EPA-600/4-75-016b
December 1975
Environmental Monitoring Series
TURBULENCE MODELING AND
ITS APPLICATION TO ATMOSPHERIC DIFFUSION
Part II
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, N.C. 27711
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development,
U.S. Environmental Protection Agency, have been grouped into
five series. These five broad categories were established to
facilitate further development and application of environmental
technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in
related fields. The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ENVIRONMENTAL MONITORING
series. This series describes research conducted to develop
new or improved methods and instrumentation for the identifi-
cation and quantification of environmental pollutants at the
lowest conceivably significant concentrations. It also in-
cludes studies to determine the ambient concentrations of
pollutants in the environment and/or the variance of pollutants
as a function of time or meteorological factors.
This document is available to the public through the National
Technical Information Service, Springfield, Virginia 22161.
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EPA-600/4-75-016b
December 1975
TURBULENCE MODELING AND ITS APPLICATION TO ATMOSPHERIC DIFFUSION
PART II: CRITICAL REVIEW OF THE USE OF INVARIANT MODELING
by
W. S. Lewellen and M. Teske
Aeronautical Research Associates of Princeton, Inc.
Princeton, New Jersey 08540
Contract No. 68-02-1310
Project Officer
Kenneth L. Calder
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
U.S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AND DEVELOPMENT
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratory, U.S. Environmental Protection Agency, and approved for
publication. Approval does not signify that the contents necessarily
reflect the views and policies of the U.S. Environmental Protection
Agency, nor does mention of trade names or commercial products constitute
endorsement or recommendation for use.
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/4-75-016b
3. RECIPIENT'S ACCESSION-NO.
4. TITLE ANDSUBTITLE
TURBULENCE MODELING AND ITS APPLICATION TO ATMOSPHERIC
DIFFUSION. PART II: CRITICAL REVIEW OF THE USE OF
INVARIANT MODELING
5. REPORT DATE
December 1975
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
W. S. Lewellen and M. Teske
8. PERFORMING ORGANIZATION REPORT NO,
A.R.A.P. Report 254, Part II
9. PERFORMING ORGANIZATION NAME AND ADDRESS
10. PROGRAM ELEMENT. NO.
Aeronautical Research Associates of Princeton, Inc.
50 Washington Road
Princeton, New Jersey 08540
1AA009
11. CONTRACT/GRANT NO.
EPA 68-02-1310
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
13. TYPE OF REPORT AND PERIOD COVERED
Interim
14. SPONSORING AGENCY CODE
EPA-ORD
15. SUPPLEMENTARY NOTES
Issued as Part II of 2 Parts
16. ABSTRACT
A method for the calculation of turbulent shear flows based on closure of
the equations for second-order correlations of fluctuating quantities is
reviewed. Various model possibilities for closure are described and detailed
evaluation of coefficients for a simple model is outlined. Comparisons of
model predictions and experimental data for a wide variety of laboratory
and atmospheric flows are presented.
7.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
c. COS AT I Field/Group
* Turbulence
* Turbulent flow
* Atmospheric diffusion
* Mathematical models
12A
20D
04A
8. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (ThisReport)
UNCLASSIFIED
21. NO. OF PAGES
55
20. SECURITY CLASS (Thispage)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
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PREFACE
This critical review of our A.R.A.P. turbulence model
was presented as a three-hour lecture in the one-week course,
"Fundamentals and Applications of Turbulence," at the
University of Tennessee Space Institute, Tullahoma, Tennessee,
during the week of April 21-25, 1975. It includes a discussion
of the development, validation tests, limitations, and some of
the applications of the model. The work reviewed has been
supported by a number of government agencies over the past
four years. In addition to EPA support since 1971, the work
has been partially supported by NASA, AFOSR, and the Navy in
addition to internal corporate support. Detailed contractual
acknowledgements are provided in the referenced reports.
ii
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TABLE OP CONTENTS
Preface ii
Abstract 1
1. Introduction 1
2. Model Development 2
1. Closure Requirements 2
2. Dissipation Terms , 4
3. Pressure Correlations 5
4. Third-Order Velocity Correlations 7
5. Modeled Equations 8
6. Scale Determination 9
3. Evaluation of Model Coefficients 12
1. Dissipation Coefficient (b) 12
2. Diffusion Coefficient (vc) 13
3. Scale Determination 13
4. Low Reynolds Number Dependence (a) 16
5. Additional Coefficients Required to
Compute Temperature Fluctuations (A, s, Sc)l6
4. Model Verification 18
1. Axisymmetric Free Jet 19
2. Free Shear Layer 20
3. Two-Dimensional Wake 21
4. Axisymmetric Wake 21
5. Flat Plate Boundary Layer 22
6. Flow Over an Abrupt Change in Surface
Roughness 27
7. Temperature Fluctuations in the Plane
Turbulent Wake 27
8. Stability Influence in the Atmospheric
Surface Layer 27
9. Shear Layer Entrainment in a Stratified
Fluid 31
10. Free Convection 31
11. Planetary Boundary Layer for Neutral
Steady State 33
5. Local Equilibrium Approximations 34
6. Applications 37
1. Diurnal Variations in the Planetary
Boundary Layer 38
2. Stratified Wake 39
3. Pollutant Dispersal 4l
iii
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7. Concluding Remarks
References
iv
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USE OF INVARIANT MODELING
W. S. Lewellen
Aeronautical Research Associates of Princeton, Inc.
Princeton, New Jersey
Abstract
A method for the calculation of turbulent shear flows
based on closure of the equations for second-order correlations
of fluctuating quantities is reviewed. Various model possibil-
ities for closure are described and detailed evaluation of
coefficients for a simple model is outlined. Comparisons of
model predictions and experimental data for a wide variety of
laboratory and atmospheric flows are presented.
1. Introduction
Previous lectures in this series have contained several
different methods for calculating turbulent flow with varying
degrees of complexity and probable computational success. Here
I will describe the development and some applications of a
method which has been pursued by Donaldson and his colleagues
at A.R.A.P. for several years. The name of the model, an
invariant model, can be interpreted in two ways. It refers to
the constraints imposed on the choice of model terms required
for closure. That is, any model term must exhibit the same
tensor symmetry and dimensionality as the term it replaces.
But the goal of the approach can also be described as an invar-
iant model in the sense that our goal is a model which,although
it is semi-empirical, has no varying constants which must be
determined for each new flow.
This review has been partially funded with federal funds from
the Environmental Protection Agency under Contract No. EPA 68-
02-1310. The content of this paper does not necessarily
reflect the views or policies of the U.S. Environmental Pro-
tection Agency.
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W. S. LEWELLEN
Our starting point is the exact equation for the Reynolds
stress. In choosing to attempt closure at the level of the
equations for the second-order correlations, we are assuming
that this will yield a more general model than first-order
closure does. Certainly the second-order equations contain a
great deal of physical information regarding the dynamics of
the turbulent fluctuations. Providing this information is not
lost by inappropriate modeling of the terms required for
closure, this approach should be more general than first-order
closure. In fact, it will be seen in a later section that a
relatively rational, first-order closure model is obtained as
a subset of our invariant model.
No attempt will be made to give a complete review of the
literature surrounding turbulence transport modeling. Some
surveys have been given by Bradshaw,! Mellor and Herring,2 and
Reynolds.3 A convenient collection of some of the most impor-
tant literature has been made by Harlow. I will attempt to
relate the model described here to those under development by
other investigators. A limited number of comparisons between
the numerical results of various models will be made.
