Calculation Of Probability Density Function Research Articles

Calculation of probability density function of particle velocity in turbulent flow

The random motion of inertial particles in uniform isotropic turbulence is considered. Fluctuations of the gas velocity at the particle trajectory are modelled as a random Gauss process with a finite decay time of the autocorrelation function. A closed equation for the probability density function (PDF) of the random particle velocity is derived. An analytical solution of the equation for the PDF is found. The equation for the PDF is solved by two numerical methods. The first method of solution is based on the finite-difference approximation of the equation for the PDF. The second method is based on the calculation of empirical PDF, which is obtained by averaging over an ensemble of random trajectories of particles. The results of the comparison of analytical numerical solutions are presented.

Application of the Stochastic Field Method to Multi‐Fluid Flow

AbstractThe Stochastic Field Method was originally derived in the field of combustion and firstly applied to a cavitation problem by our group [1]. Randomly distributed variables exist in both combustion and cavitating problems. Calculating strong nonlinear processes using mean values as input, leads to unphysical results. The consideration of the variable's probability density function (PDF) becomes mandatory. The Stochastic Field Method approximates this PDF by samples similar to methods of Lagrangian particles where imaginary particles are introduced to the flow. In contrast to Lagrangian particles methods, these samples are represented by Eulerian fields described by stochastic partial differential equations. This pure Eulerian interpretation makes the method attractive for CFD‐codes: A coupling of an Eulerian and a Lagrangian perspective becomes obsolete which results in efficient computation times ‐ especially in the presence of a vast number of bubbles. In contrast to other Eulerian methods, the calculation of an arbitrary PDF is possible at low computational cost. The representation of samples as fields allows the visualization of PDFs within every computational cell. The first implementation into a commercial CFD‐Code is presented. In addition, industrial examples, like cavitation in an automotive injection nozzle, are shown.

A LES/PDF simulator on block-structured meshes

A block-structured mesh large-eddy simulation (LES)/probability density function (PDF) simulator is developed within the OpenFOAM framework for computational modelling of complex turbulent reacting flows. The LES/PDF solver is a hybrid solution methodology consisting of (i) a finite-volume (FV) method for solving the filtered mass and momentum equations (LES solver), and (ii) a Lagrangian particle-based Monte Carlo algorithm (PDF solver) for solving the modelled transport equation of the filtered joint PDF of compositions. Both the LES and the PDF methods are developed and combined to form a hybrid LES/PDF simulator entirely within the OpenFOAM framework. The in situ adaptive tabulation method [S.B. Pope, Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation, Combust. Theory Model. 1 (1997), pp. 41–63; L. Lu, S.R. Lantz, Z. Ren, and B.S. Pope, Computationally efficient implementation of combustion chemistry in parallel PDF calculations, J. Comput. Phys. 228 (2009), pp. 5490–5525] is incorporated into the new LES/PDF solver for efficient computations of combustion chemistry with detailed reaction kinetics. The method is designed to utilise a block-structured mesh and can readily be extended to unstructured grids. The three-stage velocity interpolation method of Zhang and Haworth [A general mass consistency algorithm for hybrid particle/finite-volume PDF methods, J. Comput. Phys. 194 (2004), pp. 156–193] is adapted to interpolate the LES velocity field onto particle locations accurately and to enforce the consistency between LES and PDF fields at the numerical solution level. The hybrid algorithm is fully parallelised using the conventional domain decomposition approach. A detailed examination of the effects of each stage and the overall performance of the velocity interpolation algorithm is performed. Accurate coupling of the LES and PDF solvers is demonstrated using the one-way coupling methodology. Then the fully two-way coupled LES/PDF solver is successfully applied to simulate the Sandia Flame-D, and a turbulent non-swirling premixed flame and a turbulent swirling stratified flame from the Cambridge turbulent stratified flame series [M.S. Sweeney, S. Hochgreb, M.J. Dunn, and R.S. Barlow, The structure of turbulent stratified and premixed methane/air flames I: Non-swirling flows, Combust. Flame 159 (2012), pp. 2896–2911; M.S. Sweeney, S. Hochgreb, M.J. Dunn, and R.S. Barlow, The structure of turbulent stratified and premixed methane/air flames II: Swirling flows, Combust. Flame 159 (2012), pp. 2912–2929]. It is found that the LES/PDF method is very robust and the results are in good agreement with the experimental data for both flames.

