Filtered Density Function Transport Equation Research Articles

Scalar Mixing Study at High-Schmidt Regime in a Turbulent Jet Flow Using Large-Eddy Simulation/Filtered Density Function Approach

Mixing of a passive scalar in a high-Schmidt turbulent round jet was studied using large-eddy simulation (LES) coupled to filtered density function (FDF). This coupled approach enabled the solution of the continuity, momentum, and scalar (concentration) transport equations when studying mixing in a confined turbulent liquid jet discharging a conserved scalar (rhodamine B) into a low-velocity water stream. The Monte Carlo method was used for solving the FDF transport equation and controlling the number of particles per cell (NPC) using a clustering and splitting algorithm. A sensibility analysis of the number of stochastic particles per cell as well as the influence of the subgrid-scale (SGS) mixing time constant were evaluated. The comparison of simulation results with experiments showed that LES/FDF satisfactorily reproduced the behavior observed in this flow configuration. At high radial distances, the developed superviscous layer generates an intermittency phenomenon leading to a complex, anisotropic behavior of the scalar field, which is difficult to simulate with the conventional and advanced SGS models required by LES. This work showed a close agreement with reported experimental data at this superviscous layer following the FDF approach.

Scalar Filtered Density Function for Large Eddy Simulation of a Bunsen Burner

The scalar filtered density function methodology is employed for large eddy simulation of a turbulent stoichiometric premixed methane-air flame. The scalar filtered density function accounts for the subgrid-scale chemical reaction by considering the mass-weighted probability density function of the subgrid-scale scalar quantities. A transport equation is derived for the scalar filtered density function in which the effects of chemical reactions appear in closed form. The subgrid-scale mixing is modeled via the linear mean square estimation model, and the convective fluxes are modeled via a subgrid-scale viscosity. The modeled scalar filtered density function transport equation is solved by a hybrid finite difference and Monte Carlo scheme. A novel irregular Monte Carlo portioning procedure is developed that facilitates efficient simulations with realistic flow parameters. Combustion chemistry is modeled via five-step, nine-species reduced chemical kinetics. Simulated results are assessed by comparisons against laboratory data. Good agreements are observed, capturing several important features of the flame as observed experimentally.

Application Of The Two-phase Filtered DensityFunction Approach For LES Of A 2D DropletLaden Turbulent Mixing Layer

In this study, the two-phase velocity-scalar filtered density function transport equation for large eddy simulation is considered in the limit of a continuum-dispersed phase two-phase flow. For a sufficiently small dispersed particulate phase, all quantities conditionally filtered within the dispersed phase may be disregarded leaving only terms conditionally filtered on the phase interface. These conditionally surface-filtered terms account for phase-coupling between the dispersed and continuum phases of the flow. Closure models are presented and implemented for a two-phase system consisting of a water droplet laden 2d temporally developing mixing layer. Marginal FDF transport equations are presented for each phase and a statistically equivalent set of Ito stochastic differential equations (SDE) are derived from each marginal FDF equation. Simulations are conducted via a full stand-alone Lagrangian particle Monte-Carlo method with closure models to account for subgrid scale (SGS) mixing and inter-phase conversion processes. The effect of variable Stokes number on turbulent dispersion of evaporating and non-evaporating droplets is discussed. 1 Marginal FDF transport equations Two-phase velocity-scalar filtered density function (TVSFDF) methods have recently been developed for large eddy simulation (LES) of separated two-phase flows [1]. In this approach, unclosed terms arise in the marginal FDF transport equations in three classes: quantities conditionally filtered within the phase, quantities conditionally filtered on the phase interface, and inter-phase conversion terms © 2005 WIT Press WIT Transactions on Engineering Sciences, Vol 50, www.witpress.com, ISSN 1743-3533 (on-line) Computational Methods in Multiphase Flow III 283

Formulation Of A Two-phase Filtered DensityFunction Approach For Large Eddy Simulation

Two-phase velocity-scalar filtered density function (FDF) methods have recently been extended for large eddy simulation (LES) of separated two-phase flows [1]. Several terms in the two-phase FDF transport equation are unclosed and must be modeled for practical application of the two-phase FDF theory. Some terms, such as conditional dissipation in momentum, scalar, and momentum-scalar phase spaces are analogous to those in the single-phase velocity-scalar FDF formulation for which closures are readily available. Phase-coupling and interphase conversion terms, however, which are unique to the two-phase FDF formulation are unclosed and must b e modeled. There are, then, th ree d istin ct types o f unclo sed terms in th e two-phase FDF formulation: conditional dissipation terms, phase-coupling terms and interphase conversion terms. In this study, the mathematical formulation of this approach is presented and conditional dissipation, phase-coupling and interphase conversion terms are identified and their connection to standard LES closures identified . 1 Two-phase FDF transport equation Two-phase probability density function (PDF) methods have recently been formulated and successfully applied for dispersed/continuum phase flows [2, 3, 4]. Probability density function methods are based on defining a suitable set of flow realizations over which the PDF may be defined in the ensemble statistics context and therefore in practice are devoid of any spatial information about the two-phase flow. That is to say, turbulent flow field information in the PDF approach must be given in the time-averaged or RANS context, and consequently coupling to a RANS calculation for the carrier gas field is common [4, 2]. Filtered density func© 2005 WIT Press WIT Transactions on Engineering Sciences, Vol 50, www.witpress.com, ISSN 1743-3533 (on-line) Computational Methods in Multiphase Flow III 275

Large Eddy Simulation (2D) of a Reacting Plane Mixing Layer Using Filtered Density Function Closure

The filtered density function (FDF) is implemented for a two-dimensional, large eddy simulation (LES) of a gas phase, spatially developing, reacting and non-reacting, constant-density, plane mixing layer in a flow regime prior to the mixing transition where the flow is mainly two-dimensional. The unresolved scalar fluctuations are taken into account by considering the probability density function (PDF) of subgrid scale (SGS) scalar quantities following the FDF approach. In the derived FDF transport equation, the effect of chemical reactions appears in a closed form. The Lagrangian Monte Carlo scheme is used to solve the FDF transport equation. The applicability and performance of the FDF for LES of a reacting plane mixing layer are assessed by comparisons with experimental measurements. In non-reacting flow, the calculated mean streamwise velocity profiles and mean mixture fraction profiles relax to self-similarity, which is in satisfactory agreement with the measurements. In reacting flow, the FDF calculation provided a satisfactory accuracy in comparison with measurements of mean reactant and product concentration. The increase in the total amount of product formation in the flip case demonstrates the asymmetric characteristics of the entrainment and mixing characteristics in the mixing layer.