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Analytical solutionsof the balance equation for the scalar variance
inone-dimensional turbulent flows under stationary conditions
Andrea Amicarelli1,2,4 , Annalisa Di Bernardino2 , Franco Catalano3 , Giovanni Leuzzi 2 , Paolo Monti 2
1
RSE (Research on Energy Systems), SFE (Environ ment and Sustainable Develop ment Depart ment)
2
Sapienza Un iversity of Ro me, Depart ment of Civil and Environ mental Eng ineering
3
ENEA, Italy
4
Corresponding author
andrea.amicarelli@rse-web.it
annalisa.dibernardino@uniro ma1.it
franco.catalano@enea.it
giovanni.leuzzi@uniro ma1.it
paolo.monti@uniro ma1.it
Abstract.
This study presents1D analytical solutions for the ensemble variance of reactive scalars in onedimensional turbulent flows, in case of stationary conditions, homogeneous mean scalar gradient
and turbulence, Dirichlet boundary conditions and first order kinetics reactions. Simplified solutions
and sensitivity analysis are also discussed. These solutionsrepresents both analytical tools for
preliminary estimations of the concentration variance and upwind spatial reconstruction schemesfor
CFD (Computational Fluid Dynamics)–RANS (Reynolds-Averaged Navier-Stokes) codes, which
estimate the turbulent fluctuations of reactive scalars.
Keywords.
Concentration fluctuations; concentration variance; analytical solution; upwind schemes; spatial
reconstruction schemes.
1. INTRODUCTION
Modelling the turbulent fluctuations of a transported scalar interests several industrial processes and
environmental phenomena, both in atmosphere ad water bodies, especially where the scalar fly time
(tf) is smaller than the Lagrangianintegral time scale (micro-scale dispersion) or if the target
parameter, such as damage due to high concentration levels, is non- linear with respect to the
transported scalar. In particular, scalar fluctuations are crucial in modelling pollutant reactions, as
they normally depend on the instantaneous concentrations rather than their mean values.
Concentration fluctuations are relevant in several dispersion phenomena: accidental releases,
dispersion of reactive pollutants, impact of odours, micro-scale air quality and water quality and
several industrial processes, such as combustion and process fluid treatments.
In this context, Reynolds‟ average or mean concentration (𝐶) is generally insufficient to represent
the time and spatial evolutions of the instantaneous concentration field. Thus, a series of
experimental (e.g. [1],[2],[3],[4], [5]) and numerical (e.g. [6],[7],[8]) studies have investigated
concentration fluctuations, mainly focusing on the ensemble variance of concentration (  C ), whic h
2
is also involved in the definition of the intensity of fluctuations 𝑖 𝐶 =
𝜎𝐶
𝐶
.
Several numerical methods have been developed in order to evaluate the concentration variance.
They are based on Direct Numerical Simulations (DNS; [9]), Large Eddy Simulations (LES)
coupled with Lagrangian sub- grid schemes ([10]), probability density function (pdf) models
([11],[12]), RANS models ([13],[14]) and Lagrangianmicromixing numerical models
([15],[16],[17],[18],[19]).
These methods are somewhat constrained to the respect of the balance equation of the concentration
variance, which was first derived by [20],for passive scalars. Ref. [21] further discussed its terms
and provided a similarity solution. Ref. [22] proposed a formulation for the balance equation of  C ,
depending on Lagrangian parameters. The influence of the reactive termsis discussed in [23]. So
far, the reference analytical solutions for concentration fluctuations were derived in decaying grid
turbulence, under homogeneous non-stationary conditions: [15] treated the concentration variance,
whereas [24] represented the probability density function of a transported scalar.
In this paper, 1D analytical solutions for the ensemble variance of a reactive scalarare derived,
assuming stationary conditions, homogeneous turbulence and mean scalar gradient, first order
kinetics reactions, and Dirichlet boundary conditions.These solutions aim at providing an analytical
solver for both preliminary estimations of the concentration varianceand upwind schemes for CFD
codes, potentially involving CFD models for pollutant dispersion based on the Finite Volume
Method ([13])orthe Finite Difference Method ([25], [26]).
The paper is organizedas follows. Sec.2 brieflyrevises the balance equation of the concentration
variance, according to [21], and the uniform solution provided by [15], both simply adapted to
represent reactive scalars. In Sec.3, this studyproposes severalanalytical solutionsfor the
concentration variance, under stationary conditions. Sec.4discusses a sensitivity analysis on the
main non-dimensional parameters of the analytical models of Sec.3. Finally, Sec.5resumes the main
conclusions of the study, whereas Appendix A reports a couple of analytical solutions for the mean
scalar gradient, which is one of the key input of the analytical modelsof Sec.3.
2
2
2. BALANCE EQUATION FOR THE ENSEMBLE VARIANCE OF A REACTIVE
SCALAR IN A TURBULENT FLOW
2.1. Balance equation of the ensemble variance
The balance equation for the concentration variance of a passive scalar dispersed in a turbulent flow
was first derived by [20] and then discussed in detail in [21]. Their formulation is here briefly
reported and adapted to consider reactive scalars.
The balance equation of the pollutant mass reads:
C
C
 2C
u j
 DM
T
(2.1)
t
x j
x j2
whereC is the instantaneous concentration, DMthe molecular diffusion coefficient andu the velocity
vector. Einstein notation applies to the subscript “j ” hereafter, if not otherwise stated. From left to
right, the terms of (2.1) represent the local rate of change of the instantaneous concentration, the
advective term of C, the divergence of the molecular diffusion flux of C and the instantaneous
reactive term (T).
Concentration and velocity can be expressedaccording to Reynolds decomposition. Reynolds
average (over-bar symbol) of (2.1) provides:

 



