Advances in Mathematical Physics

Volume 2015, Article ID 424827, 13 pages

http://dx.doi.org/10.1155/2015/424827

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

^{1}Environment and Sustainable Development Department (SFE), Ricerca sul Sistema Energetico (RSE) SpA, Via Rubattino 54, 20134 Milan, Italy^{2}Department of Civil and Environmental Engineering, Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy^{3}ENEA, Casaccia, Via Anguillarese 301, Santa Maria di Galeria, 00123 Rome, Italy

Received 10 November 2014; Accepted 26 December 2014

Academic Editor: Xiao-Jun Yang

Copyright © 2015 Andrea Amicarelli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 () 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 () 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 (), which is also involved in the definition of the intensity of fluctuations ().

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 , 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. Section 2 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 where is the instantaneous concentration, the molecular diffusion coefficient, and the velocity vector. Einstein notation applies to the subscript “” 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 , the divergence of the molecular diffusion flux of , and the instantaneous reactive term ().

Concentration and velocity can be expressed according to Reynolds decomposition. Reynolds average (over-bar symbol) of (1) provides Hereafter, the apex “” denotes a turbulent fluctuation. Introducing the continuity equations for an incompressible turbulent flow into (2), one can write the balance equation for the mean concentration of a passive scalar: From left to right, the terms of (4) represent the local rate of change of the mean concentration, the advective term of , the divergence of the turbulent and the molecular diffusion fluxes of and the mean reactive term.

Subtracting (4) from (1), the balance equation for the concentration fluctuation is obtained: After multiplying (5) times and considering the following equality: one can write Averaging (7), one obtains the balance equation of the concentration variance of a reactive scalar dispersed in a turbulent incompressible flow: where the terms on the left hand side represent the local rate of change of , 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 (always nonnegative), the dissipation rate of the concentration variance due to molecular diffusion ( always nonpositive), the divergence of the molecular diffusion flux of (negligible) and the reactive term (), 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 “-theory,” which is the theory of the turbulent dispersion coefficients, as reported and further developed by several authors [27]: Here, and are the vectors of the turbulent dispersion coefficients of the scalar mean and variance, respectively. They are usually assumed equal and denoted by .

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 (). In particular, this study refers to the formula of [15], here reported and discussed. First, one may notice the following equalities: Considering (4) and (5), assuming a turbulent regime and that reaction rates do not affect the dissipation term, (10) can be expressed as follows: 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: where represents the mean concentration conditioned on the Lagrangian velocity vector () and 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: Further, in case of uniform mean scalar gradient and 1D dispersion, [15] reports the following expression for the conditional mean: where 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 becomes 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 () and its dissipation rate () via Richardson constant (; [15]): where the Lagrangian integral time scale () 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 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 term is represented by where species and are reactants and is the reaction rate.

The ensemble mean of (18) is equal to and the ratio between the last two terms in (19) defines the segregation coefficient After considering the fluctuation of (18): the reactive term in the balance equation of the variance becomes The first term in the final formulation of (22) depends on the reactant covariance, the second one on the variance of the control reactant () and the last one is a triple correlation term, whose eventual parameterization can refer to [39] The advantage of (23) simply relies on the fact that, in the limit of reactants never coexisting (), (23) guarantees that is exactly zero.

1st-order kinetics can be simply represented by imposing in (22): 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., ) is equivalent to considering both and in (22). In this case, the concentration variance does not depend on the reaction rate: 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: Defining the constant “,” one can write After integration from the initial time () to the generic time , one obtains Finally, considering (38), as explained in the following section, the uniform time-dependent solution for the concentration variance becomes The solution tends to an equilibrium value, when the production and the dissipation terms equalize: 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 ), the concentration variance linearly grows with time, indefinitely: In case of a null mean scalar gradient there is no production term and the variance tends to zero, as follows:

#### 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: Introducing the parameterizations for the turbulent fluxes (9) and (17), as well as the expression of the reactive term in case of 1st order kinetics reaction (24), one obtains After assuming the definition of the turbulent kinetic energy in 1D: considering (16) and (36) and dividing by , (35) becomes 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: The system (16), (36), and (38) provides another expression for the turbulent dispersion coefficient and allows writing (37) with no explicit dependency on : Equation (40) represents a 2nd-order inhomogeneous ODE (Ordinary Differential Equation) with the following constant coefficients: Its general solution assumes the form which is completed by the following definition: 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 The value of can be derived from (44): Replacing and in (45), according to (44) and (46), one obtains which finally verifies that the system (43) and (44) is a general solution of (41): Dirichlet boundary conditions can now be imposed on the left boundary () and the right boundary () of the domain Combining (49) with (50), one can obtain the value of : Thus, the constant from (49) becomes Considering the values of and from (52) and (51), respectively, (43) assumes the following form: 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: 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.