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ARTICLE IN PRESS
Atmospheric Environment 41 (2007) 5848–5862
www.elsevier.com/locate/atmosenv
A simple network approach to modelling dispersion among large
groups of obstacles
David Hamlyna,, Trevor Hildermanb, Rex Brittera
a
Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK
Coanda Research and Development Corp., 110A-3430 Brighton Ave, Burnaby, British Columbia, Canada V5A 3H4
b
Received 5 September 2006; received in revised form 15 March 2007; accepted 19 March 2007
Abstract
A simple network approach has been developed to simulate the movement of pollutant within urban areas. The model
uses estimates of pollutant exchange obtained from velocity measurements in experiments with various regular obstacle
arrays. The transfer of tracer material was modelled using concepts of advection along streets, well-mixed flow properties
within street segments and exchange velocities (akin to aerodynamic conductances) across side and top facets of the street
segments.
The results predicted both the centreline concentration and lateral dispersion of the tracer with reasonable accuracy for a
range of packing densities and wind directions. The basic model’s concentration predictions were accurate to better than a
factor of two in all cases for the region from two obstacle rows behind a source located within the array to around eight
rows behind, a range of distances that falls into the so-called ‘‘neighbourhood-scale’’ for dispersion problems. The results
supported the use of parameterized rates of exchange between regions of flow as being useful for fast, approximate
dispersion modelling. It was thought that the effects of re-entrainment of tracer back into the canopy were of significance,
but modelling designed to incorporate these effects did not lead to general improvements to the modelling for these steadystate source experiments.
The model’s limitations were also investigated. Chief amongst these was that it worked poorly among tall buildings
where the well-mixed assumption within street segments was inadequate.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Urban canopy; Box model; Exchange velocity; Re-entrainment
1. Introduction
This paper describes the development of a simple
network approach for modelling dispersion at the
neighbourhood scale (i.e. several streets downstream of a source). This model, while remaining
Corresponding author. Tel.: +44 1223 332869.
E-mail address: [email protected] (D. Hamlyn).
URL: http://www.coanda.ca (T. Hilderman).
1352-2310/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.atmosenv.2007.03.047
fast to run, aims to take account of the important
advective and diffusive flow processes within the
obstacle arrays, rather than representing the obstacles simply as a spatially averaged roughness.
The model might be particularly attractive for
assessing pollutant spread from hotspots such as
busy intersections to surrounding parts of a city, or
to assess potential danger areas after an accidental
or intentional release of hazardous material. The
model has been developed and tested against
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D. Hamlyn et al. / Atmospheric Environment 41 (2007) 5848–5862
a range of experiments using regular obstacle
arrays, which provided basic flow data for model
development and dispersion data for model testing.
Flow among arrays of cubes has been examined
in experiments (e.g. Davidson et al., 1995, 1996;
Mavroidis and Griffiths, 2001). These studies
commented on the relevance for dispersion of
obstacle wakes and longitudinal channelling along
streets. Computational studies by Kim and Baik
(2004) and Hamlyn and Britter (2005) have also
examined the form of the flow structures, finding a
range of large recirculatory structures present in the
gaps between obstacles, with the form of these
structures dependent on the wind direction and
obstacle array packing density.
The model concept is that a series of short streets
form a network of boxes that comprise the canopy
space. Tracer material is advected along the streets
and is assumed to be well mixed within each box, or
‘‘cell’’ due to vigorous mixing from recirculatory
structures and locally generated turbulence around
the obstacles. The transfers of tracer between cells,
and transfers between the canopy and the air above
are governed by the magnitude of the resistance to
transfer (via advection or turbulent diffusion)
between the cells, or between the canopy and the
air above. The model is designed to capture the
important flow physics in a simple, fast-to-run
approach applicable to the neighbourhood scale
and requiring only very limited inputs regarding
flow velocities. Techniques and references are
suggested for estimation of these inputs. The
network model approach reflects that taken by
Soulhac’s model SIRANE (2000) but the methods
of parameterizing exchanges differ, as does our
more simplified approach of assuming local incanopy vertical homogeneity.
Computational fluid dynamics (CFD) simulations by Hamlyn and Britter (2005) focused on
quantifying the exchanges within simple obstacle
arrays. The concept of ‘‘exchange velocity’’, uE was
described by Bentham and Britter (2003) and in
essence is a characteristic velocity relating the
momentum flux across a plane to spatially averaged
reference velocities either side of the plane. The
fluxes that contribute to momentum exchange
across the canopy top are turbulent and advective
fluxes, the latter often referred to as dispersive
stresses. For pollutant exchange, the same exchange velocity concept may be applied to quantify
the exchange of pollutant mass across the canopy
top.
5849
The introduction of an exchange velocity has
come from the realization that parameterizations
like surface roughness length z0 and displacement
height d are of limited usefulness when dealing with
flow near and within an urban canopy. They only
have meaning in the context of boundary layer flow
over a very long, statistically homogeneous fetch.
