An Introduction To Single-Point Closure Methodology

The material presented in Section 1 will replicate that covered in earlier chapters. It may nevertheless be helpful to include it to familiarize the reader both with the nomenclature adopted and the assumptions being made.

Modeling Of Turbulent Transport Equations

The high-Reynolds-number turbulent flows of technological importance contain such a wide range of excited length and time scales that the application of direct or large-eddy simulations is all but impossible for the foreseeable future. Reynolds stress models remain the only viable means for the solution of these complex turbulent flows. It is widely believed that Reynolds stress models are completely ad hoc, having no formal connection with solutions of the full Navier-Stokes equations for turbulent flows. While this belief is largely warranted for the older eddy viscosity models of turbulence, it constitutes a far too pessimistic assessment of the current generation of Reynolds stress closures. It will be shown how secondorder closure models and two-equation models with an anisotropic eddy viscosity can be systematically derived from the Navier-Stokes equations when one overriding assumption is made: the turbulence is locally homogeneous and in equilibrium. A brief review of zero equation models and one equation models based on the Boussinesq eddy viscosity hypothesis will first be provided in order to gain a perspective on the earlier approaches to Reynolds stress modeling. It will, however, be argued that since turbulent flows contain length and time scales that change dramatically from one flow configuration to the next, two-equation models constitute the minimum level of closure that is physically acceptable. Typically, modeled transport equations are solved for the turbulent kinetic energy and dissipation rate from which the turbulent length and time scales are built up; this obviates the need to specify these scales in an ad hoc fashion. While two-equation models represent the minimum acceptable closure, second-order closure models constitute the most complex level of closure that is currently feasible from a computational standpoint. It will be shown how the former models follow from the latter in the equilibrium limit of homogeneous turbulence. However, the two-equation models that are formally consistent with second-order closures have an anisotropic eddy viscosity with strain-dependent coefficients - a feature that most of the commonly used models do not possess.

Introduction To Renormalization Group Modeling Of Turbulence

The renormalization group (RNG) and related e-expansion methods are a powerful technique that allow the systematic derivation of coarse-grained equations of motion for turbulent flows and, in particular, the derivation of sophisticated turbulence models based on the fundamental underlying physics. The RNG method provides a convenient calculus for the analysis of complex physical effects in complex flows. The details of the RNG method applied to fluid mechanics differ in some crucial respects from how renormalization group techniques are applied to field theories in other branches of physics. At the present time, the RNG methods for fluid dynamics are by no means rigorously justified, so their utility must be based on the quality and quantity of results to which they lead. In this paper we discuss the basis for the RNG method and then illustrate its application to a variety of turbulent flow problems, emphasizing those points where further analysis is needed. The application of a field-theoretic method like the RNG technique to turbulence is based on the fundamental assumption of universality of small scales in turbulent flows. Such universal behavior was first suggested over 50 years ago in the seminal work of A. N. Kolmogorov who argued that the small-scale spectrum of incompressible turbulence is universal and characterized by two numbers, the rate of energy dissipation ε per unit mass and the kinematic viscosity v.

Fundamental Aspects Of Incompressible And Compressible Turbulent Flows

Turbulence generally can be characterized by a number of length scales: at least one for the energy containing range, and one from the dissipative range; there may be others, but they can be expressed in terms of these. Whether a turbulence is simple or not depends on how many scales are necessary to describe the energy containing range. Certainly, if a turbulence involves more than one production mechanism (such as shear and buoyancy, for example, or shear and density differences in a centripetal field) there will be more than one length scale. Even if there is only one physical mechanism, say shear, a turbulence which was produced under one set of circumstances may be subjected to another set of circumstances. For example, a turbulence may be produced in a boundary layer, which is then subjected to a strain rate. For a while, such turbulence will have two length scales, one corresponding to the initial boundary layer turbulence, and the other associated with the strain rate to which the flow is subjected. Or, a turbulence may have different length scales in different directions. Ordinary turbulence modeling is restricted to situations that can be approximated as having a single scale of length and velocity. Turbulence with multiple scales is much more complicated to predict. Some progress can be made by applying rapid distortion theory, or one or another kind of stability theory, to the initial turbulence, and predicting the kinds of structures that are induced by the applied distortion. We will talk more about this later. For now, we will restrict ourselves to a turbulence that has a single scale of length in the energy containing range.

