Environmental Monitoring Series
A NUMERICAL AND EXPERIMENTAL STUDY OF
STABLY STRATIFIED FLOW AROUND
COMPLEX TERRAIN
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
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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 five series. These five broad
categories were established to facilitate further development and application of
environmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ENVIRONMENTAL MONITORING series.
This series describes research conducted to develop new or improved methods
and instrumentation for the identification and quantification of environmental
pollutants at the lowest conceivably significant concentrations. It also includes
studies to determine the ambient concentrations of pollutants in the environment
and/or the variance of pollutants as a function of time or meteorological factors.
This document is available to the public through the National Technical Informa-
tion Service. Springfield, Virginia 22161.
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EPA-600/4-76-021
May 1976
A NUMERICAL AND EXPERIMENTAL STUDY
OF STABLY STRATIFIED FLOW AROUND
COMPLEX TERRAIN
by
J. J. Riley, H. T. Liu and E. W. Geller
Flow Research, Inc.
Kent, Washington 98031
Contract No. 68-02-1293
Project Officer
William H. Snyder
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, N.C. 27711
U.S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AMD DEVELOPMENT
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
RESEARCH TRIANGLE PARK, N.C. 27711
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DISCLAIMER
This report has been reviewed by the Environmental Sciences Research Laboratory,
U. S. Environmental Protection Agency, and approved for publication. Approval does
not signify that the contents necessarily reflect the views and policies of the U. S.
Environmental Protection Agency, nor does mention of trade names of commercial pro-
ducts constitute endorsement or recommendation for use.
11
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PREFACE
Assessment of the environmental impact of releasing pollutants into
the atmosphere involves understanding, modeling, and predicting the pollutant
diffusion patterns. Prediction is particularly difficult when the pollutants
are released into regions of complex terrain features, especially when the
ambient atmosphere is stably stratified. Such conditions are also most
conducive to severe pollution episodes.
In the past Flow Research has been under contract with the Environmental
Protection Agency to conduct laboratory investigations of plume dispersion'
in stably stratified flows over complex terrain. Under the present contract
(No. 68-02-1293) further laboratory studies have been conducted. The results
are presented in a final report entitled, "Plume Dispersion in Stably
Stratified Flows Over Complex Terrain: Phase 2." Also under this contract, we
have developed a computer model to predict plume dispersion around complex
terrain for the case of strong stratification and have performed experiments
to make preliminary validation of this model. The results of the latter work
are presented in this report.
iii
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ABSTRACT
A computer program has been developed, based on an expansion suggested
by Drazin (1961) and Lilly (1973), to compute three-dimensional stratified
flow around complex terrain for the case of very strong stratification
(small internal Froude number). Also, laboratory experiments were performed
for strongly stratified flow past three different terrain models. Preliminary
comparisons of the results of the computer program and the laboratory modeling
indicate that the computed results are in fair agreement with the experiments.
Discrepancies are probably attributable mainly to the separated wake in the
lee of the models. Other possible sources of error are discussed in some
detail.
IV
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CONTENTS
Preface ill
Abstract iv
List of Figures vi
List of Tables vii
List of Symbols viii
I. Introduction 1
II. Summary, Conclusions and Recommendations 3
III. Brief Description of the Theory 5
IV. Description of the Experiment 11
V. Comparisons of Experimental and Numerical Results . . 17
References 29
Appendix 30
v
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LIST OF FIGURES
Number Page
1 Problem Geometry .................. 6
2 The Flow Patterns Traced by Neutrally Buoyant Dye
in the Vicinity of a Three-Dimensional Gaussian-
Shaped Model 12
3 The Crossing of Streamlines in the Vertical .... 16
4 Sketch of the Instantaneous Streak-Line Pattern . . 18
5 Comparison of Experimental and Numerically Computed
Streak Lines in the Horizontal Plane Defined by
x^ = 14 cm for Case Vc (Gaussian Model, Fh = .97). . 20
6a Apparent Body Shape ................ 21
6b Model for the Separated Wake ............ 21
7 Comparison of Experimental and Numerically Computed
Streak Lines in the Horizontal Plane Defined by
X3 = 14 cm for Case IVa (Conical Model, Fh = .37). . 23
8 Experimental Results for Vertical Displacement for
Case IVa (Conical Model, Fh = .37) . 24
9 Comparison of Experimental and Numerically Computed
Results for Vertical Displacement for Case IVb
(Conical Model, Fh = . 74) 25
10 Comparison of Experimental and Numerically Computed
Results for Vertical Displacement for Case Vc
(Gaussion Model, Fh = -97) 26
11 Comparison of Experimental and Numerically Computed
Streak Lines in the Horizontal Plane Defined by
x3 = 11.5 cm (Idealized Terrain Model, ?h = .137). . 28
vi
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LIST OF TABLES
Number Page
1 Experimental Parameters ... 14
2 Upstream Positions of Dye Release 15
vii
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LIST OF SYMBOLS
Symbol Description
D diameter of circular cylinder
dPc characteristic scale of the ambient density gradient
dx2
F = , internal Froude number
Nn
F, = , internal Froude number based on the Brunt-Vaisala frequency
in cycles/sec.
