United States
Environmental Protection
Agency
Environmental Sciences Research
Laboratory
Research Tnang>leffinfr NC 27711
EPA-600/4-78-041
July 1978
Research and Development
Flow Structure and
Turbulent Diffusion
Around a Three-
Dimensional Hill
Fluid Modeling Study
on Effects of
Stratification
Part I. Flow Structure
PROPERTY Op
WSK«
OF
X
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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 Iriteragency Energy-Environment Research and Development
8 Special Reports
9 Miscellaneous Reports
This report has been assigned to the ENVIRONMENTAL PROTECTION TECH-
NOLOGY series This series describes research performed to develop and dem-
onstrate instrumentation, equipment, and methodology to repair or prevent en-
vironmental degradation from point and non-point sources of pollution This work
provides the new or improved technology required for the control and treatment
of pollution sources to meet environmental quality standards
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EPA-600/4-78-041
July 1978
FLOW STRUCTURE AND TURBULENT DIFFUSION
AROUND A THREE-DIMENSIONAL HILL
Fluid Modeling Study on
Effects of Stratification
Part I. Flow Structure
by
Julian C.R. Hunt
Department of Applied Mathematics
and Theoretical Physics
University of Cambridge
Cambridge, England CB3 9EW
William H. Snyder
Meteorology & Assessment Division
Environmental Sciences Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
and
Robert E. Lawson, Jr.
Northrop Services, Inc.
Research Triangle Park, NC 27709
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NC 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. Mention of trade names or commercial products does not
constitute endorsement or recommendation for use.
William H. Snyder is a physical scientist in the Meteorology and
Assessment Division, Environmental Sciences Research Laboratory,
U.S. Environmental Protection Agency, Research Triangle Park, North
Carolina. He is on assignment from the National Oceanic and Atmospheric
Administration, U.S. Department of Commerce.
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ABSTRACT
This research program was initiated with the overall objective of
gaining understanding of the flow and diffusion of pollutants in complex
terrain under both neutral and stably stratified conditions. This report
covers the first phase of the project; it describes the flow structure
observed over a bell shaped hill (polynomial in cross section) through
neutral wind tunnel studies and stably stratified towing tank studies.
It verifies and establishes the limits of applicability of Drazin's theory
for flow over three-dimensional hills under conditions of small Froude
number. At larger Froude number, a theory is developed, and largely
verified, to classify the types of lee wave patterns and separated flow
regions and to predict the conditions under which they will be formed.
Flow visualization techniques are used extensively in obtaining both qual-
itative and quantitative information on the flow structure around the hill.
Representative photographs of dye tracers, potassium permanganate dye
streaks, shadowgraphs, surface dye smears, and hydrogen bubble patterns
are included. While emphasis centered on obtaining basic understanding
of flow around complex terrain, the results are of immediate applicability
by air pollution control agencies. In particular, the location of the
surface impingement point from an upwind pollutant source can be identified
under a wide range of atmospheric conditions. Part II, to be printed as a
separate report, will describe the concentration field over the hill result-
ing from plumes released from upwind stacks and will further quantify the
results obtained in Part I.
This report covers a period from April 1, 1977, to March 31, 1978, and
work was completed as of March 31, 1978.
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CONTENTS
ABSTRACT i i i
FIGURES vi
TABLES ix
SYMBOLS x
ACKNOWLEDGEMENTS xii
1. INTRODUCTION 1
2. CONCLUSIONS 4
3. REVIEW OF THEORY 7
3.1 Small Froude Number Theory 7
3.2 Lee Waves and Separation 9
3.3 Kinematic Theory 12
3.4 Estimating the Velocity and Streamline Displacement
Over the Hill in Neutral Flow 13
4. APPARATUS 17
4.1 Large Towing Tank 17
4.1.1 Polynomial hill 18
4.1.2 Flow visualization 19
4.1.3 Measurement of plume impingement points and
streamline deflections 22
4.2 Small Towing Tank Experiment^ 22
4.2.1 Surface stress patterns 23
4.2.2 Hydrogen bubble technique 24
4.3 Wind Tunnel Experiments 25
4.3.1 Velocity measurements 25
4.3.2 Flow visualization 26
5. PRESENTATION AND DISCUSSION OF RESULTS 28
5.1 Presentation of Results 28
5.2 Qualitative Description of the Flow 30
5.3 Compari son wi th Theory 33
5.4 Flow Variations with Reynolds Number 40
REFERENCES 42
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FIGURES
Number Page
1 Sketch of plume behavior over three-dimensional hill
in neutral and stable stratification 45
a. Neutral stratification
b. Stable stratification
2 Small Froude number theory for flow over three-dimen-
sional hills 46
a. Definition of regions
b. Streamline patterns
c. Plan view
3 Stratified flow over two-dimensional hills in a channel 47
a. Supercritical Froude number; no waves possible
b. Hill with low slope; subcritical Froude number
c. Hill with moderate slope; supercritical Froude number
d. Subcritical Froude number; lee wave induced separation
on lee slope
4 Singular points of the surface shear stress lines and
mean stream! ines 48
a. Saddle points in surface shear stress lines
b. Node points in surface shear stress lines
c. Singular points in the mean streamline pattern
5 Potential flow over an ellipsoidal hill 49
a. Coordinates and notations for analysis
b. Calculated speed-up for b>L0
c. Calculated speed-up for L0^h
6 The large stratified towing tank 50
a. The EPA water channel/towing tank
b. The filling system
c. Typical density profiles
7 Details for polynomial hill model 51
a. Sketch of hill or baseplate
b. Profile shape of hill
c. Plan view of hill or, baseplate
d. Coordinates of sampling tubes on hill
VI
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Number Pac
8 Detail showing suspension of model hill, lighting
and photographic arrangement, and sampling and dye
injection system 53
9 Experimental apparatus for work in small towing tank 54
a. Small water channel/stratified towing tank
b. Schematic diagram of current source for generating
hydrogen bubbles
c. Plan view of set-up for photographing hydrogen
bubbles
10 Details of wind tunnel measurements 55
a. The EPA Meteorological Wind Tunnel
b. Placement of model in wind tunnel
c. Reynolds stress correction factor for boundary
layer probe
11 Visualization of surface flow patterns from injection
of tracers 56
a. Dye release from 180° ports, F=0.2
b. Dye release from 0° ports, F=0.2
c. Dye release from 180° ports, F=0.4
d. Dye release from 0° ports, F=0.4
e. Dye release from 180° ports, F=0.9
f. Dye release from 180° ports, F=1.6
g. Smoke release from 180° ports, F=°°
h. Smoke release from 0° ports, F=»
i. TiCl. smoke at downwind base of hill, F=»
j. TiCl? smoke at downwind base of hill, F=°°
12 Visualization of surface shear stress patterns 60
a. Side views of KMnO, streamers in large tank
b. Top views of KMnO, streamers in large tank
c. Top views of gelatin/dye and KMnO, streamers in
small tank
13 Shadowgraphs of flow over hill 63
14 Experimental observations of centerplane streamlines
from multilevel tracer injection 64
a. Side views in large tank
b. Top views in large tank
c. Tracings of smoke plume center!ines in wind tunnel,
F=.
Vi i
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Number Page
15 Hydrogen bubble photographs indicating speedup over
top of hill and the downwind lee wave pattern 67
a. F=0.4
b. F=1.0
c. F=1.7
d. F=co
16 Plumes from upwind stacks at various elevations and
Froude numbers in large tank 68
a. F=0.2
b. F=0.4
c. F>Q.8
17 Wind tunnel measurements of the flow over and in the
absence of the hill 71
a. Mean velocity profiles
b. Mean velocity vectors
c. Local longitudinal turbulence intensity profiles
d. Local vertical turbulence intensity profiles
e. Reynolds stress profiles
18 Derived centerplane streamline and surface shear stress
patterns 76
a. F=0.2
b. F=0.4
c. F=1.0
d. F=1.7
e. F=»
19 Plume impingement on the hill surface 81
a. Test of hypothesis F=1-H /h
b. Impingement height as a function of source height
20 Streamline distributions over Elk Mountain, Wyoming 82
a. Estimated from aircraft observations
b. Estimated by computer simulations based on shallow
water model
21 Displacement of streamlines above hill surface 83
viii
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TABLES
Number Page
1 Classification and Location of Singular Points on
Polynomial Hill 29
2 Comparison of Observed and Calculated Streamline
Deflections 34
3 Lee Wave Estimates and Separated Flow Classification 40
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SYMBOLS
b horizontal half-width of hill perpendicular to flow direction
[B] flow region at base of hill when F«l
D depth of towing tank or channel
D diffusivity of heat or salt
f(r) height of hill as a function of the radius
F Froude number = U/Nh
F. Froude number = U/NL
g acceleration due to gravity
h height of hill
HS stack height or height of streamline far upstream of hill
[H,] flow region below top of hill when F«l
[H2] flow region above top of hill when F«l
L half-length of hill at half-height [=RQ(h/2)]
L half-length of hill in x-direction
n distance of streamlines from z=0 axis far upstream of hill
n distance of streamlines from surface of hill
N Brunt-Vaisala frequency C=(g(dp/dz)/P)1/2]
{N} node point
{N1} half-node point
p pressure
/\
p perturbation pressure
P perimeter of ellipsoidal hill on x=0
P. singular points of surface shear stress
r radial coordinate in horizontal plane
RQ(z) radius of hill
S speedup factor [KJ^/Ujh)]
{S} saddle point
{S1} half-saddle point
[T] flow region at top of hill when F«l
x
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Um*v, maximum velocity over hill
i MaX
Ur horizontal velocity in r-direction
Ue horizontal velocity in e-direction
U^ upstream velocity, constant with elevation
U^U) upstream velocity, varying with elevation
W vertical velocity
x horizontal, streamwise coordinate
y horizontal, spanwise coordinate
z vertical coordinate
z. elevation of plume impingement point (height of maximum concentration)
otp coefficient denoting thickness of [B]
u-r coefficient denoting thickness of [T]
$0 speedup integral (Eq. 3.13)
A vertical deflection of streamline
e angle of "longitude" in (r,e,z) system
\ aspect ratio of hill (=b/L )
A wavelength of lee wave
p density
P (z) upstream density profile
i H| number of node points
EC- number of saddle points
y streamline
{}'a' attachment point
O^s' separation point
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ACKNOWLEDGEMENTS
JCRH is grateful to his research student Dr. P.W.M. Brighton for teaching
him so much about stratified flow over obstacles and to Dr. Phillip Mason of
the Meteorological Office (U.K.) for an enlightening conversation about lee
waves. We are grateful to Messrs. Roger Thompson and Daniel Dolan for help
with the photographs, to Messrs. Lewis Knight, Leonard Marsh, and the late
Karl Kurfis for help with running the experiments and to Miss Tameria Bass
for typing the manuscript. Financial support for JCRH was provided through
an appointment as Associate Professor, Department of Geosciences, N.C. State
University, through EPA Grant R804653.
xii
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1. INTRODUCTION
The flow patterns and velocity distributions around three-dimensional
hills need to be better understood before one can even interpret measurements,
let alone make reasonable predictions of the dispersion of pollution from
sources in this kind of complex terrain. The aspects of the structure of these
stratified flows that most affect dispersion are:
(a) Whether streamlines from upwind impinge on the hill, go round the
hill, or go over the top;
(b) The size and location of internal hydraulic jumps and the region of
the separated or recirculating flow in the lee of the hill, as the
stratification varies; and
(c) The effects of heating and cooling of the surface.
