Joint Velocity-scalar Probability Density Function Research Articles

Turbulent scalar fluxes from a generalized Langevin model: Implications on mean scalar mixing and tracer particle dispersion

A Generalized Langevin Model (GLM) formulation to be used in transported joint velocity-scalar probability density function methods is recalled in order to imply a turbulent scalar-flux model where the pressure-scrambling term is in correspondence with standard Monin's return-to-isotropy term. The proposed non-constant C0 formulation is extended to seen-velocity models for particle dispersion modeling in dispersed two-phase flows. This allows us to correct the wrong turbulent scalar-flux modeling in the limit of tracer particles. Moreover, this allows us to have a more general formulation in order to consider advanced Reynolds-stress models. The cubic model of Fu, Launder, and Tselepidakis is considered, together with the model of Merci and Dick for turbulent dissipation. Results are presented for different swirling and recirculating single-phase and two-phase flows, showing the capabilities of the proposed non-constant C0 GLM formulations compared to the standard GLM.

Large Eddy Simulation of a supersonic lifted flame using the Eulerian stochastic fields method

Scramjet propulsion systems can be the key to deliver the next generation of hypersonic planes. The high costs and complexity of gathering experimental data is a limiting factor in the development of such engine. In this context, numerical simulation has become increasingly popular to investigate supersonic combustion phenomena that otherwise would be prohibitively expensive. Despite recent progress, the simulation of high-speed compressible and reactive flows is still very challenging and presents many associated challenges. The chemical source term is highly non-linear and most combustion models are designed to operate in low-Mach number conditions. The present work investigates the use of Probability Density Function (PDF) in the context of Large Eddy Simulation models under supersonic conditions. Two approaches are considered: an extension of the joint scalar-enthalpy PDF for high-speed flows and a novel joint velocity-scalar-energy PDF model. Both formulations use the Eulerian stochastic fields approach implemented in a fully compressible density-based CFD code. The performance of the models are investigated in a supersonic lifted flame, comparing the stochastic formulations with traditional models that neglect sub-grid fluctuations. The results show that sub-grid contributions are important at coarse meshes and the stochastic fields approach can reproduce the experimental data and the scatter observed. The simulations suggest that the scalar-enthalpy PDF is the most robust formulations and the sub-grid closures of the joint velocity-scalar PDF need further investigation.

Transported PDF Modeling of Ethanol Spray in Hot-Diluted Coflow Flame

This paper presents a numerical modeling study of one ethanol spray flame from the Delft Spray-in-Hot-Coflow (DSHC) database, which has been used to study Moderate or Intense Low-oxygen Dilution (MILD) combustion of liquid fuels (Correia Rodrigues et al. Combust. Flame 162(3), 759–773, 2015). A “Lagrangian-Lagrangian” approach is adopted where both the joint velocity-scalar Probability Density Function (PDF) for the continuous phase and the joint PDF of droplet properties are modeled and solved. The evolution of the gas phase composition is described by a Flamelet Generated Manifold (FGM) and the interaction by exchange with the mean (IEM) micro-mixing model. Effects of finite conductivity on droplet heating and evaporation are accounted for. The inlet boundary conditions starting in the dilute spray region are obtained from the available experimental data together with the results of a calculation of the spray including the dense region using ANSYS Fluent 15. A method is developed to determine a good estimation for the initial droplet temperature. The inclusion of the “1/3” rule for droplet evaporation and dispersion models is shown to be very important. The current modeling approach is capable of accurately predicting main properties, including mean velocity, droplet mean diameter and number density. The gas temperature is under-predicted in the region where the enthalpy loss due to droplet evaporation is important. The flame structure analysis reveals the existence of two heat release regions, respectively having the characteristics of a premixed and a diffusion flame. The experimental and modeled temperature PDFs are compared, highlighting the capabilities and limitations of the proposed model.

Accurate and computationally efficient mixing models for the simulation of turbulent mixing with PDF methods

Different simulation methods are applicable to study turbulent mixing. When applying probability density function (PDF) methods, turbulent transport, and chemical reactions appear in closed form, which is not the case in second moment closure methods (RANS). Moreover, PDF methods provide the entire joint velocity-scalar PDF instead of a limited set of moments. In PDF methods, however, a mixing model is required to account for molecular diffusion. In joint velocity-scalar PDF methods, mixing models should also account for the joint velocity-scalar statistics, which is often under appreciated in applications. The interaction by exchange with the conditional mean (IECM) model accounts for these joint statistics, but requires velocity-conditional scalar means that are expensive to compute in spatially three dimensional settings. In this work, two alternative mixing models are presented that provide more accurate PDF predictions at reduced computational cost compared to the IECM model, since no conditional moments have to be computed. All models are tested for different mixing benchmark cases and their computational efficiencies are inspected thoroughly. The benchmark cases involve statistically homogeneous and inhomogeneous settings dealing with three streams that are characterized by two passive scalars. The inhomogeneous case clearly illustrates the importance of accounting for joint velocity-scalar statistics in the mixing model. Failure to do so leads to significant errors in the resulting scalar means, variances and other statistics.

