Scalar ﬂuxes from urban street canyons. Part II: Modelsws99jfb/publications/HBBpart2_final.pdf · 2003-09-17 · Scalar ﬂuxes from urban street canyons. Part II: Model Ian N

Scalar fluxes from urban street canyons. Part II: Model

Ian N. Harman∗, Janet F. Barlow and Stephen E. BelcherDepartment of Meteorology, University of Reading, Earley Gate, P.O. Box 243,Reading, RG6 6BB, U.K.

Abstract. A practical model is developed for the flux of a scalar, such as heatfrom an urban street canyon that accounts for variations of the flow and turbulencewith canyon geometry. The model gives the magnitude and geometric dependenceof the flux from each facet of the urban street canyon that is shown to agree wellwith wind tunnel measurements described in Part I.

The geometric dependence of the flux from an urban street canyon is shown tobe determined by two physical processes. Firstly, as the height to width ratio ofthe street canyon increases, so does the roughness length and displacement heightof the surface. This increase leads to a reduction in the wind speed in the inertialsub-layer above the street canyons. Since the speed of the circulations in the streetare proportional to this inertial sub-layer wind speed, the flux then reduces withthe inertial sub-layer wind speed. This process is dominant at low height to widthratios. Secondly, the character of the circulations within the street canyon also varyas the height to width ratio increases. The flow in the street is partitioned into arecirculation region and a ventilated region. When the street canyon has high heightto width ratios the recirculation region occupies the whole street canyon and thewind speeds within the street are low. This tendency decreases the flux at high heightto width ratios. These processes tend to reduce the flux density from the individualfacets of the street canyon, when compared to the flux density from a horizontalsurface of the same material. But the street canyon has an increased total surfacearea which means that the total flux from the street canyon is larger than from ahorizontal surface.

The variations in scalar flux from an urban street canyon with geometry is overa factor of two, which means that the physical mechanisms responsible should beincorporated into energy balance models for urban areas.

Keywords: energy balance, sensible heat flux, turbulent transfer, urban streetcanyon, wind tunnel

1. Introduction

The turbulent fluxes of heat, moisture and momentum play a key rolein determining the energy balance of any surface, and therefore in de-termining the surface temperature and the vertical profiles of the windand temperature in the boundary layer. The building configuration ofurban areas is known to affect the radiative component of the energybalance (Arnfield, 2003) and the turbulent flux of momentum (Raupachet al., 1991; Grimmond and Oke, 1999b). The sensible heat flux from

∗ Corresponding author e-mail: [email protected]

c© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

respaper_final.tex; 8/08/2003; 17:36; p.1

2 Ian N. Harman et al.

an urban area depends on the surface type and usage (Grimmond andOke, 2002). How this dependence relates to differences in the buildingconfiguration remains unknown. This knowledge is needed for studiesof the near-surface urban air quality and the formulation of the sensibleand latent heat fluxes in surface energy balance models of urban areas.

There are a range of existing models for the turbulent fluxes fromurban areas, primarily as part of models of the surface energy balance.Grimmond et al. (1991) and Grimmond and Oke (1999a) present anurban surface energy balance model in which the flux of energy into thebuilding fabric and then the turbulent fluxes are related to the evolutionof the net radiation. Any influence of the building configuration onthe turbulent fluxes is dealt with empirically by training the modelwith observations to each location. Best (1998; 1999) takes a similarapproach by modifying the ground heat flux and allowing the turbulentfluxes to adjust accordingly. What these approaches lack is treatmentof how urban geometry affects the physical processes governing theturbulent fluxes.

Many models of the atmospheric processes in an urban area arebased on a generic unit of an urban area. One such generic unit is theurban street canyon (Nunez and Oke, 1977). This generic unit is usedas the basis of many energy balance models for urban areas (Johnsonet al., 1991; Mills, 1993; Sakakibara, 1996; Arnfield and Grimmond,1998; Masson, 2000; Kusaka et al., 2001; Martilli et al., 2002). Urbanstreet canyon models parameterise the turbulent exchange of heat fromthe canyon facets to the air above the canyon. The parameterisationis commonly empirical (Kusaka et al., 2001) or based on the turbulenttransfer from a horizontal but rough surface (Masson et al., 2002).Most models formulate the fluxes in terms of the wind speed withinthe street canyon. A range of methods are employed to parameterisethis wind speed. These include invoking results from the studies of theflow through vegetation canopies (Masson, 2000), invoking continuity(Mills, 1993) and determining the vertical profile of the wind throughthe accumulated effects of drag from a series of street canyons (Martilliet al., 2002). In all cases, the parameterisation of the turbulent fluxfrom the individual facets and the flux from the entire canyon unitneed to be validated.

The flow (Oke, 1988; Baik et al., 2000) and turbulence (Johnsonand Hunter, 1995) within an urban street canyon varies with canyongeometry. The turbulent flux of a scalar from the facets of a singleurban street canyon is therefore expected to be influenced by canyongeometry. Barlow and Belcher (2002) develop a wind tunnel method formeasuring the facet-averaged flux of a passive scalar in a neutral at-mosphere from the street facet of an urban street canyon. Barlow et al.


Scalar fluxes from urban street canyons. Part II. 3

(2003), hereafter referred to as Part I, used this method to measure thefacet-averaged flux from each of the facets of a series of urban streetcanyons. These measurements showed a dependence of the fluxes oncanyon geometry. The purpose of this paper is to develop a quantitativemodel for the turbulent flux of a scalar from an urban street canyonthat explains the dependence of the fluxes on the geometry of the streetcanyon that was observed in Part I.