Existing theory (see Sect. 3) and observations of air flow over mountains
(Queney et al., 1960) all indicate that, when the stratification is strong
enough, the air flows in approximately horizontal planes around the topography
(see Figure 1). This observation is used in estimating surface concentrations
caused by upwind sources of pollution in the EPA Valley model (Burt and Slater,
1977). But hitherto there has been little firm laboratory or atmospheric
data as to how strong the stratification must be for any given streamline
starting below the top of the hill to pass round rather than over the hill.
A criterion for this changeover to occur is suggested in Sect. 3 on the basis of
the low-Froude number theory of Drazin (1961), which the experiments described
in Sect. 5 confirm. As a scale for comparison between these and other measure-
ments of the increase in wind speed over three-dimensional hills in neutral
or weakly stratified flows, we summarize the results of potential flow
calculations of the wind speed and streamline displacement over ellipsoidal
hills in neutral flow.
1. The adverb "round" is used in the sense of following a circular path,
whereas "around" is used in the sense of nearby or about.
1
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It is well known that a small amount of stratification can have a strong
effect on the separated flow downwind of bluff obstacles such as hills or
escarpments (see, for example, Scorer, 1954, 1968). In the case of two-
dimensional hills, the separation behind hills with moderate slopes can be
suppressed, but "rotors", in which there is reverse flow, may be found farther
downwind as part of the lee wave pattern. Internal hydraulic jumps are found
downwind of hills with moderate or large slopes in which the streamlines jump
abruptly upwards, the horizontal component of velocity decreases, and a large
amount of energy is dissipated. Despite its practical importance to air
pollution dispersion and aeronautics, the distinctions and connections among
these phenomena are still far from clear, even in the case of two-dimensional
hills. There has not even been a laboratory study of two-dimensional flows
where all these phenomena occur, although some valuable insights were provided
by the wind tunnel study of Kitabayashi et al. (1971) and the aircraft
observations of Connell (1976) around Elk Mountain, Wyoming. If the slope
of the hill is small enough, separation may occur without hydraulic jumps.
For that case, Brighton (1977) has, for the first time, calculated and
confirmed experimentally, in a stratified water channel at Cambridge, the
dependence of laminar separation on Froude number for two-dimensional hills.
In this report, we describe laboratory studies of the flow patterns,
with particular emphasis on the separated flow regions and the internal
hydraulic jumps, around an axisymmetric, three-dimensional hill. A fourth
order polynomial shape has been studied in a large stratified towing tank
(2.4 x 1.2 x 25 m), in a small, stratified towing tank (0.20 x .10 x 2.0 m),
and in an unstratified wind tunnel.
The method we adopt for analyzing and presenting a large quantity of flow
visualization data of this kind is to first locate the points where the surface
shear stresses are zero or where the velocity in a cross section through the flow
is zero. These points are then characterized in terms of whether they are
separation or attachment points and, more unusually, in terms of their
topological nature as saddle points or node points, because topological
theory (Hunt et al., 1978) shows that the number of saddle points must equal
the number of node points. This provides a kinematic check on the inferred
flow from the flow visualization data. This minimal attempt at systematizing
visualization data provides a more concrete basis for qualitative comparisons
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between (a) full scale and model scale observations of the flow around hills
and (b) the theory and experiment.
Some quantitative results are also obtained from the flow visualization
experiments: namely, at low Froude number, the vertical deflection of stream-
lines; at moderate to large Froude numbers, the distance of streamlines from
the surface of the hill and the connection between the lee waves and the
separated flow downwind of the hill; and at all Froude numbers, the increase
in velocity or speedup over the hill.
As Scorer has pointed out, laboratory studies of stratified flows tend to
overemphasize the effect of the upwind stratification rather than the equally,
perhaps more, important effects of the local heating and cooling of the surface
of the hills. These effects are not only local but may also have a large
effect on the flow by inhibiting or promoting separation (Scorer, 1968,
p. 113; Brighton, 1977). There have been some attempts to heat the surface
of model terrain, but more to simulate fumigation of elevated plumes than
katabatic or anabatic winds (Liu and Lin, 1976). Any comparison between
full scale and model experiments, where such thermally generated winds are
not simulated, must be made with great caution.
The main objective of Part I of this report is to elucidate, by theory
and experiment, the structure of the flow, the velocity field, and the pattern
of the streamlines of the stably stratified and neutral flow over a model hill,
so that the concentration measurements described in Part II can be understood.
Secondly, we hope that some general conclusions can be drawn about stably
stratified flows around three-dimensional hills; and thirdly, we hope that
these experiments will act as a stimulus for further experimental, theoretical,
and computational investigations of these flows.
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2. CONCLUSIONS
1. A flow structure to describe stably stratified flow over three-dimensional
hills was developed in sections from physical scaling laws, some primitive
analyses, and analogy with two-dimensional hills. The experiments broadly con-
firmed the general predictions about the flow structure, the important features
of which are summarized below:
(a) When F«l (say F<0.3 for hills with moderate slope), the flow
is approximately horizontal except in narrow regions at the top [T]
and bottom [B] of the hill. These regions have thicknesses of
about Fh, and streamlines starting in region [T] tend to pass over
the top of the hill. A small lee wave or internal hydraulic jump
may be found on the lee side of the hill in region [T]. Below region
[T] the flow separates on the lee side of the hill, where a mainly
horizontal recirculating flow is found. However, there is some
vertical mixing caused by streamlines, deflected downwards by about
hF2 outside the wake, entering the wake and reverting to their orig-
inal height.
(b) When F is of the order of 1 but not very much less than 1 (say
>0.3), then it is less useful to think in terms of perturbations
from the asymptotic state of F->0. Rather, it proves better to think
in terms of the response of the flow over the hill to the wavelength
A of the lee wave pattern set up by the hill. Experiments on a
number of shapes of hills show that there is a range of the Froude
number FL based on the length of the hill, in which 2UA^5L, where
the lee wave is neither so long as to allow the boundary layer to
separate (supercritical flow), nor so short as to force the lee wave
to generate a rotor on the lee face (subcritical).
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(c) When F is not much greater than 1, say, of the order of 1.5 or
greater, overall, but not in detail, the flow structure around the
hill becomes similar to that of neutral flow. But far downstream
the stratification may have a strong effect, even when F is as
large as 10.
2. The potential effects of the stratification on a plume from an upwind
source are summarized below. The implications for surface concentrations
will be further amplified in Part II.
(a) The stronger the stratification, the narrower will be the vertical
width and the wider will be the horizontal width of the plume. If
the plume starts below the hill top, yet goes over the top, it will
pass very close to the hill surface and be further contracted in the
vertical and expanded in the horizontal.
(b) The criterion for determining whether the plume will impact on the
hill surface and go round the sides or go over the top is, roughly,
H =h(l-F), where H is the dividing streamline height, h is the hill
O o
height, and F is the Froude number characterizing the stratification.
If the plume height (upstream) is smaller than this streamline height,
it will impact on the hill surface; otherwise, it will go over the
top.
(c) In moderately stratified flow (say, 0.4
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(e) From conclusions l(a) and 2(c), we arrive at the "2-1/2 times rule",
long ago established from neutral wind tunnel studies of flow from
stacks close to obstructions and amply demonstrated as adequate from
field observations (mostly from the lack of complaints!) over many
years.
3. Since our experiments were all conducted in an approach flow without
shear, the application of the results of modeling to the atmosphere has to
be more than usually justified. Turbulent unstratified flows around bluff
bodies all have much the same structure of separation regions, singular points,
etc., for large variations in the approaching shear because the body's own
wake flow has such a strong controlling effect (Hunt et al., 1978). The
actual position of upstream vortices can be changed, but the structure of the
wake flows are remarkably similar. Thus, in a stratified flow where the
stratification somewhat "decouples" the flow at different heights, as one can
see exactly when F«l (Eq. 3.1), the effect of the upstream shear on the flow
structure is likely to be small. But, of course, quantitative results such
as the coefficient ay defining the thickness of the [T] region, or the speed-
up factor S will be considerably influenced; we do not even know yet whether
they will be increased or decreased!
4. A more fundamental limitation of our experiments may be the fact that
they were conducted in finite depth tanks. The approximate argument of Sect.
3.2 suggests that the effect of the finite depth on three-dimensional lee
waves is not very marked. Investigation of the lee wave pattern in the large
tank may help elucidate this question. Experiments are certainly needed in
stratified flows where the density gradient varies with depth, as it always
does in the atmosphere; but we hope that the general ideas about the relation
between the wavelengths of lee waves A and the horizontal length scale of the
hill L may still be useful for estimating when, where, and what kind of
separated flows will occur.
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3. REVIEW OF THEORY
3.1 Small Froude Number Theory
The only theory giving quantitative results for stratified flow around a
three-dimensional mountain or hill which is not merely a small perturbation
of a plane surface is that due to Drazin (1961), which has been extended and
largely confirmed experimentally by Riley et al. (1976) and Brighton (1978).
The theory is valid asymptotically as the Froude number F(=Uoo(h)/Nh)-^0,
where U(h) is the upstream velocity at the level of the top of the hill, N is
the Brunt-Vaisala frequency (i.e., proportional to the square root of the
density gradient), and h is the height of the hill (see Figure 2a). The
essential results from our present understanding of the theory are these:
(a) The fluid moves approximately horizontally in two regions: [H,],
above the plane z=0 and below the summit of the hill, and [Hp], above
the summit (see Figure 2).
(b) Regions [B] and [T] exist at the base and the top of the hill with
thicknesses aDFh, and aTFh , where the a's are factors of order one
o 1
and depend on the shape of the hill. (We assume that the a's are
1.0 unless otherwise stated, and that the extent of [T] is
symmetrical above and below the summit.) In these regions, the
flow is not constrained to move horizontally.
(c) In the lee of the hill there may be a wake in which the flow also
moves approximately horizontally, but the streamlines in the wake
do not emanate from the same levels upstream.
(d) In [H-|], to the first approximation as F+0, the horizontal velocity
UH=(Ur,UQ) is described by potential theory; for example, if the
mountain is axisymmetric about the z axis, with a radius RO(Z), and
if the wake is ignored,
1. Note that Fh=Uoo/N, so if a^ and a-j- are independent of F, the thicknesses
of [B] and [T] may be independent of the hill height h.
7
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Ur = Ujz)(l - R2(z)/r2)cos e, and (3.la)
Ue = -UjzMl + R2(z)/r2)sin e . (3.1b)
Riley et al. (1976) showed how the effect of the wake can be
estimated by using free streamline potential theory.
(e) It follows from Eqs. 3.1 that the horizontal pressure gradients
in [H.J] vary with z due to U^ and R variations, and, consequently,
the vertical pressure gradient (3p/3z) is perturbed. For example,
/s
on the stagnation line this perturbation (denoted by 9p/3z) is
negative because dRQ/dz is negative. In strongly stratified flows,
the perturbation pressure gradient is balanced by perturbations to
the density gradient, which are produced by vertical displacements
A(r,e) of streamlines starting from upstream. Generalizing Drazin's
theory to allow for variations in L)(z), Brighton (1978) showed that
A(r,e) = -(ap/3z)/[g 3pQ/3z], or (3.2)
A(r,e) = -
1
9
°- (U 2 - (U 2 + U 2))
az v « v r e ''
For the simple flow given by Eqs. 3.1,
dUjU) R2
A(r,e) = -
(cos 20 - R 2/2r2)
dz r^ ^ ° u o
U2 d(R2)
(cos 2e - R2/r2) . (3.3)
The mean vertical velocity W is the time rate of change of A of a
fluid particle, so that
Note that W/Uoo(h) is of 0[F2 x (local slope)]. The deflection of a
typical streamline v is shown in Figures 2b and c. Note that for
8
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all shapes of obstacles
A(r,e) = W = 0 (3.5)
at the upwind stagnation line.