 
 C C '
 C C '
2 C C '
 u j  u j'
 DM
 T T '
t
x j
x j2

(2.2)
Hereafter the apex “ „“ denotes a turbulent fluctuation.Introducing the continuity equationsfor an
incompressible turbulent flow
u j
x j
 0,
u j
x j
 0,
u j '
x j
0
(2.3)
into(2.2), one can write the balance equation for the mean concentration of a passive scalar:
C
C
C '
 2C
uj
uj '
 DM
T
t
x j
x j
x j2
(2.4)
From left to right, the terms of (2.4) represent the local rate of change of the mean concentration,
the advective term of C , the divergence of the turbulent and the molecular diffusion fluxes of C
and the mean reactive term.
Subtracting (2.4) from (2.1), the balance equation for the concentration fluctuationis obtained:
C '
C '
C
C '
C '
 2C '
uj
uj '
uj '
uj '
 DM
T '
(2.5)
t
x j
x j
x j
x j
x j2
After multiplying(2.5) times 2C‟ and considering the following equality:
 
C ' 

2C '



x j 
x j  x j
2
 C '2 



  2C '  C2'  2 C ' 
 x 
 x 
x j
j 

 ji 
2
(2.6)
one can write:
2
2
2
2
  2 C '2
 C '  
C '
C '
C
 C '
C '



  2C 'T '
uj
 2u j 'C '
uj '
 2C 'u j '
 DM
2
2
 x  
t
x j
x j
x j
x j
 x j
 j  

3
(2.7)
Averaging(2.7), one obtains the balance equation of the concentration variance of a reactive scalar
dispersed in a turbulent incompressible flow:
2
2 2
 C ' 
 C2
 C2 u j ' C '
C
  D M   C  2C 'T '
u j

 2u j 'C '
 2D M 
 x 
t
x j
x j
x j
x j2
 j 
2
(2.8)
where the terms on the left hand side represent the local rate of change of  C , the advective term
and the divergence of the turbulent flux of the concentration variance, respectively. On the right2
hand side, (2.8)shows the production term of  C (always non- negative), the dissipation rate of the
2
2
 C ' 
 always non-positive), the
concentration variance due to molecular diffusion (  C  2D M 
 x 
 j
divergence of the molecular diffusion flux of  C (negligible) and the reactive term R 2  2C 'T ',
which quantifies the direct effects of chemical and physical reactions.
The production term in (2.8)represents the increase in concentration variance due to nonhomogeneous conditions of the mean concentration field.
The dissipation rate of the concentration variance is ruled by molecular diffusion. Let us consider a
fixed point and time: close fluid particles exchange pollutant mass due to molecular diffusion and
thus homogenize the instantaneous concentration field. The concentration variance then decreases.
This term is not negligible even at very high Reynolds numbers (Re) as the gradient of the
instantaneous concentration would tend to infinity.
2
2.2. Parameterizations ofthe turbulent fluxes
The turbulent fluxes in the balance equation of the concentration variance(2.8)assume
approximatedand simpler formulations, according to the “K-theory”, which is the theory of the
turbulent dispersion coefficients, as reported and further developed by several authors ([27]):
C
u j 'C '  K T , j
x j
(2.9)
 C2
2
u j ' C '  K s , j
x j
HereKT ,jand K s,jare the vectors of the turbulent dispersion coefficients of the scalar mean and
variance, respectively. They are usually assumed equal and denoted by KT ,j.
These parameterizationsare commonly used in Euleriannumerical models, although they introduced
several limitations. The most important shortcomings emerge in case of strongly non- linear
relationships between the turbulent flux and the mean/variance gradient. Further, (2.9) loses the
information about the concentration-velocity covariance and the velocity autocorrelation. The
resulting errors are not negligible at the micro-scale, even if we just compute the mean
concentration.
2.3. Parameterizations of the dissipation rate of the concentration variance
Several parameterizations are available for the dissipation rate of the concentration variance (𝜀𝐶 ). In
particular, this study refers to the formula of [15], here reported and discussed.First, one may notice
the following equalities:
4
2
2
 C ' 
  2C ' DM  C2'
C  2DM 

x j
 x j 
(2.10)
Considering(2.4)and (2.5), assuming a turbulent regime and that reaction rates do not affect the
dissipation term, (2.10)can be expressed as follows:

dC
dC
 dC '
  dC
 C  2C ' 
T '   2C '
C '
 C 'T '   2C '
(2.11)

dt
dt
 dt
  dt

IECM (Interaction by the Exchange with the Conditional Mean; [15];[28])represents a major
micromixingformulation, which is alternative to other simpler schemes used in Lagrangian
stochastic modelling for pollutant dispersion (e.g. [29]). IECM relies on the following expression
for the Lagrangian derivative of the instantaneous concentration:
C  Cu
dC

dt
tm
(2.12)
where C U represents the mean concentration conditioned on the Lagrangianvelocity vector (U)
and tm is the mixing time (i.e. the time scale of the dissipation rate of the concentration variance),
which is defined by [15], [30] or [31].
Combining (2.11) and (2.12), [15] obtains the following expression:
C  C u
 tm

 C  2C ' 
2


C' Cu
  2 C 'C   C 

 tm
tm
tm




 2 2  C' C u

tm C



(2.13)
Further, in case of uniform mean scalar gradientand 1D dispersion, [15] reports the following
expression for the conditional mean:
C 'u '
C u C  2 u
(2.14)
u
where  u stands for the standard deviation of velocity.This parameter can be provided by diagnostic
tools, RANS codes (e.g. [32],[33],[34], [35]) or LES models, which allow a detailed
characterization of the turbulent structure of the atmospheric boundary layer (e.g.[36],[37]. [38]).
Considering both (2.14) and (2.9), the dissipation rate of  C becomes:
2
2
2  2 C 'u ' 
2
 C    C  2   
t m 
tm
 u 

K 2  C
 C2  T2 
 u  x





2
 2 K T2  C
 ,  C  2 
 u  x





2
(2.15)
Aswe consider 1D dispersion phenomena, the plume spread has the same dimension as the
boundarylayer height. Under these conditions, the mixing time can be related to the turbulent
kinetic energy (q) and its dissipation rate () via Richardson constant (C [15]):
2 2
2q 3C 0
tm 

