## Abstract

This paper provides a discussion of several aspects of the construction of approaches that combine statistical (Reynolds-averaged Navier–Stokes, RANS) models with large eddy simulation (LES), with the objective of making LES an economically viable method for predicting complex, high Reynolds number turbulent flows. The first part provides a review of alternative approaches, highlighting their rationale and major elements. Next, two particular methods are introduced in greater detail: one based on coupling near-wall RANS models to the outer LES domain on a single contiguous mesh, and the other involving the application of the RANS and LES procedures on separate zones, the former confined to a thin near-wall layer. Examples for their performance are included for channel flow and, in the case of the zonal strategy, for three separated flows. Finally, a discussion of prospects is given, as viewed from the writer's perspective.

## 1. Introduction

The often perplexing predictive variability of Reynolds-averaged Navier–Stokes (RANS) models and their persistent failure in complex three-dimensional flows has encouraged the general view that large eddy simulation (LES) is the only credible route to computing three-dimensional flows, in general, and separated flows, in particular. Unsurprisingly, in view of the fact that LES resolves all important large-scale dynamics of the turbulent motion, LES is indeed observed to be able to provide a far greater degree of predictive realism and consistency than even the most elaborate RANS models. However, LES is far from infallible, and it also poses substantial challenges that are often underrated and understated. It requires non-diffusive—hence, potentially unstable—discretization schemes; it imposes highly restrictive cell aspect-ratio limits; it requires a full spectral description of any turbulent inflow into the solution domain; and it is very costly if performed in such a manner as to render subgrid-scale modelling dynamically uninfluential. Particularly problematic are flows at high Reynolds numbers in which frictional wall effects exercise a significant influence on the gross flow features. This is the case in separation from continuous surfaces having weak to moderate curvature. The problem posed by such flows is shown in figure 1, a redrawn version of one presented by Piomelli & Balaras (2002). The figure may be taken to be representative of the requirements for a channel flow, a jet or a boundary layer, for example, with the thickness of the shear layer used to form the Reynolds number. When *Re*_{L} exceeds roughly 10^{5}, the resource requirements are entirely dominated by the need to resolve the near-wall region (the inner layer), with the number of nodes rising roughly as . Because the resolution requirements depend greatly on the definition of the Reynolds number and the geometric complexity of the flow, figure 1 must be viewed with considerable caution; in many circumstances, the resolution will have to be significantly higher than indicated by figure 1. As an example, at *Re*_{L}=200 000, figure 1 suggests that a wall-resolved LES would require around 5M nodes. However, for the example of the wing shown in figure 1 and operating at the same Reynolds number, based on the mid-span chord, a 24M-node, block-structured grid used by Li & Leschziner (2007*b*) still involved wall-normal cell dimensions of order *y*^{+}=20–50 in the near-wall cells. For this case, a wall-resolved LES would require of the order of 500M nodes. Simulations for practical configurations would not, in most circumstances, be undertaken today with meshes exceeding 50 million nodes, corresponding to *Re*_{L}≈5×10^{5} in figure 1, with a much lower value (perhaps as low as 10^{5}) applicable to complex conditions. Thus, if the near-wall layer is the region of prime interest, or if this region affects significantly the gross characteristics of the outer flow, wall-resolving LES quickly becomes economically untenable as the Reynolds number rises.

The general approach taken in recent years towards addressing this problem has been to combine RANS-type modelling near the wall with LES in the outer flow. The key premise underpinning this strategy is that it should allow numerical cells to be used that have far higher aspect ratios than those required by LES. To take a channel flow or a boundary layer as an example, LES resolution constraints at the wall typically impose aspect-ratio limits of the order of (where *x* and *z* are the streamwise and spanwise directions, respectively), while (steady) RANS computations are known to tolerate aspect-ratio values of 1000 and higher. The implication is thus that substantial savings could be made by using much coarser streamwise and spanwise meshes than are dictated by the LES constraints.

Several methodologies of the above type have been proposed over the past few years, some reviewed later. All involve significant uncertainties, however, which may be claimed to limit their usefulness, sometimes to the extent of seriously compromising the very reason for adopting a LES in preference to a RANS method. Some of the major questions and issues that require consideration are listed below.

First, although the RANS and LES equations appear to have the same form, this superficial similarity is misleading, as the underlying principles of their derivation, and thus the physical rationale, are very different. The RANS equations arise from temporal integration, while the LES equations emerge from a spatial filtering operation that excludes, in a predetermined way, high-order components from the turbulence spectrum. Thus, the Reynolds stresses in the RANS equations are very different in nature from the subgrid-scale stress in the filtered LES equations.

Second, the use of RANS models as part of a highly unsteady LES environment is questionable, as RANS models were formulated and calibrated by reference to steady, close-to-equilibrium conditions. Such models are designed to represent, statistically, the

*totality*of turbulence effects, not merely a portion of them.Third, as noted already, the basic rationale of using a RANS method in the near-wall region, as part of a RANS–LES strategy, rests on the observation that RANS computations of

*steady*boundary-layer-type flows can tolerate very high aspect ratios of the near-wall cells, without loss of accuracy, because wall-parallel convection is low. However, this does not extend to highly unsteady conditions, in which case unsteady wall-parallel fluctuations result in (instantaneously) large velocity gradients parallel to the wall and thus substantial wall-parallel convection.Finally, there are a whole host of practical issues associated with numerical and model-related coupling or blending between the RANS and LES subdomains.

