The theoretical foundations and potential for large-eddy simulation (LES) in fluvial geomorphic and sedimentological research

Earth-Science Reviews 71 (2005) 271 – 304 www.elsevier.com/locate/earscirev

The theoretical foundations and potential for large-eddy simulation (LES) in fluvial geomorphic and sedimentological research C.J. Keylocka,T, R.J. Hardyb, D.R. Parsonsc, R.I. Fergusonb, S.N. Laneb, K.S. Richardsd a

Earth and Biosphere Institute and School of Geography, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK b Department of Geography, Science Laboratories, South Road, Durham DH1 3LE, UK c Earth and Biosphere Institute and School of Earth and Environment, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK d Department of Geography, Downing Place, Cambridge CB2 3EN, UK Received 20 February 2004; accepted 22 March 2005

Abstract Large-eddy simulation (LES) is a method for resolving the time-dependent structure of high Reynolds number, turbulent flows. With LES it is possible to model and track the behaviour of coherent turbulent structures and study their effect on the flow field. Hence, LES is potentially an important research tool in the fluvial sciences where flow mixing, sediment entrainment and sediment transport are all affected by the presence of coherent vortices and their interactions with channel boundaries and other flow structures. This paper introduces the LES methodology, discusses a variety of ways for representing small-scale processes within LES (the subgrid-scale modelling problem), and provides some examples of early work into the use of LES in a fluvial context. A number of advances in computational power and numerical methods are required before LES can be effectively applied at the river reach scale. This paper considers some recent developments and their potential for providing validated large-eddy simulations of river flow at the channel scale. D 2005 Elsevier B.V. All rights reserved. Keywords: fluvial environment; numerical methods; turbulence; geomorphology; sedimentology

1. Introduction In an everyday sense rivers are near-ubiquitous features of landscapes, generally acting as valuable resources and ecological habitats but occasionally as deadly hazards. Their ubiquity and practical importance has led to a long history of attempts to quantify, T Corresponding author. E-mail address: [email protected] (C.J. Keylock). 0012-8252/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.earscirev.2005.03.001

model, and predict flow in rivers. This is not a simple problem. Geophysically, rivers are unsteady nonuniform flows with high Reynolds numbers over rough boundaries with complicated three-dimensional geometry. The complexity is ultimately because river channels are self-formed, with a two-way interaction between channel configuration and river processes. Flood flows drive entrainment and transport of bed and bank sediment. Any divergence in sediment flux causes the channel geometry to change through local

272

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

erosion or deposition, thus altering the boundary conditions for future flows. The bed roughness depends on the calibre and packing of the bed sediment, and on the nature and dimensions of any bedforms; these too are subject to adjustment by the flow which in the short term is affected by them. The outcome of these interactions, over a sequence of floods of varying magnitude and duration, is commonly a channel that varies in width and depth over short distances, is rougher in some places than others, is meandering rather than straight, and may be locally divided by bars. Vortical structures of varying types and scales may be present, including Kelvin–Helmholtz vortex streets induced by lateral or vertical velocity shear, helicity about a streamwise axis because of plan curvature or bed-bank corner effects, and vertical-axis recirculation induced by flow separation at channel junctions or in sharp bends. These complexities affect the geomorphological behaviour of the river system and the sedimentary structures that result. Historically, fieldwork and laboratory experimentation have been the most commonly employed methods to study river systems. The mapping of sedimentary structures in fluvial environments has long been supplemented by descriptions of emplacement, arising from a careful consideration of field context (e.g. Harms et al., 1963; Smith, 1971; Miall, 1976), while the pioneering work of Gilbert (1914) demonstrated the role that experimentation can play in enhancing process understanding. Many studies since have used the laboratory to enhance understanding of the depositional history of observed stratigraphic sequences (Middleton, 1965) or from a more geomorphologic perspective, to gain an understanding of the controls upon erosion and deposition (Friedkin, 1945; Mosley, 1976). A background to much of the relevant work in sedimentology can be found in Friedman and Sanders (1978). More recently, there has been a move to supplement these classic methodologies using process-based modelling. Thus, for example, process-based models of sediment entrainment, movement and deposition (Wiberg and Smith, 1985; Bridge, 1992; Schmeeckle and Nelson, 2003) have the potential to quantify the process rates required in qualitative models of deposition in braided gravel-bed rivers (Bridge, 1993) and help interpret observed fluvio-sedimento-

logical architectures. This has resulted in research such as that by Lane et al. (1999), who combined photogrammetric mapping of volumetric change in a braided river system (Lane et al., 1994a,b) with numerical model predictions of boundary shear stresses and hence, propensity for sediment movement. Modelling is particularly useful in these contexts because intensive programmes of fieldwork do not give data for a variety of flow conditions, or are difficult to interpret due to changes in discharge during the measurement period. In addition, the difficulty in working in rivers at high flows means that this approach can rarely provide the required information at those times of greatest concern. Thus, an alternative is to couple fieldwork to computational modelling of the flow conditions, where the former provide boundary conditions and a validation dataset for the latter (Ferguson et al., 2003). The use of flow modelling in fluvial geomorphology and hydrology, as well as fish habitat research, is a rapidly growing area. Hydraulic modelling of rivers for practical purposes has generally used one-dimensional (1D; i.e. width- and depth-averaged) simplifications of the fundamental equations (Horritt, 2002), but 2D (depth-averaged) models are increasingly used (Bates and Anderson, 1993; Hardy et al., 1999, 2000) and there has been much recent research on the application of 3D computational fluid dynamics (CFD) codes to natural rivers (Hodskinson and Ferguson, 1998; Booker et al., 2001; Lane et al., 2002, 2004). Lane and Ferguson (2005) review this whole subject area, including the coupling of sediment transport models to flow dynamics models. However, the lack of explicit representation of turbulence in these approaches is coming to be seen as a limitation for some purposes. As already noted, large vortical structures are present in many rivers. If these are central to the application (as for example if the interest is in dispersion and mixing), a timeaverage treatment is unlikely to capture their effects adequately. Lack of explicit consideration of turbulence is also regarded by some authors as a limitation in standard procedures for computing the entrainment and transport of bed material by a given flow. The driving variable generally used is the shear stress on the bed, computed in turn from water depth and slope (in 1D models) or the square of the time-average velocity (in 2D or 3D models). A number of studies

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

have shown that entrainment actually fluctuates at turbulence timescales, with maximum entrainment of fine material during bursts and of coarse material during sweeps. It can be argued that this is allowed for in the calibration of the threshold time-average shear stress for entrainment, but if the interest is in the details of the process, or in the bedforms produced by it, an explicit treatment of the turbulence would evidently be preferable. This review examines the potential for using largeeddy simulation (LES) as a fluvial modelling tool. Applications of LES have increased greatly in the last 10 years due to improved computational power and theoretical refinements to the method itself. We argue that while LES will not be necessary or appropriate for all problems, there are situations where its ability to provide information on the time-dependent flow structure may be of importance for gaining an enhanced understanding of fluvial processes. At present, computational demands restrict LES applications to the modelling of small-scale processes or laboratory flows. However, recent theoretical developments in LES modelling as well as improved numerical methods for representing complex boundary conditions mean that future applications of LES at much larger scales may be envisaged. The types of problems where LES may be of greatest relevance are for those where it is necessary to treat flow structures explicitly. For example, in a variety of sizes of bed material, Heathershaw and Thorne (1985), Best (1992), Nelson et al. (1995) and Mazumder (2000) have all noted the importance of turbulent flow structures (Lesieur et al., 2003) for sediment movement. Hence, detailed bedload entrainment studies may find LES to be a very useful tool. In addition, the dynamics of these eddies play an important role in flow mixing processes in rivers (Best and Roy, 1991; Gaudet and Roy, 1995) and these are of concern to researchers interested in the dispersion of fines and other pollutants. For these important research topics, more typical fluid dynamics modelling methods, which generally provide the timeaveraged flow structure, will not be able to resolve the velocity and shear stress fluctuations that can be so important for both mixing and sediment movement. However, for problems where a bulk quantity such as discharge is the dominant variable (such as flood risk assessment for example) the computational cost of

273

LES will more than offset any benefit that is gained from its use. For other problems, such as the prediction of erosion and deposition over a multitude of reaches, current approaches based on time-averaged modelling of the flow field might be enhanced in the future by using detailed LES of representative morphological environments (diffluences, confluences, pools and riffles) to characterise the distribution of instantaneous velocities. Hence, in those locations where the average stresses are below a sediment entrainment threshold, LES might give improved estimates for threshold exceedance and hence, erosion potential. In this paper we begin by introducing various flow modelling approaches. We then describe the LES equations and move on to examine subgrid-scale modelling methods in LES. The numerical implementation of LES, including the use of boundary conditions is described before an assessment of the various methods is attempted. The validation of LES results is then discussed before some applications of relevance to fluvial and sedimentological research are described. Finally, we consider the important practical issue of scaling up LES simulations to tackle practical problems, before briefly concluding.

2. Modelling approaches The classic approach in three-dimensional modelling is to close the Navier–Stokes equations by Reynolds decomposition of the velocity components into mean and fluctuating components (Reynolds, 1895), and then to employ a Boussinesq approximation (Boussinesq, 1877) to link the resulting Reynolds stresses to properties of the time-averaged flow. This is known as the Reynolds-averaged Navier–Stokes (RANS) approach. The mixing-length model (Prandtl, 1925) is one such closure scheme where the characteristic length and timescales of the turbulence are prescribed a priori. The so-called k–e model (Harlow and Nakayama, 1968; Jones and Launder, 1972; Launder and Spalding, 1974) is the earliest and still the most popular way of determining these length and timescales from properties of the flow. In this approach, transport equations are written for turbulent kinetic energy (k) and viscous energy dissipation (e). The values for the constants in these equations may be

274

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

determined by experiment or by using renormalization methods (Yakhot and Orszag, 1986; Smith and Woodruff, 1998). Since the 1970s, additional closure schemes have been developed as the limitations of applying the standard k–e model to complex flows near boundaries have become better understood. Such methods include: non-linear versions of the k–e model that permit inequalities in the values for the normal Reynolds stresses (Yoshizawa, 1984; Speziale, 1987); and second-order schemes that no longer assume there is a clear cut separation of scales at the level of the Reynolds stress tensor. These latter methods can, consequently, deal with history and nonlocal effects (Launder et al., 1975; Speziale et al., 1991). In the geophysical literature a particularly popular second moment approach is that of Mellor and Yamada (1982), which is incorporated into the Princeton Ocean Model (Ezer and Mellor, 1997). This is based on the energy redistribution hypothesis of Rotta (1951) and the small-scale isotropic dissipation hypothesis of Kolmogorov (1941) but extended to the case of stratified flows. Lien and Leschziner (1994) evaluated the performance of a variety of turbulence closure schemes for the flow over a backward-facing step where the flow complexities include the presence of a wall, disruption to the boundary layer, the development of a shear layer and recirculation of flow behind the step. When compared to experimental data, the more sophisticated closure schemes were better able to capture the basic flow features of the flow such as the distance along the wall from flow separation to reattachment. However, the approaches mentioned above focus their attention upon an accurate representation of the mean flow field. Thus, while flow quantities may be evaluated spatially, fluctuations in velocity and hence, coherent flow structures (Smith, 1996; Haller and Yuan, 2000; Farge et al., 2003) cannot be resolved explicitly. Resolving such structures is important for a variety of fluvial problems. Mixing in turbulent flows is a result of molecular diffusion and advection. The latter stretches and folds the surface of a scalar quantity (such as a pollutant), increasing the surface area and hence, the rate of diffusion. The folding takes place on various scales, which results in a complex and fractal structure to the interface the nature of which affects the type of mixing law (Vassilicos,

2002). Steady-state or unsteady RANS representations of the flow field will not enable such phenomena to be modelled in a way that provides an insight into process mechanisms. Bedform development from an initially planar bed is a problem that has received a great deal of attention in process sedimentology (Engelund and Fredsoe, 1982; McLean, 1990). While mathematical investigations have considered initiation from the perspective of stability analysis (Richards, 1980), process-based investigations have focussed on the role played by hairpin vortices and sweep events that extend the mathematical approaches to cases where the bedform is sufficiently large to modify the flow (McLean et al., 1994, 1999; Bennett and Best, 1995; Williams et al., 2003a). Flow structures are also important for sediment transport more generally. Heathershaw and Thorne (1985) and Nelson et al. (1995) noted the importance of outward interactions for sediment movement at the bed and hence, the difficulty of using Reynolds stresses to predict bedload transport. This suggests a more explicit resolution of the flow near the bed is preferable to an approach based on Reynolds stress closure. In addition, studies have shown that turbulent structures generated by dunes transport a significant amount of suspended sediment (Kostachuk, 2000), resolution of which requires some capacity to represent turbulence structure explicitly. The complex flow conditions in environments with significant bedforms alter correlations between mean and fluctuating flow, invalidating approaches based on this method for estimating instantaneous forces for predicting sediment movement (Naden, 1987). Either field data or more sophisticated, high-resolution modelling studies are needed to derive these relations. In principle, one method of dealing with flow fluctuations is to solve the Navier–Stokes equations using direct numerical simulation or DNS (Moin and Mahesh, 1998). However, with current technology, it is only possible to apply direct numerical simulation to a flow with a low ratio of inertial to viscous forces (low Reynolds number). This is because as the Reynolds number (Re) increases, the range of eddy sizes with significant energy dissipation also increases. The number of mesh points required to resolve a flow to a particular level of accuracy in three dimensions is proportional to Re 9/4 (Hirt, 1969). Thus, to simulate a river flow at Re = 10,000, which would

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

arise for an average depth and velocity of just 0.1 m and 0.1 m s
1, respectively, would require of the order of 109 mesh points. Consequently, alternative methods are needed for modelling flows that are of interest to geomorphologists. An additional benefit that comes from modelling the Navier–Stokes equations rather than simulating them directly is that, by exploring various modelling strategies, it may be possible to gain insights into the importance processes controlling turbulence production, transport and dissipation in particular environments. The simplest way to incorporate some information about the time-dependent flow field is by using unsteady RANS methods. An ensemble-averaging of the Navier–Stokes equations is used, with the turbulent fluctuations represented by the Reynolds stresses. However, large-scale motions (variations in the mean flow field) are modelled by retaining a time derivative for the mean flow. This approach can be extremely useful when interest is focussed upon eddyshedding at scales greater than the integral timescale and when computational limitations restrict the utility of LES, which requires integration of the flow field over a longer period to obtain statistically robust average solutions (Iaccarino and Durbin, 2000). Large-eddy simulation (LES) has an advantage as it resolves the turbulence above a particular filter scale, rather than resolving variations greater than the integral timescale as occurs in unsteady RANS. This can yield accurate results in situations where turbulent structures of importance to the modeller are generated at a variety of scales (that are perhaps not known a priori). LES has been developed extensively in the last 15 years in the turbulence and fluid engineering communities and can be viewed as an intermediate case between DNS and RANS approaches. While DNS deals with all eddies larger than the smallest (dissipation) scale and steady RANS methods deal with the mean flow characteristics, LES calculates the properties of all eddies larger than the filter size and models those smaller than this scale by a subgrid-scale (SGS) turbulence transport model. Other recently developed approaches (Wang and Moin, 2001; Davidson and Peng, 2001) are essentially a hybrid of LES and RANS, such as detached eddy simulation or DES (Spalart et al., 1997; Nikitin et al., 2000; Schmidt and Thiele, 2002).

275

The purpose of this paper is to provide a description of the LES approach and consider its applicability and potential in fluvial science. Although there are a number of published introductions to LES (Lesieur and Me´tais, 1996; Lesieur et al., 1997; Rodi et al., 1997; Sagaut, 1998; Frfhlich and Rodi, 2002; Moin, 2002), none of these is written from the perspective of the earth sciences. The environmental applications of LES commenced with the work of Deardorff (1970a,b) and the technique has been used a great deal in the atmospheric literature. In both the fluvial and atmospheric cases there is a need to consider the effect that the solid boundaries have on the flow and how these effects can be represented numerically. However, in general, this is perhaps more important in the fluvial context due to the presence of channel sidewalls and the greater relative roughness of the lower surface compared to the depth of the flow. For example, Schmeeckle and Nelson (2003) noted that in the atmospheric literature, it has been possible to scale the fluctuating velocities on the shear velocity (Stull, 1988) as a function of relative height within a boundary layer. However, work by McLean et al. (1994) and Nelson et al. (1995) indicates that such a scaling is less appropriate in fluvial flows where the roughness length cannot be considered constant. The next sections of this paper provide a theoretical introduction to the LES equations and the subgrid-scale modelling techniques used to represent the small-scale behaviour of the flow.

