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.

Abstract

As more high-resolution observations become available, our view of ocean mesoscale turbulence more closely becomes that of a “sea of eddies.” The presence of the coherent vortices significantly affects the dynamics and the statistical properties of mesoscale flows, with important consequences on tracer dispersion and ocean stirring and mixing processes. Here we review some of the properties of particle transport in vortex-dominated flows, concentrating on the statistical properties induced by the presence of an ensemble of vortices. We discuss a possible parameterization of particle dispersion in vortex-dominated flows, adopting the view that ocean mesoscale turbulence is a two-component fluid which includes intense, localized vortical structures with non-local effects immersed in a Kolmogorovian, low-energy turbulent background which has mostly local effects. Finally, we report on some recent results regarding the role of coherent mesoscale eddies in marine ecosystem functioning, which is related to the effects that vortices have on nutrient supply.

Introduction

The ocean transports heat, salt, momentum and vorticity, nutrients and pollutants, and many other material and dynamical quantities across its vast spaces. Some of these transport processes are at the heart of the mechanisms of climate variability and of marine ecosystem functioning. In addition, a large portion of the available data on ocean dynamics are in the form of float and drifter trajectories. These provide a Lagrangian view of the ocean circulation which is not always easy to disentangle.

This book has been motivated by the recent surge in the density and availability of Lagrangian measurements in the ocean, recent mathematical and methodological developments in the analysis of such data to improve forecasts and transport characteristics of ocean general circulation models, and numerous applications to dispersion of biological species. Another source of motivation has been the Lagrangian Analysis and Prediction of Coastal and Ocean Dynamics (LAPCOD) workshops (www.rsmas.miami.edu/LAPCOD/meetings.html).

The main purpose of this book is to conduct a review of Lagrangian observations, analysis and assimilation methods in physical and biological oceanography, and to present new methodologies on Lagrangian analysis and data assimilation, and new applications of Lagrangian stochastic models from biological dispersion studies. Some of the chapters included in this volume were presented at LAPCOD workshops, while others have been specifically written for this collection. Given the size of the Lagrangian field, the present work cannot be considered as an exhaustive effort, but one which is aimed to cover many of the central research topics. It was our intent to maintain a good balance between historical and state-of-the-art developments in Lagrangian-based observations, theory, numerical modeling and analysis techniques.

This book seems to be a first of its kind because the central theme is the Lagrangian viewpoint for studying the transport phenomena in oceanic flows. Another unique and timely aspect of this book is its multidisciplinary nature with contributions from experimentalists, theoreticians, and modelers from diverse fields such as physical oceanography, marine biology, mathematics, and meteorology.

Introduction

In the last 20 years, the deployment of surface and subsurface buoys has increased significantly, and the scientific community is now focusing on the development of new techniques to maximize the use of these data. As shown by Davis (1983, 1991), oceanic observations of quasi-Lagrangian floats provide a useful and direct description of lateral advection and eddy dispersal. Data from surface drifters and subsurface floats have been intensively used to describe the main statistics of the general circulation in most of the world ocean, in terms of mean flow structure, second-order statistics and transport properties (e.g. Owens, 1991; Richardson, 1993; Fratantoni, 2001; Zhang et al., 2001; Bauer et al., 2002; Niiler et al., 2003; Reverdin et al., 2003). Translation, swirl speed and evolution of surface temperature in warm-core rings, which are ubiquitous in the oceans, have also been studied with floats by releasing them inside of the eddies (Hansen and Maul, 1991). Trajectories of freely drifting buoys allow estimation of horizontal divergence and vertical velocity in the mixed layer (Poulain, 1993). Also, data from drifters allows investigation of properties and statistics of near-inertial waves, which provide much of the shear responsible for mixing in the upper thermocline and entrainment at the base of the mixed layer (Poulain et al., 1992). Drifters have proved to be robust autonomous platforms with which to observe ocean circulation and return data from a variety of sensors.

Introduction

The prediction particle trajectories in the ocean is of practical importance for problems such as searching for objects lost at sea, tracking floating mines, designing oceanic observing systems, and studying ecological issues such as the spreading of pollutants and fish larvae (Mariano et al., 2002). In a given year, for example, the US Coast Guard (USCG) performs over 5000 search and rescue missions (Schneider, 1998). Even though the USCG and its predecessor, the Lifesaving Service, have been performing search and rescue operations for over 200 years, it has only been in the last 30 years that Computer Assisted Search Planning has been used by the USCG. The two primary components are determining the drift caused by ocean currents and the movement caused by wind. The results presented in this review are motivated by the drift estimation problem.

