# Analytical Solutions of the Balance Equation for the Scalar Variance in One-Dimensional Turbulent Flows under Stationary Conditions

Authors: Andrea Amicarelli; Annalisa Di Bernardino; Franco Catalano; Giovanni Leuzzi; Paolo Monti
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

This study presents 1D analytical solutions for the ensemble variance of reactive scalars in one-dimensional 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 solutions represent both analytical tools for preliminary estimations of the concentration variance and upwind spatial reconstruction schemes for CFD (Computational Fluid Dynamics)—RANS (Reynolds-Averaged Navier-Stokes) codes, which estimate the turbulent fluctuations of reactive scalars.


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## Body

## 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 Lagrangian integral time scale (microscale dispersion) or if the target parameter, such as damage due to high concentration levels, is nonlinear 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, microscale 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., [1–5]) and numerical (e.g., [6–8]) studies have investigated concentration fluctuations, mainly focusing on the ensemble variance of concentration (σC2), which is also involved in the definition of the intensity of fluctuations (iC=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 subgrid schemes [10], probability density function (pdf) models [11, 12], RANS models [13, 14], and Lagrangian micromixing numerical models [15–19].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. Reference [21] further discussed its terms and provided a similarity solution. Reference [22] proposed a formulation for the balance equation of σC2, depending on Lagrangian parameters. The influence of the reactive terms is discussed in [23]. So far, the reference analytical solutions for concentration fluctuations were derived in decaying grid turbulence, under homogeneous nonstationary 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 scalar are 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 variance and upwind schemes for CFD codes, potentially involving CFD models for pollutant dispersion based on the Finite Volume Method [13] or the Finite Difference Method [25, 26].The paper is organized as follows. Section2 briefly revises 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 Section 3, this study proposes several analytical solutions for the concentration variance, under stationary conditions. Section 4 discusses a sensitivity analysis on the main nondimensional parameters of the analytical models of Section 3. Finally, Section 5 resumes 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 inputs of the analytical models of Section 3.


## 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(1)∂C∂t+uj∂C∂xj=DM∂2C∂xj2+T,
where C is the instantaneous concentration, DM the molecular diffusion coefficient, and u_ the velocity vector. Einstein notation applies to the subscript “j” hereafter, if not otherwise stated. From left to right, the terms of (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 expressed according to Reynolds decomposition. Reynolds average (over-bar symbol) of (1) provides
(2)∂C¯+C′∂t¯+uj¯+uj′∂C¯+C′∂xj¯=DM∂2C¯+C′∂xj2¯+T¯+T′¯.
Hereafter, the apex “′” denotes a turbulent fluctuation. Introducing the continuity equations for an incompressible turbulent flow
(3)∂uj∂xj=0,∂uj¯∂xj=0,∂uj′∂xj=0
into (2), one can write the balance equation for the mean concentration of a passive scalar:
(4)∂C¯∂t+uj¯∂C∂xj¯+uj′∂C′∂xj¯=DM∂2C¯∂xj2+T¯.
From left to right, the terms of (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 (4) from (1), the balance equation for the concentration fluctuation is obtained:
(5)∂C′∂t+uj¯∂C′∂xj+uj′∂C¯∂xj+uj′∂C′∂xj-uj′∂C′∂xj¯=DM∂2C′∂xj2+T′.
After multiplying (5) times 2C′ and considering the following equality:
(6)∂∂xj2C′∂C′∂xj=∂∂xj∂C′2∂xj=2C′∂2C′∂xj2+2∂C′∂xji2
one can write
(7)∂C′2∂t+uj¯∂C′2∂xj+2uj′C′∂C¯∂xj+uj′∂C′2∂xj-2C′uj′∂C′∂xj¯=DM∂2C′2∂xj2-2∂C′∂xj2+2C′T′.
Averaging (7), one obtains the balance equation of the concentration variance of a reactive scalar dispersed in a turbulent incompressible flow:
(8)∂σC2∂t+uj¯∂σC2∂xj+∂uj′C′2¯∂xj=-2uj′C′¯∂C¯∂xj-2DM∂C′∂xj2¯+DM∂2σC2∂xj2+2C′T′¯,
where the terms on the left hand side represent the local rate of change of σC2, the advective term and the divergence of the turbulent flux of the concentration variance, respectively. On the right-hand side, (8) shows the production term of σC2 (always nonnegative), the dissipation rate of the concentration variance due to molecular diffusion (εC=-2DM∂C′/∂xj2¯ always nonpositive), the divergence of the molecular diffusion flux of σC2 (negligible) and the reactive term (R2¯≡2C′T′¯), which quantifies the direct effects of chemical and physical reactions.The production term in (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. Parameterizations of the Turbulent Fluxes

