6)
vIMM vI 1 b j MM ¼ þu Mij Mij IMM T vt vxj
(7)
2. Numerical methods 2.1. Governing equations The wind speed in urban areas is relatively low and the air can be treated as incompressible. In large eddy simulation, the continuity equation and momentum equations for filtered velocity ui and filtered pressure p read
vui ¼0 vxj vui vui uj 1 vp v þ ¼ þ r vxi vxj vt vxj
(1)
tij þ v
vui vxj
! þ fi
(2)
where xi (i ¼ 1,2,3) are streamwise, spanwise and vertical coordinates respectively, and ui (i ¼ 1,2,3) are filtered velocity components in corresponding directions, fi are external forces. The air density is r ¼ 1.208 kgm3, and the kinetic viscosity coefficient is n ¼ 1.5 105 m2 s1. Transport equation of filtered pollutant concentrationc reads:
vc vc ui v n vc þ tci þ þQ ¼ vt vxi Sc vxi vxi
The time scale T is set to equal 1.5D(ILMIMM)1/8 as proposed by Meneveau et al. (1996), and superscripts and ~ represent filtering at D1 and D2 respectively. Equation (6) and equation (7) can be easily discretized and coupled with governing equations, thus model coefficient Cs2 ¼ ILM =IMM can be obtained. The Lagrangian dynamic model is suitable for heterogeneous turbulent flow and previous investigation (Shi et al., 2008) indicates that dynamic model performs better than the standard Smagrinsky model in solving turbulent wind field. The details of the sub-grid scale model, including the Lagrangian dynamic model, can be found in Sagaut (2002). The sub-grid scale mass fluxtci is taken as a gradient model and 2 D2 b the sub-grid mass diffusivityDt ¼ Cc;s 1 S , where Cc,s is the model
(3)
In the above equation Sc is the molecular Schmidt number, Q is the pollutant source term, in this study the point source will be inserted in some position in equation (3) and Q ¼ qd(x) in which d(x) is the Dirac function. 2.2. Sub-grid scale model In equations (2) and (3), tij ¼ ui uj ui uj and tci ¼ ui c ui c are sub-grid scale stress and sub-grid scale mass flux which should be closed by models. In this study the sub-grid eddy viscosity and subgrid eddy diffusivity model are accepted respectively for tij and tci which can be written astij ¼ 2vt Sij þ tkk dij =3, tci ¼ Dt vc=vxi whereSij ¼ ðvui =vxj þ vuj =vxi Þ=2 is the filtered strain tensor. Usually the sub-grid eddy viscosity is formulated asvt ¼ Cs2 D2 S in which Cs is the model coefficient and D is the grid size.
Fig. 3. Distribution of cubes in verification case.
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Fig. 4. Spanwise distribution of pollutant concentration, compared with experiment. (a) 2 h behind the source, z ¼ 0.25 h; (b) 6 h behind the source, z ¼ 0.25 h; (c) 2 h behind the source, z ¼ 1.25 h; (d) 6 h behind the source, z ¼ 1.25 h.
coefficient and can also be determined by the dynamic procedure. 2 e e 2 g In this caseLi ¼ ueie c uf i c andMi ¼ D2 Svc=vxi D1 S vc=vxi . The two time-integral values Ic,LM and Ic,MM read:
Z0
Ic;LM ¼
Li Mi ðzðt 0 Þ; t 0 ÞWðt t 0 Þdt 0
(8)
∞
Ic;MM ¼
Z0
Mi Mi ðzðt 0 Þ; t 0 ÞWðt t 0 Þdt 0
2.3. Numerical scheme Finite volume method (FVM) is employed to discretize governing equations (1) through (3). The QUICK scheme is used in spatial discretization and the third order RungeeKutta scheme is utilized for time advancement. The discretized equations are solved using SIMPLE methods on non-staggered grids with momentum interpolation. In the lateral and longitudinal direction, the periodic boundary condition is applied for each velocity component, and the
(9)
∞
which satisfy following equations with the weigh function mentioned above
vIc;LM vI 1 b j c;LM ¼ L M Ic;LM þu T i i vt vxj
(10)
vIc;MM vI 1 b j c;MM ¼ Mi Mi Ic;MM þu T vt vxj
(11)
Finally the model coefficient of sub-grid mass diffusivity is 2 Cc;s ¼
Ic;LM Ic;MM
With the sub-grid viscosity and the sub-grid diffusivity both determined by the Lagrangian dynamic procedure, the usually assumed sub-grid Schmidt number is abandoned.
Fig. 5. Streamlines around an isolated 3D roughness element by the present computation, the color level indicates the height z/h.
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Fig. 6. Distribution of pollutant concentration in isolated roughness flow. (a) Horizontal plane, z/h ¼ 0.1; (b) spanwise plane, (x-xs)/h ¼ 2; (c) spanwise plane, (x-xs)/h ¼ 5.
non-slip condition is applied at the cube surface and ground. A free slip and impermeable condition is applied at the upper boundary. The concentration is set to zero at the inlet and lateral boundaries of the domain. The zero normal gradient of concentration is
imposed at the outlet and the upper boundary as well as all the solid surfaces. In each of the present simulations, the mesh has 12 grid points along each dimension of the cube mounted on the wall with the
Fig. 7. Spanwise distribution of non-dimensional pollutant in isolated roughness flow, compared with double Gaussian fitting results. z/h ¼ 0.1.
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the flow on the Reynolds number can be neglected according to Snyder (1972). A total of 1000 T* is simulated after the flow approaching statistically steady state, where T* ¼ h/ut. 2.4. Verification of the numerical method and computing code
Fig. 8. Flow pattern of external wake interference flow.
distance from the first grid point to the boundary refined. Since vortices and wakes of scales above h/10 contributed more significantly to mass and momentum transfer in the core region (Xie et al., 2008), the present resolution is sufficient for large eddy simulations in terms of the average flow field (Kanda et al., 2004). A constant longitudinal pressure gradient of u2t =Lz is imposed to counterbalance the drag from cubes and the ground, where ut is the total wall friction velocity and Lz is the domain height. The roughness Reynolds number based on the friction velocity and height of the cube Ret ¼ uth/n ¼ 500, which is large enough so that the dependence of
The numerical code has been verified by comparison of numerical results of testing case with corresponding experimental results provided by Hilderman and Chong (2007) from a fully developed turbulent boundary layer in a water tunnel. In the experiment, there are three layouts of cubes one of which is a square array of 16 16 cubes with a height h equaling 31.75 mm. The dye is injected into the array after the first row of cubes. According to the experiments of Hilderman and Chong (2007), the molecular diffusivity of the fluorescent dye used in the water channel is D ¼ 5.3 1010 m2 s1, and the molecular Schmidt number is about 1920 with the kinematic viscosity of water n ¼ 1.0 106 m2 s1. The molecular diffusivity of the fluorescent dye is much lower than that of air; however, the molecular diffusivity is negligible in turbulent flow since the turbulent diffusivity plays a dominant role in scalar dispersion. The turbulent diffusivity is in the same order of magnitude of turbulent viscosity whether the turbulent flow is in air or water, and it is usually accepted that turbulent Schmidt number is around 1.0 in simulation of turbulent dispersion. Hence the water channel experiment using the dye of a low diffusivity can serve as a reasonable approximation of urban atmosphere layer and also a reliable validation of numerical simulations. The water channel experiments mentioned above was numerically simulated through Direct Numerical Simulation (DNS) with an array of 8 8 cubes by Branford et al. (2011), where the molecular Schmidt number was set as 1.0 and three orders less than in the water channel, and the DNS results of mean concentration agree with those from the experiments. The layout of the array in present
Fig. 9. Distribution of pollutant concentration in external wake interference flow. (a) Horizontal plane, z/h ¼ 0.1; (b) streamwise plane, (y-ys)/h ¼ 0; (c) spanwise plane, (x-xs)/h ¼ 2; (d) spanwise plane, (x-xs)/h ¼ 5.
