Conservation of particle identity

The essence of Lagrangian fluid dynamics is fluid particle identity acting as an independent variable. The identifier or label may be the particle position at some time, but could for example be a triple of the thermodyamic properties of the particle at some time. Time after labeling is the other independent variable. The fluid particle may not actually have been released into the flow at the time of labeling, but merely labeled with position or with some other properties at that time. Nevertheless, “time of release” will be used interchangeably with “labeling time.” The subsequent position of the particle is a dependent variable, even though it may coincide with the independently chosen position of an Eulerian observer at the subsequent time. The Eulerian observer also employs time, after some convenient initial instant, as the other independent variable. Of course, a particle path can be calculated in the Eulerian framework by integrating velocity on the path, with respect to time. Indeed, the suppression or implicitness of this detailed path information is the basis of the relative simplicity of the Eulerian formulation. On the other hand, fluid velocity is readily calculated from the particle position in the Lagrangian framework by the local operation of particle differentiation with respect to time after labeling.

Conservation of particle identity is not an immediately compelling consideration in the Eulerian framework, but is fundamental in the Lagrangian.

Viscous stresses and heat conduction

It has been assumed to this point that there are no viscous stresses, nor any heat conduction. Thus, the dynamics of the ideal fluid of the preceding chapters are compatible with an isotropic distribution of molecular velocities. In fact, anisotropy is always present in a real assembly of molecules, owing to the walls of the fluid container, fields of force or sources of heat. The Navier-Stokes equations for a real fluid may be derived from Boltzmann's equation for a dilute gas using the Chapman-Enksog expansion (Chapman and Cowling, 1970), which assumes a molecular velocity distribution close to an isotropic equilibirum. A simpler derivation, requiring less physical insight, follows from the general principles of continuum mechanics by adopting Newton's and Fourier's laws as the constitutive relations. The essential aspect of these constitutive relations is that they are local in the Eulerian framework: the viscous stress tensor is proportional to the Eulerian rate of strain tensor, while the heat flux is proportional to the Eulerian temperature gradient. The Navier-Stokes equations are accordingly expressed naturally in Eulerian form, while the Lagrangian form can only be derived by “cheating.” That is, it cannot be derived from Boltzmann's equation. Cheating can be minimized (see Aside in Section 3.2), but in the interest of moving forward, let us cheat in full.

Book's scope

Large-eddy simulations (LESs) of turbulent flows are extremely powerful techniques consisting in the elimination of scales smaller than some scale Δx by a proper low-pass filtering to enable suitable evolution equations for the large scales to be written. The latter maintain an intense spatio-temporal variability. Large-eddy simulation (LES) poses a very difficult theoretical problem of subgrid-scale modeling, that is, how to account for small-scale dynamics in the large-scale motion equations. LES is an invaluable tool for deciphering the vortical structure of turbulence, since it allows us to capture deterministically the formation and ulterior evolution of coherent vortices and structures. It also permits the prediction of numerous statistics associated with turbulence and induced mixing. LES applies to extremely general turbulent flows (isotropic, free-shear, wall-bounded, separated, rotating, stratified, compressible, chemically reacting, multiphase, magnetohydrodynamic, etc.). LES has contributed to a blooming industrial development in the aerodynamics of cars, trains, and planes; propulsion, turbo-machinery; thermal hydraulics; acoustics; and combustion. An important application lies in the possibility of simulating systems that allow turbulence control, which will be a major source of energy savings in the future. LES also has many applications in meteorology at various scales (small scales in the turbulent boundary layer, mesoscales, and synoptic planetary scales). Use of LES will soon enable us to predict the transport and mixing of pollution. LES is used in the ocean for understanding mixing due to vertical convection and stratification and also for understanding horizontal mesoscale eddies. LES should be very useful for understanding the generation of Earth's magnetic field in the turbulent outer mantle and as a tool for studying planetary and stellar dynamics.

In 1949, in an unpublished report to the U.S. Office of Naval Research, John von Neumann remarked of turbulence that

Few people today would disagree with these comments on the importance of understanding turbulence and, as implied, of its prediction. And, although the turbulence problem has still yet to be “solved,” our understanding of turbulence has significantly advanced since that time; this progress has come through a combination of theoretical studies, often ingenious experiments, and judicious numerical simulations. In addition, from this understanding, our ability to predict, or at least to model, turbulence has greatly improved; methods to predict turbulent flows using large-eddy simulation (LES) are the main focus of the present book.