The turbulent fluxes in the balance equation of the concentration variance (8) assume approximated and 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]:
(9)uj′C′¯=-KT,j∂C¯∂xj,uj′C′2¯=-Ks,j∂σC2∂xj.
Here, KT,j and Ks,j are 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 parameterizations are commonly used in Eulerian numerical models, although they introduced several limitations. The most important shortcomings emerge in case of strongly nonlinear relationships between the turbulent flux and the mean/variance gradient. Further, (9) loses the information about the concentration-velocity covariance and the velocity autocorrelation. The resulting errors are not negligible at the microscale, 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:
(10)εC≡-2DM∂C′∂xj2¯=2C′DM∂2C′∂xj2¯.
Considering (4) and (5), assuming a turbulent regime and that reaction rates do not affect the dissipation term, (10) can be expressed as follows:
(11)εC=2C′dC′dt-T′¯=2C′dCdt¯-C′dC¯dt¯-C′T′¯=2C′dCdt¯.
IECM (Interaction by the Exchange with the Conditional Mean; [15, 28]) represents a major micromixing formulation, 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:
(12)dCdt=-C-C∣u_tm,
where C∣U_ represents the mean concentration conditioned on the Lagrangian velocity 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 (11) and (12), [15] obtains the following expression:
(13)εC=-2C′C-C∣u_tm¯=-2C′C¯tm+σC2tm-C′C∣u_tm¯=-2tmσC2-C′C∣u_¯.
Further, in case of uniform mean scalar gradient and 1D dispersion, [15] reports the following expression for the conditional mean:
(14)C∣u=C¯+C′u′¯σu2u,
where σu stands for the standard deviation of velocity. This parameter can be provided by diagnostic tools, RANS codes (e.g., [32–35]), or LES models, which allow a detailed characterization of the turbulent structure of the atmospheric boundary layer (e.g., [36–38]).Considering both (14) and (9), the dissipation rate of σC2 becomes
(15)εC=-2tmσC2-C′u′¯2σu2=-2tmσC2-KT2σu2∂C¯∂x2,hhhhhhhhhhhhhhhhhhhhhhhhhσC2≥KT2σu2∂C¯∂x2.
As we consider 1D dispersion phenomena, the plume spread has the same dimension as the boundary layer height. Under these conditions, the mixing time can be related to the turbulent kinetic energy (q) and its dissipation rate (ε) via Richardson constant (Cφ; [15]):
(16)tm=2qCϕε=3C02CϕTL,TL=2σu2C0ε,
where the Lagrangian integral time scale (TL) is introduced. Its relationship with the dissipation rate of the turbulent kinetic—last equation in (16)—is reported by several authors (e.g., [15]). This relationship can be derived applying Taylor analysis and comparing the formulas of the plume spread alternatively depending on the Lagrangian structure function or the Lagrangian integral time scale.Taking into account (16), the dissipation term becomes
(17)εC=-4Cϕ3C0TLσC2-KT2σu2∂C¯∂x2.
In conclusion, we can refer to a couple of simplified formulations for the dissipation rate of the concentration variance: a 3D generic formulation (13) and a simplified 1D expression in case of uniform mean scalar gradient (17).


### 2.4. Representation of the Reactive Terms

The role of the reactive term in the balance equation of the concentration variance is here discussed, by alternatively considering 2nd-order kinetics, 1st-order kinetics, and 0th-order kinetics.In case of 2nd order kinetics, the reactive termT is represented by
(18)T=-rCACB,CA≡C,
where species A and B are reactants and r is the reaction rate.The ensemble mean of (18) is equal to
(19)T¯=-rCACB¯=-rCA¯+CA′CB¯+CB′¯=-rCA-CB-+CA′CB′¯
and the ratio between the last two terms in (19) defines the segregation coefficient
(20)Is≡CA′CB′¯CA¯CB¯.
After considering the fluctuation of (18):
(21)T′=T-T¯=-rCACB-rCA¯CB¯+CA′CB′¯=-rCA¯+CA′CB¯+CB′-CA¯CB¯+CA′CB′¯=-rCA¯CB′+CA′CB¯+CA′CB′-CA′CB′¯
the reactive term in the balance equation of the variance becomes
(22)R2¯=2CA′T′¯=-2rCA¯CA′CB′+CA′2CB¯+CA′2CB′-CA′CA′CB′¯¯=-2rCA¯CA′CB′¯+σCA2CB¯+CA′2CB′¯.
The first term in the final formulation of (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]
(23)CA′2CB′¯≈ISCA′CB′¯CA¯+σCA2CB¯.
The advantage of (23) simply relies on the fact that, in the limit of reactants never coexisting (Is=-1), (23) guarantees that R2¯ is exactly zero.1st-order kinetics can be simply represented by imposingCB=1 in (22):
(24)R2¯=2CA′T′¯=-2rσCA2.
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; thus, it locally decreases the concentration variance (provided the same mean scalar gradient) according to (24).Finally, a 0th order kinetics (e.g.,T=-r) is equivalent to considering both CA=1 and CB=1 in (22). In this case, the concentration variance does not depend on the reaction rate:
(25)R2¯=0.
In conclusion, the expressions (22), (24), and (25) alternatively represent the reactive term in the balance equation of the concentration variance, in case of 2nd-order, 1st-order, and 0th-order kinetics reactions, respectively.


### 2.5. 1D Analytical Solution of Sawford (2004) under Uniform and Nonstationary Conditions

Sawford [15] reported a 1D analytical solution of the concentration variance under homogeneous and nonstationary 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 (8), (9), (15), and (24), provided a uniform concentration variance:
(26)∂σC2∂t=2KT∂C¯∂x2+2KT2σw2tm∂C¯∂x2-2σC2tm=4KT∂C¯∂x2-2σC2tm.
Defining the constant “a,” one can write
(27)∂σC2∂t=-2σC2-atm/2tm,a≡4KT∂C¯∂x2.
After integration from the initial time (t=0) to the generic time t, one obtains
(28)∫σC20σC2tdσC2σC2-atm/2=∫0t-2dttm⟹ln⁡σC2t-atm2-ln⁡σC20-atm2=-2ttm⟹σC2-atm2=σC20-atm2e-2(t/tm)⟹σC2=σC20e-2(t/tm)+atm21-e-2(t/tm)⟹σC2=σC20e-(4Cϕ/3C0)(t/TL)+3C0CϕKTTL∂C¯∂x21-e-4Cϕ/3C0t/TL.
Finally, considering (38), as explained in the following section, the uniform time-dependent solution for the concentration variance becomes
(29)σC2=σC20e-(4Cϕ/3C0)(t/TL)+3C0Cϕσu2TL2∂C¯∂x21-e-4Cϕ/3C0t/TL.
The solution tends to an equilibrium value, when the production and the dissipation terms equalize:
(30)σC2t⟶∞=3C0Cϕσu2TL2∂C¯∂x2.
This formula highlights the importance of modelling the dissipation term in (26). When this term is absent (this is the case of Lagrangian models without any micromixing scheme or Eulerian models without any dissipation term for σC2), the concentration variance linearly grows with time, indefinitely:
(31)∂σC2∂t=2KT∂C¯∂x2⟹σC2=2KT∂C¯∂x2t=2σu2TLt∂C¯∂x2.
In case of a null mean scalar gradient
(32)∂σC2∂t=-σC22tm+r
there is no production term and the variance tends to zero, as follows:
(33)σC2=σC20e-2/TL+r.


