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This chapter discusses numerical simulations of the incompressible Navier-Stokes equations. Exact Navier-Stokes solutions are presented that are used as benchmarks to validate new codes and evaluate the accuracy of a particular discretization. Some aspects of direct numerical simulation (DNS) and large-eddy simulation (LES) are discussed. The issue of stabilization at high Reynolds number is then presented using the concepts of dynamic subgrid modelling, over-integration, and spectral vanishing viscosity. A new parallel paradigm based on multi-level parallelism is introduced that can help realize adaptive refinement more easily. The final section includes a heuristic refinement method for Navier-Stokes equations.

In many-body problems there is generally an interest in reducing the number of degrees of freedom necessary to describe a system. Traditionally this reduction was accomplished in one precipitous step by means of a global ensemble average. But, in recent times, there has been a growing interest in more progressive methods, such as decimation schemes, multilevel methods and renormalization group. In turbulence, large-eddy simulation (LES), with its associated subgrid-scale (SGS) modelling problem, lies somewhere between these two extremes. In the particular case of homogeneous, isotropic turbulence, theis chapter interprets the Fourier modes as the degrees of freedom. So the interest here lies in reducing the number of these modes. For LES this means replacing the high-wavenumber modes by some effective turbulence response. This chapter does not seek to identify a satisfactory SGS model. Nor does it compare the merits of one model with another. Instead, the chapter aims to reflect fundamental research, over the past few decades, which has been aimed at understanding the underlying physics of momentum and energy transfer between explicit and subgrid scales, along with its consequent implications for subgrid-scale models. The intention is to set out and elucidate a framework based on current and recent research which will provide guidelines for the formulation of SGS models, at least for the restricted case of homogeneous turbulence. One general outcome of all this research is the increasingly widespread recognition that the turbulence response cannot be represented by the traditional eddy viscosity alone.

Chapter 7 discusses the role of numerical simulations in turbulence. The emphasis is on the new physical insights these numerical experiments have yielded, as well as some of the limitations of numerical methods.

This chapter assesses the effects of deep convective clouds in the perturbed climate system. To understand the role of deep convection in the climate system, model simulations are required across all scales. These models include large eddy simulation models, cloud models, or cloud system-resolving models (CSRMs). The chapter discusses the limitations of current cloud simulations and observational approaches, and also offers suggestions for future research.

Chapter 4 discusses turbulent shear flows. This includes boundary layers, homogeneous shear, and free shear flows, such as jets and wakes. The physical characteristics of these flows are reviewed, including the log-law of the wall for momentum and heat. In addition, simple models of shear flows are discussed, such as the mixing-length model and so-called one-point closure models, including the k-ɛ and Reynolds stress models. Large-eddy simulation is also introduced.

This chapter addresses the shortcomings that emerge in global climate models attributable to the interactions between resolved and parametized unresolved cloud-related processes (convection, turbulence, clouds and radiation). It also discusses alternative modeling techniques to study perturbed clouds, which include the numerical weather prediction model, large eddy simulation models, cloud-resolving models, and super-parameterized single column models.

The renormalization group (RNG) and related e-expansion methods are a powerful technique that allow the systematic derivation of coarse-grained equations of motion for turbulent flows and, in particular, the derivation of sophisticated turbulence models based on the fundamental underlying physics. The RNG method provides a convenient calculus for the analysis of complex physical effects in complex flows. The details of the RNG method applied to fluid mechanics differ in some crucial respects from how renormalization group techniques are applied to field theories in other branches of physics. At the present time, the RNG methods for fluid dynamics are by no means rigorously justified, so their utility must be based on the quality and quantity of results to which they lead. In this paper we discuss the basis for the RNG method and then illustrate its application to a variety of turbulent flow problems, emphasizing those points where further analysis is needed. The application of a field-theoretic method like the RNG technique to turbulence is based on the fundamental assumption of universality of small scales in turbulent flows. Such universal behavior was first suggested over 50 years ago in the seminal work of A. N. Kolmogorov who argued that the small-scale spectrum of incompressible turbulence is universal and characterized by two numbers, the rate of energy dissipation ε per unit mass and the kinematic viscosity v.

Chapter 6 focuses on isotropic turbulence, with an emphasis on its real-space (rather than Fourier-space) description. After introducing the language of isotropic turbulence (i.e., kinematics), the dynamical consequences of the Karman-Howarth equation are explored. The dynamics of both the large and small scales are discussed, as well as the effects of intermittency. The shortcomings of the standard statistical tools are emphasized.