vvEPA
United States
Environmental Protection
Agenc,
Research and Development
Environmental Sciences Research EPA-600, 4-79-053
Lut." story September 1 979
ReseareJh Triangle Park NC 27711
^*«b **f S i»
Ar\.
Basic Studies of
Flow and Diffusion
Over Hills
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U S Environmental
Protection Agency, have been grouped into nine series These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are
1. Environmental Health Effects Research
2 Environmental Protection Technology
3 Ecological Research
4 Environmental Monitoring
5 Socioeconomic Environmental Studies
6. Scientific and Technical Assessment Reports (STAR)
7 Interagency Energy-Environment Research and Development
8 "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the ENVIRONMENTAL MONITORING series.
This series describes research conducted to develop new or improved methods
and instrumentation for the identification and quantification of environmental
pollutants at the lowest conceivably significant concentrations It also includes
studies to determine the ambient concentrations of pollutants in the environment
and/or the variance of pollutants as a function of time or meteorological factors.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/4-79-053
September 1979
BASIC STUDIES OF FLOW AND
DIFFUSION OVER HILLS
by
S. P. S. Arya
Department of Geosciences
North Carolina State University
Raleigh, North Carolina 27650
and
J. C. R. Hunt
Department of Applied Mathematics
and Theoretical Physics
University of Cambridge
Cambridge, CB3 9EW, England
Grant Number R-804653
Project Officer
William H. Snyder
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratory, U.S. Environmental Protection Agency, and approved for publication.
Approval does not signify that the contents necessarily reflect the views and
policies of the U.S. Environmental Protection Agency, nor does mention of
trade names or commercial products constitute endorsement or recommendation
for use.
ii
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ABSTRACT
This research program was initiated with the overall objectives of gain-
ing a better understanding of flow and diffusion of pollutants in complex
terrain under both neutral and stably stratified conditions, providing a sound
data base for testing existing theories and developing new theories of flow
and diffusion around isolated hills and ridges. To this end, experiments were
conducted with models of a bell shaped hill and a 2-D steep ridge in EPA's
meteorological wind tunnel and salt-water stratified towing tanks. Measure-
ments were made of the flow structure, as well as the concentration patterns
around the hills due to point sources located at different heights and posi-
tions relative to the hills.
The experiments on stably stratified flow over a 3-D hill verify and
establish the limits of applicability of Drazin's theory for small Froude
numbers. The location of the surface impingement point from an upwind
pollutant source can be identified under a wide range of atmospheric
conditions.
The experiments on the neutral boundary layer flow over a 2-D ridge
show that significant ridge effect is felt by turbulence structure to
distances greater than eighty ridge heights downstream. Ground-level concen-
trations in the lee of the ridge are very sensitive to the source height and
position relative to the ridge.
This report was submitted in fulfillment of Grant No. R-804653 by North
Carolina State University under the sponsorship of the U.S. Environmental
Protection Agency. This report covers a period from September 1, 1976 to
August 31, 1977.
iii
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CONTENTS
Abstract iii
Figures vi
Symbols vii
Acknowledgments viii
1. Introduction 1
2. Conclusions 3
3. Experiments and Apparatus 5
3.1 Large Towing Tank Experiments 5
Flow visualization , , . , , 5
Concentration measurements ,.......,., 6
3.2 Small Towing Tank Experiments 6
Surface stress patterns ..... , 7
Hydrogen bubble visualization . . . , 7
3.3 Wind Tunnel Experiments 7
Mean velocity and turbulence measurements ........ 8
Flow visualization 8
Concentration measurements 8
4. Results and Discussion 9
4.1 Flow Around A 3-D Hill 9
Flow visualization , . , 9
Criteria for separation 10
4.2 Diffusion Around A 3-D Hill 10
Plume growth and impingement 10
Concentration distributions . 11
4.3 Boundary Layer Flow Over A Steep Ridge 12
Mean flow 12
Turbulence structure 12
4.4 Diffusion Over A Steep Ridge 13
Source upwind of the ridge 13
Source at the ridge top 13
Source downwind of the ridge 13
References 15
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FIGURES
Number Page
1 Shadowgraphs of flow over 3-D hill (N = 1.33 rad/s) ....... 16
2 The variation of the maximum deficit in mean velocity in the
wake with distance behind the ridge .............. 17
3 Variations of the maximum perturbations in Reynolds stress and
velocity variances in wake with distance behind the ridge. . . 18
4 The iBflTHmiim ground-level concentration as a function of source
height and position relative to the ridge ........... 19
5 Distance of the maximum g.l.c. from the source as a function
of source height and position relative to the ridge ...... 20
vi
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SYMBOLS
f(r)
F
g
h
Hd
H
L
N
r
u
s
u1
U(h)
U
CO
w1
x
xs
z
A^
max
A(w'2)
e
x
p
x
6
'max
'max
max
height of hill as a function of radius
Froude number = U /Nh
00
Froude number = U /NL
OO
acceleration due to gravity
maximum height of hill or ridge
height of the dividing streamline or plume impingement
source height above the level surface
half-length of the hill in x-direction
Brunt-Vaisala frequency = (•*• -r^-)
radial coordinate in horizontal plane
mean velocity in x-direction
velocity fluctuation in x-direction
mean velocity at z = h in the absence of hill
ambient uniform velocity
velocity fluctuation in z-direction
longitudinal coordinate axis; also the distance from the hill
distance of the source from the ridge
vertical coordinate
maximum perturbation in mean velocity
maximum perturbation in the Reynolds stress
maximum perturbation in the variance of longitudinal velocity
fluctuations
maximum perturbation in the variance of vertical velocity
fluctuations
angle in the polar (r, 0, z) coordinate system
wavelength of lee wave
density
normalized concentration
boundary layer thickness
vii
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ACKNOWLEDGMENTS
We are grateful to Messrs. Roger Thompson and Daniel DoIan for help with
the photographs, and to Messrs. Mike Shlpman, Robert Lawson, Lewis Knight,
Leonard Marsh, and the late Karl Kurfis for help with running the experiments.