The exchange velocity directly addresses the aspect
of the flow that is of direct importance in the urban
environment, the ventilation of the urban canopy.
Thus, it could be interpreted as the dominant link in
any network model of the urban canopy. It should
not be confused with various mass, momentum or
heat conductances that connect surface variables
with variables within the canopy or above it. The
exchange velocity is not a particularly novel concept
(it is similar to the concept of aerodynamic
conductances e.g. see Monteith and Unsworth,
1990) but, for its importance, it has been very little
studied or parameterized in the urban context when
compared with the efforts on z0 , d and conductances
between surfaces and the air.
The exchange velocity is probably best parameterized in terms of the friction velocity u and
Bentham and Britter (2003) developed a simple
analysis to connect the two. However u is not an
easily determinable parameter in the urban context.
Unrealistically (see Cheng and Castro, 2002a) long
fetches that have statistical homogeneity are required if the velocity profile is to be used to
determine u , z0 and d. Additionally, the considerable sensitivity of the determined u , d and z0 to the
velocity profile is reflected in the relative insensitivity of the wind speed at a particular height to a
(consistent) set of estimates for u , d and z0 . The
determination of the friction velocity from measurements of the Reynolds stress is also quite problematic in an urban context from measurements
within or above the roughness sub-layer for a
variety of reasons. Consequently, the wind speed
at a particular height, measured, calculated or
estimated, can be a preferable reference velocity
to the more fundamentally correct variable, the
friction velocity, u . For this reason, we choose
a reference velocity towards the top of the roughness sub-layer, at 2.5 times the average obstacle
height, H.
It might be considered that in obstacle arrays,
knowledge of the canopy-top exchange velocities
uEtop , and a mean in-canopy exchange velocity
between sheltered and unsheltered regions of flow
uEside could be sufficient to produce a simple model
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D. Hamlyn et al. / Atmospheric Environment 41 (2007) 5848–5862
for urban dispersion. For wind directions where
there are not distinct sheltered regions, a spatially
averaged velocity can be used to characterize
advection in that street.
The well-mixed assumption within cells is a major
assumption of this work. Flow visualization studies
suggested that it is a good starting point for a simple
model of this kind, and may be adequate in the
context of fast, approximate operational modelling.
This is particularly so for application to real cities
where the wind directions vary and will rarely be
aligned with streets containing a source; which
would be expected to be the worst case for
applicability of well-mixed assumptions. Care
should however be taken when considering low
plan area density development where streets may
be wide.
The model simulates dispersion from a point
source within a canopy, though this could easily be
extended to line or area sources. It is tested here for
regular cube arrays, using the water channel work of
Hilderman and Chong (2004a, b) (Coanda R&D
Corporation for Eugene Yee, Defence Research and
Development Canada—Suffield, hereafter referred
to as the DRDC-S data set) and data from
Macdonald et al. (2000, 2001) and Hall et al.
(1998) as additional test cases. Estimates of the
exchange velocities used in the model were made
from separate Laser Doppler Velocimetry (LDV)
measurements carried out in the Coanda water
channel. The cube arrays considered are described
using the parameters ‘‘plan area density’’, lp and
‘‘frontal area density’’, lf . These represent the total
built area divided by total lot area and the total
frontal area of buildings divided by total lot area,
respectively. For cube arrays, these are identical,
and so will be referred to simply as l, or packing
density.
2. The model
2.1. Model concept and structure
2.1.1. Network structure and in-canopy flow
Fig. 1a shows an aerial view of some real urban
geometry at the neighbourhood scale (Marylebone
Rd., London, UK) and Fig. 1b shows part of the
geometry within a regular obstacle array similar to
the test cases, indicating the similarities and
differences between regular obstacle arrays and real
urban form. Fig. 1c shows schematically the
exchange velocity applied to a single intersection
region that forms an element of the model’s network
structure displayed in Fig. 1b.
The dashed lines in Fig. 1b surround network
‘‘cells’’ (of three types, A, B and C) which enclose
the fluid volume shown up to the canopy top. The
fluid velocity is the only detail of the flow considered
within a cell. In the case of a 0 wind direction, the
in-canopy averaged velocity uCb in type B and C
cells below is a user input and the velocity uCa in
type A cells is taken as zero. In cases where the wind
direction is not 0 , the user can input in-canopy
velocities uCa and uCb along the two street directions. Alternatively, for angled cases, the user can
specify a velocity appropriate for the 0 case. If this
option is chosen, the model estimates along-street
velocities by multiplying this by cosðyÞ where y is the
direction of each street relative to the wind, an
approach used by Soulhac (2000).