(f) Streamlines in [H2L such as ¥H2, are not deflected.
(g) As a consequence of the result (b), streamlines such as ¥, in
Figure 2b pass through region T and therefore over the top if
they originate upstream at a height
- F) . (3.6)
3.2 Lee Haves and Separation
When the Froude number is of the order of 1, then hMJ/N; in other
words, the length of waves (e.g., lee waves) set up in the stratified
flow by the hill are of the same order as the height of the hill, and
they can then begin to control the flow (See Figure 3).
Either an exact or an order of magnitude analysis of stratified flow
over two-dimensional hills with moderate slopes shows that the character
of the flow depends primarily on the ratio of the wavelength of the lee
waves (2irU/N) to the total length (2LQ) of the hill, rather than the height.
The reason is that streamlines and isopycnals are deflected by the hill
by a distance of order h (i.e., h is characteristic of the amplitude of the
waves) up to a height above the hill of the order of 2L (rather than
h), producing a change in potential energy of the order of ghLQ8p/9z;
the change in kinetic energy is of order pU^h/L^, and so the dimensionless
ratio determining the flow must be (Uoo/(L()N)). This ratio is proportional
to the wavelength of a lee wave (2TrUoo/N) divided by the length of the
hill LQ.
Some general conclusions about the way this ratio determines the
flow can be drawn from the recent analytical and experimental work of
Brighton (1977), the computational work of Sykes (1978) on laminar flow
over two-dimensional hills, and the general observations of Scorer (1968).
It appears that when the stratification is such that 2ir(J /N=2L , separation
oo Q '
of the flow over a rounded hill will be suppressed, if it can be, at any
Froude number. (Clearly, for very steeply sided hills separation always
occurs.) But if these two lengths differ considerably from each other,
-------
separation must occur on hills of moderate slope. If 2L »2irUoo/N (i.e.,
F«l), separation is controlled by the pressure distribution produced
by the lee wave pattern, and, if 2l_0«2TrUoo/N (i.e., F»l), separation
on the lee slope of the hill is controlled by the boundary- layer flow.
It is useful to designate the highest Froude number at which separation
is first suppressed as the critical Froude number for separation, F ../ N
rr ----- c - cnt(s),
and to describe flows for which F is greater than or less than F .. / x as
V.« I I w \ o J
super- or subcritical, respectively. (Normally, these terms are used to
denote the existence or nonexistence of waves; hence the suffix s for
separation.)
In practice any experiment has to be conducted in a flume or towing
tank of finite depth D. In that case, the lee wave pattern can be quite
different, as may be seen in the case of two-dimensional hills by comparing
the computed streamlines of Huppert (1968) for a semicircular hill in an
infinite fluid, and of Davis (1969) in a finite channel (or see Turner,
1973, pp. 61-62). Brighton (1977) has shown theoretically how, for laminar
flow over two-dimensional hills with low slopes, separation is determined
by the Froude number based on the depth of the tank Fp, and the Froude
number F, (>FD) based on a suitable length L of the hill, such as the semi-
length at the half height. When Fn
-------
For such a hill, D/(Lrr) is the critical Froude number. This theory was
confirmed by experiment in a small stratified flume at Cambridge on a
hill with maximum slope of 0.3. (See Figures 3a, b, and d.).
When the hill has a moderate but not large slope, the internal wave
motion may not be strong enough to suppress the separation on the hill
until A^2L , the overall length of the hill. This is similar to the case
of an infinite fluid, where the separated flow can also be suppressed
when A^2L . Brighton's calculations suggest that the critical wavelength
(See Figure 3c). Consequently, we might expect that, for a moderate
slope, separation is suppressed only over a narrow range of wavelengths,
namely,
2L < A < 5L, (3.7b)
or
0.3 = 2/(2Tr) * FL(1 - u2FL2L2/D2)~1/2 * 5/2rr = 0.8 . (3.7c)
Thus, if L«D, the critical Froude number is about 0.8, and the lower critical
Froude number is about 0.3. The flow over the sharp-edged triangular hill
used by Davis (1969) roughly satifies this inequality.
Can these ideas be applied to three-dimensional hills? The definitive
theoretical calculation of Crapper (1959) of an infinite stratified flow
over an axisymmetric hill with small slope shows that the lee waves:
(a) have a wavelength in the flow direction of about Zirl'/N,
(b) are confined in the spanwise (y) direction to a strip of width 2L
(rather than in a wedge like shipwaves as was thought at first
(Queney et al., I960)),
(c) do not decrease significantly with height (z) above the hill, and
(d) can have a greater amplitude than those of a two-dimensional hill.
These results suggest that:
(a) The lee waves created by a three-dimensional hill must be just as
much affected by the depth of the flume or tank as those of a two-
dimensional hill. But the side walls are not likely to have much
effect, provided the width of the tank is somewhat greater than
2L , which it must be to avoid the normal "blockage" effects.
11
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(b) We can estimate the streamwise wavelength A of three-dimensional
waves in a tank of depth D (assuming D»L) by assuming the spanwise
wavelength over an axisymmetric hill is about 2L and using the
linearized wave equation (e.g., the Boussinesq form of Crapper's
Eq. 31); then
A =
- FL2/4
8D'
if FL2/4 « 1.
(c) Since lee waves over a three-dimensional hill are just as strong,
they should be able to suppress separation just as much as they
do over a two-dimensional hill.
(d) Assuming that the general criterion for the suppression of
separation is also A=2L0=5L for three-dimensional hills, then
it follows from Eqs. 3.7c and 3.8a that separation on the lee
side of a hill with moderate slope will be suppressed when
0.3 <; FL(1 - FL2L2/8D2) s 0.8 . (3.8b)
3.3 Kinematic Theory
Although little dynamical theory exists that predicts the velocity
and streamline pattern of stratified flow over hills, some useful kinematic
theory provides a limitation on the large number of possible streamline
patterns which can be drawn from a limited amount of experimental information.
(See Lighthill, 1963, and, for the most recent results, Hunt et al., 1978.)
When identifying and examining the "singular" points on the surface
of the hill and the surrounding plane where the shear stress is zero, there
are advantages in describing these points in terms of their topological
characteristics such as saddle {S} or node {N} points, rather than in
the usual fluid dynamical categories of separation and attachment points,
which are here denoted by superscripts {}^s' and {}'a . Figure 4 shows
examples of these singular points. The main advantage is the topological
constraints on the total number of saddle points (z^) and node points (E^)-
12
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For any shape of hill (provided that it does not have any holes through the
middle, which a building might have), the surface flow pattern inferred from
the observations must satisfy:
zs - £N = 0 . (3.9)
Another advantage of this approach is that, by identifying the positions of
singular points, one provides a more succinct way of describing the flow (as
opposed to defining the whole streamline pattern). One experiment can be
more easily compared with another, and computations can also be tested this way.
The mean streamline pattern in a plane y=constant can also be analyzed
in terms of the topological properties of the points where the mean velocity
in the plane is zero, denoted by the letters, {S}, {N}. However, half saddles
{S1} and half nodes {N1} also have to be defined where the singular point
is on the surface (See Figure 4c.) The flow pattern must now satisfy
(zs + %zs.) - (ZN + %ZN.) = 0 . (3.10)
There is an additional constraint in deriving the flow pattern in stratified
flows. If the diffusivity D of heat (if temperature stratified) or salt (as
in our towing tank experiments) is small enough (i.e., zy(Uh)«l), if the
turbulent mixing in the wake is weak, and if the velocity, pressure, and
density distributions far downstream are the same as upstream, then fluid
particles starting at a height z upstream must return to a height z downstream.
This implies, where mean streamlines and mean particle paths (from upstream
sources) are close together (only true in weak turbulence), that no streamline
starting at height z>0 can be the streamline which attaches to the plane z=0
at the downstream attachment point, as will become clearer in Sect. 5.
3.4 Estimating the Velocity and Streamline Displacement over the Hill
in_ Neutral Flow
For flow over a three-dimensional hill, a useful way of estimating the
increase of velocity and, of greater importance in pollution studies, the
displacement of streamlines relative to the surface, is to consider potential
flow over an ellipsoid (see Figure 5a). For an ellipsoid described by
13
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x2 v2 z2
*9 + *9 + % = 1»
if b^ h*
(3.11)
Milne-Thompson (1960, p. 518) has proven (in fact he quotes the result in an
examination question!) that the surface velocity is the same at all points
on the central plane of the ellipsoid (x=0), is a maximum, and is given by
(3.12)
where
Lo
h
b
h
du
2T/9 9 1/0 1/9
• •• ,~, J. i i \~/ f-l 1 i /U\£ , ,,\l/^/1 i ,,\'/^
ULQ/nj + u) ((b/h) + u) (1 + u)
• (3.13)
The upwind flow Uro is uniform with height. We shall define the term "speed-
up" as
S =
S =
(3.14)
From Eq. 3.12, we can first find how the transverse aspect ratio of the
hill (x=b/LQ), i.e., the ratio of its horizontal width perpendicular to the
wind to its length parallel to the wind, affects the speedup S. We shall
assume for convenience that h=LQ. Evaluating the integral in Eq. 3.13 with
the help of Gradshteyn & Ryzhik (1965, p. 78),
1
- 1)
3/2
1
whence as
so that
x =
A
n (X + /X2 - 1 ), (3.15)
(3.15b)
2(1 -
(3.15c)
X2
Thus if x=10, S=1.96; the speedup is reduced by 2% below the value of an
infinitely long circular cylinder. When x-»l, BQ->-2/3(l+0(x-l)), so that
, the speedup over a sphere.
14
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Thus if x=10, 5=1.96; the speed-up is reduced by 2% below the value of an
infinitely long circular cylinder. When X+l, B -*-2/3(l+0(A-1)), so that
S\,3/2, the speed-up over a sphere.
If the wind is parallel to the longest side of the hill, or if the flow
over a spherical hill separates, so that the effective length of the hill in
the direction of the wind increases, then, assuming h=b, and X<1,
o
= x
2
(1 - x2) (1 - x2)3/2
; (3.16a)
when x+l, g = 2/3 as before, and when x«l,
B0 * X2 [ln(4/x2) - 2], (3.16b)
so that p -> 0, and S -> 1 as x+0, i.e., there is no increase in velocity
over the hill.
From the graphs presented in Figures 5b and c, we note that if x=2.5,
the speed-up is 1.75, i.e., half-way between a cylinder and a sphere. If
XAjO.6, the speed-up is half-way between that of a sphere and undisturbed
flow.
To estimate the magnitude of the distance of streamlines from the surface
of the ellipsoid, we define an average displacement n"s of a surface stream-
line in the plane x=0. (See Figure 5a.) Far upstream, these streamlines
all touch an ellipse with minor axis n and major axis (b/h) n in the
y-direction. (By far upstream we mean that |x| is very much greater than
whichever is the largest of b, h or L .) Then, by flow continuity
F) I «,\- <3-17>
where P is the perimeter of the semi-ellipsoid on the plane x=0.
