Transactions on Ecology and the Environment vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
Group analysis for bidimensional steady
turbulent diffusion equation for removal
process
P. Barrera
Dip. di Meccanica ed Aeronantica*
T. Bmgarino
Dip. di Matematica ed Applicazioni*
U. Bmnelli & L. Pignato
Dip. di Energetica ed Applicazioni di Fisica*
Italia
Abstract
The group analysis for the steady bidimensional turbulent diffusion equation
for removal process is presented. We assume that the wind speed and eddy
diffusivity coefficients depend on the height in monomial form. Some analytical solutions are presented. They can be used to test numerical solutions.
1
Introduction
In the last three decades, devoted attention has been attracted for a
large variety of physical phenomena governed by nonlinear equations.
Nonlinear diffusion mechanisms is of great interest in many physics
and engineering applications,
Nowadays, the influence of the vehicular traffic on the degradation of urban atmosphere is focused on by researchers belonging to
different fields so important for public health. The internal combustion engines in today's cars and trucks are the major source of urban
Transactions on Ecology and the Environment vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
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air pollution.
In the planning of highways it is necessary to conform to procedures of Environmental Impact Assessment. The assessment of atmospheric environment is a very important parameter in this procedure
according to the law in EEC (Economic European Community).
Then it is necessary to have suitable atmospheric dispersion models
for predicting the concentration distributions of CO, NO and NO%
emitted from the traffic of vehicles moving in a long street.
For this problem, in particular atmospheric conditions like very
slow wind, it is necessary to examine the transformation of CO and
NO along the path introducing in the diffusion equation a term well to
describe the removal process; e.g. Khairul Alam & Seinfeld/ Shukla
& Chauhan,^ Galmarini, Vila-Guerau De Arellano & Duynkerke^.
2
Mathematical formulation
The turbulent transport diffusion for incompressible flow field for a
removal process is assumed to be governed by the differential equation:
oc
oc
oc
oc _
, .
m*™fa*"dy'*'™dz'~
^ '
d
where:
• c(x>y,z,t) is the mean concentration
• kx> ky, kz are the coefficients of eddy diffusivity;
• u, v, w are the mean velocity components along x, %/, z axis;
• A is the rate constant of the removal process;
# # is the order of the removal process.
In the steady case, if:
a) the mean wind velocity is along x-axis;
b) the convective term in x direction is greater than the diffusive
term;
Transactions on Ecology and the Environment vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
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723
c) the diffusive term in y direction is negligible;
the governing eqn. (1) becomes:
^(x,.)
(2)
We assume that the wind speed and eddy diffusivity coefficient depend on the height in monomial form:
For our purposes it is suitable to write the eqn. (2) in the following
form:
#(z)c(%, 4* - K(z)c(%, z), - c(z, z),, + M(z)c"(%, 4 = 0 (3)
where:
• \z = dlog/dz.
The derivatives with respect to variables x, z and c are denoted by
subscripts %, z and c.
3
Group analysis
Similarity solutions are generally yielded by dimensional analysis
which is a particular case of group analysis. Using this it is often
possible to provide exact solutions to linear and nonlinear, ordinary
and partial differential equations. By solving systems of partial differential equations, similarity solutions can be obtained by recasting
the original system into a system of equations with a smaller number
of independent variables, (see, e.g. Bluman & Cole,* Ovsiannikov,^
Barrera, Brugarino & Pignato,^).
Transactions on Ecology and the Environment vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
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The group analysis of the eqn. (3) is performed through the one
parameter Lie group of transformations:
c* = c + eC(c x z] + O(e^)
x* — x -f eX(c x z} 4- O(e^)
(4)
where C, Z and % are the infinitesimal generators of transformations.
Eqn. (3) is invariant with respect to the group (4) of transformations, if c* is the solution of eqn. (3) in the star variables.
To put
6 - C— + Z— + X— + fcJ—- + [cJ—
+
(cU—-
then eqn. (3) is invariant with respect to the Lie group of transformations (4) if the following condition holds; Bluman & Cole/
Ovsiannikov,^:
6(F) - 0
(5)
where [c^], [c^] and [c^] are the infinitesimal generators of the transformed derivatives determined from eqn. (4). Because eqn. (4) determine the transformed derivatives, then [c%], [c^], and [GZZ] can be
expressed in terms of C, Z and X.
We obtain, for example,
DC
DZ
DX
where ^ is the total derivative. Similarly, we proceed for [c^]. The
transformed derivative [c^] can be derived by recurrence formula.
If the solution is invariant under the group of transformations, the
solution must map into itself, i.e.,
c*-c(z*,%*,r) = 0
(6)
Expanding the invariance condition, eqn. (5), of eqn. (6), we get
C = c,Z + c*X
(7)
Eqn. (7) is the invariant surface condition. The general solution of
eqn. (7) is obtained by solving the characteristic equation
dz
d%
dc
Transactions on Ecology and the Environment vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
725
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In principle, the general solution of eqn. (8) can be found, and
in this case the number of independent variables of the equation can
be reduced. The new variables are the similarity variables, and the
new function of the similarity variables is the similarity solution.
A considerable difficulty in determining the generators of the
group of invariance lies in the amount of the auxiliary calculations
involved. We performed the calculations of the generators of the
transformation group on a PC using the REDUCE package.
After imposing the invariance condition eqn. (6) we obtain from
eqn. (3) that C, Z and X must satisfy the determining equations:
- #2% = 0
+
=0
=0
-1=0
where A\(x,z) and B\(x,z) are arbitrary functions of x and z.