The impact of von Neumann is still felt today in the prediction of turbulent flows, both in his work on numerical methods and in the people and the research he has influenced. The genesis of the method of large-eddy simulation (or possibly more appropriately, “simulation des grandes échelles”) was in the early 1960s with the research of Joe Smagorinsky. At the time, Smagorinsky was working in von Neumann's group at Princeton, developing modeling for dissipation and diffusion in numerical weather prediction.

Compressible turbulence has extremely important applications in subsonic, supersonic, and hypersonic aerodynamics. More generally, and even at low Mach numbers, strong density differences caused by intense heating (in combustion for instance) may have profound consequences on the flow structure and the associated mixing. Heating a wall may, for instance, completely destabilize a boundary layer, as will be shown for some applications in this chapter. The chapter is organized as follows. We will first present the compressible LES formalism for an ideal gas in a simple way, allowing us to generalize the use of incompressible subgrid models. This is possible using the concept of density-weighted Favre filtering together with the introduction of a macropressure and a macrotemperature related by the ideal-gas state equation. Then we will study compressible mixing layers at varying convective Mach numbers. Afterward we will consider low or moderate Mach numbers in boundary layers, channel, cavities, and separated flows and also a transonic rectangular cavity. A supersonic application relating to the European space shuttle Hermés rear-flap heating during atmospheric reentry will be discussed in detail. This problem, studied in Grenoble in 1993, has acquired a tragic topicality with the loss of the American Columbia shuttle on February 1, 2003. The latter disintegrated during reentry at an approximate elevation of 60 km and a speed of 21,000 km/h while making a turn at an angle of 57°. It seems that the left wing overheated, possibly because of damage to the protection tiles during takeoff.

We have clearly shown in the former chapters the advantages of the spectral eddy-viscosity models with, in particular, the possibility of accounting for local or semilocal effects in the neighborhood of the cutoff. More details on this point may be found in Sagaut, which contains many advanced aspects on LES modeling. However, in most industrial or environmental applications, the complexity of the computational domain prohibits the use of spectral methods. One thus has to deal with numerical codes written in physical space and employing finite-volume or finite-differences methods often with unstructured grids. This last point will not be considered in this book, although it is crucial for practical applications. We will present, however, simulations on orthogonal grids of mesh size varying in direction and location and sometimes in curvilinear geometry. This chapter will mainly be devoted to models of the structure-function family with applications to isotropic turbulence, free-shear and separated flows, and boundary layers. We will also present in less detail alternative models such as the dynamic structure-function model, hyperviscosity model, mixed structure-function/hyperviscous model, and the mixed model.

Structure-function model

Formalism

The structure-function (SF) model is an attempt to go beyond the Smagorinsky model while keeping in physical space the same scalings as in spectral eddy-viscosity models. The original SF model is due to Métais and Lesieur.

As already stressed, the large-eddy simulation (LES) concept was developed by the meteorologists Smagorinsky, Lilly, and Deardorff. In fact, geophysical and astrophysical fluid dynamics contain an innumerable list of processes (generally three-dimensional) that can be understood experimentally only via laboratory and in situ experiments and numerically mostly by LES. We recall, for instance, that the Taylor-microscale-based Reynolds number for smallscale atmospheric turbulence is larger than 104, and thus implementation of DNS does not seem feasible in this case even with the unprecedented development of computers. As far as Earth is concerned, these processes are part of the extraordinarily important issue of climate modeling and prediction involving a very complex system that dynamically and thermodynamically couples the atmosphere (with water vapor, clouds, and hail), the oceans (with salt and plankton), and ice for periods of time from seconds to hundreds of thousands of years. The issue of global warming, which requires our being able to predict the evolution, under the action of greenhouse-effect gases, of Earth temperature (in the average or in certain particular zones), is vital for the survival of populations living close to the oceans and seas. Indeed, global warming induces ice melt, which implies a sea-level elevation. It also increases evaporation, resulting in heavy rains and floods.

We first provide in this chapter a general introduction to geophysical fluid dynamics (GFD). Then we will concentrate on two problems for which LES and DNS provide significant information. The first is the effect of a fixed solid-body rotation on a constant-density free-shear or wall-bounded flow. The second is the generation of storms through baroclinic instability in a dry atmosphere.