In Section 2, the requirements for closure are outlined
and some of the various model choices that have been made by
various investigators are presented. Following a discussion of
the alternatives, a relatively simple second-order closure
model is chosen. The determination of the required empirical
coefficients is described in Section 3. Detailed comparisons
between the simple model's predictions and available data are
made in Section 4 for a wide spectrum of flows. Although there
are discrepancies in some of the variables for some of the
flows, the overall verification is highly encouraging. In
Section 5, the simplfications to the transport equations under
local equilibrium conditions are explored. As already mention-
ed, this provides a bridge between first- and second-order
closure. Section 6 contains some of the results of calcula-
tions for three applications of practical interest: a strati-
fied wake, diurnal variations in the planetary boundary layer,
and pollutant dispersal in the atmosphere.
2. Model Development
2.1 Closure Requirements
When the flow variables are decomposed into a mean and a
fluctuating quantity, the mean flow equations may be written
as
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USE OF INVARIANT MODELING
3t j 3x 3x 3x 3x p 3x G
3U
^T=0 (2.2)
M+ v |®_= _ -_±_ + ^_ | k ^_| (2.3)
j 4- *1 fl Y »!-- Jwl -J__ I \ - /
The flow has been assumed incompressible but with small varia-
tions in density due to changes in temperature. Following the
Boussinesq approximation,^ the only effect of the density vari-
ation is in the gravitational body force term in the momentum
equation (2.1). The last term in (2.1) is that due to coriolis
forces present in a coordinate system rotating with angular
velocity fi . It is included here in anticipation of atmospher-
ic applications. Equation (2.3) is a diffusion equation for
the temperature perturbation.
This system of equations is not complete due to the
presence of the Reynolds stress terms. Exact equations for
these correlations of the fluctuating variables may be derived
as outlined in the introductory lectures or by Donaldson.
- 2e., na unu, - 2e
J£kVkul " 3T(Viuj)
2 - -
____
p 3x. p 3Xi 2 9Xk
k (2.4)
3u u.9
. fl a u a.
1 . i J2_ _ V6 - t-ku — (2.5)
p 3x. VD . 2 KUi a 2 U.'3''
i 3x 3x.
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W. S. LEWELLEN
»2,u
9t +U
362
""I
^O
"„ fl du
/U . U ~
j ox
V2
,, 39 36 ,_ ,,
- 2k T— -5— (2.6)
3x. 3x
This still does not permit the system to close. In fact,
for each new equation added, several more unknown flow varia-
bles are added. Clearly, continuing the process by deriving
exact expressions for these correlations will add further new
correlations in ever-increasing numbers. This is the closure
problem.
Closure of the system of equations at the level of (2.4)
to (2.6) is called second-order closure. In (2.4) this
requires neglecting or modeling in terms of the other varia-
bles three terms:
u. r, \ 3u. 3u
Let us consider these three terms in some detail. The analo-
gous terms in (2.5) and (2.6) can then be modeled in a similar
manner.
2.2 Dissipation Terms
The last term in (2.7) measures the effect of viscous
decay on the structure of the Reynolds stresses. Even in high
Reynolds number flow, we expect viscous dissipation to be the
major loss mechanism for turbulent kinetic energy. Due to the
nonlinear terms in the Navier-Stokes equation, a reduction in
viscosity is compensated by a reduction in scale of the small-
est eddies in the flow. The turbulent dissipation process is
often compared to that of a cascading waterfall with energy
flowing from the mean flow to ever-decreasing scale eddies.
The cascade continues until a scale sufficiently small to per-
mit viscosity to dissipate the energy as heat is reached.
When the dissipation eddies are much smaller than the eddies
which receive their energy directly from the mean flow (i.e.,
when the Reynolds number is large), one may assume that they
are statistically independent of the mean flow geometry. The
scale of these isotropic dissipative eddies can then only
depend on the dissipation rate e and the viscosity v . This
Kolmogorov microscale^ is
^ 1/4
n = (v/erf (2.8)
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USE OF INVARIANT MODELING
Just as the end of the cascade contains no information on
the scale of the large eddies, the breakup rate of the large
eddies should be independent of v . Therefore, for high
Reynolds number, it appears dimensionally correct to have
9u 8u
where q is the rras value of the total velocity fluctuation
and A is a macroscale of the eddies. Since the viscous dis-
sipation process is an isotropic process, most investigators
model the dissipation terms as an isotropic term; i.e. ,
3u 9u
8 9 10
Some ' ' have modeled it as an anisotropic term. A form
which goes to (2.10) in the limit of Re -»• °° , but reflects the
probability that at low Re dissipation will be anisotropic, is
3u 3 C
<2'u)
* iij j
The first term of the form adopted in (2.11) corresponds to
taking the Taylor microscale X proportional to
1/2
A/ (a + bqA/v) where a and b are constants, as suggested
by Rotta.11
Several investigators prefer to calculate e from a
dynamic equation obtained by modeling its governing equation.
This will be discussed in Section 2.6 since, through (2.9), it
is equivalent to an equation for A .
2.3 Pressure Correlations
The correlations involving pressure fluctuations are more
difficult to model than the dissipation terms. These terms
redistribute the turbulent energy produced by the leading term
on the right-hand side of (2.4). This can perhaps be more
readily seen by rewriting the pressure correlations as
ij p 3x p 3xt
Since the flow is incompressible, there will be no contribution
from the last term in (2.12) to the kinetic energy of the
turbulence, q /2 = u u /2 . It contributes only to a
-------
W. S. LEWELLEN
redistribution of energy between the Reynolds stresses. A vol-
ume integral of the first two terms on the right-hand side of
(2.12) over any finite region of turbulence bounded by laminar
flow will yield zero. Thus these two terms must only contri-
bute to a spatial redistribution, i.e., are diffusion terms.
All investigators appear to follow Rotta^- in modeling at
least one contribution of the pressure correlation as a return-
to-isotropy term proportional to the extent to which the flow
is anisotropic; i.e.,
Y^ - 6« £) (2'13)
The term in parentheses, the departure from isotropy, provides
the right tensor symmetry and q/A provides the correct dimen-
sionality. Donaldson ' adds to this a spatial diffusion term
with proper symmetry and dimensionality to represent the first
two terms on the right-hand side of (2.12):
(2.14)
\ J
It is possible to show that the pressure correlation
should probably depend on the mean flow strain. By taking the
divergence of the fluctuating Navier-Stokes equations, a
Poisson equation for p may be obtained:
3U 9u
- V
p
= - 2
3x3x
N A i 39
-uiV+lT3x-
- 2e
n
9uk
J
(2.15)
Equation (2.15) can be formally integrated to obtain p which
may then be differentiated and correlated with u^ to form
TT-ji . The integral cannot be solved so the procedure can only
be used to suggest the form of the modeled term. The simplest
addition to (2.14) to incorporate an influence of mean strain
is a term of the form
'3U.
3U.
3x
(2.16)
J
In fact, a numerical value for C^ (= 0.2) has been obtained
by Rotta-^ and by Crow-^ for the limiting case of isotropic
turbulence in a weak, homogeneous mean strain. However, such
a large value appears to make the term too overpowering for
any practical calculations. A term such as (2.16) has been
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USE OF INVARIANT MODELING
included by several modelers-^' » ^ with €5 empirically
determined.
A form for the mean strain term with a good deal of appeal
is that suggested by Launder^'
where P^^ represents the production terms for the Reynolds
stress in (2.4). A similar term is included in the more in-
volved model of Naot et al.18 Adding (2.17) to (2.14) would
be equivalent to assuming that the pressure correlations not
only drive the turbulence towards isotropy with a term linearly
proportional to the departure from isotropy, but also redistri-
bute the production terms with a term linearly proportional to
the anisotropy of the production terms.
Neither (2.16) nor (2.17) is sufficient to account for the
observation that the mean strain effect is often not symmetric
between U2 and U3 when the flow is one-dimensional in the
x-^ direction. More complicated model variations for the
pressure correlation terms may be found in the work of Lumley
and Kha j eh-Nouri,19 Hanjalic and Launder,20 Wolfshtein, et al,
Zeman and Tennekes , 2^ and Varma.