## 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(1)∂C∂t+uj∂C∂xj=DM∂2C∂xj2+T,
where C is the instantaneous concentration, DM the molecular diffusion coefficient, and u_ the velocity vector. Einstein notation applies to the subscript “j” hereafter, if not otherwise stated. From left to right, the terms of (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 expressed according to Reynolds decomposition. Reynolds average (over-bar symbol) of (1) provides
(2)∂C¯+C′∂t¯+uj¯+uj′∂C¯+C′∂xj¯=DM∂2C¯+C′∂xj2¯+T¯+T′¯.
Hereafter, the apex “′” denotes a turbulent fluctuation. Introducing the continuity equations for an incompressible turbulent flow
(3)∂uj∂xj=0,∂uj¯∂xj=0,∂uj′∂xj=0
into (2), one can write the balance equation for the mean concentration of a passive scalar:
(4)∂C¯∂t+uj¯∂C∂xj¯+uj′∂C′∂xj¯=DM∂2C¯∂xj2+T¯.
From left to right, the terms of (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 (4) from (1), the balance equation for the concentration fluctuation is obtained:
(5)∂C′∂t+uj¯∂C′∂xj+uj′∂C¯∂xj+uj′∂C′∂xj-uj′∂C′∂xj¯=DM∂2C′∂xj2+T′.
After multiplying (5) times 2C′ and considering the following equality:
(6)∂∂xj2C′∂C′∂xj=∂∂xj∂C′2∂xj=2C′∂2C′∂xj2+2∂C′∂xji2
one can write
(7)∂C′2∂t+uj¯∂C′2∂xj+2uj′C′∂C¯∂xj+uj′∂C′2∂xj-2C′uj′∂C′∂xj¯=DM∂2C′2∂xj2-2∂C′∂xj2+2C′T′.
Averaging (7), one obtains the balance equation of the concentration variance of a reactive scalar dispersed in a turbulent incompressible flow:
(8)∂σC2∂t+uj¯∂σC2∂xj+∂uj′C′2¯∂xj=-2uj′C′¯∂C¯∂xj-2DM∂C′∂xj2¯+DM∂2σC2∂xj2+2C′T′¯,
where the terms on the left hand side represent the local rate of change of σC2, the advective term and the divergence of the turbulent flux of the concentration variance, respectively. On the right-hand side, (8) shows the production term of σC2 (always nonnegative), the dissipation rate of the concentration variance due to molecular diffusion (εC=-2DM∂C′/∂xj2¯ always nonpositive), the divergence of the molecular diffusion flux of σC2 (negligible) and the reactive term (R2¯≡2C′T′¯), which quantifies the direct effects of chemical and physical reactions.The production term in (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. Parameterizations of the Turbulent Fluxes

The turbulent fluxes in the balance equation of the concentration variance (8) assume approximated and 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]:
(9)uj′C′¯=-KT,j∂C¯∂xj,uj′C′2¯=-Ks,j∂σC2∂xj.
Here, KT,j and Ks,j are 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 parameterizations are commonly used in Eulerian numerical models, although they introduced several limitations. The most important shortcomings emerge in case of strongly nonlinear relationships between the turbulent flux and the mean/variance gradient. Further, (9) loses the information about the concentration-velocity covariance and the velocity autocorrelation. The resulting errors are not negligible at the microscale, 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:
(10)εC≡-2DM∂C′∂xj2¯=2C′DM∂2C′∂xj2¯.
Considering (4) and (5), assuming a turbulent regime and that reaction rates do not affect the dissipation term, (10) can be expressed as follows:
(11)εC=2C′dC′dt-T′¯=2C′dCdt¯-C′dC¯dt¯-C′T′¯=2C′dCdt¯.
IECM (Interaction by the Exchange with the Conditional Mean; [15, 28]) represents a major micromixing formulation, 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:
(12)dCdt=-C-C∣u_tm,
where C∣U_ represents the mean concentration conditioned on the Lagrangian velocity 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 (11) and (12), [15] obtains the following expression:
(13)εC=-2C′C-C∣u_tm¯=-2C′C¯tm+σC2tm-C′C∣u_tm¯=-2tmσC2-C′C∣u_¯.
Further, in case of uniform mean scalar gradient and 1D dispersion, [15] reports the following expression for the conditional mean:
(14)C∣u=C¯+C′u′¯σu2u,
where σu stands for the standard deviation of velocity. This parameter can be provided by diagnostic tools, RANS codes (e.g., [32–35]), or LES models, which allow a detailed characterization of the turbulent structure of the atmospheric boundary layer (e.g., [36–38]).Considering both (14) and (9), the dissipation rate of σC2 becomes
(15)εC=-2tmσC2-C′u′¯2σu2=-2tmσC2-KT2σu2∂C¯∂x2,hhhhhhhhhhhhhhhhhhhhhhhhhσC2≥KT2σu2∂C¯∂x2.
As we consider 1D dispersion phenomena, the plume spread has the same dimension as the boundary layer height. Under these conditions, the mixing time can be related to the turbulent kinetic energy (q) and its dissipation rate (ε) via Richardson constant (Cφ; [15]):
(16)tm=2qCϕε=3C02CϕTL,TL=2σu2C0ε,
where the Lagrangian integral time scale (TL) is introduced. Its relationship with the dissipation rate of the turbulent kinetic—last equation in (16)—is reported by several authors (e.g., [15]). This relationship can be derived applying Taylor analysis and comparing the formulas of the plume spread alternatively depending on the Lagrangian structure function or the Lagrangian integral time scale.Taking into account (16), the dissipation term becomes
(17)εC=-4Cϕ3C0TLσC2-KT2σu2∂C¯∂x2.
In conclusion, we can refer to a couple of simplified formulations for the dissipation rate of the concentration variance: a 3D generic formulation (13) and a simplified 1D expression in case of uniform mean scalar gradient (17).


