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Development of a k–ε model for the determination of air exchange rates for street canyons
ARTICLE IN PRESS
Atmospheric Environment 39 (2005) 7285–7296 www.elsevier.com/locate/atmosenv
Development of a k2 model for the determination of air exchange rates for street canyons Xian-Xiang Li, Chun-Ho Liu, Dennis Y.C. Leung Department of Mechanical Engineering, The University of Hong Kong, 7/F., Haking Wong Building, Pokfulam Road, Hong Kong, China Received 30 April 2005; received in revised form 15 August 2005; accepted 2 September 2005
Abstract A two-dimensional renormalized-group (RNG) k2 turbulence model was developed to calculate the air exchange rate (ACH) of idealized street canyons. This model was validated against other modelling (large-eddy simulation, LES, and k2) and wind tunnel results. It was demonstrated that the current k2 model was able to capture the complicated flow structures inside street canyons whose outputs agreed reasonably well with other modelling and wind tunnel results available in literature. This k2 model was then applied to calculate the ACH for street canyons of aspect (building-heightto-street-width) ratios 0:5, 1:0, and 2:0. The values of ACH calculated by the current k2 model compared favourably with those calculated by LES in which their differences were less than 20%. Moreover, there was a remarkable saving in computer resources and computation time by using the current k2 model. r 2005 Elsevier Ltd. All rights reserved. Keywords: Air exchange rate (ACH); Two-dimensional; Renormalized-group (RNG) k2 turbulence model
1. Introduction Flow and pollutant transport in urban street canyons have attracted great public concern and have been extensively studied during the past two decades. Field measurements, laboratory-scale physical modelling and computational fluid dynamics (CFD) techniques are common tools adopted to study the wind flow and pollutant distributions in urban street canyons (Meroney et al., 1996; Baik et al., 2000; Louka et al., 2000; Jeong and Andrews, 2002; Liu and Barth, 2002; Xie et al., 2003). The Corresponding author. Tel.: +852 2859 7911; fax: +852 2858 5415. E-mail addresses: [email protected] (X.-X. Li), [email protected] (C.-H. Liu), [email protected] (D.Y.C. Leung).
1352-2310/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.atmosenv.2005.09.007
flow patterns inside a street canyon are characterized by three regimes depending on the streetcanyon geometry: isolated roughness flow, wake interference flow, and skimming flow (Oke, 1988). The street-canyon geometry is mainly characterized by the building-height-to-street-width (aspect) ratio h=b, where h is the building height and b the street width. Most of the studies available in literature focus on the skimming flow regime, in which a stable and isolated recirculation develops at the street-canyon centre. This isolated nature of flow recirculation results in poor air ventilation and pollutant dilution in street canyons. With the ever-increasing power of computer hardware, numerous sophisticated CFD models have been developed to study the flow field and pollutant transport in street canyons. In CFD
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modelling, the two-equation k2 model and its variants are widely employed as turbulence closure schemes due to their computational robustness and efficiency. They have been extensively tested and calibrated for industrial flow around bluff bodies and structures (Berkowicz, 1998). Many high quality studies using k2 turbulence models have improved our understanding of street-canyon flow and pollution problems, which include the effects of aspect ratio (Sini et al., 1996; Baik and Kim, 1999; Jeong and Andrews, 2002), street-canyon geometry (Assimakopoulos et al., 2003), ambient wind direction (Kim and Baik, 2004), thermal effects (Sini et al., 1996; Kim and Baik, 2001), and source strength (Chan et al., 2002). Some of the aforementioned studies also compared the performance of three versions of k2 turbulence models, including standard, renormalized-group (RNG), and realizable, when they were utilized to simulate streetcanyon flow and pollution problems. Using the commercial CFD code FLUENT, Chan et al. (2002) evaluated the three versions of k2 turbulence models by simulating the flow over an isolated street canyon. They suggested that the RNG k2 turbulence model was the optimum one amongst the three versions. However, also using FLUENT, Sagrado et al. (2002) found that the realizable k2 turbulence model with the two-layer zonal approach appeared to be the most reliable one. This contradiction may have been due to the different criteria adopted for selecting the most accurate turbulence model. To select the most appropriate k2 model, Sagrado et al. (2002) compared the reattachment lengths behind a backward-facing step calculated by k2 turbulence models with those calculated by direct numerical simulation (DNS) (Le et al., 1996). They found that the error between realizable k2 model and DNS results was around 3%. On the other hand, Chan et al. (2002) validated their k2 model results against an extensive experimental database collected by wind-tunnel measurements. They found that the pollutant concentration predicted by RNG k2 model was the most accurate. However, these studies only validated the mean quantities, but not the turbulence kinetic energy (TKE) handling the mixing processes. Recently, Liu et al. (2005) adopted the concepts of air exchange rate (ACH) and pollutant exchange rate (PCH) to study the air ventilation and pollutant dilution in street canyons. Making use of the detailed large-eddy simulation (LES) database (Liu and Barth, 2002; Liu et al., 2004), they calculated
the ACH and PCH for street canyons of aspect ratios 0.5, 1.0, and 2.0 at a Reynolds number ðReÞ of 12,000 and a Schmidt number ðScÞ of 0.72. The ACH could be taken as a measure to compare the ventilation efficiency of street canyons. It was found that the air ventilation and pollutant dilution were generally governed by the roof-level vertical turbulence. The air removal from the street canyon was counterbalanced by the air entrainment from the free surface layer. The ACH for the street canyon of aspect ratio 0:5 was largest. Moreover, the ACH for the street canyon of aspect ratio 2:0 was 50% smaller than those of the other two street canyons. Because of its large computational load, LES is a research tool rather than a tool for engineering applications. In particular, the lengthy CPU time for comprehensive sensitivity tests is impractical. It is thus necessary to develop an efficient tool for reliable ACH determination for street canyons. In this study, a two-dimensional (2D) CFD model with RNG k2 turbulence closure scheme for street-canyon flow problems was developed. This model particularly focused on the quick calculation of ACH values. It was also validated against wind tunnel measurements together with numerical outputs from other k2 turbulence models and LES studies. 2. Mathematical model and methodology The computational domain consists of 13 identical street canyons aligned in the streamwise direction for developing periodic boundary conditions (Fig. 1). The flow properties of the street canyon in the middle (the seventh) of the domain are presented in the following discussion. The height of the street canyon of unity aspect ratio (h ¼ b ¼ H) and the free-stream inflow speed U were selected as the reference length and velocity scales, respectively. To facilitate the comparison with previous studies, the Reynolds number (Re ¼ rUH=m, where r and m are, respectively, the air density and molecular viscosity) was prescribed at ¼ 12; 000. Since the flow direction was perpendicular to the street axis, a 2D computational domain was adopted. The flow was treated as an incompressible, isothermal, and pseudo steady-state turbulence. The governing equations for the problem consist of the continuity equation qui ¼0 qxi
(1)
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Free Stream Flow U
U inflow
outflow
target canyon
(a)
Free Stream Flow U hf = 3h bu = b
Leeward Building
bd = b
Street Canyon
Windward Building
h
b (b) Fig. 1. Schematic diagram of the computational domain and boundary conditions: (a) the whole domain; and (b) the enlarged view of the target street canyon.