2.4 Third-Order Velocity Correlations
This term represents a process by which the Reynolds
stress is transferred from one part of the flow to another
without any net production or loss. Again this can be demon-
strated, as it was for the pressure diffusion term of (2.12),
by integration over a finite volume. When the volume is
bounded by either laminar or by homogeneous isotropic flow, the
integral will vanish. The most popular modeling of this trans-
port term is as gradient diffusion, although Bradshaw^- has
suggested that it could well be algebraic.
A number of different gradient diffusion forms have been
used as a model for this term. Donaldson proposed a form
which satisfies the tensor symmetry of u^u^u^ with a scalar
diffusion coefficient
Dij • t£ (ViV - c< - ' - + - + ^^ I (2'18)
Hanjalic and Launderzu used a form with tensorial diffusivity
9 A
C5
They obtained this form by a "firm pruning" of the exact
-------
W. S. LEWELLEN
equation for u,u.u.. The simplest possible form is given by
k 1 j
9u u
(2.20)
Although this form satisfies the tensor symmetry of D^j , it
does not satisfy the symmetry of u,u u. itself.
10 J 19
Wolfshtein et al. and Lumley and Khajeh-Nouri propose
forms that have considerably more terms and involve a number of
coefficients. These currently appear to have too many coeffi-
cients for manageable computations.
2.5 Modeled Equations
Our philosophy here is to first attempt calculations with
the simplest possible second-order closure models. We choose
to use (2.11) for the dissipation term, (2.13) for the pressure
velocity interaction, and the pressure contribution to diffu-
sion incorporated with the velocity diffusion for one diffusion
term like (2.20).
3
a^^kVj^axT
du. b6 q aVu u.
= -lA- + —H (2'21)
(2.22)
(2.23)
In this restatement of the modeled terms, C^ has been set
equal to 1 by using (2.13) to define the macroscale A •
other coefficients have been assigned the symbols used in the
most recent A.R.A.P. publications.
Equations (2.21) - (2.23) provide the minimum requirements
of any second-order closure. The combination of terms in (2.21)
and (2.22) provides a destruction term for each of the Reynolds
stress components which allows an equilibrium value to be
reached in a time long compared with A/q . The term in (2.23)
prevents any excessively sharp gradients in Reynolds stresses
from occurring.
When similar terms are used in the heat flux and tempera-
ture variance equations, (2.5) and (2.6), the modeled set of
second-order correlation equations may be written as
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USE OF INVARIANT MODELING
A 9 + "a" + -0
2e.l0fl.u0u. - 2e OULU. +v -f^
ikx. k i j jfck Jl K i c 3x
...
'24)
4
j 3x
- u±i
+ V
c
- sn
1 oX .
J j
a A 9u-,9
3qA i
3x 3x
3U. 8,62
,, ft 1 i 1
V V e
•-f¥+k
n— O -. A
- - 2eij kVke
o
3"u 6 aku 6
^- i 1
2 2
j (2.25)
38 . „ 36 0—5- 39 , 3gA 36 . . 36 2bsg6
——— -4- TT •...— — _ 9n H ——- -4- \r * —. 4- if .— -. . *
3t Uj 3x ^Ujb 3x Vc 3x 3x 3x2 A
(2.26)
This is the set we choose to make calculations with here.
Rather than use models with a large number of coefficients that
are finely adjusted to fit a few particular flows, we choose to
work with a relatively minimum number. Only as experience
proves that more terms are required will other terms and coef-
ficients be added. Two additional coefficients were included
in (2.25) and (2.26), A and s . Both of these were first
set equal to one, but as will be seen in the next section, it
was later found desirable to add flexibility.
2.6 Scale Determination
To complete closure, it is necessary to provide some means
for determining the turbulent scale A involved in the modeled
terms. This is approached in different ways by various invest-
igators. It may be specified empirically based on the gross
features of the particular flow geometry. Or it may be pre-
dicted from a semi-empirically modeled, dynamic differential
equation. Each of these approaches has some advantages and
disadvantages.
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W. S. LEWELLEN
The macroscale A is defined by (2.21)-(2.23) . It is
expected to be related to the integral scale discussed in
earlier lectures but, due to our choice of normalization, not
equal to it. It is also related to the mixing length discussed
in Deissler's lecture. As such, there is empirical information
which can be used in our determination of this model parameter.
It appears fairly clear that the scale cannot exceed some
fraction of the total spread of the region of turbulence and
that, in some neighborhood of the wall, it should be propor-
tional to the distance from the wall. These two simple ideas,
together with empirical information used to determine the two
implied constants of proportionality, are sufficient to permit
the system to close with relatively good numerical results for
many problems. 6,23,24 Qthers2->,26 ^ave specified a completely
empirically determined distribution of A across the region of
interest in the same manner as is done for first-order mixing
length approaches .
In an attempt to remove some of the arbitrariness of the
specification of A for different flows, a number of investi-
gators have resorted to a modeled dynamic equation for A or
its equivalent. The starting points for such attempts have
varied widely. A two-point-velocity-correlation equation forms
the basis for the work of Rotta,-^ Naot et al. ° Donaldson,**
and Rodi.15 By forming an equation for the two-point velocity
correlation u^(x)u^(x+r) and integrating to form an integral
scale, it appears appropriate to take
q2A = cJJJ ui(x)ui(x+r) ^ (2.27)
v
For a flow without body forces, this leads to, following
Wolfshtein,10
Dt
(x)u.,(x+r)
3U. (x + r) 9U (x)
- ui(x)uk(x+r) ^- + uk(x)ui(x+r)
- ui(x)uk(x)ui(x+r)]
10
-------
USE OF INVARIANT MODELING
9p(x)ui(x+r)
3u (x)u (x+r)
- 3
k
32u (x)u (x + r)
+ 2v — = - - (2.28)
*"\
3rk
The difficulty with this approach is readily apparent. None of
the terms on the right-hand side of (2.28) can be integrated
exactly. All of the terms must be modeled. This is also true
if one starts with the equation for the dissipation e as is
favored by many (e.g., Lumley and Khaleh-Nouri , 19 Harlow and
Nakayama, and Hanjalic and Launder ) or with the equation
for the vorticity fluctuations as suggested by Daly and Harlow*
or Wilcox and Alber.28 With the model chosen in Section 2.5,
e = bq-VA . Also the vorticity fluctuation can be taken as
proportional to q/A . Thus with the aid of the energy equa-
tion, any of these approaches may be reduced to an equation for
A . As Bradshawl and Mellor and Herring2 have pointed out, all
of the resulting A equations have the same form
3U
- s.bq + diffusion terms (2.29)
The difference in the various expressions lies in the construc-
tion of the turbulent diffusion terms. Unfortunately, these
turn out to be more important in the scale equation than in the
Reynolds stress equation. The scale equation we have experi-
mented with at A.R.A.P.29'30 starts with a rather general
diffusion term of the form
\ 2
JJ^/T 3 ; A 3A \ . A 3 / . 3q
diffusion terms = S-. T— v qA T— + s_ —TT -5— qA •~3~
0 3x. \ c 3x.I 32 3x. P 3a
i V i' q i \
(2.30)
2
As pointed out by Mellor and Herring this covers the possibil-
ity of starting with equations for the quantity qmAn and in
combination with the energy equation reducing it to (2.29). As
in the formulations of Refs. 6 and 9, we will add a term
11
-------
W. S. LEWELLEN
proportional to (A/q )g u 0/0 to permit the direct effect of
stratification.
It is immediately obvious that the scale equation contains
much more arbitrariness than the Reynolds stress equations
where many of the terms were determined precisely without re-
course to modeling or coefficients. With such a large number
of coefficients in the scale equation, a correspondingly large
number of different experiments must be matched concurrently if
the resulting coefficients are to have any invariant validity.