2.1.2. Pollutant sources and pollutant exchanges
The steady-state source considered in this study
was of strength q ðkg s1 Þ located in a cell at the
front and lateral centre of the array. It is modelled
as an additive term in this cell’s concentration
balance such that at time step n:
X
C sourcecell;n :V ¼ qn dt þ C sourcecell;n1 :V fi;n dt.
i
(1)
Here, V is the source cell volume and fi represents
the pollutant mass fluxes ðkg s1 Þ out of the cell
through each cell facet, i. While the model is
demonstrated here with single steady sources, its
data storage methods allow for multiple and/or
unsteady sources.
To characterize the pollutant exchanges between
cells and across the canopy top, exchange velocities
are input by the user. In all cell types, the exchange
between the cell and the air above is as follows
(written here for the cell with position index x ¼ p,
y ¼ q). Here, the net pollutant mass leaving the
cell through the canopy top (facet 1) in a time step
dt is f1 , where
f1 ¼ ðCðp; qÞ C ext ðp; qÞÞ:A1 :uE dt.
(2)
C ext ðp; qÞ represents the concentration of the air above
the cell at z ¼ H and is zero unless re-entrainment
modelling is included (see Section 2.1.3). Cðp; qÞ is the
cell’s concentration and A1 represents the area for
exchange at the cell top. The concentrations used are
those at the previous time step. A small time step is
needed to maintain accuracy. In the model runs
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D. Hamlyn et al. / Atmospheric Environment 41 (2007) 5848–5862
5851
Fig. 1. (a) The real level of urban complexity near Marylebone Road, London. (b) The model concept for a simple urban-like array,
showing the role of the exchange velocity. (c) The concept of exchange velocity applied to an intersection region.
carried out, dt ¼ 0:1 s was used for the Macdonald
et al. and Hall et al. test cases and dt ¼ 0:05 s for
DRDC-S test cases.
The methods of calculating mass exchange across
interfaces between cells (within the canopy) are
explained below for a type C cell, as this type of cell
is involved in all in-canopy exchanges.
In this cell, for the 0 wind direction case, the
exchanges occur at the ends of the street to the type
A cells with exchange velocity uEside . The exchanges
into the cell from the upstream type B cell and out
of the cell to the downstream type B cell are
assumed to be purely advective in nature, involving
the upstream concentration and the appropriate instreet velocity. For example, the fluxes through
facets 2 and 4 are
f2 ¼ ðCðp; qÞ Cðp; q þ 1ÞÞ:A2 :uEside dt,
(3)
f4 ¼ ðCðp 1; qÞÞ:A4 :uCb dt.
(4)
For angled cases, all of the exchanges are assumed
to be advective rather than diffusive in nature. So,
for example, the flux through facet 2 becomes
f2 ¼ ðCðp; q þ 1ÞÞ:A2 :uCa dt.
(5)
There also exists a ‘‘type D’’ cell, i.e. the volume
within the building. The concentration here is
initially zero and remains zero as no tracer
exchanges into it are permitted. However, given
information about pollutant infiltration rates into
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D. Hamlyn et al. / Atmospheric Environment 41 (2007) 5848–5862
buildings, the concentrations inside buildings might
also be modelled.
2.1.3. The effects of re-entrainment
Re-entrainment effects are transfers of pollutant
from the air above the canopy into the canopy
below. Simple additional modelling was created to
account for re-entrainment, involving an extra
concentration variable C ext . This represents the
concentration of the air above the canopy at the
canopy-top height and is calculated in a similar
manner to that in SIRANE (Soulhac, 2000),
although each cell is discretized as a single point
source rather than a line source for faster computation. The use of the exchange velocity concept
allows for a very simple method of finding the net
flux through the canopy top. Thus, at each time and
each cell (written here at time i for cell p; q) this
source strength is
qi ðp; qÞ ¼ ðC i ðp; qÞ C ext;i ðp; qÞÞ:uE :A1 .
(6)
The contribution of this source (located at point a,
say) to the external concentration at point b is the
reflected (at the canopy-top) Gaussian profile:
2
2
C ext;ab ¼ qi ðp; qÞ=ðp:uH :sy :sz Þ:eðyab =2:sy Þ ,
(7)
where uH is a user input representing the velocity of
the air above the canopy close to the canopy top,
and xab and yab are the longitudinal and lateral
distances between a and b. Superposition is used to
estimate the resultant downstream roof level concentrations.
To make Eq. (7) suitable for real geometries and
wind tunnel models to both be simulated by the
model, the non-linear terms were removed from the
Briggs (1973) expressions used for the dispersion
parameters, giving sy ¼ 0:16:xab and sz ¼ 0:14:xab .
For an average geometrical repeat unit (or city
block spacing) of 40 m, this would give less than
20% error in s values for the largest source–receptor distances considered. This is acceptable in the
context of other approximations made.
