A lagrangian based scalar pdf method for turbulent combustion models

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

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

Transported probability density function modelling of the vapour phase of an n-heptane jet at diesel engine conditions

An aging progress variable (APV) is proposed as a convenient tool for conditioning quantities used to calculate reaction rates in premixed turbulent combustion. The APV is defined to obey an advection-diffusion-reaction equation where the source term is linearly related to the fuel consumption rate when the APV is less than a threshold representative of the trailing edge of the fuel consumption layer. Above this threshold, the APV has a constant source term. To test the proposal, three-dimensional fully compressible direct numerical simulation (DNS) is performed in an inflow-outflow configuration with inlet turbulence forcing using a modified version of the DNS code SENGA. A model chemical mechanism involving two steps and four species is used which has an order of magnitude difference between the time scales associated with the two steps. The inlet Taylor scale Reynolds number is 51 and the Damköhler numbers are 2.0 and 0.29 for the two steps. When conditionally averaged on the proposed APV the scalar fluctuations about the conditional average are negligible. Further, it is shown that the probability density function of the APV can be reasonably approximated based on the first two moments of the APV and the fuel mass fraction. The APV probability density function (PDF) is approximated on rectangular slabs of cells normal to the flow direction using only these moments as a test case. Convolution of the PDF so approximated with conditional mean reaction rates—calculated from conditionally averaged scalar fields where the averaging was carried out over the entire domain—leads to an approximation for the unconditional mean reaction rates on each of these slabs typically within 10% of the true value for both steps. That the correlation between the reaction rates and the APV is strong, and that the PDF can be approximated for a situation where approximating the PDF for a product based progress variable is nontrivial, makes the proposed APV a strong alternative to traditional progress variables for both flamelet models and premixed conditional moment closure (CMC) approaches.

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

On the relationship between Lagrangian micromixing models and computational fluid dynamics

Present work is concerned with the flow field analysis inside an annular gas turbine combustor both under non-reacting and reacting conditions. Three-dimensional gas turbine combustor of 20-degree sector has been modeled using the pre-processor GAMBIT. Flow through the combustor has been simulated using FLUENT code by solving the appropriate governing equations viz., conservation of mass, momentum and energy. RNG κ-ε turbulence model is used for physical modeling. Initially prediffuser optimization has been carried out with respect to angle, length and contours. Flow through holes is modeled using porous jump boundary condition as well as modeling real holes themselves to study the efficacy of real hole modeling. Total pressure loss has been calculated to evaluate the cold flow as well as hot flow losses. Combustion has been modeled using the Probability Density Function (PDF) approach. Temperature and species concentrations are predicted.

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

A new method for the approximation of multivariate scalar probability density functions (PDFs) in turbulent reacting flow by means of a joint presumed discrete distribution (jPDD) is presented. The jPDDs can be generated with specified mean values and variances as well as covariances. Correlations between variables – e.g. fluctuating mixture fractions and/or reaction progress – can thereby be taken into account. In this way the new approach overcomes an important limitation of ordinary presumed PDF methods, where statistical independence between the variables is often assumed. Different methods are presented to generate discrete distributions, based either on biased random number generators or on mixing models familiar from PDF transport models. The new approach is extensively validated on a turbulent flow configuration with simultaneous mixing and reaction. Large eddy simulation data as well as results from a transported PDF model are used for the validation of the jPDD approach. The comparison shows that in particular distributions generated with mixing models are able to predict mean reaction rates accurately. For the configuration considered, the neglect of correlations results in significant underestimation of reaction rates. Moreover it is found that higher statistical moments (e.g. the skewness) can influence reaction rates. The consequences for the generation of jPDDs are discussed. In summary, the new jPDD model has the potential to be significantly more accurate than established presumed PDF methods, because correlations between fluctuating variables can be taken into account. At the same time, the new approach is nearly as efficient as standard presumed PDF formulations, since mean rates are computed in a pre-processing step and stored in look-up tables as a function of the first and second moments of the relevant variables.

Assumed PDF turbulence-chemistry closure with temperature-composition correlations

This paper presents an experimental and numerical study of the flow field and heat release (HRL) zone of a six-nozzle FLOX®-based combustor at atmospheric pressure. The combustor is suitable for the use in a micro gas turbine (MGT) based combined heat and power (CHP) system with an electrical power output of 3 kW. The velocity field was measured using stereoscopic particle image velocimetry (PIV). The heat release zone was visualized by OH*-chemiluminescence (OH* CL) and the flame front by OH planar laser-induced fluorescence (OH PLIF). The results are compared with CFD simulations to evaluate the quality of the applied numerical turbulence and combustion models. The simulations were performed using Reynolds-averaged Navier-Stokes equations in combination with the k-ω-SST-turbulence model. Since the FLOX®-based combustion is dominated by chemical kinetics, a reaction mechanism with detailed chemistry, including 22 species and 104 reactions (DRM22), has been chosen. To cover the turbulence-chemistry interaction, an assumed probability density function (PDF) approach for species and temperature was used. Except for minor discrapancies in the flow field, the results show that the applied models are suitable for the design process of the combustor. In terms of the location of the heat release zone, it is necessary to consider possible heat losses, especially at lean operating conditions with a distributed heat release zone.

Comparison of transported scalar PDF and velocity-scalar PDF approaches to ‘Delft flame III’

Scalar and joint velocity–scalar PDF modelling of near-wall turbulent heat transfer

A transported probability density function (PDF) approach closed at the joint scalar level was used to model a bluff body stabilised turbulent diffusion flame (HM2) investigated experimentally by Masri and co-workers. The current effort extends a previous study of HM1 (Re = 15,800) to a flame with a higher degree of local extinction (Re = 23,900). The impact of an algebraic model that accounts for local Damkohler number effects on the time-scale ratio of scalar to mechanical turbulence is also evaluated along with the impact of improved thermochemistry. The computations have been performed using a hybrid Monte Carlo/finite volume algorithm and a systematically reduced H/C/N/O mechanism featuring 300 reactions, 20 solved and 28 steady-state species. The joint scalar PDF equations were solved using moving particles in a Lagrangian framework and the velocity field was closed at the second moment level. The redistribution terms were modelled using the Generalized Langevin model of Haworth and Pope. Results show that scalar fields are reproduced with encouraging accuracy and that the revised time scale model improves agreement with experimental data. A high sensitivity to the NO chemistry was observed and encouraging agreement was obtained for the first two moments following adoption of updated reaction rates proposed in an earlier study.

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