Abstract

The flow past bluff bodies, which occurs in many engineering situations, is very complex, involving often unsteady behaviour and dominant large-scale structures; it is therefore not very amenable to simulation by the RANS method using statistical turbulence models. The large-eddy simulation technique is more suitable for these flows. In this section work in the area of large-eddy simulations of bluff body flows is summarised, with emphasis on work by the author's research group as well as on experiences gained from two LES workshops. Results are presented and compared for the vortex-shedding flow past square and circular cylinders and for the flow around surface-mounted cubes. The performance, the cost and the potential of the LES method for simulating bluff body flows, also vis-à-vis RANS methods, is assessed.

Introduction

In many engineering situations, bluff bodies are exposed to flow, generating complex phenomena such as flow separation and often even multiple separation with partial reattachment, vortex shedding, bi-modal flow behaviour, high turbulence level and large-scale turbulent structures which contribute considerably to the momentum, heat and mass transport. For solving practical problems, there is a great demand for methods for predicting such flows and associated heat and mass-transfer processes, in particular the loading, including dynamic loading on the bodies and the scalar transport in the vicinity of structures. Usually the Reynolds number is high in practical problems so that turbulent transport processes are important and must be accounted for in a prediction method in one way or another. Until recently, mainly RANS based methods were used in which the entire spectrum of the turbulent motion is simulated by a statistical turbulence model. In vortex-shedding situations, unsteady RANS equations are solved to determine the periodic shedding motion and only the superimposed stochastic turbulent fluctuations are simulated with the turbulence model. So far, mainly variants of the k-ϵ eddy-viscosity turbulence model have been used for calculating the flow around bluff bodies, but some results have been reported that were obtained with Reynolds-stress models.

The RANS calculations have shown that statistical turbulence models have difficulties with the complex phenomena mentioned above, especially when large-scale eddy structures dominate the turbulent transport and when unsteady processes like vortex shedding and bistable behaviour prevail and dynamic loading is of importance.

Non-local wall effects

The elliptic nature of wall effects was recognized early in the literature on turbulence modeling (Chou 1945)and has continued to influence thoughts about how to incorporate non-local influences of boundaries (Launder et al. 1975). In the literature on closure modeling the non-local effect is often referred to as ‘pressure reflection’ or ‘pressure echo’ because it originates with the surface boundary condition imposed on the Poisson equation for the perturbation pressure, p.

The Poisson equation is (we are considering constant density flow ρ ≡ 1); the boundary condition is usually taken to be ∂p/∂xn = 0, ignoring a small viscous contribution. The boundary condition influences the pressure of the interior fluid through the solution to (1.1). Mathematically this is quite simple: the solution to the linear equation (1.1) consists of a particular part, forced by the right-hand side, and a homogeneous part, forced by the boundary condition. The fact that the boundary condition adds to the solution interior to the fluid can be described as a non-local, kinematic effect.

Figure 1 schematizes non-locality in the Poisson equation as a reflected pressure wave, but for incompressible turbulent fluctuations the wall effect is instantaneous, though non-local. Pressure reflection enhances pressure fluctuations; indeed, Manceau et al. (2001)sho w that pressure reflection can increase redistribution of Reynolds stress anisotropy. Redistribution is due to the pressure-strain correlation: the notion that it is increased by the wall effect is contrary to most second moment closure (SMC)models, which represent pressure echo as a reduction of the redistribution term.

The idea of associating inviscid wall effects with pressure reflection is natural, because the pressure enters the Reynolds stress transport equation through the velocity-pressure gradient correlation. Suppression of the normal component of pressure gradient by the wall should have an effect on the rate of redistribution of variance between those components of the Reynolds stress tensor that contain the normal velocity component – i.e., unui, where n denotes the wall-normal direction. This effect enters the evolution equation for the Reynolds stress (equation (1.3)below).

However, there is another notion about how anisotropy of the Reynolds stress tensor is altered non-locally by the presence of a wall. The inviscid boundary condition on the normal component of velocity is the no-flux condition u · n= 0. This constraint on the normal velocity produces another non-local, elliptic influence of the boundary.

