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.

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.