EPA-R4-73 016b
March 1973 Environmental Monitoring Series
Derivation of a Non-Boussinesq Set
of Equations for an
Atmospheric Shear Layer
,^°ST^
Office of Research and Monitoring
U.S. Environmental Protection Agency
Washington, D.C. 20460
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EPA-R4-73-016b
Derivation of a Non-Boussinesq
Set of Equations
for an
Atmospheric Shear Layer
by
Coleman duP. Donaldson
Aeronautical Research Associates of Princeton, Inc.
50 Washington Road
Princeton, New Jersey 08540
Contract No. 68-02-0014
Program Element No. A-11009
EPA Project Officer: Kenneth L. Calder
Meteorology Laboratory
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Prepared for
OFFICE OF RESEARCH AND MDNITORING
U. S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460
March 1973
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This report has been reviewed by the Environmental Protection Agency
and approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of the Agency,
nor does mention of trade names or commercial products constitute
endorsement or recommendation for use.
11
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TABLE OF CONTENTS
1. Introduction
2. Derivation of Basic Equations
3. Discussion, of the Steady Atmosphere
4. Derivation of the Equations for a Turbulent
Shear Layer
5. Modeling of Terms in Basic Equations to Achieve
Closure
6. An Invariant Model of the Atmosphere
7. Equations Used in A.R.A.P. Report No. 169
8. Discussion
9. References
111
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1-1
1. INTRODUCTION
Volume I of this report (prepared for the Environmental
Protection Agency in connection with EPA Contract No. 68-02-
0014) presented a detailed discussion of the method of invariant
modeling as it is applied to the problem of the computation of
turbulence in an atmospheric shear layer and to the dispersal
of pollutants in such a layer. In Volume I, the basic equations
used are the familiar Boussinesq approximations. These
equations are derived by considering the atmospheric motion to
be a small departure from an adiabatic atmosphere at rest and
further assuming that the extent of the shear layer being
investigated is small compared to the scale of the atmosphere -
this latter scale being of order p /(9p /8z) where p is the
density of the undisturbed adiabatic atmosphere and z is the
altitude. Within the Boussinesq approximation, the divergence
of the- velocity field of the motion under study may be neglected.
Since this divergence may not be considered zero for motions
whose scale is of the order of the atmosphere, i.e., 7000 meters,
we at A.R.A.P. have used equations for much of our work that are
approximate equations, akin to the Boussinesq approximation, but
which do not consider that the divergence of the velocity field
is zero. This set of equations was used in Reference 1 and was
also used in A.R.A.P. Report No. 169 [Ref. 2], A.R.A.P.'s first
report to EPA on the application of invariant modeling to the
dispersal of pollutants in the atmosphere.
In this report (Volume II of A.R.A.P. Report No. 186),
the equations which represent a non-Boussinesq set of approxi-
mate equations for the motion of the atmosphere and which are
used in References 1 and. 2 are derived.
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2-1
2. DERIVATION OF BASIC EQUATIONS
The equations which we shall take to govern the motion
of a compressible, nonchemically-reacting perfect gas are
given below. They are:
. the perfect gas law
p = pRT (2.1)
the continuity equation
P_i_ / **""" ^ _. r\ ^oo^
the momentum equation
3t
J -L J
where the stress due to molecular diffusion is given by
3u. 3u,\ 3u_
the energy equation
+
+ a 1L.-+
+ u +
.
3t 3x j 9t j 3x. 3x 3x
J J J \ J
the species conservation equation
("f\ p \ ± ^ ^ n vi r* ^ I x pf r* 1 (o f\ \
^ t \ PW / * ^v ^ ^ * ' "" Ci v I Mo^ ^ L/ I . \ £ U /
o T/ Ot oX, ,) Ot o A , \ dX. Ot /
J j \ j /
In what follows, we will be interested in the development
of turbulence and the transport of matter in the atmosphere.
-------
2-2
We will assume that the matter to be transported in the. atmos-
phere is not too highly concentrated, so that it makes no
first-order effect upon the heat capacity or gas constant of
the air in which it is carried. In this case, if we designate
the heat capacity as Cp and the gas constant as R , we may
write (2.1) as ' ".
P = PRQT (2.7)
and (2.5) as
Ci+- ? Ci v Ci v Civ
OO '.1 O A , O A. . u A, *
J 0 J
+ ?i0- 1^7 (2'8)
j
Since the Mach number of the flow in which we are interested is
small, we may neglect the last term on the right-hand side of
(2.8) since this term represents the heat generated by the
dissipation of the motion and is of order M2 compared with
the other terms in the equation. The final form of the energy
equation is, then,
J J
Following the usual practice for obtaining the equations
for the motion of an atmospheric shear layer [Refs. 3 and 4],
we will consider, the atmosphere to be in a state slightly
removed from an adiabatic atmosphere at rest. We consider then
an expansion of the equations presented above according to the
following scheme:
= Ca0 + Ca
p = PO + P
T = TQ + T u* = \i* + y* (2.10)
u . = 0 + u. ic = k+k
-------
2-3
If we expand the gas law (2.7) according to this scheme, we have
= poRoTo
(2.11)
and
p = R0(PQT + pTQ + pT)
(2.12)
If we neglect the second-order term on the right-hand side of
(2.12), we have
p = RQ(poT + PTo)
The continuity equation (2.2) yields
(2.13)
3t
= 0
(2.14)
which agrees with our assumption of a steady state, and
If + k7(l>ouj + 'V
J
which, to first-order, can be written
If * Ir- <»oV
J
The momentum equation (2.3) yields
(2.15)
(2.16)
o&i
(2.17)
and
[(
3u. 3u
9u
(2.18)
-------
To first order, this may be written
2-4
ft "o
ui> + ITT
J
. 8
3XJ
3u
3u
m
3x
m
(2.19)
Expansion of the energy equation (2.9) yields
3T
_
3t 3x
3p T 3p
o o
3t
3x
(2.20)
J \ " "M
which, according to .our assumption of steady base flow, requires
3T
k
'o
3x.