g acceleration due to gravity
H subscript denoting the horizontal component of a vector
h characteristic vertical scale of the model
L characteristic horizontal scale of the model
/e Ufjc
- -a- - , characteristic Brunt-Vaisala frequency, rad./sec.
P0 dx3
n vortex shedding frequency, rad./sec.
p nondimensional pressure perturbation from the ambient
/\
p dimensional pressure perturbation from the ambient
p(n) n^h order expansion variable for p
R = , Reynolds number based on the body diameter
S = , Strouhal number
U free-stream horizontal wind speed
u = (u,, u~, u,), nondimensional velocity vector
u = (u, , u~, u,), dimensional velocity vector
/\ s \ / \ +-T-,
u^n) = (u^^ , u^ , u 3 ), n order expansion variable for u
VI11
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LIST OF SYMBOLS (CONT'D)
x = (x-p X2> XT) , nondimensional coordinate vector
>N /S *>. /-v
x = (x,, x9, xO , dimensional coordinate vector
"* -L Z. _j
V nondimensional gradient operator
/N
V dimensional gradient operator
9
£ = F » expansion parameter
V kinematic viscosity
p nondimensional density perturbation from the ambient
p nondimensional ambient density
/^
p dimensional density perturbation from the ambient
/v
p dimensional ambient density
p^n^ n order expansion variable for p
p constant characteristic density scale
ty nondimensional vertical displacement
/N,
ip dimensional vertical displacement
j n order expansion variable for
IX
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SECTION I
INTRODUCTION
Assessment of the environmental impact of the release of pollutants into
the atmosphere involves the estimation of diffusion patterns under atmospheric
conditions ranging from average to extreme. A detailed knowledge of the wind
field is important in the estimation of diffusion patterns, especially if the
region of release is characterized by complex terrain. Thus, in the assessment
of pollution effects, the understanding and prediction of local wind fields is
often very important.
One approach to understanding and predicting local wind fields is numeri-
cal simulation. However, the numerical simulation of three-dimensional
stratified flow over complex terrain is a very difficult task. This difficulty
is a result of numerical complications associated with stratification effects
and complex boundaries, and also of the limitations imposed by the core size
and cycle time of present day computers. Thus, exploration of certain limit-
ing conditions under which the physical, mathematical and numerical problems
can be simplified is useful. One such limit is that of very large internal
Froude number, i.e., weak stratification, where the tools of three-dimensional
potential-flow theory are often available. Another limit is very small
internal Froude number, or strong stratification.
When fluid is strongly stably stratified, vertical motions are heavily
constrained and fluid elements tend to remain in their horizontal planes.
The degree to which they do remain is measured by the ratio of their initial
1 ?
kinetic energy, ~JPOU > to the P°tem
element over or around the obstacle,
1 ?
kinetic energy, ~JPOU , to the potential energy required to lift the fluid
dx2
^h2
Here, h is the characteristic vertical scale of the obstacle, g is the
acceleration due to gravity, and ^ is a characteristic ambient stratification.
The ratio is
dp \NhI
-------
the square of the internal Froude number, where
is a characteristic Brunt-Vaisala frequency of the ambient fluid. For
strongly stratified flows (F -> 0), Drazin (1961) and, later, Lilly (1973)
o
have proposed a formal expansion in F , which predicts that, to the lowest
order, the flow resembles two-dimensional (horizontal) flow around contours
of terrain at a given level. The deviations from this two-dimensional flow
can be determined from the higher order terms in a power series.