Kike Shipman was primarily responsible for collecting and reducing the wind
tunnel data. The cooperation and help of Dr. William Snyder in planning and
day to day conduct of experiments is gratefully acknowledged.
viii
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SECTION 1
INTRODUCTION
A series of laboratory experiments on neutral and stably stratified flows
over isolated topographical features was conducted in 1977 under a grant from
the Environmental Protection Agency. For these experiments, we used the
meteorological wind tunnel, salt-water stratified towing tank and other
support facilities and instrumentation of the EPA Fluid Modeling Facility in
Research Triangle Park, NC. This report gives a summary of the research work
done. More detailed and fuller descriptions are given in the following
technical reports and papers:
1. "Flow structure and turbulent diffusion around a three-dimensional
hill - Fluid modeling study on effects of stratification, Part I -
Flow Structure". By J.C.R. Hunt, W.H. Snyder and R.E. Lawson, Jr.
U.S. EPA Report EPA-600/4-78-041, Environmental Sciences Research
Laboratory, Research Triangle Park, NC.
2. "Flow structure and turbulent diffusion around a three-dimensional
hill - Fluid modeling study on effects of stratification, Part II -
Surface concentrations due to upstream sources". By J.C.R. Hunt and
W.H. Snyder> EPA report under preparation.
3. "A model study of boundary layer flow and diffusion over a ridge".
By S.P.S. Arya and M.S. Shipman; Preprints, Fourth Symposium on
Turbulence, Diffusion and Air Pollution, January 15-18, 1979, Reno,
Nevada, American Meteorological Society, pp. 584-591, 1979.
In addition to the work reported above, Dr. J.C.R. Hunt also authored or co-
authored the following papers while working on this project:
4. "A review of the theory of rapidly distorted turbulent flows and its
applications". By J.C.R. Hunt, paper presented at XIII Biennial Fluid
Dynamics Symposium - Advanced Problems and Methods in Fluid Dynamics,
Warsaw, Poland, September 5-10, 1977.
5. "Turbulent diffusion from sources near obstacles with separated wakes.
Part I. An eddy diffusivity model". By J.S. Puttock and J.C.R. Hunt.
J. Fluid Mech.. 1978.
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6. "Distortion of turbulence by a circular cylinder." By R. E. Britter,
J. C. R. Hunt and J. C. Mumford. J. Fluid Mech., 92. 269-301, 1979.
S7. "A Lagrangian statistical analysis of diffusion from a ground-level sorce
in a turbulent boundary layer." By J. C. R. Hunt and A. H. Weber.
Quart. J. Roy. Meteor. Soc., 105, 423-443, 1979.
8. "Highway modeling. Part I: Prediction of velocity and turbulence fields
in the wake of vehicles." By R. E. Eskridge and J. C. R. Hunt. J. Appl.
Meteor., 18, 387-400, 1979.
9. "Highway modeling. Part II: Advection and diffusion of SF, tracer gas.
By R. E. Eskridge, F. S. Binkowski, J. C. R. Hunt, T. L. Clark and K. L.
Demerjian. J. Appl. Meteor., 18, 401-412, 1979.
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SECTION 2
CONCLUSIONS
The most significant conclusions that can be drawn from our physical
modeling studies of flow and diffusion over an isolated bell-shaped hill and
a 2-D steep triangular ridge are as follows.
Flow and Diffusion Around a 3-D Hill
1. For the stratified flow past an isolated hill, the criteria for the
occurrence and location of separation on the lee side are governed by the
gross slope of the hill and the ratio of the wavelength of lee waves to the
length of the hill in the direction of flow. For hills of moderate slopes,
the Froude number (F) essentially governs the phenomena of separation,
hydraulic jump, lee waves and rotors.
