EPA-45a/4-79-0ia-
DRAFT.
for
Public Comment
GUIDELINE FOR aUID MODELING
OF ATMOSPHERIC DIFFUSION
June 1975
* U.S. ENVIRONMENTAL PROTECTION AGENCY
Office of Air, Noise* and Radiation
! Office of Air Quality Planning and Standards
» Research Triangle Park,. Nortfr Carolina 27711
DRAFT
-------
EPA-450/4-79-016
DRAFT
GUIDELINE FOR FLUID MODELING
OF ATMOSPHERIC DIFFUSION
by
William H. Snyder
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
U.S. ENVIRONMENTAL PROTECTION AGENCY
Office of Air, Noise, and Radiation
Office of Air Quality Planning and Standards
Research Triangle Park, MC 27711
June 1979
DRAF1
-------
DISCLAIMER
Mention of trade names or commercial products does not constitute
endorsement or recommendation for use.
The author, 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 Cceanic and Atmospheric Administration,
U.S. Department of Commerce.
-------
PREFACE
The U.S. Environmental Protection Agency is charged by Congress with
establishing and enforcing air pollution control standards to protect
the public health and welfare. To accomplish its mission, it is essential
to be able to describe and predict the transport and diffusion of pollutants
in the atmosphere. Present mathematical models are not yet adequate for
calculating concentrations of contaminants when the plume is affected
by obstructions such as hills and buildings. Field programs to obtain
adequate data are very expensive and time consuming. Small scale models
immersed in the flow of wind tunnels and water channels, i.e., fluid models,
can frequently be of use in simulating atmospheric transport and diffusion
in a timely and relatively inexpensive manner.
It is the aim of this guideline to point out the capabilities and
limitations of fluid modeling and to recommend standards to be followed
in the conduct of such studies. The guideline is intended to be of use,
both to scientists and engineers involved in operating fluid modeling
facilities and to air pollution control officials in evaluating the quality
and credibility of the reports resulting from such studies.
The fundamental principles of fluid modeling are well-established,
but when decisions must be made concerning a particular model study, the
fundamental principles frequently do not provide specific guidance. There
is a need for basic and systematic modeling studies to provide more specific
guidance. This guideline will be periodically revised as more specific
experience is gained, new techniques are developed, and old ones refined.
-------
TABLE OF CONTENTS
PREFACE 111
LIST OF FIGURES vii
NOMENCLATURE x
ACKNOWLEDGEMENTS xiv
1. INTRODUCTION 1
2. FUNDAMENTAL PRINCIPLES 3
2.1 The Equations of Motion 3
2.2 The Dimension]ess Parameters 7
2.2.1 The Rossby Number 8
2.2.2 The Reynolds Number v 14
2.2.2.1 The Laminar Flow Analogy 14
2.2.2.2 Reynolds Number Independence 17
2.2.2.3 Dissipation Scaling 27
2.2.3 The Peclet Number and the Reynolds-Schmidt Product 29
2.2.4 The Froude Number 31
2.3 Boundary Conditions 34
2.3.1 General 34
2.3.2 Jensen's Criterion and Fully Rough Flow 36
2.3.3 Other Boundary Conditions 38
2.4 Summary and Recommendations 39
3. PRACTICAL APPLICATIONS 41
3.1 Plume Rise and Diffusion 41
3.1.1 Near-Field Plume Behavior 43
3.1.2 Summary and Recommendations on Modeling Near Field
Plumes 56
3.1.3 Far-Field Plume Behavior 57
3.1.3.1 Ignoring the Minimum Reynolds Number. . . 58
3.1.3.2 Raising the Stack Height 60
3.1.3.3 Distorting the Stack Diameter 60
3.1.4 Summary and Recommendations on Modeling Far-
Field Plumes 63
3.2 The Atmospheric Boundary Layer 65
3.2.1 Characteristics of the Atmospheric Boundary
Layer 67
3.2.1.1 The Adiabatic Boundary Layer 68
3.2.1.2 Summary of the Adiabatic Boundary
Layer Structure 80
3.2.1.3 The Diabatic Boundary Layer 82
3.2.1.4 Summary of the Diabatic Boundary
Layer Structure 105
3.2.2 Simulating the Adiabatic Boundary Layer 107
3.2.3 Simulating the Diabatic Boundary Layer 120
3.2.4 Summary on Simulating the Atmospheric Boundary
Layer 121
3.3 Flow Around Buildings 124
3.3.1 Discussion 124
-------
3.3.2 Recommendations -...-... 130
3.4 Flow Over Hilly Terrain r 133
3.4.1 Neutral Flow 133
3.4.2 Stratified Flow 136
3.4.3 Recommendations 139
3.5 Relating Measurements to the Field 141
3.6 Averaging Times and Sampling Rates in the Laboratory .... 143
4. THE HARDWARE 147
4.1 Visual Observations 147
4.2 Quantitative Measurements 148
4.3 Producing Stratification 149
4.4 Air Versus Water 152
4.5 Summary 154
5. CONCLUDING REMARKS 155
6. REFERENCES 158
VI
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LIST OF FIGURES
NUMBER TITLE PAGE
1 Schematic of diffusion in the Ekman Layer 9
2 Turbulent jets illustrating Reynolds number
i ndependence 18
3 Shadowgraphs of the jets shown in Figure 2 18
4 Filter function in Equation 2.12 20
5 Spectrum of wind speed at 100m 21
6 Form of turbulence spectrum 22
7 Change of spectrum with Reynolds number 23
8 Plume downwash in the wake of a stack 44
9 Variation of plume rise with Keynolds number b5
10 Laminar plume caused by low Reynolds number effluent 59
11 Effects of wind shear on the flow round a building 65
12 The depth of the adiabatic boundary layer according
to the geostrophic drag law compared with other
schemes 70
13 Typical wind profiles over uniform terrain in neutral
flow 75
14 Variation of power law index, turbulence intensity,
and Reynolds stress with roughness length in the
adiabatic boundary layer 75
15 Shear stress distributions measured at various down-
wind positions in a wind tunnel boundary layer 76
16 Variation of longitudinal turbulence intensity with
height under adiabatic conditions 78
17 Variation of integral length scale with height and
roughness 1 ength 79
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NUMBER TITLE PAGE
18 Empirical curves for spectra and cospectrum for
neutral conditions 80
19 Typical nonadiabatic boundary layer depths from
the geostrophic drag relations 89
20 Variation of friction velocity with stability from
the geostrophic drag relations 89
21 Theoretical variation of the power-law exponent as
a function of z and L for z equal to 100m 91
22 Variation of the power-law exponent, averaged over
layer from 10 to 100m, as a function of surface
roughness and Pasquill stability class 92
23 Typical surface layer velocity profiles under
nonadiabatic conditions 95
24 Typical temperature profiles in the surface layer 95
25 The relationship between Ri and z/L 98
26 Variation of and <$a with z/L in the surface layer 98
w 9
27 Variation of $u with z/L in the surface layer 100
28 Variation of $v with z/L 100
29 Universal spectral shape 101
30 Location of spectral peak for u,v,w and e plotted
against z/L 101
31 Upstream vi ew of a 1 ong wi nd tunnel 109
32 Vortex generators and roughness in a short wind tunnel..110
33 Schematic representation of the counter-jet technique...111
34 Development of boundary layer in a long wind tunnel Ill
35 Development of mean velocity profiles along the smooth
f 1 oor of a 1 ong tunnel 114
36 Thickness parameters for boundary layer of Figure 35 114
viii
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NUMBER TITLE PAGE
37 Spectrum of the longitudinal component of "velocity 116
38 _ Velocity profiles above crest of triangular ridge
indicating effect of blockage 131
39 Contour map of three-dimensional hill showing
inappropriate choice of area to be modeled 138
40 Averaging time requirements for wind tunnel
measurements 145
ix
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NOMENCLATURE
A constant or area
B constant
c constant
C constant or concentration
C cospectrum of Reynolds stress
UW
d zero-plane displacement
D stack diameter
E spectrum function
f nondimensional frequency or arbitrary function
f Coriolis parameter
fm nondimensional frequency corresponding to spectral peak
F^ Lagrangian spectrum function
Fr Froude number
g acceleration due to gravity
G geostrophic wind speed
h hill, building or obstacle height
r roughness element height
H stack or building height
I turbulence integral scale
k von Karman constant
K eddy viscosity or diffusivity
£B buoyancy length scale
£ momentum length scale
m
L characteristic length scale or Monin-Obukhov length
-------
Lu integral length scale of longitudinal velocity in « - direction
oc
Lwa integral length scale of vertical velocity in ^-direction
n frequency
p pressure or power-law index
Pe Peclet number
Q pollutant emission flow rate
Re Reynolds number
Ri Richardson number
Rio bulk Richardson number
Ri^ flux Richardson number
Ro Rossby number
S spectrum -function
Sc Schmidt number
t time
T averaging time, time of travel from source, or fluid
temperature
u fluctuating velocity in x-direction (streamwise)
u* friction velocity
U mean wind speed
U. instantaneous flow velocity in i-direction
v fluctuating velocity in y-direction (cross-streamwise)
w fluctuating velocity in z-direction (vertical)
W effluent speed
x Cartesian coordinate (streamwise)
x. coordinate in i-direction
y Cartesian coordinate (cross-streamwise) or particle displacement
xi
-------
z Cartesian coordinate (vertical)
ZQ roughness length
a molecular mass diffusivity
6 blockage ratio (area)
5 boundary layer depth
6.j. Krorecker's delta
<5P deviation of pressure from that in neutral atmosphere
<5T deviation of temperature from that in neutral atmosphere
Ah plume rise
Ap density difference
e dissipation rate of turbulence, roughness element
height, or fractional error
e..,, alternating tensor
n Kolmogoroff microscale
e potential temperature
< thermal diffusivity or wavenumber
\m wavelength corresponding to spectral peak
u stability parameter
v Kinematic viscosity
5 time separation
P fluid density or Lagrangian autocorrelation function
a Standard deviation (x, y, z subscripts refer to puff or plume
widths; u, v, w to velocity flucuations)
T fluctuating temperature (deviation from mean)
ij Kolmogoroff velocity
* nondimensional potential temperature gradient
h
$H nondimensional horizontal (u plus v) turbulence intensity
XI T
1 I
-------
$ nondimensional wind speed gradient
m
$ nondimensional longitudinal turbulence intensity
$ nondimensional lateral turbulence intensity
$ nondimensional vertical turbulence intensity
w
$ nondimensional intensity of temperature fluctuations
0
x nondimensional concentration
co earth's rotation rate
Subscripts and Special Symbols
( ) ambient value
ct
( ) equilibrium value
eq
( ) field value
( ) geostrophic value
( ). Lagrangian value
( ) model value
( ) maximum value
II 1A
( ) value of quantity in neutral atmosphere, except as noted
( ) prototype value
( ) reference quantity
( ) stack value
( ) value of quantity in x-direction
A
( ) value of quantity in y-direction
( ) value of quantity in z-direction
( )(o freestream value
( ) nondimensional quantity
( ) average value
( ) vector quantity
X1 i 1
-------
--] . INTRODUCTION
The present mathematical models of turbulent diffusion in the lower atmos-
pheric layer tend to ignore the fundamental fluid-dynamical processes involved
in the dispersion of materials. This results from the fact that the memory
size of the latest computer is far too small to keep track of the large number
of "eddies" in a turbulent flow. Corrsin (1961a), in speculating on the fu-
ture role of large computing machines in following the consequences of the Na-
vier-Stokes equations under random initial conditions, estimated a required
13
memory size of "10 bits, then asked if "the foregoing estimate, is enough to "
_*
suggest the use of analog instead of digital computation; ifTparticular, how
about an analog consisting of a tank of water?" (emphasis added). In spite of
the tremendous advances in computer memories in the past tw^Kdecades, Corrsin's
remark is still appropriate.
Fluid models of various aspects of atmospheric motion have been described -
in the literature many times. The necessity of studying the dispersion of at-
mospheric pollutants, especially in urban areas, has further directed thoughts
of meteorologists towards fluid modeling.
Many factors affect the dispersion of pollutants in the atmosphere; ther-
mal effects, the topography, the rotation of the Earth, etc. Fluid modeling
studies are desirable mostly because essential variables can be controlled at
will, and the time and expense are greatly reduced from that required in full-
scale studies. It is not usual, however, for all the factors influencing at-
mospheric dispersion to be included in a model. Normally, the similarity crl-
1
-------
teria are conflicting in some sense; it may be necessary to model one physical
process at the expense of not being able tcr model another.
For correct modeling, certain nondimensional parameters in the prototype
must be duplicated in the model. Almost invariably, duplication of these non-
dimensional parameters is impractical or impossible. Hence, a decision must
be made as to which parameters are dominant. The less important ones must be
ignored. This decision will generally depend upon the scale in which the in-
vestigator is interested. For example, when studying the upper air flow above
a city, the waffle-like topography may be treated as surface roughness. The
heat: island effect may be modeled by using a heated plate. If the city is
large enough, Coriolis forces may be important. If, however, the interest is
in dispersion in the immediate vicinity of buildings, the topography cannot be
treated as surface roughness. The heat-island effect would require a detailed
distribution of heat sources, and Coriolis forces could be ignored because the
aerodynamic effects of the flow around the buildings would dominate.
Chapter 2 reviews the fundamental principles for fluid modeling relevant
to air pollution meteorology and evaluates the usefulness of such models from
both scientific and engineering viewpoints. Because many detailed decisions
must be made during the design and execution of each model study, and because
the fundamental principles frequently do not provide enough guidance, discus-
sions of the details of the most common types of modeling problems are provid-
ed in Chapter 3. Air and water are most commonly used as media for the simu-
lation of atmospheric motions. The potentials of both of these fluids are re-
viewed in Chapter 4.
-------
2. FUNDAMENTAL PRINCIPLES
A discussion of the fundamental principles for fluid modeling of atmosphe-
ric phenomena is presented here. The dynamics of the flow in the fluid
model must accurately simulate those in the field. Similarity criteria
are derived through analysis of the equations of motion. This analysis
shows that various nondimensional parameters must be matched between the
model and field flows. The significance of each of these parameters is
discussed in detail. Additionally, effects in the field upstream of the
modeled area must be accounted for in the fluid model by developing appropriate
boundary conditions. Hence, some discussion of boundary conditions is included
at the end of the chapter.
2.1 THE EQUATIONS OF MOTION
The equations of motion are the starting point for the similarity
analysis. With the Earth as a reference frame rotating at an angular velocity
n, the fluid motion is described by the following equations (Lumley and
Panofsky, 1964):
Conservation of Momentum
dU, U.8U, 1 ddP g vd2U,
1 * J ' i -> jj n i 3 x-rx < ..' / ? 1 ^
at cx} QO oxi T0 ox
Continuity
^ = ° (2.2)
Energy
cST 86T o25T ,
= K ^- (i =1,2.3)
ct dx,
where the x3 axis is taken vertically upward, U. is instantaneous velocity,
<$P and 6T are deviations of pressure and temperature from those of a neutral
atmosphere, pQ and TQ are density and temperature of a neutral atmosphere
(functions of height), v is kinematic viscosity, K is thermal diffusivity,
eijk is the alternating tensor (if any two of the indices i, j, k, are equal,
3
-------
the component is 0; if i, j, and k are all unequal and are in cyclic order,
the component is +1; if not in cyclic order, the component is -1), <5. . is Kro-
neker's delta (<5.. = 1 if the two indices are equal and 0 if unequal), and the
' J
summation convention is used here (whenever a suffix is repeated in a term,
it is to be given all possible values and the terms are to be added for all).
Equation 2.1 shows that the vector sum of the forces per unit mass acting
on a parcel of fluid must balance the acceleration of that parcel. The first
term represents the unsteady acceleration of the fluid element. The second
represents the advective acceleration. The remainder are, respectively, the
Coriolis force, the pressure gradient force, the buoyancy force, and the fric-
tional force per unit mass.
Equation 2.2 is, of course, the continuity equation, which expresses the
conservation of mass in an incompressible fluid. Equation 2.3 expresses the
conservation of thermal energy; the time rate of change of thermal energy
(first term) equals the convection (or advection) of energy by the flow (sec-
ond term) plus the conduction of energy (third term).
The assumptions made in deriving the above equations are:
(1) The atmosphere is composed of a perfect gas of constant
composition,
(2) the deviations of pressure, temperature, and density are
small compared with the neutral (adiabatic) values,
(3) the density is independent of the fluctuating pressure
(small Mach number),
(4) variations of v and < are negligible,
(5) the generation of heat through viscous stresses is
negligible, and
(6) there are no sources of any kind.
The second step in the similarity analysis is to nondimensionalize the
4
-------
eauaticns cf rcticn through the use cf appropriate reference Quantities.
Reference ruantities assumed to be supplied through the boundary corditiers
are: L, length; U0, velocity; pR, density; cTR, temperature deviatior; and
:.,., angular velocity. The dimersicnless variables are
oP 6T
c>P' = - -, 6T' = -
ir = ^ .
Using these definitions to nondimensicna1ize Eqs. 2.1 to 2.3 yields
(?L'' : ,-n' ' C'6P> ^ ' sr-s
, . . -f KnklkQj =- , . ; 4- dTcK, -r ,, ,
ex. Ro <> i-X. Fi- Re ex .ex.
\r /
(2.5)
and
~6T' C6T'
where Ro=UD/l.Q is the Rossby number,
R
1 /?
aLdTp/T ^ ' is the densimetric Froude number,
- '
rnp
R' v- R' o
Re=L'DL/^ is the Reynolds rurrber,
K
and Pe=URL/K is the Feclet number.
Ccrcerninq the philosophy of rcdeling, Eqs. 2.4 to 2.6 with appropriate
boundary conditions completely determine the flow. The cuesticr cf uniqueness
-------
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-------
increased by 1C implies that e"TR must be increased by a factor of 1COC, which
is, cf course, highly impractical. Ir general, a length scale reduction much
greater than 10 is desired.
A factor c^ 15 ir the Reynolds number nay be gained by modeling with
water as the fluid medium, but then the Peclet number and Deynolds rumber
criteria cannot be satisfied simultaneously. The Peclet number can be writ-
ten as the product of the Reynolds number and the Frandtl number. Even if
the Reynolds rumber can be matched, the Prandtl number cannot, because it is
a fluid property and differs by a factor cf 10 between air and water. The
Prandtl number is net, however, a critical parameter (see later discussion).
I^'any examples of this type can be shown. All modelers recognize that
rigorous modeling with significant reduction in scale is impossible. Under
certain circumstances, however, some cf the criteria may be relaxed. In the
first example, if the atmospheric flow \\ere cf neutral stability, the Frcude
number would be infinite. This is easily accomplished by making the model
flew isothermal. (The vertical dimension cf a typical wind tunnel is small
enough that the temperature differences between isothermal and strictly neu-
tral conditions is extremely small. Hence, l:neutral'' ard "isothermal" are
used interchangeably when referring to wind tunnel flows.) Hence, both Rey-
nolds number and Froude number criteria may be satisfied simultaneously.
It is instructive now to examine the nonairensional parameters in de-
tail.
2.2.1 The P.ossby Number, Up/LcR
The Rossby number represents the ratio of advective cr local accelera-
2
tiers (Up/L) to CcricMs accelerations (proportional to l:r,r.p). Local accel-
-------
erations may result from unsteadiness or non-uniformities in the velocity
field. Coriolis accelerations, of course, result from the fact that the earth
rotates. The importance of the Rossby number criterion for modeling of atmos-
pheric diffusion is described as follows.
In the planetary boundary layer, or "Ekman" layer, which extends from
the Earth's surface to a height of one to two kilometers, the combined effects
of the Coriolis acceleration, the pressure gradient, and surface friction cause
the wind vector to change direction or spiral with increasing height from the
surface. The geostrophic wind is parallel to the isobars, whereas the surface
wind blows to the left across the isobars, typically at an angle of 20° to 40°.
The maximum rate of change of wind direction with height occurs at the surface.
Imagine a cloud of material released at ground level in an Ekman layer.
Its transport and dispersion are illustrated in Figure 1. The surface wind is
directly into the paper. The crosswind velocity profile is as shown. The ini-
CROSSWIND VELOCITY
(SURFACE WIND INTO PAGE)\-
I. INITIAL CLOUD
-2. INITIAL DIFFUSION
\\ '3 CROSSWIND TILTING
\\ \ (NO DIFFUSION)
4 DIFFUSION (NO
FURTHER TILTING)
\
Fig. 1. Schematic of diffusion in the Ekman layer.
tial cloud (Step 1), being small, is transported mainly by the surface wind.
Its size increases mainly by turbulent dispersion (Step 2). £t this point,
the upper levels of the cloud will be advected in a different direction frcm
that of the surface wind. Conceptually, the tilting of the clcuc is imagined
to occur independently of diffusion, whereas, in reality, tilting and dirfi;-
-------
sicn occur simultaneously. The clouc is tilted by the crcssv/ind (Step 3),
and the simultaneous turbulent diffusion (Step 4) increases the width of the
clouc at ground level over what it would have been by turbulent diffusion a-
lone. In fact, the center of gravity of a slice of the clcud at ground level
will net follow the surface wind.
"he F.cssby number describes the relative importance of the Coriolis ac-
celerat'cns when compared with advective, or local accelerations. If the
Rossby number is large, Coriolis accelerations are snail, so that enhanced
dispersion due to directional wind shear may be ignored, Equivalently, a
near infinite Rossby number is automatically matched in a model.
Tc date; all v\ind and water tunnel modelers have assumed the Rossby num-
ber to be large and discarded the terms "involving it in the equations of mo-
tion, or, equivalency, ignored that particular parameter as a criterion for
modeling. Cermak et a!. (1966) mace the rather broad statement that, provided
the typical length in the horizontal plane is less than "EC km. the Rossby
number can generally be eliminated from the requirements for similarity. Hi-
dy (1967) made similar statements. f-'cVehil et al. (1967) ignored the Rossby
number v,hen modeling atmospheric motions on the scale of one kilometer -'n the
vertical and several tens of kilometers in the horizontal. Ikeguchi et al .
(1967) claimec that the cut-off was AC to 50 km. Pery (^$£9) claimed that the
Ccrio'lis force may be neglected if the characteristic length is less than 15
km. """he present discussion shows that the cut-off point is on the order of
5 t-m for modeling diffusion under appropriate atmospheric conditions (i.e..
neutral or stable conditions in relatively flat terrain).
"he criterion is based en. a length scale rather than on the Rossby num-
ber itself because the angular rotation cf the Earth, r.0, is a constant
10
-------
(Ro = UD/'Lr< ), and the characteristic velocity of the atmospheric flow does
K 0
not vary by more than an order cf magnitude, so that the characteristic length
is primarily responsible for determining the Rossby number.
Several papers have examined the effect of crosswind shear on dispersion.
Pasquill (1962a) measured horizontal spread both in the longitudinal and cross-
wind directions for medium-range dispersion. His data, however, were insuf-
ficient to allow firm conclusions to be drawn about the relative importance
of turbulence and shear in promoting horizontal spread.
Corrsin (1953) showed that anj' in a uniform shear flew (a is stream-
x ^
wise puff width), by considering Lagrangian particle motions. Saffman (1962)
applied the concentration-moment (von Karman integral) method to the classical
diffusion equation (he did net consider turning of wind with height, although
similar considerations are involved). For a semi-infinite flow, ground-level
3/2
source (puff), and linear velocity profile, he also found <^«t . For a com-
*> / o
pletely unbounded flow, Smith (1965) showed that cx*t , using statistical
techniques.
S"'nce the contribution to the spread from the turbulence alone is
0 xt^^, it is clear that the shear effect will eventually dominate the dis-
x
persion process. These solutions are valid only for constant ciffusivity and
large times; they do not provide any indication cf the early development.
Hence, they are of no help in determining at what distance the shear effect
becomes dominant.
Tyldesley and Wellington (1965) used a numerical scheme and an analog
computer to study the effect of crosswind shear en dispersion. They used an
1c
8-step Ekrcan spiral except with the surface wind 22 frcm ^e ceostrophic
wind. They claimed that the 3/2 power law cces not apply because the crcss-
11
-------
wind shear is net constant with height, "heir estimates indicate that the
vine shear becomes dominant around 4 to 6 krr from the source. For much larg-
er distances, the turbulence will again dominate because the shear gees tc ze-
rc for large heights.
Kocstrom (1?£4) and Smith (1965) used statistical approaches to the
crossvlrd shear problem. A horogenecus field of turbulence, a rear wind
speed which was constant with height, and a crcsswird component which varied
linearly with height were assumed. They obtained expressiors valid for all
times cf travel, but did not estimate tires or distances at which the shear
v.ould dominate.
Csanady (1969) attempted to confirm analytically the numerical results
of Tyldesley and Wellington. Because of analytical difficulties, he confined
his investigation to a slice at ground level cf a cloud ^eleased from a source
at ground level. He found that, indeed, the centro-'d of the slice at ground
level did not follow the surface wind. By the time the clouc occupied 1/3 cf
the Ekman layer (~60Cm), its distance from a lire para!1el tc the surface
wind was cf order 1G km. The contribution to the spread from the turbulence
and from the shear were found tc be ecual at one kilometer from the source.
