Development of a grid-dispersion model in a large-eddy-simulation--generated planetary boundary layer

Abstract

Numerical simulations of dispersion experiments within the planetary
boundary layer are actually feasible making use of Large Eddy
Simulations (LES). In Eulerian framework, a conservation equation for a
passive scalar may be superimposed on LES wind/turbulence fields to get
a realistic description of time-varying concentration field. Aim of this
work is to present a numerical technique to solve the Eulerian
conservation equation. The technique is based on Fractional Step/Locally
One-Dimensional (LOD) methods. Advection terms are calculated with a
semi-Lagrangian cubic-spline technique, while diffusive terms are
calculated with Crank-Nicholson implicit scheme. To test the grid model,
the dispersion of contaminants emitted from an elevated continuous point
source in a convective boundary layer is simulated. Results show that
the calculated concentration distributions agree quite well with
numerical and experimental data found in the literature.

Full text

IL NUOVO CIMENTO Vol. 26 C, N. 3 Maggio-Giugno 2003
Development of a grid-dispersion model
in a large-eddy-simulation–generated planetary boundary layer
U. Rizza(1), G. Gioia(1)(2),C.Mangia(1)andG. P. Marra(1)(3)
(1)CNR, I SAC, S ezi one di Lecce - Lecce, I taly
(2)Dipartimento di Scienze dei Materiali, Universit`a di Lecce - Lecce, It aly
(3)Dipartimento di Ingegneria dell’Innovazione, Universit`a di Lecce - Lecce, It aly
(ricevuto il 29 Ottobre 2002; approvato l’11 Aprile 2003)
Summary. — Numerical simulations of dispersion experiments within the plan-
etary boundary layer are actually feasible making use of Large Eddy Simulations
(LES). In Eulerian framework, a conservation equation for a passive scalar may be
superimposed on LES wind/turbulence fields to get a realistic description of time-
varying concentration field. Aim of this work is to present a numerical technique
to solve the Eulerian conservation equation. The technique is based on Fractional
Step/Locally One-Dimensional (LOD) methods. Advection terms are calculated
with a semi-Lagrangian cubic-spline technique, while diffusive terms are calculated
with Crank-Nicholson implicit scheme. To test the grid model, the dispersion of con-
taminants emitted from an elevated continuous point source in a convective bound-
ary layer is simulated. Results show that the calculated concentration distributions
agree quite well with numerical and experimental data found in the literature.
PAC S 92.60.Ek – Convection, turbulence, and diffusion.
PAC S 92.60.Sz – Air quality and air pollution.
1. – Introduction
Large eddy simulation has become a well-established technique to study the 3D tur-
bulent characteristics of Planetary Boundary Layer (PBL) [1]. The physical basis of LES
is the separation of the flow into large-scale motions which contain most of energy and
are dependent on the flow environment, and small-scale motions which are believed to be
more universal in character. The proper distinction between flow scales is accomplished
by applying to the Navier-Stokes equations a high-pass filter with a cut-off length ∆f.
This means that the small scales are parameterized, while the large ones are solved ex-
plicitly by computing numerically the filtered equations. It is clear that LES will provide
excellent simulations when the cut-off length ∆fis much smaller of the scale of Energy
Containing Eddies (ECE) of the turbulence.
The velocity and turbulence fields provided by LES may be used to calculate the
transport and dispersion of contaminants. In this way one would obtain, at the same
c
Societ`a Italiana di Fisica 297