In the work discussed in this report, we have developed computer
programs to solve the equations resulting from the expansion suggested by
Drazin and Lilly. We have also performed laboratory studies of stratified
flows past simple terrain configurations to validate the numerical programs.
Finally, we have made preliminary comparisons of theory and experiment.
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SECTION II
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
We have developed a computer program to predict three-dimensional stably
stratified flow around complex terrain. The method is based on scaling
analysis suggested by Drazin (1961) and Lilly (1973), and is valid for the
case of very strong stratification (small internal Froude number). In
order to validate the computer model, we have also conducted laboratory
studies of flow past isolated terrain features. Three different terrain
models were used, and simulations were performed at several Froude numbers
for each model. Then those streamline locations measured in the experiments
were compared to those predicted by the computer model.
This study was intended to be a very preliminary examination of the use
of the scaling suggested by Drazin and Lilly for the case of very low-Froude-
number, three-dimensional flow over complex terrain. Preliminary results
indicate the following:
(i) For the Froude number regime studied, the basic scaling suggested
was appropriate, and the flow did resemble two-dimensional flow
around contours of the terrain model at the appropriate level.
(ii) To improve the numerical model, the inclusion of at least two
effects is of primary importance. They are: (a) the displacement
effect of the boundary layer, and, more importantly; (b) the dis-
placement effect of the separated wake.
(iii) The prediction of vertical displacement was roughly valid for the
case computed, considering that the effects discussed in (ii) were
not modeled.
(iv) The accuracy of the lowest order solution depends strongly on the
type of terrain feature considered, as well as the vertical level,
the lateral distance from the terrain feature, and, of course, the
Froude number. For a given Froude number, the agreement between
the numerical model and experiment was much better for the idealized
complex peak than for the Gaussian and conical models.
(v) The computed vertical displacement can be used to estimate regions
of applicability of the lowest order solution.
(vi) Slight unsteadiness in the oncoming flow may be a result of turbu-
lent vortex shedding in the lee of the models.
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One obvious improvement can be made in the numerical model. The separated
wake can be crudely modeled by standard techniques used in aerodynamics to com-
pute flow past two-dimensional bodies. (Note, however, that one may have to
take into account the three-dimensional nature of the boundary layer.)
Several other improvements are also possible. First, one could include
the vertical and horizontal shearing of the free-stream flow. Second, the
atmospheric boundary layer could be modeled. This modeling can be accomplished
most simply by using the computed inviscid flow to drive a turbulent boundary
layer model. A more sophisticated approach would allow the computed boundary
layer to react back on the inviscid flow. Third, the scaling analysis could
include the effect of atmospheric compressibility, although this effect
shouldn't be too important because the vertical motions are weak. Fourth,
Coriolis forces could be included. Fifth, as discussed by Drazin and
Lilly, the scaling breaks down near the model peaks (because the local
scale height is very small, and, therefore, the local Froude number is
very large) . An investigation of the coupling of the present numerical
model with some other model near the mountain peaks could be performed.
Sixth, turbulent diffusion could be modeled in the plume dispersion process
with the turbulent diffusivity related to the local Richardson number,
Finally, it is implicit in the scaling analysis that ,Pc characterize the
_
complete density profile. For example, in regions where ^ is very small
d"pr d^-3
compared to -5°-, the expansion will probably break down. So the case of a
QX o
two-layer fluid (each layer having a different constant density) cannot be
treated with the present scaling. Thus, rescaling the equations to include
these more general cases would be useful.
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SECTION III
BRIEF DESCRIPTION OF THEORY
Consider the steady-state flow past a three-dimensional terrain feature
with a typical vertical scale, h, and horizontal scale, L, (see figure 1,
where the coordinate system is also defined) . We assume the oncoming (free-
stream) flow has characteristic velocity, U, and characteristic stratification,
dp /dx_, which is a constant. We will also make the Boussinesq approximation
and will neglect viscous (turbulent) effects. The equations of motion are (see
Phillips, 1966):
/-s
u . ^ = - i- Vp +-f-g g = (0,0, -g) (3-D
Mo Mo ~
V G = 0 (3.2)
and
s\
/\ ^ s\ AO
u Vp + u, 7^- = 0 . (3.3)
j uX«
Here, u is the velocity vector, p is the density fluctuation about the ambient
/\ "^
p, and p is the pressure perturbation about the ambient. The vertical displace-
ment of a fluid element, $, is given by
G V$ = G3 . (3.4)
Following Lilly (1973), we scale the variables as follows:
(3.5a)
, (3.5b)
L '
X3
- , (3.5c)
and P = ^T ' (3.5d)
-------
u
L
Figure 1. Problem Geometry.