He estimated that, in the actual atmosphere, the shear effect would overtake
the turbulence effect at 3 to 4 km from the source. He showed that for small
i /^
times, a *t!/ (i.e., turbulent diffusion dominates). For intermediate times,
O f '"* ^ / '**
c <*f''~ (i.e., shear-induced diffusion, dominates). For large times, ? "tl'L
again, because the cloud height is the same as the Ekman layer depth, and the
flow is, in effect, bounded (a is cross-streamwise puff w'dth).
Thus far, no ciffusion experiments have been reported that have been spe-
cifically designed to examine the relative effects cf turbulence and shear,
12
-------
but PasquiT! (1969) reexaim'ned two independent studies which contain
irfcrmatior of interest in this connection. He looked at Hcgstrom's (1964)
data on the behavior of srroke puffs released from an elevated source under
neutral and stable stmcspheric conditions. The data for the crosswird spread
shew the onset of a rrore rapid than linear growth at 2.5 km. These data are
somewhat misleading because they indicate the total puff width rather than the
width at a given level. Hence, they indicate the bodily cistortion (tilting)
of the puff but do not show directly the enhanced spread at a given level. In
accounting for this, Pasquill concluded that the enhanced spread at a given
level as a result of shear becomes important around 5 km from the source.
In analyzing the Hanford data of Fuquay, Simpscn, and Kinds (1964) (con-
gruous ground release of a tracer), Pasquill concluded that the effect on
spread at ground level under stable atmospheric conditions appeared to have
set in significantly at about 12.8 km. He summarized:
...a bodily crosswind cistortion of the plume from a point
source (either elevated or on the ground) sets in between 2
and 2 km. However, the form of the crosswind growth curves
suggests strcngly that the communication of the distortion
to the spread at a given level was not of practical impor-
tance below about 5 krr in the case of the e"!evated source
and about 12 km in the case of the ground level source.
Thereafter the implication is that the shear contribution is
dominant.
It is evident that, if it is desired to model diffusion in a prototype with e
length scale greater than about 5 km, under neutral or stable atmospheric con-
ditions, in relatively flat terrain, the Rossby criterion should be considered,
One encouraging note is that Harris'(Toe' high wind (neutral stability) re-
sults show no systematic variation in. wind direction with height over r1at
terrain up to z-20Crr.
13
-------
2.2.2 The Reynolds Number, ILL/ v>
TV -r, r_ _ -!. j-^
The physical significance o^ the Reynolds number becomes apparent by not-
A
ing that it measures the ratio of inertia! forces (l'£/L) to viscous or fric-
K
i
ticnal forces (vLL/L'") in the equations of motion. It imposes very strong lim-
itat-;ons on rigorous simulation; it is the most abused criterion in models of
atmospheric flows. The scale reductions commonly used result in model Reynolds
numbers three to four orders of magnitude smaller than found in the atmosphere.
The viscous forces are thus relatively more important in the model than they
are in the prototype. If strict adherence to the Peyno^s number criterion
were required, no atmospheric flows could be modeled.
Various arguments have teen presented which attempt to justify the use of
smaller Reynolds numbers in a model (i.e., to justify the neglect of the Rey-
nolds number criterion). These arguments nay be divided into three general
categories; the laminar flow analogy, Reynolds number independence, and dissi-
pation scaling. Each of these is discussed below.
2.2.2.1 The Laminar Flow Analogy
Abe (19^1) was the first to introduce this concept. If the instantane-
ous velocity, temperature, and pressure in Eq . 2.1 are written as the sum of
mean and fluctuating parts (U,. = U. -* u.), and the equation is then averaged,
the following equation is obtained (after minor manipulation):
+ v + . (2.9)
a* * O ox'CXj 5xj
An eddy viscosity is defined to relate the Reynolds stress to the mean veloci
ty, -(u^u.) = K(3U.j/?x ) The nondimensicra1 equation is then
-------
cf' ~" l ^P ' ~<- ' ^'^-' __ L
^ U ' ~ = ~ '~~ : " °3'~
-
Ro
where Re^ = URL/K is called a "turbulent" Reynolds number. Now if K/v is of
order 1C", the term containing the turbulent Reynolds number is nuch larger
than the term containing the ordinary Reynolds r. urr.be r. If the nondimensior.al
equation for laminar flow were now written, it would appear identical to Equa-
tion 2. 1C, with the exception that the term containing the turbulent Reynolds
number would fce absent. Assuming that the prototype flow is turbulent and
that the model flow is laminar, the scale ratio is of the order l:iCJ, and Up
is the saire order of magnitude in model and prototype, then,
( Dc,\ I Of. }
'mocel = xeK'prototype.
Hence, similarity may be established by modeling a turbulent prototype flow
by a laminar model flow when the scale ratio is on the order o* 1:1C~, all
else being equal in a nondimensional sense.
This scheme is fundamentally incorrect for the same reasons that K-theo-
ries are fundamentally incorrect. Eddy sizes scale with distance from the
ground, with the size of the obstacle, or: generally, with the scale cf sub-
stantial variation in the mean flow. Turbulent diffusion is a flow property,
rot a fluid property. The laminar flow analogy assumes unrealistically that
eddy sizes are very small compared with the scale of variation cf the proper-
ty being diffused. Perhaps under very restrictive conditions, when there ex-
ists a small upper bound to the sizes of atmospheric eddies (i.e., extremely-
stable conditions), there may be seme realistic modeling possib-'l ities, but
-------
the chances of that being the case without ary doubt are small.
Perhaps qualitative use of this technique is the answer to those (non-
diffusive) problems where the spread of a contaminant is control "led primarily
by advective transport (mean flow). Two previous experimental studies give
guidance here. Abe (1941) attempted to model the flow around Mt. Fuji, Japan,
at a scale ratio of 1:5C,COO using this analogy. Cermak et al. (1966) claimed
that ''the model flow patterns obtained were not ever, qualitatively close to
those observed in actual field tests'1 (the original paper was not available
fcr verification). Cermak and Peterka (1966) made a second study of the wind
field over Point £rguellc, California (a peninsula jutting into the Pacific
Ccean). Cermak et al. (1966) claimed that:
Comparison of the surface flow directions and smoke traces for
neutral and inversion flows established an excellent agreement
in wind flow patterns over the Point Argue!Ic area fcr flows ap-
proaching from the northwest.
After careful study of the figures presented, the present author is not con-
vinced of the validity of this statement. Large scatter of concentration lev-
els in the field data prevented firm conclusions concerning diffusion charac-
teristics of the two flow fields. Rather surprisingly, a logarithmic plot of
concentration versus downwind distance showed that rates of decrease of con-
centration with distance were grossly similar in model and prototype. In view
of the d-'ssirr.ilarity in surface flow patterns, this agreement is regarded as
fortuitous.
Since kinematic viscosity is a fluid property, it is rot an adjustable
parameter. Turbulent eddy viscosity varies strongly with height, stability,
and direction. This severely limits the use of this laminar-turbulent analo-
gy for fluid modeling. Mathematical modeling techniques are superior to fluic
modeling techniques in the sense that K is a controllable variable in the math-
16
-------
statical model (e.g., a function of height, stability and direction). The
quantity !< is not a controllable variable in this sense in the fluid model.
2.2.2.2 Reynolds Number Independence
This approach is based on the hypothesis that in the absence of thermal
and Coriolis effects and for a specified flew system, whose boundary conditions
are expressed nondimensicnally in terms of a characteristic length L and ve-
locity Up, the turbulent flov,' structure is similar at all sufficiently high
Reynolds numbers (Tov/nsend, 1956). Most ncndimensional near-value functions
depend only upon ncndimer.sional space and time variables and not upon the Reyn-
olds number, provided it is large enough. There are two exceptions: (1) those
functions which are concerned with the very small-scale structure of the turbu-
lence (i.e., those responsible for the viscous dissipation of energy), and
(2) the flow very close to the boundary (the r.o-slip condition is a viscous
constraint). The viscosity has very little effect en the main structure of
the turbulence in the interior of the flow; its major effect is limited to
setting the size of the small eddies which convert mechanical energy to heat.
One way to avoid the effects of viscosity at the boundaries is to roughen the
surface of the model (see discussion of Boundary Conditions).
This hypothesis of Reynolds number independence was put forth by Town-
send (1956). He called it Reynolds number similarity. There now exists a
large amount of experimental evidence supporting this principle. Townsend
stated it simply: "geometrically similar flows are similar at all sufficient-
ly high Reynolds numbers." This is an extremely rcrtunate phenomenon from the
standpoint of modeling. The gross structure of the turbulence is similar over
a very vside range of Reynolds numbers, "his concept is usec1 fcy nearly all rrcc1-
17
-------
elers. It is graphically illustrated in Figures 2 and 3. The twc jets shown
in each figure are identical in every way except for the viscosity of the flu-
ids, and therefore the Reynolds numbers, which differ by a factor of 50.
F i c u re 2
Ficure 3,
: Re=20,000
Re=400
Turbulent jets showing that the Reynolds number does not
much affect the large scale structure, so long as it is
sufficiently large that the jet is indeed turbulent. The
upper jet has a Reynolds number 5G times that of the low-
er. (Courtesy of National Committee for Fluid Mechanics
Films and R.W. Stewart).
Shadowgraphs of the jets shown in Fig. 2. Note how much
finer grained is the structure in the high Reynolds rum-
ber jet than that in the low Reynolds number jet. (Cour-
tesy of rational Committee for Fluid Mechanics Films and
R. W. Stewart).
18
-------
To obtain a better idea of the contributions to dispersion from the vari-
ous scales of turbulence, it is convenient to examine the spectral form of the
"Taylcr (1921) diffusion equation. Taylor's expression for the mean-square
fluid, "article displacement in a stationary homogeneous turbulence is given by
T t
,v-(T) = 2t-
where '/" is the variance of particle velocities, p(s}Ev(t)v(t+0 is the Lag-
rangian autocorrelation of particle velocities with tine separation ?, and T
is the time c* travel from the source. Kampe de Feriet (1939) ana Batchelcr
(19^-9) applied the Fourier-transform between the autocorrelation and the cor-
responding Lagrangian spectrum function, F^(n),
FL(n) = [ 0(t) cos Ont) dt,
tc obtain
= t-r-
PL\n\
\\here n is the frequency. The squared term under the integral, illustrated
in Figure 4, is very small when n>l/T and virtually unity v:hen n
-------
ly ircrease Its v;icitr.
Sin (ml)
-nT
^
I
, 1
0.01/T
±
-j
~1
-i
i
4-
o.v
1/T 10/T
Frequency, n (hertz)
Figure 4. Filter function in Eq. 2.12: As the travel time T in-
creases, the contribution of srrall scale (high frequency)
motions to dispersion diminishes rapidly.
It tray be helpful here to examine typical energy spectra so that 1) "weather"
and "turbulence" may be defined as separate and distinct entities, and 2) the
influence of Reynolds number upon the shape of the spectrum will be mere easi-
ly understood.
Figure 5 shews a spectrum
n) = 4 /0 u(t) u(t+t'~) cos 27rnt' dt'
of v;ir,d speed near the grcund frcm a study by Van der Hover. (1957). It is ev-
ident that wind effects can be separated roughly irtc two scales of motion:
20
-------
large scale (lev.' frequency) rroticns lasting longer than a few hours, and small
scale (high frequency) motions that last considerably less than an hour. The
larae scale motions are due to diurnal fluctuations,
l-leather
Spectral Gap
Turbulence
." -^
Cycles/hr
Hours
10
0.1
100
001
1COO
0001
Fiaure 5. Spectrum of wind speed at ICOin, from a study by-
Van der Hoven (1957).
pressure systems, passage of frontal systems, seasonal and annual changes,
etc., and are generally called weather. The small scale motions are associat-
ed with roughness elements, topographical features, and differential surface
heating in the boundary layer and are called turbulence. The spectral gap
(low energy region) separating weather frcm, turbulence is a very fortunate oc-
curence, both frcn an analytical viewpoint and frcm a fluid modeling view-
point. Because of this gap, it is possible tc consider these regions inde-
pendently and tc execute proper mathematical operations tc determine the sta-
tistical properties of the tv/o regions. It is the smaller scales of motion,
the turbulence, which are simulated in a fluid modeling facility. Steady
state averages of fluctuating quantities in the model atmosphere correspond
to approximately one-hour time periods in the real atmosphere (during which
the mean wind is steady in speed and direction). From results cf model ex-
21
-------
periments conducted at different mean wind speeds and directions, the low fre-
quency contribution can be constructed analytically from distribution charts
of wind speed and direction (wind roses).
Figure 6 is a definition sketch of a turbulent energy spectrum from Wyn-
gaard (1973), which will be helpful in understanding future discussions. (It
is net intended here to present a detailed discussion of turbulent energy
spectra. Only those features of direct interest will be covered. The ardent
student should consult Batchelor, 1953a, hinze, 1975, or Tennekes and Lumley.
1972.) In Figure 6, we have used wave number < instead of frequency n so that
In fC
Fig. 6. Form of Turbulence spectrum.
we will be more inclined to think in terms of length scales. The relation-
ship is K-=2-rn/U. (The spectrum function S in Figure 5 is ore-dimensional,
whereas that of Figure 6, E, is three-dimensional. The differences need net
concern us here.) Note that an integral scale I and a nicrcscale "i are de-
fined. The integral scale may be thought cf as the characteristic size of the
-------
energy containing eddies and is located at the peak cf tue three-dimensicna1
spectrum, "he frieze scale ~ay be thought cf as characteristic cf the smal lest
eddies ~'P a turbulent flcv-. ~hey are the ones prirarily responsible fcr the
ciss;pat-'cn u) of turbulent kinetic energy, "he ratio cf the integral scale
to the T'crc scale, then, is a measure of the width cf the spectrurr or the
range or edcy s;zes in. the turbu1erce.
A pertinent cuesticn at this point is: hew aces the spectrum cf turbu-
lence in a simulated atrrcspher-'c boundary layer in, say, a wind tunnel compare
v-'f that -> the >"eal atmospheric boundary laye^-?
cu
ance cr scectrutr v,-
~h,e tr~ectr'jrr snace "'s a^~ected .* chanc'rc the ^.i
-------
g-'ver c'ass of -urbulent r1cv. a decrease cf the °evrclds rur-
e ra^ge of the rich-frecoercy end c* tue spectrum, whereas the
i'ze c- :re er.erg; ccrta-'n-'rc sccres charges crly very slowly v.ith Reynolds
"urber.
".: :e screwhat rcre Quantitative, it ':s useful re examine the trenas :~[-p," a - - ~ -? ~- j " c-cer er a tec -""cvis s^cv" t^at c^'v tre i""cr~--v"ecuerc . ere c~ "he
-------
spectrum is cut out, so that this reduction in spectral width has insignificant
<: fleets. It is found empirically that I/M at a fixed distance x/V downstream
from a grid -'s nearly independent of Reynolds number (Corrsin, 1963). Similarly,
it tray be expected in other flow geometries that I/I at corresponding geometrical
locations wil1 be roughly independent of Reynolds number, i.e.,
IP
p m (2.15)
Indeed, the integral scale is found to be roughly half the size of the charac
teristic length and independent of Reynolds number in a wide variety of class
es of flows.
Combining Eqs. 2.13 and 2.15 yields
(2.16)
In summary, "integral scales reduce with the first power of the geometrical
scale ratio (as desired), whereas Kolmoccroff micrcscales reduce with only the
cne-fourth power of the geometrical scale ratio. fs we have seen in our pre-
vious discussion, the largest eddies contribute to the spread of ? plume and
the ones smaller than the plume width have little dispersive effect; hence, the
mismatch of Reynolds number between the model and the prototype is insignifi-
cant.
A practical example here will make the point clear. The Koltr.ogorcff mi-
croscale in the atmosphere is about one millimeter between 1 and ICO m above
ground (Lumley and Fanofsky, 1964). As indicated by Eq. 2.14, at a scale ra-
25
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tic of 1:500, the spectral width in a model would be approximately two orders
of magnitude smaller than desired. The Kolmcccrcff micrcscale in the model
would be about ICO times larger than required by rigorous similarity. This
would correspond to a Kolmcgorcff micrcscale of 10 cm in the atmosphere. It
is difficult to imagine a practical atmospheric diffusion problem where eddies
smaller than 1C cm would contribute significantly to the spread of a contami-
nant.
It should be noted that the arguments beginning with Eq. 2.13 have been
concerned with Eulerian scales and spectra, v;hereas they would best have been
posed in terrrs of Lagrangian coordinates, as required in EC. 2.12. However,
it is reasonable to assume that if the Eulerian spectra c.nd scales of model
and prototype are similar, then so will be the Lagrangiar spectra and scales;
more appropriately, if Eulerian spectra differ in certain respects between
model and prototype, then the Lagrsngian spectra will te dissimilar in like
fashion. Hence, conclusions drawn in the Eulerian framework are expected to
be valid in the Lagrangian framework (to at least within the same order of
macnituce).
Concerning puff (relative) diffusion, the problem is somewhat different.
Ccrrsin (1961b) has argued that the principal contribution to two-particle
relative diffusion comes from eddies of roughly the same size as the particle
pair separation. This contrasts with single particle diffusion, where the
principal contribution comes from eddies of the same size and larger. Hence,
puff diffusion in a model will depend somewhat rcre strongly upon the model
Reynolds number. Intuitively, it appears that reasonable results would be ob-
tained if the Kolmogorcff micrcscale were small compared with the initial puff
width.
26
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The discussion of the Reynolds number criterion in the modeling litera-
ture normally centers arcurd sharp-edged geometry where it is usually stated
that the mean flow patterns hill not be much affected by changing the Reynolds
number. While this is true, it does not make full use of the concept. Most
mean-value functions, including those describing the main turbulence structure,
will be nearly independent of Reynolds number, prov'dir.g it is sufficiently
high; the only exceptions are those two discussed at the beginning of this
section.
The question is: how high must the Reynolds number be to be high enough?
A precise answer would depend upon the geometrical shape of the boundaries,
the roughness of the model surface, the accuracy desired, the type of informa-
tion desired from the model, and possibly other effects (e.g., these charac-
terized by the Rossby and Froude numbers). The answer to this question is rea-
scrably well known for simple flow classes such as jets and cylinder wakes,
but is largely unknown for models of atmospheric notions. Specific recommen-
dations on minimum Reynolds numbers to be achievec -in various classes of flews
will be made in Section 3, Practical Applications.
2.2.2.3 Dissipation Scaling
A hypothesis or, the similarity of the detailed turbulence structure of
model and prototype flows was proposed by remote (1S68). Again, a basic as-
sumption is that thermal and Coriclis effects are negligible. He reasoned
that mean flow patterns of both the model ana prototype would be similar if
the turbulent structure of the two flows were geometrically similar. Two as-
sumptions were made:
(1) the turbulence of both model and prototype flows was 'locally
isotropic' (Kolrcogcrof*. 1C<11),
27
-------
(2) the Kolmogoroff velocity, u , and microscale, n. characterize the
turbulence at each Dclnt in the flow.
Assumption "' is satisfied if the Reynolds number is very large (Hinze,
197E). Using assumption 2, Nerroto reasoned that, the turbulence structures of
model ana prototype flows would be similar when
>1m = Lm
n~ Lp (2.17)
and
"m _ LKm
'Jp=LV (2.18)
where the subscripts m and p refer to model and prototype, respectively.
From the definitions of n and u, the following equation may be established:
(2. IS)
From Fqs. ?.17, 2.18 and 2.19, n rray be deduced that:
(2 2C)
Fq. 2.20 is the similarity criterion proposed by Nerr.oto. He has also shown
how the above relationship may be obtained from a special ronch'rr.ensiona! iza-
tion of the turbulent energy equation.
It is agreed that the turbulence structures of model and prototype fcws
would be similar i-" EqS . 2.17 and 2.18 could be satisfied. However,
it is impossible to satisfy Fq . 2.17 using typical length scale reductions
(i.e., 1:300 to 1:1000). £s mentioned previously, the Kclmogoroff m'croscale
*' n the lower "iayer of the atmosphere is about Irrm. Typical values of
n in "laboratory flews are C. 075mm (Wyngaard, "!Q67), and 0.5mm (Snyder and
28
-------
Lumley, 1971). measurements in very high Reynolds number laboratory flows
f ^istler and v'rebalovicb, 1966) show that the smallest Kolmogoro" r.icroscale
which can reasonably be generated in the laboratory is C.OSmm. Thus, the
largest ratio of microscales is of order 20, which is far short of the
typical ratio required (i.e., 3CO to 1000).
Generally speaking, the satisfaction of Eq. 2.17 requires identical Reyn-
olds numbers of the model and prototype flows, as shown in the discussion of
Reynolds number independence. This is also easily shewn by the usual nor.di-
rensional Nation of t^e turbulent energy equation (Cernak et a!., 1966).
C . 2 . 3 The Pec 1 e t Number and the Reynolds -Schmidt Product
The Peclet number is most easily discussed by writing it as
where Pr is the Prandtl number.
~he Ceynolds-Schrri'dt product may be written as
L"RL URL v
R = R = Re-Sc.
j. v a
coth of these dimensicrless parameters have the same form (i.e., the product
of a Deynolds number and a ratio of molecular transport coefficients). Both
the Prandtl and Schmidt numbers are fluid properties and not flow properties.
^he Drandtl number is the ratio of the momentum difftsivity (kinematic vis-
cosity) to the thermal difrusivity. T^e Schmidt number is the ratio of the
momentum diffusivity to mass diffusivity.
For air, the Prandt1 number does not van strongly with temperature.
When air is used as the medium for modeling, the prandtl number is nearly the
same in model and prototype, and, if the Reynolds numbers were the same, the
-------
Peclet number criterion would be nearly satisfied. However, if water is used
as the medium for modeling, the Prandtl number at ordinary room temperatures
is a factor of about 1C larger than it is in air, and it varies rather great-
ly with temperature. Thus, from the standpoint of rigorous similarity, it
dees not appear that water would be a suitable medium in which to do model
stud-ies.
The Schmidt number for most gases in air is about one. Thus, the Schmidt
number for an effluent plume (which contains only minor fractions of gases
ether than air) diffusing in the atmosphere is about one. If air is used as
the medium for modeling, the Schmidt number (for nearly any foreign gas intro-
duced) will be nearly the same in model and prototype. If, at the same time,
the Reynolds number were the same, the Reynolds-Schmidt product criterion
hould be nearly satisfied.
When water is used as the medium for modeling, salt water or alcohol are
typically used to simulate the buoyancy of a plume. The Schmidt number for
sodium chloride or alcohol in water is approximately 800. Thus, it appears
that strict similarity using water as the modeling medium would be difficult
to obtain.
The basic problem, however, in matching of the Peclet number or Reynolds-
Schm'dt product, is not in the Prandtl or Schmidt numbers, but rather in the
Reynolds number. Arguments similar to those constructed for Reynolds number
independence may be used to justify the neglect of the Peclet number and Reyn-
olds-Schmidt product as modeling criteria. The term on the right-hand side
of Eq. 2.6 represents the molecular diffusion of heat. The term on the right-
hand side of Eq. 2.8 represents the molecular diffusion of mass. In this
connection, both heat and mass are regarded as passive scalar contaminants.
30
-------
If the flew is of a sufficiently high Reynolds number, then the main struc-
ture of the turbulence will be almost totally responsible for the transport
of the contaminant (heat or mass). Molecular diffusion will contribute very
little to the bulk contaminant transfer; its main function is to smooth out
the very small-scale discontinuities of concentration or temperature (i.e.,
it acts as a low-pass filter on the concentration or temperature fluctuations).
Indeed, arguments of this nature have been used to postulate the form of con-
centration or temperature spectra at large wave numbers (see Corrsin, 1964;
Pao, 1965). The main effect of the diffusivities is confined to setting the
high wave-number cutoff of the temperature or concentration spectrum. Since
turbulent diffusion strongly dominates molecular diffusion in turbulent air
flows, especially at high Reynolds numbers, and since molecular diffusion is
even less important for Prandtl or Schmidt numbers larger than unity, the
effect of not matching the Prandtl or Schmidt numbers of the prototype in the
model is unimportant. Generally speaking, the Peclet number and Reynolds-
Schmidt product may be neglected as modeling criteria if the flow exhibits
Reynolds number independence. Both air and water are suitable media fcr mod-
eling, from this standpoint. Further discussions of air versus water are giv-
en in Chapter £.
2.2.4 The Froude Number. UR/(gL5TR/To)1/2
The square of the Froude number represents the ratio of inertia! forces
to buoyancy forces. A large value of the Froude number implies that buoyancy
forces are snail compared to inertia! forces. Thus, thermal effects become
important as the Frcude number approaches unity. Batchelor (1953b) has shown
how this parameter is related to the Richardson number. In the absence of a
clearly defined length in the atmospheric boundary layer, it is convenient to
31
-------
2
replace UR/L by a representative velocity gradient and <5TR/L by a representa-
tive temperature gradient. Substitution of these gradients into the expres-
sion for the Froude number yields,
, _ r0 l/i _L = TO (dOjdz}2R
Fr~~l,L2 5TR~ g (cdTlo-.}R
Thus, the Froude number may be regarded as the Averse square root of a Rich-
ardson number. It is also related to the Monin-Cbukhov (1954) length. Al-
though any of these parameters may be used as similarity criteria, the Froude
number is used here because it appeared naturally through the non-dimensionali-
zation of Eq. 2.1. Eatchelor (1952b) has also discussed the conditions under
which this parameter is the sole governing criterion for dynamical similarity
of rrotions of a perfect gas atmosphere.