P = 2b E(90%x), (3.18a)
where E(/) is a complete elliptic integral of the second kind, and
a = sin-1((b2 - h2)1/2/b), (b>h) (Abramowitz & Stegun, 1965, pp. 590-592).
To within 10%, P can be approximated by
15
-------
P = 2L0X[Tr/2 - sin~](/l - 1/x^Ml - 2/ir)], where x-,=b/h (3.18b)
whence, if h=L (x=x,), as X -> <*>, P+2 LQX = 2b; as X
If b=h when X<1 ,
P = un = 2TrL0/x . (3.18c)
Thus, from (3.17), the ratio of the distance of a streamline from the
'ellipsoid over the central plane to its distance upstream is, when X>1 :
ns TrX,n (2 - B )
_ = - ] o o - (3.19a)
nQ 8l_oxI>/2 - sin"'(/ 1 - l/x
Thus, in the limit x-*» (assuming L =h); since (3=1,
When x+i, BQ->2/3, and
o
ns/n0 = U/8)(n0/L0). (3.19b)
ns/no = no/(3Lo) = n
When x0,
ns/n0 = n0/(2h). (3.20b)
The practical implication of (3.19b), (3.19c), and (3.20b) is that, for a wide
range of shapes (where h=b or LQ), the ratio ns/nQ is roughly 1/2 to 1/3 of
the initial ratio of the streamline height to the hill height.
However, if the hill's width is very much greater than LQ, one may be
interested in the ratio of n"s to nQ, where nQ is calculated when
b = (xL ) »x» LQ. In this case "UP-21"1. where x»l » so that
which is the two-dimensional result.
16
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4. APPARATUS
4.1 Large Towing Tank
The experiments were conducted in both towing tanks and in the wind
tunnel of the EPA Fluid Modeling Facility. Details of the experimental
apparatus are shown in Figures 6 to 10. The large towing tank is 1.2 m in
depth, 2.4 m in width, and 25 m in length (Figure 6a). It has an aluminum
framework, and the sides and bottom are lined with acrylic plastic for view-
ing purposes. A towing carriage allows models to be towed the length of the
tank at variable speeds. Rails on the side walls of the tank support the
carriage across the tank. (See Figure 6, or for additional details,
Thompson and Snyder, 1976.) The carriage is pulled by cables driven by a
motor at the end of the tank. The lowest reasonably uniform speed that can
be attained is about 5 cm/s, and the highest is about 50 cm/s.
Coupled with the tank is a filling system (similar to that described by
Pao et al., 1971) that allows the tank to be filled with an arbitrary stable
density profile using salt water (specific gravity from 1.0 to 1.2). The
filling system is composed of a brinemaker, five large tanks, and numerous
pumps and valves as shown in Figure 6b. The filling operation is started
with the tank empty. Saturated salt water (sodium chloride) is pumped
continuously from tank A through valve 5, pump 1, and valve 27 into head tank
F, with the overflow returning to tank A through valve 19. Similarly, fresh
water is pumped from tank C through valve 4 and pump 2 into head tank G, with
overflow returning to tank C through valve 22. This procedure maintains a
constant pressure on the mixing valve, which proportions the amounts of fresh
and salt water going into the towing tank. Four longitudinal tubes extending
the length of the towing tank are manifolded to the supply line at one end.
These longitudinal tubes are supported approximately 3 mm above the floor
and contain 3 mm holes spaced approximately 15 cm apart. These tubes, then,
distribute the water along the floor of the towing tank. Initially, a 4-cm
17
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layer of fresh water is pumped into the tank. The mixing valve is adjusted
to admit a small amount of salt water which is to be mixed with the fresh
water in the supply line. A 4-cm depth of this mixture is then admitted
to the towing tank. This mixture, being heavier, flows under the fresh
water, lifting it. This procedure is repeated, each time increasing the
specific gravity of the mixture, until the tank is full. Since the specific
gravity of the mixture and the layer depths are variable, it is possible
to produce nearly any stable density profile shape. The specific gravity
of the mixture is monitored with a Princo Densitrol specific gravity indicator
(Model W 747) mounted on a bypass on the supply line.
The final density profiles were measured by withdrawing samples from
various depths in the tank and measuring their specific gravity with precision
hydrometers or an electronic balance. Figure 6c shows one of the density
profiles measured during this set of experiments and compares it with another
measured after 13 days and more than 15 tows of the hill through the tank.
••
The initial profile is quite linear with a Brunt-Vaisala frequency of
N=1.33 rad/s. The later profile shows only a slight erosion of the gradient
near the surface, i.e., the top 8 cm of water is near neutral in stability,
with the bulk of the water matching the original stratification.
4.1.1 Polynomial hill
The model hill was made of 6 mm acrylic plastic sheet by vacuum molding
onto a wooden former. Ideally, the shape was to be a fourth order polynomial
[f(r)=h/(l+(r/L) )] that would avoid sharp gradients at the top of the
hill (similar to a Gaussian shape), but would decay reasonably rapidly as
r became large, in order that the hill would fit onto the baseplate. Due
to imperfect construction techniques, the final shape was closer to
f(r) = h . -- - . 0.03 , (4.1)
[l + (r/L)4 1 +[(r-r1)/L1]Z J
where h=L=22.9 cm, r^=20.3 cm, and L-|=7.6 cm. This equation describes
the shape of the hill to within +_ 2 mm. The maximum slope was about 1.0.
18
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This hill was mounted on a flat baseplate, which was constructed of
acrylic plastic attached to a framework of rectangular aluminum tubing.
This framework was in turn suspended from the carriage through four jackscrews,
which permitted leveling of the baseplate on the water surface. The hill
was thus towed upside down across the water surface. The baseplate was
immersed approximately 4 mm for each tow. Details of the model are provided
in Figure 7. A sketch of the hill mounted on the baseplate is provided
in Figure 7a. The upstream edge of the model was bevelled at 45° in order
to reduce the abruptness of the step. Twenty-eight sampling ports (i.d.=1.6;
o.d.=2.4 mm) were fixed on the surface of the hill along each of the radial
lines e=180°, -165°, -90°, and 0° (see Figure 7b, c and d). These sampling
ports were not flush with the surface, but protruded a distance of 2.5 mm
from the surface to simulate a full scale sampler height of, say, 3 m.
On a full scale hill, surface roughness and convection would diffuse a
plume to the ground if the plume were a small distance above it; on the
ultra-smooth model hill at much lower Reynolds number, surface roughness
and convection were absent, so that flush surface measurements might indicate
zero surface concentrations in highly stable flows. This may have occurred
in some of the model towing experiments at Flow Research (Liu and Lin, 1975).
Figure 7d defines the coordinate system. The origin (right-handed
system) is on the baseplate directly under the top center of the hill.
4.1.2 Flow visualization
As part of our study of the flow over the polynomial hill, dye was
emitted isokinetically from a "stack" of 0.635 cm o.d. The stack was bent
through 90° parallel to the flow to avoid any cross-stream momentum in
the effluent. It was located 91.8 cm (4h) upstream of the hill center
and 97.9 cm downstream of the platform's leading edge (see Figure 7c).
The stack height H$ was varied from 0 to 1.2 hill heights. The effluent
used was blue food dye (Warner Jenkinson No. 393) diluted with the proper
amount of salt water to obtain a neutrally buoyant plume of 1 part dye
and 15 parts salt water. The density of this dye solution was made equal
to the local density of the static fluid at the emission height of the stack.
19
-------
The effluent was pumped into the stack through a positive displacement
pump (Fluid Metering Inc. Model RP-D) from a large graduated cylinder.
A damper downstream of the pump, in addition to flexible tubing, minimized
pulsations in the flow from the stack. The flow rate was preset and also
double-checked during each test.
In other tests, dye was released from the surface sampling ports on
the windward and leeward hill lines (e=180°, 0°) to study the surface flow
patterns, or alternately, through an injection rake emitting dye at 11
levels above ground at the stack position to study the centerplane streamlines.
To ensure that the dye solution at each port or tube would have the same
density as the static fluid at the same height as the dye port with the
hill stationary, salt solution was drawn through each port into small jars
by application of vacuum pressure to a second tube fixed into the lid of
the jars (see Figure 8). Small amounts of concentrated dye were then added
to each of the jars. Four different colors of dye were used in an alternating
pattern for the surface ports to distinguish different levels. Then, with
the carriage moving, the "vacuum" tank was pressurized, forcing the dye/salt
solution back through the ports into the flow around or upstream of the
hill. Color photographs (see Figure 11) and motion pictures were taken
of the surface releases using this process at Froude numbers of 0.2, 0.4,
0.9, and 1.7. Black and white photographs of the upstream multilevel dye
release were taken at Froude numbers of 0.2, 0.4, 1.0 and 1.7.
The multicolored dye solution, the streamers from the upstream multilevel
injection tubes, and the plumes from the upstream stacks were photographed
using 35mm cameras from the side and bottom of the towing tank (top view
of the hill). The photographic and lighting arrangements are sketched in
Figure 8. Both cameras used Kodak Tri-X film, shutter speeds of 1/125 s,
and f/8 aperture settings. Back diffusive lighting for the side views was
provided by a fluorescent light panel. This panel consisted of six 2.4 m
long fuorescent tubes (75 W each) mounted on a white backboard and a 3 mm x
1.2 m x 2.4 m sheet of white, translucent acrylic plastic placed between the
tubes and the towing tank to diffuse the light. Reflective lighting for the
top views was provided by banks of fluorescent tubes placed just under the
floor of the towing tank. Each of these banks contained three 76 cm and two
20
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51 cm long fluorescent tubes (total wattage: 130 W/bank) and covered an area
72 x 118 cm. These banks lined the outside thirds of the floor of the towing
tank over its entire length (see Figure 8). The model and baseplate were
painted white to provide a suitable background against which to photograph
the dye. For the side views of the multicolored dye experiments, the still
camera, motion picture camera, and the diffusive light panel were attached
to the carriage and towed with the hill. For top views and side views of
the plumes from the upstream stack, all the cameras and the light panel were
stationary relative to the towing tank. The refraction of light by the salt
solution resulted in some distortion and blurring of the photographs, so that
quantitative information was sometimes difficult to derive directly from the
photographs.
To obtain a better understanding of the surface flow patterns, granules
(2- to 3-mm diameter) 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. Whereas the technique was highly useful for the intended
purpose, it was undesirable from a number of viewpoints: (a) After a few
minutes of contact with the water, the KMnO, stained the model surface a
dark brown, making the dye streamers difficult to distinguish from the brown
surface stains in the black and white photographs. At towing speeds in
excess of about 15cm/s, the streamers were exceedingly difficult to photo-
graph because they became too thin. To the eye, however, they were easily
distinguishable, (b) The KMnO. colors the entire water body in a short
amount of time, making further visualization more difficult. It can be
neutralized with sodium sulfite, and the resulting precipitate filtered out;
whereas this could be useful for filtered recirculating water channel, it
was not feasible in our stratified tank, (c) It is a strong oxidizer; it
reacts with numerous materials used in the construction of the tank and is
somewhat hazardous to handle, (d) It is tedious and cumbersome to apply
and cleanup in a tank as large as ours.