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where the subscript, H, denotes the horizontal component of a vector. To
-, r\ /\ /N
scale p, we assume that -r? and -f- g are in approximate (hydrostatic) bal-
o "^3 PO
ance in the vertical momentum equation. Then,
2\ / / \ (3'5e)
p U w ' ' x
o
J~" r\ ^\
To scale u,,, we assume that Go ~nr- and either u, -?* or u0 -^ (or both) are
J j dx3 1 dxj_ 2 3x2
in approximate balance in equation (3.3), which results in
' (3'5f)
U
where
F = -jr (3.6a)
Nh
is the Froude number,
dp.
is the characteristic Brunt-Vaisala frequency, and where we scale the ambient
stratification (assumed horizontally uniform) as
- - / dp
(3.6c)
dx_ dx., / dx-
From (3.4), the scaling for ty is
hF
-------
Substituting (3.5) into the horizontal component of (3.1), we find
The vertical component of (3.1) becomes
", a^ '-£-- P <3°8)
3
Continuity is now expressed as
and the incompressibility condition is
Vp + F2 u. |£- - uq ^_ = 0 . (3.10)
3 dx_ 3 dx
Finally, the equation for ip5 the displacement, becomes
u Vijj + F u T^ = u . (3.11)
~H 3 3x3 3
Next, we expand the independent variables in the powers of the small
2
parameter, e = F , i.e.,
, uf >
n=0
The resulting equations to the lowest order are
(o)
" ' V ' (3.13)
-------
v . *» .
^ - °'
(3.14)
= - P
(o)
(3.15)
and
(3.16)
(3.17)
Combining (3.16) and (3.17) gives
(0)
dx.
Assuming p and ty are zero in the free stream, then, since
dent of x , one obtains
dx.
is indepen-
dp_
or with (3.15)
(3.18)
(o)
(o)
Note that, from equations (3.13) and (3.14), the equations for LL: ' and
are those for an inviscid, two-dimensional flow in a horizontal plane.
In particular, if the incoming (horizontal) flow is irrotational, then the
entire horizontal flow field is irrotational. Thus, in this case the tools
of the potential-flow theory can be employed to compute the flow. In a given
horizontal plane, the resulting solution would be that of a two-dimensional
flow about an obstacle defined by the contour of the terrain at the vertical
level of that terrain. The vertical displacement can be computed from (3.18),
where it is a result of the pressure difference in the flow between two
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adjacent horizontal layers. A calculation of the vertical displacement can
also be used to estimate the region of validity of the results.
Equations (3-13), (3.14) and (3.18) were programmed on the computer for
flow past somewhat arbitrarily shaped terrain features. The free-stream flow
was assumed irrotational, and standard numerical procedures for computing the
two-dimensional potential flow past arbitrarily shaped bodies were used.
Details of the numerics are presented in the appendix.
10
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SECTION IV
DESCRIPTION OF THE EXPERIMENT
The experimental setup was basically the same as that discussed in Flow
Research Report No. 57 (Liu and Lin, 1975), which describes Task I of this
study. In addition to the idealized terrain model, which has been used for
detailed studies in the past (see Flow Research Report No. 29) and is defined
by
17.8
exp -
.0008513(Xl - 61)2 - ,01197(x - 18.82)2
+ exp -.0008513(xx - 61)2 - .01197 (x2 + 18.82)2
+ 16
exp
[-.01171(xx - 61)2 - .002314 x2
we used a conically shaped model,
MI-^) r<
/ ^ '
15
r > 15 ,
(4.1)
(4.2)
and a Gaussian-shaped model,
hG(x15x2) = 30 exp
1/2
(4.3)
Here, r = (x + x ) and all distances are measured in centimeters.
The two latter (new) models were designed to be interchangeable with the
idealized model. Neutrally buoyant dyes, each of a different color, were
released through small stainless-steel tubes (.3 mm I.D.) at three levels up-
stream of the model. Three plumes spaced in the horizontal were released in
each level. The plume trajectories were photographed, and then analyzed for
later comparisons with analytical results. Figure 2 shows a typical side view
of the streak-line patterns for flow past the Gaussian peak.