2. The Froude number based on the hill height (h) also determines the
criterion of whether the approach flow at some level will go over the top of
the hill, or it will go around the sides. According to our flow visualization
studies, the height of the dividing streamline H^ = h(l-F), which gives a
simple criterion for the impingement of plumes from upwind sources. If the
source height (Hs) is smaller than h(l-F) and the source is located on the
stagnation streamline, then the plume will impact on the hill surface,
bifurcate, and go around the sides, causing probably the highest ground-
level concentrations (g.l.c.) along the area of impaction. However, slow
oscillations in the flow direction will cause the area of impingement to
increase and hence lower the maximum concentration.
3. When F < 1, and Hg - h, the ratio of the maximum concentration on the
surface of the hill to the maximum concentration in the plume in the absence
of the hill lies between 0.5 and 1.2; for F > 1, the above ratio decreases
rapidly with increasing Froude number.
4. If a plume goes over the hill, its vertical width is reduced by the con-
verging streamlines in the vertical plane, and the lateral width is amplified
by the divergence in the horizontal plane. There is also an indirect effect
due to the density gradient being increased by the convergence of stream-
lines in the vertical plane.
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Flow and Diffusion Over a Steep Ridge
1. The separation bubble or cavity region behind the ridge extends to a dis-
ttance of 13h in the longitudinal direction and 2.5h in the vertical and is
characterized by greatly reduced but circulating mean flow and very high
intensities of turbulence.
2. Beyond the cavity region there is an extensive wake region whose height
varies as x**. It is also characterized by reduced mean flow and increased
turbulence, but the perturbations caused by the ridge decay monotonically
with distance (x) behind the ridge.
3. The maximum perturbations in the mean velocity, the Reynolds stress and
variances of velocity fluctuations in the wake all decay with distance as
x~l, while the heights of maxima vary as x*5. Even for this ridge of low
height (about 1/10 of the boundary layer thickness), the perturbations caused
by the ridge are very large in the near wake region and remain significant to
a distance of as large as 80h.
4. When an elevated source of height _>_ h is located upwind of the ridge or
at the ridge top, the effect of the ridge is generally to reduce g.l.c. down-
wind of the ridge. For the ground sources, however, a good part of the plume
may impinge on the separation streamline and get trapped in the cavity region,
so that the maximum g.l.c. in the cavity region may become much larger than
the g.l.c. at the same distance in the absence of the ridge. Thus the ground-
level concentrations downwind of the ridge are very sensitive to both the
source height and its position relative to the ridge.
5. When the source is located downwind of the ridge, the increased turbulence
in the wake results in much lower concentrations aloft, but higher concen-
trations at the ground-level. The highest ground-level concentrations occur
when the source is located within the cavity region near the base of the
ridge. The peak g.l.c. decreases and Its position shifts farther downwind
from the source, as the source height and its distance from the ridge
Increase.
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SECTION 3
EXPERIMENTS AND APPARATUS
The experiments were conducted in the towing tanks and the meteorological
wind tunnel of the EPA Fluid Modeling Facility. In towing tanks, models of
three-dimensional polynomial hills were towed at various speeds and for vari-
ous degrees of stable stratification. In the wind tunnel two types of experi-
ments were conducted to study (1) the flow and diffusion over and around a
three-dimensional hill of height much larger than the surface boundary layer
thickness, and (2) the flow and diffusion over a two-dimensional ridge of
height much less than the surface boundary layer thickness. The wind tunnel
experiments corresponded to neutral stability.
3.1 LARGE TOWING TANK EXPERIMENTS
The large towing tank is 25m long, 2.4m wide and 1.2m deep. A towing
carriage mounted on rails allows models to be towed the length of the tank at
variable speeds between 5 and 50cm/sec. The tank can be filled layer by layer
with salt water to obtain a desired stable density profile in about four hours
(Thompson and Snyder, 1976). The density profiles are measured by with-
drawing samples from various depths in the tank and measuring their specific
gravity with precision hydrometers or an electronic balance.
The 3-D model hill was made of an acrylic plastic sheet by vacuum mold-
ing onto a wooden former. The hill profile was close to a fourth order poly-
nomial
f(r) = h/[1-K^) ],
where h is the maximum height in the center and L is a horizontal length
scale of the hill such that at r = L, f(r) = h/2. With our particular choice
of h = L a 0.23m, the bell-shaped hill had a maximum slope of about unity.
The hill was mounted on a flat base plate and towed upside-down across the
water surface. Twenty-eight sampling ports were fixed on the surface of the
hill along each of the radial lines 6=0, -90, -165 and 180°.
Flow visualization
As part of our study of diffusion and dispersion over the polynomial
hill, a neutrally buoyant dye was emitted from a model stack located at 4h
upstream of the center of the hill. The stack height HS was varied from 0 to
1.2h. In other tests, dye was released from the surface sampling ports on the
windward and leeward hill lines (6 = 180° and 0°) to study the surface flow
patterns, or alternately, through an injection rake emitting dye at several
levels above ground to study the centerplane streamlines.
