Elsevier

Chemical Engineering Science

Volume 127, 4 May 2015, Pages 334-343
Chemical Engineering Science

Evaluation of scalar dissipation rate sub-models for modeling unsteady reacting jets in engines

https://doi.org/10.1016/j.ces.2015.01.055Get rights and content

Highlights

  • Evaluation of filtered scalar dissipation rate models for reacting LES.

  • Strain rate tensor model is the most suitable model.

  • Exponential and log‐normal PDFs for the scalar dissipation rate are evaluated.

  • A model equation is formulated for the scalar dissipation rate variance.

Abstract

The filtered scalar dissipation rate is an important variable required in many turbulent combustion models used in large eddy simulations (LES). In the present study, direct numerical simulations (DNS) of reacting and non-reacting turbulent mixing layers are used to evaluate different models for the filtered scalar dissipation rate. This study is conducted at elevated temperature and pressure conditions relevant to engine applications. The models evaluated are the turbulent diffusivity model, kε model, the strain rate tensor (SRT) model and the subfilter kinetic energy (SKE) model. It is found that the SRT model is the best choice when considering the performance and ease of implementation in LES codes. DNS results are also used to evaluate the marginal PDF for scalar dissipation rate. It is found that an exponential PDF works well for low values of scalar dissipation rate and smaller filter sizes whereas a lognormal PDF works well for larger values of scalar dissipation rate and larger filter sizes. A model is also formulated for the variance of the scalar dissipation rate to be used when employing the lognormal PDF by relating it to the mean and variance of the mixture fraction.

Introduction

With recent advances in computational capabilities, large eddy simulation of turbulent reacting flows has started to become feasible for conditions where the pressure and temperature are elevated such as in engine applications. The sub-grid scale modeling of turbulence–chemistry interactions is, however, a continuing subject of inquiry. For turbulent non-premixed combustion, flamelet-based models are attractive when the combustion occurs in reaction zones that are smaller than the Kolmogorov scale. This class of models includes the unsteady flamelet model (Pitsch et al., 1998), steady flamelet progress variable model (Pierce and Moin, 2004) and unsteady flamelet progress variable model (Ihme et al., 2005, Ihme and Pitsch, 2008, Ihme and See, 2010). When the reactions occur in thin zones, the effect of the turbulence is to locally stretch as well as extinguish these reaction zones. Under these conditions, the flame can be assumed to be locally one-dimensional in the mixture fraction (Z) space. Within these reaction zones, which are called as flamelets, the evolution of the species mass fractions are governed by the unsteady flamelet equations given byϕt=χ22ϕZ2+ω̇ϕ,where ϕ is a vector representing the set of all reactive scalars, which includes the temperature and mass fractions of all the species, and ω̇ϕ is the corresponding source term due to chemical reactions. The symbol χ is the instantaneous scalar dissipation rate defined asχ=2D|Z|2,where D is the molecular diffusivity. Notice that this fundamental definition cannot be employed in LES (or RANS) because only filtered (or Reynolds-averaged) values of Z are available. In a mixing layer, the functional form of the dependence of χ on Z is typically assumed to follow an error function profile (Peters, 2000, Mukhopadhyay and Abraham, 2012).χ=χstexp{2[erfc1(2Z)]2}exp{2[erfc1(2Zst)]2}.By using this assumption, the value of the scalar dissipation rate at any Z can be related to its value at the stoichiometric mixture fraction, χst.

The unsteady flamelet equations (Eq. (1)) can be solved for different values of χst and the solution is usually tabulated as a function of 3 independent variables Z, χst and Λ, where Λ is called the progress variable and it is an indicator of how much the reactions have progressed in the flamelet (Pierce and Moin, 2004, Ihme et al., 2005, Ihme and Pitsch, 2008, Ihme and See, 2010). These models are only valid if a single flamelet is present in an LES computational cell. Typically, every computational cell contains multiple flamelets, and the filtered reaction rate in a computational cell is given by averaging the reaction rate over all these flamelets.ω̇ϕ˜=ω̇ϕP(Z,χst,Λ)dZdχstdΛ=ω̇ϕP(Z)P(χst)P(Λ)

In the above equation, P(Z,χst,Λ) is the joint-PDF of Z, χst and Λ, and P(Z), P(χst) and P(Λ) are the marginal PDFs of Z, χst and Λ respectively. The implicit assumption is made that these 3 variables are statistically independent so that the joint PDF can be written as the product of the marginal PDFs. Mukhopadhyay and Abraham (2012) have verified this assumption for reacting mixing layers.

In this study, models for the filtered scalar dissipation rate, χst˜, and its marginal PDF are studied using DNS. The usual reference in the literature to DNS is to computations that resolve the flow scales, but not necessarily the flame scales. In the present study, both flow and flame scales are resolved. This makes the computations with complex kinetics computationally intensive. The complex kinetics is required because the interest is in autoigniting mixing layers of relevance to fuel-injected compression-ignition engines. Furthermore, the computations are carried out under high-pressure and high-temperature conditions relevant to engines. For this reason, the DNS simulations performed in this work are 2D. In prior 3D DNS studies, single-step and two-step kinetics, or other simplifications using tabulated chemistry have been employed (van Oijen et al., 2007). 3D DNS of reacting flows in which the flame scales are also resolved and multi-step kinetics is employed are still rare (Yoo et al., 2011, Yoo et al., 2013). They are generally restricted to atmospheric conditions.

