United States EPA-600/8-81 -009
Environmental Protection April 1981
Agency
vvEPA Research and
Development
Guideline for
Fluid Modeling of
Atmospheric Diffusion
Prepared for
Office of Air Quality
Planning and Standards
Prepared by
Environmental Sciences Research
Laboratory
Research Triangle Park NC 27711
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600/8-81-009
April 1981
GUIDELINE FOR FLUID MODELING
OF ATMOSPHERIC DIFFUSION
by
William H. Snyder
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NC 27711
us Environmental Protection Agency.
Slo^Sor
ChLgo, Illinois 60604
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DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratory, U.S. Environmental Protection Agency, and approved for
publication. Approval does not signify that the contents necessarily reflect
the views and policies of the U.S. Environmental Protection Agency, nor does
mention of trade names or commercial products constitute endorsement or
recommendation for use.
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 Oceanic and Atmospheric Administration,
U.S. Department of Commerce.
ii
U,S. Environmental Protection Agency
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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 fre-
quently 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 cred-
ibility 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 fundamen-
tal 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.
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TABLE OF CONTENTS
Page
PREFACE iii
LIST OF FIGURES AND TABLES vii
NOMENCLATURE x
ACKNOWLEDGEMENTS x1v
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 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 32
2.3 Boundary Conditions 35
2.3.1 General 35
2.3.2 Jensen's Criterion and Fully Rough Flow 37
2.3.3 Other Boundary Conditions 39
2.4 Summary and Recommendations 40
3. PRACTICAL APPLICATIONS 42
3.1 Plume Rise and Diffusion 42
3.1.1 Near-Field Plume Behavior 44
3.1.2 Summary and Recommendations on Modeling Near-Field
Plumes 63
3.1.2.1 The Stack Downwash Problem 63
3.1.2.2 The Near-Field Non-downwashed Plume
Problem 64
3.1.3 Far-Field Plume Behavior 65
3.1.3.1 Ignoring the Minimum Reynolds Number. . . 65
3.1.3.2 Raising the Stack Height 67
3.1.3.3 Distorting the Stack Diameter 67
3.1.4 Summary and Recommendations on Modeling Far-
Field Plumes 70
3.2 The Atmospheric Boundary Layer 72
3.2.1 Characteristics of the Atmospheric Boundary
Layer 74
3.2.1.1 The Adiabatic Boundary Layer 75
3.2.1.2 Summary of the Adiabatic Boundary
Layer Structure 87
3.2.1.3 The Dlabatic Boundary Layer 89
3.2.1.4 Summary of the Dlabatic Boundary
Layer Structure 112
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3.2.2 Simulating the Ad1abat1c Boundary Layer 114
3.2.3 Simulating the D1abat1c Boundary Layer 127
3.2.4 Summary on Simulating the Atmospheric Boundary
Layer 129
3.3 Flow Around Buildings 132
3,3.1 Discussion 132
3.3.2 Recommendations 140
3.4 Flow Over HUly Terrain 141
3.4.1 Neutral Flow 141
3.4.2 Stratified Flow 144
3.4.3 Recommendations 147
3.5 Relating Measurements to the Field 148
3.6 Averaging Times and Sampling Rates in the Laboratory. . . 151
4. THE HARDWARE 156
4.1 General Requirements 156
4.1.1 The Speed Range and Scale Reductions 156
4.1.2 Test Section Dimensions 159
4.2 A1r versus Water 161
4.2.1 Visual Observations 161
4.2.2 Quantitative Measurements 163
4.2.3 Producing Stratification 164
4.2.4 Examples 166
4.2.5 Summary 168
5. CONCLUDING REMARKS 169
6. REFERENCES 172
V1
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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
independence 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 100 m 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 Reynolds number 59
10 Laminar plume caused by low Reynolds number effluent 66
11 Effects of wind shear on the flow round a building 72
12 The depth of the adiabatic boundary layer according
to the geostrophic drag law compared with other
schemes 77
13 Typical wind profiles over uniform terrain in neutral
flow 82
14 Variation of power law index, turbulence intensity,
and Reynolds stress with roughness length in the
adiabatic boundary layer 82
15 Shear stress distributions measured at various down-
wind positions in a wind tunnel boundary layer 83
16 Variation of longitudinal turbulence intensity with
height under adiabatic conditions 85
17 Variation of Integral length scale with height and
roughness length 86
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NUMBER TITLE PAGE
18 Empirical curves for spectra and cospectrum for
neutral conditions 87
19 Typical nonadiabatic boundary layer depths from
the geostrophic drag relations 96
20 Variation of friction velocity with stability from
the geostrophic drag relations 96
21 Theoretical variation of the power-law exponent as
a function of z and L for z equal to 100 m 98
22 Variation of the power-law exponent, averaged over
layer from 10 to 100m, as a function of surface
roughness and Pasquill stability class 99
23 Typical surface layer velocity profiles under
nonadiabatic conditions 102
24 Typical temperature profiles in the surface layer 102
25 The relationship between R1 and z/L 105
26 Variation of » and « with z/L in the surface layer 105
W 6
27 Variation of * with z/L in the surface layer 107
28 Variation of *y with z/L 107
29 Universal spectral shape 108
30 Location of spectral peak for u,v,w and e plotted
against z/L 108
31 Upstream view of a long wind tunnel 116
32 Vortex generators and roughness in a short wind tunnel..117
33 Schematic representation of the counter-jet technique...118
34 Development of boundary layer in a long wind tunnel 119
35 Development of mean velocity profiles along the smooth
floor of a long tunnel 121
36 Thickness parameters for boundary layer of Figure 35 121
viil
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NUMBER TITLE PAGE
37 Spectrum of the longitudinal component of velocity 123
38 Velocity profiles above crest of triangular ridge
i ndi cati ng effect of bl ockage 139
39 Contour map of three-dimensional hill showing
inappropriate choice of area to be modeled 146
40 Averaging time requirements for wind tunnel
measurements of turbulent energy 153
41 Example of limits of wind tunnel simulation of buoyant
effluent dispersal in the atmospheric boundary layer..158
LIST OF TABLES
NUMBER TITLE PAGE
1 Typical Parameters for Modeling Plume Downwash 54
2 Techniques used for Simulation of Buoyant Plumes
at Various Fluid Modeling Facilities 62
3 Values of Surface Roughness Length for Various
Types of Surfaces 80
4 Typical Values for the Various Stability Parameters 93
5 Dimensionless Length Scales as Functions of Ri 109
ix
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NOMENCLATURE
A constant or area [L2]
B constant
c constant
C constant or concentration [M/L3]
c,,u, cospectrum of Reynolds stress [L2/T'4]
uw
d zero-plane displacement [L]
D stack diameter [L]
E spectrum function [L3/T2]
f nondimensional frequency or arbitrary function
f Coriolis parameter (^ 10" /sec in mid-latitudes)
f nondimensioral frequency corresponding to spectral peak
F source buoyancy flux = g(D2/4)(Ap/pJ [L1*/!3]
c
F. Lagrangian spectrum function [T]
Fm source momentum flux = (ps/Pa)(D2/4)(W$) [L"/T2]
Fr Froude number
g acceleration due to gravity [L/T2]
G geostrophic wind speed [L/T]
h hill, building or obstacle height [L]
h roughness element height [L]
H stack or building height [L]
I turbulence integral scale [L]
* *
k von Karman constant
K eddy viscosity or diffusivity [L2/T]
£n buoyancy length scale [L]
am momentum length scale [L]
L characteristic length scale or Monin-Obukhov length [L]
-------
Lu integral scale of longitudinal velocity in a-direction [L]
Lw integral scale of vertical velocity in a-direction [L]
a
n frequency [T ]
N Brunt-Vaisala frequency = ((g/p)(dp/dz))1/2 [T"1]
p pressure [K/LT2] or power-law index
Pe Peclet number
Q pollutant emission flow rate [M/T]
Re Reynolds number
Ri Richardson number
Rio bulk Richardson number
Ri,. flux Richardson number
Ro Rossby number
S spectrum function [L2/T] or scale reduction factor
Sc Schmidt number
t time [T]
T averaging time [T], time of travel from source [T], or temperature [T]
u fluctuating velocity in x-direction (streamwise) [L/T]
u* friction velocity [L/T]
U mean wind speed [L/T]
U. instantaneous flow velocity in i-direction [L/T]
v fluctuating velocity in y-direction (cross-streamwise) [L/T]
w fluctuating velocity in z-direction (vertical) [L/T]
W effluent speed [L/T]
x Cartesian coordinate (streamwise) [L]
Xj coordinate in i-direction [L]
y Cartesian coordinate (cross-streamwise) or particle displacement [L]
xi
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z Cartesian coordinate (vertical) [L]
z roughness length [L]
a molecular mass diffusivity [L2/T]
B blockage ratio (area)
6 boundary layer depth [L]
6.. Kronecker's delta
' J
6P deviation of pressure from that in neutral atmosphere [M/LT2]
<5T deviation of temperature from that in neutral atmosphere [T]
Ah plume rise [L]
Ap density difference [M/L3]
E dissipatition [L2/T3] roughness element height [L], or fractional error
eijk alternating tensor
n Kolmogoroff microscale [L]
e potential temperature [T]
K thermal diffusivity (air:0.21 cm2/s; water:0:0014 cm2/s)
A resistance coefficient for pipe flow
A wavelength corresponding to spectral peak [L]
v stability parameter
v kinematic viscosity (air:0.15 cm2/s; water:0.01 cm2/s)
£ time separation [T]
p density (air: 1.3 g/1 ; water:! g/cm3) or autocorrelation function
o standard deviation
T fluctuating temperature (deviation from mean) [T]
u Kolmogoroff velocity [L/T]
*h nondimensional potential temperature gradient
*u nondimensional horizontal (u plus v) turbulence intensity
xii
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$ nondimensional wind speed gradient
$ nondimensional longitudinal turbulence intensity
$ nondimensional lateral turbulence intensity
$ nondimensional vertical turbulence intensity
w
*. nondimensional intensity of temperature fluctuations
o
x nondimensional concentration
uj earth's rotation rate [T ]
Subscripts and Special Symbols
( )„ ambient value
a
( ) equilibrium value
( )- field value
( ) geostrophic value
( )L Lagrangian value
( ) model value
( LY maximum value
IHA
( ) value of quantity in neutral atmosphere, except as noted
( ) prototype value
( ) reference quantity
( ) stack value
( ) value of quantity in x-direct1on
^
( ) value of quantity in y-direction
( )2 value of quantity in z-direction
( )at free stream value
( )' nondimensional quantity
IT average value
( ) vector quantity
xiti
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ACKNOWLEDGEMENTS
The author wishes to acknowledge the many helpful discussions with his
colleagues and, in particular, with Dr. Ian Castro, University of Surrey, Dr.
Rex Britter and Dr. Julian Hunt, University of Cambridge, Dr. Igor Nekrasov,
University of Moscow, and Dr. Gary Briggs, National Oceanic and Atmospheric
Administration, all of whom spent extended periods of time at the Fluid Model-
ing Facility. Mr. Alan Huber's persistent prodding saw this work to fruition.
Mr. Roger Thompson very willingly engaged in a large number of discussions and
expositions over the course of several years. Mr. Robert Lawson kept the lab-
oratory running smoothly, allowing me to concentrate on the manuscript. Many
people provided comments on the drafts dated March or June 1979; especially
thorough or thought-provoking reviews were provided by Dr. Alan Robins, March-
wood Engineering Laboratories, Central Electricity Generating Board, Dr. Robert
Meroney, Colorado State University, Dr. Gary Ludwig and Dr. George Skinner,
Calspan Corporation, Dr. James Halitsky, Croton-on-Hudson, NY, Dr. John
Wyngaard, National Center for Atmospheric Research, and Dr. Frank Binkowski,
National Oceanic and Atmospheric Administration. Mr. Mike Shipman, Northrop
Services, constructed many of the graphs as well as a library program that
made the list of references a breeze. Miss Laurie Lamb, Miss Tammy Bass and
Ms. Carolyn H. Coleman painstakingly typed various versions of the drafts and
final manuscript. Finally, my wife Hazel and children JB and Jennifer endured
through it all. To each of these, I express my sincere thanks.
xi v
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1. INTRODUCTION
Most mathematical models of turbulent diffusion in the lower atmospheric
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 future
role of large computing machines in following the consequences of the Navier-
Stokes equations under random initial conditions, estimated a required memory
size of ~1013 bits, then asked if "the foregoing estimate is enough to suggest
the use of analog instead of digital computation; in particular, how about an
analog consisting of a tank of water?" (emphasis added). In spite of the tre-
mendous advances in computer memories in the past two decades, Corrsin's remark
1s 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
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atmospheric dispersion to be included in a model. Normally, the similarity
criteria are conflicting in some sense; it may be necessary to model one phys-
ical process at the expense of not being able to 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 provided
in Chapter 3. Air and water are most commonly used as media for the simulation
of atmospheric motions. The potentials of both of these fluids are reviewed in
Chapter 4.
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2. FUNDAMENTAL PRINCIPLES
A discussion of the fundamental principles for fluid modeling of atmos-
pheric 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 ac-
counted for in the fluid model by developing appropriate boundary conditions.
These are discussed 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
^L (2.1)
« CXj Q0 CXf TO CX»C.Vk
Continuity
^ = 0 (2.2)
C.Vj
Energy
cdT cST c2dT ln „»
— + — Vt = K^^ (i =1,2,3) (2.3)
ct cxj CA-.C.Y,
where the x., axis is taken vertically upward, U. is instantaneous velocity,
6P 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,
eiik 1S ^e alternatin9 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), 6^. is Kro-
neker's delta (fi^. = 1 if the two indices are equal and 0 if unequal), and the
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
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equations of motion through the use of appropriate reference quantities.
Reference quantities assumed to be supplied through the boundary conditions
are: L, length; UR, velocity; pR, density; 6TR, temperature deviation; and
nR, angular velocity. The dimensionless variables are
, *i v,
x = - U = —
L ' UR
<'=^t Q>,«>
L QR
.
o,
Using these definitions to nondimensionalize Eqs. 2.1 to 2.3 yields
2rr'
8U,' 2 , , 1 ddP' 1 , 1 d2Ul
(2.5)
and
ddT ddT 1 d26T
where RO=UP/LQD is the Rossby number,
K R
0} is the densimetric Froude number,
ResURL/v is the Reynolds number,
and PeHURL/K is the Peclet number.
Concerning the philosophy of modeling, Eqs. 2.4 to 2.6 with appropriate
boundary conditions completely determine the flow. The question of uniqueness
-------
of solutions of the Navier-Stokes equations is avoided here (see Lumley, 1970).
These five equations contain five unknowns, U.J, Ui, 111, 6P1 and 6T1, so that,
in principle, their solutions may be determined. Any two flows within the
same general category (i.e., insofar as they are governed by the above
equations) will be similar if and only if they are described by identical
solutions to the given set of Eqs. 2.4 to 2.6. Solutions to this set of
equations will be identical if and only if the coefficients Ro, Fr, Re, and
Pe and the nondimensional boundary conditions are identical (Birkhoff, 1950;
Batchelor, 1953a). The final statement, then, as it applies to laboratory
modeling of atmospheric motions becomes: any atmospheric flow which can be
described by Eqs. 2.4 to 2.6 may be modeled by any other flow which can
also be described by the same set of equations, provided that the Rossby,
Froude, Reynolds, and Peclet numbers are identical, and provided that the
nondimensional boundary conditions (to be discussed later) are identical.
Equations 2.4 to 2.6 apply to both laminar and turbulent flows. It is not
necessary to determine a priori whether the flow is laminar or turbulent.
Thus far, only the requirements for similarity of flow patterns have
been discussed. The dispersion of a pollutant within the system will now
be considered. The contaminant is assumed to be completely passive in the
sense that it is without effect on the governing equation and undergoes no
transformations in the fluid.
Prediction of the dispersion of this contaminant in space and time is
desired. Since a passive contaminant is specified, its dynamic behavior must
be the same as that of the air; hence, the similarity criteria describing
its dynamic behavior have already been specified. One additional parameter,
however, remains to be specified. This is obtained through nondimensionali-
zation of the molecular diffusion equation:
6
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where x represents the instantaneous concentration and a is the molecular
mass diffusivity. Nondimensionalization of this equation requires
x1 = X/XR, and yields
W+ ' ~dx'., ~ ReSc dx-dx'i , (2.8)
where Sc = v/a is the Schmidt number.
To summarize, Equations 2.4 to 2.6 and 2.8 form a complete set of equa-
tions which govern the dispersion of a dynamically passive contaminant in
the atmosphere and in a model. If and only if the nondimensional coefficients
in these equations and the boundary conditions are identical, the dispersion
of the contaminant in a model will be identical to that in the atmosphere.
2.2 THE DIMENSIONLESS PARAMETERS
It is generally impossible to simultaneously match all of the dimensionless
parameters when the ratio of the length scales (i.e., prototype to model) is
greater than about 10. As an example, consider the Reynolds and Froude
numbers: UL UR
Re = , Fr = , .
If one models in a wind tunnel, the values of v and g are identical to values
in the atmosphere, and T is roughly the same. Thus, decreasing the length
scale by 10 requires an increase of 10 1n the velocity to satisfy the Reynolds
number criterion. Considering the Froude number, L decreased by 10 and U~
-------
increased by 10 implies that 6TR must be increased by a factor of 1000, which
is, of course, highly impractical. In general, a length scale reduction much
greater than 10 is desired.
A factor of 15 in the Reynolds number may be gained by modeling with
water as the fluid medium, but then the Peclet number and Reynolds number
criteria cannot be satisfied simultaneously. The Peclet number can be writ-
ten as the product of the Reynolds number and the Prandtl number. Even if
the Reynolds number can be matched, the Prandtl number cannot, because it is
a fluid property and differs by a factor of 10 between air and water. The
Prandtl number is not, however, a critical parameter (see later discussion).
Many 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 of the criteria may be relaxed. In the
first example, if the atmospheric flow were of neutral stability, the Froude
number would be infinite. This is easily accomplished by making the model
flow isothermal. (The vertical dimension of a typical wind tunnel is small
enough that the temperature differences between isothermal and strictly neu-
tral conditions is extremely small. Hence, "neutral" and "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 nondimensional parameters in de-
tail.
2.2.1 The Rossby Number, UR/LnR
The Rossby number represents the ratio of advective or local accelera-
P
tlons (UR/L) to Corlolis accelerations (proportional to URnR). 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 r- 4. DIFFUSION (NO
-2. INITIAL DIFFUSION \ FURTHER TILTING)
\3. CROSSWIND TILTING
(NO DIFFUSION)
Figure 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). At this point,
the upper levels of the cloud will be advected in a different direction from
that of the surface wind. Conceptually, the tilting of the cloud is imagined
to occur independently of diffusion, whereas, in reality, tilting and diffu-
9
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sion occur simultaneously. The cloud is tilted by the crosswind (Step 3), and
the simultaneous turbulent diffusion (Step 4) increases the width of the cloud
at ground level over what it would have been by diffusion alone. The center of
gravity of a slice of the cloud at ground level will not follow the surface wind.
The Rossby number describes the relative importance of the Coriolis ac-
celerations when compared with advective, or local accelerations. If the
Rossby number is large, Coriolis accelerations are small, so that enhanced
dispersion due to directional wind shear may be ignored. Equivalently, a
near infinite Rossby number is automatically matched in a model.
To date, all modelers have assumed a large Rossby number and discarded
terms involving it in the equations of motion, or, equivalently, ignored it as
a criterion for modeling of diffusion. (Howroyd and Slawson, 1975, and others
have simulated the Ekman spiral using annular wind tunnels and rotating tanks,
but diffusion in such boundary layers has not been attempted. Such facilities
are not yet practical for the types of studies discussed here.) Cermak et al.
(1966) and Hidy (1967) made the rather broad statement that, provided the typ-
ical length in the horizontal plane is less than 150km, the Rossby number can
generally be eliminated from the requirements for similarity. McVehil et al.