298 U. RIZZA, G. GIOIA, C. MANGIA and G. P. MARRA
time, detailed dispersion data and complete information on meteorological and turbulent
parameters. Several studies with LES for atmospheric dispersion have been reported
in the literature and concern both Eulerian and Lagrangian dispersion simulation ap-
proaches [2-5]. Providing a detailed knowledge of turbulence fields, Lagrangian particle
models are very successful in describing turbulent dispersion of passive contaminants
because they are able to take into account essential aspects of turbulence, but they are
limited to simplified set of reacting species. On the other side, Eulerian approach being
based on conservation equation, can incorporate the numerous second- and high-order
chemical kinetic equations necessary to describe photochemical smog generation, which
represents, presently, a challenging problem. The critical point is the numerical scheme
for the conservation equation which can generate nonphysical results.
Aim of this paper is to present a numerical technique to solve the Eulerian conserva-
tion equation using the gridded wind/turbulence field generated by Moeng’s LES code [6].
The technique is based on Fractional Step/Locally One-Dimensional methods [7-9]. Ad-
vection terms are calculated with a semi-Lagrangian cubic-spline technique [10], while
diffusive terms are calculated with Crank-Nicholson implicit scheme. To avoid unwanted
numerical noises produced by the advection numerical scheme, a non-linear filter [11] is
used. The main advantage of this scheme with respect to those ones involved in most
used air pollution grid models [12], consists in its better adaptability to domains with
irregular grid-spacing.
To test the grid model, the dispersion processes of passive contaminants emitted from
an elevated source in a convective boundary layer are simulated. This represents the
first step before to extend the technique to more complex cases and to reactive species.
Simulation results are compared with laboratory data of Willis and Deardoff [13, 14].
2. – The large eddy model
The large eddy model used in this study is the same as described by Moeng [6] with
the further developments described in [1,15]. Here, we will only give a short outcome.
In the LES technique, the smallest eddies in a large Reynolds number PBL flow are
removed by applying a spatial filter function to the Navier-Stokes equations [16]. For
each turbulent quantity f, the filtered (or resolved) variable, denoted by an overbar, is
defined as
(1) f(x,t)=
D
f(y,t)G(x−y)dy=
D
f(x−y,t)G(y)dy.
Integration is over the flow volume D. The function Gis a three-dimensional low-pass
filter that removes the subgrid scale fluctuations (or small eddies: f(x,t)).
Applying the filtering operator to the incompressible Navier-Stokes equations and
making the substitution f(x,t)=f(x,t)+f(x,t), we obtain the governing equations
for the filtered variables:
(2) ∂ui
∂t +∂uiuj
∂xj
=−1
ρ
∂p
∂xi
−∂Tij
∂xj
+ν∂2ui
∂xj∂xj
+gi
θ
θ0
−2εijk Ωjuk,
where i≡(x, y, z), ρis the density, pis the pressure term, ν=µ/ρ is the kinematic
viscosity, µis the dynamic viscosity, the gravitational acceleration giis non-zero only
in the zdirection, θis the virtual potential temperature, θ0is the temperature of some

DEVELOPMENT OF A GRID-DISPERSION MODEL ETC. 299
reference state, εijk is the permutation tensor and Ωjis the angular vector of the earth’s
rotation. The terms
(3) Tij =u
iuj+uiu
j+u
iu
j
are the subgrid scale fluxes which represent the effect of the subgrid scale on the resolved
field.
The tensor Tij is modeled following the Smagorinsky-Lilly hypothesis [17, 18]
(4) Tij =−KM∂ui
∂xj
+∂uj
∂xi.
The eddy viscosity coefficient KMis expressed as
(5) KM=Ckl(e1/2),
where Ckis a diffusion coefficient to be determined, lis the mixing length and eis the
subgrid turbulent energy.
For unstable stratification, lis related to the grid size l=∆=(∆x∆y∆z)1/3, while
for a stable temperature stratification, the length scale is related to the Brunt-Vaisala
frequency [19]
(6) l=0.76e1/2g
θ0
∂θ
∂z.
The SGS energy eis determined by the following prognostic equation:
(7) ∂e
∂t +ui
∂e
∂xj
=P+B−ε+D,
where the different terms on the right-hand side are shear production P, buoyancy B,
dissipation εand diffusion D[1].
3. – Eulerian dispersion of a passive contaminant
The filtered conservation equation for a generic scalar Cis given by
(8) ∂C
∂t =−∂uiC
∂xi
−∂τci
∂xi
+SQ,
where SQ=Qδ(x)δ(y)δ(z−HS) is the source term, where Qis the rate emission, HSis
the source height, τci are the SGS turbulent scalar fluxes. The closure model for τci is
(9) τci =−KC
∂C
∂xi
,
where KCis the eddy diffusivity for a scalar quantity. Introducing the SGS Schmidt
number
(10) Sc=KM
KC
,