There is also the question as to what extent it is really
appropriate to have the integral quantity A determined by
point values of the other variables. It seems that spatial
boundary conditions may be expected to play a much more impor-
tant role in the determination of A than they do for the
Reynolds stress.
3. Evaluation of Model Coefficients
With the exception of the model scale variable A , the
simple model chosen in Section 2.5 contains five coefficients:
a, b, v , A, and s . Ideally these coefficients should be
constants, but conceptually the approach is still valid if they
turn out to be unique functions of the dimensionless parameters,
For the flow specified in (2.1)-(2.3), there are three such
dimensionless parameters: a Reynolds number, a Richardson
number, and a Rossby number. The coefficients might also be a
function of some dimensionless number characterizing the state
of the turbulence, such as that proposed by Lumley and Khajeh-
. 2, N/ 4
^ UjU.. - fi^q /3)/q
We will initially, optimistically, assume that the coefficients
are constant. If the coefficients have to be functions of the
dimensionless variables, then the second-order approach will
lose some of its advantage over first-order closure. Insofar
as is possible, each coefficient should be determined from a
critical flow experiment that involves only that coefficient.
3.1 Dissipation Coefficient (b^
For high Reynolds number turbulence, the most sensitive
model coefficient in (2.24) is the coefficient b . The value
of this coefficient can be isolated by considering steady,
homogeneous turbulence with no body forces. When a one-dimen-
sional mean flow in the x-^ direction is considered with a
gradient in only one direction, x^ , (2.24) reduces to
12
-------
USE OF INVARIANT MODELING
3U, 3U,
0 = -
(3.1)
From these separate component equations, one can obtain several
different expressions for b independent of A .
U'U" 3 /I U3U3\ 3 /I
-1 •Vl-'llT-^rl 0.2)
q
31
The data of Champagne et al. for such a homogeneous shear
flow in a wind tunnel yield values of 0.12, 0.14, and 0.08,
respectively, for the three expressions in (3.2). With the
exception of the last value, these are all close to the value
of 1/8 chosen by Donaldson by comparison with other data. A
closer fit to all the data will require additional model terms
as discussed in Section 2.3.
The computational results to be described here have all
been made with b = 0.125.
3.2 Diffusion Coefficient (v_)
The diffusion coefficient v is difficult to decouple
from the determination of the scale A . We have chosen a
O ^
value of 0.3 based on computer studies of jets and wakes.
However, due to the relative insensitivity of our results to
this coefficient, it is considered known only to ±.05. This
is the basic reason we have chosen to use the simplest form of
diffusion, as given in (2.23). Critical measurements may show
one of the more complicated forms to be preferable. The diffi-
culty of measuring the pressure correlation contribution to
diffusion makes the diffusion terms particularly difficult to
isolate empirically. The best chance for finer detailed model-
ing of those terms probably lies in the possibility of detailed
comparisons with ensemble averages from the type of simulated
turbulence calculation described in Orzsag's lecture.
3.3 Scale Determination
Section 2.6 discussed two approaches to the determination
of A . Let us first determine the two constants needed to fix
the bounds on A . Close to a wall, we will assume A = az
with z the distance normal to the wall. The coefficient a
is related directly to von Karman's constant K . In the
"constant flux," surface layer region of a flat plate boundary
layer, (3.1) holds with the subscript 1 denoting the directim
13
-------
W. S. LEWELLEN
of the free stream flow and XT = z . When the definition of K
-1
(3.3)
is used to eliminate the mean flow gradient, (3.1) can be
arranged to yield
I -3/4
3 L (3.4)
With our previously assigned value of b = 0.125, (3.4) yields
a = 1.68K (3.5)
The generally accepted value of < from laboratory flows is
0.40, but rather extensive measurements in the atmospheric
surface layer at much higher Reynolds numbers^ give K = 0.36.
The corresponding values of a are 0.67 and 0.60. We have
used 0.65 in most of the calculations to follow, although some
have been performed with a = 0.60.
The other bound of A is that tied to the spread of the
region. This we have picked by computer optimization. For an
axisymmetric wake or jet, it is 0.20 times the radius at which
the turbulent energy falls to one-fourth its maximum value.
For a two-dimensional boundary layer or wake, it is 0.3 times
the similar measure of the turbulent spread. For a two-dimen-
sional shear layer, it might be expected to be 0.6. It is
clear that the more complicated the flow geometry is, the more
difficult it becomes.to specify A .
The second approach is an attempt to circumvent this
difficulty by using a dynamic A equation with coefficients
that are independent of flow geometry. Let us attempt to
determine the coefficients in (2.29) and (2.30).
The coefficient 82 can be estimated from the decay of
homogeneous grid turbulence. If homogeneous turbulence is
assumed to decay as
q2 ~ x"n (3.6)
then from the contraction of (3.1)
] (3.7)
Equation (2.29) then leads to
c _ U 1A _ (2-n) ,, „.
S» = - 7 — T~ - - - (-J'O
2 bq 9x n
A recent review of grid turbulence experiments by Gad-el-Hak &
14
-------
USE OF INVARIANT MODELING
Corrsin-" shows values of n predominantly between 1 and 1.3
with more of the values lying near 1.25. This value of n
gives a value of s« = -0.6.
To reduce the number of diffusion coefficients, it is
desirable to keep the coefficient of the direct diffusion term
SQ the same as in the Reynolds stress equation. This calls
for SQ = 1 . It is compatible with this assumption to take
33 = 0 , since with SQ = 1 any combination of m and n for
an expression of the form
3 / artmAn \
d / , dq A I ,_ _.
v "5— qA T^ (3.9)
c 9x ln 9x /
would lead to an equation for DA/Dt with 83 = 0 .
A simple relationship between SQ, s^, 82, and s^ may be
obtained from the reduced form of the scale equation in the
steady, neutral, constant shear stress layer near a solid
boundary where A = otz . In this region, the scale equation
reduces to
2
0 = BI ^- || - S2bq + vcqd2 + s4qa2 (3.10)
q
which, with the aid of the energy equation, may be reduced to
- SA = vc + (b/a2)(Sl - s2) (3.11)
This leaves Sj , Sg , and s-j to be determined by computer
optimization. Prior to doing this, we must determine boundary
conditions for the scale.
The boundary conditions on the scale equation are not as
straightforward as those on the Reynolds stress equations,
since the scale need not go to zero at the free boundary of a
region of turbulence. In fact, the eddies extending the
farthest are expected to be the largest eddies present in the
region. At a free boundary, we therefore set A equal to
twice the bound established by the previous approach to A .
The edge boundary condition appears simpler if one chooses
to use the dissipation equation, since e = bq /A clearly
approaches zero. However, this leaves A free to approach any
value from zero to infinity as q -»• 0 , and no independent
information is gained.
Since the production term is small for a momentum!ess
wake,22 this provides a flow for estimating s, and s, to
lie between 0.3 and 1. To reduce the uncertainty in these
coefficients, we have rather arbitrarily set s^ = s^ = Sj/2 .
15
-------
W. S. LEWELLEN
The last coefficient, s^ , may now be determined by computer
fit with the free jet result to be ~ - .35 . This is in the
middle of the range of values used by Rodi^ who found it
necessary to vary his analogous coefficient from 1.2 to 2
(corresponding to varying s^ in our case from -1 to 0.6).
With these choices for the coefficients, our proposed
scale equation for no body forces becomes
DA ~ oc A i . n e. v «j_no 9 A 9A
— = 0.35 — u u g— + 0.6 — A + 0.3 -r— qA 3—-
Ut t- 1 J °X . ,Z OX, OX
. 0,375 /M\2 (
Q
As noted earlier, we cannot assign as high a confidence level
to this equation as to our modeled Reynolds stress equations.