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Fig. 10. Spanwise distribution of non-dimensional pollutant in external wake interference flow, compared with Gaussian and double Gaussian fitting results. z/h ¼ 0.1.
LES is similar to that in the DNS shown in Fig. 3; the domain size is 16 h in longitudinal and lateral directions and 8 h in vertical direction. It has been shown by many authors (Cheng and Castro, 2002; Coceal et al., 2006; Wang et al., 2014) that the fully developed turbulent boundary layer flow over an array of cubes can be simulated numerically by the semi-channel model of flow over the same array appropriately as long as the height of the semi-channel is large enough, some author accepted a smaller height, e.g. Lz ¼ 4h~6 h (Coceal et al., 2006) while Wang et al. (2014) suggested Lz ¼ 8 h. In this paper Lz ¼ 8 h is used. In Hilderman and Chong's experiment (Hilderman and Chong, 2007), a point source is located at the distance from ground zs ¼ 0.0625 h and xs ¼ 2 h, ys ¼ 7 h (Fig. 3). The comparison is made between numerical results and experimental data. Fig. 4 shows the comparison of spanwise distribution of time average dimensionless concentration C* ¼ Cuth2/q between numerical results and Hilderman and Chong's experiment measurements at distance 2 h and 6 h behind pollutant source. Both the concentration distribution inside canopies (z ¼ 0.25 h) and that above canopies (z ¼ 1.25 h) show fairly good agreement between numerical simulations and experimental measurements. It is concluded that the present numerical code can be used to investigate the flow pattern and pollutant transportation and dispersion in urban canopies.
determined by the flow Reynolds number and the size of cubes. If the Reynolds number is high enough the flow is Reynolds number independent, or so called self-similarity (Snyder, 1972). In following numerical simulations h equals 0.02 m, the velocity of free-stream flow U equals 10 m/s, and corresponding Reynolds number Re(¼ Uh/n) is estimated as 104. The flow over an isolated 3D roughness element has been investigated by many authors. The flow pattern and geometrical parameters are presented in Snyder and Lawson (1994). Martinuzzi and Tropea (1993) also found that
3. Flow pattern and pollutant dispersion in 3D urban canopy The urban building canopy is composed of buildings which are simplified as cuboids in this study. Like isolated roughness in 2D street canyon, the isolated roughness element is the simplest form of 3D urban canopy in which the flow pattern is completely
Fig. 11. Flow pattern of near wake interference flow.
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Fig. 12. Distribution of pollutant concentration in near wake interference flow. (a) Horizontal plane, z/h ¼ 0.1; (b) streamwise plane, (y-ys)/h ¼ 0; (c) spanwise plane, (x-xs)/h ¼ 2; (d) spanwise plane, (x-ys)/h ¼ 5.
the recovery length behind a three dimensional obstacle is shorter than in the case of a two-dimensional flow, whose value close to unity scaled by building height for obstacles with a small aspect ratio (l/h~1). Besides the complex flow geometry, the peculiar
feature of the flow pattern is that there exist a horseshoe vortex around the lower part of the roughness element and a recirculation cavity behind the roughness element (Fig. 5). In this study the flow region containing the horseshoe vortex tails is called the external
Fig. 13. Spanwise distribution of non-dimensional pollutant in near wake interference flow, compared with Gaussian and double Gaussian fitting results. z/h ¼ 0.1.
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Fig. 14. Flow pattern in skimming flow. (a) non-staggered array; (b) staggered array.
wake area and the flow region inside the recirculation cavity is called the internal wake area. At a high Reynolds number, the spacing of horseshoe vortex tails Wex can be approximated as a function of the streamwise distance from the windward face of buildings (Schulman et al., 2000)
Wex ¼
R x 1=3
w þ 2 3 R
(12)
in which w is the width of roughness element and R is the geometrical parameter of the roughness element defined by R ¼ (BS)2/3(BL)1/3, BS ¼ Min(w, h), BL ¼ Max(w, h), where h is the height of the roughness element; in the cubical roughness element case R ¼ w ¼ h. Equation (12) indicates that the width of horseshoe vortex increases moderately along the downstream direction. The lateral size of the cavity zone Win is approximately equal to the
width of the roughness element and the streamwise size Lin equals the height of the element approximately.
Win zw; Lin zh
(13)
Based on the flow characteristics around the roughness element we can classify the 3D building canopy flow into the following five types. 3.1. Isolated roughness The first type of 3D building canopy is defined as the isolated roughness when the lateral building interval WL is much greater than Wex and the streamwise interval WS is much greater than the longitudinal size of internal wake Lin for cubical roughness. In this case neighboring buildings do not influence the flow around each
Fig. 15. Distribution of pollutant concentration of skimming flow in non-staggered layout.
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Fig. 16. Distribution of pollutant concentration of skimming flow in staggered layout.
individual building; hence the flow pattern is similar to the flow over an isolated roughness element. To simulate the flow developing along the longitudinal direction, the inflow data is generated according to Xie and Castro (2008) and the normal gradient of each velocity component u, v, w is set to zero at the outlet with flux correction. A point source of pollutant is located at zs ¼ 0.05 h from the ground and xs ¼ 0.5 h ahead of the upwind face in the centerplane. The distributions of concentration in horizontal and vertical cross-sections are illustrated in Fig. 6 which shows that the pollutant is mainly distributed in the horseshoe vortex region. To further explore the pollutant dispersion in longitudinal direction Fig. 7 shows spanwise distribution of non-dimensional concentration C*¼Cu*h2/q in six downstream locations shown in Fig. 7a. Two peaks of the pollutant concentration can be viewed obviously in Fig. 7 and C* can be best fitted by the double Gaussian or biGaussian expression (MacDonald et al., 1998):
vortex tails cannot expand along streamwise direction but is equal to the lateral distance of buildings. The internal wake region in Fig. 8 is similar to that of isolated roughness flow in Fig. 5. Fig. 9 illustrates the distribution of pollutant concentration for the external wake interference flow on horizontal and vertical cross-sections. The point source is located in the head region of the horseshoe vortex; however the pollutant is not concentrated in the horseshoe vortex region but spread in to the internal wake region because the spacing of horseshoe vortex tails is narrower. Fig. 9a clearly shows this pattern. Fig. 10 shows distribution of C* on six streamwise locations marked in Fig. 10a for the external wake interference flow. Both single Gaussian and double Gaussian fitting results are presented. The single Gaussian expression reads:
. C* ¼ Cexp ðy ys Þ2 2s2y :
. . o n C* ¼ C exp ðy ys Þ2 2s2y þ exp ðy þ ys Þ2 2s2y (14) where ys is the displacement of the plume center and sy is the spanwise width of each plume. Double Gaussian fitting results show fine agreement with the simulated pollutant concentration. 3.2. External wake interference regime When the lateral distance between buildings is less than the length of external wake (WL < Wex) whereas the streamwise distance still much larger than the size of near wake (WS [ Lin), the flow over building canopy is called the external wake interference. In this case the individual building can no longer be viewed as isolated. The external wake of neighboring buildings, i.e. the horseshoe vortex, interacts with each other, although the internal wake is not affected by neighboring buildings. An example of this flow regime is illustrated in Fig. 8 in which flow is simulated numerically over a square array with both streamwise and spanwise interval of the buildings equaling 4 h. Obviously the horseshoe vortex from individual building cannot be spreading freely but restricted by other neighboring external wakes that the spacing of
Fig. 17. Flow pattern in street network, flow direction is parallel to street.
(15)
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Fig. 18. Distribution of non-dimensional pollutant in street network. (a) Horizontal plane, z/h ¼ 0.2; (b) streamwise plane, (y-ys)/h ¼ 0.
