-------
2 -
2
Ŗ + m (a
+ m (a
if HI > a, z = 0 ;
1/2
where m
H r ( h} i
-Ŗ-+[[ n J + 1 J ,
h = height (depth) of valley, a = half-width of valley, and Ŗ = arbitrary
parameter. x is directed along the approach flow direction (origin at center
of valley), and z is directed vertically upward. Since the surface is two-
dimensional, the equations do not include the y variable, and the models
extended across the width of the test section of the wind tunnel. These
valley shapes are smooth, symmetric about the z-axis, and smoothly merge into
a flat plane at the points x = ą a. They describe a two-parameter family of
surfaces, the parameters being h and the aspect ratio n = a/h.
The valleys constructed for this study had aspect ratios of 3, 5, and 8
(maximum slopes of 26°, 16°, and 10°, respectively). All had heights (depths)
of 117 mm and, therefore, half-widths of a = 351 mm, 585 mm, and 936 mm.
Figure 1 shows the construction details of the three valleys and Fig. 2 shows
the valley shapes and compares the ideal shapes with those measured after they
were installed in the wind tunnel. The actual shapes differed from the ideal
ones by less than ą 5 mm. All the models were covered with gravel to match
the roughness of the wind-tunnel floor. The difference between the ideal and
actual shapes shown in Fig. 2 is due primarily to the uncertainty in measuring
to the top of the gravel roughness.
The leading edges of all the valleys were placed 8 m downwind of the
fence at the entrance to the test section.
2.3 Velocity measurements
Two basic types of instruments were used to measure the flat-terrain
boundary layer and the flow structure within the valleys, a hot-wire
anemometer (HWA) and a pulsed-wire anemometer (PWA). The HWA is a convenient
instrument to use where the flow is reasonably well-behaved, that is, where
the turbulence intensities are relatively low (less than, say, 20%), or where
the instantaneous velocity vector remains within a cone with a total angle of,
-------
^- 1 X10 lumber (15 required)
Spacers to raise to level of test-section floor
SIDE VIEW
sponge rubber - 3/4 X11n.
3/16 In. unfinished luan paneling
ffSfffffSffS
10 In.
10 In.
END VIEW
10 In.
(a) Valleys
Figure 1. Construction details of models.
-------
00
60 In.
46-1/16
1X10 lumber (15 required)
3/16 In. unfinished luan paneling
Spacers to raise to level of test-section floor
SIDE VIEW
/sponge rubber - 3/4 X 1 In
10 In.
END VIEW
(b) Valleys
Figure 1. Construction details of models.
-------
(0
1X10 lumber (15 required)
3/16 In. unfinished luan paneling
Spacers to raise to level of test-section floor
10 In.
END VIEW
(c) Valleys
Figure 1. Construction details of models.
-------
-120
-1000
-500
0
x, mm
500
1000
Figure 2. Shapes of valleys. Lines: ideal shapes; symbols: measured shapes.
10
-------
say, 30°. The PWA, on the other hand, is ideally suited for use in flows of
very high turbulence intensities, and even in reversing flows (Bradbury and
Castro, 1971). The PWA is less suited for measurements in low-intensity
flows, especially for measuring components perpendicular to the mean flow
vector. Thus, in valley 8, where indicated turbulence intensities were
generally less than 25%, the HWA was used for the bulk of the measurements.
In valleys 3 and 5, however, the flows were highly turbulent and reversing, so
the HWA measurements within the valley were supplemented with PWA
measurements.
The HWA was a model IFA-100 from TSI, Inc. Three types of X-array
hot-wire probes were used. The model 1241-T1.5 (end-flow style) was used for
a few measurements in the flat terrain. Because of the geometric
configuration, this probe was inconvenient to use within the valleys, so the
bulk of measurements was made with the model 1243-T1.5 (boundary-layer style)
probe. The orientation of this probe is such as to measure the longitudinal
and vertical components (u and w) of the flow. A few measurements (primarily
in Valley 8) were made with a model 1243L-T1.5 probe. This is also a
boundary-layer style probe, but the orientation is such as to measure the
longitudinal and lateral components (u and v) of the flow. All of these
probes were fabricated using tungsten wire with diameter of 1.5 Aģm. For a
detailed examination of the response of these probes, see Khurshudyan er a/.
(1981).
The analog output signals from the HWA were digitized to 12-bit precision
at a rate of 500 Hz. The digitized data were subsequently linearized and
processed on a personal computer (80286 microprocessor). A sampling duration
of 2 min yielded reasonably repeatable results (generally within ą5% on the
measurement of turbulence intensity).
The probes were traversed about the test section of the wind tunnel by
using an automated carriage. The carriage control system was also driven by
the microcomputer through the use of the software program HOT. Positioning
accuracy of the system is ą 1.0 mm.
The hot-wire probes were calibrated against a Pitot-static tube mounted
in the free-stream of the wind tunnel above the boundary layer (typically at a
height of 1200 mm above the floor). These calibrations were checked at least
once each day and recalibrations were made as deemed necessary. Calibrations
were made over the velocity range of interest, typically for 6 to 8 points
11
-------
over the range of 0.5 to 5 m s"1, and "best fit" to King's law
E2 = A + BUa,
where E = output voltage of anemometer. U = wind speed indicated by the
Pitot-static tube (manometer), and A, B and a are constants that were
determined through an iterative least-squares procedure. The calibrations
were accomplished using the program HCALX, developed inhouse (Shipman, 1988).
A typical calibration curve is shown in Rg. 3. Corrections for varying
ambient temperatures were made according to the method of Bearman (1971),
temperatures being monitored continuously.
The PWA was manufactured by PELA Row Instruments, Ltd. The principle of
operation of the PWA is quite simple. The probe consists of three fine wires,
two outside wires being parallel to one another and a central wire being
perpendicular to the outer ones (see Fig. 4). The central wire is pulsed with
a high current for a few microseconds duration, which raises the temperature
of the wire to several hundred degrees Celsius. Thus, a tracer of heated air
is released into the flow and is convected away with the instantaneous
velocity of the air stream. The two outside wires are the sensors, which are
operated as simple resistance thermometers. They are used to measure the time
of arrival of the heated air parcel. Under ideal conditions, the time taken
for the heated parcel to reach the sensor wire is
t = x/|U|cos 6,
where x is the distance between the pulsed and sensor wires, U is the
magnitude of the flow velocity vector, and 8 is the angle between the
direction normal to the probe (i.e., perpendicular to all three wires) and the
instantaneous velocity vector. The use of two sensor wires, one on either
side of the pulsed wire, ensures that the flow direction is determined
unambiguously.
The PWA probe can be oriented to measure the velocity components in all
three coordinate directions. Because of the finite wire lengths, the probe
has a yaw response up to about 70°, so that, for reasonable measurements of
transverse components of the flow, the turbulence intensity must be relatively
high, e.g., above 20 to 25%.
The electronics of the PWA provide for triggering the pulsed wire up to
60 times per second and for measuring the transit time (and direction) of the
heat pulse. The instrument provides a 12-bit binary output indicative of the
12
-------
1.35
PROBE CALIBRATION E VS U° 01-19-89
i l i i i I i i i l i i i l i i i I i i i I i i i I i ,, I i
1.2
1.05
CM
UJ
0.9
PROBE ID
SLOPE =
INT =
ALPHA a
CAUB.TEMP.=
WIRE TEMP. -
B63SS-2
0.298
0.669
0.480
25.3
2SO.
0.75 ;:
Q CALIBRATION POINTS
ZERO FLOW VOLTAGE
0.6
0 0.2 0.4 0.6 0.8
U
1.2
,0
1.4 1.6 1.8
2.2
Figure 3. Typical calibration curve of hot-wire anemometer.
13
-------
SENSOR
WIRES
PULSED
WIRE
PROBE
AXIS
Figure 4. Sketch of pulsed-wire anemometer.
14
-------
time of flight of the heated air parcel each time the sensor wire is
triggered. The PWA was controlled with the personal computer which provided
triggering pulses 30 times per second. Calibrations were performed, as with
the HWA, against a Pitot-static tube mounted in the free-stream of the wind
tunnel, over a typical velocity range of 0.5 to 5 m s" . These calibration
points were "best-fit" with an inhouse software program PWACAL to the equation
,
T T2 T3
where U = wind speed indicated by the Pitot-static tube (manometer), T is the
time of flight of the heated air parcel (from the pulsed wire to the sensor
wire), and A, B and C are constants that are determined through an iterative
least-squares procedure. A typical calibration curve is shown in Fig. 5. As
with the HWA, a sampling duration of 2 min was found to yield reasonably
stable mean velocity and turbulence intensity values.
To maintain consistency in the measurement of probe elevations above the
gravel surface, a square flat plate 10 cm on each side was placed on the
gravel surface below the probe; the bottom of the plate was regarded as the
origin of the z-coordinate (positive upwards), and probe elevations are
reported with reference to this system.