This paper therefore focusses on the geometric dependence of turbu-lent exchange from an urban street canyon and the physical processesresponsible for this dependence. The urban area is represented by aseries of street canyons oriented normally to the wind. Considerationis focussed on the case of forced convection. A model for the facet-averaged fluxes from each facet of an urban street canyon in neutralconditions is presented for range of canyon geometries. The modelis developed from ideas on the turbulent transfer across developingboundary layers (Appendix A) and ideas on the geometric dependenceof the flow within and above the street canyon (Sections 2 and 3). Acomparison between the model predictions and wind tunnel measure-ments of Part I then allows an assessment of which physical processesdetermine the geometric dependence of the turbulent flux of a scalarfrom an urban surface.

2. Flow patterns in an urban street canyon

The sensible heat flux from an urban area is studied here by consideringthe flux from a two-dimensional street canyon. A street canyon consistsof two parallel, infinitely long buildings that are uniform in height andhave flat roofs. The sensible heat flux from the urban street canyon ismodelled here by considering the heat flux from each constituent facet.This is done by constructing a network of resistances to the transport ofheat between the constituent canyon facets and the air in the inertialsub-layer (Garratt, 1992). The flow patterns within the urban streetcanyon are considered now in order to develop the resistance network.

The flow within the urban street canyon can be decomposed into tworegions, see Figure 1. Firstly, a recirculation region forms in the nearwake of each building. Secondly, when the street is sufficiently wide,there is a ventilated region downstream of the recirculation region. Thedifferent flow characteristics in these two regions mean that the fluxesfrom these two regions scale differently. In addition, the partitioningof the flow into the two regions depends on geometry, as shown inFigure 1. Hence the model for the turbulent flux of a scalar from anurban street canyon developed here is based on a parameterisation of



the flows in these two regions. Figure 2 shows the dimensions of thetwo regions together with the nomenclature used.

Figure 1. Schematic of the streamlines in the three flow regimes. a) Lr < W iso-lated roughness flow regime; b) Lr/2 > W > Lr wake interference flow regime; c)W > Lr/2 skimming flow regime (flow regimes from Oke (1987) ).

The recirculation region is taken here to have a trapezoidal cross sec-tion (Hertel and Berkowicz, 1989). Measurements show that the lengthof the recirculation region, Lr, scales on the height of the building. Thevalue of the ratio Lr/H depends somewhat on the turbulence levelsin the boundary layer above and the shape of the buildings and roof.Oke (1987) suggests Lr/H ≈ 2−3; Castro and Robins (1977) suggestLr/H ≈ 2 for cubes; Okamoto et al. (1993) suggest Lr/H ≈ 3.5 for thetwo-dimensional geometry considered here. Here we therefore take thelength of the recirculation region to be Lr = 3H.

Oke (1987) describes how three flow regimes arise as the canyonaspect ratio is varied. Figure 1 shows schematically how the divisionof the street canyon flow varies with geometry and hence how the flowregimes can be related to the length scales of the two flow regions. For



L / 2

L

R

H

W

U

regionventilated

region

r

recirculation

r

Figure 2. Schematic cross-section of an urban street canyon together with charac-teristic dimensions of the recirculation region.

wide street canyons, Lr < W , so that H/W < 1/3, the recirculationregion does not impinge on the downstream building (Figure 1a). This isthe isolated flow regime of Oke (1987). For intermediate street canyons,Lr/2 < W < Lr, so that 1/3 < H/W < 2/3, the recirculation regionbegins to impinge on the downstream building (Figure 1b). This is thewake interference flow regime of Oke (1987). For narrow street canyons,W < Lr/2, so that H/W > 2/3, the entire canyon canyon volume isoccupied by the recirculation regime (Figure 1c). This is the skimmingflow regime of Oke (1987).

Within the ventilated region, when it exists, high speed air fromabove roof level is brought down to street level. An internal boundarylayer then develops along the street surface, the vertical profile of thewind adjusts to a log-layer in equilibrium with the underlying streetsurface.

The flow within the recirculation region is driven by the intermittentinjection of a high momentum jet associated with the shear layer that isshed off the upstream roof. This jet decelerates as it progresses aroundthe recirculation region due to the entrainment of slower moving air anddue to drag of solid boundaries in a similar way to a rough-boundarywall jet (Townsend, 1976). This picture is supported by observations.Observations by Louka et al. (2000) show that the mean vertical ve-locity at the top of a narrow street canyon is negative in a narrowregion next to the downstream wall. The compensating region of posi-tive vertical velocities adjacent to the upstream wall was broader andweaker than the region of negative vertical velocities. A similar patternof mean vertical velocities at canyon mid-height was found by Catonet al. (2003) in observations of street canyon flow in a water flume.Brown et al. (2000) show that turbulent intensity varies monotonicallyacross a street canyon with low values adjacent to the upstream walland high values adjacent to the downstream walls. Finally, Part I of



this paper showed that the facet averaged turbulent flux from thedownstream wall facet of an urban canyon was, on average, a factorof 2.2 higher than that from the upstream wall facet. All features areconsistent with a decelerating wall jet. The strength of the flow withinthe recirculation region therefore depends on the path length of the jetwhich in turn depends on the dimensions of the recirculation region.These arguments explain why wind speeds decrease as the jet circulatesfrom the downstream wall across the street, and then up the upstreamwall.

At high canyon aspect ratios, the jet may not reach the street sur-face and multiple counter rotating vortices can form within the canyoncavity (e.g. Sini et al., 1996; Baik et al., 2000). This further increasesthe deceleration of the flow around the street canyon.

3. Resistance network for an urban street canyon

The previous section described a partitioning of the canyon air flowinto recirculating air and ventilated air. This partitioning leads tothree pathways for the turbulent transport from the surface to theboundary layer above. These pathways are, firstly, from the wall andstreet surfaces within the recirculation region to the recirculating airand then aloft, secondly, from the wall and street surface within theventilation region to the ventilated air and then aloft, and finally, fromthe roof facet to the air aloft. The ventilated air and recirculating airare each assumed to be independently well mixed and so property Xtakes a single, but different, value within each of these two air volumes(Nakamura and Oke, 1988). Figure 3 shows the resistance network thatwe use to represent transport from the facets of an urban surface tothe inertial sub-layer along these three pathways.

