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A new Lagrangian method for modelling the buoyant plume rise

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A new Lagrangian method for modelling the buoyant plume rise
Stefano Alessandrini
a
,
*
, Enrico Ferrero
b
,
c
,
1
, Domenico Anfossi
c
,
2
a
RSE, Ricerca Sistema Energetico, via Rubattino 54, 20134 Milano, Italy
b
Università del Piemonte Orientale, Dipartimento di Scienze e Innovazione Tecnologica, viale Teresa Michel 11, 15121 Alessandria, Italy
c
ISAC-CNR, Istituto di Scienze dell
’
Atmosfera e del Clima, Corso Fiume 4, 10133 Torino, Italy
h i g h l i g h t s
The paper deals with the problem of the plume rise in the dispersion model.
A new technique based on temperature difference is used for plume rise computation.
Comparison of the model with experiments shows a satisfactory agreement.
a r t i c l e i n f o
Article history:
Received 5 October 2012Received in revised form12 April 2013Accepted 26 April 2013
Keywords:
Plume risePollution dispersionLagrangian model
a b s t r a c t
A new method for the buoyant plume rise computation is proposed. Following Alessandrini and Ferrero(Phys A 388:1375
e
1387, 2009) a scalar transported by the particles and representing the temperaturedifference between the plume and the environment air is introduced. As a consequence, no more par-ticles than those inside the plume have to be released to simulate the entrainment of the background airtemperature. A second scalar, the vertical plume velocity, is assigned to each particle. In this way theentrainment is properly simulated and the plume rise is calculated from the local property of the
ow.The model has been tested against data from two laboratory experiments in neutral and stable strati
ed
ows. The comparison shows a good agreement.Then, we tested our new model against literature analytical formulae in a simple uniform neutralatmosphere, considering either the case of a single plume or the one of two plumes from adjacent stackscombining during the rising stage. Finally, a comparison of the model against an atmospheric tracerexperiment (Bull Run), characterized by vertically non-homogeneous
elds (wind velocity, temperature,velocity standard deviations and time scales), was performed. All the tests con
rmed the satisfactoryperformance of the model.
2013 Elsevier Ltd. All rights reserved.
1. Introduction
The computation of plume rise is one of the basic aspects for acorrect estimation of the transport and dispersion of airbornepollutants and for the evaluation of ground level concentration.A buoyant plume rises under the action of its initial momentumand buoyancy. It experiences a shear force at its perimeter, wheremomentum is transferred from the plume to the surrounding air,and ambient air is entrained into the plume. This phenomenon,entrainment, is responsible of the plume diameter increase, of thedecrease of its mean velocity and of the average temperature dif-ference between air and plume as well. In the
rst stage the plumealso spreads under the action of the buoyancy-generated turbu-lence but progressively the effect of ambient turbulence becomespredominant. In a calm atmosphere, plumes rise almost vertically,whereas in windy situations they bend over. In this case, the ve-locity of any plume parcel is the vector composition of horizontalwindvelocityandverticalplumevelocityinthe
rststageandthenapproaches the horizontal wind velocity.In the Eulerian dispersion models, the calculation of plume riseis based on the
uid dynamic equations, namely on the mass,momentum and energy conservation equations. A complete,exhaustive theory is not yet available. These equations are closedusing the entrainment assumption proposed by Morton et al.(1956), which prescribes that the entrainment velocity, i.e. therate at which ambient air is entrained into the plume, is propor-tional to the mean local rise velocity, i.e.:
*
Corresponding author. Tel.:
þ
39 0239924624.
E-mail addresses:
stefano.alessandrini@rse-web.it, alessandrini@rse-web.it(S. Alessandrini), enrico.ferrero@unipmn.it (E. Ferrero), d.anfossi@isac.cnr.it
(D. Anfossi).
1
Tel.:
þ
39 0131360151.
2
Tel.:
þ
39 0116606376.
