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transcript
Proceedings of the 22nd
National and 11th
International ISHMT-ASME Heat and Mass Transfer Conference
December 28-31, 2013, IIT Kharagpur, India
HMTC1300151
ON SOLUTION OF HYPERBOLIC EQUATION MODEL FOR ATMOSPHERIC DISPERSION
B. Ghosh Bhabha Atomic Research Centre Mumbai, Maharashtra, 400085
India bghosh@barc.gov.in
H.G. Lele Bhabha Atomic Research Centre Mumbai, Maharashtra, 400085
India hglele@barc.gov.in
R. K. Singh Bhabha Atomic Research Centre Mumbai, Maharashtra, 400085
India rksingh@barc.gov.in
ABSTRACT The dispersion of pollutant in the atmospheric boundary
layer is governed by varied degree of the fluctuating velocity
fields generated by atmospheric stability conditions. The
resulting propagation of the pollutant is more suitably
governed by the hyperbolic diffusion equation that the
conventional parabolic differential equation. In this study, the
dispersion of pollutant has been modeled under the boundary
layer parameterization by Monin-Obukhov (MO) stability
theory. Towards this end, the atmospheric velocity profile has
been obtained from MO stability parameterization. The
boundaries of the plume have been evaluated under different
atmospheric conditions. The vertical profiles of the
concentration distribution have been calculated from the
governing hyperbolic dispersion equation.
NOMENCLATURE β Isobaric thermal expansion coeffient
cp Isobaric specific heat
χ Pollutant concentration
D Molecular diffusion coefficient
ε Turbulent kinetic energy dissipation rate
g Acceleration due to gravity
H Height
k von Karman constant
K Exchange coefficient
L Monin-Obukhov length
Pr Prandtl number
q Ground heat flux
Q Quantity of release for instantaneous source
Ri Richardson number
ρ Ambient air density
t Time
T Temperature
τ Shear stress at ground
u Velocity vector
u Horizontal velocity in x-direction
w Vertical velocity in z-direction
x Position vector in space
ξ Non-dimensional x-coordinate
z Vertical coordinate
ζ Nondimensional z-coordinate
INTRODUCTION
There has been a great deal of work on turbulent diffusion
(see the treatise by Monin and Yaglom [1]). Turbulent
diffusion of a passive additive in an incompressible fluid is
usually described by the parabolic advection–diffusion
equation, proposed by Boussinesq [2] and Taylor [3],
( ){ }D.
DDD K S
tχ
χχ− ∇ + = (1)
where χ(x, t) is the concentration, D is the molecular diffusion
coefficient, KD(x, t) is the turbulent diffusion coefficient, Sχ is
the strength of the source of additive, D/Dt = ∂/∂t + u.∇ is the
advective time derivative and u(x, t) is the mean velocity of
the ambient fluid.
The parabolic character of the semi-empirical equation of
diffusion means that the contamination, upon exit from the
source, is instantaneously propagated in all directions and can
be discovered quickly, even though in completely negligible
quantity, at any large distance from the source. Usually, this
inadequacy is acceptable, since the volume inside which the
contamination concentration is not too small is always limited,
and the concentration distribution inside this volume is
generally satisfactorily described by the parabolic diffusion
equation (Monin [4]). However, in some cases (in particular,
close to the real boundary of the contamination cloud), use of
the parabolic diffusion equation can lead to significant errors.
For example, smoke issuing from a chimney of height h
reaches the ground at a distance from the pipe not less than
uh/w, where u is the wind velocity, v is the maximum velocity
of the propagation of the smoke along the vertical. In this
same time, according to the solution of the parabolic equation
of diffusion, the smoke is found at the surface of the earth
arbitrarily close to the chimney.
The generalization of the diffusion equation, which would
give this equation a hyperbolic character was initially
proposed Davydov [5], Goldstein [6], Davies [7] and Monin
[8-9]. Batchelor and Townsend [10] suggest that ‘a description
of the diffusion by some kind of integral equation is more to
be expected’ (p 360).