Flux balances may now be formed for each of the pathways, asfollows. Let Fi denote the flux per unit area of scalar X across the ithresistor in Figure 3. The flux from the upstream wall and the portion ofthe street that lies in the recirculation region into the recirculating air(denoted point B in Figure 3) equals the flux out of the recirculationregion into the boundary layer aloft, i.e.

HF3 + LrF4 =Lr

2F5. (1)

Similarly, the flux from the downwind wall and the portion of the streetthat lies in the ventilated region to the ventilated air (which is denotedpoint C in Figure 3) equals the flux from the ventilation region to the



r

rrr 8

r

4r3

r

r

2

1 5

6

7

2< u ( z ) >

C

.

B

A

a

5

2

1r r

r3 r4r6

< u ( z ) >2r

B

A

b

Figure 3. The resistance network for a) a wide canyon, when there are distinctrecirculation and ventilation regions, and b) a narrow canyon, when there is only arecirculation region. The resistance network for the wake interference regime is thelinear interpolation between these two networks.

boundary layer aloft, i.e.

HF8 + (W − Lr)F6 =

(

W −Lr

2

)

F7. (2)

The total flux density across the top of the street canyon, Fc, whichcan be written in terms of a transfer velocity wc, i.e. Fc = wc∆Xc, isthen given by,

Wwc∆Xc =

(

W −Lr

2

)

F7 +Lr

2F5. (3)

Similarly the flux density from the roof, Fr = wr∆Xr, is given by

(R − W )wr∆Xr = (R − W )F2 = (R − W )F1. (4)

Finally the total flux density from the street canyon, Ft = wt∆Xt, isgiven by

Rwt∆Xt = RFt = (R − W )F2 +Lr

2F5 +

(

W −Lr

2

)

F7. (5)

In Equations (3)–(5) ∆X represents the difference between the valueX takes at the surface, averaged over surface area of the individual



facets concerned, and the value X takes at the atmospheric referencelevel. This formulation shows how the value X takes at one facet caninfluence the flux from the other facets by changing the value of X atthe intermediate points A, B and C.

The values of the resistances in the network shown in Figure 3 fallinto either of two generic types. The resistance to transport from eachfacet to the intermediate points A, B or C in the figure is determined bythe across the internal boundary layers that develop along each facet.The resistance to transport between the intermediate points A, B or Cand the reference height z2 represents the resistance to transport acrossa free shear layer. Appendix A develops formulae for the resistance totransport across an internal boundary layer (Equation (30)) and acrossa free shear layer (Equation (32)).

Appendix A formulates each of the resistances to transport in termsof the local wind speeds. Therefore all that remains is the specifica-tion of wind speeds to parameterise each of the fluxes Fi in Equations(1)–(5). Figure 4 shows the location and nomenclature used for theserepresentative winds. The geometric dependence of the forcing windspeed 〈u(z2)〉 and these representative wind speeds is considered next.

rf

uw

ct

us

U

re

dw

dsUU

rf UU

UU

U

< u (z ) >2

Figure 4. Schematic of the representative winds and their positions.

3.1. Wind profile in the inertial sub-layer

The vertical profile of the wind in the inertial sub-layer of the at-mospheric boundary layer is characterised by a roughness length anddisplacement height for the underlying urban surface, namely

〈u(z)〉 =u∗

κln

(

z − dT

z0T

)

, (6)

where κ is the von Karman constant taking a value of 0.4 andu∗ = (τ/ρ)1/2 is the friction velocity. The dependence of the flow in theinertial sub-layer on the characteristics of the urban area can be repre-sented by an effective roughness length, z

0T, and displacement height,

dT

(Grimmond and Oke, 1999a). MacDonald et al. (1998) calculate the



vertically integrated drag on an array of cubes and deduce analyticalforms for the roughness length and displacement height as functionsof the morphological characteristics of the surface. These expressionshave many of the observed features of urban roughness lengths anddisplacement heights (Grimmond and Oke, 1999a). These functionsare used here with the morphological characteristics appropriate fortwo-dimensional street canyons. Figure 5 shows the effective roughnesslength and displacement height, normalized by the building height, fora series of street canyons with the buildings represented by square barsoriented normally to the wind at varying separations. Figure 5 alsoshows the wind speed at twice the building height normalized by thefree-stream wind speed, Uδ, which we take to be the externally imposedforcing to the scalar transport. To facilitate comparison between themodel and observations (Section 4) the free-stream wind is taken at theheight of the reference wind in the wind tunnel in Part I, i.e. zδ = 9.5H.Note how, as the canyon aspect ratio increases, there is a rapid dropand subsequent slow rise in the wind speed at z2 = 2H due to thecombined effects of roughness and displacement.

0.00.1 0.2 0.3 0.4 0.5 0.6 0.7

λf

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

canyon aspect ratio - H/W

Figure 5. Normalized effective parameters for the inertial sub-layer over a sequenceof street canyons calculated from the method of Macdonald et al. (1998). Solid line,roughness length for momentum, z

0T/H ; dashed-dotted line, displacement height of

flow, dT

/H ; dashed line, wind speed at twice building height normalized by the freestream flow, 〈u(z2)〉/Uδ .

Cheng and Castro (2002) observe in wind tunnel studies over anurban-type roughness that when the vertical wind profile is spatiallyaveraged the log-layer observed in the inertial sub-layer extends rightdown to roof level. The wind speed at the reference level, z2 = 2H, and



wind speed at canyon top, uct, are then,

〈u(z2)〉 =u∗

κln

(

2H − dT

z0T

)

, (7)

uct

Uδ= ln

(

H − dT

z0T

) /

ln

(

zδ − dT

z0T

)

. (8)

We shall see that this variation of wind speed at canyon top withgeometry exerts one of the strong controls on scalar transport fromthe surface.

