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

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.