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.

The subject of hydrodynamic stability theory is concerned with the response of a fluid system to random disturbances. The word “hydrodynamic” is used in two ways here. First, we may be concerned with a stationary system in which flow is the result of an instability. An example is a stationary layer of fluid that is heated from below. When the rate of heating reaches a critical point, there is a spontaneous transition in which the layer begins to undergo a steady convection motion. The role of hydrodynamic stability theory for this type of problem is to predict the conditions when this transition occurs. The second class of problems is concerned with the possible transition of one flow to a second, more complicated flow, caused by perturbations to the initial flow field. In the case of pressure-driven flow between two plane boundaries (Chap. 3), experimental observation shows that there is a critical flow rate beyond which the steady laminar flow that we studied in Chap. 3 undergoes a transition that ultimately leads to a turbulent velocity field. Hydrodynamic stability theory is then concerned with determining the critical conditions for this transition.

For both types of problem, we can view the mathematical problem as one of determining the consequence of adding an initial perturbation in the velocity, pressure, temperature, or solute concentration fields to a basic unperturbed state. If the perturbation grows in time, the original unperturbed state is said to be unstable.

In the preceding chapters, we focused mainly on fluid dynamics problems, with only an occasional problem involving heat or mass transfer. In this chapter, we change our focus to problems of heat (or single-solute mass) transfer. Specifically, we address the problem of heat (or mass) transfer from a finite body to a surrounding fluid that is moving relative to the body. In this chapter, we concentrate on problems in which the fluid motion is viscous in nature, and thus is “known” (or can be calculated) from creeping-flow theory. Later, after we have considered flows at nonzero Reynolds number, we will also consider heat (or mass) transfer for this situation.

In all of the fluid mechanics problems that we have considered until now, the nonlinear inertia terms in the equations of motion were either identically zero or small compared with the viscous terms. We begin this chapter by considering the corresponding heat (or mass) transfer problem, in which the fluid motion is “slow” in a sense to be described shortly, so that convection effects are weak and the transport process is dominated by conduction. When convection terms in the thermal energy equation can be neglected altogether, the resulting pure conduction problem is mathematically and physically analogous to the creeping flows that we have been studying in the preceding two chapters. The transport of heat is purely “diffusive” in this limit, i.e., conduction, just as the transport of momentum (or vorticity) in a creeping flow is also “diffusive.”

Although the application of the “thin-film” approximation to analyze lubrication problems is one of its most important successes, there is an even larger body of problems in which the thin-film approximation can still be applied but in which the upper surface (or in some cases both surfaces of the thin film) is an interface. Examples include such diverse applications as gravity currents in geological phenomena, such as the gravitationally driven spread of molten lava; the dynamics of foams and or emulsions for which the thin films between bubbles (or drops) play a critical role in the dynamics; the dynamics of thin films in coating operations, and a variety of other materials processing applications; and thin films in biological systems, such as the coatings of the lung. Not only are the areas of application very diverse, but such films can and do display an astonishing array of complex phenomena, in spite of the limitations inherent in the thin-film assumptions. In part this is a consequence of the wide variety of physical effects that can play a role, including the capillary and Marangoni phenomena associated with surface tension, the possibility of a significant role for nonhydrodynamic effects such as van der Waals forces across the thin film and the possibility of transport processes such as evaporation/condensation.

In this chapter, we derive the governing equations for this class of thin films and show how they can be modified to account for the presence or absence of the various physical phenomena that were mentioned above.

We are now in a position to begin to consider the solution of heat transfer and fluid mechanics problems by using the equations of motion, continuity, and thermal energy, plus the boundary conditions that were given in the preceding chapter. Before embarking on this task, it is worthwhile to examine the nature of the mathematical problems that are inherent in these equations. For this purpose, it is sufficient to consider the case of an incompressible Newtonian fluid, in which the equations simplify to the forms (2–20), (2–88) with the last term set equal to zero, and (2–93).

The first thing to note is that this set of equations is highly nonlinear. This can clearly be seen in the term u · grad u in (2–88). However, because the material properties such as ρ, Cp, and k are all functions of the temperature θ, and the latter is a function of the velocity u through the convected derivative on the left-hand side of (2–93), it can be seen that almost every term of (2–88) and (2–93) involves a product of at least two unknowns either explicitly or implicitly. In contrast, all of the classical analytic methods of solving partial differential equations (PDEs) (for example, eigenfunction expansions by means of separation of variables, or Laplace and Fourier transforms) require that the equation(s) be linear. This is because they rely on the construction of general solutions as sums of simpler, fundamental solutions of the DEs.

In Chap. 9 we considered strong-convection effects in heat (or mass) transfer problems at low Reynolds numbers. The most important findings were the existence of a thermal boundary layer for open-streamline flows at high Peclet numbers and the fundamental distinction between open- and closed-streamline flows for heat or mass transfer processes at high Peclet numbers. An important conclusion in each of these cases is that conduction (or diffusion) plays a critical role in the transport process, even though Pe → ∞. In open-streamline flows, this occurs because the temperature field develops increasingly large gradients near the body surface as Pe → ∞. For closed-streamline flows, on the other hand, the temperature gradients are O(1) – except possibly during some initial transient period – and conduction is important because it has an indefinite time to act.

In this chapter we continue the development of these ideas by considering their application to the approximate solution of fluid mechanics problems in the asymptotic limit Re → ∞, with a particular emphasis on problems in which boundary layers play a key role. Before embarking on this program, however, it is useful to highlight the expected goals and limitations of the analysis in which we formally require Re → ∞ but still assume that the flow remains laminar. In practice, of course, most flows will become unstable at a large, but finite, value of Reynolds number and eventually undergo a transition to turbulence, and this is the flow we will see in the lab.

This book represents a major revision of my book Laminar Flow and Convective Transport Processes that was published in 1992 by Butterworth-Heinemann. As was the case with the previous book, it is about fluid mechanics and the convective transport of heat (or any passive scalar quantity) for simple Newtonian, incompressible fluids, treated from the point of view of classical continuum mechanics. It is intended for a graduate-level course that introduces students to fundamental aspects of fluid mechanics and convective transport processes (mainly heat transfer and some single solute mass transfer) in a context that is relevant to applications that are likely to arise in research or industrial applications. In view of the current emphasis on small-scale systems, biological problems, and materials, rather than large-scale classical industrial problems, the book is focused more on viscous phenomena, thin films, interfacial phenomena, and related topics than was true 14 years ago, though there is still significant coverage of high-Reynolds-number and high-Peclet-number boundary layers in the second half of the book. It also incorporates an entirely new chapter on linear stability theory for many of the problems of greatest interest to chemical engineers.

The material in this book is the basis of an introductory (two-term) graduate course on transport phenomena. It starts with a derivation of all of the necessary governing equations and boundary conditions in a context that is intended to focus on the underlying fundamental principles and the connections between this topic and other topics in continuum physics and thermodynamics.