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Boundary-Layer Meteorol (2007) 124:43–60DOI 10.1007/s10546-007-9176-7

ORIGINAL PAPER

Transfer processes in a simulated urban street canyon

E. Solazzo · R. E. Britter

Received: 2 August 2006 / Accepted: 27 February 2007 / Published online: 13 April 2007© Springer Science+Business Media B.V. 2007

Abstract The transfer processes within and above a simulated urban street canyon wereinvestigated in a generic manner. Computational fluid dynamics (CFD) was used to aidunderstanding and to produce some simple operational parameterisations. In this study weaddressed specifically the commonly met situation where buoyancy effects arising from ele-vated surface temperatures are not important, i.e. when mechanical forces outweigh buoyancyforces. In a geophysical context this requires that some suitably defined Richardson numberis small. From an engineering perspective this is interpreted as the important case when heattransfer within and above urban street canyons is by forced convection. Surprisingly, thisparticular scenario (for which the heat transfer coefficient between buildings and the flowis largest), has been less well studied than the situation where buoyancy effects are impor-tant. The CFD technique was compared against wind-tunnel experiments to provide modelevaluation. The height-to-width ratio of the canyon was varied through the range 0.5–5 andthe flow was normal to the canyon axis. By setting the canyon’s facets to have the same ordifferent temperatures or to have a partial temperature distribution, simulations were carriedout to investigate: (a) the influence of geometry on the flow and mixing within the can-yon and (b) the exchange processes within the canyon and across the canyon top interface.Results showed that the vortex-type circulation and turbulence developed within the canyonproduced a temperature distribution that was, essentially, spatially uniform (apart from arelatively thin near-wall thermal boundary layer) This allowed the temperatures within thestreet canyon to be specified by just one value Tcan , the canyon temperature. The variation ofTcan with wind speed, surface temperatures and geometry was extensively studied. Finally,the exchange velocity uE across the interface between the canyon and the flow above wascalculated based on a heat flux balance within the canyon and between the canyon and theflow above. Results showed that uE was approximately 1% of a characteristic wind velocityabove the street canyon. The problem of radiative exchange is not addressed but it can, ofcourse, be introduced analytically, or computationally, when necessary.

E. Solazzo (B)· R. E. BritterDepartment of Engineering, University of Cambridge,Trumpington Street, Cambridge CB2 1PZ, UKe-mail: [email protected]

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44 Boundary-Layer Meteorol (2007) 124:43–60

Keywords Forced convection · Heat transfer processes · Temperature distribution ·Urban street canyon

1 Introduction

Modifications to the flow in urban areas due to building geometry have been extensivelystudied with field and wind-tunnel experiments, numerical and theoretical models (Sini et al.1996; Bentham and Britter 2003; Kastner-Klein and Rotach 2004). However, there are fewerstudies addressing the thermal aspects of the urban environment and the spatial temperaturedistribution within urban street canyons. Due to flow separation from buildings and the con-sequent asymmetric pressure field around them, the exchange of momentum in the urbanenvironment is more efficient than the exchange of scalar quantities, such as temperatureor moisture. The knowledge obtained from analysis of the momentum exchange cannot bedirectly applied to the study of the transfer of scalars: this requires separate study and analysis(Barlow et al. 2004).

Understanding and predicting the temperature distribution within the street canyon isessential for several reasons, each having a direct impact on people living in urban areas.These include (a) the effects of higher temperatures on public health, (b) the influence ofhigher temperatures in the city on pollution levels, and (c) the increase in energy consump-tion of buildings due to increased temperature and the consequent increased demand for airconditioning.

Generally there are two cases of interest. The first is when the temperature differencesare small enough and the wind speed is large enough so that there are no, or insignificant,effects of buoyancy on the flow field. In a geophysical context this requires that some suitablydefined Richardson number is small. The second case is when the wind speed is low andthe temperature differences are large; in this case buoyancy will produce changes to the flowfield and a consequent dynamic interaction between the temperature (or density) distributionand the flow field. In engineering parlance, the first case is called forced convection and thesecond case is called free or natural convection (when the wind speed is zero) and mixedconvection (when the wind speed is non-zero). In many urban areas the most commonlymet case is when the wind speed is sufficiently large to outweigh the influence of thermally-generated buoyancy effects on the flow field. It is these situations that produce the largestheat transfer coefficients between the building surfaces and the surrounding flow. Somewhatsurprisingly, this case has been little studied, with most emphasis being placed on buoyancyinfluenced scenarios.

The temperature distribution within the street canyon and on the building surfaces mustbe consistent with the heat fluxes from the building surfaces and the flow patterns withinthe street canyon. It is this required consistency that links urban meteorology with thermalresponse models developed for buildings.

1.1 Thermal effects within urban street canyons: results overview

Literature addressing thermal effects within the urban street canyon has mainly focused onthe influence of buoyancy on the flow field, typically expressed in term of, for example, aRichardson number.