Contents lists available at SciVerse ScienceDirect
Atmospheric Environment
journal homepage: www.elsevier.com/locate/atmosenv
1352-2310/$
e
see front matter
2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.atmosenv.2013.04.070
Atmospheric Environment 77 (2013) 239
e
249
d
R
d
t
¼
b
d
z
d
t
¼
u
e
(1)
where
R
is theplume radius,
z
the distancefromthe source,
b
is thedimensionless entrainment parameter and
u
e
the entrainmentvelocity; alternatively:
R
¼
b
z
(2)
which expresses the observed fact that the plume radius linearlyincreaseswiththedistancefromthesource.Thevalueof
b
hastobeempirically established: although there are slightly different sug-gestions for it, the following values are generally accepted (Briggs,1975; Hoult and Weil, 1972):
b
¼
0.6 in the bent over buoyantplumes (in which the angle
f
between the horizontal and thecentreline
f
/
0) and
b
¼
0.11 for vertical buoyant plumes(
f
/
90
). The simple entrainment assumption (eq. (1) or 2) wasgenerally used in semi-empirical analytical formulations (Briggs,1975), whereas more complex three-dimensional expressions,which account at each plume position for the effect on theentrainmentof the directions of both wind and plume, are insertedin the integral models (see, for instance:Ooms, 1972; Glendening
et al.,1984). An example is the following:
u
e
¼
b
u
s
U
a
cos
j
p
j
a
cos
f
p
f
a
þ
a
U
a
h
1
cos
2
j
p
j
a
cos
2
f
p
f
a
i
1
=
2
(3)
in which:
a
,
p
refer to air and plume, respectively,
U
a
is the windvelocity,
b
¼
0.6and
a
¼
0.11aretheentrainmentconstants,
f
and
j
are the angles that the
x
and
z
axes make with the wind (
a
) andplume (
p
) velocities.In Lagrangian Particle Models (LPM) plume rise can be dynam-icallycomputed, i.e. each particle, at each time step, can respond tolocal conditions: wind speed and direction, ambient stability andturbulence (boththe self-generated and ambientones). This allowsobtaining a high degree of resolution. Moreover, it allows simu-lating the interaction of a plumewith acapping inversion layerandthe mixture of different plumes in a
“
natural
”
way. However, theintroduction of the plume rise computation in an LPM is notstraightforward. In fact, the buoyant forces acting on the plumeportions depend on the difference between their temperature andthat of the background air. This can be computed only consideringthe entrainment phenomenon. To do that, these models shouldtake into account all the
uid simultaneously, i.e. by
lling thewhole domain with a large amount of particles. This is practicallyimpossible due to the huge amount of particles that it should benecessary to track in the computational domain. To overcome thisproblem three different
“
hybrid
”
techniques (i.e., that considertogether Lagrangian and Eulerian properties) have been proposed(WebsterandThomson,2002;AnfossiandPhysick,2005):particles
emission occurs at the
nal plume height computed by analyticalmodels (such as the Briggs, 1975; ones), an integral plume risemodel is used at each time step for each particle and the derivedvelocitiesareaddedtotheLagrangianstochasticparticlesvelocities(see, for instance, Anfossi et al., 1993) and a set of differentialequations describing the time and space evolution of bulk plumequantities are solved at each time step (Webster and Thomson,2002; Anfossi et al., 2010). An interesting method was proposed
by van Dop (1992) in which a Langevin equation for the particletemperature is solved and the buoyancy of the particle is includedintheLangevin equationfortheparticle velocityevolution.Thoughhaving advantages from the turbulence closure point of view, thisapproach has problems in the entrainment treatment.InthepresentpaperthemethodsuggestedbyAlessandrinietal.(2012) and Alessandrini and Amicarelli (2012) for the buoyant
plume rise computation is proposed again in a revised version andvalidated with further tests. It is based on the same idea describedin recent papers related with chemical reactions in Lagrangianparticle models (Alessandrini andFerrero, 2009; Alessandrini et al.,
2011) for simulating the ozone background concentration. A
cti-tiousscalartransportedbytheparticles,thetemperaturedifferencebetween the plume portions and the environment air temperature,isintroduced.Asaconsequencenomoreparticlesthanthoseinsidethe plume have to be released to simulate the entrainment of thebackground air temperature. In this way the entrainment is prop-erly simulated and the plume rise is calculated from the localpropertyof the
ow.ThisnewapproachiswhollyLagrangianinthesense that the Eulerian grid is just used to compute the property of aportionoftheplumefromtheparticlescontainedineverycell.Noequation of the bulk plume is solved on a
xed grid. Withoutconsidering the simpli
cation of the model, this method has theadvantage of being applicable also in those situations where theanalytical equation for the vertical velocity is dif
cult to de
ne, i.e.where several plumes, released by different stacks located close toeach other, mix themselves together.In the next Section the new plume rise model is presented. ThethirdSectiondealswiththemodelvalidationinfoursituations:twowater tank experiments (one in neutral and the other in stableconditions), an idealized neutral atmosphere (both single and twostacks cases) and,
nally, an atmospheric tracer experiment, BullRun (Hanna and Paine,1987, 1989).