In fact, it appears that unless the turbulence is far out of
equilibrium, the approach of simply limiting the A to be
equal to the lower of the two bounds previously described is
adequate for a simple model.
3.4 Low Reynolds Number Dependence (a)
The coefficient a in (2.24) is only important at low
Reynolds numbers, i.e., when bqA/aV <_ 0(1). The best measured
low Reynolds turbulent flow is the transition region in a
boundary layer just outside the so-called laminar sublayer. In
this region, (2.1) reduces to
2
constant = u, (3.13)
*
Equation (3.13) and the four nonzero Reynolds stress component
equations from (2.24) may be solved in this region and compared
to Cole's empirical law of the wall.3^ This is done in Figure
3.1 for three values of a for the previously chosen values
of a, b, and v . The best choice appears to be a = 3 .
3.5 Additional Coefficients Required to Compute Temperature
Fluctuations (A, s, 85)
The coefficient A in (2.25) can be determined by again
considering the constant flux region of a high Reynolds number
boundary layer with no body forces, i.e., the limit of small
temperature fluctuations. The normal component of (2.25) then
reduces to
-iCl A_
= 0 (3.14)
16
-------
USE OF INVARIANT MODELING
20
16
12
Coles' Law of the Wall
34
10
yu*/i/
100
Fig. 3.1 Influence of a on model predictions of law of the
wall
while the shear stress equation from (3.1) for the same
conditions reduces to
9U1 a n
-U3U3 3x7 - A U1U3 = °
(3.15)
Equations (3.14) and (3.15) may be used to form the ratio of
eddy diffusivity for momentum to that for heat. This is some-
times called the turbulent Prandtl number
U1U3
30
Pr =
9U
= A
(3.16)
Based on the measurements of Businger et al.^2 in the neutral
atmospheric surface layer, we chose a value of 0.75 for this
coefficient.
The coefficient of the temperature dissipation term s
can be evaluated by comparing the frequency spectrum of the
temperature fluctuations with the spectrum of the turbulent
kinetic energy. In the inertial subrange, it is experimentally
observed that the three-dimensional energy spectrum in scalar
wave-number space is given by'
17
-------
W. S. LEWELLEN
E(k) = ae2/3k 5/3 (3.17)
with a a constant.
The temperature variance spectrum is similarly given by
EQ(k) = 6Ne"1/3k"5/3 (3.18)
with (3 a constant and N the dissipation of temperature
variance. By integrating (3.17) to form q2/2 and (3.18) to
form 1^/2 , it is possible to show that the coefficient s
in (2.26) should be equated to a/6 . With the values of
a = 1.5 and B = 0.5 given by Tennekes and Lumley7 this
would yield s = 3.0 . But some values of the ratio are
quoted as low as 0.6.35 we have chosen a value of 1.8.-™
Mellor1^ chooses a value of 15/8 while Wyngaard et al.37'3
chose a value of 2.8.
An estimate for the coefficient of the stratification
term in the scale equation can be obtained from the stable,
constant flux layer, for Richardson number equal to its critic-
al value « 0.21. Assuming A and q constant, and picking
the value of A that leads to Ric = 0.21,36 we find a coeff-
icient of 0.8 on the buoyancy term. Thus the form of the
scale equation used here is
3U
DA . ,, A i . . , vA 9gA 9A
— = 0.35 —T u. u. -r-— + 0.6 —r- + 0.3 V1— -5—
Dt q2 i j 8x. X2 9Xi 3Xi
0.375/9qA\2 0.8A
/q
|
4. Model Verification
In viewing the comparisons between model predictions and
experiments in this section, the reader should remember our
goal of making reasonable predictions for a wide class of
flows rather than very accurately predicting any one flow.
Flows for which significant discrepancies occur between model
predictions and observations for some of the variables will be
noted. If one is interested in a relatively narrow class of
flows, these discrepancies could be reduced by changes in the
model coefficients. A better general model would require more
sophisticated modeling of the individual terms, as discussed
in Section 2.
18
-------
USE OF INVARIANT MODELING
4.1 Axisymmetric Free Jet
Comparisons between model predictions and the experimental
data of Wygnanski and Fiedler-*" for the self-similar region of
an axisymmetric free let are given in Figure 4.1. The computa-
tions have been made »30 by starting with some arbitrary
velocity and turbulence distributions and letting the jet
develop until self-similar distributions are obtained. The
model predictions have been made using both of the approaches
to the scale determination discussed in Sections 2.6 and 3.3.
The scale variations for the two approaches are compared with
the measured longitudinal integral scale in Figure 4.1(c). The
variation predicted by (3.12) is closer to the observations as
expected, but the improvement, if any, in the predicted flow
variables is not significant. The fact that the predicted
energy is low while the predicted shear stress is high is the
Wygnanski 8 Fiedler
39
Fig. 4.1 Similarity profiles for axisymmetric free jet with
constant A (—) and dynamic A ( )>
(a) mean velocity and shear stress
(b) normal Reynolds stress components
19
-------
W. S. LEWELLEN
1.2
1.0 -
Ar=2.4r.
o Wygnonski a Fiedler
0.2 r*
Dynamic Eq.
39
.4 .8
1.6 2.0 2.4
Fig. 4.1(c) scale variation
type of discrepancy one might expect to overcome by incorpora-
ting additional terms to the model of pressure correlations
discussed in Section 2.3.
4.2 Free Shear Layer
Results for a two-dimensional, self-similar shear layer
are compared with observations^0>***• in Figure 4.2. The agree-
ment with the mean flow profile is good, but there exists
1.2
1.0
A.R.A.P. prediction
o A Wygnanski 8 Fiedler 40
v a Childs 4I
-2.4 -2 -1.6 -1.2 -.8
z-z
50
Z25-Z75
Fig. 4.2 Similarity profiles for free shear layer
20
-------
USE OF INVARIANT MODELING
considerable scatter in the data for the shear stress distribu
tion. The components of the normal stress predicted by the
model were compared with Wygnanski and Fiedler's data by
Donaldson. Because of some uncertainty surrounding the data,
these comparisons are not shown here.
4.3 Two-Dimensional Wake
/ o
Predictions are compared with Townsend's self-similar
wake measurements in Figure 4.3. Lumley and Khajeh-Nouri's^-'
model predictions are also included for comparison. Their
• A.R.A.P. prediction
-U/U
Lumley 8 Khajeh-Nouri
19
}Townsend data
o uwj
42
1.6
2.0
2.4
Z/Z
50
Fig. 4.3
Similarity profiles for two-dimensional wake;
(a) mean velocity and shear stress
model, which contains many more terms and coefficients, makes
a marginally better prediction for the normal components of the
Reynolds stress for this flow. The lateral and transverse
velocity fluctuations are equal in the present model predic-
tions while the observations are significantly different.,.
Again, there is some uncertainty in the data since Thomas
reports measurements of the longitudinal rms turbulence inten-
sity which are about 33% higher than Townsend's reported
values.
4.4 Axisymmetric Wake
The capability of the model to predict the development of
a wake was tested by matching the velocity and Reynolds stress
distributions as observed by Chevray^ at one axial station and
then comparing the predicted and observed development. ^ jhe
decay oj_the maximum velocity defect wj) , the maximum shear
stress uwmax , and the maximum axial velocity fluctuation
21
-------
W. S. LEWELLEN
A.R.A.R prediction
Lumley 81 Khojeh-Nouri
o uu i
A vv j Townsend data
a w"w
• 19
2.4
Fig. 4.3(b) normal Reynolds stress components
wwma are given in Figure 4.4. The largest discrepancy is in
the decay of the axial velocity fluctuations. The model pre-
dicts an increase of 35% over the initial value at z/D = 3
which the observations do not show.
The wake spread as measured by the radius for which the
defect in the mean velocity is equal to half of its maximum
value is given in Figure 4.5. The agreement is good. Although
Chevray fits a 1/3-power law growth rate to his data eighteen
diameters behind the body,.the model indicates that v 100
diameters is required for the wake to reach self similarity.