= 0
and
cP0t
+ kr [CPO + P)UJTO+ (po + P)UJT]
J
- u
u
3t j
3x
+ k)
3T
3x
k
To first order, this may be written
CP0t (P0T)
(2.21)
(2.22)
= u
3t
3x
23L. t !M
3x
3x_. I
(2.23)
-------
2-5
In view of the continuity equation (2.16), we may neglect the
second term on the left-hand side of (2.23). Also, because of
our assumption that the undisturbed atmosphere is adiabatic, i.e.,
9p 9T
2. = n r 2. (?
ax. PO°PO 9x u
J J
we may neglect the first term on the right-hand side of (2.23).
Finally, the terms 9p/9t and u. 9p/9x. taken together repre>
J J
sent the compressional heating due to the disturbed motion
2
considered apart from gravitational effects, are of order M ,
and may be neglected. Therefore, (2.23) may be written
/!_ T) + 9
\9t Po + 9Xj
Cn ^r (P.T)' + ^r- (P u.T) = f k fi- + k -,-£ (2.25)
o J J 9x^o 9x. 9x.
Substitution of the expansion (2.10) in the species conservation
equation (2.6) yields
9p
->(i\
.26)
3t
J \ J
which is satisfied since we have assumed a steady base atmos-
pheric condition in which Ca is a constant, and
ft (Po +'P>V+
(2 27)
To first order, this equation becomes '
9p C a
H2-^ + Isr;
J J
We return now to the equations that result from the perfect
gas law, namely, (2.11) and (2.13). If we divide (2.11) by (2.13),
we obtain
-------
2-6
2
Now we note that the changes in p will be of order p u. , so
that
<2-3o>
We will, therefore, neglect the effects of motion-induced
pressure changes on the variation of density and temperature and
take
p T
P = - - (2.3D
o
This assumption is related to, and entirely consistent with, our
neglect of the pressure work terms in the energy equation.
We may now derive an equation for the divergence of the
velocity field. If (2.3D is substituted into the continuity
equation (2.15), the result is, after some manipulation,
(2.32)
o x OJ - o i
J J \
The energy equation (2.25) may be written
3P.T .a i a / am ' . 3?,,
iy
(2.33)
'aT ~ ~ ^(P^Un^) ' P "iw I ^ ^ ' ^ a^T
ou ox. o J ^n dxi V Oox. ox.
J -^o J \ J J
Equating (2.32) and (2.33) results in
3 , . u T_ 9To + ±_ 3 /k 3T . + k 9To
J T0 J o o j \ j
(2.3^)
or
3T
+ k °
_ _
3x, " p 3x. T T 3x. Cn p 3x. I o 3x, 3x.
J o J o o j P0po J \ j j
(2.35)
-------
2-7
Since (1/p )(3p /3'x.) is of the same order as (1/TQ) (3To/8x,),
that is, of the order of an inverse atmospheric scale L~ , we
cL
may, by virtue of the T/T in the second term on the right-
hand side of (2.35), neglect this term relative to the first.
Resort to a high Reynolds number argument [Ref. 5] shows that we
may also neglect the third term on the right-hand side of (2.35)
in comparison with the first. Thus, the equation for the
divergence of the velocity field may be written
3u u 3p
83E1 = - ^TT (2'36)
8xj po axj
This equation may now be used to replace the continuity equation
(2.16).
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3-1
3. DISCUSSION OP THE STEADY ATMOSPHERE
In the previous section, the equations that were obtained
for the basic atmosphere, which we wish to be adiabatic and in
a steady state, were:
the perfect gas law
continuity
3p
3iT = ° (3'2)
momentum
3x. = - P0g.j _ (3.3)
energy
3t 3t ,3x, \ o 3x.
or, in view of (3.1) and (3-2)
3T \
o ^T (3.5)
o 3Xj j
where v = C~ /Cv In addition we had the adiabatic condi-
'o P0 o
tion
Substitution of (3-3) into (3.6) establishes the adiabatic
lapse rate
3T g
83T = - C^ (3'7)
8XJ CP0
or, if g. = 6o-g = constant and (x,,x2,x.,) is defined as
(x,y,z)
3T 3T
o _ o .
3x 3y U
(3.8)
3T _ g .
