Filtered Probability Density Function Research Articles

Simulation of turbulent reactive flows using a FDF methodology – Advances in particle density control for normalized variables

The present work is devoted to the development and implementation of a computational framework to perform numerical simulations of low Mach number turbulent reactive flows. The numerical algorithm designed for solving the transport equations relies on a semi implicit predictor-corrector integration scheme. The final computational model relies on a hybrid Large Eddy Simulation / transported Filtered Probability Density Function (FDF) framework. The TVD method CUBISTA is used in the discretization of the advective terms of the transport equations. The dynamic Smagorinsky turbulence closure model if retained and the influence of residual sub-grid scale scalar fluctuations on the filtered chemical reaction rate is taken into account within the Lagrangian PDF framework. The corresponding PDF model makes use of a Monte Carlo technique: Stochastic Differential Equations (SDE) equivalent to the Fokker-Planck equations are solved for the progress variable of chemical reactions. Due to the stochastic nature of the Monte-Carlo method and to the turbulent motions, the number of particles present in a given cell may significantly change in time. We present an algorithm for conservation of statistical moments up to second and suggestion of how one may reduce drastically a notional particle population and yet maintain an acceptable PDF shape. The TVD algorithm capabilities are demonstrated in the one-dimensional advection of different pulses, the energy equation is validated through the simulation of a natural convection flow in a cavity. Finally, the new cloning and annihilation particles algorithm is tested on known (given) PDF functions and then applied to LES of reactive turbulent channel flows of methane/air mixtures.

Large Eddy Simulation of Spark Ignition in a Gas Turbine Combustor

Ignition in an aircraft gas turbine combustion chamber is simulated using Large Eddy Simulation (LES) in conjunction with the filtered probability density function (pdf) equation approach, which is solved using the Eulerian stochastic field method. The LES-pdf methodology is used for both dispersed (liquid) and gas phases. The liquid phase is described using a Lagrangian formulation whilst an Eulerian approach is employed for the gas phase. The spark energy deposition was mimicked by a distributed energy source term added to the enthalpy equation. Unsuccessful and successful ignition sequences have been simulated and the results suggest that spark ‘size’ is an important parameter in the ignition of kerosene fuelled combustion chambers.

LES-pdf simulation of a spark ignited turbulent methane jet

A spark ignited turbulent jet flame is simulated using Large Eddy Simulation (LES) in conjunction with the filtered probability density function (pdf) equation approach, which is solved using the Eulerian stochastic field method. The spark energy deposition is mimicked by a gaussian distributed source term in the enthalpy equation. A dynamic model for the sub-grid stresses together with a simple gradient diffusion approximation for the scalar fluxes is adopted. The chemistry is represented by a global 4-step chemistry involving seven species. The different stages of the ignition sequence, namely the flame kernel growth, the triple flame propagation against the flow and the stabilisation were in good agreement with the experimental data.

Large Eddy Simulation of the Sandia Flame Series (D–F) using the Eulerian stochastic field method

Three turbulent piloted methane jet flames with increasing levels of local extinction (Sandia Flames D–F) have been computed using Large Eddy Simulation. The smallest unresolved scales of the flow, in which combustion occurs, are represented using the filtered probability density function method where the corresponding evolution equation is solved directly. A dynamic model for the sub-grid stresses together with a simple gradient diffusion approximation for the scalar fluxes is applied in conjunction with the linear mean square estimation closure for sub-filter scale mixing. An augmented reduced mechanism (ARM) derived from the full GRI 3.0 mechanism is incorporated to describe the chemical reaction. The results demonstrate the ability of the method in capturing quantitatively finite rate effects such as extinction and re-ignition in turbulent flames.

An Eulerian approach for large eddy simulation of particle transport in turbulent flows

The objective of this paper is to advance an Eulerian continuum approach for modeling the transport of inertial particles in the framework of large eddy simulation. This approach is based on a kinetic equation for the filtered probability density function of particle velocity in turbulent flow. Some models developed for the subgrid-scale turbulent stresses of the particulate phase are presented. The effect of filter width on the averaged subgrid particle kinetic energy is explored. It is revealed that, in the limit of zero filter width, the subgrid kinetic stress and energy of inertial particles can have nonzero values.