## 2.4. Representation of the Reactive Terms

The role of the reactive term in the balance equation of the concentration variance is here discussed, by alternatively considering 2nd-order kinetics, 1st-order kinetics, and 0th-order kinetics.In case of 2nd order kinetics, the reactive termT is represented by
(18)T=-rCACB,CA≡C,
where species A and B are reactants and r is the reaction rate.The ensemble mean of (18) is equal to
(19)T¯=-rCACB¯=-rCA¯+CA′CB¯+CB′¯=-rCA-CB-+CA′CB′¯
and the ratio between the last two terms in (19) defines the segregation coefficient
(20)Is≡CA′CB′¯CA¯CB¯.
After considering the fluctuation of (18):
(21)T′=T-T¯=-rCACB-rCA¯CB¯+CA′CB′¯=-rCA¯+CA′CB¯+CB′-CA¯CB¯+CA′CB′¯=-rCA¯CB′+CA′CB¯+CA′CB′-CA′CB′¯
the reactive term in the balance equation of the variance becomes
(22)R2¯=2CA′T′¯=-2rCA¯CA′CB′+CA′2CB¯+CA′2CB′-CA′CA′CB′¯¯=-2rCA¯CA′CB′¯+σCA2CB¯+CA′2CB′¯.
The first term in the final formulation of (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]
(23)CA′2CB′¯≈ISCA′CB′¯CA¯+σCA2CB¯.
The advantage of (23) simply relies on the fact that, in the limit of reactants never coexisting (Is=-1), (23) guarantees that R2¯ is exactly zero.1st-order kinetics can be simply represented by imposingCB=1 in (22):
(24)R2¯=2CA′T′¯=-2rσCA2.
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; thus, it locally decreases the concentration variance (provided the same mean scalar gradient) according to (24).Finally, a 0th order kinetics (e.g.,T=-r) is equivalent to considering both CA=1 and CB=1 in (22). In this case, the concentration variance does not depend on the reaction rate:
(25)R2¯=0.
In conclusion, the expressions (22), (24), and (25) alternatively represent the reactive term in the balance equation of the concentration variance, in case of 2nd-order, 1st-order, and 0th-order kinetics reactions, respectively.


## 2.5. 1D Analytical Solution of Sawford (2004) under Uniform and Nonstationary Conditions

Sawford [15] reported a 1D analytical solution of the concentration variance under homogeneous and nonstationary 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 (8), (9), (15), and (24), provided a uniform concentration variance:
(26)∂σC2∂t=2KT∂C¯∂x2+2KT2σw2tm∂C¯∂x2-2σC2tm=4KT∂C¯∂x2-2σC2tm.
Defining the constant “a,” one can write
(27)∂σC2∂t=-2σC2-atm/2tm,a≡4KT∂C¯∂x2.
After integration from the initial time (t=0) to the generic time t, one obtains
(28)∫σC20σC2tdσC2σC2-atm/2=∫0t-2dttm⟹ln⁡σC2t-atm2-ln⁡σC20-atm2=-2ttm⟹σC2-atm2=σC20-atm2e-2(t/tm)⟹σC2=σC20e-2(t/tm)+atm21-e-2(t/tm)⟹σC2=σC20e-(4Cϕ/3C0)(t/TL)+3C0CϕKTTL∂C¯∂x21-e-4Cϕ/3C0t/TL.
Finally, considering (38), as explained in the following section, the uniform time-dependent solution for the concentration variance becomes
(29)σC2=σC20e-(4Cϕ/3C0)(t/TL)+3C0Cϕσu2TL2∂C¯∂x21-e-4Cϕ/3C0t/TL.
The solution tends to an equilibrium value, when the production and the dissipation terms equalize:
(30)σC2t⟶∞=3C0Cϕσu2TL2∂C¯∂x2.
This formula highlights the importance of modelling the dissipation term in (26). When this term is absent (this is the case of Lagrangian models without any micromixing scheme or Eulerian models without any dissipation term for σC2), the concentration variance linearly grows with time, indefinitely:
(31)∂σC2∂t=2KT∂C¯∂x2⟹σC2=2KT∂C¯∂x2t=2σu2TLt∂C¯∂x2.
In case of a null mean scalar gradient
(32)∂σC2∂t=-σC22tm+r
there is no production term and the variance tends to zero, as follows:
(33)σC2=σC20e-2/TL+r.