and the momentum conservation equation 2
qui qui 1 qp m q ui q 00 00 þ uj ¼ þ u u þ gi , r qxi r qxj qxj qxj i j qt qxj (2) which are expressed in tensor notation and the usual summation convention on repeated indices (i, j ¼ 1; 2) are employed. The spatial coordinates xi are the Cartesian coordinates in the streamwise ðxÞ and vertical ðzÞ directions. The overbars represent ensemble averaged flow properties commonly employed in k2 turbulence models. Here, ui are the streamwise (u) and vertical (w) velocity components, p the pressure, mt qui quj 2 00 00 þ ui uj ¼ kdij 3 r qxj qxi the Reynolds stress, gi the gravitational acceleration, mt the turbulent viscosity, k the turbulent kinetic energy, and dij the Kronecker symbol. To close Eq. (2), the RNG k2 turbulence model is introduced (Yakhot and Orszag, 1986) by employing two additional transport equations for the turbulent kinetic energy qk qk 1 q qk Pk þ ui (3) ¼ ak meff þ qt qxi r qxi qxi r
and the dissipation rate q q 1 q q 1 þ ui ¼ a meff þ C 1 Pk qt qxi r qxi qxi r k 2 3 C m rZ ð1 Z=Z0 Þ , C 2 þ 1 þ bZ3 k
ð4Þ
where Pk is the turbulent kinetic energy production, ak and a the inverse effective Prandtl numbers for k and , respectively, meff the effective turbulent viscosity, Z ¼ Sk=, and S the scalar measure of the deformation tensor. The turbulent Prandtl number is determined by an analytical formula in the current RNG k2 turbulence model rather than a constant value in the standard k2 model. With the calculated turbulent kinetic energy and dissipation rate, the turbulent viscosity is then calculated by mt ¼ rC m k2 =. The modelling constants adopted in the current RNG k2 turbulence model are fC 1 ; C 2 ; C m ; Z0 ; bg ¼ f1:42; 1:68; 0:0845; 4:38; 0:012g. The aforementioned mathematical procedures were implemented using the commercial CFD code FLUENT 6.1. The computational domain was discretized into non-overlapping unstructured triangular elements whose properties are summarized in Table 1. The velocity profile was defined along the
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Table 1 The number of (triangular) elements and spatial resolution (in term of element area) adopted in discretizing the (two-dimensional) street canyons of different aspect ratios h=b Aspect ratio h=b Number of elements Minimum element area 0.5 1.0 2.0
2,127,592 1,141,878 2,498,420
1:47 103 H 2 1:94 103 H 2 3:83 104 H 2
Here H is the reference length scale which is defined as the building height of street canyon of unity aspect ratio.
inflow boundary. Zero gradient boundary conditions were prescribed at the outflow and upper boundaries. Similar to Sagrado et al. (2002), the two-layer zonal wall treatment was applied on all rigid walls. Initially all the quantities in the computational domain are set to zero and the inflow velocity profile will develop over the upstream buildings. The finite volume method (FVM) and the SIMPLE algorithm (Patankar, 1980) were employed to solve the transport equations. 3. Results and discussions This section discusses the results obtained from the current RNG k2 turbulence model, which will be compared with the LES results, other k2 model outputs, and wind tunnel measurements available in literature. Afterwards the mathematical procedure, results, and accuracy of the proposed ACH calculation will be discussed. 3.1. Model validation The accuracy of the current k2 turbulence model was first compared with the LES outputs collected by Liu et al. (2004). Flow for three street canyons of aspect ratios 0:5; 1:0, and 2.0 at a Reynolds number of 12,000 was considered. The model validation exercise is described in detail in this section. Fig. 2 shows the spatial contours of the mean streamwise velocity u calculated by the current RNG k2 turbulence model and the LES. The current k2 model captured well the main flow structures in street canyons that were comparable to those calculated by the LES. In particular, the complicated vertically aligned three-layer structures in the street canyon of aspect ratio 2.0 were calculated well (Fig. 2c). More detailed analysis showed that the current k2 model slightly under-
predicted the roof-level and near-ground mean streamwise velocities compared with those calculated by the LES. This discrepancy was mainly due to the different wall treatments adopted. In the LES, perfectly smooth walls were assumed and the processes in near-wall regions, including both surface sublayers and buffer regions, were calculated explicitly at the expense of a highly refined spatial resolution. On the other hand, a wall model was used to represent rigid walls in the current k2 model that prescribed the near-wall velocity components as functions of wall roughness with affordable near-wall spatial resolution. Since the detailed near-wall flow properties were not our main concern, the wall function was considered accurate enough to represent the wall effects in the current study. Fig. 3 shows the spatial contours of the mean vertical velocity w in the street canyons. While the extents of upward and downward flow calculated by the current k2 model generally agreed well with those of the LES results, the mean vertical velocity obtained by the k2 model, similar to the mean streamwise velocity, was also underpredicted in the near-wall regions. In particular, the vertical downward mean flow near the windward wall was underpredicted by as much as 30%. This underprediction was partially attributed to the different near-wall treatments adopted in the current k2 model and LES. Moreover, a more stable upwind scheme was employed in the current k2 model, which would unavoidably introduce numerical dissipation, compared to the less dissipative Galerkin finite element method being employed in the LES (Liu et al., 2004). The flow recirculations within street canyons are depicted in Fig. 4 in the form of streamfunctions. It was evident that the flow recirculations calculated by the current k2 model were comparable to those calculated by the LES. In the street canyon of aspect ratio 1.0, the two counterclockwise-rotating secondary recirculations at ground-level corners were well resolved, which had not been predicted by some previous studies using k2 models (Hassan and Crowther, 1998; Huang et al., 2000; Chan et al., 2002). However, analogous to other k2 models, namely Jeong and Andrews (2002), the current k2 model did not calculate the secondary recirculations at the roof-level leeward corner for street canyons of aspect ratios 1.0 and 2.0. This roof-level secondary recirculation had been uniquely determined by the LES (Liu and Barth, 2002; Liu et al., 2004). It thus
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Fig. 2. Spatial contours of the dimensionless mean streamwise velocity u=U in street canyons of different aspect ratios h=b at a Reynolds number of 12,000 calculated by the current k2 turbulence model (left) and LES (Liu et al., 2004) (right): (a) h=b ¼ 0:5; (b) h=b ¼ 1:0; and (c) h=b ¼ 2:0.
suggested that the k2 turbulence models were unable to capture this kind of secondary recirculation most likely due to its limitation of modelling the broad range of turbulence motion by only using a single length scale. The centres of the primary recirculations predicted by several numerical models and experiments are tabulated in Table 2. It was observed that the
centres of the primary recirculations predicted by the current k2 model showed reasonable agreement with most of the other k2 models, LES, as well as experimental results, whereas the elevations of the centres of the primary recirculations determined by Jeong and Andrews (2002) showed large discrepancies compared with other modelling and experimental findings.
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Fig. 3. Spatial contours of the dimensionless mean vertical velocity w=U in street canyons of different aspect ratios h=b at a Reynolds number of 12,000 calculated by the current k2 turbulence model (left) and LES (Liu et al., 2004) (right): (a) h=b ¼ 0:5; (b) h=b ¼ 1:0; and (c) h=b ¼ 2:0.