A different interpretation of the Frcude number is quite useful in
considering stably stratified flow over hi^ly terrain. Suppose the flow approach-
ing an isolated hill has a uniform velocity profile and a linear density gradient
The appropriate form of the Froude number is then
Fr^ = Pl| /(ghAp),
where the characteristic length L has been replaced by the height h of the
hill and the density difference Ap is that between the base and top of the
hill. The square of the Froude number represents the ratio of kinetic
to potential energy, i.e., it represents the ratio of the kinetic energy in the
approach flow to the potential energy required to ra^se a fluid element from tKe
base to the top of the hill. It is clear that if the Froude number is ruch less
than unity (very strong stratification), there is insufficient kinetic energy in
the approach flow to raise fluid from the base to the top of the hill. With a
two-dimensional hill perpendicular to the wind direction, this would result
-------
-'r upstream blocking o* the flow below the 'r'1! top (Long, 1972). For a
three-dimensional hill, the fluid, rather than being blocked, can go round the
hill (Hunt and Snyder, 1579). Hunt et al. (1978) and Snyder et al. (1979) have
shown in more quantitative terms how the streamline patterns (hence, plume
trajectories) change drastically with changing Froude number. It is thus
evident that the Froude number is ar essential parameter to be matched when
modeling stably stratified flow over hilly terrain (see further discussion
in Section 2.4).
The Froude number is not, by itself, a difficult parameter to duplicate
'n a fluid model. It is likely to be the most important individual parameter
to be matched when the model is to simulate atmospheric diffusion. When the
modeling medium is air, provisions must be made for heating or cooling of the
air stream to obtain the temperature stratification. However, in order to natch
small Froude numbers of the prototype in a model with a typical scale reduction
(i.e., 1:300 to 1:1000), and in order to maintain reasonable temperatures
n'.e., maximum temperature difference of 200°C) in the model, it is necessary
to decrease the mean flow speed. Tc match the Reynolds number between the
model and the prototype requires that the mean -How speed be increased, THS
conflict is resolved by matching the Froude number while insuring that a
°eyno1ds-number-independent flow is established. This is not always possible.
When considering water as the medium for modeling, it is necessary to
define the Froude number in terms of density, rather than temperature, i.e.,
Fr =
s
33
-------
vhere p represents density. The cotnrion method of producing stable density
stratification in water is by producing thin layers of various concentrations
of salt in water. In view of the very small mass diffusivity of salt in water,
an undisturbed stable mass of salt water will remain that way for several weeks
before the density gradient is changed substantially by molecular diffusion.
Maximum density differences are limited (about 2C% in the dimensionless densi-
ty difference), so that flow speeds must be reduced as was the case with air
as the modeling medium. Recirculating systems using this technique have been
impractical because of resulting mixing within a pump. However, Odell and Kc-
vaszray (1971) have recently designed a rotating cisk pump which maintains the
gradient; this device may permit the use of recirculating salt water systems,
although thus far it has only been used in very small channels. An interest-
ing technique is reported by Homma (1969) wherein fresh and saline water are
mixed to produce stable density gradients at the entrance of a once-through
open water channel. This technique offers the pcssiblity of providing the
proper boundary conditions of turbulent flow (see next section), which is
quite difficult in the still tank.
2.3 BOUNDARY CONDITIONS
2.3.1 General
A statement was made earlier that it was not necessary to determine
a priori whether the flow was laminar or turbulent in order to apply Eqs. 2.4
to 2.6 to the determination of the similarity parameters. It is certainly
necessary to determine whether or not the flew is turbulent in order to
specify the boundary conditions. It is assumed here that the atmospheric
flow is always turbulent. Furthermore, it was stated that the model flow
would be identical to the prototype flow if, among ether things, the
-------
ron-dimersional boundary conditions were identical.
Batchelor (1953b) points out:
Regarding the boundary conditions, we do rot know enough about
the differential equations concerned to be able to say with
certainty what conditions must be specified to make the problem
determinate, but it is a plausible inference from physical
experience that u_' and p1 must be given as functions of t' at
all spatial boundaries and uj o', p' must be given as functions
of x_' at an initial value of t1; it seems certain that such a set
of boundary conditions is sufficient, although in some circum-
stances the conditions may well be over-sufficient.
(The prime here indicates a nondimensional quantity and the underline signifies
a vector.) In the problem considered in the present paper, it is clear that
,5T' and ;(' also must be specified at an initial value of t' at all values of
x_' and at all spatial boundaries as functions of t1. This information is nev-
er available; even if it were, it could not be applied tc the physical model.
In the same sense, such detail is not required nor even desired as the output
frcm the model. What is desired are quantities characteristic of the ensemble
of realizations of turbulent flow. Evidently, what is required are properties
characteristic of the ensemble of realizations of boundary conditions. This
would require the specification of aT! the statistical properties (ell cf the
moments) of the velocity, temperature, pressure, and density fields both ini-
tially everywhere and on the boundaries for all times. Even this much infor-
mation is not available.
The most complete information which can be supplied at the present time
are the first few moments, at least the near, and the variance. It is not
known if the specification of only the first few moments is sufficient; it
is plausible, from physical experience, that such a specification is sufficient.
Indeed, it is dubious that any moments above the first twc could be controlled
at will.
-------
Nearly all modelers have considered the specification of boundary condi-
tions from a different viewpoint, that is, through the spectrum of turbulence
-'n the approach flow. £rmitt and Counihan (1968) have given qualitative argu-
ments which suggest that, for the study of plume dispersal, not only must the
turbulence intensity components be properly modeled, but also the spectrum of
each component is required, particularly the low-frequency end of the spec-
trum. This idea is in agreement with the previous discussion on the cortribu-
tions of the various scales of turbulence to the dispersal of contaminants.
Some control of the turbulence spectra in the approach flew is possible (see
Section 3.2).
2.3.2 Jensen's Criterion and Fully Rough Flow
T^e specification of the velocity on solid boundaries is simple; it is
zero, and all of its moments are zero. Hence, geometrical similarity of model
and prototype is required. This raises another question; how nuch detail is
necessary? From the stancpoint of rigorous similarity, of course, every detail
of the prototype must be duplicated in the model. However, in view of the fact
that the Reynolds number will rot be duplicated, the fine detai1 is unnecessary,
Jensen (1958) has suggested that, if the roughness length o^ the prototype.
z , may be determined (or at least estimated), then it should be sca"!ed accord-
ing to
z L
Offi m .
Z°P LP
(the roughness length is a fictitious length scale characterizing flew ever a
rough surface. For uniformly distributed sand grains of size e, the roughness
length is typically e/20.)
36
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This equation is known as Jensen's criterion, and has been widely used. It
implies that elements or details smaller than will have very little effect
on the overall flew; hence they need not be matched in the model. Details a-
bout the same size as s. need to be matched only approximately (for example,
randomly distributed grains of sand). It may be necessary, with large reduc-
tions in scale, to abandon Jensen's criterion, as discussed below.
The flow of fluid close to a smooth boundary is not Reynolds number indepen-
dent. The ro-slip condition at the surface is a viscous constraint. A vis-
cous sublayer exists immediately adjacent to the wall where viscous stresses
dominate. If the surface is roughened such that the irregularities are larger
than the thickness of the viscous sub-layer which would have existed on a
smooth surface under otherwise identical flew conditions, viscous stresses be-
come negligible. The irregularities then behave l^ke bluff bodies whose re-
sistance is predominantly fcrm drag, i.e., the resistance is due to the pres-
sure difference across the obstacle rather than to viscous stresses. Such a
rough surface is said to be aerodynamically rough; the flow ever an aerodynam-
ical ly rough surface is Reynolds-number independent. The criterion which in-
sures that the flow is aerodynamically rough is u*z As-2.5 (Suttcn, 19^?),
where u* is the friction velocity.
This is extremely fortunate from a mode"1 ing standpoint, because atmos-
pheric flows are almost always aerodynamically rough (Suttcn, 1949). If model
flow conditions are chosen such that u*zJ'^2.S, one can be certain that the
L«
boundary layers are turbulent, sc that such things as separation 'bubbles' and
wakes behind obstacles and transition, separation, ard reattachment of bounda-
ry layers on topographical surfaces will change very little with Reynolds num-
ber. The critical roughness Reynolds number, then, is that at which the bound-
ary layer en the model becomes Qualitatively comparable to that on the prcto-
37
-------
type.
For large reductions in scale, the simultaneous satisfaction of Jensen's
criterion and the critical Reynolds number may not be possible. The critical
Reynolds number criterion is undoubtedly the more important of the two crite-
ria, because it controls the quality of the flow. Over-roughening of the mod-
el surfaces, thereby ignoring the Jensen criterion, will merely limit the
resolution of the few over the model (details about same size as e will ret
be capable of being resolved), but, since the Reynolds criterion is met, the
over-all flow patterns will most likely be matched.
2.3.3 Other Boundary Conditions
Specification of the detailed temperature distributions at the solid
boundaries is rarely discussed from the modeling standpoint. IP current prac-
tice, the solid boundary is maintained at constant temperature. It is plausi-
ble that the amount of detail in the temperature distribution should be deter-
mined on the same basis as the amount of detail in the geometrical boundaries.
This has never been done, although elementary attempts have been made by
Chaudhry and Cermak (1971). Similar considerations apply tc the specification
of the boundary conditions of the density distributions when the modeling medi-
um is salt water.
Specification of boundary conditions on concentration distributions is,
in principle, easy. In practice, the difficulty would depend on the type cf
problem to be studied. For example, if the problem were to determine the ef-
fect cf a single source, the boundary conditions could be x'=0 initially every-
where and x'=ccnstant at the location of the source for all time thereafter.
Very little is known about the boundary specification of the pressure, p'
Normally, the mean pressure gradient in a wind tunnel is adjusted to zero.
38
-------
Fluctuating pressures are, 'n any event, not controllable parameters in a
model flcv/.
In current practice, the upstream boundary conditions on velocity and
temperature ere specified to be reasonably similar to some theoretical formula
such as the logarithmic velocity distribution. Cermak et al. (1966) argue that
boundary layers (kinetic and thermal) grown naturally over long lengths of
rough ground must be inherently similar tc those ir the atmosphere. Others
(Ludwig and Sundaram, 1969; rrmitt and Ccunihan, 1968; Mery, 1969) use artifi-
cial techniques for generating thick boundary layers over short distances.
Only Mery (1969) has attempted to model both the velocity and temperature pro-
files using artificial techniques. Any of the present techniques for boundary-
layer generation appears to be suitable; all of them come reasonably close to
matching the first two moments of the velocity and temperature distributions.
Boundary-"!ayer heights must, of course, correspond with the geometrical scale
ratio. Practical goals and techniques for simulating the atmospheric bound-
ary layer are discussed in Section 3.2.
2.4 SUMMARY AND RECOMMENDATIONS
Similarity criteria fcr modeling atmospheric flows in air and water have
been derived. Rigorous similarity requires that five rondimensicnal parame-
ters plus a set of nondimensional boundary conditions must be matched in both
model and prototype. It has been determined that the Rossby number sjou](j be
considered when, model ing prototype f 1 cws with a_ l_encjth scale Greater than about
5_ kmu under neutral or stable atmospheric conditions, j_n_ relatively flat
terrain. It is concluded that none work needs to be dene to determine under
what conditions the prototype length scale may be extended while still ignoring
the Rossby number criterion, it is recommended that study be started or
39
-------
methods for simulating Coriolis forces in a model.
The concept of Reynolds number independence has been found to be extreme-
ly useful and powerful. Heuristic arguments have been given through the use
of this concept that i_t_ jjs_ not necessary to. match Reynolds number, Peclet
number or. Reynolds-Schmidt product between model and prototype, provided the
model Reynolds number is sufficiently large. Current practice indicates that
sufficiently large Reynolds numbers are attainable at least for sharp-edged
geometrical structures in ordinary meteorological wind tunnels. More work
needs to be done to determine if sufficiently high Reynolds numbers may be
obtained in the laboratory for the simulation of flow over more streamlined
surfaces. The Froude number j_s_ the^ most important single parameter describing
the prototype flow which must be duplicated in the model. The specification
of boundary conditions was found to be nebulous both in terms of how many
variables are necessary and sufficient and also in terms of the type of
statistical information required (i.e., is the specification of only a few
lower order moments of each variable sufficient?) Geometrical similarity
(ncndistorted models) jjs_ required from the specification p_f zero velocity
ajb the sol id boundaries. It was decided that details cjf the_ prototype of. size_
smaller than the roughness length need not be_ reproduced in_ the model. Objects
about the same s i z e as_ the. roughness length need not be_ reproduced vn_ geometrical
form but an. equivalent roughness must. be_ established. Over-roughening may JD§_
requ i red to. satisfy the roughness Peynol_ds_ number criterion.
Boundary conditions in the fluid model are set by simulating the atmos-
pheric boundary layer. Practical goals and techniques for simulating the
atmospheric boundary layer are summarized in Section 3.2.4.
40
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3. PRACTICAL APPLICATIONS
The fundamental principles of fluid modeling have teen discussed ">n the
previous chapter. When it come to the particular details of a model study,
however, many decisions must be made, and the fundamental principles fre-
quently do net provide enough guidance. It is the aim of this chapter to
cover the most common types of problems encountered by a modeler when
designing a particular model study, and to provide rational guidelines where
possible or to cite common practice where there is no rationale.
The following sections discuss in detail the special problems encountered
r modeling plume rise, the atmospheric boundary layer, flew around buildings,
and flow ever complex terrain. Each of these sections is summarized with a
set of recommendations.
3.1 PLUME RISE AND DIFFUSION
Numerous investigators have studied the rise of plumes from model stacks.
Many different kinds of facilities, including wind tunnels, water tanks, tow-
ing tanks, water channels, and even the calm stably-stratified environment of
an ice-skating rink, have been used. The water tanks and the ice-skating rink
have been used to study the behavior of plumes in calm environments, both
-------
stratified and unstratified. The wind tunnels, water channels and towing tanks
have been used to study the behavior of plumes issuing from stacks into cross-
winds. The effluent has ranged from pure jets to strongly buoyant plumes.
The crosswinds have ranged from neutrally stratified with uniform velocity
profiles to simulated atmospheric boundary layers (stable and unstable strati-
fication with, for example, logarithmic velocity profiles).
The historical development of modeling techniques concerning plume rise
is analogous to the historical development of theoretical formulas for the
prediction of plume rise, i.e., the effluent buoyancy was thought to be
negligible in comparison with its momentum. Sherlock and Stalker (1940)
appear to have done the first wind tunnel study relating to plume behavior.
Specifically, their experiments established the rule that the effluent speed must
exceed one-and-one-half times the wind speed in order to avoid downwash in the
lee of the stack. This ''one-and-one-half-times" rule is still widely applied
today. They worked with a 1:200 scale model, but chose to use essentially
identical model and full-scale values of wind speed and effluent temperature.
Ore of their conclusions was "...the temperature of the [stack] gas is rela-
tively unimportant as a means of control line the downwash...". £t the present
time, it is still not clear exactly what effect buoyancy does have on the one-
and-cne-half-times rule, but it is evident that the buoyancy was not properly
scaled in the Sherlock and Stalker experiments. Their experiments, ever, with
the very hot (£00° F) effluent, were highly momentum-dominated plumes (effec-
tively, jets), and corrections were made only for the change of momentum due
to change in temperature (density). I^cre recent work (cf. Huber et al .
1979) indicates that the buoyancy per se of the lighter effluent is ineffectual
in preventing downwash; instead, the decrease -'n density alone contributes to
42
-------
downwash because it reduces the effluent momentum.
Numerous model studies have been conducted since those of Sherlock and
Stalker, but very few of the results have been compared with atmospheric data
or even with other model results. Much worse, there has not been a uniform
application of similarity criteria. Each investigator appears to apply a dif-
ferent set of rules which ensure that his experiment models the rise of a
plume ir the atmosphere. It is evident after only a little study that some of
the rules are conflicting and that all of them cannot be correct.
3.1.1 Near-Field Plume Behavior
Let us consider the simplest conceivable problem of a plume dcwnwashing
in the lee of a stack because the effluent contains insufficient momentum to
overcome the low pressure suction due to the crosswind (See Figure 8). We will
suppose that the stack walls are thin, the stack is tall and the effluent has
the properties of air at the same temperature as the surroundings, i.e.,
D /D.=l, H /D»l, p /p =1, ip=0 and Fr=«. Further, since we are concerned only
with the local flow field near the top of the stack, we have purposely omitted
shear in the crosswind (su/3z«U/D) as well as stratification and turbulence
in the approach flow. To model this problem, we must match cnly two parameters,
the ratio of effluent speed to wind speed and the Reynolds number:
WS/U, WsD/v .
As discussed in Section 2.2.2.2, provided the Reynolds number is larger than
seme critical value, its precise magnitude is irrelevant. There appears, hew-
ever, to be considerable disagreement concerning the particular value of thjs
"critical" Reynolds number. Ricou and Spaulding (1961) have shown that the
entrainment rate of momentum-dominated jets in calm surroundings is essentially
constant for Reynolds numbers in excess of 25000. Only minor variations were
-------
observed between 15,000 and 25,000. Substantial variations were observed belcw
10,000; the entrapment rate was increased by more than 20%. Hence, if minor
errors are acceptable, the critical Reynolds number is 15,000. This simplified
problem would be relatively easy to model even in a fairly srrall wind tunnel
(say, 0.5m square test section) with moderate wind speeds (~2C m/s) and a
small stack (Ten dia.). We will shortly see, however, that if the effluent is
buoyant, the problem becomes much more complicated. It will not be so easy to
obtain such a large Reynolds number, and we must look harder to determine if
the 15,000 value for the critical Reynolds number can be reduced. We will re-
turn to our discussion of critical Reynolds numbers later in this section.
Figure 8. Plume dcwnwash in the wake of a stack.
Notice that, provided the Reynolds number exceeds 15,000, there is cnl^
one parameter of importance, WS/U. Since ful1 scale stacks and effluent
speeds (even fairly small stacks and low speeds) result in huge Reynolds num-
bers (a typical small value might be 1C6), the full scale flow is Reynolds
number independent. This implies that the size and shape of the wake behind
-------
the stack and the amount of downwash depend on only one parameter, WS/U, and
not on wind speed per se. Similarly, provided the Reynolds number is large
enough, the flow structure in a model in a wind tunnel is similar to that of
the prototype and is independent of wind speed per se. (This discussion may
appear obvious and therefore trivial, but in demonstrating wind tunnel experi-
ments to novices, or even frequently to accomplished experimentalists, the
inevitable question is: to what full scale wind speed does this flow correspond?
The correct answer, i.e., all wind speeds above the barest minimum, invokes
puzzled glances and disbelief!)
Let us now complicate the problem, one step at a time, to see what addi-
tional issues arise in the modeling. There are really two Reynolds
numbers in this problem, corresponding to two different classes of flow: one
for the flow inside the stack WsD/r and one for the flow around the outside of
the stack UD/v. The critical Reynolds numbers may differ because the classes
of the flows differ. The critical Re for pipe flow assuredly differs from
that of a two-dimensional wake.
It is useful here to consider the changes that occur in the flow pattern
around a circular cylinder as the Reynolds number is increased (for additional
details, see Goldstein, 1965). At very low Re (<1), the streamline patterns
are symmetrical fore and aft of the cylinder. As Re is increased (~10), two
symmetrical standing vortices are formed at the back; they grow in size and
are stretched farther and farther downstream until at Re~100, they break down
and are shed alternately at regular intervals from the sides of the cylinder.
7 c
This type of flow persists over a very wide range of Re (10 sReslO ). Townsend
(1956, p 134-5) showed that various mean quantities such as the velocity defect,
the turbulence intensity, and the width of the wake of a two-dimensional cir-
cular cylinder were invariant with Re in the range of 800 to 8000, but they
45
-------
are most likely invariant over a much wider range. At an Re of about 10 , the
boundary layer on the cylinder becomes turbulent and leaves the cylinder far-
ther back on the surface, reducing the drag coefficient from a nearly constant
value of 1.1-10% in the range 102
-------
half the ambient air density. In a wind tunnel, it is usually easier and more
practical to use a lighter gas to sirrulate this high temperature field effluent
then it is to heat the model effluent. Similarly, in a water channel or tank,
it is usually easier to use salt water or alcohol than to heat or cool the
model effluent. This low density manifests itself in two opposite ways: first,
at a fixed effluent speed, the effluent momentum ^lux is reduced, tending to
make the plume more easily bent-over, thus promoting downwash, and second, the
buoyancy of the effluent is increased, tending to inhibit the downwash. It is
not clear which is the more important effect. Cvercamp and Moult (1971) showed
rather convincingly that the effect of the increased buoyancy was to inhibit
1 / "7
dcwnwash of cooling tower plumes, where the Froude number Ws/(gD/Wca)
ranged from 0.2 to 2. Huber et al. (1979), however, observed enhanced
downwash as the effluent density was decreased. The Frcude numbers in their
experiments were greater than 4, which is more typical of power plant plumes.
It would appear that the crossover point where the effect of the lower density
switches from inhibiting downwash to enhancing it occurs at a Frcude number
around 3; however, it is likely also to be a function of the effluent momentum tc
to crosswind momentum ratio.
Most investigators would agree that the following set of parameters to te
matched for this mere complex problem are sufficient (although, perhaps, net
all are necessary):
w
Since products of similarity parameters are themselves similarity parameters,
the following set is fully equivalent to that above:
-------
. Ps
r, £ Re,
p a (gDWP-)"£ (3.2)
a a
The first parameter expresses the ratio of effluent momentum flux to crcss-
vind momentum flux, and must be matched if the initial bending cr the rise due
to the initial momentum of the plume is important. The last parameter is the
Frcude number, which expresses the ratio of inertia! to buoyancy forces in the
effluent. (Note the different interpretation of Fr here as opposed to its
characterizing the stratification in the approach flov; as it was introduced in
Section 2.2.4.)
A questionable parameter is the density ratio PS/P, per se. As mentioned
previously, the density difference manifests itself through its effects on the
effluent momentum and the effluent buoyancy, which are expressed in the first
and last parameters of Eq. 3.2. Of course, it is perfectly acceptable to match
the density ratio between model and prototype, but if it is not an essential
parameter, then the full capabilities of modeling facilities will not te real-
ized. It is frequently advantageous to exaggerate the density differences in
the irodel in order to achieve lew Froude numbers. Ricou and Spaldirg (1961)
showed very convincingly that the rate of entrainment dm/dx in a highly
momentum-dominated jet (no crosswird) obeys the relation
]_ dm
P. dx
1/2
-£| WC, (3.3)
a ~" !paj
where C is a constant. Hence the entrainment rate is a direct function of the
density ratio. For small density differences, the entrainment rate is ret
much affected. However, if helium (S.G.=0.14) were used as the buoyant efflu-
ent from a stack in a wind tunnel, rraintainirg geometric similarity ard the
48
-------
ratio of effluent to wind speed W /U, tc simulate a full scale effluent with
specific gravity of 0.7, then the entrainnent rate would be halved. Hence, its
rise due to initial momentum would not be correctly modeled. Considerations
such as these are evidently what lead Hcult (1973) to state that the density
difference WP must not exceed 0.4. But such a statement cannot be made un-
equivocally. If the density difference in the field were 0.5, as might be the
case for a gas turbine exhaust, then it would certainly be desirable tc exceed
0.4 i-n the model. More importantly, as shown by Eq. 3.3, it is certainly pos-
sible to exaggerate the density difference to 0.8 by using essentially pure
helium as the model effluent and still maintain the same entrapment rate by-
increasing the effluent flow rate Wg. But Eq. 3.3 of Ricou and Spa!ding applies
cnly beyond a few diarreters beyond the stack exit (say >10D). In order tc avoid
an "impedance mismatch" in our dcwnwash problem, where we are concerned with
the flow behavior right at the top of the stack, it is necessary tc match
the density ratio. Beyond a few diameters, it is orly essential to match the
momentum flux ratio.
A major point of disagreement among investigators concerns the cefiniticn
of the Froude number. Approximately half the investigators define the Froude
number with the effluent density as the reference density, Fr ; the ether half
define it with the ambient density (at stack top) as the reference density,
Fr . Yet, nowhere is any reason given for the particular choice. It might
a
appear at first glance that the choice is completely arbitrary. However, giver
that two plumes have the same Froude number based on effluent density, 1-t is
not necessarily sc that they have the same Froude number based on anb-'ent den-
sity. Consider the following example cf a typical power plant plume being
modeled at a scale of 1;£CC using helium as the buoyant effluent in L V'ird
tunnel :
49
-------
TABLE 1: Typical Parameters for Modeling Plume Downwash.