Due to the influence of external wake region, the spanwise distribution of pollutant concentration still shows double peaks clearly near the head of horseshoe vortex, yet the double peaks are gradually reduced in downstream location (Fig. 10d). The spanwise spacing of
double peak is also reduced because the spanwise spacing of the horseshoe vortex is reduced, which illustrates the higher concentration in the recirculation zone compared to the isolated roughness flow. The quantitative analyses will be presented in Section 4.
Fig. 19. Streamwise decay of pollutant.
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Fig. 20. Streamwise development of the spacing of the double peaks.
3.3. Internal wake interference When the streamwise interval of buildings is in the same order of the longitudinal size of internal wake, i.e. Ws ~ Lin, the development of internal wake behind each individual building will be influenced by downstream buildings. A case with both streamwise and spanwise distances of the buildings equaling 2 h is simulated to represent such kind of flow. As viewed from streamlines in Fig. 11, external wake region is similar to the last case while flow in the recirculation zone is altered in the vicinity of downstream buildings. This flow pattern is named the internal wake interference flow. Fig. 12 illustrates the distribution of pollutant concentration for the internal wake interference flow in horizontal and vertical crosssections. The pollutant spreads quickly to neighboring columns, which is different from dispersion in external wake interference flow. As shown in Fig. 12a and b, although the point source is located before the second row, there is high concentration behind the first row since a proportion of the pollutant released from the point source is trapped in the internal wake region belonging to the first building row. Fig. 13 illustrates the spanwise distribution of C* for the internal wake interference flow at different downstream locations marked in Fig. 13a. The double peaks of the pollutant concentration profile still exist (Fig. 13b) with narrow spacing and the double peaks nearly coalesce in further downstream locations (Fig. 13c,d) due to
the enhanced mixing by the interference of internal wake (see Fig. 12c,d). 3.4. Skimming flow If building intervals of both longitudinal and lateral directions approach h, then both internal and external wakes are restricted. This case is named the skimming flow. Two different layouts of cubes are investigated corresponding to staggered and nonstaggered arrangements of buildings, see Fig. 14a and b. In the non-staggered case (Fig. 14a), the flow field is divided into two regions named ST1 and ST2 respectively. In the region ST1 the flow is almost parallel to the street with greater speed, while the complicated recirculation flows are generated in the ST2, perpendicular to ST1, by lateral and vertical large shears. This is the fundamental difference between the 3D skimming flow and the 2D skimming flow that there is only vertical shear in the 2D skimming flow in street canyons. The difference between 3D and 2D skimming flows can be clearly shown in the distribution of pollutant concentration in Fig. 15 for the non-staggered case. The pollutant is transported along streamwise direction in ST1, meanwhile it is dispersed in lateral direction in ST2. The pollutant dispersion in the 3D skimming is similar to the 2D skimming flow only in the central section at (y-ys)/h ¼ 0. In the staggered case the 3D skimming flow is more complicated as shown in Fig. 14b in which the flow field is also divided into ST1
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Fig. 21. Streamwise development of the plume width.
and ST2. In ST1 the incoming flow down-washes against the obstacle and enters into ST2. Fig. 16 presents the distribution of pollutant concentration for the staggered case. By comparing between Figs. 15 and 16 it can be seen that the pollutant concentration is lower inside building canopy in staggered case due to the strong dispersion in ST1. In both staggered and non-staggered cases the external and internal wake cannot be distinctly identified due to the dense-packing layout and the flow is at high speed above building canopies with low speed circulation inside canopies. This is similar to the skimming flow in 2D street canyon except that the flow is inherently three dimensional inside the canopy.
channel flow while the flow in streets perpendicular to the free stream direction is similar to the 2D skimming flow. The pollutant dispersion is demonstrated in Fig. 18. The pollutant is dispersed downstream and gradually decaying, and it is restricted mostly in streets parallel to free stream direction since lateral dispersion is weak. For building canopies with streets in oblique direction from the free stream, the flow pattern is more complicated but can still be simplified by network models as elaborated by Hamlyn et al. (2007) and Soulhac et al. (2011). We will not introduce street network models in detail. We just note that the street network is one of the accepted simplifications of building canopy with the streamwise and lateral interval between buildings much less than the building height, i.e. WS≪h and WL≪h.
3.5. Street network 4. Discussions and conclusion When the spacing between buildings is much less than the building height, flow within the canopy is nearly homogeneous along streets direction, whether parallel or perpendicular to the free stream direction, and behave like 2D flow. Hence it can be approximated by a street network model and can be called network flow. In this case both external and internal wakes disappears. The street network is somewhat simpler building canopy and has been studied by Hamlyn et al. (2007) and Soulhac et al. (2011). Fig. 17 shows the flow pattern in the street network where the flow in streets parallel to the free stream direction can be simplified as
The numerical investigation by LES shows identifiable patterns of flow and pollutant dispersion in 3D building canopies. When there exist the external wake (the horseshoe vortex around each individual building) and the internal wake (the downstream recirculating), the flow pattern is similar to the flow over isolated roughness element and the pollutant dispersion shows double peak in the lateral direction (Figs. 7, 10 and 13). The distribution of pollutant concentration can be best fitted with double Gaussian function, Equation (14)
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. . o n C ¼ C exp ðy ys Þ2 2s2y þ exp ðy þ ys Þ2 2s2y : In this expression, C, ys and sy are varying in the streamwise direction and are of interest for modeling. C denotes the decay of pollutant concentration; ys stands for the spanwise displacement of each plume and sy represents the plume width in terms of standard deviation. Fig. 19 show that the streamwise decay of pollutant concentration is similar among isolated roughness element flow, external wake interference flow and internal wake interference flow. However, the decay is slower for the isolated roughness flow since both external wake and internal wake are not interfered by neighboring buildings. The fastest decay of pollutant occurs in the internal wake interference flow since the flow is restricted in streamwise direction while spanwise transfer of mass is enhanced in the building canopy (compare Fig. 9a with Fig. 12a). The peak spacing and the plume width is presented respectively in Figs. 20 and 21 for the three cases. The spacing of double peaks is larger in the isolated roughness element flow since the external wake is not restricted. In Fig. 20a it also shows that the development of double peak spacing agrees with the power law ysf(x-xs)1/3. This indicates that the pollutant is mostly trapped in the external wake (see Equation (12)). The development of the peak spacing and the plume width is quite similar between the external and internal wake interference flow, see Fig. 20b, c and Fig. 21b, c, since the flow is restricted in space by building intervals in both cases. The 3D skimming flow is more complicated that the flow pattern and pollutant dispersion are dependent on the layout of the buildings, e.g. staggered or non-staggered. However the street network flow is a peculiar 3D building canopy and it has been studied by a couple of authors (Hamlyn et al., 2007; Soulhac et al., 2011). In summary flow patterns and pollutant dispersion in 3D building canopies can be identified as five regimes which depend on geometrical parameters of building canopy and characteristic length scales of flow over isolated roughness element. The essential feature between different flow patterns is the building wake effect which is varying with the packing density. For low packing density the flow is similar to the isolated roughness flow and the pollutant dispersion is governed by the downwash effect of the building wake (Schulman et al., 2000). For higher packing density the flow is quasi-two-dimensional and the street network model (Soulhac et al., 2011; Hamlyn et al., 2007) applies appropriately. The external wake interference and internal wake interference flows are 3D complex turbulent flow, where the characteristics of pollutant concentration vary with the building density as well. Therefore practical model for these two flow types is expected to be reconstructed after more detailed quantitative study. Acknowledgements This study was sponsored by National Natural Science Foundation of China (NSFC Grant 11132005, 11490551). References Bou-Zeid, E., Overney, J., Rogers, B.D., Parlange, M.B., 2009. The effects of building representation and clustering in large-eddy simulations of flows in urban canopies. Bound. Layer Meteorol. 132 (3), 415e436. Branford, S., Coceal, O., Thomas, T.G., Belcher, S.E., 2011. Dispersion of a point-source release of a passive scalar through an urban-like array for different wind directions. Bound. Layer Meteorol. 139 (3), 367e394.