11
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a. U = 1.5 cm/s, F = .37
b.
U = 4 cm/s,
= .97
Fig. 2 The Flow Patterns Traced by Neutrally Buoyant Dye
in the Vicinity of a Three-Dimensional Gaussian-
Shaped Model. N = .135 Hz.
12
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Note that in each horizontal plane, three streak lines were released.
However, in the side view, the three are difficult to distinguish, especially
in the lowest Froude number cases. This difficulty is somewhat compounded
by the slight vertical spread of each streak line. However, in a given hori-
zontal plane, the innermost streak line is displaced more than the others, so
that its displacement is easily detectable in the photographs. Thus, when
comparisons were made, we used the innermost streak line.
Other cases are presented in a 16 mm movie, which is a part of the Task
I report. Table 1 presents the conditions for the runs, and table 2 shows
appropriate upstream positioning of the dye release. In these tables,
F, = ~^r~ = 27TF is the defining Froude number.
h Nh
The choice of the model conditions was based on the following criterion.
The expansion can only remain valid as long as streamlines do not cross in
/s /\
the vertical. This crossing would occur, for example, if i^-, > ^7 + Ax
/*\ x\
(figure 3). Thus, -Ax_ > ^ - ij; , or in the limit, as Ax -> 0,
When one considers both upward and downward displacements, this condition
generalizes to
84;
> 1 .
Thus, in nondimensional terms, a necessary condition for the validity of
the expansion is
F2 1^- < 1 (4.4)
We selected the various parameters in the experiments so that (4.4) was
satisfied over a large portion of the vertical region of interest.
13
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TABLE 1. EXPERIMENTAL PARAMETERS
Run Number
Terrain Model
IVa Conical Model
(eq. (4.2))
IVb Conical Model
IVc Conical Model
Va Gaussian Model
(eq. (4.3))
Vb Gaussian Model
Vc Gaussian Model
Via Idealized Terrain Model
(eq. (4.1))
VIb Idealized Terrain Model
Stratification Profile
1-Layer,
Constant N
1-Layer,
Constant N
1-Layer,
Constant N
1-Layer,
Constant N
1-Layer,
Constant N
1-Layer,
Constant N
1-Layer,
Constant N
1-Layer,
Constant N
N (cycle/sec) U (cm/sec) F
.135 1.5 .37
135
.135
.138
,138
.138
,135
.135
3.0
6.0
1.5
3.0
4.0
3.0
6.0
,74
1.48
,36
,72
.97
1.37*
2.72*
Froude No. Based on the Interior Ridge Height
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TABLE 2. UPSTREAM POSITIONS OF DYE RELEASE
Terrain Model
Vertical Level (cm)
(Measured From Terrain
Model Base)
Horizontal Position (cm)
(Measured From Terrain
Model Centerline, x = 0)
Conical Model
(eq. 4.2))
Gaussian Model
(eq. (4.3))
x.
Idealized Terrain Model
(eq. (4.1))
14.0
14.0
14.0
7.6
7.6
7.6
13.2
13.2
13.2
20.3
20.3
20.3
11.5
11.5
11.5
1.0
5.0
13.0
3.3
8.4
19.0
2.1
6.8
17.1
0.6
7.0
17.5
1.0
5.0
12.0
15
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. ^
~ + Ax
X,
Figure 3. The Crossing of Streamlines in the Vertical,
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SECTION V
COMPARISONS OF EXPERIMENTAL AND NUMERICAL RESULTS
As mentioned in the previous section, tables 1 and 2 give the experimental
conditions. The general behavior of the streak lines can be seen by examina-
tion of figure 2b. In the middle horizontal plane, consider the inner streak
line, which exhibits the largest vertical displacement. As a fluid element
approaches the model, it slows down in a manner similar to an element in a two-
dimensional flow about a cylinder. Simultaneously, the element experiences an
upward pressure force, causing it to rise upward (see equation 3.18). As the
element starts around the mountain, it accelerates, the vertical pressure force
changes direction, and the element is displaced downward. The flow separates
near the point of maximum lateral extension of the model. Past the midpoint
of the mountain, the elements return to their equilibrium levels, and are
entrained into the wake of the model. For the Froude number range in the
experiments, the wake flow in the lee of the model appeared to consist of tur-
bulent, quasi-horizontal eddies, whose vertical velocity fluctuations were
rapidly decaying with downstream distance.
