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Modelling air flow regimes in urban canyons
Energy and Buildings, 15 - 16 (1990/91) 315 - 324
315
Modelling Air Flow Regimes in Urban Canyons L. J. HUNTER, I. D. WATSON and G. T. JOHNSON
School of Mathematics, Physics, Computing and Electronics, Macquarie University, Sydney, N S W 2109 (Australia)
ABSTRACT
The geometry of urban canyons is an important determinant of near-surface air flow patterns in cities. Characteristic canyon geometries, expressed in terms of height-to-width ( H / W ) and length-to-height (L/H) ratios, are known to produce three principal air flow regimes: 'isolated roughness', 'wake interference' and 'skimming flow'. However, there remain uncertainties about the relationship between canyon geometry and transition from one flow regime to another. This paper illustrates the potential for mathematical modelling to resolve some of the current uncertainties. We use a numerical model which solves the Reynolds momentum equations to simulate the major canyon air flow regimes and to illustrate the relationship between canyon geometry and transition from one flow regime to another.
INTRODUCTION
The climate of the urban canopy layer is primarily controlled by the micrometeorological effects of canyon geometry and composition rather than the mesoscale forces controlling the climate of the urban boundary layer [1]. A significant component of the city canyon climate, in the sense that air flow often directly affects the well-being of city inhabitants, is its characteristic wind pattern. For example, air pollution within urban canyons is of concern to human health, and a number of studies have sought to identify optimum urban canyon H / W ratios for efficient dispersion of pollutants [2, 3]. Engineers and town planners are also interested in air flow so that problems of channelling and turbulence at the base of buildings can be eliminated, or at least reduced to acceptable levels [4, 5]. What has emerged from such studies is 0378-7788/91/$3.50
that it is important to understand the nature of the air flow regimes within urban canyons. Recent work has established that air flow regimes vary with urban canyon geometry. Consistent patterns have been found by field observation [1, 3] and in wind tunnel studies [4]. When the above-roof wind direction is perpendicular to the canyon, air flow within the canyon exhibits one of three regimes, 'isolated roughness', 'wake interference' or 'skimming' flow (following the nomenclature of Oke [6]). The mechanisms by which each of the flow regimes occurs may be summarized as follows. When the H / W ratio of a canyon is less t h a n 0.3, i.e., the buildings are well spaced, they act essentially as individual buildings (or 'isolated roughness elements') since the air travels a sufficient distance downwind of the first building before encountering the next obstacle (Fig. l(a)). As buildings become more closely spaced and H / W ratios increase, the disturbed air flow has insufficient distance to readjust before encountering the next obstacle. The result is 'wake interference' flow (Fig. l(b)). With reduced building spacing, the mesoscale flow skims over the top of the canyon (Fig. l(c)). In this case, mesoscale flow is almost decoupled from the within-canyon flow (hence the term 'skimming' flow), but provides a weak tangential force which drives a lee vortex cell within the canyon. The vortex can be explained in terms of pressure difference [7]. Air flow encountering a canyon is obstructed so that the resulting mass convergence creates an area of increased pressure above the upwind building. Subsequent divergence in the lee of the building creates an area of relatively low pressure within the canyon so t h a t flow tends toward the area of lower pressure from the high-pressure area above the building. As air descends, it is deflected downwards © Elsevier Sequoia/Printed in The Netherlands
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Fig. 1. Flow regimes associated with different canyon H/W ratios (after Oke [6]). (a) Isolated roughness flow; (b) wake interference flow; (c) skimming flow.
Fig. 2. Thresholds for flow regimes in urban canyons as functions of urban canyon H/W and L/H ratios (after Oke [7]).
by the windward face of the downwind building. There remains an area of low pressure in the lee of the upwind building, resulting in flow across the canyon floor in the opposite direction to the above-canyon flow. On reaching the face of the building, air flow is deflected up the building wall, conservation of mass requiring an upward vertical flow to compensate for the downward flow. Stronger mesoscale flow across the top of the canyon gives horizontal velocity to the (now weak) vertical flow. A similar explanation for the occurrence of skimming flow is given by Nicholson [8]. The approach is based on the theory of conservation of mass and is used as the basis of the Scalar Budget-Box Diffusion Model developed by Nicholson. As air flows over the building, its velocity at roof level is zero. Decreased surface stress over the canyon cavity results in acceleration of the layers of air near the roof level. The acceleration of upper layers draws air from within the canyon upwards. Because the flow accelerates between the upwind building and the centre of the canyon at roof level, horizontal mass t r a ns por t in the layer between roof level and the height at which the buildings have no effect is greater than that above the centre of the canyon. Continuity of mass necessitates a positive vertical motion (i.e., mass tr an s p o r t out of the canyon) over the lee half of the canyon. Thus a compensating downward flow occurs over the downwind half of the canyon.
regarding the critical thresholds of canyon geometry (H/W and L/H) t hat mark the shift from one flow regime to anot her remains limited. Much of our c u r r e n t understanding is derived from wind tunnel studies which have concent rat ed upon architectural applications: the emphasis has been on investigating the behaviour of air flow around individual buildings. The urban climatologist, however, focuses upon city-wide problems, in which case the urban canyon may be a more useful city unit for investigation. In this case, the length of buildings may be a more significant determinant of air flow regime than their height or width. The relationships between the three principal air flow regimes and canyon H/W and L/H ratios have been summarized in a nomogram by Oke [7] (Fig. 2), but the wind tunnel observations from which the nomogram is compiled did not extend to L/H ratios greater t han about four: our current knowledge of flow patterns in long urban canyons is thus uncert ai n for the want of data. In the absence of appropriate field and wind tunnel observations, we will use a numerical model of air flow in urban canyons for a wide range of canyon geometries in an attempt to answer the following questions: (i) What are the threshold H~Wratios which distinguish isolated roughness, wake interference and skimming flow in urban canyons? (ii) How do different L/H ratios modify characteristic threshold H~ W ratios?
THE PROBLEM
METHOD
While the mechanisms described above are reasonably well understood, the information
Solution of the problem described above requires a turbulence model in which the
317 Navier-Stokes equations are solved by a set of differential equations to represent air flow in three dimensions. We have used a k-~ model (see ref. 9 for a review of approaches to solution of the momentum equations) as implemented by Paterson [10].