Parameter Prototype Value Model Value
Ws 20m/s 1.67m/s
g 9.8m/s2 9.8m/s2
D ICm 2.5cm
oa(T ) 1.2g/l(20°c) 1.2g/l(20°c)
G 3
cs(Ts) 0.83g/l(150cc) 0.17g/l(20°c)
Fr 3.6 3.6
a
Frs 3.C 1.37
P.e 13xl06 36C
The Froude numbers differ by a factor of the square root of the density
ratios, i.e., Fr =(p /p ) Fr , so that unless P_/P, is the same in model
a a s s s a
and prototype, the choice of the definition of the Froude number is not arbi-
trary. Yet, almost all investigators exaggerate the density differences in
the model in crder to obtain large enough buoyancy in the plumes (low Frcude
numbers). They do not match PS/P, as required by Eq. 3.2, so that it is not
possible to match both Froude numbers simultaneously.
This is a particularly vexing problem because most plume rise theories
are founded on the assumption of small density differences, so that the two
Froude numbers are essentially equivalent. There are only two places in the
literature providing guidance on which Froude number is the most appropriate.
Hoult et al. (1977) state that two complete, independent wind tunnel tests
were run, one using the ambient density, the second using effluent density,
as the reference density. The tests involved the modeling of gas turbine
exhausts, which generally involve large effluent velocities and high effluent
50
-------
temperatures. They claim (unfortunately, without presenting any data to
support their claim) that the better choice is ambient density and that the
error between model and field observations (presumably, of far-field plume
rise) when using effluent density was about 9C?'>, which was nearly 1C times
the error incurred using ambient density. It is important to rote that using
ambient density as a reference corresponds to using effluent temperature as
a reference, i.e.,
Ta Vcs - 13. i}
due to the perfect gas law at constant pressure, p7=const.
A physical interpretation of the difference between the two definitions
is suggested in a footnote by Eriggs (1972) (his comments applied specifically
tc alternate definitions of buoyancy flux, but are equally applicable here):
"the difference amounts to different approximations for the effective den-
sity (inertia per unit volume) of the fluic being driven by the buoyant force:
(1) that the effective density is approximately constants , which is
reasonable very close tc the stack, say within a few stack5 diameters
downwind ;
(2) that the effective density is approximately constart=p , which is a
better approximation at all larger distances." a
It is clear, then, that in our stack dcwnwash problem, we should use the
effluent density as the reference density in matching of Froude numbers. It
is also clear that, if we were attempting to model far f^eld plunge rise, we
should use the cmbient density as the reference density, in agreement with
Fcult et a" . (1977) .
A second point of disagreement among modelers has tc do with whether the
51
-------
relevant parameter is the ratio of effluent speed to wind speed W /U or the
? ?
ratic of effluent momentum flux to crosswir.d momentum flux p Vr/(p U ). This
s s a
may alsc be thought of as a ratic of dynamic pressures. Again, if p /p. is
matched between model and prototype, the choice is arbitrary. As discussed
above, however, rost modelers drop the requirement of matching the density
ratic, and the choice is nc longer arbitrary. Sherlock and Stalker (IS^C)
found that the behavior of the plume depended upon the ratio of the momenta
and that the ratic of the two speeds was a close approximation, provided that
both velocities were reduced to equivalent velocities a_t a common temperature.
Their one-arid-one-half-times rule was thus based or the momentum ratic, a
feet not appreciated by most authors who quote the rule. The recommendation
made here, then, is that the relevant parameter is the momentum ratio, and
not the speed ratio per se.
A third problem is that in water tanks or channels, a heavy salt solution
is commonly used to simulate a buoyant effluent by inverting the stack and
exhausting the salt solution into lighter fresh water. The same principle
could conceivably be used In a wind tunnel by using a heavier-than-air gas
such as freon with an inverted stack. There is a subtle question here that
has not been fully answered. In the field, it is a lighter effluent entrain-
ing heavier air, whereas in the water tank, it is a heavier effluent entraining
a lighter ambient fluid. It is conceivable that the er.trainmer.t mechanisms
could be significantly altered due to this interchange of heavier and lighter
fluids. EQ. 3.3 indicates that a heavy fluid issuing from an inverted stack-
can be used to simulate a lighter fluid from an upright stack, if the effluent
speed W is appropriately reduced. As argued previously, there may be a
subtle effect on the entrainment very near the stack, but this disappears
52
-------
quickly as the density difference is rapidly diluted. The total rise of the
plume is not highly sensitive to the entrainment parameter; the forced plumes
cf Hoult and Weil (1972) and Lin et a1. (1974), both using salt water effluent,
appear to simulate field results quite well. It is clear that the problem is
rot yet completely answered and requires detailed systematic study. A tenta-
tive conclusion, in view cf other inherent inaccuracies in modeling at small
scales, is that this subtle difference may be overlooked.
It was shown in the early part of this section that a critical Reynolds
number of 15,000 was not difficult to achieve per se. However, the introduction
cf buoyancy rakes this Pe much more difficult to attain. In order
to match Froude numbers, it was essential to introduce helium as the effluent
(which incidentally, has a kinematic viscosity approximately 8 times that of
air) and to reduce the effluent speed by a factor of 12 (see Table 1),
so that the effluent Reynolds number is cnly 36C. Thus, we must determine
whether a lower critical Reynolds number can be justified.
The data of Ricou and Spalding (1961), which suggestec Re =15,COG, was
applicable tc momentum-dominated jets in calm surroundings. Mcst investigators
would agree that for a bent-over plume, the critical value may be substantially-
lower, of the order of 200C, i.e., a value that is wel1-establishea for the
maintenance of turbulent flow in a pipe. This is equivalent to saying that
the plume behavior is independent cf Reynolds number provided that the effluent
Hew is fully turbulent at the stack exit. Lin, et al. (1974) have taken this
ore step further. They tripped the flow to ensure that the effluent was fully
turbulent at the stack exit at a Re cf 530 by placing an orifice with opening
D/2 inside the stack and located 3D from the exit. Their data for (1) the
terminal rise cf a buoyant plume in a calm and stably-stratified environment
and (2) the trajectory of a buoyant plume in a stably-stratified crosswind
E3
-------
compared reasonably well with other laboratory and field data. Moot, et al.
(1973) used a similar technique. Liu and Lin (1975), however, indicate that
the placement of the orifice relative to the top of the stack was critical at
a stack Re of 200. If the distance was smaller than required, the effluent
flow was governed by the orifice diaireter and not the stack diameter; if
larger, the flow tripped by the orifice would laminarize before it reached
the stack exit.
Briggs and Snyder (1979) shewed a critical Reynolds number of 2CCC fcr
jets and 200 for buoyant plumes in a calm, stably stratified salt water tank
(see Figure 9) for M=0.06, where M is a dimensionless source momentum para-
meter; M=p W N/gAp, and N is the Brunt-Vaisala frequency characterizing the
density stratification. It was argued on theoretical grounds that the critical
Reynolds number fcr a buoyant plume is proportional to H" . Below these
critical values, the plumes were laminar at the stack ex-it with resulting
rises too high. A few attempts at tripping the few inside the stack yieldec
unpredictable results.
Experiments with buoyant plumes ^'n neutrally stable crosswinds, conducted
by h'oult and Weil (1972), show: (1) at a Reynolds number greater than SCO, the
plume appears to be fully turbulent everywhere; at lower Reynolds numbers,
the plune becomes turbulent only some distance dcwnstrean of the exit (there
was apparently no tripping of the flow inside the stack),(2) ignoring
scatter in the data, no dependence of far field plume trajectory on Reynolds
number was observed for Reynolds numbers between 28 and 2800, (3) the vertical
plume width was substantially reduced close to the stack exit (within 1C stack
diameters downwind) for Reynolds numbers be'ow 200.
It is difficult to reconcile the results of the various sets of experi-
ments. There are numerous possible reasons why the critical Reynolds numbers
54
-------
100.0
1000.0 10000.0
REYNOLDS NUMBER
(a) Neutrally buoyant plumes
1000000
'-
0
10.0
(F/N3)1/4
(F/N3)1/4
"eq
1000 10000
REYNOLDS NUMBER
(b) Buoyant pluires
10000.0
Figure ?. Variation of plume n'ie with Reynolds
55
-------
are different. The effluent flows differ (morrentum-dcmirated versus buoyancy
dominated), the stratification of the ambient fluids differ (neutral versus
stable stratification), and in one case the jet issued into a calm environment
whereas in the other twc, the plumes were bent over by a crosswird. The two-
crder-of-nagritude difference in critical Reynolds number, however, is
difficult to explain. It is evident that a basic systematic study needs to
be undertaken to establish, that Reynolds number (perhaps different ones for
different sets of conditions) above which the rise end spread of model plumes
is independent of Reynolds number.
3.1.2 Summary and Recommendations on_ Modeljng [[ear-fj_ejd_ Plumes
In summary, to model the near-field rise of a buoyant plume from a stack,
is recommended that:
1. The effluent Reynolds number be as large as possible.
(a) Fix the effluent Reynolds number to be as large as possible,
preferably greater than 15,OGO.
(b) If it is necessary to reduce the effluent Reynolds numbe^ be^cw
2000, it may be necessary to trip the flow to ensure a fully
turbulent exhaust.
(c) If H is desired to reduce the effluent Reynolds number below
2CO, it will be necessary to do some experimentation to deter-
mine under what conditions the plume will simulate the behavior
cf a plume in the field.
2. The set of parameters to be matched (equal in model and prototype) is
either
(a) For a scale ratio less than about ^CO, matching of all the
following parameters is generally possible and certa-'riy adequate:
56
-------
VA . JL 1/2
u pa
For a scale ratio greater than about 4CC, it is generally
rot possible to natch p /c and it is probably safe to ignore
s a
it. It will then be essential to rratch:
c U ,
a o
Notice that in Condition 2a, the choice of the reference density in the
Frcude nurber does not natter: both will be matched. Ir Condition 2b, however,
it is important to natch the Frcude number based on effluent density. Since
there is only one possible reference length in this problem, the stack diam-
eter, it determines the scale ratio, and geometric scaling is implicit, i.e.,
all lengths should be referenced to the stack diameter. Other lengths that
nay be important are the stack wall thickness and the stack height. Obviously
these lengths should be scaled with the stack diameter. Also in Condition 2b,
it should be cautioned that if WPS is made very large, the initial entrain-
ment mechanism may be altered due to "impedance mismatch1'.
These recommendations are subject to change pending future work.
3.1.3 Far-Field Plume Behavior
The previous section reviewed the criteria to be met for modeling plumes
close to the top of the stack. With one exception, the use of the ambient
density as the reference in. the definition of the Froude number, these same
criteria also apply to modeling plumes far downwind of the stack. "It is obvi-
ous, however, that this implies an even greater reduction in scale (larger
geographical area to be modeled), and even those criteria will be difficult
if not impossible to satisfy. The cuesticn to be answered in this section,
then, is whether further compromises can be made without making the results
unduly suspect.
-------
Ps an example, suppose we wish to rrodel a power plant in complex terrain,
where the scale reduction factor is 1:5CCO. Typical conditions from the plant
operations record might be TS=4CO°K, T =2CO°K, W =25m/s, U=10m/s, D =%. Sup-
pose we try to match conditions 2b from the previous section, basically the
momentum ratio, the stack Frcude number (with ambient density reference), and
a minimum Reynolds number, using pure nethane as the model effluent
(ps/Pa=0.56) in a wind tunnel. The model stack diameter would be C.Smm. The
Froude number of the full scale effluent is 8, which implies a model effluent
speed of Q.£6m/s. These conditions yield a stack Reynolds number of
Re = !^ = 46crn/s x 0.08cm = 2~
s y .16cm2/s
considerably below the value recommended in the previous section.
There are several directions available at this point, notice that the
problem was not created because of the matching of the momentum ratio, al-
though that requirement may later cause problems in obtaining a minimum
Reynolds number based en the roughness of the underlying terrain (see Section
3.2). The problem was caused because of the matching of Frcude numbers. This
problem has been attacked in a variety of ways.
3.1.3.1 Ignoring the Minimum Reynolds Number
Ludv/ig and Skinner (1976) ignored the minimum Reynolds number requirement; thus,
their plumes were laminar in the immediate vicinity cf the stack (see Figure
10). Discussion in their report admitted that the rise of ar initially laminar
plume would exceed that of an initially turbulent plume because the turbulent
one mixes more rapidly with the ambient air. They felt, however, that this
-------
stfcVri*^'''-> ." -Y^--*^"^
Figure 10. Laminar plume caused by low Reynolds rumber effluent (from Ludwig
and Skinner, 1976).
was not a serious limitation in their model because their initial plume rise
v\as quite small before atmospheric turbulence began to dominate the mixing
process. It is evident from Figure 1C, however, ^hat the scale of the atmospheric
turbulence is considerably larger than the initial plume diameter, so that the
plume Trajectory is highly contorted, but little rea1 mixing of effluent with
ambient air occurs for many stack heights downstream. If the effluent
plume were turbulent, it would be diluted very rapidly (within a few stack
diameters) by ambient air. The resulting plume rise could be substantially
different in the two cases, depending on the precise effluent parameters.
Ludwig and Skinner did rot feel that tripping of the flow within the stacks was
possible because the stack diameters ranged from 0.25 to 1.2 mm. Liu and Lin
(1975), however, were able to use a sapphire nozzle of 0.18 mm dia. to trip
the flow in their stack. As mentioned in Section 2.1.1, the size and place-
ment of the orifice is evidently critical and will require special experimenta-
tion.
-------
3.1.3.2 Raising the Stack Height
racy (1971) ignored the p'une buoyancy, per se, but irstead extended the
stack and bent-over the top such that the effluent was emitted at the same
elevation as that calculated frorr plume rise formulas. This technique has the
advantage that flow Reynolds numbers can be made as large as desirable. The
disadvantages, however, are obvious. Since the plume rise is a function of
wind speed, there is a contribution to vertical dispersion due to both long-
tudinal ana vertical fluctuations in the wind speed that cannot be simulated
via this method. Also, the physical stack height in the model must be changed
to simulate different wind speeds. But the most serious limitation is that
the complex trajectory of the plume, which may be the most useful information
obtained from the model, cannot be obtained using this method. It is frequently
desired to determine whether a plume goes over the top, is diverted around,
or impacts on the surface of a hill. If the plume is emitted ->to a
different mean streamline, its resulting trajectory could be entirely different.
Acding momentum to the effluent to obtain the same rise as for a buoyant p^ume
is objectionable for similar reasons, ^his technique, however, might be
acceptable under certain circumstances; for example, if the problem were to
determine concentrations on an isolated hill far downwind of the source (beyond
the point of maximum plume rise), then it might be acceptable to inject the
plume at its terminal rise height. An additional problem is that ore must
presume to know the plume rise. This may be acceptable for stable flows,
but is an unsettled matter for neutral and unstable flows.
3.1.2.3 Distorting the Stack Diameter
Briggs (1969) equation for the trajectory of a plume possessing both
initial momentum and buoyancy, valid only for downwind distances considerably
smaller than that to the point of maximum rise is rewritten here *r, a differ-
ent form:
60
-------
-------
essentially the same point, but suggested additionally that Eq. 3.9 cculd be
met by exaggerating the stack diarreter. They performed an experimental run
on a complex terrain model at a scale of IrlCOCO using a stack diameter
exaggeration factor of 2. Photographs show that the plume was turbulent at
the stack exit, even though the effluent Re was only 68, but no experiments
were made to validate the use of this method.
It is conceivable from inspection of Eq. 3.5 that, by clever manipula-
tion, we could vary any and all of the parameters p , p,, V , D, or U in such
s a s
a fashion that the coefficients would not be changed, and, therefore, that the
plume trajectory would be unchanged, i.e., it would net be necessary to ignore
the momentum termwe could include it toe. This is equivalent to reducing
the momentum and buoyancy lengths by the geometric scale reduction factor.
Obviously, however, if we change D, we will also change the plume width at
the stack exit (=D). This violates our previous requirement of geometric
similarity; it may or may not have serious consequences, dependent upon the
amount of the exaggeration and on the particular flow field. It is not en-
tirely clear what extraneous effects may be introduced by the manipulation of
the other variables.
Basically, the coefficients of the x/H£ terms in Eqs. 3.5 and 3.6 are
products of similarity parameters, i.e.,
1/2
(3.10)
and
' 1
, 1
'D
Hs
I J
'psKV
k, ,
w/ "-
'Pal
ps
/
3/2
(3.11)
62
-------
If we insist en geometric similarity, then F.q. 2.1C is identical tc cur
previous momentum matching requirement, arc Eq. 3.11 is only very slightly
different from our previous Froude number matching requirement. There is re
reason, a priori, to favor one or the other; the choice will require experi-
mental verification.
In a rather unusual derivation in a recent report, Skinner and Lucwig
'1978) have arrived at scaling laws that are essentially equivalent to
matching the ratios of momentum and buoyancy length scales to the stack height,
i.e., Eqs. 3.10 and 3.11. They also conducted some experimental v/crk showing
that "enhanced" scaling (exaggerated density differences) produced the same
results as "restricted" scaling (matched density differences/, with
both tests done in the wind tunnel. Further, they have pointed out the
possibility of exaggerating the stack diameter, but they have not conducted
tests to verify this.
2.1.£ Summary and Recommendations or^ N!odejni£ Far-fje]_d_ P.} ume_s_
.''ost likely more important than the decision on matching the momentum
and buoyancy length scales versus diameter exaggeration versus matching of
the Froude number and the momentum ratio are the effects of the approach flow,
i.e., the stratification and ambient turbulence to which the plume is sub-
jected, as well as the effects of downwash and flow diversion or channeling
caused by buildings and terrain. These will be discussed further in later
sections. For the present, the recommendation is to avoid a rcnturbulen.t
effluent flow ard to avoid raising the stack, either physically or through
the addition of momentum. Instead, the most advantageous of the methods
discussed in Section 3.1.3.3 should be used.
-------
Thus, to model the far-field rise of a buoyant plume from a stack, it
is recommended that the modeler:
1. Insure a fully turbulent effluent flow and
2. Either (in order of decreasing ''correctness1'):
o
,fi. ,,
a) match ps 's s and I?
9 ' 1 /? > >
Pair (gDip/pa)I/£ HS
b) natch lm/Hs and TB/H ,
1} following geometric similarity or
11) exaggerating the stack diameter, but avoiding stack riownwash,
or c) match, lr/H ,
i) following geometric similarity or
11) exaggerating the stack diameter, but avoiding stack downwash.
Obviously, if the stack diameter is exaggerated, other lengths are to be
referenced to the stack height and not the stack diameter.. It is implicit
above that the simulated atmospheric boundary layer is matched and that geo-
metric similarity is followed everywhere, with the possible exception cf
exaggerating the stack diameter as noted. Notice that uncer conditions 2b
and c, that an exaggeration in stack diameter will generally be accompanied
by a reduction in the momentum ratio. It. must be remembered that the momentum
ratio should not be reduced to the point where the plume is downwashed in the
wake of the stack.
It is obvious that there are many unresolved problems concerning the
modeling of plume rise, in spite of nearly 40 years of such modeling. Because
of the lack of basic, systematic studies on these fundamental problems, the
above recommendations are tentatively proposed ard are subject to change
pending future developments.
64
-------
3.2 THE ATMOSPHERIC BOUNDARY LAYER
In the early wind tunnel studies of flow around buildings (Strom et al. ,
1957), complex terrain (Strom and Halitsky, 1953), and urban areas (Kalir.ske
et al., 1945), care was taken to insure that the flow approaching the models
was uniform and of low turbulence across the wind tunnel test section. Jensen
(1958) was the first to suggest that the simulation of the atmospheric bound-
ary layer was important: he was also the first to produce a simulated
atmospheric boundary layer approach flow by matching the ratio of the rough-
ness length to the building height between model and prototype. Strong
variations in the surface pressure coefficient were observed along with
variations in the cavity size and shape downwind from a building with differen"
depths of boundary layers in wind tunnels by Jensen and Frank (1965) and
Kalitsky (1968). Tan-atichat and Nagib (1974) and Castro and Robins (1975)
have shewn that the nature, strength, and locations of vertices in the flow
pattern around buildings differs markedly with and without a thick boundary
layer approach flow. The wind sjiear and the presence of the ground produce a
downward flow on the front face of a building, a reverse flow and an increase
in speed upwind of the building, and high winds near the sides as sketched in
Figure 11 (Hunt, 1975). It is now generally agreed that a thick boundary
layer is essential if similar concentration fields are to be observed dcv.n-
wind of a model.
UPWIND
VEU3CITY PROFILE
SEPARATED
FLOW ON ROOF
MEAN VEUXJTV IN
REVERSE DIRECTION
INCREASE IN SPEED
NEAR SIDES
Figure H Effects of wind shear on the flow round
' ' a building (from Hunt, 1975).
-------
Further, not just any thick boundary laver will dc. It must simulate
the atmospheric boundary layer structure, including as a rrinimum, the mean
velocity profile and the intensity and spectral distribution of the turbulence.
That the simulation of the spectrum is essential is evidenced in a report by
Dean (1977). He attempted to duplicate the results of Snyder and Lav/son (1976)
at the somewhat smaller scale of 1:5CC (compared with 1:2CC). The boundary
layer depth was scaled properly and the mean velocity and turbulence profiles
were reasonable facsimilies of those of Snyder and Lawscn (S&L). However,
when neasuHng concentration profiles in the boundary layer downwind from an
isolated stack, he found a vertical plume width over 2 times that of S&L and
a maximum concentration l/9th as large, which was characteristic of Pasquill
diffusion category A, highly unstable. A removal of the vortex generators,
leaving the roughness strips intact, did rot change the velocity cr turbulence
intensity profiles appreciably, but produced a marked change in the energy
spectra, which in turn brought the plume width and maximum concentration to
within a few percent of those of S&L, more nearly characteristic of category D,
neutral stability.
If the atmospheric boundary layer is tc be simulated in a wind tunnel or water-
channel, it is necessary tc decide at seme point just what characteristics car
and should be matched. If adequate data are available describing the atmos-
pheric boundary layer structure for the specific site to be modeled, it is.
of course, more appropriate to use these data. But, generally, sufficient data
are not available, so that some model must be chosen. For example, if we want
to simulate the dispersion of pollutants from a stack in the atmospheric
boundary layer, we need first tc answer the question of what the approach flow
should look like. Hhat is a typical atmospheric boundary layer depth? What
are appropriate parameters that describe atmospheric stability and what are
66
-------
typical values fcr these parameters? How dc the turbulence spectra vary with
stability, height above ground, etc.? It is necessary to establish a goal to
be rret, say, e simplified analytical description, of the flow in the atmos-
pheric boundary layer. The first part of this section (3.2.1) suggests some
goals which the modeler should atterr.pt to achieve. What we attempt to
describe is the tarotropic planetary boundary layer under steady-state and
horizontally homogeneous conditions. The second part (Section 3.2.2} reviews
the most promising techniques that have been tried for generating thick, neutral
boundary layers simulating the atmospheric boundary layer. The third part
(Section 3.2.3) reviews methods which appear promising for simulating strati-
fied boundary layers. Finally, ar attempt is made (Section 3.2.4) to summarize
the previous sections and to establish guidelines for modeling of the atmospheric
boundary layer. Because the discussion of Sections 3.2.1.1 and 3.2.1.3 go into
considerable detail, the disinclined reader may wish to skip to the summaries
(Section 3.2.1.2 and 3.2.1.4) for the essentials.
3.2.1 Characteristics of the Atmospheric Boundary Layer
The atmospheric boundary layer, alternately referred to as the Ekman
layer, the friction layer, or the planetary boundary layer, is concerned with
that portion of the atmosphere where the aerodynamic fricticn due to the motion
of the air relative to the earth's surface is of prime importance. £bove the
boundary layer, the air motion is geostrophic, reflecting a balance between
the horizontal pressure gradient and the CorloMs fcrce, and the velocity
obtained there is the gradient velocity. rhe cepth of the boundary layer is
highly variable, although it is typically between 1/2 and 2 km under neutral
conditions. The overall boundary layer rray be divided into at least 2 sub-
67
-------
layers, principally the surface layer, also misnamed the constant stress layer1,
and a transition region above, wherein the shear stress diminishes from the
nearly constant value in the surface layer tc a near-zero value at the gradient
height. The surface layer is typically 10 to 20 percent of the planetary
boundary layer. Generally, the mean velocity profile in the surface layer is
described by a logarithmic law. Above the surface layer there are numerous
analytical expressions for describing the velocity profile. If the entire
boundary layer is :o be described by one expression, it is common engineering
practice to use a power law.
Since Coriolis forces cannot be modeled in an ordinary type of facility,
i.e., wind tunnel or water channel, modeling efforts should be restricted to
those classes of problems where Coriolis forces are unimportant. As discussed
in Section 2.2.1, Coriolis forces may be important under neutral or stably
stratified conditions in relatively flat terrain when the length of the model
exceeds approximately 5 km. It appears that, if the length scale of the field
situation to be modeled is less than 5 km, Coriolis forces may be ignored.