Because the streamers and the dye from the multilevel injection tubes
were difficult to photograph at the higher tow speeds (higher Froude numbers),
the tank was filled with a much weaker stratification (fKO.5 rad/s as opposed
to ~1.3 rad/s) so that surface flow and centerplane streamline patterns
21
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could be photographed at high Froude numbers and low towing speeds. This
provided an additional benefit in that it provided an opportunity to observe,
at least to a limited extent, the variation of the flow patterns with
Reynolds number alone (see Sect. 5.4). Photographs of the surface flow
patterns are presented in Figure 12.
Shadowgraphs (see Figure 13) were also photographed using a 35mm camera
placed 9 m from the viewing screen. The light source for the shadowgraphs
was a 750 W lantern slide projector, placed 9 m from the model center!ine.
A large sheet of translucent drafting paper was taped to the sidewalls of
the tank for a viewing screen.
4.1.3 Measurement of plume impingement points and streamline deflections
One series of tests was run at Froude numbers of 0.2, 0.4, 0.6, and 0.8
and stack heights of 0.2, 0.4, 0.6, and 0.8 hill heights in order to
determine (a) whether the plume went over the hill or around it, (b) the
impingement point z-, defined as the height of the maximum concentration
on the 180° line (upstream centerline on the hill surface), and (c) the
streamline deflection at e=-90°. Samples were drawn during the tow simultan-
eously through all the surface ports using the sampling system described
previously (Figure 8). The collected sample jars were then visually inspected
to determine which contained the highest concentration of dye (generally
an easy task).
The multilevel dye injection photographs (Figure 14) were also analyzed
to obtain streamline deflections over the top of the hill.
4.2 Small Towing Tank Experiments
The disadvantage of the large stratified towing tank is that the
residual flow takes about an hour to settle down, and it takes at least
2 hours to change a model. Consequently, if some quick qualitative experi-
ments using a number of differently shaped models are required, or if the
models need to be withdrawn from the fluid often, a small tank is desirable.
Another advantage is that hydrogen bubble visualization experiments can
be more easily conducted when there is not a large volume of refracting
fluid to distort the image, as happens in the large tank.
22
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These are some of the reasons for also using a small stratified towing
tank 2.0m x 0.20m x 0.10m deep in these experiments. The tank was filled
by the usual "two-tank" method of Oster and Yamamoto (1963), with a stratified
salt solution having specific gravity varying from 1.0 at the surface to
as much as 1.2 at the bottom. Model hills were mounted on a baseplate
suspended from a carriage driven by a cable and electric motor similar in
principle to that of the large tank (see Figure 9a). The practical speed
range was 5 to 25 cm/s. The model hill and baseplate could be lowered into
the tank and fixed simply by fastening two nuts. The polynomial hill used
here had about the same shape as the large hill; its height was 2 cm.
4.2.1 Surface stress patterns
Shear stress patterns were observed on the surfaces of several model
mountains by a technique similar to the surface oil flow technique commonly
used in wind tunnel studies. The technique itself consisted of coating
the models with a gelatinous solution of dye and KNOX brand gelatin (100 cm
blue liquid dye to 10 g gelatin). The models were then clamped into place
on the towing carriage and drawn through the tank (about 2 m distance).
As shown in Figure 12c, 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.
This technique is fraught with difficulties, making its use less
satisfactory than the corresponding wind tunnel technique using zinc oxide
powder and oil. Because the residual motion of the water destroyed the
surface pattern after towing ceased, it was essential to photograph the
pattern during the last instant of the tow. The consistency of the dye/gelatin
mixture was critical; if the solution was too gelatinous, no pattern formed;
if it was too thin, the dye was quickly diluted and washed away. Temperature
also played a role in obtaining proper consistency. A solution temperature
of 30 ° to 32 °C permitted uniform surface coating of the gelatin mixture
for towing in the water at 24 ° to 25 °C. Finally, as the gelatin slowly
dissolved, the dye separated from the surface with the flow at the separation
points; these dye streaks, away from the surface, although weak, were super-
imposed on the surface flow patterns in the photographs.
23
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4.2.2 Hydrogen bubble technique
A hydrogen bubble wire system (Schraub et al., 1965) was developed
to study the streamline pattern and the velocity field on the centerline
of the hills (see Figure 9a). The wire was 6 cm long and fixed 5.5 cm
from the leading edge of the baseboard. The wires used in this study
were 0.025-mm diameter chrome! thermocouple wires. Several other types and
diameters were tried, but most produced poor bubble quality (i.e., large
bubbles which rapidly rose to the surface) or were rapidly contaminated.
Time markers were produced by pulsing the voltage applied to the wires,
generating distinct lines of bubbles. To generate uniformly spaced bubble
streaks, the wire was kinked by running it between two small gears (Clutter
and Smith, 1961). This provided a very uniform spacing of streaks, but
was usable only at speeds in excess of about 8 cm/s because of problems
with bubble rise. The bubbles that formed on the wire tended to be swept
back to the apex of the kink and coalesce before they were swept off by
the flow. Even finer wire would have been helpful in this case. Current
flow through the wire was also found to be significant in altering the bubble
size, lower current producing smaller bubbles.
The power supply used (see Figure 9b) was a general purpose unit with
output voltage of 15 to 30 V at 1 A maximum current. It could be used in
the continuous mode or pulsed to provide time markers. The oscillator section
provided variable pulse rate and pulse width. The output current was adjustable
over a wide range, which was found to be important in adjusting bubble size.
Provision for reversing polarity was also included for cleaning purposes.
Polarity reversal removed any contaminants that had been electrolytically
deposited. Photographs of the pulsed bubble patterns are shown in Figure 15.
The lighting and photographic arrangement used is shown in Figure 9c.
Backlighting was used to illuminate the bubbles. This consisted of two
750 W studio lights arranged at an angle of 25° to the test section. A
flat black background was placed immediately behind the test section to
provide adequate contrast for the illuminated bubbles. Photographs were
recorded with a 35mm camera equipped with a 135mm lens to reduce distortion
due to parallax. Tri-X film was used with an aperture setting of f/5.6
and exposure time of 1/250 s.
24
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Velocities were obtained from the photos by determining the distance
between successive bubble streaks. This was accomplished by digitizing
the centerpoints of successive streaks, then computing the straight line
distance between the two points. Conversion of the measured distances
to velocity values was accomplished by scaling the measured displacements
by the displacement of a set of reference streaks located in the free stream
upstream of the model. It was assumed that these freestream reference
points represented the actual towing speed for the streamline.
While many photographs were taken, not all were of sufficiently good
quality for analysis. The poor quality photographs were mostly at low
velocities where bubble rise was a serious problem. Also, as the bubbles
rose they tended to cling to the model and coalesce, forming bubbles large
enough to alter the flow patterns (cf. Figure 15a). In some of the photographs,
bubble quality was not good enough to allow for complete analysis, though
the gross flow pattern was quite visible.
4.3 Wind Tunnel Experiments
The identical polynomial hill used in the towing tank was also used
in the wind tunnel for flow studies under neutral stability conditions.
The EPA Meteorological Wind Tunnel (Thompson and Snyder, 1976) has a test
section 3.7 m wide, 2.1 m high, and 18.3 m long (see Figure lOa). The
air speed in the test section may be varied from 0.5 to 10 m/s. The flow
uniformity in the core (outside the wall boundary layers) is good, i.e.,
deviations from the average flow speed are less than 1% across the test
section at 3 m/s. The turbulence intensity is typically 0.5% at this speed.
The hill was placed such that its apex was 424 cm from the entrance
to the test section (Figure lOb).
4.3.1 Velocity measurements
Turbulence measurements were made with Thermo-Systems Model 1054A
anemometers in conjunction with Model 1243-20 cross-film (boundary layer
style) probes. The output signals from the anemometers were digitized
at the rate of 1000 samples/s and linearized and processed on a POP 11/40
minicomputer. Sampling (averaging) times of 1 min were found to yield
reasonably repeatable results.
25
-------
Probes were calibrated next to a pi tot tube in the free stream flow
in the test section of the wind tunnel. A computer program calculated
the parameters (A, B, and a) to yield a "best fit" to Kings law, E2=A+Bl)a,
from the six input calibration points. Other programs calculated the
average velocity U, the longitudinal and vertical turbulence intensities
I /p I /O
(u2") ' and (w2") ' , the Reynolds stress -uw, and the flow angle,
4>=tan (U^-UgJ/tUn+Up)). where U, and Up are the mean velocities indicated
by film numbers 1 and 2, respectively. Because the hot-film was sensitive
to ambient temperature drift, the sampling program was set up to correct
o
for temperature drift according to E =(A+BUa)(T.p-T , )(T-r-T_al), where
T FiGW T Co I
Tf Tnew' and Tcal are the fl"lm» new roorn' and calibration temperatures,
respectively (Bearman, 1971).
The Reynolds stress measurements were corrected for errors due to
sensor yaw response using the correction factor graphed in Figure lOc.
This correction factor was obtained by measuring shear stress profiles
in a pipe flow apparatus over a range of velocities from 2 to 17 m/s.
The resulting profiles were extrapolated to the wall of the pipe, and
the value obtained was compared with the wall shear stress calculated
from the measured pressure drop along the length of the pipe. The ratio
of the measured wall shear stress to the extrapolated value was then used
as the Reynolds stress correction factor. It was found to be dependent
upon both velocity and the geometry of the sensor support; hence, the
correction in Figure lOc is specific to the boundary layer style probe.
A comparison of the transverse and longitudinal components of turbulence
intensity measured in the pipe flow apparatus showed good agreement with
those measured by other investigators, hence, no corrections were applied
to those measurements.
4.3.2 Flow visualization
A paraffin oil-fog generator (Kenney Engineering Model 1075SG) was
used to produce smoke for the qualitative 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.
26
-------
One phase of this study involved visualization of smoke emitted from
the stack upstream of the hill. Plume centerlines were traced from photo-
graphs at various stack heights in order to obtain an idea of the center!ine
streamline 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 which was also helpful for understanding the
surface flow pattern.
27
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5. PRESENTATION AND DISCUSSION OF RESULTS
5.1 Presentation of Results
Figures 11 through 16 present the visualization photographs. Figure 11
shows the flow patterns observed from the injection of tracers through the
surface ports, multicolored dye in the towing tank, and smoke in the wind
tunnel. Figure 12 shows the surface shear stress patterns observed from
the gelatin/dye solution in the small towing tank and from the potassium
permanganate in both the large and small tanks. The shadowgraphs of Figure
13 show the change in the location of the separation point downstream of the
hill as the Froude number changes. Figure 14 shows the streamers from the
multilevel injection tubes upstream of the hill and tracings of smoke plume
center!ines from the wind tunnel tests. The speedup over the top of the
hill and the downstream lee wave patterns are illustrated in the hydrogen
bubble photographs of Figure 15. Figure 16 shows how plumes from upwind
stacks impinge on the hill surface and/or go round the hill, depending on
the Froude number.
Figure 17 presents the quantitative measurements of the flow structure
with and without the hill in the wind tunnel. We have attempted to collate
all the observations by drawing the surface shear stress patterns and the
mean center!ine streamline patterns for a number of different Froude
numbers in Figure 18. The classification of the surface singular points,
denoted by P., are tabulated in Table 1.
The plume impingement height is plotted as a function of the source
height and the hypothesis F=l-H$/h is shown to be valid in Figure 19.