A number of authors (e.g., Aref, 1984; Samelson, 1996) have shown that prediction of particle motion is an intrinsically difficult problem because Lagrangian motion often exhibits chaotic behavior, even in regular and simple Eulerian flows. In the ocean, the combined effects of complex time-dependence (Samelson, 1992; Meyers, 1994; Duan and Wiggins, 1996) and three-dimensional structure (Yang and Liu, 1996) are likely to induce chaotic transport. Chaos implies strong dependence on the initial conditions, which are usually not known with great accuracy, so that the task of predicting particle motion is often extremely difficult.

Introduction

A complete description of a dynamical system must include information about two things: its state and its kinetics. The first part defines its condition or state at some instant in time, but nothing about its motion. The latter does the opposite, it tells us how the system is evolving, but nothing about its state. Thus, for a full understanding of a dynamical system, we need information on both. If we consider the ocean as such a system, its state would be determined by the distribution of mass while the kinetics of the system would be given by the distribution of currents. Since the birth of modern oceanography, we have developed an increasingly accurate picture of the state of the ocean, more specifically the distribution of heat and salt: the two properties that determine the mass field and hence the internal forces acting on it. Progress has been much slower – and more recent – with respect to a corresponding description of the kinetics of the ocean. Indeed, our view of the ocean circulation is still incomplete and depends to a significant extent upon assumptions about its internal dynamics in order to estimate ocean currents from the observed mass field. We have employed this methodology out of convenience and necessity because for a very long time we did not have the tools to observe the ocean in motion directly.

Introduction

Plankton have inhabited the Earth's oceans for hundreds of millions of years as evidenced by the fossil record. The exterior covering of identifiable dinoflagellates, for example, are well preserved in Mesozoic rock strata. Pelagic diatoms possess siliceous frustules with identifiable species dating from early Cretaceous sediments (see Falkowski et al., 2004). Given the extent of fossil plankton, it is apparent that a drifting mode of life has been a successful means for survival in the sea for much of life's history.

With the importance of plankton in marine ecosystems, it is surprising that biological oceanographers have only recently begun to use drifting, or more formally Lagrangian, techniques. However, as with other aspects of biological oceanography, the Lagrangian ‘tools’ for studying plankton are relatively recent, and have often followed technique development by physical oceanographers and engineers. The main goal of this chapter is to summarize how biological oceanographers have applied Lagrangian and related methods to further our understanding of oceanic plankton distributions and dynamics, as well as biogeochemical processes. Our target audience is physical oceanographers and mathematicians who will hopefully gain some benefit from this exercise, while biological oceanographers may also be encouraged to further consider Lagrangian approaches in their field studies. We include studies on bacterio-, phyto-, zoo-, and ichthyoplankton and discuss the advances made in specific sub-disciplines of biological oceanography through the use of Lagrangian techniques. This review is timely in that new, low power sensors are now being adapted for deployments on a variety of Lagrangian platforms.

Introduction

We produce the population dynamics of a stage structured population, where the stages are defined by sharp biological events (egg hatching, molt, adult emergence, beginning and end of oviposition, death), by means of a stochastic individual-based model that simulates the life histories of its individuals (Judson, 1994; Berec, 2002; Buffoni et al., 2002; Buffoni et al., 2004). Aspects of the life history of an individual, such as survival probabilities, development rates and egg production, depend on its “status,” on the population size, and on external factors such as the environmental conditions (e.g. physical factors, food availability). In general, the status of an individual can be identified by means of a number of physiological variables or biometric descriptors, which describe the behavior of an individual in a given situation, and define its physiological age. The physiological age of an individual is generally described only by a variable. Here the status of an individual is individuated by its stage and its physiological age in the stage. The physiological age is defined as the percentage of development for non-reproductive individuals, and as the percentage of the potential reproductive effort for an adult female. The life history is obtained by the time evolution of the status of an individual, from birth to death, following its development and, when the individual is an adult female, the production of eggs.