3.2. Transport from the roof

The resistance to transport from the roof, and indeed each facet, to theintermediate point A represents the resistance to transport across theinternal boundary layer that develops along its length. Since the depthof the internal boundary layer grows as the length of the roof facetincreases, it follows that the resistances increase with facet length. Forsimplicity, the depth of the internal boundary layer is taken to be 10%of the facet length.

As for the wind speed at canyon top the representative wind speedfor the roof facet, urf , is obtained by extrapolating the wind profile inthe inertial sub-layer down to the depth of the internal boundary layeralong the roof facet, namely

urf

〈u(z2)〉= ln

(

H + δrf − dT

z0T

) /

ln

(

z2 − dT

z0T

)

. (9)

The resistance to transport from the roof facet to this point, A in Figure3, is then given using Equation (30) and the resistance from A to theair aloft using Equation (32), which yield

r1 = ln

(

δrf

z0m

)

ln

(

δrf

z0X

)

/

κ2urf , (10)

r2 =〈u(z2)〉 − urf

u2∗

, (11)

where z0m and z

0Xare the roughness lengths for momentum and X

respectively for the surface material of the roof facet, and u∗ is calcu-lated from the wind profile in the inertial sub-layer. finally δrf is thethickness of the internal boundary layer developed along the roof facet,taken as δrf = min[0.1(R − W ), z2 − H].



3.3. Transport from the recirculation region

The measurements of Part I and the fluid dynamical ideas developed inSection 2 motivated a description of the flow in the recirculation regionas that of a jet that decelerates exponentially as it travels around thecanyon. First consider the wind speed of the jet when it first impingeson the street facet, ure. This wind speed is then the wind speed atcanyon top, uct, scaled down exponentially to account for entrainment,i.e.

ure = uct exp {−α1Lse/H} , (12)

where Lse is the length of the sloping edge of the recirculation region.The strongest flow along the canyon facets is then located at the endof the recirculation region as observed by Okamoto et al. (1993).

The jet then circulates along the street and wall facets in the recir-culation region, and the wind speed varies as

u(x) = ure exp {−α2x/H} , (13)

where x is the total distance travelled by the jet from the end of therecirculation region. The two exponents α1 and α2 are different due tothe different physical processes which they represent.

The wind speeds representative of the turbulent flux from the up-stream wall, uuw, from the fraction of the street facet in the recir-culation region, uus, and from the downstream wall if it is in therecirculation region, udw, are taken as the average of the flow u(x)along each facet, namely

u† =ure

b

a+b∫

a

exp {−α2x/H} dx, (14)

where † is one of uw, us or uw, the total distance travelled by the jetto the start of the facet in question is a and b is the length of the facet.For instance, when considering the transfer from the upstream wall, inthe isolated roughness regime a = Lr and b = H, in the skimming flowregime a = W + H and b = H.

Internal boundary layers of a fixed depth of 0.1H are then takenon each facet. The resistances to transport, r3, r4, r5 and r8 (in theskimming flow regime) are then calculated from Equations (30) and



(32) as

r3 = ln

(

0.1H

z0m

)

ln

(

0.1H

z0X

)

/κ2uuw, (15)

r4 = ln

(

0.1H

z0m

)

ln

(

0.1H

z0X

)

/κ2uus, (16)

r5 =〈u(z2)〉 − uus

u2∗

, (17)

r8 = ln

(

0.1H

z0m

)

ln

(

0.1H

z0X

)

/κ2udw, (18)

where u∗ is the friction velocity calculated from the wind profile in theinertial sub-layer. uus is used in Equation (17) and not uuw or udw asthese are representative wind speeds for the turbulent transfer in thenarrow regions next to each wall whereas r5 represents the transferacross the top of the recirculation region.

3.4. Transport from the ventilated region

Part of the jet that impinges on the street facet moves along the streetfacet into the ventilated region. Within the ventilated region high mo-mentum air is mixed downwards so that the jet is only deceleratedsomewhat. This mechanism is not available in the recirculation regionwhich therefore has lower wind speeds. The wind speeds representativeof the turbulent transport from the downstream fraction of the street,uds, and the downstream wall in the isolated roughness regime case,udw, are given as

uds =ure

W − Lr

W−Lr∫

0

exp {−α2x/H} dx, (19)

udw =ure

H

W−Lr+H∫

W−Lr

exp {−α2x/H} dx. (20)

As before these wind speeds are taken to be located 0.1H away fromthe facets.

As explained above, the mixing of high momentum air downwardsin the ventilated region street places a minimum bound on the valuethat this wind speed can take. Here we represent this this process byplacing a minimum bound on the wind speed computed using Equation(19. This bound is taken to be the wind speed in a boundary layer inequilibrium with the underlying surface when forced by the wind speed



uct at a height z = H. The wind speed at z = 0.1H takes the value

minuds = uct ln

(

0.1H

z0m

)

/ ln

(

H

z0m

)

. (21)

Similarly, the wind profile is then incident on the downstream wall. Theresulting turbulent transport from the wall then scales as the verticalaverage of the wind profile. It follows that there is a minimum boundon the wind speed udw in the isolated roughness and wake interferenceflow regimes which is

min udw =uct

H − z0m

∫

z0m

ln

(

0.1H

z0m

)

/ ln

(

H

z0m

)

dz. (22)

The resistances to transport r6, r7 and r8 (in the isolated roughnessregion flow regime) are then given as

r6 = ln

(

0.1H

z0m

)

ln

(

0.1H

z0X

)

/κ2uds, (23)

r7 =〈u(z2)〉 − uds

u2∗

, (24)

r8 = ln

(

0.1H

z0m

)

ln

(

0.1H

z0X

)

/κ2udw, (25)

where u∗ is the friction velocity calculated from the wind profile in theinertial sub-layer.

The resistance to transport r8 in the wake interference flow regimeis calculated by taking a weighted average of udw calculated in the tworegions and then linking r8 to both points B and C in Figure 3.