TL ,
TL  u
(2.16)
C   2C 
C 0
wherethe Lagrangian integral time scale (TL) is introduced. Its relationship with the dissipation rate
of the turbulent kinetic - last equation in (2.16)- is reported by several authors (e.g. [15]). This
relationship can be derived applying Taylor analysis and comparing the formulas of the plume
5
spread alternatively depending on the Lagrangian structure function or the Lagrangian inte gral time
scale.
Taking into account (2.16), the dissipation term becomes:
4C   2 K T2  C
 C  2 
C  
3C 0TL 
 u  x





2




(2.17)
In conclusion, we can refer to a couple of simplified formulations for the dissipation rate of the
concentration variance: a 3D generic formulation (2.13) and a simplified 1D expression in case of
uniform mean scalar gradient(2.17).
2.4. Representation of the reactive terms
The role of the reactive term in the balance equation of the concentration varianceis here discussed,
by alternatively considering a 2nd-order kinetics, a 1st-order kinetics and a 0th order kinetics.
In case of 2nd order kinetics, the reactive term T is represented by:
(2.18)
T  rC AC B , C A  C
where the species A and B are reactants and r is the reaction rate.
The ensemble mean of (2.18)is equal to:




T   rC AC B  r C A C A' C B  C B'  r C AC B  C A' C B'

(2.19)
and the ratio between the last two terms in(2.19)defines the segregation coefficient:
C 'C '
Is  A B
C AC B
After considering the fluctuation of (2.18):

 

(2.20)

 
T ' T T  rC AC B  r C AC B C A' C B'  r C A C A' C B C B'  C AC B  C A' C B' 

 r C AC B C AC B C AC B C AC B
'
'
'
'
'
'

the reactive term in the balance equation of the variance becomes:


R 2  2C A' T '  2r C AC A' C B'  C A'  C B  C A'  C B' C A' C A' C B' 
2
2
 
2
 2r C A C A' C B'   C2A C B  C A' C B' 


(2.21)
(2.22)
The first term in the final formulation of (2.22)depends on the reactant covariance, the second one
on the variance of the control reactant (A) and the last one is a triple correlation term, whose
eventual parameterization can refer to [39]:
C  C
' 2
A
'
B

 I S C A' C B' C A   C2A C B

(2.23)
The advantage of (2.23) simply relies on the fact that, in the limit ofreactants never coexisting (Is =1), (2.23)guarantees that R 2 is exactly zero.
A 1st -order kinetics can be simply represented by imposingCB=1 in (2.22):
R 2  2C A' T '  2r C2A
6
(2.24)
This kind of reactions always decreases both the instantaneous and the mean concentration. Further,
a first order kinetics formula is linear with respect to the reactant concentration;thusit locally
decreases the concentration variance (provided the same mean scalar gradient) according to (2.24).
Finally, a 0-th order kinetics (e.g. T=-r) is equivalent to consideringboth CA=1 and CB=1 in (2.22).
In this case, the concentration variance does not depend on the reaction rate :
(2.25)
R2  0
In conclusion, the expressions (2.22), (2.24)and (2.25)alternatively represent the reactive term in the
balance equation of the concentration variance, in case of 2 nd-order, 1 st -order and 0-th order kinetics
reactions, respectively.
2.5. 1D analytical solution of Sawford (2004) under uniform and non-stationary conditions
Sawford (2004, [15]) reported a 1D analytical solution of the concentration variance under
homogeneous and non-stationary conditions. Here, this solution is reported and adapted to reactive
scalars.
First, consider the 1D balance equation of the concentration variance, as resulting from the
combination of (2.8), (2.9), (2.15) and (2.24), provided a uniform concentration variance:
 C
 C2
 2K T 
t
 x
2

2K 2
  2T
  t
w m

 C

 x

2

 C
 C2
 2
 4K T 

tm

 x
2

 C2
 2

tm

(2.26)
Defining the constant “a”, one can write:
 C
 2
t
2
 C2  a
tm
tm
2 ,
 C
a  4K T 
 x




2
(2.27)
After integration from the initial time (t=0) to the generic time t, oneobtains:
C2 t 
d C2
 
t
C 0   2  a m
 C
2

2
t



 
0
2dt
tm
t 
t 
t


ln  C2 t   a m   ln  C2 0  a m   2 
2
2
tm



t
t
t  2

  C  a m   C2 0  a m e t m
2 
2
2
  C   C 0e
2
2

4C  t
3C 0 TL
  C   C 0e
2
2
 C
C
 3 0 K T TL 
C
 x




2
t
tm
2
t
a m
2
4C  t


 1  e 3C 0 TL


t
2

1  e t m






(2.28)




Finally, considering(3.5), as explained in the following section, the uniform time-dependent
solution for the concentration variance becomes:
 C   C 0e
2
2

4C  t
3C 0 TL
 C
C
 3 0  u2TL2 
C
 x




2
4C  t


 1  e 3C 0 TL






(2.29)
The solution tends to an equilibrium value, when the production and the dissipation terms equalize:
 C
C
 C t     3 0  u2TL2 
C
 x
2
7




2
(2.30)
This formula highlights the importance of modelling the dissipation term in (2.26). When this term
is absent (this is the case of Lagrangianmodels without anymicromixing schemeor Eulerian models
without any dissipation term for  C ), the concentration variance linearly grows with time,
indefinitely:
2
 C
 C2
 2K T 
t
 x
2

 


 C
  2K T 
 x
2
C
2

 C
 t  2 u2TL t 

 x


In case of a null mean scalar gradient:
 2

 C2
  C2 
 r 
t
t m





2
(2.31)
(2.32)
there is no production term and the variance tends to zero, as follows:
 C   C 0e
2
2
 2


r 
 TL

8
(2.33)
3. 1D SOLUTIONS FOR THE BALANCE EQUATION OF THE CONCENTRATION
VARIANCE UNDER STATIONARY CONDITIONS
3.1. Main solution unde r stationary conditions
Provided 1D stationary conditions and homogeneous turbulence, the balance equation of the
concentration variance(2.8) assumes the following form:
 C2 u ' C '
C
 C ' 