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Britter, R.E., Hanna, S.R., 2003. Flow and dispersion in urban areas. Annu. Rev. Fluid Mech. 35, 469e496. Cheng, H., Castro, I.P., 2002. Near-wall flow development after a step change in surface roughness. Bound. Layer Meteorol. 105 (3), 411e432. Coceal, O., Dobre, A., Thomas, T.G., Belcher, S.E., 2007. Structure of turbulent flow over regular arrays of cubical roughness. J. Fluid Mech. 589, 375e409. Coceal, O., Thomas, T.G., Castro, I.P., Belcher, S.E., 2006. Mean flow and turbulence statistics over groups of urban-like cubical obstacles. Bound. Layer Meteorol. 121 (3), 491e519. Germano, M., 1992. Turbulence: the filtering approach. J. Fluid Mech. 238, 325e336. Hamlyn, D., Hilderman, T., Britter, R., 2007. A simple network approach to modelling dispersion among large groups of obstacles. Atmos. Environ. 41 (28), 5848e5862. Hanna, S.R., Briggs, G.A., Hosker, R.P.J., 1982. Handbook on Atmospheric Diffusion. U.S. Department of Energy DOE/TIC-11223. Hilderman, T., Chong, R., 2007. A Laboratory Study of Momentum and Passive Scalar Transport and Diffusion within and above a Model Urban Canopy. DRDC Suffield CR 2008-025, Defence R&D Canada, Suffield, Ralston, Alberta. Holmes, N.S., Morawska, L., 2006. A review of dispersion modelling and its application to the dispersion of particles: an overview of different dispersion models available. Atmos. Environ. 40 (30), 5902e5928. Inagaki, A., Castillo, M.C.L., Yamashita, Y., Kanda, M., Takimoto, H., 2012. Large-eddy simulation of coherent flow structures within a cubical canopy. Bound. Layer Meteorol. 142 (2), 207e222. Kanda, M., Moriwaki, R., Kasamatsu, F., 2004. Large-eddy simulation of turbulent organized structures within and above explicitly resolved cube arrays. Bound. Layer Meteorol. 112 (2), 343e368. Li, X.X., Liu, C.H., Leung, D.Y.C., 2008. Large-eddy simulation of flow and pollutant dispersion in high-aspect-ratio urban street canyons with wall model. Bound. Layer Meteorol. 129 (2), 249e268. Liu, Y.S., Cui, G.X., Wang, Z.S., Zhang, Z.S., 2011. Large eddy simulation of wind field and pollutant dispersion in downtown Macao. Atmos. Environ. 45 (17), 2849e2859. Liu, Y.S., Miao, S.G., Zhang, C.L., Cui, G.X., Zhang, Z.S., 2012. Study on microatmospheric environment by coupling large eddy simulation with mesoscale model. J. Wind Eng. Ind. Aerodyn. 107, 106e117. MacDonald, R.W., Griffiths, R.F., Hall, D.J., 1998. A comparison of results from scaled field and wind tunnel modelling of dispersion in arrays of obstacles. Atmos. Environ. 32 (22), 3845e3862. Martinuzzi, R., Tropea, C., 1993. The flow around surface-mounted, prismatic obstacles placed in a fully-developed channel flow. J. Fluids Eng. Trans. Asme 115 (1), 85e92. Meneveau, C., Lund, T.S., Cabot, W.H., 1996. A Lagrangian dynamic subgrid-scale model of turbulence. J. Fluid Mech. 319 (1), 353e385. Oke, T.R., 1988. Street design and urban canopy layer climate. Energy Build. 11 (1e3), 103e113. Sagaut, P., 2002. Large-eddy Simulation for Incompressible Flows - an Introduction, second ed. Springer-Verlag. Schulman, L.L., Strimaitis, D.G., Scire, J.S., 2000. Development and evaluation of the PRIME Plume Rise and building downwash model. J. Air & Waste Manag. Assoc. 50 (3), 378e390. Shi, R.F., Cui, G., Wang, Z., Xu, C., Zhang, Z., 2008. Large eddy simulation of wind field and plume dispersion in building array. Atmos. Environ. 42 (6), 1083e1097. Snyder, W.H., 1972. Similarity criteria for the application of fluid models to the study of air pollution meteorology. Bound. Layer Meteorol. 3 (1), 113e134. Snyder, W.H., Lawson, R.E., 1994. Wind tunnel measurements of flow fields in the vicinity of buildings. In: 8th Joint Conference on Applications of Air Pollution Meteorology. American Meteorological Society and the Air and Waste Management Association, AMS, Boston, MA. Soulhac, L., Salizzoni, P., Cierco, F.X., Perkins, R., 2011. The model SIRANE for atmospheric urban pollutant dispersion; part I, presentation of the model. Atmos. Environ. 45 (39), 7379e7395. Tseng, Y.H., Meneveau, C., Parlange, M.B., 2006. Modeling flow around bluff bodies and predicting urban dispersion using large eddy simulation. Environ. Sci. Technol. 40 (8), 2653e2662. Walton, A., Cheng, A.Y.S., 2002. Large-eddy simulation of pollution dispersion in an urban street canyon - Part II: idealised canyon simulation. Atmos. Environ. 36 (22), 3615e3627. Walton, A., Cheng, A.Y.S., Yeung, W.C., 2002. Large-eddy simulation of pollution dispersion in an urban street canyon - Part I: comparison with field data. Atmos. Environ. 36 (22), 3601e3613. Wang, B.B., Wang, Z.S., Cui, G.X., Zhang, Z.S., 2014. Study on the dynamic characteristics of flow over building cluster at high Reynolds number by large eddy simulation. Sci. China Phys. Mech. Astron. 57 (6), 1144e1159. Xie, Z.T., Castro, I.P., 2006. LES and RANS for turbulent flow over arrays of wallmounted obstacles. Flow Turbul. Combust. 76 (3), 291e312. Xie, Z.T., Castro, I.P., 2008. Efficient generation of inflow conditions for large eddy simulation of street-scale flows. Flow Turbul. Combust. 81 (3), 449e470. Xie, Z.T., Coceal, O., Castro, I.P., 2008. Large-eddy simulation of flows over random urban-like obstacles. Bound. Layer Meteorol. 129 (1), 1e23.
