Probability density function approach for compressible turbulent reacting flows

An accurate prediction of the probability density function (PDF) of the mixture fraction is crucial to the prediction of combustion since mixing plays an important role in turbulent non-premixed and partially premixed flames. This work provides an assessment of the large-eddy simulation (LES)/filtered density function (FDF) method for the prediction of the PDF of the mixture fraction. The advantage of the LES/FDF method is that it provides the full predictions of the statistical distribution of scalars including the mixture fraction. The predictive accuracy of the method for the PDF is yet to be fully validated. Assessing the prediction of the PDF of the mixture fraction, a conserved scalar, is an important starting point. The Sydney/Sandia inhomogeneous inlet jet flame is used as a test case. A quick comparison shows that the LES/FDF predicted PDF shapes of the mixture fraction deviate significantly from the commonly presumed Beta-PDF as well as from the experimental data in the flame. To examine the source of the discrepancy, we clarify the different PDF definitions used in the comparison among the predictions, measurements, and presumed shape PDFs. The discrepancy observed from the comparison is largely reconciled by clarifying the difference between the PDFs that are examined. The PDF of the resolved mixture fraction is shown to be close to the Beta-PDF in both the measurements and predictions, while the PDF directly deduced from the LES/FDF particles deviates significantly from the Beta-PDF. A multimodal PDF analysis and a pseudo convergence analysis are conducted to provide plausible evidence to support the predicted multimodal PDF shapes. The sub-filter scale FDF is shown to be close to the Beta-PDF too through the construction of a synthesised PDF, which supports the common presumed Beta-PDF assumption used in the presumed PDF methods when combined with LES.

Combustion instability due to thermo-acoustic interactions in high-speed propulsion devices such as gas turbines and rocket engines result from pressure waves with very large amplitudes propagating back and forth in the combustion chamber. Exposure to the pressure fluctuations over a long period of time can lead to a cataclysmic failure of engines. The underlying physics governing the generation of the thermo-acoustic instability is a complex interaction among heat release, turbulence, and acoustic waves. Currently, it is very difficult to accurately predict the expected level of oscillations in a combustor. Hence development of strategies and engineering solutions to mitigate thermo-acoustic instability is an active area of research in both academia and industry. In this work, we carry out numerical modeling of thermo-acoustic instability in a self-excited, laboratory scale, model rocket combustor developed at Purdue University. Two different turbulent combustion models to account for turbulence-chemistry interactions are considered in this study, the flamelet model and the transported probability density function (PDF) method. <br>In the flamelet modeling approach, detailed chemical kinetics can be easily incorporated at a relatively low cost in comparison to other turbulent combustion models and it also accounts for turbulence-chemistry interactions. The flamelet model study is divided into two parts. In first part, we examine the effect of different numerical approaches for implementing the flamelet model. In advanced modeling and simulations of turbulent combustion, the accuracy of model predictions is affected by physical model errors as well as errors that arise from the numerical implementation of models in simulation codes. Here we are mainly concerned with the effect of numerical implementation on model predictions of turbulent combustion. Particularly, we employ the flamelet/progress variable (FPV) model and examine the effect of various numerical approaches for the flamelet table integration, with presumed shapes of PDF, on the FPV modeling results. Three different presumed-PDF table integration approaches are examined in detail by employing different numerical integration strategies. The effect of the different presumed-PDF table integration approaches is examined on predictions of two real flames, a laboratory-scale turbulent free jet flame, Sandia Flame D and the self-excited resonance model rocket combustor. Significant difference is observed in the predictions both of the flames. The results in this study further support the claims made in previous studies that it is imperative to preserve the laminar flamelet structure during integration while using the flamelet model to achieve better predictions in simulations. In the second part of the flamelet modeling study, computational investigations of the coupling between the transient flame dynamics such as the ignition delay and local extinction and the thermo-acoustic instability developed in a self-excited resonance combustor to gain deep insights into the mechanisms of thermo-acoustic instability. A modeling framework that employs different flamelet models (the steady flamelet model and the flamelet/progress variable approach) is developed to enable the examination of the effect of the transient flame dynamics caused by the strong coupling of the turbulent mixing and finite-rate chemical kinetics on the occurrence of thermo-acoustic instability. The models are validated by using the available experimental data for the pressure signal. Parametric studies are performed to examine the effect of the occurrence of the transient flame dynamics, the effect of artificial amplification of the Damkohler number, and the effect of neglecting mixture fraction fluctuations on the predictions of the thermo-acoustic instability. The parametric studies reveal that the occurrence of transient flame dynamics has a strong influence on the onset of the thermo-acoustic instability. Further analysis is then conducted to localize the effect of a particular flame dynamic event, the ignition delay, on the thermo-acoustic instability. The reverse effect of the occurrence of the thermo-acoustic instability on the transient flame dynamics in the combustor is also investigated by examining the temporal evolution of the local flame events in conjunction with the pressure wave propagation. The above observed two-way coupling between the transient flame dynamics (the ignition delay) and the thermo-acoustic instability provides a plausible mechanism of the self-excited and sustained thermo-acoustic instability observed in the combustor.<br>The second turbulent combustion model considered in this study is the transported PDF method. The transported PDF method is one of the most attractive models because it treats the highly-nonlinear chemical reaction source term without a closure requirement and it is a generalized model for a wide range of turbulent combustion problems.Traditionally, the transported PDF method has been used to model low-Mach number, incompressible flows where the pressure is assumed to be thermodynamically constant. Since there is significant pressure fluctuations in the model rocket combustor, the flow is highly compressible and it is necessary to account for this compressibility in the transported PDF method. In the past there has been very little work to model compressible reactive flows using the transported PDF and no effort has been made to model thermo-acoustic instability using the transported PDF method. There is a pressing need to further examine and develop the transported PDF method for compressible reactive flows to broaden our understanding of physical phenomenon like thermo-acoustic instability, interaction between combustion and strong shock and expansion waves, coupling between acoustic and heat release which are observed in high-speed turbulent combustion problems. To address this, a modeling framework for compressible turbulent reactive flows by the using the transported PDF method is developed. This framework is validated in a series of test cases ranging from pure mixing to a supersonic turbulent jet flame. The framework is then used to study the thermo-acoustic interactions in the self-excited model rocket combustor.