(1967) ignored the Rossby number when modeling atmospheric motions on the scale
of one kilometer in the vertical and several tens of kilometers in the horizon-
tal. Ukeguchi et al. (1967) claimed that the cutoff was 40 to 50km. Mery (1969)
claimed that the Coriolis force may be neglected if the characteristic length
is less than 15km. The present discussion shows that the cutoff point is on the
order of 5km for modeling diffusion under appropriate atmospheric conditions,
i.e., neutral or stable conditions in relatively flat terrain.
The criterion is based on a length scale rather than on the Rossby num-
ber itself because the angular rotation of the Earth, n , is a constant
w
-------
(Ro = UD/Lfi ), and the characteristic velocity of the atmospheric flow does
K 0
not vary by more than an order of 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 (1962) 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 a <*t3' in a uniform shear flow (a is stream-
X A
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 not consider turning of wind with height, although
similar considerations are involved). For a semi-infinite flow, ground-level
source (puff), and linear velocity profile, he also found o «t ' . For a com-
A
3/2
pletely unbounded flow, Smith (1965) showed that a «t , using statistical
/\
techniques.
Since the contribution to the spread from the turbulence alone is
1 /o
o «t ' , it is clear that the shear effect will eventually dominate the dis-
J\
persion process. These solutions are valid only for constant diffusivity and
large times; they do not provide any indication of 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 on dispersion. They used an
1°
8-step Ekman spiral except with the surface wind 22*- from the geostrophic
wind. They claimed that the 3/2 power law does not apply because the cross-
11
-------
wind shear is not constant with height. Their estimates indicate that the
wind shear become? dominant around 4 to 6 km from the source. For much larg-
er distances, the turbulence will again dominate because the shear goes to
zero for large heights.
Hogstrom (1964) and Smith (1965) used statistical approaches to the
crosswind shear problem. Homogeneous turbulence, a mean wind that was constant
with height, and a crosswind component that varied linearly with height were
assumed. They obtained expressions valid for all times of travel, but did
not estimate times or distances at which the shear would dominate.
Csanady (1969) attempted to confirm analytically the numerical results
of Tyldesley and Wallington. Because of analytical difficulties, he confined
his investigation to a slice at ground level of a cloud released from a source
at ground level. He found that, indeed, the centroid of the slice at ground
level did not follow the surface wind. By the time the cloud occupied 1/3 of
the Ekman layer (~600m), its distance from a line parallel to the surface
wind was of order 10 km. The contribution to the spread from the turbulence
and from the shear were found to be equal 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
1 /2
times, a «t ' (i.e., turbulent diffusion dominates). For intermediate times,
3/2 1/2
a «t (i.e., shear-induced diffusion dominates). For large times, a at '
•J J
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 width). Maul (1978)
extended Csanady's approach and derived expressions to include effects of
source height: and finite depth of mixing.
Only a few diffusion experiments have been reported that have been spe-
cifically designed to examine the relative effects of turbulence and shear.
12
-------
PasquHl (1969) reexarcinec! tv/o independent studies that contain information
of interest in this connection. He looked at Hogstrom's (1964) data on the
behavior of smoke puffs released from an elevated source under neutral and
stable atmospheric conditions. The data for crosswind spread show the onset
of a more rapid than linear growth at 2.5 km. These data are somewhat mis-
leading because they indicate the total puff width rather than the width at a
given level. Hence, they indicate bodily distortion (tilting) of the puff,
but do not show directly enhanced spread at a given level. In accounting for
this, Pasquill concluded that 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, et al (1964) (continuous 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 sum-
marized:
"...a bodily crosswind distortion of the plume from a point source
(either elevated or on the ground) sets in between 2 and 3 km. How-
ever, the form of the crosswind growth curves suggests strongly that
the communication of the distortion to the spread at a given level
was not of practical importance below about 5 km in the case of the
elevated source and about 12 km in the case of the ground level source.^
Thereafter, the implication is that the shear contribution is dominant."
Limited field measurements by Brown and Michael (1974) suggested that
directional wind shear was the dominant factor in plume dispersion at great
distances downwind (^50 km) during stable conditions. More extensive experi-
mental measurements by Howroyd and Slawson (1979) showed qualitative and
quantitative agreement with the analysis of Pasquill (1969).
The Rossby criterion should be considered, therefore, if diffusion model-
ing is desired in a prototype with a length scale greater than about 5 km,
under neutral or stable atmospheric conditions, in relatively flat terrain.
One encouraging note is that Harris' (1968) high wind (neutral stability)
results show no systematic variation in wind direction with height over flat
13
-------
terrain up to z*200m.
2.2.2 The Reynolds Number. UrL/v
The physical significance of the Reynolds number becomes apparent by not-
ing that it measures the ratio of inertial forces (UJ^/L) to viscous or fric-
2
tional forces (vUR/L ) in the equations of motion. It imposes very strong lim-
itations 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 Reynolds number criterion
were required, no atmospheric flows could be modeled.
Various arguments have been 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 may 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 (1941) 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.. = IT- + u..), and the equation is then averaged,
the following equation is obtained (after minor manipulation):
80t .fjdU: 1 ddP g — S2Ut d^Tj
——h Uj — \- 2eijklljUk = 1 oTo3i + v . /o n \
at dXj QO oXf T0 dXjdXj dXj \£- -y I
An eddy viscosity is defined to relate the Reynolds stress to the mean veloci-
ty, -(u.jU.)=K(3U../9x.). The nondimensional equation is then
14
-------
du; ,du: i _ i ddp1 i — i d2u; i a2t//
where ReK = URL/K is called a "turbulent" Reynolds number. Now if K/v is of
3
order 10 , the term containing the turbulent Reynolds number is much larger
than the term containing the ordinary Reynolds number. If the nondimensional
equation for laminar flow were now written, it would appear identical to Equa-
tion 2.10, with the exception that the term containing the turbulent Reynolds
number would be absent. Assuming that the prototype flow is turbulent and
3
that the model flow is laminar, the scale ratio is of the order 1:10 , and UR
is the same order of magnitude in model and prototype, then,
(Re)model = (ReKJprototype.
Hence, similarity may be established by modeling a turbulent prototype flow
by a laminar model flow when the scale ratio is on the order of 1:10 , 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 of sub-
stantial variation in the mean flow. Turbulent diffusion is a flow property,
not a fluid property. The laminar flow analogy assumes unrealistically that
eddy sizes are very small compared with the scale of variation of 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 some realistic modeling possibilities, but
15
-------
the chances of that being the case without any doubt are small.
Perhaps qualitative use of this technique is the answer to those (non-
diffusive) problems where the spread of a contaminant is controlled 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 1:50,000 using this analogy. Cermak et al. (1966) claimed
that "the model flow patterns obtained were not even qualitatively close to
that (sic) observed in actual field tests" (the original paper was not avail-
able for verification). Cermak and Peterka (1966) made a second study of the
wind field over Point Arguello, California (a peninsula jutting into the
Pacific Ocean). Cermak et al. (1966) stated 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 Arguello area for 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 dissimilarity in surface flow patterns, this agreement is regarded as
fortuitous.
Since kinematic viscosity is a fluid property, it is not 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 fluid
modeling techniques in the sense that K is a controllable variable in the math-
16
-------
ematlcal model (e.g., a function of height, stability and direction). The
quantity K 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 flow system, whose boundary conditions
are expressed nondimensionally in terms of a characteristic length L and ve-
locity UR, the turbulent flow structure is similar at all sufficiently high
Reynolds numbers (Townsend, 1956). Most nondimensional mean-value functions
depend only upon nondimensional 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 no-slip condition is a viscous
constraint). The viscosity has very little effect on 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 and Section 3.3.1).
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 fortunate phenomenon from the
standpoint of modeling. The gross structure of the turbulence is similar over
a very wide range of Reynolds numbers. This concept is used by nearly all mod-
17
-------
elers. It is graphically illustrated in Figures 2 and 3. The two 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.
Figure 2. 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 50 times that of the lower. (Reproduced with
permission from Illustrated Experiments in Fluid Mechanics,
National Committee for Fluid Mechanics Films, copyright 1972
Education Development Center, Inc., Newton, MA).
Figure 3. Shadowgraphs of the jets shown in Fig. 2. Note how much finer
grained is the structure in the high Reynolds number jet than
that in the low Reynolds number jet. (Reproduced with per-
mission from Illustrated Experiments in Fluid Mechanics, National
Committee for Fluid Mechanics Films, copyright 1972, Education
Development Center, Inc. Newton, MA).
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
Taylor (1921) diffusion equation. Taylor's expression for the mean-square
fluid particle displacement in a stationary homogeneous turbulence is given by
T t
= 2? J J
0 0
(2.11)
where v2" is the variance of particle velocities, pU)=v(t)v(t+c)/vz(t) is the
Lagrangian autocorrelation of particle velocities with time separation £, and
* *
T is the time of travel from the source. Kampe de Feriet (1939) and Batchelor
(1949) applied the Fourier-transform between the autocorrelation and the cor-
responding Lagrangian spectrum function
FL(n) E f p(t) cos (2irnt) dt,
o
to obtain
.. (2.12)
where n is the frequency. The squared term under the integral is the filter
function illustrated in Figure 4; it is very small when n>l/T and virtually
unity when n<0.1/T. For very small travel times, the filter function is
virtually unity, so that all scales of turbulence contribute to the dispersion
with the same weight that they contribute to the total energy. For larger
travel times, larger scales of turbulence progressively dominate the dispersion
process. This means that eddies with diameter smaller than about one-tenth
the plume width do not significantly affect the spread of the plume; only
those eddies with diameter about the same size as the width of the plume and
larger substantially increase its width.
19
-------
0.01/T
0.1/T 1/T
FREQUENCY, n (Hz)
10/T
Figure 4. Filter function in Eq. 2.12: As the travel time T in-
creases, the contribution of small scale (high frequency)
motions to dispersion diminishes rapidly.
It may 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 more easily understood. Figure 5 shows a spectrum
Su(n) - 4
u(t) u(t+t') cos 2imt' dt1
of wind speed near the ground from a study by Van der Hoven (1957). It is
evident that wind effects can be separated roughly into two scales of motion:
20
-------
large scale (low frequency) motions lasting longer than a few hours, and small
scale (high frequency) motions that last considerably less than an hour. The
large scale motions are due to diurnal fluctuations, pressure systems, passage
1 1 1 1 1 I I
SPECTRAL GAP * TURBULENCE
Cycles/hr
Hours
100
0.01
1000
0.001
Figure 5.
Spectrum of wind speed at 100m. (Reprinted with permission
from J. Meteorology, American Meteorological Society, van
der Hoven, 1957.)
of frontal systems, seasonal and annual changes, etc., and are generally
called weather. The small scale motions are associated 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 from turbulence is a very fortunate occurrence,
both from an analytical viewpoint and from a fluid modeling viewpoint.
Because of this gap, it is possible to consider these regions independent-
ly and to execute proper mathematical operations to determine the statistical
properties of the two 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 of 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 not 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 K instead of frequency n so that
ENERGY
CONTAINING
-RANGE
INERTIAL
- SUBRANGE -
I
I
I DISSIPATION!
RANGE j
I
in
Figure 6.
Form of turbulence spectrum. (Reprinted with
permission from Workshop on_ Micrometeorology,
American Meteorological Society, Wyngaard, 1973.)
we will be more inclined to think in terms of length scales. The relation-
ship is K=2irn/Lf. (The spectrum function S in Figure 5 is one-dimensional,
whereas that of Figure 6, E, is three-dimensional. The differences need not
concern us here.) Note that an integral scale I and a microscale n are de-
fined. The integral scale may be thought of as the characteristic size of the
22
-------
energy-containing eddies and Is located at the peak of the three-dimensional
spectrum. The microscale may be thought of as characteristic of the smallest
eddies in a turbulent flow. They are the ones primarily responsible for the
dissipation (e) of turbulent kinetic energy. The ratio of the integral scale
to the microscale, then, is a measure of the width of the spectrum or the
range of eddy sizes in the turbulence.
A pertinent question at this point is: how does the spectrum of turbu-
lence in a simulated atmospheric boundary layer in, say, a wind tunnel compare
with that in the real atmospheric boundary layer?
1nE(K)
RE
INCREASING
I/I 1/T? Inx
Figure 7. Change of spectrum with Reynolds number.
Figure 7 shows how the spectrum shape 1s affected by changing the Reynolds
23
-------
number. For a given class of turbulent flow, a decrease of the Reynolds num-
ber decreases the range of the high-frequency end of the spectrum, whereas the
size of the energy containing eddies changes only very slowly with Reynolds
number.
To be somewhat more quantitative, it is useful to examine the trends ob-
served experimentally and theoretically in grid-generated turbulence. A good
measure of the width of a turbulent energy spectrum is the ratio of the inte-
gral scale, I, to the Kolmogoroff micrbscale, n. It may be shown, through ar-
guments presented by Corrsin (1963), that this ratio is
^Re3/4, (2.13)
where Re is the grid Reynolds number (based upon upstream velocity, U, and
mesh size, M). Ideally, both I and n would be reduced in the same proportion
in a model (i.e., the geometrical scale ratio), so that the width (number of
"decades") of the spectra would be identical in model and prototype. It is
clear, however, that this would require identical Reynolds numbers in model
and prototype. Eq. 2.13 may be used to estimate the comparative widths of
model (m) and prototype (p) spectra:
3/4
3/4
(2.14)
assuming that the ratio of flow speed to viscosity is roughly the same in mod-
el and prototype. (L is a characteristic length in the flow, for example, the
mesh size or the height of a building.) Hence, at a scale ratio of 1:1000, a
seven-decade-wide atmospheric spectrum is "modeled" by a 4i-decade-wide labora-
tory spectrum. This appears to be a drastic reduction in spectral width, but
observations of grid-generated flows show that only the high-frequency end of the
24
-------
spectrum is cut out, so that this reduction in spectral width has insignificant
effects. It is found empirically that I/M at a fixed distance x/M downstream
from a grid is nearly independent of Reynolds number (Corrsin, 1963). Similarly,
it may be expected in other flow geometries that I/L at corresponding geometrical
locations will be roughly independent of Reynolds number, i.e.,
'i~-'t. <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
3/4 /r \l/4
„ T /T \3/4 /r \
n'~ MM ~[-?i
^~~T\ j I ~\T I
fm Jm \Lp/ \LmJ
In summary, integral scales reduce with the first power of the geometrical
scale ratio (as desired), whereas Kolmogoroff microscales reduce with only the
one-fourth power of the geometrical scale ratio. As we have seen in our pre-
vious discussion, the largest eddies contribute to the spread of a 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 Kolmogoroff mi-
croscale in the atmosphere is about one millimeter between 1 and 100 m above
ground (Lumley and Panofsky, 1964). As indicated by Eq. 2.14, at a scale ra-
25
-------
tio of 1:500, the spectral width in a model would be approximately two orders
of magnitude smaller than desired. The Kolmogoroff microscale in the model
would be about 100 times larger than required by rigorous similarity. This
would correspond to a Kolmogoroff microscale of 10 cm in the atmosphere. It
is difficult to imagine a practical atmospheric diffusion problem where eddies
smaller than 10 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, whereas they would best have been
posed in terms of Lagrangian coordinates, as required in Eq. 2.12. However,
it is reasonable to assume that if the Eulerian spectra and scales of model
and prototype are similar, then so will be the Lagrangian spectra and scales;
more appropriately, if Eulerian spectra differ in certain respects between
model and prototype, then the Lagrangian spectra will be 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
magnitude).
Concerning puff (relative) diffusion, the problem is somewhat different.
Corrsin (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 more strongly upon the model
Reynolds number. Intuitively, it appears that reasonable results would be ob-
tained if the Kolmogoroff microscale were small compared with the initial puff
width.
26
-------
The discussion of the Reynolds number criterion in the modeling litera-
ture normally centers around sharp-edged geometry where it is usually stated
that the mean flow patterns will 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, providing 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., those charac-
terized by the Rossby and Froude numbers). The answer to this question is rea-
sonably well known for simple flow classes such as jets and cylinder wakes,
but is largely unknown for models of atmospheric motions. Specific recommen-
dations on minimum Reynolds numbers to be achieved in various classes of flows
will be made in Section 3, Practical Applications.
2.2.2.3 Dissipation Scaling
A hypothesis on the similarity of the detailed turbulence structure of
model and prototype flows was proposed by Nemoto (1968). Again, a basic as-
sumption is that thermal and Coriolis effects are negligible. He reasoned
that mean flow patterns of both the model and 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
isotropic1 (Kolmogoroff, 1941),
27
-------
(2) the Kolmogoroff velocity, u, and microscale, n» characterize the
turbulence at each point in the flow.
Assumption 1 is satisfied if the Reynolds number is very large (Hinze,
1975). Using assumption 2, Nemoto reasoned that the turbulence structures of
model and prototype flows would be similar when
Lp (2.17)
and
o. Ua
r "f
(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:
_»*_
* (2.19)
From Eqs. 2.17, 2.18 and 2.19, it may be deduced that:
URm AmY/3 /LmY/3
UR" ^£p' ^Lp' (2.20)
Eq. 2.20 is the similarity criterion proposed by Nemoto. He has also shown
how the above relationship may be obtained from a special nondimensionaliza-
tion of the turbulent energy equation.
It is agreed that the turbulence structures of model and prototype flows
would be similar if Eqs. 2.17 and 2.18 could be satisfied. However,
it is impossible to satisfy Eq. 2.17 using typical length scale reductions
(i.e., 1:300 to 1:1000). As mentioned previously, the Kolmogoroff microscale
in the lower layer of the atmosphere is about 1mm. Typical values of
n in laboratory flows are 0.075mn (Wyngaard, 1967), and 0.5mm (Snyder and
28
-------
Lumley, 1971). Measurements 1n very high Reynolds number laboratory flows
(Klstler and Vrebalovich, 1966) show that the smallest Kolmogoroff microscale
which can reasonably be generated in the laboratory is 0.05 mm. Thus, the
largest ratio of microscales is of order 20, which is far short of the
typical ratio required (i.e., 300 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 shown by the usual nondi-
mensionalization of the turbulent energy equation (Cermak et al., 1966).
2.2.3 The Peclet Number and the Reynolds-Schmidt Product
The Peclet number is most easily discussed by writing it as
URL URL v
V K
where Pr is the Prandtl number.
The Reynolds-Schmidt product may be written as
v a
Both of these dimenslonless parameters have the same form (i.e., the product
of a Reynolds number and a ratio of molecular transport coefficients). Both
the Prandtl and Schmidt numbers are fluid properties and not flow properties.
The Prandtl number is the ratio of the momentum diffusivity (kinematic vis-
cosity) to the thermal diffusivity. The Schmidt number is the ratio of the
momentum diffusivity to mass diffusivity.
For air, the Prandtl number does not vary strongly with temperature.
When air is used as the medium for modeling, the Prandtl number is nearly the
same 1n model and prototype, and, 1f the Reynolds numbers were the same, the
29
-------
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 10 larger than it is in air, and it varies rather great-
ly with temperature. Thus, from the standpoint of rigorous similarity, it
does not appear that water would be a suitable medium in which to do model
studies.
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
other 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
would 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-
Schmidt 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 flow 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 for mod-
eling, from this standpoint. Further discussions of air versus water are giv-
en in Chapter 4.
A cautionary note is advised here. The foregoing comments were primarily
intended to apply to diffusion of a plume from a smokestack, and the implication
was that if the Reynolds number was large enough, then so would be the Peclet
number and Reynolds-Schmidt product. However, experiments involving molecular
diffusion over smooth surfaces, such as heat transfer from model buildings
(Meroney, 1978) and water evaporation from a simulated lake (Cermak and Koloseus,
31
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1954), suggest that Peclet numbers and Reynolds-Schmidt products need to be
quite large for similarity, i.e., 500,000 or greater. This is evidently due to
the existence of laminar sublayers beneath the turbulent boundary layers in the
models, where molecular diffusion may have been the controlling parameter. It
is conceivable that this problem could have been overcome by roughening the
model surface (see further discussion in Section 3.3.1), although this is not
a straightforward solution because the roughness would, of course, increase the
surface area for heat and mass transfer, and possibly provide undesireable
insulation.