The model The k-~ model involves the solution of the following set of six partial differential equations: three momentum equations
Vj ~vi~xj=~[vt ~x~ j~gi] ~xiOP
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the continuity equation ~U~ = 0
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the equation for transportation of turbulent kinetic energy
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and the equation for dissipation of turbulent kinetic energy
+clerk
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where U i, i = 1,2,3, is a fluid velocity P ( = ( P / p ) + (2k/3)) is augmented pressure
k ( = [u~[/2) is turbulent kinetic energy ( = v(aui/~xj)(~ui/~xj)) vt ( = c, k2/e) is turbulent viscosity and c,, cl, c2, ak, a~ are constants [10]. The model is limited to neutral conditions (unless other terms are added to the equations) but, for the present purposes, the assumption of neutral conditions is not a significant problem. Nakamura and Oke [1] observed t h a t although air within the urban canyon is often unstable, conditions are near neutral when vortex circulation is strongest (in the late afternoon). Another limitation of the model is the need for large Reynolds numbers for convergence to a solution. For air flow around buildings with wind speeds greater t h a n 1 m s -1, the
threshold Reynolds number is exceeded. The restriction is not relevant in practical terms, since the threshold velocity for the formation of the vortex is greater than 2 m s -1 [3].
Numerical solution In order to solve the set of partial differential equations shown above (eqns. (1)-(4)), it is necessary to obtain a substitute set of equations for solution by standard numerical techniques. Paterson's implementation employs finite differencing using the control volume method with hybrid upwinding. The resultant linear equations are then solved by the alternating direct implicit (ADI) method. An iterative method is suitable when the solution cannot be estimated with any accuracy, but a good initial guess, such as the logarithmic wind profile or power law profile, is available.
Model parameters It is necessary to initialize the model with canyon dimensions, vertical wind profiles at grid boundaries, and roughness characteristics of building and ground surfaces. For the present purposes, building heights and widths are assumed to be 20 m so that variations in H/W and L/H ratios for different model runs are obtained by changing canyons widths and lengths. Upwind profiles were chosen to approximate flow in medium-density suburban areas [6] by using the logarithmic wind profile (suitable for neutral conditions) with a roughness length of 0.7 m. Building surfaces were assumed to be smooth, with a roughness length of 0.004m, and the ground was assumed to have a roughness length of 0.1 m. Wind speed was specified as 5 m s - ' at a height of 20m at upwind and lateral boundaries, ensuring t h a t the minimum threshold speed of 2 m s -1 for development of canyon vortices observed by DePaul and Sheih [3] was comfortably exceeded.
RESULTS AND DISCUSSION In an attempt to determine thresholds for a shift from one flow regime to another, as in Oke's nomogram (Fig. 2), model runs were undertaken with four L/H ratios (1, 3, 5 and 7, where 5 and 7 represent canyon arrangements for which wind tunnel data were unavailable) and an appropriate range of H/W for each of
318
the four L / H ratios. Table I summarizes the canyon geometries used, and the anticipated flow (after Oke [7]). Identification of the three flow patterns is on the following basis. Skimming flow is characterized by a clear single vortex within the canyon. Wake interference flow is characterized by a reverse (with respect to upwind flow direction) horizontal flow in the lower canyon and forward flow along the top of the canyon. A small vortex may appear behind the upwind building but is not dominant. An area of low wind speed appears in the centre of the canyon. Isolated roughness flow is characteristic of a flow around a single building. The distinguishing features are a strong forward horizontal flow and an increase in velocity with height downwind of the lee vortex. The results for each of the four canyon L / H classes are outlined below.
Cubic canyon (L/H = 1) The geometry represented by a L / H ratio of 1 is a building height of 20m and a canyon
length of 20 m. Canyon widths are shown in Table 1.
H~ W ratio of 1 (anticipated flow, skimming) The modelled flow pat t ern is shown in Fig. 3(a) and is characterized by a vortex circulation typical of skimming flow. The centre of the vortex is at 0.8 H and 0.3 W. The maximum wind speed within the canyon is 2 . 7 m s 1, which occurs at the top of the canyon. Relatively high wind speeds also occur with the descending flow against the downwind building.
H / W ratio of O.74 (anticipated flow, transition to wake interference) Figure 3(b) still shows a strong vortex circulation although the centre of the vortex has now moved to 0.75 H and 0.37 W. The maximum wind speed within the canyon has increased to 3 m s -1 across the top of the canyon, with relatively high wind speeds occurring down the face of the downwind building.
TABLE 1 Modelled urban canyon geometries and anticipated flow regimes
L/H ratios Cubic canyon 1 (20 m/20 m)
Short canyon 3 (60 m/20 m)
Medium-length canyon 5 (100 m/20 m)
Long canyon 7 (140 m/20 m)
H/W ratios
Type of flow regime anticipated (after Oke [7])
20 27 40 50 60 80
1.0 0.74 0.5 0.4 0.33 0.25
skimming flow transition to wake interference wake interference transition to isolated roughness isolated roughness isolated roughness
20 40 60 80
1.0 0.5 0.33 0.25
skimming flow wake interference transition to isolated roughness isolated roughness
20 34 40 60 100 140
1.0 0.6 0.5 0.33 0.2 0.14
skimming flow transition to isolated roughness wake interference wake interference isolated roughness isolated roughness
20 40 74 100 140
1.0 0.5 0.27 0.2 0.14
skimming wake interference transition to isolated roughness isolated roughness isolated roughness
Width of canyon (m)
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H/W ratio of 0.5 (anticipated flow, wake interference) The flow is shown in Fig. 3(c) and is similar to that shown in Fig. 3(b), but shows evidence of wake interference. A weak vortex circulation is present behind the upwind building, centred at 0.8 H and 0.2 W. Wind speeds are relatively high towards the centre and top of the canyon, the maximum being 3.7 m s-l. H/W ratio of 0.4 (anticipated flow, transition to isolated roughness) Figure 3(d) shows characteristic wake interference flow. A small vortex exists in the lee of the upwind building and areas of relatively low velocity and abrupt changes in direction occur within the canyon. By approximately 0.75 W the horizontal component reverts to a logarithmic profile, although the horizontal flow is still in the reverse direction lower in the canyon. The combination of some of the characteristics of both wake interference and isolated roughness flow indicated that this is a transitional regime. HI W ratio of 0.33 (anticipated flow, isolated roughness) The isolated roughness flow shown in Fig. 3(e) is similar to that described by Oke [ 61 (Fig. 1). There is evidence of lee eddy circulation and a return to the upwind velocity profile. After the flow has adjusted, the downwind building begins to have an effect on the flow by
deflecting it down the face of the building. The maximum wind speed is 4.4 m s-l across the top of the canyon, which is close to the wind speed of the upwind profile at this level (4.9 m s-l). Flow down the face of the downwind building is weaker and a stronger flow sweeps down into the canyon from above the roof of the upwind building. HI W ratio of 0.25 (anticipated flow, isolated roughness) Figure 3(f) clearly shows a return to isolated roughness flow. There is evidence of the lee eddy circulation, but the flow has returned to a forward direction throughout the depth of the canyon by approximately 0.5 W. By 0.75 W, the flow resembles a logarithmic profile, the profile used for the initial conditions. The downwind building affects only air flow in close proximity to it; air is deflected down the face of the building. There is evidence of a ‘bolster’ eddy [ 61 near the face of the downwind building. Short canyon (Z/H = 3) The geometry of this canyon arrangement is a height of 20 m and a length of 60 m, for the H/W ratios listed in Table 1. HI W ratio of 1 (anticipated flow, skimming) Figure 4(a) shows skimming flow, although the vortex pattern is different from that shown in Fig. 3(a). The vortex is now centred about
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0.8H a n d 0.95 W. T h e c i r c u l a t i o n is w e a k , with little d o w n w a r d flow w i t h i n t h e c a n y o n . S t r o n g l a t e r a l flow s h o w n in Fig. 4a(i) e x p l a i n s the u p w a r d m o v e m e n t . M a x i m u m wind speeds a t the top of the c a n y o n are u p w a r d o u t of t h e c a n y o n a n d r e l a t i v e l y weak: 1.2 m s -~ as comp a r e d to u p w i n d wind speed of 5 m s -~ at a h e i g h t of 20 m. T h e m a x i m u m l a t e r a l wind speed b e h i n d the b u i l d i n g s o c c u r s n e a r the c e n t r e of the b u i l d i n g a n d is 0.9 m s -1.