A probability density or distribution function of turbulence has been thought to be symmetric due to the symmetry of the partial differential equations from the first principles. However, the experimental data have been shown otherwise by a so-called Taylor correlation function, and this is an unresolved issue. A recent study shows that this probability density function can be asymmetric analytically by introducing negative eigenvalues. Here, in this work, we show the mathematical basis for this asymmetry, although the partial differential equations follow the symmetry of the Lie groups. We also demonstrate a complete solution of partial differential equations, including the exponential terms and negative eigenvalues, which plays a vital part in transient phenomena. Our analysis shows that the asymmetry is from the partition of velocities of the same or opposing direction, not from the negative eigenvalues. Fundamentally, the loss of rotational symmetry is caused by the exponential terms for the transient solution, which we demonstrate by the derivation of the complete solution. The new correlation produces excellent agreement with the experimental data. The universality and limitations of the correlation function are discussed, and through the parameter study, the variations and statistical nature of the probability function are clarified. The asymmetry probability function should have wider applicability than the symmetric Gaussian distribution, which is the special case of the asymmetry probability function.

The statistics of laser scintillation have been studied for many years. The random fluctuations are describedby the probability density function (PDF). A rigorous solution for the PDF exists only for weak scattering (lognormal PDF) and very strong scattering (negative exponential PDF). Many models for a general PDF have beenproposed, and a common method of analysis has been the comparison of the moments of the PDF with experi-mental data. The accuracy of these experimental estimates for the moments is difficult to determine theoreti-cally. The statistical behavior of these estimates for the moments is determined by simulation of actual experi-mental procedures assuming various models for the true PDF. Realizations for histograms of digitized data is gen- erated using binomial random deviates. The only parameters required are the total number of statistically independent data samples collected and the average value for each data sample, which depends on the modelPDF. Each realization for the histogram generates a realization for the moments. The PDF of these estimates forthe moments is approximated by producing a histogram of a large number of such realizations. The PDF of theestimates for the moments then provides reliable confidence intervals. The behavior of the confidence limits ispresented for common models for the PDF of laser scintillation.

11 - Are my results significant?

A flamelet-based Eulerian transported PDF method for the modeling and simulation of supersonic combustion

Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods often found wanting when sampling from difficult distributions (e.g., high-dimensional PDFs, PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. First, the statistical efficiency of the algorithm is enhanced by use of the systematic resampling scheme. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated using three examples with irregularly shaped posterior PDFs.

Reactive flow simulations using large-eddy simulations (LES) require modelling of sub-filter fluctuations. Although conserved scalars like mixture fraction can be represented using a beta-function, the reactive scalar probability density function (PDF) does not follow an universal shape. A one-point one-time joint composition PDF transport equation can be used to describe the evolution of the scalar PDF. The high-dimensional nature of this PDF transport equation requires the use of a statistical ensemble of notional particles and is directly coupled to the LES flow solver. However, the large grid sizes used in LES simulations will make such Lagrangian simulations computationally intractable. Here we propose the use of a Eulerian version of the transported-PDF scheme for simulating turbulent reactive flows. The direct quadrature method of moments (DQMOM) uses scalar-type equations with appropriate source terms to evolve the sub-filter PDF in terms of a finite number of delta-functions. Each delta-peak is characterized by a location and weight that are obtained from individual transport equations. To illustrate the feasibility of the scheme, we compare the model against a particle-based Lagrangian scheme and a presumed PDF model for the evolution of the mixture fraction PDF. All these models are applied to an experimental bluff-body flame and the simulated scalar and flow fields are compared with experimental data. The DQMOM model results show good agreement with the experimental data as well as the other sub-filter models used.