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Color photographs and motion pictures were taken of the surface dye re-
leases and black and white photographs were taken of the upstream multi-level
dye releases. In some cases, shadowgraphs were taken of the flow patterns be-
hind the hill.
In order to visualize the surface flow patterns, granules of potassium
permanganate (KMnO^) were cemented to the hill surface. These granules
dissolved rather slowly as the hill was towed through the tank, yielding
bright purple streamers indicating the surface flow patterns.
One series of tests was run for various combinations of stratifications
and stack heights in order to determine (a) whether the plume went over the
hill or around it, (b) the impingement height (H^) where the maximum concen-
tration on the upstream centerline on the hill surface is expected to occur,
and (c) the streamline deflection at the side of the hill (9 = -90'). Sam-
ples were drawn during the tow simultaneously through all the surface ports.
The collected sample jars were then visually inspected to determine which
contained the highest concentration of dye.
Concentration measurements
In our studies of diffusion and dispersion from a model stack upstream
of the hill, the ground-level concentrations on the hill were measured by
drawing samples of the dye mixed fluid through the various ports on the hill
surface. For concentration measurements in the wake of the hill, the samples
-were collected by drawing the fluid through sampling ports of the rake, whose
height and lateral position were adjusted as desired. Dye concentrations in
the collected samples were measured by passing a beam of light through a test
tube filled with the sampled fluid and measuring the intensity of the light
coming on the other side of the tube by means of a photo cell.
In order to bring out the effect of the hill, the concentration measure-
ments were also made in the absence of the hill for the same stratifications
and for the same source heights as used In the hill study.
3.2 SMALL TOWING TANK EXPERIMENTS
In the large towing tank the residual flow requires about an hour to set-
tle down after a tow, and it takes at least two hours to change a model. For
some quick qualitative experiments a small towing tank was found to be more
desirable. Smaller tank was also considered more suitable for flow visuali-
zation with hydrogen bubbles. We used a 2.0m long, 0.20m wide and 0.10m deep
tank. The tank was filled with a stratified salt solution with specific
gravity varying from 1.0 at the surface to 1.2 at the bottom. Model hills
were mounted on a base plate suspended from a carriage and towed at speeds
ranging from 2 to 25cm/sec. The polynomial hill used here had about the same
shape as the large hill, but only 2cm in height.
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Surface stress patterns
Shear stress patterns on the surface were observed Ly coating the hill
model with a gelatinous solution of dye and KNOX brand gelatin. As the model
was towed, the dye was sheared away in regions of high stress and tended to
collect along the stagnation areas, leaving a visual record of the surface
flow patterns. Due to certain difficulties, however, this technique was found
to be much less satisfactory than the corresponding wind tunnel technique
using zinc oxide powder and oil.
Hydrogen bubble visualization
A hydrogen bubble wire system was developed to study the streamline
pattern and the velocity field on the centerline of the hill. A 0.025mm
diameter chromel thermocouple wire was kinked by running it between two gears.
This provided a very uniform spacing of streaks. The bubble size could be
varied by varying the current flow through the wire. Photographs were taken
with a 35mm camera, and velocities were obtained from these photos by
determining the distance between successive bubble streaks.
3.3 WIND TUNNEL EXPERIMENTS
The EPA meteorological wind tunnel (Thompson and Snyder, 1976) has a test
section 18m long, 3.7m wide and 2.1m high. The air speed in the test section
may be varied from 0.5 to lOm/sec. The tunnel ceiling was adjusted to obtain
a zero pressure gradient in the core region.
Two basically different types of experiments were conducted in the wind
tunnel. In the first category a three-dimensional polynomial hill identical
to that used in the large towing tank was placed near the entrance to the
test section. The boundary layer over the smooth tunnel floor was approxi-
mately 65mm thick at the center of the hill (but in the absence of it) of
height 230mm.
In the second category of experiments in the wind tunnel, a symmetric
triangular ridge of the height to base ratio of unity was placed within an
artificially thickened turbulent boundary layer developed on the rough tunnel
floor. The ridge height of O.lm was about 1/10 the undisturbed boundary
layer thickness at the location of the ridge (8.2m from the entrance). The
boundary layer was artifically thickened by placing a 0.15m fence at the
entrance to the test section.
In both cases, measurements were made of the mean velocity, variances of
the longitudinal and vertical velocity fluctuations and the Reynolds stress
at various positions in the boundary layers with respect to the hill. With a
point or line source located at various positions upstream and downstream of
the ridge, measurements were also made of the longitudinal ground-level con-
centrations as well as of the lateral and vertical concentration distribu-
tions. Similar measurements were made in the absence of the hill or the
ridge.
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Mean velocity and turbulence measurements
Turbulence measurements were made with two hot-film anemometers whose
outputs were digitized and processed on a PDF 11/40 minicomputer. Averaging
time of one minute was found to yield reasonably repeatable results. Probes
were calibrated next to a pitot tube in the free stream flow in the core
section of the wind tunnel. A computer program calculated the average
1 TJ»2
velocity u, the longitudinal and vertical velocity variances u'z and w'z, the
Reynolds stress -u'w1 and the flow angle.