" "
-------
3-2
Substitution of (3.8-) into (3.5) yields
9T
(3-9)
Since the thermal conductivity is a function of the temperature
T , we see from (3.9) that the basic atmosphere we have assumed
cannot be in true equilibrium. However, an order-of-magnitude
analysis shows that the time constant for the rise in To with
time is so long compared to most motions in which we are inter-
ested that for all practical purposes it is legitimate to
consider 9T /3t =0.
o
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4-1
4. DERIVATION OF THE EQUATIONS FOR A TURBULENT SHEAR LAYER
Collection of the equations for an atmospheric motion that
were derived in Section 2 results in the following set:
Continuity
9u. u. 3p
i = - -J- 2.
(4.1)
Momentum
t (pou±)
gip
Energy
3u.
3u
(4.2)
3T
r
J
Species Mass Fraction
It
where p is given by
f
J
P = -
o
(4.5)
The set of equations used in A.R.A.P. Report No. l69[Ref. 2] was
derived from this set as follows .
First, it was assumed that the molecular transport coeffi-
cients y , \i* , k j and p £r were constant.
-------
4-2
Second, it was assumed that the Prandtl and Schmidt
numbers, y Cp /k and yo/Po/.r > of tne medium were close
enough to one so that we could write
ko = yoCP0 and k = ° (i<-6)
and
p <& = y (4.7)
Equations. (4.2), (4.3), and (4.4) then simplify to
_
8t
32u.
3x . i m
J
It
It
J
We now assume the velocity, pressure, temperature, density,
and species mass fraction fields to be composed of mean and
fluctuating parts according to the following scheme:
p = p + p'
T = T + T'
P = P + Pf
C = C" + C'
a a a
-------
4-3
Substitution of this scheme into (4.1) yields
9x. 9x. pQ 9x. PQ
Averaging this equations yields
9u. u, 3p^
or, alternatively,
| (p u.) = o (4.14)
Subtracting (4.13) from (4.12) gives
9u! u! 9p^
(4.15)
or, alternatively,
IT- (P u') = 0 (4.16)
-j ° j
Substituting the scheme (4.11) in (4.5) results in
P = - 5^ T (4.17)
o
and p .
P' = - ip2 T' . (4.18)
When (4.11) is substituted in (4.8), the result is
3 / - s . 9
t (poai} + t (pouP + rrpoaiaj + posiuj + pouisj +.pouiuj)
2-
. p.s +M !
p S "
9u'
o
o o m
-------
Averaging this result yields
^ - IX
-
o o 3x±
Subtracting (4.20) from (4.19) yields an equation for the
fluctuating velocity u! , namely,
It (poui) + IxT (Po5iuj + pouiaj + pouiuj -
J
j 3x. i
J
If this equation is multiplied by u' and the resulting equation
written again with the indices i and k reversed and then the
two equations added together and averaged, the result is
It (po^P + IxT 'PoVFE'
(j
'2"i +
XT' + U
i 3x2
9XJ
u
o o x k x
o o I i 3xk k 3X;L m
-------
4-5
We may modify (4.20) by using (4.13) and (4.17) and write
oi\ 3 / ^ \ _
at
J
u 3p
In the same way, we may modify (4.22) through use of (4.15) and
(4.18) to obtain, after noting that
32u' 32u» 9u~; 3u
t -
ut _ + ut _ = - _ 2 _ - _ -
Uk 2 + Ui -a 2 9 2 ^ 3x 3x
3X 3X 3X. j J
- poujuk HT - poujui w: - IT: <"ouiujui>
J J J
. _ a _ ,
Ir- «p'ui'-
rC
p p
O '
+ T ' u ' e + T 'u ' s
T k si T i Sk
o o
92uTuJ 3u.' 3u^.
+ y.
_
o 2 o 3x. 3x
X j
- (1J0
o
-------
4-6
The momentum and Reynolds stress equations, (4.23) and (4.25),
given above are the same as Eqs. (30) and (3^) of Reference 2.
We now'derive equations for the mean temperature field
and the variance of the temperature fluctuations from (4.9).
Substitution of the scheme (4.11) into (4.9) yields
It <»o?> + ft (po + H7 (po3/ + poV' + POUJ? + P0UJT')
J
p p
a T1 % T I
Averaging this equation gives
It <'o?) + fer (o0> «o ?
If we subtract (4.2?) from (4.26), we .obtain an equation for
'the temperature fluctuation, namely,
It (poT?) + k-(poV' + VJJT + poujT' - poujT')= Uo
J
(4.28)
To obtain an equation for T'2, (4.28) may be multiplied by 2T'
and the resulting equation averaged, with the'result that
2 2
t
J . J
"^ -
82T'2 3T
n p o
o 2 o
-------
4-7
The mean temperature equation and the variance equation, (4.27)
and (4.29) are the same as Eqs . (3D and (36) of Reference 2.
An equation for the heat transport correlation uIT" can
be obtained from (4.21) and (4.28).
T1 and then multiply (4.28) by u'±
equations are then added and averaged; the result is
First, multiply (4.21) by
. The resulting two
£_
3t
3u
> 11 t m i ±.
>oV 9x,
_
oji 3x 3x
J J
3
3x
+ u*) T
T
3"u.[T'
2 "Uc
J
3ui 3T«
5 3xj 3xj
u
, ^
' - _ _ -
(4.30)
This equation is identical to Eq. (35) of Reference 2.