A fast and objective multidimensional kernel density estimation method: fastKDE

Numerous facets of scientific research implicitly or explicitly call for the estimation of probability densities. Histograms and kernel density estimates (KDEs) are two commonly used techniques for estimating such information, with the KDE generally providing a higher fidelity representation of the probability density function (PDF). Both methods require specification of either a bin width or a kernel bandwidth. While techniques exist for choosing the kernel bandwidth optimally and objectively, they are computationally intensive, since they require repeated calculation of the KDE. A solution for objectively and optimally choosing both the kernel shape and width has recently been developed by Bernacchia and Pigolotti (2011). While this solution theoretically applies to multidimensional KDEs, it has not been clear how to practically do so.A method for practically extending the Bernacchia–Pigolotti KDE to multidimensions is introduced. This multidimensional extension is combined with a recently-developed computational improvement to their method that makes it computationally efficient: a 2D KDE on 105 samples only takes 1 s on a modern workstation. This fast and objective KDE method, called the fastKDE method, retains the excellent statistical convergence properties that have been demonstrated for univariate samples. The fastKDE method exhibits statistical accuracy that is comparable to state-of-the-science KDE methods publicly available in R, and it produces kernel density estimates several orders of magnitude faster. The fastKDE method does an excellent job of encoding covariance information for bivariate samples. This property allows for direct calculation of conditional PDFs with fastKDE. It is demonstrated how this capability might be leveraged for detecting non-trivial relationships between quantities in physical systems, such as transitional behavior.

Modeling of heat transfer and differential diffusion in transported PDF methods

In order to model the conditional diffusive heat and mass fluxes in the joint probability density function (PDF) transport equation of the thermochemical variables, the diffusive fluxes are decomposed into their Favre mean and fluctuation. While the mean flux appears to be closed, the contributions of fluctuating fluxes are modeled with the interaction by exchange with the mean (IEM) model. Usually, the contribution of the Favre averaged diffusive fluxes is neglected at high Reynolds numbers. Here, however, this term is included to account for molecular mixing in regions, where turbulent mixing is negligible. This model approach is applied in steady state Reynolds Averaged Navier–Stokes (RANS)/transported PDF calculations to simulate the heat transfer of wall bounded flows as well as the stabilization of a hydrogen–air flame at the burner tip. For both flow problems it is demonstrated that molecular transport is recovered in regions where turbulent mixing vanishes. In wall bounded flows this is the case in the viscous sublayer. Heat transfer studies show, that “mixing models” based on the high Reynolds number assumption fail to compute correctly the temperature field and the heat flux close to the wall. A similar situation occurs at the flame root of the investigated turbulent hydrogen-air jet flame, where turbulent mixing is still too weak to achieve a fast mixing of reactants. In this area differential diffusion effects are observed in the experiment, i.e. superequilibrium temperatures and nonlinear relations between the elemental mixture fractions of hydrogen and oxygen. It will be shown, that the presented model can successfully reproduce these effects, which underlines the necessity to include Favre averaged molecular diffusive fluxes in transported PDF methods.

An investigation of mixing in a three-stream turbulent jet

Mixing models are required to account for the effects of molecular diffusion in probability density function (PDF) methods for reacting flows, and these models entail one of the greatest sources of modeling uncertainty in PDF calculations. In this work, we perform PDF calculations of a turbulent coaxial jet using both Reynolds-Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES) based methods. The configuration of the coaxial jet comes from a recently developed experiment, in which the conditional diffusion and scalar dissipation rate of two conserved scalars are measured. These measurements present the unique opportunity to compare directly the conditional diffusion implied by PDF mixing models to the measured conditional diffusion. In the RANS-PDF calculations, three classic mixing models are evaluated: Interaction by Exchange with the Mean (IEM), Modified Curl (MC), and Euclidean Minimum Spanning Tree (EMST). Though calculations with all three mixing models yield time-averaged mean and RMS statistics of the two scalars in good agreement with the experimental measurements, the joint PDFs of the two scalars show a wide variability from the measurements. The EMST mixing model is unique in that it produces compositions which lie very close to the slow manifold identified in the experimental work. Next, LES calculations of the same flow are performed, and the dissipation rate and conditional diffusion are calculated. The resolved scalar dissipation rate is found to be in good agreement with the experimental data, but depends strongly on the resolution; the total dissipation rate from the RANS-PDF and LES calculations indicates significantly larger scalar dissipation rates than those measured experimentally. Finally, LES-PDF calculations of the same flow yield joint PDFs in very good agreement with the experimental data. The attenuation of variance production model is introduced and tested in this flow: this new model is found to yield calculations of improved accuracy in LES-PDF calculations on coarse grids.

