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.

Introduction

In this chapter we extend our study of the underlying justification of implicit large eddy simulation (ILES) to the numerical point of view. In Chapter 2 we proposed that the finite-volume equations, found by integrating the governing partial differential equations (PDEs) over a finite region of space and time, were more appropriate models for describing the behavior of discrete parcels of fluid, including computational cells in numerical simulation. However, effective simulation of turbulent flows must consider not only issues of accuracy but also those of computational stability. Here we introduce and apply the machinery of modified equation analysis (MEA) to identify the properties of discrete algorithms and to compare different algorithms. We then apply MEA to several of the nonoscillatory finite-volume (NFV) methods described in Chapter 4, with the goal of identifying those elements essential to successful ILES. In the process we make connections to the some of the explicit subgrid models discussed in Chapter 3, thus demonstrating that many subgrid models implicit within NFV methods are closely related to existing explicit models. MEA is also applied with the methods description in Chapter 4a.

We consider the answer to this question: What are the essential ingredients of a numerical scheme that make it a viable basis for ILES? Many of our conclusions are based on MEA, a technique that processes discrete equations to produce a PDE that closely represents the behavior of a numerical algorithm (see Hirt 1968; Fureby and Grinstein 2002; Margolin and Rider 2002; Grinstein and Fureby 2002; Margolin and Rider 2005).

Introduction

The importance of investigating nonlinear bifurcation phenomena in fluid mechanics lies in enabling a clearer understanding of hydrodynamic stability and the mechanism of laminar-to-turbulent flow transition. Bifurcation phenomena have been observed in a number of laboratory flows, with incompressible flow in sudden expansions being one of the classical examples. At certain Reynolds numbers, these flows present instabilities that may lead to bifurcation, unsteadiness, and chaos (Mullin 1986).

For example, the existence of symmetry-breaking bifurcation in suddenly expanded flows has been demonstrated (Chedron, Durst, and Whitelaw 1978; Fearn, Mullin, and Cliffe 1990). This is manifested as an asymmetric separation that occurs beyond a certain value of Reynolds number. Similarly, Mizushima et al. (Mizushima, Okamoto, and Yamaguchi 1996; Mizushima and Shiotani 2001) have conducted experimental investigations to extend suddenly expanded flows to suddenly expanded and contracted channel flow. They found that this type of geometry exhibits similar flow effects to the simpler suddenly expanded channel, with instabilities manifesting as asymmetric separation at Reynolds numbers within a critical range. In the experiments, the instabilities were triggered by geometrical imperfections and asymmetries in the inflow conditions upstream of the expansion. In a symmetric numerical setup, however, these asymmetries can only be generated by the numerical scheme and are associated with dissipation and dispersion properties of the numerical method employed. In the past, computational investigations have been conducted for unstable separated flows through sudden expansions (Alleborn et al. 1997; Drikakis 1997). In particular, numerical experiments by Patel and Drikakis (2004) using explicit (symmetric) solvers and different highresolution schemes were conducted to show that symmetry breaking depends solely on the details of the numerical scheme employed for the discretization of the advective terms.

Introduction

Large eddy simulation (LES) has emerged as the next-generation simulation tool for handling complex engineering, geophysical, astrophysical, and chemically reactive flows. As LES moves from being an academic tool to being a practical simulation strategy, the robustness of the LES solvers becomes a key issue to be concerned with, in conjunction with the classical and well-known issue of accuracy. For LES to be attractive for complex flows, the computational codes must be readily capable of handling complex geometries. Today, most LES codes use hexahedral elements; the grid-generation process is therefore cumbersome and time consuming. In the future, the use of unstructured grids, as used in Reynolds-averaged Navier–Stokes (RANS) approaches, will also be necessary for LES. This will particularly challenge the development of high-order unstructured LES solvers. Because it does not require explicit filtering, Implicit LES (ILES) has some advantages over conventional LES; however, numerical requirements and issues are otherwise virtually the same for LES and ILES. In this chapterwe discuss an unstructured finite-volume methodology for both conventional LES and ILES, that is particularly suited for ILES. We believe that the next generation of practical computational fluid dynamics (CFD) models will involve structured and unstructured LES, using high-order flux-reconstruction algorithms and taking advantage of their built-in subgrid-scale (SGS) models.

ILES based on functional reconstruction of the convective fluxes by use of high-resolution hybrid methods is the subject of this chapter. We use modified equation analysis (MEA) to show that the leading-order truncation error terms introduced by such methods provide implicit SGS models similar in form to those of conventional mixed SGS models.