2.1.4. Transient behaviour
The concentration field takes time to develop
within the array due to the need for pollutant to
travel downstream within the initially ‘‘clean’’
domain. The time needed to reach a steady-state
concentration field was examined. It was found that
20 cells directly downstream of the steady-state
source, around 400 time steps, or 20 s was needed to
observe a steady-state in-canopy concentration (for
the l ¼ 0:25 cube array at 0 to the wind). Scaling
for a real building height of 20 m and a wind speed
of 5 m s1 , this would represent a timescale of the
order of 10 min in the full-size geometry. This is
around five times longer than the advection time at
velocity uC would be in the absence of hold-up of
pollutant in recirculation regions.
3. Test cases, results and discussion
As a source of flow data to develop the model and
dispersion data to test the model, laboratory
experiments provide the fullest and most reliable
data sets. Several sets of experiments on cube arrays
exist, encompassing different packing densities and
flow directions relative to the array. The geometries
investigated in such studies also crudely replicate
some of the geometric features of urban areas
(in terms of highly three-dimensional structure,
possible grid layout and appropriate plan area
densities). As such, they provide a consistent set of
data for assessment of how the model performs
under various conditions of geometry and flow
angle. The approach taken was to use experimental
velocity data to provide velocity input (in a highly
parameterized format) to the dispersion model. The
dispersion model was then tested against experimental data from dispersion experiments.
3.1. Test cases
The range of test cases considered for dispersion
modelling is detailed in Table 1, which also includes
some cases used later to investigate the assumptions
of the model. The DRDC-S geometries of particular
interest were an in-line cube array with l ¼ 0:25
(case URB001), the angled l ¼ 0:25 cube arrays at
30 and 45 (URB031 and URB037), and the in-line
cube array with obstacle height doubled (URB003).
Also considered were two other cube packing
densities, l ¼ 0:16 and l ¼ 0:44 from Macdonald
et al. (2000, 2001) and Hall et al. (1998).
We take as our datum case the in-line l ¼ 0:25
array with the source located within a sheltered
region behind an obstacle. This case is referred to
using DRDC-S data set notation as URB001-D, the
‘‘D’’ representing a source in the centre of a
sheltered region, rather than ‘‘C’’, representing a
source in the centre of a region of channelled flow,
as illustrated in Fig. 2. By default, the source was
located in row 1 of the array, although case
URB001-8D was also examined, where the source
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Table 1
Details of obstacle array configurations, source and measurement positions
Array ID data
source and
lp
Layout
Wind angle
(deg)
Other array
characteristics
Source location
Source height
Cell
concentration
measurement
height
DRDC-S
URB001C
DRDC-S
URB001D
DRDC-S
URB001-8D
DRDC-S
URB002C
DRDC-S
URB002D
DRDC-S
URB003D
DRDC-S
URB010D
0.25
Square
0
Cubes
Row 1
Ground level
z=H ¼ 0:5
0.25
Square
0
Cubes
Row 1
Ground level
N/A
0.25
Square
0
Cubes
Row 8
Ground level
z=H ¼ 0:5
0.25
Staggered
0
Cubes
Row 1
Ground level
N/A
0.25
Staggered
0
Cubes
Row 1
Ground level
N/A
0.25
Square
0
2H obstacles
Row 1
Ground level
z=H ¼ 1:0
0.25
Square
0
Row 1
Ground level
N/A
DRDC-S
URB031D
0.25
Square
30
Alternating rows of
2H and 3H
obstacles
Cubes
Ground level
z=H ¼ 0:5
DRDC-S
URB037C
0.25
Square
45
Cubes
Ground level
z=H ¼ 0:25
Macdonald et al.
0.16
Square
0
Cubes
Approx. 6H
downwind into
array
Approx. 1
downwind repeat
unit into array
0:5H Ahead of first
obstacle
z=H ¼ 0:5
z=H ¼ 0:4
Case 1
Macdonald et al.
0.16
Square
0
Cubes
Row 3, 0:5H ahead
of obstacle
z=H ¼ 0:5
z=H ¼ 0:4
Case 2
Hall et al. Case 1
Hall et al. Case 2
0.16
0.16
Square
Square
0
0
Cubes
Cubes
Ground level
Ground level
Ground level
Ground level
Hall et al. Case 3
Hall et al. Case 4
0.44
0.44
Square
Square
0
0
Cubes
Cubes
Within array
Ahead of first row
of array
Within array
Ahead of first row
of array
Ground level
Ground level
Ground level
Ground level
was located in a sheltered region 8 rows within the
array.
The model requires as inputs in-canopy velocities
and exchange velocities. These results are shown in
Tables 2–4, and were measured from experiments,
or derived from existing experimental data where
possible.
Starting with the exchange velocities at the
canyon top (Table 2), results were estimated from
measurements for all the DRDC-S test cases, except
for the 30 flow case, where measurements did not
permit this. Thus, for this case the model was run
with three exchange velocities; these were the
exchange velocity for the same geometry with 0
and 45 wind directions, and their arithmetic mean.