3. The Navier–Stokes and LES equations In this study, the LES equations are derived by applying a filter to the Navier–Stokes equations. This is the most common way in which to proceed, but there are alternatives, such as vorticity-based methods, where the filter is applied to the vorticity transport equation instead (Mansfield et al., 1998; Gharakhani, 2003). The continuity equation for an incompressible fluid, which is a mathematical expression of the principle of mass conservation, is given for a threedimensional flow by: Bu Bv Bw þ þ ¼0 Bx By Bz

ð1Þ

276

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

where u, v and w are the velocities in the orthogonal directions x, y and z. More typically, Eq. (1) is written using Einstein summation notation as: Bui ¼0 Bxi

ð2Þ

where u and x represent generalised velocities and directions, which are indicated more precisely by the value for i, where u 1, u 2 and u 3 would be equal to u, v and w in Eq. (1). The appropriate equation for the conservation of momentum is given in non-conservative form by:
  B ui uj Bui Bð
p þ sii Þ Bsji Bski q þ þ þ ¼ Bxi Bt Bxj Bxk Buj ð3Þ where q is the fluid density, p is the pressure, and s is the viscous stress. In a Newtonian fluid, the viscous stresses are proportional to the rates of deformation and the constant of proportionality is the dynamic viscosity (l). Thus, it is possible to substitute deformation terms for stress terms. When dealing with an incompressible fluid, the volumetric deformation is zero by definition and can be neglected. Substituting expressions for deformation into the stress terms yields: sii ¼ 2l

Bui Bxi

ð4aÞ 

sij ¼ sji ¼ l

Buj Bui þ Bxj Bxi

 ð4bÞ

Substituting Eqs. (4a) and (4b) into Eq. (3) and expressing the viscosities in their kinematic form (m = l/q) gives:
    B ui uj Buj Bui 1 Bp B Bui þ ¼
þ m þ : q Bxi Bxj Bxj Bt Bxj Bxi ð5Þ 3.1. Using filters to derive the LES equations Whilst RANS approaches to modelling the Navier– Stokes equations decompose the velocity in to mean and fluctuating components, the typical re-organisation in LES is based upon a length scale for a filter (D), often taken to be equal to the grid size employed

in the solution of the equations. (For the purpose of illustration and to simplify the derivation we assume here that the grid is cubic.) More formally, the largescale or directly resolvable scale of a quantity h is given in one dimension by convolution with a filter function G D of width D: Z

 ¯h ð xÞ ¼ GD x
xV h xV dxV ð6Þ where the quantity h has been decomposed into gridscale (h¯ ) and subgrid-scale (hV) components: h ¼ h¯ þ hV

ð7Þ

Leonard (1974) defined some appropriate filters. Of these, the box (top-hat) filter is the classic choice (Deardorff, 1970a; Clark et al., 1979; Silveira Neto et al., 1993), although Gaussian filters are also appropriate (Vreman et al., 1994). In three dimensions the box filter may be defined as: (  

 ð1=DÞ3 ; xi
xVi V 12 D V GDx x
x ¼ ð8Þ
 0; jxi
xVi jN 12 D Applying such a filter to the continuity and Navier– Stokes equations gives the filtered Navier–Stokes equations for continuity and momentum conservation: B¯u i ¼0 Bxi
 B u¯ i u¯ j B¯u i 1 B¯p B þ ¼
þ q Bxi Bxj Bxj Bt     B¯u j B¯u i  m þ
sSG ij Bxj Bxi

ð9Þ

ð10Þ

where the term s ijSG is a tensor containing the subgridscale (SGS) fluctuations in the flow field: PP ¯ u¯ sSG i j ij ¼ ui uj
u

ð11Þ

In addition to the box and Gaussian filters, spectral cutoff filters have also been used in LES (Ha¨rtel and Kleiser, 1998). According to Ha¨rtel (1996) and Ha¨rtel and Kleiser (1997) the choice of filter makes little difference to the solution. However, Vreman et al. (1994) defined a set of realizability conditions for LES, in the spirit of work by Schumann (1977), and found that the spectral filter does not guarantee that

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

the subgrid-scale shear stress tensor, s ijSG, is positive. This is important because if s ijSG is positive, the generalised turbulent kinetic energy of a LES (Germano, 1992) is constrained to be positive everywhere. This is often a necessary requirement of a subgridscale model. For comparison with Eq. (10), the RANS momentum equation is given by:     BUi 1 BP B BUi BUj Uj ¼
þ m þ
qhu˜ i u˜ j i q Bxj Bxj Bxj Bxj Bxi ð12Þ where the Reynolds decomposition into the timeaveraged and fluctuating components for an instantaneous velocity u in the ith direction is given by: ui ¼ Ui þ u˜ i

ð13Þ

and the angled braces (as well as capitalisation) indicate temporal averaging. A comparison between Eqs. (10) and (12) reveals qualitative similarities between the Reynolds-averaged equations and the LES-filtered equations. The important differences are that the LES equations retain a time derivative (as opposed to Eq. (12), which has no such derivative, or the unsteady RANS approach, where only a derivative of the ensemble-averaged field is retained) and that the additional stress term in the LES equations (s ijSG) contains more components than the Reynolds stresses seen on the far right-hand side of Eq. (12), as shown in Eq. (11). The former difference explains why LES can be employed to give time-transient solutions. The latter can be more clearly demonstrated by expanding Eq. (11) to reveal additional terms, which include a term analogous to the Reynolds stress:
PPP  PPP PPP  PPP ¯ i u¯ j
u¯ i u¯ j þ u¯ i uVj þ u¯ j uVi þ uVi uVj sSG ð14Þ ij ¼ u Following Clark et al. (1979), the three groups of terms on the right-hand side of Eq. (14) are known as the Leonard term, the cross term and the SGS Reynolds stress, respectively. All the terms that appear in the Leonard term are defined at the grid scale and so this term may be evaluated explicitly. It is necessary to model the other terms. In practice the decomposition of the SGS stress tensor (Eq. (14)) is rarely performed and the tensor is

277

modelled as a whole instead. An important exception is the work of Clark et al. (1979), who adopted the Smagorinsky approach for the SGS Reynolds stress and used a tensor diffusivity to model the sum of the Leonard and cross terms. Another is the recent advocation of explicit filtering methods for the LES equations (Vasilyev et al., 1998; Ghosal, 1999; Gullbrand, 2001; Marsden et al., 2002). The conventional convolution filter used in Eq. (6) is coupled to the mesh so that as the mesh size tends to zero, the results tend to a DNS. With explicit filtering, the mesh and the filter are decoupled, so that mesh refinement leads to a solution that is independent of the mesh, but dependent upon the chosen scale of the filter. The spatial filters used in explicit filtering are closely related to the wavelet transform, which has seen numerous applications in the earth and environmental sciences (e.g. Kumar and Foufoula-Georgiou, 1994; Smith et al., 1998; Torrence and Compo, 1998) and through the lifting transform (Sweldens, 1997) has been used recently to develop an alternative approach to LES (Goldstein and Vasilyev, 2004).

4. Subgrid-scale modelling The standard terminology in LES discusses processes in terms of the resolved grid scale processes and the subgrid processes that require modelling in some way. However, the key distinction is really between the filter scale and subfilter scale processes. More recent developments in LES have meant that the grid and filter scales are less likely to be identical. However, in this study to simplify exposition, we assume that the grid and filter scales are identical unless it is explicitly stated otherwise. Subgrid models act to provide an appropriate exchange of energy between the grid and subgrid scales. On average, the direction of this energy transfer is from the larger to the smaller eddies (the energy cascade). However, this energy transfer is not continuous and the situation can arise where there is an inverse transfer of energy between scales. This inverse energy cascade or dbackscatterT effect is particularly important in areas where eddies impinge upon solid surfaces (the wall region) as well as in shear layers. Furthermore, Piomelli et al. (1990) have shown the importance of this effect for the growth of

278

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

perturbations during the transition to turbulence in wall-bounded flows. An ideal SGS model should be able to capture both the forward and inverse energy transfers. This has proven to be a major stumbling block for the development of SGS models that are effective in all situations. Hence, SGS modelling has been an important topic for research and we devote some space in this article to this important topic. For the purposes of detailed scientific study involving simple flow configurations (e.g. Silveira Neto et al., 1991) it is possible to use highresolution LES, meaning that a greater proportion of the flow is resolved explicitly. This becomes too computationally expensive for either applications or for flows within highly anisotropic domains, and a coarser grid becomes necessary. It is also the case that as the Reynolds number of the flow increases, an increased proportion of the energy of the flow is contained in relatively smaller and smaller scales. The consequence of this is that for a given grid size, as Reynolds number increases so does the importance of the SGS model. As applications of LES develop in the earth and environmental sciences, thought will have to be given to whether, for a given problem, refinement of the mesh or the introduction of a more sophisticated SGS model is the best way forward. From a theoretical perspective, improving SGS modelling is essential. For environmental applications upon coarse meshes, more sophisticated SGS models are likely to be highly beneficial compared to the simple models, which are commonly found to be too diffusive. However, at present the simple models are still perhaps the most commonly used, with the Smagorinsky model often the classic choice. 4.1. The Smagorinsky SGS model Work by Deardorff (1970a,b) represented an early application of LES to turbulent channel flow and the planetary boundary layer. The SGS model adopted by Deardorff was that of Smagorinsky (1963) and this model has proved to be particularly popular in a range of literatures (Bedford and Babajimopoulos, 1980; Moin and Kim, 1982; Cai and Steyn, 1996). This is an eddy-viscosity model and is absolutely dissipative: it can only account for energy transfer from large to small eddies.

Following Boussinesq (1877), the Reynolds stresses can be linked to mean rates of deformation via an eddy-viscosity parameter. For a two-dimensional flow, this is:   BU1
hu˜ 1 u˜ 3 i ¼ mt ð15Þ Bx3 The mixing-length model of Prandtl (1925) was developed for two-dimensional turbulence and assumes that the eddy viscosity is proportional to a length scale and a velocity scale. That is, as a fluid particle moves from a region of lower to higher velocity over a distance ˜l , the velocity difference between the particle and the local mean flow is: BU1 u˜ 1 ¼ U1
l˜ Bx3

ð16Þ

Therefore, the equation for the Reynolds stress may be written as: BU1
hu˜ 1 u˜ 3 i ¼
hu˜ 3 l˜i Bx3

ð17Þ

Assuming from the continuity of mass that on average
u˜ 3cu˜ 1, and defining a root-mean-square length (the mixing length) as l m2=hl˜m2i then (17) may be written as: 2
hu˜ 1 u˜ 3 i ¼ lm

jBU1 j BU1 jBx3 j Bx3

ð18Þ

From Eqs. (15) and (18), the eddy viscosity can be linked to the mixing length as: 2 m ¼ lm

jBU1 j : jBx3 j

ð19Þ

This relatively simple relation, when generalised into three dimensions, defines the Smagorinsky SGS stress model. In this case, the mixing length is taken to be equal to the characteristic grid size D¯, or filter width, which is usually defined as:
1 D¯ ¼ Dx  Dy  Dz 3 ð20Þ where the terms in the bracket represent the length of the filter in different directions. In fact, Ferziger (1993) noted that, theoretically, a better length scale would combine the filter size with the integral length scale of the turbulence (the distance over which velocities are appreciably correlated). However, the

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

latter is extremely difficult to determine and, consequently, this alternative formulation is very rarely used in practice. Returning to the SGS stress tensor s ijSG (Eq. (14)) and making an eddy-viscosity assumption gives: 1 SG ¯ sSG ij
skk ¼
2ms S ij 3

ð21Þ

where m s is the SGS eddy viscosity and S¯ ij is the deformation rate tensor for the filtered flow field:   B¯u j 1 B¯u i þ S¯ ij ¼ ð22Þ 2 Bxj Bxi Thus, the first term on the right-hand side of Eq. (21) is the SGS version of Eq. (4b). The mixing-length approach is used to evaluate m s in an equation that is similar in form to Eq. (19):
2 ms ¼ Cs D¯ jS¯ j ð23Þ where C s is the Smagorinsky constant and |S¯ | is given by:
1 jS¯ j ¼ 2S¯ ij S¯ ij 2 : ð24Þ The value for C s can be obtained theoretically by linking it to the Kolmogorov constant C k (the constant in the Kolmogorov energy cascade of turbulence from large to small scales). Schumann (1991) obtained 0.17 using this type of approach, although Deardorff (1970a) and Moin and Kim (1982) have found that a lower value (0.1) is required for turbulent channel flow. In addition, Rogallo and Moin (1984) emphasise that C s is a dynamic quantity (a function of time and space) with a value that can vary between 0.07 and 0.24. Canuto and Cheng (1997) noted that the value for C s will vary with the type of flow considered and explained how under conditions of plane and homogeneous shear different processes contribute to the value of C s. Employing the Smagorinsky model to evaluate the SGS stress tensor (Eq. (11)) in the LES momentum equation (Eq. (10)) yields:
 B u¯ i u¯ j B¯u i 1 B¯p B þ ¼
þ q Bxi Bxj Bxj Bt    B¯u j B¯u i  ð m þ ms Þ þ ð25Þ Bxj Bxi

279

Some time has been spent outlining the Smagorinsky SGS model because it is the most commonly used subgrid-scale treatment and is also much simpler than many of the more recent models. One particular disadvantage of this model is that the SGS viscosity is still present in regions very close to the wall where the flow is no longer turbulent. To account for this, it is common to use a damping function close to the wall (Van Driest, 1956). 4.2. Scale-similarity and spectral SGS models One of the simpler alternative SGS models is the scale-similarity model proposed by Bardina et al. (1980), which is based on the assumption that the most important unresolved eddies are those of a size just smaller than the filter size. A double filter is used in this procedure, which results in a smoother field. The scale-similarity model does not dissipate sufficient energy and is therefore often employed in combination with the Smagorinsky model. Furthermore, it may also be employed within the dynamic model framework introduced below (Horiuti, 1997). The scale-similarity model has the advantage that it is potentially possible for backscatter effects to be mimicked at the subgrid scales, an important weakness of the Smagorinsky model. Instead of operating in physical space, it is possible to define a spectral eddy viscosity. Chollet and Lesieur (1981) developed such a model using the theoretical ideas of Kraichnan (1976). The effective eddy viscosity was obtained as a function of the cutoff wave number (given by the filter size) and the wave number under consideration by assuming a spectrum with a slope of
5/3 and a sharp cutoff at the boundary between the resolved and unresolved scales. This model had a limited inclusion of the inverse energy cascade at the small wave numbers close to the cutoff (Sagaut, 1998). See Lesieur et al., 1999 for a recent review of this approach. These types of model are unlikely to be of much use in fluvial science because of the difficulty of using them for flows through complex and irregular geometries (as opposed to a more idealised wavenumber space). This led Me´tais and Lesieur (1992) to produce a modified form that was applicable in physical space—the so-called dstructure–function modelT. The method used by Chollet and Lesieur (1981) was

280

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

simplified by deriving an effective viscosity that was just a function of the cutoff wave number. The energy at the cutoff was evaluated using a second-order structure function, which can be linked to turbulent kinetic energy using an expression for the Kolmogorov energy spectrum. Silveira Neto et al. (1993) applied this model to a backward-facing step flow and found its performance was superior to the Smagorinsky model with a value for C s of 0.2, although Ha¨rtel and Kleiser (1998) found that the Smagorinsky model was superior in straight channel flows at higher Reynolds numbers. Additional models exist that are based upon this approach such as the selective structure function model (David, 1993) and the modified selective structure function model (Ackermann and Me´tais, 2001). 4.3. Dynamic SGS models The major development in SGS modelling in the last 15 years has been the introduction of dynamic methods. The major assumption of the Smagorinsky model is that subgrid-scale turbulence is isotropic and conforms to a Kolmogorov energy cascade. When studying turbulence and turbulence-generated shear (as occurs at river confluences for example) these assumptions are questionable, particularly close to solid boundaries. Furthermore, Rogallo and Moin (1984) found that values for C s vary between 0.07 and 0.24 over the flow field. Recognition of this difficulty has resulted in dynamic models, where a second dtestT filter is employed to evaluate C s as a function of time and space (Germano et al., 1991; Germano, 1992; Meneveau and Katz, 1999). Initially, the Smagorinsky method was taken as the base model to which the dynamic procedure could be applied. Subsequently, other SGS models have also been used to underpin dynamic SGS calculations. More recent work has noted some of the limitations of this method (Brun and Friedrich, 2001) and suggested improvements that are particularly relevant to environmental applications (Ghosal et al., 1995; Piomelli and Liu, 1995; Meneveau et al., 1996). The dynamic approach uses the smallest scales of the resolved turbulence to provide information about the local value for the SGS model coefficient (taken to be the Smagorinsky coefficient C s in the discussion below). If a new test filter is introduced to the

equations, then Eq. (20) can be supplemented by a test filter width:  13

ð26Þ Dˆ ¼ Dˆx  Dˆy  Dˆz where the hat operator represents filtering by the test filter and the filter size in the x and y directions is double that given in Eq. (20), with the size in the z direction unchanged. Hence, in practice, the test filter is only applied in the wall-parallel directions. The Germano identity relates the resolved turbulent stresses to the subgrid and subtest-scale (TijST) stresses: Lij ¼ TijST
sˆ SG ij