The self-similar profiles obtained by the model are compared
in Figures 4.6(a) and (b).
Comparisons between predictions and the momentumless wake
measurements of Naudascher^ have also been made^ with
results similar to the comparisons shown here with Chevray's
measurements. In this case, no true self similarity exists.
4.5 Flat Plate Boundary Layer
Predicted velocity distributions in the neighborhood of
the wall were compared with Coles' law of the wall^ in Figure
3.1 in order to choose (a). Comparison with Klebanoff's^°
velocity and Reynolds stress measurements are given in Figure
4.7 for a Reynolds number of 5 x 10^. The agreement with U
22
-------
10
10"
10"
USE OF INVARIANT MODELING
Chevroy data
44
10
SO 100
K/D
Fig. 4.4 Downstream decay of axisymmetric wake with constant
A (—) and dynamic A (—)
23
-------
W. S. LEWELLEN
5.0r
x/0
Fig. 4.5 Wake radius growth for conditions of Figure 4.4
|/2
Fig. 4.6 Similarity profiles for conditions of Figure 4.4;
(a) velocity defect and shear stress
-------
USE OF INVARIANT MODELING
I* -6
E .4 -
.5
1.0 1.5 2.0 2.5
r/r..,
Fig. 4.6(b) normal Reynolds stress components
I.Or
U/U,
Klebonoff dota
Fig. 4.7 Similarity profiles for flat plate boundary layer
(a) mean velocity
-------
W. S. LEWELLEN
A.R.A.P. prediction
Klebanoff data46
Fig. 4.7(b) Reynolds stress components
2U
CD
Coles correlation
34
Fig. 4.8 Surface shear stress on a flat plate
26
-------
USE OF INVARIANT MODELING
and uw is good but the sharp peaks observed in uu" and vv
very near the wall are not predicted. There also appears to be
a little too much diffusion in the outer region.
A plot of surface shear stress versus Reynolds number
(Rex = Ux/v) is given in Figure 4.8 along with the laminar,
Blasius value, and Coles-^ compilation for fully turbulent flow
For this comparison, the calculation is started with the lami-
nar velocity profile at a relatively small value of Rex . All
turbulent correlations are initially zero except for a small
spot of turbulent energy. The model does a creditable job at
predicting transition.
4.6 Flow Over an Abrupt Change in Surface Roughness
A calculation was made in simulation of the atmospheric
surface data of Bradley^' who recorded velocities and surface
shear stress at a little-used airfield in Australia. Transi-
tions are shown in Figure 4.9 for rough-to-smooth and smooth-
to- rough, with the roughness measured in terms of the aerodyn-
amic roughness ZQ . The calculations^' begin with the
velocity and turbulence components in equilibrium for one value
of ZQ . The surface boundary condition is changed to that
appropriate for the other value of ZQ at x = 0 . When both
runs are allowed to continue far downstream, the turbulence
comes into equilibrium with the new ZQ value. Figure 4.9
also includes the predictions of Rao et al.^^ who used the
second-order closure model developed in Reference 38.
4.7 Temperature Fluctuations in the Plane Turbulent Wake
Measurements behind a slightly heated cylinder have been
made by Freymuth and Uberoi and LaRue and Libby. Figure
4.10 shows a comparison between the measured distributions far
downstream of the cylinder and the self-similar profiles
predicted by the model. The normalizing length Z5Q is fixed
for the mean velocity profile given in Figure 4.3. The agree-
ment is quite good, particularly in the concurrence of the
amplitude and position of the maximum temperature fluctuation.
4.8 Stability Influence in the Atmospheric Surface Layer
The atmospheric surface layer extending a few tens of
meters above the earth's surface is the most extensively
studied example of a turbulent shear layer incorporating the
influence of stratification on the dynamics of turbulence.
Following Monin and Obukhov,^ considerable success has been
achieved in experimentally determining similarity functions
which describe the dependence of the mean turbulence character-
istics on height, shear stress, and heat flux. In this layer,
the shear stress and heat flux may be considered constant.
27
-------
W. S. LEWELLEN
SMOOTH-TO-ROUGH TRANSITION
z'0 = .002cm
iOr z0 =.25cm
T/T-
Bradley's data
47
0 2 4 6 8 10 12 14 16
Fetch, m
ROUGH-TO-SMOOTH TRANSITION
z'0 = .25cm
o Bradley's data
Fig. 4.9 Step change in surface roughness. A.R.A.P. predic-
tion (—); prediction by Rao et al. ( )
l.6r
1.2
1.0
o A LaRue-Libbey
•A.R.A.P. prediction
.8 1.2 1.6 2.0 2.4
Fig. 4.10 Temperature similarity profiles in a two-dimensional
wake
28
-------
USE OF INVARIANT MODELING
From these turbulent transport constants, a characteristic
length may be found
OuT
L = -
kgw6
This is the Monin-Obukhov length. At altitudes below which any
crossflows appear due to the coriolis force caused by the
earth's rotation and above which the direct viscous contribu-
tions to momentum transport are important, z/L is the only
variable determining the normalized turbulence quantities for
a steady homogeneous layer. Equations (2.24)-(2.26) have been
integrated for this case by Lewellen and Teske.36 Results for
the normalized wind and temperature gradients are compared with
atmospheric data in Figure 4.11.
In these computations, the scale was set equal to 0.6z
with an upper bound of 0.20L in stable flow. This was chosen
to yield the observed critical Richardson number of 0.21. That
is, no matter how stable the flow becomes in terms of large z/L,
Ri never exceeds 0.21. This bound was also used in determining
the coefficient of the buoyant term in the scale equation in
Section 3.3.
-2 -I
0
Z/L
0
z/L
Fig. 4.11 Normalized atmospheric surface layer gradients as a
function of stability. Data from Businger et al.
The vertical velocity and temperature fluctuations are
compared with the data of Wyngaard et al. in Figures 4.12 and
4.13. The agreement for the vertical velocity is very good but
that for the temperature fluctuations leaves a bit to be
desired. As pointed out by Wyngaard et al.^2 there is
29
-------
W. S. LEWELLEN
Fig. 4.12 Vertical velocity fluctuation in the atmospheric
surface layer. Data from Wyngaard et al.
ffl2) u*
-2.0 -1.5
Fig. 4.13 Temperature fluctuation in the atmospheric surface
layer. Data from Wyngaard et al.^
— 1/2
considerable uncertainty in (6^) /w6 at z/L = 0 because
both variables go to zero.
Mellor-'-" made a somewhat similar calculation of these
surface layer similarity functions. The major difference is
that Mellor eliminated the diffusion terms so that the differ-
ential equations reduce to an algebraic set. In this case the
length scale may be normalized out of the problem, eliminating
any need for, or any possibility of, incorporating an influ-
ence of stratification on the scale. His model coefficients
30
-------
USE OF INVARIANT MODELING
must then be chosen so that they will match the critical
Richardson number.
4.9 Shear Layer Entrainment in a Stratified Fluid
s ^
Kato and PhillipsJJ measured turbulent entrainment in an
annular tank of density-stratified water with a shear stress
applied at the surface by a rectangular mesh screen. The rate
of spreading of the turbulent fluid into the linear density
field was measured visually by injected dye. Their experimen-
tal results of entrainment rate versus flow Richardson number
are given in Figure 4.14. The two model prediction curves-^
in the figure represent calculations using the two different
approaches to the scale discussed in Section 2.6. The dynamic
scale rounds the apparently straight-line prediction of the
gross scale model. Neither model predicts quite as steep a
decrease of entrainment with Richardson number as the data
indicate.