300 U. RIZZA, G. GIOIA, C. MANGIA and G. P. MARRA
we can express the eddy diffusivity in terms of eddy viscosity:
(11) τci =−KM
Sc
∂C
∂xi
.
Substitution of eq. (11) into eq. (8) leads to
(12) ∂C
∂t =−∂uiC
∂xi
+1
Sc
∂
∂xiKM
∂C
∂xi+SQ.
Following [20] we assumed Sc=0.33.
The filtered wind components uiand the SGS eddy diffusivity KMin each grid point
are provided by LES.
4. – Numerical method
Numerical solution of eq. (12) is not trivial because of numerical diffusion and conse-
quent generation of non-physical results.
The numerical method consists in splitting eq. (12) into a set of time-dependent
equations, each one Locally One-Dimensional (LOD) [7], [8] and [9]:
(13) ∂C
∂t =
3

i=1
ΛiC,
where Λiis the sum of the advective and diffusive operators
(14) Λi=Ai+Di≡−ui
∂
∂xi
+1
Sc
∂
∂xiKM
∂
∂xi.
Using Crank-Nicholson time integration we have
(15) Cn+1 =
3

j=1 I−∆t
2Λj−1I+∆t
2ΛjCn=
3

j=1
Tn
jCn,
where Iis the unity matrix.
To obtain second-order accuracy, it is necessary to reverse the order of the operators
at each alternate step to cancel the two non-commuting terms. Therefore, we replace the
scheme (15) with the following double-sequence equations:
Cn=
3

j=1
Tn
jCn−1,(16a)
Cn+1 =
1

j=3
Tn
jCn.(16b)
In order to develop a scheme that preserves peaks, retain positive quantities, and
does not severely diffuse sharp gradients, after each advective step a filtering procedure

DEVELOPMENT OF A GRID-DISPERSION MODEL ETC. 301
is applied. This is necessary for damping out the small-scale perturbations before they
can corrupt the basic solution.
So, the effective system utilized is the following:
Cn=
3

i=1
[AiFD
i]Cn−1,(17)
Cn+1 =
3

i=1
[DiAiF]Cn,(18)
where the operator Frepresents the filter operation described in [11].
The advective terms (operators Ai), which are usually the most difficult to implement,
are solved using a method based on cubic-spline interpolations, while a Crank-Nicholson
implicit scheme is used for the diffusive terms (operators Di).
4.1. Details of the fractional step/locally one-dimensional method. – The finite dif-
ference algorithm for eq. (17) (or its reverse eq. (18)) contains three steps, one for each
direction. In the following we only show the numerical scheme for the xdirection, the
scheme being the same for the other directions with the appropriate boundary conditions.
We use the notation j∈[1,N
x],k ∈[1,N
y],m∈[1,N
z] for increments in the (x, y, z)
Cartesian space, so we have
xj=x0+j∆x,
yk=y0+k∆y,
zm=z0+m∆z,
where ∆x,∆y,∆zare the grid sizes and (Nx,N
y,N
z) are the number of grid points
along the x,y,zdirections, respectively.
For each direction, the scheme (eqs. (17) and (18)) contains three sub-steps: a) the
advective part, b) the filtering procedure, c) the diffusive part.
a) The advection is computed using a quasi-Lagrangian cubic-splines method [10,21],
so for the operator Axwe have
Ch+1
2
j,k,m =Sh(xj−α∆x),if uh
j,k,m 0,(19)
Ch+1
2
j,k,m =Sh(xj+α∆x),if uh
j,k,m <0,(20)
with α=uh
j,k,m ∆t
∆x, the superscript hdenotes an intermediate fictitious time step between
nand n+ 1. This is called Fractional Steps (FS) technique.
The interpolation function (cubic-spline) Smay be expressed in terms of the spline
derivatives Ph
j=∂C
∂x h
j,k,m as
Sh(x)=Ph
j−1
(xj−x)2(x−xj−1)
h2
j
−Ph
j
(x−xj−1)2(xj−x)
h2
j
+(21)
+Ch
j−1
(xj−x)2[2 (x−xj−1)+hj]
h3
j
+Ch
j
(x−xj−1)2[2 (xj−x)+hj]
h3
j
,

