The different types of uncertainty and the theories developed to account for them are presented in this chapter. The concepts of risk and reliability are introduced and defined following which basic probability ideas are discussed. The methods used to determine structural reliability are described (the direct integration method, Level II reliability methods and the Level I method). Level II reliability methods include the mean value first-order second-moment method and variants of it based on the Hasofer-Lind reliability index used in the case of nonlinear limit state functions and the Rackwitz-Fiessler procedure that has to be followed in the case of non-normal distributions. The Level I reliability method is discussed and the approach followed to determine partial safety factors described. In the next section fuzzy logic and fuzzy set theory are described. They are introduced by distinguishing between classical logic and fuzzy logic, following which fuzzy sets and fuzzy inference are described. In the last section the steps in the fuzzy inference system are presented and examples of such systems mentioned.

The nonlinear response of the hull girder to global loads is treated in this chapter. These include torsional loads, the result of major damage leading to loss of longitudinal strength of part of the hull girder, and hull girder collapse. In the case of torsional loads, of critical importance is the position of the shear centre, and this depends on hull girder geometry (closed or open section). The effect of structural arrangements is then described in relation to longitudinal warping. The effect of discontinuities is discussed and design issues are considered. Combined and coupled horizontal bending and torsion are treated next. The next section deals with the determination of reserve strength of the hull girder following damage. The approach followed by a classification society to calculating residual strength is described and the use of IACS Common Structural Rules in calculating residual strength of oil tankers is presented. The topic of the last part of the chapter is the ultimate strength of the hull girder in longitudinal bending. The need to calculate ultimate strength is discussed, followed by the calculation of ultimate strength using a simplified, upper bound approach. Progressive collapse analysis is presented and this allows for the gradual spread of elasto-plastic behaviour in individual stiffened plate elements of the hull girder.

Quantifying and assessing the computational accuracy of coarse-graining simulations of turbulence is challenging and imperative to achieve prediction – computations and results with a quantified and adequate degree of uncertainty that can be confidently used in projects without reference data. Verification, validation, and uncertainty quantification (VVUQ) provide the tools and metrics to accomplish such an objective. This chapter reviews these methods and illustrates their importance to coarse-graining models. Toward this end, we first describe the sources of computational errors and uncertainties in coarse-graining simulations of turbulence, followed by the concepts of VVUQ. Next, we utilize the modified equation analysis and the physical interpretation of a complex problem to demonstrate the role of VVUQ in evaluating and enhancing the fidelity and confidence in numerical simulations. This is crucial to achieving predictive rather than postdictive simulations.

Using high-order simulations, we have shed light on complex chemically reacting flow processes and identified new mechanisms of the supersonic combustion process. We have employed 11th-order accurate implicit large eddy simulation (ILES) in conjunction with a finite-rate (Arrhenius) thermochemistry model using a reduced reaction mechanism for the combustion of hydrogen and air. We compare the coarse-grained computations with available experiments from the German Aerospace Centre (DLR) and discuss the accuracy and uncertainties. A supersonic combustion chamber can be accurately modelled using high-order ILES without a specific turbulence-chemistry model. The simulations reveal that the flame intermittently propagates upstream behind the wedge-shaped flame holder, alternating between the upper and lower turbulent free shear layers at a frequency of ≃ 7,990 Hz. This can be a leading cause of unsteady pressure loadings on the interior surfaces downstream of the combustion chamber and is a crucial structural design parameter. Furthermore, the simulations reveal that high temperatures are sustained long distances downstream of the combustion onset. A barycentric map for the Reynolds stresses is employed to analyze the turbulent anisotropy. The results correlate the axisymmetric contraction and expansion of turbulence with the interaction of the reflected shock waves and the supersonic combustion hydroxyl production regions. The physics insights presented in this study could potentially lead to more efficient supersonic combustion and engineering designs.