While the mean flow quantities calculated by the current k2 turbulence model were in line with those of the LES results, the turbulence quantities were further examined to obtain more reliable calculation of the ACH. Fig. 5 illustrates the spatial contours of dimensionless turbulent kinetic energy (TKE). The TKE calculated by the current k2 model exhibited characteristics similar to the LES
results. Large TKE was observed at the upper street canyons for all the aspect ratios tested. This was mainly due to the enlarged TKE production by mechanical wind shear around the roof-level leeward and windward corners. This finding generally agreed with the results collected by Jeong and Andrews (2002) though their local maximum TKE determined was much larger than that of the current
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study. This was probably caused by the larger inflow turbulence intensities used in Jeong and Andrews (2002). Apart from the qualitative comparison with the results calculated by LES and other k2 turbulence models discussed above, the outputs from the current k2 model were compared with previous wind tunnel measurements (Gayev and Savory,
1999; Kastner-Klein et al., 2001). The profiles of dimensionless mean velocity and turbulence intensities are shown in Fig. 6. The vertical profiles of mean streamwise velocity at x=b ¼ 0:33 and 0.67 (Figs. 6a and b) calculated by the current k2 model were in good agreement with the LES and the wind tunnel results. It is worth mentioning that the streamwise velocity in the vicinity of roof level
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Table 2 Locations of the centre of primary recirculations ðxc =H; zc =HÞ in street canyons of different aspect ratios being determined by numerical models and wind tunnel experiment Aspect ratio h=b
Current study LES (Liu et al., 2004) k2 turbulence model (Jeong and Andrews, 2002) k2 turbulence model (Baik et al., 2000) Wind tunnel experiment (Baik et al., 2000)
0.5
1.0
ð1:32; 0:52Þ ð1:30; 0:52Þ — — —
2.0
ð0:54; 0:53Þ ð0:54; 0:53Þ ð0:56; 0:73Þ ð0:57; 0:57Þ ð0:54; 0:58)
Upper
Lower
ð0:55; 1:41Þ ð0:54; 1:38Þ (—, 1:74) ð0:58; 1:56Þ ð0:67; 1:56Þ
ð0:52; 0:37Þ ð0:53; 0:44Þ (—, 0:44) ð0:54; 0:64Þ ð0:59; 0:34Þ
Here, xc and zc are, respectively, the horizontal and vertical distance measuring from the ground-level leeward corner. The reference length scale H is the building height of the street canyon of unity aspect ratio.
exhibited large spatial variation. As discussed above, the wall model employed in the current k2 model led to a lower but more realistic prediction of mean streamwise velocity near the roof level and ground level of street canyons at high Reynolds number than did LES. The reliability of the wall model employed in the current study was confirmed by the wind tunnel results (Gayev and Savory, 1999). The LES slightly overpredicted the velocity near ground level, while the current k2 model outputs agreed well with the wind tunnel results. On the other hand, the current k2 model showed a more reliable prediction of streamwise velocity at and above the roof level than that of LES. The relatively larger roof-level mean streamwise velocity calculated by the LES could lead to other consequences during comparison of quantities normalized by roof-level mean velocity uðHÞ, which will be discussed in the upcoming sections. The comparison of streamwise turbulence inten1=2 sities u00 u00 =uðHÞ (Figs. 6c and d) was more complicated. The current k2 model reported reasonably good roof-level turbulence intensities when compared with the wind tunnel results (Gayev and Savory, 1999), while the agreement inside the street canyon was rather poor. The wind tunnel results of Gayev and Savory (1999) showed a local maximum of TKE at the street-canyon centre (z ¼ 0:5H). However, neither the current k2 model, the LES study (Liu et al., 2004) nor the wind tunnel measurements (Kastner-Klein et al., 2001) supported this experimental finding. Although the results of Kastner-Klein et al. (2001) were obtained by averaging the TKE at seven points evenly distributed across the street canyon (different from our methods), the profiles presented here could still signify that the turbulence intensity was
relatively constant below the street-canyon centre. The LES of Walton and Cheng (2002) seemed to be the only study that calculated a trend similar to Gayev and Savory (1999), but with a much smaller magnitude. These large discrepancies would be partially attributed to, as Kastner-Klein et al. (2001) suggested, the sensitivity of the normalization by uðHÞ, because the velocity near the roof level varied sharply, as noted above. Due to the larger predicted roof-level mean streamwise velocity, the LES by Liu et al. (2004) showed a much lower dimensionless streamwise turbulence intensity. 3.2. Air exchange rate (ACH) The concept of ACH used in street-canyon pollution problems, which represents the volumetric air exchange per unit time, was outlined in Liu et al. (2005). Positive (ACHþ) and negative (ACH) ACH represent, respectively, the air removal (with pollutant) and entrainment (fresh air). Because of continuity, the magnitudes of positive and negative ACH are equal. The temporal average of positive ACH is obtained by integrating the time-dependent ACH þ ðtÞ along the air ventilation area (a line in the current 2D case) G ¼ b across the street in accordance with Z T ACHþ ¼ ACH þ ðtÞ dt 0
Z
T
Z
w00þ ðt; GÞjroof dG dt Z Z T 00 wþ ðt; GÞjroof dt dG, ¼
¼
0
G
G
0
ð5Þ
where w00þ ðt; GÞ is the positive vertical velocity RT fluctuation. The integral 0 w00þ ðt; GÞjroof dt is the
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Fig. 5. Spatial contours of the dimensionless turbulent kinetic energy k=U 2 in street canyons of different aspect ratios h=b at a Reynolds number of 12,000 calculated by the current k2 turbulence model (left) and LES (Liu et al., 2004) (right): (a) h=b ¼ 0:5; (b) h=b ¼ 1:0; and (c) h=b ¼ 2:0.
ensemble average of positive roof-level fluctuating vertical velocity. It is noteworthy that the removal and entrainment of air are transient processes that cannot be represented in the current k2 turbulence model in an ensemble average manner. Hence, it is assumed that the transient air exchange is divided evenly into removal and entrainment processes. Moreover, the amount of air removal is equal to its
entrainment counterpart. The positive ACH can then be obtained by Z 1 1=2 ACHþ ¼ w00 w00 roof dG. (6) 2 G Isotropic turbulence (u00 ¼ v00 ¼ w00 ), which was a reasonable approximation in the current highReynolds-number flow in the regions far from the
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walls, was assumed in this study. The wind tunnel results of Kastner-Klein et al. (2001) also suggested isotropy in the street-canyon wind flow at such a high Reynolds number. Thus, k ¼ ðu00 u00 þ v00 v00 þ 1=2 w00 w00 Þ=2 ¼ 3w00 w00 =2 that yields naturally w00 w00 ¼ ð2=3kÞ1=2 . Hence, qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 (7) w00 w00 jroof ¼ 23 kjroof . Substituting Eq. (7) into Eq. (6) leads to the mathematical expression of ACH determination Z pffiffiffiffiffiffiffiffiffiffiffi 1 kjroof dG. (8) ACHþ ¼ pffiffiffi 6 G
The calculated ACHþ by Eq. (8) was tabulated in Table 3 together with those calculated by LES (Liu et al., 2005). The ACH values were non-dimensionalized by the volume of street canyon of unity aspect ratio V and the reference time scale T ð¼ H=UÞ. It was observed that the current k2 model showed a reasonably accurate prediction of the average ACH values as they were overpredicted by less than 20% compared with the LES-calculated values. This range of errors was certainly within acceptable tolerance from a practical engineering point of view. In fact, the ACH calculation by the current k2 model was much faster than that by LES, which would markedly speed up the analyses.