 2u 'C '
 2D M 
  2C 'T '
x
x
x
 x 
2
2
u
(3.1)
Introducing the parameterizations for the turbulent fluxes (2.9)and  C (2.17), as well as the
expression of the reactive term in case of 1-st order kinetics reaction (2.24), one obtains:
 C
 C2
 2 C2
u
KT
 2K T 
2
x
x
 x
2
4C 

 
 3C T
0 L


K 2  C
 C2  T2 
 u  x






2

  2r C2


(3.2)
After assuming the definition of the turbulent kinetic energy in 1D:
3
2
q   u2
(3.3)
considering(2.16) and (3.3) and dividing by KT , (3.2) becomes:
 2  C 
 2 C2 u   C2
1 C 

 C  2 



2
r
 x 
K T  q
x 2 K T x

 
2
 K T C 
 1 
2q u2


  0

(3.4)
It is convenient to introduce the relationship between the turbulent dispersion coefficient and the
Lagrangian integral time scale, as derived from a simple analysis of these turbulent scale
parameters:
K T   u2T L
(3.5)
The system (2.16), (3.3) and (3.5) provides another expression for the turbulent dispersion
coefficient:
8 q

9 C 0
2
KT
(3.6)
andallows writing (3.4) with no explicit dependency on K T :
 C 

 2 C2 9 uC 0   C2 9 C 0  C  




 2r  C2  2
2
2
2 

8 q
8 q  q
x
x

 x 
2

2C 
 1 
  0
 3C0 
(3.7)
Eq.(3.7) represents a 2nd-order inhomogeneousODE (Ordinary Differential Equation)
 2 C2
  C2
a
 b C2  c  0,
2
x
x
  a 2  4b  0
(3.8)
withthe following constant coefficients:
9 uC 0
a 
,
8 q2

9 C  C 
b   02    2r ,
8q  q

9
 C
c  2
 x



2
 2C 
 1 

3
C
0 

(3.9)
Its general solution assumes the form:
 C2  c 1e  x  c 2e  x 
1
2
c
b
(3.10)
which is completed by the following definition:
𝜆 1,2
−𝑎 ± 𝑎2 − 4𝑎𝑏 1 9 uC 0 
9 𝐶0 𝜀 𝜀 9 𝑢 2 𝐶0
≡
=
±
+ 𝐶𝜙 + 2𝑟
2
2 8 q2
2 𝑞 2 𝑞 32 𝑞
(3.11)
One can verify that the system (3.10) and (3.11) is a general solution of (3.8), as briefly described in
the following. According to (3.10), the left hand side of (3.8) becomes::
c

21c 1e 1x  22c 2e 2x  a 1c 1e 1x  2c 2e 2x  b  c 1e 1x  c 2e 2x    c
(3.12)
b


The value of
21,2

2i can be derived from (3.11):
  a  a 2  4b


2

2
 a 2 a 2  4b a
a2
a 2
 


a 2  4b   b 
a  4b

4
4
2
2
2

(3.13)
Replacing iand i2 in (3.12),according to (3.11) and (3.13), one obtains:
a2

a2

a 2
a 2
1x


b 
a  4b c 1e    b 
a  4b c 2e 2x 
2
2
 2

 2

  a  a 2  4b
 a 
2

  x   a  a 2  4b
c e 1  
 1

2


 x
c e 2   b c e 1x  c e 2x  c   c
1
2
 2
b



(3.14)
which finally verifies that the system (3.10) and (3.11) is a general solution of (3.8):
 a 2
   a 2  a a 2  4b  
a
c 1e 1x   b  a 2  4b   
 b 
 