Free Turbulent Reacting Jet Simulation Based on Combination of Transport Equations and PDF

In this paper, a method based on the combination of transport equations and PDF (Probability Density Function) calculations is tested through the study of a free axisymmetric turbulent jet and non-premixed flames. This method solves the difficulties related to the closure of the chemical source term. PDF calculations provide the statistical properties of the flow by which the mean values of physical quantities are calculated. In addition, Favre averaged quantities, i.e. temperature, species concentration and velocity fields, can be obtained by using PDF. A comparison has been made with previous experimental and numerical studies in the same field. The model is then applied to the investigation of a diffusive turbulent flame for which experimental data exist. The results obtained show good accuracy for velocities, main species and temperature and are very encouraging for use in combustion simulations.

Turbulent dispersion from line sources in grid turbulence

Probability density function (PDF) calculations are reported for the dispersion from line sources in decaying grid turbulence. The calculations are performed using a modified form of the interaction by exchange with the conditional mean (IECM) mixing model. These flows pose a significant challenge to statistical models because the scalar length scale (of the initial plume) is much smaller than the turbulence integral scale. Consequently, this necessitates incorporating the effects of molecular diffusion in order to model laboratory experiments. Previously, Sawford [Flow Turb. Combust. 72, 133 (2004)] performed PDF calculations in conjunction with the IECM mixing model, modeling the effects of molecular diffusion as a random walk in physical space and using a mixing time scale empirically fit to the experimental data of Warhaft [J. Fluid Mech. 144, 363 (1984)]. The resulting transport equation for the scalar variance contains a spurious production term. In the present work, the effects of molecular diffusion are instead modeled by adding a conditional mean scalar drift term, thus avoiding the spurious production of scalar variance. A laminar wake model is used to obtain an analytic expression for the mixing time scale at small times, and this is used as part of a general specification of the mixing time scale. Based on this modeling, PDF calculations are performed, and comparison is made primarily with the experimental data of Warhaft on single and multiple line sources and with the previous calculations of Sawford. A heated mandoline is also considered with comparison to the experimental data of Warhaft and Lumley [J. Fluid Mech. 88, 659 (1978)]. This establishes the validity of the proposed model and the significant effect of molecular diffusion on the decay of scalar fluctuations. The following are the significant predictions of the model. For the line source, the effect of the source size is limited to early times and can be completely accounted for by simple transformations. The peak centerline ratio of the rms to the mean of the scalar increases with the Reynolds number (approximately as Rλ1/3), whereas this ratio tends to a constant (approximately 0.4) at large times independent of Rλ. In addition, the model yields a universal long-time decay exponent for the temperature variance.

MS1-3: Numerical simulation of diesel combustion with high EGR rate(MS: Modeling and Simulation,General Session Papers)

A series calculation method from injector nozzle internal flow to in-cylinder combustion in diesel engines has been developed. In this study, the predicted in-cylinder pressure, heat release rate and NO concentration are compared with the experimental results on several conventional operating conditions and a higher EGR rate condition with high boost pressure and high injection pressure. The pressure and heat release rate histories are in reasonable agreement with the experiments, but the NO concentration is underestimated especially on the higher EGR rate condition. Therefore the investigation for improving the quantitative accuracy of NO concentration is performed. The PDF (probability density function) method is able to estimate the real mean reaction rate containing the reaction rate fluctuation by turbulence. However the PDF calculation is prohibitive on the computer resources. In this study, the PDF method only for the calculation of NO formation is developed for striking a balance between improvement of calculation accuracy and calculation cost. The result shows that the effect of reaction rate fluctuation on NO formation has a little on conventional operating conditions, but increases with decreasing the combustion temperature.

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

Numerical simulation results are presented for the turbulent piloted jet diffusion flame ‘Delft Flame III’. In this flame, which is one of the target flames of the International Workshop on Measurements and Computations of Turbulent Nonpremixed Flames (TNF), effects of turbulence-chemistry interaction are strong and modelling the turbulence-chemistry interaction is a challenge. After an outline of the theoretical framework, a comparison is presented of results with on the one hand the velocity-scalar transported probability density function (PDF) approach with reduced chemistry (ILDM) and on the other hand the scalar PDF approach with detailed chemistry (C 1-mechanism). The same micromixing model (modified coalescence-dispersion model, CD) is used in both studies. The reasons for the significantly better prediction of the mean temperature field by the scalar PDF calculations are discussed. Results of other micromixing models are briefly mentioned. To cite this article: D. Roekaerts et al., C. R. Mecanique 334 (2006).