ARTICLE IN PRESS X.-X. Li et al. / Atmospheric Environment 39 (2005) 7285–7296 Table 3 Dimensionless air exchange rate (ACH) calculated by the current RNG k2 turbulence model and the LES (Liu et al., 2005) for street canyons of aspect ratios h=b ¼ 0:5; 1:0, and 2.0 Aspect ratio h=b
0.5 1.0 2.0
ACH=ðV=TÞ
%Deviation
RNG k2 turbulence model
LES
0.126 0.058 0.059
0.120 0.050 0.050
5.23 16.9 17.5
The speedup was roughly about one to two orders of magnitude. The use of steady-state models (instead of transient models in LES) as well as excluding the statistical analysis (post processing) were the major factors contributing to the marked speedup of the current proposed ACH calculation by k2 model. The possible sources of error in the proposed ACH calculation of k2 model could be the mathematical assumptions adopted. Firstly the assumption of isotropic turbulence would be invalid in large-scale flow or near-wall turbulence (e.g. roof level of buildings). However, only the small subgridscale motions were assumed to be isotropic in LES. Secondly, a 2D model was employed in this study that neglected all the motions in the spanwise direction. This negligence resulted in the inaccuracy of transport equations for k and . Finally, LES resolved all the transient fluid motions while k2 models only calculated the ensemble-averaged values. This difference made the current k2 model unable to capture some important transient phenomena, in particular, the turbulence quantities such as TKE. 4. Conclusion Based on the RNG k2 turbulence closure, a reasonably accurate numerical model for quick determination of ACH was established and validated for street canyons of aspect ratios 0:5; 1:0, and 2.0 at a Reynolds number of 12,000 against other numerical models (Liu et al., 2004) and wind tunnel results. This model was capable of capturing the complicated flow structures inside street canyons. The mean velocities as well as turbulence intensities calculated by the current k2 model agreed well with the wind tunnel results available in literature. However, at the centre core of street canyons, the
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comparison was more complicated due to the contradiction between the two sets of wind tunnel data. Following the ACH concept outlined in Liu et al. (2005), the determination of ACH was derived based on the current 2D RNG k2 model outputs. The average values of ACH for street canyons of aspect ratios 0:5; 1:0, and 2.0 were compared with those calculated by the LES (Liu et al., 2005). The ACH calculated by the current k2 outputs exhibited a slight overprediction but the deviation was less than 20%. This agreement was reasonably good in view of the assumptions made and was certainly within the tolerance of practical engineering applications. In addition, The major contribution of the current 2D k2 turbulence model is its computational efficiency which is faster than that of LES by one to two orders of magnitude. Acknowledgements This study is supported by a research grant from the Hong Kong Research Grant Council (Project No.: HKU 7196103E).
References Assimakopoulos, V.D., ApSimon, H.M., Moussiopoulos, N., 2003. Numerical study of atmospheric pollutant dispersion in different two-dimensional street canyon configurations. Atmospheric Environment 37, 4037–4049. Baik, J.-J., Kim, J.-J., 1999. A numerical study of flow and pollutant dispersion characteristics in urban street canyons. Journal of Applied Meteorology 38, 1576–1589. Baik, J.-J., Park, R.-E., Chun, H.-Y., Kim, J.-J., 2000. A laboratory model of urban street-canyon flows. Journal of Applied Meteorology 39, 1592–1600. Berkowicz, R., 1998. Street scale models. In: Fenger, J., Hertel, O., Palmgren, F. (Eds.), Urban Air Pollution: European Aspects. Kluwer Academic Publishers, Dordrecht, pp. 223–252. Chan, T.L., Dong, G., Leung, C.W., Cheung, C.S., Hung, W.T., 2002. Validation of a two-dimensional pollutant dispersion model in an isolated street canyon. Atmospheric Environment 36, 861–872. Gayev, Y.A., Savory, E., 1999. Influence of street obstructions on flow processes within street canyons. Journal of Wind Engineering and Industrial Aerodynamics 82, 89–103. Hassan, A.A., Crowther, J.M., 1998. Modelling of fluid flow and pollutant dispersion in a street canyon. Environmental Monitoring and Assessment 52, 281–297. Huang, H., Akutsu, Y., Arai, M., Tamura, M., 2000. A twodimensional air quality model in an urban street canyon: evaluation and sensitivity analysis. Atmospheric Environment 34, 689–698.
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Jeong, S.J., Andrews, M.J., 2002. Application of the k2 turbulence model to the high Reynolds number skimming flow field of an urban street canyon. Atmospheric Environment 36, 1137–1145. Kastner-Klein, P., Fedorovich, E., Rotach, M.W., 2001. A wind tunnel study of organised and turbulent air motions in urban street canyons. Journal of Wind Engineering and Industrial Aerodynamics 89, 849–861. Kim, J.-J., Baik, J.-J., 2001. Urban street-canyon flows with bottom heating. Atmospheric Environment 35, 3395–3404. Kim, J.-J., Baik, J.-J., 2004. A numerical study of the effects of ambient wind direction on flow and dispersion in urban street canyons using the RNG k2 turbulence model. Atmospheric Environment 38, 3039–3048. Le, H., Moin, P., Kim, J., 1996. Direct numerical simulation of turbulent flow over a backward-facing step. Report No. TF58, Thermosciences Division, Department of Mechanical Engineering, Stanford University and Department of Mechanical, Aerospace and Nuclear Engineering, University of California, Los Angeles. Liu, C.-H., Barth, M.C., 2002. Large-eddy simulation of flow and scalar transport in a modeled street canyon. Journal of Applied Meteorology 41 (6), 660–673. Liu, C.-H., Barth, M.C., Leung, D.Y.C., 2004. Large-eddy simulation of flow and pollutant transport in street canyons of different building-height-to-street-width ratios. Journal of Applied Meteorology 43, 1410–1424. Liu, C.-H., Leung, D.Y.C., Barth, M.C., 2005. On the prediction of air and pollutant exchange rates in street canyons of
different aspect ratios using large-eddy simulation. Atmospheric Environment 39, 1567–1574. Louka, P., Belcher, S.E., Harrison, R.G., 2000. Coupling between air flow in streets and the well-developed boundary layer aloft. Atmospheric Environment 34, 2613–2621. Meroney, R.N., Pavageau, M., Rafadalis, S., Schatzmann, M., 1996. Study of line source characteristics for 2D physical modelling of pollutant dispersion in street canyon. Journal of Wind Engineering and Industrial Aerodynamics 62, 37–56. Oke, T.R., 1988. Street design and urban canopy layer climate. Energy and Buildings 11, 103–113. Patankar, S.V., 1980. Numerical heat transfer and fluid flow. McGraw-Hill, New York, pp. 126–131. Sagrado, A.P.G., van Beeck, J., Rambaud, P., Olivari, D., 2002. Numerical and experimental modelling of pollutant dispersion in a street canyon. Journal of Wind Engineering and Industrial Aerodynamics 90, 321–339. Sini, J.-F., Anquetin, S., Mestayer, P.G., 1996. Pollutant dispersion and thermal effects in urban street canyons. Atmospheric Environment 30, 2659–2677. Walton, A., Cheng, A.Y.S., 2002. Large-eddy simulation of pollution dispersion in an urban street canyon—part II: idealised canyon simulation. Atmospheric Environment 36, 3615–3627. Xie, S., Zhang, Y., Qi, L., Tang, X., 2003. Spatial distribution of traffic-related pollutant concentrations in street canyons. Atmospheric Environment 37, 3213–3224. Yakhot, V., Orszag, S.A., 1986. Renormalization group analysis of turbulence. Journal of Scientific Computing 1 (1), 1–51.
