The main research topic in this dissertation is the development of the closure method of multiple testing procedures. Considering a general procedure that allows the underlying test statistics as well as the associated parameters to be dependent, we first propose a step-down procedure controlling the FWER, which is defined as the probability of committing at least one false discovery. Holm (1979) first proposed a step-down procedure for multiple hypothesis testing with a control of the familywise error rate (FWER) under any kind of dependence. Under the normal distributional setup, Seneta and Chen (2005) sharpened the Holm procedure by taking into account the correlations between the test statistics. In this dissertation, the Seneta-Chen procedure is further modified yielding a more powerful FWER controlling procedure. We then advance our research and propose another step-down procedure to control the generalized FWER (k-FWER), which is defined as the probability of making at least k false discoveries. We compare our proposed k-FWER procedure with the Lehmann and Romano (2005) procedure. The proposed k-FWER procedure is more powerful, particularly when there is a strong dependence in the tests. When the proportion of true null hypotheses is expected to be small, the traditional tests are usually conservative by a factor associated with pi0, which is the proportion of true null hypotheses among all null hypotheses. Under independence, two procedures controlling the FWER and the k-FWER are proposed in this dissertation. Simulations are carried out to show that our procedures often provide much better FWER or k-FWER control and power than the traditional procedures. / Statistics

The accurate description of the turbulence chemistry interactions that can determine chemical conversion rates and flame stability in turbulent combustion modelling is a challenging research area. This thesis presents the development and implementation of a model for the treatment of fluctuations around the conditional mean (i.e., the auto-ignition and extinction phenomenon) of realistic turbulence-chemistry interactions in computational fluid dynamics (CFD) software. The wider objective is to apply the model to advanced combustion modelling and extend the present analysis to larger hydrocarbon fuels and particularly focus on the ability of the model to capture the effects of particulate formation such as soot. A comprehensive approach for modelling of turbulent combustion is developed in this work. A direct quadrature conditional moment closure (DQCMC) method for the treatment of realistic turbulence-chemistry interactions in computational fluid dynamics (CFD) software is described. The method which is based on the direct quadrature method of moments (DQMOM) coupled with the Conditional Moment Closure (CMC) equations is in simplified form and easily implementable in existing CMC formulation for CFD code. The observed fluctuations of scalar dissipation around the conditional mean values are captured by the treatment of a set of mixing environments, each with its pre-defined weight. In the DQCMC method the resulting equations are similar to that of the first-order CMC, and the “diffusion in the mixture fraction space” term is strictly positive and no correction factors are used. Results have been presented for two mixing environments, where the resulting matrices of the DQCMC can be inverted analytically. Initially the DQCMC is tested for a simple hydrogen flame using a multi species chemical scheme containing nine species. The effects of the fluctuations around the conditional means are captured qualitatively and the predicted results are in very good agreement with observed trends from direct numerical simulations (DNS). To extend the analysis further and validate the model for larger hydrocarbon fuel, the simulations have been performed for n-heptane flame using detailed multi species chemical scheme containing 67 species. The hydrocarbon fuel showed improved results in comparison to the simple hydrogen flame. It suggests that higher hydrocarbons are more sensitive to local scalar dissipation rate and the fluctuations around the conditional means than the hydrogen. Finally, the DQCMC is coupled with a semi-empirical soot model to study the effects of particulate formation such as soot. The modelling results show to predict qualitatively the trends from DNS and are in very good agreement with available experimental data from a shock tube concerning ignition delays time. Furthermore, the findings suggest that the DQCMC approach is a promising framework for soot modelling.