Flow visualisation
A paraffin oil-fog generator was used to produce smoke for the qualita-
tive flow visualization studies. In this generator, paraffin oil is aspirated
onto a heating element which creates a fine oil-fog. A separate air supply
then carries the smoke into the wind tunnel.
One phase of this study involved visualization of smoke emitted from the
stack upstream of the hill. Plume centerlines were traced from photographs
at. various stack heights in order to obtain an idea of the centerline stream-
line pattern over the hill.
Photographs were also taken of the smoke being emitted at low speed
through the surface sampling ports to obtain an idea of the surface flow
pattern. Finally, drops of titanium tetrachloride were placed at various
positions on the hill surface. This created a dense white smoke that was also
helpful for understanding the surface flow pattern.
Concentration measurements
The source used in the wind tunnel was an air-methane or air-ethylene
mixture emitted as a tracer gas located at various heights and positions with
respect to the hill. Concentrations were measured at each selected position
by passing a sample of air mixed with tracer gas through a Beckman Model 400
Hydrocarbon Analyzer. The sampling rate and source parameters were fixed on
the basis of previous experiments (Huber and Snyder, 1976; Huber et al, 1976)
using the same system.
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SECTION 4
RESULTS AND DISCUSSION
Here we present only a summary of our experimental results, which have
been discussed in considerably more detail in our other reports referred to
earlier in section 1.
4.1 FLOW AROUND A 3-D HILL
Flow Visualizations
Hunt et al (1978) have presented photographs of various types of flow
visualizations around the polynomial or bell-shaped hill for various values
of Froude number in the range 0.2 to °°. Here the Froude number, F = U^/Nh, is
based on the ambient flow speed or towing speed U,,,, the hill height h and the
Brunt-Vaisala frequency
••«?£>*•
For illustration purposes, shadowgraphs of flow at various values of F are
shown in Figure 1.
At low Froude numbers (F = 0.1 - 0.2), the flow is constrained to move
in essentially horizontal planes around the hill, except in a narrow region
near the hill top where it can go over the top. In the middle region, the
centerline plume bifurcates after its impingement on the upstream face of the
hill, it dips slightly as it goes around the sides, then rises again as it
separates from the surface of the hill. Separation occurs at an angle of
approximately 110° from the upstream stagnation line. The wake region behind
the hill has more of a 2-D rather than 3-D character; it has a symmetric pair
of more or less vertically oriented vortices. The flow structure and the
observed streamline deflections in going around the sides of the hill are
found to be consistent with Drazin's (1961) theory, which is asymptotically
valid for F -»• o. This theory breaks down, however, within a distance of the
order of Fh from the submit. In this top region the flow is largely three
dimensional and the streamline passing over the submit separates within a
short distance on the lee side followed by the appearance of a slight hydrau-
lic jump (see Figure 1).
As the Froude number is increased, beyond 0.2 the region where the flow
goes over the top becomes increasingly broader, the separation of the center-
line streamline passing over the hill top occurs farther and farther down-
stream, and the hydraulic jump becomes more prominent (Figure 1). At F = 0.9,
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almost all the flow in the center-plane goes over the top of the hill, and
does not separate until well beyond the lee side base of the hill. At
F = 1.0, however, the flow is again at the verge of separating just past the
top of the hill.
As the Froude number is increased farther (F > 1), the flow separates near
the top of the hill, the size of the cavity or recirculating region grows and
the wake region also grows in both the lateral and vertical directions. At
F = 1.7, there is a strong resemblance with the neutral flow (F = °°) except
that in the slightly stratified case the streamlines are more closely spaced
and they lose elevation much faster in the wake region than those in the
neutral flow.
Smoke visualization in the wind tunnel (F = °°) show the plumes to spread
broadly to cover the entire hill surface. Flow separates slightly upwind of
the top and there is up-slope flow on the lee side of the hill. The mean
velocity vectors show significant vertical components as far as 2h upstream
and well above the level of the summit. Directly above the hill top and on
the sides, however, the vertical components are essentially zero.
Criteria for Separation
An important consideration is the existence and position of the separa-
tion of flow on the lee side of hill. Flow separation is expected to be
governed by the gross slope (h/L) of the hill, as well as by the ratio of the
fundamental wavelength of the lee waves (X = 2irU /N) to the half length of
the hill (typically, 2L). The latter ratio is proportional to the Froude
number F^ = U^/LN, defined on the basis of the length scale L.
When the gross slope is small (h/L « 1), separation is not likely to
occur and when it is large (h/L » 1), separation is almost certain to occur.
For hills of moderate slopes (h/L -1), as in our case, the occurrence and
location of separation is critically dependent on the Froude number FL = F.
