This book presents a general conceptual and basic quantitative analysis of the mechanical efficiency of heat engines. Typically, treatment of the mechanical efficiency of heat engines has been performed on a case-by-case basis. In ordinary practice, kinematic analysis and computer simulation of specific engine mechanisms coupled with calculated or measured pressure–volume cycles usually can indeed be effectively used for evaluating and locally optimizing engine designs. However, going beyond the specific and local requires broader insights that only a general theory can provide.

No general approach to mechanical efficiency of heat engines had been available until recently. This is in sharp contrast to the situation regarding the thermal efficiency of heat engines. Classical thermodynamics treats the subject of thermal efficiency in great generality. Its results, although obtained in a highly idealized setting, are of profound importance to engine theorists, designers, and practitioners. This book presents a theory of mechanical efficiency at a similar level of ideality and generality.

The first results in this area were published in 1985 and further developed in a series of papers up to the writing of this book. The work modeled the interaction between the mechanical section of an engine and its thermal section at a level compatible with that of classical thermodynamics.

Introduction

In Chapter 4e, a computational method based on vorticity confinement (VC) is described that has been designed to capture thin vortical regions in high-Reynolds-number incompressible flows. The principal objective of the method is to capture the essential features of these small-scale vortical structures and model them with a very efficient difference method directly on an Eulerian computational grid. Essentially, the small scales are modeled as nonlinear solitary waves that “live” on the lattice indefinitely. The method allows convecting structures to be modeled over as few as two grid cells with no numerical spreading as they convect indefinitely over long distances, with no special logic required for merging or reconnection. It can be used to provide very efficient models of attached and separating boundary layers, vortex sheets, and filaments. Further, the method easily allows boundaries with no-slip conditions to be treated as “immersed” surfaces in uniform, nonconforming grids, with no requirements for complex logic involving “cut” cells.

There are close analogies between VC and well-known shock- and contact-discontinuity-capturing methodologies. These were discussed in Chapter 4e to explain the basic ideas behind VC, since it is somewhat different than conventional computational fluid dynamics (CFD) methods. Some of the possibilities that VC offers toward the very efficient computation of turbulent flows, which can be considered to be in the implicit large eddy simulation (ILES) spirit, were explored. These stem from the ability of VC to act as a negative dissipation at scales just above a grid cell, but that saturates and does not lead to divergence.

This book represents the combined efforts of many sponsors. Most of the basic planning and organization was carried out while one of us (F. F. Grinstein) was the 2003–2004 Orson Anderson Distinguished Visiting Scholar at the Institute for Geophysics and Planetary Physics (IGPP) at Los Alamos National Laboratory (LANL). It is very important thatwe acknowledge the critical role played by the implicit large eddy simulation (ILES) workshops at LANL in January and November of 2004. These workshops took place under the auspices of IGPP and with partial support from the Center for Nonlinear Studies at LANL. They provided us with an ideal forum to meet and exchange ILES views and experiences, and to extensively discuss their integration within the book project. At the personal level, special thanks go to IGPP's Gary Geernaert and to the U.S. Naval Research Laboratory's (NRL's) Jay Boris and Elaine Oran for their continued encouragement and support. Last but not least, continued support of F. F. Grinstein's research on ILES during his tenure at NRL from the U.S. Office of Naval Research through NRL and from the U.S. Department of Defense High-Performance Computing Modernization Program is also greatly appreciated.

This book has evolved far beyond the early plan of merely putting together a collection of review papers on ILES authored by the lead researchers in the area. Several very useful collaborations have quite spontaneously occurred in the process of integrating the material, and we now have an active ILES working group that is focusing on a variety of timely research projects.

Introduction

In this chapter we make a connection between the filtering approach (Leonard 1974) and the averaged-equation approach (Schumann 1975) to large eddy simulation (LES). With the averaged-equation approach, the discrete system for evolving a grid-function approximation of the continuous solution is considered directly as a truncated representation of the continuous system. With the filtering approach, a continuous filtered system is considered as an approximation; the numerical error in solving this continuous system is considered to be negligibly small. The filtering approach provides an analytic framework for deriving LES equations and commonly is employed as a basis for the development of functional and structural models (Sagaut 2005) and Chapter 3 of this book. In practice, models derived on the basis of the filtering approach were plagued by the problem that the numerical error in most cases was nonnegligible. The effect of discretizing the filtered continuous equations on the subgrid-scale (SGS) force was analyzed in detail for the first time by Ghosal (1996). It was revealed that, over a large wave-number range, the truncation error of commonly employed nonspectral discretizations can be as large as the SGS stress, if not larger.

During the attempt of improving eddy-viscosity-based models, it was revealed that the correlation of predicted SGS stresses with the exact SGS stresses is much less than unity. This fact is reviewed by Meneveau and Katz (2000) on the basis of experimental data. A much larger correlation is achieved by the scale-similarity model (Bardina, Ferziger, and Reynolds 1983), which does, however, underpredict SGS dissipation.

The numerical simulation of turbulent fluid flows is a subject of great practical importance to scientists and engineers. The difficulty in achieving predictive simulations is perhaps best illustrated by the wide range of approaches that have been developed and that are still being used by the turbulence modeling community. In this book, we describe one of these approaches, which we have termed implicit large eddy simulation (ILES).