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Atmospheric Environment journal homepage: www.elsevier.com/locate/atmosenv
Flow pattern and pollutant dispersion over three dimensional building arrays Zhi Shen, Bobin Wang, Guixiang Cui*, Zhaoshun Zhang Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
h i g h l i g h t s Accomplishment of LES investigation of flow and pollutant dispersion in building arrays. Discovery of five identifiable flow patterns which is closely related to building density. Exploration of the relationship between flow patterns and pollutant dispersion.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 October 2014 Received in revised form 29 April 2015 Accepted 12 June 2015 Available online 15 June 2015
The flow pattern and pollutant dispersion in urban canopies is investigated by large eddy simulation of flow over an array of cubes. It had been found that the pattern of flow over an isolated cubical obstacle can be characterized by an external wake of horseshoe vortex around the lower part of the windward face and an internal wake of recirculation cavity leeward. The width of the external wake Wex and the size of the internal wake Lin in the isolated roughness flow are used as key parameters to determine the wake effects on the flow and dispersion in the canopy of the same roughness height h in the isolated roughness flow. Flow patterns are categorized into five types based on the packing density as a result. The five types of urban canopy flow are introduced as (1) Isolated roughness flow when the lateral building interval WL is much greater than Wex and the streamwise building interval WS is much greater than Lin; (2) External wake interference flow when WL is less than Wex while WS is greater than Lin; (3) Internal wake interference flow when WS is in the same order of the size of Lin; (4) Skimming flow when WL is less than Wex and WS is less than Lin (5) Street network flow when WL and WS are much less than the Wex and Lin respectively. Results of time-averaging velocity field and pollutant concentration contours are demonstrated for each type of flow patterns. It is concluded that the behavior of flow pattern and pollutant dispersion is governed by the packing density from a very low packing density case, approximated as the flow around an isolated roughness element, to a high packing density case, resembled as the network flow. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Large-eddy-simulation Urban canopy Pollutant dispersion Flow pattern
1. Introduction The pollutant dispersion in urban atmosphere is an important topic in urban environment and has been paid great attention since late last century. It is well known that the pollutant dispersion is governed by flow patterns that consist of complex turbulence in the urban canopy, of which the geometric configuration is complicated and different from case to case. For two dimensional (2D) urban canopies Oke (1988) proposed a street canyon model that the flow
* Corresponding author. E-mail address: [email protected] (G. Cui). http://dx.doi.org/10.1016/j.atmosenv.2015.06.022 1352-2310/© 2015 Elsevier Ltd. All rights reserved.
pattern over 2D street canyon is dependent on the ratio of the building interval of l to its height h. The flow pattern is denoted as isolated roughness if l/h > 2 (Fig. 1a), wake interference if l/h~1 (Fig. 1b) and skimming when l/h < 1/2 (Fig. 1c). The pollutant dispersion is closely related to the flow pattern in flow over two dimensional canyons. If the pollutant source is located in the isolated roughness canyon it is spreading far behind roughness element and high concentration exists in the recirculation zone behind the building. When the pollutant source is situated in the skimming canyon the pollutant concentration is great in the leeward surface of the canyon and the pollutant is transported to the next canyon with considerable concentration (Walton and Cheng, 2002). The classification of flow regimes for 2D street
Z. Shen et al. / Atmospheric Environment 116 (2015) 202e215
Fig. 1. The flow regimes in 2D street canopy (Oke, 1988).
canyon gives great help for understanding pollutant dispersion in building canopy and casting engineering models (Walton et al., 2002). However in real urban canopies the geometrical configuration is three dimensional (3D) and flow patterns are different from those in 2D street canyon. For instance the flow pattern of 3D isolated roughness, shown in Fig. 2 (Hanna et al., 1982), is completely different from the flow pattern over 2D isolated roughness elements (Fig. 1a). Results of flow visualization (Martinuzzi and Tropea, 1993) indicate that there exist a horseshoe vortex around the roughness element and a pair of circulating vortex behind the roughness element. In this case the pollutant will be entrained into the horseshoe vortex with high concentration if there is a pollutant source at the front of the roughness element. Obviously the flow pattern around 3D roughness element is completely different from that in 2D street canyon for the pollutant dispersion is limited in the cavity in 2D isolated roughness flow without any lateral dispersion which is the dominant dispersion mechanism in 3D
203
roughness case. Building wake effects on dispersion of pollutant and particles have been taken account of in urban and regional dispersion models. For instance, Holmes and Morawska (2006) and Schulman et al. (2000) developed a model considering plume captured by recirculation cavity in near wake and re-emitted to downstream far wake bounded by horseshoe vortex envelop. The 3D flow pattern and pollutant dispersion will be presented in detail in Section 3. The purpose of this paper is to explore whether flow patterns can be classified into some identifiable regimes in 3D urban building canopy and the essential feature of pollutant dispersion in the classifications. In 2D street canyon the idea of the classification of flow regimes is based on the geometrical parameters of street canyon, i.e. the interval and height of the building (Oke, 1988). The geometrical parameters are also important criteria for identifying flow patterns in 3D building canopy but they are not the unique set of parameters. The characteristic length scales of average flow structures around each individual roughness elements are important as well. In concrete the spacing of horseshoe vortex tails and the streamwise and lateral size of the recirculation cavity behind roughness elements (see Fig. 2) are also key parameters in classification of 3D building canopy flow patterns. The criteria of the classification are established based on the comparison between geometrical parameters of urban building canopy and the characteristic length scales in isolated roughness flow. The flow pattern over 3D building canopies can be classified into five types and will be demonstrated in Section 3. The tool of this investigation is modern computational fluid dynamics (CFD). The computational fluid dynamic method has been developed quickly for investigating complex turbulent flows, including atmospheric environmental flows, since late last century. It has been convinced that Large Eddy Simulation (LES) (one of modern CFD methods) is an appropriate method for investigating flow and pollutant dispersion in complex atmospheric environment flow (Britter and Hanna, 2003). In particular a number of authors applied LES to investigate the flow and dispersion in 2D
Fig. 2. Flow pattern around 3D isolated roughness element (Hanna et al., 1982).
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street canyons (Walton and Cheng, 2002; Walton et al., 2002; Li et al., 2008; Inagaki et al., 2012). Numerous studies on the flow and dispersion in 3D building canopy have been bloomed since late last century and various numerical models have been proposed. For instance a number of authors applied LES to investigate flows in simplified building canopy, e.g. cubical roughness arrays (Coceal et al., 2006; Tseng et al., 2006; Xie and Castro, 2006; Coceal et al., 2007; Shi et al., 2008). LES has also been applied to investigating atmospheric environment flow in real cities, for instance Liu et al. (2011, 2012) investigated the traffic pollutant dispersion in City Macao and a residence area of City Beijing, Tseng et al. (2006), BouZeid et al. (2009) applied LES to investigate the urban atmospheric flow of City Baltimore and Lausanne respectively. The LES is used in this paper for investigation of the flow pattern and pollutant dispersion, for easier computation and analyses the building canopy is simplified as a cubical roughness array while the cubes are fully resolved in computation. The paper is organized as follows. Section 2 presents the numerical method of LES and verification of the code. Section 3 will illustrate the five identifiable flow patterns in exemplified 3D urban canopies and the pollutant dispersion formulation. Analyses and discussions of the numerical results are given in Section 4 with brief concluding remarks.