For comparison with the experiments of Macdonald
et al. and Hall et al. (cube arrays of different
packing densities at 0 to the flow direction), the
canopy-top exchange velocities were estimated from
measurements made in the Coanda water channel
using LDV within arrays of the appropriate packing
densities.
Considering the exchange velocities at the interface between sheltered regions and channelled flow
for 0 flow cases (Table 3), results for datum in-line
l ¼ 0:25 geometry and the double-height case were
derived from DRDC-S data set measurements,
although estimates for the spatially averaged uEside
values were based on data from three vertical
profiles only, and thus may lack good streamwise
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5854
Fig. 2. Array layout (part of a 16 obstacle row by 16 obstacle column array) and obstacle shape for the URBXXX test cases. Terminology
for source location (C or D) shown; shading indicates source location for this test case.
Table 2
Exchange velocities at canopy top used as inputs to the model
Exchange
velocity
uE (% uref )
URB001
URB003
URB037
l ¼ 0:16
l ¼ 0:44
Turbulent
Advective
Total
1.3
1.0
2.3
0.7
0.1
0.9
1.3
0.4
1.7
1.3
0.6
1.8
1.2
0.4
1.6
Table 3
Exchange velocities between sheltered regions and channelled
regions used as inputs to the model for cases of flow direction
aligned with the array
Exchange velocity
uEside (% uref )
URB001
URB003
Turbulent
Advective
Total
2.7
3.6
6.3
2.4
0.3
2.7
resolution. In the absence of other measurements,
the datum in-line cube array (uEside ) estimate was
used for other packing densities at 0 to the wind.
In-canopy and reference velocities are shown in
Table 4. In these cases the values of uC used in the
uE calculation were estimated from the vertical
average from z ¼ 0 to H from two profiles, at the
canyon centre and at the channel centre in the
intersection. While the velocities here were measured values for the purposes of evaluating the
model concept, it would be possible instead to
estimate in-canopy velocities using an approach
such as that suggested by Bentham and Britter
(2003) or via Macdonald’s (2000) exponential
profile approach.
3.2. Results and discussion
For the DRDC-S arrays, experimental results and
model predictions of cross-sectional concentration
profiles at various distances x=H downstream of the
source were compared. However for the doubleheight, Macdonald et al. and Hall et al. experiments, only centreline concentration measurements
were available. The comparisons for the DRDC-S
data set cases are shown in Figs. 3–5, with
comparisons for Macdonald et al. and Hall et al.
test cases shown in Figs. 6 and 7. Where experimental concentration measurements were carried
out at various heights within the canopy, the results
from the closest measurement location to the
canopy mid-height are shown here.
In these figures, the results from two model
variants are shown. One included the re-entrainment modelling described, while the other did not
model re-entrainment. Thus, it could be tested
whether or not this (computationally intensive)
aspect of the modelling gave significant improvements to the model predictions.
3.2.1. Centreline concentrations
For the DRDC-S arrays, centreline concentrations are well predicted in most of the cases and
downstream distances considered. Fig. 3 shows the
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5855
Table 4
Reference velocities used in the model
Reference velocities ðm s1 Þ
URB001
URB003
URB031
URB037
l ¼ 0:16
l ¼ 0:44
uC
uCa
uCb
uref
uH
0.185
N/A
N/A
0.326
0.160
0.110
N/A
N/A
0.376
0.172
N/A
0.072
0.042
0.313
0.159
N/A
0.062
0.062
0.297
0.126
0.0294
N/A
N/A
0.0636
0.0310
0.0152
N/A
N/A
0.0636
0.0126
uH is only used for purposes of re-entrainment modelling.
Fig. 3. Comparison of model and experimental lateral concentration profiles at various downstream distances x=H for case URB001 with
source located in row 8 of the array.
results for the datum, in-line geometry with the
source well inside the array (URB001-8D). Apart
from at one row downstream ðx=H ¼ 2Þ, where
concentrations are underestimated by just under a
factor of two in the model with re-entrainment
effects, the centreline concentrations are predicted
with reasonable accuracy up to eight rows downstream. With the model at 45 to the flow, the
maximum error in predicted concentrations (Fig. 4)
is acceptable for fast, approximate modelling, being
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D. Hamlyn et al. / Atmospheric Environment 41 (2007) 5848–5862
double-height building case (Fig. 5). The specific
(and operationally rare) case of wind direction
aligned completely with a regular grid layout of
obstacles exacerbates these difficulties. These issues
will be discussed further in Section 4.5. However,
even in this case discrepancies between the prediction and the measured concentration (at z ¼ H)
may be reasonable estimates from an operational
perspective, with centreline concentration errors
from around 15% (with re-entrainment modelling)
at x=H ¼ 2 to around 55% at x=H ¼ 16. Even at 13
rows downstream, the worst model prediction is an
overestimate by a factor of 2.3.