2. The new plume rise model
The new Lagrangian plume rise module was introduced in theLagrangian stochastic particle model SPRAY (Tinarelli et al., 2000;Alessandrini et al., 2005a,b; Alessandrini and Ferrero, 2011). In the
new module each particle carries two quantities that specify thedifference between the temperature and the momentum of theplume air and the environment. To this aim we assign to any i-themitted particle in the time interval
D
t
the
“
temperature mass
”
m
T
,de
ned as follows:
m
T
i
¼
T
p
init
T
a
ð
H
s
Þ
j
w
u
j
A
D
t N
p
(4)
where
T
p
init
is the initial plume temperature,
T
a
(
H
S
) is the environ-ment air temperature at the stack height
H
s
,
A
is the stack exitsection and
N
P
is the total number of particles released in the timeinterval
D
t
and
w
u
the plume exit velocity. Note that
m
T
does nothave the dimension of a mass but can be considered a
“
mass
”
if thetemperature difference plays the role of a density. Considering thedomain divided in
xed regular cubic cells, the air-plume temper-ature difference for the generic cell,
D
T
c
, is:
D
T
c
ð
t
0
Þ ¼
P
M i
m
T
i
ð
t
0
Þ
V
c
;
(5)
where
M
is the number of particles in the cell
c
and
V
c
is the cellvolume.In order to take into account the momentum
ux we de
ne themomentum mass
m
wi
, which is assigned to each particle. Also inthis case,
m
wi
has not the dimension of a mass but it can beconsidered so when
w
u
plays the role of a density. At the beginningof the simulation we have:
S. Alessandrini et al. / Atmospheric Environment 77 (2013) 239
e
249
240
m
w
i
¼
w
u
j
w
u
j
A
D
t N
p
(6)
Thenthecellverticalvelocity
w
c
(
t
0
)atthetime
t
0
iscomputedas
w
c
ð
t
0
Þ ¼
P
M i
m
w
i
ð
t
0
Þ
V
c
(7)
The new temperature difference
D
T
C
(
t
1
) at the time
t
1
(where
t
1
¼
t
0
þ
D
t
) is calculated by the equation:
D
T
c
ð
t
1
Þ ¼
D
T
c
ð
t
0
Þ þ
G
ð
z
c
Þ
$
w
c
ð
t
0
Þ
$
D
t
0
:
0098
$
w
c
ð
t
0
Þ
$
D
t
(8)
where
z
c
is the cell height and
G
(
z
C
) is the lapse rate of the ambientair at cell height
z
c
. The second term on the right side updates thetemperature difference between the cell and the ambient consid-ering the vertical inhomogeneity of the atmosphere temperature.The third term on the right takes into account the adiabaticexpansion due to the plume ascending motion. Clearly, in case of neutral temperature pro
le, these two terms delete each other.-Equation(8)isoriginallysuggestedinthepresentpaperanditaimsat simulating the variation in time of plume-ambient air temper-ature difference due to the ascending motion.Afterwards, the value of
w
c
at the time
t
1
is computed for everycell using the following equation:
w
c
ð
t
1
Þ ¼
w
c
ð
t
0
Þ þ
D
T
c
ð
t
0
Þ
T
a
ð
z
c
Þ þ
D
T
c
ð
t
0
Þ
g
D
t
0
:
5
$
c
D
$
S
$
w
2
c
ð
t
0
Þ
$
r
a
r
p
$
V
c
$
D
t
(9)
where
T
a
(
z
c
) is the ambient air temperature at the same cellheight
z
c
,
S
the cell horizontal surface area,
c
D
the drag coef
cientand
r
a
and
r
p