Full scale measurements have shown little influence of buoyancy effects on the flow. Loukaet al. (2002) measured a thin thermal boundary close to a heated wall, as there must be, butdid not measure velocities in this region, so the flow direction was not clear. Observations of

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Boundary-Layer Meteorol (2007) 124:43–60 45

balloon trajectories in the same field experiment (Mestayer, personal communication, 2006)suggested some regions of upward flow very close to the wall. Most recently, Idczak et al.(2006) conducted a controlled field experiment using a 1/5th scale model of a street canyonwith an aspect ratio of H/W = 2.5. Measurements gathered on 2 days with similar referencewind conditions (speed and direction) but significantly different solar radiation intensitieswere compared. The two cases were characterised by different bulk Richardson number Rb

(Rb = gH(T − Tre f )/(U 2re f Tre f )), where g is the acceleration due to gravity, H is the

canyon height, T is the temperature of the wall at a specified height, Tre f is the referencetemperature and Uref is the mean wind speed measured at a reference height). On the firstday Rb was mainly positive, reaching values greater than 1.0 while on the second day Rb waspredominantly smaller than 0.1. These two conditions were selected in order to determinewhether a buoyancy-driven upward air flow or a mechanically driven downward air flow tookplace in the region near the facet that was hotter and on the downwind side of the canyon.Temperature measurements indicated a small region of elevated temperatures very close tothe wall, with a thickness of 0.025 times the canyon width. Measurements of the verticalvelocity components (taken at 0.16 of the canyon width) showed that, even in the case ofhigh Rb, no upward motion was observed. The wind flow structure measured within thecanyon remained unaltered by any thermal effect but, it must be remembered that no velocitymeasurements were made very close to the wall. The conclusions were that, in the range ofthe Rb examined, temperature variations were only observed very close to the heated facet,as found by Louka et al. (2002), and that no evidence of a dynamic effect of the temperature(density) was detected.

Wind-tunnel studies carried out by Kovar-Panskus et al. (2002) examined the effects ofthe buoyancy forces generated by heated walls in a simulated urban street canyon. Resultsshowed that, for low Froude numbers (equivalent to the inverse of the bulk Richardsonnumber) Fr(Fr ≈ 0.3), a secondary and buoyancy induced circulation near the grounddeveloped, but for Fr > 1, the free stream reference velocity was the driving force for theflow features within the canyon, with very little influence of the wall temperatures on theflow, except for a very thin layer near the heated wall.

Sini et al. (1996) carried out Computational fluid dynamics (CFD) simulations using thestandard k − ε model for a two-dimensional (2D) street canyon. The influence of the facettemperatures on the air flow pattern and vertical exchanges between the flow within the streetand that above the roof was studied. Results showed a strong effect on the flow structure whenthe downstream facet was heated; an upward buoyancy-induced flow opposed the downwardadvective flow along that facet, dividing the flow structure into two contra-rotating cells, in thesimulated case of a canyon with H/W = 0.89, Rb ≈ 0.15. When the street or the upstreamfacet was heated, a negligible effect on the flow was observed. Kim and Baik (2001) carriedout 2D simulations using a k − ε model to investigate the flow field in a street canyon withbottom heating. By varying the canyon aspect ratio and the bottom temperature it was foundthat the thermal heating played a significant role in determining the flow field within the streetcanyon for Rb ranging from 0.25 to 12.5. Numerical studies carried out by Bohnenstengelet al. (2004) showed that large scale stability effects may indirectly influence the street scalecirculation by changing the wind speed at the roof level, but only for large Rb(∼ 10) was theflow regime within the canyon no longer dominated by the external wind field.

Thus, in contrast to the field results and the wind-tunnel experiments, the CFD studiesoverestimated the effect of the buoyancy arising from heated facets, resulting in a strongerinfluence on the flow patterns. This behaviour could be due to the incorrect implementation ofthe thermal wall function or to an insufficient grid resolution near the walls in the numericalmodels.

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In summary, three generic flow fields are apparent when the canyon facets are heated (orcooled):

• A relatively thin thermal boundary layer is evident but there is no change in the velocityfield anywhere

• There is a buoyancy-driven upward flow presumably coinciding with the thin thermalboundary layer, but there is no change in the bulk motion within the street canyon

• There is a change in the bulk motion within the street canyon

Because of the possible over-emphasis on the buoyancy influenced scenarios the aim ofthis paper was to characterise the spatial temperature distribution within the urban streetcanyon and to quantify the scalar exchange through the canyon top in the commonly metsituation when the flow will not be influenced by buoyancy i.e. the small Richardson number,forced convection scenario; the first of the three flows above.

