LES-based Eulerian PDF approach for the simulation of scramjet combustors

Reactive flow simulations using large-eddy simulations (LES) require modelling of sub-filter fluctuations. Although conserved scalars like mixture fraction can be represented using a beta-function, the reactive scalar probability density function (PDF) does not follow an universal shape. A one-point one-time joint composition PDF transport equation can be used to describe the evolution of the scalar PDF. The high-dimensional nature of this PDF transport equation requires the use of a statistical ensemble of notional particles and is directly coupled to the LES flow solver. However, the large grid sizes used in LES simulations will make such Lagrangian simulations computationally intractable. Here we propose the use of a Eulerian version of the transported-PDF scheme for simulating turbulent reactive flows. The direct quadrature method of moments (DQMOM) uses scalar-type equations with appropriate source terms to evolve the sub-filter PDF in terms of a finite number of delta-functions. Each delta-peak is characterized by a location and weight that are obtained from individual transport equations. To illustrate the feasibility of the scheme, we compare the model against a particle-based Lagrangian scheme and a presumed PDF model for the evolution of the mixture fraction PDF. All these models are applied to an experimental bluff-body flame and the simulated scalar and flow fields are compared with experimental data. The DQMOM model results show good agreement with the experimental data as well as the other sub-filter models used.

LES/PDF based modeling of soot–turbulence interactions in turbulent flames

The behavior of the probability density function (PDF) transport equation at the limits of the probability space is studied from the point of view of fluid mechanics. Different boundary conditions are considered depending on the nature of the variable considered (velocity, scalar, and position). A study of the implications of entrance and exit conditions is performed, showing that a new term should be added to the PDF transport equation to preserve normalization in some nonstationary processes. In practice, this term is taken into account naturally in particle methods. Finally, the existence of discontinuities at the limits is also investigated.

An investigation of a turbulent spray flame using Large Eddy Simulation with a stochastic breakup model

This work presents a direct and transparent interpretation of two concepts for modelling turbulent combustion: generalised Multiple Mapping Conditioning (MMC) and sparse-Lagrangian Large Eddy Simulation (LES). The MMC approach is presented as a hybrid between the Probability Density Function (PDF) method and approaches based on conditioning (e.g. Conditional Moment Closure, flamelet, etc.). The sparse-Lagrangian approach, which allows for a dramatic reduction of computational cost, is viewed as an alternative interpretation of the Filtered Density Function (FDF) methods. This work presents simulations of several turbulent diffusion flame cases and discusses the universality of the localness parameter between these cases and the universality of sparse-Lagrangian FDF methods with MMC.

Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach

Theoretical analysis and experimental investigations have shown that the mean heat fluxes in turbulent gaseous flows are influenced not only by the mean scalar fields (temperature and molar fraction of the species), but also by the scalar fluctuations. It is widely recognized that the increase of radiative fluxes in comparison with laminar flows may exceed 100%. This interaction between turbulence and radiation is mainly due to the non-linearity between radiative emission and temperature. It is particularly important in reactive flows, since temperature fluctuations are typically higher in these flows than in non-reactive ones. In this article, a survey of the theory concerning turbulence-radiation interaction (TRI) is presented, along with applications in numerical simulations. We firstly present experimental and theoretical fundamentals on TRI. Then, direct numerical simulation and stochastic methods are addressed. Although they provide reliable information on TRI, they are too computationally demanding for practical applications. We will then focus on methods based on the solution of the time-averaged form of the conservation equations. Although many different approaches are available, we will concentrate on two methods. One is based on the solution of the time-averaged form of the radiative transfer equation using the optically thin fluctuation approximation, and a combustion model based on a prescribed probability density function (pdf) approach. The second one is based on the photon Monte Carlo method for radiative transfer calculations in media represented by discrete particle fields, and a combustion model based on the Monte Carlo solution of the transport equation for the joint pdf of scalars. Finally, the role of TRI in large eddy simulation is discussed, and the main consequences of TRI in combustion systems are summarized.

A large eddy simulation based on filtered vorticity transport equation has been coupled with filtered probability density function transport equation for scalar field, to predict the velocity and passive scalar fields. The filtered vorticity transport has been formulated using diffusion-velocity method and then solved using the vortex method. The methodology has been tested on a spatially growing mixing layer using the two-dimensional vortex-in-cell method in conjunction with both Smagorinsky and dynamic eddy viscosity subgrid scale models for an anisotropic flow. The transport equation for filtered probability density function is solved using the Lagrangian Monte-Carlo method. The unresolved subgrid scale convective term in filtered density function transport is modelled using the gradient diffusion model. The unresolved subgrid scale mixing term is modelled using the modified Curl model. The effects of subgrid scale models on the vorticity contours, mean streamwise velocity profiles, root-mean-square velocity and vorticity fluctuations profiles and negative cross-stream correlations are discussed. Also the characteristics of the passive scalar, i.e. mean concentration profiles, root-mean-square concentration fluctuations profiles and filtered probability density function are presented and compared with previous experimental and numerical works. The sensitivity of the results to the Schmidt number, constant in mixing frequency and inflow boundary conditions are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

