As indicated in Chapter 18, ensemble theory is especially germane when calculating thermodynamic properties for systems composed of dependent rather than independent particles. Potential applications include real gases, liquids, and polymers. In this chapter, we focus on the thermodynamic properties of nonideal gases. Our overall approach is to develop an equation of state using the grand canonical ensemble. From classical thermodynamics, equilibrium properties can always be determined by suitably operating on such equations of state. As shown later in this chapter, typical evaluations require an accurate model for the intermolecular forces underlying any macroscopic assembly. This requirement is endemic to all applications of ensemble theory, including those for liquids and polymers. As a matter of fact, by mastering the upcoming procedures necessary for the statistics of real gases, you should be prepared for many pertinent applications to other tightly-coupled thermodynamic systems.

The Behavior of Real Gases

As the density of a gas rises, its constituent particles interact more vigorously so that their characteristic intermolecular potential exercises a greater influence on macroscopic behavior. Accordingly, the gas becomes less ideal and more real; i.e., its particles eventually display greater contingency owing to enhanced intermolecular forces. This deviation from ideal behavior is reflected through a more complicated equation of state for real gases.

An equation of state, you recall, describes a functional relation among the pressure, specific volume, and temperature of a given substance.

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.

Introduction

In the study of transport phenomena in moving fluids, the fundamental laws of motion (conservation of mass and Newton's second law) and energy (first law of thermodynamics) are applied to an elemental fluid. Two approaches are possible:

1. a particle approach or

2. a continuum approach.

In the particle approach, the fluid is assumed to consist of particles (molecules, atoms, etc.) and the laws are applied to study particle motion. Fluid motion is then described by the statistically averaged motion of a group of particles. For most applications arising in engineering and the environment, however, this approach is too cumbersome because the significant dimensions of the flow are considerably bigger than the mean-free-path length between molecules. In the continuum approach, therefore, statistical averaging is assumed to have been already performed and the fundamental laws are applied to portions of fluid (or control volumes) that contain a large number of particles. The information lost in averaging must however be recovered. This is done by invoking some further auxiliary laws and by empirical specifications of transport properties such as viscosity µ, thermal conductivity k, and mass diffusivity D. The transport properties are typically determined from experiments. Notionally, the continuum approach is very attractive because one can now speak of temperature, pressure, or velocity at a point and relate them to what is measured by most practical instruments.

Introduction

In practical applications of CFD, one often encounters complex domains. A domain is called complex when it cannot be elegantly described (or mapped) by a Cartesian grid. By way of illustration, we consider a few examples.

Figure 6.1 shows the smallest symmetry sector of a nuclear rod bundle placed inside a circular channel of radius R. There are nineteen rods: one rod at the channel center, six rods (equally spaced) in the inner rod ring of radius b1, and twelve rods in the outer ring of radius b2. The rods are circumferentially equispaced. The radius of each rod is r0. The fluid (coolant) flow is in the x3 direction. The flow convects away the heat generated by the rods and the channel wall is insulated. It is obvious that a Cartesian grid will not fit the domain of interest because the lines of constant x1 or x2 will intersect the domain boundaries in an arbitrary manner. In such circumstances, it proves advantageous to adopt alternative means for mapping a complex domain.

These alternatives are to use

1. curvilinear grids or

2. finite-element-like unstructured grids.

Curvilinear Grids

It is possible to map a complex domain by means of curvilinear grids (ξ1, ξ2) in which directions of ξ1 and ξ2 may change from point to point. Also, curvilinear lines of constant ξ1 and constant ξ2 need not intersect orthogonally either within the domain or at the boundaries. Figure 6.2 shows the nineteen-rod domain of Figure 6.1 mapped by curvilinear grids.

Structure of the Code

The 1D conduction code is divided into two parts:

1. a user part containing files COM1D.FOR and USER1D.FOR and

2. a library part containing file LIB1D.FOR.

The user part is problem dependent. Therefore, the two files in this part are used to specify the problem to be solved. In contrast, the library part is problem independent. Thus, the LIB1D.FOR file remains unaltered for all problems. In this sense, the library part may be called the solver whereas the user part may be called the pre- and postprocessor.

This structure is central to creation of a generalised code. To execute the code, USER1D.FOR and LIB1D.FOR files are compiled separately and then linked before execution. The COM1D.FOR is common to both parts and its contents are brought into each subroutine or function via the “INCLUDE” statement in FORTRAN. Variable names starting with I, J, K, L, M, and N are integers whereas all others are real by default. The list of variable names with their meanings is given in Table B.1. The listings of each file are given at the end of this appendix.

File COM1D.FOR

In this file, logical, real, and integer variables are included. The PARAMETER statement is used to specify the maximum array dimension IT and values of π, GREAT, and SMALL. The latter are frequently required for generalised coding.