1.2 Methodology

The analysis carried out in this work was focused on the differential temperatures betweenthe canyon bounding facets, the spatial temperature distribution within the canyon and thetemperature of the air above the canyon. In this work ‘heat’ was not used simply as a passivetracer; it was also used to study the heat transfer (and, implicitly the mass transfer) processes.

The computational study was divided into two parts. The first dealt with the evaluationof the CFD approach through a comparison of our results with recent wind-tunnel data fromBarlow et al. (2004). The second, which was the main investigation of this work, dealt witha simple and idealised geometry from which generic conclusions might be drawn. The twoparts were generally very similar but with some slight differences.

The evaluation study was based on the recently available wind-tunnel data addressingtransfer processes within idealised street canyons. These experiments of Barlow et al. (2004)consisted of rows of two-dimensional (2D) obstacles that formed a series of street canyonswith particular attention placed on the last canyon downstream. For this evaluation study itwas important to mimic the wind tunnel flow as correctly as possible with a CFD calculation.Considerable care was made to match the inlet and boundary conditions between the windtunnel and the CFD calculation.

The main study was undertaken on a similar flow: the 2D cavity interpreted as a simpli-fied street canyon. This was conducted in the simplest manner possible to assist in the laterinterpretation of the results. In particular the inlet velocity profile was taken to be uniformrather than with a boundary-layer profile. This removed the complexity and arbitrariness ofthe use and selection of a specific velocity profile. Of course, this may have removed one ofthe important aspects of the interaction between the canyon flow, the heat transfers and theflow above the canyon, but it does allow for a simple and relevant flow that would allow forgeneric interpretation. The flow patterns developed from the two configurations were not thesame, just similar, there being no specific reason that the flow for the evaluation study andfor the generic study undertaken needed to be precisely the same. Neither flow particularlyreflects real urban street canyons but they both are useful in improving our understanding anddescription of the flows and in providing ideas as to what assumptions and approximationsmight be permissible when trying to develop simpler operational models.

In Sect. 2 the computational programme is presented. The common features to the eval-uation and main study are discussed in Sect. 2.1. Sections 2.2 and 2.3 are dedicated to thediscussion of the specific conditions used for the two studies. Results of the evaluation study

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Fig. 1 Conceptual description of the exchange velocity uE within and above a simulated street canyon (seetext)

are discussed in Sect. 3, while results of the main calculations and their analysis are discussedin Sect. 4.

In Sect. 5 the methodology adopted to study the exchange process at the canyon top inter-face is discussed together with the results relevant to the exchange velocity uE . Figure 1shows the transfer and exchange processes of heat (or mass) taking place within and abovea simulated 2D street canyon. The heat flux emanating from the heated facets (small whitearrows) was shown to be well mixed within the canyon to produce a near uniform tempera-ture. Heat is transported out of the canyon (large white arrow) by the exchange of air acrossthe canyon top interface (blended black arrows at the canyon top). The heat flux densityacross the canyon top normalised with the temperature difference between the canyon andthe layer above is what, in the present work, is defined as the exchange velocity uE .

2 Computational programme

2.1 Conditions common to all the computational runs

The computational study was conducted using FLUENT, a finite-volume based commercialcode (Fluent Inc. 2003). The standard k − ε turbulence model was adopted. The simulationswere performed for a steady state scenario only because the time scale for the forced con-vective heat transfer processes within the canyon was much shorter than that for the thermalstorage process of the fabric of the city. Thus, it was possible to study the flow and heattransfer processes within the street canyon in isolation from other thermal aspects of urbanclimatology.

A 2D computational domain was modelled with the wind direction set normal to thestreet canyon axis. This simple configuration, although not frequently met in real situations,allowed us to produce simple and useful results that could be extended to more realisticconfigurations.

Boundary conditions. In Figs. 2 and 3 the domain configurations and the boundary con-ditions for the model evaluation and the main analysis respectively, are shown; symmetrycondition at the top of the domain, outflow condition at the downstream edge of the domain

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Fig. 2 Computational domain and boundary conditions for the evaluation study

Fig. 3 Computational domain and boundary conditions for the main study

and no-slip conditions at the side facets, street and roof of the buildings. The temperature dis-tribution was set at the component facets of the canyon. In this work, the terms ‘heated facet’refer to a solid boundary facet where a fixed temperature was set as the thermal boundarycondition.

Grid. In modelling urban flows, small grid sizes are required near buildings, but awayfrom the buildings larger sizes are permitted (Kim and Baik 2004). An unstructured meshwas used, setting the smaller cells size equal to 2% of the buildings height H and using anexpansion factor equal to 1.08 to keep a similar grid configuration for all the simulated cases.Grid independency analyses were conducted to ensure that the results were not dependenton the particular grid configuration.