Introduction

A grand challenge for computational fluid dynamics (CFD) is the modeling and simulation of the time evolution of the turbulent flow in and around different engineering applications. Examples of such applications include external flows around cars, trains, ships, buildings, and aircrafts; internal flows in buildings, electronic devices, mixers, food manufacturing equipment, engines, furnaces, and boilers; and supersonic flows around aircrafts, missiles, and in aerospace engine applications such as scramjets and rocket motors. For such flows it is unlikely that we will ever have a really deterministic predictive framework based on CFD, because of the inherent difficulty in modeling and validating all the relevant physical subprocesses, and in acquiring all the necessary and relevant boundary condition information. On the other hand, these cases are representative of fundamental ones for which whole-domain scalable laboratory studies are extremely difficult, and for which it is crucial to develop predictability as well as establish effective approaches to the postprocessing of the simulation database.

The modeling challenge is to develop computational models that, although not explicitly incorporating all eddy scales of the flow, give accurate and reliable flowfield results for at least the large energy-containing scales of motion. In general terms this implies that the governing Navier–Stokes equations (NSE) must be truncated in such a way that the resulting energy spectra is consistent with the |k|-5/3 law of Kolmogorov, with a smooth transition at the high-wave-number cutoff end. Moreover, the computational models must be designed so as to minimize the contamination of the resolved part of the energy spectrum and to modify the dissipation rate in flow regions where viscous effects are more pronounced, such as the region close to walls.

Introduction

The use of the piecewise parabolic method (PPM) gas dynamics simulation scheme is described in detail in Chapter 4b and used in Chapter 15 (see also Woodward and Colella 1981, 1984; Collela and Woodward 1984; Woodward 1986, 2005). Here we review applications of PPM to turbulent flow problems. In particular, we focus our attention on simulations of homogeneous, compressible, periodic, decaying turbulence. The motivation for this focus is that if the phenomenon of turbulence is indeed universal, we should find within this single problem a complete variety of particular circumstances. If we choose to ignore any potential dependence on the gas equation of state, choosing to adopt the gamma law with γ = 1.4 that applies to air, we are then left with a one-parameter family of turbulent flows. This single parameter is the root-mean-square (rms) Mach number of the flow. We note that a decaying turbulent flow that begins at, say, Mach 1 will, as it decays, pass through all Mach numbers between that value and zero. Of course, we will have arbitrary possible entropy variations to deal with, but turbulence itself will tend to mix different entropy values, so that these entropy variations may not prove to be as important as we might think. In all our simulations of such homogeneous turbulence, we begin the simulation with a uniform state of density and sound speed unity and average velocity zero. We perturb this uniform state with randomly selected sinusoidal velocity variations sampled from a distribution peaked on a wavelength equal to half that of our periodic cubical simulation domain.

Many important industrial applications, as well as insight into the phenomena of nature, rely crucially on knowledge about fluid phase behavior. In space and other high-temperature industries, as well as in combustion processes, the properties of gases manifesting various types of reaction, including dissociation and ionization, are required. In chemical and environmental science and technology, phase and reaction equilibria of multicomponent mixtures form the basis of understanding the phenomena and designing synthesis, separation, and purification processes. Biotechnological downstream processing relies on the distribution properties of biomolecules in different phases of aqueous and organic solutions. Even in standard mechanical engineering equipment technology, such as refrigerator design, lack of data for new environmentally friendly refrigerants has proved to be a severe obstacle to technological progress. In all these cases, and many others, fluid phase properties form the basis of modern technological processes and detailed and quantitative knowledge of their properties is the premise of innovation. Experimental studies alone, although indispensible in the field of fluid system science, cannot serve these needs. The project of studying the fluid phase behavior of a multicomponent system experimentally is hopeless in view of the large number of data that would be needed. Instead, molecular models, which can be evaluated on a computer and make use of the limited data available to predict the fluid phase behavior in the full range of interest, are needed. Due to the broad availability of high-speed computers, such models can be quite ambitious, including use of quantum-chemical and molecular simulation computer codes.