This chapter gives an overview of data-driven methods applied to turbulence closure modeling for coarse graining. A non-exhaustive introduction of the various data-driven approaches that have been used in the context of closure modeling is provided which includes a discussion of model consistency, which is the ultimate indicator of a successful model, and other key concepts. More details are then presented for two specific methods, one a neural-network representative of nontransparent black-box approaches and one specific type of evolutionary algorithm representative of transparent approaches yielding explicit mathematical expressions. The importance of satisfying physical constraints is emphasized and methods to choose the most relevant input features are suggested. Several recent applications of data-driven methods to subgrid closure modeling are discussed, both for nonreactive and reactive flow configurations. The chapter is concluded with current trends and an assessment of what can be realistically expected of data-driven methods for coarse graining.

A nuclear detonation’s energy release can be approximately broken up into blast (50%), thermal (35%), and radiation (15%). If a detonation occurs significantly above ground (airburst) and various factors are favorable, for example, few clouds and no snow on the ground, then thermal radiation can ignite surface fires. These fires will first commence within fine fuels, such as paper and leaves on vegetation, but given time, these small-scale fires can upscale to larger fires that burn entire houses, trees, and possibly a city. Depending on weather conditions, the fires may continue to spread within a city and impact first responders or civilians sheltering in place to avoid fallout. This chapter highlights the coarse-graining of turbulence, combustion, and cloud physics associated with ignition, spread, and possible interaction of fires with nuclear fallout plumes. In particular, examples are given to illustrate the complex relationship between fallout and fires, an idealized detonation over Dallas (Texas, USA) and Hiroshima (Japan). For both examples, even though the nuclear airburst was at a fallout-free height of burst, the complex and turbulent interaction of the fires with clouds induced significant fallout on the ground.

With high-speed turbulent combustion applications, we here mean airbreathing engine systems capable of powering aircraft at supersonic and low hypersonic flight speeds between 3 < Ma < 8. Such aircraft are most likely to be designed differently compared to today’s aircraft, being centered around a common engine duct embracing different engine systems activated at different flight speeds. For takeoff and landing, conventional turbojet engines will likely be used, whereas for cruise conditions, a dual-mode ramjet engine, capable of transi-tioning between pure ramjet and scramjet modes is preferred. Such engines do not yet exist, but experimental and computational research is currently generating data and information, paving the way toward further understanding of the aerothermodynamics. This will generate the basis for more advanced experiments that, together with high-fidelity simulations, can lead toward the realization of hypersonic flight vehicles. Here, coarse-grained reacting large eddy simulation and hybrid Reynolds-averaged Navier–Stokes or LES, together with small comprehensive reaction mechanisms, conjugate heat transfer, and thermal radiation modeling, play an important role. In this chapter, the necessary modeling steps and methods, as well as chemical reaction mechanisms, are scrutinized, and results from a few selected cases are presented to illustrate the key physical processes as well as the accuracy of present LES-based prediction methods and the remaining challenges.

We live in a turbulent world observed through coarse-grained lenses. Coarse graining (CG), however, is not only a limit but also a need imposed by the enormous amount of data produced by modern simulations. Target audiences for our survey are graduate students, basic research scientists, and professionals involved in the design and analysis of complex turbulent flows. The ideal readers of this book are researchers with a basic knowledge of fluid mechanics, turbulence, computing, and statistical methods, who are disposed to enlarging their understanding of the fundamentals of CG and are interested in examining different methods applied to managing a chaotic world observed through coarse-grained lenses.

We explore the treatment of near-wall turbulence in coarse-grained representations of wall-bounded turbulence. Such representations are complicated by the fact that at high Reynolds number the near-wall effects occur in an asymptotically thin layer. Because of this, many near-wall models are posed as effective boundary conditions, essentially eliminating the thin wall layer that is too thin to resolve. This is commonly referred to as wall-modeled large eddy simulation, and the viability of this approach is supported by the weakness of the interaction between the near-wall turbulence and that further away. Such models are generally informed by known characteristics of near-wall turbulence, such as the log-layer in the mean velocity and the so-called law-of-the-wall. In this chapter, we consider such coarse-grained near-wall models and the approximations implicit in their formulation from the perspective of thin-layer asymptotics.