ð27Þ

where s ijSG is given by Eq. (11) and hence, sˆijSG is given by ˆ PP ˆ PP sˆ SG ij ¼ ui uj
ui uj

ð28Þ

TijST has an analogous form for the test filtered flow field: ˆ PP ˆ ˆ TijST ¼ ui uj
P ui P uj

ð29Þ

Hence ˆ PP ˆ ˆ Lij ¼ ui uj
P ui P uj

ð30Þ

Appropriate alternative expressions for s ijSG and TijST are found by substituting the appropriate relations given by or related to those in Eqs. (21)–(24): 1 SG ¯2 ¯ ¯ sSG ij
dij skk ¼
2Cd D jS jS ij 3

ð31Þ

1 ST ¼
2Cd Dˆ 2 jS¯ˆ jS¯ˆ ij TijST
dij Tkk 3

ð32Þ

where C d is the dynamic Smagorinsky constant. Hence, ˆ 1 2 2 Lij
dij Lkk ¼
2Cd Dˆ jS¯ˆ jS¯ˆ ij m 2Cd D¯ jS¯ jS¯ ij 3 ¼ C a
ˆ Cb d ij

d ij

ð33Þ

It is not possible to evaluate Eq. (33) directly because of the test filtering of C d in the last term. Thus, in the original dynamic formulation the coefficient is moved outside of the filtering operation: ˆ Cd bij ¼ Cd bˆ ij

ð34Þ

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

The value locally chosen for C d is that minimising the residual error E ij that arises from the simplification in Eq. (34), which is given by: 1 Eij ¼ Lij
dij Lkk
Cd aij þ Cd bˆ ij 3

ð35Þ

Germano et al. (1991) proposed constraining Eq. (35), which consists of five independent algebraic expressions and hence, values for the constant, by a single expression: BEij S¯ ij ¼ 0: BCd

ð36Þ

Since it was proposed there have been a number of suggested modifications to Germano’s dynamic procedure. Perhaps the most significant was that of Lilly (1992) who introduced a simplification to the procedure, where a least-squares technique is used to evaluate C d to give the local dynamic model. This method has become more popular than the original approach. For example, Vreman et al. (1997) used the Lilly method when comparing LES results to a filtered DNS of a turbulent mixing layer for a variety of SGS formulations at low Reynolds number. Although the dynamic models gave much more accurate estimates of the total kinetic energy, subgrid-scale dissipation and streamwise energy spectra than other models, such as the original Smagorinsky method, there were still errors in the prediction of turbulence intensities and the momentum thickness. Because the Germano identity is not specific to a particular SGS, various authors have used the dynamic framework with other base SGS models. For example, Zang et al. (1993) replaced the Smagorinsky model with the Bardina et al. (1980) method, while Vreman et al. (1996a, 1997) used the Clark et al. (1979) model. 4.4. Enhanced dynamic SGS modelling There are some important theoretical and practical difficulties with the original dynamic model formulation that have been addressed by more recent work. The theoretical difficulty arises from the assumption made in Eq. (34). By moving the coefficient outside of the filter, the relation between neighbouring values for the coefficient is eliminated, which leads to too much variability in the coefficient field. In addition, negative eddy viscosities are produced by the local

281

model, which although potentially useful in that they allow for the possibility of representing inverse energy transfer, tend to destabilise the numerical solution because regions of negative viscosity persist and grow, and the method does not maintain an overall kinetic energy budget to counteract this. In practice, it was possible to deal with this latter problem by bclippingQ negative values to positive (Zang et al., 1993) and/or assuming at least one homogeneous direction in the flow and averaging the coefficient over this direction to generate a more smoothly varying field for C d that was less likely to become negative (Moin et al., 1991). These ad hoc procedures were formalised more appropriately by Ghosal et al. (1995), who also dealt with the problems raised by the simplification in Eq. (34) by keeping the coefficient within the filtering operator. This meant that the simplification to a set of algebraic equations could not be made and instead, the minimisation problem was recast in terms of the resulting integral equations. Having removed this mathematical simplification, the instability arising from negative eddy viscosity was dealt with by ensuring that the eddy viscosity depends on the SGS kinetic energy, constraining the amount of inverse energy transfer that can take place. By introducing an equation for the transport of the SGS kinetic energy k sgs, the inverse cascade can be permitted while maintaining numerical stability. This was achieved by replacing Eq. (23) with 1=2

m ¼ 2Cd4 D¯ kSGS

ð37Þ

where C*d is a new value for the dynamic constant, and then introducing a transport equation for the SGS kinetic energy following the form used for kinetic energy equations in RANS modelling. This important advance in dynamic SGS modelling is known as the dynamic localization approach. It should be noted that another method of incorporating the inverse energy cascade is to add a stochastic term into the SGS modelling. Early work in this regard was undertaken by Mason and Thomson (1992), but as noted by Ghosal et al. (1995) if the backscatter is modelled as a simple noise that is not temporally autocorrelated, there is an implicit scale separation between the smallest resolved eddies and the eddies just smaller than the filter size. Stochastic representation of the inverse cascade within the

282

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

framework of dynamic localization was developed by Carati et al. (1995) by introducing an eddy force at both the grid and test filter scales, which was modelled as a zero-centred white noise process. To prevent the global minimisation becoming a stochastic problem, the errors resulting from using modelled as ˆ opposed to known values for s ijSG and TijST in Eq. (27) were averaged over the possible realisations of the noise for a particular velocity field. This removed any effect due to mean behaviour, but retained an effect upon the flow energy. The minimisation of an energy error functional gave the values for the eddy force parameterisation. Carati et al. gave an example of the computational costs that result from these more sophisticated dynamic procedures compared to a simple simulation with a specified value for the Smagorinsky constant. In terms of memory costs, the dynamic model (DM), the dynamic localization model with the eddy viscosity constrained to be positive (DLM+) and the stochastic dynamic localization model (SDLM) had an increase in memory overhead of between 25% and 29%. However, the dynamic localization model with a kinetic energy budget (DLMK) had a 70% increase in memory cost. In terms of CPU time, DM was 4% more expensive, DLM+ 16%, SDLM 20% and DLMK 67%. Hence, compared with DLMK, the stochastic procedure permitted a representation of backscatter at a much-reduced computational cost. These computational issues motivated Piomelli and Liu (1995) to develop an approximation to the integral required in the dynamic localization method based on time extrapolation. Another important development was the work by Meneveau et al. (1996) who noted that because the Smagorinsky model is of an eddyviscosity type, ensemble-averaging is needed to determine the effect of the small scales on the modelled flow. Their approach was to define this temporal averaging based on the Lagrangian trajectory of fluid packets. Averaging was defined using a timescale based on the smallest resolved eddies, but with the definition sufficiently flexible to allow a shorter time frame for averaging in regions of high strain and to permit less weight to be attached to values representing backscatter (although clipping was still required on occasions). This Lagrangian approach had similar memory requirements to DM,

with an increase in computational cost of 9%, meaning that the improved modelling relative to the standard dynamic approach was obtained at less cost than the DLM methods. In addition, simulations by Meneveau et al. (1996) of eddy structures showed that the use of Lagrangian averaging reduced the dissipation for ejection events, giving more realistic results than standard averaging. 4.5. Current developments A recent approach to improving subgrid-scale models has been to try to optimise the scales that are available in the LES. Such developments have been termed dinverse modellingT (Geurts, 1997) or dapproximate deconvolutionT (Domaradzki and Saiki, 1997; Domaradzki and Loh, 1999; Stolz and Adams, 1999; Stolz et al., 2001). For example, Domaradzki and Loh (1999) used an extrapolation procedure to develop the SGS velocities and stresses from the resolved scales of the flow. Stolz et al. (2001) used regularized deconvolution of the velocity field to estimate the unfiltered flow field. In addition, Hughes et al. (1998, 2001a,b) and Collis (2001) have shown that better LES results can be obtained if the solved equations are split into small- and large-scale representations and only the shear stresses extracted at the smaller and unresolved scales are modelled. Hughes et al. termed this the variational multiscale method and Collis (2001) and Vreman (2003) discussed some alternative approaches where large-scale stresses are modelled, but in a different manner to the small and unresolved scales. Vreman (2003) extended this technique to the filtering approach, outlined above. Again, a test filter is needed to supplement the primary filter and this test filter allows the large-scale quantities to be derived. When these values are subtracted from those obtained from the primary filter, the small scales are isolated. Vreman performed calculations for a channel flow and showed that the vertical profiles of downstream channel velocity and Reynolds stress were predicted as well using this technique as was possible with a dynamic Smagorinsky model, and better than with a standard Smagorinsky formulation. In terms of the future development of SGS models, a possible way forward has recently been suggested by Geurts and Holm (2003). This approach reasserts

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

283

the primary role of the convolution filtering operation used in Eq. (6) to derive the filtered Navier–Stokes equations (Eqs. (9) and (10)), over and above the details of the SGS parameterisation. The method applies a filter and its formal inverse, together with an appropriate regularization procedure, to give an implied subgrid-scale model. This approach has been tested against DNS data and applications to rotating shear layers and other test cases are beginning to emerge. However, from the perspective of environmental and earth science applications, where the state of the art is rarely more sophisticated than the simple Smagorinsky model (although see Porte´-Agel et al., 2000), the Lagrangian averaging method proposed by Meneveau et al. (1996) would seem to be an important development. This is mainly due to its enhanced representation of eddy structures compared to the simple Smagorinsky and dynamic models, and its relatively low computational cost compared to the dynamic localization methods.

object diameters downstream of a cylinder, with that of Beaudan and Moin (1994) yielding the better shape of the stress profile. The second-order central differencing scheme was found to be preferable if the grid resolution was enhanced and if very limited mesh distortion was introduced. Thus, mesh distortion is an important concern. A regular mesh will have greater stability but it may be computationally unfeasible to implement for a natural river channel, with complex geometry. Consideration of the appropriate mesh should also be related to the complexity of the solver. In addition, the relationship between the filter and mesh resolutions should be evaluated. Vreman et al. (1996b) compared second-order and fourth-order central differencing solvers for the modelling of a temporal mixing layer using LES. They also examined cases where the filter size was equal to the mesh size or twice as large. In the former case, the secondorder scheme performed best, while in the latter case it was the fourth-order method.

5. Implementing LES

5.2. Flow and wall boundary conditions

5.1. Solving the equations

For simulations of turbulent channel flow, a decision is required concerning the appropriate inlet velocity and physical boundary conditions. For the former, one might prescribe a mean velocity with random fluctuations at the upstream boundary (Silveira Neto et al., 1993) or define these fluctuations with a precise energy spectrum to give a periodic boundary condition (Le et al., 1997). An alternative inlet boundary condition is to recycle the flow field that is produced at the outlet of the computational domain. Such conditions can be of great utility when studying idealised representations of landforms. For example, instead of simulating the flow through a train of meander bends with the same geometry, one could recycle the boundary conditions through one bend. This permits a concomitant increase in the resolution of the modelled solution. An alternative use of such boundary conditions is to study flows that have not completely evolved within the simulated flow domain. An experimental campaign, where high-resolution measurements of the effect of a pebble cluster upon the flow field have been undertaken (Buffin-Be´langer and Roy, 1998),

Once the method for formulating the LES equations and approximating the SGS terms has been decided upon, it is necessary to determine how to solve the equations. Mittal and Moin (1997) noted that flow simulations have been undertaken successfully with dynamic SGS models when spectral solvers have been used (Piomelli, 1993). However, as has already been mentioned, spectral methods are of little use for flows in complex geometries where finite difference or finite volume methods are preferable (see Ferziger and Peric, 1999, for a detailed discussion of these numerical methods). Beaudan and Moin (1994) noted that when using a high-order upwind-biased finite difference scheme (Rai and Moin, 1991) to simulate a complex flow, numerical dissipation could outweigh the contribution from the SGS model. A second-order central difference scheme (Choi et al., 1994) gave a much better representation of the streamwise velocity spectra because there was no numerical dissipation, and hence greater energy, at smaller scales. However, both simulations underpredicted the stress levels 10

284

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

could be used to validate an initial simulation of flow over a pebble cluster. Using cyclic boundary conditions, it would then be possible to explore the effect of a train of pebble clusters on the flow (similar to the experimental studies of Hassan and Reid, 1990). Applying appropriate boundaries to the walls of the channel and to the interface between the surface of the flow and the air above (the free surface) is a further issue. The most obvious boundary condition close to the wall is a no-slip condition, but this can mean that most of the solution grid is used to resolve the flow close to the wall (Piomelli and Balaras, 2002). Alternatives include use of wall functions (Launder and Spalding, 1974) or nesting finer grids within a coarser mesh (Boersma et al., 1997). SGS models are not designed to be able to deal with the presence of near-wall coherent structures (Kline et al., 1967; Grass, 1971; Robinson, 1991; Paiement-Paradis et al., 2003) and the associated effects upon energy production and diffusion. Thus, the use of wall functions, which assume a shape for the velocity profile between the lowest computational nodes and the wall, is of great importance. Schumann (1975) attempted to supply LES with stresses developed at the wall, while Cabot and Moin (1999) and Piomelli and Balaras (2002) have reviewed a number of wall boundary approximation methods. The simplest two-layer method (LL2 in Table 1) assumes a linear velocity profile within the viscous sublayer and Table 1 Results from Temmerman et al. (2003) for the sensitivity of separation and reattachment positions to the wall approximation adopted No. of mesh cells ( 106)

Wall function

(x/h)sep

(x/h)reat

0.66 0.66 0.66 0.66 0.66 0.66 0.66 4.60

NS WW WW-p LL2 LL2-i LL3 LLK NS

1.12 0.46 0.52 0.54 0.41 0.53 0.49 0.22

2.17 4.00 3.06 2.95 3.95 2.98 3.38 4.72

The abbreviation NS stands for a no-slip boundary condition and no wall function. The WW wall functions are based on those of Werner and Wegle (1991), with the p indicating a point-wise form. The LL functions are log-law implementations of various types (the numeral indicates the number of layers, i is a cell-integrated form and LLK uses two layers and the resolved turbulent kinetic energy). All simulations were performed with the WALE SGS model.

a logarithmic velocity profile outside this to predict the wall shear stress: þ þ u1;2=3 x  if x3þ V11 uþ ¼ ¼ 13
ð38Þ þ if x3 N11 u4 j log 9:8x3 xþ 3 ¼

x 3 u4 m

ð39Þ

where u 1,2/3 is the horizontal velocity at a distance x 3 above the bed (the computational node nearest the wall), u * is the shear velocity, j is the von Ka´rma´n constant and m is the viscosity. The data taken from the simulation are either instantaneous or averaged velocities from the first computational cell that lies within the logarithmic region. The logarithmic velocity profile may be applied in the average downstream direction, or can be aligned with the actual horizontal velocity vector (Mason and Callen, 1986). A more sophisticated version of LL2 uses three layers to provide an explicit representation of the transitional regime in the boundary-layer profile (Breuer and Rodi, 1996). It is also possible to substitute the shear velocity for an expression involving turbulent energy (u * = C l1/4k 1/2; where C l = 0.09 and k is the turbulent energy at the computational node nearest the wall). This substitution provides an alternative method for calculating Eqs. (38) and (39) and although it is only strictly valid where there is turbulence energy equilibrium, it has been applied to LES (Murakami et al., 1993). In Table 1 (below), this model is referred to as LLK. This simple method is computationally efficient and Reynolds number independent. More complex methods based on solving unsteady boundary layer equations (simplified forms of the Navier–Stokes equations) are more accurate, particularly in the presence of flow separation and recirculation, but are more expensive computationally and exhibit a Reynolds number dependence for a given mesh size. Furthermore, in natural channels, where specification of the boundaries for the flow is not trivial, such methods may be difficult to justify. However, these more complex methods (e.g. Balaras et al., 1996) naturally lead to the detached eddy simulation DES method, where a full RANS model provides the stresses near the wall for the LES that is applied to the flow above. Recently, Nicoud et al. (2001) adopted a strategy where the wall stresses were used as a control in forcing