2.2.4 The Froude Number, Un/(gl-STn/T.)1/2
~~~~ — ~—~— ~ " K K 0
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 small compared to inertial forces. Thus, thermal effects become
important as the Froude 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
2
replace U^/L by a representative velocity gradient and 6TR/L by a representa-
tive temperature gradient. Substitution of these gradients into the expres-
sion for the Froude number yields,
g L2 5TR g (ddTldz)R
Thus, the Froude number may be regarded as the inverse square root of a Rich-
ardson number. It is also related to the Monin-Obukhov (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-dimensional i
zation of Eq. 2.1. Batchelor (1953b) has also discussed the conditions under
32
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which this parameter is the sole governing criterion for dynamical similarity
of motions of a perfect gas atmosphere.
A different interpretation of the Froude number is quite useful in
considering stably stratified flow over hilly 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
Fr2 = PU2 /(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 raise a fluid element from the
base to the top of the hill. It is clear that if the Froude number is much 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
in upstream blocking of the flow below the hill top (Long, 1972). For a
three-dimensional hill, the fluid, rather than being blocked, can go round the
hill (Hunt and Snyder, 1980). 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 an essential parameter to be matched when
modeling stably stratified flow over hilly terrain (see further discussion
in Section 3.4).
The Froude number is not, by itself, a difficult parameter to duplicate
in 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
33
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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 match
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
(i.e., maximum temperature difference of 200°C) in the model, it is necessary
to decrease the mean flow speed. To match the Reynolds number between the
model and the prototype requires that the mean flow speed be increased. This
conflict is resolved by matching the Froude number while insuring that a
Reynolds-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. The
common 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 20% in the dimensionless density 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 Kovasznay
(1971) have designed a rotating disk pump that 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 interesting
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 possiblity of providing the
proper boundary conditions of turbulent flow (see next section), which is
quite difficult in a still tank wtth a towed model.
34
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Hydraulic engineers and others (e.g., Halitsky, 1979) frequently define
the Froude number as U/(gL)^2 or U2/gL; at first glance, it would appear they
have omitted the dimensionless density difference Ap/p. However, the Froude
number was originally introduced in the field of hydraulics and was used to
describe the behavior of the free surface of rivers and channels, where the
density difference was that across the free surface, i.e.,
Ap/p = (pwater -pair)/pwater * K
P
The parameter U /gL thus has a very meaningful interpretation in the field
of hydraulics; it does include a density term, albeit one that degenerates
to unity. However, density differences in the vast majority of atmospheric
2
simulation problems are much less than unity, so that the parameter U /gL
does not have a meaningful interpretation in and of itself; it has a mean-
ingful interpretation only when combined with the density term. Further
support of this view comes from an examination of the nondimensionalization
of the equations of motion (e.g., see Eq. 2.4 or Halitsky, 1979); the para-
meter U /gL appears only in combination with the density term.
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 flow 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 other things, the
35
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non-dimensional boundary conditions were identical.
Batchelor (1953b) points out:
Regarding the boundary conditions, we do not 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 t1 at
all spatial boundaries and ui p1, p1 must be given as functions
of x1 at an initial value o7 t1; it seems certain that such a set
of Foundary 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
-------
Nearly all modelers have considered the specification of boundary condi-
tions from a different viewpoint, that Is, through the spectrum of turbulence
1n the approach flow. Armltt and Coum'han (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 1s required, particularly the low-frequency end of the spec-
trum. This idea 1s in agreement with the previous discussion on the contribu-
tion of the various scales of turbulence to the dispersal of contaminants.
Some control of the turbulence spectra 1n the approach flow is possible (see
Section 3.2).
2.3.2 Jensen's Criterion and Fully Rough Flow
The 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 1s required. This raises another question; how much detail is
necessary? From the standpoint of rigorous similarity, of course, every de-
tail of the prototype must be duplicated in the model. However, in view of
the fact that the Reynolds number will not be duplicated, the fine detail is
unnecessary. Jensen (1958) has suggested that, 1f the roughness length of
the prototype, ZQ, may be determined (or at least estimated), then it should
be scaled according to
zom
(the roughness length is a fictitious length scale characterizing flow over a
rough surface. For uniformly distributed sand grains of size e, the roughness
length Is typically e/30.)
37
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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 e will have very little effect
on the overall flow; hence they need not be matched in the model. Details a-
bout the same size as e 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 no-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 flow conditions, viscous stresses be-
come negligible. The irregularities then behave Hke bluff bodies whose re-
sistance is predominantly form 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 over an aerodynam-
ical ly rough surface is Reynolds-number independent. The criterion which in-
sures that the flow is aerodynamically rough is u*zQ/v>2.5 (Sutton, 1949),
where u^ is the friction velocity.
This is extremely fortunate from a modeling standpoint, because atmos-
pheric flows are almost always aerodynamically rough (Sutton, 1949). If model
flow conditions are chosen such that u*z^/v>2.5, one can be certain that the
o —
boundary layers are turbulent, so that such things as separation 'bubbles' and
wakes behind obstacles and transition, separation, and 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 on the model becomes qualitatively comparable to that on the proto-
38
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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 flow over the model (details about same size as z will not
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. In 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 basic attempts have been made by Chaudhry
and Cermak (1971). Similar considerations apply to the specification of the
boundary conditions of the density distributions when the modeling medium is
salt water.
Specification of boundary conditions on concentration distributions is,
in principle, easy. In practice, the difficulty would depend on the type of
problem to be studied. For example, if the problem were to determine the ef-
fect of a single source, the boundary conditions could be x'=0 initially every-
where and x'=constant 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.
39
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Fluctuating pressures are, of course, not controllable parameters in a model.
In current practice, the upstream boundary conditions on velocity and
temperature are 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 to those in the atmosphere. Others
(Ludwig and Sundaram, 1969; Armitt and Counihan, 1968; Mery, 1969) use artifi-
cial techniques for generating thick boundary layers over short distances. Mery
(1969) and Ogawa et al. (1980) have attempted to model both velocity and temp-
erature profiles using artificial techniques. Any of the present techniques
for boundary-layer generation appears to be suitable; all of them come reason-
ably close to matching the first two moments of the velocity/temperature distri-
butions. Boundary-layer heights should correspond with the geometric scale
ratio, but, as various studies have shown, if the depth of the boundary layer
is large compared with the model height, there is some scope for selecting
boundary-layer heights for convenience. Practical goals and techniques for
simulating the atmospheric boundary layer are discussed in Section 3.2.
2.4 SUMMARY AND RECOMMENDATIONS
Similarity criteria for modeling atmospheric flows in air and water have
been derived. Rigorous similarity requires that five nondimensional parameters
plus a set of nondimensional boundary conditions must be matched in both model
and prototype. It has been determined that the Rossby number should be con-
sidered when modeling prototype flows with a_ length scale greater than about
5_ km. under neutral or_ stable atmospheric conditions, i_n_ relatively flat
terrain. It is concluded that more work needs to be done 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 continued on
40
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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 rt is_ 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 jhs^ 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
(nondistorted models) ij^ required from the specification of_ zero velocity
ajb the solid boundaries. It was decided that details cif the prototype p_f_ size
smaller than the roughness length need not be_ reproduced j_n_ the model. Objects
about the same size as_ the roughness length need not be_ reproduced jm^ geometrical
form but a.n_ equivalent roughness nuisj^ be_ established. Over-roughening may be_
required to_ satisfy the roughness Reynolds 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.
41
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3. PRACTICAL APPLICATIONS
The fundamental principles of fluid modeling have been discussed in the
previous chapter. When it comes to the details of a particular model study,
however, many decisions must be made, and the fundamental principles frequen-
tly do not 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
1n modeling plume rise, the atmospheric boundary layer, flow around buildings,
and flow over 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
42
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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 neg-
ligible in comparison with its momentum. Sherlock and Stalker (1940) appear
to have done the first wind tunnel study relating to plume behavior. Specif-
ically, 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 1:300 scale model, but chose to use essentially
identical model and full-scale values of wind speed and effluent temperature.
One of their conclusions was "...the temperature of the stack gas is relativ-
ely unimportant as a means of controlling the downwash...". At the present
time, it is still not clear exactly what effect buoyancy does have on the one-
and-one-half-times rule, but is evident that the buoyancy was not properly
scaled 1n the Sherlock and Stalker experiments. Their experiments, even with
the very hot (400°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). More 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 1n density alone contributes to
43
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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 in the atmosphere. It is evident after only a little study that some of
these rules are conflicting and that all of them cannot be correct. Indeed,
this is currently a highly controversial topic that has generated much discus-
sion, correspondence and further experimental work over the past 3 to 4 years.
Several groups of experimenters claim to have done "the definitive tests", but
as yet, a consensus on modeling buoyant plumes has not emerged.
3.1.1 Near-Field Plume Behavior
Let us consider the simplest conceivable problem of a plume downwashing
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).
Figure 8. Plume downwash in the wake of a stack.
44
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We will suppose that the stack walls are thin and smooth, 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, Ap=0 and Fr=°°. Further, since
01 s s a
we are concerned only -with the local flow field near the top of the stack, we
have purposely omitted shear in the crosswind (9u/8z«U/D) as well as strat-
ification and turbulence in the approach flow. (Indeed, turbulence in the
approach flow may well affect the flow characteristics in the wake of a cyl-
inder (Goldstein, 1965, p. 430), and for a proper model study, both the
intensity and scale of the turbulence in the approach flow near the top of
the stack should be matched, but for purposes of the present discussion, this
effect is ignored.) To model this problem, we must match only 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
some critical value, its precise magnitude is irrelevant. There are really,
however, three Reynolds numbers in this problem, corresponding to three dif-
ferent classes of flow: one for the flow inside the stack W D/v, one for the
jet issuing from the stack and entraining ambient air (also WsD/v), 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 cylinder wake.
There appears, however, to be considerable disagreement concerning the partic-
ular values of these "critical" Reynolds numbers.
45
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Let us consider first the flow inside the stack and ascertain how we
would obtain similar velocity profiles at the stack exit. Because full scale
stacks result in very large Reynolds numbers (a typical small valve might be
10 ), and if we assume the flow is fully developed (our H /D»l assumption),
we can predict the shape of the velocity profile very closely (i.e., from
Schlichting, 1968, Fig. 20.2). By reducing the Reynolds number in our model,
it may be seen that the shape of the velocity profile will change only mar-
ginally; even though the resistance coefficient \ changes from 0.012 at Re=106
to 0.04 at Re=4X103 (ibid, Fig. 20.1), the power law exponent a in the velocity
profile U/Umx=(y/R)a changes from 0.12 to 0.17. (From another viewpoint, the
ratio of the mean to maximum velocity changes from 0.85 at Re=106 to 0.79 at
o
Re=4X10 .) In view of other possible uncertainties in the problem, such as
knowing the real shape of the velocity profiles Inside the stack (i.e., do
we have the 25 to 40 diameters required for full development of the velocity
profile), these small changes with Reynolds number are regarded as insignif-
icant. Hence, we conclude that similar exit velocity profiles will be obtained
at model Reynolds numbers of 4000. Indeed, the lower limit would appear to be
2300, i.e., the value required for the maintenance of turbulent flow in a pipe.
Note that, if our full scale stack had had rough internal walls, we could
have come even closer to matching the full scale velocity profile by roughen-
ing the inside of our model stack. For example, for a value of R/k =30.6 (R
is the pipe radius, k$ is the Nikuradse equivalent sand grain roughness), the
resistance coefficient \ would vary from 0.045 at Re=106 to 0.040 at Re=4X103
(ibid, Fig. 20.18) and the shape of the velocity profiles would have been es-
sentially identical.
46
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Let us consider second the flow immediately outside the stack. Ricou
and Spaulding (1961) have shown that the entrainment rate of momentum-domin-
ated jets in calm surroundings is essentially constant for Reynolds numbers
in excess of 25,000. Only minor variations were observed between 15,000 and
25,000. Substantial variations were observed below 10,000; the entrainment
rate was increased by more than 20%. Hence, if minor errors are acceptable,
this critical Reynolds number is 15,000.
Let us consider third the ambient flow around the outside of the stack.
It is useful here to consider the changes that occur in the flow pattern
around a circular cylinder as the ,-ynolds 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^lOO, they break down
and are shed alternately at regular intervals from the sides of the cylinder.
O C
This type of flow persists over a very wide range of Re (10 •
cylinder: one where the Karman vortex street (shedding vortices) is formed
47
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(lower) and one where the boundary layer becomes turbulent (upper). At still
larger Reynolds numbers, the transition to turbulence occurs earlier in the
boundary layer, so that the separation point moves forward on the cylinder,
and the drag coefficient increases gradually.
The drag coefficient in this instance is quite important, because it is
a reflection of the pressure distribution on the lee side of the stack and
will have fairly strong influence on the downwash of the plume (i.e., we can-
not ignore large variations in the drag coefficient as we did in the case of
the resistance coefficient for pipe flow). If the full scale Re is less than
about 105, then it is only necessary that the model Re exceed the lower cri-
tical Re of about 400, because the drag coefficient is essentially constant.
For a rectangular stack, because of the sharp corners forcing separation, the
lower critical Re would be lower still.
Modeling downwash around a full scale stack where Re^lO5, however, is
much more difficult. It may be accomplished by ensuring that the model Re
exceeds the upper critical Re of 10 which would require a large and/or fair-
ly high speed wind tunnel (i.e., a stack diameter of 10cm and wind speed of
15m/s). It is also possible to simulate a higher Re using the common wind
tunnel practice of tripping the boundary layer, either through use of a trip
wire or by roughing the surface of the cylinder, thereby forcing the boundary
layer to become turbulent (Goldstein, 1965, Figs. 162 & 163). This technique,
however, would gain at most a factor of two in Reynolds number; as the size
of the roughness elements or wire diameters increases, the sharp drop in drag
coefficient occurs at lower Re, but the magnitude of the drop is progressively
diminished; at Re £ 30,000, the roughness and wires are completely ineffective.
For a rectangular stack, because of the sharp corners forcing separation, the
upper critical Re would be much higher, possibly non-existent.
48
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First-hand experience with this problem was gained by Tunstall and
Batham (Robins, private communication). They followed on from Armitt's (1968)
work with cooling towers and investigated the feasibility of surface roughen-
ing to reproduce very high Reynolds number flow around model chimneys in sim-
ulated boundary layers. Their aim was to reproduce field measurements of the
pressure field on the Fawley Power Station chimney (200m x 20m, Re ^ 3x10 ).
This was achieved, but for scale reductions between 250 and 500, it was found
necessary to roughen the model with sand of about 0.5 mm dia. and to operate
c
the tunnel so that Re ^ 10 .
It should be noted that if we were not concerned with entrainment of
effluent into the wake, i.e., if we wanted to model a non-downwashed plume,
then the cylinder Re would be relatively unimportant in any event, so that
we need be concerned only with the critical Re (15000) for the effluent as
it exits the stack under ordinary circumstances. This simplified problem
would be relatively easy to model even in a fairly small wind tunnel (say,
0.5m square test section) with moderate wind speed (^20m/s) and a small stack
(1 cm 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
15000 value for the critical Reynolds number can be reduced. We will return
to our discussion of critical Reynolds numbers later in this section.
Notice that, provided the Reynolds number exceeds 15,000, there is only
one parameter of importance, WS/U. Since full scale stacks and effluent
speeds (even fairly small stacks and low speeds) result in huge Reynolds
numbers, 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, W /U, and not on wind speed per se.
49
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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 indep-
endent of wind speed per se. (This discussion may appear obvious and there-
fore trivial, but in demonstrating wind tunnel experiments 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 1)
Now let us complicate the problem, one step at a time, to see what addit-
ional issues arise in the modeling. Suppose the effluent is of high tempera-
ture, so that its density is, say, half the ambient air density. In a wind
tunnel, it is usually easier and more practical to use a lighter gas to simu-
late this high temperature field effluent than 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 flux 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. Overcamp and Hoult (1971) showed rather convincingly that the effect
of the increased buoyancy was to inhibit downwash of cooling tower plumes,
1 /2
where the Froude number W /(gDAp/p ) ' ranged from 0.2 to 2. Huber et al.
s a
(1979), however, observed enhanced downwash as the effluent density was
so
-------
decreased. The Froude 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 Froude number around 3; however, it must also be
a function of the effluent momentum to crosswind momentum ratio.
Most investigators would agree that the following set of parameters to
be matched for this more complex problem are sufficient (although, perhaps,
not all are necessary):
ws PS w
5> •* Do - ,- „
_,_,Re, - -r/2 (3J)
Since products of similarity parameters are themselves similarity
parameters, the following set is fully equivalent to that above:
_
5" > ~
p
.
(gDAP/pa)
(3.2)
The first parameter expresses the ratio of effluent momentum flux to cross-
wind momentum flux, and must be matched if the initial bending or the rise due
to the initial momentum of the plume is important. The last parameter is the
Froude 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 flow as it was introduced
in Section 2.2.4.)
A questionable parameter is the density ratio Ps/Pa per se. As mention-
ed previously, the density difference manifests itself through its effects on
51
-------
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 be realized. It is frequently advantageous to exaggerate the density
differences in the model in order to achieve low Froude numbers. Ricou and
Spalding (1961) showed very convincingly that the rate of entrainment dm/dx
in a highly momentum-dominated jet (no crosswind) obeys the relation
1 dm _ r
~~ j,, ~~ \f
P
a
dx
1/2
WSD , (3.3)
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
not much affected. However, if helium (S.G. = 0.14) were used as the buoyant
effluent from a stack in a wind tunnel, maintaining geometric similarity and
the ratio of effluent to wind speed WS/U, to simulate a full scale effluent
with specific gravity of 0.7, then the entrainment rate would be halved. (We
will refer to this phenomenon as an "impedance mismatch", analogous to its
usage in acoustic engineering as the ratio of pressure to volume displacement
in a sound-transmitting medium.) Hence, its rise due to initial momentum
would not be correctly modeled. Considerations such as these are evidently
what lead Hoult (1973) to state that the density difference Ap/p must not
a
exceed 0.4. But such a statement cannot be made unequivocally. If the den-
sity difference in the field were 0.5, as might be the case for a gas turbine
exhaust, then it would certainly be desirable to exceed 0.4 in the model.
52
-------
More importantly, as shown by Eq. 3.3, it is certainly possible to exaggerate
the density difference to 0.8 by using essentially pure helium as the model
effluent and still maintain the same entrainment rate by increasing the ef-
fluent flow rate W . But Eq. 3.3 of Ricou and Spalding applies only beyond a
few diameters beyond the stack exit (say >10D). In order to avoid an "imped-
ance mismatch" in our downwash problem, where we are concerned with the flow
behavior right at the top of the stack, it is necessary to match the density
ratio. Beyond a few diameters, it is only essential to match the momentum
flux ratio.
A major point of disagreement among investigators concerns the definition
of the Froude number. One group of investigators define the Froude number
with the effluent density as the reference density, Frg; another group defines
it with the ambient density (at stack top) as the reference density, Frfl.
Yet, nowhere is any reason given for the particular choice. It might appear
at first glance that the choice is completely arbitrary. However, given that
two plumes have the same Froude number based on effluent density, it is not
necessarily so that they have the same Froude number based on ambient density.
Consider the example (Table 1) of a typical power plant plume being modeled
at a scale of 1:400 using helium as the buoyant effluent in a wind tunnel.
The Froude numbers differ by a factor of the square root of the density
1 /2
ratios, i.e., Fr =(p /p) ' Fr , so that unless PC/P. is the same in model
a a s s bo
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 order to obtain large enough buoyancy in the plumes (low Froude
numbers). They do not match P../P,, as required by Eq. 3.2, so that it is not
s a
possible to match both Froude numbers simultaneously.
53
-------
TABLE 1: TYPICAL PARAMETERS FOR MODELING PLUME DOWNWASH.