3.5. Model parameters

The model has five parameters. The surface material roughness lengths,z

0m and z0X

, are determined by the underlying surface material. Here,for simplicity, z

0X= 0.1z

0m is used to represent the relation betweenthe roughness lengths of momentum and heat for the facet surfaces(Garratt, 1992). The two exponents, α1 and α2, represent the decelera-tion of the jet due to the entrainment of slower moving fluid by the jetand frictional effects from the canyon facets. These exponents are foundby comparison to the observations and take the values α1 = 0.1875 andα2 = 0.15 × max[1, 1.5H/W ]. The second factor in α2 represents theincrease in the deceleration of the jet in the skimming flow regime dueto the formation of multiple vortices within the canyon cavity in theskimming flow regime. α1 is a property of the fluid; α2 incorporates



a dependence on the surface material roughness. Finally, as explainedin Section 2, Lr/H is taken to be 3. The precise value of the ratio isfound to have little impact on the model results provided Lr falls inthe reasonable range of 1–3 building heights.

4. Comparison with observations

The results from the model are now compared with the wind tunnelmeasurements from Part I. The buildings are represented as squarebars. The variation of the transfer is then measured as the buildingseparation is varied. For this geometry, the planar area index, λp, andfrontal area index, λf , used by Grimmond and Oke (1999b) and othersto classify urban areas are given by λp = λf = H/R. Results are shownhere as the transfer velocity for the flux density across a horizontalplane normalised by the wind speed at the top of the boundary layer,Uδ, plotted against the canyon aspect ratio (bottom axis) and thefrontal area index (top axis). The comparisons are made by calculatingthe transfer velocity from the model and comparing directly with theobservations. This is done is preference to a comparison of resistancescalculated from the observations as this requires prior knowledge of themodel structure. The roughness length for momentum has one value forall four canyon facets determined by comparison with the observationsto be z

0m = 5 × 10−5m.

4.1. Flux off the roof

Figure 6 shows the normalized transfer velocity for the flux densityfrom the roof facet (wr/Uδ). The solid line is the model prediction andthe symbols are observations from Part I. For H/W greater than about0.2, both the model and observations indicate that the flux from theroof facet varies little with canyon geometry.

The model shows how the flux varies largely because of the varia-tion of the wind speed in the inertial sub-layer with canyon geometry(see Section 3.1 and Figure 4). In the range of the observations, thisvariation in the wind speed is small resulting in the flat profile. Atsmall H/W , when the building separation becomes large, the windspeed in the inertial sub-layer increases, and the flux from the roofcorrespondingly increases. The transfer from the roof is higher thanthe transfer from a horizontal surface of the same material calculatedusing Equations (28) and (32) which yields w0t/Uδ = 2.50 × 10−3.At large H/W , as the buildings approach each other and the surfaceresembles a horizontal surface at building height the flux from the roof



0.00.1 0.2 0.3 0.4 0.5 0.6 0.7

λf

0 0.5 1 1.5 2 2.52.5

3

3.5

4

4.5

5

5.5

6


103 ×

tran

sfer

vel

ocity

wr /

Uδ

Figure 6. Variation of the transfer velocity for the canyon top flux density for theroof facet normalized by the free stream velocity, wr/Uδ , with canyon aspect ratio.Solid line, model prediction; triangles, equivalent wind tunnel measurements takenfrom Part I. Roughness lengths have the values z

0m = 5×10−5m, z0X

= 5×10−6m.Dashed line, equivalent prediction from scaled version of Masson (2000).

facet asymptotes to that off a horizontal surface of the same materialbut displaced to roof level (wr/Uδ = 2.93 × 10−3 ).

4.2. Flux off the street

Figure 7 shows the normalised transfer velocity across canyon top forthe flux density from the street facet (wc/Uδ) and its variation withcanyon aspect ratio. The solid line is calculated from the model. Thesymbols are observations from a single street canyon (stars, Barlow andBelcher, 2002) and a series of street canyons (crosses, Part I). The modelsuccessfully captures several features in the observations including thedip at H/W ' 0.3, the peak at H/W ' 0.6 and the almost lineardecrease as the canyon aspect ratio increases further. The transfer fromthe street is notably lower than from the roof facet (Figure 6) for allcanyon aspect ratios.

The model of the flux from the street facet has the correct limitas the canyon aspect ratio tends to zero, namely that of a horizontalsurface of the same material located at z = 0, which yields w0t/Uδ =2.50 × 10−3. The initial decrease and subsequent general increase inwc/Uδ as the canyon aspect ratio increases from zero to 0.6 follows thevariation in the wind speed in the inertial sub-layer (shown in Figure5). The flow pattern within the street canyon also influences the flux



0.00.1 0.2 0.3 0.4 0.5 0.6 0.7

λf

0 0.5 1 1.5 2 2.50.5

1

1.5

2

2.5

3


103 ×

tran

sfer

vel

ocity

wc /

Uδ

Figure 7. Variation of the transfer velocity for the canyon top flux density fromthe street facet normalized by the free stream velocity, wc/Uδ, with canyon aspectratio calculated from the weighted average of the flux through the recirculation andventilated regions. Solid line, model prediction; symbols, equivalent wind tunnelmeasurements taken from Barlow and Belcher (2002) (squares) and Part I (crosses).Roughness lengths have the values z

0m = 5 × 10−5m, z0X

= 5 × 10−6m. Dashedline, equivalent prediction from scaled version of Masson (2000).

as explained in Section 3. firstly, the jet within the recirculation regiondecelerates as it progresses round the canyon cavity. The decelerationis particularly marked at high canyon aspect ratios, H/W > 0.6, due toproduction of multiple vortices. This explains the reduction in wc/Uδ

when H/W > 0.6.

4.3. Flux off the walls

Figure 8 shows the normalised transfer velocities across a horizontalplane at canyon top for the flux from the two wall facets (wc/Uδ).The transfer velocities are expressed in terms of the flux density acrosscanyon top for ease of comparison with the flux from the street facet.This normalisation introduces a dependence on the relative surfaceareas of the walls and canyon top, i.e. the ratio H/W , which leadsto wc/Uδ being proportional to H/W .