Due to the growing concern on the issues of global warming, climate change, and energy shortage, more rigorous requirements are imposed on the energy conversion process of combustion. Due to the relative safety and convenience in transportation and storage, liquid fuels are commonly used in many practical combustion systems such as diesel engines, gas turbines and industrial furnaces. In these combustion devices, turbulent spray flows are involved. The character of spray dispersion, evaporation, mixing and combustion of fuel droplets strongly determines the performance of these systems with respect to the combustion efficiency, stability, and emissions. Therefore, an improved understanding of turbulent spray combustion and the development of predictive models are needed for a better design of more efficient combustion systems. The present work focuses on the development of a computational methodology based on the transported joint probability density function (PDF) method for the modeling and simulation of two-phase turbulent spray flows without and with chemical reactions. In the non-reacting situation, the dependent variables of the joint PDF include the gas velocity and the mixture fraction. For the simulation of reacting spray flows, a three-variate joint PDF transport equation is derived and modeled in order to account for the pre-vaporization effects and partially premixed regime in turbulent spray flames. The dependent variables include the gas phase mixture fraction, the reaction progress variable and gas enthalpy. Detailed combustion chemistry is considered through an extended spray flamelet model by including a reaction progress variable in addition to the classical formulation. The dilute spray is simulated using a Lagrangian discrete parcel method for the description of droplet motion, heating and evaporation. The infinite conductivity model with consideration of non-equilibrium effects based on the Langmuir-Knudsen law is considered. The spray evolution and flame structures in the frame of the polydisperse reacting spray flows are investigated. Numerical results are compared with experimental data provided by Prof. Masri at the University of Sydney, Australia, and the test cases include three different turbulent non-reacting acetone spray flows in air and turbulent spray flames with the liquid fuel ethanol. For the acetone spray flows, computational results generally show good agreement with experimental data in terms of the droplet size and mean velocity distribution, as well as the liquid volume flux. The results show that the inflow liquid mass loading hardly affects the droplet diameter distribution, whereas the inlet turbulence level has a pronounced effect. The tendency of droplet accumulation near the jet centerline is found with a somewhat overprediction of liquid volume flux at downstream locations. A more sophisticated turbulence model is expected to eliminate this discrepancy. Moreover, the local joint PDF of the gas velocity and the mixture fraction is analyzed. A linear correlation of the gas velocity and the mixture fraction exists close to the nozzle exit outside the main spray jet, and no regions are found where statistical independence prevails. For the simulation of spray flames, computations with the newly developed spray flamelet/progress variable approach and with the previous standard spray flamelet formulation are carried out and compared with the experiments by Prof. Masri at the University of Sydney. A good agreement between the computations with the new formulation and the experiments for gas temperature and droplet size and velocity is achieved. The major spray and combustion properties are correctly captured using this new formulation, which is compared with an unphysically attached flame near the nozzle exit predicted by the previous model. A partially premixed combustion prevails in this piloted turbulent spray flame. Due to the prevaporization of the ethanol droplets near the nozzle exit, a lean premixed gas mixture is found at the inner side of spray jet. Moving downstream, the lean-sided diffusion flame is promoted towards the inner fuel-rich side by heating up the inner premixed core that is controlled by the droplet evaporation. Additionally, it is observed that in the far-field region, the diffusion flame becomes the dominant combustion mode. In summary, an efficient computational model based on the transported joint PDF method is developed to two-phase turbulent spray flows. The combined transported joint PDF and a newly proposed spray flamelet/progress variable approach shows an improved performance in the prediction of complex turbulent spray flames, and new insights on the local flame structure influenced by evaporating sprays are obtained.

Communicating editor: K.J. Flynn Molting rate is a key life history parameter in copepods. Since copepod population growth is an inherently exponential process, accurate formulation of molting rate is of critical importance. Many experiments have been conducted to culture different copepod species under varying temperatures and food concentrations. Probability density functions (PDFs) then were used to estimate the median development time (MDT) of different copepod stages from the experimental data. These MDTs are used in copepod population models. Asymmetrical PDFs are widely used to model molting rate, because the shapes of these curves are similar to laboratory data on cohort development. In this paper, we developed an individual stochastic model (ISM) to simulate the molting rate with different PDFs. We showed that there was no connection between the asymmetry of cohorts and the asymmetry of the molting PDF. Although age-within-stage models have been widely used to simulate copepod population dynamics, we found that none had used the correct formulation of molting rate. The population model requires the probability of molting at each time step, whereas the laboratory-derived PDF is the frequency distribution of stage duration. Therefore, the PDF cannot be applied directly to the population model. We present here a corrected formula based on the PDF for use in copepod population models, termed the probability of molting for remaining individuals (PMR). Despite emphasis on use of the gamma function for copepod molting, we found simpler functions work equally well, but that prior use of incorrect molting rate functions in copepod models can seriously overestimate generation time.