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

the LES towards a desired solution. The method was sub-optimal in that instead of storing a large number of 3D flow fields, the equations were discretised and the control strategy was implemented over short time periods. The approximate boundary conditions of Piomelli et al. (1989) were adopted and the Smagorinsky SGS was used within the Lilly (1992) version of the dynamic framework. Because the flow was simple, the mean velocity profile was expected to be logarithmic. Hence, the cost function to be minimised was given by the difference between the actual velocities and Eq. (38). Although it was noted that this method could be used to produce accurate wall boundary conditions in LES, the computational costs were considered prohibitive. Consequently, this simulation was used as a benchmark dataset to which more costeffective wall models could be compared. A model based on linear stochastic estimation of the wall stresses from local velocities was found to give a good match with the dataset when velocity information from more than one wall-parallel plane was included in the estimate. 5.3. Free-surface effects Just as there are complications in representing the flow at the bed, problems also arise at the free surface. The main complication with the upper surface is that its position will vary depending upon the pressure, which varies through time and over space. Variations in the position in the free surface may be induced due to boundary conditions (channel curvature at meander bends) or by short-lived turbulence events. Shi et al. (2000) identified a number of important phenomena occurring at or close to the free surface: (1) the turbulent kinetic energy of vertical velocity fluctuations is redistributed into the horizontal plane; (2) coherent structures exist that are normal and nearly parallel to the free surface; (3) the surface-normal vortices deform the free surface; and (4) an inverse energy cascade may be present at the free surface when surface-normal vortices of the same sign merge. In addition, Shen et al. (1999) discussed how hairpin vortices interact with the free surface to give persistent, counter-rotating vortices connected to the free surface. It may be possible for models of straight rivers flowing over smooth beds, with no input from tributaries, to ignore these issues and to apply a rigid

285

lid to the solution (i.e. assume that the upper surface maintains a constant location at the upper boundary). More sophisticated treatments introduce pressure terms into the momentum equations that can be treated dynamically to represent fluctuations in water surface elevations. However, problems can arise in this approach due to errors in mass conservation. An explicit free-surface treatment treats the flow depth as a variable to be solved for. When modelling the flow around a meander bend, super-elevation of the water surface occurs on the outer bend (Ikeda and Parker, 1989). If the upper surface was fixed by a rigid lid, then energy that would have been expended in elevating the flow is redirected downwards, enhancing secondary flow structures and, consequently, providing inaccurate flow predictions. Thus, some form of free-surface treatment is required in many practical situations, even if this is only through pressure corrections. Shi et al. (2000) used LES to investigate the properties of the free surface and adopted a kinematic expression for the movement of the surface, where the rate of change of elevation of the surface is proportional to the flux of fluid over the surface. Dynamic, external boundary conditions acting on this surface are a zero net pressure, normal stress and tangential stress. The difficulty arises in knowing where the free surface is and thus, where to apply these conditions. One method that can be used for this problem is the volume-of-fluid approach (Hirt and Nichols, 1981), where one solves an equation for the void fraction in a cell. In order for such methods to operate, appropriate numerical schemes must also be chosen (Leonard, 1997). Shi et al. used the Smagorinsky model as their SGS model and then implemented a hybrid method for representing the free surface. However, it is possible to go further than this and actually implement SGS models that are designed to account for free surface processes. Shen and Yue (2001) have discussed and evaluated a variety of these techniques.

6. An evaluation of different wall functions and SGS models Temmerman et al. (2003) evaluated the performance of various SGS models and wall function approximations for a turbulent channel flow with a wavy bed. LES studies of flows over such beds have

286

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

been previously undertaken by Henn and Sykes (1999) and Salvetti et al. (2001). The Reynolds number based on channel depth and bulk velocity was 21 560, with a bed wave crest height h, a flow depth of 3.035h, a channel width of 4.5h, a downstream distance between bed wave crests of 9h and a distance from crest to toe of 1.93h. Flow separation at the wave crests was observed, making this test case of broad relevance to dune and ripple flow environments (Bridge, 1981; Bridge and Best, 1988; Dinehart, 1989; Bennett and Best, 1995; Bennett and Bridge, 1995; LeClair and Bridge, 2001). The majority of the SGS models investigated by Temmerman et al. (2003) have been described above (Smagorinsky, 1963; Germano et al., 1991; Zang et al., 1993; Piomelli and Liu, 1995). However, additional models included were the mixed-scale model of Sagaut (1996) and the wall-adapted local eddyviscosity (WALE) model of Ducros et al. (1998). The selected wall functions were mainly of the loglaw type (see above), with the major differences involving the number of layers used. However, the Werner and Wegle (1991) law assumes a 1/7 power law outside the viscous sublayer. In their study, Temmerman et al. compared an approximately orthogonal mesh, with ~4.6  106 cells to two coarser meshes (~0.66  106 and ~1.04  106 cells), where the latter had mesh refinement in the lower wall and close to the wave crest. The sensitivity of two important flow parameters (the dimensionless time-averaged separation (x/h)sep and reattachment (x/ h)reat positions) to the type of wall approximation used is given in Table 1. In all cases, the numerical code used was STREAMLES (Lardat and Leschziner, 1998) and the SGS was the WALE model. The results presented in Table 2 are for the same flow parameters and numerical code, but with the wall function held constant and the SGS model varied. From these two tables it is clear that increasing mesh resolution has a more dramatic effect on the prediction of the recirculation length than the choice of wall function or SGS model. Furthermore, for a constant mesh resolution, SGS model and choice of wall function, the choice of code can still have an effect upon the predicted flow properties. Adopting the LESSOC code (Breuer and Rodi, 1996; Mathey et al., 1999) and using the SM + WD SGS and the WW wall function gave values for (x/h)sep and (x/h)reat of 0.45

Table 2 Results from Temmerman et al. (2003) for the sensitivity of separation and reattachment positions to the SGS model used No. of mesh cells ( 106)

SGS model

(x/h)sep

(x/h)reat

0.66 0.66 0.66 0.66

SM + WD MSM LDSM WALE

0.50 0.45 0.47 0.46

3.59 4.18 3.56 4.00

The abbreviation SM + WD stands for a Smagorinsky model with wall damping, MSM is the mixed-scale model and LDSM is the localized, dynamic, Smagorinsky model. All simulations were performed with the Werner and Wegle (1991) wall function.

and 3.60. The change in recirculation length compared to the second row of Table 2, while not as great as that found by using the MSM or WALE SGS, is greater than that found by using the LDSM SGS. Hence, the method of solving the equations has an important effect on the final results obtained. Temmerman et al. stated that for flow problems involving separation, the mesh resolution at the expected point of separation is highly influential, which has important implications for the modelling of these processes in a fluvial context. In addition, using wall functions would appear to be essential when modelling complex flows with relatively coarse meshes as a comparison between the no-slip (NS) and other conditions shown in Table 1 shows. Temmerman et al. also found that the WW wall approximation gave better results than the nonintegrated log-law models and that this was primarily due to the integration involved in the implementation. The variability introduced by varying the wall approximation would appear to be greater than that induced by changing the numerics or the SGS model. Thus, it would seem that how we implement our LES solutions to fluvial problems (mesh resolution and wall functions) outweighs concerns over the best way to formulate or numerically model the equations. Such conclusions are similar to those of Rodi et al. (1997) who noted the importance of appropriate meshing and problems introduced by unstructured meshes for even relatively simple engineering test-case flows.

7. Validation Recently, there has been concern in the fluvial and hydrologic modelling literature over the ability of a

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

model to yield appropriate results due to errors involved in the specification of model parameters and physical boundary conditions (Beven and Freer, 2001). This has led to the adoption of the Generalised Likelihood Uncertainty Estimation (GLUE) methodology (Freer et al., 1996) by some fluvial modellers (e.g. Hankin et al., 2001) as a way to determine appropriate parameter sets for a particular flow problem. Such an approach is likely to be prohibitively expensive in terms of computer time for LES, which take much longer to run than RANS models. Furthermore, for three-dimensional computations, model results should be less dependent upon parameterisation than for the two-dimensional, depth-averaged simulations considered by Hankin et al. Hence, increasing mesh resolution and thus, reducing the dependency of the solution upon the subgrid-scale model, or using a better SGS modelling procedure are likely to be more appropriate ways of dealing with uncertainty, meaning that traditional validation methods would appear to be more reasonable. A common method of assessing the performance of a simulation is to check that the solution is mesh independent. Roache (1997) outlined the grid convergence index approach to this issue. Hardy et al. (2003) applied this technique to a LES of a parallel-channel confluence flow (e.g. Best and Roy, 1991; Bradbrook et al., 1998), with a time series obtained for 1024 s at 10 Hz. The ability of the LES to generate time-dependent flow structures was clearly mesh resolution dependent, with only the highest resolution mesh producing such patterns. However, the grid convergence index showed that there was good agreement between the two higher resolution meshes in terms of mean flow quantities. Hence, this study highlighted that the standard guidelines for verifying and validating numerical flow models (e.g. AIAA, 1998) may need to be modified for the case of LES studies. Because for many river flow problems, such as sediment erosion and transport and flow mixing, the fluctuations in the velocity are of great importance, validation will need to be based upon more than mean flow quantities. For example, where periodic flow structures are significant, the LES should accurately mimic the velocity power spectra above a particular cutoff frequency that delimits these structures from background turbulence.

287

Validation of LES also requires more than a consideration of spatial discretisation. The temporal discretisation must be sufficient for velocity fluctuations to be represented adequately. In fact, time-step (or relaxation) resolution is also important for steadystate RANS modelling. This topic seems to be treated in the literature more sparsely than spatial discretisation, although it is clearly of great importance for LES. Roache (1997) discussed one method by which n temporal error can be evaluated. If f i,j,k indicates the value for a variable at cell i,j,k at time-step n, then a n+1 predicted value for f i,j,k can be found by  n;nþ1

Dt nþ1 n n n
1 Fi;j;k ¼ fi;j;k þ n
1;n fi;j;k
fi;j;k ð40Þ Dt where Dt n,n+1 is the time between time-steps n and n + 1. A point-wise error estimator E for the appropriate time-step and grid cell can then be written as nþ1 nþ1 nþ1 Ei;j;k ¼ Fi;j;k
fi;j;k

ð41Þ

From this, it is possible to produce a statistic characterising the error over the whole spatial domain at a particular time-step nþ1 Edomain ¼ 100

nþ1 maxi;j;k jEi;j;k j nþ1
f nþ1 fmax min

ð42Þ

This statistic indicates the maximum error in the domain at a given time-step and the temporal resolution can be refined until this error is acceptable. The definition of acceptability here will vary with the application under consideration or the scientific goals of the study. A rather different approach to LES validation was suggested by Langford and Moser (1999). Instead of determining appropriate criteria for mesh and timestep resolution, they proposed a more holistic approach. An ideal LES was defined, which gave the deterministic part of the time-evolving flow field at large scales. Hence, the difference between this and the actual flow field evolution is a measure of the significance of small-scale stochastic effects upon the evolution of the large-scale dynamic. This could prove an important validation tool because, although the limiting errors of the ideal LES will not be known, the errors of different models can be compared to see which is closest to the ideal model. This approach has been applied to isotropic turbulence where the results

288

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

suggested that the form of the subgrid-scale model might not be that important. However, Langford and Moser cautioned that the role of the subgrid-scale model would probably be much more significant in anisotropic flows with complex boundary conditions. Another validation technique developed within the LES community for refining the values for the SGS coefficients in an optimal way was introduced by Meyers et al. (2003) and extended to the dynamic modelling procedure by Meyers et al. (2005). Using a database of simulations, an accuracy chart was developed based on the spatial resolution of the computational mesh and the Smagorinsky length, which is a function of the Smagorinsky constant and the LES filter width (not necessarily the same as the mesh spacing). A comparison with DNS leads to a path through this space where the model may be refined in an optimal way to minimise errors as a function of scale. As applications of LES in the earth and environmental sciences begin to occur at larger scales, these types of methods, supplemented by more conventional grid convergence indices, may be extremely useful for guiding model development.

8. Applications River beds are complex, porous and irregular surfaces (Kirchner et al., 1990; Butler et al., 2002; Lane et al., 2004). Consequently, solving the flow accurately in this region is difficult, but essential if fluctuating forces acting upon the bed are needed to predict bedload sediment entrainment (Nelson et al., 1995) or flow mixing processes (Gaudet and Roy, 1995). There have been many investigations into the flow over rough beds that may be relevant to fluvial studies (e.g. Smith and McLean, 1977; Nowell and Church, 1979; Schlichting, 1979; Raupach et al., 1991; Kirkbride, 1993; Robert et al., 1993; Kim and Chung, 1995; Belcher et al., 2003). However, modelling such flows is problematic (Patel, 1998) and only recently have attempts been made to model such flows using LES (Ciofalo and Collins, 1992; Lee, 2002). Cui et al. (2003) produced a LES of a turbulent channel flow over surface-mounted transverse ribs using a dynamic Smagorinsky SGS model and the finite-volume code of Zang et al. (1994). They

distinguished between two types of roughness. The first is d type, where the ribs are sufficiently close together that vortex shedding from the ribs is insignificant. This type of roughness was defined by a ratio of distance between ribs to rib height that was less than 4.0. For the more widely spaced k type roughness, eddy shedding from between the ribs is important and the flow structure is dependent upon roughness height. In the latter case, flow separation and reattachment (Simpson, 1989) were observed between the elements. The rib height was 10% of the depth of the flow and 66 cells were used in the vertical, 20 to the top of the rib and 46 from the top of the rib to the channel top. A close agreement between measured and calculated mean streamwise velocities was found over the whole flow depth. For k type roughness, stresses fluctuated between negative and positive values depending on the vortex structure between the ribs. Hence, there was a small positive value immediately behind the rib due to the small corner vortex. Values became strongly negative within the recirculation region and more strongly positive as the flow reattached and recovered. The LES was able to detect the presence of favourable and adverse pressure gradients at the wall as the flow expanded and contracted over the ribs. Turbulence intensities were highest for k roughness, where penetration of the outer flow into the cavity occurred. Highest values occurred at the rib top where a shear layer had developed. The k type roughness produced a drag that was nearly four times that for d roughness. Hence, by coupling a sophisticated SGS to a high-resolution mesh, the important time-averaged and instantaneous characteristics of flow over a rough boundary can be obtained. Work on flow over gravel-bed rivers (e.g. Kirkbride and Ferguson, 1995) has led to a growing awareness of the complexity of the flow field close to the bed in natural river channels (Roy et al., 1999; Buffin-Be´langer et al., 2000). From Cui et al.’s study it would appear that LES is a tool that can provide a real breakthrough in the modelling of these processes. However, the mesh resolution required would appear to be an important constraint on studying larger scale river flow patterns (e.g. Best and Ashworth, 1997). Flow around bed obstacles introduces complex turbulent structures to channel flow (Buffin-Be´langer and Roy, 1998; Lawless and Robert, 2001) and can also have implications for local scour (Paola et al.,

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

1986) and bedform formation (Best, 1992). Flow around idealised objects also forms an important test case for LES (Rodi, 1993, 2002). Fig. 1 shows a LES of flow around a 4 cm cube undertaken at 10 Hz. The images shown are snapshots taken at 0.3 s at half the height of the cube and show the flapping of the recirculation region to the rear of the cube. A higher resolution simulation would resolve individual vortices within the shear layers. A steady-state numerical method would not resolve these aspects of the flow, leading to inaccuracies in the estimation of recirculation zone statistics and the effect of the bed-mounted obstacle on downstream flow mixing and instantaneous bed forces.

289

The potential of large-eddy simulation for investigating the shear layer and flow mixing processes in large areas of separated flows was demonstrated by Parsons (2003a). A laboratory experiment of a 908 double-width expansion was constructed and then numerically modelled (Fig. 2)). The average inflow velocity was set at 0.5 m s
1 with an inflow width of 0.15 m, which then doubled rightwards to 0.3 m, creating a large region of recirculating flow in a separation zone that extended downstream from the point of width expansion. Both particle imaging velocimetry or PIV (Westerweel, 1997; Williams et al., 2003b) and acoustic Doppler velocimetry (ADV) were used to validate both the time-averaged and

(a)

(b)

(c)

Fig. 1. Planform view of the flapping recirculation zone behind a wall-mounted cube obtained using LES. The fluctuations in the downstream velocity are shown (units of m s
1). The cube is 0.04 m across and the snapshots are taken 0.02 m above the bed. The simulation was performed at 10 Hz, with the three images shown obtained at 0.3 s intervals.

290

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

X-dist (cm)

30

15

0 0

50

Y-dist (cm)

100

150

Fig. 2. Planform geometry for the flume experiment and numerical simulation of the double-width expansion, with flow from left to right. The post-expansion channel was 0.3 m wide, with the inflow half this. The location of the ADV sample points taken a several flow heights (diamonds) and the area observed by the PIV at 0.035 and 0.115 m from the bed (grey square) are indicated. The dotted circle shows the position of the time series in Fig. 3.