Parameter
Ws
g
D
pa(V
PS(TS)
Fra
Frs
Re
Prototype value
20m/ s
9.8m/s2
10m
1.2g/l(20°C)
0.83g/l(150°C)
3.6
3.0
13xl06
Model value
1.67m/s
9.8m/s2
2.5cm
1.2g/l(20°C)
0.17g/l(20°C)
3.6
1.37
360
A third group of modelers define the Froude number using the wind speed
at stack top rather than the effluent speed. We will designate this Froude
number as Fr;; or FrjJ. Note that if Wg/U is matched (Eq. 3.1), then matching
Fra is equivalent to matching Fr and matching Fr is equivalent to matching
a as
22 U
Fr : if p,W_/p U is matched, then matching Fr is equivalent to matching
s s s a a
Fr and matching Fr is equivalent to matching (p_/pj ' Fr . which is not
s s s a s
the same as Fr.
a
This is a particularly vexing problem because most plume-rise theories
are founded on the assumption of small density differences, so that all
Froude numbers are essentially equivalent. There are only a few places in
the literature providing guidance on which Froude number is the most approp-
riate. Hoult et al. (1977) stated that two complete, independent wind tunnel
tests were run, one using the ambient density and the second using effluent
density as the reference. The tests involved the modeling of gas turbine
exhausts, which generally involve large effluent velocities and high effluent
temperatures. They claim (unfortunately, without presenting any data to
54
-------
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 90$, which was nearly 10 times
the error incurred using ambient density. It is important to note that using
ambient density as a reference corresponds to using effluent temperature as
a reference, i.e.,
VTa pa"ps , (3.4)
Ts = pa
due to the perfect gas law at constant pressure, pT=const.
A physical interpretation of the difference between the two definitions
is suggested in a footnote by Briggs (1972) (his comments applied specifically
to alternative 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 fluid being driven by the buoyant force:
(1) that the effective density is approximately constant=p , which is
reasonable very close to the stack, say within a few stack diameters
downwind;
(2) that the effective density is approximately constant=p , which is a
better approximation at all larger distances."
It is apparent, then, that in our stack downwash problem, we should use
the effluent density as the reference density in matching of Froude numbers.
It is also apparent that, if we were attempting to model far field plume rise,
we should use the ambient density as the reference density, in agreement with
Hoult et al. (1977). Fr is used as a scaling parameter by Isyumov et al.
(1976), Cermak (1971, 1975), Melbourne (1968) and Barrett (1973).
A second point of disagreement among modelers has to do with whether the
relevant parameter is the ratio of effluent speed to wind speed HS/U or the
55
-------
2 2
ratio of effluent momentum flux to crosswlnd momentum flux peW_/(p,U ). This
5 b a
may also be thought of as a ratio of dynamic pressures. Again, if PP/P, is
S a
matched between model and prototype, the choice is arbitrary. As discussed
above, however, most modelers drop the requirement of matching the density
ratio, and the choice is no longer arbitrary. Sherlock and Stalker (1940)
found that the behavior of the plume depended upon the ratio of the momenta
and that the ratio of the two speeds was a close approximation, provided that
both velocities were reduced to equivalent velocities at a common temperature.
Their one-and-one-half-times rule was thus based on the momentum ratio, a
fact 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, a lighter effluent entrains
heavier air, whereas in the water tank, a heavier effluent entrains a lighter
ambient fluid. It is conceivable that the entrainment 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
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
56
-------
of Hoult and Weil (1972) and Lin et al. (1974), both using salt water effluent,
appear to simulate field results quite well. It is clear that the problem is
not yet completely answered and requires detailed systematic study. A tenta-
tive conclusion, in view of 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 introdu-
tion of buoyancy makes this Re much more difficult to attain. In order to
\f
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 was only 360. Thus, we must determine
whether a lower critical Reynolds number can be justified.
The data of Ricou and Spalding (1961), which suggested Re =15,000, was
\f
applicable to momentum-dominated jets in calm surroundings. Most investi-
gators would agree that for a bent-over plume, the critical value may be
substantially lower, of the order of 2300, i.e., a value that is well-estab-
lished for the maintenance of turbulent flow in a pipe. This is equivalent
to saying that the plume behavior is independent of Reynolds number provided
that the effluent flow is fully turbulent at the stack exit. Lin et al.
(1974) have taken this one step further. They tripped the flow to ensure
that the effluent was fully turbulent at the stack exit at a Re of 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 of a buoyant plume in a calm and
stably-stratified environment and (2) the trajectory of a buoyant plume in a
stably-stratified crosswind compared reasonably well with other laboratory
and field data. Hoot, et al. (1973) and Nakai and Shikata (1977) have used
57
-------
similar techniques. Liu and Lin (1975), however, indicated that the place-
ment of the orifice relative to the top of the stack was critical at a stack
Re of 290. If the distance was smaller than required, the effluent flow was
governed by the orifice diameter and not the stack diameter; if larger, the
flow tripped by the orifice would laminarize before it reached the stack exit.
Isyumov and Tanaka (1979) reported on the influence of the shape of the
velocity distribution on subsequent plume behavior. The normally parabolic
(laminar) velocity profile of the effluent stream was "flattened" using a
somewhat larger diameter stack with a short contraction to the required in-
side diameter at the top of the stack. Experimental data presented showed
the effects of this improvement in the shape of the velocity profile to be
small. The Reynolds numbers, however, were approximately 30, so that the
"flattened" profile was assuredly laminar. In line with our previous dis-
cussion (see also discussion in Section 3.1.3.1) requiring a turbulent flow
at the exit, this technique is not recommended.
Wilson (private communication) reported that the shape of the stack exit
velocity profile as well as its turbulence level was important in determing
near-stack effects. He inserted a perforated plug in the stack three dia-
meters from the exit to help produce a flat velocity profile at the exit with
a turbulance intensity of about 20%. His experiments were conducted in a
simulated neutral atmospheric boundary layer.
Briggs and Snyder (1980) tested the rise of jets and buoyant plumes in
a calm, stably stratified salt water tank. Data relevant to establishing
critical Reynolds numbers are presented in Figure 9, where it may be seen
that both the maximum (overshoot) heights and equilibrium (final heights
are Reynolds number independent for large enough Reynolds numbers. For jets,
58
-------
0
100.0
10.0
1000.0 10000.0
REYNOLDS NUMBER
(a) Neutrally buoyant plumes
T
Ahn
(F/N3)1/4
Aheq
Ah
1
100.0 1000.0
REYNOLDS NUMBER
100000.0
10000.0
(b) Buoyant plumes
Figure 9. Variation of plume rise with Reynolds number (Ah =max1mum
niA
height reached by plume, Ah =equ111br1um plume height, Resource
momentum flux, F=source buoyancy flux, N=Brunt-Va1sala frequency).
53
-------
the critical Reynolds number was approximately 2000; for buoyant plumes, it
was approximately 200. Below these critical values, the plumes were laminar
at the stack exit, with resulting rises too high (the upward trend of the
curves at low Reynolds numbers). Limited attempts at tripping the flow in-
side the stacks yielded unpredictable results: sometimes the trips were
ineffective in establishing turbulent flow; in other cases, the trips were
overly effective: they enhanced entrainment to such a degree that the re-
sultant plume rise was too small (the, lower bound of the shaded areas in
Figure 9a).
Experiments with buoyant plumes in neutrally stable crosswinds, conducted
by Hoult and Weil (1972), show: (1) at a Reynolds number greater than 300, the
plume appears to be fully turbulent everywhere; at lower Reynolds numbers,
the plume becomes turbulent only some distance downstream 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 10 stack dia-
meters downwind) for Reynolds numbers below 300.
It is difficult to reconcile the results of the various sets of experi-
ments. There are numerous possible reasons why the critical Reynolds numbers
are different. The effluent flows differ (momentum-dominated versus buoyancy-
dominated), the stratification of the ambient fluids differ (neutral versus
stable stratification), and in some cases, the jets issued into calm environ-
ments, whereas in others, the plumes were bent over by a crosswind. The two-
order-of-magnitude difference in critical Reynolds number, however, is
difficult to explain. It is evident that a basic systematic study needs to
60
-------
be undertaken to establish that Reynolds number (perhaps different ones for
different sets of conditions) above which the rise and spread of model plumes
is independent of Reynolds number.
Equation 3.1 provides 4 basic parameters that, most investigators would
agree, provide for a complete or "exact" simulation of buoyant plume rise,
provided that a minimum Reynolds number requirement is met. However, if the
density difference is exaggerated, a "Pandora's box" is opened, allowing a
multitude of different interpretations and different sets of similarity par-
rreters. Table 2 provides a list of these sets of parameters as used (or at
least suggested) in various laboratories. Even though each of these sets is
unique, there are many more possible sets.
Some of these techniques have been "proven" through comparison with field
data, albeit on very limited bases, and others have been "proven" or "dispro-
ven" by comparison with "exact" simulations, i.e., matching the 4 parameters
of Eq. 3.1, although it is not always clear that minimum Re requirements have
been met. Notice that the CALSPAN technique does not match any Froude numbers;
they have combined the 4 basic parameters of Eq. 3.1, in a different fashion
to match buoyancy and length scales as used in Briggs (1969) plume rise eq-
uations (this will be discussed further in Section 3.1.3). In our near-field
problem, the flow at the mouth of the stack is important, so that exaggeration
of the stack diameter is unacceptable. Of the remaining techniques, only 3
have been "proven" (Robins, 1980; Isyumov and Tanaka, 1979; Hoult et al., 1977;
Wilson, private communication), but these tests have consisted primarily of
far field comparisons, i.e., maximum ground-level concentrations or plume rise
at x-lOH . The one near-field comparison of the CALSPAN technique "disproved"
its validity (Isyumov and Tanaka, 1979), but these results are questioned as
61
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TABLE 2: TECHNIQUES USED FOR SIMULATION OF BUOYANT PLUMES
AT VARIOUS FLUID MODELING FACILITIES.
SCALING
PARAMETER
Density ratio
WP,
a
Fra
Ws/(gDAp/p.)}1/2
^
lV(9DAp/P )]/2
d
Frs
ws/(gD/iP/Ps)1/2
^
U/(gDAp/ps)'/2
Velocity ratio
ws/u
Momentum ratio
pswf/(P/)
Momentum ratio
for distorted stack
"sD2wf/(PaH2U2)
Mass flow rate
P/Ws/(paHs2U)
Volumetric flow rate
D2Ws/(H2U)
Buoyancy length
VHs=gDipWsD/(4U3paHs)
Buoyancy length
^/Hs=gDAPWsD/(4U3psHs)
Momentum length
VHs=psWsD/<4Pau2V
Geometric scale
D/HS
Proven
Disproven
"EXACT"
SIMULATION
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
X
,
APPROXIMATE SIMULATIONS
UWO1.. CSU2/MIT3 CALSPAtl4 MONASH5 BRISTOL6 CERL7 CERL8 CERL9 CERL10
(DMA)" (DVA) (BMA) (EMS) (DMS) (DVS) (BMD)
1
10 no no no no no no no yes
no yes no no no no no no no
yes yes no yes yes no no no yes
yes no no yes yes no no yes no
no no no no no no yes yes yes
no yes no no no no no yes no
yes no yes yes yes yes yes no no
yes no yes no no yes yes no yes
no no no yes no no no no no
no yes no no yes no no yes no
no yes yes no no no no no yes
no no no no no yes no yes yes
yes no yes no no yes yes no no
yes yes yes no no yes yes yes no
X XXX X
XX XX X X X XX
1. University of Western Ontario, Isyumov et al., 1976. 2. Colorado State University, Cermak, 1971, 1975.
3. Massachusetts Institute of Technology, Hoult et al., 1977. 4. Calspan Corp., Skinner and Ludwig, 1978.
5. Monash University, Melbourne, 1968 (interpreted from Isyumov and Tanaka, 1980). 6. Bristol University,
Barrett, 1973 (interpreted from Isyumov and Tanaka, 1980). 7-10. Central Electricity Research Laboratory,
Robins, 1980. 11. Three-letter codes in parentheses are notations of Robins, 1980.
62
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indicated in our previous remarks; they are questioned for other reasons by
Ludwig (private communication). Because of these uncertainties, none of
the approximate techniques can be recommended for modeling the near-field
rise of buoyant plumes. The various techniques will be discussed further in
Section 3.1.3.
3.1.2 Summary and Recommendations on Modeling Near-Field Plumes
3.1.2.1 The Stack Downwash Problem
In summary, to model the problem where the effluent is downwashed into
the wake of a cylindical stack, it is recommended that:
1. If the full-scale Reynolds number based on the wind speed and out-
side diameter of the stack exceeds 10, it will be necessary to
5
exceed a Reynolds number of 10 in the model also. Some small
relaxation of this requirement (at most, a factor of 2) may be pos-
sible through the use of roughness or trip wires to force a turbulent
boundary layer on the cylinder surface.
2. If the full-scale Reynolds number (as in 1) is less that 10 , it will
be necessary that the model Re exceed 400.
1 /p
3. The parameters W_/U, p/pa and Wc/(gD/Ap ) ' must be matched between
5 S cl S o
model and prototype. It is expected that if Fr >4, the density dif-
a
ference tends to enhance downwash, so that a mismatch of Froude
numbers would lead to conservative results; if Fr <4, the density
a
difference will tend to inhibit downwash, so that matching all para-
meters is important.
Since there is only one possible reference length in this problem,
the stack diameter, it determines the scale ratio, and geometric scaling
is implicit, i.e., all lengths should be referenced to the stack diameter.
63
-------
Other lengths that may be important are the stack wall thickness and the
stack height. Obviously, these lengths should be scaled with the stack
diameter.
3.1.2.2 The Near-Field, Non-Downwashed Plume Problem
To model the problem where the plume is expected to disperse in the pres-
ence of aerodynamic effects of buildings, etc. (but not in the wake of the
stack itself), it 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 15,000.
(b) If it is necessary to reduce the effluent Reynolds number below
2300, it may be necessary to trip the flow to ensure a fully
turbulent exhaust.
(c) If it is desired to reduce the effluent Reynolds number below
300, it will be necessary to do some experimentation to deter-
mine under what conditions the plume will simulate the behavior
of a plume in the field.
2. The set of parameters to be matched (equal in model and prototype) is:
WS/U, ps/pa, Ws/(gDAP/ps)1/2.
This "exact" simulation will generally limit the scale reduction to less
than 400. Should the scale reduction exceed 400 or other techniques be des-
ireable, they should be "proven" by detailed comparisons with "exact"
techniques.
Again, the stack diameter determines the geometric scale ratio, and all
other lengths, such as building height, should be reduced by the same fraction.
64
-------
3.1.3 Far-Field^ Plume Behavior
The previous section reviewed criteria to be met for modeling plumes
close to the top of the stack (say, less than a few stack heights downwind).
To model plumes farther downwind of the stack, it is obvious that even greater
reductions in scale are required (larger geographical areas to be modeled),
and "exact" scaling will be difficult if not impossible to satisfy. The
question to be answered in this section, then, is whether further compromises
can be made without making the results unduly suspect.
As an example, suppose we wish to model a power plant in complex terrain,
where the scale reduction factor is 1:5000. Typical conditions from the plant
operations record might be T =540°K, T =300°K, W =25m/s, U=10m/s, D =10m. Sup-
s a s s
pose we try to match the "exact" conditions from the previous section using
pure methane as the model effluent (P./P =0.56) in a wind tunnel. The model
S a
stack diameter would be 2mm. The Froude number of the full scale effluent is
8, which implies a model effluent speed of 0.35m/s. These conditions yield a
stack Reynolds number of
w n
D« s 35cm/s x 0.2cm ..
K6 - — x tH
s v O.lGcnT/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 density or velocity ratios,
although the velocity requirement may later cause problems in obtaining a
minimum Reynolds number based on the roughness of the underlying terrain (see
Section 3.2). The problem was caused because of the matching of Froude num-
bers. This problem has been attacked in a variety of ways.
3.1.3.1. Ignoring the Minimum Reynolds Number
Ludwig and Skinner (1976) ignored the minimum Reynolds number require-
65
-------
ment; thus, their plumes were laminar in the immediate vicinity of the stack
(see Figure 10). Discussion in their report admitted that the rise of an
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 was not a serious limitation in their model because their
initial plume rise was quite small before atmospheric turbulence began to
dominate the mixing process. It is evident from Figure 10, however, that the
scale of the atmospheric turbulence is considerably larger than the initial
plume diameter, so that the plume trajectory is highly contorted, but little
real mixing of effluent with ambient air occurs for many stack heights down-
stream. 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 not feel that tripping of the flow within
the stacks was possible because the stack diameters ranged from 0.25 to 1.3mm
and there were 49 separate stacks in the model. Liu and Lin (1975), however,
were able to use a sapphire nozzle of 0.18mm dia. to trip the flow in their
one stack. As mentioned in Section 3.1.1, the size and placement of the ori-
fice is evidently critical and will require special experimentation.
•• -As
Figure 10. Laminar plume caused by low Reynolds number effluent
(from Ludwig and Skinner, 1976).
66
-------
3.1.3.2 Raising the Stack Height
Facy (1971) ignored the plume buoyancy, per se, but instead extended the
stack and bent-over the top such that the effluent was emitted at the same
elevation as that calculated from 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-
itudinal and 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 into a different
mean streamline, its resulting trajectory could be entirely different. Adding
momentum to the effluent to obtain the same rise as for a buoyant plume is ob-
jectionable for similar reasons. This technique, however, might be acceptable
under some circumstances; for example, if the problem were to determine concen-
trations on an isolated hill far downwind of the source (beyond the point of
maximum rise), then it might be acceptable to inject the plume at its terminal
height. Another problem is that one must presume to know the rise. This may be
acceptable for stable flows, but is an unsettled matter otherwise.
3.1.3.3 Distorting the Stack Diameter
Briggs1 (1969, 1975) equation for the trajectory of a plume with both
initial momentum and buoyancy, valid only for distances considerably smaller
than that to the point of maximum rise, is rewritten here in a different form:
67
-------
Ah
*T
4HS
4p/Hs
(3.5)
(3.6)
where 3-j and 82 are entrainment coefficients (3^1/3+U/Wg and B2=0.6),
1 is a momentum length scale and K is a buoyancy length scale, defined by
Briggs (1975) as
P* LJ
- w.
(3.7)
IT
-,2 W.
and
(3.8)
A physical interpretation of these length scales is that they represent the
initial radius of curvature of the plume due to momentum and buoyancy respec-
tively. In our example problem, 1 =9,3m and lp=2.72m. It is evident from
examination of Eq. 3.6 that close to the stack, the first term on the right
hand side of this equation will dominate and far from the stack, the second
term will dominate. That is, close to the stack, the initial momentum will be
important, whereas, ultimately, the buoyancy will dominate.
Hoult (1973) suggested that we ignore the initial momentum, provided only
that we avoid stack downwash, and take as our requirement
«W.-Wp • (3-9)
where subscripts m and p refer to model and prototype, respectively. He fur-
ther suggested that Eq. 3.9 could be met by exaggerating the density difference
and/or by reducing the effluent speed Ws- Lin and Liu (1976) started at
68
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essentially the same point, but suggested additionally that Eq. 3.9 could be
met by exaggerating the stack diameter. They performed an experimental run
on a complex terrain model at a scale of 1:10000 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_ al_l_ of the parameters p$, pa, W$, D, or U in such
a fashion that the coefficients would not be changed, and, therefore, that the
plume trajectory would be unchanged, i.e., it would not be necessary to ignore
the momentum term—we could include it too. This is equivalent to reducing
the momentum and buoyancy lengths by the geometric scale reduction factor.
Obviously, however, if we change D, we will also change the plume width at
the stack exit (=D). This violates our previous requirement of geometric
similarity; it may or may not have serious consequences, dependent upon the
amount of the exaggeration and on the particular flow field. It is not en-
tirely clear what extraneous effects may be introduced by the manipulation of
the other variables.
Basically, the coefficients of the x/H$ terms in Eqs. 3.5 and 3.6 are
products of similarity parameters, i.e.,
1/2
and
B.I
>au'
3/2
3/2
(3.10)
(3.11)
69
-------
If we insist on geometric similarity, then Eq. 3.10 is identical to our
previous momentum matching requirement, and Eq. 3.11 is a product of our
previous Froude number, momentum and density ratio matching requirements.
There is no reason, a priori, to favor one or the other; the choice will
require experimental verification.
In a rather unusual derivation, Skinner and Ludwig (1978) have arrived at
scaling laws that are essentially equivalent to matching the ratios of momen-
tum and buoyancy length scales to the stack height, i.e., Eqs. 3.10 and 3.11.