## 3. 1D Solutions for the Balance Equation of the Concentration Variance under Stationary Conditions

### 3.1. Main Solution under Stationary Conditions

Provided 1D stationary conditions and homogeneous turbulence, the balance equation of the concentration variance (8) assumes the following form:
(34)u¯∂σC2∂x+∂u′C′2¯∂x=-2u′C′¯∂C¯∂x-2DM∂C′∂x2¯+2C′T′¯.
Introducing the parameterizations for the turbulent fluxes (9) and εC  (17), as well as the expression of the reactive term in case of 1st order kinetics reaction (24), one obtains
(35)u¯∂σC2∂x-KT∂2σC2∂x2=2KT∂C¯∂x2-4Cϕ3C0TLσC2-KT2σu2∂C¯∂x2-2rσC2.
After assuming the definition of the turbulent kinetic energy in 1D:
(36)q=32σu2
considering (16) and (36) and dividing by KT, (35) becomes
(37)∂2σC2∂x2-u¯KT∂σC2∂x-1KTCϕεq+2rσC2+2∂C¯∂x21+KTεCϕ2qσu2=0.
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:
(38)KT=σu2TL.
The system (16), (36), and (38) provides another expression for the turbulent dispersion coefficient
(39)KT=89q2C0ε
and allows writing (37) with no explicit dependency on KT:
(40)∂2σC2∂x2-98u¯C0εq2∂σC2∂x-98C0εq2Cϕεq+2rσC2+2∂C¯∂x21+23CϕC0=0.
Equation (40) represents a 2nd-order inhomogeneous ODE (Ordinary Differential Equation)
(41)∂2σC2∂x2+a∂σC2∂x+bσC2+c=0,Δ≡a2-4b≥0
with the following constant coefficients:
(42)a≡-98u¯C0εq2,b≡-98C0εq2Cϕεq+2r,c≡2∂C¯∂x21+23CϕC0.
Its general solution assumes the form
(43)σC2=c1eλ1x+c2eλ2x-cb
which is completed by the following definition:
(44)λ1,2≡-a±a2-4ab2=1298u¯C0εq2±92C0εq2εq932u¯2C0q+Cϕ+2r.
One can verify that the system (43) and (44) is a general solution of (41), as briefly described in the following. According to (43), the left hand side of (41) becomes
(45)λ12c1eλ1x+λ22c2eλ2x+aλ1c1eλ1x+λ2c2eλ2x+bc1eλ1x+c2eλ2x-cb+c.
The value of λi2 can be derived from (44):
(46)λ1,22=-a±a2-4b22=a24+a2-4b4∓a2a2-4b=a22-b∓a2a2-4b.
Replacing λi and λi2 in (45), according to (44) and (46), one obtains
(47)a22-b-a2a2-4bc1eλ1x+a22-b+a2a2-4bc2eλ2x+a-a+a2-4b2c1eλ1xhhhHhh+-a-a2-4b2c2eλ2x+bc1eλ1x+c2eλ2x-cb+c
which finally verifies that the system (43) and (44) is a general solution of (41):
(48)c1eλ1xa22-b-a2a2-4b+-a2+aa2-4b2+b+c2eλ2x·a22-b+a2a2-4b+-a2-aa2-4b2+b-c+c=0.
Dirichlet boundary conditions can now be imposed on the left boundary (x=0)
(49)σC2x=0≡σC02⟹σC02=c1+c2-cb⟹c1=σC02-c2+cb
and the right boundary (x=L) of the domain
(50)σC2x=L≡σCL2⟹σCL2=c1eλ1L+c2eλ2L-cb⟹c2eλ2L=σCL2-c1eλ1L+cb.
Combining (49) with (50), one can obtain the value of c2:
(51)c2eλ2L=σCL2-σC02-c2+cbeλ1L+cb⟹c2eλ2L-eλ1L=σCL2-σC02+cbeλ1L+cb⟹c2=σC02eλ1L-σCL2+c/beλ1L-1eλ1L-eλ2L.
Thus, the constant c1 from (49) becomes
(52)c1=σCL2-σC02eλ2L-(c/b)eλ2L-1eλ1L-eλ2L.
Considering the values of c1 and c2 from (52) and (51), respectively, (43) assumes the following form:
(53)σC2=σCL2-σC02eλ2L-(c/b)eλ2L-1eλ1L-eλ2Leλ1x+σC02eλ1L-σCL2+c/beλ1L-1eλ1L-eλ2Leλ2x-cb.
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:
(54)σC2=-cbeλ1xeλ2L-1-eλ2xeλ1L-1hhhhhhh+eλ1L-eλ2L+σC02eλ1L+λ2x-eλ1x+λ2Lhhhcbeλ1xeλ2L-1-eλ2xeλ1L-1+σCL2eλ1x-eλ2x1eλ1L-eλ2L,λ1,2≡1298u¯C0εq2±92C0εq2εq932u¯2C0q+Cϕ+2r,b≡-98C0εq2Cϕεq+2r,c≡2∂C¯∂x21+23CϕC0.
It is immediate to verify that (54) satisfies the boundary conditions imposed.The solution (54) is widely discussed and analysed in Section 4, where several examples are available (e.g., Figure 1), in terms of both nondimensional and dimensional physical quantities. Hereafter, we just provide some clarification about the role of the reactive term and the mean scalar gradient.Figure 1