2.2 Conditions specific to the evaluation study

The results from the CFD methodology were compared with the wind-tunnel measurementsof mass transfer described in Barlow and Belcher (2002) (experiment B) and Barlow et al.(2004). In the experiments a neutral boundary layer was generated and this was followed

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with an array of eight street canyons downwind (Fig. 2). For the computational analysis,the boundary conditions were set as shown in Fig. 2. At the inlet boundary the profiles forwind speed U (z), turbulent kinetic energy k(z) and dissipation rate ε(z) were provided as inRichards and Hoxey (1993):

U (z) = u∗κ

ln

(z

z0

)(1a)

k(z) = u2∗√Cµ

(1 − z

δ

)(1b)

ε(z) = u3∗κz

(1 − z

δ

). (1c)

The logarithmic velocity profile at the inlet was provided by Barlow et al. (2004) as was theheight of the boundary layer δ = 0.125 m and the related free stream velocity Uδ = 4.5 m s−1,the friction velocity u∗ (given as u∗/Uδ = 0.10) and the roughness length z0 = 0.0025 m;Cµ = 0.09 and κ = 0.40 are parameters. The facets were modelled as smooth walls, thisbeing appropriate for modelling the wind-tunnel experiments. The comparison between theevaluation analysis and the experimental data is discussed in Sect. 3.

2.3 Conditions specific to the main study

For the main study (Fig. 3), the domain inflow length was set equal to 5H , the output lengthequal to 10H , and the domain height, from the street level to the domain top, equal to 10H ,where H is the canyon height. In such a configuration, the inflow was not influenced bythe presence of the canyon downstream, and the flow features were not dependent on theboundary conditions at the top and at the outflow plane (Sini et al. 1996).

A uniform wind profile Uin was set as the velocity inlet boundary condition. This simpleand clear configuration allowed the focusing of effort directly on to the main fluid mechanicalprocesses governing the problem, independent of the inflow wind profile. The inlet turbu-lent kinetic energy profile was set equal to k = IU 2

in (Kim and Baik 2004) where I is theturbulence intensity (set equal to 0.1 to ensure that molecular diffusion does not play anysignificant role in the mass and heat transfer processes), and the inlet turbulent dissipationprofile was ε(z) = C0.75

µ k1.5κ−1z−1 with Cµ and κ parameters specified in Sect. 2.2. Allfacet surfaces were taken to be smooth-walled for these generic studies.

The Reynolds number was varied over the range 5 × 105 − 20 × 105 (by using threedifferent inlet velocities Uin = 2.5, 5.0, 10.0 m s−1). The reference length for the Reynoldsnumber calculation is the canyon height H , set arbitrarily to 3 m. Variation of canyon aspectratio H/W in the range 0.5–5, was simulated.

The simulation runs were carried out by setting three facet temperatures Tw, equal to 310 K,320 K and 330 K. These values were imposed in order to simulate the case of uniformly orpartially heated facets, (discussed in Sects. 4.1 and 4.2). The ambient air temperature Tamb

was equal to 300 K. The resulting Rb was between 0.01 and 0.1, allowing buoyancy effectson the flow to be neglected, so that the results must be only a function of the wind velocity,canyon geometry and facet temperatures.

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3 Evaluation study: results and discussion

In the wind-tunnel experiments by Barlow et al. (2004) the naphthalene sublimation tech-nique was used to study the scalar exchange between the facets of the last street canyondownstream and the flow above, at a specified reference height. The technique consistedin coating the canyon facets (i.e. side walls, street, and roof) with naphthalene. The masstransfer rate was estimated by weighing the canyon at the start and end of the experiment.The ‘transfer velocity’ was defined as wt = Fρ−1

s , where F is the spatially averaged fluxdensity of naphthalene vapour, sublimating from a canyon facet and ρs is the naphthalenesaturated vapour density at the temperature at the reference height. wt was also expressedas wt = CUδ , where C is a non-dimensional ‘transfer coefficient’. These measurementswere of direct use when heat transfer processes are investigated, because the process of masstransfer of naphthalene vapour can be considered the same as the process of heat transfer.

To evaluate the computational methodology adopted in this work the mass transfer mea-surements were compared with the heat transfer computations. The transfer velocity wasdetermined as the balance between the heat flux density from the specified facet and thetemperature difference between the facet and the ambient temperature:

C = wt

Uδ

= Q

ρC p (Tw − Tamb) Uδ

. (2)

Q is the spatially averaged surface heat flux density (in W m−2) from the facet, computedas an area-weighted average surface integral; Tw = 305 K is the temperature of the heatedfacet; Tamb = 300 K is the reference temperature; ρ = 1.225 kg m−3 is the density of theair at 300 K and C p = 1006 J kg−1 K−1 its specific heat capacity at constant pressure. Thearea-weighted average of a quantity was computed by dividing the summation of the productof the selected field variable and facet area by the total area of the surface (computed bysumming the areas of the facets that define the surface); 1

A

∑i Qi |Ai | = 1

A

∫A Qd A′, where

Qi are the fluid-side local surface heat flux densities (Fluent Manual 2003). The referencewind speed Uδ was specified at a longitudinal position in line with the centre of the last streetcanyon downstream. To ensure that buoyancy had no influence on the flow, Rb = 10−4 wasmaintained during the simulations. The same inflow boundary conditions were kept for allthe simulated cases.