When the stratification is such that X - 2L, separation of the flow over a
rounded hill (h/L ~ 1) will be suppressed. If these two lengths (X and 2L)
differ considerably from each other, separation would occur on hills of
moderate slope. If X « 2L (or, FL « 1), separation is controlled by the
pressure distribution produced by the lee wave pattern. But, if X » 2L
(or, F^ » 1) , separation on the hill is controlled by the boundary layer
flow. These criteria for the occurrence of separation have been suggested by
recent experimental and theoretical work on stratified flow over 2-D hills
(Brighton, 1977; Sykes, 1978). Our experimental results for the 3-D poly-
nomial hill with h/L = 1 are found to be in general agreement with the above
criteria.
4.2 DIFFUSION AROUND A 3-D HILL
Plume Growth and Impingement
The flow patterns not only indicate the path of the center line of the
plume but also its growth in the vertical and the lateral directions.
10
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Observations of plumes in the absence of the hill clearly show that the
stronger the stratification, the narrower becomes its vertical dimension and
the wider becomes its lateral dimension.
Over the hill the vertical plume width is reduced by the streamlines
converging in the vertical plane, and the horizontal width is amplified by
the divergence in the horizontal plane. There is also an indirect effect due
to the density gradient being increased by the convergence of the streamlines
in the vertical plane. For a plume starting upstream the latter effect mainly
reduces the growth of the plume width (a ), while the former effect actually
reduces the width of the plume.
In stably stratified flows around 3-D hills, of great practical impor-
tance is the height H, of the dividing streamline, above which the flow goes
over the hill and below which it goes around the sides. The criterion for
determining whether the plume will impact on the hill surface and go around
the sides is given by the relation H^ * h(l - F), which was suggested by Hunt
et al (1978) on the basis of Drazin's (1961) theory and confirmed by our
towing tank experiments. If the plume height upstream, which is the same as
the source height Hg, is smaller than h(l - F), the plume will impact on the
hill surface; otherwise it will go over the top. The former situation is
likely to result in the highest ground-level concentrations.
Although no source was placed downstream of the hill in our experiments,
it is evident from the mainly horizontal circulation patterns in strongly
stratified flow (F < 0.3) that a plume below h(l - F) would also impinge on
the hill surface. For weaker stratifications, the plume could also be
brought down to the ground as a result of lee waves and rotors induced by the
hill.
Concentration Distributions
Because it was of considerable interest to compare concentrations mea-
sured on the hill with those in the plume in the absence of the hill, the
baseline concentration distributions in the plume were measured for the same
source heights and Froude numbers. The lateral concentration profiles are
very close to Gaussian. The vertical profiles for large stack heights (Hg >
0.5h) are roughly Gaussian, but for the smaller stack heights they are not
Gaussian, as one would expect. Since there is a turbulent boundary layer
growing along the base plate, the concentration profiles for the sources in-
side and outside this boundary layer are quite different.
With the hill in place, concentrations were measured on the hill surface
along various sectors. Concentrations were also measured in the wake region
downstream of the hill, the height and the lateral position of the sampling
rake being adjusted to measure maximum concentrations. At low Froude
numbers, the lateral concentration profiles in the wake show two distinct
maxima - one on each side of the wake, due to the plume impinging on the
surface, going around the sides and separating from the surface. At large
Froude numbers there is greater vertical and horizontal mixings and the con-
centration is approximately constant in the whole cavity region.
11
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Of considerably greater interest is the ratio of the maximum concentra-
tion Xmax on the surface of the hill to the concentration x i-n the center of
the plume in the absence of the hill. The broadest conclusion we could draw
is that when F < 1 and HS < h, Xmax/X0 lies between 0.5 and 1.2. When F > 1
and HS > 0.5h, the ratio Xmax/X decreases rapidly with increasing F.
At low values of F, when Hg < H^ (the height of impingement), the maximum
concentration occurs at the upstream face. A small displacement of the source
off the stagnation streamline does not change the magnitude of X^Y* but gen-
erally moves its location to the side of the hill. Thus slow oscillations in
the flow direction, whether caused by the wake or upstream eddying, affect
the plume upwind of the hill so as to increase its area of impingement on the
hill and hence lower the maximum concentration.
4.3 BOUNDARY LAYER FLOW OVER A STEEP RIDGE
Distributions of mean velocity, the Reynolds stress and variances of
velocity fluctuations in the absence of the ridge were quite typical of a
well-developed equilibrium boundary layer. The surface stress did not change
noticeably with distance along the tunnel floor.
Mean Flow
When a symmetric ridge of slope 2:1 was placed at the floor in the middle
of the test section, the boundary layer thickness (6) increased somewhat, but
it remained more or less constant (6 - lOh) behind the ridge. As expected,
the flow separated at the ridge top and reattached to the surface at a dis-
tance of about 13h. The recirculating cavity or bubble region is comparable,
in size and shape, to that behind a normal bluff plate.