H / W ratio of 0.5 (anticipated flow, wake interference) T h e flow s h o w n in Fig. 4(b) is r e v e r s e a n d u p w a r d o u t of t h e c a n y o n . A c r o s s the top of the c a n y o n the flow r e t u r n s to the u p w i n d direction. T h e c h a n g e of d i r e c t i o n a n d dec r e a s e in v e l o c i t y is c o n s i s t e n t w i t h the w a k e i n t e r f e r e n c e p a t t e r n . T h e m a x i m u m wind speed w i t h i n the c a n y o n is 2.1 m s 1, n e a r the centre. H / W ratio of 0.33 (anticipated flow, transition to isolated roughness) T h e p a t t e r n s h o w n in Fig. 4(c) is r e p r e s e n t a t i v e of w a k e i n t e r f e r e n c e flow. T h e flow c o m p o n e n t at the top of the c a n y o n is no l o n g e r u p w a r d o v e r the w i d t h of the c a n y o n b u t b e c o m e s h o r i z o n t a l , w i t h a m a x i m u m wind s p e e d of 2 . 4 m s -1 n e a r t h e c e n t r e of the c a n y o n . An a r e a of low w i n d speeds s u r r o u n d s
the e l o n g a t e d v o r t e x c i r c u l a t i o n at approxim a t e l y 0.75 H.
H / W ratio of 0.25 (anticipated flow, isolated roughness) T h e p a t t e r n p r o d u c e d is c h a r a c t e r i s t i c of w a k e i n t e r f e r e n c e flow (Fig. 4(d)), a l t h o u g h in the u p p e r a r e a of the c a n y o n n e a r the downwind b u i l d i n g t h e r e is e v i d e n c e of a n adjustm e n t to i s o l a t e d r o u g h n e s s flow, w i t h f o r w a r d h o r i z o n t a l flow d o w n to a h e i g h t of 13 m. Medium length canyon ( L / H = 5) T h e c a n y o n g e o m e t r y for a L / H r a t i o of 5 is a h e i g h t of 20 m a n d a l e n g t h of 100 m. H~ W ratio of 1 (anticipated flow, skimming) F i g u r e 5(a) shows a n air flow p a t t e r n cons i s t e n t w i t h s k i m m i n g flow. A v o r t e x circulat i o n is e v i d e n t a n d c e n t r e d a b o u t 0 . 5 H , 0.75 W. T h e m a x i m u m wind speed w i t h i n the c a n y o n is 2.3 m s -~ n e a r r o o f level a n d o v e r the v o r t e x centre. H / W ratio of 0.6 (antipicated flow, transition to wake interference) F i g u r e 5(b) shows a small v o r t e x c e n t r e d at 0 . 7 H a n d 0.9 W. T h e u p w i n d h a l f of the c a n y o n is b e g i n n i n g to show c h a r a c t e r i s t i c s of w a k e i n t e r f e r e n c e flow w i t h a r e v e r s e horizon-
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t a l flow i n d i c a t i n g a c h a n g e of flow regimes. T h e m a x i m u m w i n d s p e e d of 2.3 m s -1 is n e a r t h e t o p of t h e c a n y o n .
H/W ratio of 0.5 (anticipated flow, wake interference) T h e p a t t e r n s h o w n in Fig. 5(c) does n o t c o n f o r m to w a k e i n t e r f e r e n c e flow as is expected. Since t h e r e is no e v i d e n c e of a v o r t e x or a c h a n g e to f o r w a r d h o r i z o n t a l flow, it is n e i t h e r s k i m m i n g n o r i s o l a t e d r o u g h n e s s flow. T h e i r r e g u l a r n a t u r e of w e a k i n t e r f e r e n c e flow m a k e s a n a c c u r a t e definition difficult, b u t the v a r i a t i o n in d i r e c t i o n of this flow m a y well indicate a further transitional stage towards w a k e i n t e r f e r e n c e flow.
H/W ratio of 0.33 (anticipated flow, wake interference) F i g u r e 5(d) s h o w s w a k e i n t e r f e r e n c e flow. F l o w w i t h i n t h e c a n y o n h a s a n u p w a r d vertical c o m p o n e n t , due to the s t r o n g l a t e r a l flow (Fig. 5(d)i), w i t h a m a x i m u m v e l o c i t y of 1.2 m s -1, T h e m a x i m u m h o r i z o n t a l v e l o c i t y is 2 m s -x a c r o s s t h e top of t h e c a n y o n a n d in t h e middle of t h e c a n y o n a t a h e i g h t of approxim a t e l y 5 m.
H/W ratio of 0.2 (anticipated flow, isolated roughness) A l t h o u g h Fig. 5(e) is still c h a r a c t e r i s t i c of w a k e i n t e r f e r e n c e flow, t h e r e is e v i d e n c e of a c h a n g e to i s o l a t e d r o u g h n e s s flow w i t h the i n c r e a s e d d e p t h of f o r w a r d h o r i z o n t a l flow in the d o w n w i n d h a l f of the c a n y o n .
H/W ratio of 0.14 (anticipated flow, isolated roughness) I n v e s t i g a t i o n of a r e t u r n to a l o g a r i t h m i c profile d o w n w i n d of the s e c o n d b u i l d i n g suggests t h a t a r e t u r n to a f o r w a r d h o r i z o n t a l flow does n o t o c c u r for a p p r o x i m a t e l y 5 H d o w n w i n d . F i g u r e 5(f) s h o w s t h a t t h e downw i n d b u i l d i n g influences the flow for approxim a t e l y 20 m into the c a n y o n : t h u s a c a n y o n w o u l d n e e d to be o v e r 120 m wide (i.e., h a v e a H/W r a t i o less t h a n 0.17) in o r d e r to r e e s t a b lish a l o g a r i t h m i c v e l o c i t y profile. In t h e p r e s e n t case, t h e r e t u r n to a l o g a r i t h m i c profile is evident, a l t h o u g h r e v e r s e flow still o c c u r s a l o n g t h e c a n y o n floor. T h e m a x i m u m w i n d s p e e d is 3 m s -1 a l o n g the top of the c a n y o n , a r e d u c t i o n of 39% of t h e initial w i n d s p e e d a t 19 m.