10"
10
° Data of Kato & Phillips
--- Dynamic A equation
10
50 100
500
Fig. 4.14 Shear layer entrainment into a stratified fluid
4.10 Free Convection
Willis and Deardorff have measured the velocity and
temperature fluctuations in an unstable layer bounded above by
a stable density gradient and below by a surface with a posi-
tive heat flux. Their experiment called for establishing an
initial stable temperature gradient between two surfaces and
then applying a heat flux to the region through the lower sur-
face. With mean velocities restricted to a minimum, free
convection occurs in a mixed layer above the lower surface.
The thickness of this mixed layer increases with time, but the
31
-------
W. S. LEWELLEN
fluctuating velocity distributions exhibit self-similarity when
normalized by the characteristic velocity
* =
w:
where w6g is the surface heat flux and
the mixed layer.
is the depth of
Figure 4.15 shows our model predictions for the normalized
vertical velocity fluctuations and the experimental observa-
tions. For the model predictions, we assumed no mean velocity
to simulate the laboratory experiment and a constant surface
heat flux. The agreement between predictions and observations
is heartening, particularly since no model constant adjustments
were involved.
.75
.5
.25
.2 .4
WW/W* 2
Fig. 4.15
Similarity profile of vertical velocity fluctuation
in an unstable mixed layer. Data from Willis and
Deardorff5^
The horizontal velocity fluctuations are given in Figure
4.16. Here the comparison with experiment is not as close
since the observations show a peak near the lower surface that
is not evidenced in the model results. The distributions of
temperature fluctuations are shown in Figure 4.17. The agree-
ment between model and experiment is very good except at the
top of the mixed layer where the observations show a much
stronger local maximum than is predicted. This may be due to
the fact that inertial oscillations exist in this region.
Overall, the comparison between model predictions and experi-
mental observations is quite favorable.
32
-------
USE OF INVARIANT MODELING
.75
Z/Zi
.5
.2 .4
LUJ/W*2
Fig. 4.16 Horizontal velocity fluctuations for the conditions
of Figure 4.15
.75
Z/Z
10
15 20
Fig. 4.17 Temperature fluctuations for the conditions of
Figure 4.15
4.11 Planetary Boundary Layer for Neutral Steady State
Height variation of the mean velocities and Reynolds
stress components is given in Figure 4.18 as predicted by our
model for the atmospheric Ekman layer. The distributions are
compared with Deardorff 's^-> results of ensemble averages of
integrations of the unsteady equations of motion for all the
turbulent fluctuations above a prescribed grid size. The mean
33
-------
W. S. LEWELLEN
2400
U.V,10*Q CMPS]
Fig. 4.18 Steady neutral atmospheric boundary layer profiles.
A.R.A.P. prediction (—); prediction from Deardorff"
velocity distributions are in good agreement. The value of the
surface shear stress (uA/Ug = .036) is within the scatter of
field data for the same value of Rossby number. At the surface
q is 14% lower than Deardorff's result for the same value of
u*. A large part of this discrepancy occurs in the lateral
velocity fluctuations. This tends to reinforce the need for
some refinement of our pressure correlation modeling, as pre-
viously discussed. Near the top of the boundary layer, it
appears that the accuracy of Deardorff's results is affected by
his choice of a low value for his fixed boundary layer height.
5. Local Equilibrium Approximations
If the second-order correlations are assumed to be in
local equilibrium so that there is no time evolution or spatial
diffusion of the correlations, (2.24)-(2.26) may be reduced to
-------
USE OF INVARIANT MODELING
0 = - u
0
0
30
- - U1UJ 3x.
- 2"u~e 9G
2UJ6 9x.
3U
11 n i
u .U - 1
j 3x..
2bsq92
A
s/
0
The laminar terms have also been neglected since low Reynolds
number is incompatible with equilibrium turbulence. Equations
(5.1)-(5.3) form a closed set of algebraic relationships
between the second-order correlations and the gradients of the
mean velocity and temperature. This is what Donaldson** calls
the super equilibrium approximation.
The relationships between the second-order correlations
and the mean flow gradients determined by (5.1)-(5.3) form a
first-order closure or K theory for turbulence. This will be
a valid approximation, provided 1) any changes in the mean flow
are very slow compared with the characteristic time of the
turbulence A/q and 2) spatial variations in the turbulence
are small over the scale length A . Note that only rarely are
both conditions satisfied because A is usually related to
spatial variations in the flow. One particular region where
both conditions are satisfied is in the constant flux region of
the boundary layer when A/q is approaching zero.
The functional dependence of the correlations on the mean
flow gradients obtained from (5.1)-(5.3) for the case where
Ui = (U(z),0,0); g± = (0,0,-g); and 0 = 0(z) is shown in
Figure 5.1. The only independent variable is the Richardson
number of the mean flow
R, - s. li /(Jin2
Rl ~ 0 9z /Uzj
A critical Ri is reached, above which no turbulence can
exist. This critical value of 0.56 given in Figure 5.1 is
higher than the value 0.21 observed in the atmospheric surface
layer by Businger et al.32 as reported in Section 4. However,
35
-------
W. S. LEWELLEN
.4 .6
Fig.5.1(a)Superequilibrium Reynolds stress profiles normalized
by A2(9u/8z)2
in this surface layer, the turbulent correlations vary with
height as stability varies with height, so one should expect
the diffusion terms to have some influence.
Superequilibrium relationships have also been computed for
constant density swirling flows."'",58 jn this case, there
is both a lower bound and an upper bound on the range of values
the swirl parameter may have for turbulence to exist.
Other approximations to (2.24)-(2.26) may be tried short
of reducing them to (5.1)-(5.3). Mellor and Yamada26 have
experimented with four levels of approximation. We have had
some success with a type of energy equation approach we call
quasi-equilibrium.2^»30 This consists of carrying the energy
equation formed by the contraction of (2.24) in full but using
(5.1)-(5.3) to relate the individual turbulent correlations to
q2 . This should be considerably more accurate than super-
36
-------
USE OF INVARIANT MODELING
equilibrium because overall levels of time evolution and spa-
tial diffusion are included through the turbulent energy
equation. It cannot accurately predict flows when the trans-
port of a quantity is counter to its gradient.
Fig. 5.1(b) Superequilibrium temperature correlation profiles
normalized by A2(90/9z)(9U/8z), etc.
6. Applications
We believe the verifications of the model predictions made
in Section 4 sufficiently accurate over a wide variety of
conditions to justify detailed calculations of flow situations
for which no detailed measurements are available. A.R.A.P. is
currently applying the model to a number of such areas. A few
interesting results of these applications are presented here.
Progress is also being made on applications to other areas
including: compressible boundary layers,•>* compressible shear
layers,60,61 three-dimensional isolated vortex,°2 chemically
reacting wakes" and jets,°^ swirling flow in an annulus.
37
-------
W. S. LEWELLEN
and moisture in the atmospheric boundary layer.
6.1 Diurnal Variations in the Planetary Boundary Layer
Over homogeneous terrain, the atmospheric boundary layer
may be considered a function of time and altitude only. The
results of calculations for a fixed geostrophic wind and upper
level temperature lapse rate with a cyclic surface heat flux
approximating conditions over the Midwestern plains during
summer^ are presented in Reference 24. These results were
obtained using the quasi-equilibrium approach outlined in
Section 5.2 and the gross feature approach to the scale. The
results presented here were obtained using the full equations.
There is relatively little difference in the numerical results.
The predicted contours of constant turbulent fluctuations
are presented in Figure 6.1 as a function of time and altitude.
3500 T
3000..
2500..
5
N
2000. .
1500 .
1000 ..
500.