are the ambient air and plume density, respectively.The second term on the right represents the buoyancy verticalacceleration while the last term on the right represents theaerodynamic drag. Equation (9) (srcinally proposed in the pre-sent paper) simulates the plume vertical ascending velocityvariation in time due to the buoyancy acceleration and theaerodynamic drag.Then, the
“
temperature difference and velocity masses
”
at thetime
t
1
,
mT
i
t
1
ð Þ
and
mw
i
t
1
ð Þ
are computed for each particlefollowing the two equations:
m
T
i
ð
t
1
Þ
m
T
i
ð
t
0
Þ
$
D
T
c
ð
t
1
Þ
D
T
c
ð
t
0
Þ
(10)
m
w
i
ð
t
1
Þ ¼
m
w
i
ð
t
0
Þ
$
w
c
ð
t
1
Þ
w
c
ð
t
0
Þ
(11)
ThismethodwasproposedbyChockandWinkler(1994a,b)andapplied for a different purpose. In fact, in their papers, the masseswererepresentingtheactualmassesof differentsubstancescarriedby the particles in a chemically reactive plume. In our algorithmthey carry the information, for each particle, relative to the twoscalars introduced, the difference between the plume and envi-ronment temperatures and the vertical momentum. The wholealgorithm, for sake of clearness, can be summarized with thefollowing sequential steps.1) Attheemission(
t
¼
t
0
),twoscalarsareassignedtoeachparticleaccording to equations (4) and (6).2) For every cell
c
of the Eulerian grid the vertical velocity
w
c
(
t
0
)and temperature difference
D
T
c
(
t
0
) (between the plume andbackground air) at the time
t
0
are computed by equations (5)and (7).3) For every cell
c
again, the vertical velocity
w
c
(
t
1
) and temper-ature difference
D
T
c
(
t
1
) are updated at the time
t
1
¼
t
0
þ
D
t
byequations (9) and (8).4) At time
t
1
the two scalars are updated by equations (10) and(11) for each i-th particle.5) Finally the set of three dimensional Langevin equations forparticle velocity
uctuation normally used by the modelindependently from the plume rise scheme are used.6) The vertical Lagrangian particle displacement,
D
z, is
nallycomputed multiplying by the time step
D
t
the sum of the windvelocity component (if different from zero), the stochastic
uctuation and the buoyancy vertical velocity,
w
c
(
t
1
),computed at the 3rd step for every i-th particles in everycell
c
.To compute the horizontal Lagrangian particle displacement(
D
x
,
D
y
)thesumofthewindvelocitycomponentandstochastic
uctuation is considered only.Subsequently, at the time
t
1
, thealgorithm can be repeatedfromthe step n
2 for the particles alreadyemitted and from the step n
1for the new released particles.Regarding the choice of the drag coef
cient
c
D
in eq. (8) we didnot
nd a generally accepted value in the literature. As a matter of fact, Webster and Thomson (2002) in their model set this value to0.21.Ooms (1972)proposedinhisintegralmodelfortheplumerise
c
D
¼
0.3 on the basis of both theoretical considerations and theresults of only one experiment. Other suggestions can be found butin other applications like the rise of updraft or cumulus clouds.Evenifthereisnotauniqueanswer,variablesvaluesfor
c
D
between0.045 and 0.5 depending on the vertical velocity intensity and onthe Reynold number of the
uid (Bunker, 1952; Kon, 1981), are
suggested. Consequently, we considered
c
D
as an adjustableparameter.