The transfer coefficient was found to have different values, depending on the canyonfacet. It was approximately 0.1–0.2% for the street and upstream wall and 0.2–0.3% for thedownstream wall and downstream roof. These values did not change significantly when thecanyon aspect ratio was varied. The evaluation study using the CFD technique reproduced thisbehaviour very well, with good agreement between experimental mass transfer data and theheat transfer CFD simulations (Fig. 4). The comparison is shown for the canyon componentfacets, changing the aspect ratio.

Generally, the comparison was very satisfactory. In particular, that for the street andupstream wall was excellent. These results also support the analogy between the transfer ofmass and other scalar quantities, such as heat.

4 Main study: results and discussion

Three thermal boundary settings were simulated: (a) with the canyon facets at uniform tem-perature, with all the facets at the same temperature, (b) with the facets at uniform but differenttemperatures, and (c) with canyon facets partially heated, where only a portion of the facets

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Fig. 4 Comparison between the transfer coefficient C from wind-tunnel data of Barlow et al. (2004) (stars)and the results from CFD calculations (triangles). Results are shown for the canyon facets: (a) street,(b) upstream wall, (c) downstream wall and (d) roof

temperature was different to ambient. Results for cases (a) and (b) are discussed in Sect. 4.1and for case (c) in Sect. 4.2. All the cases were systematically investigated by changing theinput wind velocity, the facet temperatures and the canyon aspect ratio.

4.1 All canyon component facets uniformly heated

Figure 5 shows the flow patterns within the simulated street canyon for aspect ratios thatranged from 0.5 to 5, Uin = 5 m s−1 and Tw = 310 K for all three facets. Skimming flow wasseen in all configurations. A main vortex structure developed within the canyon, changingthe position of its centre as the aspect ratio varied. For H/W = 5 (Fig. 5f) a second vortexin the lower part of the canyon developed, induced by the vortex in the upper part.

Figure 6a shows the temperature contours within the canyon for H/W = 1, Tw = 310 Kand Uin = 5.0 m s−1. The distribution of temperature within and above the canyon was sur-prisingly simple in comparison with, for example, the velocity field. Over nearly the entirestreet canyon the temperature was approximately uniform except for a relatively thin regionclose to the heated facets, where the temperature changed rapidly from the facet temperatureto the near uniform temperature in the canyon. This thin ‘thermal’ boundary layer close to anyparticular heated facet need not be coincident with the ‘velocity’ boundary layer developingalong the facets, as the velocity boundary layer will develop over all facets but the thermalboundary layer will only develop over heated facets.

In Fig. 6b the vertical profiles at the mid-section and 0.5 m distant from the upstream anddownstream facets had similar shape and magnitude. The temperature reduced from 310 Kto about 303.8 K over a distance of about 0.1 m (with an estimated temperature gradient of

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Fig. 5 Velocity path lines for canyon aspect ratio H/W of: (a) 0.5, (b) 1, (c) 2, (d) 3, (e) 4 and (f) 5

order 50 K m−1). The shape of the vertical profiles showed a strong temperature gradient(of about 15 K m−1) across the turbulent shear layer at roof level, where the temperatureapproached the ambient temperature. The vertical profiles of temperature closer to the sidefacets showed some slight, though interesting, differences. Close to the downwind facet thebulk entrainment into the canyon of cooler air from the shear layer produced a local reductionin temperature. The profile close to the upwind facet had a higher temperature, increasingeven more near the canyon top. This was consistent with a vortical flow in the canyon produc-ing a velocity and thermal boundary layer growing in a clockwise sense along the downwindside facet, bottom facet and upwind side facet.

These results were in contrast to the small variations in temperature of less than ± 0.5 Kover the rest of the canyon height and among all five vertical profiles (with maximum tem-perature gradients of about 1.0 K m−1 but typically much smaller). In this sense the spatialtemperature distribution was considered to be ‘near uniform’.

