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.

During the last three decades, computational fluid dynamics (CFD) has emerged as an important element in professional engineering practice, cutting across several branches of engineering disciplines. This may be viewed as a logical outcome of the recognition in the 1950s that undergraduate curricula in engineering must increasingly be based on engineering science. Thus, in mechanical engineering curricula, for example, the subjects of fluid mechanics, thermodynamics, and heat transfer assumed prominence.

I began my teaching career in the early 1970s, having just completed a Ph.D. degree that involved solution of partial differential equations governing fluid motion and energy transfer in a particular situation (an activity not called CFD back then!). After a few years of teaching undergraduate courses on heat transfer and postgraduate courses on convective heat and mass transfer, I increasingly shared the feeling with the students that, although the excellent textbooks in these subjects emphasised application of fundamental laws of motion and energy, the problem-solving part required largely varied mathematical tricks that changed from one situation to another. I felt that teachers and students needed a chance to study relatively more real situations and an opportunity to concentrate on the physics of the subject. In my reckoning, the subject of CFD embodies precisely this scope and more.

The introduction of a five-year dual degree (B. Tech. and M. Tech.) program at IIT Bombay in 1996 provided an opportunity to bring new elements into the curriculum.

Structure of the Code

The structure of the 2D Cartesian code is similar to that of the 1D conduction code. The code is again divided into two parts: The problem-dependent user part containing files COM2D.FOR and USER2D.FOR and the problem independent library part that contains the LIB2D.FOR file. The listings of each file are given at the end of this appendix. The list of variable names with their meanings is given in Table C.1.

File COM2D.FOR

In this file, again logical, real, and integer variables are included. All other contents of this file bear the same description as the COM1D.FOR file.

File USER2D.FOR

This is the main control file at the command of the user. The first routine PROGRAM MAIN is the command routine from where subroutine MAINPR is called. The latter is the first subroutine of the LIB2D.FOR file. Here, listings of USER files are given for three problems solved in Chapter 5. These are (a) 1D porous body flow, (b) turbulent flow in an axisymmetric pipe expansion, and (c) natural convection evaporation.

BLOCK DATA This routine at the end of the USER2D.FOR file specifies all the problem-dependent data such as control parameters, relaxation parameters, Prandtl numbers, flow conditions, equations to be solved, and convection scheme used.