Meyers [11] has derived a three dimensional hyperbolic
differential equation based on finite correlated particle
velocities. This is appropriate for modeling anisotropic
turbulent diffusion in the atmosphere. Cauchy initial data, the
mean wind, the Reynolds stress tensor, and a typical frequency
of pulsation are required for complete solution. The outlines of
plumes and puffs may be obtained with only knowledge of the
Reynolds stress tensor and mean wind velocity. The classical
parabolic diffusion equations are a limiting form of this
hyperbolic model.
Ghoshal and Keller [12] derived hyperbolic equation,
analogous to the telegrapher’s equation in one dimension,
from an integro-differential equation for the mean
concentration which allows it to vary rapidly. If the mean
concentration varies sufficiently slowly compared with the
correlation time of the turbulence, the hyperbolic equation
reduces to the advection–diffusion equation. However, if the
mean concentration varies very rapidly, the hyperbolic
equation should be replaced by the integro-differential
equation.
Kasanski and Monin [13] filmed smoke plumes under
different stability conditions. Figure 1 shows the behaviour of
a smoke plume under neutral stability condition atmospheric
boundary layer. It is obvious from this figure that the smoke
cloud is confined with a bounded region which grows in the
down wind direction in a regular manner.
Fig.1 Smoke Plume under Neutral Atmospheric Stratification
The behaviour under unstable stratification is shown in
figure 3. Here the smoke plume appears to be more mixed up.
But, even then, the upper boundary of the cloud is clearly
identifiable.
Fig.2 Smoke Plume under Unstable Atmospheric Stratification
Thus, the diffusion of pollution in the atmosphere is due to
the turbulent pulsations of the wind velocity. The magnitude
of these pulsations is limited (for example, it does not exceed
the sound velocity); therefore, according to Monin [14], the
following guidelines may be observed: Propagation of the
pollution through space due to atmospheric diffusion occurs
with a limited velocity.
In this work, we analyse the behavior of pollution
dispersion in the atmospheric conditions characterised by
Monin-Obukhov (MO) similarity theory. Towards this goal,
we first study the atmospheric velocity and temperature
profiles under different atmospheric stability conditions as
dictated by Monin-Obukhov (MO) similarity principle. Then
we explore the limit of the boundary of smoke plume. Finally,
we undertake the calculation of concentration distribution
within the plume.
MO SIMILARITY THEORY AND ABL PROFILES
The portion of the Planetary Boundary Layer (PBL)
immediately adjacent to the surface, typically upto 100 m
from the ground, is called surface layer, and that above the
surface layer is called the Ekman layer. Within the surface
layer, the vertical turbulent fluxes of momentum and heat are
assumed constant with respect to height, and indeed they
define the extent of this region.
When the turbulence in the atmosphere is maintained by
buoyant production, the boundary layer is said to be in a
convective state. The source of buoyancy is the upward heat
flux originating from the ground heated by solar radiation.
Convective turbulence is relatively vigorous and causes rapid
vertical mixing in the atmospheric boundary layer.
Depending on state of turbulence in the atmospheric
boundary layer, the Atmospheric Boundary Layer (ABL) is
classified as: (1) neutral, (2) stable and (3) unstable (or
convective).
The stationary turbulent regime in the surface layer, when
the turbulence is homogeneous in the horizontal plane, obeys
the similarity theory developed by Monin and Obukhov [15].
ABL Similarity Parameters: The turbulent regime is completely determined by the
external parameters,
* /u τ ρ= and ( )pq cρ …(2)
which do not vary with altitude in the surface layer and by the
universal parameter g/T0 characterizing the effect of
Archimedes forces. T0 is the potential temperature. u* is the
friction velocity.
According to Monin-Obukhov (MO) similarity theory, in
the surface layer, the only scale of velocity is *u and the only
scale of length is
( )0
3
*
p
g q
T c
uL
k ρ
=−
,
0 for 0, unstable
for 0, neutral
0 for 0, stable
q
q
q
< >
→ ∞ →
> <
…(3)
where k is von Karman constant. The resulting length scale, L,
is known as Monin-Obukhov length.