The mean velocity profiles at various positions relative to the ridge
indicate that, in the lower layer, the flow decelerates slightly as it
approaches the ridge, accelerates by about 10% while passing over the ridge
and then considerably decelerates in the wake region behind the ridge. With
increasing distance (x) behind the ridge, the height of the wake grows as x^,
and so does the height of the maximum deficit (as compared to the case of no
ridge) in the mean velocity. The velocity deficit in the wake decreases in
inverse proportion to the distance from the ridge and, surprisingly, it takes
a distance of more than 80h for a full recovery of the undistrubed velocity
profile (see Figure 2).
Turbulence Structure
Changes in the Reynolds stress and the variances of velocity fluctuations
are rather insignificant upwind of the ridge. But, in the wake behind the
ridge turbulence is greatly enhanced. The vertical profiles of the Reynolds
stress and velocity variances in the wake are all characterized by a maxi-
mum whose position moves gradually upwards with increasing distance from the
ridge. The maximum perturbation in each of these quantities are found to
decrease in inverse proportion to the distance (see Figure ?}, similar KJ the
variation of the mean velocity deficit. Even at a distance of 80h, the
12
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maximum perturbations in Reynolds stress and velocity variances caused by
the ridge remain as large as the maximum values of these quantities in the
undisturbed boundary layer. Thus the effect of a steep ridge on turbulence
structure in its wake may persist for long distances.
The vertical distributions of the perturbation stress and variances are
qualitatively very similar. The positions of maxima in their profiles at a
given distance from the ridge lie within one ridge height of each other.
4.4 DIFFUSION OVER A STEEP RIDGE
Diffusion measurements were made in both the disturbed and the undis-
turbed boundary layers using the same point source. The ground-level
concentrations (g.l.c.) downstream of the ridge are found to be very sensitive
to the source height (H6) and its location (x,,) relative to the ridge.
a S
Source Upwind of the Ridge
For a point source or stack located upwind of the ridge, the surface
concentration increases as the source is brought closer to the ridge and they
decrease with the increase in source height. The concentrations in the cavity
region are by no means uniform and seem to depend on where and how much of the
plume comes in contact with the separation streamline.
Source at the Ridge Top
With a source located at the ridge top, the lateral diffusion on the lee
side of the ridge is enhanced considerably. The width of the plume at the
ground level at a distance lOh downwind of the ridge is about twice the width
of the plume at the same distance from the source without the ridge.
The vertical concentration profiles are also considerably modified by
the ridge. For an elevated source, the vertical profiles downwind show a
substantial plumerise. The concentration profiles in the cavity region are
remarkably uniform (see also Huber et al, 1976). The maximum concentrations
in the presence of ridge are 1/3 to 1/2 of the maximum values in the absence
of the ridge.
Source Downwind of the Ridge
For a source located in the cavity region, significant ground-level
concentrations are found both upwind (relative to the overall flow direction)
and downwind of the source. The position of the maximum g.l.c. is generally
found to be some distance (depending on the source height) downwind of the
source. For a near ground-level source, however, the recirculating flow
towards the ridge in the lower part of the cavity may cause the maximum g.l.c.
to occur slightly upwind of the source. The maximum concentration decreases
rapidly as the source height is increased.
When the source is located outside the cavity region, the maximum g.l.c.,
for a given source height, decreases with increasing distance of the source
13
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from the ridge and the position of the maximum shifts farther away from the
source. Thus the main effect of the ridge is to considerably enhance the
ground-level concentrations.
Figure 4 summarizes the results for the maximum g.l.c. for various source
heights and positions, and Figure 5 gives the distance of the maximum g^l.c.
from the source. It is seen that the ridge has the most effect when the
source is located about 5h downwind of the same and that the ridge effect
becomes insignificant when the source is located beyond a distance of about
30h from the ridge.
The vertical concentration profiles in the plume from an elevated source
located at about lOh downwind from the ridge are quite different from the
concentration profiles taken in the absence of the ridge. The comparison of
the two shows that, while the concentrations near the surface are enhanced,
those at higher levels are considerably reduced. Thus the increased turbu-
lence in the wake is very effective in diffusing the material down towards
the wall region, where there is actually slight reduction in turbulence.
14
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REFERENCES
1. Brighton, P. W. M., 1977: Boundary Layer and Stratified Flow Over
Obstacles. Ph.D. Thesis, University of Cambridge, Cambridge, England.
2. Drazin, P. G., 1961: On the Steady Flow of a Fluid of Variable Density
Past an Obstacle. Tellus, 13, 239-251.
3. Huber, A. H., W. H. Snyder, R. S. Thompson, and R. E. Lawson, Jr., 1976:
Stack Placement in the Lee of a Mountain Ridge. U.S. Environmental
Protection Agency Report No. EPA-600/4-76-047, Environmental Sciences
Research Laboratory, Research Triangle Park, North Carolina.