302 U. RIZZA, G. GIOIA, C. MANGIA and G. P. MARRA
where hj=xj−xj−1[22].
The spline derivatives are obtained by solving the tridiagonal algebric system [22,23]:
(22) 1
2Ph
j−1+2Ph
j+1
2Ph
j+1 =3
2∆xCh
j−Ch
j−1+3
2∆xCh
j+1 −Ch
j.
Using eqs. (21), (22), eqs. (19), (20) become, respectively,
Ch+1
2
j,k,m =Ch
j,k,m −Ph
jhjα+(23a)
+Ph
j−1hj+2Ph
jhj+3Ch
j−1,k,m −Ch
j,k,mα2−
−Ph
j−1hj+Ph
jhj+2Ch
j−1,k,m −Ch
j,k,mα3,
Ch+1
2
j,k,m =Ch
j,k,m +Ph
jhj+1α−(23b)
−Ph
j+1hj+1 +2Ph
jhj+1 +3Ch
j,k,m −Ch
j+1,k,mα2+
+Ph
jhj+1 +Ph
j+1hj+1 +2Ch
j,k,m −Ch
j+1,k,mα3.
b) After each advective step, a filter operation is applied to the intermediate field to
remove any negative concentration which is usually produced by the advection
Ch+1
2
j,k,m =FCh+1
2
j,k,m.
c) Finally, the diffusive step (operator Dx) is computed by means of Crank-Nicholson
implicit scheme
Ch+1
j,k,m −Ch+1
2
j,k,m
∆t=1
2Sc


K
MCh+1
2
j+1,k,m −Ch+1
2
j,k,m−K
MCh+1
2
j,k,m −Ch+1
2
j−1,k,m
(∆x)2

+(24)
+1
2Sc

K
MCh+1
j+1,k,m −Ch+1
j,k,m−K
MCh+1
j,k,m −Ch+1
j−1,k,m
(∆x)2
,
where K
M=KM(xj+1/2), K
M=KM(xj−1/2).
4.2. Boundary conditions. – Equations (23a), (23b) and (24) are the numerical schemes
for advective/diffusive terms in xdirection. As we said, similar schemes are obtained for
yand zdirections with the appropriate boundary conditions.
Boundary conditions for the advective terms
We assume, for every time step, zero gradient boundary conditions and zero outflow

DEVELOPMENT OF A GRID-DISPERSION MODEL ETC. 303
boundary conditions, i.e.
for j=1: Ph
0=0,
Ch
0,k,m =0;
(25a)
for j=Nx:Ph
Nx=0,
Ch
Nx+1,k,m =0.
(25b)
Analogous conditions are applied along yand zdirections.
Boundary conditions for the diffusion terms
We assumed, for every time step, the zero outflow boundary conditions, i.e.
for j=1: Ch+1
0,k,m =Ch+1
2
0,k,m =0,(26a)
for j=Nx:Ch+1
Nx+1,k,m =Ch+1
2
Nx+1,k,m =0.(26b)
Analogous conditions are applied along ydirection.
For diffusion along zdirection, the boundary conditions are the zero gradient condi-
tions, that is
for m=1: Ch
j,k,0=Ch
j,k,1,(27a)
for m=Nz:Ch
j,k,Nz+1 =Ch
j,k,Nz.(27b)
5. – Simulations
In order to test the grid model described, a dispersion experiment in a convective
turbulent regime is simulated. The procedure is the following. First we run the large
eddy model for a sufficient time period to reach a quasi-stationary turbulence field, then
we inject a contaminant into the numerical domain and integrate simultaneously the
large eddy model and the conservation equation.
5.1. Experiment characteristics . – It is well known that turbulence within the PBL
is generated by two main forcing mechanisms: mechanical and convective. The former
is related to wind shear, and is governed by geostrophic wind. The latter is directly
related to the surface heat flux and it is generally responsible for convective transport of
momentum, heat and any other scalar.
Our simulation concerns a convective PBL. The calculations are performed in a rect-
angular domain arranged in a way that it could comprise several updrafts at a given
time. The dimensions of the box are 10 ×10 km in the horizontal directions and 2 km
in the vertical direction.
The resolution is 128 grid points in xand ydirections, 96 grid points in zdirection.
The simulation started from a laminar flow, with the geostrophic wind constant through-
out the whole numerical domain. Turbulence is generated by heating the surface at a
constant rate. External parameters—extension of domain , grid size, geostrophic winds,
surface heat flux, initial capping inversion height—are summarised in table I. Time
step is set equal to 2 seconds. After some time a layer with quasi-stationary convective
turbulence establishes itself. This condition is obtained after the LES run for 5000 time