The length scale L, first introduced by Obukbov [16], is an
important physical characteristic of the state of the surface
layer and can be called the height of the sub-layer of dynamic
turbulence.
The other derived external parameter is the temperature scale,
*
*
1
p
qT
ku cρ= − …(3)
Atmospheric Boundary Layer Profiles The nondimensional characteristics of the averaged field of
velocities and temperatures ( dd
kz uu zτ
and *
dd
TzT z
) should be
definite functions of the “external parameters” and of
coordinate z. The only non-dimensional combination which
we can make from g/T0, uτ, q/(ρcp), and z is z/L, from which it
follows that
d
dv
uu z
z kz L
τ ϕ
=
and *d
dT
T T z
z z Lϕ
=
…(4)
The equality of the resulting exchange coefficients implies
that
( ) ( ) ( )z z zv TL L L
ϕ ϕ ϕ= = …(5)
Keeping z and decrease magnitude q indefinitely, one
approaches the conditions of neutral stratification, which
correspond to infinite growth of the scale L (with respect to
absolute magnitude). Obviously, in this limit, we should retain
the relation velocity gradient for neutral condition, from which
it follows that
( )0 1ϕ = …(6)
It follows that in a stationary turbulent surface layer, the
wind and temperature profiles can be described using one
universal function of z/L.
The average velocity and temperature profiles are given
by,
0*( )zu z
u z f fk L L
= −
and 0*0( )
zT zT z T f f
k L L
= + −
…(7)
where z0 is the roughness and f(ζ) is the universal function.
where the universal function can be identified as,
( )( ) df
ζ ϕ ςζ ς
ς= ∫ …(8)
f(ζ) is, moreover, connected to the local Richardson number
through the relation,
Ri 1
Ri ( )c f ζ=
′ …(9)
Thus, exact functional form of f(ζ) depends on atmopheris
stability condition.
Surface Layer Profiles: In view of the limiting value in
equation (6) of φ(ζ), in the case |z/L| < 1 we can limit
ourselves to the first terms of the function φ(z/L) expanded in
a power series. As a consequence, f(ζ) takes the following log-
linear form
0( ) lnf cζ ζ ζ≈ + where c0 = 0.6. …(10)
For large positive ζ (the case of a stable stratification), the
function is asymptotically proportional to ζ, while for large
negative ζ (case of thermal convection), it is asymptotically
approaches a constant according to the law
f(ζ) = c1ζ-1/3
+ const. …(11)
In subsequent subsection, we will further elaborate the
causes and implications of these limiting behaviours.
Convective Condition: In this case, due to the lack of an
averaged wind, the friction stress, on average will be zero (uτ
= 0), while the turbulence regime is characterized by only the
parameters q > 0 and g/T0. The regime of purely thermal
turbulence is automodular (self-patterning) and rom
dimensional considerations we get
2/3
1/3
1 0
p
qT T c T
c gzρ∞
= +
where C is the non-dimensional (universal) constant and T∞ is
a constant which has a temperature dimension.
Therefore, in the case of purely thermal turbulence, the
universal function f(z/L) (determined to within an additive
constant) has the form
f(z/L) = c1(z/L)−1/3
+ const.; when z/L << -1.
It can easily be shown that the consequent augmentation of
the turbulent elements with an increase in height and
simultaneous increase in the intensity of the fluctuations
causes rapid increase in exchange coefficients with height.
Stable Stratification: Turbulence decays in this limiting
case of abrupt inversion with a vanishingly weak wind.
Moreover, in this case turbulent exchange between different
atmospheric layers is hampered and turbulence takes on a
local character; at rather high altitudes z >> L (or, to put it
another way, with strong stability, that is with small L > 0) the
turbulence characteristics evidently cannot be functions of the
distance z to the ground. Thus, we may consider that in the
case of stable stratification with an increase in height z, (or,
with an increase in stability, i.e., a decrease in L) the
coefficient of mixing K and the Richardson number Ri tend
toward certain constant values. Accordingly, there is a
(universal) value Ri* of the Richardson number, which is such
that when z/L >> 1,
Ri ~ Ri*.
