In this chapter we consider the behavior of essentially 2D membranes. The membranes may be linear or nonlinear and are generally considered to undergo arbitrarily large deformations. More specifically, the membranes considered here are modeled after biological membranes such as those that comprise cell walls or the layers that exist within biomineralized structures, e.g., shells or teeth. The discussion is preliminary and meant to provide a brief introduction to the basic concepts involved in the constitutive modeling of such structures.

Biological Membranes

Figure 33.1a illustrates an idealized view of the red blood cell that shows its hybrid structure consisting of an outer bilipid membrane and an attached cytoskeleton; Fig. 33.1b shows a micrograph of a section of the cytoskeleton that is illustrated schematically in Fig. 33.1a. The cell membrane is a hybrid, i.e., composite structure consisting of an outer bilipid layer that is supported (i.e., reinforced) by a network attached to it on the cytoplasmic side, which is on the inside of the cell. The cytoskeleton is built up from mostly tetramers, and higher order polypeptides, of the protein spectrin attached at actin nodes. We note that the spectrin network has close to a sixfold nodal coordination. It has been known that the nonlinear elastic properties of the membrane depend sensitively on the details of the topology that includes, inter alia, nodal coordination, spectrin segment length, and the statistical distribution of such topological parameters.

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.