A large amount of information was collected on the flow structure over
the valleys. In general, measurements were made at 16 locations (longitudinal
positions) from x/a = -2.0 to x/a > 5.0 for each valley. HWA measurements
were made with the uw-probe at several longitudinal positions in flat terrain
and at all locations for each valley. A complete set of HWA measurements was
made with the uv-probe over valley 8 only. Approximately 75 individual
profiles were measured with the HWA, each profile consisting of approximately
20 measurement points. Each measurement provided information on the mean
velocity, the angle of the mean velocity, two components of turbulence
intensity (u and w or u and v), and Reynolds stress.
The HWA measurements were supplemented by PWA measurements within the
high-turbulence regions in the valleys. Only the longitudinal component was
measured with the PWA in valley 8, but in valleys 5 and 3, all three
components were measured (u, v, and w, one at a time), and each of these at
between 5 and 9 locations within each valley. Approximately 60 PWA profiles
were measured, each profile consisting of approximately 10 points. Each
measurement provided information on the mean velocity, the turbulence
15
-------
4 -
3
2
s
O
-1 -
-2
-3 -
-4
-5
U(CALC) a 1.287x10""
+ 1.440x1O"6/!2- 2.458x1 o"10fT
1/T
s'1
283
284
443
584
788
985
1169
1335
1492
1628
U(MEAS)
ms"1
0.46
0.43
0.83
1.21
1.82
2.46
3.06
3.67
4.29
4.88
U(CALC)
ms
0.47
0.48
0.83
1.19
1.79
2.43
3.08
3.70
4.31
4.85
ERROR
ms'1
0.02
0.05
0.00
-0.01
-0.03
-O.03
0.02
0.03
0.02
-0.03
zero velocity indicated
U(MEAS)
ms"1
s
-239
-242
-391
-542
-753
-947
1125
1291
1447
1579
U(CALC)
ms'1
-O.47
-0.47
-0.82
-1.21
-1.82
-2.44
-3.07
-3.68
-4.30
-4.84
ERROR
ms'1
-0.04
0.00
0.01
-0.01
0.03
0.02
-0.01
-0.01
-0.02
0.02
U(CALC) = 1.719x1 O*3/! + 9.980x10*7/T2- 9.286x1 O*11 /T 3
-2000 -1500 -1000 -500
500 1000 1500 2000
Figure 5. Typical calibration curve of pulsed-wire anemometer.
16
-------
intensity, and the skewness and kurtosis of the velocity distributions.
Additionally, time-series data were recorded at several elevations above each
of the valley centers (a total of about 40 time series). These data were
analyzed to obtain more detail on the nature of the velocity fluctuations;
that is, the probability density distributions were analyzed from each of
these time series.
2.4 The source
In accordance with the purposes of the present investigation, the tracer
was to be neutrally buoyant and released from a point source. The tracer used
was ethane (C2Hg), which . has a molecular weight of 30 and is only slightly
heavier than air. In combination with typical flow speeds of 2 to 4 m s"1 at
the release point and the generally small release rates (typically 4000
cm min"1), this tracer may be regarded as neutrally buoyant.
The ethane was emitted from a "stack* that was adjustable in height. It
contained a 90° bend and a perforated hollow plastic sphere of 10 mm diameter,
as shown in Fig. 6. This spherical source was used because the wind was not
always horizontal (as on the windward sides of the valleys) and possibly in
the reverse direction (as in the separated region within the steep valley),
and this configuration should provide the closest practical approximation to
an ideal point source. Note that in our earlier study (RUSHIL), the source
used was constructed of porous stone and was 15 mm in diameter. That source,
however, was quite fragile and tended to "plug up" with usage. The present
hollow sphere is regarded as an improvement because it is rugged, stable, and
smaller in diameter. Approximately 200 holes of 0.1-mm diameter were drilled
through the surface to provide for a "uniform" release rate in all directions.
To determine whether the source emission rate had an influence on the
downwind concentration field, a special test was run wherein lateral and
vertical concentration profiles were measured a short distance downwind of the
source, and the source flow rate was varied by a factor of 3. The
nondimensional plots of concentration (see next section) are shown in Fig. 7;
since the data at the two different emission rates coincide using this
nondimensionalization, the emission rate is clearly inconsequential.
The ethane flow rate was monitored continuously using a Meriam laminar
flow element (model 50MJ10-1/2) and micromanometer (model 23FB2TM-20, null
type) as shown in Fig. 6. The laminar flow element was calibrated (and
checked periodically) using a volumetric flow calibrator (Brooks model 1050A
17
-------
u
.223 mm
8175 mm
STRAIGHT FENCE
153mm high
Ethane
Pressure
Regulator
10-mm gravel
Perforated
Hollow
Sphere
DIA. = 10 mm
Laminar
Flow
Element
Figure 6. Diagram of wind-tunnel setup, source, and flow-measurement apparatus.
18
-------
100
E
N"
90
80
70
60
50
40
30
20
10
S-
DA
AD
A Q = 4000 cm3min"1
D Q =
AQ
AD
AD
DA
DA
0.1
10
100
(a) Vertical profiles
Figure 7. Concentration profiles measured downwind of sources with different
flow rates. Hg = 29 mm, xg= 234 mm, (Jn= 4 m s".1
19
-------
100
10 -
1
0.1
0.01
A
D
m
A Q = 4000 cm3min
a
Q = 1133cm3min'1
A
D
D
A
a
a
A
-150 -100 -50 0 50
y, mm
100 150
(b) Lateral profiles
Figure 7. Concentration profiles measured downwind of sources with different
flow rates. Hs= 29 mm, xg= 234 mm, \Jn= 4 m s*.1
20
-------
1J1) which has a rated accuracy of 0.5%. These calibrations were performed in
situ, i.e., the entire plumbing system was calibrated as used during
measurements; the "stack* was enclosed in a sealed container that collected
the ethane released from the source and diverted it to the volumetric flow
calibrator.
2.5 Concentration Measurements
Concentration profiles were obtained by collecting samples through 1.6-mm
OD tubes that were fastened to the instrument carriage. These samples were
drawn through Beckman model 400 hydrocarbon analyzers (flame ionization
detectors, FIDs) operating in the continuous sampling mode for analysis.
Typically, five FIDs were used simultaneously in conjunction with sampling
rakes which contained sampling tubes of fixed spacing. As with the velocity
measurements, elevations of the tubes above the surface were measured with
reference to the bottom of a square plate placed on the gravel surface. For
surface concentration measurements, however, a special spring-loaded sampling
rake was used with flat circular disks on the bottom of each tube (see Fig. 8)
to ensure that the samples were drawn consistently from an elevation of 5 mm
above the tops of the rocks". No probe interference effects were observed
during the measurement of surface (streamwise along the ground) profiles, even
though sampling tubes were in the wakes of one another during the sampling
process (see Fig. 9).
The FIDs were initially calibrated and checked for linear response using
certified gases of (nominally) 1.0, 0.5, 0.05, and 0.005% ethane in hydro-
carbon-free air from Scott Environmental Technology, Inc. A typical calib-
ration curve is shown in Fig. 10.
The "span" (full scale) and "zero" settings were adjusted before
beginning measurements and checked after finishing measurements each day.
Background (room) hydrocarbon level measurements were made at the beginning
and end of each profile. The computer program HCA subtracted a background
level from each sample measurement by assuming a linear change in background
with time between sequential background measurements. Typical background
levels were 5 ppm at the beginning and 30 ppm at the end of the day.
The output signals from the FIDs were digitized at the rate of 20 Hz
(each unit) and processed on the personal computer. With 2-min sampling
times, reasonably repeatable values of concentration were obtained, generally
within ą 5% on mean concentrations. The HCA program controlled the sampling
21
-------
STOP
-SPRING
(rubber
band)
FRAME
SAMPLING TUBE
(may slide vertically
relative to frame)
Figure 8. Detail of spring-loaded sampling rake.
22
-------
0.1
0.01
0.001
100
D
A
A Analyzer 1
D Analyzer 2
O Analyzer 3
o Analyzer 4
v Analyzer 5
4-
1000
, mm
10000
Figure 9. Surface longitudinal concentration profile measured with spring-
loaded sampling rake.
23
-------
10000
1000 -
I
o
o
Ŗ
s
1
100
10 -
A Analyzer 1
D Analyzer 2
O Analyzer 3
O Analyzer 4
v Analyzer 5
H-
10 100 1000
Actual Cone., ppm
Figure 10. Calibration of flame lonlzation detectors.
10000
24
-------
and range-switching of the FIDs and rake positioning through the instrument
carriage. It also processed the data (scaling, averaging, etc.) and stored
the data from each of the instruments in separate files. These files contain
all possible information collected, including "span", "zero" and background
readings. HCA optionally permits these "raw" files to be stripped of this
superfluous information and merged together to form a "merged" profile.
The "corrected" concentrations (i.e., with background subtracted) were
normalized as follows:
X = CUooH^Q .
where x is the normalized concentration (used in all graphs), C is the
corrected concentration (in ppm by volume), U^ is the free-stream wind speed
(nominally 4 m s"1), H is a convenient length scale (H = 234 mm was used
throughout this study for ease in comparison with earlier RUSHIL data), and Q
is the ethane flow rate (cm min*).