Flow within building arrays is heterogeneous in horizontal plane and the eddy viscosity coefficient Cs is determined by a dynamic procedure which is based on the scale similarity between two filter scale D1 and D2 with D1 ¼ D and D2 ¼ 2D respectively. Unlike Germano's method (Germano, 1992) which is valid for turbulent flows with homogeneous directions, Meneveau et al. (1996) proposed a Lagrangian dynamic model which takes averages along trajectories of fluid particles and results in two time-integral values
Z0
ILM ¼
Lij Mij ðzðt 0 Þ; t 0 ÞWðt t 0 Þdt 0 ;
(4)
∞
IMM ¼
Z0
Mij Mij ðzðt 0 Þ; t 0 ÞWðt t 0 Þdt 0 :
(5)
∞ 2 e e eu e ug In equations (4) and (5)Lij ¼ u u , M ¼ 2½D i j i j ij 2 SSij g D21 SSij , and W(tet’) is a weight function. If the weight function takes the exponential form, i.e.Wðt t 0 Þ ¼ expððt t 0 Þ=TÞ=T, then ILM and IMM can be determined by following equations:
vILM vI 1 b j LM ¼ þu L M ILM T ij ij vt vxj
(6)
vIMM vI 1 b j MM ¼ þu Mij Mij IMM T vt vxj
(7)
2. Numerical methods 2.1. Governing equations The wind speed in urban areas is relatively low and the air can be treated as incompressible. In large eddy simulation, the continuity equation and momentum equations for filtered velocity ui and filtered pressure p read
vui ¼0 vxj vui vui uj 1 vp v þ ¼ þ r vxi vxj vt vxj
(1)
tij þ v
vui vxj
! þ fi
(2)
where xi (i ¼ 1,2,3) are streamwise, spanwise and vertical coordinates respectively, and ui (i ¼ 1,2,3) are filtered velocity components in corresponding directions, fi are external forces. The air density is r ¼ 1.208 kgm3, and the kinetic viscosity coefficient is n ¼ 1.5 105 m2 s1. Transport equation of filtered pollutant concentrationc reads:
vc vc ui v n vc þ tci þ þQ ¼ vt vxi Sc vxi vxi
The time scale T is set to equal 1.5D(ILMIMM)1/8 as proposed by Meneveau et al. (1996), and superscripts and ~ represent filtering at D1 and D2 respectively. Equation (6) and equation (7) can be easily discretized and coupled with governing equations, thus model coefficient Cs2 ¼ ILM =IMM can be obtained. The Lagrangian dynamic model is suitable for heterogeneous turbulent flow and previous investigation (Shi et al., 2008) indicates that dynamic model performs better than the standard Smagrinsky model in solving turbulent wind field. The details of the sub-grid scale model, including the Lagrangian dynamic model, can be found in Sagaut (2002). The sub-grid scale mass fluxtci is taken as a gradient model and 2 D2 b the sub-grid mass diffusivityDt ¼ Cc;s 1 S , where Cc,s is the model
(3)
In the above equation Sc is the molecular Schmidt number, Q is the pollutant source term, in this study the point source will be inserted in some position in equation (3) and Q ¼ qd(x) in which d(x) is the Dirac function. 2.2. Sub-grid scale model In equations (2) and (3), tij ¼ ui uj ui uj and tci ¼ ui c ui c are sub-grid scale stress and sub-grid scale mass flux which should be closed by models. In this study the sub-grid eddy viscosity and subgrid eddy diffusivity model are accepted respectively for tij and tci which can be written astij ¼ 2vt Sij þ tkk dij =3, tci ¼ Dt vc=vxi whereSij ¼ ðvui =vxj þ vuj =vxi Þ=2 is the filtered strain tensor. Usually the sub-grid eddy viscosity is formulated asvt ¼ Cs2 D2 S in which Cs is the model coefficient and D is the grid size.
Fig. 3. Distribution of cubes in verification case.
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Fig. 4. Spanwise distribution of pollutant concentration, compared with experiment. (a) 2 h behind the source, z ¼ 0.25 h; (b) 6 h behind the source, z ¼ 0.25 h; (c) 2 h behind the source, z ¼ 1.25 h; (d) 6 h behind the source, z ¼ 1.25 h.
coefficient and can also be determined by the dynamic procedure. 2 e e 2 g In this caseLi ¼ ueie c uf i c andMi ¼ D2 Svc=vxi D1 S vc=vxi . The two time-integral values Ic,LM and Ic,MM read:
Z0
Ic;LM ¼
Li Mi ðzðt 0 Þ; t 0 ÞWðt t 0 Þdt 0
(8)
∞
Ic;MM ¼
Z0
Mi Mi ðzðt 0 Þ; t 0 ÞWðt t 0 Þdt 0
2.3. Numerical scheme Finite volume method (FVM) is employed to discretize governing equations (1) through (3). The QUICK scheme is used in spatial discretization and the third order RungeeKutta scheme is utilized for time advancement. The discretized equations are solved using SIMPLE methods on non-staggered grids with momentum interpolation. In the lateral and longitudinal direction, the periodic boundary condition is applied for each velocity component, and the
(9)
∞
which satisfy following equations with the weigh function mentioned above
vIc;LM vI 1 b j c;LM ¼ L M Ic;LM þu T i i vt vxj
(10)
vIc;MM vI 1 b j c;MM ¼ Mi Mi Ic;MM þu T vt vxj
(11)
Finally the model coefficient of sub-grid mass diffusivity is 2 Cc;s ¼
Ic;LM Ic;MM
With the sub-grid viscosity and the sub-grid diffusivity both determined by the Lagrangian dynamic procedure, the usually assumed sub-grid Schmidt number is abandoned.
Fig. 5. Streamlines around an isolated 3D roughness element by the present computation, the color level indicates the height z/h.
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Fig. 6. Distribution of pollutant concentration in isolated roughness flow. (a) Horizontal plane, z/h ¼ 0.1; (b) spanwise plane, (x-xs)/h ¼ 2; (c) spanwise plane, (x-xs)/h ¼ 5.
non-slip condition is applied at the cube surface and ground. A free slip and impermeable condition is applied at the upper boundary. The concentration is set to zero at the inlet and lateral boundaries of the domain. The zero normal gradient of concentration is
imposed at the outlet and the upper boundary as well as all the solid surfaces. In each of the present simulations, the mesh has 12 grid points along each dimension of the cube mounted on the wall with the
Fig. 7. Spanwise distribution of non-dimensional pollutant in isolated roughness flow, compared with double Gaussian fitting results. z/h ¼ 0.1.
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207
the flow on the Reynolds number can be neglected according to Snyder (1972). A total of 1000 T* is simulated after the flow approaching statistically steady state, where T* ¼ h/ut. 2.4. Verification of the numerical method and computing code
Fig. 8. Flow pattern of external wake interference flow.
distance from the first grid point to the boundary refined. Since vortices and wakes of scales above h/10 contributed more significantly to mass and momentum transfer in the core region (Xie et al., 2008), the present resolution is sufficient for large eddy simulations in terms of the average flow field (Kanda et al., 2004). A constant longitudinal pressure gradient of u2t =Lz is imposed to counterbalance the drag from cubes and the ground, where ut is the total wall friction velocity and Lz is the domain height. The roughness Reynolds number based on the friction velocity and height of the cube Ret ¼ uth/n ¼ 500, which is large enough so that the dependence of
The numerical code has been verified by comparison of numerical results of testing case with corresponding experimental results provided by Hilderman and Chong (2007) from a fully developed turbulent boundary layer in a water tunnel. In the experiment, there are three layouts of cubes one of which is a square array of 16 16 cubes with a height h equaling 31.75 mm. The dye is injected into the array after the first row of cubes. According to the experiments of Hilderman and Chong (2007), the molecular diffusivity of the fluorescent dye used in the water channel is D ¼ 5.3 1010 m2 s1, and the molecular Schmidt number is about 1920 with the kinematic viscosity of water n ¼ 1.0 106 m2 s1. The molecular diffusivity of the fluorescent dye is much lower than that of air; however, the molecular diffusivity is negligible in turbulent flow since the turbulent diffusivity plays a dominant role in scalar dispersion. The turbulent diffusivity is in the same order of magnitude of turbulent viscosity whether the turbulent flow is in air or water, and it is usually accepted that turbulent Schmidt number is around 1.0 in simulation of turbulent dispersion. Hence the water channel experiment using the dye of a low diffusivity can serve as a reasonable approximation of urban atmosphere layer and also a reliable validation of numerical simulations. The water channel experiments mentioned above was numerically simulated through Direct Numerical Simulation (DNS) with an array of 8 8 cubes by Branford et al. (2011), where the molecular Schmidt number was set as 1.0 and three orders less than in the water channel, and the DNS results of mean concentration agree with those from the experiments. The layout of the array in present
Fig. 9. Distribution of pollutant concentration in external wake interference flow. (a) Horizontal plane, z/h ¼ 0.1; (b) streamwise plane, (y-ys)/h ¼ 0; (c) spanwise plane, (x-xs)/h ¼ 2; (d) spanwise plane, (x-xs)/h ¼ 5.
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Fig. 10. Spanwise distribution of non-dimensional pollutant in external wake interference flow, compared with Gaussian and double Gaussian fitting results. z/h ¼ 0.1.