To summarize the relationship between the basic equations
given in Reference 2 and the equations derived here, one may
refer to Table 4.1 below:
Table
Equation Number
Reference 2
on _ _ _ _ _
ju
01 _ _ _ _ _
0-L
DO _
J^-
3'3 _ _ _ _ _ _
JJ
qll _______
34 _______
3^ _ _ _ _ _
50
3fi _ _ _ _ _ _
JO
4.1
Equation Number
This Report
_ _ _ _ il P"5
- - - i\ ?7
-----ill^
- - iJ 1 S
ll PR
_____ ii ^n
_ _ _ _ ii ?q
-------
4-8
We now derive a set of equations consistent with those just
presented for the mean species mass fraction C , for the
transport correlation u.'C' , and for the correlations C'T'
Q ' 1 Ot OC
and C^ .
If the scheme (4.11) is substituted in (4.10), one obtains
IT (P C ) + IT- (P C') + | (p u.C + p u.C' + p u!C~ + p u!C')
3t o or 3t Ko a 3x. \ o j a o J a Ko j a o J a/
j
P ?
3^0 30'
Averaging this equation yields
3p C . _ 32C~ ~
ZT^- + IT" ?
9 C' 9C'. 3C'
a ^ a a
-------
4-9
To obtain an equation for u'C1 , we multiply (4.33) by u!
-L vjt> . J-
and (4.21) by C' and add the two equations. If the resulting
equation is averaged, one obtains
It
3u. 3C"
_ _
'= - p u'.C' Tr-- - p u!u! ~- - | (p u.'u'.C')
Ho j a 3x. Ko j i 3x. 3x. VKo i j a
J J J
p
. ax.,
j t
m
m
The equation for T'C' is obtained by multiplying (4.33) by T;
and (4.28) by C' and adding the two equations. The resulting
equation, when averaged, yields
ft «"0'ptoi'
Pouj a
f rn t "*
o
3^T'(T
2
n 1 1 ' P ' ".. .
P u.u
o j a ox .
, 3T- 3Ca
yo 3x. 3x.
J J
3x.
J
U,0OJ. U f \i ~> C. \
U0 5 2y^ ^^ ^^ (4.36)
Equations (4.32), (4.34), (4.35), and (4.36) are the equations
necessary to describe the dispersal of a passive pollutant in
an atmospheric shear layer governed by the equations given in
Reference 2. These equations are the basic equations from
which those discussed in A.R.A.P. Report No. 169 were derived
by invariant modeling.
-------
iJ-10
We will now turn to a discussion of the modeling of the
terms in the equations derived above that must be modeled to
form a closed set of equations.
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5-1
5. MODELING OF TERMS IN BASIC EQUATIONS TO ACHIEVE CLOSURE
First let us list the basic equations derived in the previous
section which we will need to enable us to calculate the genera-
tion of turbulence in an atmospheric shear layer and the dispersal
of a passive pollutant in this layer. In rewriting these
equations, the forms of some will be slightly changed by use of
the two continuity equations [(4.13) and (4.15)] and the condi-
tion that 9pQ/9t = 0.
9u\ _ 9ui 9- pQT
po 3t PoUj 9x . 9x. T ^i
J 10
32u.
u 9p
9T , - 9T 92T 9
9G H 9p
1 = _ _il 2.
9x. p 9x.
J o j
9u! u! 9p
i = _ _J 2.
3x. pQ 8Xj
-------
5-2
3u!u'
3u'u'
o 3t
- puu
- puu
ok 377 - oi 377
J u
3x.
j
iP - §77 (p'uk} - fer
k
78Ui + _U_i
\9xk *xi,
-3U
o 3x . 3x .
J J
9P,
3p,
3u
3u
m , .. , m
p 3x 1 ui 3x, + uk 3x. ,
o m \ k i /_l
(5.5)
3u!T
3u!T
3u
o 3t
+ P
.1 3x
- p u'.T'
o J
ir 9T
)oujui 3x
J
3 u.'T' 3u! 3T?
Y 2y.
3x .
J
o 3x . 3x.
J J
T'u
m 9x.
3p 3u'
± O. m t ^
p 9x 9x.
Ko m i J
(5.6)
31
p
o
- 3 T'
'oUj 37-= * 2poT'Uj
i ( o u'T'2Nl
3x, VpoUjT )
32T'2 3T' 31"
2y
3xf
J
j j
(5.7)
-------
9C"
9C 92C
J SET ' "o rrr - 5
J . dX,-
J
5-3
(5'8)
9u.'C'
9u!C'
9u.
9C
- puu
oji 9
H7
"
517 ax.
3C" Pn
317 + T;
(5.9)
9C'T'
9C 'T1
J
n 11 t P "T » 1
9C
n 1,1011 P1. n n
pou1T 9x, pou
J J
->
9"T'C'
-l-ii .... . _ On
'xm
f p 1 ^ -^-
j a 3Xj
Mm 3C1
9T ' a
a
- (5.10)
9C
'2
and, finally,
9C'2
9Xj
9C ,
O «. f /"l f ^* "
c~i "i rv ^i Y ^ Y
LJ.I VA O A . O A .
J J J
^ i
3e-C1'' 9C'
a ^ a
° 9x2 ° Xj
J
KV.
