11 - Are my results significant?

This work newly proposes an uncertainty quantification (UQ) method named sparse approximation of moment-based arbitrary polynomial chaos (SAMBA PC) that offers a single solution to many current problems in turbomachinery applications. At the moment, every specific case is characterized by a variety of different input types such as histograms (from experimental data), normal probability density functions (PDFs) (design rules) or fat tailed PDFs (for rare events). Thus, the application of UQ requires the adaptation of ad hoc methods for each individual case. A second problem is that parametric PDFs have to be determined for all inputs. This is difficult if only few samples are available. In gas turbines, however, the collection of statistical information is difficult, expensive, and having scarce information is the norm. A third critical limitation is that if using nonintrusive polynomial chaos (NIPC) methods, the number of required simulations grows exponentially with increasing numbers of input uncertainties: the so-called “curse of dimensionality.” It is shown that the fitting of parametric PDFs to small data sets can lead to large bias and the direct use of the available data is more accurate. This is done by propagating uncertainty through several test functions and the computational fluid dynamics (CFD) simulation of a diffuser, highlighting the impact of different PDF fittings on the output. From the results, it is concluded that the direct propagation of the experimental data set is preferable to the fit of parametric distributions if data is scarce. Thus, the suggested method offers an alternative to the maximum entropy theorem to handle scarce data. SAMBA simplifies the mathematical procedure for many different input types by basing the polynomial expansion on moments. Its moment-based expansion automatically takes care of arbitrary combinations of different input data. It is also numerically efficient compared to other UQ implementations. The relationship between the number of random variables and number of simulation is linear (only 21 simulations for ten input random variables are required). It is shown in this paper that SAMBA's algorithm can propagate a high number of input distributions through a set of nonlinear analytic test functions. Doing this, the code needs a very small number of simulations and preserve a 5% error margin. SAMBA's flexibility to handle different forms of input distributions and a high number of input variables is shown on a low-pressure turbine (LPT) blade-based on H2 profile. The relative importance of manufacturing errors in different location of the blade is analyzed.

In recent decades, analysis of structures considering variability of some parameters for more reliable design has attracted the attention of researchers. In this paper, the stochastic analysis of a cantilever deep beam made of large deformable neo-hookean material is carried out. For this purpose, the meshless local Petrov-Galerkin (MLPG) method is developed to obtain the geometrically non-linear equilibrium equations. The radial point interpolation method is used for generating the shape functions. The incremental iterative Newton-Raphson method with suitable load steps is used to solve the non-linear governing equations. The results of deterministic analysis obtained with proposed method are compared with the finite element results and good agreement is achieved. The initial elasticity modulus of neo-hookean material is considered to be uncertain variable. To generate random field of uncertain variable with normal, lognormal and uniform probability density functions (PDFs), the Monte Carlo Simulation (MCS) technique was employed. The sufficient number of simulations for convergence the results was determined experimentally. The effect of elasticity modulus, PDF and coefficients of variation (COV) on maximum vertical displacement, PDF and COV of results are studied in details. Comparing the stochastic and deterministic results shows that the uncertainty in mechanical properties has significant effect on results.

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.

This article proposes a novel probabilistic Distribution Feeder Reconfiguration (DFR) based method to consider the uncertainty impacts into account with high accuracy. In order to achieve the set aim, different scenarios are generated to demonstrate the degree of uncertainty in the investigated elements which are known as the active and reactive load consumption and the active power generation of the wind power units. Notably, a normal Probability Density Function (PDF) based on the desired accuracy is divided into several class intervals for each uncertain parameter. Besides, the Weiball PDF is utilised for modelling wind generators and taking the variation impacts of the power production in wind generators. The proposed problem is solved based on Fuzzy Adaptive Modified Particle Swarm Optimisation to find the most optimal switching scheme during the Multi-objective DFR. Moreover, this paper holds two suggestions known as new mutation methods to adjust the inertia weight of PSO by the fuzzy rules to enhance its ability in global searching within the entire search space.

The problem of statistical pattern recognition with noisy or imprecise feature measurements is considered. An exact analytical expression is found for the probability of misclassification under this condition, for multiclass multivariate systems. The probability of error exceeds that of the ideal case for the special case of two classes, the a priori conditional probability density functions are assumed to be normal, along with the two cases of feature measurement error, namely normal and uniform probability density functions. Monotonicity of the misclassification probability with measurement error variance is shown. Numerical results are presented for both cases over a workable range of parameters. The study is useful in practical pattern recognition problems.

A new method for the approximation of multivariate scalar probability density functions (PDFs) in turbulent reacting flow by means of a joint presumed discrete distribution (jPDD) is presented. The jPDDs can be generated with specified mean values and variances as well as covariances. Correlations between variables – e.g. fluctuating mixture fractions and/or reaction progress – can thereby be taken into account. In this way the new approach overcomes an important limitation of ordinary presumed PDF methods, where statistical independence between the variables is often assumed. Different methods are presented to generate discrete distributions, based either on biased random number generators or on mixing models familiar from PDF transport models. The new approach is extensively validated on a turbulent flow configuration with simultaneous mixing and reaction. Large eddy simulation data as well as results from a transported PDF model are used for the validation of the jPDD approach. The comparison shows that in particular distributions generated with mixing models are able to predict mean reaction rates accurately. For the configuration considered, the neglect of correlations results in significant underestimation of reaction rates. Moreover it is found that higher statistical moments (e.g. the skewness) can influence reaction rates. The consequences for the generation of jPDDs are discussed. In summary, the new jPDD model has the potential to be significantly more accurate than established presumed PDF methods, because correlations between fluctuating variables can be taken into account. At the same time, the new approach is nearly as efficient as standard presumed PDF formulations, since mean rates are computed in a pre-processing step and stored in look-up tables as a function of the first and second moments of the relevant variables.

In this paper we discuss non-parametric estimation of the probability density function (PDF) of a univariate random variable. This problem has been the subject of a vast amount of scientific literature in many domains: while statisticians are mainly interested in the analysis of the properties of proposed estimators, engineers treat the histogram as a ready-to-use tool for dataset analysis. By considering histogram data as a numerical sequence, a simple PDF estimator is presented in this paper. It is based on basic notions related to the reconstruction of a continuous-time signal from a sequence of samples and it is as accurate as kernel-based estimators, widely adopted in the statistical literature. The major properties of the proposed PDF estimator are discussed and then verified by simulations related to the common case of a normal density function.

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.

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.

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.

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.

The objective of the present work is to extend the probability density function (PDF) tubulence model to compressible reacting flows. The proability density function of the species mass fractions and enthalpy are obtained by solving a PDF evolution equation using a Monte Carlo scheme. The PDF solution procedure is coupled with a compression finite-volume flow solver which provides the velocity and pressure fields. A modeled PDF equation for compressible flows, capable of treating flows with shock waves and suitable to the present coupling scheme, is proposed and tested. Convergence of the combined finite-volume Monte Carlo solution procedure is discussed. Two super sonic diffusion flames are studied using the proposed PDF model and the results are compared with experimental data; marked improvements over solutions without PDF are observed.

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.