2
2
 2
 
 
 a 2
   a 2  a a 2  4b
a 2
 c 2e 2x   b 
a  4b   
2
2
 2
 
 
  b c  c  0
 
 
(3.15)
Dirichlet boundary conditions can now be imposed on the left boundary (x=0)
𝑐
𝑐
2
2
2
𝜎𝐶2 𝑥 = 0 ≡ 𝜎𝐶0
⇒ 𝜎𝐶0
= 𝑐1 + 𝑐2 − ⇒ 𝑐1 = 𝜎𝐶0
− 𝑐2 +
𝑏
𝑏
andthe right boundary (x=L) of the domain:
2
2
𝜎𝐶2 𝑥 = 𝐿 ≡ 𝜎𝐶𝐿
⇒ 𝜎𝐶𝐿
= 𝑐1 𝑒 𝜆1 𝐿 + 𝑐2 𝑒 𝜆2 𝐿 −
𝑐
𝑐
2
⇒ 𝑐2 𝑒 𝜆2 𝐿 = 𝜎𝐶𝐿
− 𝑐1 𝑒 𝜆1 𝐿 +
𝑏
𝑏
Combining (3.16) with (3.17), one can obtain the value of c2 :
10
(3.16)
(3.17)
𝑐 𝜆 𝐿 𝑐
𝑒 1 + ⇒
𝑏
𝑏
𝑐
𝑐
2
− 𝜎𝐶0
+ 𝑒 𝜆1 𝐿 + ⇒
𝑏
𝑏
2
2
𝑐2 𝑒 𝜆2 𝐿 = 𝜎𝐶𝐿
− 𝜎𝐶0
− 𝑐2 +
2
⇒ 𝑐2 𝑒 𝜆2 𝐿 − 𝑒 𝜆1 𝐿 = 𝜎𝐶𝐿
(3.18)
𝑐
⇒ 𝑐2 =
2 𝜆1 𝐿
2
𝜎𝐶0
𝑒
− 𝜎𝐶𝐿
+ 𝑏 𝑒 𝜆1 𝐿 − 1
𝑒 𝜆1 𝐿 − 𝑒 𝜆2 𝐿
Thus, theconstant c1 from (3.16)becomes:
𝑐
2
2 𝜆2 𝐿
𝜎𝐶𝐿
− 𝜎𝐶0
𝑒
− 𝑏 𝑒 𝜆2 𝐿 − 1
𝑐1 =
𝑒 𝜆1 𝐿 − 𝑒 𝜆2 𝐿
(3.19)
Consideringthe values of c1 and c2 from (3.19) and (3.18), respectively, (3.10)assumes the following
form:
𝑐
𝑐
2
2 𝜆2 𝐿
2 𝜆1 𝐿
2
𝜎𝐶𝐿
− 𝜎𝐶0
𝑒
− 𝑒 𝜆2 𝐿 − 1
𝜎𝐶0
𝑒
− 𝜎𝐶𝐿
+ 𝑒 𝜆1 𝐿 − 1
𝑐
𝑏
𝑏
2
𝜆1 𝑥
𝜎𝐶 =
𝑒
+
𝑒 𝜆2 𝑥 −
(3.20)
𝜆
𝐿
𝜆
𝐿
𝜆
𝐿
𝜆
𝐿
𝑒 1 −𝑒 2
𝑒 1 −𝑒 2
𝑏
Other few and simple algebraic passages finally provide a complete 1D solution of the balance
equation of the concentration variance for a reactive scalar in a turbulent flow, under stationary
conditions:
𝑐
− 𝑒 𝜆1 𝑥 𝑒 𝜆2 𝐿 − 1 − 𝑒 𝜆2 𝑥 𝑒 𝜆1 𝐿 − 1 + 𝑒 𝜆1 𝐿 − 𝑒 𝜆2 𝐿 +
𝑏
1
𝜎𝐶2 =
2
,
+𝜎𝐶0
𝑒 𝜆1 𝐿+𝜆2 𝑥 − 𝑒 𝜆1 𝑥 +𝜆2 𝐿 +
𝜆
𝐿
𝑒 1 − 𝑒 𝜆2 𝐿
2
+𝜎𝐶𝐿
𝑒 𝜆1 𝑥 − 𝑒 𝜆2 𝑥
𝜆1,2 ≡
1
2
9 uC 0 
8 q
2
±
9 𝐶0 𝜀 𝜀
2 𝑞2

9 C  C 
b   02    2r ,
8 q  q

2
9 𝑢 𝐶0
+ 𝐶𝜙 + 2𝑟 ,
𝑞 32 𝑞
 C
c  2
 x




2
(3.21)
 2 C 
 1 

3
C
0 

It is immediate to verify that (3.21) satisfies the boundary conditions imposed.
The solution (3.21) is widely discussed and analysed in Sec.4, where several examples are available
(e.g. Figure 4.1),both in terms of non-dimensional and dimensional physical quantities. Hereafter,
wejust provide some clarification about the role of the reactive term and the mean scalar gradient.
The reference solution is derived considering 1st order kinetics reactions. In this case, however, one
cannot obtain an exact uniform mean scalar gradient (stationary regime), in the absence of a source
term, as we can appreciate from the balance equation of the mean concentration:
𝜕𝐶 𝐷𝑇 𝜕2 𝐶 𝑟
=
− 𝐶
(3.22)
𝜕𝑥
𝑢 𝜕𝑥 2 𝑢
This means that (3.21)can only consider a uniform mean gradient as an approximated condition for
both 1 st and 2nd order kinetics reactions.
On the other hand, an exact uniformscalar mean gradientcan relate to a 0-th order kinetics:
11
𝜕𝐶
𝜕𝐶
𝜕2 𝐶
𝜕𝐶 𝐷 𝑇 𝜕2 𝐶 𝑟
𝑟
+𝑢
= 𝐷𝑇 2 − 𝑟 ⇒
=
− =−
2
𝜕𝑡
𝜕𝑥
𝜕𝑥
𝜕𝑥
𝑢 𝜕𝑥
𝑢
𝑢
(3.23)
However, in this case there is no need for a reactive term in the balance equation of the variance,
according to (2.25).
The mean scalar gradient is the key parameter in the production term of the concentration variance
and affects the constant c in (3.21).
C
should be approximately uniform and can be calculated by
x
means of simplified analytical solutions (Appendix A) or using the measured or simulated values of
the mean concentration, where they are available in the domain of interest.
3.2. Solution under stationary conditions, in the absence of mean velocity
In the absence of mean velocity,(3.11) becomes:
𝜆 1,2 = ±
3
2
𝐶0 𝜀 𝜀𝐶𝜙
+ 2𝑟
2𝑞 2 𝑞
,
Δ = 0 = 𝑢, 𝜆 ≡ 𝜆 1 = −𝜆 2
(3.24)
Under these conditions, we can derive a simpler and alternative formulation of the general solution
(3.21).Considering that𝜆 ≡ 𝜆 1 = −𝜆 2 , the first line of (3.21)becomes:
𝑐
− 𝑒 𝜆𝑥 𝑒 𝜆𝐿 − 1 − 𝑒 −𝜆𝑥 𝑒 𝜆𝐿 − 1 + 𝑒 𝜆𝐿 − 𝑒 −𝜆𝐿 +
𝑏
1
1
2
+𝜎𝐶0
𝑒 𝜆 𝐿−𝑥 − 𝜆 𝐿 −𝑥 +
𝜎𝐶2 =
(3.25)
𝜆𝐿
𝑒
𝑒 − 𝑒 −𝜆𝐿
2
+𝜎𝐶𝐿
𝑒 𝜆𝑥 −
1
𝑒 𝜆𝑥
Introducing the following hyperbolic functions (and their derivatives):
𝑐𝑜𝑠ℎ 𝑥 =
𝑒 𝑥 + 𝑒 −𝑥
𝑒 𝑥 − 𝑒 −𝑥
, 𝑠𝑖𝑛ℎ 𝑥 =
, 𝑐𝑜𝑠ℎ ′
2
2
𝑥
= 𝑠𝑖𝑛ℎ 𝑥 , 𝑠𝑖𝑛ℎ′ 𝑥 = 𝑐𝑜𝑠ℎ 𝑥
into(3.25), one can obtain the following expression:

c 
e L  1 x
1

C2,0senh L  x   C2,L senhx 
 C2  2 1  2L
e  e  L x   


  e 1
senh

L

This change in the reference system:
x ' x 
L
2
 x  x '
(3.26)
(3.27)
L
2
(3.28)
implies that:
L