From above condition and general relation (9) between profile
gradient and Richardson number, it follows that when z/L >>
1,
*
1const.
Ri
z zf
L L
≈ +
Figure 3 shows that characteristics of vertical variation of
horizontal velocity under different stability conditions and
roughness of the ground.
-4 -2 0 2 4
0
2
4
6
8
10
ζ = - 0.1
ζ = - 0.01
ζ = - 0.001
ζ = 0.001
ζ = 0.01
Norm
alis
ed V
elo
city,
ku/u
* = f
(ζ)
- f(
ζ 0)
Normalised Height, ζ = z/L
Stable Stratification
Unstable Stratification
ζ = 0.1
Fig. Stability Dependent ABL Velocity Profiles
It is observed from figure 3 that for unstable ABL
condition, the velocity distribution is relatively uniform with
height. Though the non-dimensional velocity values in this
case appear less compared to stable condition, but the
multiplying dimensional factor (u*) is order of magnitude
higher in former than in the latter situation.
DETERMINATION OF PLUME BOUNDARY
Here we explore the extent of pollutant cloud boundary
under the premises that: The turbulent diffusion in a
horizontally-homogeneous stationary surface layer of air
obeys the similarity theory in which the values L and *u are
the only scales of length and velocity.
The diffusion of pollution in the atmosphere is due to the
turbulent pulsations of the wind velocity. The magnitude of
these pulsations is limited (for example, it does not exceed the
sound velocity); therefore, according to Monin [14], it follows
that: Propagation of the pollution through space due to
atmospheric diffusion occurs with a limited velocity.
Accordingly, the space occupied by a smoke flowing out of
any source, has a very distinct boundary beyond which there is
no smoke. Such boundaries have been observed in
experiments conducted by Kasanski and Monin[13].
In conformity with MO similarity, the maximum velocity
of the vertical propagation of diffusing smoke is given by
*
* ( )w uλ φ ζ= …(12)
where ϕ(ζ) is a universal function, which can be subjected to
the condition ϕ (0) = 1 so that λ will be equal to *
* uw
under neutral stratification.
Kanaski and Monin [13] carried out experiment under
neutral condition from which, w* = 0.12 m/s and u* = 0.16
m/s, so that λ = 0.75.
In order to get the form of the function ϕ(ζ), let us use the
turbulent energy equation 2
0
d d
d dM H
u g TK K
z T zε
− =
…(13)
Rewriting this equation (13) in the form
2 2
0
d / d Ri1 1
(d / d ) (d / d ) Pr
H
M M M t
K g T z
K u z K T K u z
ε= − = −
and using the exchange relation, 2
*
d
dM
uu K
z=
4
*
Ri1
PrM tu K
ε= − …(14)
In accordance with similarity theory, 3*
t
w
lε ∝ and *
M tK w l∝ …(15)
where lt is the scale of turbulence.
Substituting these scaling relations and using the limiting
condition: w* ≈ 0 at Ri = Ric, we get
4
*
*
Ri Ri1 1
Pr Rit c
w
u
= − = −
As Prt can be equated to Ric, it becomes 4
*
*
Ri1
Ric
w
u
= −
…(16)
from which (using the relations (12) and (9)) 1/4 1/4
Ri 1( ) 1 1
Ri ( )c fφ ζ
ζ
= − = − ′
…(17)
When ζ is small and log-linear law of velocity profile is
valid,
( )1/4
1/41 1( ) 1 1 1
1/ 4φ ζ ζ ζ
ζ β
= − ≈ − ≈ −
+
The equations of motion of the smoke particles at the upper
boundary of the plume have the form
d
d
xu
t= and
*d
d
zw
t= …(18)
Substituting the u and w* profiles, the shape of the upper
boundary smoke plume is given by,
( )
( )0
* 1/4
/ ( / )d 1
d 1 1/ ( / )
f z L f z Lx u
z w k f z Lλ
−= =
′− …(19)
Integrating this differential relation (19) for plume
boundary, we arrive at the relation
−
=
L
z
L
hF
L
z
L
zF
k
Lx 00 ,,
λ; …(20)
where ( ){ }∫ ′−
−=
ζ
ζ
ζζ
ζζζζ
0
d)(/11
)()(,
4/1
00
f
ffF
Thus, the shape of the boundary of the smoke plume (in
particular, their inclination to horizon) does not depend upon
the wind velocity, but does depend upon the stratification of
the atmosphere.
