A DNS evaluation of mixing models for transported PDF modelling of turbulent nonpremixed flames

Abstract

Transported probability density function (TPDF) methods are well suited to modelling turbulent, reacting, variable density flows. One of the main challenges to the successful deployment of TPDF methods is accurately modelling the unclosed molecular mixing term. This study examines three of the most widely used mixing models: the Interaction by Exchange with the Mean (IEM), Modified Curl (MC) and Euclidean Minimum Spanning Tree (EMST) models. Direct numerical simulation (DNS) data-sets were used to provide both initial conditions and inputs needed over the course of the runs, including the mean flow velocities, mixing frequency, and the turbulent diffusion coefficient. The same chemical mechanism and thermodynamic properties were used, allowing the study to focus on the mixing model. The simulation scenario was a one-dimensional, nonpremixed, turbulent jet flame burning either a syngas or ethylene fuel stream that featured extinction and reignition. This test scenario was selected because extinction and reignition phenomena are sensitive to the mixing model. Three DNS cases were considered for both the syngas and ethylene cases with a parametric variation of Reynolds and Damköhler numbers, respectively. Extinction events became more prevalent with increasing Reynolds number in the syngas cases and with decreasing Damköhler number in the ethylene cases. The model was first tested with the mixing frequency defined from the dissipation rate and variance of mixture fraction. With this definition, for the syngas cases this study finds that the TPDF method is successful at predicting flame extinction and reignition using all three mixing models for the relatively lower and intermediate Reynolds number cases, but that all models under-predict reignition in the relatively higher Reynolds number case. In the ethylene fuelled cases, only the EMST mixing model correctly predicts the reignition event for the two higher Damköhler number cases, however, in the lowest Damköhler number case the EMST model over-predicts reignition and the IEM and MC models under-predict it. Mixing frequency was then modelled based on the turbulence frequency and a model constant $C_{\varphi }$, the ratio of scalar to mechanical mixing rates. The DNS cases were reexamined with this definition and the results suggested that the optimal value for $C_{\varphi }$ is mixing model and case dependent. In particular, it was found in the ethylene case considered that reignition could be achieved with the IEM and MC models by adjusting the value of $C_{\varphi }$.

Introduction

Nonpremixed turbulent flames are the predominant combustion mode in aero gas turbine engines, diesel engines, industrial burners and fires. At the mixing rates encountered in practical operation, these flames are challenging to model due to their strong turbulence chemistry interactions.

Transported probability density function (TPDF) methods [1], [2] provide a computationally tractable approach to modelling nonpremixed turbulent flames [3], [4]. In TPDF methods, single point statistics of flow and thermochemical state variables are evaluated using transport equations for their joint probability density functions. This approach has the important advantage that the nonlinear chemical source term appears in closed form [1].

The so-called composition transported probability density function approach (C-TPDF) has received the most attention to date due its relatively simple implementation in conventional computational fluid dynamics solvers [1], [4]. In this approach, the joint PDF of chemical species composition and an energy variable, such as enthalpy, is evolved, requiring the following components:

• a model for the unclosed molecular mixing term, which requires input mixing frequency;

• a model for turbulent flow, for example a $k-∊$ model or a large-eddy simulation model, providing the modelled mixing frequency, turbulent diffusion coefficient, and mean or filtered velocities;

• physical models for the thermodynamic properties and chemical reaction rates;

• boundary and initial conditions; and

• numerical algorithms to implement the modelling.

The main closure challenge is to develop an accurate model of molecular mixing [5], [6]. A number of mixing models have been proposed. Some of the more frequently considered include the Interaction by Exchange with the Mean (IEM) [7], Modified Curl (MC) [8], [9], and the Euclidean Minimum Spanning Tree (EMST) models [10]. However, the existing literature does not clearly identify the optimal choice of a mixing model. Studies of experimental flames performed with the IEM [11], [12], [13], MC [14], [15], [16], and EMST [17], [18], [19] models have all obtained good agreement for experimental results. There are three points which make comparing and evaluating these studies difficult:

• There is uncertainty in the accuracy of the chemical kinetic and other physical models, and they are not frequently the same between studies.