L
L
   x ' L2 
  x '  
 

1 
x '


2

2

e e
 e
e
 e  e  x '   e 2 coshx ',


e 



 L
 L


senh L  x   senh    x ' ,
senhx   senh    x ' 


 2
 2

x
 L  x 

12
(3.29)
Combining (3.27), (3.28) and (3.29), the simplified solution of the 1D stationary balance equation
of the concentration variance, in the absence of mean velocity, assumes the following expression:

c 
e L  1  L2
 C2  2 1  2L
e coshx ' 
  e
1

 2
 L
 L

 
2

 C ,0senh   x '    C ,L senh   x '   , 𝜆 =
senhL  

 
 2
 2
1
3𝜀
𝐶0 𝐶𝜙
2𝑞
2𝑞
(3.30)
This solution, represented in Figure 4.1 (e.g. the green plot), is verified, as described in the
following. One may first notice that:
 2 C2  2 C2
,
b  2 ,

x 2 x '2
(3.31)
 2
 L
 L

 
2
2
 C  A  B coshx '  C  C ,0senh   x '    C ,L senh   x '  

 
 2
 2

where B and C are constants, directly defined by comparison with (3.30).
One can consider the second derivative of the variance in the new reference system:

 L
 L
 2 C2
c

  
 2B coshx '  2C  C2,0senh   x '    C2 ,L senh   x '     C2  2
2
b
x '

  
 2
 2

and then write (3.8), according to (3.30) and (3.32):
 2 C2
c
 2 C2  c  2  c  0
2
b
x '
(3.32)
(3.33)
whichfinally verifies that (3.30) is the solution of (3.8), in the absence of mean velocity.
Under these conditions, we can simply write the derivative of the concentration variance:

 L
 L
 C2

 
 Bsenhx '  C  C2 ,0 cosh   x '   C2 ,L cosh   x '  
x '

 
 2
 2

(3.34)
Although it is not immediate to get a general expressionto locate the maxima/minima, a simple


2
2
solution is obtained in case C ,0  C ,L , as plotted in Figure 4.1 (blue line). This assumption
provides a maximum or a minimum at the domain centre(x‟=0), in case

2
C ,0


2
C ,0

 C2,eq or
 C2,eq , respectively, according to the definition of the equilibrium variance (3.35), as
provided in Sec.3.3.
3.3. Solution under uniform and stationary conditions
The balance equation of the concentration variance shows a simple formulation under uniform and
stationary conditions. In this case, thedissipation term  C and the reactive term (2.24)exactly equals
the production termof  C (“equilibrium solution”):
2
13
2
 C2 ,eq
 C   2 C  