1D analytical solution for the balance equation of the concentration variance, under stationary conditions. Legend: + solution with reference parameters; × solution with reference parameters in the absence of mean velocity andrN=0; ∗ solution with reference parameters in the absence of mean velocity, null variance at boundaries, and rN=0.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:(55)∂C¯∂x=DTu¯∂2C¯∂x2-ru¯C¯.
This means that (54) can only consider a uniform mean gradient as an approximated condition for both 1st and 2nd order kinetics reactions.On the other hand, an exact uniform scalar mean gradient can relate to a 0th-order kinetics:(56)∂C¯∂t+u¯∂C¯∂x=DT∂2C¯∂x2-r⟹∂C¯∂x=DTu¯∂2C¯∂x2-ru¯=-ru¯.
However, in this case there is no need for a reactive term in the balance equation of the variance, according to (25).The mean scalar gradient is the key parameter in the production term of the concentration variance and affects the constantc in (54). ∂C¯/∂x should be approximately uniform and can be calculated by 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, (44) becomes
(57)λ1,2=±32C0ε2q2εCϕq+2r,Δ=0=u¯,λ≡λ1=-λ2.
Under these conditions, we can derive a simpler and alternative formulation of the general solution (54). Considering that λ≡λ1=-λ2, the first line of (54) becomes
(58)σC2=-cbeλxeλL-1-e-λxeλL-1+σC02eλL-x-1eλL-x+σCL2eλx-1eλxhhhhh+eλL-e-λL+σC02eλL-x-1eλL-x+σCL2eλx-1eλx·1eλL-e-λL.
Introducing the following hyperbolic functions (and their derivatives)
(59)cosh⁡x=ex+e-x2,sinh⁡x=ex-e-x2,cosh⁡′x=sinh⁡x,sinh⁡′x=cosh⁡x
into (58), one can obtain the following expression:
(60)σC2=cλ21-eλL-1e2λL-1eλx+eλL-x+1senhλLσC,02senhλL-x+σC,L2senhλx.
This change in the reference system
(61)x′=x-L2⟹x=x′+L2
implies that
(62)eλx+eλL-x=eλx′+L/2+eλL/2-x′=eλ(L/2)eλx′+1eλx′=eλ(L/2)cosh⁡λx′,senhλL-x=senhλL2-x′,senhλx=senhλL2+x′.
Combining (60), (61), and (62), the simplified solution of the 1D stationary balance equation of the concentration variance, in the absence of mean velocity, assumes the following expression:
(63)σC2=cλ21-eλL-1e2λL-1eλ(L/2)cosh⁡λx′+1senhλL·σC,02senhλL2-x′+σC,L2senhλL2+x′,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhλ=32εqC0Cϕ2q.
This solution, represented in Figure 1 (e.g., the green plot), is verified, as described in the following. One may first notice that
(64)b=λ2,∂2σC2∂x2=∂2σC2∂x′2,σC2≡A+Bcosh⁡λx′+CσC,02senhλL2-x′hhhhh+σC,L2senhλL2+x′,
where B and C are constants, directly defined by comparison with (63).One can consider the second derivative of the variance in the new reference system:(65)∂2σC2∂x′2=λ2Bcosh⁡x′+λ2CσC,02senhλL2-x′hhhhhh+σC,L2senhλL2+x′=σC2+cbλ2
and then write (41), according to (63) and (65):
(66)∂2σC2∂x′2-λ2σC2+c=λ2cb+c=0
which finally verifies that (63) is the solution of (41), in the absence of mean velocity.Under these conditions, we can simply write the derivative of the concentration variance:(67)∂σC2∂x′=λBsenhλx′-λCσC,02cosh⁡λL2-x′hhhhh+λσC,L2cosh⁡λL2+x′.
Although it is not immediate to get a general expression to locate the maxima/minima, a simple solution is obtained in case σC,02=σC,L2, as plotted in Figure 1 (blue line). This assumption provides a maximum or a minimum at the domain centre (x′=0), in case σC,02<σC,eq2 or σC,02>σC,eq2, respectively, according to the definition of the equilibrium variance (68), as provided in Section 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, the dissipation termεC and the reactive term (24) exactly equal the production term of σC2 (“equilibrium solution”):
(68)σC,eq2=-cb=169∂C¯/∂x21+2/3Cϕ/C0C0ε/q2Cϕε/q+2r.
This expression also represents the value of the horizontal inflection points in the general solution (54). Under these conditions, the divergence of both the turbulent and the mean transport scalar fluxes is null.