ILES is remarkable for its simplicity and general applicability. Nevertheless, it has not yet received widespread acceptance in the turbulence modeling community. We speculate that this is the result of two factors: the lack of a theoretical basis to justify the approach and the lack of appreciation of its large and diverse portfolio of successful simulations. The principal purpose of this book is to address these two issues.

One of the complicating features of turbulence is the broad range of spatial scales that contribute to the flow dynamics. In most examples of practical interest, the range of scales is much too large to be represented on even the highest-performance computers of today. The general strategy, which has been employed successfully since the beginning of the age of computers, is to calculate the large scales of motion and to introduce models for the effects of the (unresolved) small scales on the flow. In the turbulence modeling community, these are called subgrid-scale (SGS) models.

In ILES, we dispense with explicit subgrid models. Instead, the effects of unresolved scales are incorporated implicitly through a class of nonoscillatory finite-volume (NFV) numerical fluid solvers.

Introduction

High-Reynolds' number turbulent flows contain a broad range of scales of length and time. The largest length scales are related to the problem geometry and associated boundary conditions, whereas it is principally at the smallest length scales that energy is dissipated by molecular viscosity. Simulations that capture all the relevant length scales of motion through numerical solution of the Navier–Stokes equations (NSE) are termed direct numerical simulation (DNS). DNS is prohibitively expensive, now and for the foreseeable future, for most practical flows of moderate to high Reynolds' numbers. Such flows then require alternate strategies that reduce the computational effort. One such strategy is the Reynolds-averaged Navier–Stokes (RANS) approach, which solves equations averaged over time, over spatially homogeneous directions, or across an ensemble of equivalent flows. The RANS approach has been successfully employed for a variety of flows of industrial complexity. However, RANS has known deficiencies when applied to flows with significant unsteadiness or strong vortex-acoustic couplings.

Large eddy simulation (LES) is an effective approach that is intermediate in computational complexity while addressing some of the shortcomings of RANS at a reasonable cost. An introduction to conventional LES is given in Chapter 3. The main assumptions of LES are (1) that the transport of momentum, energy, and passive scalars is mostly governed by the unsteady features in the larger length scales, which can be resolved in space and time; and (2) that the smaller length scales are more universal in their behavior so that their effect on the large scales (e.g., in dissipating energy) can be represented by using suitable subgrid-scale (SGS) models.

Introduction

Prediction of the Earth's climate andweather is difficult in large part because of the ubiquity of turbulence in the atmosphere and oceans. Geophysical flows evince fluid motions ranging from dissipation scales as small as a fraction of a millimeter to planetary scales of thousands of kilometers. The span in time scales (from a fraction of a second to many years) is equally large. Turbulence in the atmosphere and the oceans is generated by heating and by boundary stresses – just as in engineering flows. However, geophysical flows are further complicated by planetary rotation and density–temperature stratification, which lead to phenomena not commonly found in engineering applications. In particular, rotating stratified fluids can support a variety of inertia-gravity and planetary waves. When the amplitude of such a wave becomes sufficiently large (i.e., comparable to the wavelength), the wave can break, generating a localized burst of turbulence. If one could see the phenomena that occur internally in geophysical flows at any scale, one would be reminded of familiar pictures of white water in a mountain stream or of breaking surf on a beach. The multiphase thermodynamics of atmosphere and oceans – due to ubiquity of water substance and salt, respectively – adds complexity of its own.

Because of the enormous range of scales, direct numerical simulation (DNS) of the Earth's weather and climate is far beyond the reach of current computational technology. Consequently, all numerical simulations truncate the range of resolved scales to one that is tractable on contemporary computational machines. However, retaining the physicality of simulation necessitates modeling the contribution of truncated scales to the resolved range.

Introduction

Rayleigh–Taylor (RT) instability (see Sharp 1984) occurs when the interface between two fluids of different density is subjected to a normal pressure gradient with a direction such that the pressure is higher in the less dense fluid. The related Richtmyer–Meshkov (RM) process (see Holmes et al. 1999) occurs when a shock wave passes through a perturbed interface. These instabilities are currently of concern for researchers involved in inertial confinement fusion (ICF). RT and RM instabilities can degrade the performance of ICF capsules, where high-density shells are decelerated by lower-density thermonuclear fuel. In these applications and in many RT or RM laboratory experiments, the Reynolds number is very high. Turbulent mixing will then occur. Direct numerical simulation (DNS) is feasible at a moderate Reynolds number. However, for most experimental situations, the calculation of the evolution of turbulent mixing requires some form of large eddy simulation (LES).

The flows of interest here involve shocks and density discontinuities. It is then highly desirable to use monotonic or total variation diminishing (TVD) numerical methods, either for calculating the mean flow or the development of instabilities. Hence, for three-dimensional turbulent flows, monotone-integrated LES (MILES) is very strongly favored.

My purpose in this chapter is to show that a particular form of MILES gives good results for RT and RM mixing. I consider the mixing of miscible fluids, and I assume the Reynolds number to be high enough for the effect of the Schmidt number to be unimportant.