The paper describes the development and validation of an efficient and cost effective method for the prediction of the NOx emissions of turbulent gas turbine burners in the early burner design phases which are usually focused on the optimization of the swirler aerodynamics and the fuel air mixing. As the method solely relies on non-reacting tests of burner models in the water channel it can be applied before any test equipment for combustion experiments exists. In order to achieve optimum similarity of fuel air mixing in the water channel tests with engine operation the model is operated at the engine momentum ratio. During the LIF measurements the water flow representing the fuel is doped with fluorescent dye, a plane perpendicular to the length axis near the burner exit plane is illuminated with a 5W Ar-Ion Laser and the fluorescence is recorded with a video camera from downstream. From the video sequences the local probability density functions (PDF) of the dye concentration fluctuations are calculated from the data. Furthermore, the time mean velocity fields are measured with PIV. From the LIF data the PDFs of the local equivalence ratio are derived. Assuming flamelets, the NOx generation in the entire equivalence ratio range observed in the water channel tests is computed using the unstrained freely propagating one-dimensional flame model in Cantera and the GRI3.0 reaction scheme. Although neither flame stretch nor post flame NOx generation were considered the NOx values computed were in excellent agreement with the experimental data from perfectly premixed combustion experiments. The local time averaged NOx mole fraction is obtained by integrating the flamelet NOx over the mixture PDF. Finally the global NOx emission of the burner at the considered operating point is obtained by spatial integration considering the measured velocity field. The method was validated using a conical swirl burner with two fuel injection stages, allowing the degree of premixedness to be adjusted over a wide range depending on the specific fuel injection scenario. For the case with fuel injection along the air inlet slots NOx values slightly above the minimum NOx limit for perfectly premixed combustion were computed. This is consistent with the emission measurements and indicates finite mixing quality of this injection method. In the partially premixed regime the configurations with potential for low NOx emissions were reliably identified with the LIF and PIV based water channel method. The method also shows the steep increase of the NOx emissions with decreasing degree of premixing observed in the experiments but quantitative predictions would have required a postprocessing of the data from the LIF mixing study with a higher spacial resolution than available.

The generation of continuous random processes with jointly specified probability density and covariation functions is considered. The proposed approach is based on the interpretation of the simulated process as a stationary output of a nonlinear dynamic system, excited by white Gaussian noise and described by a system of a first-order stochastic differential equations (SDE). The authors explore how the statistical characteristics of the equation's solution depends on the form of its operator and on the intensity of the input noise. Some aspects of the approximate synthesis of stochastic differential equations and examples of their application to the generation of non-Gaussian continuous processes are considered. The approach should be useful in signal processing when it is necessary to translate the available a priori information on the real random process into the language of its Markov model as well as in simulation of continuous correlated processes with the known probability density function. >

A novel approach to degree-hour calculation: Indoor and outdoor reference temperature based degree-hour calculation

We propose the suitable use of the Post-Widder inversion formula for Laplace transforms – coupled with the Wynn’s ρ-algorithm for accelerating sequences – in order to evaluate (up to the desired accuracy) the probability density function and the distribution function of a large collection of random variables. The method is illustrated on the Tweedie law and the tempered positive Linnik law. In addition, a further application to some laws arising in the context of Brownian motion is considered.

New, exact expressions for the probability density and distribution functions of a bivariate Hoyt (Nakagami-q) process with arbitrary correlation pattern in a nonstationary environment are derived, a solution to a longstanding unsolved problem. More specifically, the following are obtained: joint probability density function, joint cumulative distribution function, power and envelope correlation coefficients, and some statistics related to the signal-to-noise ratio at the output of the selection combiner, namely, outage probability and probability density function. The exact expressions are given in infinite series form, but are mathematically tractable, easy to evaluate, and flexible enough to accommodate a myriad of correlation scenarios, useful in the analysis of a more general fading environment. The power correlation coefficient is found in an exact, simple, closed-form formula, whereas the envelope one, also exact, is more mathematically elaborate. Approximate, simple closed-form expressions for the joint distribution are also obtained that yield (i) excellent results for small to medium correlation coefficients and (ii) reasonable ones for high correlation coefficient.