Velocity (cm/s)

LES model results, which were then used for visualization and investigation into the flow dynamics. The LES results demonstrate the expected coalescence of vortices along the shear layer between the recirculating fluid and the downstream flow, which then intermittently impinge on the true right wall, as the region of separated flow re-attaches. This intermittent impingement results in large parcels of flow pulsing up the length of the separation zone, close to the true right wall (Fig. 3). This parcel of flow is then forced towards the true left as the return wall at the upstream limit of separation is reached. This in turn creates a dbulgeT of flow that interacts and deforms the shear layer, which creates an instability that propagates downstream to the reattachment point. This experiment has highlighted the significant potential of LES to analyse the ways in which flows mix into and out areas of flow separation, providing a new methodology for the examination of dispersion through river reaches more generally. At a larger scale, the flow through river channel confluences has recently been an important area of investigation in fluvial science. Parsons (2003b) 5 4 3 2 1 0 -1 -2 -3 -4

50

briefly reviews some of the important debates and the role for LES in this work. Bradbrook et al. (2000) used LES to study the flow structures in a laboratory parallel-channel confluence with a significant bed discordance (Best and Roy, 1991; Bradbrook et al., 2001). The Smagorinsky model was used, together with: a standard wall function; 70  44  25 cells for a domain of size 1.0 m  0.30 m  0.1 m; and a flow depth of 0.1 m in the deeper channel and 0.05 m in the shallow tributary. Flow periodicities derived from the LES were compared to those obtained in the flume. Based on the successful validation of these results, the confluence of the Bayonne and Berthier rivers (Biron et al., 1993; De Serres et al., 1999) was modelled using 82  56  12 cells for a domain of size 30 m  20 m  0.75 m. On the basis of this LES, it was possible to elucidate the processes affecting the tilting of the mixing layer between the two flows. The extent to which fluid from the deeper channel could penetrate beneath that from the shallower was related to variability in the velocity patterns in the mixing layer. The timescale for these fluctuations was of the order of 60 s in the simulations, similar

100

150

200

Time (seconds)

Fig. 3. LES velocity time series from within the dead zone close to the expansion wall (see Fig. 2 for location). The downstream velocity component is the heavy black line, the dotted line is the lateral velocity component and the grey line shows the vertical velocity component.

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

to that of b 0.013 Hz identified by Biron et al. (1993) for the flapping of the shear layer between the flows. Keylock (2002) developed some aspects of this work by comparing experimental and LES results of flow periodicities in experimental and natural confluences based on a Strouhal number (St) scaling: St ¼

fH ux ðref Þ

ð43Þ

where f is frequency, H is the step height or bed discordance and u x (ref) is the reference input velocity upstream of the confluence, which is typically the mean velocity. Although the lowfrequency periodicity of the shear layer flapping and the high-frequency periodicity of the Kelvin– Helmholtz vortices were successfully identified in experiments, LES and nature, the LES results were too smooth to detect fluctuations within individual vortices. However, both the numerical and experimental studies in parallel-channel laboratory confluences detected an additional periodicity that was not observed at the Bayonne–Berthier confluence. It was hypothesised that this was a consequence of the narrow geometry of the experimental, idealised confluence (Table 3). LES of a wider confluence with a similar discharge ratio confirmed this to be the case (Fig. 4). The greater extent of the recirculation zone in the wider confluence resulted in weaker pressure fluctuations that decoupled the shear layer over the step from the shearing between the two flows. Thus, both simulations showed evidence for Kelvin–Helmholtz vortex shedding at St c 0.06, but the emergent structure at St = 0.03 was no longer evident in the wider confluence, which

291

also had a much poorer correlation between the spectra for different velocity components. The lack of spectral peak combined with a decorrelation in the velocity spectra strongly implied that this structure is an emergent property of a narrower confluence system, where the shear layer between the flows is coupled to that associated with separation over the step. So far, there has been a limited attempt to integrate LES with models for suspended or bedload sediment entrainment and transport in fluvial environments. However, this is likely to be an important area of research in the next decade. Given the significance of instantaneous forces for bedload sediment movement (Heathershaw and Thorne, 1985; Wiberg and Smith, 1987; Nelson et al., 1995; Schmeeckle and Nelson, 2003), LES has important advantages over RANS methods for fluvial sedimentological study. In addition, the evolution of bedform structures (e.g. Paola and Seal, 1995; Tribe and Church, 1999) will affect more moments of the flow statistics than just the mean. LES has the potential to resolve these additional moments, which will feedback into further evolution of the bed and banks (Roy and Lane, 2003). An initial foray into this area was undertaken by Zedler and Street (2001) who used LES to study the flow over bed ripples. Zedler and Street used the Zang et al. (1994) numerical code, with a dynamic SGS (Salvetti and Banarjee, 1995) and a bed topography involving ripples with variability in both two and three dimensions. Sediment entrainment was modelled using the approach of Van Rijn (1984), which is a shear stress approach that employs the Shields parameter. By using LES, Zedler and Street were able

Table 3 Description of the flow domain for a simulation comparing the flow structures in parallel-channel confluences with channels of a differing width Domain length  width  height (m) No. of computational cells in the downstream  cross-stream  vertical Step height (m) Width of channel with bed discordance (m) Width of channel without bed discordance (m) Average input velocity in channel with bed discordance (m s
1) Average input velocity in channel without bed discordance (m s
1) Discharge ratio [( Q raised
Q unraised)/((1/2)( Q raised + Q unraised))] Discharge ratio ( Q raised/Q unraised)

Narrow confluence

Wider confluence

1.0  0.30  0.12 200  60  24 0.05 0.145 0.145 0.548 0.523
0.482 0.611

1.0  0.445  0.12 200  89  24 0.05 0.29 0.145 0.274 0.523
0.482 0.611

292

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

(a)

Spectral Power

100.000 10.000 1.000 0.100 0.010 0.001 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.07

0.08

0.09

Strouhal number

(b)

Spectral Power

10.000 1.000 0.100 0.010 0.001 0.00

0.01

0.02

0.03

0.04

0.05

0.06

Strouhal number Fig. 4. Power spectra from two LES experiments of a parallel-channel confluence flow. The heavy black line is the downstream velocity component u x , the heavy grey line is the lateral velocity component u y and the thin black line is the vertical velocity component u z . The spectra were taken in the mesh point above the bed a similar distance downstream of the reattachment zone. (a) shows the flow in the narrow tributary and (b) the wider tributary.

to evaluate instantaneous shear stresses and use an instantaneous version of the Shields approach. Bed shear stresses were extrapolated from the two grid cells closest to the bed. Erosion and deposition were switched on in the model once the flow field was in a statistically steady state. The shear stress distribution was as expected (McLean et al., 1994), with high positive values on the upslopes of the ripple crests. High sediment concentrations were observed to occur in regions where the vertical velocity was high (either regions of fluid ejection or where counter-rotating vortices were present). Shear stresses in the recirculation zone were stronger for the two-dimensional ripples, resulting in more sediment movement in this region. These observed links between the modelled flow field and the suspension of sediment demonstrate the importance of coherent structures within the flow

field (Soulsby et al., 1994; Mazumder, 2000; Chang and Scotti, 2003). Zedler and Street’s work, together with the other studies described above, has demonstrated the potential of LES for studying fluvial sedimentological and geomorphological problems. However, there are at least two important practical issues that will need resolving if LES is to fulfil this potential. First, if simulations are performed for timescales significantly greater than that used by Zedler and Street, erosion and deposition will significantly alter the bed topography. The traditional way to deal with this using body fitted co-ordinates would be to periodically remesh the computational domain and recalculate the flow field. This is computationally demanding, may require manual interference to check that the computational domain is appropriate

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

and also prevents direct comparison or validation of flow fields obtained at different time intervals. It is perhaps worth noting that suitable modification to the recently proposed immersed boundary method may have some potential in this respect (Fadlun et al., 2000). The second problem is that actual (as opposed to idealised) flow domains are highly complex. To apply LES at the river reach scale in an attempt to replicate the results of Zedler and Street would require careful consideration of the type of SGS model, method of filtering, type of sediment transport model, type of mesh, mesh resolution and specification of boundary conditions. Some of these points are developed below.

9. Scaling up to environmental applications of LES The fluvial scientist requires results from LES that are at a sufficient resolution to represent important processes, yet can be applied to a large flow volume, which is much greater in planar extent than it is in the vertical, and where topography is complex. In order to fulfil these requirements, compromises need to be made. Significant mesh distortion or body-fitted meshes may be needed or process representation may have to be sacrificed for spatial coverage. This section of the paper outlines some recent developments that may enhance the suitability of LES for the study of river flow problems by reducing the extent of such compromises, while also noting the constraints that currently exist. Instead of using LES, some of the modern hybrid methods (Tucker and Davidson, 2004), which are computationally more efficient, may permit modelling over larger scales. The rationale for detached eddy simulation (DES) is that resolving the flow close to the wall is computationally expensive. Spalart et al. (1997) and Mellen et al. (2003) discussed some of the difficulties of using LES to simulate flow over an airfoil, which, like many river flows, involves high Reynolds numbers and flow separation. By using a RANS method near the wall and coupling it to LES for the far field, significant gains in computation time can be obtained. The advantage of DES is that there is no a priori specification of the regions where RANS and LES should apply. Instead, a one-equation turbulence model (Spalart and Allmaras, 1994) is

293

adopted, with the modification that the wall normal distance d is replaced by: d˘uminðd; CDES DG Þ

ð44Þ

where D G is a mesh-based scale given by (contrast with Eq. (20)): DG ¼ maxðD1 ; D2 ; D3 Þ

ð45Þ

and C DES is a new constant and was found to have a value of 0.65 by Shur et al. (1999), although these authors noted that the value is dependent upon the differencing scheme. Near the wall, D G is greater than d and the standard Spalart and Allmaras model is applied. Further away, the model acts as a LES, with an assumption that the mixing length is related to the mesh size. Hence, a model with some similarities to the Smagorinsky SGS is adopted (Eqs. (20) and (21)). Schmidt and Thiele (2002) showed that DES was able to accurately capture the frequency of eddy shedding from a wall-mounted cube despite a cruder mesh resolution and a shorter computational time than a LES. However, the authors concluded their study by stating that in what they term internal flows where the majority of the flow is affected by external boundaries, DES does not offer a major improvement over conventional LES. Owing to the importance of external boundary conditions for many fluvial environments, this would suggest that DES does not offer any major advantages for fluvial science because of the use of the weaker RANS methodology close to boundaries. This conclusion raises the question as to whether the DES philosophy can be reversed, with the LES used in the near-wall region and a RANS approach adopted for the external flow field. This approach was adopted by Hamba (2001). Two overlapping meshes were used: a one-dimensional RANS grid and a three-dimensional LES grid, with the SGS model of Yoshizawa and Horiuti (1985) adopted for the near-wall flow. The standard coefficients in the k– e RANS model were altered to make the values compatible with the LES in the near-bed region. Good predictions of downstream velocity profiles and turbulence intensities were obtained using this method. Hence, it may be the case that hybrid approaches have some potential for modelling fluvial processes at larger scales, with the approach adopted by Hamba

294

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

(2001) of particular relevance in situations where accurate representation of the flow near boundaries is required. However, it should be noted that, at present, there are still some important issues that need resolving with DES and hybrid approaches. For example, Hamba (2003) noted that in many of the studies undertaken so far (Nikitin et al., 2000; Davidson and Peng, 2001; Hamba, 2001), there is a mismatch in mean velocity profiles at the join between the RANS and LES simulations. Geurts and Leonard (2002) provide a recent attempt to review the applicability of LES to complex flow problems. They argued that despite significant progress in LES over the last decade, there are still a number of outstanding issues that need addressing. They considered the recent developments of explicit filtering methods (Vasilyev et al., 1998; Marsden et al., 2002) to offer important advantages for the modelling of complex flows due to the potential to modify the filter size in accordance with changes in turbulence intensity. This would result in computational savings because the filter resolution could be deteriorated in reaches where the flow was relatively simple, with significant enhancement occurring in regions of more complex flow dynamics (e.g. confluences). However, the energy transfer between scales is affected by the variable nature of the filter size and this produces new terms in the equations that need to be accounted for. Thus, there is still research that needs to be done on explicit filtering methods until they are at a stage that allows their application in fluvial science. A key issue for the application of LES to large flow domains, where explicit filtering may also be able to help in the future, is the relative importance of errors introduced by the discretisation of the flow field compared to other sources of error such as the accuracy of the subgrid-scale model. Vreman et al. (1994, 1996b) found that an explicit filter with the filter width greater than the mesh spacing gave a subgrid error contribution that was significantly larger than the discretisation error. However, when using an implicit filtering approach, with the mesh and filter size the same, the error due to discretisation was greater than the contribution from the subgrid term. Hence, the adoption of more sophisticated SGS models is only likely to provide significant gains for LES of environmental flows

when explicit filtering methods come into common usage. Geurts and Leonard (2002) made a number of recommendations for a successful LES of a complex flow. These included the use of isotropic meshes and the avoidance of numerical dissipation to disguise any shortcomings in the mesh resolution or modelling of the equations. In addition, they suggested performing simulations at different filter width to mesh size ratios and favoured the use of dynamic SGS modelling together with explicit filtering, and optimising the scales available for LES. They concluded by stating that effective modelling of the near-wall region is vital and that the number of nodes used in the solution should be sufficiently larger than the number needed to predict a quantity to a particular level of accuracy. The adoption of isotropic meshes can be difficult for flows with complex geometries such as those found in many environmental flows, including rivers. The traditional approach to the discretisation of channels with complex geometry involves boundary fitted co-ordinates (BFCs), with parameterisation of smaller scale aspects of topography using a roughness parameter (e.g. Hodskinson and Ferguson, 1998; Nicholas and Smith, 1999; Lane et al., 1999). Although this enables a stable RANS solution to be obtained, problems arise when LES simulations are attempted. This is because most LES methods couple the filter size to the computational grid cell. Hence, variation in the size of the grid cells means that the length scale of the filter (D) is not constant throughout the domain. This can cause numerical difficulties (a lack of convergence and excessive numerical diffusion) as well as create problems with the calculation of terms based on fluctuating components of velocity (Reynolds stresses and turbulence intensity) because the velocity estimates are representative of varying flow volumes. Furthermore, the variation in the length scale that delimits grid scale and subgrid-scale behaviour can affect the representation of the inverse energy cascade because different parts of the computational grid will be able to resolve eddies at varying resolutions. In an attempt to deal with this problem, a porosity algorithm has been developed (Lane et al., 2002, 2004; Hardy et al., in press), which enables complex topography to be included within a regular Cartesian mesh. This maintains a constant grid cell size and

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

allows a uniform length scale filter (D) to be maintained throughout the domain. The porosity treatment is based upon the Olsen and Stokseth (1995) approach, and has been extended to modify the drag terms introduced due to the effect of porosity. For a given regular (i.e. cuboid) Cartesian mesh, the topography is transformed into a porosity within the mesh. In this situation, we assume that the topographic data takes the form of a digital elevation model (DEM), with a density of topographic data equal to or greater than the grid density. We also assume that the numerical grid is defined with vertices that are exactly collocated to the DEM. This means that either: (1) the DEM elevations map directly onto grid cells (equal density); or (2) the grid cells each contain a unique set of DEM elevations (greater topographic density), which can be used to determine the average elevation. In the latter case, with a regular grid, there is a geometric series of possible DEM elevations in a given grid cell (i.e. 1, 4, 9, 16, 25, 36. . .). These two situations need slightly different treatments in terms of porosity. Consider a structured, orthogonal Cartesian grid, with directions i and j in the planform and k in the vertical. The equal density case is simplest because each porosity value ( P ijk ) can be defined for each column of data at planform location ij, which has a singular value of bed elevation, E ij . Thus, if Z ijk is the centre of a cell and we assume cuboid cells so that the thickness of the cell (DZij ) is the same throughout the column, then there are three rules: If Eij zZijk þ 0:5DZij ;

Pijk ¼ 0

If Zijk
0:5DZij bEij bZijk þ 0:5DZij ;  
 Eij
Zijk Pijk ¼ þ 0:5 DZij If Eij VZijk
0:5DZij ;

Pijk ¼ 1

ð46aÞ

ð46bÞ ð46cÞ

For the case given by Eq. (46b) there are a number of solutions of varying complexity. The simplest is based upon deriving the average elevation for a given grid cell from the DEM values within that grid cell. However, if there are 9 (i.e. a 33 elevation matrix) or more elevations being assigned to a single grid cell, it is possible to determine a more realistic value of

295

porosity by fitting surfaces to the elevation matrix and then working out the area-integrated average elevation as the basis of the porosity value. Following Olsen and Stokseth (1995), and noting that, for rough, gravelly surfaces, dealing with the effects of topography upon mass conservation should be a prime concern, we assume that the source term represented by the drag terms in the momentum equation is so great that it dominates over the turbulent diffusion terms. Therefore, the only modification to the momentum equations is a scaling of the drag term derived from the law-of-the-wall to represent the effective exposed area of the surface. To do this, we recognize that the effective drag on an element will depend upon the direction of flow. With a first approximation of flow direction, the drag exerted by a given cell is augmented according to the change in porosity in the flow direction. For a given cell, the four differences in porosity between cell (i,j,k) and cells (i
1,j,k), (i + 1,j,k), (i,j
1,k) and (i,j + 1,k) are calculated. If there is a component of flow out of the cell in the + or
i-direction or + or
j-direction and porosity decreases in that direction (i.e. the adjacent cell has a higher elevation) then there is an effective djumpT and the drag coefficient must be scaled accordingly. Otherwise, no scaling is applied. The scaling is based upon the effective (equivalent) surface area. We assume that all drag is exerted along the base of cells other than for the sidewalls of the channel. A surface is then fitted to the 33 horizontal matrix of elevations centred on cell (i,j,k). The area (A ijk ) of this surface for cell (i,j,k) can then be determined for the four vertices of the cell: ( i + 0 . 5 DX i j k , j + 0 . 5 DY i j k , k ) ; ( i + 0 . 5 DX i j k , j
0.5DYijk ,k); (i
0.5DX ijk , j
0.5DYijk ,k); and (i
0.5DX ijk , j + 0.5DYijk ,k); where DX ijk and DYijk are the grid resolutions in the i- and j-directions, respectively, for cell (i,j,k). With a cubic grid, DX ijk = DYijk = DZ ijk and the scaling factor (s ijk ) is then applied to the roughness height in the wall function using: sijk ¼

Aijk : 2 DZijk

ð47Þ

An examination of Eq. (47) shows that the drag augmentation takes place by increasing the effective elevation at which the planform component of

296

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

velocity becomes zero, through an increase in the roughness height. The algorithm also provides a framework for LES where a constant length scale filter is maintained throughout the domain. The cube simulation (Fig. 1) is an example of initial testing of this algorithm. Fig. 5 shows a simulation of flow over a bed of water-worked sediment. This dataset was obtained by digital photogrammetry and is described in more detail by Butler et al. (1998). The two images show the vertical velocity at the same cross-section (flow direction towards the reader) separated by time intervals of 1 s. The simulation is able to capture some of the complexity of the near-bed flow, including regions of upwelling and fluctuations in the strength of such zones. If some of the bed was eroded at this location, flow properties obtained before and after erosion could be directly compared as the mesh is not distorted to fit the bed. This would be much more problematic if body-fitted co-ordinate methods were adopted.