Atmospheric Environment 39 (2005) 7285–7296 www.elsevier.com/locate/atmosenv
Development of a k2 model for the determination of air exchange rates for street canyons Xian-Xiang Li, Chun-Ho Liu, Dennis Y.C. Leung Department of Mechanical Engineering, The University of Hong Kong, 7/F., Haking Wong Building, Pokfulam Road, Hong Kong, China Received 30 April 2005; received in revised form 15 August 2005; accepted 2 September 2005
Abstract A two-dimensional renormalized-group (RNG) k2 turbulence model was developed to calculate the air exchange rate (ACH) of idealized street canyons. This model was validated against other modelling (large-eddy simulation, LES, and k2) and wind tunnel results. It was demonstrated that the current k2 model was able to capture the complicated flow structures inside street canyons whose outputs agreed reasonably well with other modelling and wind tunnel results available in literature. This k2 model was then applied to calculate the ACH for street canyons of aspect (building-heightto-street-width) ratios 0:5, 1:0, and 2:0. The values of ACH calculated by the current k2 model compared favourably with those calculated by LES in which their differences were less than 20%. Moreover, there was a remarkable saving in computer resources and computation time by using the current k2 model. r 2005 Elsevier Ltd. All rights reserved. Keywords: Air exchange rate (ACH); Two-dimensional; Renormalized-group (RNG) k2 turbulence model
1. Introduction Flow and pollutant transport in urban street canyons have attracted great public concern and have been extensively studied during the past two decades. Field measurements, laboratory-scale physical modelling and computational fluid dynamics (CFD) techniques are common tools adopted to study the wind flow and pollutant distributions in urban street canyons (Meroney et al., 1996; Baik et al., 2000; Louka et al., 2000; Jeong and Andrews, 2002; Liu and Barth, 2002; Xie et al., 2003). The Corresponding author. Tel.: +852 2859 7911; fax: +852 2858 5415. E-mail addresses: [email protected] (X.-X. Li), [email protected] (C.-H. Liu), [email protected] (D.Y.C. Leung).
1352-2310/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.atmosenv.2005.09.007
flow patterns inside a street canyon are characterized by three regimes depending on the streetcanyon geometry: isolated roughness flow, wake interference flow, and skimming flow (Oke, 1988). The street-canyon geometry is mainly characterized by the building-height-to-street-width (aspect) ratio h=b, where h is the building height and b the street width. Most of the studies available in literature focus on the skimming flow regime, in which a stable and isolated recirculation develops at the street-canyon centre. This isolated nature of flow recirculation results in poor air ventilation and pollutant dilution in street canyons. With the ever-increasing power of computer hardware, numerous sophisticated CFD models have been developed to study the flow field and pollutant transport in street canyons. In CFD
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modelling, the two-equation k2 model and its variants are widely employed as turbulence closure schemes due to their computational robustness and efficiency. They have been extensively tested and calibrated for industrial flow around bluff bodies and structures (Berkowicz, 1998). Many high quality studies using k2 turbulence models have improved our understanding of street-canyon flow and pollution problems, which include the effects of aspect ratio (Sini et al., 1996; Baik and Kim, 1999; Jeong and Andrews, 2002), street-canyon geometry (Assimakopoulos et al., 2003), ambient wind direction (Kim and Baik, 2004), thermal effects (Sini et al., 1996; Kim and Baik, 2001), and source strength (Chan et al., 2002). Some of the aforementioned studies also compared the performance of three versions of k2 turbulence models, including standard, renormalized-group (RNG), and realizable, when they were utilized to simulate streetcanyon flow and pollution problems. Using the commercial CFD code FLUENT, Chan et al. (2002) evaluated the three versions of k2 turbulence models by simulating the flow over an isolated street canyon. They suggested that the RNG k2 turbulence model was the optimum one amongst the three versions. However, also using FLUENT, Sagrado et al. (2002) found that the realizable k2 turbulence model with the two-layer zonal approach appeared to be the most reliable one. This contradiction may have been due to the different criteria adopted for selecting the most accurate turbulence model. To select the most appropriate k2 model, Sagrado et al. (2002) compared the reattachment lengths behind a backward-facing step calculated by k2 turbulence models with those calculated by direct numerical simulation (DNS) (Le et al., 1996). They found that the error between realizable k2 model and DNS results was around 3%. On the other hand, Chan et al. (2002) validated their k2 model results against an extensive experimental database collected by wind-tunnel measurements. They found that the pollutant concentration predicted by RNG k2 model was the most accurate. However, these studies only validated the mean quantities, but not the turbulence kinetic energy (TKE) handling the mixing processes. Recently, Liu et al. (2005) adopted the concepts of air exchange rate (ACH) and pollutant exchange rate (PCH) to study the air ventilation and pollutant dilution in street canyons. Making use of the detailed large-eddy simulation (LES) database (Liu and Barth, 2002; Liu et al., 2004), they calculated
the ACH and PCH for street canyons of aspect ratios 0.5, 1.0, and 2.0 at a Reynolds number ðReÞ of 12,000 and a Schmidt number ðScÞ of 0.72. The ACH could be taken as a measure to compare the ventilation efficiency of street canyons. It was found that the air ventilation and pollutant dilution were generally governed by the roof-level vertical turbulence. The air removal from the street canyon was counterbalanced by the air entrainment from the free surface layer. The ACH for the street canyon of aspect ratio 0:5 was largest. Moreover, the ACH for the street canyon of aspect ratio 2:0 was 50% smaller than those of the other two street canyons. Because of its large computational load, LES is a research tool rather than a tool for engineering applications. In particular, the lengthy CPU time for comprehensive sensitivity tests is impractical. It is thus necessary to develop an efficient tool for reliable ACH determination for street canyons. In this study, a two-dimensional (2D) CFD model with RNG k2 turbulence closure scheme for street-canyon flow problems was developed. This model particularly focused on the quick calculation of ACH values. It was also validated against wind tunnel measurements together with numerical outputs from other k2 turbulence models and LES studies. 2. Mathematical model and methodology The computational domain consists of 13 identical street canyons aligned in the streamwise direction for developing periodic boundary conditions (Fig. 1). The flow properties of the street canyon in the middle (the seventh) of the domain are presented in the following discussion. The height of the street canyon of unity aspect ratio (h ¼ b ¼ H) and the free-stream inflow speed U were selected as the reference length and velocity scales, respectively. To facilitate the comparison with previous studies, the Reynolds number (Re ¼ rUH=m, where r and m are, respectively, the air density and molecular viscosity) was prescribed at ¼ 12; 000. Since the flow direction was perpendicular to the street axis, a 2D computational domain was adopted. The flow was treated as an incompressible, isothermal, and pseudo steady-state turbulence. The governing equations for the problem consist of the continuity equation qui ¼0 qxi
(1)
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b (b) Fig. 1. Schematic diagram of the computational domain and boundary conditions: (a) the whole domain; and (b) the enlarged view of the target street canyon.