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Fig. 6 (a) Temperature contours for the simulated case with H/W = 1, Uin = 5.0 m s−1, Tw = 310 K. Themeaning of the arrows is the same as in Fig. 1. (b) Temperature profiles within the canyon for five verticalsections: the mid-section and four sections (0.3+ and 0.5+) near to the downstream facet (respectively 0.3 mand 0.5 m from the downstream facet) and 0.3− and 0.5− near to the upstream facet (at the same distances)

Thus, the vortical flow and mixing within the canyon was found to have an important rolein the heat transfer processes such that the temperature is, effectively, uniform throughout thewhole canyon and this allowed the canyon to be characterised by a single temperature, thecanyon temperature Tc. A suitable non-dimensionalisation of Tc was provided by the ratio:

R = Tc − Tamb

Tw − Tamb(3)

(where Tamb is the reference temperature, Tw is the facet temperature and Tc is the canyontemperature evaluated along the middle section of the cavity). This ratio was calculated as a

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Fig. 7 Sensitivity analysis of R ratio to (a) the inflow wind speed, and (b) to the facet temperatures as afunction of the canyon aspect ratio. The straight line is the linear fit of the data

function of the canyon aspect ratio for three different inflow wind speeds Uin , keeping thefacet temperatures constant (Tw = 310 K), and for two different facet temperatures, keepingthe wind speed constant (Uin = 5 m s−1). For a given geometry, R was found (Fig. 7b) to beTw-independent, and, on reflection, this might have been deduced directly from dimensionalreasoning alone. R was also found (Fig. 7a) to be independent of Uin , although some dis-crepancies were observed (particularly for H/W = 3, probably due to the use of a differentgrid). This independence of R from Uin was a little surprising but is equivalent to R beingindependent of the Reynolds number of the flow. This is less surprising; some dependence onReynolds number is to be expected but this should be slight provided the Reynolds numberwas large.

For H/W = 5 the temperature distribution was found to be spatially non-uniform, andstrongly influenced by the weaker canyon penetration of the inflow wind. It was not possibleto sensibly define a canyon temperature for this case.

Interestingly the data suggested a linear relationship between R and the aspect ratio. Fora fixed H , a decrease in W will produce a reduction in both the area for the venting of the

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cavity W and a reduction in the area W +2H for heat transfer from the facets and the formerwill outweigh the latter. Thus Tc should approach Ta , or R approach zero, for H/W smalland Tc should approach Tw , or R approach unity for large H/W .

Thus, it is anticipated that R should increase monotonically with increasing aspect ratioH/W . It could be argued further that R should be proportional to (W + 2H)/W and, con-sequently R will tend to be linearly dependent on H/W for large H/W and be constantfor H/W small. Figure 7b has some evidence for this behaviour though Fig. 7a shows noevidence of a constant R at small H/W . Our general approach is less applicable at H/W of 5and above due to the non-uniformity of the temperature distribution and also less applicablefor H/W less than around 0.5 when the flow will be more like a wake interference flow thanthe assumed skimming flow (Oke 1988). However our observations will be applicable to theoperationally useful range of H/W in between 0.5 and 5.0.

Linear regression analyses to the datasets in Fig. 7a, b were performed with the data forcedto pass through the origin. The linear best fit correlating R with the canyon aspect ratio H/Wcould be expressed as R = 0.11 H/W . Thus Tc can be determined from

Tc = 0.11H

W(Tw − Tamb) + Tamb. (4)

Equation (4) is a simple operational parameterisation derived from analysis of generic CFDcalculations that can be employed to obtain a first estimate of the canyon air temperaturewhen the bounding facet temperatures and the ambient temperature are known. It should beremarked that it was derived for the dominantly important skimming flow regime and for asmooth walled canyon.

Harman et al. (2004) analysed the fluxes from an urban street canyon’s (rough wall)facets in term of the height-to-width ratio of the canyon, and of the roughness lengths formomentum and heat. Thus the numerical factor ‘0.11’ in (4), though directly applicable tosmooth wall flows, should be thought of as replacing a more detailed dependence of thecanyon temperature on the flow developed within the canyon and the momentum and heattransfer properties of the canyon facets. Nevertheless, application of (4) to the cases dis-cussed in Idczak et al. (2006) gave quite sensible agreement, demonstrating the potential ofsuch a simple parameterisation. This possibly reflects the weaker dependence of convectiveheat transfer on surface roughness compared with the dependence of surface shear stress onsurface roughness.

4.2 Canyon component facets partially heated

In order to simulate the effects of solar radiation, the case of a street canyon with singleheated facets was also simulated. For a given elevation angle of the sun with respect to thehorizontal, only a fraction of the canyon facets are heated, the other fraction being in shadow.This configuration might produce an asymmetry to the distribution of temperature within thecanyon. The aim here was to investigate this possibility.

Two idealised sunlit configurations were simulated corresponding to incoming solar radi-ation inclined at 45 or 135 degrees to the horizontal. The H/W ratio (varied in the range0.5–5) governed the fraction of solar radiation that can penetrate within the canyon: whenthe aspect ratio value is unity, one of the side facet was insolated; when the aspect ratio istwo, one half of the side facet was insolated, etc. It was supposed that the facets in the shadedarea were at the same temperature as the ambient flow.