10. Conclusion In this article we have outlined the basic philosophy underpinning LES and described some of the commonly used SGS models needed to connect the resolved and unresolved scales of turbulence. Methods of dealing with the presence of boundaries (i.e. wall functions) have also been introduced and evaluated. Some applications of LES to flows that may be of interest to geomorphologists and sedimentologists have been presented, including an initial attempt to couple a LES with a sediment transport model. For this research to develop there are a number of important issues related to the feasible size of the computational domain and the scales of important processes that require some work. In particular, scaling current LES work to the channel reach scale may well mean that simulations become more reliant upon the subgrid-scale model. In this case, there is a

Fig. 5. Flow over a river gravels using LES. The vertical velocity component is shown. The two images are obtained at the same location but separated by 1 s. Note the change in location of the region of upwelling towards the right of the images.

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

need to test different subgrid-scale models for flows through complex topography in order to analyse performance as a function of model resolution. The research reviewed in this article would lead to the conclusion that the logical directions to pursue would be the introduction of the dynamic modelling procedure and, in particular, the enhanced dynamic procedure using Lagrangian averaging developed by Meneveau et al. (1996). This approach seems to be able to resolve eddy structure more accurately, which will be advantageous to those studies that consider the detailed interaction between turbulent flows and sediment transport. It is too early to say how significant some of the more recent developments in SGS modelling will be for environmental applications, but the regularization method of Geurts and Holm (2003) is one interesting and emerging avenue. One difficulty with using LES in sediment transport studies is the need to adjust the boundary conditions as erosion and deposition take place (remeshing of the computational domain). We have outlined one method of dealing with this based on a porosity treatment (Lane et al., 2002, 2004). While large-scale studies of flow and sediment transport in rivers are unlikely to benefit from the direct application of LES due to the high computational cost, smallscale process modelling should benefit directly. In addition, at intermediate scales, flow statistics derived from local LES studies can perhaps be used to supplement RANS methods. Large-eddy simulation is a modelling technique that offers much potential for studies where the resolution of flow details is important. For a number of practical problems existing approaches based on RANS methods may be considered acceptable. However, hybrid methods that bring some of the advantages of LES at a reduced computational cost may have the potential to resolve aspects of the flow that are currently ignored in modelling studies at scales larger than have hitherto been attempted using LES.

Acknowledgements This work was undertaken while CK was in receipt of Nuffield Award for New Lecturers NAL/00495/G, NERC Studentship GT04/97/55/FS and JSPS ShortTerm Fellowship PE 04511, and was supported by

297

NERC grants GR9/5059 and GR3/9715, and NERC Studentship NER/A/S/2001/00445. We thank James Brasington, the editor and two anonymous referees for some insightful comments on the work contained herein.

References Ackermann, C., Me´tais, O., 2001. A modified selective structure function subgrid-scale model. Journal of Turbulence 2 (011), 1 – 27. AIAA, 1998. Guide for the verification and validation of computational fluid dynamics simulations. AIAA Report G-077-1998. Balaras, E., Benocci, C., Piomelli, U., 1996. Two-layer approximate boundary conditions for large-eddy simulations. AIAA Journal 34, 1111 – 1119. Bardina, J., Ferziger, J.H., Reynolds, W.C., 1980. Improved subgrid models for large eddy simulation. AIAA Journal 80, 1357. Bates, P.D., Anderson, M.G., 1993. A two dimensional finiteelement model for river flow inundation. Proceedings of the Royal Society of London. Series A 440, 481 – 491. Beaudan, P., Moin, P., 1994. Numerical Experiments on the Flow Past a Circular Cylinder at Sub-Critical Reynolds Numbers. Dept. of Mechanical Engineering, Stanford University, Stanford, CA. Report TF-62. Bedford, K., Babajimopoulos, C., 1980. Verifying lake models with spectral statistics. Journal of the Hydraulics Division, ASCE 106, 21 – 38. Belcher, S.E., Jerram, N., Hunt, J.C.R., 2003. Adjustment of a turbulent boundary layer to a canopy of roughness elements. Journal of Fluid Mechanics 488, 369 – 398. Bennett, S.J., Best, J.L., 1995. Mean flow and turbulence structure over fixed, 2-dimensional dunes—implications for sediment transport and bedform stability. Sedimentology 42, 491 – 513. Bennett, S.J., Bridge, J.S., 1995. The geometry and dynamics of low-relief bed forms in heterogeneous sediment in a laboratory channel, and their relationship to water-flow and sediment transport. Journal of Sedimentary Research. Section A, Sedimentary Petrology and Processes 65, 29 – 39. Best, J.L., 1992. On the entrainment of sediment and initiation of bed defects: insights from recent developments within turbulent boundary layer research. Sedimentology 39, 797 – 811. Best, J.L., Ashworth, P.J., 1997. Scour in large braided rivers and the recognition of sequence stratigraphic boundaries. Nature 387, 275 – 277. Best, J.L., Roy, A.G., 1991. Mixing layer distortion at the confluence of channels of different depth. Nature 350, 411 – 413. Beven, K., Freer, J., 2001. Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. Journal of Hydrology 249, 11 – 29. Biron, P.M., De Serres, B., Roy, A.G., Best, J.L., 1993. Shear layer turbulence at an unequal depth channel confluence. In: Clifford, N.J., French, J.R., Hardisty, J. (Eds.), Turbulence: Perspectives

298

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

on Flow and Sediment Transport. John Wiley and Sons, Chichester, UK, 360 pp. Boersma, B.J., Kooper, M.N., Nieuwstadt, F.T.M., Wesseling, P., 1997. Local grid refinement in large-eddy simulations. Journal of Engineering Mathematics 32, 161 – 175. Booker, D.J., Sear, D.A., Payne, A.J., 2001. Modelling threedimensional flow structures and patterns of boundary shear stress in a natural pool–riffle sequence. Earth Surface Processes and Landforms 26, 553 – 576. Boussinesq, J., 1877. The´orie de l’e´coulement tourbillant. Me´moires Pre´sente´s par divers Savants Acadamie Scientifique Institut Franc¸aise 23, 46 – 50. Bradbrook, K.F., Biron, P.M., Lane, S.N., Richards, K.S., Roy, A.G., 1998. Investigation of controls on secondary circulation in a simple confluence geometry using a three-dimensional numerical model. Hydrological Processes 12, 1371 – 1396. Bradbrook, K.F., Lane, S.N., Richards, K.S., Biron, P.M., Roy, A.G., 2000. Large-eddy simulation of periodic flow characteristics at river channel confluences. Journal of Hydraulic Research 38, 207 – 215. Bradbrook, K.F., Lane, S.N., Richards, K.S., Biron, P.M., Roy, A.G., 2001. Role of bed discordance at asymmetrical river confluences. ASCE Journal of Hydraulic Engineering 127 (5), 351 – 368. Breuer, M., Rodi, W., 1996. Large eddy simulation of complex turbulent flows of practical interest. In: Hirschel, E.H. (Ed.), Flow Simulation with High Performance Computers II. Notes on Numerical Fluid Mechanics, vol. 52. Viweg, Braunschweig, pp. 258 – 274. Bridge, J.S., 1981. Bed shear-stress over subaqueous dunes, and the transition to upper-stage plane beds. Sedimentology 28, 33 – 36. Bridge, J.S., 1992. A revised model for water flow, sediment transport, bed topography and grain size sorting in natural river bends. Water Resources Research 28, 999 – 1013. Bridge, J.S., 1993. The interaction between channel geometry, water flow, sediment transport and deposition in braided rivers. In: Best, J.L., Bristow, C.S. (Eds.), Braided Rivers. Special Publication Geological Society of London, vol. 75, pp. 13 – 71. Bridge, J.S., Best, J.L., 1988. Flow, sediment transport and bedform dynamics over the transition from dunes to upper-stage plane beds—implications for the formation of planar laminae. Sedimentology 35, 753 – 763. Brun, C., Friedrich, R., 2001. Modeling the test SGS tensor Tij : an issue in the dynamic approach. Physics of Fluids 13, 2373 – 2385. Buffin-Be´langer, T., Roy, A.G., 1998. Effects of a pebble cluster on the turbulent structure of a depth-limited flow in a gravel-bed river. Geomorphology 25, 249 – 267. Buffin-Be´langer, T., Roy, A.G., Kirkbride, A.D., 2000. On large scale flow structures in a gravel bed river. Geomorphology 32, 417 – 435. Butler, J.B., Lane, S.N., Chandler, J.H., 1998. Assessment of DEM quality characterising surface roughness using close range digital photogrammetry. Photogrammetric Record 16, 271 – 291. Butler, J.B., Lane, S.N., Chandler, J.H., Porfiri, K., 2002. Through water close range digital photogrammetry in flume and field environments. Photogrammetric Record 17, 419 – 439.

Cabot, W., Moin, P., 1999. Approximate wall boundary conditions in the large eddy simulation of high Reynolds number flow. Flow, Turbulence and Combustion 63, 269 – 291. Cai, X.-M., Steyn, D.G., 1996. The von Karman constant determined by large eddy simulation. Boundary-Layer Meteorology 78, 143 – 164. Canuto, V.M., Cheng, Y., 1997. Determination of the Smagorinsky– Lilly constant C s. Physics of Fluids 9, 1368 – 1378. Carati, D., Ghosal, S., Moin, P., 1995. On the representation of backscatter in dynamic localization models. Physics of Fluids 7, 606 – 616. Chang, Y.S., Scotti, A., 2003. Entrainment and suspension of sediments into a turbulent flow over ripples. Journal of Turbulence 4 (019), 1 – 22. Choi, H., Moin, P., Kim, J., 1994. Direct numerical simulation of flow over riblets. Journal of Fluid Mechanics 255, 503 – 539. Chollet, J.P., Lesieur, M., 1981. Parameterization of small scales of three-dimensional isotropic turbulence utilizing spectral closures. Journal of the Atmospheric Sciences 38, 2747 – 2757. Ciofalo, M., Collins, M.W., 1992. Large-eddy simulation of turbulent flow and heat transfer in plane and rib-roughened channels. International Journal for Numerical Methods in Fluids 15, 453 – 489. Clark, R.A., Ferziger, J.H., Reynolds, W.C., 1979. Evaluation of subgrid models using an accurately simulated turbulent flow. Journal of Fluid Mechanics 91, 1 – 16. Collis, S.S., 2001. Monitoring unresolved scales in multiscale turbulence modelling. Physics of Fluids 13, 1800 – 1806. Cui, J., Patel, V.C., Lin, C.-L., 2003. Large-eddy simulation of turbulent flow in a channel with rib roughness. International Journal of Heat and Fluid Flow 24, 372 – 388. David, E., 1993. Mode´lisation des e´coulements compressibles et hypersoniques. The`se de Doctorat de l’INPG, Grenoble, France. Davidson, L., Peng, S.-H., 2001. A hybrid LES–RANS model based on a one equation SGS model and a two-equation k–x model. In: Lindborg, E., Johansson, A., Eaton, J., Humphrey, J., Kasagi, N., Leschziner, M., Sommerfield, M. (Eds.), Turbulence and Shear Flow Phenomena. Second International Symposium, pp. 175 – 180. Stockholm, KTH. Deardorff, J.W., 1970a. A numerical study of three-dimensional turbulent channel flow at large Reynolds number. Journal of Fluid Mechanics 41, 453 – 480. Deardorff, J.W., 1970b. A three-dimensional numerical investigation of the idealized planetary boundary layer. Geophysical Fluid Dynamics 1, 377 – 410. De Serres, B., Roy, A.G., Biron, P.M., Best, J.L., 1999. Threedimensional structure of flow at a confluence of river channels with discordant beds. Geomorphology 26, 313 – 335. Dinehart, R.L., 1989. Dune migration in a steep, coarse-bedded stream. Water Resources Research 25, 911 – 923. Domaradzki, J.A., Loh, K.-C., 1999. The subgrid-scale estimation in the physical space representation. Physics of Fluids 11, 2330 – 2342. Domaradzki, J.A., Saiki, E.M., 1997. A subgrid-scale model based on the estimation of unresolved scales of turbulence. Physics of Fluids 9, 2148 – 2164.

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304 Ducros, F., Nicoud, F., Poinsot, T., 1998. Wall-adapted local eddyviscosity models for simulations in complex geometries. 6th ICFD Conference on Numerical Methods for Fluid Dynamics, pp. 293 – 299. Engelund, F., Fredsoe, I., 1982. Sediment ripples and dunes. Annual Review of Fluid Mechanics 14, 1 – 13. Ezer, T., Mellor, G.L., 1997. Simulations of the Atlantic Ocean with a free surface sigma coordinate ocean model. Journal of Geophysical Research 102, 15647 – 15657. Fadlun, E.A., Verzicco, R., Orlandi, P., Mohd-Yusof, J., 2000. Combined immersed-boundary/finite difference methods for three-dimensional complex flow simulations. Journal of Computational Physics 161, 35 – 60. Farge, M., Schneider, K., Pellegrino, G., Wray, A.A., Rogallo, R.S., 2003. Coherent vortex extraction in three-dimensional homogeneous turbulence: comparison between CVS-wavelet and PODFourier decompositions. Physics of Fluids 15 (10), 2886 – 2896. Ferguson, R.I., Parsons, D.R., Lane, S.N., Hardy, R.J., 2003. Flow in meander bends with recirculation at the inner bank. Water Resources Research 39, 1322. Ferziger, J.H., 1993. Subgrid-scale modelling. In: Galperin, B., Orszag, S.A. (Eds.), Large Eddy Simulation of Complex Engineering and Geophysical Flows. Cambridge University Press, Cambridge. Ferziger, J.H., Peric, M., 1999. Computational Methods for Fluid Dynamics. Springer-Verlag, Berlin. 389 pp. Freer, J., Beven, K.J., Ambroise, A., 1996. Bayesian estimation of uncertainty in runoff prediction and the value of data: an application of the GLUE approach. Water Resources Research 32, 2161 – 2173. Friedkin, J.F., 1945. A Laboratory Study of the Meandering of Alluvial Rivers. US Waterways Experimental Station, Vicksburg, Mississippi. Friedman, G.M., Sanders, J.E., 1978. Principles of Sedimentology. John Wiley & Sons, Chichester. Frfhlich, J., Rodi, W., 2002. Introduction to large eddy simulation for turbulent flows. In: Launder, B.E., Sandham, N.D. (Eds.), Closure Strategies for Turbulent and Transitional Flows. Cambridge University Press, Cambridge. 768 pp. Gaudet, J.M., Roy, A.G., 1995. Effect of bed morphology on flow mixing length at river confluences. Nature 373, 138 – 139. Germano, M., 1992. Turbulence: the filtering approach. Journal of Fluid Mechanics 238, 325 – 336. Germano, M., Piomelli, U., Moin, P., Cabot, W.H., 1991. A dynamic subgrid-scale eddy viscosity model. Physics of Fluids. A, Fluid Dynamics 3, 1760 – 1765. Geurts, B.J., 1997. Inverse modeling for large-eddy simulation. Physics of Fluids 9, 3585 – 3587. Geurts, B.J., Holm, D.D., 2003. Regularization modeling for largeeddy simulation. Physics of Fluids 15, L13 – L16. Geurts, B.J., Leonard, A., 2002. Is LES ready for complex flows? In: Launder, B.E., Sandham, N.D. (Eds.), Closure Strategies for Turbulent and Transitional Flows. Cambridge University Press, Cambridge, 768 pp. Gharakhani, A., 2003. Application of the vorticity redistribution method to LES of incompressible flow. Journal of Turbulence 4 (004), 1 – 7.