A large number of concentration profiles (approximately 170) were
measured. Primary stack heights Hs were 29, 59, 117, and 176 mm (Hs/h = 0.25,
0.50, 1.0, and 1.50, respectively). Primary stack positions were at the
upstream edge (x/a = -1.0), center (x/a = 0.0) and downstream edge (x/a = 1.0)
of each valley. Full surface concentration profiles were measured with each
of these stack heights and locations within each valley (and in flat terrain).
Abbreviated surface profiles, sufficient to determine the value and location
of the maximum ground-level concentration were made at intermediate stack
heights and locations. Lateral and vertical concentration profiles were
measured at only a few downwind positions for a very limited number of stack
positions and heights.
25
-------
3. PRESENTATION AND DISCUSSION OF EXPERIMENTAL RESULTS
3.1 Boundary-Layer Structure in Flat Terrain
Sufficient measurements were made of the flat-terrain boundary-layer
structure and its dispersive characteristics to ascertain that it was similar
to that used during the RUSHIL study and also a reasonable simulation of the
neutral atmospheric boundary layer. The free-stream wind speed was 4 m s*1,
and the ceiling height was adjusted to obtain a zero longitudinal pressure
gradient.
3.1.1 Velocity
Figure 11 shows local mean speed u normalized by the freestream speed U^,
turbulence intensity (au/u,
-------
1000
-I-
as
E
100 -
Distance from position of
upwind edge of valley, mm
A -1075
n -175
O 475
o 3825
v 7025
0 (RUSHIL data)
where u *= 0.19 m s'1,
d = -2 mm, and ZQ= 0.16 mm.
10
.1 .2 .3 .4 .5 .6 .7 .8 .9
u/UM
(a) Mean velocity profiles
Figure 11. Structure of the flat-terrain boundary layer.
27
-------
1000
E
N"
100 -
10 -
Distance from position of
upwind edge of valley, mm
A -1075
D -175
O 475
o 3825
* 7025
0 (RUSHIL data)
V<2>
V O O 13
.1
0-U/U
.3
(b) Longitudinal turbulence intensity profiles
Figure 11. Structure of the flat-terrain boundary layer.
28
-------
IUUU -
100 -
E
E
N"
10 -
1 -
c
r ix US* 1 1 1 i 1 1 1 I 1 r 1 1
x
ss^ Distance from position of
[ XT upwind edge of valley, mm '
^ A -1075
\0aD a -175
' O 475
Wi ° *B2S
Vo v 7025
: fŽ 0 (RUSHIL data) :
; ^CCD ;
'. *
i
^ T> D
) 0.1 0.2 0.
CTW/U
(c) Vertical turbulence intensity profiles
Figure 11. Structure of the flat-terrain boundary layer.
29
-------
IUUU -
100 -
E
N"
10 -
1
1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 viaa LJ
D T2> _A' -
D O O XT"'
X
A O D O ,X?
ADO o\" " v
\
A D OO V /
DA ŠV ,''
/
DA *& O i
O AVD^
l& **& (
; OB ov i '.
a oo' v
a asov 'N
V
\
\& Q^ V '
Si OO ^7
Distance from position of \
: upwind edge of valley, mm D A °° \v :
A -1075 \ -
D -175 A O D <> V
0 475
o 3825
v 7025
0 (RUSHIL data)
2 -1.5 -1 -0.5 C
TPwYu*
(d) Reynolds stress distributions
Figure 11. Structure of the flat-terrain boundary layer.
30
-------
100
10
X 1 :
.1 :
.01
10
100
1000
10000
mm
Figure 12. Longitudinal surface concentration profiles in flat
terrain. Dashed lines are RUSHIL data.
31
-------
100
10 -
x
E 1
0.1 -
0.01
10
D current data
A RUSHIL data
X oc Hi2
-+-
100
, mm
1000
Figure 13. Maximum ground-level concentration as a function
of source height.
32
-------
On the basis of these comparisons of flow structure and dispersion
characteristics, we were confident that the boundary layer was the same in
every substantial way as that used in the RUSHIL study. Figure 13, then,
served as the flat terrain or reference data for later evaluations of TAFs.
A comparison is also made .between the dispersion characteristics in this
simulated atmospheric boundary layer and those estimated using Pasquill-
Gifford stability categories C and D (Turner, 1970), as shown in Fig. 14. The
Pasquill-Gifford stability C describes a slightly convective atmosphere and is
used here for comparison only; the wind tunnel models neutral flow. A scale
factor of 1:1250 was used in this comparison. As is common with wind-tunnel
simulations, the dispersion was approximated as between stability classes C
and D near the source, but farther downstream, it was closely approximated by
stability class D. Also shown is the dispersion scheme suggested by Hosker
(1974), Gifford (1975). and Smith (1973) for stability class D and a 40 cm
roughness length (scaling up the wind tunnel roughness length gives z0 = 20
cm, but their scheme uses discrete values and 40 cm is the next largest
choice). This scheme uses a distance-dependent roughness-length correction and
allows much larger values of roughness length than the Pasquill-Gifford
approach; it matches the wind-tunnel data much better than the
Pasquill-Gifford scheme. In summary, we conclude that the wind tunnel
boundary layer provides a reasonable simulation of the dispersion
characteristics of the neutral atmospheric boundary layer.
3.2 Flow Structure over the Valleys
Figures 15 through 17 show vertical profiles of longitudinal mean
velocity, turbulence intensity, and Reynolds stress, respectively, over the
three valleys. Although not all are shown, 16 profiles were measured for each
valley, spanning the range from one valley half-width (a) upstream to at least
6a downstream. Note that these profiles contain a mixture of hot-wire and
pulsed-wire data. Because the hot-wire data are inaccurate at high-turbulence
intensities and the pulsed-wire data are inaccurate at low intensities (at
least for components perpendicular to the mean flow), the profiles taken with
each instrument were merged by selecting the most accurate data for each
range. Some region of overlap was allowed, as may be evident in the figures.
Also, the Reynolds stresses have been converted to the "natural" coordinate
system, that is, parallel and perpendicular to the mean streamlines in
33
-------
10-5
E
O
D
U
10
-6 '
10
-7
Hosker-Gifford-Smith
D stability
Pasquill-Gifford
C stability
.1
1
xg, km
Pasquill-Gifford
D stability
10
Figure 14. Surface longitudinal concentration profile in flat terrain,
scaled up 1250:1 from wind-tunnel measurements, compared with
Gaussian plume predictions using Pasquill-Gifford stabilities C and
D and Hosker-Gifford-Smith stability D with zo = 40 cm.
H = 146 m (117 mm in wind tunnel).
s
34
-------
-1.25
1.25
5
4
3 -
2 -
1 -
0 -
-1 -
0 1
-1.25 -1 -.75 -.5 -.25
0
x/a
.25 .5 .75 1 1.25
-1.25 -1 -.75 -.5 -.25 0 .25 .5 .75 1 1.25
x/a
Figure 15. Longitudinal mean velocity profiles over the valleys (u/Uģ).
Solid lines are flat-terrain profiles.
35
-------
-1.25
-1.25
-.75 -.5 -.25
0
x/a
.25
.75
-.75 -.5
.25
0
x/a
.25
.75
1.25
1.25
-1.25 -1 -.75 -.5 -.25 0 .25 .5 .75 1 1.25
x/a
Figure 16. Longitudinal turbulence Intensity profiles over the
valleys (^U/UW). Solid lines are flat-terrain profiles.
36
-------
.c
N
-1.25 -1 -.75 -.5 -.25
0
x/a
.25 .5
.75
1 1.25
-1.25 -1 -.75 -.5 -.25
0
x/a
.25 .5
.75
1 1.25
Figure 17. Reynolds-stress distributions over the valleys (-u'w' x
Solid lines are flat-terrain profiles.
37
-------
accordance with the angle of the mean velocity vector. The profiles measured
in flat terrain are shown for reference as solid lines in the figures.
Mean streamlines were calculated from the mean velocity measurements.
For valleys 5 and 8, mass-consistent wind fields were computed using the model
that is described in Section 4. For valley 3, the mean velocity profiles were
simply integrated to find the elevations of specific values of the stream
functions. These streamline patterns are displayed in Rg. 18.
At first glance, the streamline pattern over valley 8 is reminiscent of
potential flow, but closer examination reveals it is clearly asymmetrical,
with the lower streamlines being considerably closer to the surface on the
downwind slope than on the upwind one. The streamline pattern over valley 5
is clearly asymmetrical and, because the streamlines diverge strongly away
from the surface, it is clear that the velocity is reduced remarkably at the
valley center; indeed, it appears that a stagnation region exists in the
valley bottom. In valley 3, the streamline pattern clearly shows a
recirculation region, with separation occurring a short distance down the
upwind slope and reattachment occurring about halfway up the downwind slope
from the valley center. The three valley shapes thus result in three
fundamentally different flow patterns. We believe these basic flow structures
are fairly typical and cover the range of patterns to be observed at full
scale, albeit in neutral stratification.