Lifted methane–air jet flames in a vitiated coflow

The present vitiated coflow flame consists of a lifted jet flame formed by a fuel jet issuing from a central nozzle into a large coaxial flow of hot combustion products from a lean premixed H 2/air flame. The fuel stream consists of CH 4 mixed with air. Detailed multiscalar point measurements from combined Raman–Rayleigh–LIF experiments are obtained for a single base-case condition. The experimental data are presented and then compared to numerical results from probability density function (PDF) calculations incorporating various mixing models. The experimental results reveal broadened bimodal distributions of reactive scalars when the probe volume is in the flame stabilization region. The bimodal distribution is attributed to fluctuation of the instantaneous lifted flame position relative to the probe volume. The PDF calculation using the modified Curl mixing model predicts well several but not all features of the instantaneous temperature and composition distributions, time-averaged scalar profiles, and conditional statistics from the multiscalar experiments. A complementary series of parametric experiments is used to determine the sensitivity of flame liftoff height to jet velocity, coflow velocity, and coflow temperature. The liftoff height is found to be approximately linearly related to each parameter within the ranges tested, and it is most sensitive to coflow temperature. The PDF model predictions for the corresponding conditions show that the sensitivity of flame liftoff height to jet velocity and coflow temperature is reasonably captured, while the sensitivity to coflow velocity is underpredicted.

PDF modelling of turbulent non-premixed combustion with detailed chemistry

Although the significant advantage for the probability density function (PDF) methods of the exact treatment of chemical reactions in turbulent combustion problems, a detailed chemistry mechanism (e.g., the GRI mechanism) has not been implemented in the practical calculations by now due to the prohibitive computation of PDF methods. In this work, a detailed mechanism (GRI-Mech 3.0, consisting of 53 species and 325 elemental reactions) is firstly incorporated into the PDF calculation of a turbulent non-premixed jet flame (Sandia Flame D). The flow is formulated in the boundary layer form. The joint composition PDF closure level is applied and a multiple-time-scale (MTS) k– ε turbulence model is combined for the closure of turbulent transport terms. The molecular mixing process is modelled by the Euclidean minimum spanning tree (EMST) mixing model. The solutions are obtained by using the space marching algorithm for turbulence equations and node-based Monte Carlo particle method for PDF evolution equation. The chemical reaction source terms are integrated directly. Extensive comparisons between the predictions and the measurements are made, which involve radial profiles of mean and rms (root mean square), conditional mean, scatter plots of scalars and conditional PDF distribution etc. The flame structures are well represented by the present calculation, including intermediate species (e.g. CO and H 2) mass fractions, pollutant NO emission and local extinction.

Numerical Prediction of Diluted Hydrogen Turbulent Jet Diffusion Flame Using a Scalar Probability Density Function Method Considering Detailed Chemical Kinetics

A scalar probability density function (PDF) approach considering a detailed chemical kinetics is applied to a diluted hydrogen jet diffusion flame to evaluate the performance. The flame is formed on the nozzle of an inner diameter of 6 mm, with a fuel exit velocity of 30 m/s, surrounded by two annular pipes issuing airs of higher and lower velocities of 30 m/s and 3 m/s, respectively. The flow field has been solved on the basis of the k-ε two equation model. The modeled PDF transport equation has been solved by an Eulerian Monte-Carlo method. The results are compared with those of two flamelet calculations, which are based on the conventional laminar flamelet model method and a scalar PDF method based on the conserved scalar approach. The present PDF method predicts the maximum temperature on the center axis close to the experimental one than the other methods. The temperature downstream of the maximum temperature position has however been overestimated due to the underestimate of the mixing. The present PDF method has reproduced the extinction and reignition phenomena, which cannot be easily predicted by the conventional flamelet calculation. The variation in the diffusion flame structure due to the flame stretch, which was experimentally confirmed by Barlow et al., has been also reproduced. Moreover, the PDF calculation has revealed that intermediate radical species are created largely under the flame stretch effect.

Experimental and numerical study of a highly diluted turbulent diffusion flame close to blowout

The nature of stability, finite rate chemistry, and differential diffusion inherent in a highly diluted turbulent non-premixed hydrogen flame close to blowout is investigated. Major-species concentrations and temperature are measured by Raman/Rayleigh spectroscopy. OH concentration is simultaneously measured by laser induced fluorescence (LIF), and laser Doppler velocimetry (LDV) is used to determine velocities. The experimental data are used to assess the performance of a probability density function (PDF) simulation with a five-step reduced chemistry mechanism and differential diffusion effects. Comparison is made in terms of averaged quantities, scatter plots, and conditional averaged data. First- and second-order statistics of predictions and experimental data are found in reasonably good agreement. The significant amount of finite rate chemistry observed in the flame as well as the relaxation to equilibrium with downstream distance is well reproduced by the PDF calculation. PDF model predictions of differential diffusion compare favorable with experimentally determined values. Numerical results reveal that differential diffusion may have a major impact on the flame stability, although the average deviation from the equal diffusivity limit is small. Computed laminar extinction limits of laminar flame calculations with multicomponent diffusion and equal diffusivity are in agreement with this observation.

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.