Examining the l ¼ 0:16 (Fig. 6) and l ¼ 0:44
(Fig. 7) arrays, the model with re-entrainment gives
an overall good match with centreline concentrations for the Macdonald et al. and Hall et al. test
cases. The accuracy varies between cases, although
some significant scatter in experimental results
should also be noted.
For the Hall et al. l ¼ 0:16 array, the predictions
are accurate to within about 25% (suggesting that
this geometry is sufficiently dense to allow wellmixed assumptions), except at one repeat unit
downstream of the source, where concentrations
are underestimated severely, as for the datum
geometry. The worse predictive performance at this
point than in the Macdonald et al. test case may be
as a result of different source vertical positions in
the two cases, resulting in differences in how well
mixed the tracer becomes within the cells near the
source. The model predictions for the l ¼ 0:44 array
case were less accurate overall. Yet even here, at the
largest
measurement
distance
downstream
ðx=H ¼ 21Þ, the model variants with and without
re-entrainment effects resulted in, respectively, overand under-estimates by factors of approximately 2.3
and 1.5. These may often be adequate in the context
of operational modelling.
Fig. 4. Comparison of model and experimental lateral concentrations at various distances x=H along the 0 direction aligned
with the obstacle grid for case URB037 (array at 45 to wind
direction).
only around 45% (with the model including
re-entrainment effects) at the furthest downstream
distance compared.
The difficulty encountered one row downstream
in the datum geometry can be attributed to issues of
how well mixed the tracer is within a cell, and the
same issue results in the relatively large errors in the
3.2.2. Lateral dispersion
For the in-line datum geometry (Fig. 3), predictions of lateral dispersion were generally good
throughout the domain. For the cases with the
flow at an angle to the array, inflections in the
profiles around street intersections were well captured by the model and overall predictions were
reasonable, but some discrepancies were noted
between model predictions and experimental
results. These will be discussed with reference to
the case with 45 flow angle relative to the array
(Fig. 4).
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Fig. 5. Comparison of model and experimental lateral concentrations at various downstream distances x=H for case URB003.
Firstly, the model over-estimated the lateral
spread in the positive y direction, beginning at short
x=H distances, although the predictions become
closer to the experimental measurements with
increasing distance downstream. Secondly, in the
negative y direction, the model without re-entrainment predicts zero concentrations. This is only
incorrect close to the source where lateral dispersion
occurs over one row or so. The model with reentrainment actually predicts these concentrations
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Fig. 6. Comparison of model and experimental centreline concentrations at various downstream distances in l ¼ 0:16 cube arrays.
4. Further discussion
Aside from the more straightforward discussion
of the model results, there are other, more subtle
issues raised by the work carried out. These, and the
major assumptions of the model are discussed in
this section.
4.1. Sensitivity to source location
Fig. 7. Comparison of model and experimental centreline
concentrations at various downstream distances in a l ¼ 0:44
cube array.
rather well close to the source though possibly for
the wrong reasons, as some of the real transport to
y=H ¼ 1 was due to gusting within the canopy
rather than re-entrainment via the canopy top.
Further downstream, the lateral spread in the
negative y-direction is overestimated increasingly
severely by the model with re-entrainment, indicating potential errors in the re-entrainment modelling
there.
When comparing model results with experiments
(Figs. 3–7), the most appropriate comparisons were
with experiments with the source well within the
array. Macdonald et al. (2000) suggested that the
mean flow and turbulence fields within the canopy
reached a near equilibrium after three obstacle rows
and Cheng and Castro (2002b) suggested approximate similarity by the fifth row of a l ¼ 0:25 cube
array. For sources closer to the front of the array,
the canopy-top exchanges near the source would be
enhanced by increased advective and turbulent
fluxes, due to the flow taking a few rows to adapt
to the canopy’s presence. Thus, increased tracer
losses through the canopy top would be expected
and would be reflected in lower experimentally
recorded concentrations. However, the differences
are not very large, so it was felt acceptable to make
comparisons when the source was in row 1.
Results for a source upstream of the array have
been included (Figs. 6 and 7) to indicate the (often
significant) sensitivity of the results to source
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location. Experiments where this source location
has been used are less representative of real
urban dispersion problems where the source is
usually located in a street well within a developed
area.
4.2. Re-entrainment modelling
In most of the arrays studied, using the reentrainment modelling led to some improvements to
the predicted centreline concentrations near to the
source, but increased the overestimation of concentrations further downstream. The re-entrainment
modelling also has some effect at smoothing out the
sudden concentration drop between the first and
second street rows and the slower decay thereafter.
This provides evidence for the physical relevance of
these effects to urban dispersion problems, although
their predicted magnitude appears inaccurately
modelled.