Numerical simulation of flame propagation and localized preflame autoignition in enclosures

A novel computational approach based on the coupled 3D Flame-Tracking–Particle (FTP) method is used for numerical simulation of confined explosions caused by preflame autoignition. The Flame-Tracking (FT) technique implies continuous tracing of the mean flame surface and application of the laminar/turbulent flame velocity concepts. The Particle method is based on the joint velocity–scalar probability density function approach for simulating reactive mixture autoignition in the preflame zone. The coupled algorithm is supplemented with the database of tabulated laminar flame velocities as well as with reaction rates of hydrocarbon fuel oxidation in wide ranges of initial temperature, pressure, and equivalence ratio. The main advantage of the FTP method is that it covers both possible modes of premixed combustion, namely, frontal and volumetric. As examples, combustion of premixed hydrogen–air, propane–air, and n-heptane–air mixtures in enclosures of different geometry is considered. At certain conditions, volumetric hot spots ahead of the propagating flame are identified. These hot spots transform to localized exothermic centers giving birth to spontaneous ignition waves traversing the preflame zone at very high apparent velocities, i.e., nearly homogeneous preflame explosion occurs. The abrupt pressure rise results in the formation of shock waves producing high overpressure peaks after reflections from enclosure walls.

Generalised Langevin Model in Correspondence with a Chosen Standard Scalar-Flux Second-Moment Closure

In the context of transported joint velocity-scalar probability density function methods, the correspondence between Generalised Langevin Models (GLM) for Lagrangian particle velocity evolution and Eulerian Reynolds-stress turbulence models has been established in the 1990’s by S.B. Pope. It was shown that the GLM representation of a given Reynolds stress model is not unique. It was also shown that a given GLM together with a given mixing model for particle composition evolution implies a differential scalar-flux model. In this paper, we study how extra constraints can be applied on the choice of the GLM coefficients in order to imply a chosen scalar-flux model. This correspondence between GLM-implied and standard scalar-flux models is based on the linear relaxation term and on the mean velocity gradient contributions in the rapid term. In general, GLM-implied models possibly involve more terms (including anisotropy effects in the scalar-flux decay rate and some high-order terms in the rapid-pressure-scrambling term). The proposed form of the GLM supposes a non-constant value for the diffusion coefficient C 0, originally identified as a Kolmogorov constant. Here, the value of C 0 is determined in order to yield the Monin model for linear relaxation of the scalar-flux, and the constant in the rapid-pressure contribution is related to the choice of the parameter β ∗ in the GLM. We finally show how GLM-implied scalar-flux models are in general dependent on the choice of the mixing model and how the proposed GLM can reduce this dependency. These developments are illustrated by results obtained from calculations of the Sydney bluff-body stabilised flame HM1.

Conditional statistics for passive-scalar mixing in a confined rectangular turbulent jet

Experimental results for the conditional statistics, such as the velocity conditioned on a conserved scalar and the scalar conditioned on velocity, in a confined liquid-phase rectangular jet are presented and analyzed for a data set collected using simultaneous particle image velocimetry and planar laser-induced fluorescence. The joint velocity-scalar probability density function (PDF) is not joint Gaussian in this flow, as the PDF of the conserved scalar is accurately described by a beta-PDF. The conditional mean velocity is found to agree with a linear model when the scalar is close to its local mean value. A gradient PDF model is found to give poor predictions for the streamwise conditional velocity. However, the improved gradient PDF model predicts both the streamwise and transverse conditional velocities well. A linear model for the scalar fluctuation conditioned on velocity is also tested against the experimental data, showing that this model only obtains good approximations when the joint velocity-scalar PDF approaches a joint Gaussian at farther downstream locations in this flow.

