CFD-Based Prediction of Combustion Dynamics and Nonlinear Flame Transfer Functions for a Swirl-Stabilized High-Pressure Combustor

Thermoacoustic instabilities in gasturbine combustor systems can be predicted in the design phase with a thermoacoustic network model. In this model, the coupling between acoustic pressure fluctuations and the combustion rate is described by the Flame Transfer Function. The present paper introduces a new, efficient, and robust method for deriving the FTF from CFD predictions by means of a discrete multi-frequency sinusoidal fuel flow excitation method. The CFD-based FTF result compares well with experimental data for the time delay, but for the gain, only up to 400 Hz. Above 400 Hz, the CFD result reveals a smooth low-amplitude gain, which is not found in the measured data. A novel, accurate continuous correlation function for the FTF gain is computed based on the results for discrete frequencies. When this is implemented into a 1D acoustic network model, the stability map shows, below 600 Hz, two eigenfrequencies, by both the experiment and CFD-based FTF, that are identical. The CFD-based FTF correctly predicts marginal activity at the highest eigenfrequency, while the experimentally based FTF suggests an unstable operation. The unstable operation is not observed in the experiments. This suggests that the CFD-based FTF is also correct for high frequencies.</p

Coupled/combined compact IRBF schemes for fluid flow and FSI problems

The thesis is concerned with the development of compact approximation methods based on Integrated Radial Basis Functions (IRBFs) and their applications in fluid flows and FSI problems. The contributions include (i) new compact IRBF stencils where first- and second-order derivatives are included; (ii) a preconditioning technique where a preconditioner to enhance the stability of the flat IRBF solutions; and, (iii) the incorporation of the proposed stencils into the immersed boundary methods. Numerical experiments show the present schemes generally produce more accurate solutions and better convergence rates than existing methods (e.g. FDM, high-order compact FDM and compact IRBF methods)

Restarted Hessenberg method for solving shifted nonsymmetric linear systems

It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual (GMRES) method in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. Many variants of them have been proposed to enhance their performance. We show that another restarted method, the restarted Hessenberg method [M. Heyouni, M\'ethode de Hessenberg G\'en\'eralis\'ee et Applications, Ph.D. Thesis, Universit\'e des Sciences et Technologies de Lille, France, 1996] based on Hessenberg procedure, can effectively be employed, which can provide accelerating convergence rate with respect to the number of restarts. Theoretical analysis shows that the new residual of shifted restarted Hessenberg method is still collinear with each other. In these cases where the proposed algorithm needs less enough CPU time elapsed to converge than the earlier established restarted shifted FOM, weighted restarted shifted FOM, and some other popular shifted iterative solvers based on the short-term vector recurrence, as shown via extensive numerical experiments involving the recent popular applications of handling the time fractional differential equations.Comment: 19 pages, 7 tables. Some corrections for updating the reference

Asymptotic theory for Bayesian nonparametric inference in statistical models arising from partial differential equations

Partial differential equations (PDEs) are primary mathematical tools to model the behaviour of complex real-world systems. PDEs generally include a collection of parameters in their formulation, which are often unknown in applications and need to be estimated from the data. In the present thesis, we investigate the theoretical performance of nonparametric Bayesian procedures in such parameter identification problems in PDEs. In particular, inverse regression models for elliptic equations and stochastic diffusion models are considered. In Chapter 2, we study the statistical inverse problem of recovering an unknown function from a linear indirect measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty. We prove a semiparametric Bernstein–von Mises theorem for a large collection of linear functionals of the unknown, implying that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a frequentist point of view. The general result is applied to three concrete examples that cover both the mildly and severely ill-posed cases: specifically, elliptic inverse problems, an elliptic boundary value problem, and the recovery of the initial condition of the heat equation. For the elliptic boundary value problem, we also obtain a nonparametric version of the theorem that entails the convergence of the posterior distribution to a prior-independent infinite-dimensional Gaussian probability measure with minimal covariance. As a consequence, it follows that the Tikhonov regulariser is an efficient estimator, and we derive frequentist guarantees for certain credible balls centred around it. Chapter 3 is concerned with statistical nonlinear inverse problems. We focus on the prototypical example of recovering the unknown conductivity function in an elliptic PDE in divergence form from discrete noisy point evaluations of the PDE solution. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate, algebraic in inverse sample size, for the estimation error of the associated posterior means. Finally, in Chapter 4 we extend the posterior consistency analysis to dynamical models based on stochastic differential equations. We study nonparametric Bayesian models for reversible multi-dimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and p-exponential priors, which are shown to converge to the truth at the minimax optimal rate over Sobolev smoothness classes in any dimension. Chapter 1 is dedicated to introducing the statistical models considered in Chapters 2 - 4, and to providing an overview of the theoretical results derived therein. The main theorems of Chapter 2 and Chapter 3 are illustrated via the results of simulations, and detailed comments are provided on the implementation.Richard Nickl’s ERC grant No. 647812; EPSRC grant EP/L016516/1 for the Cambridge Centre for Analysi