A streamline pattern over Elk Mountain, Wyoming, is presented in Figure 20
for a crude comparison with the present observations. Figure 21 shows the
displacement of streamlines above the hill top as a function of Froude number.
28
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TABLE 1: CLASSIFICATION AND LOCATION OF SINGULAR POINTS ON POLYNOMIAL HILL
Froude PI P2 P3 P4 Pg
No. (x,y) (x,y) (x,y) (x,y) (x,y)
(-2.4,0) (-.4,0) (.4,0) (3.4,0.9) (5.8,0)
(-1.8,0) (-1.0,0) (1.0,0) (2.4,0.5) (3,0)
(0.5,0) (1.2,0.4) (1.6,0)
1.7 (N'}^ {N1}^ {S1}^
(0.4,0) (1.8,0.7) (3.4,0)
(0.8,0)
(1.2,0)
(-0.3,0) (1.3,0.7) (3.5,0)
(0.2,0)
(0.6,0)
Notes: (i) Singular points denoted by P.. are shown in Figure 18.
(ii) {S1}, half-saddle point; {N1}, half-node point;
{Ka', attachment point; {}^s's separation point.
(iii) Coordinates of P. are given relative to the center of the hill
and are normalized in terms of the hill height, h.
(iv) The singular points P3 may, in fact, be three singular points
as found at F=1.7 and F=°°.
29
-------
5.2 Qualitative Description of the Flow
At the lowest Froude number (F=0.2, Figures lla, b, 12a, b, 13a, b,
14a, b, 16, and 18a), the flow is, in large measure, constrained to move
in horizontal planes. In a narrow region near the top of the hill, the
flow has enough kinetic energy to overcome the potential energy and, thus,
to go over the top. In the middle region, [H,] of Figure 2, the plume rises
slightly before it impacts on the surface, dips slightly from its upstream
elevation as it goes round the sides, then rises again as it separates from
the surface of the hill. Separation occurs as the flow goes round the sides
at an angle of approximately 110° from the upstream stagnation line. A
slight hydraulic jump occurs just downstream of the top of the hill. The
surface shear stress patterns show a symmetric pair of more or less vertically
oriented vortices downstream causing an upstream flow on the center!ine.
The downstream dye release, however, shows the surface flow to be primarily
perpendicular to the free-stream flow direction and to oscillate from one
side to the other, broadly filling the wake with dye. There is only a very
weak upslope component to the flow on the leeward slope. Frequently,
different colors of dye, emanating from different elevations on the surface,
would be observed flowing in opposite directions. This oscillation of the
flow in the wake also caused the plumes upstream of the hill to oscillate
from side to side at low frequency. (See, for example, Figure 14b, at a
lower Froude number of 0.1.) A time sequence of photographs showed the
plume (from a stack of height H =0.4h) oscillation to be irregular in amplitude
and frequency. The lateral displacement of the plume impingement point
reached as far as one-third the local hill radius on each side of the center-
line at a Froude number of 0.2. There was a significant component of energy
at a vortex shedding period of 33 s (calculated from the Strouhal number
using the hill radius at the height of the stack), but, since this period
differs with elevation and since the flows at different levels are not dis-
connected, some overall, apparently irregular, oscillation occurs.
A weak downslope flow on the windward center!ine results in a weak,
perhaps intermittent, horseshoe vortex; its location appears to move up
and down the slope. Indeed, the streamers from the upstream multilevel
30
-------
injection tube could be seen rolling up in vortices of one sign of circula-
tion at one instant and of the opposite sign at the next. Because of the
problems with bubble rise, the hydrogen bubble technique did not yield
satisfactory results at this low Froude number.
At a Froude number of 0.4 (Figures lie, d, 12a, b, 13d, 14a, b, 15a,
and 18b), the flow has more energy to move in the vertical direction.
The region where the flow goes over the top is now broader. At a stack
height of only 0.6h, the plume is spread thinly to cover the entire upper
half of the hill (Figure 16). In the middle region [H,], as at the smaller
Froude number, the plumes rise somewhat before they impact on the hill;
they dip significantly, however, from their upstream elevation as they
go round the sides and continue losing elevation, at least for some distance
downstream. In going round the sides, the flow separates from the surface
at about the same point as it did at the smaller Froude number (110°).
In going over the top, however, the flow does not separate until it is
roughly half-way down the lee slope. The shadowgraphs (Figure 13d) show
a strong hydraulic jump just downstream of this separation point. The
height of this disturbance varied from 1.2 to 1.8 hill heights. The hydrogen
bubble photographs from the small towing tank show a weaker hydraulic jump
but illustrate the lee wave pattern more vividly.
The surface shear stress patterns again show a symmetric pair of vertically
oriented vortices downstream causing an upstream flow on the centerline.
These vortices, however, are smaller and closer to the base of the hill
(rather than farther downstream). Dye release from the downstream ports
again shows the surface flow to be perpendicular to the free-stream flow
direction, to oscillate from side to side, and to broadly fill the (now
narrower) wake. The motion picture film shows this more clearly.
The reverse flow down the windward slope is much more prominent, with
the plume from any stack less than h/2 being rolled up into an intermittent
horseshoe vortex. At a stack height of 0.2h, the location of the point
of maximum concentration was lower in elevation than at the lower Froude
number of 0.2.
At a Froude number of 0.6, separation begins near the downwind base
of the hill, and the hydraulic jump peaks approximately two hill heiahts
downstream of the hill center (see Fiaure 13f). At a Froude number of 0.8,
31
-------
separation begins even farther downstream, and the hydraulic jump peaks
somewhere beyond three hill heights downstream (Figure 13g).
At a Froude number of 0.9 (Figure lie), all the flow in the center-
plane goes over the top of the hill, but a plume starting on the ground-
level center!ine upstream of the hill would be broadly spread to cover
nearly the entire hill surface. The flow goes down the leeward slope
without separating. At a slightly higher Froude number (1.0), the hydrogen
bubble photograph (Figure 15b) and the surface shear stress patterns (Figures
12b and c) clearly show separation at the top of the lee slope. However,
neither the streamers from the multilevel injection tubes (Figure 14)
nor the shadowgraph (Figure 13) make this separation bubble visible.
Evidently, the flow is on the verge of separating from the top of the
lee side of the hill; the existence or nonexistence as well as the size
of the recirculating region is evidently critically dependent on the exact
Froude number, the Reynolds number, and/or the precise upstream flow
conditions.
As the Froude number is increased farther, the size of the recircu-
lating region grows, and the wake dimensions grow laterally and vertically.
At a Froude number of 1.7 (Figures llf, 12, 13, 15c, and 18d), the flow
separates just past the top of the hill, resulting in a large recirculating
region on the leeward slope. Again, a plume starting at ground level
on the upstream centerline would be spread broadly to cover the entire
hill surface. This flow resembles neutral flow except that the streamlines
are closer to the top of the hill and they lose elevation much faster in the
wake of the hill.
Smoke visualization in the wind tunnel (F=°°) showed the plumes to
be spread broadly to cover the entire hill surface. Figures llg and h
show oil smoke being emitted through the sampling ports. From the release
from the upwind ports, it may be seen that the flow speed is low on the
lower half of the slope and fairly high on the upper half. The release
from the downwind ports shows the flow to be u£ the slope, separating
upstream of the hill center. Although these are instantaneous photographs,
they are fairly representative of the average characteristics. Figures
Hi and j show smoke being emitted from a puddle of titanium tetrachloride
32
-------
at the downwind base of the hill. The large fluctuations in the flow
field are evident, but the up-slope flow and separation at the top are
fairly consistent. Also, the horseshoe vortex may exist intermittently
low on the upwind slope (see Figure 12c).
Measurements of the velocity fields with and without the hill in place
in the wind tunnel (F=°°) are presented in Figure 17; included are mean
velocity (speed as well as direction), local longitudinal and vertical
turbulence intensity, and Reynolds stress profiles. The free stream speed
U^ was 3 m/s. From the graphs, it may be seen that the boundary layer
over the smooth tunnel floor was approximately 65 mm thick at the center
of the hill (but in the absence of it). The free stream turbulence intensity
was typically less than 0.5%. The filling out of the velocity profiles
on the upwind slope of the hill and the overspeed regions on the top and
side are evident.
The mean velocity vectors show quite significant vertical components
as far as two hill heights upstream and up to twice as high as the hill
itself. Directly above the hill and on the side, however, the vertical
components are essentially zero. The abnormally high turbulence intensities
at the upwind base of the hill are most likely caused by the abrupt step
(6 mm) from the wind tunnel floor to the hill, only a short distance
upstream from there. The very small turbulence intensities on the top
and side of the hill are a result of the overspeed and extremely thin
boundary layers at those points.
5.3 Comparison with Theory
From all the observations at low Froude number (F<0.4), we can see
the validity of the theoretical description of the flow (given in Sect.
3) for the asymptotic region [H-j]. That is, the velocities are primarily
horizontal, and the impinging streamlines are deflected downwards (unlike
neutral flow where they mainly move upwards).
Unlike the smaller scale experiments of Brighton (1978), where the flow
was everywhere laminar, a most important feature of these experiments is
that the boundary layer on the surface of the hill is turbulent. Despite
this, the low Froude number theory is approximately valid. This gives
additional support to the validity of this scaling to the atmosphere,
33
-------
where almost always the surface flow is turbulent.
Of great practical importance is the height z. at which the plume
impinges on the hill. The asymptotic theory when F«l predicts that when
Hsh(l-F) (our hypothesis of Sect. 3.1), the plume will touch or rise
over the top of the hill. Note that if h>Hs>h(l-F), the streamlines can
pass down the lee side of the hill very close to the surface, as if the
top of the hill was a hill of height (Fh) on a plane at z=h(l-F).
The photographs of surface dye releases also support the hypothesis
that the summit region [T] has a thickness of Fh. Figures lla and c show
that dye from the ports moves upwards rather than horizontally or downward
when z/h>0.82 for F=0.2, and z/h>0.61 for F=0.4, compared with predicted
values of 0.80 and 0.60.
The maximum observed downward deflections at e=90° are compared in
Table 2 with values computed from Eq. 3.3 using potential flow theory
and from Eq. 3.2 using a free-streamline model for UQ, following Riley
et al.,(1976), which given a maximum value for Ug/U^ of about 1.4 when
e=90°. H is the upstream height of the observed streamlines.
TABLE 2. COMPARISON OF OBSERVED AND CALCULATED STREAMLINE DEFLECTIONS.
Froude
Number
0.2
0.2
0.2
0.2
0.2
0.4
0.4
0.4
Hs/h
0.20
.40
.55
.60
.80
.40
.55
.60
lObserved
>0.15
.12
.16
.21
.08
.12
.10
.38
-A/h @ 6=90°
Calculated
(Eq. 3.3) (Free
0.25
.17
.16
.17
.25
.67
.64
.67
Calculated 1
streamline theory)
0.08
.08
.32
.32
34
-------
When F=0.2, the observed vertical deflection in the middle region
[H.j], is only slightly smaller than that calculated from Eq. 3.3, but
about twice as large as that computed using the free-streamline model.
The agreement is poor near the top of the hill in region [T] as was expected.
When F=0.4, the deflections calculated from Eq. 3.3 are greater than the
upstream height of the streamlines, which is, of course, absurd; the theory
is obviously invalid at this Froude number. The free-streamline theory
overestimates the deflections by a factor of two to three.