3. Model validation
In order to validate the model four tests were carried out. Intwo of them the model was tested against water tank experiment(Huq and Stewart, 1996; Contini et al., 2011). In the third one the
rise of both a single stack and two stacks plumes in an idealizedneutral atmosphere was considered and,
nally, in the fourth onea tracer dispersion in the real atmosphere (Bull Run, Hanna andPaine, 1987) was simulated. While in the
rst three cases theplume trajectory during the rising phase was simulated, in thefourth one the observed and prescribed tracer ground level con-centrations were accounted for.In the
rst water tank experiment (Huq and Stewart, 1996) aneutral strati
ed turbulent
ow was reproduced and the buoyantplume dispersion from a point source was measured and ana-lysed. In the second one (Contini et al., 2011), the dispersion froma point source was studied in a stably strati
ed
uid at rest and arelative velocity was created by moving the source itself bymeans of a towing trolley. It can be noted that, in this case, theturbulence is not a condition of the background
ow but ismerely self-induced by the plume. This is an extremely idealizedcase. In particular, being in absence of any background turbu-lence, it may be useful to test the effect of the self-generatedturbulence of the plume and hence the internal mechanisms of the plume rise alone, without the in
uence of other externalfactors. In order to reproduce such idealized case we set theturbulence parameters of the model to a minimum valuecompatible with the stochastic differential equations. Moreover,as it is well known, the stable case is an important situation forthe pollutant dispersion.
S. Alessandrini et al. / Atmospheric Environment 77 (2013) 239
e
249
241
3.1. Water tank experiment of Huq and Stewart (1996), (neutralconditions)
Different experiments were carried out characterized bydifferent values of the
ux of buoyancy (
F
B
) and the
ux of mo-mentum (
F
M
):
F
B
¼
g
0
Q F
M
¼
Qw
s
r
s
r
a
with
Q
the source volume
ux given by
Q
¼
p
b
2
w
s
and
g
0
¼
g
r
s
r
a
r
s
where
w
s
is the source exit velocity,
b
¼
0.36 cm the source radius,
r
s
the plume density,
r
a
the water density.The buoyancy length scale is given in terms of the source con-ditions as
l
b
¼
F
B
u
3
and the Froude number is
Fr
¼
w
s
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
bg
0
p
Out of the 9 water tank experiments reported in the paper, wehave chosen three of them, which are representative for thedifferent values of
F
b
and
F
m
available. The main parameters of these experiments are listed in Table 1.The experiments were simulated with the Lagrangian modelcalculating the scaling factors based on the Froude number simili-tude and on the following hypothesis:
D
qq
¼
D
rr
and
H
m
H
¼
1000
where
H
m
H
1
(the subscript
m
refers to the model) represents theratio between model and water tunnel experiment length scalesrespectively and
q
the potential temperature. The Froude numberconservation gives the following scaling factors for velocity andtime:
u
m
u
¼
t
m
t
¼
10
1
:
5
The simulation parameters obtained applying this scaling arereported in Table 2, where
u
is the
ow velocity,
w
s
is the verticalvelocity of the plume at the stack exit and
q
p
is the potential tem-perature of the plume and
q
a
the one of the ambient air.The simulations have been carried out setting a
xed Euleriangrid for the computations of
D
T
c
(eq. (5)) and
w
c
(
t
0
) (eq. (7)). Thedimension of the cells has been set equal to 3
3
3 m
3
while theLagrangian particle displacement time step equal to 1 s. The choiceof the cell size was suggested by the stack diameter dimensionwhichis,inthiscase,equalto3.6m.Clearlyachoiceoflargervalueswould cause a non-realistic dilution of the plume in the
rst timesteps after the emission and an underestimation of plume tem-perature and momentum and, consequently, of vertical plume rise.The values of standard deviation of velocities and Lagrangian timescales (which are local decorrelation time scales) have been setvertically homogeneous and not homogeneous along the wind di-rection to reproduce the wind tunnel conditions. In particular,equations (6) and (7) of Huq and Stewart (1996) provide the
experimental trends for the standard deviation of velocities valuesandTKE(TurbulentKineticEnergy)dissipationrate.TheLagrangiantimescaleswerethenobtainedbythefollowingequation:
T
L
¼
s
2
/
3
,where
T
L
is the Lagrangian time scales,
s
is the standard deviationof velocities and
3
is the TKE dissipation rate.Twoseriesofsimulationshavebeenperformedsettingthevalueof
c
D
equal to 0.3 and 0.5 respectively. The results are shown inFigs.1 and 2.