We will examine similarity solutions with the assigned laws of k(z)
and u(z).
3.1
Infinitesimal generators and similarity equations
Without compromising with the generality we assume B\ = 0.
The integration of determining equations leads to the following form
of the X, Z and C:
( X=
where X\, X<2 and C\ are constants and:
d-b + 2
„
-8 + a
We examine the similarity equations corresponding to particular
values of the constants. We select the following.
Transactions on Ecology and the Environment vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
726
3.1.1
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Case A)
we have:
= Xi% + %2
= 5#IJ*+te)"^^-^i
(12)
=(-^i + (l-6)(^)-^^Ci + (^)Xi)c
where:
We can consider the following two particular cases.
Case AI) When:
%2 = Ci = 0,
%i ^ 0
we have:
where
6-2
'"2(4-36)'
1
' 2(1-6)
where / satisfies the ordinary similarity differential equation:
Case A2) When:
we have:
where
A
u\
and / is solution of the ordinary similarity differential equation:
=0
Transactions on Ecology and the Environment vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
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3.1.2
727
Case B)
When:
%2 = 0,
%i /O,
a = 2,
d ^0
whe have:
' = #z
,
.
.
(15)
where:
C-2
The similarity solution:
satisfies the ordinary differential equation:
3.1.3
Case C)
When:
Xi^Q,
%2 = 0,
we have:
a = 2,
C2 = 0
{-»/-rr
-A. — v\ \X
Z= &
^,
(6+rf)(2a — d) -rr
=^ —oZ3—^A^C
(17)
where:
'"
2(1 +3d)'
'
2(1 +3d)
The similarity solution:
satisfies the ordinary differential equation:
= 0
(18)
Transactions on Ecology and the Environment vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
728
4
/(/
Similarity solutions
Because the similarity differential eqns. (13, 14, 16, 18) are of the
type
= 0
(19)
we are interested in the conditions on A(£),B(£) and C(£) for the
existence of point transformations to the equations of the EmdenFowler-Bellmann type:
%,% = Kr,'V°(n)
(20)
classified in Zaitesev & Polyanin^ by means discrete-group methods.
With the invertible point transformation
for eqn. (19) we obtain the following constraints for A, B, C, g and
77:
'= 0
4.1
(22)
Some similarity solutions
For simplicity, we search to satisfy the eqns. (22) for the eqn. (14).
In this case for g(£) and r/(£) we obtain:
Mf ) = 3e-"i(,
%(() = Taie^^
(23)
where S and T arbitrary constants. For the g and rj given by eqn.
(23), the eqn. (20) becomes:
(24)
"
Among the equations listed in Zaitesev & Polyanin® we consider only
the equation corrisponding to a = — 1.
Transactions on Ecology and the Environment vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
729
4.1.1
Similarity solution for a = — 1
In this case, for suitable values of constants S and T, the solution of
eqn. (24) is, in parametric form:
| V - bc\ exp(r^) + c%
where c\ and eg are constants and the function erfi is so related to
error function: erfi(() = (-i)erf(i().
In the variables x and z the solution of the eqn. (2) is:
(26)
exp f inverfi^ f —
where:
(exp (v\ (z - x) - eg)) J j
_
p
VA7T
12 = -y-ci
and inverfi is the inverse function of the erfi.
5
Conclusions
We have considered the analytical solutions of the bidimensional turbulent diffusion equation for removal process. For the eddy diffusivity
coefficient fc and wind speed u we have selected monomial laws respect to height z.
The studied equation well describes the diffusion from a ground
level infinite line source into a turbulent flow of air moving perpendicularly to the source and it is important to study the pollution
generated from vehicular traffic moving in a long street. By using
the found solution it is possible to validate numerical results.
References
[1] Khairul Alam, M. & Seinfeld, J. H., Solution of the steady state,
three-dimensional atmospheric diffusion equation for sulfur dioxide and sulfate dispersion from point souces Atmospheric Environment, 15 A, pp. 1221-1225, 1981.
Transactions on Ecology and the Environment vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
730
Air Pollution
[2] Shukla, J. B. & Chauhan, R. S., Unsteady state dispersion of air
pollutant from a time dependent point source forming a secondary
pollutant Atmospheric Environment, 22, pp. 2573-2578, 1988.
[3] Galmarini, S., Vila-Guerau De Arellano, Y, & Duynkerke, P. G.,
The effect of micro-scale turbulence on the reaction rate in a chemically reactive plume Atmospheric Environment, 29, pp. 87-95,
1995.
[4] Bluman, G. W. & Cole, J. D., Similarity methods for differential
equations, Springer-Verlag, New York, 1974.
[5] Ovsiannikov, L. V., Group analysis of differential equations, Academic Press, New York, 1982.
[6] Barrera, P., Brugarino, T. & Pignato, L., On the application of
the group analysis for solutions of turbulent diffusion processes in
nonhomogeneous media Proceedings of the 2nd IUAPPA Regional
Conference on Air Pollution Vol. II Seoul, Korea, pp. 165-176,
1991.
[7] Barrera, P., Brugarino, T. & Pignato, L., Group analysis for
nonlinear diffusion equation in unsteady turbulent boundarylayer flow, Progress in Turbulence Research eds H. Branover, &
Y. Unger, AIAA, 162, pp. 271-281, 1995.
[8] Zaitsev, V. F. & Polyanin, A. D., Discrete-Group Methods for
Integrating Equations of Nonlinear Mechanics, CRC Press, Boca
Ratom & London, 1994.