It is instructive to examine the detailed characteristics of the velocity
fluctuations within the valleys. Figure 19 shows the probability density
distribution (or function - PDF) of the longitudinal velocity fluctuations at
half the valley height above the center of valley 3. This was measured with
the pulsed-wire anemometer. Several interesting features are to be noted from
these figures:
1. A best-fit Gaussian distribution is shown for comparison (on a
logarithmic scale, the Gaussian distribution appears parabolic). The
measured data are obviously slightly skewed, but for many practical
purposes, the data may be considered to have a Gaussian distribution.
The skewness and kurtosis are 0.18 and 2.85, respectively; these may be
compared with Gaussian values of 0 and 3.0.
2. The distribution shows that the mean velocity is positive but the
instantaneous velocity is negative perhaps 40% of the time. Thus,
whereas flow reversals in the sense of the mean flow do not occur (at
this position), instantaneous reversals occur frequently. The local
turbulence intensity au/u is very large, over 170%!
3. Abnormalities are observed at the points close to zero velocity. This is
due to a hardware limitation of the PWA. If the speed of the heat pulse
38
-------
4.5
4
3.5
3
2.5
1 -
.5
__-srSl;l (a) VALLEY 3 -:
.yt'.VA'.'.'A'.VAV1^['y.yi A |T "V \, \ \ \ \ t i
-1.25
-1
-.75
-.5
-.25
0
x/a
.25
.5
.75
1.25
4.5
1.25
-1.25
-1
-.75
-.5
-.25
0
x/a
.25
.5
.75
1.25
Figure 18. Streamline patterns derived from experimental
measurements over the valleys. Note that the vertical
scales are exaggerated.
39
-------
10
1 :
.1
.01
.001
-.5
-.25
.25
U/U
oo
.5
Figure 19. Probability density distribution of longitudinal
velocity fluctuations measured at half the valley
height above the center of valley 3.
.75
40
-------
is too low, it will not reach the sensor wire before the clock overflows,
or the heat will diffuse such that the sensor wire will not detect it.
In either event, very low velocities (typically less than 0.15 m s"1 -
positive or negative) will be indicated as zero velocities. This region
is shown on the calibration curve in Fig. 5. Depending upon how the
sorting slots are distributed about zero velocity, the results will be
distorted. These abnormalities could be reduced somewhat by clever data
manipulation, but no attempt was made to do so here.
Figure 20 shows the PDFs of the longitudinal velocity fluctuations at 5
elevations above the center of valley 3. The following characteristics of the
flow are illustrated:
1. The mean velocities are negative at the two lowest elevations, h/8 and
h/4, very close to zero at h/2, and some flow reversals occur even at the
valley top h.
2. The magnitude of the velocity fluctuations, as evidenced by the widths of
the distributions, is largest at the top of the valley. (This was also
evident from the vertical intensity profiles, but is more dramatically
illustrated here.) This is the position where the shear in the mean flow
is largest (see Fig. 11 a).
3. The reversed flow at the lowest elevation is quite steady, with only
occasional (=ą10% of the time) positive velocities indicated there.
Figures 21 and 22 show similar distributions measured above the centers
of valleys 5 and 8. At the lowest levels within valley 5, the mean velocities
were quite small, but instantaneous flow reversals were very common (up to 40%
of the time at z/h = 0.13). A few reversals occurred at elevations as high.as
the valley top h in valley 5. A very few reversals (<1% of the time) were
also observed in valley 8 between the surface and z = 0.4h.
Taken together, all of these data have important implications for the
behavior of pollutants released within these valleys. Two primary features of
two-dimensional neutral flow affect ground-level concentrations (glcs): the
displacement of the mean streamlines and the changes in turbulence. The
displacement of streamlines determines how near to the surface the centerline
of a plume will reach. The convergence and divergence of the streamlines
affect the plume width (a) directly and (b) indirectly through their
distorting effects on the velocity gradients, which in turn affect the
turbulence. The turbulence itself, of course, spreads and diffuses the plume.
All of these effects can either increase or decrease surface concentrations.
Note that, because the flow is two-dimensional, the mean streamlines remain in
vertical planes. However, the longitudinal and vertical turbulence
intensities are greatly increased through the flow distortions and, because of
41
-------
4.4
4 ..
3.6
3.2
2.8
8 2.4
2
1.6
1.2
.8
.4
.6 -.4 -.2 0 .2 .4 .6 .8 1 1.2 1.4
Figure 20. Probability density distributions of longitudinal velocity
fluctuations at various elevations above the center of
valley 3.
42
-------
-2 -1
u, m s
2
-1
Figure 21. Probability density distributions of longitudinal velocity
fluctuations at various elevations above the center of valley 5.
43
-------
1.05
0.9 -
0.75
0.6
0.45
0.3
0.15
u, m s
-1
Figure 22. Probability density distributions of longitudinal velocity
fluctuations at various elevations above the center of valley 8.
44
-------
the tendency of turbulence to distribute its energy equally in all directions
(return toward isotropy), the lateral turbulence intensities will also be
greatly increased.
It is useful to examine in some detail the flow structure within the
three valleys with a view toward anticipating the behavior of pollutants
released within them. Consider first the mean streamline pattern over valley
8 (Fig. 18c). Because it displays the smallest flow distortions of the three
valleys, we may expect the smallest changes (from flat terrain) in maximum
surface concentrations, (I.e., TAFs closest to 1.0). Because the mean stream-
lines begin to diverge at the upstream edges of the valleys, we might expect
maximum glcs from sources placed there to be less than those observed in flat
terrain (TAFs less than unity). On the upwind slope, the mean streamlines
diverge so that plumes released there would be transported farther from the
surface. However, because the turbulence is increased, these plumes would be
diffused more rapidly to the surface. With these counteracting tendencies, it
is difficult to speculate on the net effects except to state that we would not
expect the largest TAFs to occur with sources located in this position. For
sources placed above the valley center, it is easily seen that the streamlines
transport the plumes closer to the surface and the enhanced turbulence rapidly
diffuses a plume to the surface. Hence, we may expect the largest TAFs from
sources placed in the valley center. For sources placed at the downwind edge
of the valley, we again observe the counteracting effects of streamline
divergence but increased turbulence, and we expect the TAFs to be near unity
again. Note that enhanced lateral diffusion will diminish surface concentra-
tions along the plume axis, but increase the area of coverage in the lateral
direction.
For valleys 5 and 3, we may expect roughly the same behavior from sources
placed near the upstream and downstream edges of the valleys, i.e., TAFs near
unity. For low sources above the center of valley 5, however, we observe very
small mean transport speeds and very common flow reversals. We may expect the
plumes to be wafted back and forth while being diffused strongly in the
lateral and vertical directions before eventually being transported
downstream. Thus, we may expect quite large TAFs for low sources near the
center of valley 5.
In valley 3, the streamline patterns show a definite recirculation region
that extends to nearly 75% of the valley depth. Plumes released on the
separation/reattachment streamline will be transported directly to the
45
-------
surface. We may thus expect very large TAFs from such releases. Plumes
released well below the separation/reattachment streamline, say in the
reversed flow region around h/4, would be transported upstream in a more
routine manner, with the plume axis remaining nearly parallel to the surface.
Because of the very large turbulence intensities and relatively low transport
speeds, we may still expect large TAFs (but not as large as those from the
higher sources).
The general features of these flows are remarkably similar to those
observed on the lee sides of the hills in the RUSHIL study. As mentioned
earlier, those hills had exactly inverse shapes (and slopes) as the present
valleys. And as with the valleys, hill 8 exhibited no separation, hill 5
exhibited incipient separation (i.e., instantaneous flow reversals were
observed approximately 40% of the time, but the mean flow was always
downstream), and hill 3 exhibited clear separation with mean-flow reversals at
low elevations.
3.3 Concentration Measurements in the Valleys
Figure 23 illustrates some typical comparisons between surface
concentration profiles measured from sources placed within the valleys and
those from sources of the same height in flat terrain. In all cases, the
stack height Hs was equal to the valley depth h. Xg denotes the distance from
the source. In Fig. 23a, the source was located at the upstream edge of each
valley, and the changes in the maximum glcs induced by the valleys are small,
less than 15% in all cases. This figure shows that the location of the
maximum glc is beyond the downstream edge of the valley in each case and that,
where the maximum glc is higher, its location is closer to the source and vice
versa.
In Fig. 23b, the source was located at the valley center. The increased
concentrations caused by the valleys are dramatic and TAFs range from about
2.5 in valley 8 to about 15 in valley 3. Also, as the concentration
increases, the distance to the maximum decreases. Indeed, the location of the
maximum in valley 3 is only 2 stack heights downstream, compared with about 15
stack heights in flat terrain. This is obviously because the plume is
released very near to the separation/reattachment streamline.
When the source was placed at the downstream edge of the valley (Fig.
23c), virtually no change in the maximum glc was observed.
One more case is illustrated in Fig. 24, where the source location was
half the valley height above the valley center. The location of the maximum
46
-------
.1
.01
.001
Valley 3
Valley 5
Valley 8
Flat terrain
10 100
xs/h
(a) Source located at upstream edge of valleys
Figure 23. Comparison of surface concentration profiles from
sources placed within the valleys with one from a source
of the same height in flat terrain. Hs = h.
