Geometrical description of homogeneous shear turbulence using dissipation element analysis

As a still unresolved classical problem in physics, turbulent flows have many applications not only in aerodynamics, meteorology, geo- and astrophysics, but also for internal flows in combustion engines and turbo-machinery. Although the relevant research has been carried out for more than one century, the understanding of turbulence is still far from being satisfactory and many aspects remain to be explained. But, one need not apologize for or despair over the difficulty of turbulence. Scientists have made substantial progress in addressing those difficulties, as commented by G.Falkovich and K. R. Sreenivasan. The complex behavior of turbulence is the consequence of the nonlinearity of the Navier-Stokes equations, which are valid both for laminar and turbulent flows. It is still not possible to develop a perturbation theory around the linear part of these equations at high Reynolds numbers because of the absence of an appropriate small parameter. In this regard, the Navier- tokes equations are the most intractable of all of the field equations we know. The complexities reflected in the physics of turbulence can mainly be ascribed to: (1) Multiple scales and nonlocality; (2) Reynolds number dependence; (3) Effects of boundary and initial conditions; (4) Scalar mixing problem. In this thesis a new method to investigate turbulent flows, dissipation element analysis, is introduced. If a scalar field which is diffusion controlled at the small scales is such that the scalar field satisfies the property of a Morse function, then any local extremal point can be jointed by a gradient trajectory of finite length. The three components of the velocity vector and the passive scalar, for instance,l may serve as such a field variable. Thus starting from each material point in the scalar field, trajectories in ascending and descending directions along scalar gradients will inevitably reach the maximal and minimal ending points. The ensemble of material points sharing the same pair of maximal and minimal points cover a spatial region which is named a dissipation element, within which the scalar varies monotonously. All of the turbulence field will be filled by dissipation elements and therefore this decomposition is space-filling and nonoverlapping. Therefore this geometrical decomposition is nonlocal and serves as a natural way to mark units' in turbulence. This thesis will focus to a large extent on passive scalar turbulence and dissipation element analysis will be carried out both numerically and theoretically. This thesis is organized as the following. After a background introduction in chapter 1, chapter 2 discuss the algorithm of Direct Numerical Simulation (DNS) of homogeneous shear turbulence. Depending on different purposes, mean passive scalar gradients will also be imposed to study passive scalar problems. Chapter 3 discusses some general concepts of extremal points. A new trajectory-tracing algorithm with the second order accuracy is introduced to calculate trajectories. In chapter 4 some geometrical properties of dissipation elements are presented. For the fluctuating passive scalar field, as an example, the two most important characteristic parameters describing each dissipation element will be, the difference of the fluctuating scalar at the two extremal points, and the linear distance connecting these two points. The joint PDF of these two parameters will play a central role for the statistics of dissipation element theory. From this joint PDF the conditional mean shows the scaling of the mean conditional scalar difference with respect to the length, which is another representation of structure function. In chapter 5 a stochastic evolution equation for the PDF of length of elements, based on a Poisson random cutting-reconnection model, is derived. The modeling hypotheses introduced is quite generic and is believed to fit to different field variables. Chapter 6 focuses on the structure functions from dissipation element analysis. Based on the modeling for the marginal PDF of length, an appropriate joint PDF equation is presented in chapter 7. In addition, the effect of the dimensionality of the space, in which dissipation elements are considered, is discussed in chapter 8. Another part of this thesis is a preparative work for the passive scalar mixing modeling in turbulence. Chapter 9 shows that the representation of scalar dissipation in mixture fraction space becomes more organized than that in physical space. Along the trajectories in passive scalar fields, strain rate and scalar dissipation are closely coupled, Intermittency and anisotropy, the central topics of chapter 10, belong to the important properties of turbulence at small scales and can be newly understood form dissipation element analysis