and the momentum conservation equation 2
qui qui 1 qp m q ui q 00 00 þ uj ¼ þ u u þ gi , r qxi r qxj qxj qxj i j qt qxj (2) which are expressed in tensor notation and the usual summation convention on repeated indices (i, j ¼ 1; 2) are employed. The spatial coordinates xi are the Cartesian coordinates in the streamwise ðxÞ and vertical ðzÞ directions. The overbars represent ensemble averaged flow properties commonly employed in k2 turbulence models. Here, ui are the streamwise (u) and vertical (w) velocity components, p the pressure, mt qui quj 2 00 00 þ ui uj ¼ kdij 3 r qxj qxi the Reynolds stress, gi the gravitational acceleration, mt the turbulent viscosity, k the turbulent kinetic energy, and dij the Kronecker symbol. To close Eq. (2), the RNG k2 turbulence model is introduced (Yakhot and Orszag, 1986) by employing two additional transport equations for the turbulent kinetic energy qk qk 1 q qk Pk þ ui (3) ¼ ak meff þ qt qxi r qxi qxi r
and the dissipation rate q q 1 q q 1 þ ui ¼ a meff þ C 1 Pk qt qxi r qxi qxi r k 2 3 C m rZ ð1 Z=Z0 Þ , C 2 þ 1 þ bZ3 k
ð4Þ
where Pk is the turbulent kinetic energy production, ak and a the inverse effective Prandtl numbers for k and , respectively, meff the effective turbulent viscosity, Z ¼ Sk=, and S the scalar measure of the deformation tensor. The turbulent Prandtl number is determined by an analytical formula in the current RNG k2 turbulence model rather than a constant value in the standard k2 model. With the calculated turbulent kinetic energy and dissipation rate, the turbulent viscosity is then calculated by mt ¼ rC m k2 =. The modelling constants adopted in the current RNG k2 turbulence model are fC 1 ; C 2 ; C m ; Z0 ; bg ¼ f1:42; 1:68; 0:0845; 4:38; 0:012g. The aforementioned mathematical procedures were implemented using the commercial CFD code FLUENT 6.1. The computational domain was discretized into non-overlapping unstructured triangular elements whose properties are summarized in Table 1. The velocity profile was defined along the
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Table 1 The number of (triangular) elements and spatial resolution (in term of element area) adopted in discretizing the (two-dimensional) street canyons of different aspect ratios h=b Aspect ratio h=b Number of elements Minimum element area 0.5 1.0 2.0
2,127,592 1,141,878 2,498,420
1:47 103 H 2 1:94 103 H 2 3:83 104 H 2
Here H is the reference length scale which is defined as the building height of street canyon of unity aspect ratio.
inflow boundary. Zero gradient boundary conditions were prescribed at the outflow and upper boundaries. Similar to Sagrado et al. (2002), the two-layer zonal wall treatment was applied on all rigid walls. Initially all the quantities in the computational domain are set to zero and the inflow velocity profile will develop over the upstream buildings. The finite volume method (FVM) and the SIMPLE algorithm (Patankar, 1980) were employed to solve the transport equations. 3. Results and discussions This section discusses the results obtained from the current RNG k2 turbulence model, which will be compared with the LES results, other k2 model outputs, and wind tunnel measurements available in literature. Afterwards the mathematical procedure, results, and accuracy of the proposed ACH calculation will be discussed. 3.1. Model validation The accuracy of the current k2 turbulence model was first compared with the LES outputs collected by Liu et al. (2004). Flow for three street canyons of aspect ratios 0:5; 1:0, and 2.0 at a Reynolds number of 12,000 was considered. The model validation exercise is described in detail in this section. Fig. 2 shows the spatial contours of the mean streamwise velocity u calculated by the current RNG k2 turbulence model and the LES. The current k2 model captured well the main flow structures in street canyons that were comparable to those calculated by the LES. In particular, the complicated vertically aligned three-layer structures in the street canyon of aspect ratio 2.0 were calculated well (Fig. 2c). More detailed analysis showed that the current k2 model slightly under-
predicted the roof-level and near-ground mean streamwise velocities compared with those calculated by the LES. This discrepancy was mainly due to the different wall treatments adopted. In the LES, perfectly smooth walls were assumed and the processes in near-wall regions, including both surface sublayers and buffer regions, were calculated explicitly at the expense of a highly refined spatial resolution. On the other hand, a wall model was used to represent rigid walls in the current k2 model that prescribed the near-wall velocity components as functions of wall roughness with affordable near-wall spatial resolution. Since the detailed near-wall flow properties were not our main concern, the wall function was considered accurate enough to represent the wall effects in the current study. Fig. 3 shows the spatial contours of the mean vertical velocity w in the street canyons. While the extents of upward and downward flow calculated by the current k2 model generally agreed well with those of the LES results, the mean vertical velocity obtained by the k2 model, similar to the mean streamwise velocity, was also underpredicted in the near-wall regions. In particular, the vertical downward mean flow near the windward wall was underpredicted by as much as 30%. This underprediction was partially attributed to the different near-wall treatments adopted in the current k2 model and LES. Moreover, a more stable upwind scheme was employed in the current k2 model, which would unavoidably introduce numerical dissipation, compared to the less dissipative Galerkin finite element method being employed in the LES (Liu et al., 2004). The flow recirculations within street canyons are depicted in Fig. 4 in the form of streamfunctions. It was evident that the flow recirculations calculated by the current k2 model were comparable to those calculated by the LES. In the street canyon of aspect ratio 1.0, the two counterclockwise-rotating secondary recirculations at ground-level corners were well resolved, which had not been predicted by some previous studies using k2 models (Hassan and Crowther, 1998; Huang et al., 2000; Chan et al., 2002). However, analogous to other k2 models, namely Jeong and Andrews (2002), the current k2 model did not calculate the secondary recirculations at the roof-level leeward corner for street canyons of aspect ratios 1.0 and 2.0. This roof-level secondary recirculation had been uniquely determined by the LES (Liu and Barth, 2002; Liu et al., 2004). It thus
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suggested that the k2 turbulence models were unable to capture this kind of secondary recirculation most likely due to its limitation of modelling the broad range of turbulence motion by only using a single length scale. The centres of the primary recirculations predicted by several numerical models and experiments are tabulated in Table 2. It was observed that the
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0 0
0.5
0.05
0.05 -0.05
-0.05 0
0 -1
(c)
-0.5
0
0
0
0
0.5
x/H
1
0 -0.5
0
0
0
0.5
1
1.5
x/H
Fig. 3. Spatial contours of the dimensionless mean vertical velocity w=U in street canyons of different aspect ratios h=b at a Reynolds number of 12,000 calculated by the current k2 turbulence model (left) and LES (Liu et al., 2004) (right): (a) h=b ¼ 0:5; (b) h=b ¼ 1:0; and (c) h=b ¼ 2:0.