The experimental measurements indicated in the previous
g-ures refer to Fig. 4 of Huq and Stewart (1996) in which all the ex-
periments with different buoyancy and momentum
uxes areplotted together. Furthermore, these experiments were conductedboth in laminar and turbulent
ows, while we considered only thecase with turbulence as it represents a situation closer to the realatmosphere.Allthethreesimulationsshowtwodifferenttrendsforshort distance and large distance from the source. In the
rst casethe results approximately follow the 1/3 power law slope, while atfarthest distances they approach the 2/3 power law slope. Thesetwoasymptotictrendsarethoseindicatedbythewell knowncurvegiven by the Briggs (1975) formula for neutral thermal conditions:
D
h
ð
t
Þ ¼
30
:
6
2
F
M
t u
þ
32
,
0
:
6
2
F
B
t
2
u
1
=
3
(12)
where
u
is the horizontal wind velocity and
t
the time from therelease.Thechoiceof
c
D
¼
0.3seemstoproduceabetteragreementwiththe measurements especially in the two runs number 3 and 9 andfor larger distances from the source. In fact, the simulations with
c
D
¼
0.5 show an underestimation of the vertical rise when
x
/
l
b
isapproximately around 1000. Run 6 underestimates the measure-ments for low
x
/
l
b
values and for both the
c
D
choices. For larger
x
/
l
b
values the behaviour of the simulated data seems to be correct. Itreproduces rather well the 2/3 power law trend.Itcanbenotedthat,consideringthe
rststageaftertheemissionof the plume ascending motion, the momentum
ux should bepredominant, it is not likely that measured plume heights, charac-terizedbydifferentbuoyancy
uxes,shouldcollapsetogetherwhen
Table 1
Huq and Stewart (1996) water tank experiments parameters for the three chosen experiments.Run
u
(cms
1
)
w
s
(cms
1
)
Q
(cm
3
s
1
)
D
r
/
r
F
b
(cm
4
s
3
)
F
m
(cm
4
s
2
)
l
B
(cm) Fr3 8.3 25 2.62 0.0143 36.66 66.34 0.0641 11.136 7.8 30 3.14 0.0658 202.71 199.81 0.4272 6.229 8.3 20 2.09 0.0155 31.80 31.80 0.0556 8.5
Table 2
Simulations parameters for the Huq and Stewart (1996) case.Run
u
(ms
1
)
w
s
(ms
1
)
q
p
(K)
q
a
(K)3 2.62 7.9 304.4 3006 2.47 9.5 322.7 3009 2.62 6.3 304.7 300
S. Alessandrini et al. / Atmospheric Environment 77 (2013) 239
e
249
242
normalizeby
l
b
.Thismaybethereasonforwhichwefounddifferentbehavioursintheresultsoftherun6withrespecttotheonesofruns3and9.As amatterof fact,these last two experimentshavesimilarvalues for
l
b
, which is much larger in the run 6 (see Table 1).