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Long canyon ( L / H = 7) The g e o m e t r y r e p r e s e n t e d by a L / H r a t i o of 7 is a building h e i g h t of 20 m and a l e n g t h of 140 m. C a n y o n widths are given in Table 1. H~ W ratio of 1 (anticipated flow, skimming) F i g u r e 6(a) shows a v o r t e x c i r c u l a t i o n c h a r a c t e r i s t i c of skimming flow. The c e n t r e of the v o r t e x is 0 . 5 H and 0.6 W. A m a x i m u m wind speed of 2.6 m s -1 o c c u r s across the top of the c a n y o n with r e l a t i v e l y high wind speeds down the face of the d o w n w i n d building. F i g u r e 6(a)i shows the lateral flow a r o u n d the building, but with a c a n y o n of this length, the lateral c o m p o n e n t in the c e n t r e of the building is effectively zero. H / W ratio of 0.5 (anticipated flow, wake interference) F i g u r e 6(b) shows a v o r t e x c e n t r e d at 0.45 H and 0.75 W. In the lee c o r n e r of the c a n y o n the flow has c h a n g e d d i r e c t i o n and m a y i n d i c a t e a t r a n s i t i o n to w a k e i n t e r f e r e n c e flow. Maximum wind speeds o c c u r across the top of the c a n y o n and r e l a t i v e l y high wind speeds o c c u r down the face of the d o w n w i n d building.
H~ W ratio of 0.27 (anticipated flow, transition to isolated roughness) F i g u r e 6(c) shows t h a t the flow is now typical of w a k e interference. S t r o n g flows are e v i d e n t across the top and along the lower p a r t of the canyon. The m a x i m u m wind speed is 2.9 m s -1. H / W ratio of 0.2 (anticipated flow, isolated roughness) F i g u r e 6(d) shows t h a t the flow is still typical of w a k e interference, a l t h o u g h t h e r e is some evidence of a t r a n s i t i o n to isolated r o u g h n e s s flow. The maximum wind speed w i t h i n the c a n y o n has i n c r e a s e d to 3.1 m s -1 across the top of the canyon. H / W ratio of 0.14 (anticipated flow, isolated roughness) F i g u r e 6(e) shows isolated r o u g h n e s s flow with a r e t u r n to a f o r w a r d h o r i z o n t a l flow at a p p r o x i m a t e l y 0.84 W. Again, the flow along the floor of the c a n y o n is still reverse. The m a x i m u m wind speed across the top of the c a n y o n is 3.3 m s -1 and an a r e a of r e l a t i v e l y high wind speed exists in the lee eddy.
323 0.17
0.20 -
The results of the threshold value for the change from skimming to wake interference flow using the k-~ model verify the results presented by Oke [7]. The results from the model also confirm the observation of Hussain and Lee [4] t h a t the length of the building has little effect on this threshold value. For the change from wake interference to isolated roughness flow, the difference between the results of Oke [7] and those from the k-~ model is most pronounced. The methods agree for cubic geometry only. Threshold values calculated by the k - s model are significantly greater, as shown in Fig. 7(b). The threshold value appears to change slightly at a H/W ratio of 0.19 for L/H ratios greater than 5, whereas Oke suggests a H/W ratio of 0.32.
Oke ( 1 9 8 8 )
-
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(b) Fig. 7. Threshold H / W values for t r a n s i t i o n from (a) s k i m m i n g to wake interferences a n d (b) wake interference to isolated r o u g h n e s s flow in u r b a n canyons.
Summary Threshold values for the change of flow regimes have been evaluated in detail above and are shown in summary form in Fig. 7 along with the threshold values presented by Oke [7]. Threshold values estimated from the k - e model of turbulence are similar to those of Oke [7], which are based on those of Hussain and Lee [4] for skimming to wake interference flow (Fig. 7(a)), but differ somewhat for wake interference to isolated roughness flow (Fig. 7(b)). For a cubic geometry, and a threshold value for the transition from skimming to wake interference flow, Oke suggests a H/W ratio of 0.71 and the k-~ model a H/Wratio of 0.74. For the change to isolated roughness flow, both the results presented by Oke [7] and those of the k-~ model suggest a H/W threshold of 0.6. Thus the results using the k-~ model for a cubic geometry compare favourably with the results of wind tunnel studies.
The results of the modelling exercises described present some interesting preliminary findings regarding the relationship between canyon geometry and prevailing air flow regime. In the range of canyon L/H ratios where observations are available, the k-~ model produces similar H/W thresholds for changes in flow regime. For greater L/H ratios where there is a lack of data, the model suggests that thresholds occur at smaller H~W ratios than those estimated by Oke [7]. It should be stressed that the findings are tentative and require validation by field observation. Nevertheless, it is clear that the k-~ model is capable of simulating the kind of flow regimes which have been observed in both field and wind tunnel studies. Furthermore, and subject to verification, the model offers the sort of flexibility than enables flow patterns in urban canyons to be investigated in a way that has hitherto been beyond the scope of field measurement programmes.
REFERENCES 1 Y. N a k a m u r a and T. R. Oke, Wind, t e m p e r a t u r e and stability conditions in an e a s t - w e s t oriented u r b a n canyon, Atmos. Environ., 12 (1988) 2691 - 2700. 2 W. B. J o h n s o n , F. L. Ludwig, W. F. Dabberdt and R. J. Allen, An u r b a n diffusion simulation model for carbon monoxide, J. A i r Pollution Control Assoc., 27 (1973) 490 - 498.
324 3 F. T. DePaul and C. M. Sheih, Measurements of wind velocities in a street canyon, Atmos. Environ., 20 (1986) 455- 459. 4 M. Hussain and B. E. Lee, An investigation of wind forces on three-dimensional roughness elements in a simulated atmospheric boundary layer flow. Part II. Flow over large arrays of identical roughness elements and the effect of frontal and side aspect ratio variations, Department of Building Sciences, University of Sheffield, Report No. BS 56, 1980. 5 N. J. Cook, On simulating the lower third of the urban adiabatic boundary layer in a wind tunnel, Atmos. Environ., 7 (1973) 691 - 705.
6 T. R. Oke, Boundary Layer Climates, Methuen, New York, 2nd edn., 1988. 7 T. R. Oke, Street design and urban canopy layer climate, Energy Build., 11 (1988) 103- 113. 8 S. E. Nicholson, A pollution model for street-level air, Atmos. Environ., 9 (1975) 19.31. 9 B. E. Launder and D. B. Spalding, The numerical computation of turbulent flows, Comp. Meth. Appl. Mech. Eng., 3 (1974) 269- 289. 10 D. A. Paterson, Computation of wind flows over three dimensional buildings, Thesis, University of Queensland (submitted in partial fulfillment of the requirement of Doctor of Philosophy), 1986.