LES is similar to that in the DNS shown in Fig. 3; the domain size is 16 h in longitudinal and lateral directions and 8 h in vertical direction. It has been shown by many authors (Cheng and Castro, 2002; Coceal et al., 2006; Wang et al., 2014) that the fully developed turbulent boundary layer flow over an array of cubes can be simulated numerically by the semi-channel model of flow over the same array appropriately as long as the height of the semi-channel is large enough, some author accepted a smaller height, e.g. Lz ¼ 4h~6 h (Coceal et al., 2006) while Wang et al. (2014) suggested Lz ¼ 8 h. In this paper Lz ¼ 8 h is used. In Hilderman and Chong's experiment (Hilderman and Chong, 2007), a point source is located at the distance from ground zs ¼ 0.0625 h and xs ¼ 2 h, ys ¼ 7 h (Fig. 3). The comparison is made between numerical results and experimental data. Fig. 4 shows the comparison of spanwise distribution of time average dimensionless concentration C* ¼ Cuth2/q between numerical results and Hilderman and Chong's experiment measurements at distance 2 h and 6 h behind pollutant source. Both the concentration distribution inside canopies (z ¼ 0.25 h) and that above canopies (z ¼ 1.25 h) show fairly good agreement between numerical simulations and experimental measurements. It is concluded that the present numerical code can be used to investigate the flow pattern and pollutant transportation and dispersion in urban canopies.
determined by the flow Reynolds number and the size of cubes. If the Reynolds number is high enough the flow is Reynolds number independent, or so called self-similarity (Snyder, 1972). In following numerical simulations h equals 0.02 m, the velocity of free-stream flow U equals 10 m/s, and corresponding Reynolds number Re(¼ Uh/n) is estimated as 104. The flow over an isolated 3D roughness element has been investigated by many authors. The flow pattern and geometrical parameters are presented in Snyder and Lawson (1994). Martinuzzi and Tropea (1993) also found that
3. Flow pattern and pollutant dispersion in 3D urban canopy The urban building canopy is composed of buildings which are simplified as cuboids in this study. Like isolated roughness in 2D street canyon, the isolated roughness element is the simplest form of 3D urban canopy in which the flow pattern is completely
Fig. 11. Flow pattern of near wake interference flow.
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209
Fig. 12. Distribution of pollutant concentration in near wake interference flow. (a) Horizontal plane, z/h ¼ 0.1; (b) streamwise plane, (y-ys)/h ¼ 0; (c) spanwise plane, (x-xs)/h ¼ 2; (d) spanwise plane, (x-ys)/h ¼ 5.
the recovery length behind a three dimensional obstacle is shorter than in the case of a two-dimensional flow, whose value close to unity scaled by building height for obstacles with a small aspect ratio (l/h~1). Besides the complex flow geometry, the peculiar
feature of the flow pattern is that there exist a horseshoe vortex around the lower part of the roughness element and a recirculation cavity behind the roughness element (Fig. 5). In this study the flow region containing the horseshoe vortex tails is called the external
Fig. 13. Spanwise distribution of non-dimensional pollutant in near wake interference flow, compared with Gaussian and double Gaussian fitting results. z/h ¼ 0.1.
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Fig. 14. Flow pattern in skimming flow. (a) non-staggered array; (b) staggered array.
wake area and the flow region inside the recirculation cavity is called the internal wake area. At a high Reynolds number, the spacing of horseshoe vortex tails Wex can be approximated as a function of the streamwise distance from the windward face of buildings (Schulman et al., 2000)
Wex ¼
R x 1=3
w þ 2 3 R
(12)
in which w is the width of roughness element and R is the geometrical parameter of the roughness element defined by R ¼ (BS)2/3(BL)1/3, BS ¼ Min(w, h), BL ¼ Max(w, h), where h is the height of the roughness element; in the cubical roughness element case R ¼ w ¼ h. Equation (12) indicates that the width of horseshoe vortex increases moderately along the downstream direction. The lateral size of the cavity zone Win is approximately equal to the
width of the roughness element and the streamwise size Lin equals the height of the element approximately.
Win zw; Lin zh
(13)
Based on the flow characteristics around the roughness element we can classify the 3D building canopy flow into the following five types. 3.1. Isolated roughness The first type of 3D building canopy is defined as the isolated roughness when the lateral building interval WL is much greater than Wex and the streamwise interval WS is much greater than the longitudinal size of internal wake Lin for cubical roughness. In this case neighboring buildings do not influence the flow around each
Fig. 15. Distribution of pollutant concentration of skimming flow in non-staggered layout.
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Fig. 16. Distribution of pollutant concentration of skimming flow in staggered layout.
individual building; hence the flow pattern is similar to the flow over an isolated roughness element. To simulate the flow developing along the longitudinal direction, the inflow data is generated according to Xie and Castro (2008) and the normal gradient of each velocity component u, v, w is set to zero at the outlet with flux correction. A point source of pollutant is located at zs ¼ 0.05 h from the ground and xs ¼ 0.5 h ahead of the upwind face in the centerplane. The distributions of concentration in horizontal and vertical cross-sections are illustrated in Fig. 6 which shows that the pollutant is mainly distributed in the horseshoe vortex region. To further explore the pollutant dispersion in longitudinal direction Fig. 7 shows spanwise distribution of non-dimensional concentration C*¼Cu*h2/q in six downstream locations shown in Fig. 7a. Two peaks of the pollutant concentration can be viewed obviously in Fig. 7 and C* can be best fitted by the double Gaussian or biGaussian expression (MacDonald et al., 1998):
vortex tails cannot expand along streamwise direction but is equal to the lateral distance of buildings. The internal wake region in Fig. 8 is similar to that of isolated roughness flow in Fig. 5. Fig. 9 illustrates the distribution of pollutant concentration for the external wake interference flow on horizontal and vertical cross-sections. The point source is located in the head region of the horseshoe vortex; however the pollutant is not concentrated in the horseshoe vortex region but spread in to the internal wake region because the spacing of horseshoe vortex tails is narrower. Fig. 9a clearly shows this pattern. Fig. 10 shows distribution of C* on six streamwise locations marked in Fig. 10a for the external wake interference flow. Both single Gaussian and double Gaussian fitting results are presented. The single Gaussian expression reads:
. C* ¼ Cexp ðy ys Þ2 2s2y :
. . o n C* ¼ C exp ðy ys Þ2 2s2y þ exp ðy þ ys Þ2 2s2y (14) where ys is the displacement of the plume center and sy is the spanwise width of each plume. Double Gaussian fitting results show fine agreement with the simulated pollutant concentration. 3.2. External wake interference regime When the lateral distance between buildings is less than the length of external wake (WL < Wex) whereas the streamwise distance still much larger than the size of near wake (WS [ Lin), the flow over building canopy is called the external wake interference. In this case the individual building can no longer be viewed as isolated. The external wake of neighboring buildings, i.e. the horseshoe vortex, interacts with each other, although the internal wake is not affected by neighboring buildings. An example of this flow regime is illustrated in Fig. 8 in which flow is simulated numerically over a square array with both streamwise and spanwise interval of the buildings equaling 4 h. Obviously the horseshoe vortex from individual building cannot be spreading freely but restricted by other neighboring external wakes that the spacing of
Fig. 17. Flow pattern in street network, flow direction is parallel to street.
(15)
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Fig. 18. Distribution of non-dimensional pollutant in street network. (a) Horizontal plane, z/h ¼ 0.2; (b) streamwise plane, (y-ys)/h ¼ 0.
Due to the influence of external wake region, the spanwise distribution of pollutant concentration still shows double peaks clearly near the head of horseshoe vortex, yet the double peaks are gradually reduced in downstream location (Fig. 10d). The spanwise spacing of
double peak is also reduced because the spanwise spacing of the horseshoe vortex is reduced, which illustrates the higher concentration in the recirculation zone compared to the isolated roughness flow. The quantitative analyses will be presented in Section 4.