We live in a turbulent world observed through coarse-grained lenses. Coarse graining (CG), however, is not only a limit but also a need imposed by the enormous amount of data produced by modern simulations. Target audiences for our survey are graduate students, basic research scientists, and professionals involved in the design and analysis of complex turbulent flows. The ideal readers of this book are researchers with a basic knowledge of fluid mechanics, turbulence, computing, and statistical methods, who are disposed to enlarging their understanding of the fundamentals of CG and are interested in examining different methods applied to managing a chaotic world observed through coarse-grained lenses.

The present work is intended as a proposition for a new research program for rigorous physical subgrid-scale (SGS) modeling on the combined basis of (i) the Germano identity (ii) and Lie symmetries, which are the axiomatic foundation of classical mechanics. First, new results are presented in this regard. The basic idea here is based on the Germano identity and the fundamental assumption in that the SGS model is just a functional of the resolved scales $\overline{U}$, that is, in the usual notation ${\tau }_{ij}^{\text{SGS}}={\tau }_{ij}^{\text{SGS}}(\overline{U})$, although this can of course also be generalized.

This alone defines a new functional equation of the form ${\tau }_{ik}[\tilde{\tilde{U}}]-\widetilde{{\tau }_{ik}[\overline{U}]}-\widetilde{{\overline{U}}_i{\overline{U}}_k}+{\tilde{\overline{U}}}_i{\tilde{\overline{U}}}_k\equiv 0$ for the SGS model, if the residual error in the Germano identity is set exactly to zero. This is in contrast to the usual dynamic procedure, where a given SGS model is introduced into the Germano identity and the residual error is minimized according to a given norm. The resulting functional equation for ${\tau }_{ij}^{\text{SGS}}(\overline{U})$ defines a new class of model equations. The solution of the aforementioned equation for the SGS model ${\tau }_{ij}^{\text{SGS}}$ using homogenization transform and Fourier transform shows an extremely large variety of potential solutions, that is, SGS models, which at the same time addresses the classical question of how the shape of the test filter as well as the SGS model are related to each other. The analysis quite naturally shows that the proposed analysis focuses solely on the nonlinear term of the Navier–Stokes equations. For physically realizable SGS models, the very large variety of solutions is restricted by means of the classical as well as statistical Lie symmetries of the filtered Navier–Stokes equations. The latter describes the intermittency and non-Gaussian behavior of turbulence. The symmetries can be used decidedly here, that is, it can be selected quite specifically which symmetries are to be fulfilled. A number of models are presented as examples, some of which have similarities to classical models, and also new nonlocal models emerge. As an additional new result we find that the Germano identity can be extended by a divergence-free tensor. The physical meaning of this previously overlooked term needs to be further investigated, but in the classical dynamical procedure the term does not vanish and may be employed profitably, for example, for model optimization. We conclude the presented formulation of a mathematical work program for the development of SGS models based on Lie symmetries and the Germano identity with an extensive outlook for potential further research directions.

Most turbulence theory is derived in a theorized asymptotic state. But real engineering problems almost never reach such a state; in the real world, the route to turbulence leaves its fingerprints on the observed flow. Any coarse-grained simulation must handle this, either by resolving the transition process or modeling some or all of it. Either approach faces significant challenges. If the transition is to be resolved, then a suitable mechanism for turning on and off any turbulence model in the appropriate places is needed. If it is to be modeled, then the model must be capable of handling subfilter fluctuations that may have very different properties than those of fully developed turbulence. All of these approaches have been tried in the literature, and a complete solution is still an active research problem. This chapter reviews the approaches that have been used for coarse-grained simulation of transition.