They also conducted some experimental work 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 dia-
meter, but they have not conducted tests to verify this.
A review of the literature shows that a wide variety of approximate
techniques have been proposed and used (see Table 2). The only technique
that has been independently tested and "proven" in different laboratories is
the CALSPAN technique, and it is recommended for that reason.
3.1.4 Summary and Recommendations on Modeling Far-Field Plumes
Most 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 nonturbulent
effluent flow and to avoid raising the stack, either physically or through
70
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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, 1t
1s recommended that the modeler:
1. Insure a fully turbulent effluent flow and
2. Either (in order of decreasing "correctness"):
a) match lm/Hs and IB/HS,
1) following geometric similarity or
11) exaggerating the stack diameter, but avoiding stack downwash,
or b) match IB/HS,
1) 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 of
exaggerating the stack diameter as noted. Notice 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. Be-
cause of the lack of basic, systematic studies on these fundamental problems,
the above recommendations are tentatively proposed and are subject to change
pending future developments.
71
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3.2 THE ATMOSPHERIC BOUNDARY LAYER
In early wind tunnel studies of flow around buildings (Strom et al., 1957),
complex terrain (Strom and Halitsky, 1953), and urban areas (Kalinske et al.,
1945), care was taken to insure that the approach flow 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 boundary layer was important; he
was also the first to produce a simulated atmospheric boundary layer approach
flow by matching the ratio of roughness length to building height between model
and prototype. Strong variations in surface pressure coefficient were observed
along with variations in cavity size and shape downwind from a building with
different depths of boundary layers in wind tunnels by Jensen and Franck (1963)
and Halitsky (1968). Tan-atichat and Nagib (1974) and Castro and Robins (1975)
have shown that the nature, strength, and locations of vortices in the flow
pattern around buildings differs markedly with and without a thick boundary
layer approach flow. Wind shear and the presence of the ground produce a down-
ward flow on the front face of a building, a reverse flow and an increase in
speed upwind, 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 downwind of a model.
UPWIND
VELOCITY PROFILE
SEPARATED
FLOW ON ROOF
MEAN VELOCITY IN
REVERSE DIRECTION
INCREASE IN SPEED
NEAR SIDES
Figure 11.
Effects of wind shear on the flow round a building. (Reprinted
with permission from Models and Systems in Architecture and
Building, Construction Press, Ltd., Hunt, 1975.)
72
-------
Further, not just any thick boundary layer will do. It must simulate
the atmospheric boundary layer structure, including as a minimum, 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 Lawson (1976)
at the somewhat smaller scale of 1:500 (compared with 1:300). The boundary
layer depth was scaled properly and the mean velocity and turbulence profiles
were reasonable facsimilies of those of Snyder and Lawson (S&L). However,
when measuring concentration profiles in the boundary layer downwind from an
isolated stack, he found a vertical plume width over 3 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 not change the velocity or 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 to be simulated in a wind tunnel or water
channel, it is necessary to decide at some point just what characteristics can
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 to answer the question of what the approach flow
should look like. What is a typical atmospheric boundary layer depth? What
are appropriate parameters that describe atmospheric stability and what are
73
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typical values for these parameters? How do the turbulence spectra vary with
stability, height above ground, etc.? It is necessary to establish a goal to
be met, say, a 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 attempt to achieve. What we attempt to
describe is the barotropic 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, an 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 friction due to the motion
of the air relative to the earth's surface is of prime importance. Above the
boundary layer, the air motion is geostrophic, reflecting a balance between
the horizontal pressure gradient and the Coriolis force, and the velocity
obtained there is the gradient velocity. The depth of the boundary layer is
highly variable, although it is typically between 1/2 and 2 km under neutral
conditions. The overall boundary layer may be divided into at least 2 sub-
layers, principally the surface layer, also misnamed the constant stress
74
-------
layer , and a transition region above, wherein the shear stress diminishes from
the nearly constant value in the surface layer to a near-zero value at the grad-
ient height. The surface layer is herein defined to be the lower 10 to 20% of
the planetary boundary layer. Generally, the mean velocity profile in the
surface layer is described by a logarithmic law, although deviations can be
large. Above the surface layer, there are numerous analytical expressions
for describing the velocity profile. If the entire boundary layer is to be des-
cribed by one expression, it is common engineering practice to use a power law.
Since Coriolis forces cannot be modeled in an ordinary wind tunnel or
water channel, modeling efforts should be restricted to those classes of prob-
lems 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 mod-
eled is less than 5 km, Coriolis forces may be ignored. Likewise, if the
terrain is rugged, so that the flow is highly dominated by local (advective)
forces, Coriolis forces may be ignored. This restricted class of flows limits
the usefulness of fluid modeling facilities, but there still exists a very
large range of problems in which it is not at all unreasonable to ignore these
effects. Orgill et al. (1971), for example, suggested that diffusion over
complex terrain could be reasonably simulated over distances of 50 km.
3.2.1.1 The Adiabatic Boundary Layer
Picking a depth for the adiabatic (neutrally stable) boundary layer is no
simple task. After an extensive review of the literature on adiabatic boundary
layers, Counihan (1975) concluded that the boundary layer depth is 600m,
1. Strictly speaking, a constant stress layer exists only in a boundary layer
with zero pressure gradient, which is seldom the case in the atmosphere.
75
-------
practically independent of wind speed and surface roughness. Davenport's
(1963) scheme, previously accepted as the "standard" for wind tunnel studies
of wind forces on buildings, specified the depth as a function of the rough-
ness length z only, varying from 6=300 m at zQ=0.03 m to 6=600 m at zQ=3 m.
Another popular scheme which is claimed to fit observations quite well is:
6 =cu*/fc , (3.12)
where 6 is the boundary layer depth, u* is the friction velocity (=/TQ/P),
and f is the Coriolis parameter Ztosine, where u is the earth's rotation
-4 -1
rate and e is the latitude. In mid-latitudes, fca10 sec . Typical values
for c range from 0.2 (Hanna, 1969) to 0.3 (Tennekes, 1973b). It is common to
use the geostrophic drag law, which relates the "drag coeffient" u*/G to the
surface Rossby number G/f z :
m rr-A + ln u:
CO *
k2G2 R2
7T ~ 0
1/2
(3.13a)
where G is the geostrophic wind speed (with components U and V ), k is
^ ^
von Karman's constant (0.4), and A and B are "constants" which differ consider-
ably from one author to the next. From Blackadar and Tennekes (1968), A is
about 1.7 and B about 4.7. For the sake of completeness, we also write the
expression for the angle a between the surface stress and the geostrophic
wi nd :
sin a = . (3.13b)
These three schemes for specifying the boundary layer depth are compared in
Figure 12, where it may be seen that they yield drastically different results.
In view of the uncertainties involved and also because Counihan's (1975)
literature review showed np_ measurements of depths in excess of 600m, if
specific measurements to the contrary are unavailable, the boundary layer
76
-------
should be assumed to be approximately 600m 1n depth. (In modeling, the
depth of the boundary layer is usually large compared with the model height,
so that the precise depth chosen is not usually critical; within limits,
there is room for choice.) 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 not unreas-
onable to use as a model. Also, many of Counihan's (1975) as opposed to
Davenport's (1963) conclusions are repeated here because they are represen-
tative of a wider range of data and they are more thorough in the sense that
more kinds of statistics are covered.
2500
2250
2000
1750
1500
1250
1000
750
500
250
20
11 il
DAVENPORT (1963)
. . ..nl i
i i i 11 i
.001
.01
.1
z0.m
10
Figure 12.
The depth of the adlabatic boundary layer according to
the geostrophic drag law compared with other schemes.
77
-------
The depth of the surface layer, in which the mean velocity profile follows
a logarithmic law, and from which the roughness length may be defined, is
generally stated to be 10 to 20% of the boundary layer depth. Counihan (1975)
suggests a value of 100m as a reasonable average depth for the surface layer.
The roughness length ZQ may be derived from the mean velocity profiles in the
range 1.5h
-------
elements! He suggested that the cause may be due to the more vigorous turbu
lence scouring the buildings, with the air stream "penetrating" more deeply
between buildings, 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 neglected for terrain
types where the roughness length is less than about 0.2m. It is suggested
by Simiu and Scanlan (1978) that reasonable values of d in cities may be
estimated using the formula
(3J5)
^ ^
where ff is the general roof-top level and k is the von Karman constant (0.4).
The mean velocity profile throughout the entire depth of the boundary
layer is adequately represented by a power law:
U/Uw = (z/6)p , (3.16)
where U is the mean velocity at the top of the boundary layer of depth 6 and
79
-------
TABLE 3: VALUES OF SURFACE ROUGHNESS LENGTH FOP VARIOUS TYPES OF SURFACES'
Type of Surface
Sand
Sea Surface
Snow surface
Mown Grass (-U3.01 m)
Low grass, steppe
Fallow field
High grass
Palrretto
Pine forest (Mean height of trees: 15 m;
one tree per 1C m2; 2^=12 m)
Outskirts of towns, suburbs
Centers of towns
Centers of large cities
(a) From Simiu and Scanlan (1978).
(b) Wind speed at 10 m above surface = 1.
(c) Wind speed at 10 m above surface > 15
(d) These values are exceptionally small;
7.
0
(cm)
0.01 - 0.1
O.C003b - 0.5C
0.1 - 0.6
0.1 - 1
1 - 4
2 - 3
4 - 1C
10 - 30
90 - 100
20 - 40d
35 - 45d
60 - 80d
5 m/sec.
m/sec.
see text.
80
-------
p is the power law index. This form is popular in engineering practice and is
highly useful frorr. a practical viewpoint. Davenport (1963) claims that the
overall reliability of the power law is at least as good as much more sophis-
ticated expressions and it is recommended here for that reason. It was shown
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
z (from Counihan, 1975). The power law index varies from about 0.1 in excep-
tionally smooth terrain such as ice to about 0.35 in very rough terrain such
as built-up urban areas.
As shown by Counihan, the turbulence intensity at a 30 m elevation follows
the same (empirical) formula as the power law index; their numerical values
as functions of roughness length are identical.
p = (/uVO")30m = 0.24 + 0.096 log1Qz0 + 0.016(log1()z0)2 , (3.17)
where z is to be specified in meters. The scatter in the Reynolds stress
_ o
measurements was considerable, and -100uw/lr 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
-ITvr/uf = uJ/U^ = 2.75xlO"3 + 6xlO"4 log1Qz0 , (3.18)
which is also shown in Figure 14. Counihan does not suggest how uw varies with
81
-------
45§!-
3M
A
Figure 13: Typical wind profiles over uniform terrain ir neutral flew.
.4
.35
3
I lll(
X
=2.75xir3 + fair4 ut,,1
Jtl
Figure 14: Variation of power law irdex, turbulence intensity, and
Reynolds stress with roughness length ir the adiabatic
boundary layer (fror Counihar, 197E).
82
-------
height; he irplies 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 2=6.
- uw(z) =
(3.19)
Thus, at heights less than C.lf, the stress is within 10? of its surface value
(see Figure 15).
u
1.1
!\
l°n
i
>— \
\ \A
I \
I \ "
1 |
U00=1l-/i
= STATION Ifa
- STATION 13
: STATION It
• STATION 19
• STATION 22
i : STATION 24
\
i
—i
14
I-
\
Oo£ A
^.n
4
i.5
1J
1.5
Figure 15. Shear stress distributions measured at various dottmrind posi-
tions in a wind tunnel boundary layer (neutral flow). Adapted
fror, Zoric and Sandborn (1972). (Courtesy of Boundary-Layer
Meteorology, D. Reidel Publishing Co.)
83
-------
These figures may be used for model design purposes in a number of ways.
In a general type of study, such as diffusion over an urban area, they can be
used directly to determine appropriate values for z and p. Or, in a specific
study, once U is chosen, one has only to determine zn (by measurement or
ft
estimation) and, since p,"\uVll, and -uw/lr are principally functions of z ,
to obtain them from Figure 14. If it is desired to match the wind speed U-, at
a particular height z-j, say, at the top of a stack, Eq. 3.16 may be rewritten
as
and the free stream wind speed (gradient wind) may be determined.
The variation of the longitudinal turbulence intensity in the surface
layer is given by
V?/U"= p ln(30/zQ)/ln(z/z0) . (3.20)
(This formula is slightly different from that of Counihan, but it is consistent
with his data and other formulas. Counihan's formulation did not match Eq. 3.17
for the turbulence intensity at z=30 m for low values of z0 and was somewhat
ambiguous in the range 0.1 m
-------
1.0
0.9
0.8
0.7
0.6
°-5
0.4
0.3
0.2
0.1
0.001
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
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
n
decreased with height. A summary showing the variation of Lu with elevation
A
and roughness length is given in Figure 17. For other integral length scales,
Counihan has concluded:
UJ - 0.3~0.4 Lu¥
y x
Lu = 0.5~0.6 Lu
10 m
-------
1000
'«
5
i
10
TERRAIN PRE POST
TYPE -M '40
J~ 4 O •
- v'lsfl.O ••
0.4
(r • MTERMITTENCY ••
• FACTOR) \,
LENGTH SCALE - U. <•)
FOR: 10 < Km) < 240. UM • C (t)l/B
Figure 17. Variation of integral length scale with height and roughness
length. (Reprinted with permission from Atmos. Envir., v. 9,
Counihan, Copyright 1975, Pergamon Press, Ltd.)
nSu(n)/u* = 105f/(l+33f)5/3 , (3.23a)
nSv(n)/u* = 17f/(H-9.5f)5/3 , (3.23b)
nSw(n)/u* = 2f/(l+5.3f5/3) , (3.23c)
and -nC^nJ/u* = 14f/(l+9.6f)2'4 , (3.23d)
where f=nz/D" is a nondimensional frequency (see Figure 18 for plots).
It may be seen that these spectral functions are dependent on z insofar
as u* and U are functions of ZQ. These expressions may be used to estimate
integral scales (e.g., Kaimal, 1973), but scales thus derived are not consist-
ent with those in Figure 17 (scales derived from Counihan's suggested spectral
86
-------
10
CM*
,3
^ .1
.01
.001
T T
a=u
.001
.01
.1
10
100
f = nz/U
Figure 18. Empirical curves for spectra and cospectrum for neutral
conditions (from Kaimal et al., 1972).
forms are even less consistent with Figure 17 -- such is our knowledge of the
neutral atmospheric boundary layer!). This is very unfortunate, because, as
was pointed out 1n the previous section, the larger scales of 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
to use those data as the target to simulate in the wind tunnel. If not, as
is usually the case, it is recommended that the following model be used.
For the sake of conciseness, justifications for the particular choices are
87
-------
omitted here. The interested reader may consult the previous section and
the references given there. Listed here are the main features of the
steady state adiabatic boundary layer over horizontally homogeneous terrain
(uniform roughness).
1. The depth 6 of the boundary layer is 600m, independent of
surface roughness and wind speed.
2. The mean velocity profile is logarithmic in the surface layer,
which is 100m deep.
3. The roughness length ZQ and the friction velocity u* may be
derived from the mean velocity profile in the range l.S
where h is the general height of the roughness elements, and
d is the displacement height (neglected for z <0.2m and given
by Eq. 3.15 for z >0.2m). Typical values for ZQ are given in
Table 3.
4. The mean velocity profile through the entire depth of the
boundary layer is represented by a power law U/U^ = (z/6)P.
The power law index p is a function of ZQ alone and may be
obtained from Figure 14 or Eq. 3.17. It varies from 0.1 over
smooth ice to 0.35 in built-up urban areas.
5. The Reynolds stress in the surface layer may be calculated as
a function of z from Eq. 3.18. Its vertical variation may
be approximated as a linear decrease with height from its
surface value (Eq. 3.18) to zero at z=600m (Eq. 3.19).
88
-------
6. The variation of the local longitudinal turbulence intensity
with z and elevation is shown in Figure 16. The vertical
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
may be obtained from Eqs. 3.22.
8. Spectral shapes are given by Eqs. 3.23 and shown in Figure 18.
3.2.1.3 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 more 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
(Wyngaard, 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
cooling generates a strongly stably stratified layer very close to the surface
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; Caughey et al., 1979).
89
-------
It 1s convenient at this point to discuss various paramenters that charac-
terize the stratification. (This discussion closely follows that of Businger,
1973.) The dlabatlc surface layer differs from the neutral one, of course*
because of the presence of the heat flux that creates the stratification that
very markedly affects the turbulence structure. This 1s clearly seen by exam-
ination of the turbulent energy budget equation (see, for example, Busch, 1973),
where a very Important production term appears that 1s proportional to the heat
flux. Another production term is, of course, the mechanical term due to wind
shear. Richardson (1920) introduced a stability parameter that represented the
ratio, hence, the relative Importance of these two production terms:
R1 .
6
where e 1s mean potential temperature (e=T+yz, where T 1s actual temperature
and y 1s the adlabatic decrease of temperature with height). This parameter
Is known as the gradient Richardson number or, simply, Richardson number.
In deriving this stability parameter, it was assumed that the eddy transfer
coefficients for heat and momentum were equal (K^K^). Since this assumption
1s not quite valid, it is better to leave the flux terms in the form in which
they appear 1n the energy equation, instead of assuming the fluxes are pro-
portional to the gradients of the mean quantities. Hence, we have a flux
Richardson number
Ri = S. :Z-£L - • (3.24b)
T T uw 3U/3Z
where T represents temperature fluctuations.
90
-------
This parameter is rather difficult to determine because of the covariance
terms, whereas 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 3U/3z=u*/kz. Substituting this expression into the flux
Richardson number yields a dimensionless height
_z. c[ wT kz
L =" T 4 '
T "*
where L = - ••- (3.24c)
9 k wT
is the Monin-Obukhov (M-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 constant stress layer in the neutral boundary layer). L there-
fore is a characteristic height that determines the structure of the surface
layer. It has been found that many features of the turbulence in the surface
layer depend solely upon the dimensionless height z/L. Such dependence is
referred to as M-0 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 diabatic scaling in the entire boundary layer, much as z/L governs
diabatic scaling in the surface layer (Tennekes, 1973).
ku
(3.24d)
91
-------
A final stability parameter is the Froude number
(3.24e)
which was discussed in Section 2.2.4. The Froude 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. More common in the meteorological liter-
ature is the inverse square of this Froude number, which is called the bulk
Richardson number
"HU
To give the reader a "feel" for the magnitudes of these various stability
parameters, we have listed typical values 1n Table 4. These values are not to
be taken as definitions or as absolute 1n any sense. Particular values depend
on the height chosen for specification of the wind 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 of the diabatlc boundary layer (albeit one that is steady and
horizontally homogeneous). Let us first quantify our rather qualitative des-
cription earlier in this section of the boundary layer depth.
According to Hanna (1969), the formula
0.75 11
6 =
tf 4H
where Ae/Az 1s the average vertical gradient of potential temperature through
the boundary layer, agrees well with observed boundary layer thicknesses.
92
-------
TABLE 4: TYPICAL VALUES FOR THE VARIOUS STABILITY PARAMETERS.
Qualitative
description
Highly
unstable
Unstable
Slightly
unstable
Neutral
Slightly
stable
Stable
Highly
stable
(a)
(b)
Pasquill-Gifford
category
A
B
C
D
E
F
G
L,m z/L
-5
-10
-20 -0
00
100 0
20 0
10
-2
-1
.5
0
.1
.5
1
Rif
-5
-2
-1
0
0.07
0.14
0.17
Ri
n -3
-2 -0.03
-1 -0.02
-0
0.
0.
0.
The assumed height of the anemometer and
the lower thermometer: 2m. A roughness
assumed in the calculations.
The
friction velocity
listed
is
that
Fr
P «..*
- -4000
- -120
.5 -0.01 -
0
07 0.
14 0
17 0
upper
length
value used
0
004
.05
.17
00
16
7
5
thermometer
of 0.01 m
in
-60
0
12
40
100
was 10 m
was also
sb
3
3
3
3
3
2
1
*
calculating y.