315
Modelling Air Flow Regimes in Urban Canyons L. J. HUNTER, I. D. WATSON and G. T. JOHNSON
School of Mathematics, Physics, Computing and Electronics, Macquarie University, Sydney, N S W 2109 (Australia)
ABSTRACT
The geometry of urban canyons is an important determinant of near-surface air flow patterns in cities. Characteristic canyon geometries, expressed in terms of height-to-width ( H / W ) and length-to-height (L/H) ratios, are known to produce three principal air flow regimes: 'isolated roughness', 'wake interference' and 'skimming flow'. However, there remain uncertainties about the relationship between canyon geometry and transition from one flow regime to another. This paper illustrates the potential for mathematical modelling to resolve some of the current uncertainties. We use a numerical model which solves the Reynolds momentum equations to simulate the major canyon air flow regimes and to illustrate the relationship between canyon geometry and transition from one flow regime to another.
INTRODUCTION
The climate of the urban canopy layer is primarily controlled by the micrometeorological effects of canyon geometry and composition rather than the mesoscale forces controlling the climate of the urban boundary layer [1]. A significant component of the city canyon climate, in the sense that air flow often directly affects the well-being of city inhabitants, is its characteristic wind pattern. For example, air pollution within urban canyons is of concern to human health, and a number of studies have sought to identify optimum urban canyon H / W ratios for efficient dispersion of pollutants [2, 3]. Engineers and town planners are also interested in air flow so that problems of channelling and turbulence at the base of buildings can be eliminated, or at least reduced to acceptable levels [4, 5]. What has emerged from such studies is 0378-7788/91/$3.50
that it is important to understand the nature of the air flow regimes within urban canyons. Recent work has established that air flow regimes vary with urban canyon geometry. Consistent patterns have been found by field observation [1, 3] and in wind tunnel studies [4]. When the above-roof wind direction is perpendicular to the canyon, air flow within the canyon exhibits one of three regimes, 'isolated roughness', 'wake interference' or 'skimming' flow (following the nomenclature of Oke [6]). The mechanisms by which each of the flow regimes occurs may be summarized as follows. When the H / W ratio of a canyon is less t h a n 0.3, i.e., the buildings are well spaced, they act essentially as individual buildings (or 'isolated roughness elements') since the air travels a sufficient distance downwind of the first building before encountering the next obstacle (Fig. l(a)). As buildings become more closely spaced and H / W ratios increase, the disturbed air flow has insufficient distance to readjust before encountering the next obstacle. The result is 'wake interference' flow (Fig. l(b)). With reduced building spacing, the mesoscale flow skims over the top of the canyon (Fig. l(c)). In this case, mesoscale flow is almost decoupled from the within-canyon flow (hence the term 'skimming' flow), but provides a weak tangential force which drives a lee vortex cell within the canyon. The vortex can be explained in terms of pressure difference [7]. Air flow encountering a canyon is obstructed so that the resulting mass convergence creates an area of increased pressure above the upwind building. Subsequent divergence in the lee of the building creates an area of relatively low pressure within the canyon so t h a t flow tends toward the area of lower pressure from the high-pressure area above the building. As air descends, it is deflected downwards © Elsevier Sequoia/Printed in The Netherlands
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Fig. 1. Flow regimes associated with different canyon H/W ratios (after Oke [6]). (a) Isolated roughness flow; (b) wake interference flow; (c) skimming flow.
Fig. 2. Thresholds for flow regimes in urban canyons as functions of urban canyon H/W and L/H ratios (after Oke [7]).
by the windward face of the downwind building. There remains an area of low pressure in the lee of the upwind building, resulting in flow across the canyon floor in the opposite direction to the above-canyon flow. On reaching the face of the building, air flow is deflected up the building wall, conservation of mass requiring an upward vertical flow to compensate for the downward flow. Stronger mesoscale flow across the top of the canyon gives horizontal velocity to the (now weak) vertical flow. A similar explanation for the occurrence of skimming flow is given by Nicholson [8]. The approach is based on the theory of conservation of mass and is used as the basis of the Scalar Budget-Box Diffusion Model developed by Nicholson. As air flows over the building, its velocity at roof level is zero. Decreased surface stress over the canyon cavity results in acceleration of the layers of air near the roof level. The acceleration of upper layers draws air from within the canyon upwards. Because the flow accelerates between the upwind building and the centre of the canyon at roof level, horizontal mass t r a ns por t in the layer between roof level and the height at which the buildings have no effect is greater than that above the centre of the canyon. Continuity of mass necessitates a positive vertical motion (i.e., mass tr an s p o r t out of the canyon) over the lee half of the canyon. Thus a compensating downward flow occurs over the downwind half of the canyon.
regarding the critical thresholds of canyon geometry (H/W and L/H) t hat mark the shift from one flow regime to anot her remains limited. Much of our c u r r e n t understanding is derived from wind tunnel studies which have concent rat ed upon architectural applications: the emphasis has been on investigating the behaviour of air flow around individual buildings. The urban climatologist, however, focuses upon city-wide problems, in which case the urban canyon may be a more useful city unit for investigation. In this case, the length of buildings may be a more significant determinant of air flow regime than their height or width. The relationships between the three principal air flow regimes and canyon H/W and L/H ratios have been summarized in a nomogram by Oke [7] (Fig. 2), but the wind tunnel observations from which the nomogram is compiled did not extend to L/H ratios greater t han about four: our current knowledge of flow patterns in long urban canyons is thus uncert ai n for the want of data. In the absence of appropriate field and wind tunnel observations, we will use a numerical model of air flow in urban canyons for a wide range of canyon geometries in an attempt to answer the following questions: (i) What are the threshold H~Wratios which distinguish isolated roughness, wake interference and skimming flow in urban canyons? (ii) How do different L/H ratios modify characteristic threshold H~ W ratios?
THE PROBLEM
METHOD
While the mechanisms described above are reasonably well understood, the information
Solution of the problem described above requires a turbulence model in which the
317 Navier-Stokes equations are solved by a set of differential equations to represent air flow in three dimensions. We have used a k-~ model (see ref. 9 for a review of approaches to solution of the momentum equations) as implemented by Paterson [10].
The model The k-~ model involves the solution of the following set of six partial differential equations: three momentum equations
Vj ~vi~xj=~[vt ~x~ j~gi] ~xiOP
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(1)
the continuity equation ~U~ = 0
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k ( = [u~[/2) is turbulent kinetic energy ( = v(aui/~xj)(~ui/~xj)) vt ( = c, k2/e) is turbulent viscosity and c,, cl, c2, ak, a~ are constants [10]. The model is limited to neutral conditions (unless other terms are added to the equations) but, for the present purposes, the assumption of neutral conditions is not a significant problem. Nakamura and Oke [1] observed t h a t although air within the urban canyon is often unstable, conditions are near neutral when vortex circulation is strongest (in the late afternoon). Another limitation of the model is the need for large Reynolds numbers for convergence to a solution. For air flow around buildings with wind speeds greater t h a n 1 m s -1, the
threshold Reynolds number is exceeded. The restriction is not relevant in practical terms, since the threshold velocity for the formation of the vortex is greater than 2 m s -1 [3].