Likewise, if the terrain is rugged, so that the Tow is highly dominated by
'oca! ('advectice) forces, Ccriolis forces may be ignored. This restricted
class o-' flows limits the usefulness of fluid modeling facilities, but there
s:m exists a very large range of problems in which it is net at all
unreasonable to ignore these effects.
3.2.1.1 The Adiabatic Boundary Layer
Picking a depth for the adiabatic (neutrally stable) boundary layer is nc
s^nle task. After an extensive review of the literature on adiabatic boundary
7ayers, Counihan (1975) concluded that the boundary ^ayer depth is SOCm,
1. Strictly speaking, a constant stress -ayer exists only in a boundary 1
witi zero pressure gradient, wh^'ch is seldom the case for" the a-mo*
, .... _,_-.. ?ayer> - -- ' -
68
-------
practically independent of v;ind speed and surface roughness. Davenport's
(1963) scheme, which was previously accepted as the "standard" for wind tunnel
studies of wind forces on buildings, specified the depth as a function cf the
roughness length z only, varying from 6=3CCm at z = C.C3rr to f=6CCm at ZQ = 3rr.
Another popunar scheme which is claimed tc fit observations quite well is:
where 5 is the boundary layer depth, u* is the friction velocity ( = >' T /p, ) ,
0
and f is the Corio"lis parameter 2u sir. f , where u is the earth's rotation
-4 1
rate and e is the latitude. In mid-latitueds, f ~10 sec . Typical values
for c range from 0.2 (Hanna, 1969) to 0.3 (Tennekes, 1973). Tc determine u*
using this scheme, it is customary to use the geostrophic drag law. which
relates the 'drag coeffient" u*/G tc the surrace Rossty number G/TZ:
p r IX'-P'- ? i T /"
G_ft,i_G ^ r- ' '
co *
2
, i
- B" '"- , (3.;3a)
where G is the cecstrophic wind speed (with ccnponents l'c and V ), k is
von Karmar's constant (0.4), and A and E are "constants'' which differ consider
ably from one author to the next. From Elackadar and Terr.ekes (1968), ,- is
about 1.7 and B about
-------
literature rev lev/ shewed n_c_ measurements of depths In excess of 6COm, if
specific measurements to the contrary are unavailable, the boundary layer
should be assumed to be approximately 6CCm in depth. "This value, as well as
recommendations that follow, are not in any sense to be taken as absolute.
They are recommended in the sense that in the absence of other data, these
values are net unreasonable to use as a model. Also, many of Courihan's
(1975) as opposed to Davenport's (1963) conclusions are repeated here because
they are representative of a wider range of data and they are more thorough
in the sense that mere kinds of statistics are covered.
2SC
Counihar, (1975)
"0*
Fioure 12: The depth of the &diabetic boundary layer ecccrdirg tc the
geostrophic drag lav, compared w'th ether schemes.
7G
-------
The depth of the surface layer, in which the mean velocity profile follows
a logarithmic lav., and from which the roughness length may be defined,
is generally stated to be 10 tc 2C* of the boundary layer depth.
Counihan (197F) suggests a value of 1CCm as a reasonable average depth for
t>e surface layer. The roughness length z may be derived from the mean
velocity profiles in the range 1.Eh i A \
u* k ZK , \j.i4;
* o
where h is the height of the roughness elements, and d is the zero plane
displacement, "he logarithmic profile is well established and has been amply
demonstrated in wind tunnel boundary layers and in atmospheric profile studies
over ideal sites with roughness varying from smooth ice to trees and buildings.
! fact, Thcmpson (1978) has reported logarithmic wind profiles measured over
highly rugged terrain and was able tc extrapolate these profiles to obtain a
roughness length of 25'. The logarithmic law is, in fact, frequently found
to be a close approximation to the velocity profile to heights considerably
beyond the "constant, stress" layer (Panofsk.y, 1974).
Typical values o^ z for various types of surfaces are given in Table 2
(^rom Simiu and Scanlar, 1978). It should be noted that the values quoted in
Table 2 for suburbs, towns, and large cities are exceptionally small. Numerous
authors (see Davenport, 1962, or Hogstrom and Hogstrom, 1978, for example)
suggest values of 1 to 5m for urban areas. Cuchene-Marullaz (1975) suggested
values greater than 2m, but on the assumption that the zero plane displacement
d was zero. Secondly, it is veil known that z ma> vary with wird speed over
the open ocean and over long grass and trees, but Pasquill (1971) cited
evidence of z increasing (as well as d decreasing) with increasing wind speed
-------
over Central London, i.e., large and rigid roughness elements! He suggested
that the cause may be due to the mere vigorous turbulence scouring the
buildings, with the air stream "penetrating" more deeply between buddings,
thereby increasing both the inter-building wind speed and the depth of the
building contributing effectively to the drag.
The zero plane displacement d may generally be negleted for terrain
types where the roughness length is less than about 0.2m. It is suggested
by Simiu and Scan!an (1978) that reasonable values of d in cities may be
estimated using the formula
_ z
d = H - ~Y- ' (3.15)
where F is the general roof-top level and k is the von Karman constant (0.4).
72
-------
TABLE 2: Values of Surface Roughness Length (z ) for Various Types of
Surfaces (from Simiu and Scanlar, 1978).
Type of Surface
2o
/ ^
I en1 ;
Sand
Sea Surface
Snow Surface
Mown Grass (~C.C1m)
Low Grass. Steppe
Fa1 low Field
High Grass
Palmetto
Pine Forest (Fean height of trees: 15m;
s\
one tree per ICmS z , 12m)
Outskirts of Towns, Suburbs
Centers of Towns
Centers of Large Cities
0.01
0.0003a
0.1
0.1
1
o
L.
4
1C
90
20
35
60
0.1
- 0.5b
0.6
1
4
3
10
30
100
40
45
60
aWind speed at 1C m above surface =1.5 rc/sec.
Wind speed at 10 rr< abcve surface > 15 m/sec.
"""he mean velocity profile throughout the entire depth of the bouncary
layer is adequately represented by a power law:
L'/L^ = (z/^)p , (3.16)
where U is the mean velocity at the top of the boundary layer cf depth c ard
-------
p is the power law index. This form is popular in engineering practice and is
highly useful from a practical viewpoint. Davenport (1963) claims that the
overall reliability of the power law is at least as good as much mere sophis-
ticated expressions and it is recommended here for that reason. It was shov/n
by Davenport to work quite well for high geostrophic winds. It should also
work well for light winds as long as the atmosphere is neutral. Even for light
winds in the atmosphere, Reynolds numbers are very large. The problem is not
that the power law will not work for light winds, but that, especially under
light winds, the atmosphere is seldom neutral. Figure 13 shows typical mean
wind profiles and Figure 14 shows the variation of p with the roughness length
ZQ (from Ccunihan, 197E). The power law index varies from about 0.1 in excep-
tionally smooth terrain such as ice to about C.35 in very rough terrain such
as built-up urban areas.
*$ shewn by Counihan, the turbulence intensity at a 3C m elevation follows
the same formula as the power law index; their numerical values as functions of
roughness length are identical.
P = (-^/U)30ir = 0.24 + 0.096 lcglczo + O.C16(loglczc)2 , (3.17)
where ZQ is to be specified in meters. The scatter in the Reynolds stress
measurements was considerable, and -ICOuw/U2 could have been represented by
the identical formula (3.17), but Counihan felt that would underestimate the
stress in moderately rough terrain. Hence, he proposed, for the surface
layer -ITw~/U2 = u2/U2 = 2.75x10"3 + 6x10"4 lcg,nzrt (3J8)
or co I U 0
which is also shown in Figure 14. Counihan does not suggest how uw" varies with
74
-------
I
l ;
n
Z/c
Figure 13: Typical wind profiles ever uniform terrain in neutral flew.
CM
o
o
o
00
CSJ
= 2.75x10 w + 6x10 '
, m
Figure 14: Variation of power lav; index, turbulence intensity, and
Reynolds stress with roughness length in the adiabatic
boundary layer (from Counihan, 197E).
-------
height; he implies that Eq. 3.18 gives its "constant" value in the "constant
stress layer. A convenient approximation is a linear decrease with height from
its surface value to zero at z=6.
- uw(z) =
(3.19)
Thus, at heights less than C.lc, the stress is within 10% cf its surface value
(see Figure 15).
1.0
.8
.6
z_
c
_ 16 m/ttc
0 Station 98T
» 13
0-16
. 19
-22
* 24
l.C
1 .5
-uw/u*
Figure 15. Shear stress distributions measured at various downwind posi
tions in a wind tunnel boundary layer (neutral flew). Data
from Zoric and Sandborn (1972).
76
-------
These figures may be used for model design purposes in a number of ways.
In a general type cf study, such as diffusion ever an urban area, they can be
used directly to determine appropriate values for ZQ and p. Cr, in a specific
study, once U is chosen, on>e has only to determine ZQ (by measurement or
estimation) and, since p, u2/U, and -m7/U2 are principally functions of ZQ>
to obtain them from Figure 14. If it is desired to match the wind speed U, at
a particular height z]4 say, at the top of a stack, Eq. 3.16 may be rewritten
as
Vu~= (zi/f)P
and the free stream wind speed (gradient wind) may be determined.
^he variation of the longitudinal turbulence intensity in. the surface
layer is given by
J7/U = p ln(30/z)/(ln(z/z) . (3.2C)
c
(This formula is slightly different from that of Counihar, but it is consistent
with his data and other formulas. Ccunihan's formulation did not match Eq. 3.17
for the turbulence intensity at z=3C m for low values of ZQ and was somewhat
ambiaucus in the range 0.1 m
-------
Figure 16. Variation of longitudinal turbulence intensity with height
under adiabatic conditions.
decreased with increase of surface roughness and increased with height up to
200-300 m. Above this level, Lu was independent of surface roughness and
A
decreased with height. A summary showing the variation of Lu with elevation
/\
and roughness length is given in Figure 17. For other integral lergth scales,
Counihan has concluded:
Luy = 0.3-0.4 Lux,
Lu = 0.5-0.6 Lu
10m
-------
TERRAIN PRE POST
TYPE 40 60
Figure 17. Variation of integral length scale with height and roughness
length (from Ceunihar, 1975).
nSv(n)/u*
105f/(H33f)5/3 ,
17f/(l+9.5f)5/3 ,
2f/(H5.3f5/3) ,
and -nC
= 14f/(H9 .6f )
2 '
(3.23a)
(3.23b)
(3.23c)
(3.23d)
where f=nz/U is a nondlmensional frequency (see Figure 18 for plots).
It may be seen that these spectral functions are dependent en ZQ insofar
as u* ana I' are functions of z . These expressions may be used to estimate
integral scales (e.g., Kaimal , 1973), but scales thus derivec are not consist-
ent with those in Figure 17 (scales derived from Counihan's suggested spectral
forms are even less consistent with Figure 17 -- such is our knowledge of the
79
-------
CM
W UW
f=nz/U
Figure 18. Empirical curves for spectra and cospectrum ""or neutral
conditions (from Kaimal et al., 1972).
neutral atmospheric boundary layer!) This is very unfortunate, because, as
was pointed out in the previous section, the larger scales cf the turbulence
are highly important in simulating diffusion.
3.2., 1.2 Summary of the Adiabatic Boundary Layer Structure
If specific site data are available giving adequate information on the
structure of the adiabatic boundary layer, it is, of course, most desirable
tc use those data as the target to simulate in the wind tunnel. If not, ss
is usually the case, it is recommended that the following model be used.
For the sake of conciseness, justifications for the particular choices are
80
-------
emitted here. The interested reader may consult the previous section and
the references given there. Listed here are the n:a^r features of the
steady state adiahatic boundary layer ever horizontally homogeneous terrain
'uniform roughness).
1. The depth c of the boundary layer is 6COm, independent of
surface roughness and wind speed.
2. The mean ve^city profile is logarithmic in the surface layer,
which is 100m deep.
3. the roughness length z and the friction velocity u^ may be
derived from the mean velocity profile in the range 1 .Eh ^_z<_0.156
U* Zo
where h is the general height of the roughness elements, and
c: is the displacement height (neglected fcr zt<".2m and giver
by Eq. 3.15 for z
-------
6. The variation of the local longitudinal turbulence intensity
with z and elevation is shown in Fiaure 16. The vertical
o
and lateral turbulence intensities are approximately half
and three-quarters, respectively, of the longitudinal turbu-
lence intensity.
7. The variation of the longitudinal integral length scale with
z and elevation is shown in Figure 17. Other integral scales
o
may be obtained from Eqs. 3.22.
3. Spectral shapes are given by Eqs. 3.23 and shown in Figure 18.
3.2.1.2 The Diabatic Boundary Layer
In many ways, our knowledge of diabatic boundary layers, at least in the
surface layers, is more extensive than that of adiabatic boundary layers.
This is so because diabatic boundary layers are far mere common and because the
change in the surface heat flux is generally slow enough that the surface
layer turbulence is able to track it, i.e., the boundary layer is stationary
long enough that reasonably stable averages are more readily obtainable
(Wyrgaard, 1975). The depth of the boundary layer is highly dependent upon
the stratification. During the day over land, the effective top of the bound-
ary layer may usually be defined as the inversion height, i.e., a layer with
stable density stratification exists at some height that is typically in the
range of 0.5 to 2 km. On a cloudless night with light winds, the ground
cooking generates a strongly stably stratified layer very close to the ground
that suppresses the turbulence; the effective boundary layer, then, may be very
shallow indeed, as low as a few tens of meters or even meters (Businger and
Arya, 1974).
32
-------
It is convenient at this pcirt tc discuss various parameters that charac-
terize the stratification. (This discussion closely follows that of Ousinger,
1?72.) The diabetic surface layer differs ^rcm the neutral o^e, cf course,
because a* the presence cf the heat flux that creates the stratification that
very markedly affects the turbulence structure. TMs is clearly seen by exam-
inaf'cn of the turbulent energy budget equation (see, for example, Euscr, 1972),
where a very important production term appears that is proportional tc the heat
flux. Another production term is, cf course, the mechanical term due tc wind
shear. Richardson (192C) introduced a stability parameter t:*at represented the
ratio, hence, the relative importance of these two production terms:
Ri = , (3.24a)
where r: is meer potential temperature (p=T-tYz, where T is actual temperature
and -f is the ach'abatic decrease of temperature with height). This parameter
is kncwr, as the gradient Picrardson number cr, simply, Richardson number.
In deriving this stability parameter, it was assumed that the eddy transfer
coefficients 'or heat and momentum were equal (Kh=K ). Since this assumption
Js not quite valid, it is better to leave the flux terms ^'n the form in.
which they appear in the energy equation, instead of assuming the fluxes
are proportional to the gradients cf the mean quantities. Hence, \ve have
a flux Richardson number
P1-f = -f- WT (3.24b)
uw jU/~z
where - represents temperature fluctuations.
83
-------
This parameter is rather difficult to determine because of the covariance
terms, >,hereas the determination of Ri involved only the measurement of mean
temperature and mean velocity separately as functions of height.
If we differentiate the expression for the logarithmic velocity profile
(Eq. 3.14), we obtain eU/3z=u+/kz. Substituting this expression into the flux
Pichardson number yields a dimensicnless height
_z_ a WT kz
L = " T 3
Where L = "T- ~h= (3.24c)
is the Monin-0bukhcv (^-0) length. This length is a very useful stability
parameter. It contains only constants and fluxes that are approximately con-
stant throughout the surface layer (also called the constant: flux layer,
analogous to the corstant stress layer in the neutral boundary layer). L there-
""ore is a characteristic height that determines the structure of the surface
ayer. It has been found that many features of the turbulence in the surface
layer depend solely upon the dimersionless height z/L. Such dependence is
referred to as f'-C similarity, which we will return to later in this section.
Another stability parameter is the Ekman-layer equivalent of z/L, i.e.,
it governs diabetic scaling in the entire boundary layer, much as z/L governs
diabat1':. scaling ^'n the surface layer (Tennekes, 1973).
84
-------
A final stability parameter is the Froude number
Fr = -
(3.24e)
which was discussed in Section 2.2.4. The Frcude number might characterize the
stratification in the surface layer or that of the entire boundary layer,
depending upon the height H chosen for specifying the velocity and the upper
level for the temperature difference, fore common in the meteorological liter-
ature is the inverse square of this Frcude number, which is called the bulk
Richardson number
RiD =
^ eHU
To give the reader a "feel" for the magnitudes cf these various stability
parameters, we have listed typical values in Table 3. These values are not to
be taken as definitions or as absolute in any sense. Particular values depend
en the height chosen for specification of the v«ird speed and temperature and
there is not, in any event, a one to one correspondence between the parameters.
With these stability parameters in hand, we will be able to specify many
of the features cf the diabatic boundary layer (albeit one that is steady and
horizontally homogeneous). Let us first quantify cur rather qualitative des-
cription earlier in this section of the boundary layer depth.
According to Hanna (1969), the formula
0.75 U
r .3 / - ,-x r- \
where 15/iz is the average vertical gradient of potential temperature through
the boundary layer, agrees well with observed boundary layer thicknesses.
/-.
This equation implies that the bulk Richardson number gfie/jl^) equals C.56
85
-------
fl
h
B
C
D
£
F
-5
-10
-20
CO
100
20
o
-1
-0.5
0
C.I
0.5
-5
-2
-1
0
0.07
0.14
r*
"L.
1
~ i
-0.5
0
0.07
0.14
-0.03
-0.02
-0.01
n,
0.004
C.02
- -4000
- -120
-60
0
16 12
7 40
Table 3: Typical Values for the Various Stability Parameters.
Qualitative Pasquill-Gifford L,m z/l Pir Pi Rip Fr
Description Category
Hiohly
Unstable A -5 -2 -5 -2 -0.03 - -4000 .3
Unstable B -10 -1 -2 -1 -0.02 - -120 .3
Slightly
Unstable
Neutral
Slightly
Stable
Stable F 20 0.5 0.14 0.14 C.02 7 40 .2
Highly
Stable G 10 1 C.17 0.17 C.C4 5 <1Q .1
(a) The assumed height cf the anemometer and upper thermometer was 1C m;
the lower thermometer: 2 n. A roughness length cf C.C1 m was also
assumed in the calculations.
(b) 'he friction velocity listed is that value used in ca1culating u.
for all stable boundary layers, i.e.. the boundary layer adjusts itself until
this criterion is met.
Arya (1977), to the contrary, claims that observations indicate that the
bulk Richardson number increases with increasing stratification and may
approach a constant (critical) value only under extremely stable conditions.
He, Businger and Arya (1974), Kyrgaard (1975), Erost and Wyngaard (1978) and
others using widely differing theoretical approaches all arrive at the simple
form for the height where the stress is some specified small fraction of the
surface value:
1/2
/L = au* or £= a(Lu*/f ) (3.25b)
86
-------
where a is a constant and p*=u*/fcL is a stability parameter rented to
Eq. 3.24d through i:=kv*. The constant a, however, is highly dependent upon the
value chosen for the stress criterion. For a V,'. stress criterion, Businger and
Arya (1974) find a=0.72, whereas for 5?, a^0.4. The latter value is supported
by the second order closure model of Brost and Kyr.gaard (1978) ever a wide
range of cooling rates. Comparisons with Wangara data (Arya, 1977) show very
large scatter, and that a=l would be a much better fit.
To estimate u*, the geostrophic drag relation (Eq. 3.13) is used, where
the "constants" A and B are functions of the stability parameters. In a
critical review, Arya (1977) has suggested
A = ln(c:/L) - 0.96(6/L) + 2.5 (3.26a)
and B = 1.156/L + 1.1 , (3.26b)
where i *s determined from Eq. 3.25b with a=l. (This is obviously an itera-
tive procedure in that Eqs. 3.13, 3.25, and 3.26 all involve u*, which we are
attempting to determine.)
Mr.ally, an interpolation formula suggested by Ceardcrff (1972), i.e.,
'ix f x i H
6 = !_l_ + _s_+ i_j (3_2r
in which ZT is the height of the trcpopause, does not suffer from "blowing up
under neutral conditions (where L-«) near the equator (where f^C);
6-»0.25u*/f under neutral conditions in mid-latitudes (i.e., Eq. 3.12) and
5-30L under very stable conditions and/or in lew latitudes. Eq. 3.27 yields
results comparable to Eq. 3.25b in mid-latitudes (see Figure IS).
The unstable boundary layer is almost always capped by an inversion at
some elevation. It is now generally agreed that the height of this boundary
87
-------
layer is determined by the height of the base of the inversion, i.e., S=z,.
(Deardcrff, 1972, Wyngaard et a 1., 1974). The height of the inversion, base
varies from day to day, but its diurnal trend is quite similar. Kain.al et al.
(1976) describe their observations in the Minnesota experiments as follows:
"2etv,'een sunrise and local noon (1300 GDI) z. crew rapidly in response
to the steadily increasing heat flux (Q ). The growth of z. slowed down
between 13CC and 1600 CDT as 0 reached°its maxirrum value. ^But as Q
decreased through the late afternoon, z. began to level off to a neaf'ly
constant value which it maintained even after Q turned negative."
Even though this connective boundary layer depth changes rather rapidly
with tire, there is justification for treating its midday structure as if it
were in steady state, or at least in a condition of moving equilibrium or
quasi-steady state (Kainal et al., 1976). To predict the height of this
boundary layer, Ceardorff (1974) and Arya (1977) recommended a rate equation.
Fcr purposes cf fluid modeling, it is sufficient to pick typical values,
i.e., ;>il to 2 km, as the typical rrax-'mum height for the inversion base is
1 to 1 km. Cnce a boundary layer height is chosen, we car estimate '* from
the geostrophic drag relation, Eq. 3.13, where the parameters A and P are
f>:rcticns o^ the stability parameters 6/L and f c./u* (Arya, 1977).
\,'
A = ln(-c/L) + lp(fcS/O + 1.5 (3.2'Sa)
E = k(f5/uj~' + 1.8(ff/u*) exp(0.2f/l.) . (3.28b)
Figure 19 shows predicted boundary layer depths from Eqs. 3.25b and 3.27. It
may be seen why the neutral boundary layer depth is so difficult to determine;
only slight departures from neutrality effect drastic changes in its depth.
F1'cure 20 shows how the friction velocity u* varies vsith stability es predicted
88
-------
!!~ l'/'
"'''' f /
:!" i'/'
/ \
/ ' \
/ / /\
/ i
5 ,m
"\
6=(lu*/fcJ
.
1/2
3.27
, m
-1
Figure 19. Typical nonadiabatic boundary layer depths fr:- the -jc-ostrophic
-4
drag relations (G=10m/s, z =0.01m,zTlCkm, fc*i0" /s, a-1).
1/L , m
-1
Figure 20.
Variation cf friction velocity vvith stability from the
geostrophic drag relations (Eqs. 3.13, 3.26, 3.28, G=lGrr./s,
z =.01m, zT=1
W I
39
-------
from the geostrophic drag relations using Eqs. 3.26 and 3.28.
Regarding the mean wind profile under non-neutral conditions, DeMarrais
(195S) has measured the power law index p and has drawn the following general
conclusions:
"During the day, v.nen superadiabatic conditions and neutral lapse rates
prevail, the values of p vary from C.I to C.3. This variation is princi-
pally in proportion to the roughness of the terrain. At night, when
stable, isothermal, and inversion conditions exist, the value of p
generally varies from 0.2 to 0.8; this variation is proportional to the
degree of stability and the roughness of the underlying terrain."
Panofsky et al. (I960) have used a formula due to Ellison (1957) to derive
a theoretical relationship for p as a function of z and 1/L (L is the f-'-O
length). Irvvin (1978a) has, analogously to Pancfsky et al., used results of
Nickerson and Sr^'ley (1975) to establish a theoretical relationship between p,
2 and 1/L. The results, shown in Figure 21, support DeMarra^s' (1959) con-
C>
elusions reasonably well.
Air pollution meteorologists frequently use Pasquili stability classes
(cr similar groupings) to categorize atmospheric diffusion. Golder (1972) has
related the qualitative Pasquill classes to more definitive measures of sta-
bility through analysis of a large number of observations at 5 sites. Irv/in
(1975a) has taken Golder's results relating Pasquill classes to the
'"onin-Obukhov length and roughness length and overlaid them as shown in
Figure 21. Irvn'n (1978b) has further plotted the variation of p with z
where the Pasquill stability class is a parameter (see Figure 22). It may
be seen from Figures 21 and 22 that the shape of the wind profile is much more
strongly dependent on stability than on the roughness length under stable ccn-
-------
Q.001
0.06
0.02
-0.02
-0.06
-0.10
-0.14
Stability Length ,1/L (m"1)
Figure 21. Theoretical variation of the power-law exponent as a functicn
of 2 and L for z equal to lOCm. The dashed curves cverplct-
o
ted are the limits defined by Gclder (1972) of the Pasquill
stability classes as adapted by Turner (196*) (from Irwin,
1978a) .
ditions. It is relatively insensitive to stability but mere dependent upon
roughness under unstable conditions. Comparisons (and tailoring) of Irv/in's
results with field data of CeKarrais (1959), Touma (1977) and Izumi (1971)
agreed well and explained reported differences in exponent values. His theo-
retical predictions compare very well with Counihan's (1975) results for
neutral conditions, i.e., stability class D (see Figure 22).