Abstract

Extensions of the frontiers of rapid distortion theory (RDT) and multi-point closures are discussed, especially developments leading towards inhomogeneous turbulence. Recent works related to zonal RDT and stability analyses for wavepacket disturbances to non-parallel rotational base flows are presented. Application of linear theories to compressible flows are touched upon. Homogeneous turbulence is revisited in the presence of dispersive waves, taking advantage of the close relationship between recent theories of weakly nonlinear interactions, or ‘wave-turbulence’, and classical two-point closure theories. Among various approaches to multi-point description and modelling, a review is given of multi-scale or multi-tensor transport models, which use, more or less explicitly, a spectral formulation.

Inhomogeneous turbulence

Multi-point formulations are not nearly as well-developed for inhomogeneous as for homogeneous turbulence. An assumption of weak inhomogeneity, in which variations of the flow statistics take place over distances greater than O(ℓ), the size of the large turbulent eddies, allows some progress to be made, as, to a lesser extent, does the RDT limit of weak turbulence.

Linear theories

The solution of RDT with a known mean flow Ui, arbitrarily varying in space, is a difficult problem in general, but becomes somewhat simpler if the mean flow is irrotational, as in the classical case of high Reynolds number flow past a body, outside the wake and boundary layer. Consider a particle convected by the flow, having position x′ at time t′ and x at time t. The deformation of fluid elements is characterised by the Cauchy tensor and the evolution of vorticity for inviscid incompressible flow is then described by the Cauchy solution The above formulation is exact and expresses the classical theory of inviscid vortical dynamics. However, in the context of RDT for irrotational mean flows, one can neglect the fluctuating part of the velocity compared with the mean part, x and x′ are related by mean flow convection and Fij becomes the deformation tensor of the mean flow alone. Since the mean flow is assumed irrotational, there is no mean vorticity, and (2) describes the fluctuating vorticity giving the curl of the fluctuating velocity u′i.

Introduction

This chapter is meant as an introduction to Large-Eddy Simulation (LES) for readers not familiar with it. It therefore presents some classical material in a concise way and supplements it with pointers to recent trends and literature. For the same reason we shall focus on issues of methodology rather than applications. The latter are covered elsewhere in this volume. Furthermore, LES is closely related to direct numerical simulation (DNS) which is also widely discussed in this volume. Hence,w e concentrate as much as possible on those features which are particular to LES and which distinguish it from other computational methods.

For the present text we have assembled material from research papers,earlier introductions and reviews (Ferziger 1996,Härtel 1996,Piomelli 1998),and our own results. The selection and presentation is of course biased by the authors’ own point of view. Supplementary material is available in the cited references.

Resolution requirements of DNS

The principal difficulty of computing and modelling turbulent flows resides in the dominance of nonlinear effects and the continuous and wide spectrum of observed scales. Without going into details (the reader might consult classical text books such as Tennekes and Lumley (1972)) we just recall here that the ratio of the size of the largest turbulent eddies in a flow, L,to that of the smallest ones determined by viscosity, η, behaves like L/η ∼ Re3/4u′. Here, Reu′= u′ L/ν with u′ being a characteristic velocity fluctuation and ν the kinematic viscosity. Let us consider as an example a plane channel,a prototype of an internal flow. Reynolds (1989) estimated Re′ ∼ Re0.9 from u′ ∼ u c1/2f, cf ∼ Re−0.2, where Re is based on the center line velocity and the channel height. In a DNS no turbulence model is applied so that motions of all size have to be resolved numerically by a grid which is sufficiently fine. Hence, the computational requirements increase rapidly with Re. According to this estimate a DNS of channel flow at Re = 106 for example would take around hundred years on a computer running at several GFLOPS. This is obviously not feasible. Moreover,in an expensive DNS a huge amount of information would be generated which is mostly not required by the practical user. He or she would mostly be content with knowing the average flow and some lower moments to a precision of a few percent.

In this chapter the joint velocity-scalar PDF approach is described. This approach was mainly developed by S.B. Pope and includes from the start the complete one-point joint statistics of velocity and scalars. This is conceptually appealingb ecause it delivers in one framework closure models for Reynolds stresses, Reynolds fluxes and chemical source terms. The present text is based on the PhD thesis of H.A. Wouters (Wouters 1998), which can be consulted for further details and other applications.