Numerical solution In order to solve the set of partial differential equations shown above (eqns. (1)-(4)), it is necessary to obtain a substitute set of equations for solution by standard numerical techniques. Paterson's implementation employs finite differencing using the control volume method with hybrid upwinding. The resultant linear equations are then solved by the alternating direct implicit (ADI) method. An iterative method is suitable when the solution cannot be estimated with any accuracy, but a good initial guess, such as the logarithmic wind profile or power law profile, is available.
Model parameters It is necessary to initialize the model with canyon dimensions, vertical wind profiles at grid boundaries, and roughness characteristics of building and ground surfaces. For the present purposes, building heights and widths are assumed to be 20 m so that variations in H/W and L/H ratios for different model runs are obtained by changing canyons widths and lengths. Upwind profiles were chosen to approximate flow in medium-density suburban areas [6] by using the logarithmic wind profile (suitable for neutral conditions) with a roughness length of 0.7 m. Building surfaces were assumed to be smooth, with a roughness length of 0.004m, and the ground was assumed to have a roughness length of 0.1 m. Wind speed was specified as 5 m s - ' at a height of 20m at upwind and lateral boundaries, ensuring t h a t the minimum threshold speed of 2 m s -1 for development of canyon vortices observed by DePaul and Sheih [3] was comfortably exceeded.
RESULTS AND DISCUSSION In an attempt to determine thresholds for a shift from one flow regime to another, as in Oke's nomogram (Fig. 2), model runs were undertaken with four L/H ratios (1, 3, 5 and 7, where 5 and 7 represent canyon arrangements for which wind tunnel data were unavailable) and an appropriate range of H/W for each of
318
the four L / H ratios. Table I summarizes the canyon geometries used, and the anticipated flow (after Oke [7]). Identification of the three flow patterns is on the following basis. Skimming flow is characterized by a clear single vortex within the canyon. Wake interference flow is characterized by a reverse (with respect to upwind flow direction) horizontal flow in the lower canyon and forward flow along the top of the canyon. A small vortex may appear behind the upwind building but is not dominant. An area of low wind speed appears in the centre of the canyon. Isolated roughness flow is characteristic of a flow around a single building. The distinguishing features are a strong forward horizontal flow and an increase in velocity with height downwind of the lee vortex. The results for each of the four canyon L / H classes are outlined below.
Cubic canyon (L/H = 1) The geometry represented by a L / H ratio of 1 is a building height of 20m and a canyon
length of 20 m. Canyon widths are shown in Table 1.
H~ W ratio of 1 (anticipated flow, skimming) The modelled flow pat t ern is shown in Fig. 3(a) and is characterized by a vortex circulation typical of skimming flow. The centre of the vortex is at 0.8 H and 0.3 W. The maximum wind speed within the canyon is 2 . 7 m s 1, which occurs at the top of the canyon. Relatively high wind speeds also occur with the descending flow against the downwind building.
H / W ratio of O.74 (anticipated flow, transition to wake interference) Figure 3(b) still shows a strong vortex circulation although the centre of the vortex has now moved to 0.75 H and 0.37 W. The maximum wind speed within the canyon has increased to 3 m s -1 across the top of the canyon, with relatively high wind speeds occurring down the face of the downwind building.
TABLE 1 Modelled urban canyon geometries and anticipated flow regimes
L/H ratios Cubic canyon 1 (20 m/20 m)
Short canyon 3 (60 m/20 m)
Medium-length canyon 5 (100 m/20 m)
Long canyon 7 (140 m/20 m)
H/W ratios
Type of flow regime anticipated (after Oke [7])
20 27 40 50 60 80
1.0 0.74 0.5 0.4 0.33 0.25
skimming flow transition to wake interference wake interference transition to isolated roughness isolated roughness isolated roughness
20 40 60 80
1.0 0.5 0.33 0.25
skimming flow wake interference transition to isolated roughness isolated roughness
20 34 40 60 100 140
1.0 0.6 0.5 0.33 0.2 0.14
skimming flow transition to isolated roughness wake interference wake interference isolated roughness isolated roughness
20 40 74 100 140
1.0 0.5 0.27 0.2 0.14
skimming wake interference transition to isolated roughness isolated roughness isolated roughness
Width of canyon (m)
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H/W ratio of 0.5 (anticipated flow, wake interference) The flow is shown in Fig. 3(c) and is similar to that shown in Fig. 3(b), but shows evidence of wake interference. A weak vortex circulation is present behind the upwind building, centred at 0.8 H and 0.2 W. Wind speeds are relatively high towards the centre and top of the canyon, the maximum being 3.7 m s-l. H/W ratio of 0.4 (anticipated flow, transition to isolated roughness) Figure 3(d) shows characteristic wake interference flow. A small vortex exists in the lee of the upwind building and areas of relatively low velocity and abrupt changes in direction occur within the canyon. By approximately 0.75 W the horizontal component reverts to a logarithmic profile, although the horizontal flow is still in the reverse direction lower in the canyon. The combination of some of the characteristics of both wake interference and isolated roughness flow indicated that this is a transitional regime. HI W ratio of 0.33 (anticipated flow, isolated roughness) The isolated roughness flow shown in Fig. 3(e) is similar to that described by Oke [ 61 (Fig. 1). There is evidence of lee eddy circulation and a return to the upwind velocity profile. After the flow has adjusted, the downwind building begins to have an effect on the flow by
deflecting it down the face of the building. The maximum wind speed is 4.4 m s-l across the top of the canyon, which is close to the wind speed of the upwind profile at this level (4.9 m s-l). Flow down the face of the downwind building is weaker and a stronger flow sweeps down into the canyon from above the roof of the upwind building. HI W ratio of 0.25 (anticipated flow, isolated roughness) Figure 3(f) clearly shows a return to isolated roughness flow. There is evidence of the lee eddy circulation, but the flow has returned to a forward direction throughout the depth of the canyon by approximately 0.5 W. By 0.75 W, the flow resembles a logarithmic profile, the profile used for the initial conditions. The downwind building affects only air flow in close proximity to it; air is deflected down the face of the building. There is evidence of a ‘bolster’ eddy [ 61 near the face of the downwind building. Short canyon (Z/H = 3) The geometry of this canyon arrangement is a height of 20 m and a length of 60 m, for the H/W ratios listed in Table 1. HI W ratio of 1 (anticipated flow, skimming) Figure 4(a) shows skimming flow, although the vortex pattern is different from that shown in Fig. 3(a). The vortex is now centred about
320
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0.8H a n d 0.95 W. T h e c i r c u l a t i o n is w e a k , with little d o w n w a r d flow w i t h i n t h e c a n y o n . S t r o n g l a t e r a l flow s h o w n in Fig. 4a(i) e x p l a i n s the u p w a r d m o v e m e n t . M a x i m u m wind speeds a t the top of the c a n y o n are u p w a r d o u t of t h e c a n y o n a n d r e l a t i v e l y weak: 1.2 m s -~ as comp a r e d to u p w i n d wind speed of 5 m s -~ at a h e i g h t of 20 m. T h e m a x i m u m l a t e r a l wind speed b e h i n d the b u i l d i n g s o c c u r s n e a r the c e n t r e of the b u i l d i n g a n d is 0.9 m s -1.