3.2. Water tank experiment of Contini et al. (2011) (stableconditions)
TheContinietal.(2011)experimentswerecarriedoutinawatertank where the
uid was at rest and the source was moving. Thewater was stably strati
ed.In order to test our model in stable conditions, we consideredthreedifferentcases,Runs1,15and16,whoseparametersarelistedin Table 3
r
a
indicates the cross-
ow density at the stack height,
r
s
the emission density, the densitygradient
grad(
r
)
characterizes thelinear strati
cation of the stable cross
ow, Q is
ow-rate and
D
isthe internal diameter of the stack.In order to perform the numerical simulation of the three caseslisted in Table 3, we rescaled the experiment parameters in thesame way as in case of the Huq and Stewart (1996) data. The pa-rameters used for the simulations are summarized in Table 4,where
q
HS
is the air potential temperature at the stack height (
H
s
),
Dq
is the temperature difference between plume and the ambientair,
U
s
is the relative
ow velocity and
D
s
is the stack diameter.Asinthe previouscase, the simulationshave been carriedout bysettinga
xedEuleriangridforthecomputationsof
D
T
c
(eq.(5))and
w
c
(
t
0
) (eq. (7)). The dimension of the cells has been set equal to7
7
7 m
3
according to the source diameter, and the Lagrangianparticle displacement time step was equal to 1 s. In this case weconsidered
c
D
¼
0.3 and
c
D
¼
0.5 as done for the Huq and Stewart(1996) simulations. As already mentioned, even if the ambient
uidis in laminar condition, there is a presence of turbulence self-generated by the motion of the plume. These conditions cannot befound in the real atmosphere. On the contrary, our model needsturbulence parameters as an input for the particles velocities in theLangevin equations. To consider that, we have set the standard de-viationsofvelocitiesandLagrangiantimescalestoaminimumvalue,compatible with the stochastic differential equations and constantinside the computational domain, equal to 0.2 ms
1
and 250 srespectively. It can be noted that small variations of these values donot produce remarkable effects on the plume centreline values.In Fig. 3 a,b,c, the plume centreline height as a function of thetime is depicted for the three considered experiments. Bothsimulation results with
c
D
¼
0.3,
c
D
¼
0.5 and the Contini et al.(2011) measurements are shown together with the
nal plumerise
ð
D
hF
in
Þ
given by the Briggs equation (1975):
D
h
F
in
¼
2
:
6
F
B
u
$
s
1
=
3
(13)
where
s
is the stability parameter. In the run 15 the measurementsare quite well reproduced by both the simulations with a slightoverestimation with
c
D
¼
0.3 and a slight underestimation with
c
D
¼
0.5.As it is well known, in stable conditions plumes generallyovershoot their equilibrium heights and display a quickly dampedoscillation. The presence of descending motion appearing in theexperiments is more evident in the two simulations, which canshow data for a longer time from the release. In run 1 there is aslight overestimate of the equilibrium height in both simulatedplumeswhileinrun16theequilibriumheightofexperimentaldatais well reproduced by both simulations. The descending motionexhibited by the measurements after the
rst maximum in the run1 and 16 is only slightly hinted by the model results in run 1. Thiscan be due to numerical problems. In fact, the limited cell dimen-sion is not able to catch the small temperature differences betweenambient and plume far from the release point wherethe dilution ishigh. In other simulations (not shown here), performed setting thecell size at 30 m, we observed a plume oscillation, around theequilibrium level, as in the experiment. However, in these simu-lations, due to a too slow ascent velocity in the
rst step after therelease, there is a clear underestimation of the
nal measuredplume height. This is caused, as already mentioned, by an arti
cialplume dilution due to the cell dimension, which is, in this case,greater than the source diameter. Clearly an optimal choice couldbe an adaptive grid enlarging its cells as the plume broadness in-creases, but this possibility has not been tested in this paper.
Fig. 2.
As in Fig. 1 but for
c
D
equal to 0.5.
Fig. 1.
Normalized elevation of the plume centreline as a function of the normalizeddownwind distance from the source. Triangles indicate the measurements from Huqand Stewart (1996) in turbulent conditions only. The coloured symbols refer to thethree simulations of the different experiments. The two continuous black lines indicatethe 1/3 and 2/3 power-law slopes.
c
D
was set equal to 0.3.
S. Alessandrini et al. / Atmospheric Environment 77 (2013) 239
e
249
243

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