For the angled cases, such as in Fig. 4, the lateral
spread prediction also becomes weaker further
downstream if re-entrainment modelling is included.
These differences suggest that the re-entrainment
has been over-estimated, at least at x=HX4,
reinforcing the above conclusions.
Based on these observations, the additional
computational time and model complexity introduced by the re-entrainment modelling included
here does not appear to lead to significant consistent
improvements for these steady-state experiments.
However, it is anticipated that re-entrainment
modelling will be of consequence when dealing with
transient scenarios, particularly when the source has
stopped.
4.3. Model limits of applicability
The model was not, nor was expected to be very
accurate close to the source within a few obstacle
heights where the well-mixed assumption is likely to
be invalid.
Using the data of Macdonald et al., some
investigation was made into how far downstream
the model is appropriate. At large distances downstream the concentrations outside and inside the
canopy will be similar. As a result the exchanges
calculated may contain large percentage errors if no
re-entrainment modelling is included. Even with reentrainment modelling, the accuracy will degrade
with distance downstream from the source, as errors
build up in the in-canopy concentrations.
5859
It is possible to estimate a rough downwind limit
of applicability. Macdonald et al. (2001) report zc ,
the elevation of the plume centreline from Hall
et al.’s (1998) results. For the l ¼ 0:16 cube array, zc
rose to 1.0 by x=H15 downstream. This might be
taken as an approximate limit of when the negative
concentration gradient qC=qz at z ¼ H exists until,
and hence an indication of the point by which the
model may have lost applicability. Beyond this
downstream distance, the approach outlined here
should be replaced by conventional dispersion
approaches. This estimate seems to be approximately borne out by the results for the DRDC-S
arrays.
4.4. Sensitivity to value of exchange velocity
Three canopy-top exchange velocities were investigated for the 30 angled flow case as explained
earlier. The closest overall fit to the experimental
data was for the intermediate value, although
differences between the results in each case were
found to be very small indeed. This suggests that
(given the overall range of exchange velocities is not
much higher than that of the three values used here)
it may be possible just to use a single value of uE for
all square arrays within the range of packing
densities considered. This would be very useful for
operational modelling purposes.
4.5. Known assumptions and limitations
4.5.1. Appropriateness of well-mixed assumptions
The model described here assumes that in-canopy
concentrations are essentially uniform throughout a
cell. Though this might be expected several rows
downstream, its validity is less clear near the source.
In particular, tracer might be well-mixed vertically
in some areas (e.g. sheltered regions behind buildings with recirculations) but not in others (e.g.
faster-moving flow in streets parallel to the abovecanopy wind direction).
To investigate the mixing effectiveness, the ratio
of maximum to minimum concentrations measured
along the vertical profile up to the canopy top was
calculated for several arrays from the DRDC-S data
set, checking the data manually for outlier data
points. In cases where the building height varied, the
canopy-top height was taken as the taller of the
surrounding buildings. It should be noted that many
of the cases examined here have one of the two
street directions aligned with the above-canopy
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wind and are probably the ‘‘worst cases’’ for spatial
inhomogeneity of concentrations as a result. For the
majority of the time, when no such alignment
occurs, effects such as helical flow (Hosker, 1987)
along streets will aid mixing.
For the arrays with a predominantly uniform
canopy height of 1H, Fig. 8 shows that the max/min
concentration ratio within these canopies is relatively small, being less than two even at just 2H
downstream of the source. Further downstream, the
situation becomes increasingly well mixed, suggesting that flow features in the sheltered regions
promote rapid vertical mixing. In contrast, sources
located in streets exactly aligned with the external
wind (a somewhat special case) would not show
such strong vertical mixing. This is because the
predominant flow is longitudinal channelling until
the plume has spread enough laterally to become
mixed vertically within sheltered regions and reejected back into the channelled regions. At x=H ¼ 2
the max/min concentration ratio is around 6 for
these cases, although it reaches a factor of 2 by
x=H ¼ 4.
For the double-height array and the case of
alternating rows of 2H and 3H obstacles
(URB010D), where the source is in a sheltered
region, the max/min concentration ratios are
around three to five at x=H ¼ 2, and only reach
two after more than 10H downstream. However,
while a vertical variation of concentrations by
around a factor of four might appear large, if this
vertical variation could be reasonably approximated
as linear, then the maximum error between any
concentration measurement made in the cell and the
Fig. 8. Max/Min concentration ratio within canopies of obstacles
with lp ¼ 0:25. URB001 and URB002: 1H tall obstacles in
square and staggered arrangements, respectively. URB011-014:
Square arrangement, as in URB001, but with some obstacles
replaced by obstacles of height 3H.
cell-average concentration would only be around a
factor of two. Thus simple modelling efforts
invoking well-mixed approximations might work
acceptably well in these cases, albeit with some
performance degradation.