4. Huber, A. H. and W. H. Snyder, 1976: Building Wake Effects on Short
Stack Effluents. Proc. Third Symposium on Atmospheric Turbulence,
Diffusion and Air Quality; Oct. 26-29, 1976; Raleigh, North Carolina,
235-242.
5. Hunt, J. C. R., W. H. Snyder, and R. E. Lawson, Jr., 1978: Flow Struc-
ture and Turbulent Diffusion Around a Three-Dimensional Hill - Fluid
Modeling Study on Effects of Stratification, Part 1. Flow Structure.
U.S. Environmental Protection Agency Report No. EPA-600/4-78-041,
Environmental Sciences Research Laboratory, Research Triangle Park, North
Carolina, 84 pp.
6. Sykes, R. I., 1978: Stratification Effects in Boundary Layer Flow Over
Hills. Proc. Roy. Soc., Series A.
7. Thompson, R. S. and W. H. Snyder, 1976: EPA Fluid Modeling Facility.
Proc. Conference on Environ. Modeling and Simulation, Cincinnati, Ohio,
April 19-22, EPA 600/9-76-016.
15
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(A) F = 0.1, Re = 6870
(B) F = 0.2, Re = 13740
(C» F = 0,3, Re = 20610
(E) F = 0.5, Re = 34400
(I) F = 1.2, Re = 81600
'•' <•*
(D) F = 0.4, Re = 27480
(F) F = 0.6, Re = 41200
CG) F = 0.8, Re = 55000
(J) F = 1,7/80 = 117000
Figure 1. Shadowgraphs of flow over hill (N = 1.33 rad/sec).
16
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Q.2+
U(h)
30 40 50 60 80 \
Figure 2. The variation of the maximum deficit in mean velocity
in the wake with distance behind the ridge.
17
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Q A(-u'w')max/U2(h)
© A(^)max/U2(h)
.005+
.003-f
50 60 80 100
x/h
Figure 3. Variations of the maximum perturbations in
Reynolds stress and velocity variances in the
wake with distance behind the ridge.
18
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0.2-
max
0.1+
2.0+ Hs * 0.5h
O
1.0+
0.6
0.4
4. x
\ Hs = l.5h
\ 8
Hs = 2.0h \
O
\
\
N\ H« = 2.5H
N
.04+
.02
10
xs/h
15
20
Figure 4. The maximum g.l.c. as a function of source
height and its position relative to the
ridge.
19
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20--
16-
12-
xmax ~ *s
h
8-
0.
-4-
\
\^ ^ Hs = 2.5h
^ ^-^*
o /S
Hs = 2.0h ,' H. = l.5h
< A'''
\
\ A
XN^ ^--^'""
Ol 1 1 I
5 10 15 20
Hs = 0.5h
xs/h
Figure 5. The distance of the maximum g.l.c. from the
source as a function of source height and its
position relative to the ridge.
20
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/4-79-053
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
BASIC STUDIES OF FLOW AND DIFFUSION OVER HILLS
5. REPORT DATE
September 1979
6. PERFORMING ORGANIZATION CODE
7. AUTHORIS)
S.P.S. Arya and J.C.R. Hunt
8. PERFORMING ORGANIZATION REPORT NO
Fluid Modeling Report No. 5
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Department of Geosciences
North Carolina State University
Raleigh, NC 27650
10. PROGRAM ELEMENT NO.
1AA603 A8-34 (FY-76A)
11. CONTRACT/GRANT NO.
Grant No. R-804653
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory—RTF, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final 9/1/76-8/31/77
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
1. University of Cambridge, England
16. ABSTRACT
This research program was initiated with the overall objective of gaining a bettejr-
understanding of flow and diffusion of pollutants in complex terrain under both neutra
and stably stratified conditions, providing a sound data base for testing existing
theories and developing new theories of flow and diffusion around isolated hills and
ridges. To this end, experiments were conducted with models of a bell shaped hill
and a 2-D steep ridge in EPA's meteorological wind tunnel and salt-water stratified
towing tanks. Measurements were made of the flow structure, as well as the concen-
tration patterns around the hills due to point sources located at different positions
relative to the hills.
The experiments on stably stratified flow over a 3-D hill verify and establish
the limits of applicability of Drazin's theory for small Froude numbers. The locatior
of the surface impingement point from an upwind pollutant source can be identified
under a wide range of atmospheric conditions.
The experiments on the neutral boundary layer flow over a 2-D ridge show that
significant ridge effect is felt by the turbulence structure to distances greater thar
eighty ridge heights downstream. Ground-level concentrations in the lee of the ridge
are very sensitive to the source height and position relative to the ridge.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COS AT I Field/Group
Air Pollution
*Wind (Meteorology)
*Wind Tunnel Models
*Hills
*Atmospheric Diffusion
*Stratification
13B
04B
14B
08F
04A
8. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (ThisReport)
UNCLASSIFIED
20. SECURITY CLASS (Thispage)
UNCLASSIFIED
21. NO. OF PAGES
29
22. PRICE
EPA Form 2220-1 (9-73)
21
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