The flux from the downstream wall is greater than the flux from theupstream wall by approximately a factor of two for all canyon aspectratios. The difference between the two fluxes, taking the relative surfacearea into account, is increased as the canyon aspect ratio increases.These two features relate to the variation in turbulent intensity acrossthe street canyon. The downstream wall experiences higher wind speeds



0.00.1 0.2 0.3 0.4 0.5 0.6 0.7

λf

0 0.5 1 1.5 2 2.50

1

2

3

4

5


103 ×

tran

sfer

vel

ocity

wc /

Uδ

Figure 8. Variation of the transfer velocities for the canyon top flux density from thetwo wall facets normalized by the free stream velocity, wc/Uδ , with canyon aspectratio. Lower solid line, transfer from the upstream wall; upper solid line, transferfrom the downstream wall calculated as a weighted average of the flux through therecirculation and ventilated regions; Symbols, equivalent wind tunnel measurementsadapted from Part I for the upstream wall (stars) and downstream wall (circles).Roughness lengths have the values z

0m = 5 × 10−5m, z0X

= 5 × 10−6m. Dashedline, equivalent prediction for both walls from scaled version of Masson (2000).

and a higher turbulent intensity than the upstream wall for all canyonaspect ratios. The deceleration of the flow around the recirculationregion results in reduced flow and hence reduced turbulent transportfrom the upstream wall. As for the flux from the street facet, thisdeceleration is more marked as the canyon aspect ratio increases beyondH/W ' 0.6 which increases the difference in the flux from the two walls.

4.4. Total flux

Figure 9 shows the contribution to the flux through a horizontal planein the inertial sub-layer from each of the canyon facets, normalised bythe transfer velocity from a horizontal surface of the same materiallocated at z = 0, which has w0t/Uδ = 2.50 × 10−3. Also shown is thetransfer velocity associated with the total flux from the canyon surface,calculated assuming that X takes the same value on all facets. HenceFigure 9 shows how much more efficient an urban street canyon is atreleasing scalars by turbulent transport than is a flat surface.

The fluxes from each individual facet are smaller than off a hori-zontal surface for almost all canyon aspect ratios (the transfer fromthe roof asymptotes ' 1.2). The reason is that the canyon geometry



0.00.1 0.2 0.3 0.4 0.5 0.6 0.7

λf

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2


tran

sfer

vel

ocity

w

t / w

0t

Figure 9. Transfer velocities from each of the facets and the whole canyon surfacenormalized by the transfer velocity from a flat surface of equivalent surface materialproperties and total planar area (wt/w0t). Circles, downstream wall; stars, upstreamwall; crosses, street; triangles, roof; solid line; total transport.

reduces the near-surface flow and hence the flux. The total flux ishowever greater than off a horizontal surface. This increase is due to theincreased surface area, indicated by the approximately linear increasein the total flux as the canyon aspect ratio increases from zero to aboutH/W = 0.75. At high canyon aspect ratios, H/W > 0.75, the reductionin the near-surface flow is sufficient to reduce the total flux.

Figures 6–8 also show values for the normalized transfer velocitiescalculated using the model of Masson (2000) with a building heightof 10 m (dashed lines). The order of magnitude from the two modelsagree due to the similarity in the values common parameters take. Themodel presented here accounts for the the additional physical processesof variation of the wind speed in the inertial sub-layer as the geometryvaries and the systematic variation in the flow field within the streetcanyon. Only when these processes are accounted for do the the modelpredictions show the variation with geometry found in the observations.

5. Conclusions

Barlow and Belcher (2002) and Barlow et al. (2003) showed that theflux of a passive scalar from an urban street canyon under transverseflow depends on the geometry of the canyon. The variation with canyongeometry is as much as 50% and differs for each facet of the canyon.



This variation is sufficient to warrant inclusion in models of the urbansurface energy balance.

The model developed here is based on a partitioning of the canyonair into two distinct regions namely a recirculation region and a venti-lated region. The partitioning ensures the correct limit for the transfervelocities as the canyon aspect ratio becomes very small or very large.The individual facets of the canyon flux different amounts implyingthe need for different representative wind speeds. A single canyon windspeed is unable to reproduce the geometric dependence of all of thefacets.

We have shown that there are two principal effects of urban geometryon the flux of a passive scalar from the surface. Firstly, canyon geometryacts to reduce the flow in the inertial sub-layer and thence the windspeeds within the vicinity of the street canyon, which then reduces theflux densities off the canyon facets. Secondly, the total surface areaof a street canyon is increased compared to a flat surface which actsto increase the total flux from the surface. These two processes occurregardless of the specific geometry used to represent the urban area.

The model also accounts for streets and buildings made from differ-ent materials through the roughness lengths of the underlying surfacematerial. The transfer velocities from the model depend on the ratiosz

0m/H and z0X

/H. Increasing either roughness length increases theturbulent transport and increases the magnitude of the peak in thetransfer from the street facet. The roughness of one facet can alsoinfluence the transfer from the other facets. Increasing the roughnesslength of the street facet by a factor of ten to simulate the presence oftrees, for instance, increases the transfer off the two walls by 0–5%.

The urban street canyon is only one generic unit of an urban area.Other generic units will have differing low level flow and turbulencefields. The model presented can be extended to other generic arraysof buildings provided the wind speeds representative of the turbulenttransfer are adapted to include the appropriate physical processes oc-curring such as corner vortices or flow separation due to peaked roofs.Such a generalisation remains a task for future work.

The sensible heat flux is a key term in the surface energy balance.The present work shows how this sensible heat flux varies with geome-try. We suggest that a model for the urban energy balance that is validover the full range of urban areas will need to account for this variation.