9C'
a
Xj
and
P =- 7 T
o
P' = - ip T'
o
(5.12)
(5.13)
-------
It will be noted that Eqs. (5.1) through (5-7) do not
depend on the passive scalar C . Thus one can, in principle,
solve for the turbulence in an atmospheric shear layer using
(5.1) through (5.7) and with this information in hand solve for
the dispersal of the pollutant field given by C .To do this,
(5.8) through (5.10), which are independent of (5.11), may be
solved for C . Finally, (5.11) may be solved for the vari-
ance C' . The mean density field and the variance of the
density fluctuations may be obtained from (5-12) and (5-13) from
2
the previous solutions for T and T' .
What is needed to perform these calculations is a closure
of the correlation equations given above. We give next a list
of the terms that must be modeled, ordered according to the
type of term.
a. Tendency-to-isotropy terms
In the equation for u\u\<- > (5.5)
p.
p
In the equation for u.'T' , (5.6)
3T'
n i
P
In the equation for u'C , (5-9)
3C1
n'
P
3x±
b . Velocity diffusion terms
In the equation for u.'u' , (5-5)
pouiujuk
In the equation for u.'T' , (5.6)
-------
In the. equation for T'2 , (5-7)
Pou!T'2
In the equation for uC , (5-9)
p u.'u'.C'
o i j a
In the equation for T'CM , (5-10)
p u'.T'C'
o j a
In the equation for C'2 , (5-11)
p u'.C'2
o j a
c. Pressure diffusion terms
In the equation for u!u' , (5-5)
p'u-.' and P'u'
In the equation for uJT' , (5.6)
p'T
In the equation for u.'C1 , (5-9)
p'C'
^ a
d. Dissipation terms
In the equation for u.'u' , (5.5)
9x. 9x .
J J
In the equation for u!T' , (5.6)
9T
9x. 9x
J J
5-0
c;
-------
5-6
In the equation for T'2 , (5-7)
91" 91"
9x . 9x.
J J
In the equation for ujCM , (5-9)
9u! 9C'
i a
In the equation for T'CM , (5-10)
91" 3C'
o^
9x. 9x.
J J
o
In the equation for Cf , (5.11)
9C1 9C'
o_ o_
9x. 9x.
J J
e. Divergence/strain terms
In the equation for u.'u' , (5-5)
9u' 9u
u' and u'
i 9x, k
In the equation for u'1" , (5.6)
9x.
In the equation for u.'C' , (5-9)
1 (X
3u
The modeling we will adopt for these terms follows the
detailed discussion of modeling presented in Ref. 5- We will
start with the tendency-towards-isotropy terms.
The model used in Ref. 5 in the u'uv equation was
9u! 9u.' \ p q / 2
-------
5-7
This equation satisfies the condition for incompressible flows
that when i = k the model vanishes as it must, since the basic
expression vanishes as a result of the velocity fields being
nondivergent. In the case considered here, the divergence of
the turbulent velocity field does not vanish, but is given by
(5.4), namely,
9u! u! 9p
9^ = - ^ 93T (5'4)
9XJ Po 9xj
We may change the model so as to agree with this result in two
simple ways. We may write
k
p'u'
or
9u! 3u.' \ pa
o ,, __.
" PO 9xi " po 8xk 5 5
The first model above allows the divergence of the turbu-
lence field to contribute only to turbulent energy directly;
while the second model allows contribution of the divergence
only to terms in an actual atmosphere which contains the verti-
cal fluctuation since only 9p /9z exists. The second expression
(5.15) is the modeling given in Ref. 2. After considerable
thought, it is now felt that (5.14) might have been a more appro-
priate choice since a simple divergence of a fluid element will
affect all velocities equally.
Following the model for the tendency towards isotropy in the
u.'u,* equation, we choose
-------
5-8
and
3C' P q
The models for the velocity diffusion terms can be taken
over directly from our previous work. We choose
9u!u! 3u!u' 9u!u'
iJ. + _iJl + _.!_« ) . (5.18)
8u!T'
(5.19)
3uJC' 9u!C'
(5.20)
kj.
2M 3x.
J
8C'T"
(5.21)
Pou!C'^ = - pQA q j^~ (5.23)
o j a o Xj
The pressure diffusion terms may also be taken over from
our previous work. We write, therefore,
9uTuT
-------
5-9
ST^-uT
(5'25)
(5'26)
If we follow our previous work on incompressible turbulent
shear flows, the modeling of the dissipation terms need not be
changed. We have, therefore,
'
uu
_ _
9x. 3x. ~ ,2
J J A
3u! 3T' u!T»
3^377= -T5- (5'28)
J J A
3T1 3T1 _ '!
3x. 3x. " .2
,2
3u! 3CT u.'C'
x a
3x. 3x, .2
J J A
3T' 3C' T'C1
__
3x. 3x. " .2
J J A
3C' 3Cf C1
a. a _ a /,- o^
3TT 33T - 72" (5'32)
J J A
The divergence/strain terms did not appear at all in the
case of an incompressible shear layer. They are new and must
be modeled for the first time. Consider the term u!(3u'/3x. ).
1 m k
-------
5-10
There are two simple ways in which it might be modeled.
9u' u.'u! 9p
and
u!u'
Both of these models satisfy the continuity equation (5.^)- At
the present time, (5.33) is preferred; however, (5.3*0 was the
modeling used in A.R.A.P. Report No. 169 [Ref. 2].