## 3.1. Main Solution under Stationary Conditions

Provided 1D stationary conditions and homogeneous turbulence, the balance equation of the concentration variance (8) assumes the following form:
(34)u¯∂σC2∂x+∂u′C′2¯∂x=-2u′C′¯∂C¯∂x-2DM∂C′∂x2¯+2C′T′¯.
Introducing the parameterizations for the turbulent fluxes (9) and εC  (17), as well as the expression of the reactive term in case of 1st order kinetics reaction (24), one obtains
(35)u¯∂σC2∂x-KT∂2σC2∂x2=2KT∂C¯∂x2-4Cϕ3C0TLσC2-KT2σu2∂C¯∂x2-2rσC2.
After assuming the definition of the turbulent kinetic energy in 1D:
(36)q=32σu2
considering (16) and (36) and dividing by KT, (35) becomes
(37)∂2σC2∂x2-u¯KT∂σC2∂x-1KTCϕεq+2rσC2+2∂C¯∂x21+KTεCϕ2qσu2=0.
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:
(38)KT=σu2TL.
The system (16), (36), and (38) provides another expression for the turbulent dispersion coefficient
(39)KT=89q2C0ε
and allows writing (37) with no explicit dependency on KT:
(40)∂2σC2∂x2-98u¯C0εq2∂σC2∂x-98C0εq2Cϕεq+2rσC2+2∂C¯∂x21+23CϕC0=0.
Equation (40) represents a 2nd-order inhomogeneous ODE (Ordinary Differential Equation)
(41)∂2σC2∂x2+a∂σC2∂x+bσC2+c=0,Δ≡a2-4b≥0
with the following constant coefficients:
(42)a≡-98u¯C0εq2,b≡-98C0εq2Cϕεq+2r,c≡2∂C¯∂x21+23CϕC0.
Its general solution assumes the form
(43)σC2=c1eλ1x+c2eλ2x-cb
which is completed by the following definition:
(44)λ1,2≡-a±a2-4ab2=1298u¯C0εq2±92C0εq2εq932u¯2C0q+Cϕ+2r.
One can verify that the system (43) and (44) is a general solution of (41), as briefly described in the following. According to (43), the left hand side of (41) becomes
(45)λ12c1eλ1x+λ22c2eλ2x+aλ1c1eλ1x+λ2c2eλ2x+bc1eλ1x+c2eλ2x-cb+c.
The value of λi2 can be derived from (44):
(46)λ1,22=-a±a2-4b22=a24+a2-4b4∓a2a2-4b=a22-b∓a2a2-4b.
Replacing λi and λi2 in (45), according to (44) and (46), one obtains
(47)a22-b-a2a2-4bc1eλ1x+a22-b+a2a2-4bc2eλ2x+a-a+a2-4b2c1eλ1xhhhHhh+-a-a2-4b2c2eλ2x+bc1eλ1x+c2eλ2x-cb+c
which finally verifies that the system (43) and (44) is a general solution of (41):
(48)c1eλ1xa22-b-a2a2-4b+-a2+aa2-4b2+b+c2eλ2x·a22-b+a2a2-4b+-a2-aa2-4b2+b-c+c=0.
Dirichlet boundary conditions can now be imposed on the left boundary (x=0)
(49)σC2x=0≡σC02⟹σC02=c1+c2-cb⟹c1=σC02-c2+cb
and the right boundary (x=L) of the domain
(50)σC2x=L≡σCL2⟹σCL2=c1eλ1L+c2eλ2L-cb⟹c2eλ2L=σCL2-c1eλ1L+cb.
Combining (49) with (50), one can obtain the value of c2:
(51)c2eλ2L=σCL2-σC02-c2+cbeλ1L+cb⟹c2eλ2L-eλ1L=σCL2-σC02+cbeλ1L+cb⟹c2=σC02eλ1L-σCL2+c/beλ1L-1eλ1L-eλ2L.
Thus, the constant c1 from (49) becomes
(52)c1=σCL2-σC02eλ2L-(c/b)eλ2L-1eλ1L-eλ2L.
Considering the values of c1 and c2 from (52) and (51), respectively, (43) assumes the following form:
(53)σC2=σCL2-σC02eλ2L-(c/b)eλ2L-1eλ1L-eλ2Leλ1x+σC02eλ1L-σCL2+c/beλ1L-1eλ1L-eλ2Leλ2x-cb.
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:
(54)σC2=-cbeλ1xeλ2L-1-eλ2xeλ1L-1hhhhhhh+eλ1L-eλ2L+σC02eλ1L+λ2x-eλ1x+λ2Lhhhcbeλ1xeλ2L-1-eλ2xeλ1L-1+σCL2eλ1x-eλ2x1eλ1L-eλ2L,λ1,2≡1298u¯C0εq2±92C0εq2εq932u¯2C0q+Cϕ+2r,b≡-98C0εq2Cϕεq+2r,c≡2∂C¯∂x21+23CϕC0.
It is immediate to verify that (54) satisfies the boundary conditions imposed.The solution (54) is widely discussed and analysed in Section 4, where several examples are available (e.g., Figure 1), in terms of both nondimensional and dimensional physical quantities. Hereafter, we just provide some clarification about the role of the reactive term and the mean scalar gradient.Figure 1
1D analytical solution for the balance equation of the concentration variance, under stationary conditions. Legend: + solution with reference parameters; × solution with reference parameters in the absence of mean velocity andrN=0; ∗ solution with reference parameters in the absence of mean velocity, null variance at boundaries, and rN=0.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:(55)∂C¯∂x=DTu¯∂2C¯∂x2-ru¯C¯.
This means that (54) can only consider a uniform mean gradient as an approximated condition for both 1st and 2nd order kinetics reactions.On the other hand, an exact uniform scalar mean gradient can relate to a 0th-order kinetics:(56)∂C¯∂t+u¯∂C¯∂x=DT∂2C¯∂x2-r⟹∂C¯∂x=DTu¯∂2C¯∂x2-ru¯=-ru¯.
However, in this case there is no need for a reactive term in the balance equation of the variance, according to (25).The mean scalar gradient is the key parameter in the production term of the concentration variance and affects the constantc in (54). ∂C¯/∂x should be approximately uniform and can be calculated by 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, (44) becomes
(57)λ1,2=±32C0ε2q2εCϕq+2r,Δ=0=u¯,λ≡λ1=-λ2.
Under these conditions, we can derive a simpler and alternative formulation of the general solution (54). Considering that λ≡λ1=-λ2, the first line of (54) becomes
(58)σC2=-cbeλxeλL-1-e-λxeλL-1+σC02eλL-x-1eλL-x+σCL2eλx-1eλxhhhhh+eλL-e-λL+σC02eλL-x-1eλL-x+σCL2eλx-1eλx·1eλL-e-λL.
Introducing the following hyperbolic functions (and their derivatives)
(59)cosh⁡x=ex+e-x2,sinh⁡x=ex-e-x2,cosh⁡′x=sinh⁡x,sinh⁡′x=cosh⁡x
into (58), one can obtain the following expression:
(60)σC2=cλ21-eλL-1e2λL-1eλx+eλL-x+1senhλLσC,02senhλL-x+σC,L2senhλx.
This change in the reference system
(61)x′=x-L2⟹x=x′+L2
implies that
(62)eλx+eλL-x=eλx′+L/2+eλL/2-x′=eλ(L/2)eλx′+1eλx′=eλ(L/2)cosh⁡λx′,senhλL-x=senhλL2-x′,senhλx=senhλL2+x′.
Combining (60), (61), and (62), the simplified solution of the 1D stationary balance equation of the concentration variance, in the absence of mean velocity, assumes the following expression:
(63)σC2=cλ21-eλL-1e2λL-1eλ(L/2)cosh⁡λx′+1senhλL·σC,02senhλL2-x′+σC,L2senhλL2+x′,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhλ=32εqC0Cϕ2q.
This solution, represented in Figure 1 (e.g., the green plot), is verified, as described in the following. One may first notice that
(64)b=λ2,∂2σC2∂x2=∂2σC2∂x′2,σC2≡A+Bcosh⁡λx′+CσC,02senhλL2-x′hhhhh+σC,L2senhλL2+x′,
where B and C are constants, directly defined by comparison with (63).One can consider the second derivative of the variance in the new reference system:(65)∂2σC2∂x′2=λ2Bcosh⁡x′+λ2CσC,02senhλL2-x′hhhhhh+σC,L2senhλL2+x′=σC2+cbλ2
and then write (41), according to (63) and (65):
(66)∂2σC2∂x′2-λ2σC2+c=λ2cb+c=0
which finally verifies that (63) is the solution of (41), in the absence of mean velocity.Under these conditions, we can simply write the derivative of the concentration variance:(67)∂σC2∂x′=λBsenhλx′-λCσC,02cosh⁡λL2-x′hhhhh+λσC,L2cosh⁡λL2+x′.
Although it is not immediate to get a general expression to locate the maxima/minima, a simple solution is obtained in case σC,02=σC,L2, as plotted in Figure 1 (blue line). This assumption provides a maximum or a minimum at the domain centre (x′=0), in case σC,02<σC,eq2 or σC,02>σC,eq2, respectively, according to the definition of the equilibrium variance (68), as provided in Section 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, the dissipation termεC and the reactive term (24) exactly equal the production term of σC2 (“equilibrium solution”):
(68)σC,eq2=-cb=169∂C¯/∂x21+2/3Cϕ/C0C0ε/q2Cϕε/q+2r.
This expression also represents the value of the horizontal inflection points in the general solution (54). Under these conditions, the divergence of both the turbulent and the mean transport scalar fluxes is null.


