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.

Numerical investigations of convective flow and heat transfer in two different engineering applications, namely cross-corrugated channels for heat exchangers and rib-roughened channels for gas turbine blade cooling, using wall-modeled large eddy simulations (LES), are presented in this chapter. Mesh resolution requirements for LES, subgrid model dependence, and heat transfer and friction factor characteristics are investigated and compared with previously published experimental data. The LES computations form a coherent suite of monotonically behaving predictions, with all aspects of the results converging toward the predictions obtained on the finest grids. Various subgrid and Reynolds-averaged Navier–Stokes equations (RANS) models are compared to account for their reliability and efficiency in the prediction of hydraulic and thermal performances in the presence of complicated flow physics. Results indicate that subgrid models such as wall-adapting local eddy viscosity model (WALE) and localized dynamic kinetic energy model (LDKM) provide the most accurate results, within 201b of Nusselt number and Darcy’s friction factor, compared to selected RANS models, which presents up to 3501b deviation from experimental data. The conclusion is that both LES and RANS have their strengths and weaknesses, and the choice between them depends on the specific application requirements and available computational resources.

An overview is presented of the filtered density function (FDF) methodology as a closure for large eddy simulation (LES) of turbulent reacting flows. The theoretical basis and the solution strategy of LES/FDF are briefly discussed, with the focus on some of the closure issues. Some of the recent applications of LES/FDF are reviewed, along with some speculations about future prospects for such simulations.

The Kolmogorov scale-by-scale equilibrium cascade and concepts related to it have provided the physical basis for explicit large eddy simulation subgrid models since the mid-twentieth century. However, mounting evidence and theory have been accumulating over the past ten years for scale-by-scale nonequilibrium in a variety of turbulent flows with some new general nonequilibrium laws. One of the resulting challenges now is to translate these new nonequilibrium physics into predictive turbulence modeling.

Particle-resolved (PR), Euler–Lagrange (EL), and Euler–Euler (EE) formulations are the three widely used computational approaches in multiphase flow. In PR formulation, the focus is on the flow physics at the microscale and all the details are resolved at the microscale. However, due to computational limitations, the PR approach cannot reach the length and time scales needed to explore the meso and macroscale multiphase phenomenon. In the EL formulation of a dispersed multiphase flow, the continuous phase is averaged (or filtered), and all the microscale details of the flow on the scale of individual particles are coarse-grained. If all the dispersed phase elements (i.e., all the particles, drops, or bubbles) are tracked then there is no averaging of the dispersed phase. In the EE formulation, both the continuous and dispersed phases are averaged/filtered. We will discuss systematic coarse graining to obtain the governing equations of the EL and EE approaches. The coarse-graining process introduces two interesting challenges: (i) the unavoidable closure problem where the Reynolds stress and flux terms must be expressed in terms of filtered meso/macroscale variables, and (ii) the coupling between the continuous and the dispersed phases must be appropriately posed in terms of the filtered variables. Recent innovations on both these fronts are discussed.

Scale-resolving simulation (SRS) methods of practical interest are coarse-graining formulations widely used in science and engineering. These methods aim to efficiently predict complex flows by only resolving the phenomena not amenable to modeling, unleashing the concept of accuracy on demand. This chapter provides an overview of the SRS methods best suited for engineering applications: hybrid and bridging models. It starts by reviewing basic turbulence modeling concepts. Following on from that is an overview of hybrid and bridging models, discussing their main advantages and limitations. The challenges to the predictive application of these models are enumerated, as well as possible strategies to solve or mitigate them. Several examples are provided to illustrate the potential of these classes of SRS methods. Overall, the chapter intends to help new and experienced SRS modelers and users obtain predictive turbulence computations.

Longstanding design and reproducibility challenges in inertial confinement fusion (ICF) capsule implosion experiments involve recognizing the need for appropriately characterized and modeled three-dimensional initial conditions and high-fidelity simulation capabilities to predict transitional flow approaching turbulence, material mixing characteristics, and late-time quantities of interest – for example, fusion yield. We build on previous coarse-graining (CG) simulations of the indirect-drive national ignition facility (NIF) cryogenic capsule N170601 experiment – a precursor of N221205 which resulted in net energy gain. We apply effectively combined initialization aspects and multiphysics coupling in conjunction with newly available hydrodynamics simulation methods, including directional unsplit algorithms and low Mach-number correction – key advances enabling high fidelity coarse-grained simulations of radiation-hydrodynamics driven transition.

The filtering approach is a simple deterministic way to formalize analytically coarse-grained representations of a given turbulent flow. By their own nature, turbulence and coarse graining (CG) are multiscaled, and in this chapter, we discuss the specific question of the relations between turbulence, coarse graining, and filtering in a unified operational form, with particular interest to multiscale properties and aspects. Reynolds averaged Navier–Stokes (RANS) averaging, explicit convolutional large eddy simulation (LES) filtering formulations (Leonard, 1975), implicit LES and scale resolving simulations (SRS) approaches (Grinstein et al., 2010; Grinstein, 2016; Pereira et al., 2021), functional and structural LES modeling procedures (Sagaut, 2006) and hybrid RANS/LES methods (Fr¨ohlich and von Terzi, 2008), are revisited and discussed from the point of view of a multiscale operational filtering approach (OFA) (Germano, 1992) based on the multiscale properties of the generalized central moments (GCM). Some recent results are presented both as regards analysis, modeling, and post-processing of turbulent flows, and finally, some conclusions and some personal recalls are provided.

Accurate predictions with quantifiable uncertainty are essential to many practical turbulent flows in engineering, geophysics, and astrophysics typically comprising extreme geometrical complexity and broad ranges of length and timescales. Dominating effects of the flow instabilities can be captured with coarse-graining (CG) modeling based on the primary conservation equations and effectively codesigned physics and algorithms. The collaborative computational and laboratory experiments unavoidably involve inherently intrusive coarse-grained observations – intimately linked to their subgrid scale and supergrid (initial and boundary conditions) specifics. We discuss turbulence fundamentals and predictability aspects and introduce the CG modified equation analysis. Modeling and predictability issues for underresolved flow and mixing driven by underresolved velocity fields and underresolved initial and boundary conditions are revisited in this context. CG simulations modeling prototypical shock-tube experiments are used to exemplify relevant actual issues, challenges, and strategies.

Originating from irreversible statistical mechanics, the Mori–Zwanzig (M–Z) formalism provides a mathematical procedure for the development of coarse-grained models of complex systems, such as turbulence, that lack scale separation. The M–Z formalism begins with the application of a specialized class of projectors to the governing equations. By leveraging these projectors, the M–Z procedure results in a reduced system, commonly referred to as the generalized Langevin equation (GLE). The GLE encapsulates the system’s behavior on a macroscopic (resolved) scale. The influence of the microscopic (unresolved) scales on resolved scales appears as a convolution integral – often referred to as memory – and an additional noise term. In essence, fully resolved Markovian dynamics is transformed into coarse grained non-Markovian dynamics. The appearance of the memory term in the GLE demonstrates that the coarse-graining procedure leads to nonlocal memory effects, which have to be modeled. This chapter introduces the mathematics behind the projection approach and the derivation of the GLE. Beyond the theoretical developments, the practical application of the M–Z procedure in the construction of subgrid-scale models for large eddy simulations is also presented.