LES–Lagrangian particle method for turbulent reactive flows based on the approximate deconvolution model and mixing model
Introduction
Turbulent mixing with chemical reactions can be observed in engineering and environmental flows (e.g., in chemical reactors and for pollutants emitted into the atmosphere). Large eddy simulation (LES) has been one of the most common tools to predict mixing and scalar transfer in turbulent flows, and it is expected to be applicable to engineering and environmental flows. In LES, velocity and scalar fields are divided into large-scale and small-scale parts by a filtering procedure. LES resolves large-scale features in flows, whereas the effects of unresolved small-scale (subgrid scale (SGS)) features have to be modeled. In LES of turbulent reactive flows, SGS scalar fluctuations have to be modeled to close the filtered reaction source terms, which appear in an unclosed form in the filtered scalar transport equations [1]. LES of turbulent reactive flows using models for the SGS scalar fluctuations has been developed in several studies [2], [3], [4]. However, LES is often combined with another numerical method to simulate reactive scalars (e.g., conditional moment closure [5] and the probability density function (PDF) method [6], [7]) because it is difficult to model the contribution of SGS scalar fluctuations to the filtered reaction source terms.
Most of LES for reactive flows has been developed based on the Smagorinsky model [8] (the dynamic Smagorinsky model [9]) for the SGS Reynolds stress and the gradient diffusion model for the SGS scalar flux. The gradient diffusion model is widely used with a dynamic procedure to determine the SGS turbulent Schmidt (Prandtl) number [10]. SGS models used in LES can be classified into two main types: eddy viscosity models (e.g., the Smagorinsky model, the dynamic Smagorinsky model, and the structure-function model [11]), and models based on the reconstruction of the information in small scales from a resolved field (e.g., the scale similarity model [12]). Recently, Stolz and Adams proposed the approximate deconvolution model [13] (ADM), in which the information in the small scales is approximately reconstructed by using an approximate deconvolution filter. LES based on the ADM has been applied to various turbulent flows [14], [15], [16], [17]. The ADM has some advantages over the Smagorinsky model or the dynamic Smagorinsky model. It is well known that the Smagorinsky model is too dissipative [16]. The dynamic Smagorinsky model overcomes some of problems in the Smagorinsky model by dynamically determining the model parameter. However, the procedure to determine the parameter often includes spatial averaging to prevent numerical instability [18]. Spatial averaging might be inadequate for intermittent flows, which consist of turbulent, non-turbulent, and transition regions [19], because the SGS eddy viscosity changes between turbulent and non-turbulent regions [20]. It was also reported that the eddy viscosity model can reduce the effective Reynolds number of the flow [21]. Some of these problems relating to the eddy viscosity model do not exist in LES based on the ADM. The ADM can be applied to both turbulent and non-turbulent flows without adjusting the model [16]. In LES based on the ADM, the flow and scalar fields can be divided into three scales: a filtered scale, for which the spatial filtering procedure has no explicit effects, a subfilter scale, which can be represented on a computational grid of LES but is removed or suppressed by the filter, and a subgrid (unresolved) scale, which cannot be represented on the computational grid [22]. The filtered and subfilter scales can be collectively referred to as the resolved scale. The ADM can reconstruct well the subfilter scale although the SGS is implicitly modeled by a relaxation term, which emulates the dissipations in the SGS.