While the mean flow quantities calculated by the current k2 turbulence model were in line with those of the LES results, the turbulence quantities were further examined to obtain more reliable calculation of the ACH. Fig. 5 illustrates the spatial contours of dimensionless turbulent kinetic energy (TKE). The TKE calculated by the current k2 model exhibited characteristics similar to the LES
results. Large TKE was observed at the upper street canyons for all the aspect ratios tested. This was mainly due to the enlarged TKE production by mechanical wind shear around the roof-level leeward and windward corners. This finding generally agreed with the results collected by Jeong and Andrews (2002) though their local maximum TKE determined was much larger than that of the current
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1
1 -0.04 -0.06 -0.08
0.5
0 0.002 0.004
-0.1
z/H
z/H
-0.02
-0.11
0
0.5
-1
-0.5
0
0.5
0
0.006
0.006
0 -1.5
-0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16
0.002 0.004
0
1
0 -0.5
1.5
0
0
0.5
1
x/H
(a)
1
0. 001 0
-0.08
z/H
z/H
-0.06 -0.08
-0.09
0.5 0
0
0 0.001
0.001
0.001
-0.5
0
0.5
1
0.001
0
0 -0.5
0
z/H
1.5
2
-0.01 0 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09
0
1.5
1
0 0.001
0.001 0.01
0.5 0.01 0.015
(c)
-0.5
0.015
0
0
0 -1
-0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08
1 0
0.5
1.5
x/H
z/H
2
1
0.5
x/H
(b)
2.5
-0.06
-0.04
0 -1
2
-0.02 -0.04
-0.02
0.5
1.5
x/H
1 0
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0
0.5
x/H
0
1
0 -0.5
0
0
0.5
1
1.5
x/H
Fig. 4. Spatial contours of the dimensionless mean streamfunction c in street canyons of different aspect ratios h=b at a Reynolds number of 12,000 calculated by the current k2 turbulence model (left) and LES (Liu et al., 2004) (right): (a) h=b ¼ 0:5; (b) h=b ¼ 1:0; and (c) h=b ¼ 2:0.
study. This was probably caused by the larger inflow turbulence intensities used in Jeong and Andrews (2002). Apart from the qualitative comparison with the results calculated by LES and other k2 turbulence models discussed above, the outputs from the current k2 model were compared with previous wind tunnel measurements (Gayev and Savory,
1999; Kastner-Klein et al., 2001). The profiles of dimensionless mean velocity and turbulence intensities are shown in Fig. 6. The vertical profiles of mean streamwise velocity at x=b ¼ 0:33 and 0.67 (Figs. 6a and b) calculated by the current k2 model were in good agreement with the LES and the wind tunnel results. It is worth mentioning that the streamwise velocity in the vicinity of roof level
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Table 2 Locations of the centre of primary recirculations ðxc =H; zc =HÞ in street canyons of different aspect ratios being determined by numerical models and wind tunnel experiment Aspect ratio h=b
Current study LES (Liu et al., 2004) k2 turbulence model (Jeong and Andrews, 2002) k2 turbulence model (Baik et al., 2000) Wind tunnel experiment (Baik et al., 2000)
0.5
1.0
ð1:32; 0:52Þ ð1:30; 0:52Þ — — —
2.0
ð0:54; 0:53Þ ð0:54; 0:53Þ ð0:56; 0:73Þ ð0:57; 0:57Þ ð0:54; 0:58)
Upper
Lower
ð0:55; 1:41Þ ð0:54; 1:38Þ (—, 1:74) ð0:58; 1:56Þ ð0:67; 1:56Þ
ð0:52; 0:37Þ ð0:53; 0:44Þ (—, 0:44) ð0:54; 0:64Þ ð0:59; 0:34Þ
Here, xc and zc are, respectively, the horizontal and vertical distance measuring from the ground-level leeward corner. The reference length scale H is the building height of the street canyon of unity aspect ratio.
exhibited large spatial variation. As discussed above, the wall model employed in the current k2 model led to a lower but more realistic prediction of mean streamwise velocity near the roof level and ground level of street canyons at high Reynolds number than did LES. The reliability of the wall model employed in the current study was confirmed by the wind tunnel results (Gayev and Savory, 1999). The LES slightly overpredicted the velocity near ground level, while the current k2 model outputs agreed well with the wind tunnel results. On the other hand, the current k2 model showed a more reliable prediction of streamwise velocity at and above the roof level than that of LES. The relatively larger roof-level mean streamwise velocity calculated by the LES could lead to other consequences during comparison of quantities normalized by roof-level mean velocity uðHÞ, which will be discussed in the upcoming sections. The comparison of streamwise turbulence inten1=2 sities u00 u00 =uðHÞ (Figs. 6c and d) was more complicated. The current k2 model reported reasonably good roof-level turbulence intensities when compared with the wind tunnel results (Gayev and Savory, 1999), while the agreement inside the street canyon was rather poor. The wind tunnel results of Gayev and Savory (1999) showed a local maximum of TKE at the street-canyon centre (z ¼ 0:5H). However, neither the current k2 model, the LES study (Liu et al., 2004) nor the wind tunnel measurements (Kastner-Klein et al., 2001) supported this experimental finding. Although the results of Kastner-Klein et al. (2001) were obtained by averaging the TKE at seven points evenly distributed across the street canyon (different from our methods), the profiles presented here could still signify that the turbulence intensity was
relatively constant below the street-canyon centre. The LES of Walton and Cheng (2002) seemed to be the only study that calculated a trend similar to Gayev and Savory (1999), but with a much smaller magnitude. These large discrepancies would be partially attributed to, as Kastner-Klein et al. (2001) suggested, the sensitivity of the normalization by uðHÞ, because the velocity near the roof level varied sharply, as noted above. Due to the larger predicted roof-level mean streamwise velocity, the LES by Liu et al. (2004) showed a much lower dimensionless streamwise turbulence intensity. 3.2. Air exchange rate (ACH) The concept of ACH used in street-canyon pollution problems, which represents the volumetric air exchange per unit time, was outlined in Liu et al. (2005). Positive (ACHþ) and negative (ACH) ACH represent, respectively, the air removal (with pollutant) and entrainment (fresh air). Because of continuity, the magnitudes of positive and negative ACH are equal. The temporal average of positive ACH is obtained by integrating the time-dependent ACH þ ðtÞ along the air ventilation area (a line in the current 2D case) G ¼ b across the street in accordance with Z T ACHþ ¼ ACH þ ðtÞ dt 0
Z
T
Z
w00þ ðt; GÞjroof dG dt Z Z T 00 wþ ðt; GÞjroof dt dG, ¼
¼
0
G
G
0
ð5Þ
where w00þ ðt; GÞ is the positive vertical velocity RT fluctuation. The integral 0 w00þ ðt; GÞjroof dt is the
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0.025
1
0.03
0.015
0.5
0.04
0.5
0.01
0.01
0 -1.5
-1
-0.5
0.01
0.005
0.005
0
(a)
0.5
1
0 -0.5
1.5
0
0.02
0.5
1
1.5
2
2.5
x/H
0.025
0.03
0.02
0.01
x/H
1
0.03 0.03
0.02
z/H
z/H
0.02
0.025
1
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0.025
1
0.02 0.025
0.025
z/H
z/H
0.02
0.5
0.01
0.5
0.01
0.005
0.005 0.005
0 -1
-0.5
0
(b)
0.5
0 -0.5
1
0
0.5
x/H 0.02
0.02
0.035
2
0.03
2
0.03 0.02
0 . 0 0 25
1.5
0.001 0.03
0.03 0.02
0.0025
1.5
1.5
0.01
0.01
z/H
z/H
1
x/H
1
1
0.5
0.5
0.005
0.005 0.0025
0.0025 0.001
0 -1
(c)
-0.5
0
0.5
x/H
1
0 -0.5
0
0.5
0.001
1
1.5
x/H
Fig. 5. Spatial contours of the dimensionless turbulent kinetic energy k=U 2 in street canyons of different aspect ratios h=b at a Reynolds number of 12,000 calculated by the current k2 turbulence model (left) and LES (Liu et al., 2004) (right): (a) h=b ¼ 0:5; (b) h=b ¼ 1:0; and (c) h=b ¼ 2:0.