Riley et al. (1976) and Brighton (1978) investigated the flow around
model hills that did not slowly widen at their base; their hills all had
gradients greater than or equal to one near the base. The flow pattern
produced by this type of hill is quite different from one whose gradient
tends to zero near its base (i.e., dR /dz-*>° as z/h+0). This slope tends
to produce a large deflection when z/h<0.1. This shape of hill also pro-
duces a downflow on the upstream center!ine of the hill, resulting in
a large negative deflection at 9=180 , whereas the theory predicts zero
deflection on the upstream stagnation line. This is part of the explanation
for the vortex on the lower upstream side of the hill seen in Figure 18a.
This vortex exists in the base region [B] (see Figure 2a), where the
asymptotic theory described in Sect. 3 is invalid. The vortex is necessary
kinematically if the other singular point classifications in Figures 18a
and b and Table 1 are correct.
Another possible explanation for the differences between theory and
experiment is that the plume diffuses downwards more than upwards when
z/h
-------
Figure 18a and b show some aspects of the complicated separated flow
in the rear of a bluff three-dimensional hill when F<1. The mainly
horizontal motion around the hill region [H-,] induces a boundary layer
motion on the hill which separates, much as on a circular cylinder. The
angle of separation is about 110° from the upstream stagnation line.
This compares to about 80° for a circular cylinder at the same Reynolds
4
number of 10 , or, since the surface flow is very turbulent, separation
might occur at 100° or even 110°.
When F=0.2, Figures lla and 16 show that centerline streamlines in
region [H,] starting upstream at a height H (
-------
we observe (Figure 14a) that streamlines emanating from a height H above
about h/3 pass over the top, while those below that impinge onto the hill
(the hypothesis of Sect. 3.1 for z^ is not valid when M).5).
The flow patterns shown in Figures 11 and 14 not only indicate the
path of the center line of the plume but also the mechanisms controlling
the width of the plume. 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. From the photo-
graphs it can be deduced that the vertical density gradient is about five
times its upstream value, so locally N increases by five. 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
~" """" Z
plume. From the photograph in Figure 14, the distance between streamlines
about 0.2h above the hill top is seen to be about half its upstream value;
for comparison purposes it is worth noting that in neutral potential flow
over a hemisphere, on a streamline 0.2 of a radius above the hill top,
the vertical plume width is about 0.6 of its upstream value.
In Figure 20a the streamline pattern over Elk Mountain, Wyoming,
estimated from aircraft observations by Marwitz et al. (1969) is shown
for comparison with Figures 14 and 18c. The density distribution was
different from that in our tank, there being a well-mixed layer below
-1 /2
a fairly sharp inversion, but the gross Richardson number (=F~ ' ) was
about 1.0. It is interesting that these observations indicate that if
H /h>l/3 the streamlines pass over the hill top, as in our experiments;
for a streamline starting at H /h^l.O, these observations indicate that
the distance ns from the hill top to the streamline is O.lh, whereas
we found 0.2h (see Figure 21). Although these are of the same order the
difference is significant and is probably due to the different upstream
density gradient.
Figure 20b shows streamline computations by Marwitz et al. (1969)
from a two-dimensional inviscid shallow water model; the model evidently
gives some of the main qualitative features of uniformly stratified flows
(probably only for F>_1). The surface streamline patterns shown in Figures
37
-------
18c and d are similar to those inferred by Connell (1976) from his aircraft
observations.
The main difference between the flow at Froude number 1.7 and neutral
flow (F=°°) is the location of the separation point on the top of the hill
P3 (Figures 18d and e and 12c). When F=1.7, P3 is slightly downwind
of the centerline, and, when F=°°, P, is upwind. The character and location
of the other singular points is approximately the same. Another difference
is that the streamlines are closer to the top when F=1.7 (Figure 21).
The mean velocity and turbulence over the hill in neutral flow measured
in the wind tunnel are shown in Figures 17a through e. Note particularly
in Figure 17a that the speedup factor S over the hill is the same on the
top as on the side of the hill, as would be expected by potential flow
theory (Sect. 3.4). Defining S on the top as S=U(1.03h)/U (h), we find
oo
S=1.27. To allow for the separated region in the lee of the hill, as
a reference value we might compute S for an ellipsoid elongated in the
direction of the flow such that h=b, and L0/h=3/2; from Figure 5c, S=1.28,
which suggests that the measured value is of the right order. The
turbulence measurements indicate the thickness of the upwind boundary
layer and how its thickness is reduced by the large negative pressure
gradient over the hill. There is effectively very little turbulence outside
the boundary layer.
Figure 21 shows the displacement of streamlines above the hill surface
as a function of upwind streamline height, derived from the multilevel
tracer injection photographs (Figures 14a and c). The neutral flow data
from the towing tank and from the wind tunnel match fairly closely the
upper limits predicted by potential flow theory (Eqs. 3.19 and 3.20),
except for the larger values of Hg. The theory predicts n «H|, but is
obviously valid only when Hs«h.
Evidently because of the wake, the flow over this body more closely
resembles that over one very long in the streamwise direction (compared
to its width) than it does that over an axisymmetric body.
It is evident from the figure that as the stratification increases,
the streamlines become much more closely packed. The implications for
concentration will be discussed in Part II.
38
-------
The data from the small towing tank consistently show smaller displace-
ments than those from the larger facilities; this may be due to problems
with bubble rise (the model was towed upside down) or it may be a Reynolds
number effect (the small towing tank is 1/12th the size of the large tank).
How do the experimental flow patterns compare with the speculative
predictions of Sect. 3.2 about the lee waves and the existence and position
of separation? We first examine the nature of the separated flows at
the four primary Froude numbers of our tests, and classify these as sub-
or supercritical, according to the nature of the separation. For subcritical
flows (F.Fcr ),
separation on the centerline is determined by the boundary layer behavior
on the lee slope under the action of the pressure gradient outside the
boundary layer, which may, of course, be affected by the lee waves. We
conclude without any doubt that, when F=0.2 and 0.4, the separated flows
are subcritical, and when F=1.0 and 1.7 the flows are supercritical; though
in the larger tank, when F=1.0, the separated flow region is very small,
and the flow is only just supercritical.
When F~F. =0.3 separation does in fact occur on the lee side of the
hill at about a value of x=L/2, the separation being caused by the rapid
upward movement in the standing lee wave behind the hill. This corresponds
approximately to the lower limit of F. at which separation occurs downstream
of the hill. Thus the observed lower critical Froude number of about
0.3 compares well with the rough estimate of 0.3 suggested in Eq. 3.8b.
When F. is less than the lower critical Froude number, then F«l, and
the lee waves have a small amplitude (MJ/N^hF) and, near the hill, are
confined to the [T] layer. This is the pattern observed when F=0.2.
Secondly, we can use the hydrogen bubble photographs in Figure 15a,
b and c to estimate the wavelength A of the lee waves, to compare these
with the natural wavelengths of the internal waves in the stratified flow,
and to note the value A/L as the separated flow changes from a sub- to
a supercritical character. We tabulate these parameters below:
39
-------
TABLE 3: LEE WAVE ESTIMATES AND SEPARATED FLOW CLASSIFICATION
Theoretical (3-
dimensional int-
ernal waves)
A/L = 2TrF,_
2.5
6.3
Measured (at
about z=2h
from Figs. 15a&b)
2.0 + 0.5
6.0 + 1.5
Character of
separated flow
Predicted Observed
(based on
eq. 3.8b)
subcritical subcritical
supercritical supercritica
0.4
1.0
Note that : (a) in our experiments FL=F because L=h, and (b) the ratio
of the hill height h to that of the tank D is about 0.2 (for the large
or small tank), so that the wavelength of the natural waves, given by
Eq. 3.8b, is about equal to that of waves in an infinite medium, i.e.
2TTU/N.
5.4 Flow Variations with Reynolds Number
Most of the runs were done in the large towing tank at a fixed strati-
fication (N^l.33 rad/s), and in order to increase the Froude number, the
towing speed was increased; this resulted in a corresponding increase in
the Reynolds number. Systematic studies to examine the influence of
Reynolds number on the flow patterns around the hill were not conducted.
However, some experiments were done in the large tank with a weaker strati-
fication (N^0.5 rad/s), and also in the small tank, thus, yielding infor-
mation at the same Froude number, but different Reynolds numbers. For
example, at F=0.4, we have Reynolds numbers of 400 and 1900 from the small
tank, 10,000 from the large tank with weak stratification, and 27,500 from
the large tank with strong stratification. In the large tank, no variation
with Reynolds number was observed; allowing for differences in visualization
techniques and for slight differences in the shape of the hill and the
absence of the "square flange" in the small tank, it is also safe to say
that only slight variations of the flow patterns with Reynolds number
40
-------
were observed. Our conclusions on the flow patterns were thus more
heavily weighted by the results from the large tank. Extrapolations of
these results to the atmosphere, where the Reynolds number may be several
orders of magnitude larger is a much more serious question, and no apologies
are offered. It has been shown by numerous authors that many of the flow
patterns observed in the atmosphere, in particular lee waves and hydraulic
jumps, can be simulated at much smaller Reynolds numbers (even considerably
smaller than in the current experiments) in the laboratory.
41
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Brighton, P.W.M., 1977: Boundary Layer and Stratified Flow over Obstacles,
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Brighton, P.W.M. 1978: Strongly Stratified Flow Past Three-Dimensional
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Clutter, D.W., and Smith, A.M.O., 1961: Flow Visualization by Electrolysis
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Connell, J.R., 1976: Wind and Turbulence in the Planetary Boundary Layer
around an Isolated 3-D Mountain as Measured by a Research Aircraft, Int.
Conf. on Mountain Meteorol. and Biometeorol., Interlaken, Switz., 28 pp.,
June.
Crapper, G.D., 1959: A Three-Dimensional Solution for Waves in the Lee of
Mountains, J. Fluid Mech., v. 6, p 51.
Davis, R.E., 1969: The Two-Dimensional Flow of a Fluid of Variable Density
over an Obstacle, J. Fluid Mech., v. 36, p 127.
Drazin, P.G., 1961: On the Steady Flow of a Fluid of Variable Density Past
an Obstacle, Tellus, v. XIII, No. 2, p 239-51.
Gradshteyn, I.S. and Ryzhik, I.M., 1965: Table of Integrals, Series and
Products, 4th ed., Academic Press, New York, N.Y., 1086 pp.
Hunt, J.C.R., Abell, C.J., Peterka, J.A. and Woo, H., 1978: Kinematical
Studies of the Flows around Free or Surface Mounted Obstacles; Applying
Topology to Flow Visualization, to be published in J. Fluid Mech.
42
-------
Huppert, H.E., 1968: Appendix to paper by J.W. Miles, J. Fluid Mech.,
v. 33, p 803.
Kitabayashi, K. , Orgill, M.M. and Cermak, J.E., 1971: Laboratory Simulation
of Airflow and Atmospheric Transport-Dispersion over Elk Mountain, Wyoming,
Fluid Dyn. and Diff. Lab. Report No. CER 70-71 KK-MMO-JEC-65, Colo. State
Univ. Fort Collins, Colo., 90 p., July.
Lighthill, M.J., 1963: Laminar Boundary Layers, p. 48-88 (Ed. L. Rosenhead),
Oxford Univ. Press.
Liu, H.T. and Lin, J.T., 1975: Laboratory Simulation of Plume Dispersion
from Lead Smelter in Glover, Missouri, in Neutral and Stable Atmosphere,
EPA-450/3-75-066, U.S. Environmental Protection Agency, Research Triangle
Park, N.C., April, 47 pp.