Acknowledgements

It is a pleasure to acknowledge useful conversations with Martin Bestand Peter Clark. This work is part of the UWERN Urban MeteorologyProgramme (web site: www.met.reading.ac.uk/Research/urb met/).This work is funded by the Natural Environmental Research Councilunder the URGENT programme (GST/02/2231) and a NERC CASEstudentship with The Met Office (NER/S/A/2000/03555).

Appendix

A. The bulk aerodynamic formulation for surface fluxes

The time- and spatiallyaveraged flux density of property X in theinertial sub-layer, F

X, is given by

FX

= 〈w′X ′〉, (26)

where the primes indicate the instantaneous departure from the meanvalue of X, the over-bar denotes a time average and the angle bracketsa spatial average. If the flux of X is mediated entirely by turbulence,dimensional analysis shows that the flux is proportional to the differ-ence of the mean values X takes at the surface and at some heightabove that surface multiplied by a transport velocity. This approach ofrelating the flux of X to the mean values of X is the bulk aerodynamicformulation (Garratt, 1992). Denoting the area-averaged value of X atthe surface by 〈Xs〉 and at any height, z, in the inertial sub-layer, 〈X〉,the flux density of X at the surface, F

X, is given by

FX

= wX

(

〈Xs〉 − 〈X〉)

=〈Xs〉 − 〈X〉

rX

, (27)

where wX

is the transfer velocity for property X and its reciprocal, rX

,is the resistance to transport.

For a neutrally stratified rough-wall boundary layer with zero dis-placement height that is in equilibrium with the underlying surface,the mean wind profile, 〈u(z)〉, and profile of X, 〈X(z)〉, vary as

〈u(z)〉 =u∗

κln

(

z

z0m

)

, (28)

〈X(z)〉 − 〈Xs〉 =X∗

κln

(

z

z0X

)

, (29)



where u∗ = (τ/ρ)1/2 is the friction velocity, X∗ = −FX

/u∗ is a scalingfor X ′, κ is the von Karman constant and z

0m and z0X

are the roughnesslengths for momentum and property X respectively. The resistance tothe transport of X therefore takes the form

rX

(z) = ln

(

z

z0m

)

ln

(

z

z0X

)

/

κ2〈u(z)〉 . (30)

Within the inertial sub-layer of the atmospheric boundary layer thearea-averaged flux density is uniform with height and therefore takesthe same value as at the surface. Using Equation (30) at two heights,z1 < z2, within the inertial sub-layer, the flux density of X at thesurface can be related to the mean profile of X by

FX

= wX∆

(

〈X(z1)〉 − 〈X(z2)〉)

=〈X(z1)〉 − 〈X(z2)〉

rX

(z2) − rX

(z1). (31)

Using Equations (28), (29) and (31) the resistance to transport betweenthe two atmospheric levels, r

X∆= r

X(z2)− r

X(z1), can be expressed as

rX∆

=∆〈X〉

u∗X∗. (32)

In a boundary layer where the mean vertical profiles of all proper-ties can be described by Equation (29), as determined by the relevantroughness lengths, r

X∆takes the same value regardless of the property

considered.A boundary layer undergoing an adjustment to new surface proper-

ties forms an internal boundary-layer. The profile of X in such a regioncan be considered to form three sub-layers. The inner-most layer isin full equilibrium with the underlying surface, the outer-most layerremains unaffected by the change in surface properties with the middlelayer blending the two profiles together. The depth of the internalboundary layer relates to the distance downstream from the change insurface properties; this dependence is taken to be linear. The turbulenttransport from the surface to a height in the outer layer depends onboth the resistance to transport off the surface, r1, and the resistanceto transport across the internal boundary layer, r12, as shown schemat-ically in Figure 10. If the wind speed at height z1 is known, r1 is givenby Equation (30). The resistance to transport r12 is given by Equation(32) where now u∗ and X∗ are the local scaling terms. These localscaling terms are approximated by those of the outer layer as transportis dominated by the larger eddies which will have this characteristicscaling.



z 2

r1

12r

z 1

z 0m1 z 0m2

blending layer

outer layer

inner layer

x

α x

U

Figure 10. Schematic of the resistance network across a developing internal bound-ary layer.

In an adjusting boundary layer it is not immediately obvious thatEquation (32) takes the same value for all properties X. In fact, assum-ing Equation (32) holds for the transport of momentum, the resistanceto transport of property X depends on the ratio r

X∆/ru∆

- when con-sidering heat this ratio is the turbulent Prandtl number. For small scaleturbulence and turbulence in a free shear layer the turbulent Prandtlnumber is approximately 1 (Tennekes and Lumley, 1997, pp.51). Itfollows that the resistance to transport of heat (temperature) across afree shear layer is given by r

T∆= ∆u/u2

∗, where the friction velocity isthat of the outer layer.

References

Arnfield, A. J.: 2003, ‘Two decades of urban climate research: A review of turbulence,exchanges of energy and water, and the urban heat island’. Int. J. of Climatol.23, 1–26.

Arnfield, A. J. and C. S. B. Grimmond: 1998, ‘An urban canyon energy budget modeland its application to urban storage heat flux modelling’. Energ. Buildings 27,61–68.

Baik, J.-J., R.-S. Park, H.-Y. Chun, and J.-J. Kim: 2000, ‘A laboratory model ofurban street-canyon flows’. J. Appl. Meteorol. 39(9), 1592–1600.

Barlow, J. F. and S. E. Belcher: 2002, ‘A windtunnel model for quantifying fluxesin the urban boundary layer’. Boundary-Layer Meteorol. 104, 131–150.

Barlow, J. F., I. N. Harman, and S. E. Belcher: 2003, ‘Scalar fluxes from urban streetcanyons: Part I: Wind tunnnel measurements’. submitted to Boundary-LayerMeteorol.

Best, M.: 1998, ‘A model to predict surface temperatures’. Boundary-Layer Meteorol.88, 279–306.

Best, M.: 1999, ‘Can we represent urban areas in operational numerical weather pre-diction models?’. In: Proceedings of the Third Urban Environment Symposium.pp. 70–71.