When fluid phase behavior over a large region of states is considered, excess function models are no longer appropriate. They are designed to address mixing effects at constant density or pressure. Effects of varying density are most conveniently treated in terms of an equation of state. The thermodynamic relations for computing fluid phase behavior from an equation of state in combination with ideal gas properties are well established; cf. Section 2.1. Although they are more demanding computationally than excess function models, there are now many well-tested computer codes available that allow the computation of fluid phase behavior from an equation of state. Basically, this approach is free from any of the restrictions associated with the use of excess function models. In principle, an equation of state model is generally applicable, including simple and complicated molecules, and, in particular, mixtures of small and large molecules, as in polymer solutions. In practice, different equation of state models are used for different applications and no general model suitable for all applications has yet emerged. The generality of the equation of state approach requires full generality of the potential energy model. A formulation in terms of contact energies, adequate for excess function models, is unsufficient. Rather, the potential energy will in principle depend on the distances between the molecular centers, on the orientations of the molecules, and, in the most general case, also on their internal coordinates. In this book we shall concentrate on equation of state models for systems composed of small molecules, such as those typically encountered in the gas industries.

The book deals with the prediction of the macroscopic behavior of fluids from the properties of their molecular constituents. The basis of such prediction is the availability of molecular models. Designing molecular models for fluids has an interdisciplinary background of foundations. Their thorough understanding is the basis for developing new models and appreciating the promise as well as the limitations of those that are established.

Molecular models are formulated in terms of the energy of a system of three-dimensional flexible bodies, i.e., the molecules. This molecular energy depends on the geometrical structures of the molecules and the force field they are moving in. The relation between the geometrical structure of a body and its energy is defined in mechanical terms. The force field results from the electrical properties of the molecules. On this level the models are thus based on classical mechanics and electrostatics. Classical theory, although most powerful even on the molecular scale, is, however, incomplete in the sense that it does not provide information on the geometry of the molecules or on their charge distributions as the origin of the electrical force field. This gap is closed by quantum mechanics, which also gives important corrections to the classical results in order to make them ultimately applicable to molecules. The link between the molecular energy of a system and its macroscopic thermodynamic functions is provided by statistical mechanics and by computer simulation. By using this link the molecular model leads to numerical data for the thermodynamic functions, from which the macroscopic behavior of a fluid can be calculated by the laws of classical thermodynamics.

Our society relies on the use of energy and matter in a plenitude of different forms. They are produced from natural resources by technical processes of energy and matter conversion that have to be designed in an economically and ecologically optimum way. In these processes it is the fluid state of matter that dominates the relevant phenomena. In particular, the properties of fluid systems in equilibrium enter into the fundamental process equations and control the feasibility of the various process steps. Models for fluids in equilibrium are thus a prerequisite for any scientific process analysis. Although fluid models can be constructed entirely within the framework of a macroscopic theory on the basis of experimental data, it is clear that this approach is limited to those few systems for which enough data can be obtained. Typical examples are the working fluids of the standard power generation and refrigeration processes. The vast majority of technically relevant processes are, however, concerned with complex fluid systems that cannot be analyzed experimentally in sufficient detail with a reasonable effort. In such cases one must turn to the microscopic basis of matter and design a theory based on the molecular properties of a fluid that requires only few data or is even fully predictive. In this introductory chapter we present an overview of the challenges of this approach by presenting a review of macroscopic fluid phase behavior in equilibrium, along with the problems associated with obtaining the necessary information from data. We also give a first introduction to the primary concepts of the microscopic world, including a brief glance at the properties of real molecules and the philosophy behind formulating molecular models.

To derive an expression for the relative probability Pi of a microstate i we first consider a canonical ensemble, i.e., a collection of systems, each with fixed values of N, V, and T; cf. Figure A 3.1. The ensemble as a whole is isolated adiabatically so that the total energy of the canonical ensemble has a fixed value, Uc. Each of the πc systems of the canonical ensemble finds itself in a large heat bath at temperature T provided by the other systems.

Many different microstates or quantum states are associated with each fixed macrostate of the canonical ensemble. Different quantum states of a canonical ensemble may be represented by different distributions of its systems over the many possible energy states of the single system.We denote by {Ei} the various possible energy values of a system in the canonical ensemble. In principle, the spectrum of energy values of the system follows from its Schrödinger equation. We assume in the following that the spectrum of energy values is known. It is identical for each system of the canonical ensemble, because the values of V and {Nα} are identical. Thus {Ei} is available as a set of values (E1, E2, E3, …, Ej), if we assume a total of j quantum states for each system. At each moment of time each system of the canonical ensemble will be found in one of these energy states, whose number j is extremely large.