There have been a few other studies reported in the literature, which also evaluate models for the filtered scalar dissipation rate. Girimaji and Zhou (1996) performed DNS of isotropic non-reacting turbulence and used the results to analyze and model the effect of subgrid scales on the filtered scalar dissipation rate by accounting for two physical phenomena: (i) the interaction among the subgrid scales and (ii) the interaction between the resolved and unresolved scales. Using spectral arguments, they derived a model, which, to the leading order, involves enhancing the molecular diffusivity with a turbulent diffusivity. They also included a model for “backscatter” of energy from the small scales to the large scales. Although this model performed well when compared to the DNS results, its performance for reacting simulations have not been evaluated. The implementation of this model for LES is challenging, as there is a requirement to solve additional transport equations. Jimenez et al. (2001) performed DNS of inert scalar mixing in homogenous turbulence to model the filtered scalar dissipation rate using a characteristic time-scale model. Model parameters were obtained by filtering the DNS data and the model was shown to predict the decay of the scalar variance accurately. Balarac et al. (2008) performed DNS of forced isotropic turbulence and used the principle of optimal estimators to evaluate different models for filtered scalar dissipation rate. They found that a model based on the ratio of turbulent-to-scalar time scale, where the turbulent time scale is evaluated from the subfilter kinetic energy gives the lowest irreducible error. These studies have all been performed using non-reacting DNS and it is not known whether their conclusions can be extended to reacting flows. A recent study by Mukhopadhyay and Abraham (2012) has shown that combustion and the associated heat release modifies the scalar dissipation rate significantly, especially near the flame front. So, the conclusions made regarding the models using non-reacting DNS may not hold when these models are applied for reacting flows.

To the authors׳ best knowledge, the only DNS study for modeling the filtered scalar dissipation rate in reacting flows is the one performed by Knudsen et al. (2012). They performed DNS of autoigniting jets and used the results to evaluate an algebraic model and a transport equation model for the filtered scalar dissipation rate and showed that the transport equation model performed superior to the algebraic model. They only evaluated one algebraic model – the commonly used turbulent diffusivity model. Although the transport equation model performed well, its computational overhead is greater as there is a need to solve additional transport equations. It is the purpose of the present study to evaluate other algebraic closure models for the filtered scalar dissipation rate and test their performance for a variety of conditions. In this study, we assess the performance of the scalar dissipation rate models by using data from DNS of both non-reacting and reacting flows. We also analyze how the model performance changes with changes in filter sizes and turbulence intensities. Note that the reacting flows of specific interest in this work are those generated through autoignition of fuel/air mixing layers at elevated temperature and pressure conditions with application to compression-ignited fuel-injected engines.

The outline for the rest of the paper is as follows. In the next section, the details about the computational setup and model are described. The different models for the filtered scalar dissipation rate evaluated in this study are discussed in Section 3. The models are assessed using the DNS results for a variety of conditions. In Section 4, two models for the PDF of the scalar dissipation rate are described and the conditions under which each PDF works well are examined. The variance of the scalar dissipation rate is an important parameter that is required to determine the PDF. In Section 5, a simple model for the scalar dissipation rate variance is derived and it is shown to be accurate for a range of conditions. The paper closes with summary and conclusions.

Section snippets

Computational setup

The numerical code employed in this work solves the compressible form of the Navier–Stokes equations for multicomponent gaseous mixtures with chemical reactions. The sixth-order compact finite-difference scheme of Lele (1992) is employed to spatially discretize the governing equations. The resulting discretized equations are solved using the tridiagonal matrix algorithm. A fourth order Runge–Kutta scheme is employed to perform the time-integration. Boundary conditions are implemented using the

Modeling the filtered scalar dissipation rate

Fig. 2 shows the instantaneous profile of the scalar dissipation rate at a time of 0.7 ms for the (a) non-reacting and (b) reacting mixing layers. The regions of high χ are localized to small regions in the computational domain. It is seen that the peak values of scalar dissipation rate are approximately two times higher for the non-reacting mixing layer. This is expected as chemical reactions cause an increase in temperature leading to local expansion and thus reduction in the gradients of the

Modeling the PDF of filtered scalar dissipation rate

As shown in Eq. (4), in addition to the accurate modeling of the filtered scalar dissipation rate χ˜, the marginal PDF of χ is also required for use in flamelet models. Fig. 14, Fig. 15 show the PDFs obtained from the DNS simulations for the baseline reacting case. When the filter size Δ is small, the entire filtered cell can be expected to have an almost uniform value of χ. In other words, the variance of χ can be expected to be very small. Under these conditions, the PDF of χ is expected to

Modeling scalar dissipation rate variance

When using the SRT or SKE model, a transport equation for the variance of mixture fraction, i.e. Zv, has to be solved. Knowing Zv, the variance of the scalar dissipation rate can be modeled as shown below. The variance of the scalar dissipation rate is needed to employ the lognormal PDF. In this section, a simple model is derived for the variance, which is found to be applicable for all the cases considered here. Using Eq. (2), the variance of the scalar dissipation rate can be expressed asχvar=

Summary and conclusions

In this study, direct numerical simulations of non-reacting and reacting mixing layers have been carried out to generate databases, which are then employed to assess the accuracy of four models for the filtered scalar dissipation rate in LES. The pressure and temperature conditions selected are relevant to compression-ignition combustion engines. N-heptane, often used as a surrogate for diesel fuel, is used as the fuel. The four models assessed are the turbulent diffusivity model, the kε

Acknowledgments

The authors would like to thank the US National Institute of Computational Sciences (NICS), eResearch SA (eRSA) and the Australian National Computational Infrastructure (NCI) for providing the computing resources for this work, and Caterpillar Inc. (grant number 204421) and Purdue Research Foundation (grant number 204533) for financial support of this work. The contributions of Professor Vinicio Magi in the development of the numerical codes are gratefully acknowledged.

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