Liu, H.T. and Lin, J.T., 1976: Plume Dispersion in Stably Stratified Flows
over Complex Terrain; Phase 2, EPA-600/4-76-022, U.S. Environmental Pro-
tection Agency, Research Triangle Park, N.C., May.
Marwitz, J.D., Veal, D.L., Auer, A.H., Jr., and Middleton, J.R., 1969:
Prediction and Verification of the Air flow over a Three-Dimensional
Mountain, Tech. Rpt. No. 60, Natural Resources Res. Inst., Univ. of Wyo.
Milne-Thomson, L.M., 1960: Theoretical Hydrodynamics, 4th ed., Macmillan
Publ. Co., New York, N.Y.
Oster, G. and Yamamoto, M., 1963: Density Gradient Techniques, Chem. Reviews,
v. 63, p 257-68.
Pao, Y.H., Lin, J.T., Carlsen, R.L., and Smithmeyer, L.P.C., 1971: The
Design and Construction of a Stratified Towing Tank with an Oil-Lubricated
Carriage, Flow Research Rpt. No. 4, Flow Res., Kent, Wash., 31 p.
Queney, P., Corby, G.A., Gerbier, N., Koschmieder, H., and Zierep, J., 1960:
The Airflow over Mountains, World Meteorol. Org. Tech. Note No. 34, Geneva,
Swi tz.
Riley, J.J., Liu, H.T. and Geller, E.W., 1976: A Numerical and Experimental
Study of Stably Stratified Flow around Complex Terrain, EPA-600/4-76-021,
U.S. Environmental Protection Agency, Research Triangle Park, N.C., May.
Shraub, F.A., Kline, S.J., Henry, J., Runstadler, P.W., Jr., Littel, A.,
1965: Basic Engr., v. 87, p 429-44.
Scorer, R.S., 1954: Theory of Airflow over Mountains, III: Air Stream
Characteristics, Q.J.R. Meteorol. Soc., v. 80, p. 417-28.
Scorer, R.S., 1968: Air Pollution, Pergamon Press, New York, N.Y.
43
-------
Sykes, R.I., 1978: Stratification Effects in Boundary Layer Flow over
HUls, Proc. Roy. Soc. A. (in press).
Thompson, R.S. and Snyder, W.H., 1976: EPA Fluid Modeling Facility, Proc.
Conf. on Envir. Modeling and Simulation, Cincinnati, Ohio, April 19-22,
EPA-600/9-76-016, U.S. Environmental Protection Agency, Washington, D.C.
Turner, J.S., 1973: Buoyancy Effects in Fluids. Cambridge Univ. Press,
Cambridge, England.
44
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Fh
(A) DEFINITION OF REGIONS
(B) STREAMLINE PATTERNS
(C) PLAN VIEW
Figure 2. Small Froude number theory for flow over three-dimensional hills.
46
-------
(A) SUPERCRITICAL FROUDE NUMBER; NO WAVES POSSIBLE. FL>D/(L7r) SEPARATION IS BOUNDARY
LAYER CONTROLLED.
(B) HILL WITH LOW SLOPE; SUBCRITICAL FROUDE NUMBER; NO SEPARATION ON LEE SLOPE;
SEPARATION CAUSED BY LEE WAVE ROTOR.
(1/77)(1-7i2FL2L2/D2)1/2
-------
SEPARATION
POINT
(A) SADDLE POINTS IN SURFACE SHEAR STRESS LINES.
ATTACHMENT
POINT
ATTACHMENT
SEPARATION
SEPARATION
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ATTACHMENT
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SADDLES (S'l ON THE SURFACE.
Figure 4. Singular points of the surface shear stress lines and mean streamlines.
48
-------
(A) COORDINATES AND NOTATION FOR ANALYSIS
(B) CALCULATED SPEEDUP FOR b>L0.
10 15 20 25 30
(b/h)/(L0/h)
(C) CALCULATED SPEEDUP FOR L0>h.
Figure 5. Potential flow over an ellipsoidal hill.
49
-------
(A) THE EPA WATER CHANNEL/TOWING TANK.
FRESH TANK
WATER TRUCI
DRAIN DR
(B) THE FILLING SYSTEM.
1 1 02 1 04 1 06 1 08 1 10 112 1 14 116 1 18 1.20
SPECIFIC GRAVITY
(C) TYPICAL DENSITY PROFILES
Figure 6. The large stratified towing tank.
50
-------
SAMPLING
RAKE
(A) SKETCH OF HILL ON BASEPLATE.
(B) PROFILE SHAPE OF HILL.
116.8
ALL DIMENSIONS IN cm.
(C) PLAN VIEW OF HILL ON BASEPLATE.
Figure 7. Details of polynomial hill model.
51
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BUBBLE WIRE
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Figure 9. Experimental apparatus for work in small towing tank.
54
-------
(A) THE EPA METEOROLOGICAL WIND TUNNEL.
DIMENSIONS IN cm
23.5
213
424
(B) PLACEMENT OF MODEL IN WIND TUNNEL.
1
1 + k2
= t7j<2" UWMEASURED
0123456789 10
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(C) REYNOLDS STRESS CORRECTION FACTOR FOR BOUNDARY LAYER PROBE.
Figure 10. Details of wind tunnel measurements.
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F = 0.2, Re = 5000
F = 0.4, Re = 10000
F = 1.0. Re = 25000
F = 1.7, Re = 42600
F=°°, Re =34400
(A) SIDE VIEWS OF KMn04 STREAMERS IN LARGE TANK.
Figure 12. Visualization of surface shear stress patterns.
60
-------
F = 0.2, Re = 5000
F = 0.4, Re =10000
F = 1.0, Re =25000
F = 1.7, Re = 42600
F = oo Re = 34000
(B) TOP VIEWS OF KMn04 STREAMERS IN LARGE TANK
Figure 12 (continued). Visualization of surface shear stress patterns.
61
-------
F = 0.2, Re = 160
F = 1.0, Re = 800
F = 1.7, Re =1400
F = oo Re = 1000 F=°°, Re = 2000
(C) TOP VIEWS OF GELATIN/DYE AND KMn04 STREAMERS IN SMALL TANK.
Figure 12 (continued). Visualization of surface shear stress patterns.
62
-------
(A) F = 0.1, Re = 6870
(B) F = 0.2, Re = 13740
(C) F = 0.3, Re = 20610
(D) F = 0.4, Re = 27480
(E) F = 0.5, Re = 34400
(F) F = 0.6, Re = 41200
(G) F = 0.8, Re = 55000
(H) F = 1.0, Re = 68700
(I) F = 1.2, Re = 81600
(J) F = 1.7, Re =117000
Figure 13. Shadowgraphs of flow over hill (N = 1.33 rad/sec).
63
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HYDRAULIC JUMP
(A) F=0.2
Figure 18. Derived centerplane streamline and surface shear stress patterns.
76
-------
HYDRAULIC JUMP
(B) F=0.4
Figure 18 (continued). Derived centerplane streamline and surface shear stress patterns.
77
-------
(C) F=1.0
Figure 18 (continued). Derived centerplane streamline and surface shearstress patterns.
78
-------
(D) F=1.7
Figure 18 (continued). Derived centerplane streamline and surface shear stress patterns.
79
-------
(E) F=°°
Figure 18 (continued). Derived centerplane streamline and surface shear stress patterns.
80
-------
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
F
(A) TEST OF HYPOTHESIS F=1-Hs/h.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Hs/h
(B) IMPINGEMENT HEIGHT AS A FUNCTION OF SOURCE HEIGHT.
Figure 19. Plume impingement on the hill surface.
81
-------
13
12
11
10
9
MT 18 MAR. 1969
'/2 0 1
—-
mile
(A) ESTIMATED FROM AIRCRAFT OBSERVATIONS.
12
11
10
9
8
7
Vs.XXNN.XN.
(B) ESTIMATED BY COMPUTER SIMULATION BASED ON SHALLOW WATER MODEL.
Figure 20. Streamline distributions over Elk Mountain, Wyoming (Diagrams
taken from Kitabayashi etal., 1971, in turn from Marwitz etal., 1969).
F ~ 1.0.
82
-------
0.50
0.45
T~T
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
O LARGE TOWTANK
D WIND TUNNEL, F = °°
A SMALL TOW TANK, F = 1.7
OSMALL TOW TANK, f = 1.0
v SMALL TOW TANK, F = °°
--LIMITS OF POTENTIAL
THEORY (NEUTRAL FLOW)
o /
/ / / F =
//• //
,2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
Hs/h
Figure 21. Displacement of streamlines above hill surface.
83
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/4-78-041
2.
3. RECIPIENT'S ACCESSION NO.
4.TiTLEANDsuBTiTLEFLOW STRUCTURE AND TURBULENT DIFFUSION
AROUND A THREE-DIMENSIONAL HILL
Fluid Modeling Study on Effects of Stratification
Part I> Flow Structure
5. REPORT DATE
July 197ft
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
J.C.R. Hunt1, W.H. Snyder2, R.E. Lawson, Jr.3
8. PERFORMING ORGANIZATION REPORT NO.
Fluid Modeling Report No. 4
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park. N.C. 27711
10. PROGRAM ELEMENT NO.
1AA603 AB-20 (FY-78)
11. CONTRACT/GRANT NO.
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory __RJP, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, N.C. 27711
13. TYPE OF REPORT AND PERIOD COVERED
In-house 4/1/77 - 3/31/78
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
1. University of Cambridge, England. 2. On assignment from National Oceanic and
Atmospheric Administration, Dept. of Commerce. 3. Northrop Services, Inc.
16. ABSTRACT
This research program was initiated with the overall objective of gaining understand-
ing of the flow and diffusion of pollutants in complex terrain under both neutral and
stably stratified conditions. This report covers the first phase of the project; it de-
scribes the flow structure observed over a bell shaped hill (polynomial in cross sectior
through neutral wind tunnel studies and stably stratified towing tank studies. It
verifies and establishes the limits of applicability of Drazin's theory for flow over
three-dimensional hills under conditions of small Froude number. At larger Froude
number a theory is developed, and largely verified, to classify the types of lee wave
patterns and separated flow regions and to predict the conditions under which they will
be formed. Flow visualization techniques are used extensively in obtaining both
qualitative and quantitative information on the flow structure around the hill. Repre-
sentative photographs of dye tracers, potassium permanganate, dye streaks, shadowgraphs,
surface dye smears, and hydrogen bubble patterns are included. While emphasis centered
on obtaining basic understanding of flow around complex terrain, the results are of
immediate applicability by air pollution control agencies. In particular, the location
of the surface impingement point from an upwind pollutant source can be identified under
a wide range of atmospheric conditions. Part II, to be printed as a separate report,
will describe the concentration field over the hill resulting from plumes released from
upwind stacks and will further quantify the results obtained in Part I.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TER'MS c. COSATI F-'ield/Group
Air pollution
* Wind(meteorology)
* Wind tunnel models
* Hills
* Atmospheric diffusion
* Stratification
* Flow distribution
13B
04B
14B
08F
04A
20D
18. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19 SECURITY CLASS (This Report)
UNCLASSIFIED
21. NO. OF PAGES
96
20 SECURITY CLASS (This page)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (Rev. 4-77)
PREVIOUS EDITION IS OBSOLETE
84
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