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.

Introduction

Almost all flows of practical interest are turbulent, and thus the simulation of turbulent flow and its diversity of flow characteristics remains one of the most challenging areas in the field of classical physics. In many situations the fluid can be considered incompressible; that is, its density is virtually constant in the frame of reference, moving locally with the fluid, but density gradients may be passively convected with the flow. Examples of such flows of engineering importance are as follows: external flows, such as those around cars, ships, buildings, chimneys, masts, and suspension bridges; and internal flows, such as those in intake manifolds, cooling and ventilation systems, combustion engines, and applications from the areas of biomedicine, the process industry, the food industry, and so on. In contrast to free flows (ideally considered as homogeneous and isotropic), wall-bounded flows are characterized by much less universal properties than free flows and are thus even more challenging to study. The main reason for this is that, as the Reynolds number increases, and the thickness of the viscous sublayer decreases, the number of grid points required to resolve the near-wall flow increases.

The two basic ways of computing turbulent flows have traditionally been direct numerical simulation (DNS) and Reynolds-averaged Navier–Stokes (RANS) modeling. In the former the time-dependent Navier–Stokes equations (NSE) are solved numerically, essentially without approximations. In the latter, only time scales longer than those of the turbulent motion are computed, and the effect of the turbulent velocity fluctuations is modeled with a turbulence model.

Introduction to monotone integrated large eddy simulation

Turbulence is proving to be one of nature's most interesting and perplexing problems, challenging theorists, experimentalists, and computationalists equally. On the computational side, direct numerical simulation of idealized turbulence is used to challenge the world's largest computers, even before they are deemed ready for general use. The Earth Simulator, for example, has recently completed a Navier–Stokes solution of turbulence in a periodic box on a 4096 × 4096 × 4096 grid, achieving an effective Reynolds number somewhat in excess of 8000. Such a computation is impossible for nearly every person on the planet. Further, periodic geometry has little attraction for an engineer, and a Reynolds number of 8000 is far too small for most problems of practical importance.

The subject of this chapter is monotone integrated large eddy simulation (LES), or MILES – monotonicity-preserving implicit LES (ILES), a class of practical methods for simulating turbulent high-Reynolds-number flows with complicated, compressible physics and complex geometry. LES has always been the natural way to exploit the full range of computer power available for engineering fluid dynamics. When the dynamics of the energy-containing scales in a complex flow can be resolved, it is a mistake to average them out. Doing so limits the accuracy of the results, because uniform convergence to the physically correct answer, insofar as one exists, is automatically voided at the scale where the averaging has been performed. Even if the computational grid is refined repeatedly, the answer can get no better. At the same time, the overall resolution of a computation suffers when many computational degrees of freedom are expended unnecessarily on unresolved scales.

Introduction

In this chapter we present the rationale behind, the validation of, and the results from our numerical models of the turbulent flow of gas expected in the convection zones of various types of stars. We review both local area models of convection expected near the surface of solar types of stars and global models of the convection zones of red giant stars. We made the local area models in slab geometry on Cartesian meshes. Even though the geometry of the full convection zone of a red giant star is basically spherical, we also made these models on Cartesian meshes. We made these two sets of models by using variants of the piecewise parabolic method (PPM) described in detail in Chapter 4b and applied to ideal turbulent flow in Chapter 7. Both the use of an Euler-based code such as the PPM to do first-principle studies of turbulent convection as well as the use of Cartesian meshes to do calculations of spherical regions motivated quite a few tests. Our focus in this chapter is to describe the effectiveness and limits of using the PPM to study turbulent stellar convection in a variety of geometries in terms of surprising yet verifiable results.

Rewards and challenges

Both simple and sophisticated one-dimensional models of stars, as well as observation, indicate that there are regions inside many types of stars that are unstable to convection. These regions are called convection zones. The convection zone in the nearby starwe call “the Sun” spans roughly the outer third of its radius. The uppermost regions of the solar convection zone are seen on the surface as granulation.

In this final section, we summarize the contributions to this book and briefly discuss new and open issues.

Précis

Our goal in this book has been to introduce a relatively new approach to modeling turbulent flows, which we term implicit large-eddy simulation (ILES). Simply stated, the technique consists of employing a fluid solver based on nonoscillatory finite volume (NFV) approximations and allowing the numerical truncation terms to replace an explicit turbulence model. NFV techniques have been a mainstream direction in the broader computational fluid dynamics community for more than 25 years, where they are known for their accuracy, efficiency, and general applicability. The application of NFV methods to turbulent flows has been more recent, but it already has produced quality results in a variety of fields.

Despite these computational advantages and simulation successes, the turbulence modeling community has been slow to accept the ILES approach. We hope to promote this acceptance by the present gathering of individual contributions of ILES pioneers and lead researchers, providing a consistent framework for and justification of this new approach.