Following the modeling given above, we .may give two models
for T' (3um/9xi) , namely,
9u' T'u'. 9p
T' 3 ' - VL T H (5'35)
and
9u' T'u' 9p^
^= - ^ 93T (5'36)
3xi . po 3xi
and two models for C'(9u'/9x.), namely,
9u' C'u! 9p
ca air = - 6mi -Tr1 -5T- (5>37)
a 9Xl mi PQ 3x -
9u' C'u' 9p
p., m _ Ot m 0 /R
Ca3Xl-- 3X, (5
Here again, the models presently preferred are (5-35) and (5-37)
while those used in A.R.A.P. Report No. 169 [Ref. 2] were
(5.36) and (5-38).
With the modelings given above, we are in a position to
write a closed set of equations for the generation of turbulence
in atmospheric shear layers and the dispersal of passive pollu-
tants in these layers.
-------
6-1
6. .AN INVARIANT MODEL OF THE ATMOSPHERE
If the models given in Section 5 are placed in the set of
equations (5.1) through (5-13), a set of closed equations for
the motion of the atmosphere and the dispersal of pollutants in
this atmosphere is obtained. The equations are not the usual
Boussinesq equations, for we have retained the divergence of the
mean and turbulent velocity fields, and we have not made the
thin layer assumption that is necessary to obtain the Boussinesq
approximation. The assumptions that have been made are that the
molecular transport coefficients are constant, that the Prandtl
and Schmidt numbers are one, and that the departure of the
atmosphere from an adiabatic state at rest is small. Since we
wish to exhibit the equations used in A.R.A.P. Report No. 169,
we will adopt the modelings that were used in that report in
the rest of this section. If (5.15) through (5-32), (5-34),
(5.36), and (5.38) are substituted into the basic set [(5.1)
through (5-13)], one obtains
3u\ _ 3u\ 3- pQT
po 3t + Pouj 3x. 3x, + Tsi
J, J i o
y
n 9 3 Y v K/~\ 1
\J r. £- v A . Ol,
3x. j
+ *) L__/!!m!!°
o o 3x. I p 3x
i \ o m
3T . . - 3T 32T 3
3u . u . 3p
- -
3x. po ax
3u! ul 3p
_
3x. po 3x.
-------
6-2
9uiuk
) U. . 7\^
o j 9x.
9u
p u'.u,'. P~UI-U,!
^T'u; g, + ^ T'u' g
oujuk 177 - poujui 9TT T TO ^ "k &i T
J ^
'o
r
+ 9T7 j poA2q I
J L
9u!u'. 9u!u.'
^Li x i ^
^k
3x.
3u!u!
pAq
9xk r° ^ -j
9p
o
u!u,' - 6.. ^- I + ,
ik ik 3 ] 3x.
92uju;
3uj"k\
9XJ. /
9u'.u
u
9x
k
1 o
0 9x2
J
2u
ik
o
'o + ^>[uium
9x
m
' 11'
u,'u.
k m
(6.5)
-------
3u!T
3u.'T'
o 3t
6-3
3u.
9x
9uI
3u
El
j J
u!T
u!T' + y^
i o
3x<
9xm
(6.6)
3T
3t
,2
3T
3x
,2
2 2
l^rp T ^
- 2y
rn I '
° ^
(6.7)
3C
32C
kJ 3x. -j
(6.8)
-------
9uiCa
'o 9t
oJ
9u
- p u'.C
u '. u!
9C
o_
9x.
J
9C'u!
; 'u!
9C'ut
g J
!oi
A-,
a
2'
9u!C
Ot
u!C'
J. CX
.9)
gc'T1
a
at
9C'T'
a
9x .
J
9C
U'
g
- p u!C'
9T
o j a 9x.
9C'T'
9x£
Q irpt
(6.10)
9C
3t
9C
and, finally.
and
9x .
J
9C
a 9
, 2
9x^7
J
P = -
o
9x. 9x.
J J
9C
,2
a
9x.
J
(6.11)
(6.12)
(6.13)
Equations (6.1) through (6.13) are the starting point for the
model equations given in A.R.A.P. Report No. 169.
-------
7-1
7. EQUATIONS USED IN A.R.A.P. REPORT NO. 169
In A.R.A.P. Report No. 169, the equations used in the
previous section were written out for a special case of atmos-
pheric motion. If we adopt the convention that (x, } x?, xO
is (x, y, z) and (u-, , Up, u.-.) is (u, v, w) and take the
direction z as perpendicular to the earth's surface so that
g. = 6 . ~g , we may express this special case of atmospheric
motion as
u=u(z3t)j v=w=0
T = T(z,t)
u'u' = u'u'(z,t)
v'v1 = v'v1(z,t)
w'w' = w'w'(z,t)
u'w'=u'w'(z,t)
(7.D
v'w1 = 0
u'v' = 0
u'T1 = u~?!rr(zjt)
v'T1 = 0
= w'T1(z,t)
T'2 = T'2(z3t)
Equations (6.1) through (6.7) of the previous section may then
be written
O o 13 o X
3z'
(7.3)
-------
3u'u
o 3t
2pou'w' §+MPoM
9u'u'
1^ U'U' - ~
A, 3
3v'v' 9
o 9t
9v'v'
A-
V
p n
9w'w' o
"O
3
7-2
Pc
3z = T"
(7.4)
i
k
(7.5)
a
= 0
(7.6)
3x^ pQ 9z
(7.7)
+ y,
92u'u'
u'u'
0 X2
(7.8)
92v'v'
° 9z2
0 X2
(7.9)
An
I w I
w 'w' -
2A3q
9^"^ 9pc
9 w' w'
0 X2
- 2(y
Q
Fz"
(7.10)
-------
7-3
o u w , . ou . u -. ,.,-.-
rx , U W W ~ T J.' 1,1 t
o 3t o 3z 1Q .