The right hand side of relation (20) has been evaluated
through numerical integration under different atmospheric
stability conditions employing corresponding universal
functions. The numerical results are plotted in figure 4.
0 5 10 15 20
0
1
2
3
4
5
ζ0 = 0.1
ζ0 = 0.01
ζ0 = 0.001
ζ0 = - 0.001
ζ0 = - 0.01
Nondim
ensio
nal P
lum
e H
eig
ht,
ζ =
z/L
Nondimensional Distance, ξ = kλx/L
Stable Stratification
Untable Stratification
ζ0 = - 0.1
Fig.4: Plume Boundaries under Different ABL Conditions
It is obvious from figure 4 that vertical growth of the
plume boundary is much faster under unstable condition than
under stable stratifications. In the former case the growth
behavior is superlinear, whereas in the latter case it is
sublinear in nature.
CONCENTRATION PROFILE WITHIN THE PLUMES Guided by the principle of finite propagation velocity, we
are denied of the routine parabolic diffusion equation
corresponding to an infinitely rapid pollution propagation in
space. The transport equation corresponding to the limited
propagation velocity should be hyperbolic: such a hyperbolic
system of equations was obtained by Monin [8-9] in the form
0t z
χ ψ∂ ∂+ =
∂ ∂ and
**
2w
wt z
ψ χνψ
∂ ∂+ = −
∂ ∂ …(21)
where ψ is turbulent flux of the diffusing pollutant and ν is a
typical frequency of turbulent pulsations which in accordance
with the similarity principle can be written as 2 1/2* 2
*
*
1 d 1 1( ) 1
2 d 2 ( )
w u uf
u z k L f
λν ζ
ζ
′= = −
′ …(22)
The stationary solution of the hyperbolic system is given by, *
* ( )2
ww
z
χνψ
∂= −
∂
Approximating this relation as,
2
* *
d d
d d
u f
z u z ku z
χ ψ ψ∂= − = −
∂ …(23)
Integration of (23) lead to the relation,
11
*
( ) ( )z z
z z f fku L L
ψχ χ
− = − −
The full hyperbolic system can only be integrated numerically.
However, in the case of neutral stratification
*
* uw λ= and z
u
k
*
2
2
1 λν = …(24)
Then the hyperbolic system simplifies to,
0=∂
∂+
∂
∂
zt
ψχ and
zu
z
u
kt ∂
∂−=+
∂
∂ χλψ
λψ 2
*
2*
2
…(25)
The continuity and flux relation can be combined to yield
2 2
2 2t k t
ψ λ ψ ψ
ς ς
∂ ∂ ∂+ =
∂ ∂ ∂ …(26)
a telegraph equation in flux.
In terms of a transformed variable,
0( , )d
t
z t tψΨ = ∫ …(27)
The equation (26) transforms to 2 2
2 2t k t
λ
ς ς
∂ Ψ ∂Ψ ∂ Ψ+ =
∂ ∂ ∂ …(28)
Introducing a similarity variable, t/ςη = , the above
equation can be integrated to yield,
( )2 d1 const .
d k
λη
η
Ψ− − Ψ = …(29)
For instantaneous surface point source of intensity Q,
( ,0) δ( 0)z Q zψ = − …(30)
and for tutwz *
* λ=≥ , i.e., ς ≥ t or ,1≥η ψ = Ψ= 0.
Using the boundary condition, Ψ(η = 1) = 0,
( )2 d1
d k
λη
η
Ψ− = Ψ …(31)
Rearranging and carrying out further integration we obtain,
2
* *
1 1 .kz z
u t u t
λ
λ λ
Ψ ∝ − +
From continuity condition,
(0, )t QΨ =
we get.