This equation can be used most of the time, because the atmosphere is usually
stable, on average, throughout the boundary layer. This equation implies that
the bulk Richardson number g6A6/(TU2) equals 0.56 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), Wyngaard (1975), Brost 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:
6/L - ayl/2 or 6 = a(Lu*/f J1/2, (3.25b)
93
-------
where a is a constant and y*=u*/fcL is a stability parameter related to
Eq. 3.24d through v=ky*. The constant a, however, is highly dependent upon the
value chosen for the stress criterion. For a 1% 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 Wyngaard (1978) over 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(6/L) - 0.96(6/L) + 2.5 (3.26a)
and B = 1.156/L + 1.1 , (3.26b)
where <5 is 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.)
Finally, an interpolation formula suggested by Deardorff (1972), i.e.,
« - f ] + fc + 1 I"1 /, 97 \
6 " [3UT O5U7 ZjTj > (3<27)
in which ZT is the height of the tropopause, does not suffer from "blowing up"
under neutral conditions (where L-*») near the equator (where fc-*0)»
6+0.25u*/fc under neutral conditions in mid-latitudes (i.e., Eq. 3.12) and
6+30L under very stable conditions and/or in low latitudes. Eq. 3.27 yields
results comparable to Eq. 3.25b in mid-latitudes (see Figure 19).
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
94
-------
layer is determined by the height of the base of the inversion, i.e., 6=z.
(Deardorff, 1972*. Wyngaard et al . , 1974). The height of the inversion base
varies from day to day, but its diurnal trend is quite similar. Kaimal et al .
(1976) describe their observations in the Minnesota experiments as follows:
"Between sunrise and local noon (1300 CDT) z. grew rapidly in response
to the steadily increasing heat flux (Qft). the growth of z. slowed down
between 1300 and 1600 CDT as Q reached0 its maximum value. But as Q
decreased through the late afternoon, z. began to level off to a neaPly
constant value which it maintained even after Q turned negative."
Even though this convective boundary layer depth changes rather rapidly
with time, 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 (Kaimal et al., 1976). To predict the height of this
boundary layer, Deardorff (1974) and Arya (1977) recommended a rate equation.
For purposes of fluid modeling, it is sufficient to pick typical values,
i.e., 6sl to 2 km, as the typical maximum height for the inversion base is
1 to 2 km. Once a boundary layer height is chosen, we can estimate u* from
the geostrophic drag relation, Eq. 3.13, where the parameters A and B are
functions of the stability parameters 6/L and f 6/u* (Arya, 1977).
A = ln(-6/L) + ln(fc«/uj + 1.5 (3.28a)
B = Mfa/uJ' + 1.8(f<5/u*) exp(0.26/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.
Figure 20 shows how the friction velocity u* varies with stability as predicted
95
-------
1500
1350
1200
1050
900
750
600
450
300 -
150 -
-0.2 -0.15 -0.1
.05 .1 .15 .2
•0.05 0
1/L, m'1
Figure 19. Typical nonadiabatic boundary layer depths from the geostrophic
drag relations (G=10m/s, z =0.01m, zT=10km, f =10 /s, a=l).
O 1C
*
-0.2 -0.15 -0.1
.15 .2
•0.05 0 .05
1/L. nr1
Figure 20. Variation of friction velocity with stability from the
geostrophic drag relations (Eqs. 3.13, 3.26, 3.28, G=10m/s,
zQ=0.01m, zT=10km).
96
-------
from the geostrophic drag relations using Eqs. 3.26 and 3.28.
Regarding the mean wind profile under non-neutral conditions, DeMarrais
(1959) has measured the power law index p and has drawn the following general
conclusions:
"During the day, when superadiabatic conditions and neutral lapse rates
prevail, the values of p vary from 0.1 to 0.3. 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. (1960) 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 M-0
length). Irwin (1978) has, analogously to Panofsky et al., used results of
Nickerson and Smiley (1975) to establish a theoretical relationship between p,
z and 1/L. The results, shown in Figure 21, support DeMarrais1 (1959) con-
clusions reasonably well.
Air pollution meteorologists frequently use Pasquill stability classes
(or 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. Irwin
(1978) has taken Golder's results relating Pasquill classes to the
Monin-Obukhov length and roughness length and overlaid them as shown in
Figure 21. Irwin (1979) 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 con-
97
-------
0.001
106 0.02 -OJ2 -0.06
STABILITY LENGTH. 1/L (m'1)
•0.10
•0.14
Figure 21, Theoretical variation of the power-law exponent as a function
of z and L for z equal to 100m. The dashed curves overplot-
ted are the limits defined by Colder (1972) of the Pasquill
stability classes as adapted by Turner (1964). (from Irwin,
1978).
ditions. It is relatively insensitive to stability but more dependent upon
roughness under unstable conditions. Comparisons (and tailoring) of Irwin's
results with field data of DeMarrais (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 ignored the influence of the
earth's rotation because this feature, in general, cannot be simulated realis-
-------
0.6
;0.4
I
oc
Ul
o
UJ
0.2
zt = 10.0
z2 = 100.0
0 = 2.0
0.01 0.10
SURFACE ROUGHNESS LENGTH, z., meters
1.0
Figure 22. Variation of the power-law exponent p, averaged over layer
from 10m to 100m, as a function of surface roughness and
Pasquill stability class. Dashed curve is result suggested
by Counihan (1975) for adiabatic conditions which should
agree with stability class D. (Reprinted with permission
from Atmos. Envir., v.13, Irwin, Copyright 1979, Pergamon
Press, Ltd.)
tically in laboratory facilities in any event. On the other hand, many fea-
tures of the surface layer can be well simulated. Panofsky (1974) has
suggested that we further subdivide the Ekman layer (overall boundary layer)
into a tower layer, i.e., below 150m or so in neutral or unstable conditions.
The surface layer proper extends to approximately 30m, but many of the relat-
ionships 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
99
-------
lower heights. In the discussion to follow, then, we will discuss 1n detail
various properties of the surface layer that may possibly be extended to 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 e*=-wt/u*. It has been found that these nondimensional gradients are
functions of z/L only (M-0 similarity). In unstable air,
z/L<0 (3.30)
fits surface observations quite well (Panofsky, 1974). The expression can be
Integrated to obtain the mean velocity profile (Paulson, 1970):
U/u* = (l/k)(ln(z/z0) - 2 ln[Jd+l/fm)] - ln(J(l+l/«J;))
+ 2 tan'1 (I/O - it/2] (3.31)
ni j •
This formulation is consistent in that under neutral conditions L-*«, «m-»-l,
and Eq. 3.31 reduces to the familiar log law. Panofsky (1974) showed that
« •O-lSz/L)"1^3 fit various data sets better than Eq. 3.30 for large values
of |z/L|.
In stable air
*m = 1+Bz/L , z/L>0 (3.32)
integrates to the familiar log-11near wind profile:
wo
-------
U/u* = (l/k)(ln Z/ZQ + 5z/L). (3.33)
Figure 23 shows typical velocity profiles as predicted by Eqs. 3.31 and 3.33.
The behavior of the nondimensional temperature gradient 4^ is somewhat
controversial. For simplicity we will here list the forms given by Panofsky
(1974)
, for z/L<0 , (3.34a)
and *h = l+5z/L , for z/L>0 , (3.34b)
These expressions integrate to
(e-e )/e* = In (Z/ZQ) - 21n{(l+l/*h/2>, for z/L<0 (3.35a)
(e-e0)/e* = In (Z/ZQ) + 5z/L, for z/L>0, (3.35b)
where e is the extrapolated temperature for z=z (not necessarily the actual
surface temperature). Typical temperature profiles as predicted by Eqs. 3.35
are shown in Figure 24. It is useful in interpreting Figure 24 to note that
e* 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 1n 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:
Ri = (z/L) (*h/»2) (3.36a)
101
-------
10000
1000 -
Figure 23. Typical surface layer velocity profiles under nonadiabatic
conditions (from Eqs. 3.31 and 3.33 with z = 0.01m).
10000
1000 -
10
8 10 12 14
(0 - 00) /0»
Figure 24. Typical temperature profiles in the surface layer (from
Eqs. 3.35 with ZQ = 0.01m).
702
-------
or Ri = z/L , for z/L<0, (3.36b)
Ri = ' for Z/L>0 (3.36c)
This relationship is shown in Figure 25.
The mean squares of the fluctuations of various turbulence quantities
are also found to obey similarity theory. The variance of the vertical
velocity fluctuations o= jvr follows Monin-Obukhov scaling, so that
W
(3.37a)
where » is a universal function. According to Panofsky (1974)
W
(VI. 25 for z/L>-0.3(including all z/L>0)/
*w - 1/3 . (3.37b)
w h.9(-z/L)l/<3 for
The variance of temperature also follows M-0 similarity, so that
(3.38a)
703
-------
where * is also a universal function which, according to Panofsky (1974)
W
is given by
"° for z/L<-0.1
(3.38b)
.1.8 for z/L>0
* and * are shown in Figure 26.
W V
The variances of the horizontal velocity components a and o do not appear
at present to follow any discernable pattern and do not obey M-0 similarity
(but see later discussion in this section of the convective boundary layer).
This is evidently due to the low-frequency contributions to these variances
that are possibly due to large scale terrain features or circulation systems
of large horizontal extent, unaccounted for in M-0 theory. Part of the problem
may also be due to the difficulty in separating fluctuations from means in the
original time series, i.e., the "spectral gap" may not be so clearly defined.
Nevertheless, various authors have attempted to force their observations
to fit M-0 scaling
ou = u,»u(z/L) , ov = u**v(z/L) . (3.39)
The "constants" $..(0) and * (0) for neutral stratification vary from 1.5 to 3
with "mean" values of 2.5 and 1.9, respectively. Observations of the variation
of a and o with height often show little attenuation, but there are notable
exceptions where slow and rapid decreases have been observed even in unstable
air (see Panofsky, 1974).
104
-------
-0.25
-1 -0.8 -0.6 -0.4 -0.2 0 .2 .4 .6 .8 .10
-0.5 -
-0.75 -
Figure 25. The relationship between Ri and z/L (Eqs. 3.36).
4 -3 -2 -1
3 4
Figure 26. Variation of *., and * with z/L in the surface layer
W v
(Eqs. 3.37 and 3.38).
105
-------
Binkowski (1979) has derived expressions for $u and $y that fit through
the middle of the very wide scatter of the Kansas and Minnesota data. (The
"wide scatter", however, most likely results from plotting in incorrect sim-
ilarity coordinates, not from scatter in the usual sense; see later discussion
in this section on the convective boundary layer.) Due to the complexity of
the formulas, they are not repeated here, but are shown in Figures 27 and 28.
Spectral forms are taken from Kaimal et al. (1972) and Kaimal (1973).
In stable air, spectral shapes of all velocity and temperature fluctuations
were found to be M-0 similar, and to have universal forms when appropriate-
ly normalized. Hence,
nS (n) 0.16 f/f
-Z- P , (3.40)
2 l+0.16(f/fn)5/0). As an engineering approximation, vertical
velocity and temperature fluctuations may be assumed to fit the universal
form (Eq. 3.40). This universal spectral shape is shown in Figure 29, and
the variation of the peak frequency with z/L is shown in Figure 30.
The integral scales are difficult to evaluate directly from the spectra;
706
-------
I '
x = KANSAS DATA
+ = MINNESOTA DATA
I
•2
0
z/L
Figure 27. Variation of »u with z/L in the surface layer (Reprinted
with permission from Atmos. Envir. v. 13, Binkowski, Copy-
right, 1979, Pergamon Press, Ltd.).
i , •
x = KANSAS DATA
+ = MINNESOTA DATA
3 -
2 ~
1 -
Figure 28. Variation of *y with z/L (Reprinted with permission from
Atmos. Envir., v. 13, Binkowski, Copyright, 1979, Pergamon
Press, Ltd.).
707
-------
0.01
10
100
1000
Figure 29. Universal spectral shape (Eq. 3.40)
Figure 30.
-25 -20 -15 -10 -O5 0 +05 +10 +15 +2.0
*'L
Location of spectral peak for u, v, w and e plotted against
z/L. Curves shown are fitted by eye (from Kaimal et al., 1972;
Reprinted with permission from the Quarterly Journal of the
Royal Meteorological Society).
108
-------
a length scale that can be obtained directly is x^, the wavelength corres-
ponding to the peak in the logarithmic spectrum nS(n). Using Taylor's
hypothesis,
• <3-42'
where n and f are the cyclic and reduced frequencies at the spectral peaks.
mm
This length scale is used extensively (as opposed to the integral scale) in
the interpretation of atmospheric spectra. Kaimal (1973) has derived a simple
expression relating these two length scales under stable conditions
Lax = 0.041z/f0a = O.lGz/f^ = Xm(«)/2ir . (3.43)
His findings for the variation of these length scales with Richardson number
are listed in Table 5.
TABLE 5. DIMENSIONLESS LENGTH SCALES AS FUNCTIONS OF Ri(Q.05
-------
with other empirical relationships. No expressions are available for the
variation of A in unstable conditions, but values for X (w) and X (e) may be
deduced from Figure 30.
Little is known about the variation of X with roughness. Wamser and
•*
Muller (1977) noted that their data showed a decrease in X (w) with increasing
roughness under neutral and convective conditions, but could not draw any con-
clusions for stable conditions. They also noted that there was no systematic
dependence of ^(u) on roughness. Higher order statistics such as cospectra
and structure parameters are beyond the scope of this review. The interested
reader is referred to Wyngaard and Cote (1972), Wyngaard et al. (1971).
Above the surface layer, the turning of wind with height generally becomes
highly important and is not amenable to simulation in the usual laboratory
facility. But one case, in fact one that is fairly typical of daytime convec-
tive conditions, deserves mention. Kaimal 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 O.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 "mixed
layer" where the mean wind is essentially uniform and the wind direction
changes little with height. In the "worst case" run, the wind direction
varied by only 15° between the surface and the top of the boundary layer;
it was typically only a few degrees.
110
-------
It is conceivable that the entire depth of this convective boundary
layer could be simulated in a laboratory facility, albeit at very low Reynolds
number. Deardorff and Willis (1974) have done the limiting case of pure con-
vection (no wind) and Schon et al. (1974) and Rey et al. (1979) have done an
unstable boundary layer, but without a capping inversion. That the two ap-
proaches can be merged appears promising.
For details of the boundary layer structure (variances, scaling, spectra,
etc.), the reader is referred to the papers by Kaimal et al. (1976), Kaimal
(1978), and Panofsky et al. (1977). The latter authors show, for example, by
using observations from several data sets over uniform surfaces, that $u and
* depend not upon z/L, but instead upon z^/L. Also, there were no signifi-
cant differences between the lateral and longitudinal components. Their ex-
pression fitted to the horizontal velocity data is
*H = (12-0.5z./L)1/3 , -400z.,)
and a transition region (z
-------
ity, where 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 Diabatic Boundary Layer Structure
Listed here are the main features of the steady-state diabatic boundary
layer over 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 to use those data as a
target to simulate.
1. The depth of the stable boundary layer may be estimated from
Eq. 3.27, where the friction velocity u* is obtained from the
geostrophic drag law (Eq. 3.13), and the "constants" 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. Once 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^O.OSU^ in unstable conditions and
u*=0.02Uoo in stable conditions.
3. The power law exponent p characterizing the shape of the mean
velocity profile over the depth of the boundary layer may be
obtained from Figure 21 or 22. In unstable conditions, it is
112
-------
dependent primarily on the roughness length and essentially Indep-
endent of the degree of Instability, varying 1n the range of 0.1 to
0.2. Under stable conditions, it 1s 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 con-
ditions, the surface layer may be even thinner than 10 to 20 m in
depth. The Monln-Obukhov length L is currently the most popular
stability parameter because most of the surface layer properties
can be described solely in terms of the dimenslonless height z/L
(M-0 similarity theory). Given L and u*, we can predict the shapes
of the mean velocity profile (Eqs. 3.30, 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). We can
also predict spetral shapes (Eq. 3.40) and scales (Eq. 3.41 through
3.45, Fig. 29 and Table 5).
5. Little 1s known of the boundary layer characteristics above the
surface layer except that generally the turning of the wind
with height 1s Important. Flow above the surface layer is thus
not usually amenable to simulation 1n 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 1n wind direction with height
113
-------
is typically only a few degrees over its typically 1 km
depth. For additional details, the reader is referred to the
original papers.
We have seen in our review of the atmospheric 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 "simplest" characteristics, its depth, is a horrendous problem. We
have attempted to assimilate the results of the most recent theories, but
they continue to 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 of those classical types as they are known at the present
time.
3.2.2 Simulating the Adi abatic Boundary Layer
In the previous sections, we have established at least the main character-
istics of the adiabatic and nonadiabatic atmospheric boundary layer. In this
section, we will examine several techniques commonly used to simulate the
neutral atmospheric boundary layer 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 towing tanks.
114
-------
The techniques can be broadly divided into three categories:
1. Long tunnels, in which 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
non-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 machine-driven 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, to 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
115
-------
Figure 31. Upstream view of a long wind tunnel (Courtesy
of the Boundary Layer Wind Tunnel Laboratory,
University of 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 sidewall boundary layers as not
116
-------
Figure 32. Vortex generators and roughness in a short wind
tunnel (Courtesy of Marchwood Engineering Labora-
tory, Central Electricity Generating Board, England)
representing steady and horizontally homogeneous atmospheric boundary
layers (Nagib et al., 1974). Recently, however, drag-producing elements
have been used in the long tunnels as well, and many techniques for generating
117
-------
FREE-STREAM
ENTERING
TEST SECTION
AM
ON
^
.
-- _.
r~~d tz -'-""
j ,•-'
~^
^
_.-:
„- _ .,
r_7
^
.
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,8
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//////,
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•x\ FLOW REGION
/ VIEW
THICKNESS OF
GENERATED
BOUNDARY LAYER
». • . ' ' ' ' ... . • •
I • -...•' .I''..
Figure 33. Schematic representation of the counter-jet technique
(Reprinted with permission from the American Institute
of Aeronautics and Astronautics, AIAA Paper No. 74-638,
Figure 1, Nagib et al., 1974).
DEPTH OF BOUNDARY LAYER OVER CARPET (z0~0.03 cm)
DEPTH OVER RECTANGULAR BLOCKS (2.5 TO 10 cm HIGH; zn ~0.3 cm)
BELL MOUTH-*) (4-DISTANCE FROM LEADING EDGE OF ROUGNESS (m) —»
Figure 34. Development of boundary layer in a long wind tunnel (Adapted
from Davenport and Isyumov, 1968, Proc. Int. Res. Seminar on
Wind Effects on Buildings and Structures, Univ. of Toronto
Press).
118
-------
thick boundary layers in short lengths of test section have been
developed. There is no reason, in principle, why a fully developed layer
with unchanging turbulence properties cannot be achieved in short tunnels.
An 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 estab-
lished in 5 to 10 boundary layer heights, a substantial improvement over
the long tunnels. It is not the aim here to favor one system over another,
but instead to stress the necessity of adequately measuring the boundary
layer, however generated, to be sure that it is laterally homogeneous and
non-developing (if that is what is desired) and that is meets the target
flow characteristics (which are not, in all cases, of course, those of the
atmospheric boundary layer).
Examples of the first category, long tunnels, are the Micro-Meteorologi-
cal Wind Tunnel at the Colorado State University (CSU) (Cermak, 1958; Plate
and Cermak, 1963) and the Boundary Layer Wind Tunnel at the University of
Western Ontario (UWO) (Davenport and Isyumov, 1968). Figure 34 illustrates
the development of the boundary 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 to show that the turbulence
in the boundary layer of the CSU long tunnel exhibited characteristics of the
Kolmogoroff local isotropy predictions, i.e., a large separation between the
119
-------
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 made
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).
Zoric 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, shows that the boundary layer
grows nearly linearly and still quite rapidly with downstream distance beyond
about 10 m. Zoric (1968) obtained results similar to those of Figures 35
and 36 for freestream 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 on eddy velocity and eddy size, may be estimated to be at most 2000.
Tennekes and Lumley (1972) suggest a bare minimum value for an inertia!
subrange (local isotropy) to exist is 4000. Even though a substantial
spectral region with a -5/3 slope was measured, it is doubtful that local
isotropy existed, i.e., the existence of the -5/3 slope is not a critical test
of local isotropy.
