1. INTRODUCTION

   To investigate the dispersion mechanisms of air pollutants in densely developed urban areas, numerous works have been conducted. Recently, some approaches to extend CFD (Computational fluid dynamics) simulations to predictions of the transport of reactive air pollutants within urban areas have been reported (Baker et al., 2004; Baik et al., 2006). They revealed the importance of the reactivity of pollutants on the formation of concentration fields of building scales. Turbulnce flucutuation of reactive pollutants' concentrations can affect chemical reaction rate during the dispersion in the atmosphere (Schumann, 1989; Fraigneau et al., 1995). Especially, for the precise prediction of Reynolds-averaged bimolecular reaction rate, the correlation of reactors' concentrations is the key to be modeled (Kikumoto and Ooka, 2011).

   First, the authors of the present study proposed a model to estimate a correlation of turbulent fluctuations of two reactive pollutants' concentrations, which enabled adequate evaluation of bimolecular chemical reaction rate in CFD simulations with RANS (Reynolds-averaged Navier-Stokes equations) model. Secondly, this correlation model was tested in a numerical experiment with LES (Large-eddy simulation) which focused on dispersion and bimolecular chemical reaction in a turbulent channel flow.

2. Modelling of Reaction rate

   When we have a bimolecular reaction of substances c₁ and c₂ such as

      ,                                                                                          (1)

its Reynolds-averaged ("< >" denotes) reaction rate under an isothermal condition can be described as follows.

      .                                                                        (2)

Where k_b is the second-order rate constant. The second term of right hand side of Eq. 2, or correlation term, is a new unknown which has to be modeled in RANS model simulation. A model of the correlation term defined in Eq. 3 was derived by the authors from a transport equation of the correlation assuming the situation of local equilibrium.

      .                                                                             (3)

Where k is turbulent kinetic energy, ƒÃ is dissipation rate of k, and C_cr is a model coefficient. Because this model doesn't need a new transport equation of correlation, and is composed of variables used in usual RANS model except C_cr, it can be implemented in an existing code with relative ease.

3. Numerical test with LES

   A numerical test using LES is conducted to verify the proposed model. It takes up dispersion and bimolecular chemical reaction in a turbulent channel flow. The analysis is performed under an isothermal condition. The static Smagorinsky models (Smagorinsky, 1963) in the form of eddy-viscosity and eddy-diffusivity with the van Driest damping function (van Driest, 1956) are employed to model the sub-grid scale fluid motions. Concentration transport equations of three reactive pollutants described in Eq. 1 are coupled with the governing equations of the flow field. The reaction rates are modeled employing the second-order rate constant model. Changing Reyonolds number Re_ƒÑ from 500 to 3000 and first Damköhler number D_a1 from 0.0 to 2.0, nine analytical cases are performed.

   In the results of the present study, the correlation evaulated from the prposed model has linear relations with that of LES in all analytical cases (Fig. 1). Although the model coefficient has a dependence on Reynolds number, it reaches asymptotically a finite value about 0.05 in the high Reynolds number region (Fig . 1 ii)).

   Figure 2 shows profiles of correlation in the case with Re_ƒÑ = 3000 and D_a1 = 0.0. Even though the model overestimates the correlation near wall surface due to inadequate evaluation of turbulent fluxes, its prediction coincides with LES result fairly well in the most part of the channel.

i) Cases with different D_a1 (Re_Ą = 1000)

ii) Cases with different Re_Ą (D_a1 = 0.0)

Figure 1  Relation between correlation of reactors' concentrations fluctuations in resolvable scale (longitudinal axis) and that evaluated from model (transverse axis) in the range of -0.7 < y/D < 0.7.

Figure 2  Profiles of correlation of reactors' concentrations fluctuations from LES and the proposed model (Re_ƒÑ = 3000, D_a1 = 0.0, C_cr=0.057).

4. Conclusions

A model was proposed to evaluate correlation of turbulent fluctuations of reactors' concentrations. It enables adequate prediction of the pollutant dispersion with bimolecular chemical reaction using RANS type turbulence model. It is composed of turbulent kinetic energy (TKE), dissipation rate of TKE, spatial gradients of mean concentrations, and a model coefficient. The model was validated against LES of dispersion and bimolecular chemical reaction in a turbulent channel flow. It was confirmed that the model can predict accurately the correlation except near-wall region. Although the model coefficient had a Reynolds number dependency, it reached a finite value (nearly 0.05) in high Reynolds number region.