In turbulent reactive flows, mixing at the molecular level often controls chemical reaction rates [23]. Therefore, the effects of SGS scalar fluctuations on chemical reactions should be accurately modeled in LES. However, in the ADM, it is difficult to model these effects because the ADM can reconstruct only the subfilter scales. To overcome the difficulty in modeling the chemical source terms by the ADM, we combine the LES based on the ADM with the representation of scalar fields by Lagrangian notional particles. Values of reactive scalars are assigned to each particle. This numerical method is referred to as a LES–Lagrangian particle (LES–LP) method hereafter. The LES is often combined with the filtered density function (FDF) method [24], in which Lagrangian stochastic particles are used for solving the governing equation for the FDF. The present method is derived based on the Lagrangian description of scalar transport equations rather than the FDF. Furthermore, the FDF method has been developed based on the eddy viscosity model although we use the ADM in the LES–LP methods. In this method, the LES based on the ADM is used for computing the resolved velocity and nonreactive scalar fields whereas the reactive scalar fields are computed by using the Lagrangian particles which evolve according to the scalar transport equations. Because the reactive scalars are treated by the Lagrangian particles, the models for the reaction source terms are not required in the LES. Because of the difficulty in accurately computing the spatial scalar derivative using the Lagrangian particles, the molecular diffusion term is modeled by a mixing model using multiple Lagrangian particles [1], [25]. The mixing model requires to specify a mixing timescale of the model among the Lagrangian particles. The mixing timescale is often determined using the ratio of the mechanical timescale to the scalar timescale [1], [25]. However, it was reported that this ratio changes depending on the Reynolds number and the Schmidt number [26]. Therefore, this procedure to determine the mixing timescale includes adjusting a model parameter, which has a great influence on the numerical results. In this study, we also develop a particles-interaction mixing model using a mixing volume concept (mixing volume model), in which the mixing timescale is determined so that the mixing model decays scalar variance in the mixing volume according to the scalar dissipation rate. It is shown that the present model can implicitly take into account the effect of distance among mixing particles without adjusting any model parameters.
In Section 2, we describe the detail of the LES–LP method and the mixing model. The LES–LP method based on the ADM and the new mixing model is applied to a planar jet with an isothermal second-order reaction (Section 3) for testing the numerical method. The results of LES–LP simulation are compared with the previous direct numerical simulation (DNS) data [27], [28] in Section 4 to demonstrate that the LES–LP method based on the ADM and the mixing volume model is useful for predicting turbulent reactive flows.
Section snippets
Large eddy simulation based on the ADM
An incompressible fluid with a nonreactive passive scalar transfer is treated in the LES. The governing equations for velocity and nonreactive passive scalar ψ are the continuity equation, the Navier–Stokes equations, and the transport equation for ψ: where P is the instantaneous pressure divided by the density ρ, ν is the kinematic viscosity, and D is the diffusivity coefficient relating to ψ. A spatial filtering operator G
Reactive planar jet
The LES–LP method based on the ADM and the mixing volume model is tested for a planar jet with an isothermal chemical reaction , which was previously investigated by using DNS [27], [28], [42]. The results of the LES–LP simulation are compared with the previous DNS data. Fig. 2 shows a schematic of the reactive planar jet. Reactant A is supplied from the jet inlet of width d, and reactant B is supplied from the ambient flow. Product P is produced by the second-order chemical reaction
LES results
The LES results are compared with the DNS results. In the LES, the time-averaged statistics are estimated from the unfiltered quantities. Fig. 4 shows the mean streamwise velocity () and mean scalar (). Fig. 5 compares the jet half-width ( and ) based on and among the LES, DNS [28], [42], and experiments [44] of planar jets. Although the Schmidt number is much larger than 1 in the experiments (), we use the experimental results for comparison because molecular
Concluding remarks
A numerical method for turbulent reactive flows was developed by combining the LES based on the ADM with the Lagrangian particle method (LES–LP method). In this approach, the LES computes the filtered velocity and the filtered nonreactive scalar using the ADM. The reactive scalars are computed by using the Lagrangian notional particles for precluding the problems on modeling of chemical source terms. Values of reactive scalars are assigned to each particle. The evolutions of Lagrangian
Acknowledgement
The authors would like to thank Dr. Takashi Kubo for his valuable comments on this study. Part of the work was carried out under the Collaborative Research Project of the Institute of Fluid Science, Tohoku University. This work was supported by JSPS KAKENHI Grant Number 25002531 and MEXT KAKENHI Grant Numbers 25289030, 25289031, and 25630052.
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