Full velocity-scalar probability density function computation of heated channel flow with wall function approach

A joint velocity-scalar probability density function (PDF) method is presented to model and simulate turbulent flows with passive inert scalars (here temperature). The full PDF approach is applied for wall-bounded flows. In the present work, the boundary conditions are imposed in the logarithmic region and the modeling is therefore performed in the wall-function spirit. The PDF equation is solved by a Monte Carlo method and the whole approach appears as a Lagrangian simulation using stochastic particles. The purpose of the work is to analyze the behavior of classical PDF models in the near-wall region and to develop new particle boundary conditions for the velocity and scalars attached to each particle. First of all, the logarithmic region is described as an equilibrium zone and resulting analytical formulas for second-order temperature–velocity statistics 〈θ2〉, 〈uθ〉, 〈vθ〉 are derived. Boundary conditions for scalars are then developed and formulated in terms of instantaneous particle variables. These results are useful to discuss consistency issues between the formulation of scalar mixing models and the statement of boundary conditions. Finally, heated channel flow is simulated with a stand-alone PDF code for two different heat-flux conditions and results are compared with available direct numerical simulation and experimental data.

Calculations of Swirl Combustors Using Joint Velocity-Scalar Probability Density Function Method

Calculations are reported for recirculating swirling reacting flows using a joint velocity-scalar probability density function (PDF) method. The PDF method offers signiflcant advantages over conventional finite volume, Reynolds-average-based methods, especially for the computation of turbulent reacting flows. The PDF calculations reported here are based on a newly developed solution algorithm for elliptic flows, and on newly developed models for turbulent frequency and velocity that are simpler than those used in previously reported PDF calculations. Calculations are performed for two different gas-turbine-like swirl combustor flows for which detailed measurements are available. The computed results are in good agreement with experimental data.

An Experimental and Computational Study of Swirling Hydrogen Jet Diffusion Flames

Computations using the joint velocity-scalar probability density function (pdf) method as well as benchmark quality experimental data for swirling and nonswirling hydrogen jet diffusion flames are reported. Previous studies of diffusion flames reported in the literature have been limited to nonswirling flames and have had no detailed velocity data reported in the developing (near-nozzle) region of the flames. The measurements and computations reported herein include velocities (mean and higher moments up to fourth order) and temperature (mean and variance) near the burner exit and downstream locations up to 26.5 jet diameters. The velocities were measured with a three-component laser-Doppler velocimeter (LDV) and the temperature was measured using coherent anti-Stokes Raman spectroscopy (CARS). The joint pdf method offers significant advantages over conventional methods for computing turbulent reacting flow, and the computed results are in good agreement with data. This study serves to present data that can be used for model validation as well as to validate further the joint pdf method.

Calculations of swirl combustors using joint velocity-scalar probability density function method

Calculations of swirl combustors using joint velocity-scalar probability density function method

Simulation of a bluff-body-stabilied diffusion flame using second-moment closure and monte carlo methods

A bluff-body-stabilized methane diffusion flame is studied using second-moment closure and Monte Carbo methods. A finite-volume method (FVM) is used to solve the Favre-averaged transport equations for mass and momentum using an assumed-shape probability density function (PDF) to model turbulent scalar fluctuations. A hybrid Monte Carlo method is used to solve for the Favre joint velocity-scalar PDF. Both methods are two-dimensional fully elliptic. The implications of second-mement closures and stochastic Langevin models for the predictions of Reynolds stresses, mean flow, and scalar fields are discussed. In the FVM, turbulence is modeled by the k-e model, Rotta's return-to-isotropy model (Rotta), and isotropization of production model (Basic Model). In the Monte Carlo method, the simplified Langevin model (SLM) and teneralized Langevin model (GLM) are used. Micromixing is modeled by the interaction by exchange with the mean (IEM) model and chemistry is modeled by a constrained equilibriumrium model. A bluff-body-stabilized flame was chosen as a test because the flow pattern exhibits complex phenomena such as recirculation regions and stagnation points. For the hybrid Monte Carlo method, consistency between the FVM and Monte Carlo submodels is checked for an inert and a reacting case. The combination SLM/k-e yields an inconsistent treatment of Reynolds stresses. Consistent results are obtained with the hybrid models SLM/Rotta and GLM/Basic Model. For the reacting flow, scalar transport differs substantially between the FVM and Lagrangian formulation because, in the Monte Carlo submodel, the flow and density fields are not coupled directly. Because the FVM submodel is sensitive to density variations, the hybrid method converges slowly. FVM predictions are compared to experimental data. The Basic Model predictions are superior to standard k-e results. The overall flame predictions are poor because of finite-rate kinetic effects and partial premixing that are not captured, by our conserved-scalar chemistry model.