D A a
P0A2q
p~q
0 11 'vr +
. u w T
Al
0
U ^^
O £.
f
( y +y ) < u w
o o
X
a 11 ' ']' ' ^11 Vl'
OU 1 _ imi "^ -.. cj j
P >\4 pWl « p U W «
Oot O oZ O oZ
9 / 3
0
+ 9 "u'T'
9z
3w'T' _ t , 3T Po rp,2
Po 3t PoW W 3z T S
o
P A q
0
P . T.
f rn t _L
Ai
r
0
3 ( ; 3u ' w ' \
9 z \ o ' ' 3 H 9 z 1
"1 n
c\ i f 'J P
yv^q ... r.
,-, U ' w '
Tr>. fi 3'-\>\ i ^ofjl
L3zlpo 3;'J p^3z JJJ
(7.11)
p_q
° 11 ' T1 '
. U 1
,, t rn T
nll U C 7 1 ~>)
^y_ o \ r -L <- /
O , f.
X
)/ V
, 3 / A 3w'T' 1
Z \ Z 1
r\l- TTlf Tt'T1?
0 3z2. ° A2
T
/ ^in\ /^n\
T ^ I ^ ° 1 ^ I ° 1
_3z \po 3z / p^3z / J
(7.13)
-------
am
- 2p w'T' f-±-
o 3z
3T
5- ~
9z \ o 2
o o
1"
~
0 3z2
rn I <-
(7.14)
In addition, we have
and
P = - ip^ T
o
P' = - T'
o
(7.15)
(7.16)
,2
so that p , p'u! , and p' may be found from T , T'u! ,
2 i . i
and T1 .
The equations given above are the same as those used in
A.R.A.P. Report No. 169 if one places A = A2 = A_ = A . In
Table 7-1 below,- the equation numbers of the two reports are
compared.
Table 7.1
Equation Number
A.R.A.P. 169
Equation Number
This Report
16---------- 7.2
17 7.5
18 7.8
19 7.9
20 7.10
21 7.11
.22 ----------7.12
23 7-13
24 7.11)
In A.R.A.P. Report No. 169, the turbulent structure of
an atmospheric shear layer having given time-independent
distributions u(z) and T(z) was found by solving (7.8)
through (7-14) simultaneously for the distributions of u.'u,'
Q" 1 K
u.'T' , and T' in z which came to equilibrium with the
-------
7-5
given profiles u(z) and T(z). Once these distributions were
known, the transport of a pollutant species C in such a layer
from a steady line or point source was discussed in terms of
Eqs. (6.8) through (6.11).
To get the equations used in A.R.A.P. Report No. 169, the
mean fields u = u(z), T = T(z), u'u'(z), v'v1(z), w ' w' ( z) ,
u'w'(z), u'T'(z), w'T'(z)j and T' (z) were assumed given.
In addition, u'v1 , v'w1 , and v'T' were equal to zero. The
source was assumed to be steady; therefore, one sought C (x,y,z)
To simplify the problem, the thin layer approximation was used
so that the derivative of a quantity in the streamwise direction
x could be neglected compared to the derivative of that same
quantity with respect,to y and z . Under this assumption,
(6.8) through (6.11) can be written
3C
P0U 3x
a
3
3C
a
o 3y 1 3y
- p v'C'
o a
~ / 3C
9_ / a_
3z I 3z
- p w'C'
o a
(7.17)
3C
pov'v'
a
3_
3y
po(2A2
3C'v'
3CMw'
3y
3y
3C'w
p~q
Cav'
13 C'v1 3 C'v
^s2-*^
3z
C'V
- 2 -5L_| (7.18)
-------
7-6
a
o 3x
3Cc
Jz~
'o
9_
9y
a
3y
9z
9C'w'
a
3C'v'
9y
3C'v'
A, . a
C'w!
9"C'w' 3 C'w'
a , a
C'w'
a
o
C'w' 3-
a 3z
9 II 9po
9p \'
9z
2 9z
(7.19)
9C 'T'
₯x
- pQW
3C'T'
3C'T'
)32C 'T' 32C 'T' C "I"
a , a o a
o -^ + ~^ 2
3y
3z
(7.20)
-------
7-7
9C'2 9C . / 9C'2 9C'2 \
9 C1
_L__ 2
We note here that (7.17) through (7.20) are exactly the
same equations as given In A.R.A.P. Report No. 169 (Eqs. (12)
through (15)), If one writes A, = A? = A,, = A .
-------
8. DISCUSSION
The equations given in the previous section, and which were
used in A.R.A.P. Report No. 1695 may be simplified by considering
high Reynolds number arguments. Normally, all terms containing
the viscosities y or y* can be dropped from the equations,
except for the dissipation term which is of the form
3a' 9b' a'b'. , p , ,.