In the above discussions, we have largely icncred the influence of the
earth's rotation because this feature, in general, cannot be simulated reclis-
91
-------
2
z 0.6
a
a.
X
cs
UJ
3
a
Q.4
«s
cc
(3 =
10.0
100.0
2.0
0.01
SURFACE ROUGHNESS LENGTH, zn. meterc
rigure 22. Variation cf the power-law exponent p, averaged over layer
from 10m to 100m, as a function cf surface roughness and
Pasquill stability class. Dashed curve is result suggested
by Counihar. (1975) for adiabatic conditions which should
agree with stability class D. (from Irwin, 1278b).
tically in laboratory facilities in any event. On the other hand, many fea-
tures cf the surface layer can be well simulated. Panofsky (1974) has suggested
that we further subdivide the Ekman layer (overal1 boundary layer) into a
tower layer, i.e., below 15Cm o>* so in neutral or unstable conditions. The
surface layer proper extends to approximately 30m, but nany of the relation-
ships developed for the surface layer may be extended to the tower layer;
whereas the earth's rotation may be important in the tower layer (it was not
in the surface layer), the turning of the wind can be ignored. In stable air,
this subdivision- is useless because significant turning may start at much
-------
lower heights. In the discussion to follow, then, we will discuss in detail
various properties of the surface layer that may possibly be extended tc the
tower layer in neutral and unstable conditions.
It is customary in discussing surface layer profiles and fluxes to define
nondimensional vertical gradients of wind speed and potential temperature as
where CV--WT/U*. It has been found that these nondimensicnal gradients are
functions of z/L only (M-0 similarity). In unstable air,
2/L<0 (3.3C)
j
fits surface observations quite well (Fanofsky, ^97^). The expression can be
integrated to obtain the mean velocity profile (Paulson, 1970}:
U/u* = (l/k)lr(z/z_) - 2
n *
#j
2 tan"1(l/$m) - T/2j
This formulation is consistent in that under neutral conditions L-~,
and Eq. 3.31 reduces to the familiar leg law. Panofsky (197<) showed that
G
integrates to the familiar log-linear hind profile
93
-------
= n/k)[ln z/z0+Ez/L], (3.33)
Figure 22 shows topical velocity profiles as predicted by Eqs. 3.31 arc 3.33.
"he behavior cf the nondimensional temperature gradient «h is sorrewhat
controversial. For simplicity we will here list the forms given by Panofsky
(:974)
th = (l-15z/L)"1/2 , for z/L<0 , (3.3^a)
ana G . (3.34b)
These expressions integrate to
(8-6 )/9* = In (z/zj - 2 ln.[(Ul/O/2], for i/LC , (3.25b)
0 *-*
i here 9«is the extrapolated temperature for z«z (not necessarily the actual
0 ^
surface temperature). Typical temperature profiles as predicted by Eqs. 3.35
are shown in Figure 24. It is useful -n interpreting Figure 24 to note that
s* and (e - e ) change sign simultaneously, so that the slopes of the curves
are always positive. It is also interesting to note that the limit as L-*» is
the same as the limit as L~°°, i.e., a logarithmic temperature distribution,
which is not the same as adiabatic, where the potential temperature would be
uniform with height. This is an anomaly ir the mathematics, because both
numerator and denominator of the left hand sides of Eqs. 3.35 approach zero
simultaneously as the surface temperature approaches the fluid temperature.
Another useful relationship is that between the gradient Richardson
number and z/L:
(3.36a)
-------
1
r
Figure 23. Typical surface layer velocity profiles under nor.adiabatic
conditions (from Eqs, 3.31 and 3.33 with z =0.01m).
1000C
z/z.
(e -
Figure 24. Typical temperature profiles ir the surface 'eyer (from EQS.
3.35 with z =O.Clm).
95
-------
or
Ri = z/L , for z/L , for z/L>0
(3.26c)
"his relationship is shown in Figure 25.
Variances
The variance of the vertical velocity a =
w >
scaling, so that
w follows Monin-Gbukhcv
!.37a'
where o is a universal function. According to Fanofsky (1970
.25 for z/L>-C.3(inclucing all z/L>0)
]/* for
(3.37b)
fhe variance of temperature also follows M-0 similarity, so that
ay = 6*4p(Z/L),
(3.38a)
95
-------
where 38b)
1.8 for z/UC
r and
-------
Ri
-3 -D. £ -O.i D. 4 - C. ._
z/L
Figure 25. The relationship between Ri and z/L (Eqs. 3.36).
w
\ _____
z/L
Figure 26. Variation of *w and &Q with z/L in the surface layer
(Eqs. 3.37 and 3.38).
98
-------
Eirkowski (1973) has derived expressions for $u and. 0) The universal spectral shape is shown in
Figure 29, and the variation of the peak frequency with z/L is shown ir
Figure 3C.
The integral scales are difficult to evaluate directly frcir the spectra;
99
-------
K« KRNSR8 DRTH
«« niNMESOTR OUT*
Figure 27. Variation of « with z/L in the surface layer (from Einkowski,
1979). u
X. KRMSflS
+. MINNESOTA DBTfl
-4
Figure 28. Variation of $v with z/L (from Binkcwski, 1279).
100
-------
TOT
[B
'(2Z6L
UMOUS
[C. e pue M 'A 'n aoj. >)sad
*t>3) adeq?
'62
f
I
-------
a length scale that can be obtained directly is A , the wavelength corres-
ponding to the peak in the logarithmic spectrum n,S(n). Using Taylor's
hypothesis
where nn) and f are the cyclic and reduced frequencies at the spectral peaks.
This length scale is used extensively (as opposed to the integral scale) in
the interpretation cf atmospheric spectra. Kairnal (1973) has derived a sitrple
expression relating these two length scales
His findings for the variation of these length scales with Richardson number
are listed in Table 4.
Table 4. Dimenslonless Length Scaj_es_ as_ Functions cjf R;i (0.05
-------
v.ith ether empirical relationships. No expressions are available for the
variation, of x in. unstable conditions, but values fcr x (w) arc ) (e) may be
m '" "'
deduced from Figure 3C.
Little is known about the variation of x with roughness. Wamser and
Fuller (1977) noted that their data showed a decrease in x (;;) with increasing
roughness under neutral and ccnvective conditions, but could not drav; any con-
clusions for stable conditions. They also noted that there was no systematic
dependence of X (u) on roughness. Higher order statistics such as ccspectra
end structure parameters are beyond the scope of this rev'ew. The interested
^
reader is referred to V.'yngaard and Cote (1S72) and ;-.'yngaard et a 1. (IS?!).
Above the surface layer, the turning of wind with height generally beccrres
highly important ard, therefore, is net amenable to simulation in a laboratory
^acility. But one case, in fact one that is fairly typical of daytime convec-
tive conditions, deserves mention. Kairnal et al. (1976) describe the structure
of this "mixed layer" as obtained from their extensive measurements in Minne-
sota. The surface layer is as described above, but is confined to the height
range z<|L|. Immediately above the surface layer, they describe a "free con-
vection" layer, where the surface shear stress is no longer important, but the
height z continues to be important. The upper level for this free convection
layer is approximately P.lz^, where z, is the height of the base of the lowest
inversion, and is also a good measure of the boundary layer depth (typically
1 to 2 km). The remaining 9/10 of the boundary layer, then, is the "nixec
layer" \^here the mean wind is essentially uniform and the v.ird cirection
changes V;ttle witu height. In the "worst case1' run, the wire direction,
varied by only 15° between the surface arc the top cf the boundary layei:
it was typically only a few degrees.
103
-------
It is conceivable that the entire depth of this corvective boundary
layer could be simulated in a laboratory facility, albeit at very low Reynolds
runber. Deardorff and Willis (1974) have done the limiting case of pure con-
vect-icn (no wind) and Schon et al. (1974) have done an unstable boundary layer,
but without a capping inversion. That the two approaches can be merged appears
promising.
For details cf the boundary layer structure (variances, scaling, spectra,
etc.). the reader is referred to the papers by Kaimal et a1. (1976), Kaimal
(197£), and Panofsky et al. (1977). The latter authors show, for exarrple, by-
using observations from several data sets over uniform surfaces, that « and $v
depend not upon z/L, but instead upon z,./L. Also, there were no significant
differences between the lateral and longitudinal components. Their expression
fitted to the horizontal velocity data is
Q = 02-Q.5z./L)1/3 , -400z,0
and a transition region (z
-------
Hy, vvhere z.. was the sole-governing length scale, applied. Interpolation
formulas for the transition region were derived. Further, it was shown how
these surface layer spectra (including w) evolve with height into their mixed
layer forms. As the empirical expressions are complicated and of somewhat
limited applicability, the interested reader is referred to the original paper.
3.2.1.4 Summary of the Ciabatic Boundary Layer Structure
Listed here are the main features of the steady-state diabatic boundary
layer ever horizontally homogeneous terrain. Again, if specific site data
are available giving, for example, typical strongly stable characteristics of
the boundary layer, it is, of course, most desirable tc use those data as a
target to simulate.
1. The depth of the stable boundary layer may be estimated from
Eq. 2.27, where the friction velocity u* is obtained from the
geostrophic drag law (Eq. 3.13), and the "constants'1 A and B
are determined from Eqs. 3.26 (an iterative procedure). It is
typically 100m deep. The unstable boundary layer undergoes a
diurnal trend with a typical maximum depth between 1 and 2 km.
2. Cnce the boundary layer depth is chosen, the friction velocity is
obtained from the geostrophic drag relation (Eq. 3.13), where the
"constants" A and B are obtained from Eqs. 3.26 for stable
conditions and Eqs. 3.28 for unstable conditions (again, an
iterative procedure). Typically, u*=C.05Uro in unstable conditions and
u*=O.C2U in stable conditions.
w 00
3. The power law exponent p characterizing ,.,c shape of the rrean
velocity profile over the depth of the boundary layer may be
obtained from Figure 21 or 22. In unstable conditions, it is
105
-------
dependent primarily on the roughness length and essentially
independent of the degree of instability, varying in the range
of 0.1 to C.2. Under stable conditions, it is highly dependent
upon the degree of stability and essentially independent of the
surface roughness, varying in the range of 0.2 to 0.8.
4. In neutral and unstable conditions, the surface layer properties
may be extended to a depth of approximately 150m. In stable
conditions, the surface layer is only 1C to 20 m in depth. The
Kcnin-Obukhov length L is currently the most popular stability
parameter because most of the surface layer properties car be
described solely in terms of the dimensicnless height z/L (M-0
similarity theory). Given L and u*, we can predict the shapes
of the mean velocity profile (Eqs. 3.3C, 3.31 and 3.33), the mean
temperature profile (Eqs. 3.34 and 3.35), the variance of vertical
velocities (Eqs. 3.37), the variance of temperature (Eqs. 3.38),
and to a rough approximation, the variances of the lateral and
longitudinal velocities (Eqs. 3.39 and Figs. 27 and 28). foe can
also predict spectral shapes (Eq. 3.40) and scales (Eq. 3.41 through
3.45, Fig. 29 and Table 4).
5. Little is krcv.>n of the boundary layer characteristics above the
surface layer except that generally the turning of the wird
with height is important. Flow above the surface layer is thus
not usually amenable to simulation in a laboratory facility.
One special case, however, is the convective boundary layer. It
appears that this entire boundary layer could be simulated in a
laboratory facility as the change in wind direction with height
106
-------
is typ-'cally only a fevv degrees over its typically 1 km
depth. For additional details, the reader is referred to the
original papers.
We have seen ir our review of the a tire spheric boundary layer that it is ever
changing, it is governed by a large number of parameters, and that its space-
time characteristics are difficult to determine. Even the specification of one
of the "sirrplest" characteristics, its depth, is a horrendous problem. We
have attempted to assimilate the results of the most recent theories, but
they continue tc develop and are rapidly modified as new experimental results
become available. Even the classical "universal" von Karman constant is
questioned (Tennekes and Lumley, 1972). There are few generic boundary
layers to emulate or to compare with our wind tunnel simulations. Neverthe-
less, we have classified typical types and have described the salient
characteristics o^ those classical types as they are known at the present
t i me.
3.2.2 Simulating the Adjajbatjc_ Boundary L^yer
!r the previous sections, we have established at least the main character-
istics cf the adiabatic and nonadiabatic atmospheric boundary layer. In this
section, we will examine severe! techniques commonly used to simulate the
neutral atmospheric boundary ^ayer and note, where possible, how successful
these techniques have been. Generally, such techniques have been applied
only in wind tunnels although, in principle, they could also be used in water
tunnels and tow'ng tanks.
107
-------
The techniques can be broadly divided into three categories:
1. Long tunnels, in v/hich a thick boundary layer develops naturally
over a rough floor (Figure 31}. The length of the test section
of such a tunnel is typically 30 m.
2. Short tunnels with passive devices, in which the boundary layer is
generated by a fence, screens or grids of non-uniform spacing,
spires or vortex generators, i.e., stationary devices that retard
the mean flow close to the floor and induce vorticity and turbu-
lence into the boundary layer (Figure 32). In order to maintain a
ron-developing boundary layer, it is essential to "match" the
generators with the roughness elements.
3. Short tunnels with active devices, in which the boundary layer is
generated by jets directed at angles to the main flow stream at
the entrance to the test section (Figure 33). Again, ''matched"
roughness must line the floor of the test section to obtain a
non-developing boundary layer.
Subcategories might include tunnels equipped with rr.achine-ariven shutters
or flaps or possibly a program-driven variable speed fan. In short tunnels
with active devices, it is claimed to be possible, within limits, tc vary the
turbulence structure independently of the mean velocity profile, but it is
not clear that boundary layers with different properties can be in
equilibrium with the same surface roughness.
Initially, the long tunnels were touted as superior to short ones
with devices for artificially thickening the boundary layers because in them
the boundary layers were developed "naturally': over rough ground. The long
103
-------
'~*:V^^>-V~''~ i^
'.'^~" '-'..* V^.^-^-^rV^--: v"
'-"^35^*- ^>-*f x^*-^- -* ^.^.^^i,":-^--!
; --'^^SS^-s^:-r^-v^-- ^^?^-".- -^:x-^ i
^ _ -j^«--p_-- ^ 7^~^_ '^s^"^- -^-**^"*' ~-*-*" "T^-** - --' *J- , -^-J
fc;: 5!r,2 *^»-» ^I^^r^v^^t^1 v^'-T^^*^^^!
Figure 31. Upstream view of a Icng wind tunnel (courtesy
Boundary Layer Wind Tunnel Laboratory, the
University cf Western Ontario).
tunnel advocates felt that the grids, jets and vortex generators introduced
extraneous turbulence scales and the turbulence dissipated and its structure
changed with downstream distance (Cermak and Arya, 1970). The short tunnel
enthusiasts, on the other hand, pointed to the developing boundary layer and
to the secondary flows caused by the growing sicewall boundary layers as not
representing steady and horizontally homogeneous atmospheric bouncery
layers (Nagib et al., 197^). Recently, however, dreg-producing elements
have been used in the long tunnels as well, and many techniques for generating
109
-------
Figtre 32. Vortex generators and roughness In a short wind turnel
(Ccurtesy of Warren Spring Laboratory, England)
no
-------
JOD
FREE-STREAM
ENTERING
TEST SECTION
_ _ I -
1
SECTION A-A , * -RECIRCULATING
- X s FLOW REGION
~">l '- "^' ' r '~^/ /// A////'// ///s\QE /\
VIEW
'- THICKNESS OF
GENERATED
fo q COMPRESSED BOUNDARY LAYER
4 AIR SUPPLY
^yi -- ^COUNTER -JETS L. SURFACE ROUGHNESS
/ / / /'PLAN /.
VIEW
Figure 33. Scherratic representation of the counter-jet technique (from
flagib et al., 197^).
aepti" of DOunaary iaver ove< 0 cafDCT 'i^ 5 003'-^
AoDroxtmate aeotn o' Boundary aver over rectangular DIOCK
(2 5 to 10 cm high z0 ^ 2 5 err-
^ L/i Note Lfs 'S tie tree stream or
undisturoed velocity
^ nl npfln^n-
In
n f] H
20 15 1C 5
Distance from leading eag« of rougrmess '
- Bell
Figure 34. Development of boundary layer in a long wind tunnel (flft*
A.G. Davenport and N. Isyumov, 1968).
Ill
-------
thick boundary layers in short lengths of test section have been
developed. There is no reason, in principle, why a fully cevelcped layer
with unchanging turbulence properties cannot be achieved ip short tunnels.
Ar experimentalist must only be clever (or lucky!) enough to determine the
proper size, number and arrangement of grids, vortex generators, jets,
roughness, etc. to obtain such. Also, some development length is required;
current practice indicates that an equilibrium boundary layer may be
established in 5 to 1C boundary layer heights, d substantial improvement
over the long tunnels. It is not the aim here to favor one system
over another, but instead to stress the necessity cf adequately measuring the
boundary layer, however generated, to be sure that it is laterally hcrrogeneous
and non-developing (if that is what is desired) and that it meets the target
flew characteristics (which are not, in all cases, of course, these of the
atmospheric boundary layer).
Examples of the first category, "long tunnels, are the Micro-Meteorological
Wind Tunnel at the Colorado State University (CSU) (Plate and Cermak, 1962)
and the Boundary Layer Wind Tunnel at the University cf Western Cntar-'c (I'WC)
(Davenport and Isyumov, 1?tc;. Figure 34 illustrates the development of the
bcuncary layer in a long tunnel and how the depth of the boundary layer depends
upon the roughness. Because of the growth of the boundary layers, these
tunnels generally have adjustable ceilings to control the axial pressure
distribution, and the ceiling is adjusted to give a zero pressure gradient
along the length of the test section.
Sandborn and Marshall (1965) were the first tc show that the turbulence in the
boundary layer of the CSU long tunne1 exhibited characteristics of the
Kolmogoroff local isotrcpy pred-ctions. i.e., a large separation between the
112
-------
integral scale and the microscale (see Section 2.2.2.2 and Figure 6) which is
characteristic of large Reynolds number turbulence, and is responsible for the
-5/3 power in the spectral equations (3.23). Their measurements were irade
over a coarse gravel floor approximately 20 m from the test section
entrance with a free stream wind speed of approximately 10 m/sec and boundary
layer thickness of about 60cm . Whereas this feature is necessary for
simulation of wind forces on buildings (see Simiu and Scanlan, 1978), it
is regarded as relatively unimportant (but certainly not harmful) for
diffusion studies (See Section 2.2.2.2). The results do indicate, however,
that the flow Reynolds number may be reduced somewhat without reducing the
total turbulent energy or shifting the location of the peak in the energy
spectrum significantly (See Figure 7).
7oric and Sandborn (1972) have shown that profiles of mean velocity
nondimensionalized by boundary layer depth are similar beyond 6 m from the
entrance (CSU tunnel). Figure 35 shows that they are approximated quite
well by a l/7th power law. Figure 36, however, shews that the boundary layer
grows nearly linearly and still cuite rapidly with downstream distance beyond
about 10 m. Zoric (1968) obtained results similar to those of Figures 35
and 36 for freestrean velocities between 18 and 30 m/s. Turbulence
profiles were strikingly similar in shape to those suggested by Counihan
(1975) for very small roughness (see Figure 16). The boundary layers
were developed over the smooth wind tunnel floor. Vertical turbulence
1. In fact, under these flow conditions, the turbulent Reynolds number,
based en eddy velocity and eddy size, may be estimated to be at most 2CCO.
Tennekes and Lumley (1972) suggest a bare minimum value for an inertial
subranae (local isotrcpy) to exist is 4000. Even thouah a substantial
spectral region with a -5/3 slope was measured, it is doubtful that joca
isotropy existed, i.e., the existence cf the -5/3 slope is not a cr.tica
I test
of local isotropy.
11
-------
_u
Uoo
08
06
04
0 ?
0
f
f /
a7c
ff°
f
_
i
f
i
?
Q
O
0
I i r ~~ i
i
' ' J-"" A "" 1
0 ; 1
Symbol S'o'ion
1 , 0 3l(m)
A 61
\ 091
' 12
0 |5
18
21
* < 24
2 04 06
i
.- - - 1
; l
U« (m/sec)
18 84
18 70
18 62
18 87
18 87
1850
18 55
1858
08 10
r/S
Figure 35. Pevelcpment of mean velccity profiles along the smooth
floor of a long tunnel (from Zoric and Sandborn, 1972),
30
Figure 36. Thickness parameters for boundary layer of Figure 35 (frcrr,
Zoric and Sandborn, 1972).
-------
intensities were about 50% cf the longitudinal intensities clcse to the
ground in accordance v.-ith Eq. 3.21, but were typically 7C% over the
upper 9C2' of the boundary layer depth.
Surprisingly little has been published along the lines of neutral
boundary layer turbulence characteristics over a rough floor that would
show, for example, that the flow was laterally homogeneous or hew it
wculd compare as a small scale model of the neutral atmospheric boundary
layer. Evidently, detailed basic and systematic studies cf the
turbulence structure in neutral boundary layers have rot been made in the
long tunnels.
Additionally, the above measurements were made at wine tunnel speeds
much in excess of those allowable for modeling buoyant plumes. Isyumov,
et al. (1976), for example, suggest typical tunnel wind speeds of C.5 to
C.7 m/s. They do present one spectrum, reproduced here as Figure 37, that
shows the rapid decrease of energy at frequencies in excess of the location
cf the spectral peak. Also, a relatively large amount ox energy at lower
freouencies is rather surprising in that significant energy in this part
of the spectrum is not generally produced in wind tunnels. Keasurements
of the spectrum cf lateral velocities would ascertain whether this energy
is, in fact, due to turbulence or whether it is l!pseudo-turbulercer, i.e.,
one-dimensional fluctuations caused, for example, by low frequency
oscillations in fan speed.
Low speed boundary layer development characteristics in short
tunnels with passive devices are much better documented. The most
popular of the passive devices is the barrier/vertex generator/roughness
system developed by Counihar (1969). It has been adopted with minor
variations at numerous laboratories. Castro et al. (1975) have made
115
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He
ent
= 400
ft !
U^
or
01 02
Estimated spectral
points from wind :
tunnel measurements
Davenport's spectrum
for neutral stability i
05
.1
1° 1
Figure 37. Spectrum of the longitudinal
(from Isyumov et. at., 1976).
component of velocity
116
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extensive measurements of a Lndeep boundary layer developed using
Counlhan's system, primarily for study of dispersion of chimney
emissions In neutral flcv;. They found that the turbulence in the
boundary layer reached a near-equilibrium state in approximately lh
generator heights (boundary layer heights) downstream. They were able
to draw this conclusion because they had measured the various terms
in the turbulent energy budget, unfortunately an all tco rare measurement.
Their conclusions were that the boundary layer thus developed:
1. Had characteristics similar to those of a suburban (or
somewhat rougher) layer of 600 m depth and roughness length
of 1.3 m at a scale ratio of 1:300, and that departures
from ecuilibrium were unimportant beyond about 5 bouncary
layer heights. The lower 10 to 20% of the boundary layer
could be used beyond about 3 H 6.
2. Was Reynolds number independent for free stream wind speeds
in the range of C.7 to 13 m/s.
3. Appeared to be unaffected by the proximity of the wind tunnel
ceiling to the top of the boundary layer.1 This conclusion
was drawn by comparing interim ttercy distributions with those
of "natural" wind tunnel boundary layers.
Moreover, Fobins (1978), shewed that dispersion in the above wind tunnel
boundary layer as well as that in a simulated rural boundary layer was a
reasonable model of the full scale process, i.e., it produced concentration
patterns approximating Fasquill category C-D atmospheric flows (slightly
unstable tc neutral), which is normal for such a large roughness length.
1. The present author notes that in some unpublished work, boundary layers
developed using Ccunihan's system were found to be very much dependent upon
the proximity cf the ceiling, i.e., when the ceiling was several times the
height or the vortex generators, even the mean velocity profiles differed
drastically from his. The ceiling thus appears essential; indeed, it is an
integral part cf the simulation system.
117
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Somewhat less-well-documented techniques include the '"spires" of
Tempi in (1969) (also cuite popular), the "fence" cf Ljdwig et al. (1971),
and the "coffee cups" of Cock (1973). Earlier methods employing graded
blockage, grids of rods or slats, etc., have been largely superseded in
the Western Hemisphere, but are still quite popular in Japan (Sato,
et al., 1974; Cgawa, 1976). Hunt and Fernholz (197E) provided a list of
wind tunnels (largely European) used for atmospheric boundary "v.s- :i;--.:-_
tions and included characteristics of the wind tunnels and relevant
measurements of such boundary layers, so that some comparisons of the
different techniques ray be made.