The incoming flow was slightly unsteady. We observed that the inner
streak lines slowly oscillated from one side of the mountain to the other.
This oscillation often produced an inner streak-line pattern, as sketched in
figure 4. One possible explanation of this phenomenon is the following. For
two-dimensional flow past a cylinder, turbulent vortex streets are observed in
the Reynolds number range of about 60 < R < 5000, where the Reynolds number R is
, D is the diameter of the cylinder, and V is the kinematic viscosity. For
our case, R is typically
R a 4 cm/sec I 10 cm « 4000 ,
.01 cm /sec
which is in this range. The vortex motion is accompanied by movement of the
stagnation points, which in turn causes the incoming flow to oscillate slightly.
For R in this range, the Strouhal number, defined by S = , where n is the
vortex shedding frequency in radians/sec, is approximately .21. Thus,
_n_g .21 x 4 cm/sec g >Q14 les/sec .
2ir 2ir x 10 cm
17
-------
K-WA4W'*"
'i^M±
-------
This value corresponds roughly to the frequency of the oscillations noted in
the experiments.
Figure 5 shows streak lines in the horizontal plane, x = 13.2 cm (the
middle plane) for the Gaussian model with F = .152 (or F, = .97, run number
h
Vc). Also shown are the numerical predictions. In addition, the contour of
the model at the free-stream level of the plumes is displayed. Note that the
numerical calculation tends to underpredict streak-line displacement. This
discrepancy is probably the result of mainly two effects. The first is the
displacement effect of the boundary layer, which is not taken into account in
the inviscid numerical model. The second and more important effect is the dis-
placement effect of the separated wake. These two effects together produce an
"apparent" body, as sketched in figure 6a.
Note that for a two-dimensional flow past a circular cylinder, if the
Reynolds number is subcritical (i.e., below approximately 3 x 10 ), the bound-
ary layer is laminar, and it separates at about 80° from the front stagnation
point (Schlichtling, 1960). Since the Reynolds numbers for the experiments
were an order of magnitude less than the critical value, it is reasonable to
assume that the boundary layer was laminar and separated before the point of
maximum lateral extension of the body. For two-dimensional flow past a circu-
lar cylinder, if the Reynolds number is supercritical, the boundary layer is
turbulent, and separation probably occurs just past the point where the cross
section starts to converge. Thus, in the full-scale case, where Reynolds num-
bers will usually be several orders of magnitude larger than the critical value,
separation is likely to occur just past the point of maximum lateral extent.
When viscous terms are added to the scaling arguments presented in section
3 , one finds that the lowest order solution is no longer two-dimensional. Thus,
the conclusions drawn above could be modified somewhat because of the three-
dimensional nature of the boundary layer.
The displacement effect of the separated wake was crudely modeled by
extending the body, as shown in figure 6b. Figure 5 also shows the results of
a calculation using this body shape instead of the circular shape. The modifi-
cation of the streak lines, especially the outermost ones, is noticeable.
However, from the discussion above, the model suggested in figure 6b is prob-
ably more adequate for the full-scale case than the laboratory case. Also,
the effect of the boundary layer may have to be taken into account to obtain
close agreement between the numerical and experimental results.
19
-------
Numerical Results
Numerical (With Simplified Wake Model)
Experimental Results
U
(CM)
Figure 5. Comparison of Experimental & Numerically Computed Streak
Lines in the Horizontal Plane Defined by x,. = 14 cm for
Case Vc (Gaussian Model F, = .97),
-------
u
"APPARENT" BODY
CONTOUR
Figure 6a. Apparent Body Shape,
U
/ /
X
\
\
/ /
Figure 6b. Model for the Separated Wake,
21
-------
The width of the tank is approximately 120 cm, so it is-possible that the
sidewalls influence the flow field near the models. Sidewalls were not
included in the numerical models. However, the horizontal displacement of the
streamline whose free-stream position is at the sidewalls was computed. For
the Gaussian and conical models, the displacement of this streamline was neg-
ligible for the cases computed. Thus, for these cases, we can assume that the
effect of sidewalls is unimportant.
