Developments in the Theory of Turbulence

Abstract

The first thing to be said about this book is that it is concerned with the analytical theory of turbulence at the deepest conceptual level and with mathematical analysis of a daunting complexity. Professor Leslie accepts that some may now question the need for such an arduous attack on the problem, bearing in mind the prospects of direct numerical analyses of ever increasing elaborateness, but hecontends that it is still necessary as he puts it, 'to tortureourselves like this'.In the context of practical meteorological problems the aim amounts, for example, to deriving the wind spiral in the boundary layer without recourse to the empirical device of eddy viscosity. In more general terms Professor Leslie's concern is with certain ideas and procedures, which have evolved since about the mid-I950s, for the more rigorous closure of the Navier-Stokes equations for turbulent flow. The closure difficulties arise from the fact that equations for the mean wind components involve second moments of these components (Reynolds stresses), and the equations for the second moments involve third moments, and so on, every additional equation bringing additional unknowns. The developments which are surveyed have sprung from the techniques of quantum field theory, in applications notably by R. H. Kraichnan, and to a large extent the book is an expounding and interpreting of the important advances made by Kraichnan and also by S. F. Edwards.The first three chapters provide a background of the established mathematics of turbulence and of the problem of closure. Chapter 4 then introduces the 'direct action' approximation which is the central theme of Kraichnan's approach, and the ideas are thereafter developed, partly in relation to alternative ideas for closure, in Chapters 5-12. 'Interaction' occurs between the modes of the wave numbers in the Fourier-transformed versions of the Navier-Stokes equationsand the adjective refers to the changes of energy in a certain wavenumber which arise directly from combinations of appropriately related wave numbers.Up to and including Chapter 12 the discussion is in terms of the idealised condition of homogeneous isotropic turbulence. In Chapter 13 the properties of channel flow as an example of the real flows studied by engineers, are discussed and in Chapter 14 the current methods of calculation, starting from the familiar eddy viscosity and mixing length models, are surveyed briefly. These lead in to the final Chapter (15) which deals with attempts made largely by Professor Leslie himself to bring the idea of 'direct interaction' to bear on real flows. It is interesting that the initial phase of Kraichnan's developments, which were couched in an Eulerian (fixed point) reference system, led in the late 1950s to properties of the turbulence spectrum which were inconsistent with Kolmogorov's theory and the famous minus-five-thirds power law. Kraichnan realised that the basic defect lay in the failure of the Eulerian specification to represent properly the sweeping round of small eddies large eddies...

Energy-Conserving Simulation of Incompressible Electro-Osmotic and Pressure-Driven Flow

“…Although this method can ensure positivity of the transported quantity and has been tested for several applications (Boris & Book, 1976), numerical diffusion necessitates the use of an anti-diffusion stage (Boris & Book, 1964), which must be tailored to the particular problem. However, if there is a sharp gradient in the solution, Eulerian numerical methods have difficulties (Woodward & Colella, 1984;LeVeque, 1990). We find that our parcel advection algorithm predicts the solution of an advection equation very well, even when a sharp gradient exists in the solution, as is typically the case in electrophoretic separation.…”

“…Alternative numerical advection algorithms can be found in Holt (1984); LeVeque (1990); Tannehill et al (1997). Another alternate is the flux-corrected transport algorithm of Boris & Book (1964), which is an Eulerian method based on Lagrangian considerations.…”

“…Without proper handling of advection terms, numerical solutions can go unconditionally unstable, producing negative concentrations or spurious wiggles (Tannehill et al, 1997;Press et al, 1997). Historically, the first alternative was the unconditionally stable method of characteristics introduced by Courant et al (1952) (sometimes referred to as the CIR scheme; see LeVeque (1990)). Since that method uses interpolation based on the nearest grid point, in spite of its unconditional stability, the overall performance is limited by the inherent computational damping associated with interpolation (Ritchie, 1986).…”

“…This assumption does not necessarily require that the density of the fluid be constant; it only requires that the density be independent of the concentration field. In other words, the concentration field is a passive scalar quantity (e.g., see Leslie (1973)). …”

A tutorial introduction to the statistical theory of turbulent plasmas, a half-century after Kadomtsev’sPlasma Turbulenceand the resonance-broadening theory of Dupree and Weinstock

“…Over that time the theory of turbulence in both fluids and plasmas has developed substantially, and a beginning researcher interested in turbulence will find that much conceptual background is assumed. Fluid turbulence theory has been addressed in multiple books (Leslie 1973;McComb 1990McComb , 2014Frisch 1995;Lesieur 1997) and articles (Eyink & Sreenivasan 2006;Falkovich 2006;Boffetta & Ecke 2012). The plasma literature is sparser, although a review by Yoshizawa et al (2001) ranges broadly over many conceptual and practical topics for both fluids and plasmas, a comprehensive review of analytical statistical turbulence theory with a focus on plasma issues is available (Krommes 2002), and the book by Diamond, Itoh & Itoh (2010) provides an introductory survey of various approaches to plasma turbulence.…”