304 U. RIZZA, G. GIOIA, C. MANGIA and G. P. MARRA
Table I. – External simulation parameters.
Mesh grid points Domain size Geostrophic wind Surface heat Initial inversion
flux height
(Nx,N
y,N
z)Lx,L
y,L
z(km) (Ug,Vg) (m/s) Q∗(ms−1K) (zi)0(m)
(128,128,96) (10,10,2) (10,0) 0.24 1000
steps (more than two hours of real simulated time). This represents our initial conditions
(t= 0) for dispersion experiment. The micrometeorological parameters are indicated in
table II.
At t= 0, a contaminant is injected from an elevated point source placed at half height
of the boundary layer height (HS=Hmix/2) into the box domain. The contaminant
is introduced every two time steps to satisfy the double sequence scheme (eqs. (16a),
(16b)). After the initial time, we continue the integration of the large eddy model which
is subjected to the same heating rate. The evolution of the point source is calculated
simultaneously by solving the conservation equation for the scalar.
5.2. Test of the advection-algorithm. – It is well known that any numerical scheme
which calculates the concentration should satisfy the following requirements [2]:
1) conservation of mass;
2) positive definiteness;
3) maintenance of sharp gradients.
The importance of the first two requirements is clear. In fact, it is fundamental for
any dispersion simulation that the mass of contaminant should keep a constant value
during the simulation in order to satisfy conservation principles. Moreover, since the
concentration is a positive definite variable it is clear that its value cannot go below zero.
Concerning the third point, since during the initial phase of a plume/puff dispersion the
instantaneous concentration fields are characterized by sharp concentration gradients,
each simulation of plumes or puffs has to maintain such gradients. This is, in general,
difficult to obtain, because the initial resolution is often inadequate. Moreover, it is well
known that any advection scheme may suffers from severe phase errors, which displace
the computed concentration peaks from their true locations, and/or artificial diffusion,
which smears out the concentration peaks [24].
To test the algorithm capability to satisfy the three requirements, we advected a cube
of passive contaminant by a constant wind only along xdirection [10].
In particular, to assess the conservation of mass we used the mass conservation ra-
tio [24], mcr, which is the ratio between the total mass in a generic instant and the initial
Table II. – Internal simulation parameters. ustar is the friction velocity, wstar is the free
convective velocity and LMO is the Monin-Obukhov length.
ustar (m/s) wstar (m/s) Hmix/LMO Hmix (m)
0.7 2.1 −18 1100

DEVELOPMENT OF A GRID-DISPERSION MODEL ETC. 305
0 200 400 600 800 1000
0,0
0,2
0,4
0,6
0,8
1,0
mass conservation ratio (mcr)
dispersion simulation time step (s)
Fig. 1. – Mass conservation ratio during the dispersion simulation.
mass
(28) mcr =
j,k,i
Cj,k,i (t)
j,k,i
Cj,k,i (0) .
The result of the advection test is showed in fig. 1. It is evident that the numerical
scheme conserves mass with high precision: the mcr keeps unit value until the contami-
nant is inside the box domain, it decreases when the contaminant crosses the boundary.
Figures 2a and 2b show the crosswind concentration distribution before and after
the the application of the filter. It can be seen that the filtering procedure assures the
positive definiteness of the concentration field.
The third requirement, i.e. the maintenance of sharp gradients, is strictly related to
the concentration distribution. In table III the (x, z )-locations of crosswind-integrated
peak concentration are summarised: it is evident as during the advection the variation
of maximum concentration is around 10% with respect to the maximum value.
Based on the test results, we conclude that the implemented numerical scheme works
quite satisfactory.
Table III. – Location and intensity of concentration peaks during advection.
X(m) Z(m) Cpeak (µg/m3)
1927 187.5 2325
3490 187.5 2603
4948 187.5 2508