There are several paths that contribute to the justification of the ILES approach to simulating turbulent flows. Practical demonstrations of capability are a necessary component, and they constitute the main content of this volume. Simulations in various chapters of this book, ranging from canonical flows with theoretical outcomes to more complex flows that have been investigated experimentally, serve to verify and validate the ILES approach. Further, these results show that ILES is competitive with classical large-eddy simulation (LES) approaches in terms of accuracywhile offering advantages in computational efficiency and ease of implementation.

Background

Urban airflow that is accompanied by contaminant transport presents new, extremely challenging modeling requirements (e.g., Britter and Hanna 2003). Reducing health risks from the accidental or deliberate release of chemical, biological, or radiological (CBR) agents and pollutants from industrial leaks, spills, and fires motivates this work. Configurations with very complex geometries and unsteady buoyant flow physics are involved. The widely varying temporal and spatial scales exhaust current modeling capacities. Crucial technical issues include turbulent fluid transport and boundary condition modeling, and post processing of the simulation results for practical use by responders to actual emergencies.

Relevant physical processes to be simulated include complex building vortex shedding, flows in recirculation zones, and approximating the dynamic subgrid-scale (SGS) turbulent and stochastic backscatter. The model must also incorporate a consistent stratified urban boundary layer with realistic wind fluctuations; solar heating, including shadows from buildings and trees; aerodynamic drag and heat losses that are due to the presence of trees; surface heat variations; and turbulent heat transport. Because of the short time spans and large air volumes involved, modeling a pollutant as well mixed globally is typically not appropriate. It is important to capture the effects of unsteady, buoyant flowon the evolving pollutant-concentration distributions. In typical urban scenarios, both particulate and gaseous contaminants behave similarly insofar as transport and dispersion are concerned, so that the contaminant spread can usually be simulated effectively on the basis of appropriate pollutant tracers with suitable sources and sinks. In some cases, the full details of multigroup particle distributions are required. Additional physics includes the deposition, resuspension, and evaporation of contaminants.

Introduction

Shear flows driven by Kelvin–Helmholtz instabilities such as mixing layers, wakes, and jets are of great interest because of their crucial roles in many practical applications. The simulation of shear flows is based on the numerical solution of the Navier–Stokes (NS) or Euler (EU) equations with appropriate boundary conditions. The important simulation issues that have to be addressed relate to the appropriate modeling of (1) the required open boundary conditions for flows developing in both space and time in finite-size computational domains, and (2) the unresolved subgrid-scale (SGS) flow features.

Appropriate boundary condition modeling is required because, in studying spatially developing flows, we can investigate only a portion of the flow – as in the laboratory experiments, where finite dimensions of the facilities are also unavoidable. We must ensure that the presence of artificial boundaries adequately bounds the computational domain without polluting the solution in a significant way: numerical boundary condition models must be consistent numerically and with the physical flow conditions to ensure well-posed solutions, and emulate the effects of virtually assumed flow events occurring outside of the computational domain. SGS models are needed that ensure the accurate computation of the inherently three-dimensional (3D) time-dependent details of the largest (grid-scale) resolved motions responsible for the primary jet transport and entrainment. At the high Reynolds number of practical interest, direct numerical simulation (DNS) cannot be used to resolve all scales of motion, and some SGS modeling becomes unavoidable to provide a mechanism by which dissipation of kinetic energy accumulated at high wave numbers can occur.

Large eddy simulation: From practice to theory

LES: Statement of the problem

It is well known that the direct numerical simulation (DNS) of fully developed turbulent flows is far beyond the range of available supercomputers. Indeed, the computational effort scales like the cube of the Reynolds number for the very simple case of incompressible isotropic turbulence, showing that an increase by a factor of 1,000 in the computational cost will only permit a gain of a factor of about 10 in the Reynolds number. The actual possibilities are illustrated by the results obtained on a grid of 40963 points by Kaneda et al. (2003) simulating incompressible isotropic turbulence at Reλ = 1201 (where Reλ is the Reynolds number based on the Taylor microscale).

The main consequence is that to obtain results at high Reynolds number, all the dynamically active turbulent scales cannot be simulated at the same time: some must be discarded. But, because of the intrinsically nonlinear nature of the Navier–Stokes equations, all turbulent scales are coupled in a dynamic way so that the effects of the discarded scales on the resolved scales must be taken into account to ensure the reliability of the results. This is achieved by augmenting the governing equations for the resolved scales to include new terms that represent the effects of the unresolved scales. The large eddy simulation (LES) technique computes the large scales (where the notion of “large” will be defined) of the flow, while modeling their interactions with small unresolved scales (referred to as subgrid scales) through a subgrid model.