Short test sections with active devices are a1 so numerous. Sch.cn and
Mery ]-?:i injected air perpendicular to the flew through a porous plate at
the entrance to the test section. This system may be thought of as a fluid-
mechanical fer.ce, where the fence "height'1 is adjusted by varying the
strength cf injected air, but it has the additional potential for injecting
gas of different density in order to quickly establish a non-neutral density
profile. They have shown that this technique can produce a boundary la^er
tw'ce as thick as the ''natural" one over a smccth floor and that its charac-
ter-'stics are essentially sirrilar. However, this system appears to recuire
a rather long development length compared with Counihan's system (Hunt
and Fernholz, 1975). Also, because cf the smooth floor, turbulence
intensities were somewhat lower than those in even a mildly rough field
surface. Mery et al. (1974) have shown that this technique produced dis-
persion patterns similar tc the Brookhaven experiments (Smith and Singer,
195E), but only after "adjusting1 their a 's by a 'actor of 2 to account
J
"or an equivalent wind tunnel averaging t^'me (converted to full scale) cf
-------
3 minutes compared with atmospheric averaging timescf 1 hour. This
adjustnent technique, however, appears somewhat arbitrary. The small
values of the c 's in the wind tunnel were, in the present author's:
opinion, most likely due to the small turbulence intensities as well
as to the lack of large scale lateral fluctuations in velocity.
fiagib et al. (1974) have added some flexibility to the Schcn et al.
technique, in that the injected air is input through a line cf holes in a
pipe perpendicular tc the flcu on the fleer at the entrance tc the test
se:;i.:;., The pipe may be rotated (See Figure 33} to change the jet
injection angle and the jet speed may be varied; these, cf course, change
the boundary layer characteristics. The ''counter-jet" technique, it is
claimed, avoids the objectionable introduction of extraneous turbulence
scales as from vertex generators cr grids, but this claim is contested by
Cock (1978). Nagib et al. (1974) and Tan-atichat et al. (1974) show that
this technique produces reasonable boundary layers v.ith adequate lateral
uniformity and that ecuilibriur is achieved in approximately A boundary
layer heights. Neither turbulence scales nor diffusive characteristics
of this boundary layer has been measured, however.
Other techniques in this third category include the multiple-jet
systems cf Teunissen (1975) and the "turbulence box" of Nee et a". (1974),
but neither of these systems appears to have been developed beyond the
initial stages.
119
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3.2.3 Simulating theJDiabatic Ecundary Layer
Only a few facilities exist for simulating the diabetic bouraary layer.
The oldest and best-known is the N'icrometeorolcgical Kind Tunnel at the
Colorado State University (Plate and Cermak, 1963). It has ncminal test
section dimensions of 1.8 X 1.8 X 27 m and an adjustable ceiling for elimina-
ting the pressure gradient due to growth of the boundary layers. Speeds in
the test section may be varied from 0 to 37 m,/s. A 12 m length of floor can be
heated or cccled and a heat exchanger in the return leg maintains ambient air
temperature ecuilibrium, permitting temperature differences between the cold
floor and hot air of about 65°C and between the hot floor and cold air of
about 105°C at "moderate" wind speeds. At a speed of about 6 m/s, a
boundary layer thickness between 70 and 120 cm can be obtained as the
roughness is varied (Cermak and Arya, 1970).
Arya and Plate (1969) have described many characteristics of the stable
boundary layer generated in this wind tunnel and have shown that the surface
layer characteristics are in excellent agreement with field data when scaling
is dene according to Monin-Cbukhov similarity theory, "heir data ranged from
neutral to moderately stable (C <_ z/L <_ 0.3) in the lowest 15°' of the boundary
layer, which was about 70 cm deep. To obtain this range of stabilities, the
temperature difference between the cold floor and free stream air was
maintained at 4CCC while the wind speed was varied frcn: 3 to 9 m/s.
f'easuremerts included distributions of mean velocity temperature, turbulence
intensities, shear stress, heat flux, and temperature fuctuations. Arya
(1?7E) has presented additional measurements in this stable boundary layer.
Thus far, all measurements in stratified boundary layers in the CSU tunnel
have been with a smccth floor.
120
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"The Fluid Mechanics Laboratory at the Ecole Centrale de Lyon has mace
extensive meast'rements of an unstable wind tunnel boundary layer and compared
Jts properties with the atmospheric boundary layer (Schon et a"., 1974; Mery
et a1., 197£; Schcn and f'ery, 1978). Flow speeds were typically 2 to 4 m/s
while the floor terrperature was maintained 5C°C above ambient. In general,
comparisons with the Kansas data (Eusincer et al., 1971; Haugen et al.,
1971) were quite satisfactory, but, again, these measurements were made over
the smooth wind tunnel floor; longitudinal turbulence intensities exhibited
a slight Reynolds number dependence, and the lack of energy in the high
frequency portions of the spectra v/ere quite evident, but as noted earlier,
this effect is expected to be unimportant in terms of simulating diffusion.
The most unstable flow in which diffusion was studied was characterized by a
[''onir-Cbukhov length of -1 m, which, when scaled to the atmosphere, corresponds
to -500 tc -1COC m, and is indeed only very slightly unstable (See Table 3).
Attempts b\ Rev (1977) in adding a rough floor tc this unstable boundary layer
showed substantial changes in the boundary layer structure w'th roughness.
2.2.4 Summary p_n_ Simulating the Atmospheric Boundary Layer
The point of the previous two sections (3.2.2 and 3.2.3) was to cite
evidence of our ability to simulate at least the main features of the lower
portion of the atmospheric bcundary layer in wind tunnels. No attempt was
made to include in detail all of the various techniques that have been used,
as that is beyond the scope cf this guideline. The point is orly tc show
that it can and "indeed has been done through various schemes.
Adequate simulations of the neutral atmospheric bouncary layer have been
obtained using short tunnels with either active cr passive devices and ]cng
tunnels. Strengths and veaknesses c* the three types, as far as their ability
121
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to produce adequate boundary layers is concerned, appear to be evenly
balanced. Hence, no ore technique or system is recommended ever any
ether. (Hov;ever, see further discussion in Chapter £). Cue to the large
number of permutations and combinations and to the possible large
charges ir boundary layer structure with seem'ngly snail changes in
configuration, however, it is imperative that, whatever technique is used,
the boundary layer characteristics are adequately documented.
Simulations of diabetic boundary layers have been accomplished using
wind tunnels with heated and cooled floors, but present technology allows
only small deviations from neutrality, i.e., mildly stable or mildly unstable.
/^so, because roughness elements on the foor would reduce heat transfer
even further, essentially all measurements to date have been made ever-
smooth wjnc! tunnel floors. As we have seen in Figure 22, the inability to
use a rough, surface could be important, ct least for unstable flews and
large roughness 1engths.
Adequate documentation of the boundary layer characteristics should
include, as a minimum:
1. Several vertical profiles cf mean velocity, turbulence intensity
(3 components), and Reynolds stress throughout the region c^ interest
to establish that the boundary layer is non-developing (or at least
very slowly developing), and is similar to the target atmospheric
boundary layer (zQ,d, u*).
2. Lateral surveys of mean velocity and turbulence structure at various
elevations to ascertain the two-dimensionality cf the boundary layer.
3. Spectral measurements of the turbulence to determine that the integral
scales and the shape cf the spectra are appropriate.
122
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4. Dispersive characteristics of the boundary layer (in the absence of
a model) to determine that the concentration patterns are
reasonably similar to those expected in the target atmosphere,
e.g., the appropriate Pasquill category.
Perhaps the most critical test of the boundary 'ayer is the measurement of
its dispersive characteristics to determine whether appropriate concentrator.
patterns result. This point cannot be over-emphasized. Wind tunnels are
generally extremely difficult to operate at lew wind speeds (
-------
3.3 FLOW AROUND BUILDINGS
V'e discuss here guidelines to be followec in modeling flow around
buildings, e.g., in order to determine a necessary height for a stack on
a power plant to avoid down wash of the p^ume ir the wake of the plant.
The class of problems covered includes single or small grcups of buildings,
prinan'ly isolated ones in a rural environment, i.e. scale reductions in the
range of 1:2CO to IrlCOO. It is evident .from preceding sections that the
building rrust be immersed in ar appropriate boundary layer. Geometrical
scaling implies that the ratio of the building height to boundary layer
height must be matched and, of course, that all length scales be reduced by
this same ratio. £ minimum, building Reynolds number criterion must be met
as discussed in Section 2.2.2.2 and further elaborated here. Finally, the
effluent plume behavior must be modeled as discussed in Section 3.1.
3.3.1 Discussion
Geometrical scales that ccme to mind ere stack height H and diameter P,
build-'ng height H, boundary layer depth 6, roughness length z . and, if
o
stratified, P-'cnin-Cbukhcv length L. There are, of course, many ether length
scales and geometric scaling requires that all lengths be reduced by the
same ratio. However, this brings up the question cf how much detai1 -s
required, i.e., is it necessary to include in the scale model a particular
protuterar.ee, say, from the roof of the building? The answer, cf course,
depends upon the size and shape cf the protuberance; it "s a question cf
whether or not the obstacle has a separated wake. Some guidance may be obtained
from Goldstein (1965), where it is stated that provided the size e o^ the
-------
protuberance is such that e u*/v^ A, it will have little effect on the
flow in a turbulent boundary layer en a flat plate. Hence, protuberances
smaller than e = 4v/u* need not be reproduced in the irodel.
A closely related but more deranding problem is as fellows. A giver
surface is aerodynamically smooth when the Reynolds number is below
a certain value; it is rough when Re exceeds this value. Hence, all
surfaces are rough when the Reynolds number is sufficiently large. Because
field values of Reynolds numbers are alrrcst always very large, we may
assume that surfaces of typical buildings are aercdynamically rough.
As we reduce sizes cf buildings to fit irto our wind tunnel, we also
reduce the Reynolds numbers, so that the surfaces become aerodynamically
smooth. Hence, locations of separation points, the drag coefficient,
and the general character of the flow along the model surface will be
affected.
Again, from Goldstein (1965), if P.ec = e u*/v < ICO, these flow
phenomena will be independent cf Reynolds number. "These results indicate
that small details need not be reproduced and, indeed, that mcdel surfaces
should te roughened to the point that the critical Reynolds number is at
least ICO.
Crude estimates will suffice here and an example will help tc clarify
the procedure. Suppose our model building has a height of 20 cm, and in
order to node! the buoyancy in the plume, we have reduced the wind speed to
1 m/s. The friction velocity is typically 0.05 l'^, so that size of
roughness elements with which to cover the surface of the model building
15 ?= (ICO) (0.15cm2/s)/5cm/s=3cm.
125
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This is, in general, an unacceptably large roughness size as we
should probably also restrict e/H<30. We must either increase the size
cf the model or the wind speed or rationalize as to why a smaller value
Rec is acceptable. There are, in fact, numerous reasons why considerably
icwer values may be acceptable: (a) the exterior flow is, after all, highly
turbulent, (b) the building shape is quite unlike a long flat plate from
which the criterion was derived; it has a bluff leading edge and a
strongly oscillating wake, and (c) the u* value that should be used is the
cne for the boundary layer on the building surface rather than that of the
approach flow; the former is likely to be larger. In the absence of more
supportive data, v/e will take the geometric mean of the twc extremes
(* and 100) as cur criterion:
which :s tc be interpreted both as the minimum size of protuberances that
rust be reproduced in the model anc as the size of the roughness elements
V'-'th Khich tc cover the model.
Regarding the minimum building Reynolds number U,j H/v for the flow
structure to be Reynolds number independent, a precise answer will depend
upon the geometrical shape, the surface roughness, etc. Only a few sys-
tematic studies have been made on Reynolds number independence relating
to atmospheric modeling, but none of these mentioned effects of mode"
surface roughness. It may be inferred that the model surface was aerodyna-
mical 'y smooth. Golden (19fl) measured the concentration patterns above the
roof of model cubes in a wind tunnel. Buoyant and neutrally buoyant e^fHients
were discharged into the air stream from a rlush vent in the center o* the
cube. Two sizes cf cubes v\ere used to vary the Reynolds number -roni 1CCC to
9^,000. The noncim.ersioral concentration isopleths above the cubes showed only
126
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slight variations over the entire range of Reynolds numbers w'th neutral"ly
buoyant effluent and with an e^fluent-to-free-stream velocity ratio of unity.
However, the maxirrum concentration en the roof itself was found to vary
strongly with Reynolds rubbers less than 11,000, but tc be invariant with
Reynolds numbers between 11,COO and 94,000. Thus, a critical Reynolds number
may be defined, which, with this type of geometry, appears to be 11,CCO.
Golden's value for the critical Reynolds number for flow around cubes is
frequently cited in the fluid mcaeling literature on building ccwnwash
problems. K'hereas Golden's value was established for a smooth surfaced
cube facing a uniform approach flow of very low turbulence intensity, it is
applied ''across-the-board" tc all shapes and orientations of buildings, in
all types of approach flow boundary layers, and without regard tc the building
surface roughness, all of which will affect the critical Reynolds number.
A'so, Golden's value was established through, the measurement cf ccncentratlcr.s
at only ore point on the roof cf the cute, as opposed to measurements of,
say, the concentration fields in the wakes. Far too much confidence seems
tc have been placed in his result. It is probably conservative as the
shear and high turbulence in an approach boundary layer as well as a rough
building surface are likely to reduce the critical Reynolds number, /^so,
as pointed cut by Halitsky (1968) lower values ere probably acceptable i'f
measurements are restricted to regions away from the building surface. Hence,
a critical Reynolds number of "l.CCC is a useful arc probably conservative
value for mode1 design purposes, but tests tc establish Reynolds number
independence should be an integral part cf any model study until such time
that firmer values are established.
-------
A study by Smith (1951) ray also be regarded as a test of Reynolds number
independence. He investigated the size cf the wake created by both model and
prototype sharp-edged buildings. He assumed that the flcv was independent
cr Reynolds number effects if the ratio of the length of the cavity region to
the building height was the same in both model and prototype. In the
prototype tests, he found this ratio to be constant for Reynolds numbers
^
(based upon an appropriate characteristic length) between about 2 X 1C " and
2 X 10 . Moreover, in the model tests, he found the ratio to be constant
^
for various block models over a range of Reynolds numbers from 2 X 10 to
2 X TO5.
Critical Reynolds numbers for other geometrical shapes remain to be
determined. A study by Halitsky et al. (1963) on a reactor shell (a hemis-
phere fitted on a vertical cylinder) indicated a critical Reynolds number
greater than 7S,CCO. The separation point, and, hence, the pressure
distribution for rounded buildings is affected by the Reynolds number.
Generally speaking, the more streamlined is the object, the larger is the
critical Reynolds number. It is quite likely that with rough surfaces,
critical Reynolds numbers for streamlined objects may be reduced substantially,
and systematic studies need to be done in this area.
Suppose there is another building upstream of our example power plant;
is it necessary to incorporate this building into our wind tunnel mccel? Some
guidance is provided by Hunt (1974), who reviewed experimental results of
several investigators and showed that the velocity deficit in the wakes of
cubes and cylinders is given by
AU
mx
(x/h)
3/2
128
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acwnwind of the separation bubble, where AU is the maxirr.urr; mean velocity
deficit created by the obstacle, h is the height of the obstacle, x is the
cistance downstream of the obstacle, and A is a constant which is dependent
upon the building shape, orientation, boundary layer thickness, and surface
roughness. Typically, A = 2.5, although it may vary *rcrr. that by a factcr
of 2. Hence, if we require that the velocity be within 3% of its undisturbed
value, then a cubical building as high as x/20 must be included upstream of
cur pov.er plant. This result, however, is dependent upon the aspect ratio;
a building with its width much greater than its height, for example, wculd
require inclusion if its height were greater than x/lCO (See Section 3.4).
The ratio of the cross-sectional area of a model to the cross-sectional
area of the wire1 tunnel is referred to as "blockage", e. It is easy to show,
through the principle of mass continuity, that the average speed-up S (increase
in velocity) of the air stream through the plane intersecting the model is
equal to the inverse of the blockage ratio, i.e., S = 1/8. Of course, in the
atmosphere, there are no sidewalls or roof to restrict the divergence of the
flcv, around the model, so that the average speed-up is zero. Wind tunnels
with adjustable ceilings can compensate to some extent by locally raising
the height of the ceiling above the model itself (with gentle slopes upwind
and downwind of the model). In fact, the average speed-up can be reduced to
zero by raising the ceiling such that the additional cross-sectional area
of the tunnel is exactly equal to the cross-sectional area of the model, but
it is obvious that this is not a perfect "fix'1, as that would require local
expansion of the sidewalls at the same time.
129
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Sore unpublished measurements by the present author en the flovv ever a
two-dimensional Mcge sheds light on this problem. Measurements of velocity
profiles above the crest of the ridge were made with a flat (unadjusted)
ceiling where the blockage caused by the ridge was 10°;. The ceiling
height was then adjusted until longitudinal surveys of velocity at an elevation
5 hill heights above the tunnel floor showed a nonaccelerating flew. Vertical
profiles of rrean velocity were similar in shape to those Pleasured with the
flat ceiling, but the rragnitude of the wind speed was lower by 1C« everywhere
above the crest (see Figure 38) with little change in the root-mean-square
values cf the longitudinal or vertical fluctuating velocities (turbulence).
It is apparent (but by no means proven) that blockage would reduce the
vertical width of a plume by approximately 1C^ as it traversed the ridge,
but, because its center!ine would be 1C* closer to the ridge crest, resulting
surface concentrations upstream cf the crest would be essential"!}' unchanged.
Kcwever, because the flow acceleraf'cn changes the pressure distribution around
the model, which will in turn afrect the location cf the separation point,
the effects dcwnstream of the crest are not apparent. Blockage "corrections"
ror conventional aeronautical wine tunnel mccels is a highly involved
engineering science problem. "Rules of thumb" indicate a limit cf ET
blockage ""'n the ordinary wind tunnel and somewhat higher, perhaps 10°.', in
a tunnel with an adjustable ceiling.
3.3.2 Recommendations
To model the flow and dispersion, around individual or small groups
cf birldings, it is recommended that:
130
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igure 38. Velocity profiles above crest of triangular ridge indicatina
efrect of blockage (s flat ceiling, 10%blockage;G raised
ceil-ing, nonacceleratinq free stream flow).
131
-------
1. The building be immersed in an appropriate boundary layer,
the main features of which include matching cf the ratios
cf roughness length, boundary layer depth and, if stratified,
the Monin-Obukhov length to building height.
2. The effluent plume be modeled as discussed in Section 3.1.
3. Reynolds number independence tests be conducted as an integral
part of the model study. For design purposes, a minimum building
Reynolds number [! H/v = ll,CClappears to be conservative.
<. The surface of the building be covered with gravel cf size t such
that eu*/v=10C. If this results in excessive roughness, i.e.,
e/H > 30, compromises may te mace, but in n.c case should su*/v be
less than 2C.
5. Another building or major obstruction upstream o~ the test building
te included if its height exceeds l/2Cth of Us cistance from the
test building. This recommendation applies to a roughly cubical
obstacle. An obstruction whose crcssv.ind dimension is large
compared to Us height must be included if its height is greater
than l/lCCth of its distance upstream (see text).
6. Blockage caused by a model be limited to 5% in an ordinary wind
tunnel and to 1C" in a tunnel with an adjustable ceiling.
132
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2.4 FLOW OVER HILLY TERRAIN
Guidelines for modeling neutral flow ever hilly terrain are essentially
similar tc those for modeling that around buildings; hence, only a few of the
unique features of terrain will be discussed in this section. Differences occur
primarily because terrain is generally much more streamlined than are buildings
and because the roughness is generally mere patchy. Whereas separation of the
flew from a building surface will almost always occur at a sharp corner, the
separation point, for a hill with a rounded top may fluctuate in position with
time, and ir.ay occur on the downwind slope of the hill, or for a hill with low
slope, may be absent entirely. Also, the stratification in the approach flow
can drastically change the nature as well as the location cf the separation
and may enhance or eliminate separation entirely (see Hunt and Snyder, 1979).
We will first discuss neutral flew, emphasizing the differences between
modeling the flow around hills and that arcund buildings. Because stratified
flews are so different from neutral flows, they will be discussed in a
separate section. The two sections are summarized with a set of recommendations
3.4. I Neutral Flow
In the field, the ridge Reynolds number based en a ridge height of 75 m
and wind speed of only 3 m/s is 107. For these very large Reynolds numbers,
at least for a ridge with moderate slopes, separation is certain to occur
near the apex, even for a ridge with a smooth rounded top (see Scorer, 1968,
p. 113). The Reynolds number for this model mountain ridge would lie
between 1G4 and 1Q5, much smaller than the full scale Re. It is possible tc
trip the flow at the apex (as done by Huber et a 1., 1976) or to roughen the
surface, so that the point of separation en the model will cccur at its
apex and similarity of the two flow patterns will be achieved.
133
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Appropriate roughening of the surface, as outlined below, is the "safest" of
the two techniques, because proper placement of a trip requires foreknowledge
or possibly urwarrented presumptions of the location (or indeed existence) of
separation.
Concerning the minimum size of protuberances that must be reproduced in
the model and also the size of the roughness elements that cover the model to
make the flow independent of the Reynolds number, we apply the same criterion
as established for buildings, namely t = 2Gv/u* - 400 v/LK, This may in seme
instances conflict with Jensen's criterion that h/zQ be matched between modeT
and prototype, but the minimum Reynolds number is regarded as more important.
A common oractice in constructing terrain models is to trace individual
contour lines from enlarged geographic maps onto plywood or styrofoam, then
to cut them out and stack them to form "stepped" terrain models. Some
laboratories then fill in or smooth out these irregularities, while others use
rather large steps and do not smooth them. Ore laboratory, in fact, proposed
to fill in and smooth the model, then to add randomly spaced blocks to
simulate surface roughness. Application of the criterion in the previous
paragraph shows both the desired step size and the roughness element size.
It is not necessary to fill in the steps if the step size is chosen appropriately
at the beginning; the steps double, to some extent, as roughness elements,
although it is most likely better to densely cover the model surface with
gravel of about the same diameter as the step size.
How much terrain is it necessary to include in the model upwind of a
power plant? For a two-dimensional ridge, Counihan et al. (197^) have
shown that the maximum deficit of mean velocity in the wake, normalized by
-------
the mean velocity at the height of the hill, decays as
iU B
U(h) x/h
where L(' is the difference in mean velocity created by the hill, h is the hill
mx
height, x is distance downstream or hill, and B is a constant dependent upon
surface rougnness and hill shape. Typically, B=3.0. Hence, if we insist
that the mean velocity be within 3% of its undisturbed value (i.e., its
value in the absence of the ridge), then all upwind ridges with heights as
large as x/'ICC should be included in the model. In actual practice, ore
should study the topographic maps of the area surrounding the plant, locate
prominent ridges upstream, then determine the height of each ridge and its
cistance from the plant. If its height is greater than x/lCC, all terrain
between the ridge and the plant should be included. If h
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the formulas do, not indicate the fetch required for the development of the
boundary layer.
The choice of a boundary layer depth for very rugged terrain is a difficult
task. Cur choice of 600m (Section 3.2.1.2) is obviously absurd if the heights
of the hills themselves are of the same magnitude. Cne indication from the
literature is from Thompson (1978), who examined wind profiles obtained from
pilot balloons over complex terrain in southwestern Virginia. The average
boundary layer depth, Thompson concluded, was about 800 m, or 4 (r'11 heights
under moderate to high wind speed neutral conditions.. As mentioned in
Section 2.2.1.1, he also observed a logarithmic wind profile with a z0 of
35 m.
3.4.2 Stratified Flow
We have discussed in depth the stable boundary layer in Section 3.2.1.4.
It was shown that under stongly stable conditions, the boundary layer is very
shallow, typically less than IOC m. Frequently, pollutant sources discharge
their effluent at much higher elevations, i.e., above the stable boundary
layer, where the plume may be transported long distances with little or no
dispersion (e.g., see cover photograph of AKS, 1979). Further, results
of Godowitch et al. (1979) indicate that extremely shallow stable boundary
"ayers under quite deep ^urface-based temperature inversions are
typical at sunrise at a rural site outside St. Louis, MO. The average depth of
the nocturnal inversion, for example, was 325 m (± 90 m standard deviation).
The average temperature gradient was 1.4nC/100m. Under these conditions, it
is evident that simulation of the stable boundary layer beneath an elevated
source is relatively unimportant. Far nore important is the simulation of
the stability above the boundary layer because, as shown by Lin et al. (1974),
136
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Hunt and Snyder (1979) and Snyder et al. (1979), the stability determines
the essential structure of the flow, i.e., whether plumes will impinge
on the hill surface or travel over the hill top, the size and location cf
hydraulic jumps, etc.
Under strongly stable conditions, the flow is constrained to move in
essentially horizontal planes. If a three-dimensional hill is placed in
the flew field, streamlines below the hill top will pass round the
sides cf the hill and not over the top. If a two-dimensional hill is placed
across the flow field, the fluid obviously cannot pass round the sides and,
because it has insufficient kinetic energy, it cannot pass over the top
of the hill (see Section 2.Z.4). Thus, upstream blocking of the flow below
the hill top will occur. The point is that the modeler must be very careful
in determining the amount of terrain tc duplicate in the model. An example
is shewn in Figure 39, where a portion cf a three-dimensional hill is turned
into a two-dimensional one by an inappropriate choice cf the area to be
modeled. Under strongly stable conditions, the combination of the hill and
tunnel sidewalls would result in upstream blocking of the flew beneath the hill
top, whereas, with a wider tunnel or smaller scale model, the flow would be
diverted around the sides of the hill as would certainly occur in the atmosphere.
