Introduction

Magnetic fields are ubiquitous in the universe (Parker (1979); Zeldovich et al. (1983)). Their interaction with an electrically conducting fluid gives rise to a complex system–a magnetofluid-whose dynamics is quite distinct from that of either a non conducting fluid, or that of a magnetic field in a vacuum (Cowling (1976)). The scales of these interactions vary in nature from metres to megaparsecs and in most situations, the dissipative processes occur on small enough scales that the resulting flows are turbulent. The purpose of this review is to discuss a small fraction of what is currently known about the properties of these turbulent flows. We refer the reader to several recent reviews for a broader view of the field (Biskamp (2003); Galtier (2008, 2009); Lazarian (2006); Lazarian & Cho (2005); Müller & Busse (2007); Kulsrud & Zweibel (2008), Bigot et al. 2008, Sridhar 2010, Brandenburg & Nordlund 2010). The electrically conducting fluid most commonly found in nature is ionized gas, i.e. a plasma, and its description in terms of all its fundamental constituents is extremely complex (see e.g. Kulsrud (2005)). In many circumstances, however, these complexities can be neglected in favour of a simplified description in term of a single fluid interacting with a magnetic field. Formally, this approach is justifiable when the processes of interest occur on timescales long compared with the light-crossing time, and on spatial scales much larger that any characteristic plasma length.

Introduction

This chapter deals with the dynamics of wall-bounded turbulent flows, with a decided emphasis on the results of numerical simulations. As we will see, part of the reason for that emphasis is that much of the recent work on dynamics has been computational, but also that the companion chapter by Marusic and Adrian (2012) in this volume reviews the results of experiments over the same period.

The first direct numerical simulations (DNS) of wall-bounded turbulence (Kim et al., 1987) began to appear soon after computers became powerful enough to allow the simulation of turbulence in general (Siggia, 1981; Rogallo, 1981). Large-eddy simulations (LES) of wall-bounded flows had been published before (Deardorff, 1970; Moin and Kim, 1982) but, after DNS became current, they were de-emphasized as means of clarifying the flow physics, in part because doubts emerged about the effect of the poorly resolved near-wall region on the rest of the flow. Some of the work summarized below has eased those misgivings, and there are probably few reasons to distrust the information provided by LES on the largest flow structures, but any review of the physical results of numerics in the recent past necessarily has to deal mostly with DNS. Only atmospheric scientists, for whom the prospect of direct simulation remains remote, have continued to use LES to study the mechanics of the atmospheric surface layer (see, for example, Deardorff, 1973; Siebesma et al., 2003). A summary of the early years of numerical turbulence research can be found in Rogallo and Moin (1984) and Moin and Mahesh (1998).

Introduction

Stable density stratification can have a strong effect on fluid flows. For example, a stably-stratified fluid can support the propagation of internal waves. Also, at large enough horizontal scales, flow in a stably-stratified fluid will not have enough kinetic energy to overcome the potential energy needed to overturn; therefore flows at this horizontal scale and larger cannot overturn, greatly constraining the types of motions possible. Both of these effects were observed in laboratory experiments of wakes in stably-stratified fluids (see, e.g., (Lin and Pao, 1979)). In the wake experiments, generally the flow in the near wake of the source, e.g., a towed sphere or a towed grid, consisted of three-dimensional turbulence, little affected by the stable stratification. As the flow decayed, however, the effects of stratification became continually more important. After a few buoyancy periods, when the effects of stable stratification started to dominate, the flow had changed dramatically, and consisted of both internal waves and quasi-horizontal motions. Following Lilly (1983), we will call such motions, consisting of both internal waves and quasi-horizontal motions due to the domination of stable stratification, as “stratified turbulence”. It has become clear that such flows, while being strongly constrained by the stable stratification, have many of the features of turbulence, including being stochastic, strongly nonlinear, strongly dispersive, and strongly dissipative.

A primary interest in stratified turbulence is how energy in such flows, strongly affected by stable stratification, is still effectively cascaded down to smaller scales and into three-dimensional turbulence, where it is ultimately dissipated.

Introduction

Ancient depictions of fluids, going back to the Minoans, envisaged waves and moving streams. They missed what we would call vortices and turbulence. The first artist to depict the rotational properties of fluids, vortical motion and turbulent flows was da Vinci (1506 to 1510). He would recognize the term vortical motion as it comes from the Latin vortere or vertere: to turn, meaning that vorticity is where a gas or liquid is rapidly turning or spiraling. Mathematically, one represents this effect as twists in the velocity derivative, that is the curl or the anti-symmetric component of the velocity gradient tensor. If the velocity field is u, then for the vorticity is ω = ∇ × u.