720
-------
0.4
0.2
- i..
0.2
0.4
0.6
0.8
z/5
— fto
SYMBOL
o
A
D
V
0
•
A
POWER LAW
STATION
3. Km)
6.1
9.1
12
15
18
21
24
-j
t
1
1
i
i
•j
"1
-1
J
!
Figure 35. Development of mean velocity profiles along the smooth
floor of a long tunnel (from Zoric and Sandborn, 1972;
Reproduced by courtesy of Boundary-Layer Meteorology,
D. Reidel Publishing Co.).
m cm
1.0 r
0.8 -
0.6 -
0.4 -
0.2 -
8
6
*• 4
K>
2
n
I
1 1 1 1 ! 1 ' '
o DISPLACEMENT THICKNESS, 5* (cm) ^^°^
0 MOMENTUM THICKNESS, 6 (cm) ^^ -x-
A BOUNDARY LAYER THICKNESS, 6 (meters) o*^^ ^^^***X^
' ^^^^^"^
.0-*>AA-^
-^
^ D D D ° o a c— I
i i i i i i
) 5 10 15 20 25 30
x (meters)
H
1.5
1 0
Figure 36. Thickness parameters for boundary layer of Figure 35
(from Zoric and Sandborn, 1972; Reproduced by courtesy of
Boundary-Layer Meteorology, D. Reidel Publishing Co.).
121
-------
intensities were about 50% of the longitudinal intensities close to the
ground in accordance with Eq. 3.21, but were typically 70% over the
upper 90% 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 how it
would compare as a small scale model of the neutral atmospheric boundary
layer. Evidently, detailed basic and systematic studies of the
turbulence structure in neutral boundary layers have not been made in the
long tunnels.
Additionally, the above measurements were made at wind tunnel speeds
much in excess of those allowable for modeling buoyant plumes. Isyumov,
et al. (1976), for example, suggest typical tunnel wind speeds of 0.5 to
0.7 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
of the spectral peak. Also, a relatively large amount of energy at lower
frequencies is rather surprising in that significant energy in this part
of the spectrum is not generally produced in wind tunnels. Measurements
of the spectrum of lateral velocities would ascertain whether this energy
is, in fact, due to turbulence or whether it is "pseudo-turbulence", i.e.,
one-dimensional fluctuations caused, for example, by low frequency
oscillations in fan speed.
722
-------
Height=400 ft.
0 Estimated spectral
points from wind
tunnel measurements
— Davenport's spectrum
for neutral stability
.01 .02
Figure 37. Spectrum of the longitudinal component of velocity
(from Isyumov et al., 1976; Courtesy of Air Pollution
Control Association).
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/vortex generator/roughness
system developed by Counihan (1969). It has been adopted with minor
variations at numerous laboratories. Castro et al. (1975) have made
123
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extensive measurements of a 2m deep boundary layer developed using
Counihan's system, primarily for study of dispersion of chimney
emissions in neutral flow. They found that the turbulence in the
boundary layer reached a near-equilibrium state in approximately 7%
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 too 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 equilibrium were unimportant beyond about 5 boundary
layer heights. The lower 10 to 20% of the boundary layer
could be used beyond about 3^6.
2. Was Reynolds number independent for free stream wind speeds
in the range of 0.7 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 intermittency distributions with those
of "natural" wind tunnel boundary layers.
Moreover, Robins (1978), showed 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 Pasquill category C-D atmospheric flows (slightly
unstable to neutral), which is normal for such a large roughness length.
1. The present author notes that in some unpublished work, boundary layers
developed using Counihan's system were found to be very much dependent upon
the proximity of the ceiling, i.e., when the ceiling was several times the
height of the vortex generators, even the mean velocity profiles differed
drastically from his. The ceiling thus appears essential; indeed, it is an
integral part of the simulation system.
724
-------
Somewhat less-well-documented techniques include the "spires" of Templin
(1969) (also quite popular), the "fence" of Ludwig et al. (1971), the "coffee"
cups" of Cook (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; Ogawa, 1976, 1981).
Hunt and Fernholz (1975) provided a list of wind tunnels (largely European)
used for atmospheric boundary layer simulations and included characteristics
of the wind tunnels and relevant measurements of such boundary layers, so that
some comparisons of the different techniques may be made.
Short test sections with active devices are also numerous. Schon and
Mery (1971) injected air perpendicular to the flow through a porous plate at
the entrance to the test section. This system may be thought of as a fluid-
mechanical fence, where the fence "height" is adjusted by varying the
strength of 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 layer
twice as thick as the "natural" one over a smooth floor and that its charac-
teristics are essentially similar. However, this system appears to require
a rather long development length compared with Counihan's system (Hunt
and Fernholz, 1975). Also, because of 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 to the Brookhaven experiments (Smith and Singer,
1955), but only after "adjusting" their o 's by a factor of 2 to account
for an equivalent wind tunnel averaging time (converted to full scale) of
725
-------
3 minutes compared with atmospheric averaging timesof 1 hour. This
adjustment technique, however, appears somewhat arbitrary. The small
values of the a '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.
Nagib et al. (1974) have added some flexibility to the Schon et al.
technique, in that the injected air is input through a line of holes in a
pipe perpendicular to the flow on the floor at the entrance to the test
section. The pipe may be rotated (See Figure 33) to change the jet
injection angle and the jet speed may be varied; these, of course, change
the boundary layer characteristics. The "counter-jet" technique, it is
claimed, avoids the objectionable introduction of extraneous turbulence
scales as from vortex generators or grids, but this claim is contested by
Cook (1978). Nagib et al. (1974) and Tan-atichat et al. (1974) show that
this technique produces reasonable boundary layers with adequate lateral
uniformity and that equilibrium is achieved in approximately 4 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 of Teunissen (1975) and the "turbulence box" of Nee et al. (1974),
but neither of these systems appears to have been developed beyond the
initial stages.
126
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3.2.3 Simulating the Diabatic Boundary Layer
Only a few facilities exist for simulating the diabatic boundary layer.
The oldest and best-known is the Micrometeorological Wind Tunnel at the Col-
orado State University (Cermak, 1958; Plate and Cermak, 1963). It has nominal
test section dimensions of 1.8 X 1.8 X 27 m and an adjustable ceiling for elim-
inating 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 cooled and a heat exchanger in the return leg maintains ambient
air temperature equilibrium, 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 done according to Monln-Obukhov similarity theory. Their data ranged from
neutral to moderately stable (0 <. 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 maintain-
ed at 40°C while the wind speed was varied from 3 to 9 m/s. Measurements
included distributions of mean velocity, temperature, turbulence intensities,
shear stress, heat flux, and temperature fluctuations. Arya (1975) has pre-
sented additional measurements 1n this stable boundary layer. Thus far, all
measurements in stratified boundary layers in the CSU tunnel have been with
a smooth floor.
127
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The Fluid Mechanics Laboratory at the Ecole Centrale de Lyon has made
extensive measurements of an unstable wind tunnel boundary layer and compared
its properties with the atmospheric boundary layer (Schon et al., 1974; Mery
et al., 1974; Schon and Mery, 1978). Flow speeds were typically 2 to 4 m/s
while the floor temperature was maintained 50°C above ambient. In general,
comparisons with the Kansas data (Businger et al., 1971; Haugen et al.,
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 fre-
quency portions of the spectra were 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 Monin-Obukhov length of -1 m, which, when scaled to the atmosphere, cor-
responds to -500 to -1000 m, and is indeed only very slightly unstable (See
Table 4). Attempts by Rey (1977) and Rey et al. (1979) in adding a rough
floor to this unstable boundary layer showed substantial changes in the
boundary layer structure with roughness.
A stratified and closed return wind tunnel has recently been construct-
ed at the Japanese Environmental Agency (Ogawa, et al., 1980). It has test
section dimensions of 3m x 2m x 24m and an adjustable ceiling for pressure
gradient control. Speeds may be varied from 0.1 to 11 m/s. Velocity- and
temperature-profile carts at the test section entrance establish the initial
velocity and temperature profiles. Sections of the floor (3 meter lengths)
can be independently heated or cooled to maintain stable or unstable boundary
layers, to generate an elevated inversion, or to simulate land-sea breezes.
128
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3.2.4 Summary on 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 boundary 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 of this guideline. The point is only to show
that it can and indeed has been done through various schemes.
Adequate simulations of the neutral atmospheric boundary layer have
been obtained using short tunnels with either active or passive devices
and long tunnels. Strengths and weaknesses of the three types, as far as
their ability to produce adequate boundary layers is concerned, appear to
be evenly balanced. Hence, no one technique or system is recommended over
any other. Due to the large number of permutations and combinations and to
the possible large changes in boundary layer structure with seemingly small
changes in configuration, however, 1t is imperative that, whatever technique
is used, the boundary layer characteristics are adequately documented.
Simulations of diabatic 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.
Also, because normal roughness elements on the floor would reduce heat trans-
fer even further, most measurements to date have been made over smooth wind
tunnel floors. Recent exceptions indicate experiments over cooled, rough-
ened terrain made from formed aluminium foil by Peterson and Cermak (1979)
and unstable boundary layers developed by Rey et al. (1979). As we have seen
in Figure 22, the inability to use a rough surface could be important, at
least for unstable flows and large roughness lengths.
729
-------
Adequate documentation of the boundary layer characteristics should
include, as a minimum:
1. Several vertical profiles of mean velocity, turbulence intensity
(3 components), and Reynolds stress throughout the region of
interest to establish that the boundary layer is non-develop-
ing (or at least very slowly developing), and is similar to the
target atmospheric boundary layer (zQ,d,u*). In a stratified bound-
ary layer, of course, profiles of temperature and temperature
fluctuations should also be included and the stability parameters
should be matched.
2. Lateral surveys of mean velocity and turbulence structure at
various elevations to ascertain the two-dimensionality of the
boundary layer.
3. Spectral measurements of the turbulence to determine that the
integral scales and the shape of the spectra are appropriate.
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.
130
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Perhaps the most critical test of the boundary layer is the measurement of
its dispersive characteristics to determine whether appropriate concentration
patterns result. This point cannot be over-emphasized. Wind tunnels are
generally extremely difficult to operate at low wind speeds (
-------
3.3 FLOW AROUND BUILDINGS
We discuss here guidelines to be followed in modeling flow around
buildings, e.g., in order to determine a necessary height for a stack on
a power plant to avoid downwash of the plume in the wake of the plant.
The class of problems covered includes single or small groups of buildings,
primarily isolated ones in a rural environment, i.e. scale reductions in the
range of 1:200 to 1:1000. It is evident from preceding sections that the
building must be immersed in an 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. A 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 come to mind are stack height H and diameter D,
5
building height H, boundary layer depth 6, roughness length ZQ, and, if
stratified, Monin-Obukhov length L. There are, of course, many other length
scales and geometric scaling requires that all lengths be reduced by the
same ratio. However, this brings up the question of how much detail is
required, i.e., is it necessary to include in the scale model a particular
protuberance, say, from the roof of the building? The answer, of course,
depends upon the size and shape of the protuberance; it is a question of
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 of the
732
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protuberance is such that e u*/v<_4, it will have little effect on the
flow in a turbulent boundary layer on a flat plate. Hence, protuberances
smaller than e = 4v/u* need not be reproduced in the model.
A closely related but more demanding problem is as follows. A given
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 almost always very large, we may
assume that surfaces of typical buildings are aerodynamically rough.
As we reduce sizes of buildings to fit into 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 Rec = e u*/v >_ 100, these flow
phenomena will be independent of Reynolds number. These results indicate
that small details need not be reproduced and, indeed, that model surfaces
should be roughened to the point that the critical Reynolds number is at
least 100.
Crude estimates will suffice here and an example will help to clarify
the procedure. Suppose our model building has a height of 20 cm, and in
order to model the buoyancy in the plume, we have reduced the wind speed to
1 m/s. The friction velocity is typically 0.05 U^, so that size of
roughness elements with which to cover the surface of the model building
S e « (100) (0.15cm2/s)/5cm/s=3cm.
733
-------
This is, in general, an unacceptably large roughness size, as we should
probably also restrict e/H
-------
slight variations over the entire range of Reynolds numbers with neutrally
buoyant effluent and with an effluent-to-free-stream velocity ratio of unity.
However, the concentration at a point on the roof itself was found to vary
strongly with Reynolds numbers less than 11,000, but to be invariant with
Reynolds numbers between 11,000 and 94,000. Thus, a critical Reynolds number
may be defined, which, with this type of geometry, appears to be 11,000.
Golden's value for the critical Reynolds number for flow around cubes is
frequently cited in the fluid modeling literature on building downwash
problems. Whereas Golden's value was established for a smooth surfaced
cube on a raised platform facing a uniform approach flow of very low turb-
ulence intensity, it is applied "across-the-board" to all shapes and orient-
ations of buildings, in all types of approach flow boundary layers, and without
regard to the building surface roughness, all of which will affect Re . Also,
Golden's value was established primarily through the measurement of concentra-
tions at only one point on the roof of the cube, as opposed to measurements of,
say, the concentration fields in the wakes. Far too much confidence seems
to 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. Also,
as pointed out by Halitsky (1968) lower values are probably acceptable if
measurements are restricted to regions away from the building surface. Hence,
a critical Reynolds number of 11,000 is a useful and probably conservative
value for model design purposes, but tests to establish Reynolds number
independence should be an integral part of any model study until such time
that firmer values are established.
135
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A study by Smith (1951) may also be regarded as a test of Reynolds number
independence. He investigated the sizes of the wakes created by both model and
prototype sharp-edged buildings. He assumed that the flow was independent
of 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 proto-
type tests, he found this ratio to be constant for Reynolds numbers (based
upon an appropriate characteristic length) between about 2x10 and 2x10 . In
the model tests, this ratio was constant for various block models over a range
of Re from 2x1O4 to 2x1O5.
In more recent work, Castro and Robins (1977) investigated flow around a
surface mounted cube in uniform and turbulent upstream boundary-layer flows.
Reynolds number independence was observed in the uniform flow when Re > 30,000
and in the shear flow when Re > 4000. These statements were based primarily
on measurements of mean surface pressures on the body, which are evidently
more sensitive than surface concentration measurements as made by Golden
(1961). The trend toward much lower critical Reynolds numbers in turbulent
shear flows, however, is clear. It should be noted that the simulated atmos-
pheric boundary layer of Castro and Robins was of a suburban character, i.e.,
a simulated roughness length of 1.3m, so that it was, indeed, highly turbulent.
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 79,000. 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,
136
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and systematic studies need to be done in this area.
Suppose there is another building a distance s upstream of our example
power plant; is it necessary to incorporate this building into our wind tunnel
model? 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
downwind of the separation bubble, where All is the maximum mean velocity
II IA
deficit created by the obstacle, h is the height of the obstacle, x is the
distance 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 from
that by a factor of 2. Hence, if we require that the velocity be within
3% of its undisturbed value, then a cubical building as high as s/20
must be included upstream of our power plant. This result, however,
is dependent upon the aspect ratio; a building with its width much greater
than its height, for example, would require inclusion if its height
were greater than s/100 (See Section 3.4).
The ratio of the cross-sectional area of a model to that of the tunnel is
referred to as "blockage", 3. 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 S = e/(l-e). Of course, in
the atmosphere, there are no sidewalls or roof to restrict divergence of the
flow around the model, so that the average speed-up is zero. Wind tunnels
with adjustable ceilings can compensate to some extent by locally raising
737
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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", as that would require local
expansion of the sidewalls at the same time.
Some unpublished measurements by the present author on the flow over a
two-dimensional ridge 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 flow. Vertical
profiles of mean velocity were similar in shape to those measured with the
flat ceiling, but the magnitude of the wind speed was lower by 10% everywhere
above the crest (see Figure 38) with little change in the root-mean-square
values of 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 10% as it traversed the ridge,
but, because its centerline would be 10% closer to the ridge crest, resulting
surface concentrations upstream of the crest would be essentially unchanged.
However, because the flow acceleration changes the pressure distribution around
the model, which will in turn affect the location of the separation point,
the effects downstream of the crest are not apparent. Blockage "corrections"
for conventional aeronautical wind tunnel models is a highly involved
engineering science problem. "Rules of thumb" indicate a limit of 5%
138
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i i
D A
1 1
n A
/ f
n A
n A
Q A
,,.11,,!,%,
.5 1 1.5 2 2.5 3 3.5
MEAN VELOCITY, m/s
4.5
Figure 38. Velocity profiles above crest of triangular ridge indicating
effect of blockage (A flat ceiling, 10$ blockage; a raised
ceiling, nonaccelerating free-stream flow).
blockage in the ordinary wind tunnel and somewhat higher, perhaps 10%, in
a tunnel with an adjustable ceiling. Blockage effects in stratified facili-
ties are even less well understood and are potentially much more important.
139
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3.3.2 Recommendations
To model the flow and dispersion around individual or small groups of
buildings, it is recommended that:
1. The building be immersed in an appropriate boundary layer,
the main features of which include matching of the ratios
of 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 UnH/v = 11,000 appears to be conservative.
4. The surface of the building be covered with gravel of size e such
that eu*/v=100. If this results in excessive roughness, i.e.,
e/H > 1/30, compromises may be made, but in no case should eu*/v be
less than 20.
5. Another building or major obstruction upstream of the test building
be included if its height exceeds l/20th of its distance from the
test building. This recommendation applies to a roughly cubical
obstacle. An obstruction whose crosswind dimension is large
compared to its height must be included if its height is greater
than 1/100th of its distance upstream (see text).
6. Blockage caused by a model be limited to 5% in an ordinary wind
tunnel and to 10% in a tunnel with an adjustable ceiling.
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3.4 FLOW OVER HILLY TERRAIN
Guidelines for modeling neutral flow over hilly terrain are essentially
similar to 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 more patchy. Whereas separation of the
flow 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 may occur on the downwind slope, or for a hill with low or moderate
slope, may be absent entirely. Also, the stratification in the approach flow
can drastically change the nature as well as the location of the separation
and may enhance or eliminate separation entirely (see Hunt and Snyder, 1979).
We will first discuss neutral flow, emphasizing the differences between
modeling the flow around hills and that around buildings. Because stratified
flows are so different from neutral flows, they will be discussed in a
separate section. The two sections are summarized with a set of recommendations.
3.4.1 Neutral Flow
In the field, the ridge Reynolds number based on a ridge height of 75 m
and wind speed of only 3 m/s is 10 . For these very large Reynolds numbers,
at least for a ridge with steep 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 10 and 10 , much smaller than the full scale Re. It is possible to
trip the flow at the apex (as done by Huber et al., 1976) or to roughen the
surface, so that the point of separation on the model will occur at its
apex and similarity of the two flow patterns will be achieved.
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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 unwarranted 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 e = 20 v/u* = 400 v/U^. This may in some
instances conflict with Jensen's criterion that h/z be matched between model
and prototype, but the minimum Reynolds number is regarded as more important.
A common practice 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. One 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. (1974) have
shown that the maximum deficit of mean velocity in the wake, normalized by
142
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the mean velocity at the height of the hill, decays as
AUmx * B
OW xTfi" '
where All is the difference in mean velocity created by the hill, h is the
Ill/v
hill height, x is distance downstream of the hill, and B is a constant depen-
dent upon surface roughness 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/100 should be included in the model. In actual practice, one
should study the topographic maps of the area surrounding the plant, locate
prominent ridges upstream, then determine the height of each ridge and its
distance from the plant. If its height is greater than x/100, all terrain
between the ridge and the plant should be included If h
-------
the fonrulas 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. Our choice of 600m (Section 3.2.1.2) is obviously absurd if the heights
of the hills themselves are of the same magnitude. Ore 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 hill heights
under moderate to high wind speed neutral conditions.. As mentioned in Sec-
tion 3.2.1.1, he also observed a logarithmic wind profile with z of 35m.