Fig. 19. Streamwise decay of pollutant.
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Fig. 20. Streamwise development of the spacing of the double peaks.
3.3. Internal wake interference When the streamwise interval of buildings is in the same order of the longitudinal size of internal wake, i.e. Ws ~ Lin, the development of internal wake behind each individual building will be influenced by downstream buildings. A case with both streamwise and spanwise distances of the buildings equaling 2 h is simulated to represent such kind of flow. As viewed from streamlines in Fig. 11, external wake region is similar to the last case while flow in the recirculation zone is altered in the vicinity of downstream buildings. This flow pattern is named the internal wake interference flow. Fig. 12 illustrates the distribution of pollutant concentration for the internal wake interference flow in horizontal and vertical crosssections. The pollutant spreads quickly to neighboring columns, which is different from dispersion in external wake interference flow. As shown in Fig. 12a and b, although the point source is located before the second row, there is high concentration behind the first row since a proportion of the pollutant released from the point source is trapped in the internal wake region belonging to the first building row. Fig. 13 illustrates the spanwise distribution of C* for the internal wake interference flow at different downstream locations marked in Fig. 13a. The double peaks of the pollutant concentration profile still exist (Fig. 13b) with narrow spacing and the double peaks nearly coalesce in further downstream locations (Fig. 13c,d) due to
the enhanced mixing by the interference of internal wake (see Fig. 12c,d). 3.4. Skimming flow If building intervals of both longitudinal and lateral directions approach h, then both internal and external wakes are restricted. This case is named the skimming flow. Two different layouts of cubes are investigated corresponding to staggered and nonstaggered arrangements of buildings, see Fig. 14a and b. In the non-staggered case (Fig. 14a), the flow field is divided into two regions named ST1 and ST2 respectively. In the region ST1 the flow is almost parallel to the street with greater speed, while the complicated recirculation flows are generated in the ST2, perpendicular to ST1, by lateral and vertical large shears. This is the fundamental difference between the 3D skimming flow and the 2D skimming flow that there is only vertical shear in the 2D skimming flow in street canyons. The difference between 3D and 2D skimming flows can be clearly shown in the distribution of pollutant concentration in Fig. 15 for the non-staggered case. The pollutant is transported along streamwise direction in ST1, meanwhile it is dispersed in lateral direction in ST2. The pollutant dispersion in the 3D skimming is similar to the 2D skimming flow only in the central section at (y-ys)/h ¼ 0. In the staggered case the 3D skimming flow is more complicated as shown in Fig. 14b in which the flow field is also divided into ST1
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Fig. 21. Streamwise development of the plume width.
and ST2. In ST1 the incoming flow down-washes against the obstacle and enters into ST2. Fig. 16 presents the distribution of pollutant concentration for the staggered case. By comparing between Figs. 15 and 16 it can be seen that the pollutant concentration is lower inside building canopy in staggered case due to the strong dispersion in ST1. In both staggered and non-staggered cases the external and internal wake cannot be distinctly identified due to the dense-packing layout and the flow is at high speed above building canopies with low speed circulation inside canopies. This is similar to the skimming flow in 2D street canyon except that the flow is inherently three dimensional inside the canopy.
channel flow while the flow in streets perpendicular to the free stream direction is similar to the 2D skimming flow. The pollutant dispersion is demonstrated in Fig. 18. The pollutant is dispersed downstream and gradually decaying, and it is restricted mostly in streets parallel to free stream direction since lateral dispersion is weak. For building canopies with streets in oblique direction from the free stream, the flow pattern is more complicated but can still be simplified by network models as elaborated by Hamlyn et al. (2007) and Soulhac et al. (2011). We will not introduce street network models in detail. We just note that the street network is one of the accepted simplifications of building canopy with the streamwise and lateral interval between buildings much less than the building height, i.e. WS≪h and WL≪h.
3.5. Street network 4. Discussions and conclusion When the spacing between buildings is much less than the building height, flow within the canopy is nearly homogeneous along streets direction, whether parallel or perpendicular to the free stream direction, and behave like 2D flow. Hence it can be approximated by a street network model and can be called network flow. In this case both external and internal wakes disappears. The street network is somewhat simpler building canopy and has been studied by Hamlyn et al. (2007) and Soulhac et al. (2011). Fig. 17 shows the flow pattern in the street network where the flow in streets parallel to the free stream direction can be simplified as
The numerical investigation by LES shows identifiable patterns of flow and pollutant dispersion in 3D building canopies. When there exist the external wake (the horseshoe vortex around each individual building) and the internal wake (the downstream recirculating), the flow pattern is similar to the flow over isolated roughness element and the pollutant dispersion shows double peak in the lateral direction (Figs. 7, 10 and 13). The distribution of pollutant concentration can be best fitted with double Gaussian function, Equation (14)
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. . o n C ¼ C exp ðy ys Þ2 2s2y þ exp ðy þ ys Þ2 2s2y : In this expression, C, ys and sy are varying in the streamwise direction and are of interest for modeling. C denotes the decay of pollutant concentration; ys stands for the spanwise displacement of each plume and sy represents the plume width in terms of standard deviation. Fig. 19 show that the streamwise decay of pollutant concentration is similar among isolated roughness element flow, external wake interference flow and internal wake interference flow. However, the decay is slower for the isolated roughness flow since both external wake and internal wake are not interfered by neighboring buildings. The fastest decay of pollutant occurs in the internal wake interference flow since the flow is restricted in streamwise direction while spanwise transfer of mass is enhanced in the building canopy (compare Fig. 9a with Fig. 12a). The peak spacing and the plume width is presented respectively in Figs. 20 and 21 for the three cases. The spacing of double peaks is larger in the isolated roughness element flow since the external wake is not restricted. In Fig. 20a it also shows that the development of double peak spacing agrees with the power law ysf(x-xs)1/3. This indicates that the pollutant is mostly trapped in the external wake (see Equation (12)). The development of the peak spacing and the plume width is quite similar between the external and internal wake interference flow, see Fig. 20b, c and Fig. 21b, c, since the flow is restricted in space by building intervals in both cases. The 3D skimming flow is more complicated that the flow pattern and pollutant dispersion are dependent on the layout of the buildings, e.g. staggered or non-staggered. However the street network flow is a peculiar 3D building canopy and it has been studied by a couple of authors (Hamlyn et al., 2007; Soulhac et al., 2011). In summary flow patterns and pollutant dispersion in 3D building canopies can be identified as five regimes which depend on geometrical parameters of building canopy and characteristic length scales of flow over isolated roughness element. The essential feature between different flow patterns is the building wake effect which is varying with the packing density. For low packing density the flow is similar to the isolated roughness flow and the pollutant dispersion is governed by the downwash effect of the building wake (Schulman et al., 2000). For higher packing density the flow is quasi-two-dimensional and the street network model (Soulhac et al., 2011; Hamlyn et al., 2007) applies appropriately. The external wake interference and internal wake interference flows are 3D complex turbulent flow, where the characteristics of pollutant concentration vary with the building density as well. Therefore practical model for these two flow types is expected to be reconstructed after more detailed quantitative study. Acknowledgements This study was sponsored by National Natural Science Foundation of China (NSFC Grant 11132005, 11490551). References Bou-Zeid, E., Overney, J., Rogers, B.D., Parlange, M.B., 2009. The effects of building representation and clustering in large-eddy simulations of flows in urban canopies. Bound. Layer Meteorol. 132 (3), 415e436. Branford, S., Coceal, O., Thomas, T.G., Belcher, S.E., 2011. Dispersion of a point-source release of a passive scalar through an urban-like array for different wind directions. Bound. Layer Meteorol. 139 (3), 367e394.
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