The aspect of turbulence which this chapter will focus upon is the structure, dynamics and evolution of vorticity in idealized turbulence – either the products of homogeneous, isotropic, statistically stationary states in forced, periodic simulations, or flows using idealized initial conditions designed to let us understand those states. The isotropic state is often viewed as a tangle of vorticity (at least when the amplitudes are large), an example of which is given in Fig. 2.1. This visualization shows isosurfaces of the magnitude of the vorticity, and similar techniques have been discussed before (see e.g. Pullin and Saffman, 1998; Ishihara et al., 2009; Tsinober, 2009). The goal of this chapter is to relate these graphics to basic relations between the vorticity and strain, to how this subject has evolved to using vorticity as a measure of regularity, then focus on the structure and dynamics of vorticity in turbulence, in experiments and numerical investigations, before considering theoretical explanations. Our discussions will focus upon three-dimensional turbulence.

The wide ranges of length and velocity scales that occur in turbulent flows make them both interesting and difficult to understand. The length scale range is widest in high Reynolds number turbulent wall flows where the dominant contribution of small scales to the stress and energy very close to the wall gives way to dominance of larger scales with increasing distance away from the wall. Intrinsic length scales are defined statistically by the two-point spatial correlation function, the power spectral density, conditional averages, proper orthogonal decomposition, wavelet analysis and the like. Before two- and three-dimensional turbulence data became available from PIV and DNS, it was necessary to attempt to infer the structure of the eddies that constitute the flow from these statistical quantities or from qualitative flow visualization. Currently, PIV and DNS enable quantitative, direct observation of structure and measurement of the scales associated with instantaneous flow patterns. The purpose of this chapter is to review the behavior of statistically-based scales in wall turbulence, to summarize our current understanding of the geometry and scales of various coherent structures and to discuss the relationship between instantaneous structures of various scales and statistical measures.

Introduction

Wall-bounded turbulent flows are pertinent in a great number of fields, including geophysics, biology, and most importantly engineering and energy technologies where skin friction, heat and mass transfer, flow-generated noise, boundary layer development and turbulence structure are often critical for system performance and environmental impact.

Introduction

Fully developed turbulence is a phenomenon involving huge numbers of degrees of dynamical freedom. The motions of a turbulent fluid are sensitive to small differences in flow conditions, so though the latter are seemingly identical they may give rise to large differences in the motions.1 It is difficult to predict them in full detail.

This difficulty is similar, in a sense, to the one we face in treating systems consisting of an Avogadro number of molecules, in which it is impossible to predict the motions of them all. It is known, however, that certain relations, such as the ideal gas laws, between a few number of variables such as pressure, volume, and temperature are insensitive to differences in the motions, shapes, collision processes, etc. of the molecules.

Given this, it is natural to ask whether there is any such relation in turbulence. In this regard, we recall that fluid motion is determined by flow conditions, such as boundary conditions and forcing. It is unlikely that the motion would be insensitive to the difference in these conditions, especially at large scales. It is also tempting, however, to assume that, in the statistics at sufficiently small scales in fully developed turbulence at sufficiently high Reynolds number, and away from the flow boundaries, there exist certain kinds of relation which are universal in the sense that they are insensitive to the detail of large-scale flow conditions. In fact, this idea underlies Kolmogorov's theory (Kolmogorov, 1941a, hereafter referred as K41), and has been at the heart of many modern studies of turbulence. Hereafter, universality in this sense is referred to as universality in the sense of K41

Introduction

For good practical reasons, most experimental observations of turbulent flow are made at fixed points x in space at time t and most numerical calculations are performed on a fixed spatial grid and at fixed times. On the other hand, it is possible to describe the flow in terms of the velocity and concentration (and other quantities of interest) at a point moving with the flow. This is known as a Lagrangian description of the flow ((Monin and Yaglom, 1971)). The position of this point x+(t; x0, t0) is a function of time and of some initial point x0 and time t0 at which it was identified or “labelled”. Its velocity is the velocity of the fluid where it happens to be at time t, u+ (t; x0, t0) = u(x+(t), t). We will use the superscript (+) to denote Lagrangian quantities, and quantities after the semi-colon are independent parameters. We refer to a point moving in this way as a fluid particle.

Flow statistics obtained at fixed points and times are known as Eulerian statistics. On the other hand, statistics obtained at specific times by sampling over trajectories, which at some reference times passed through fixed points, are known as Lagrangian statistics. For example, the mean displacement at time t of those particles that passed through the point x0 at time t0 is just 〈x+ (t; x0, t0) − x0〉. In both cases, the measurement time t can be earlier or later than the reference time.