Approximate solutions of the filtered radiative transfer equation in large eddy simulations of turbulent reactive flows

An analysis of the relevance of turbulence–radiation interaction in the numerical simulation of turbulent reactive flows is presented. A semi-causal stochastic model was used to generate a time-series of turbulent scalar fluctuations along optical paths of Sandia flame D, a widely studied piloted turbulent jet nonpremixed flame. The radiative transfer equation was integrated along these paths for every realization using a grid resolution typical of a direct numerical simulation. The correlated k-distribution method was employed to compute the radiative properties of the medium. The results were used to determine the ensemble average, as well as the extreme values, of quantities that indicate the importance of the turbulence–radiation interaction. Several approximate methods are then proposed to solve the filtered radiative transfer equation in the framework of large eddy simulations. The proposed methods are applicable along with combustion models that either assume the filtered probability density function of a conserved scalar or solve a transport equation for a joint scalar or joint scalar/velocity filtered density function. It is concluded that the errors resulting from neglecting the turbulence–radiation interaction in large eddy simulations are much lower than those found in Reynolds-averaged Navier–Stokes calculations. The optically thin fluctuation approximation may be extended to large eddy simulations yielding predictions in excellent agreement with the reference solution. If the turbulence–radiation interaction is accounted for using this approximation, the average relative error of the filtered total radiation intensity is generally below 0.3% for the studied flame.

Subgrid continuum modeling of particle motion in a turbulent flow

An Eulerian continuum approach to modeling the motion of dispersed particles within the framework of the large-eddy simulation method is developed. The approach is based on a kinetic equation for the filtered probability density function for the particle velocity in a turbulent flow. Models for the subgrid turbulent stresses of the dispersed phase are presented.

MODELING OF CONDITIONAL DISSIPATION RATE FOR FLAMELET MODELS WITH APPLICATION TO LARGE EDDY SIMULATION OF FIRE PLUMES

ABSTRACT Subgrid scale (SGS) combustion modeling using flamelet approximations require a model for the conditional dissipation rate. For high Reynolds number flow, statistical independence between the mixture fraction and dissipation is often invoked allowing the conditional dissipation to be expressed in terms of its mean filtered value. This assumption fails for application to pool fires because of the transitionally turbulent nature of this class of flows. In this study, an alternative closure for conditional dissipation rate is proposed based on a transport equation for the mixture fraction filtered probability density function. Application of this model for use in Large Eddy Simulation (LES) of a large, 1 m in diameter, methane-air fire plume results in greatly improved predictions over existing models that invoke the use of statistical independence when comparisons are made to experimental measurements.

Large eddy simulation (2D) of spatially developing mixing layer using vortex-in-cell for flow field and filtered probability density function for scalar field

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.

Assessment of a flame surface density-based subgrid turbulent combustion model for nonpremixed flames of wood pyrolysis gas

A flame surface density (FSD) model for closing the unresolved reaction source terms is developed and implemented in a large eddy simulation (LES) of turbulent nonpremixed flame of wood pyrolysis gas and air. In this model, the filtered reaction rate ω¯α of species α is estimated as the product of the consumption rate per unit surface area mα and the filtered FSD Σ¯. This approach is attractive since it decouples the complex chemical problem (mα) from the description of the turbulence combustion interaction (Σ¯). A simplified computational methodology is derived for filtered FSD Σ¯, which is approximated as the product of the conditional filtered gradient of mixture fraction and the filtered probability density function. Two models for flamelet consumption rate mα are proposed to consider the effect of filtered scalar dissipation rate. The performance of these models is assessed by direct numerical simulation (DNS) database where a laminar diffusion flame interacts with a decaying homogeneous and isotropic turbulent flow field. The chemistry is modeled by a four-step reduced mechanism that describes the oxidization process of gaseous fuel released from high temperature pyrolysis of wood occurring in a wildland fire. Two-dimensional (2D) and 3D LES computations based on the FSD models are conducted for the same conditions as the DNS. The comparative assessments confirm the applicability of the proposed FSD model to describe the filtered reaction rate and the time evolution of temperature and species concentration in the turbulent nonpremixed flame.

A flame surface density based model for large eddy simulation of turbulent nonpremixed combustion

A combustion model based on the flame surface density (FSD) concept is developed and implemented for large eddy simulation of turbulent nonpremixed combustion of wood pyrolysis gas and air. In this model, the filtered reaction rate ω̇̄α of species α is estimated as the product of the consumption rate per unit surface area ṁα and the filtered FSD Σ̄. This approach is attractive since it decouples the chemical problem from the description of the turbulence combustion interaction. The filtered FSD is modeled as the product of the conditional filtered gradient of mixture fraction and the filtered probability density function. This approach is validated from direct numerical simulation (DNS) using spatial filtering operation. Results show that the proposed FSD model provides a good description for the filtered reaction rate of wood pyrolysis gas. Temperature predicted by large eddy simulation agrees well with that filtered from DNS data.