 1 

c 16  x   3 C 0 
 
b 9 C 0  C  


 2r 
2 
q  q

(3.35)
This expression also represents the value of the horizontal inflection points in the general solution
(3.21). Under these conditions, the divergence of both the turbulent and the mean transport scalar
fluxes are null.
14
4. SENSITIVITY ANALYSIS
The solutions (3.21), (3.30) and (3.35) of the 1D balance equation of the concentration variance are
herediscussed and analysed by means of a sensitivity analysis.
The results are presented in terms of non-dimensional parameters: all the physical quantity are
2𝜎 2
normalized (“N ”) over the reference time, length and mass scales, respectively: t0 =TL= 𝐶 𝑢𝜀 ,
0
𝜕𝐶
x0 =L,𝑚 0 =
𝜕𝑥
2
𝐿 .Thus, the main non-dimensional parameters, LN,
𝜕𝐶
𝜕𝑥
𝑁
and TL,N, are always
equal to unity, whereas the other non-dimensional quantities vary according to the following
expressions: 𝑢 𝑁 =
𝑢𝑇𝐿
𝐿
, u,N=u*TL/L, rN=r*TL and𝜎𝐶2,𝑁 =
𝜎𝐶2
𝜕𝐶
𝜕𝑥
2
.
𝐿
The “reference solution” of this analysis is defined by the following arbitrary choice, which simply
provides a meaningful and general reference configuration:uref,N=u,ref,N=1/6, C0 =2, C=3, rN=1,
2
2
𝜎𝐶2,0,𝑁 = 0.00833 and 𝜎𝐶,𝐿,𝑁
= 0.05833. Under these conditions, it follows that 𝜎𝐶,𝑒𝑞
,𝑁 = 1/30and
𝜀𝑁 =
2 𝜎𝑢 ,𝑁
𝐶0 𝑇𝐿 ,𝑁
2
=1/36.
Discussion first considers some generic features ofthe solution (3.21), as plotted in Figure 4.1 under
three different configurations.
The reference solution (red plot) shows a central region of very low gradients (very weak turbulent
and mean transports of 𝜎𝐶2 ), where the production and the dissipation termsalmost balance each
other.
In this context, the reactive term plays an analogous role to the dissipation term  C and one may
refer to them as “the dissipation terms” of the concentration variance, just for simplicity of notation.
At the edges of the domain, the production (left boundary) or the dissipation (right boundary) terms
alternatively dominate. In these lateral regions, the turbulent scalar flux is always negative (directed
upstream) and provides a positive (right boundary) or a negative (left boundary) contribution to the
rate of change of the concentration variance, thus balancing the excess in the variance
dissipation/production. At the same time,a weak non-negative mean transport is responsible for a
local decrease of𝜎𝐶2 .
In other words, the production term is uniform because turbulence is homogeneous as well as the
mean scalar gradient. At the same time, the dissipation terms linearly grow with the concentration
variance. Thus, the turbulent and the mean transport are responsible for driving the concentration
variance away from the zone in excess of 𝜎𝐶2 production (low variance) and towards the regions in
excess of 𝜎𝐶2 dissipation (high variance), in order to establish a stationary regime.
Figure 4.1 also reports a couple of examples of the solution for passive scalars (3.30) in the absence
of mean velocity (green plot). In particular, the blue line represents the symmetric solution cited at
the end of Sec.3.2.
Figure 4.2 reports a sensitivity analysis of the main solution (3.21)on the choice of the nondimensional boundary values. Let us first examine the left image of Figure 4.2, where the left
boundary value is null and the right boundary value is alternatively equal to 0, 1 or 2 times the
equilibrium value of (3.35). In the first case(𝜎𝐶2,𝐿 ,𝑁 = 0), the production term equals the dissipation
terms in the central region of the domain. Elsewhere, the (uniform) production term is higher than
the dissipation terms and the turbulent fluxtransports the excess of the concentration variance away
from the domain. The mean velocity makes the solution to be slightly asymmetric, with a peak
located on the right half of the domain.In the second case (𝜎𝐶2,𝐿,𝑁 = 𝜎𝐶2,𝑒𝑞 ,𝑁 ), the solution is almost
uniform in the right half of the domain as the equilibrium value is already reached around the
2
domain centre. Finally, when (𝜎𝐶2,𝐿 ,𝑁 = 2𝜎𝐶,𝑒𝑞
,𝑁 ), the solution is almost symmetric to the first case
2
(𝜎𝐶 ,𝐿 ,𝑁 = 0)with respect to the equilibrium line, for x>L/2. In this region, the turbulent
fluxesbalance the deficiency in the variance production by transporting the concentration variance
from the right boundary towards the domain centre.
15
Figure 4.1. 1D analytical solution for the balance equation of the concentration variance, under stationary
conditions. Legend: + s olution with reference parameters ; × solution with reference parameters in the absence
of mean velocity and rN =0; * solution with reference parameters in the absence of mean velocity, null
variance at boundaries and rN =0.
Figure 4.2. Reference non-dimensional solution with sensitivity analysis on the boundary non-dimensional
2
values 𝜎𝐶,𝑁
, which are alternatively equal to 0, 1/30 and 1/15 (all the independent combinations are
reported).
The right panel of Figure 4.2 considers the uniform solution of (3.35), established with the
2
choice𝜎𝐶,0,𝑁
= 𝜎𝐶2,𝐿,𝑁 = 𝜎𝐶2,𝑒𝑞 ,𝑁 : the production term is everywhere balanced by the dissipation
terms, without any effects of neither turbulent fluxes nor advection. The other solutions refer to
(𝜎𝐶2,𝐿 ,𝑁 = 2𝜎𝐶2,𝑒𝑞 ,𝑁 ) and are almost identical in the right half of the domain. Let us notice that the
2
2
2
solution with (𝜎𝐶2,0,𝑁 = 𝜎𝐶,𝐿
,𝑁 = 2𝜎𝐶,𝑒𝑞 ,𝑁 ) is symmetrical to the configuration with (𝜎𝐶 ,0,𝑁 =
𝜎𝐶2,𝐿,𝑁 = 0; left panel of Figure 4.2), with respect to the line of the equilibrium value.
Figure 4.3investigates the role of the non-dimensional mean velocity. In the absence of a mean flow
(𝑢 𝑁 = 0), the equilibrium value is reached at the centre of the domain, where a horizontal inflection
pointis established (red plot). The solution is symmetric with respect to its central value. Increasing
the non-dimensional mean velocity, the relative importance of the mean transport grows and the
upstream boundary value of 𝜎𝐶2,𝑁 influences the solution more and more noticeably; further, the
solution is not symmetric anymore and the equilibrium value is locatedin the right half of the
domain.
The effects of the non-dimensional standard deviation of velocity, whose square is directly
proportional to the non-dimensional turbulent kinetic energy, are shown in Figure 4.4(left). An
increase in q determines a highest variance productio n all over the domain. Considering the highest
and the lowest values of u,N, the non-dimensional variance lies out of the range provided by the
Dirichletboundary values, in the most of the domain. As uN is relatively low, one can always detect
2
at leasta maximum, a minimum or a horizontal inflection point in the profile of 𝜎𝐶,𝑁
,in the ver y
inner domain.
16
Figure 4.3. Reference non-dimensional solution with sensitivity analysis on the non-dimensional mean
velocity, which is alternatively equal to 0 ( +) , 1/6 (×), 1/2 (*) and 5/3 (□).
Figure 4.4. Left: reference non-dimensional solution with sensitivity analysis on the non-dimensional
standard deviation of velocity, which is alternatively equal to 1/24 (+) , 1/8 (×), 1/6 (* ), 1/4 (□) and 1/3 (■) 2
𝜎𝐶,𝑒𝑞
,𝑁 is 0.002083, 0.01875, 1/30, 0.075 and 4/30, respectively-. Right: reference non-dimensional solution
with sensitivity analysis on the non-dimensional reaction rate, which is alternatively equal to 0 (+), 1/2 (×), 1
2
(* ), 2 (□) or 10 (■) - 𝜎𝐶,𝑒𝑞
,𝑁 is 1/12, 0.04762, 1/30, 0.02083 and 0.005208, respectively-.
Figure 4.5. Left: reference dimensional solution (but r=0.001/s) with sensitivity analysis on the Lagrangian
2
integral time scale, which is alternatively equal to 50s (+), 75s (×), 100s (* ), 150s (□) or 200s(■) -𝜎𝐶,𝑒𝑞
is
2
2
2
2
2
2
2
2
2
2
8.694g /m , 18.90g /m , 32.507g /m , 68.66g /m and 115g /m , respectively-. Right: reference
dimensional solution (but r=0.001/s) with sensitivity analysis on the scalar mean gradient, which is
2
alternatively equal to 0g/m2 (+), 0.1765g/m2 (×), 0.353g/m2 (* ), 0.44125g/m2 (□)or 0.5295g/m2 (■)-𝜎𝐶,𝑒𝑞
2
2
2
2
2
2
2
2
2
2
is 0g /m , 8.127g /m , 32.507g /m , 50.79g /m and 73.14g /m , respectively-.
Contrarily, the dependency of the non-dimensional variance on the non-dimensional reaction rate
has an opposite behaviour. In fact, the reactive termalways acts as a dissipation term (linear in 𝜎𝐶2 )
and its increase tends to lower down the scalar variance all over the domain (Figure 4.4, right). In
case of extreme values of rN, one may notice that the most of the domain is characterized by
𝜎𝐶2 values lying out of the interval provided by the Dirichlet boundary conditions.
17
The sensitivities on the Lagrangian integral time scale and the mean scalar gradient are finally
shown in Figure 4.5 in terms of dimensional parameters, just to provide a more physical
representation of the role of these quantities. An increase in TLor
term” with similar effects on the profile of the concentration variance.
18
C
improves the “production
x
5. CONCLUSIONS
The study presents 1D analytical solutions for the ensemble variance of reactive scalars in turbulent
flows, in case of stationary conditions, homogeneous mean scalar gradient and turbulence, Dirichlet
boundary conditions and 1st -order kinetics reactions.
A sensitivity analysis has discussed the role of the different terms in the balance equation of the
concentration variance, with focus on the production term, the dissipation rate of the scalar
variance, the reactive term, the turbulent transport and advection. The analysis hasdetected three
main scale parameters: the domain length, the Lagrangian integral time scale and the product of the
mean scalar gradient times the square of the domain length. Thus, the analysis has investigated the
dependency of the solution on the non-dimensional boundary values, reaction rate, mean and
standard deviation of velocity.
Theseanalytical solutions represent an immediate tool for preliminary estimates of the concentration
variance in several application fields, where concentration fluctuations play a relevant role:
accidental releases, pollutant reactions, odour assessment, micro-scale air quality and water quality,
several industrial processes, such as combustion. Although the solutionsreferred to 1D
configurations (e.g. unstable turbulent boundary layers of plug- flow reactors or pollutant treatment
devices), they can still provide approximate estimations for simplified 3D configurations.
Finally, the proposed solutionsrepresent upwind spatial reconstruction schemes for the
concentration variance, as they can be implementedin CFD-RANS codes, as well as in prognostic or
diagnostic meteorological and ocean models, which simulate the turbulent fluctuations of reactive
pollutants.
19
6. APPENDIX: 1D ANALYTICAL SOLUTIONS FOR THE MEAN SCALAR GRADIENT
Several authors (e.g. [40], [41]) presented analytical solutionson the scalar transport in fluid flows.
Here, some of them are reported to compute the mean scalar gradient, which is a key input
parameter for the analytical solutions of the concentration variance.
6.1. Mean trans port,0-th order kineticsand continuous source
In case of a continuous and infinite pollutant source over an horizontal plane y-z and neglecting the