47
-------
100
D
O
10 -:
AA"
Valley 3
Valley 5
Valley 8
Flat terrain
A
A
1 -:
.1 -:
D D
D
D
.01 -:
D
D
.001
.1
10
100
xa/h
(b) Source located at center of valleys
Figure 23. Comparison of surface concentration profiles from
sources placed within the valleys with one from a source
of the same height in flat terrain. Hs = h.
48
-------
0.1 -
0.01
0.001
1 10 100
xs/h
(c) Source located at downstream edge of valleys
Figure 23. Comparison of surface concentration profiles from
sources placed within the valleys with one from a source
of the same height in flat terrain. Hs = h.
49
-------
35
30 -
25
20
15
10
5 -
Valley 3
Valley 5
Valley 8
Flat terrain
8
Figure 24. Comparison of surface concentration profiles from
sources placed within the valleys with one from a source
of the same height in flat terrain. H = h/2. Source
position is at the center of the valley (x/a = 0).
50
-------
for valley 3 was actually slightly upwind of the source, and the TAP is about
12. For valley 5, the location of the maximum glc was downstream, but very
close - about 2 stack heights away - and the TAP is about 6.
Measurements such as these were made at an array of source locations and
heights in the vicinity of each of the three valleys, and the TAPs were
determined for each location. Maps of these TAFs are shown in Fig. 25, where
isopleths of constant TAP have been drawn. The first impression is that the
patterns are symmetrical about the vertical centerline, but closer examination
reveals some asymmetry. Nevertheless, the near-symmetry and the overall
similarity in shape amongst the three valleys is quite surprising in view of
the very different flow patterns observed. In contrast, the magnitudes of the
maximum TAFs differ widely, from 2.5 in valley 8 to 15 in valley 3. These
differences, of course, reflect the effects of the different flow structures.
The location of the maximum TAP seems to be independent of valley shape,
occurring above the valley center in each case (but see Section 4: the
theoretical model predicts maximum TAFs a short distance upwind of the center
of valley 8, at a position where no experimental measurements were made). The
TAFs display rather broad peaks on the vertical centerline - between h/2 and
h, the TAFs vary by only 10 to 20%.
Contours with TAP values of 1.4, 2, 4, efc., have been drawn where
appropriate. Note that these contours form "windows" within which the maximum
glc exceeds the glc that occurs in flat terrain by 40%, 100%, 300%, etc. The
longitudinal extent of the window of 40% excess concentration extends over
approximately 60% of the width of valley 8, 80% of the width of valley 5, and
more than 90% of the width of valley 3. The vertical extent of the 40% window
is 1.5, 2.0, and 2.5 valley heights above the valley fop for valleys 8, 5, and
3, respectively.
At first the vertical extent of the 40% windows may appear excessively
large. For example, the height of a stack placed near the center of valley 5
would have to equal 3 valley heights in order to meet a 40% excess criterion.
A very rough estimate will show, however, that this is quite reasonable. We
assume that for very tall stacks, the only effect of the valley is to reduce
the effective stack height Hs by the valley height h. Because the maximum
glcs may generally be considered inversely proportional to the square of the
stack height, the TAP may be calculated as
A = Xmx/Xmx = H*/(HS - h)2 ,
Where xmx is tne maximum normalized glc and Xmx is tne maximum normalized glc
51
-------
z/h
-1.5
-1.0
z/h
2.5
2.0
1.5
1.0
0.5
0.0
0.5
-1.0
1.21
1.0 _ 1-31
^-x ^
!s2\
^
i
-1.5
1.5
z/h
1.0 ^
t
1.09
1.13
^ 1.23
59 \
.80 1 ,
&' 1.11/
1.32
52
-------
from a source of the same height on flat terrain. For a stack height 3 times
the valley height, A = 2.25 (a 125% excess concentration). This is, of
course, an overestimate, because it does not account for the vertical rise
provided by the airflow (see streamline patterns at z = 2h in Fig. 18).
Application of the data in Fig. 25 is straightforward. Let us consider a
source which is located in the center of a rather broad valley, say one
similar in shape to valley 8; and the height of the source is half the valley
height. Figure 25c suggests that the maximum glc would be about 2.5 times
that expected from a source of the same height but located in flat terrain.
On the other hand, if the valley were considerably narrower, say close to
valley 5, Fig. 25b suggests the maximum glc would be about 7 times as large as
that from the same source in flat terrain. Although precise interpolation of
these results for valleys intermediate in shape and slope to those examined
here may be difficult, the results may allow us to place some useful limits on
the effects of valleys of intermediate shape and slope.
Figure 26 shows the loci of source positions leading to the same
locations of maximum glc. These loci have been identified by marking them
with the position of the maximum glc (in valley heights from the centers of
the valleys). Note that the "undisturbed" or flat-terrain loci (dotted lines)
are simply parallel, nearly straight, diagonal lines. Within the valley,
these loci are distorted, as shown by the solid lines. The diagrams may be
used as follows: for any given source position, we may plot that position on
the diagram, then follow the locus to the ground; the intersection of that
locus with the ground is, of course, the location of the maximum glc.
Conversely, from a knowledge of the location of the maximum glc, we may use
these diagrams to determine the line along which the source was positioned.
These loci become highly distorted near the valley centers, and the steeper
the valley, the higher the distortion. As the distance (both longitudinal and
vertical) from the valley center increases, these loci gradually relax to
their undisturbed or flat-terrain values.
We have thus far discussed primarily the locations and values of the
maximum glcs. It is also instructive to examine the area of coverage in the
crosswind direction. Figure 27 shows the lateral plume widths measured
downwind of sources of the same height (Hs = h/4) placed in the center of each
of the three valleys. These distributions were measured at the same downwind
distance in each case (Xg = 6HS). It may be seen that the lateral widths of
the plumes increase quite strongly as the steepness of the valley increases.
53
-------
z/h
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
84.7
20
-2
0 1
x/h
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
> -1.0
H
5
z/h
z/h
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0 L
20
-7.5
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-12
-5
-2.5
2.5
X/h
27.0
x/h
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
* -1.0
7.5
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
L -1.0
8
12
Figure 26. Distance in valley depths from the valley center to the
location of maximum ground-level concentration. Fiat
terrain values are indicated as dotted lines.
54
-------
27.5
-3
-1
0
y/h
Figure 27. Lateral concentration distributions measured 6HS
downwind of stacks of height h/4 placed in the center of
each of the three valleys. Solid lines are Gaussian curve fits.
55
-------
Whereas a lateral profile was not measured at the same downwind distance in
flat terrain, we may use an interpolation formula that fits the measured data
quite well (see Section 4) to estimate ay/h = 0.19. By comparison, the
corresponding values are nearly twice as large in valley 8, 4.5 times as large
in valley 5, and 6 times as large in valley 3. This is due, of course, to the
much reduced transport speeds and the greatly enhanced local turbulence
intensities. In valley 5, the transport speed is extremely slow and the
turbulence intensity is very large. Thus, the plume is wafted around in all
directions as it is gradually transported from the source to the measurement
point. In the case of valley 3, the measurement point is quite close to the
reattachment point, so that the plume is actually transported upstream (in the
mean) and recirculated within the separation/reattachment zone before reaching
the measurement location. Note that the best-fit Gaussian curves shown on the
figure actually fit the measured distributions quite well; this matches our
common observation that lateral concentration distributions are almost always
Gaussian in shape.
56
-------
4. NUMERICAL MODEL AND COMPARISONS WITH
EXPERIMENTAL RESULTS
One of the main purposes of the experimental study described in earlier
sections was to test the applicability of a diffusion model for the evaluation
of maximum glcs resulting from elevated continuous point sources placed in a
curvilinear neutral atmospheric boundary layer. The model described here uses
as input parameters data from wind-tunnel measurements of wind velocities and
turbulence characteristics to calculate the entire flow field. This flow
field is then applied in the numerical solution of the diffusion equation.
Such a model was previously developed by Berlyand et al. (1975), primarily for
evaluation of pollutant dispersion in complex terrain, on the basis of
measurements of flow structure in a wind tunnel where direct measurements of
pollutant dispersion were inaccessible.
The present version of the model does not incorporate a longitudinal
diffusion term, and therefore does not calculate the spread of pollutants in
separated flows. Thus, no attempt is made to apply the model to valley 3, and
calculations were made only for valleys 5 and 8.
Application of the diffusion equation to the calculation of the spread of
pollutants over flat terrain may be of interest in and of itself, because it
provides additional information on the applicability of the gradient transport
theory of turbulent diffusion (K-theory). Some of the assumptions involved in
the development of K-theory are not strictly satisfied for elevated point
sources, especially near to the source. However, since the interest here is
primarily in computing glcs, K-theory is applied to elevated point sources.
4.1. Calculations of dispersion over flat terrain
Modeling the spread of pollutants over homogeneous flat terrain is based
on a numerical solution of the diffusion equation for an elevated continuous
point source,
Uax7 = 5y\ky3yj + az\kzdz/> (1)
where x is the pollutant concentration, u is the mean velocity, ky and kz are
the eddy diffusivities in the crosswind and vertical directions, Xg is the
downwind distance from the source, and z is the height above ground.