The outline of this chapter is as follows. First the exact transport equation for the velocity-scalar PDF is introduced and the closure problem is discussed. Next the Monte Carlo solution method that is used to model and solve the PDF transport equation is described. Modellingof the unclosed terms describing acceleration in the turbulent flow are treated in some detail. The closure of the micromixingterms can be done alongparallel lines as is the case in the scalar PDF method discussed in [20] and is not elaborated here. A description of some methods for handlingcomplex chemistry is included.

In the final section results of test calculations are presented for a challengingtest case, combininga complex flow pattern with effects of high mixing rates and chemical kinetics. For this bluff-body-stabilized diffusion flame, the relative importance of modelingof the velocity, mixingand chemistry terms is studied.

PDF transport equation

From the full conservation equations given in [10], the transport equation for the joint velocity-scalar PDF fUϕ(V ,ψ; x, t) can be derived (Pope 1985).

By integrating this quantity over a range of values of V and ψ the probability that U and ϕ take values in these ranges is obtained. It is also useful to consider the joint velocity-scalar mass density function (MDF) defined as FUϕ(V,ψ, x; t) = ρ (ψ)fUϕ(V, ψ; x, t). The transport equation for the MDF reads in which the first two terms (1.1a) and (1.1b) on the right-hand side describe the evolution in velocity space and the last two terms (1.1c) and (1.1d) describe the evolution in scalar space. (The notation 〈A|B〉 denotes the conditional expectation value of A upon condition B). Terms (1.1a) and (1.1c) occur in closed form whereas the unclosed terms (1.1b) and (1.1d) contain conditional averages because these effects cannot be expressed in terms of the one-point distribution of U and ϕ.

In the first two chapters of this book we described many properties of the Euler and the Navier–Stokes equations, including some exact solutions. A natural question to ask is the following: Given a general smooth initial velocity field v(x, 0), does there exist a solution to either the Euler or the Navier–Stokes equation on some time interval [0, T)? Can the solution be continued for all time? Is it unique? If the solution has a finite-time singularity, so that it cannot be continued smoothly past some critical time, in what way does the solution becomes singular? This chapter and Chap. 4 introduce two different methods for proving existence and uniqueness theory for smooth solutions to the Euler and the Navier–Stokes equations. In this chapter we introduce classical energy methods to study both the Euler and the Navier–Stokes equations. The starting point for these methods is the physical fact that the kinetic energy of a solution of the homogeneous Navier–Stokes equations decreases in time in the absence of external forcing. The next chapter introduces a particle method for proving existence and uniqueness of solutions to the inviscid Euler equation. As is true for all partial differential evolution equations, the challenge in proving that the evolution is well posed lies in understanding the effect of the unbounded spatial differential operators. The particle method exploits the fact that, without viscosity, the vorticity is transported (and stretched in three dimensions) along particle paths.

In Chaps. 10 and 11 we introduced the notion of an approximate-solution sequence to the 2D Euler equations. The theory of such sequences is important in understanding the kinds of small-scale structures that form in the zero-viscosity limiting process and also for modeling the complex phenomena associated with jets and wakes. One important result of Chap. 11 was the use of the techniques developed in this book to prove the existence of solutions to the 2D Euler equation with vortex-sheet initial data when the vorticity has a fixed sign.

To understand the kinds of phenomena that can occur when vorticity has mixed sign and is in three dimensions, we address three important topics in this chapter. First, we analyze more closely the case of concentration by devising an effective way to measure the set on which concentration takes place. In Chap. 11 we showed that a kind of “concentration–cancellation” occurs for solution sequences that approximate a vortex sheet when the vorticity has distinguished sign. This cancellation property yielded the now-famous existence result for vortex sheets of distinguished sign (see Section 11.4). In this chapter we show that for steady approximate-solution sequences with L1 vorticity control, concentration–cancellation occurs even in the case of mixed-sign vorticity (DiPerna and Majda, 1988).