In the same arrays of taller obstacles however,
when the source is located in the channelled flow
region, the max/min concentration ratio at x=H ¼ 2
may be as much as two orders of magnitude larger
than in the shorter canopies. This ratio decays only
slowly downstream, remaining well above 10 even at
x=H ¼ 6 for both arrays. Thus, applying well-mixed
assumptions close to sources located in channelled
flow among tall buildings is not appropriate. This
restriction may make the model concept more
applicable to dispersion in many European city
centres than in North American ones. In addition,
well-mixed assumptions will also become much
weaker at low building packing densities (although
these do not normally present the most problematic
cases for urban dispersion) due to the presence of
wide streets, and the fact that recirculating flow
features that contribute to mixing may occupy a
smaller proportion of the in-canopy volume if
obstacles are far apart.
4.5.2. Other assumptions and problems
(1) In the formulae to calculate the external
concentrations at the canopy top for re-entrainment
modelling, an artificial ‘‘ground-level’’ for the plume
was assumed, with reflection at z ¼ H. In fact there
will be absorption on this pseudo-surface into the
flow below. In addition, the unstable shear layer at
the canopy top may flap and cause pollutant
entrainment into the canopy from heights above
the canopy-top level.
(2) The well-mixed assumptions create some
deficiencies in modelling lateral spread. For example, in a 0 wind direction case, CFD results
(Hamlyn and Britter, 2005) suggested that in the
l ¼ 0:16 array, the flow emitted from a type A cell
enters a type C cell towards its downstream corner
and becomes better mixed within the channelled
flow as it proceeds downstream. In contrast, in the
modelling, the exchange into the type C cell
effectively spreads emitted pollutant instantly
throughout the whole type C cell at the next time
step. This should logically lead to unrealistically
high lateral dispersion, particularly near the source
where large lateral concentration gradients exist.
The extent of lateral mixing within cells cannot be
assessed in any detail from the DRDC-S data set.
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However, flow visualization studies in the same
geometries have shown that recirculatory structures
due to separation off the edges of obstacles leads to
rapid lateral mixing within cells. This will of course
have a limited effect on flow within ‘‘channelled
regions’’ when the wind is aligned with the array,
although this case is somewhat rare.
(3) Some errors will be introduced into the
calculation of the downstream advection of tracer
material, due to the neglecting of detailed information about the vertical velocity variation within
cells. In a similar vein, errors will be introduced into
our modelling from the limited resolution of
measurements from which the uEside parameter
was calculated, and from the use of the uEside value
from the l ¼ 0:25 case for other cases.
Despite the limitations outlined, the model
appears robust and could be developed into an
operational model relatively easily, albeit with some
uncertainties. More research (experimental, CFD,
or simply via utilizing already published data)
focusing on these uncertainties would help to
improve model accuracy. In particular, future
research on the effect of more realistic, inhomogeneous urban geometry on exchanges at the canopy
top would prove valuable.
5. Conclusions
The model created requires as inputs only very
limited knowledge of the flow through obstacle
arrays. This work’s aim was to use this simple
modelling to test the applicability of the exchange
velocity concept to flow within a grid-like network
of short streets. The model uses fairly coarse, but
physically justifiable assumptions to track a pollutant release downstream. While the model (with its
coarse resolution) did not handle particularly well
the dispersion around the flow ‘‘cell’’ containing the
source, applying exchange velocity concepts at all
subsequent cells managed to model the downstream
changes in lateral concentration profiles relatively
well at all distances (up to 13 obstacle rows
downstream) where data was available. There is
evidence to suggest that re-entrainment effects are
of some physical importance, but that the method
used here to model here tended to overestimate the
magnitude of these fluxes. For the cases considered,
reasonable accuracy could be obtained without the
inclusion of such re-entrainment modelling. The
incorporation of re-entrainment effects will be more
important when dealing with transient releases.
5861
In this type of regular array, these results suggest
that characteristic exchange velocities exist which
apply throughout the canopy (at least after the
initial flow adjustment in its first few rows).
These findings held both for comparisons of the
model with water flume experiments by Hilderman
and Chong (2004a, b) and Macdonald et al. (2001)
and wind tunnel experiments by Hall et al.
(1998) for flow around cuboidal obstacles. It was
suggested that operationally, reasonable model
results across a range of geometries might be
attained with simply a single canopy-top exchange
velocity value.
While more work might be done to improve the
modelling around the source cell, and to make this
model more applicable to inhomogeneous geometries, the approach used in these simple cases
seems effective, and shows that the exchange
velocity concept may have potential as a flow
parametrization method for operational dispersion
modelling.
Acknowledgements
Thanks to the UK EPSRC for project funding
and to Prof. David Wilson of the University of
Alberta for funding assistance and expertize for
the experiments carried out over summer 2005.
Thanks also to Dr. Darwin Kiel, Ricky Chong and
others at Coanda who provided assistance with the
experiments.
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