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.

We derive the governing equations for the mean and turbulent kinetic energy and discuss simplifications of the equations for several canonical flows, including channel flow and homogeneous isotropic turbulence. A classical expression for the dissipation rate in isotropic turbulence is provided. In addition, the governing equations for turbulent enstrophy and scalar variance are derived with parallels to the derivation of the turbulent kinetic energy equation. A model for turbulent kinetic energy evolution and dissipation in isotropic turbulence is introduced. Finally, we derive the governing equations for the Reynolds stress tensor components and discuss the roles of the terms in the Reynolds stress budgets in homogeneous shear and channel flows. A crucial link between pressure-strain correlations and the redistribution of turbulent kinetic energy between various velocity components is established. Quantifying how energy is transferred between the mean flow and turbulent fluctuations is crucial to understanding the generation and transport of turbulence and its accompanying Reynolds stresses, and thus properties that phenomenological turbulence models should conform to.

Building on the governing equations and spectral tools introduced in earlier chapters, we analyze the energy cascade, which describes the transfer of turbulent kinetic energy from large to small eddies. This includes an estimate of the energy dissipation rate, as well as the characteristic length and time scales of the smallest-scale motions. Nonlinearity in the Navier-Stokes equations is responsible for triadic interactions between wavenumber triangles that drive energy transfer between scales. Empirical observations suggest that the net transfer of energy occurs from large to small scales. In systems where the large scales are sufficiently separated from the small scales, an inertial subrange emerges in an intermediate range of scales where the dynamics are scale invariant. Kolmogorov’s similarity hypotheses and the ensuing expressions for the inertial-subrange energy spectrum and viscous scales are introduced. The Kolmogorov spectrum for the inertial subrange, which corresponds to a -5/3 power law, is a celebrated result in turbulence theory. We further discuss key characteristic turbulence scales including the Taylor microscale and Batchelor scale.

We discuss properties of numerical methods that are essential for high-fidelity (LES, DNS) simulations of turbulent flows. In choosing a numerical method, one must be cognizant of the broadband nature of the solution spectra and the resolution of turbulent structures. These requirements are substantially different than those in the RANS approach, where the solutions are smooth and agnostic to turbulent structures. We focus on spatial discretization of the governing equations in canonical flows where Fourier analysis is helpful in revealing the effect of discretization on the solution spectra. In high-fidelity numerical simulations of turbulent flows, it is necessary that conservation properties inherent in the governing equations, such as kinetic energy conservation in the inviscid limit, are also satisfied discretely. An important benefit of adhering to conservation principles is the prevention of nonlinear numerical instabilities that may manifest after long-time integration of the governing equations. We end by discussing the appropriate choice of domain size, grid resolution, and boundary conditions in the context of canonical flows with uniform Cartesian mesh spacing.