## 4. Sensitivity Analysis

The solutions (54), (63), and (68) of the 1D balance equation of the concentration variance are here discussed and analysed by means of a sensitivity analysis.The results are presented in terms of nondimensional parameters: all the physical quantities are normalized (“N”) over the reference time, length, and mass scales, respectively: t0=TL=2σu2/C0ε, x0=L, and m0=∂C¯/∂xL2. Thus, the main nondimensional parameters, LN, ∂C¯/∂xN and TL,N, are always equal to unity, whereas the other nondimensional quantities vary according to the following expressions: u¯N=u¯TL/L, σu,N=σu*TL/L, rN=r*TL, and σC,N2=σC2/∂C¯/∂xL2.The “reference solution” of this analysis is defined by the following arbitrary choice, which simply provides a meaningful and general reference configuration:uref,N=σu,ref,N=1/6, C0=2, Cφ=3, rN=1, σC,0,N2=0.00833, and σC,L,N2=0.05833. Under these conditions, it follows that σC,eq,N2=1/30 and εN=2σu,N2/C0TL,N=1/36.Discussion first considers some generic features of the solution (54), as plotted in Figure 1 under three different configurations.The reference solution (red plot) shows a central region of very low gradients (very weak turbulent and mean transports ofσC2), where the production and the dissipation terms almost balance each other.In this context, the reactive term plays an analogous role to the dissipation termεC 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 nonnegative 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 ofσC2 production (low variance) and towards the regions in excess of σC2 dissipation (high variance), in order to establish a stationary regime.Figure1 also reports a couple of examples of the solution for passive scalars (63) in the absence of mean velocity (green plot). In particular, the blue line represents the symmetric solution cited at the end of Section 3.2.Figure2 reports a sensitivity analysis of the main solution (54) on the choice of the nondimensional boundary values. Let us first examine Figure 2(a), 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 (68). In the first case (σC,L,N2=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 flux transports the excess of the concentration variance away from the domain. The mean velocity makes the solution be slightly asymmetric, with a peak located on the right half of the domain. In the second case (σC,L,N2=σC,eq,N2), the solution is almost uniform in the right half of the domain as the equilibrium value is already reached around the domain centre. Finally, when (σC,L,N2=2σC,eq,N2), the solution is almost symmetric to the first case (σC,L,N2=0) with respect to the equilibrium line, for x>L/2. In this region, the turbulent fluxes balance the deficiency in the variance production by transporting the concentration variance from the right boundary towards the domain centre.Reference nondimensional solution with sensitivity analysis on the boundary nondimensional valuesσC,N2, which are alternatively equal to 0, 1/30, and 1/15 (all the independent combinations are reported).
(a)
(b)Figure2(b) considers the uniform solution of (68), established with the choice σC,0,N2=σC,L,N2=σC,eq,N2: the production term is everywhere balanced by the dissipation terms, without any effects of neither turbulent fluxes nor advection. The other solutions refer to (σC,L,N2=2σC,eq,N2) and are almost identical in the right half of the domain. Let us notice that the solution with (σC,0,N2=σC,L,N2=2σC,eq,N2) is symmetrical to the configuration with (σC,0,N2=σC,L,N2=0; Figure 2(a)), with respect to the line of the equilibrium value.Figure3 investigates the role of the nondimensional 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 point is established (red plot). The solution is symmetric with respect to its central value. Increasing the nondimensional mean velocity, the relative importance of the mean transport grows and the upstream boundary value of σC,N2 influences the solution more and more noticeably; further, the solution is not symmetric anymore and the equilibrium value is located in the right half of the domain.Figure 3
Reference nondimensional solution with sensitivity analysis on the nondimensional mean velocity, which is alternatively equal to 0 (+), 1/6 (×), 1/2 (∗), and 5/3 (□).The effects of the nondimensional standard deviation of velocity, whose square is directly proportional to the nondimensional turbulent kinetic energy, are shown in Figure4(a). An increase in q determines a highest variance production all over the domain. Considering the highest and the lowest values of σu,N, the nondimensional variance lies out of the range provided by the Dirichlet boundary values, in the most of the domain. As uN is relatively low, one can always detect at least a maximum, a minimum, or a horizontal inflection point in the profile of σC,N2, in the very inner domain.(a) Reference nondimensional solution with sensitivity analysis on the nondimensional standard deviation of velocity, which is alternatively equal to 1/24 (+), 1/8 (×), 1/6 (∗), 1/4 (□), and 1/3 (■)—σC,eq,N2 is 0.002083, 0.01875, 1/30, 0.075, and 4/30, respectively. (b) Reference nondimensional solution with sensitivity analysis on the nondimensional reaction rate, which is alternatively equal to 0 (+), 1/2 (×), 1 (∗), 2 (□), or 10 (■)—σC,eq,N2 is 1/12, 0.04762, 1/30, 0.02083, and 0.005208, respectively.
(a)
(b)Contrarily, the dependency of the nondimensional variance on the nondimensional reaction rate has an opposite behaviour. In fact, the reactive term always acts as a dissipation term (linear inσC2) and its increase tends to lower down the scalar variance all over the domain (Figure 4(b)). 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.The sensitivities on the Lagrangian integral time scale and the mean scalar gradient are finally shown in Figure5 in terms of dimensional parameters, just to provide a more physical representation of the role of these quantities. An increase in TL or ∂C¯/∂x improves the “production term” with similar effects on the profile of the concentration variance.(a) Reference dimensional solution (butr=0.001/s) with sensitivity analysis on the Lagrangian integral time scale, which is alternatively equal to 50 s (+), 75 s (×), 100 s (∗), 150 s (□), or 200 s (■)—σC,eq2 is 8.694 μg2/m2, 18.90 μg2/m2, 32.507 μg2/m2, 68.66 μg2/m2, and 115 μg2/m2, respectively. (b) Reference dimensional solution (but r=0.001/s) with sensitivity analysis on the scalar mean gradient, which is alternatively equal to 0 μg/m2 (+), 0.1765 μg/m2 (×), 0.353 μg/m2 (∗), 0.44125 μg/m2 (□), or 0.5295 μg/m2 (■)—σC,eq2 is 0 μg2/m2, 8.127 μg2/m2, 32.507 μg2/m2, 50.79 μg2/m2, and 73.14 μg2/m2, respectively.
(a)
(b)


## 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 has detected 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 nondimensional boundary values, reaction rate, mean, and standard deviation of velocity.These analytical 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, microscale air quality and water quality, and several industrial processes, such as combustion. Although the solutions referred 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 solutions represent upwind spatial reconstruction schemes for the concentration variance, as they can be implemented in CFD-RANS codes, as well as in prognostic or diagnostic meteorological and ocean models, which simulate the turbulent fluctuations of reactive pollutants.



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*Source: 424827-2015-01-27.xml*