H / W ratio of 0.5 (anticipated flow, wake interference) T h e flow s h o w n in Fig. 4(b) is r e v e r s e a n d u p w a r d o u t of t h e c a n y o n . A c r o s s the top of the c a n y o n the flow r e t u r n s to the u p w i n d direction. T h e c h a n g e of d i r e c t i o n a n d dec r e a s e in v e l o c i t y is c o n s i s t e n t w i t h the w a k e i n t e r f e r e n c e p a t t e r n . T h e m a x i m u m wind speed w i t h i n the c a n y o n is 2.1 m s 1, n e a r the centre. H / W ratio of 0.33 (anticipated flow, transition to isolated roughness) T h e p a t t e r n s h o w n in Fig. 4(c) is r e p r e s e n t a t i v e of w a k e i n t e r f e r e n c e flow. T h e flow c o m p o n e n t at the top of the c a n y o n is no l o n g e r u p w a r d o v e r the w i d t h of the c a n y o n b u t b e c o m e s h o r i z o n t a l , w i t h a m a x i m u m wind s p e e d of 2 . 4 m s -1 n e a r t h e c e n t r e of the c a n y o n . An a r e a of low w i n d speeds s u r r o u n d s
the e l o n g a t e d v o r t e x c i r c u l a t i o n at approxim a t e l y 0.75 H.
H / W ratio of 0.25 (anticipated flow, isolated roughness) T h e p a t t e r n p r o d u c e d is c h a r a c t e r i s t i c of w a k e i n t e r f e r e n c e flow (Fig. 4(d)), a l t h o u g h in the u p p e r a r e a of the c a n y o n n e a r the downwind b u i l d i n g t h e r e is e v i d e n c e of a n adjustm e n t to i s o l a t e d r o u g h n e s s flow, w i t h f o r w a r d h o r i z o n t a l flow d o w n to a h e i g h t of 13 m. Medium length canyon ( L / H = 5) T h e c a n y o n g e o m e t r y for a L / H r a t i o of 5 is a h e i g h t of 20 m a n d a l e n g t h of 100 m. H~ W ratio of 1 (anticipated flow, skimming) F i g u r e 5(a) shows a n air flow p a t t e r n cons i s t e n t w i t h s k i m m i n g flow. A v o r t e x circulat i o n is e v i d e n t a n d c e n t r e d a b o u t 0 . 5 H , 0.75 W. T h e m a x i m u m wind speed w i t h i n the c a n y o n is 2.3 m s -~ n e a r r o o f level a n d o v e r the v o r t e x centre. H / W ratio of 0.6 (antipicated flow, transition to wake interference) F i g u r e 5(b) shows a small v o r t e x c e n t r e d at 0 . 7 H a n d 0.9 W. T h e u p w i n d h a l f of the c a n y o n is b e g i n n i n g to show c h a r a c t e r i s t i c s of w a k e i n t e r f e r e n c e flow w i t h a r e v e r s e horizon-
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t a l flow i n d i c a t i n g a c h a n g e of flow regimes. T h e m a x i m u m w i n d s p e e d of 2.3 m s -1 is n e a r t h e t o p of t h e c a n y o n .
H/W ratio of 0.5 (anticipated flow, wake interference) T h e p a t t e r n s h o w n in Fig. 5(c) does n o t c o n f o r m to w a k e i n t e r f e r e n c e flow as is expected. Since t h e r e is no e v i d e n c e of a v o r t e x or a c h a n g e to f o r w a r d h o r i z o n t a l flow, it is n e i t h e r s k i m m i n g n o r i s o l a t e d r o u g h n e s s flow. T h e i r r e g u l a r n a t u r e of w e a k i n t e r f e r e n c e flow m a k e s a n a c c u r a t e definition difficult, b u t the v a r i a t i o n in d i r e c t i o n of this flow m a y well indicate a further transitional stage towards w a k e i n t e r f e r e n c e flow.
H/W ratio of 0.33 (anticipated flow, wake interference) F i g u r e 5(d) s h o w s w a k e i n t e r f e r e n c e flow. F l o w w i t h i n t h e c a n y o n h a s a n u p w a r d vertical c o m p o n e n t , due to the s t r o n g l a t e r a l flow (Fig. 5(d)i), w i t h a m a x i m u m v e l o c i t y of 1.2 m s -1, T h e m a x i m u m h o r i z o n t a l v e l o c i t y is 2 m s -x a c r o s s t h e top of t h e c a n y o n a n d in t h e middle of t h e c a n y o n a t a h e i g h t of approxim a t e l y 5 m.
H/W ratio of 0.2 (anticipated flow, isolated roughness) A l t h o u g h Fig. 5(e) is still c h a r a c t e r i s t i c of w a k e i n t e r f e r e n c e flow, t h e r e is e v i d e n c e of a c h a n g e to i s o l a t e d r o u g h n e s s flow w i t h the i n c r e a s e d d e p t h of f o r w a r d h o r i z o n t a l flow in the d o w n w i n d h a l f of the c a n y o n .
H/W ratio of 0.14 (anticipated flow, isolated roughness) I n v e s t i g a t i o n of a r e t u r n to a l o g a r i t h m i c profile d o w n w i n d of the s e c o n d b u i l d i n g suggests t h a t a r e t u r n to a f o r w a r d h o r i z o n t a l flow does n o t o c c u r for a p p r o x i m a t e l y 5 H d o w n w i n d . F i g u r e 5(f) s h o w s t h a t t h e downw i n d b u i l d i n g influences the flow for approxim a t e l y 20 m into the c a n y o n : t h u s a c a n y o n w o u l d n e e d to be o v e r 120 m wide (i.e., h a v e a H/W r a t i o less t h a n 0.17) in o r d e r to r e e s t a b lish a l o g a r i t h m i c v e l o c i t y profile. In t h e p r e s e n t case, t h e r e t u r n to a l o g a r i t h m i c profile is evident, a l t h o u g h r e v e r s e flow still o c c u r s a l o n g t h e c a n y o n floor. T h e m a x i m u m w i n d s p e e d is 3 m s -1 a l o n g the top of the c a n y o n , a r e d u c t i o n of 39% of t h e initial w i n d s p e e d a t 19 m.