977 9^7 = - 2»o -IT (8-l}
J J A
This term may not be dropped because the equation for the
dissipative scale A is viscosity- or Reynolds number-dependent
and is of the form [see Ref. 5]
a + b(p qA /y )
' '
Thus, at high Reynolds numbers, the dissipation (8.1) is of the
form
2bp q
1
_
a'b' (8.3)
and may not be dropped from the equations .
In our work we have chosen to retain the viscous terms to
enable a shear layer solution to be extended all the way to a
smooth solid boundary. It is necessary to retain most of the
viscous terms if such a solution is to be obtained, because
one must extend the solution through the viscous sublayer near
the surface where the viscous terms are the dominant ones in
the equations.
A simplification is possible, however. Since the terms
which appear in the equations that are multiplied by the sum
(y + y*) are caused by the inclusion of the divergence of
the velocity field (which may certainly be neglected in the
sublayer), these terms may, for atmospheric flows, be dropped
from the equations.
-------
8-2
If the terms containing (y + y*) are dropped, it is
evident (since these terms were the ones that contained the
divergence/strain effects) that the problem.of modeling such
terms is eliminated. Thus, the only new term that one is
required to model in considering these non-Boussinesq atmos-
pheric equations compared with the Boussinesq model is the
divergence that shows up in the tendency-towards-isotropy term
in the equation for u'u,1 . As mentioned in Section 5 where
X xC
modeling was discussed in some detail, we are considering the
replacement of the model used in Ref. 1 and in A.R.A.P. Report
No.. 169, i.e., (5.15)
p' a5T + a3T = - 3T- uiuk ' 6ik ^
I OX, dX; / A, I 1 K IK 3
^u£ SPo p'u-
PO 3x± - PO
with the simpler model (5-1*0
/ 3u' 3u' \ P q /
.[ i + k ] _ o / nn
I9xk 9xi/" " Al \ J
P'Um 9po
ik T~" 93T
o m
Since the effect of including the extra term
or
p'uk
po
I
-
3po
9xi
,' ,
m
"pr~
o
P'u'
1 -*
po
3p
o
3x
m
3po
9xk
p,
"
in the model f or . the tendency towards isotropy does not, in
most cases, have a significant effect upon the turbulent
velocity field, we do not intend to modify the equations in
Section 7 (which are currently programmed on the computer
-------
8-3
at A.R.A.P.) until such time as a detailed reexamination of
the .entire invariant model that is in use at the pi'esent time
.1 s made .
\
-------
9-1
9. REFERENCES
1. Donaldson, Coleman duP., Sullivan, Roger D., and Rosenbaum,
Harold: "A Theoretical Study of the Generation of Atmos-
pheric Clear Air Turbulence," AIAA Journal 10, 2, (1972),
pp. 162-170.
2. Donaldson, Coleman duP. and Hilst., Glenn R.: An Initial
Test of the Applicability of Invariant Modeling Methods to
Atmospheric Boundary Layer Diffusion," A.R.A.P. Report No.
169, October 1971.
3. Lumley, John L. and Panofsky, Hans A.: The Structure of
Atmospheric Turbulence, Interscience Publishers, New York,
1964.
4. Spiegel, E.A. and Veronis, G.: "On the Boussinesq Approxi-
mation for a Compressible Fluid," Woods Hole Oceanographic
Inst., 1959.
5. Donaldson, Coleman duP.: "On the Production of Atmospheric
Turbulence and the Dispersal of Atmospheric Pollutants,"
A.R.A.P. Report No. 186, Volume I, December 1972.
-------
BIBLIOGRAPHIC DATA
SHEET
1. Report No.
EPA-R*t-73-Ol6b
3. Recipient's Accession No.
4. Title and Subtitle
DERIVATION OF A NON-BOUSSINESQ SET OF EQUATIONS
FOR AN ATMOSPHERIC SHEAR LAYER
5. Report Date Approved
March 1973
6.
7. Author(s)
Coleman duP. Donaldson
Performing Organization Rept.
No- 186, Vol. II
9. Performing Organization Name and Address
Aeronautical Research Associates of Princeton,Inc
50 Washington Road
Princeton, New Jersey 085^0
10. Project/Task/Work Unit No.
Element A-11009
11. Contract/Grant No.
EPA 68-02-0014
12. Sponsoring Organization Name and Address
Environmental Protection Agency
Office of Research -and' Monitoring
Washington, D.C. 20460
13. Type of Report & Period
Covered
Interim
15. Supplementary Notes
16. Abstracts
A detailed derivation of the equations which describe the generation
of turbulence in the atmosphere and the equations which govern the
transport of pollutants in the atmosphere is given. The equations
are valid for motions which are small departures from an atmosphere
at rest and in adiabatic equilibrium, but the usual assumption that
.the motion is restricted to thin layers, which leads to the Boussinesq
approximation, is relaxed.
17. Key Words and Document Analysis. I7o. Descriptors
Turbulence
Turbulence models
Atmospheric motions
Atmospheric turbulence
Atmospheric transport
Atmospheric Surface Layer
Pollutant Dispersal
Pollutant Transport
17b. Identifiers/Open-Ended Terms
Second-order closure
Second-order modeling
Invariant modeling
17e. COSATI Field/Group
18. Availability Statement
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19. Security Class (This
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UNCLASSIFIED
21. No. of Pages
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22. Price
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