If, as assumed by Berlyand (1975), ky = k<,u (where k0 may be a constant
or a function of Xg), a separation of variables may be performed, so that Eq.
57
-------
(1) may be reduced to the two-dimensional problem of diffusion from a line
source,
u 5x1 - L / K, **: \
u ax3 ~ az \ ** az /
x' represents concentration in the two-dimensional problem.
Assuming reflection of pollutant from the ground surface, the boundary
conditions are as follows:
X' = [Q/u(Hs)] $(Z-HS) at Xg = 0 ,
kjdxV^z = 0 at z = 0 , (3)
X' * 0 when z -> oo .
Here, Q is the source flow rate, Hs is the source height, and 6 is the Dirac
delta function. The relation between concentrations of point and line sources
is given by the following expression (assuming y = 0 is a coordinate of the
source):
Y ' (x z) ( v2 1
X(xs, y. z) = * v a- i exp - fa \ , (4)
(AirfZ\l/2 \ Hva )
r
Jo
(5)
where G = k0(Ŗ)df
Jo
Equation (2) with boundary conditions (3) was solved numerically by a
finite-difference method. To obtain a solution, values of u and kz must be
specified. As shown in Fig. 11 a, the vertical profile of mean velocity may be
satisfactorily approximated by
' In[(z-d)/z0] when z < h, '
/C
u =
In^r^-dJ/Zo] when z > h,
where /c is the von Karman constant (0.4), u. is the friction velocity, d is
the displacement height, and ZQ is the roughness length. In accordance with
the wind-tunnel data, the various parameters were assumed to be u. = 0.19
m s" , d = -0.5 mm, and ZQ = 0.16 mm. (These values were obtained from
preliminary results of the wind-tunnel measurement program and differ only
slightly from the final values.) The value of h1 was chosen to achieve the
value of the free-stream velocity at the top of the boundary layer (4 m s").
For kz, it was assumed as is commonly done that kz = kh, where kh is the
eddy diffusivity of heat, and also that kh = akm, where km is the eddy
diffusivity of momentum. In earlier works, it was generally assumed that a =
1.35, but according to more recent data (Kader and Yaglom, 1972; Snijders et
a/., 1983), it appears more appropriate to assume a = 1.15. The latter value
was used in the present study.
km was related to the turbulent kinetic energy (per unit mass) through
58
-------
the following (Berlyand, 1975):
km(z) = Ģ 3" E(z) | "*. . (6)
where C, is a constant, and E = (1/2) (u' 2 + v* 2 + w' 2). C, = 0.025 was
chosen as the best-fit to the linear growth of km in the lower part of the
boundary layer with u* = 0.19 m s" .
Figure 28 shows the profile of km(z) as calculated by Eq. (6). As might
be expected. km increases almost linearly with height in the lower levels (as
is commonly observed in the neutral atmosphere), then begins to decrease
toward zero. This decrease seems to be a peculiarity of wind tunnel flows,
because atmospheric observations suggest a constant value may be more
appropriate.
km was also evaluated using the more usual relation km = -u'wy(du/d2),
where -u'w' is the Reynolds stress that was also measured in the wind tunnel.
The profile calculated using this method was approximated as follows
. ( KU.Z when z < h2 \
m = \ 0.818 /cu.(1-z) when z > h2 / '
Here, z must be measured in meters, and the value of h2 was specified as
0.45 m. Using Eq. (7), km also tends toward zero at the top of the boundary
layer, although Eq. (6) gave larger values at those heights. Because all
source heights were substantially smaller than the boundary-layer depth, this
difference has no practical significance.
Calculations of pollutant diffusion [solutions of Eq. (2) with boundary
conditions (3)] were made with the use of Eqs. (6) and (7).
Difficulties are encountered with K-theory in the specification of the
parameter ky. in this case, it is helpful to use assumptions from the
statistical theory of turbulent diffusion on the root-mean-square particle
displacement in the lateral direction, ay. ky may be related to ay through
the well-known equation ky = (1/2) d<72/dt. Taylor (1921) showed that the
mean-square particle displacement is given by
T
'0J0
~i rV
(t) = 2 V2 RL(fl dŖ df, (8)
J ftJ n
/ -
where RL(Ŗ) = v'(t) V(t + Ŗ) / V is the Lagrangian velocity auto-
correlation function. Because direct measurements of RL are very difficult,
59
-------
1000
100 -
E
N"
10 -
D Eqn. 6
Eqn. 7
.0001
.001
.01
.2-1
mģ
.1
Figure 28. Vertical profile of eddy diffusivity k In flat terrain.
60
-------
we have chosen to use known asymptotes of ay. As t -ģ 0, RL -> 1, and as t -ģ oo,
RL -ģ 0. Therefore,
avt when t -> 0 \
/ [ - 0)
avv2TLt when t -> oo >
oo
where TL = R|_(Ŗ)dŖ is the Lagrangian integral time scale.
J o
Because K-theory provides a Eulerian description of diffusion, it is
necessary to transform (9) into Eulerian form. For this purpose, it is usual
to apply the hypothesis of "frozen" turbulence and rewrite (9) as
-> 0
(10)
Here, a. = CT../U, U is the characteristic mean wind velocity, x_ = Ut is the
v
distance downwind from the source, and L = UTL. It was assumed that U =
Uf(Hg), the velocity over flat terrain at the source height.
In the full-scale, near-neutral atmosphere, for downwind distances up to
about 10 km from the source (much larger than the distance to the maximum
glc), a nearly linear growth of the crosswind spread of plumes with downwind
distance is usually observed (Hunt, 1980). Because of the finite width of the
wind tunnel, the low-frequency, large-scale lateral wind fluctuations are
absent, and a transition from linear to square-root growth in plume width
clearly begins. Therefore, in the present study, an interpolation formula was
used which accounts for both asymptotic limits:
1/2
ks
xs + 2L
(11)
From the wind-tunnel measurements, we have determined aa = 0.14 and L = 0.4 m.
a
Formula (11) is very close to the more usual equation derived by using an
exponential shape for the autocorrelation function (Pasquill, 1974):
ay = 2 a* L2 I" Xg/L - 1 + exp(-Xg/L) I . (12)
Figure 29 compares the interpolation curve corresponding to Eq. (11) with
experimental values of ay obtained from lateral concentration profiles
measured downwind of sources of different height in the wind tunnel.
For source heights less than 400 mm, the results of the numerical
solution of the diffusion equation showed only a small difference (in the
range of 4%) between values of maximum glcs, xmx- when either Eq. (6) or Eq.
61
-------
1000
E
E
100
10
A Hg = 117mm
a Hs = 29 mm
interpolation formula
100
1000
x , mm
10000
Figure 29. Growth of lateral plume width in flat terrain:
comparison with interpolation formula (Eqn. 11).
62
-------
(7) was used. The differences between the downwind distances from the source
to the point of maximum glc, x^, however, were larger, and they increased
more rapidly with height. For example, when Hs = 351 mm, the difference was
about 16%.
Comparison of the calculations [using Eq. (6)] with the experimental
results showed that the differences in xmx did not exceed 10% for the range of
source heights between 117 and 234 mm. At the lower stack heights of 29 and
59 mm, the calculations were overpredictions of 14% and 11%, respectively.
For the taller stacks of heights 293 and 351 mm, the calculations were
underpredictions of 20% and 21 %, respectively. With source heights between 29
and 293 mm, the differences in x^ were less than 12.5%. For the stack height
of 351 mm, the calculated xmx was 17% larger than measured. Comparison of
calculated and measured vertical concentration profiles downwind of an
elevated point source show that the calculations overpredict the vertical
spread of pollutants near the source. On the other hand, the experimental
source was not, in fact, a "point" source. Its diameter (10 mm) was not small
compared with the smallest source height (29 mm). Consequently, the behavior
of the plume in this case could differ significantly from what might be
considered a true point source.
Calculated and experimental values of xmx and xmx and their dependence on
source height are presented in Figures 30 and 31. Figure 32 shows the
interrelationship between xmx ancl xmx-
4.2 Calculations of Dispersion near Valleys
4.2.1 Generation of flow fields
As mentioned above, the model uses experimental data to reconstruct the
entire flow field for use in the solution of the diffusion equation. The
principal feature of the flow field is that it should be mass consistent, that
is, it should satisfy the continuity equation. This problem was solved
through the use of variational analysis techniques. A similar approach,
initially based on the work of Sasaki (1958, 1970a, 1970b), has been used in
regional-scale air-pollution modeling when the meteorological data in the
region being investigated is taken as input information (Dickerson, 1978;
Sherman, 1978; Pepper and Baker, 1979).
Before considering the mean wind field generation using wind-tunnel
measurement data, let us consider how this wind field may be used in the
solution of the diffusion equation. The diffusion equation may be written in
63
-------
100
10 -
1 -
.1 -
.01
A RUSHIL data
a current data
calculations
10
100
, mm
1000
Figure 30. Dependence of X on source height Hs
mx
in flat terrain.
64
-------
10000
E
1000 -
100
A experiment RUSHIL
a current experiments
calculations
+-
10
100
, mm
1000
Figure 31. Dependence of xmx on source height Hs in flat terrain.