An accurate prediction of the probability density function (PDF) of the mixture fraction is crucial to the prediction of combustion since mixing plays an important role in turbulent non-premixed and partially premixed flames. This work provides an assessment of the large-eddy simulation (LES)/filtered density function (FDF) method for the prediction of the PDF of the mixture fraction. The advantage of the LES/FDF method is that it provides the full predictions of the statistical distribution of scalars including the mixture fraction. The predictive accuracy of the method for the PDF is yet to be fully validated. Assessing the prediction of the PDF of the mixture fraction, a conserved scalar, is an important starting point. The Sydney/Sandia inhomogeneous inlet jet flame is used as a test case. A quick comparison shows that the LES/FDF predicted PDF shapes of the mixture fraction deviate significantly from the commonly presumed Beta-PDF as well as from the experimental data in the flame. To examine the source of the discrepancy, we clarify the different PDF definitions used in the comparison among the predictions, measurements, and presumed shape PDFs. The discrepancy observed from the comparison is largely reconciled by clarifying the difference between the PDFs that are examined. The PDF of the resolved mixture fraction is shown to be close to the Beta-PDF in both the measurements and predictions, while the PDF directly deduced from the LES/FDF particles deviates significantly from the Beta-PDF. A multimodal PDF analysis and a pseudo convergence analysis are conducted to provide plausible evidence to support the predicted multimodal PDF shapes. The sub-filter scale FDF is shown to be close to the Beta-PDF too through the construction of a synthesised PDF, which supports the common presumed Beta-PDF assumption used in the presumed PDF methods when combined with LES.

In this paper, a new "presumed" Probability Density Function (PDF) approach coupled with a Lagrangian tracking method is proposed for turbulent combustion modeling. The test and the investigation of the model are conducted by comparing the model results with DNS data for a premixed flame subjected in a decaying turbulent field. The newly constructed PDF, which incorporates the instantaneous chemical reaction term, demonstrates consistent improvement over conventional assumed PDF models. It has been found that the time evolution of the mean scalar, the variance and the mean reaction rate are strongly influenced by a parameter deduced by a Lagrangian equation which takes into account explicitly the local reaction rate. Tests have been performed for a moderate Damk?hler number, and it is expected the model may cover a broader range of Damk?hler number. The comparison with the DNS data demonstrates that the proposed model may be promising and affordable for implementation in a moment-equation solver.

Assumed PDF turbulence-chemistry closure with temperature-composition correlations

Here, we report on large-eddy simulation (LES) of supersonic flow, mixing, self-ignition, and combustion in a model scramjet combustor. The combustor has been experimentally investigated at DLR, German Aerospace Center Lampoldshausen and consists of a one-sided divergent channel with a wedge-shaped flame-holder: at the base of which, hydrogen is injected. This combustor has been extensively studied by many research groups and constitutes a good validation case for model development and physics elucidation. The LES model used is based on an unstructured finite volume discretization of the filtered mass, momentum, species, and energy equations, as well as an explicit flow solver. Three subgrid flow models, three reaction mechanisms, and three LES combustion models are employed to examine the sensitivity to these modeling parameters. The LES predictions are compared with experimental data for velocity, temperature, wall pressure at different cross sections, as well as schlieren and OH chemiluminesence images, showing good agreement for both first- and second-order statistics, as well as the instantaneous flow structures. Furthermore, these LES results are used to illustrate and explain the intrinsic flow, mixing, and combustion features of this combustor. The intrinsic relationship between the different modeling parameters is also discussed.

A flamelet-based Eulerian transported PDF method for the modeling and simulation of supersonic combustion

Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems: A Probability Density Function Approach

ABSTRACTA gas turbine model combustor is studied using Large Eddy Simulation with a transported Probability Density Function approach solved by the Eulerian stochastic field method. The chemistry is represented by a reduced methane mechanism containing 15 steps and 19 species while the subgrid scale stresses and scalar fluxes are modelled, respectively, via a dynamic Smagorinsky model and a gradient diffusion approximation. The test case comprises a partially premixed swirl flame in a complex geometry. Four stochastic fields are utilised in the simulations, which are performed for two different combustor operating conditions involving a stable and an unstable flame. Good agreement between the simulation and measurement data is shown in a comparison of mean velocity, temperature and species mass fraction profiles, as well as scatter plots of the instantaneous thermochemical properties. In conclusion, the predictive capabilities of the employed Large Eddy Simulation method are successfully demonstrated in this work.