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Long canyon ( L / H = 7) The g e o m e t r y r e p r e s e n t e d by a L / H r a t i o of 7 is a building h e i g h t of 20 m and a l e n g t h of 140 m. C a n y o n widths are given in Table 1. H~ W ratio of 1 (anticipated flow, skimming) F i g u r e 6(a) shows a v o r t e x c i r c u l a t i o n c h a r a c t e r i s t i c of skimming flow. The c e n t r e of the v o r t e x is 0 . 5 H and 0.6 W. A m a x i m u m wind speed of 2.6 m s -1 o c c u r s across the top of the c a n y o n with r e l a t i v e l y high wind speeds down the face of the d o w n w i n d building. F i g u r e 6(a)i shows the lateral flow a r o u n d the building, but with a c a n y o n of this length, the lateral c o m p o n e n t in the c e n t r e of the building is effectively zero. H / W ratio of 0.5 (anticipated flow, wake interference) F i g u r e 6(b) shows a v o r t e x c e n t r e d at 0.45 H and 0.75 W. In the lee c o r n e r of the c a n y o n the flow has c h a n g e d d i r e c t i o n and m a y i n d i c a t e a t r a n s i t i o n to w a k e i n t e r f e r e n c e flow. Maximum wind speeds o c c u r across the top of the c a n y o n and r e l a t i v e l y high wind speeds o c c u r down the face of the d o w n w i n d building.
H~ W ratio of 0.27 (anticipated flow, transition to isolated roughness) F i g u r e 6(c) shows t h a t the flow is now typical of w a k e interference. S t r o n g flows are e v i d e n t across the top and along the lower p a r t of the canyon. The m a x i m u m wind speed is 2.9 m s -1. H / W ratio of 0.2 (anticipated flow, isolated roughness) F i g u r e 6(d) shows t h a t the flow is still typical of w a k e interference, a l t h o u g h t h e r e is some evidence of a t r a n s i t i o n to isolated r o u g h n e s s flow. The maximum wind speed w i t h i n the c a n y o n has i n c r e a s e d to 3.1 m s -1 across the top of the canyon. H / W ratio of 0.14 (anticipated flow, isolated roughness) F i g u r e 6(e) shows isolated r o u g h n e s s flow with a r e t u r n to a f o r w a r d h o r i z o n t a l flow at a p p r o x i m a t e l y 0.84 W. Again, the flow along the floor of the c a n y o n is still reverse. The m a x i m u m wind speed across the top of the c a n y o n is 3.3 m s -1 and an a r e a of r e l a t i v e l y high wind speed exists in the lee eddy.
323 0.17
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The results of the threshold value for the change from skimming to wake interference flow using the k-~ model verify the results presented by Oke [7]. The results from the model also confirm the observation of Hussain and Lee [4] t h a t the length of the building has little effect on this threshold value. For the change from wake interference to isolated roughness flow, the difference between the results of Oke [7] and those from the k-~ model is most pronounced. The methods agree for cubic geometry only. Threshold values calculated by the k - s model are significantly greater, as shown in Fig. 7(b). The threshold value appears to change slightly at a H/W ratio of 0.19 for L/H ratios greater than 5, whereas Oke suggests a H/W ratio of 0.32.
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(b) Fig. 7. Threshold H / W values for t r a n s i t i o n from (a) s k i m m i n g to wake interferences a n d (b) wake interference to isolated r o u g h n e s s flow in u r b a n canyons.
Summary Threshold values for the change of flow regimes have been evaluated in detail above and are shown in summary form in Fig. 7 along with the threshold values presented by Oke [7]. Threshold values estimated from the k - e model of turbulence are similar to those of Oke [7], which are based on those of Hussain and Lee [4] for skimming to wake interference flow (Fig. 7(a)), but differ somewhat for wake interference to isolated roughness flow (Fig. 7(b)). For a cubic geometry, and a threshold value for the transition from skimming to wake interference flow, Oke suggests a H/W ratio of 0.71 and the k-~ model a H/Wratio of 0.74. For the change to isolated roughness flow, both the results presented by Oke [7] and those of the k-~ model suggest a H/W threshold of 0.6. Thus the results using the k-~ model for a cubic geometry compare favourably with the results of wind tunnel studies.
The results of the modelling exercises described present some interesting preliminary findings regarding the relationship between canyon geometry and prevailing air flow regime. In the range of canyon L/H ratios where observations are available, the k-~ model produces similar H/W thresholds for changes in flow regime. For greater L/H ratios where there is a lack of data, the model suggests that thresholds occur at smaller H~W ratios than those estimated by Oke [7]. It should be stressed that the findings are tentative and require validation by field observation. Nevertheless, it is clear that the k-~ model is capable of simulating the kind of flow regimes which have been observed in both field and wind tunnel studies. Furthermore, and subject to verification, the model offers the sort of flexibility than enables flow patterns in urban canyons to be investigated in a way that has hitherto been beyond the scope of field measurement programmes.
REFERENCES 1 Y. N a k a m u r a and T. R. Oke, Wind, t e m p e r a t u r e and stability conditions in an e a s t - w e s t oriented u r b a n canyon, Atmos. Environ., 12 (1988) 2691 - 2700. 2 W. B. J o h n s o n , F. L. Ludwig, W. F. Dabberdt and R. J. Allen, An u r b a n diffusion simulation model for carbon monoxide, J. A i r Pollution Control Assoc., 27 (1973) 490 - 498.
324 3 F. T. DePaul and C. M. Sheih, Measurements of wind velocities in a street canyon, Atmos. Environ., 20 (1986) 455- 459. 4 M. Hussain and B. E. Lee, An investigation of wind forces on three-dimensional roughness elements in a simulated atmospheric boundary layer flow. Part II. Flow over large arrays of identical roughness elements and the effect of frontal and side aspect ratio variations, Department of Building Sciences, University of Sheffield, Report No. BS 56, 1980. 5 N. J. Cook, On simulating the lower third of the urban adiabatic boundary layer in a wind tunnel, Atmos. Environ., 7 (1973) 691 - 705.
6 T. R. Oke, Boundary Layer Climates, Methuen, New York, 2nd edn., 1988. 7 T. R. Oke, Street design and urban canopy layer climate, Energy Build., 11 (1988) 103- 113. 8 S. E. Nicholson, A pollution model for street-level air, Atmos. Environ., 9 (1975) 19.31. 9 B. E. Launder and D. B. Spalding, The numerical computation of turbulent flows, Comp. Meth. Appl. Mech. Eng., 3 (1974) 269- 289. 10 D. A. Paterson, Computation of wind flows over three dimensional buildings, Thesis, University of Queensland (submitted in partial fulfillment of the requirement of Doctor of Philosophy), 1986.