• There is uncertainty in the accuracy of the turbulence modelling, and different choices are made in different studies for the turbulence closure: for example, velocity-composition TPDF, $k-∊$, or LES may be used.

• Numerical parameters such as grid sizes, time-steps, tolerances, and number of particles are practically limited by computational expense, and therefore might not always be sufficient. In addition, the choices made vary between studies.

• Experimental measurements have associated uncertainties and it is not always clear exactly what should be compared between models and experiments.

• The model constant $C_{\varphi }$ which measures the ratio of turbulent to scalar time scales may be selected arbitrarily to give the best results [20].

• The target flame may vary between studies. Flames with higher levels of extinction and reignition are more difficult targets than more stable flames.

Using direct numerical simulations (DNS) as a numerical experiment against which to compare models has advantages which can help overcome some of these difficulties. For example, a previous study by Mitarai et al. [6] compared C-TPDF results with DNS modelling decaying, isotropic, turbulence with a simple one-step chemistry model. The DNS provided the time varying mixing frequency, removing an element of modelling. It was found that the EMST model had the best performance in the mean, however the conditional statistics were inaccurate. A more recent study by Yang et al. [21] considered an LES-PDF model of a temporally evolving syngas flame (the same as that considered here). In this study, the DNS provided only the validation data-set– the turbulence modelling was provided by the LES.

In the present study, the C-TPDF approach is considered in a Reynolds-averaged context. Two DNS databases focused on extinction and re-ignition processes in nonpremixed temporally evolving plane jet flames are considered: a set of syngas cases from Hawkes et al. [22], [23] (previously modelled with LES-PDF methods [21], LES-linear eddy model [24], and one-dimensional turbulence [25]), and a set of ethylene cases from Lignell et al. [26] (previously modelled with one-dimensional turbulence [27]). In contrast to the previous study [21], the mean velocity, turbulent diffusion coefficient, and mixing frequency are taken directly from the DNS. The same models of kinetic rates and thermodynamic properties are used. These choices eliminate potential sources of modelling error.

The difference between the syngas and ethylene cases are the fuel composition and parametric sweep performed. In the syngas DNS, the Reynolds number was adjusted by changing the jet height and velocity, keeping Damköhler number fixed. In the ethylene DNS, the Damköhler number was adjusted at fixed Reynolds number by altering the dilution of the fuel and oxidant streams to change the chemical time-scales while preserving the location of the stoichiometric mixture fraction relative to the shear layer.

Three cases from each DNS study are considered. From the syngas DNS these are the: Lower (L), Moderate (M), and Higher (H) Reynolds number cases. From the ethylene DNS these are the Higher (1), Moderate (2), and Lower (3) Damköhler number cases. By design, the ethylene cases have similar parameters to the syngas case M, with case 3 most similar to case M. The Damköhler number is sufficiently low in all cases to cause local extinction. Local extinction increases with either increasing Reynolds number in the syngas cases or decreasing Damköhler number in the ethylene cases, due to increasing rates of turbulent mixing relative to the chemical timescales. All cases exhibit reignition later in the simulations as mixing rates relax.

Local extinction and reignition are challenging phenomena to model. Close to the point of extinction or reignition, small changes in mixing rates can cause large changes in the thermochemical state. Thus, results are sensitive to the mixing model. The parametric variation of turbulence levels and fuel type performed in this study provides a graded test for evaluating mixing model performance.

Parametric studies are also performed for the mixing constant, $C_{\varphi }=\tau /{\tau }_{\varphi }$, which is a measure of the ratio of turbulent to scalar mixing time-scales. Most TPDF modelling has been performed without access to a DNS data-set, in which case a value for $C_{\varphi }$ must be selected as a model-constant. Since a wide range of values for $C_{\varphi }$ are quoted in the literature [1], [5], [28], [29], [30], it is worthwhile to conduct a parametric study of $C_{\varphi }$ against the benchmark of the exact values of a passive scalar mixing frequency extracted directly from the DNS without modelling.