We go on to discuss what kinds of phenomena can occur when L1 vorticity control is not known. This topic is especially relevant to the case of 3D Euler solutions in which no a priori estimate for L1 vorticity control is known.

In the first half of this book we studied smooth flows in which the velocity field is a pointwise solution to the Euler or the Navier–Stokes equations. As we saw in the introductions to Chaps. 8 and 9, some of the most interesting questions in modern hydrodynamics concern phenomena that can be characterized only by nonsmooth flows that are inherently only weak solutions to the Euler equation. In Chap. 8 we introducted the vortex patch, a 2D solution of a weak form of the Euler equation, in which the vorticity has a jump discontinuity across a boundary. Despite this apparent singularity, we showed that the problem of vortex-patch evolution is well posed and, moreover, that such a patch will retain a smooth boundary if it is initially smooth.

In Chap. 9 we introduced an even weaker class of solutions to the Euler equation, that of a vortex sheet. Vortex sheets occur when the velocity field forms a jump discontinuity across a smooth boundary. Unlike its cousin, the vortex patch, the vortex sheet is known to be so unstable that it is in fact an ill-posed problem. We saw this expicitly in the derivation of the Kelvin–Helmholtz instability for a flat sheet in Section 9.3. This instability is responsible for the complex structure observed in mixing layers, jets, and wakes. We showed that an analytic sheet solves self-deforming curve equation (9.11), called the Birkhoff–Rott equation. However, because even analytic sheets quickly develop singularities (as shown in, e.g., Fig. 9.3), analytic initial data are much too restrictive for practical application.

Vorticity is perhaps the most important facet of turbulent fluid flows. This book is intended to be a comprehensive introduction to the mathematical theory of vorticity and incompressible flow ranging from elementary introductory material to current research topics. Although the contents center on mathematical theory, many parts of the book showcase a modern applied mathematics interaction among rigorous mathematical theory, numerical, asymptotic, and qualitative simplified modeling, and physical phenomena. The interested reader can see many examples of this symbiotic interaction throughout the book, especially in Chaps. 4–9 and 13. The authors hope that this point of view will be interesting to mathematicians as well as other scientists and engineers with interest in the mathematical theory of incompressible flows.

The first seven chapters comprise material for an introductory graduate course on vorticity and incompressible flow. Chapters 1 and 2 contain elementary material on incompressible flow, emphasizing the role of vorticity and vortex dynamics together with a review of concepts from partial differential equations that are useful elsewhere in the book. These formulations of the equations of motion for incompressible flow are utilized in Chaps. 3 and 4 to study the existence of solutions, accumulation of vorticity, and convergence of numerical approximations through a variety of flexible mathematical techniques. Chapter 5 involves the interplay between mathematical theory and numerical or quantitative modeling in the search for singular solutions to the Euler equations. In Chap. 6, the authors discuss vortex methods as numerical procedures for incompressible flows; here some of the exact solutions from Chaps. 1 and 2 are utilized as simplified models to study numerical methods and their performance on unambiguous test problems.

So far we have discussed classical smooth solutions to the Euler and the Navier–Stokes equations. In the first two chapters we discussed elementary properties of the equations and exact solutions, including some intuition for the difference between 2D and 3D and the role of vorticity. In Chaps. 3 and 4 we established the global existence of smooth solutions from smooth initial data in two dimensions (e.g., Corollary 3.3) and global existence in three dimensions, provided that the maximum of the vorticity is controlled (see, e.g., Theorem 3.6 for details). However, many physical problems possess localized, highly unstable structures whose complete dynamics cannot be described by a simple smooth model.

The remaining chapters of this book deal with mathematical issues related to non-smooth solutions of the Euler equations. This chapter addresses a type of weak solution appropriate for modeling an isolated region of intense vorticity, such as what one might use to model the evolution of a hurricane. In particular, we consider problems that have vorticity that is effectively discontinuous, exhibiting a strong eddylike motion in one region while being essentially irrotational in an adjacent region. To treat this problem mathematically, we must derive a formulation of the Euler equation that makes sense when the vorticity is discontinuous but bounded. We also assume that vorticity can be decomposed by means of a radial-energy decomposition (Definition 3.1) and in particular that it has a globally finite integral.