65
-------
100
10
x
E 1
x
0.1 -
0.01
100
B
A RUSHIL data
D current data
calculations
1000
x mx, mm
10000
Figure 32. Relation between X mx andxmv in flat terrain.
mx
66
-------
the following form
n§X + ^dx =d_( dx\ d_( dx\
u ax + w az ay \ *y ay J az \ ** az / '
where w is the vertical component of the mean velocity and all other notations
are the same as used above. This form of the diffusion equation may be used
if the slope of the terrain is small to moderate and only if the flow does not
separate. In the case of complex terrain, the concentration from a point
source may be related to the concentration from an infinite line source x'
using the same expression (Eq. 4) as in flat terrain. The concentration x'
will be the solution of the diffusion equation
To simplify the problem, it is convenient to transform the solution
domain to a half-plane. For this purpose, the following substitution of
variables may be used
x- = x. z* - H[z - h(x)]/[H - h(x)] . (15)
transforming the area (-00 < x < oo) x (h(x) < z < H] to the strip (-00 < x* < oo)
x (0 < z' < H). Here h(x) is the valley depth (<0), and H is the level above
which the influence of the valley on the flow structure is not important.
Within the strip 0 < z* < H,' Eq. (15) may be rewritten as
. H - h(x') ~ z'- H dh . r H
where u = - ^J- u ; w = w + ^ u ^ ; and ft, = H . h(x,}
The quantities u and w are defined such that they satisfy the equation
_ _ _
ax' dT ~ '
provided the initial components of velocity u and w satisfy the continuity
equation au aw n
ax + al ~ °
To generate a mass-consistent flow field with measured data, we consider
the solution of Eq. (16) for the diffusion problem formulated above. In this
case, the task is reduced to minimization of the functional
67
-------
(18)
Here, 0 is the stream function of the flow, d-^jdt = u, difr/dX = -w. The
stream function 0, which minimizes the functional I, corresponds to the flow
for which values of u and w are closest to the corresponding measured values a
and b (in the mean-squared sense) within the domain R. The latter should be
chosen in such a way that perturbations of the flow on its boundary dIR could
be assumed equal to zero.
The Euler equation for the variational problem under consideration is
a2V> . a20 _ aa ab
+ *-&-*'
Introducing the stream function of the approach flow V°(zO
quantities 0' and a' as follows
0 = 0°(z-) + W-zO: a = d0°/dz' + a' , (20)
it is possible to rewrite Eq. (19) as
32V>' 320' _ da' ab , .
z + gz,2 = az- - ax- ^
The function 0 = 0° + 0' minimizes the functional (18) if the function V
is the solution of Eq. (21) with the boundary condition
flflR-0. (22)
This boundary value problem, Eqs. (21) and (22), which is actually the
Dirichlet problem for the Poisson equation, was solved numerically using a
finite difference method. Since the domain R may be chosen to have a
rectangular shape, a direct method with the fast Fourier transform technique
(Ogura, 1969) was used for the solution. Calculations have been made for
valleys 8 and 5 described in Section 2.
In principal, the technique used above allows reconstruction of the mean
wind field for recirculating flows. However, the form of the diffusion
equation (13) is not appropriate for application to separated flows, because
the longitudinal diffusion term is omitted. From the wind-tunnel data, the
flow clearly separated and a recirculation region was formed within valley 3.
Hence, the mean wind field was not generated for valley 3.
68
-------
The results of the flow field generation for valleys 5 and 8 are
presented in Tables 1 and 2. These tables show values of longitudinal mean
velocities at some typical locations near the valleys. Measured values of
wind speed over flat terrain as calculated with the log-law wind profile (Eq.
5) are also included in the tables for comparison. All of the flow
measurements in valley 8 were made with the hot-wire anemometer; whereas those
measurements at lower levels in valley 5, where the turbulence intensities
were high, were made with the pulsed-wire anemometer.
Table 1 shows measured and calculated values for valley 8 with
differences that do not exceed 6%. Table 2 shows similar differences for
valley 5 at locations x/a = -1.0 and x/a = 1.0. However, for position x/a =
0.0, the differences are noticeably larger and average about 10%. The larger
differences at this position may be partly due to the method of obtaining the
measured values. The mean longitudinal velocities obtained from the
pulsed-wire anemometer were used in conjunction with the flow directions
indicated by the hot-wire anemometer.
Figures 18b and c showed the calculated streamlines for valleys 5 and 8,
respectively. The streamline pattern for valley 8 is asymmetric; for valley
5, the asymmetry is even stronger. A hint of a stagnation zone is seen near
the bottom of valley 5.
4.2.2 Calculation of eddy diffusivities
Crosswind dispersion was calculated by using Eq. (9). The diffusion
equation gives an Eulerian description of diffusion and, in the case of a
continuous source, time is not explicitly included in the equation; therefore,
the hypothesis of "frozen" turbulence was used as in the flat terrain
calculations. It was assumed that the typical speed of pollutant transport us
is equal to the speed of the mean flow along the streamline which passes
through the source position. Since the mean flow velocity over complex
terrain differs from that over flat terrain, the time of pollutant transport
to a given distance will also differ.
Under the above assumptions, the time t of pollutant transport to a
distance x, downwind of the source is
(23)
Substituting this into Eq. (9) gives an expression for ay as a function of
69
-------
downwind distance. The eddy d'rffusivity ky is determined through ay in the
same way as for flat terrain.
-1,
Table 1. Values of mean longitudinal velocity (m s ) near valley 8
Height above Flat
the ground (mm) terrain
5
15
30
50
75
100
1.61
2.11
2.43
2.66
2.85
2.98
upstream
edge
x/a = -1.0
meas
2.09
2.55
2.85
3.08
3.22
3.33
calc
1.97
2.46
2.79
3.00
3.16
3.25
center
x/a = 0.0
meas
1.21
1.36
1.59
1.89
2.17
2.42
calc
1.14
1.30
1.56
1.88
2.18
2.40
downstream
edge
x/a = 1.0
meas
2.12
2.86
3.05
3.17
3.21
3.30
calc
2.00
2.70
2.90
3.04
3.13
3.22
-1.
Table 2. Values of mean longitudinal velocity (m s ) near valley 5
Height above Flat upstream
the ground (mm) terrain edge
x/a=-1. 0
5 1.61
15 2.11
30 2.43
50 2.66
75 2.85
100 2.98
Equation (10) shows that
meas
2.21
2.63
2.95
3.11
3.26
3.30
for flat
calc
2.10
2.56
2.87
3.04
3.17
3.27
terrain
cent er
x/a = 0.0
meas calc
-
0.26
0.53
0.87
1.25
1.87
<7y IS
0.11
0.27
0.58
0.97
1.47
1.96
proportional
downstream
edge
x/a = 1.0
meas
2.47
2.97
3.00
3.06
3.18
3.29
to
calc
2.33
2.79
2.86
2.97
3.12
3.21
(Note that in complex terrain, U = u3). For more convenient
comparison of formulas for flat and complex terrain, Eq. (20) may be rewritten
in the form
t = a^ , (24)
where x, = f ".ffif! dx'. In this case, by using Eq. (24), Eq. (10) can be
Jo usvx i
70
-------
rewritten as
onx, when x< ģ 0
2 L x, when x1 ģ oo
(25)
When calculating pollutant dispersion over flat terrain it was assumed,
as is usually suggested for Gaussian-type diffusion models, that a. does not
a
depend on source height. Wind-tunnel measurements (Khurshudyan et a/., 1981)
support this assumption. Nevertheless, in complex terrain the distortion of
turbulent flow characteristics near the ground may noticeably change the
situation. Only a few measurements of crv had been made in the wind tunnel at
the time the calculations were made. Therefore, the values of o. used in Eq.
V
(25) were chosen by comparing values of cry calculated from this formula with
those obtained from measured lateral profiles of concentration. Direct
measurements of av showed that at the center of valley 5 and at heights below
100 mm, values of <7V are about twice those for flat terrain.
When comparing calculated and measured values of ay, it is best to
incorporate the finite size of the release. The finite size of the source can
be accounted for to some extent by determining the distance a virtual point
source would need to be located upstream of the true source position to
produce a similar plume. By measuring lateral and vertical profiles of
concentration near the source over flat terrain, this distance was determined
to be approximately 26 mm. For sources near the valley, the distance from the
real source to the virtual source would likely depend on the source location.
Since no evaluations were made of the virtual source distances for sources
within the valleys, the calculations were made by using distances from the
center of the true source. For that reason, calculated values of ay are given
below for ranges of Xg corresponding to positions where measurements were
made. For valley 8 with a source height of 29 mm at position x/a = 1.0, the
measured ay(Xg = 176 mm) = 23 mm. The calculated values of ay for this source
location are ay(176 mm) = 19 mm, <7y(199 mm) = 22 mm, and ay (211 mm) = 23 mm.