Hypersonic Boundary-Layer Stability Across a Compression Corner

Stability of a hypersonic boundary-layer over a compression corner was investigated numerically. The three-dimensional compressible Navier-Stokes equations were solved using a fifth-order weighted essentially non-oscillating (ArENO) shock capturing scheme to study the shock wave and boundary-layer interactions. The boundary-layer stability was studied in three distinct regions: upstream of the separation region, inside the separation region and downstream of the separation region. After the mean flow field was computed, linear stability theory was employed to predict the unstable disturbance modes in different flow regions and also to find the most amplified disturbance frequency across the compression corner. Gortler instability computations were performed to study the influence of the streamline curvatures on boundary-layer stability, and PSE(parabolized stability equation) method was employed to obtain the initial disturbances for direct numerical simulation. To study the boundary-layer stability by direct numerical simulation, two- or three-dimensional initial disturbances were introduced at the initial streamwise location of the computational domain. Two-dimensional disturbance evolution simulation shows that two-dimensional high frequency linear disturbances grow exponentially upstream and downstream of the separation region and remain neutral in the separation region, but two-dimensional low frequency linear disturbances only grow in a narrow area inside the separation region and remain neutral upstream and downstream of the separation region. Two-dimensional nonlinear disturbances will saturate downstream of the separation region when their amplitudes reach quit large amplitude. The three-dimensional disturbance evolution simulations show that three-dimensional linear mono-frequency disturbances are less amplified than its two-dimensional counterpart across the compression corner. The three-dimensional nonlinear mono-frequency disturbance evolution indicates that mode (0,2) is responsible for the oblique breakdown. Three-dimensional disturbances are much more amplified with the presence of two-dimensional primary disturbance due to the secondary instability. Finally, the simulations of three-dimensional random frequency disturbance evolution with the presence of a two-dimensional primary disturbance show that the secondary instability first occurs downstream of the separation region and a fundamental or K-type breakdown will be triggered by this secondary instability

Hypersonic Boundary Layer Receptivity to Acoustic Disturbances Over Cones

The receptivity mechanisms of hypersonic boundary layers to free stream acoustic disturbances are studied using both linear stability theory (LST) and direct numerical simulations (DNS). A computational code is developed for numerical simulation of steady and unsteady hypersonic flow over cones by combining a fifth-order weighted essentially non-oscillatory (WENO) scheme with third-order total-variation-diminishing (TVD) Runge-Kutta method. Hypersonic boundary layer receptivity to free-stream acoustic disturbances in slow and fast modes over 5-degree, half-angle blunt cones and wedges are numerically investigated. The free-stream Mach number is 6.0, and the unit Reynolds number is 7.8×106/ft. Both the steady and unsteady solutions are obtained by solving the full Navier-Stokes equations in two-dimensional and axisymmetric coordinates. Computations are performed in three steps. After the steady mean flow field is computed, linear stability analysis is performed to find the most amplified frequency and the unstable disturbance modes in different flow regions. Then time accurate computations are performed using slow and fast mode acoustic disturbances, and the initial generation, interaction and evolution of instability waves inside the boundary layers are studied. Receptivity computations showed that the acoustic disturbance waves propagated uniformly to downstream, interact with the bow shock, enter the boundary layer, and then generate the initial amplitude of the instability waves in the leading edge region. Effects of the entropy layer due to nose bluntness to the receptivity process are studied. It is found that transition location moves downstream and is delayed by increasing bluntness, and the role of the entropy layer in this process is revealed. Also, the effects of wall cooling to the receptivity process using slow and fast mode acoustic disturbances are studied. The effects of cooling on the first and second mode regions are investigated. It is found that the first mode is stabilized and the second mode is destabilized by wall cooling when the flow is forced by acoustic waves in the slow mode

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed

Statistical inference and computation in elliptic PDE models

Partial differential equations (PDE) are ubiquitous in describing real-world phenomena. In many statistical models, PDE are used to encode complex relationships between unknown quantities and the observed data. We investigate statistical and computational questions arising in such models, adopting an infinite-dimensional `nonparametric' framework and assuming the observed data are subject to random noise. The main PDE examples are of elliptic or parabolic type. Chapter 2 investigates the problem of sampling from high-dimensional Bayesian posterior distributions. The main results consist of non-asymptotic computational guarantees for Langevin-type Markov chain Monte Carlo (MCMC) algorithms which scale polynomially in key quantities such as the dimension of the model, the desired precision level, and the number of available statistical measurements. The bounds hold with high probability under the distribution of the data, assuming that certain `local geometric' assumptions are fulfilled and that a good initialiser of the algorithm is available. We study a representative non-linear PDE example where the unknown is a coefficient function in a steady-state Schr\"odinger equation, and the solution to a corresponding boundary value problem is observed. Chapter 3 studies statistical convergence rates for nonparametric Tikhonov-type estimators, which can be interpreted also as Bayesian maximum a posteriori (MAP) estimators arising from certain Gaussian process priors. The theory is derived in a general setting for non-linear inverse problems and then applied to two examples, the steady-state Schr\"odinger equation studied in Chapter \ref{sampling} and a model for the steady-state heat equation. It is shown that the rates obtained are minimax-optimal in prediction loss. The final Chapter 4 considers a model for scalar diffusion processes $(X_t:t\geq 0)$ with an unknown drift function which is modelled nonparametrically. It is shown that in the low frequency sampling case, when the sample consists of $(X_0,X_\Delta,...,X_{n\Delta})$ for some fixed sampling distance $\Delta>0$, under mild regularity assumptions, the model satisfies the local asymptotic normality (LAN) property. The key tools used are regularity estimates and spectral properties for certain parabolic and elliptic PDE related to $(X_t:t\geq 0)$