137
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igurs 39. Contour map of three-dimensional hill showing inappropriate
choice of area to be modeled.
158
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Similar extensions of this type of reasoning apply to valleys and ridges angled
diagonally across the flow stream. It is impossible to give firm and fixed
rules for determining the appropriate area of terrain to model because the
flow field must be known a priori. However, detailed study of topographic
maps of the area and the application of common sense will avoid most pitfalls.
As Scorer has pointed out, laboratory studies of stratified flows tend to
overemphasize the effect of stratification in the approach flow;
local heating and cooling of hill surfaces are equally, perhaps more,
important. The effects of anabatic and katabatic winds are not only local,
but may have large effects on the flow structure by inhibiting or enhancing
separation (Scorer, 1968, p. 113; Brighton, 1978). There have been some
attempts to simulate heating of terrain surfaces, but more to simulate
fumigation of elevated plumes than anabatic or katabatic winds (Liu and
Lin, 1976). Little is krown of the proper similarity criteria to be
applied to thermally-driven flows. Any comparison between field and model
experiments, where such thermally generated winds are absent, must be made
with great caution.
3.4.3 Recommendations
Recommendations for modeling flow and dispersion over hilly
terrain in neutral stability are essentially similar to those for
modeling flow around buildings. It is recommended that:
1. The terrain be immersed in a simulated atmospheric boundary
layer, matching the ratios of roughness length and boundary
layer depth to hill height.
2. The effluent plume be modeled as discussed in Section 3.1.
3. Reynolds number tests be conducted as an integral part of the study.
139
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4. The surface of the model be covered with gravel of size £ such
that. 20<_eu*/v for an ordinary wind tunnel and to 1C7
in a tunnel with an adjustable ceiling.
Additionally, in modeling dispersion from elevated sources ''n strongly
staHy-stratified fow over hilly terrain it is suggested that:
7. The simulation of the stable boundary layer, per se, is
relatively unimportant. More important is matching of the
Froude number based en the hill height and the density
difference between the base and top of the h-Hl.
Topographic maps of the area to be modeled should be studied
carefully to ""nsire that an appropriate area is modeled (see
text) .
Finally, laboratory models to simulate anabatic arc katabatic winds
rust be considered as exploratory in nature at the present time.
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RELATING MEASUREMENTS T0_ THE FIELD
Since buoyancy in a plume may be modeled using light gas as opposed to
temperature, the concentration measured in a model facility may be related to
that in the field through the nondimensional concentration X = CUH'YQ, where
Q
C = mass concentration of pollutant (ML"°),
U = wind speed (LT ) ,
H = characteristic length (L),
and Q = pollutant emission rate (Mi , e.g., grams of SOg/second).
The relation between model and field concentrations is thus
U Hm 2 Qf
r - r i _ oi> i _ nii / _ L\
Cf - Cm ( y * y ( Qj-
Mote that both C and Q are measured in mass units. More frequently, C and C
measured in volume units, in which case, they must be evaluated at ambient
(not stack) temperature, including Qf (Robins, 1975). An example may help tc
clarify the procedure. Suppose we model a buoyant plume with a mixture of
helium, air and methane as indicated in the following table:
Property Field Model
Reference wind speed lOm/s 1m/ s
Stack height EOm £0cm
Stack diameter 5m 5cm
Effluent speed 20m/ s 2m/s
Pollutant emission rate 500g/s(S02) lg/s(CH4)
Effluent temperature 117°C 20°C
Ambient temperature 20 C 20 C
Effluent specific gravity .75 .75
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At seme point downwind of the source in the wind tunnel, we might measure a
model concentration cf IGOppm (parts CH, per million parts air, by volume),
and the problem is to relate this to a field concentration value. First, the
model volume concentration must be converted to a mass concentration (relevant
densities at ZO°C are air: 1.2Sg/£; CH4: 0.74g/£; SO^: 2.95g/£):
]_1JLJJ_ -, 57 x
"
io6 £ air 1>29g ai> ] £ CH4 " 9 air
Herce, the field mass concentration will be:
57 x 10"°oCH, , , n c 2
4\ / ifn/ s \ /U.bm\
:J
"f v q air ' M0m/s; v 50m; MgCH./s
j q,
= 0.285 x 10"6gS00/g air.
""' t.
Convertir.g this mass concentration to volume concentration yields
C.285 x 10"6gSO^ , ,Q 1 «. S00
/- _ ' ^N /' f.~ air\ / c. \
~l VI (1 =iv ' \OC
f 'gai ar
= 0.125 x IO"6 i S02/£ ai> = O.lZEppm SC^.
To summarize, the relation between model and field concentrations in this
example is
1 ppm S02 -» 80C ppm Ch^
or 1 g 50,,/g air -^ 200g CH^/g air.
Whereas it is tempting to bypass some of the above steps by using volume
emission rates, shortcuts are not recommended. It is important to note that
all densities were specified and used at ambient temperature, i.e., 20 C.
Regarding the comparison of model resuHs with field results, it is
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well-known that in the field the averaging time has a definite effect on the
measured concentrations. This is not the case in model tests. (This
discussion is taken largely from Ludwig, 1974). The model results correspond
to short-time-averaged field measurements, taken over not more than 10 or 15
minutes in most cases. Briefly, what is involved here is the following. The
energy spectrum of wind gusts in the atmosphere generally shows a null, or near
null, in the frequency range of 1 to 3 cycles per hour (the "spectral gap"
discussed in Section 2.2.2.2). Thus, it is possible to separate the spectrum
intc two parts and to deal with the phenomena associated with each part
separately. The high-frequency portion, related to the roughness of the
surface and the turbulence around obstructions is well-simulated in a wind
tunnel. The low-frequency portion, related to the meandering of the wind,
diurnal fluctuations, passage of weather systems, etc., cannot be simulated in
a wind tunnel. However, a correction for meandering of the wind can be applied,
if desired, to derive longer term averages (Hino, 1968; Isyumov, et al., 1976).
Model averaging times, on the other hand, are chosen to provide data that are
repeatable tc within some specified accuracy, as discussed later. However, as
noted above, the data so obtained will correspond to field data measured while
the wind direction is essentially steady, which is generally not more than 1C
to 15 minutes. Shorter term averages obtained from the model can be related
to the short term fluctuations in the atmosphere, and instrumentation is being
developed to accomplish this (Fackrell, 1978).
3.6 AVERAGING TIf-E AND SAMPLING RATES IN THE LABORATORY
Because the flow is turbulent, essentially all of the quantities we
attempt to measure will fluctuate in time. Generally, we will deal with a
fluctuating electrical signal from a transducer, and it is not the precise
H~ ~i
0
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value at any particular instant of time that is of interest, but rather the
average values and the statistics of the fluctuations. It is necessary
at some point to determine how long an averaging period is required to obtain
a stable average. Frequently, it is convenient to digitize an analog signal
(sample it and convert the analog voltage to digital form). Sampling at too
high a rate is a waste cf resources; sampling at too low a rate may not allow
us to obtain the desired infcrmation and, in fact, may lead to incorrect
answers. Hence, it is also necessary to determine an appropriate sampling
rate .
To determine an appropriate averaging time for measuring the mean of a
-luctuating quantity F(t) in a wind tunnel, it is useful to consider the
turbulence as a Gaussian process. (Whereas turbulence is net a Gaussian
process, experience has shown that this assumption leads to quite reasonable
2
estimates cf the errors involved.) The variance a of the difference
between the ensemble (true) average and the average obtained by integration
ever time T is (Lumley and Panofsky, 1964):
a2 = 2f2 I >
where f~ is the ensemble variance of F about its ensemble mean, f=F-F and 1
is the integral scale of F. The fractional error e, then, is given by
? ~~2
e2 = £_ = 2f l
T~ FT
If, for example, it is desired to measure the turbulent energy u , it may
be shown, using the assumption that u has a Gaussian distribution, that
e2 = 4I/T.
To obtain a conservative estimate of the error, it is convenient to
144
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"igure 40, Averaging ti;
measurements .
"eloc
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" = 4<5/Uoe^ is a reasonably good estimate cf the averaging time required
for making all of the common measurements (mean velocity, turbulence intensity,
concentration, etc.). Higher order moments require considerably longer averaging
times.
To estimate an appropriate sampling rate, we begin by drawing from a
mathematical theorem (Miller, 1963):
"If a signal f(t) extending from 0 is « contains r.c
frequencies above W cycles per second, then it is
completely determined by giving its ordinates at a
seauence of points spaced 1/2W seconds apart."
Hence, in order not to loose information from our continuous signal through our
discrete sampling, it will be essential to determine the highest frequency
component in cur signal and to sample at twice that rate. The highest
frequency of any significance in the turbulence is the Kolr'cgorcff frequency,
f . = 1/2"- (see Section 2.2.2.2). Hence, assuming an excellent anemometer
(good frequency response and low electrical ncise), all information about the
turbulent signal may be obtained by sampling at a rate of £f , = U/Trn. At the
slow flow seeds typical cf fluid modeling studies (<'0m/s), che kolrrcgcrof*
micrcscale is not likely to be much smaller than 0.5 mm. Hence, a typical
sampling rate would be approximately 2 kilchertz at a flew speed of 5m./s.
Because of aliasing, lower sampling rates are unwise (fcr more information,
see Lumley and Fanofsky, 1964). Of course, if the transducer or amplifier
have slower frequency response, it is pointless to sample at twice the
Kclcmcgoroff frequency. A flame ionization detector, for example, has a
time constant of approximately Q.5 sec., so that a sampling rate in excess of
4 hertz is net necessary.
146
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4. THE HARDWARE
The choice of air versus water as the fluid medium for modeling of
atmospheric flow and diffusion cf pollutants will depend on many different
factors: the availability of the facility, economics, the type of problem to be
studied and the type of information to be obtained, to name a few. The
kinematic viscosity of water at normal room temperature is a feetor of 15 less
than that of air, so that, in principle, a factor of 15 in the Reynolds number
may be gained by modeling with water as the medium. However, because water
is so much heavier than air, structural and pumping requirements dictate that
water facilities be much smaller and run at much lower flow speeds than wind
tunnels. Thus, the full potential for obtaining larger Reynolds numbers using
water facilities is seldom realized.
If it is essential to obtain very high Reynolds numbers, water has some
advantages. Because of its incompressibility, it may be run at high speeds
while maintaining low Mach numbers, which is not possible with air. However,
a different problem appears with water at high speeds -- cavitaticn behind
obstacles. This may be overcome by maintaining large pressures inside the
water tunnel, which then requires heavy steel construction, so that compromises
must again be made.
4.1 . VISUAL OBSERVATIONS
Smoke and helium filled soap bubbles (for which a generator is new
commercially available) are about the only visible tracers for use in air. A
very much wider variety of tracer techniques is available for use in water,
making flow visualization much easier. These include different colors and
densities cf dye, hydrogen bubbles, potassium permanganate crystals,
shadowgraphs, and neutrally buoyant particles. And because Jlow speeds are
generally low, it is easy to observe and photograph flow patterns in
147
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water. The comparable smoke in a wind tunnel is difficult to regulate either
in concentration cr specific gravity (oil fog smoke generators are notor-
iously cantankerous and they occasionally explode!). Smoke is also difficult
to observe visually and photographically at flew speeds in excess of 1 to 2 m/s.
Titanium tetrachloride is relatively easy to use, but is corrosive, hazardous
to handle, and is not easily used as a stack effluent.
The importance of flow visualization should not be underestimated. Much
time and effort can be wasted searching for'a maximum ground level concentration
in. complex terrain using a probe and some sort of analyzer, whereas visual
observations of smoke or dye would narrow the area to be searched tremen-
dously. Fixed rakes are frequently positioned downwind cf a hill to sample the
vertical and lateral concentration profiles; but unless it is known a priori
about where the plume will be, the data collected will rot be highly useful and
the experiment may have to be run again. With flow visualization, it is obvious
at a glance, for example, whether a plume is going over or around a hill,
whereas extensive point-by-pcint measurements would be required otherwise.
Flew visualization can also be of great help in the interpretation and
understanding cf quantitative data. Hot wire anemometry, ir spite of its
increased sophistication and reliability in recent years, still cannot tell
us the direction cf flow (there is a +.180° ambiguity) and reverse flows
commonly exist downwind of bluff obstacles. Finally, some quantitative
results may also be obtained from flow visualization. For example, Hunt and
Snyder (197?) used flow visualization to measure the displacement of stream-
lines by a hill, the surface streamline patterns, the increase of velocity
or speed up over the top of the hill, for understanding lee waves, hydraulic
jumps, and separated flow regions downwind of the hill, and for extending
143
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Drazin's (1961) theory to determine whether plumes from upwind sources would
pass over the top or impact on the surface and pass round the sides of the
hill in stratified flow. Also, different colored dyes emitted from different
elevations on the hill surface showed oscillations in the wake that were
anticorrelated at different elevations; this kind of information would have
teen difficult to obtain through other means.
4.2 QUANTITATIVE MEASUREMENTS
Quantitative measurements of flow speeds are more difficult in water.
Wind tunnel techniques for these measurements have been developed and advanced
to a level of high reliability and accuracy (Bradshaw, 197C). For local
velocity measurements in wind tunnels, numerous instruments are available:
pitot tubes, hot wire-, hot film,- and pulse wire-anemometers. Hot. film
anemometers are used in water, but require much travail to obtain reliable
measurements. At typical low flow speeds used in water, pi tot tubes are not
very useful. Small propeller anemometers (^ 1 cm dia.) have been developed for
special studies in air and water, but are not readily available.
Highly accurate and reliable flame ionization detectors are available for
quantitative measurements of pollutant concentrations downwind from a source in
a vsind tunnel. These instruments are presently the most popular because they
have a relatively .fast response time (^ 0.5 second), the^r output is linear
with concentration ever a very wide range (about 0.5 to 10,000 ppm. methane
with proper adjustments), and they can be used with any hydrocarbon gas,
including methane, ethylene, and butane, which have specific gravities cf C.5,
' anc 2, respectively. Many other tracers and instruments have also been used
including sulfur dioxide, carbon monoxide, temperature, smoke (Motycka and
Leutheusser, 1972), helium (Isyumov et al., 1976), and radioactive gases
(Meroney, 1970), along with corresponding measuring devices. Smoke, temperature,
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and helium techniques offer possibilities for the measurement of concentration
fluctuations, but are generally limited to the measurement of small dilutions,
i.e., 1:100, as compared with the 1:1C,OOC desired. Fackre"1 (1978) has
developed a ''pepper valve" tc allow the flame ionizaf'on detector to be
used for the measurement of concentration fluctuations. Salts in conjunction
with conductivity meters, acids with pK meters, temperature with thermistors,
and dyes v/ith colorimeters and flucrorreters have teen used as tracers
for quantitative ireasurenents of concentration in water. Except for
temperature, these techniques offer a wide range in concentration detectability.
The conductivity probes and thermistors can be quite fast response devices.
They offer possibilities for the measurement of concentration fluctuations.
4.3 PRODUCING STRATIFICATION
There are two common methods fcr producing stratificatien in water. The
rest common method of producing stable density stratification in water is by
slowly fillinc a tank through distribution tubes on the bottom with thin
layers c^ salt v;ater, each layer increasing -'n specific gravity (Hunt et e.l..
1978). The heavier solutions flow under the lighter fluid above, thus lifting
it. In view of the very small mass diffusivity of salt in water, an undis-
turbed stable mass cf salt water will remain that way for weeks, even months,
before the density gradient is substantially changed by molecular diffusion.
Maximum dimensionless density differences are limited to about 20/:' using
common salt (NaCl). Recirculating systems using this technique have been
impractical because of the mixing within the pump. However, Cdell and
Kovasznay (1971) have designed a rotating disk pump that maintains the
gradient; this device permits the use cf recirculating salt water systems, but,
thus ~ar, has been used only for very small channels (^ 10 cm depth;.
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The second common method for producing stratification in water is by
heating and cooling. Frenzen (1963) had produced both stable and unstable
stratification in a towing tank using this method. Because of the large heat
capacity of water, large amounts of energy are required for heating and cooling
to produce significant stratification, so that this method 1-s generally limited
to small tanks. Deardorff and Willis (1974) and Liu and Lin (1976) have combined
heating and cooling (respectively) with stable salt, water stratification
to study inversion break-up phenomena.
Air, with its low heat capacity, is comparatively easy to stratify.
Provisions must be made, of course, for heating or cooling of the fleer of the
test section and, if it is a closed return tunnel, for cooling or heating the
return flow. In order not to exceed reasonable temperatures in the tunnel (say
100°C), the maximum dimensionless density difference is limited to about 35%.
The MicrometeorolcgicaI Wind Tunnel at the Colorado State University (Cermak,
1975) has a test section 1.8 m square and 27 m long. It has heating and cooling
capabilities for maintaining the floor temperature between 1 and 200°C and
the ambient air temperature between 5 and 95°C. Calspan (Ludwig and Skinner,
1976) used liquid nitrogen dripped onto aluminum plates upstream of a model in
their open return wind tunnel to produce stable stratification. Dry ice has
been used in a similar manner (Cermak .et a"!., 1970). The problem with the
liquid nitrogen and dry ice is that a stable boundary layer is created at the
point of contact, but a growing mixed-layer (elevated inversion)develops
downstream because of the air contact with the uncooled tunnel floor or model
surface.
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4. A AIP VERSUS WATER
Thus far in this chapter, we have discussed the comparative advantages
cf using air or v/ater as the fluid medium for modeling studies. There are
nc "hard and fast" rules for deciding which type of facility is best for
a particular study. Two example problems are giver below, one of which
appears best suitec for study in a wind tunnel and the other of which appears
best suited for study in a towing tank. However, in principle and --
depending upon the ingenuity and perseverance of the investigator -- in
practice, similar information could be obtained from either study in
either *acility.
Problem 1: We wish to determine the excess ground level concentrations
caused by an insufficiently tall chimney next to a power p"'ant in essentially
flat terrain.
Method of Solution: A plume from a power plant is generally highly buoyant,
sc that building downwash probably occurs only in high wind, hence neutral,
conditions. The advantages of a wind tunnel over a water channel here are
obvious. Thick, simulated neutral atmospheric boundary layers are easily
obtained in wind tunnels, whereas the development and testing cf such in a
water channel would be a cumbersome task. Measurement of the turbulent flow
structure in a water channel would be a very difficult task; accessability to
the model would be limited; instrumentation would be more expensive and less
reliable; concentration measurements would be more easily obtained in air
us^ng hydrocarbons (probably methane in this case) and a flame icnizaticn
detector; etc. A large enough Reynolds number can probably be obtained in a
wind tunnel even though it is necessary to simulate the buoyant effluent.
152
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Probably the only advantage to using a water channel in this case would be fcr
the ease of flcv; visualization, but smoke or helium filled soap bubbles would
probably be adequate in a wind tunnel.
Problem 2: We wish to determine the maximum ground level concentration (glc)
that may occur (at least once per year) on an isolated three-dimensional hill
2CO m high downwind of a 100 m high chimney. Typical nocturnal surface-based
inversions develop to 400 m depth vvith terr.perature gradients of 1.5°C/100m and
\ ind speeds of 2m/s at the 20C m elevation.
Method c* Solution^ The maximum glc will prcbably occur during the nocturnal
inversion. The boundary layer will be below the plume and, hence, is
probably unimportant. The most important parameter is the Froude number based
en the hill height and the density difference between the base and top of the
h^ll:
F = LI/Nh = U/(ghA6/e)i; = 2/(9.8x200x4/30CV5 = 0.4
(Notice that potential temperature instead of density has been used in
calculating the Froude number.) This problem is rather easily studied in a
towing tank of 1 m depth where the stratification is obtained using a
continuous gradient of salt water (s.g. = l.C at the top and 1.2 at the bottom,
yielding N=1.3rad/s). The reauired towing speed for a hill cf height 0.2m
woulri be l>FNh=10cm/s, which is a reasonable towing speed "for a water channel
and yields a Reynolds number Uh/v=20,000.
This type o~ flow has not yet (at least, to the author's knowledge) been
obtained in a wind tunnel, but rough calculations will easily illustrate the
differ I Me:, ~h" maximum temperature difference that could be generated is on
the order c^ 100°C. Let us suppose that the model hill height is also 20 cm
in the wind tunnel and that the 1CCCC temperature difference is imposed over a
153
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40 err, depth, so that N=2.9. The required tunnel speed is thus 23 cm/s, which
is exceedingly difficult to maintain, control, and measure in any wind tunnel,
especially when the temperature varies so drastically. The Reynolds number
would be only ^6CC (although it is most likely unimportant in this case, since
the flow will definitely not be turbulent).
£.5 SUMMARY
Tie ease and convenience of operating wind tunnels and associated measuring
equipment and the ability to adequately simulate the neutral atmospheric
boundary layer make the wind tunnel far superior to the water tunnel for smal !
scale studies where buoyancy is relatively unimportant. However, the inability
of the wind tunnel to achieve adequate buoyancy or stratification and adequate
Reynolds numbers simultaneously make the towing tank indispensable for the
study of elevated plume dispersion in stably-stratified flew in complex terrain.
Somewhere in the middle, where the interest is in low-level dispersion in
mildly stratified flows, the two types of facilities have essentially equal
capabil Hies.
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5. CONCLUDING REMARKS
The problem with simulating the neutral boundary layer is that the
atmosphere is very seldom neutral. There is "always" an inversion at some
height with surface heating or cooling from below. Perhaps, occasionally, the
atmosphere is truly neutral for a few minutes around sunrise or sunset, but such
a state cannot be considered stationary because it lasts only a few tens of
minutes (Kaimal, et al., 1976) and, because the surface heat flux is changing
so rapidly, the turbulence cannot track it (Wyngaard, 1973). Perhaps our only
hcpe is cloudy, high wind, conditions, but "cloudy" implies a temperature
inversion (at the base of the clouds), so this cannot be truly neutral either.
One might rightly ask at this pcint: "Why bother with wind tunnel modeling?
We can't simulate the rotational effects, and even if we restrict ourselves to
cases where rotational effects are relatively unimportant, the type of flew that
we have some chance of simulating well, the neutral surface la>er, hardly ever
exists in the atmosphere." Panofsky (197O rather summarily dismissed wind
tunnel modeling because of our inability to simulate the turning of wind with
height. The answer is "fluid modeling is heuristic." He have the ability to
control the flow and to independently adjust specific parameters. Tc paraphrase
Corrsin (1961a), a wind tunnel is, in effect, an analog computer and, compared
with digital computers (numerical models), it has the advantages of "near-
infinitesimal" resolution and "near-infinite" memory. The inability to achieve
large Reynolds number turbulence limits the size of the dissipative eddies. In
many ways, this situation is analogous to numerical fluid-dynamic models wherein
-------
the smell-scale turbulence is "parameterized." Whereas we have difficulty in
simulating the large scale eddies, we are no worse off than the numerical modelers
and v,e need not make any second-order closure "assumptions." Nor must we deal
with an inviscic potential flow that cannot separate from any body, let alone
a sharp corner. The point is that we need to understand the characteristics of
the flew we generate and to understand how changing those characteristics changes
the result. We must also recognize the limitations of our facilities and
interpret the results accordinglywith caution.
There are two basic categories of fluic modeling studies: (1) The
"generic" study wherein idealized obstacles and terrain are used with idealized
flows in ar, attempt to obtain basic physical understanding of the few and
diffusion mechanisms, and (2) the engineering "case" study wherein the miniature
scaled model of a specific building or hill is constructed and a specific decision
is to be made based upon the results of the tests, i.e., the necessary stack
height or the siting of a plant. Advances in the basic understanding obtained
from the generic studies will ultimately reduce the need for case studies, but
the present state-of-the-art falls far short of eliminating this need.
There are many "doubting Thcmases" concerning the applicability of fluid
modeling studies to the real atmosphere; yet those same "coubting Thcrrases" do
not hesitate to apply potential flow models with constant eddy diffusivities
in orcier to predict surface concentrations on hills under all types of stratified
flow conditions. Frequently, they appear to be unaware that many of the under-
lying physical ideas and even many of "constants" used directly in their models
were obtained from laboratory experiments. A fluid modeling study, after all,
employs a real fluid, and if a mathematical model is to be applied to the
atmosphere, it should also be applicable to a fluid model, e.g., b> eliminating
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or adjusting that portion of the model dealing with rotational effects, by re-
ducing the Reynolds number, etc. If a trathematical model cannot simulate the
results of an idealized laboratory experiment, what hopes does it have of being
applicable to the atmosphere? The point is that fluid models should be used to
bridge-the-gap between the mathematical model and its application to the field.
A well-designed and carefully executed fluid modeling study will yield
valid and useful information - information that can be applied to real environ-
mental problems - - \vith just as much and generally mere credibility than any
current mathematical models.
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