Large Eddy Simulation

Over a decade ago, the author (Ferziger, 1983) wrote a review of the then state-of-the-art in direct numerical simulation (DNS) and large eddy simulation (LES). Shortly thereafter, a second review was written by Rogallo and Moin (1984). In those relatively early days of turbulent flow simulation, it was possible to write comprehensive reviews of what had been accomplished. Since then, the widespread availability of supercomputers has led to an explosion in this field so, although the subject is undoubtedly overdue for another review, it is not clear that the task can be accomplished in anything less than a monograph. The author therefore apologizes in advance for omissions (there must be many) and for any bias toward the accomplishments of people on the west coast of North America. In the earlier review, the author listed six approaches to the prediction of turbulent flow behavior. The list included: correlations, integral methods, single-point Reynolds-averaged closures, two-point closures, large eddy simulation and direct numerical simulation. Even then the distinction between these methods was not always clear; if anything, it is less clear today. It was possible in the earlier review to give a relatively complete overview of what had been accomplished with simulation methods. Since then, simulation techniques have been applied to an ever expanding range of flows so a thorough review of simulation results is no longer possible in the space available here. Simulation techniques have become well established as a means of studying turbulent flows and the results of simulations are best presented in combination with experimental data for the same flow. There is also a danger that the success of simulation methods will lead to attempts to apply them too soon to flows which the models and techniques are not ready to handle. To some extent, this is already happening. Direct numerical simulation (DNS) is a method in which all of the scales of motion of a turbulent flow are computed. A DNS must include everything from the large energy-containing or integral scales to the dissipative scales; the latter is usually taken to be the viscous or Kolmogoroff scales.

Direct Numerical Simulation Of Turbulent Flows

The numerical simulation of turbulent flows has a short history. About 45 years ago von Neumann (1949) and Emmons (1949) proposed an attack on the turbulence problem by numerical simulation. But one could point to a beginning 20 years later when Deardorff (1970) reported on a large-eddy simulation of turbulent channel flow on a 24x20x14 mesh and a direct simulation of homogeneous, isotropic turbulence was accomplished on a 323 mesh by Orszag and Patterson (1972). Perhaps the arrival of the CDC 6600 triggered these initial efforts. Since that time, a number of developments have occurred along several fronts. Of course, faster computers with more memory continue to become available and now, in 1994, 2563 simulations of homogeneous turbulence are relatively common with occasional 5123 simulations being achieved on parallel supercomputers (Chen et al., 1993) (Jimenez et al., 1993). In addition, new algorithms have been developed which extend or improve capabilities in turbulence simulation. For example, spectral methods for the simulation of arbitrary homogeneous flows and the efficient simulation of wall-bounded flows have been available for some time for incompressible flows and have recently been extended to compressible flows. In addition fast, viscous vortex methods and spectral element methods are now becoming available, suitable for incompressible flow with complex geometries. As a result of all these developments, the number of turbulence simulations has been increasing rapidly in the past few years and will continue to do so. While limitations exist (Reynolds, 1990; Hussaini et al., 1990), the potential of the method will lead to the simulation of a wide variety of turbulent flows. In this chapter, we present examples of these new developments and discuss prospects for future developments.

Introduction

The aim of this book is to provide the engineer and scientist with the necessary understanding of the underlying physics of turbulent flows, and to provide the user of turbulence models with the necessary background on the subject of turbulence to allow them to knowledgeably assess the basis for many of the state-of-the-art turbulence models. While a comprehensive review of the entire field could only be thoroughly done in several volumes of this size, it is necessary to focus on the key relevant issues which now face the engineer and scientist in their utilization of the turbulent closure model technology. The organization of this book is intended to guide the reader through the subject starting from key observations of spectral energy transfer and the physics of turbulence through to the development and application of turbulence models. Chapter 1 focuses on the fundamental aspects of turbulence physics. An insightful analysis of spectral energy transfer and scaling parameters is presented which underlies the development of phenomenological models. Distinctions between equilibrium and nonequilibrium turbulent flows are discussed in the context of modifications to the spectral energy transfer. The non-equilibrium effects of compressibility are presented with particular focus on the alteration to the turbulent energy dissipation rate. The important topical issue of coherent structures and their representation is presented in the latter half of the chapter. Both Proper Orthogonal Decomposition and wavelet representations are discussed. With an understanding of the broad dynamic With an understanding of the broad dynamic range covered by both the turbulent temporal and spatial scales, as well as their modal interactions, it is apparent that direct numerical simulation (DNS) of turbulent flows would be highly desirable and necessary in order to capture all the relevant dynamics of the flow. Such DNS methods, in which all the important length scales in the energy-containing range and in the dissipation range are accounted for explicitly is presented in Chapter 2. Emphasis is on spectral methods for incompressible flows, including the divergence-free expansion technique. Vortex methods for incompressible bluff body flows are described and some techniques for compressible turbulent flow simulations are also discussed briefly.