Figure 7 shows a similar comparison for the conical model, in the middle
horizontal plane (x = 14 cm), for F = .059 (or F = .37, run number IVa).
The results and interpretation are similar to that for the previous case. Note
the multiple lines which describe the plume trajectories. For a particular
dye release, the different lines come from photographs taken of the streak line
at different times. The discrepancies in the experimental results are probably
a result of the unsteadiness caused by the vortex shedding discussed above.
Figure 8 shows the experimentally determined vertical displacement for the
conical model, run number IVa. Note for this case that the streak-line dis-
placement was very slightless than approximately .5 cm. The maximum displace-
ment computed numerically was about .5 cm, in approximate agreement with experi-
ment. The very small vertical displacement of the streak lines supports the
original scaling assumptions for this case.
Figure 9 shows similar experimental plots for the conical peak at F = .109
(or F = .74, run number IVb). Vertical displacements are much more noticeable
in this case. This increase in displacement is expected, since it has been
established (equation (3.5d)) that vertical displacement scales approximately
as the square of the Froude number. Also shown in this plot is the numerical
prediction for the innermost plume in the middle level. The numerical calcula-
tion predicts a very slight rise in the plume as it approaches the mountain,
which is also discernible in the experiments. The distance that the plume
drops is also predicted fairly well. However, the asymmetry of the experimental
results is missed entirely. This discrepancy is again probably attributable to
the neglect of the boundary layer and wake effects.
Figure 10 shows similar experimental plots for the Gaussian model at F =
.152 (or F^ = .97, run number Vc). Also shown are the numerical predictions
for the innermost plume for each of the three levels. The agreement and explan-
ation of the results are very similar to the previous case.
22
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Numerical Results
Experimental Results
U
-30
X,(CM)
Figure 7. Comparison of Experimental & Numerically Computed Streak Lines on the Horizontal
Plane Defined by x = 14 cm for Case IVa (Conical Model, F = ,37)
-------
fsj
-P-
10 cm
Figure 8. Experimental Results For Vertical Displacement For Case IVa (Conical Model F = .37)
-------
So-lid Lines = Experimental Results
Dotted Line = Numerical Results
U
ro
Ln
10 cm
Figure 9. Comparison of Experimental & Numerically Computed Results
For Vertical Displacement For Case IVb (Conical Model F - .74)
-------
Shaded Areas = Experimental Results
. = Numerical Results
U
10 cm
Figure 10.
Comparison of Experimental and Numerically Computed
Results for Vertical Displacement for Case Vc
(Gaussian Model, F, = .97).
n
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Finally, figure 11 shows comparisons of experimentally observed and
numerically computed streak lines for the idealized terrain case, with F, = .137
h
/*»
The dye was released at approximately x = 11.5 cm. Note that the calculation
included separation in the wake, in the manner discussed above. Comparisons
for this case are much better than for the previous cases because the crude
wake modeling for the potential flow calculation closely matched the real
flow. In the calculations for the previous cases, the assumption that the
separation stream line is straight and parallel to the upstream flow direction
was not supported by the experimental results which indicate a diverging wake
region.
An examination of the side view shows sizeable vertical displacement as
the stream lines traverse the ridge. This is accompanied by some motion towards
the ridge, resulting in the inner plume appearing to cut through the terrain in
the top view. As the fluid comes over the ridge, it tends to fall below its
ambient level, thus forcing the fluid laterally away from the ridge, and
producing the slight bulge seen in the figure above the ridge. This bulge may
also be attributable to a boundary layer separation bubble. According to the
potential flow calculation, the boundary layer is subjected to an adverse
pressure gradient near the most upstream point on the model. Therefore,
separation of the laminar boundary layer is likely.
Finally, calculations of the streamline at the location of the tank
sidewall show that its lateral displacement is significant. Thus, sidewall
effects, which were neglected in the calculation, could be of some importance
in this case.
27
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r 40
u
Numerical Results
Experimental Results
x (cm)
Figure 11. Comparison of Experimental & Numerically Computed Streak Lines
in the Horizontal Plane Defined by x = 11.5 cm (Idealized
Terrain Model, F = 1.37) .
-------
REFERENCES
Drazin, P. G. (1961), "On the Steady Flow of a Fluid of Variable Density Past
an Obstacle," Tellus 13, 239-251.