The CFD simulation results showed that with this configuration the temperature distri-bution within the canyon was essentially spatially uniform. The vortical motion and mixing

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Fig. 8 Sensitivity analysis of the R ratio to the inflow wind speed for different canyon aspect ratios andcanyon’s facet: (a) downstream facet and (b) upstream facet

were able to homogenise the temperature within the canyon, even when asymmetric tempera-ture boundary conditions were set. A sensitivity analysis of R to the wind speed and the facettemperature was carried out. The results for Tw = 310 K are shown in Fig. 8. Similar results(not shown) were obtained for Tw = 320 K and Tw = 330 K. There was slight variation ofR with the wind speed but this was irregular and quite small. Thus, R was independent ofthe facet temperature and the wind speed.

These results in Fig. 8a, with the downstream facet heated, are surprisingly different fromthose in Fig. 7 in that they show smaller magnitudes of R compared with Fig. 7 and R wasroughly constant with aspect ratio from 0.5 to 4.0. Firstly, the smaller magnitudes were thedirect consequence of the reduced fraction of surface that was heated. Secondly, the resultsfor the canyon aspect ratio of 5 should be separated from the rest as in this case there are twovortices and the temperature distribution did not really warrant the simplification of assum-ing it uniform. Finally, for the other aspect ratios the lack of influence of aspect ratio arisesbecause the heat transfer into the canyon from the walls and out across the canyon top bothhave similar dependencies on the aspect ratio. The constancy with aspect ratio arises quite

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simply because the area of the canyon facets being not in shadow decreases with increasedaspect ratio in the same way as the canyon top area decreases with aspect ratio.

The results in Fig. 8b, with the upstream facet heated, were initially more difficult to inter-pret. In this case the difference between the canyon and the ambient temperature was smallerthan the difference found when the downstream facet (Fig. 8a) was heated, although the facettemperature set was the same. This was thought to be due to the downward mass transporton the downwind facet that will ensure that all the heat from the facet will be advected ‘into’the canyon whereas on the upwind facet the heat emanates from the upper part of the facetand is then carried towards the canyon top to be mixed ‘out’ into the shear layer beforebeing able to be mixed and advected down into the canyon. For the configuration with theupstream facet sunlit, as the aspect ratio increased the fraction of facet heated decreased (dueto the decreasing solar penetration) and the upward flow developed along this facet directlytransports more of the heated air outside the canyon. As a consequence, when H/W = 3 and4, R becomes very small. For H/W = 2 the canyon temperature was larger because a largerfraction of the heat from the facet is not directly advected to the canyon top and mixed out ofthe canyon. However further investigations are required to determine why R then decreasesfor aspect ratios of 1.0 and 0.5.

In Fig. 8 the results for H/W = 5 are also shown. In this case the temperatures Tcan wasextrapolated as the average of the temperatures in the middle vertical section of the canyon(being not a vertical section for which the temperature is constant, as discussed in the previoussection). In this case the temperature found is higher than that for the case with H/W = 4.This observation may be, in part, a result of the significant non-uniformity of temperaturewithin the canyon for H/W = 5, compared with uniformity observed for H/W = 4 or less.

Similar behaviour was observed when the canyon bounding facets were uniformly heatedwith different temperatures for each facet. The results for this simulated case only confirmwhat we have already observed and discussed.

5 Exchange process

Investigation of the exchange or ventilation process between the street canyon and the flowabove is important as this controls the temperature levels in the whole canyon (Barlow et al.2004). An ‘exchange velocity’ uE may be defined as the spatially averaged mass exchangebetween the in-canyon and the above-canyon flow (Bentham and Britter 2003), interpretedas a ‘physical velocity’. From this investigation, an estimate of the exchange velocity uE wasobtained for our idealized street canyons (Fig. 6a). It should be emphasised that the phys-ical meaning of the ‘transfer velocity’ wt (see sect. 3), which, when non-dimensionalised,becomes the transfer coefficient C , is fundamentally different to the exchange velocity uE

obtained from our analysis in this section. The transfer velocity wt is the combination of twodistinct thermal resistances; that across the thermal boundary layer from the heated surface toa well mixed canyon and that from the canyon to the ambient air above the roof level. It couldbe an advantage to address and model these two processes separately. The relative magni-tudes of the two processes determine whether the canyon temperature is closer to the surfacetemperature or to the ambient temperature above. However, it is only the resistance betweenthe canyon and the air above that will depend on the fluid mechanical exchange processes atthe cavity top. The velocity characterising this second process is the velocity we define as theexchange velocity uE . Hence, the exchange velocity uE refers only to the exchange betweenthe canyon and the ambient flow above and it is a ‘real physical velocity’, in the sense that itreflects the exchange of mass across an interface rather than a transfer coefficient normalised

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Fig. 9 Sensitivity study of the normalised exchange velocity uE /Uin for different canyon aspect ratios

to have the units of velocity. A difficulty with the transfer velocity approach is that, if thetransfer velocity for three facets within a cavity is determined individually and then all facetsare set to operate at the same time, then the three individual velocities will not sum to thevelocity when all facets are heated together. On the contrary, the exchange velocity is thesame in all cases.