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 100 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 AMS, 1979). Further, results
of Godowitch et al. (1979) indicate that extremely shallow stable boundary
layers under quite deep surface-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 1.4°C/100m. Under these conditions, it
is evident that simulation of the stable boundary layer beneath an elevated
source is relatively unimportant. Far more important is the simulation of
the stability above the boundary layer because, as shown by Lin et al. (1974),
144
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Hunt and Snyder (1979) and Snyder et al. (1979), the stability deter-
mines 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 of 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 flow field, streamlines below the hill top will
pass round the sides of 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 2.2.4).
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 to duplicate in the model. An example is
shown in Figure 39, where a portion of a three-dimensional hill is
turned into a two-dimensional one by an inappropriate choice of
the area to be modeled. Under strongly stable conditions, the com-
bination of the hill and tunnel sidewalls would result in upstream
blocking of the flow 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. 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
745
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Figure 39. Contour map of three-dimensional hill showing inappropriate
choice of area to be modeled.
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 common sense will avoid most pitfalls.
As Scorer (1968) has pointed out, laboratory studies of stratified flows
146
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tend to overemphasize the effect of stratification in the approach flow; local
heating and cooling of hill surfaces are equally, perhaps more, important.
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). Some attempts have been made to simulate heat-
ing of terrain surfaces, but primarily to simulate fumigation of elevated
plumes (Liu and Lin, 1976; Keroney et al., 1975; Ogawa et al., 1975), rather
than to model anabatic or katabatic winds (Petersen and Cermak, 1979). Other
local heating and cooling problems include lake-shore breezes and urban heat
islands. Little is known 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.
147
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4. The surface of the model be covered with gravel of size e such
that eu*/v>20. The step size in "stepped" terrain models should
also be of order e.
5. An upstream ridge be included in the model if its height exceeds
l/100th of its distance upstream from the test portion of the
model. A three-dimensional hill should be included if its height
exceeds l/20th of its distance upwind.
6. Blockage be limited to 5% for an ordinary wind tunnel and to 10%
in a tunnel with an adjustable ceiling.
Additionally, in modeling dispersion from elevated sources in strongly
stably stratified flow 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
on the hill height and the density difference between the base and
top of the hill.
8. Topographic maps of the area to be modeled should be studied care-
fully to insure that an appropriate area is modeled (see text).
Finally, laboratory models to simulate anabatic and katabatic winds
must be considered as exploratory in nature at the present time.
3.5 RELATING MEASUREMENTS TO 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 " CUH2/Q, where
C = mass concentration of pollutant (ML"3),
U = wind speed (LT"1),
H = characteristic length (L),
and Q = pollutant emission rate (MT~ , e.g., grams of S02/second).
The relation between field and model concentrations beyond a few dia-
meters from the source is thus
UU •) f\
_ n_ c. \jf
148
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Note that both C and Q are measured in mass units. More frequently, C and Q
are measured in volume units, in which case, they must be evaluated at ambient
(not stack) temperature, including Qf (Robins, 1975). An example may help to
clarify the procedure. Suppose we model a buoyant plume with a mixture of
helium, air and methane as indicated below:
Property Field Model
Reference wind speed 10 m/s 1 ir/s
Stack height 50 m 50 cm
Stack diameter 5m 5 cm
Effluent speed 20 m/s 2 m/s
Pollutant emission rate 500 g/s(S02) 1 g/s(CH4)
Effluent temperature 117°C 20°C
Ambient temperature 20°C 20°C
Effluent specific gravity 0.75 0.75
At some point downwind of the source in the wind tunnel, we might measure a
model concentration of 100 ppm (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 20°C are air: 1.20 g/£; CH4: 0.68 g/£; S02: 2.72 g/£):
100 £ CH. , . . 0.68 g CH. 52.7 x 10"6g CHA
= f ±\ ( ' *• air \ I iL •
V C / \1 On n a-i*^ V
um v f I M ?f) n air' v 1 9 CM ' Q air
10 f "'"' "
Hence, the field mass concentration will be:
52.7 x 10"6g CHA ... n ,._ 2 500 g S0?/s
/ 4\ /im/s\ /u.om\ /
_ _
f " g~avr TOii/V ~^Gm 1 g CH4/s
= 0.264 x 10"6g S02/g air.
Converting this mass concentration to volume concentration yields:
J49
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0.264 x 10~6g S09 , 9n , 1 £ S09
r = t 2% fl.20 g airx / 2 }
f v g air ' v 1 £ air' V2.72 g S02'
= 0,116 x 10"6 £ S02/£ air = 0.116 ppm S02>
To summarize, the relation between model and field concentrations in this
example is
1 ppm S02 .». 858 ppm CH.
or 1 g S02/g air + 200 g CH4/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.
If model concentrations are to be related to field concentrations very
close to the source and the density difference is exaggerated, then Eq. 3.48
is not strictly correct. Considerations of geometric similarity of model
and field plumes and perfect gas laws show that
^m _ pp'pap ,
xp pm/pam
where p1 is the local density (i.e., not reduced to ambient temperature).
This equation is not very useful by itself because local densities are not
generally measured. To reduce it to useable form, however, will depend
upon the precise means of simulating the buoyant plume and the concentration
measurement technique. One such correction scheme has been derived by
Meroney (1978) in connection with liquid-natural-gas spills.
Regarding the comparison of model results with field results, it is
well-known that in the field the averaging time has a definite effect on the
/so
-------
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
into 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 to 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 10
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; 1979).
3.6 AVERAGING TIME 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
151
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value at any particular instant of time that 1s 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 of resources; sampling at too low a rate may not allow
us to obtain the desired information and, in fact, may lead to incorrect
answers . Hence, it may be necessary to determine an appropriate sampling
rate.
To determine an appropriate averaging time for measuring the mean of a
fluctuating quantity F(t) in a wind tunnel, it is useful to consider the
turbulence as a Gaussian process. (Whereas turbulence is not a Gaussian
process, experience has shown that this assumption leads to quite reasonable
estimates of the errors involved, except in extreme cases of signal inter-
n
mittency.) The variance o of the difference between the ensemble (true)
average and the average obtained by integration over time T is (Lumley and
Panofsky, 1964):
2 ~2~
a = 2ri/T ,
— x- _
where f is the ensemble variance of F about its ensemble mean, f=F-F and I
is the integral scale of F. The fractional error e, then is given by
e2 =
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
752
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1000
.5 100
C9
z
3
oc
111
< 10
1
1
e=1%
3%
10%
6 7 8 9 10
2 345
VELOCITY, m/s
Figure 40. Averaging time requirements for wind tunnel
~y
measurements of turbulent energy, u .
estimate a maximum integral scale as 6/1^, where 6 is the boundary layer
depth and U^ is the free stream velocity; the required averaging time is
T = 46/U e2 .
00
This relationship is shown in Figure 40 for a typical boundary layer
depth of 1m. For a wind tunnel speed of 4m/s and a desired accuracy of 10%,
a two-minute averaging time is required. It is readily observed that much
higher accuracies at such low wind speeds are impractical, as an accuracy of
1% would require an averaging time of over 3 hours.
To estimate averaging times required for measuring other quantities
(besides turbulent energy), respective integral scales must be known and
numerical factors are generally larger; however, experience has shown that
153
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T = 46/U e is a reasonably good estimate of the averaging time required
GO
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 to « contains no
frequencies above W cycles per second, then it is
completely determined by giving its ordinates at a
sequence of points spaced 1/2W seconds apart."
Hence, in order not to lose information from our continuous signal through our
discrete sampling, it will be essential to determine the highest frequency
component in our signal and to sample at twice that rate. The highest
frequency of any significance in the turbulence is the Kolmogoroff frequency,
f.=U/2irn (see Section 2.2.2.2). Hence, assuming an excellent anemometer
(good frequency response and low electrical noise), all information about the
turbulent signal may be obtained by sampling at a rate of 2fd=U/irn. At the
slow flow speeds typical of fluid modeling studies (<10m/s), the Kolmogoroff
microscale is not likely to be much smaller than 0.5 mm. Hence, a typical
sampling rate would be approximately 2 kilohertz at a flow speed of 5m/s.
Of course, if the transducer or amplifier have slower frequency response, it
is pointless to sample at twice the Kolmogoroff frequency. A flame ionization
detector, for example, has a time constant of approximately 0.5 sec., so that
a sampling rate in excess of 4 hertz is not necessary.
Strictly speaking, the above discussion on sampling rates applies
"across-the-board" to all types of measurements; because of aliasing, lower
sampling rates could conceivably yield incorrect results (for more information,
154
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see Lumley and Panofsky, 1964). However, unless the signal is so periodic
as to be totally atypical of atmospheric or laboratory shear flows, such high
rates are necessary only for spectral (or autocorrelation) type measurements.
Generally, it is sufficient to ensure that
1. The averaging time is long enough, and
2. A sufficient number of samples is taken to reduce statistical errors
arising from finite sample size.
For normally distributed velocity fluctuations, estimates of required
sample sizes for the determination of mean velocity and turbulence intensity
are (Bradbury and Castro, 1971; Mandel , 1964)
where n is the number of samples required to ensure that the estimates of
mean velocity and turbulence intensity are within ±AU and ±A(u2) , respect-
ively, of the ensemble values, t is a parameter called the Student's t
distribution function; it has values of 1.96 for a 95% probability of being
within the interval and 1.645 for a 90% probability. As an example, if the
turbulence intensity were 50%, then, to determine the mean velocity within 5%
of the ensemble mean velocity with 95% probability, it would be necessary to
average 384 samples. The turbulence intensity determined with 384 samples
would be within 7% of the ensemble intensity with 95% probability. These
estimates do not, of course, include experimental errors. They also assume
that the samples are taken far enough apart as to be independent or, from
another viewpoint, that the total length of record meets the averaging time
requirements discussed previously.
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4. THE HARDWARE
4.1 GENERAL REQUIREMENTS
Model experiments may be conducted using either air or water as the
medium, although air is by far the most coronon. The present section con-
siders the size and performance characteristics required of the facility;
it is written primarily with a view toward wind tunnels, but, of course, the
principles are equally applicable to water facilities. The requirements of
the facility are defined by the scaling laws that have been covered in prev-
ious chapters. In Section 4.2, the advantages and disadvantages of air versus
water will be discussed in detail.
4.1.1 The Speed Range and Scale Reductions
The roughness Reynolds number criterion which insures that the flow is
aerodynamically rough (Section 2.3.2) is
u*zQ/v>_2.5, (4.1)
which may be written as
U.6/V >.2.5(UaB/u*)(6/z0). (4.2)
where 6 is the boundary layer depth, U^ is the free stream tunnel speed, and
the nondimensional parameters in parentheses are to be matched in model and
prototype. As a typical example, consider a problem wherein it is required
to determine the height of a stack near a power plant that is located in a
suburban area where zQ=lm and 6=600m. From Figure 14, u^/U^ 0.05, so that
the free stream speed is
756
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IL6 U fif
— = -fg1130,000, (4.3)
or Uw >_0.00075S (m/s), (4.4)
where 6- is the atmospheric boundary layer depth and S is the geometric scale
reduction factor, typically 300
-------
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specifically applicable only to the particular example illustrated, the
general features are universal, i.e., there are excluded regions of the
same general shapes for all model studies. Similar graphs may also be
constructed to determine whether particular studies can be successfully
conducted in particular wind tunnels. Other factors also enter into this
decision; these are discussed further in the next subsection.
4.1.2 Test Section Dimensions
From the previous section, it is evident that boundary layer depths on
the order of 0.3 to 2m are required, so that the height of the test section
should be somewhat greater than 2m. Following Robins (1975), the length of
the test section is determined by four distances:
L-, - the length occupied by the boundary layer generation system
Lp - the fetch required to establish a homogeneous, equilibrium boundary
layer
L^ - the upwind fetch of a topographical model
L. - the distance downwind to and somewhat beyond the point of maximum
concentration.
For the Counihan (1969) system, L,=2.5<5 and Lp^S.Bfi. The required upwind
fetch of a topographical model would depend upon the details of the specific
topography, but 20 stack heights (HS) may be considered typical (in fact, L2
and L3 may overlap in some cases). The distance to the point of maximum
concentration is typically 10 to 20(Hs+Ah), where Ah is the plume rise.
Hence, the required test section length is
L = 86 + 40Hs + 20Ah.
For a 300m stack modeled at a scale ratio of 600:1, the required test
section length would be approximately 35m!
The width of a plume at the point of maximum concentration downwind of
759
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a tall chimney is typically 26. Effects of the sidewalls may extend into
the flow a distance of approximately 1 test section height. Hence, the
required width of the test section is about 46.
It should now be evident that many factors must be considered in the
process of selecting a scale factor for a particular model study or in
deciding whether a particular study can be conducted in a particular fac-
ility. We can, however, give some fairly definite guidance. Consider the
wind tunnel simulation of the dispersal of buoyant effluent from a tall
chimney in a neutrally stable atmospheric boundary layer. From the prev-
ious discussions, it is evident that scale reductions between 300 and 1000
are possible and, for a tall stack, a choice of 1000 would almost certainly
be made in order to accomodate the model within the test section. Thus,
a 600m atmospheric boundary layer implies a 0.6m model boundary layer. The
fetch required for boundary layer development, that necessary for upstream
flow "conditioning", and that required to obtain the maximum ground level
concentration within the test section imply a test section length of about
15m and, consequently, a height of 1m and a width of 3m; an operating speed
in the range of 0.7 to 3m/s is also implied, there being no point in model-
ing unrealistically high wind speeds. A wind tunnel that does not roughly
satisfy these requirements should not be used.
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4.2 AIR VERSUS MATER
The choice of air versus water as the fluid medium for modeling of
atmospheric flow and diffusion of 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 factor 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 incompresslbility, 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 -- cavitation 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.
4.2.1 Visual Observations
Smoke and helium filled soap bubbles (for which a generator is now
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 of dye, hydrogen bubbles, potassium permanganate crystals,
shadowgraphs, and neutrally buoyant particles. And because flow speeds are
generally low, it is easy to observe and photograph flow patterns in
767
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water. The comparable smoke in a wind tunnel is difficult to regulate either
in concentration or specific gravity (oil fog smoke generators are notor-
iously cantankerous and they occasionally explode!). Smoke is also difficult
to observe visually and photographically at flow 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 of 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 not 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-point measurements would be required otherwise.
Flow visualization can also be of great help in the interpretation and
understanding of quantitative data. Hot-wire anemometry, in spite of its
increased sophistication and reliability in recent years, still cannot tell
us the direction of flow (there 1s 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 (1980) 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
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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
been difficult to obtain through other means.
4.2.2 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, 1970). For local
velocity measurements in wind tunnels, numerous instruments are available:
pi tot tubes, hot-wire, hot-film, and pulsed-wire anemometers. (The pulsed-
wire anemometer is especially useful in low speed and reversing flows, as it
is capable of detecting the direction of flow; it is also relatively insens-
itive to small fluctuations in temperature, and therefore would be useful in
stratified flows.) 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 (Mem
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 wind tunnel. These instruments are presently the most popular
because they have a relatively fast response time (~0.5s), their output is
linear with concentration over a very wide range (about 0.5 to 10,000 ppm
methane with proper adjustments), and they can be used with any hydrocarbon
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gas, including methane, ethylene, and butane, which have specific gravities
of 0.5, 1 and 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 a!., 1976), and radio-
active gases (Meroney, 1970), along with corresponding measuring devices.
Smoke, temperature, and helium techniques offer possibilities for the meas-
urement of concentration fluctuations, but are generally limited to the
measurement of small dilutions, i.e., 1:100, as compared with the 1:10,000
desired. Fackrell (1978) has developed a "popper valve" to allow the flame
ionization detector to be used for the measurement of concentration fluctua-
tions. Fackrell (1979) has also modified a flame ionization detector to
enable continuous measurements of concentration fluctuations at rates of up
to 350hz. Another method suitable for measuring low concentration fluctua-
tions is laser/aerosol light scattering (Meroney and Yang, 1974). When
properly designed, this instrument can detect individual particles within
the sampling volume.
Salts in conjunction with conductivity meters, acids with pH meters,
temperature with thermistors, and dyes with colorimeters and fluorometers
have been used as tracers for quantitative measurements of concentration in
water. Except for temperature, these techniques offer a wide range in con-
centration detectability. The conductivity probes and thermistors can be
quite fast-response devices. They offer possibilities for the measurement of
concentration fluctuations.
4.2.3 Producing Stratification
There are two common methods for producing stratification in water. The
most common method of producing stable density stratification in water is by
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slowly filling a tank through distribution tubes on the bottom with thin
layers of salt water, each layer increasing in specific gravity (Hunt et al.
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 of 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). Reclrculating systems using this technique have been
impractical because of the mixing within the pump. However, Odell and
Kovasznay (1971) have designed a rotating disk pump that maintains the
gradient; this device permits the use of recirculating salt water systems, but,
thus far, has been used only for very small channels (^lOcm depth).
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 is 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 floor of the
test section and/or differential heating and/or cooling of the air entering
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 Micrometeorological Wind Tunnel at the Colorado State University
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(Cermak, 1975) has a test section 1.8m square and 27m 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 al., 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 inver-
sion) develops downstream because of the air contact with the uncooled tunnel
floor or model surface.
4.2.4 Examples
Thus far in this chapter, we have discussed the comparative advantages
of using air or water as the fluid medium for modeling studies. There are
no "hard and fast" rules for deciding which type of facility is best for
a particular study. Two example problems are given below, one of which
appears best suited 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 facility.
Problem 1: We wish to determine the excess ground level concentrations
caused by an insufficiently tall chimney next to a power plant
in essentially flat terrain.
Method of Solution: A plume from a power plant is generally highly buoyant,
so that building downwash probably occurs only in high wind, hence neutral,
conditions. The advantages of a wind tunnel over a water channel here are
766
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obvious. Thick, simulated neutral atmospheric boundary layers are easily
obtained in wind tunnels, whereas the development and testing of 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
using hydrocarbons (probably methane in this case) and a flame ionization
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.
Probably the only advantage to using a water channel in this case would be for
the ease of flow 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 (once per year) on an isolated three-dimensional
hill 200 m high downwind of a 100 m high stack. Nocturnal surface-
based inversions develop to 400 m depth with temperature gradients
of 1.5°C/100 m and wind speeds of 2 m/s at the 200 m elevation.
Method of Solution; The maximum glc will probably 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 on the
hill height and the density difference between the base and top of the hill:
F = U/Nh = U/(ghA6/e1/2 = 2/(9.8x200x5/300)1/2 = 0.35.
(Notice that potential temperature instead of density has been used in calcu-
lating 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. = 1.0 at the top and 1.2 at the bottom, yielding
N=1.3 rad/s). The required towing speed for a hill of height 0.2 m would be
U=FNh=9 cm/s, a reasonable towing speed for a water channel, yielding a Reynolds
number Uh/v=18,000.
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This type of flow has not yet (at least, to the author's knowledge) been
obtained in a wind tunnel, but rough calculations will easily illustrate the
difficulties. The maximum temperature difference that could be generated is on
the order of 100°C. Let us suppose that the model hill height is also 20cm
in the wind tunnel and that the 100°C temperature difference is imposed over a
40 cm 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 4600 (although it 1s most likely unimportant in this case, since
the flow will definitely not be turbulent).
4.2.5 Summary
The 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 small
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 flow 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
capabilities.
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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
hope 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 point: "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 flow that
we have some chance of simulating well, the neutral surface layer, hardly ever
exists in the atmosphere." Panofsky (1974) 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." We have the ability to
control the flow and to independently adjust specific parameters. To 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
169
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the small-scale turbulence is "parameterized." Whereas we have difficulty in
simulating the large scale eddies, we are no worse off than the numerical
modelers and we need not make any "closure assumptions". Nor must we deal
with an inviscid 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 flow 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 accordingly—with caution.
There are two basic categories of fluid modeling studies: (1) The
"generic" study wherein idealized obstacles and terrain are used with idealized
flows in an attempt to obtain basic physical understanding of the flow 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 Thomases" concerning the applicability of fluid
modeling studies to the real atmosphere; yet those same "doubting Thomases" do
not hesitate to apply potential flow models with constant eddy diffusivities
in order 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 the "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., by 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 mathematical model cannot simulate the
results of an idealized laboratory experiment, how can it possibly be applic-
able 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 - - with just as much and generally more credibility than any
current mathematical models.
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