1i SB 5 24
TIME CHRSJ
Fig. 6.1 Isopleths of turbulent energy in diurnal planetary
boundary layer
The boundary layer thickness grows during the day and shrinks
during the evening and morning hours. The turbulent kinetic
energy reaches a maximum of 4.8% of the geostrophic mean kine-
tic energy in the midafternoon at an altitude of approximately
500 meters. As the sun sets, the turbulence begins to decay
38
-------
USE OF INVARIANT MODELING
until in the early morning hours the maximum kinetic energy is
* 0.25% of the mean kinetic energy. The biggest difference
between the results presented here and those in Reference 24
occurs during the early morning hours. The full equations with
a dynamic scale predict a slower decay of q^ and consequently
considerably larger q^ in the altitude range from 100 m to
1 km in the morning hours. But even this larger value is still
quite small compared with afternoon values. Both model repre-
sentations predict such features as the temperature inversion
and local wind jet observed to occur nocturnally at low levels.
Surface shear stress is plotted as a function of the
stability variable Ku*/fL in Figure 6.2. Data points for
Ro ~ 10 , as taken from Tennekes' summary are included on the
plot. The model predictions show a hysteresis loop with the
surface shear stress significantly larger when the surface heat
flux is decreasing rather than increasing. The data show
considerable scatter but tend to be biased towards the upper
bound. The factor of two difference in u* at neutral condi-
tions demonstrates the influence of unsteady dynamics on the
atmospheric boundary layer.
•5 -r
-30
Fig. 6.2 Surface shear velocity vs. stability for diurnal
planetary boundary layers. Data from Tennekes
6.2 Stratified Wake
The passage of a body through a stratified fluid generates
a turbulent wake containing kinetic and potential energy. The
buildup of potential energy is caused by the co-mixing of
39
-------
W. S. LEWELLEN
heavier density fluid above the lighter density fluid. In the
initial stages of wake development in a weakly stratified med-
ium, the turbulent kinetic energy dominates the potential
energy, and the wake spreads. This spreading in turn increases
the potential energy until, at a time comparable to the natural
oscillating period of the fluid, inertial waves are generated.
Computations for this flow using the quasi-equilibrium version
of the present model have been reported in References 30 and
68.
Figure 6.3 shows typical contours of the turbulent kinetic
energy and secondary flow stream function at two instances of
time following wake initialization. Time is normalized by the
characteristic Brunt-Vaisala period of the fluid
= 2TT[(g/0)30/9z]
-1/2
•4,
t/t,=0.5
t/tc=I.O
q2mox =0.00001471^
t/tc=0.5
t/tc=I.O
^mo,=0.000976riUa)
Fig. 6.3 Contours of turbulent energy and secondary flow
streamlines for two instances of time after wake
initialization for RiQ = 0.00925.
1 = +10% of maximum value for that quadrant;
2 = -10%; 3 = +30%; 4 = -30%, etc.
The flow is symmetric about both axes so only one quadrant is
shown for each variable at each time. At t/tc = 0.5, the
secondary flow exhibits a simple collapsing motion, while at
t/tc = 1, a second wave mode has been added. At both times the
secondary flow extends well outside the region of turbulence.
Figure 6.4 is a summary plot showing the sensitivity of
the wake shape to initial Froude number, F = Utc/D > or
initial Richardson number of the turbulence, Rig =
-------
USE OF INVARIANT MODELING
Fig. 6.4 Vertical spread of stratified wakes,
Lin and Pao69
Data from
along with a qualitative comparison with laboratory flow visu-
alization results.69 There is good qualitative agreement
between the observations and predictions.
6.3 Pollutant Dispersal
By adding equations for species continuity C and the
correlations of species fluctuations u^c , cc , and c6 ,
(2.1)-(2.3) and (2.24)-(2.26) may be used to estimate pollutant
dispersal in the atmospheric boundary layer. As long as the
pollutant is assumed neutrally buoyant and nonreacting, the
equations for C , ujc" , 'cc' , and cF should be identical to
(2.3), (2.25), and (2.26), respectively.
Results of computations of pollutant dispersal in atmos-
pheric turbulence as represented by our model predictions have
been presented in References 6, 29, and 70. Figure 6.5 shows
the influence of atmospheric stability on the vertical spread
az of a plume released at ground level
r 2 If
(Cz^) dz / I
4) / •'O
C dz
The Richardson number Ri is based on the difference between
the values of temperature and velocity at 10 m altitude and at
the surface, specifically,
Ri =
A010m
0
u
where u = 0.1U at 10 meters. The spread rate for neutral
-------
W. S. LEWELLEN
.01
.001
10
xf/u
Fig. 6.5 Vertical plume spread from ground release into
planetary boundary layers of various stabilities
conditions is very close to that predicted by Pasquill.?! But
the present model predicts a stronger effect of stability under
unstable conditions and a weaker effect on the stable side.
Deardorff and Willis^^ have published results of particle
releases into an unstable mixed layer, the same experimental
flow used for comparison in Figures 4.15, 4.16, and 4.17.
Reference 70 shows comparable model predictions for a surface
pollutant released into the calculated, unstable-mixed layer
field of turbulence shown in those figures. The model predicts
that the maximum concentration will rise above the surface. As
pointed out in Reference 72, this feature could not be predic-
ted by any eddy viscosity model or a Gaussian plume model.
The model predicts several non-Gaussian features for
pollutant plumes caused by such mechanisms as wind shear in
magnitude and direction with altitude, interaction with the
inversion layer capping the boundary layer, and regions where
turbulent scales are much larger than plume scales.
7. Concluding Remarks
Second-order closure has been established as a viable
approach to practical turbulent flow computations. There are
still about as many models as there are modelers — each model
differing from the others in some detail. But this is a
-------
USE OF INVARIANT MODELING
healthy situation, since there is still much to be learned.
This is particularly true of the scale equation since in any
of the models, every term is this equation must be modeled.
The relatively simple model presented here does a commend-
able job over a wide variety of flows. Thus, practical calcu-
lations can proceed concurrently with attempts to improve the
model.
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o /
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-------
USE OF INVARIANT MODELING
O £
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-------
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W. S. LEWELLEN
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48
-------
NOMENCLATURE
A, a, b model constants
C mean species concentration
c species concentration fluctuations about the
mean
D laminar diffusion coefficient
d model constant relating plume spread to plume
turbulent scale, see Eq. (15)
f Coriolis parameter = 2ft sin <(>
g gravitational acceleration
k von Karman's constant = -u*/(z 8u/3z) in
the neutral surface layer
L Monin-Obukhov length = -0ou*3/kg(w"e")o
n decay rate for grid turbulence
P mean pressure (also Pasquill's stability
parameter)
p pressure fluctuation
q square root of twice the turbulent kinetic
energy
Q source strength
Ri Richardson number
Ri characteristic Richardson number defined in
Eq. (12)
Ro Rossby number based on geostrophic conditions
Ro characteristic Rossby number = u/fz
s model constant
s0 , S,, s2 constants in scale equation
t . time
t Brunt-Vaisala period
O
U. mean fluid velocity in the ith direction
U geostrophic wind velocity
u. fluctuating fluid velocity in the ith
direction
u* surface shear stress velocity
-------
u
vc
W
w
W*
z
z
o
550
k
characteristic surface velocity equal to 0.1
times the wind velocity at 10 meters
model turbulent diffusion coefficient
mean vertical velocity
fluctuating vertical velocity
free convective velocity scale =
[gz1(w9")o/0o]1/3
Cartesian coordinates
vertical coordinate
temperature inversion height above an unstable
layer
surface roughness height
distance to 1/2 the centerline value
plume release height
e
0
e
A
X
v
P
*
Superscript
Subscript:
o
g
P
t
,1/2
dissipation function
mean potential temperature
fluctuating potential temperature
model macroscale length
model microscale length = A/[a + bqA/v]'
kinematic viscosity
density
latitude
angular velocity of the coordinate system
plume spread
denotes ensemble average
denotes surface value
denotes geostrophic condition
denotes plume value
denotes ambient turbulence value
50
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