along-flow dispersion u  u , the steady-state solution of the balance equation for the mean
concentration reads ([40]):
𝑚𝑖
𝑟𝑥
𝐶 𝑥 =
𝑒𝑥𝑝 −
(A.1)
𝑢
𝑢
wheremi is the time rate of mass injection per unit area. The derivative of (A.1) with respect to x is:
𝜕𝐶 𝑥
𝑚𝑟
𝑟𝑥
= − 𝑖2 𝑒𝑥𝑝 −
(A.2)
𝜕𝑥
𝑢
𝑢
The curves in Fig.A.1 show the solutions of (A.1) and (A.2), normalized by the value at the origin,
for x>0. Given the exponential nature of (A.1) and (A.2), the two normalized solutions coincide.
Figure A.1. Normalized solutions of the mean concentration and its gradient for a continuous plane source,
only considering the advection term and a 0th order kinetics (U=5m s-1 , mi =0.1kg s-1 , k=0.8 *103 mol m-3 s-1 ).
6.2. Turbulent transport, instantaneous source and infinite domain
The solution to the advective-diffusion equation is here reported,in case of instantaneous point
source and infinite domain.
Considering a point source in stagnant conditions, the mean concentration can be described
asfollows ([40]):
𝑀
𝑥 − 𝑥0 2
𝐶 𝑥, 𝑡 =
exp⁡ −
(A.3)
4𝐾𝑇 𝑡
4𝜋𝐾𝑇 𝑡
where M represents the source strength and x0 is the injection point of the tracer.
For x=x0 , one can find the maximum value of the mean concentration:
𝑀
𝐶𝑚𝑎𝑥 𝑡 =
4𝜋𝐾𝑇 𝑡
If we consider t=0, the concentration tends to an infinite value :𝐶 𝑡 = 0 = ∞.
We can derive (A.4) with respect to time as:
20
(A.4)
𝜕𝐶
=
𝜕𝑡
𝑀
𝑒𝑥𝑝
3
4 𝜋𝐾𝑇 𝑡
−(𝑥 − 𝑥 0 )2
4𝐾𝑇 𝑡
𝑥 − 𝑥0
2𝐾𝑇 𝑡
Instead, the mean scalar gradient can be expressed as:
𝜕𝐶 𝑀(−2𝑥 + 2)
−(𝑥 − 𝑥 0 )2
=
exp⁡
𝜕𝑥 4𝐾𝑇 𝑡 4𝜋𝐾𝑇 𝑡
4𝐾𝑇 𝑡
2
−1
(A.5)
(A.6)
The analytical solution of (A.6) can be possibly usedas an approximate estimation of the mean
scalar gradient, which relevantly influences the analytical solutions for the concentration variance
(Sec.3). Although (A.6) referred to non-stationary conditions, it can approximate the mean scalar
gradient during a late stage of a dispersion process.
Acknowledge ments.
The first author is deeply grateful to Prof. Brian Sawford, who provided comments and advices at
the beginning of this study.
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