ensemble average of positive roof-level fluctuating vertical velocity. It is noteworthy that the removal and entrainment of air are transient processes that cannot be represented in the current k2 turbulence model in an ensemble average manner. Hence, it is assumed that the transient air exchange is divided evenly into removal and entrainment processes. Moreover, the amount of air removal is equal to its
entrainment counterpart. The positive ACH can then be obtained by Z 1 1=2 ACHþ ¼ w00 w00 roof dG. (6) 2 G Isotropic turbulence (u00 ¼ v00 ¼ w00 ), which was a reasonable approximation in the current highReynolds-number flow in the regions far from the
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1
z/H
z/H
1
0.5
0.5
0 -0.4 -0.2
0 0
0.2
0.4
0.6
0.8
1
1.2
u/U
(a)
0
0.1
0.3 u″u″
(c)
1
0.5 1/2
0.6
0.7
0.8
/u(H)
z/H
z/H
1
0.5
0 -0.4 -0.2
0.5
0
0.2
0.4
0.6
0.8
1
0
1.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1/2
(b)
u/U
(d)
u″u″
/u(H) 1=2
Fig. 6. Vertical profiles of the dimensionless mean streamwise velocity u=U and streamwise turbulence intensity u00 u00 =uðHÞ. u=U at (a) 1=2 x=b ¼ 0:33 and (b) x=b ¼ 0:67; u00 u00 =uðHÞ at (c) x=b ¼ 0:33 and (d) x=b ¼ 0:67. —— and - - - for the current k2 turbulence model with fine and coarse meshes, respectively, LES (Liu et al., 2004), K LES (Walton and Cheng, 2002), ’ wind tunnel data (Gayev and Savory, 1999), and & wind tunnel data (Kastner-Klein et al., 2001). Here x is the distance from the leeward wall, U the freestream velocity, and uðHÞ the roof-level streamwise velocity.
walls, was assumed in this study. The wind tunnel results of Kastner-Klein et al. (2001) also suggested isotropy in the street-canyon wind flow at such a high Reynolds number. Thus, k ¼ ðu00 u00 þ v00 v00 þ 1=2 w00 w00 Þ=2 ¼ 3w00 w00 =2 that yields naturally w00 w00 ¼ ð2=3kÞ1=2 . Hence, qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 (7) w00 w00 jroof ¼ 23 kjroof . Substituting Eq. (7) into Eq. (6) leads to the mathematical expression of ACH determination Z pffiffiffiffiffiffiffiffiffiffiffi 1 kjroof dG. (8) ACHþ ¼ pffiffiffi 6 G
The calculated ACHþ by Eq. (8) was tabulated in Table 3 together with those calculated by LES (Liu et al., 2005). The ACH values were non-dimensionalized by the volume of street canyon of unity aspect ratio V and the reference time scale T ð¼ H=UÞ. It was observed that the current k2 model showed a reasonably accurate prediction of the average ACH values as they were overpredicted by less than 20% compared with the LES-calculated values. This range of errors was certainly within acceptable tolerance from a practical engineering point of view. In fact, the ACH calculation by the current k2 model was much faster than that by LES, which would markedly speed up the analyses.
ARTICLE IN PRESS X.-X. Li et al. / Atmospheric Environment 39 (2005) 7285–7296 Table 3 Dimensionless air exchange rate (ACH) calculated by the current RNG k2 turbulence model and the LES (Liu et al., 2005) for street canyons of aspect ratios h=b ¼ 0:5; 1:0, and 2.0 Aspect ratio h=b
0.5 1.0 2.0
ACH=ðV=TÞ
%Deviation
RNG k2 turbulence model
LES
0.126 0.058 0.059
0.120 0.050 0.050
5.23 16.9 17.5
The speedup was roughly about one to two orders of magnitude. The use of steady-state models (instead of transient models in LES) as well as excluding the statistical analysis (post processing) were the major factors contributing to the marked speedup of the current proposed ACH calculation by k2 model. The possible sources of error in the proposed ACH calculation of k2 model could be the mathematical assumptions adopted. Firstly the assumption of isotropic turbulence would be invalid in large-scale flow or near-wall turbulence (e.g. roof level of buildings). However, only the small subgridscale motions were assumed to be isotropic in LES. Secondly, a 2D model was employed in this study that neglected all the motions in the spanwise direction. This negligence resulted in the inaccuracy of transport equations for k and . Finally, LES resolved all the transient fluid motions while k2 models only calculated the ensemble-averaged values. This difference made the current k2 model unable to capture some important transient phenomena, in particular, the turbulence quantities such as TKE. 4. Conclusion Based on the RNG k2 turbulence closure, a reasonably accurate numerical model for quick determination of ACH was established and validated for street canyons of aspect ratios 0:5; 1:0, and 2.0 at a Reynolds number of 12,000 against other numerical models (Liu et al., 2004) and wind tunnel results. This model was capable of capturing the complicated flow structures inside street canyons. The mean velocities as well as turbulence intensities calculated by the current k2 model agreed well with the wind tunnel results available in literature. However, at the centre core of street canyons, the
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comparison was more complicated due to the contradiction between the two sets of wind tunnel data. Following the ACH concept outlined in Liu et al. (2005), the determination of ACH was derived based on the current 2D RNG k2 model outputs. The average values of ACH for street canyons of aspect ratios 0:5; 1:0, and 2.0 were compared with those calculated by the LES (Liu et al., 2005). The ACH calculated by the current k2 outputs exhibited a slight overprediction but the deviation was less than 20%. This agreement was reasonably good in view of the assumptions made and was certainly within the tolerance of practical engineering applications. In addition, The major contribution of the current 2D k2 turbulence model is its computational efficiency which is faster than that of LES by one to two orders of magnitude. Acknowledgements This study is supported by a research grant from the Hong Kong Research Grant Council (Project No.: HKU 7196103E).
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