For the source at position x/a = 0.0, all other parameters the same, the
measurements give cry(88 mm) = 23 mm and the calculated values are ay(88 mm) =
17 mm, <7y(111 mm) = 22 mm, and ay(123 mm) = 24 mm. All calculated values of
av were obtained by taking a. = 0.14, as was the case for flat terrain. The
7 y
comparison of values of
-------
the case of valley 5 with a source height of 29 mm at position x/a = 0.0, the
measurements of av suggest using a. = 0.28. Corresponding calculations give
<7y(88 mm) = 78 mm, ay(99 mm) = 86 mm; the measurements gave ay(88 mm) = 78 mm.
These comparisons show that the model used for calculating crosswind pollutant
spreading predicts satisfactory values of ay.
The vertical eddy diffusivities km were calculated directly by using Eq.
(6) with the lower limit of integration changed to h(x), the local depth of
the valley. Because
-------
1000
100 -
E
N"
10 -
0.0001
0.001
0.01
0.1
Figure 33. Vertical profile of diffusivity km over valley 8.
73
-------
1000
100
E
10 -
0.0001
0.001
0.01
0.1
Figure 34. Vertical profile of diffusivity km over valley 5.
74
-------
where ,
f = exp ( - f -?- dz-1 . (27)
l> Jo K '
Eq. (26) was solved with an explicit-implicit scheme (except for a few steps
on x* near the source) which is conservative in the sense of preserving the
integral horizontal pollutant flux.
By using calculated maximum ground-level concentrations xmx 'or sources
of different heights and locations, as well as those Xmx for sources situated
over flat surfaces, terrain amplification factors A = Xmx/Xmx were derived.
Values of A depend on source height, source location relative to the valley,
and the shape of the valley. Table 3 presents the calculated values of A for
valley 8. For comparison, the values derived from the experimental data are
shown. The agreement between calculated and experimental results is quite
satisfactory in general. The discrepancies are mostly in the range of 30% and
often significantly less. The same is true for the downwind distance from the
source to the location of the maximum ground-level concentration. Consider
the calculated values of A, shown in Table 3, when the source is located at
the position one-quarter of the half-width of the valley upwind of the center
of the valley. It is seen that for low sources, A is largest when the source
is located slightly upwind of the valley center. For a hill, however, the
largest TAFs occur when the source is located just above the downwind base of
the hill. This has been discussed by Berlyand ef a/. (1983), who made
theoretical calculations of flow structure and pollutant dispersion near
two-dimensional valleys.
The numerical model is not strictly appropriate for calculating pollutant
dispersion in flows such as measured for valley 5 in the wind tunnel. The
streamlines of the flow near the bottom of the valley are severely distorted,
and the wind speed close to the ground is much less than that over flat
terrain. In this case, the diffusion equation without a term for longitudinal
turbulent diffusion might produce large errors in the predicted pollutant
spread. Moreover, the generation of the mean wind field in the stagnating
flow region near the bottom of the valley demands more refined computational
techniques than used for smoother flows. When there are large gradients of crv
in the longitudinal and vertical directions, as was suggested from measure-
ments with valley 5, the method described above for calculating ay should be
modified in some way to incorporate the dependence of av on x and z.
Nevertheless, the calculations of TAFs for valley 5 were made, and the results
75
-------
are presented in Table 4. It should be noted here that, according to measured
data, for low sources placed near the bottom of valley 5, the values of av
used in the calculation of ay were 1.5 to 2.0 times those for flat terrain.
One can see that despite the above-mentioned difficulties, the results are
reasonable from a practical point of view.
Contour maps of constant TAP as predicted by the model are shown in Fig.
35. These are to be compared with the measurements shown in Fig. 25. The
maps for valley 8 show generally similar overall patterns, but differ in
several details. The vertical extent of the 40%-excess window (the TAF = 1.4
contour) extends to about 1.5 h from the measurements, but to only 1.25 h from
the model predictions. The horizontal extent of the measured window is
Table 3. Values of TAF for sources near valley 8
Source
height x/a
(Hs/h)
0.25
0.50
1.00
1.50
2.00
meas
0.97
0.80
0.99
1.13
-
= -1
calc
0.93
0.96
0.96
0.96
0.96
x/a = -0.5
meas
2.50
2.05
1.73
1.58
1.32
calc
2.14
1.96
1.50
1.35
1.22
Source Location
x/a=-0.25 x/a = 0.0
meas calc
- 2.53
- 2.36
- 1.88
- 1.62
~ 1.46
meas
2.30
2.50
2.28
2.53
1.76
calc
2.02
1.97
1.87
1.65
1.50
x/a = 0.5
meas
-
1.42
1.60
1.61
1.45
calc
1.09
1.27
1.25
1.24
1.17
x/a=1.0
meas
0.74
0.83
1.13
1.13
-
calc
0.89
0.86
0.87
0.92
0.92
Table 4. Values of TAF for sources near valley 5
Source
height
(Hs/h)
0.25
0.50
1.00
1.50
2.00
x/a
= -1
meas calc
0.62
0.84
1.31
1.21
0.89
1.32
1.12
1.01
0.98
Source Location
x/a=-0.5 x/a = 0.0 x/a = 0.5
meas calc
3.19
3.34
2.72
2.00
1.61
4.75
3.91
2.05
1.89
1.45
meas
5.01
6.71
5.57
3.82
2.36
calc
7.71
4.67
3.35
2.98
2.06
meas
2.15
2.83
2.21
2.05
1.58
calc
2.16
2.26
1.92
1.52
1.41
x/a=l.O
meas
0
0
1
1
.80
.99
.10
.26
_
calc
0.80
0.84
0.85
0.97
1.05
76
-------
z/h
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-1.0
-0.5
0.0
x/a
0.5
1.0
1.5
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
Z/h
-1.5
-1.0
-0.5
0.0
x/a
0.5
2.5
2.0
1.5
1.0
.0.5
0.0
-0.5
-1.0
1.0
1.5
Figure 35. Contours of constant terrain amplification factor
derived from model calculations.
77
-------
somewhat larger than that of the model-predicted window. The model-predicted
window is shifted somewhat upstream and whereas the resolution of the grid
used for the experimental measurements was rather coarse, a hint of an
upstream shift is also observed there. The model generally predicts larger
TAFs to occur at lower elevations, whereas the measurements show elevated
maxima. Both predicted and observed TAFs were generally less than unity when
the source was at the upstream or downstream edge of the valley. Maximum TAF
values are quite close to one another. Generally similar statements may be
made when comparing the calculated and observed TAF maps for valley 5, but the
differences are somewhat larger.
78
-------
5. CONCLUSIONS
The laboratory work has provided a reasonable simulation of the flow
structure and diffusion characteristics of the neutral atmospheric boundary
layer. The model valleys were idealized in shape, but should cover the range
of a majority of valleys to be found at full scale, at least in terms of the
basic classes of flow structure that may be observed. Valley 8 was rather
gentle in slope, and the flow over it may be characterized as relatively
smooth and well-behaved. Valley 5, being steeper in slope, caused the flow to
separate intermittently, but not in the mean. In valley 3, the steepest, the
flow clearly separated a short distance from the upstream edge, and a
recirculating flow was formed within the valley. Pollutants released at the
same relative locations within each of these valleys behave very differently
from one another, and the resulting surface concentration patterns are
dramatically different.
The overall effects of the valleys on surface concentrations are
characterized in terms of terrain amplification factors (TAFs), defined as the
ratios of maximum ground-level concentrations from sources located within the
valleys to the maxima that would exist from identical sources located in flat
terrain. Maps of these TAFs are provided for each valley. Also provided are
maps detailing the distances to locations where these maximum ground-level
concentrations will occur. These maps allow a practitioner to quickly and
easily assess the likely impact of a source located in a valley, and the
location where that maximum impact will occur.
A two-dimensional theoretical model that uses a variational analysis
technique was applied to the wind-tunnel measurements of the flow structure
near the valleys to produce mass-consistent mean wind fields. Measurements of
the turbulent fluctuating velocities were also used to calculate vertical and
crosswind eddy diffusivities. The diffusion equation was then solved
numerically to obtain maximum ground-level concentrations from elevated point
sources of various heights near valleys 8 and 5, as well as over flat terrain.
The calculated and measured concentrations for flat terrain showed good
agreement.
Comparison of calculated and measured TAFs for valley 8 showed
satisfactory agreement. Valley 5 exhibited more severe streamline distortion
and a stagnation region with large fluctuating velocities near the bottom of
the valley, and therefore the differences between calculated and measured TAFs
79
-------
were rather significant in some cases. The absence of the longitudinal
diffusion term in the diffusion equation (turbulence intensities in the bottom
of valley 5 were extremely large) and the complexity of modeling eddy
diffusivities in such flows affect the performance of the numerical model. A
better method of incorporating the effects of crosswind dispersion should be
developed, because even in these two-dimensional mean flows, turbulent
diffusion is a three-dimensional phenomenon. These effects are more
pronounced in the case of complex terrain.
Finally, it may be concluded that, in spite of the many problems,
K-theory may be applied with a reasonable measure of success to the
calculation of maximum ground-level concentrations from sources in curvilinear
flows, even near rather steep obstacles. More sophisticated modeling of
diffusivities and use of an elliptical diffusion equation should improve upon
the present results.
80
-------
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