In a steady state the scalar flux from canyon facet(s) to the canyon must equal the scalarflux out of the canyon through the top interface. The relation between the rate of transfer andthe exchange velocity uE can be found by considering these processes and writing

hst (Tst − Tcan)W + [hdw(Tdw − Tcan) + huw(Tuw − Tcan)]H = ρC puE (Tcan − Tamb)W,

(5)

where Tst , Tdw, Tup are the temperatures of the facets (street, downstream and upstreamfacet), Tcan is the mean air temperature within the canyon, as discussed in the previous sec-tions, and the coefficients h are the heat transfer coefficients relative to the heated facets (inW m−2 K−1). The left hand side of Eq. (5) is the heat flux into the street canyon from thethree facets and this is available from the CFD output. Determining also the street canyontemperature from the CFD output allows determination of the exchange velocity.

The exchange velocity was calculated for a canyon with uniform heated facets (Tst =Tdw = Tup = 310 K) and various inflow wind speed Uin = 2.5, 5.0, 10.0 m s−1 and thennormalised with the inflow wind speed. The normalised exchange velocity uE/Uin is shownin Fig. 9 as a function of H/W . For canyon aspect ratios in the range 0.5–4 the exchangevelocity was between 0.80% and 1.10% of the inflow wind speed. These values comparefavourably with independent studies by Bentham and Britter (2003) who found uE of about2% of the reference wind velocity, based on momentum balance analysis (rather than scalarexchange as in this work) and by Hamlyn and Britter (2005) who found uE of about 1% ofthe reference wind velocity, from analysis of CFD results. For an aspect ratio of 5 there is asubstantial reduction in uE/Uin . As discussed in Sect. 4, the temperature within the streetcanyon was very non-uniform for H/W = 5 and this non-uniformity may have producedthe anomalous behaviour.

The spatially area-averaged heat transfer coefficients h were calculated with the optionsprovided by the CFD code Fluent, as discussed in Sect. 3. Typical values of h computed by

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the CFD code were between 1.5 W m−2 K−1 and 12.0 W m−2 K−1, for H/W between 2 and3 and for the wind speed adopted in this work. Field results by Idczak et al. (2006) quoted avariation of h between 10.0 W m−2 K−1 and 20.0 W m−2 K−1 for a canyon with aspect ratioH/W = 2.5 and wind speeds of 2–3 m s−1. However these results were determined solelyfrom the operational correlation of Rowley et al. (1930), that is h = 11.8 + 4.2Uin .

Finally it is important to remember that the approach adopted in this paper was very idea-lised. The approach argued that a street canyon temperature can be sensibly defined and canbe calculated. The street canyon temperature was determined by the magnitudes of two ther-mal resistances; that from the surface to the ‘uniform’ street canyon and that from the streetcanyon to the external flow above. If the former is the greater then the canyon temperature willbe closer to the ambient temperature, if the latter is the greater then the canyon temperaturewill be closer to the surface temperature. Extending this work to more realistic scenarios willentail more precision in the specification of the near facet processes, particular how best totreat real rough and complex surfaces. Secondly the exchange velocity will undoubtedly belarger when real street canyons or urban areas in general are addressed. It is likely that boththermal resistances will be reduced when transferring from the ideal to the real situation butmore work is required in order to be more precise.

6 Conclusions

The computational work presented allowed us to gain knowledge of the mixing process onthe scalar exchange within and above a simulated urban street canyon. Our generic CFDapproach was evaluated against wind-tunnel data, and the comparison showed good agree-ment. Results obtained in our simulations showed that, when the canyon’s bounding facetswere heated, uniformly (at the same or different temperatures) or partially, the vortical circu-lation and mixing produced a spatially uniform temperature distribution within the canyon.A simple parameterisation was proposed to evaluate the temperature within the cavity. Thisapproach showed the potential of a direct application in detailed energy budget models forurban canopies as well as for building energy models.

Investigations about the exchange processes between the street canyon and the flow aboveshowed that the non-dimensional exchange velocity uE/Uin was approximately constant(between 0.80% and 1.10%), independent of the canyon aspect ratio (in the range 0.5–4).

Of course the real urban environment has very complex topography with many physicalprocesses appearing including unsteadiness and the spatial variability of the flow. Despite thelimited direct applicability of our work we believe the generic nature of the approach mayfind a use in several areas of urban climatology.

Acknowledgements The present work was carried out within the EU sponsored ATREUS Project. Theauthors wish to thank D. Hamlyn (University of Cambridge, UK) and Dr. J. Barlow (University of Reading,UK) for providing valuable comments on this study.

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