299

Ghosal, S., 1999. Mathematical and physical constraints on largeeddy simulation of turbulence. AIAA Journal 37, 425 – 433. Ghosal, S., Lund, T.S., Moin, P., Akselvoll, K., 1995. A dynamic localization model for large-eddy simulation of turbulent flows. Journal of Fluid Mechanics 286, 229 – 255. Gilbert, G.K., 1914. Transportation of debris by running water. US Geological Survey Professional Paper 86, 263 pp. Goldstein, D.E., Vasilyev, O.V., 2004. Stochastic coherent adaptive large eddy simulation method. Physics of Fluids 16, 2497 – 2513. Grass, A.J., 1971. Structural features of turbulent flow over smooth and rough boundaries. Journal of Fluid Mechanics 50, 233 – 255. Gullbrand, J., 2001. Explicit filtering and subgrid-scale models in turbulent channel flow. Center for Turbulence Research, Stanford University, Annual Research Briefs 2001, pp. 31 – 42. Haller, G., Yuan, G., 2000. Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica. D 147, 352 – 370. Hamba, F., 2001. An attempt to combine large eddy simulation with the k–e model in a channel-flow calculation. Theoretical and Computational Fluid Dynamics 14, 323 – 336. Hamba, F., 2003. A Hybrid RANS/LES simulation of turbulent channel flow. Theoretical and Computational Fluid Dynamics 16, 387 – 403. Hankin, B.G., Hardy, R.J., Beven, K.J., Kettle, H., 2001. Using CFD in a GLUE framework to model the flow and dispersion characteristics of a natural fluvial dead zone. Earth Surface Processes and Landforms 26, 667 – 687. Hardy, R.J., Bates, P.D., Anderson, M.G., 1999. The importance of spatial resolution in hydraulic models for floodplain environments. Journal of Hydrology 216, 124 – 136. Hardy, R.J., Bates, P.D., Anderson, M.G., Moulin, C., Hervouet, J.-M., 2000. Initial testing of a joint two-dimensional finite element hydraulic and sediment transport model to reach scale floodplain environment. Proceedings of the Institute of Civil Engineers, Water, Maritime and Energy, vol. 142, pp. 141 – 156. Hardy, R.J., Lane, S.N., Ferguson, R.I., Parsons, D.R., 2003. Assessing the credibility of a series of computational fluid dynamics simulations of open channel flow. Hydrological Processes 17, 1539 – 1560. Hardy, R.J., Lane, S.N., Lawless, M.R., Best, J.L., Elliott, L., Ingham, D.B., in press. Development and testing of a numerical code for treatment of complex river channel topography in three-dimensional CFD models with structured grids. Journal of Hydraulics Research. Harlow, F.H., Nakayama, P., 1968. Transport of Turbulence Energy Decay Rate. Los Alamos Scientific Laboratory, University of California, Report LA 3854. Harms, J.C., McKenzie, D.B., McCubbin, D.G., 1963. Stratification in modern sands of the Red River, Louisiana. Journal of Geology 71, 566 – 580. H7rtel, C., 1996. Turbulent flows: direct numerical simulation and large-eddy simulation. In: Peyret, R. (Ed.), Handbook of Computational Fluid Mechanics. Academic Press, London. H7rtel, C., Kleiser, L., 1997. Galilean invariance and filtering dependence of near-wall grid-scale/subgrid-scale interactions in

300

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

large-eddy simulation. Physics of Fluids. A, Fluid Dynamics 9, 473 – 475. H7rtel, C., Kleiser, L., 1998. Analysis and modelling of subgridscale motions in near-wall turbulence. Journal of Fluid Mechanics 356, 327 – 352. Hassan, M.A., Reid, I., 1990. The influence of microform bed roughness elements on flow and sediment transport in gravel bed rivers. Earth Surface Processes and Landforms 15, 739 – 750. Heathershaw, A.D., Thorne, P.D., 1985. Sea-bed noises reveal role of turbulent bursting phenomenon in sediment transport by tidal currents. Nature 316, 339 – 342. Henn, D.S., Sykes, R.I., 1999. Large-eddy simulation of flow over wavy surfaces. Journal of Fluid Mechanics 383, 75 – 112. Hirt, C.W., 1969. Computer studies of time-dependent turbulent flows. The Physics of Fluids (Suppl. II), 219 – 227. Hirt, C.W., Nichols, B.D., 1981. Volume of fluid (VOF) method for dynamics of free boundaries. Journal of Computational Physics 39, 201 – 221. Hodskinson, A., Ferguson, R., 1998. Numerical modelling of separated flow in river bends: model testing and experimental investigation of geometric controls on the extent of flow separation at the concave bank. Hydrological Processes 12, 1323 – 1338. Horiuti, K., 1997. A new dynamic two-parameter mixed model for large-eddy simulation. Physics of Fluids 9, 3443 – 3464. Horritt, M.S., 2002. Stochastic modelling of 1-D shallow water flows over uncertain topography. Journal of Computational Physics 180, 327 – 338. Hughes, T.J.R., Feijoo, G.R., Mazzei, L., Quincy, J.B., 1998. The variational multiscale method—a paradigm for computational mechanics. Computer Methods in Applied Mechanics 166, 3 – 24. Hughes, T.J.R., Mazzei, L., Oberai, A.A., Wray, A.A., 2001a. The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence. Physics of Fluids 13, 505 – 512. Hughes, T.J.R., Oberai, A.A., Mazzei, L., 2001b. Large eddy simulations of turbulent channel flow by the variational multiscale method. Physics of Fluids 13, 1784 – 1799. Iaccarino, G., Durbin, P., 2000. Unsteady 3D RANS simulations using the t 2–f model. Center for Turbulence Research, Annual Research Briefs, Stanford University, pp. 263 – 269. Ikeda, S., Parker, G., 1989. River meandering. American Geophysical Union Water Resources Monograph 12, 485 pp. Jones, W.P., Launder, B.E., 1972. The prediction of laminarization with a two-equation model of turbulence. International Journal of Heat and Mass Transfer 15, 301 – 314. Keylock, C.J., 2002. The interactions between flow structures and bedload sediment at parallel channel confluences. Unpublished PhD thesis, University of Cambridge, 285 pp. Kim, B.N., Chung, M.K., 1995. Experimental study of roughness effects on the separated flow over a backward-facing step. AIAA Journal 33, 159 – 161. Kirchner, J.W., Dietrich, W.E., Iseya, F., Ikeda, H., 1990. The variability of critical shear stress, friction angle, and grain protrusion in water-worked sediments. Sedimentology 37, 647 – 672.

Kirkbride, A.D., 1993. Observations of the influence of bed roughness on turbulence structure in depth-limited flows over gravel beds. In: Clifford, N.J., French, F.R., Hardisty, J. (Eds.), Turbulence: Perspectives on Flow and Sediment Transport. John Wiley and Sons, Chichester, UK, pp. 185 – 196. Kirkbride, A.D., Ferguson, R., 1995. Turbulent flow structures in a gravel-bed river: Markov chain analysis of the fluctuating velocity profile. Earth Surface Processes and Landforms 20, 721 – 733. Kline, S.J., Reynolds, W.C., Schraub, F.A., Rundstadler, P.W., 1967. The structure of turbulent boundary layers. Journal of Fluid Mechanics 30, 741 – 773. Kolmogorov, A.N., 1941. The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Doklady Akademii Nauk SSSR 30, 299 – 301 (English translation in Proceedings of the Royal Society of London Series A 434, 15–17, 1991). Kostachuk, R., 2000. A field study of turbulence and sediment dynamics over subaqueous dunes with flow separation. Sedimentology 47, 519 – 531. Kraichnan, R.H., 1976. Eddy-viscosity in two and three dimensions. Journal of Atmospheric Science 33, 1521 – 1536. Kumar, P., Foufoula-Georgiou, E., 1994. Wavelet analysis in geophysics: an introduction. In: Foufoula-Georgiou, E., Kumar, P. (Eds.), Wavelets in Geophysics. Academic Press, New York, 372 pp. Lane, S.N., Ferguson, R.I., 2005. Modelling reach scale fluvial flows. In: Bates, P.D., Lane, S.N., Ferguson, R.I. (Eds.), Computational Fluid Dynamics: Applications in Environmental Hydraulics. Lane, S.N., Chandler, J.H., Richards, K.S., 1994a. Developments in monitoring and terrain modelling small-scale river-bed topography. Earth Surface Processes and Landforms 19, 349 – 368. Lane, S.N., Richards, K.S., Chandler, J., 1994b. Application of distributed sensitivity analysis to a model of turbulent open channel flow in a natural river channel. Proceedings of the Royal Society of London Series A 447, 49 – 63. Lane, S.N., Bradbrook, K.F., Richards, K.S., Biron, P.A., Roy, A.G., 1999. The application of computational fluid dynamics to natural river channels: three-dimensional versus two-dimensional approaches. Geomorphology 29, 1 – 20. Lane, S.N., Hardy, R.J., Elliott, L., Ingham, D.B., 2002. High resolution numerical modelling of three-dimensional flows over complex river bed topography. Hydrological Processes 16, 2261 – 2272. Lane, S.N., Hardy, R.J., Elliott, L., Ingham, D.B., 2004. Numerical modelling of flow processes over gravelly-surfaces using structured grids and a numerical porosity treatment. Water Resources Research 40. Langford, J.A., Moser, R.D., 1999. Optimal LES formulations for isotropic turbulence. Journal of Fluid Mechanics 398, 321 – 346. Lardat, R., Leschziner, M., 1998. A Navier–Stokes solver for LES on parallel computers. Technical Report, Department of Mechanical Engineering UMIST. Launder, B.E., Spalding, D.B., 1974. The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering 3, 269 – 289.

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304 Launder, B.E., Reece, G.J., Rodi, W., 1975. Progress in the development of a Reynolds-stress turbulence closure. Journal of Fluid Mechanics 68, 537 – 566. Lawless, M., Robert, A., 2001. Three-dimensional flow structure around small-scale bedforms in a simulated gravel-bed environment. Earth Surface Processes and Landforms 26, 507 – 522. Le, H., Moin, P., Kim, J., 1997. Direct numerical simulation of turbulent flow over a backward-facing step. Journal of Fluid Mechanics 330, 349 – 374. LeClair, S.F., Bridge, J.S., 2001. Quantitative interpretation of sedimentary structures formed by river dunes. Journal of Sedimentary Research 71, 713 – 716. Lee, C., 2002. Large-eddy simulation of rough wall turbulent boundary layers. AIAA Journal 40, 2127 – 2130. Leonard, A., 1974. Energy cascade in large-eddy simulations of turbulent fluid flows. Advances in Geophysics 18A, 237 – 248. Leonard, B.P., 1997. Bounded higher-order upwind multidimensional finite-volume convection–diffusion algorithms. In: Minkowycz, W.J., Sparrow, E.M. (Eds.), Advances in Numerical Heat Transfer. Taylor and Francis, New York, pp. 1 – 57. Lesieur, M., Me´tais, O., 1996. New trends in large-eddy simulations of turbulence. Annual Review of Fluid Mechanics 28, 45 – 82. Lesieur, M., Comte, P., Lamballais, E., Me´tais, O., Silvestrini, G., 1997. Large-eddy simulations of shear flows. Journal of Engineering Mathematics 32, 195 – 215. Lesieur, M., Comte, P., Dubief, Y., Lamballais, E., Me´tais, O., Ossia, S., 1999. From two-point closures of isotropic turbulence to LES of shear flows. Flow, Turbulence and Combustion 63, 247 – 267. Lesieur, M., Begou, P., Briand, E., Danet, A., Delcayre, F., Aider, J.L., 2003. Coherent-vortex dynamics in large-eddy simulations of turbulence. Journal of Turbulence 4 (016), 1 – 24. Lien, F.S., Leschziner, M.A., 1994. Assessment of turbulencetransport models including non-linear RNG eddy-viscosity formulation and second-moment closure for flow over a backward-facing step. Computers and Fluids 23, 983 – 1004. Lilly, D.K., 1992. A proposed modification of the Germano subgrid-scale closure method. Physics of Fluids. A, Fluid Dynamics 4, 633 – 635. Mansfield, J.R., Knio, O.M., Meneveau, C., 1998. A dynamic LES scheme for the vorticity transport equation: formulation and a priori tests. Journal of Computational Physics 145, 693 – 730. Marsden, A.L., Vasilyev, O.V., Moin, P., 2002. Construction of commutative filters for LES on unstructured meshes. Journal of Computational Physics 175, 584 – 603. Mason, P.J., Callen, N.S., 1986. On the magnitude of the subgridscale eddy coefficient in large-eddy simulations of turbulent channel flow. Journal of Fluid Mechanics 162, 439 – 462. Mason, P.J., Thomson, D., 1992. Stochastic backscatter in largeeddy simulations of boundary layers. Journal of Fluid Mechanics 242, 51 – 78. Mathey, F., Frfhlich, J., Rodi, W., 1999. LES of heat transfer in turbulent flow over a wall-mounted matrix of cubes. In: Voke, P., Sandham, N., Kleiser, L. (Eds.), Direct and Large-Eddy Simulation III. Kluwer Academic, pp. 51 – 62. Mazumder, R., 2000. Turbulence–particle interactions and their implications for sediment transport and bedform mechanics

301

under unidirectional current: some recent developments. Earth Science Reviews 50, 113 – 124. McLean, S.R., 1990. The stability of ripples and dunes. EarthScience Reviews 29, 131 – 144. McLean, S.R., Nelson, J.M., Wolfe, S.R., 1994. Turbulence structure over two-dimensional bedforms: implications for sediment transport. Journal of Geophysical Research 99, 12729 – 12747. McLean, S.R., Wolfe, S.R., Nelson, J.M., 1999. Predicting boundary shear stress and sediment transport over bedforms. Journal of Hydraulic Engineering ASCE 125, 725 – 736. Mellen, C.P., Frohlich, J., Rodi, W., 2003. Lessons from LESFOIL project on large-eddy simulation of flow around an airfoil. AIAA Journal 41, 573 – 581. Mellor, G.L., Yamada, T., 1982. Development of a turbulence closure-model for geophysical fluid problems. Reviews of Geophysics 20, 851 – 875. Meneveau, C., Katz, J., 1999. Dynamic testing of subgrid models in large eddy simulation based on the Germano identity. Physics of Fluids 11, 245 – 247. Meneveau, C., Lund, T.S., Cabot, W.H., 1996. A Lagrangian dynamic subgrid-scale model of turbulence. Journal of Fluid Mechanics 319, 353 – 385. Me´tais, O., Lesieur, M., 1992. Spectral large-eddy simulations of isotropic and stably-stratified turbulence. Journal of Fluid Mechanics 239, 157 – 194. Meyers, J., Geurts, B.J., Baelmans, M., 2003. Database-analysis of errors in large-eddy simulation. Physics of Fluids 15, 2740 – 2755. Meyers, J., Geurts, B.J., Baelmans, M., 2005. Optimality of the dynamic procedure for large-eddy simulations. Physics of Fluids 17, art no. 045108. Miall, A.D., 1976. Paleocurrent and paleohydrologic analysis of some vertical profiles through a Cretaceous braided stream deposit, Banks Island, Arctic Canada. Sedimentology 23, 459 – 483. Middleton, G.V., 1965. Antidune cross-bedding in a large flume. Journal of Sedimentary Petrology 35, 922 – 927. Mittal, R., Moin, P., 1997. Suitability of upwind-biased finite difference schemes for large-eddy simulation of turbulent flows. AIAA Journal 35, 1415 – 1417. Moin, P., 2002. Advances in large eddy simulation methodology for complex flows. International Journal of Heat and Fluid Flow 23, 710 – 720. Moin, P., Kim, J., 1982. Numerical investigation of turbulent channel flow. Journal of Fluid Mechanics 118, 341 – 377. Moin, P., Mahesh, K., 1998. Direct numerical simulation: a tool in turbulence research. Annual Review of Fluid Mechanics 30, 539 – 578. Moin, P., Squires, K., Cabot, W., Lee, S., 1991. A dynamic subgridscale model for compressible turbulence and scalar transport. Physics of Fluids. A, Fluid Dynamics 3, 1766 – 1771. Mosley, M.P., 1976. An experimental study of channel confluences. Journal of Geology 84, 535 – 562. Murakami, S., Rodi, W., Mochida, A., Sakamoto, S., 1993. Large eddy simulation of turbulent vortex shedding flow past 2d square cylinders. Engineering Applications of Large