Brown, M. J., R. E. Lawson, D. S. Decroix, and R. L. Lee: 2000, ‘Mean flow andturbulence measurements around a 2-D array of buildings in a wind tunnnel’. In:



11th Joint AMS/AWMA conference on the applications of air pollution. LongBeach, CA.

Castro, I. P. and A. G. Robins: 1977, ‘The flow around a surface-mounted cube inuniform and turbulent streams’. J. Fluid Mech. 79(2), 307–335.

Caton, F., R. E. Britter, and S. Dalziel: 2003, ‘Dispersion mechanisms in a streetcanyon’. Atmos. Environ. 37, 693–702.

Cheng, H. and I. P. Castro: 2002, ‘Near wall flow over an urban-like roughness’.Boundary-Layer Meteorol. 104, 229–259.

Garratt, J. R.: 1992, The Atmospheric Boundary Layer, Chapt. 8, pp. 224–257.Cambridge University Press.

Grimmond, C. S. B., H. A. Cleugh, and T. R. Oke: 1991, ‘An objective urban heat-storage model and its comparison with other schemes’. Atmos. Environ. 25B,311–326.

Grimmond, C. S. B. and T. R. Oke: 1999a, ‘Aerodynamic properties of urban areasderived from analysis of surface form’. J. Appl. Meteorol. 38, 1262–1292.

Grimmond, C. S. B. and T. R. Oke: 1999b, ‘Heat storage in urban areas: Local-scaleobservations and evaluation of a simple model’. J. Appl. Meteorol. 38, 922–940.

Grimmond, C. S. B. and T. R. Oke: 2002, ‘Turbulent heat fluxes in urban areas:Observations and a Local-scale Urban Meteorological Parameterization Scheme(LUMPS)’. J. Appl. Meteorol. 41, 792–810.

Hertel, O. and R. Berkowicz: 1989, ‘Modelling pollution from traffics in a streetcanyon. Evaluation of data and model development’. Technical Report DMULUFT-A129, National environmetal research institute, Denmark.

Johnson, G. T. and L. J. Hunter: 1995, ‘A numerical study of the dispersion ofpassive scalars in city canyons’. Boundary-Layer Meteorol. 75, 235–262.

Johnson, G. T., T. R. Oke, T. J. Lyons, D. G. Steyn, I. D. Watson, and J. A.Voogt: 1991, ‘Simulation of surface urban heat islands under ‘ideal’ conditionsat night. Part 1: Theory and tests against field data’. Boundary-Layer Meteorol.56, 275–294.

Kusaka, H., H. Kondo, Y. Kikegawa, and F. Kimura: 2001, ‘A simple single-layerurban canopy model for atmospheric models: Comparision with multi-layer andslab models’. Boundary-Layer Meteorol. 101, 329–358.

Louka, P., S. E. Belcher, and R. G. Harrison: 2000, ‘Coupling between air flow instreets and in the well-developed boundary layer aloft’. Atmos. Environ. 34,2613–2621.

MacDonald, R. W., R. F. Griffiths, and D. J. Hall: 1998, ‘An improved method forthe estimation of surface roughness of obstacle arrays’. Atmos. Environ. 32(11),1857–1864.

Martilli, A., A. Clappier, and M. W. Rotach: 2002, ‘An urban surface exchangeparameterisation for mesoscale models’. Boundary-Layer Meteorol. 104, 261–304.

Masson, V.: 2000, ‘A physically-based scheme for the urban energy budget inatmospheric models’. Boundary-Layer Meteorol. 94(3), 357–397.

Masson, V., C. S. B. Grimmond, and T. R. Oke: 2002, ‘Evaluation of the townenergy balance (TEB) scheme with direct measurements from dry districts intwo cities’. J. Appl. Meteorol. 41, 1011–1026.

Mills, G. M.: 1993, ‘Simulation of the energy budget of an urban canyon—I. Modelstructure and sensitivity test’. Atmos. Environ. 27B(2), 157–170.

Nakamura, Y. and T. R. Oke: 1988, ‘Wind, temperature and stability conditions inan east-west oriented urban canyon’. Atmos. Environ. 22(12), 2691–2700.



Nunez, M. and T. R. Oke: 1977, ‘The energy balance of an urban canyon’. J. Appl.Meteorol. 16, 11–19.

Okamoto, S., K. Nakaso, and I. Kawai: 1993, ‘Effect of rows of two-dimensionalsquare ribs of flow property along plane wall’. JSME Inter. J. Series B. - FluidTherm. Engin. 36(1), 121–129.

Oke, T. R.: 1987, Boundary Layer Climates, Chapt. 8, pp. 262–303. Routledge, 2ndedition.

Oke, T. R.: 1988, ‘Street design and urban canopy layer climate’. Energ. Buildings11, 103–113.

Raupach, M. R., R. A. Antonia, and S. Rajagopalan: 1991, ‘Rough-wall turbulentboundary layers’. Appl. Mech. Rev. 44(1), 1–25.

Sakakibara, Y.: 1996, ‘A numerical study of the effect of urban geometry upon thesurface energy budget’. Atmos. Environ. 30, 487–496.

Sini, J.-F., S. Anquetin, and P. G. Mestayer: 1996, ‘Pollutant dispersion and thermaleffects in urban street canyons’. Atmos. Environ. 30(15), 2659–2677.

Tennekes, H. and J. L. Lumley: 1997, A first course in turbulence, Chapt. 2, pp.27–58. The MIT press, 16th edition.

Townsend, A. A.: 1976, The structure of turbulent shear flow, Chapt. 7. Boundarylayers and wall jets, pp. 259–333. Cambridge University Press, 2nd editionedition.


Documents

Scalar ﬂuxes from urban street canyons. Part II: Modelsws99jfb/publications/HBBpart2_final.pdf · 2003-09-17 · Scalar ﬂuxes from urban street canyons. Part II: Model Ian N