Lilly, D. K. (1973), "Calculation of Stably Stratified Flow Around Complex
Terrain," Flow Research Note No. 40.
Lin, J. T., Liu, H. T. and Pao, Y. H. (1974), "Laboratory Simulation of Plume
Dispersion in Stably Stratified Flows Over Complex Terrain," Flow Research
Report No. 29.
Liu, H. T. and Lin, J. T. (1975), "Plume Dispersion in Stably Stratified
Flows Over Complex Terrain: Phase II," Flow Research Report No. 57.
Phillips, 0. M. (1966), The Dynamics of the Upper Ocean, Cambridge University
Press.
Schlichting, H. (1960), Boundary Layer Theory, McGraw Hill.
29
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APPENDIX: NUMERICAL METHODS
The basic numerical problem involves computing the two-dimensional poten-
tial flow about an arbitrarily shaped body. The streamlines are found using
the stream function, and the vertical displacement is determined by computing
the pressure field at two adjacent levels, and then using a simple differenc-
ing scheme (see equation (3.18)).
The two-dimensional potential flow about a closed contour is constructed
by combining a uniform flow with the velocity field of a vorticity sheet
located on the contour. The uniform flow is chosen to give the desired flow
at infinity, and the distribution of the vorticity strength is found such that
the prescribed boundary condition on the contour is satisfied. For this work
the boundary condition is that the flow be tangent to the contour (i.e., zero
velocity component normal to the contour).
The governing integral equation expressing the boundary condition is
solved numerically for the vorticity distribution by approximating the contour
with an inscribed polygon and by assuming that the vorticity varies linearly
with the arc length along each segment of the polygon. With no discontinuities
in strength at the polygon corner points, this approximating vorticity distri-
bution is defined by the corner-point values. Enforcing the boundary condition
at the midpoints of the segments gives a set of algebraic equations in terms of
the unknown vorticity corner-point values.
An additional relation is needed to obtain a unique solution to these
equations. For this work the circulation about the contour was required to
vanish. This requirement provides an additional algebraic equation in the
unknown vorcicity corner-point values.
After obtaining the approximating vorticity sheet from the numerical solu-
tion of the governing integral equation, we construct the velocity field and
the stream function field associated with the vorticity sheet and add them to
the free-stream contributions to obtain the desired results.
The pressure field is obtained from the velocity field using the incom-
pressible Bernoulli relation.
30
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TECHNICAL REPORT DATA
(Please read laamcrions on the reverse before completing)
NO.
£PA-6QO/4-76-Q21
3. RECIPIENT'S ACCESSIOI^NO.
' TITLE AND SUBTITLE
A NUMERICAL AND EXPERIMENTAL STUDY OF STABLY
STRATIFIED FLOW AROUND COMPLEX TERRAIN
5. REPORT DATE
May 1976
6. PERFORMING ORGANIZATION CODE
. AUTHOR(S)
J. J. Riley, H. T. Liu and E. W. Geller
8. PERFORMING ORGANIZATION REPORT NO.
Flow Research Report No. 58
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Flow Research, Inc.
1819 South Central Avenue
Kent, Washington 98031
10. PROGRAM ELEMENT NO.
1AA009
11. CONTRACT/GRANT NO.
68-02-1293
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory
Office of Research and Development
U. S. Environmental Protection Agency
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final Report, 5/74-3/75
14. SPONSORING AGENCY CODE
EPA - ORD
15. SUPPLEMENTARY NOTES
16. ABSTRACT
A computer program was developed to compute three-dimensional stratified
flow around complex terrain for the case of very strong stratification (small
internal Froude number). Laboratory experiments were performed for strongly
stratified flow past three different terrain models for comparison.
results are in fair agreement with the experiments for the cases of
terrain models. The discrepancies are probably attributable to the
wake in the lee of the models. The agreement was not as good for the case of
the more complex terrain. Possible sources of error are di.scussed in some detail
The computed
two simpler
separated
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
c. COSATI Field/Group
*Mathematical models
*Plumes
*Stratification
*Terrain
*Air pollution
12A
21B
14G
08F
138
13. DISTRIBUTION STATEMEN1
RELEASE TO PUBLIC
19. SECURITY CLASS I This Report)
UNCLASSIFIED
21. NO. OF PAGES
41
20. SECURITY CLASS (This page)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
31
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