302

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

Eddy Simulations 1993, ASME FED Publication, vol. 162, pp. 113 – 120. Naden, P., 1987. An erosion criterion for gravel-bed rivers. Earth Surface Processes and Landforms 12, 83 – 93. Nelson, J.M., Shreve, R.L., McLean, S.R., Drake, T.G., 1995. Role of near-bed turbulence structure in bed load transport and bed load mechanics. Water Resources Research 31, 2071 – 2086. Nicholas, A.P., Smith, G.H.S., 1999. Numerical simulation of threedimensional flow hydraulics in a braided channel. Hydrological Processes 13, 913 – 929. Nicoud, F., Baggett, J.S., Moin, P., Cabot, W., 2001. Large eddy simulation wall modeling based on sub optimal control theory and linear stochastic estimation. Physics of Fluids 13, 2968 – 2984. Nikitin, N.V., Nicoud, F., Wasisto, B., Squires, K.D., Spalart, P.R., 2000. An approach to wall modelling in large-eddy simulations. Physics of Fluids 12, 1629 – 1632. Nowell, A.R.M., Church, M., 1979. Turbulent flow in a depthlimited boundary layer. Journal of Geophysical Research 84, 4816 – 4824. Olsen, N.R.B., Stokseth, S., 1995. 3-Dimensional numerical modeling of water-flow in a river with large bed roughness. Journal of Hydraulic Research 33, 571 – 581. Paola, C., Seal, R., 1995. Grain size patchiness as a cause of selective deposition and downstream fining. Water Resources Research 31, 1395 – 1407. Paola, C., Gust, G., Southard, J.B., 1986. Skin friction behind isolated hemispheres and the formation of obstacle marks. Sedimentology 33, 279 – 293. Paiement-Paradis, G., Buffin-Be´langer, T., Roy, A.G., 2003. Scalings for large turbulent flow structures in gravel-bed rivers. Geophysical Research Letters 30, 1773. Parsons, D.R., 2003a. Flow separation in meander bends. Unpublished PhD Thesis. University of Sheffield, Department of Geography. Parsons, D.R., 2003b. Discussion of bThree-dimensional numerical study of flows in open-channel junctionsQ by Jianchun Huang, Larry J. Weber, and Yong G. Lai. Journal of Hydraulic Engineering 129, 822 – 823. Patel, V.C., 1998. Perspective: flow at high Reynolds number and over rough surfaces—Achilles heel of CFD. Journal of Fluids Engineering 120, 434 – 444. Piomelli, U., 1993. High Reynolds number calculations using the dynamic subgrid-scale stress model. Physics of Fluids. A, Fluid Dynamics 5, 1484 – 1490. Piomelli, U., Balaras, E., 2002. Wall-layer models for largeeddy simulations. Annual Review of Fluid Mechanics 34, 349 – 374. Piomelli, U., Liu, J., 1995. Large-eddy simulation of rotating channel flows using a localized dynamic model. Physics of Fluids 7, 839 – 848. Piomelli, U., Ferziger, J., Moin, P., Kim, J., 1989. New approximate boundary conditions for large eddy simulations of wall-bounded flows. Physics of Fluids. A, Fluid Dynamics 1, 1061 – 1068. Piomelli, U., Zang, T.A., Speziale, C.G., Hussaini, M.Y., 1990. On the large-eddy simulation of transitional wall bounded flows. Physics of Fluids. A, Fluid Dynamics 2, 257 – 265.

Porte´-Agel, F., Meneveau, C., Parlange, M.B., 2000. A scaledependent dynamic model for large-eddy simulation: application to a neutral atmospheric boundary layer. Journal of Fluid Mechanics 415, 261 – 284. Prandtl, L., 1925. Bericht qber Untersuchungen zur ausgebildeten Turbulenz. Zeitschrift fqr Angewandte Mathematik und Mechanik 5, 136 – 139. Rai, M.M., Moin, P., 1991. Direct simulation of turbulent flows using finite-difference schemes. Journal of Computational Physics 96, 15 – 53. Raupach, M.R., Antonia, R.A., Rajagopalan, S., 1991. Roughwall turbulent boundary layers. Applied Mechanics Reviews 44, 1 – 25. Reynolds, O., 1895. On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philosophical Transactions of the Royal Society of London. Series A 186, 123 – 164. Richards, K.J., 1980. The formation of ripples and dunes on an erodible bed. Journal of Fluid Mechanics 99, 597 – 618. Roache, P.J., 1997. Quantification of uncertainty in computational fluid dynamics. Annual Review of Fluid Mechanics 29, 123 – 160. Robert, A., Roy, A.G., De Serres, B., 1993. Space-time correlations of velocity measurements at a roughness transition in a gravelbed river. In: Clifford, N.J., French, F.R., Hardisty, J. (Eds.), Turbulence: Perspectives on Flow and Sediment Transport. John Wiley and Sons, Chichester, UK. Robinson, S.K., 1991. Coherent motions in the turbulent boundary layer. Annual Review of Fluid Mechanics 23, 601 – 639. Rodi, W., 1993. On the simulation of turbulent flow past bluff bodies. Journal of Wind Engineering and Industrial Aerodynamics 46–47, 3 – 9. Rodi, W., 2002. Large-eddy simulation of the flow past bluff bodies. In: Launder, B.E., Sandham, N.D. (Eds.), Closure Strategies for Turbulent and Transitional Flows. Cambridge University Press, Cambridge, 768 pp. Rodi, W., Ferziger, J.H., Breuer, M., Pourquie, M., 1997. Status of large eddy simulation: results of a workshop. ASME Journal of Fluids Engineering 119, 248 – 262. Rogallo, R.S., Moin, P., 1984. Numerical simulation of turbulent flows. Annual Review of Fluid Mechanics 16, 99 – 137. Rotta, J.C., 1951. Statistische Theorie nichthomogener Turbulenz. Zeitschrift fqr Physik 129, 547 – 572. Roy, A., Lane, S., 2003. Putting the morphology back into fluvial geomorphology: the case of river meander and tributary junctions. In: Trudgill, S., Roy, A. (Eds.), Contemporary Meaning in Physical Geography. Arnold, London, 292 pp. Roy, A.G., Biron, P.M., Buffin-Be´langer, T., Levasseur, M., 1999. Combined visual and quantitative techniques in the study of natural turbulent flows. Water Resources Research 35, 871 – 877. Sagaut, P., 1996. Simulations of separated flows with subgrid models. La Recherche Ae´rospatiale 1, 51 – 63. Sagaut, P., 1998. Large Eddy Simulation for Incompressible Flows. Springer, Berlin. Salvetti, M.V., Banarjee, S., 1995. A priori tests of a new dynamic subgrid-scale model for finite-difference large eddy simulations. Physics of Fluids 7, 2831 – 2847.

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304 Salvetti, M.V., Damiani, R., Beux, F., 2001. Three-dimensional coarse large-eddy simulations of the flow above two-dimensional sinusoidal waves. International Journal of Numerical Methods in Fluids 35, 617 – 642. Schlichting, H., 1979. Boundary Layer Theory, 7th edition. McGraw-Hill, New York. Schmeeckle, M.W., Nelson, J.M., 2003. Direct numerical simulation of bedload transport using a local, dynamic boundary condition. Sedimentology 50, 279 – 301. Schmidt, S., Thiele, F., 2002. Comparison of numerical methods applied to the flow over wall-mounted cubes. International Journal of Heat and Fluid Flow 23, 330 – 339. Schumann, U., 1975. Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. Journal of Computational Physics 18, 376 – 404. Schumann, U., 1977. Realizability of Reynolds-stress turbulence models. Physics of Fluids 20 (5), 721 – 725. Schumann, U., 1991. Direct and large eddy simulation of turbulence—summary of the state-of-the-art 1991. Lecture Series 1991–02. Introduction to the Modelling of Turbulence. Von Karman Institute, Brussels. Shen, L., Yue, D.K.P., 2001. Large-eddy simulation of free-surface turbulence. Journal of Fluid Mechanics 440, 75 – 116. Shen, L., Zhang, X., Yue, D.K.P., Triantafyllou, G.S., 1999. The surface layer for free-surface turbulent flows. Journal of Fluid Mechanics 386, 167 – 212. Shi, J., Thomas, T.G., Williams, J.J.R., 2000. Free-surface effects in open channel flow at moderate Froude and Reynolds numbers. Journal of Hydraulic Research 38, 465 – 474. Shur, M., Spalart, P.R., Strelets, M., Travin, A., 1999. Detached-eddy simulation of an airfoil at high angle of attack. In: Rodi, W., Laurence, D. (Eds.), Engineering Turbulence: Modelling and Experiments, vol. 4. Elsevier, Amsterdam, pp. 669 – 678. Silveira Neto, A., Grand, D., Me´tais, O., Lesieur, M., 1991. Largeeddy simulation of the turbulent flow in the downstream region of a backward-facing step. Physical Review Letters 66, 2320 – 2323. Silveira Neto, A., Grand, D., Me´tais, O., Lesieur, M., 1993. A numerical investigation of the coherent vortices in turbulence behind a backward-facing step. Journal of Fluid Mechanics 256, 1 – 25. Simpson, R.L., 1989. Turbulent boundary-layer separation. Annual Review of Fluid Mechanics 21, 205 – 234. Smagorinsky, J., 1963. General circulation experiments with the primitive equations: Part I. The basic experiment. Monthly Weather Review 91, 99 – 167. Smith, N.D., 1971. Transverse bars and braiding in the Lower Platte River, Nebraska. Geological Society of America Bulletin 82, 3407 – 3420. Smith, C.R., 1996. Coherent flow structures in smooth-wall turbulent boundary layers: facts, mechanisms and speculation. In: Ashworth, P.J., Bennett, S.J., Best, J.L., McLelland, S.J. (Eds.), Coherent flow Structures in Open Channels. Wiley and Sons, Chichester. 733 pp. Smith, J.D., McLean, S.R., 1977. Spatially averaged flow over a wavy surface. Journal of Geophysical Research 82, 1735 – 1746.

303

Smith, L.M., Woodruff, S.L., 1998. Renormalization-group analysis of turbulence. Annual Review of Fluid Mechanics 30, 275 – 310. Smith, L.C., Turcotte, D.L., Isacks, B.L., 1998. Stream flow characterization and feature detection using a discrete wavelet transform. Hydrological Processes 12, 233 – 249. Soulsby, R.L., Atkins, R., Salkeld, P., 1994. Observations of the turbulent structure of a suspension of sand in a tidal current. Continental Shelf Research 14, 429 – 435. Spalart, P.R., Allmaras, S.R., 1994. A one-equation model for aerodynamic flows. La Recherche Ae´rospatiale 1, 5 – 21. Spalart, P.R., Jou, W.H., Strelets, M., Allmaras, S., 1997. Comments on the feasibility of LES for wings and on a hybrid LES/RANS approach. In: Liu, C., Liu, Z. (Eds.), Advances in DNS/LES. Greydon Press. Speziale, C.G., 1987. On nonlinear K–l and K–e models of turbulence. Journal of Fluid Mechanics 178, 459 – 475. Speziale, C.G., Sarkar, S., Gatski, T.B., 1991. Modelling the pressure–strain correlation of turbulence: an invariant dynamical systems approach. Journal of Fluid Mechanics 227, 245 – 272. Stolz, S., Adams, N.A., 1999. An approximate deconvolution procedure for large-eddy simulation. Physics of Fluids 11, 1699 – 1701. Stolz, S., Adams, N.A., Kleiser, L., 2001. An approximate deconvolution model for large eddy simulation with application to incompressible wall-bounded flows. Physics of Fluids 13, 997 – 1015. Stull, R.B., 1988. An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, Dordrecht, 666 pp. Sweldens, W., 1997. The lifting scheme: a construction of second generation wavelets. SIAM Journal on Mathematical Analysis 29, 511 – 546. Temmerman, L., Leschziner, M.A., Mellen, C.P., Frohlich, J., 2003. Investigation of wall-function approximations and subgrid-scale models in large eddy simulation of separated flow in a channel with streamwise periodic constrictions. International Journal of Heat and Fluid Flow 24, 157 – 180. Torrence, C., Compo, G.P., 1998. A practical guide to wavelet analysis. Bulletin of the American Meteorological Society 79, 61 – 78. Tribe, S., Church, M., 1999. Simulations of cobble structure on a gravel streambed. Water Resources Research 35, 311 – 318. Tucker, P.G., Davidson, L., 2004. Zonal k–l based large eddy simulations. Computers and Fluids 33, 267 – 287. Van Driest, E.R., 1956. On the turbulent flow near a wall. Journal of Aeronautical Sciences 23, 1007 – 1011. Van Rijn, L.C., 1984. Sediment pick-up functions. Journal of Hydraulic Engineering ASCE 110, 1494 – 1502. Vasilyev, O.V., Lund, T.S., Moin, P., 1998. A general class of commutative filters for LES in complex geometries. Journal of Computational Physics 146, 82 – 104. Vassilicos, J.C., 2002. Mixing in vortical, chaotic and turbulent flows. Philosophical Transactions of the Royal Society of London Series A 360, 2819 – 2837. Vreman, B., 2003. The filtering analog of the variational multiscale method in large-eddy simulation. Physics of Fluids 15, L61 – L64.

304

C.J. Keylock et al. / Earth-Science Reviews 71 (2005) 271–304

Vreman, B., Geurts, B., Kuerten, H., 1994. Realizability conditions for the turbulent stress tensor in large eddy simulation. Journal of Fluid Mechanics 278, 351 – 362. Vreman, B., Geurts, B., Kuerten, H., 1996a. Large eddy simulation of the temporal mixing layer using the Clark model. Theoretical and Computational Fluid Dynamics 8, 309 – 324. Vreman, B., Geurts, B., Kuerten, H., 1996b. Comparison of numerical schemes in large eddy simulation of the temporal mixing layer. International Journal of Numerical Methods in Fluids 22, 297 – 311. Vreman, B., Geurts, B., Kuerten, H., 1997. Large eddy simulation of the turbulent mixing layer. Journal of Fluid Mechanics 339, 357 – 370. Wang, M., Moin, P., 2001. Wall modeling in LES of trailing-edge flow. Second International Symposium of Turbulence and Shear Flow Phenomena, vol. 2, pp. 165 – 170. Werner, H., Wegle, H., 1991. Large-eddy simulation of turbulent flow over and around a cube in a plate channel. 8th Symposium on Turbulent Shear Flows, pp. 155 – 168. Westerweel, J., 1997. Fundamentals of digital particle imaging velocimetry. Measurement Science and Technology 8, 1379 – 1392. Wiberg, P.L., Smith, J.D., 1985. A theoretical model for saltating grains in water. Journal of Geophysical Research 90, 7341 – 7354. Wiberg, P.L., Smith, J.D., 1987. Calculations of the critical shear stress for motion of uniform and heterogeneous sediments. Water Resources Research 23, 1471 – 1480.

Williams, J.J., Bell, P.S., Thorne, P.D., 2003a. Field measurements of flow fields and sediment transport above mobile bed forms. Journal of Geophysical Research 108, 3109. Williams, T.C., Hargrave, G.K., Halliwell, N.A., 2003b. The development of high-speed particle image velocimetry (20 kHz) for large eddy simulation code refinement in bluff body flows. Experiments in Fluids 35, 85 – 91. Yakhot, V., Orszag, S.A., 1986. Renormalization group analysis of turbulence: I. Basic Theory. Journal of Scientific Computing 1, 3 – 51. Yoshizawa, A., 1984. Statistical analysis of the deviation of the Reynolds stress from its eddy-viscosity representation. Physics of Fluids. A, Fluid Dynamics 27, 1377 – 1387. Yoshizawa, A., Horiuti, K., 1985. A statistically-derived subgridscale kinetic energy model for the large-eddy simulation of turbulent flows. Journal of the Physical Society of Japan 54, 2834 – 2839. Zang, Y., Street, R.L., Koseff, J.R., 1993. A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Physics of Fluids. A, Fluid Dynamics 5, 3186 – 3196. Zang, Y., Street, R.L., Koseff, J.R., 1994. A non-staggered grid, fractional step method for the time-dependent incompressible Navier–Stokes equations in curvilinear coordinates. Journal of Computational Physics 114, 18 – 33. Zedler, E.A., Street, R.L., 2001. Large-eddy simulation of sediment transport: currents over ripples. Journal of Hydraulic Engineering 127, 444 – 452.