2
* *
1 1 .kz z
Qu t u t
λ
λ λ
Ψ = − +
…(32)
From first equation of hyperbolic system,
0 0 0d d d
t t t
t t tt z z z
χ ψψ
∂ ∂ ∂ ∂Ψ= − = − = −
∂ ∂ ∂ ∂∫ ∫ ∫
or,
12
*
1* 2
*
1
( , ) .
1
k
k
z
u tQz t
ku tz
u t
λ
λ
λχ
λ
−
+
−
=
+
…(33)
which is valid for 0 ≤ tutwz *
* λ=≤ and λ ≥ 2k.
If one has a stationary linear source of pollution, perpendicular
to the wind direction, and one neglects the horizontal mixing
and change of wind velocity with height, the concentration of
the pollution is given by,
12
*
1* 2
*
1
( , )
1
k
k
uz
u xQx z
ku xuz
u x
λ
λ
λχ
λ
−
+
−
=
+
ɺ
; *0u
z xu
λ≤ ≤ …(34)
Consequently, the concentration profiles at different heights
from the surface are similar, and the maximum (surface)
concentration is inversely proportional to the distance from the
source. It is of interest to find out whether these results are at
least approximately true also in general case; i.e. under any
stratification and taking into account the change of wind
velocity with altitude. Formulating the hypothesis,
( , ) ( )( )
m
zx z x
H xχ χ ψ
=
where H(x) is the height of the smoke plume at distance x
from the source.
From the continuity condition,
0
dH
zu z Qχ =∫ ɺ
we obtain
1m
cpu Hχ ∝ ,
where cpu is the average value of the wind velocity in the
layer from z0 to H weighted by the concentration distribution
function, ψ(z/H). Taking H ∝ x and taking into consideration
that cpu vary with the distance very slowly, we obtain
m xχ ∝ .
Thus, the concentration profiles in a smoke plume at different
distances from the source are approximately similar to each
other. The maximum concentration in the smoke plume is
approximately inversely proportional to the distance from the
source.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Norm
aliz
ed C
oncentr
ation,
χku
*x/Q
Normalized Height, uz/(λu*x)
λ/k = 2.0
λ/k = 2.5
λ/k = 3.0
Fig.5: Vertical Distribution of Concentration
Figure 5 presents the vertical concentration profile for
different values constant of proportionality between the
characteristic vertical velocity and the friction velocity. The
concentrations decreases with height as observed in numerous
experiments reported in the literature. The concentration
profiles in a plume at different distances from the source are
approximately similar to each other. The maximum
concentration in the smoke plume is approximately inversely
proportional to the distance from the source.
van Ulden [17] gave an alternative method for prediction of
plume boundary under different stability conditions, based on
diffusion approximation and Businger [18] parameterization
of atmospheric boundary layer . The results predicted by this
method are plotted in figure 6.
0
1
2
3
4
5
0 5 10 15 20
ζ0 = 0.1
ζ0 = 0.01
ζ0 = 0.001
ζ0 = - 0.001
ζ0 = - 0.01
Non-Dimensional Distance, ξ = kλx/L
Nondim
ensio
nal P
lum
e H
eig
ht,
ζ =
z/L
ζ0 = - 0.1
Stable Stratification
Unstable Stratification
Fig.6: Plume Boundary Predicted by van Ulden Method
Results in Figures 4 and 6 reveal that the predictions of
hyperbolic method is comparable to that of van Ulden method.
CONCLUSIONS We have studied the behavior of atmospheric dispersion of
pollutant under different atmospheric stability conditions. The
velocity and temperature profiles in the atmospheric boundary
layer have been presented as derived from Monin-Obukhov
similarity theory. The boundary of the plume is observed to
grow faster in the case of convective condition than the stable
condition. The vertical concentration profiles of the pollutant
have been calculated. Comparison of plume boundary with
van Ulden[17] method gives a reasonable agreement.
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