306 U. RIZZA, G. GIOIA, C. MANGIA and G. P. MARRA
Fig. 2. – Concentration distribution before the filtering application (a) and after (b).
5.3. Dispersion simulation and results . – The main characteristics of passive plume
dispersion in the CBL have been demonstrated through numerical predictions [25-27],
field observations [28,29], and in a more detailed way through laboratory experiments
of [13] and [14] in late 70s. All these studies showed that in the CBL, the plume centreline,
defined as the locus of maximum concentration, from an elevated source descends until
it reaches the ground, it remains there for some distance and then it rises. The descent
of the plume centreline is due to the size, long life, and organized nature of the down-
drafts characteristics of the CBL. Because of the greater areal coverage of downdrafts
the probability of material to be released into them is higher.
In our experiment, the contaminant is injected from an elevated point, located at
HS=Hmix/2, all along the duration of simulation (release time = 5000 time steps) each
two time steps. The travel time, which is a rough estimate of how long the plume takes
to cross the longitudinal domain (10 km), is about 200 time steps. To get a proper plume
behaviour of PBL dispersion, sampling and release time should be greater than travel
time. In order to satisfy this constrain, we choose a concentration averaging time of 4000
time steps.
Let us introduce the crosswind–integrated concentration by
Cy=1
∆T
∆T
Ly
C(x, y, z)dydt,
∆Tis an arbitrary average interval. This concentration is made dimensionless by dividing

DEVELOPMENT OF A GRID-DISPERSION MODEL ETC. 307
Fig. 3. – Crosswind-integrated concentration averaged between last 3000 time steps, as a function
of dimensionless height and downwind distance.
by Q/UHmix , where Uis the mean longitudinal wind velocity. Figure 3 shows the
adimensional crosswind-integrated concentration vs. the non-dimensional distance X∗,
defined by x/U
Hmix/w∗.
Figure 4 shows the ground-level concentration vs. the adimensional distance compared
with Willis and Deardoff [13, 14] data.
It is evident by the figures that dispersion model reproduces the main characteristics
of the plume dispersion. The greatest concentration is found along a descending path
from the source. There is a good agreement both for the value and the position of
maximum concentration.
6. – Conclusions
Large Eddy Simulation represents nowadays a very powerful method in calculating
3D turbulent structures which are fundamental for describing any dispersion phenomena
in the planetary boundary layer.
In this work we utilised LES for modelling Eulerian dispersion of passive contaminants
as first step before to extend it to reactive species. To solve numerically the conservation
equation we utilised a splitting technique developed in 70s by Soviet mathematicians.
The advective terms which present, from a numerical point of view, many difficulties
in their discretisation are solved with a quasi-Lagrangian cubic-spline technique. Such
scheme has been preferred as it can be easily adapted to domains with irregular grid-
spacing [21].
To test the model, we have simulated dispersion of an elevated continuous point
source in a convective boundary layer. The non-trivial characteristics of dispersion are

308 U. RIZZA, G. GIOIA, C. MANGIA and G. P. MARRA
0.0
0.5
1.0
1.5
2.0
Cy/(Q/UHmx )
X*
0.0 0.4 0.8 1.2 1.6 2.0 2.4
Fig. 4. – Crosswind-integrated surface concentration as a function of dimensionless downwind
distance. The dashed line represents our simulation result, the solid line represents Willis and
Deardorff experimental data.
adequately captured and the crosswind-integrated concentration distribution resembles
closely the well-established numerical and laboratory experiments found in the literature.
Results confirm that LES constitutes an alternative for field experiments: it can provide
databases of dispersion data on which a wide range of dispersion models can be developed
and tested, in view of their utilisation for air quality applications.
∗∗∗
We thanks Mr. C. Elefante for his technical support.
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