These questions remain unanswered, at present, but practical needs dictate that pragmatic ‘solutions’ are formulated and tested, for, without them, LES cannot currently be applied to high Reynolds number flows that are sensitive to frictional wall characteristics. Specifically in respect of separation from curved surfaces, an important question is whether the approximate near-wall treatments seriously compromise the ability of the baseline LES scheme to capture the dynamics of the separation process, due to the inevitable loss of structural information near the wall, in addition to the inaccuracies inherent in the use of RANS models in highly unsteady conditions. This is one of the subjects discussed below.

## 2. Brief review of combined LES–RANS methodologies

Over the past few years, a whole range of approaches to the general strategy of combining LES with RANS models have been proposed, and this may be taken to reflect, as in the case of RANS, general uncertainty on the most appropriate way forward. Excluding very few exceptional strategies, alternative approximations may be grouped into the following categories:

*Wall laws or wall functions*. This is the oldest method and is based on analytical relationships of the form reflecting the assumption of equilibrium in the turbulence that is modelled. These allow the unsteady wall shear stress to be extracted from the near-wall, wall-parallel velocity that is resolved by the coarse-grid LES at a wall-normal distance*y*, which is well removed from the wall (i.e. at a universal wall distance*y*^{+}typically greater than 30). The shear stress is then used as the wall boundary condition for the LES.*Zonal schemes*. These employ the solution of (usually) parabolized RANS equations over a separate, thin near-wall-layer grid superimposed on the relatively coarse LES grid. The solution of the RANS equations, subject to boundary conditions extracted from the LES, then yields the wall shear stress, which is used, as in the case of (i), as the wall boundary condition for the LES.*Seamless schemes*. Usually these employ a single RANS-type formulation across the entire flow, but switch (discontinuously or gradually) the modelled length scale from a grid-related dimension, or a prescribed cut-off eddy size, to the macro-turbulent length scale (say,*k*^{3/2}/*ϵ*or the wall-normal distance*y*) as the wall is approached.*Hybrid schemes*. A single, continuous numerical method is used to accommodate both the LES solution and the near-wall RANS solution, but with separate subgrid-scale and RANS models used in the predefined LES and near-wall regions, respectively, with matching conditions imposed at the interface separating the two regions.

Exceptional schemes that do not fall neatly into any of the above include that of Hamba (2001), which comprises a RANS solution in the *outer* region and a LES solution *near the wall*, and the method of Uribe *et al*. (2007), which blends RANS stresses derived from the on-the-run time-averaged LES field upon its insertion into the RANS turbulence model with the subgrid-scale stresses derived from the evolving LES solution.

The use of equilibrium-flow *wall functions* goes back to early proposals by Deardorff (1970) and Schumann (1975), and a number of versions have subsequently been investigated, which are either designed to satisfy the log-law in the time-averaged field or, more frequently, involve an explicit log-law or closely related power-law prescription of the instantaneous near-wall velocity (e.g. Werner & Wengle 1991; Hoffman & Benocci 1995; Temmerman *et al*. 2003). These can provide useful approximations in near-equilibrium conditions, e.g. channel flow, but generally will not give an adequate representation in non-equilibrium and separated flows, especially at moderate Reynolds numbers, where the near-wall flow often lacks a distinct log-law region.

*Zonal schemes* have been proposed or applied by Balaras *et al*. (1996), Cabot & Moin (2000), Wang & Moin (2000) and Tessicini *et al*. (2006), among others. In all cases, unsteady forms of the boundary-layer (or thin-shear-layer) equations are solved across an inner layer of prescribed thickness, which is covered with a fine wall-normal mesh, with a conventional RANS turbulence model providing the eddy viscosity. In the simplest form of the approach, this layer is essentially decoupled from the LES region, in so far as the pressure field just outside the inner layer is imposed across the layer, and the wall-parallel diffusion is ignored, rendering the equations parabolic. The wall-normal velocity is then determined from an explicit application of the mass-continuity constraint, one consequence being a discontinuity in this velocity at the outer RANS boundary. As in the case of the wall-function approach, the principal information extracted from the RANS computation is the wall-shear stress or a velocity at a particular wall-normal distance, which is then fed into the LES solution as an unsteady boundary condition.

Different types of blending and *seamless schemes* have been proposed by Spalart *et al*. (1997), Dejoan & Schiestel (2001), Girimaji *et al*. (2003), Abe (2005) and Chaouat & Schiestel (2005). The best known realization of this strategy is the detached eddy simulation (DES) method of Spalart *et al*. (1997), recently improved by Spalart *et al*. (2006) and denoted by Delayed DES (DDES). Yet another version, the I(mproved)DDES (Travin *et al*. 2006), permits attached boundary layers ahead of separation to be included in the LES branch of the DES, but this method is, arguably, no longer a DES method, but a hybrid RANS–LES scheme. In the original scheme, the RANS and LES models are not blended, but a length-scale limiter simply switches from RANS to LES or vice versa. The interface location is dictated by the switching condition . A single turbulence model, normally the one-equation Spalart–Allmaras model (Spalart & Allmaras 1994), is used, both as a RANS model in the inner region and as a subgrid-scale model in the outer LES region. The key feature of this method is that it enforces the interface location by reference to the streamwise and spanwise grid density. In general flows, this density often needs to be high to achieve adequate resolution of complex geometric and flow features, both close to the wall (e.g. separation and reattachment) and away from it. Thus, the interface can be forced to be close to the wall, often as near as *y*^{+}=20, defeating the rationale and objective of DES. Moreover, if a streamwise grid refinement is effected rapidly in locations of high adverse pressure gradient, so that the RANS region is made to thin rapidly in accordance with the refinement, DES is observed to cause premature separation, and this has led to the introduction of additional separation-delaying fixes that are documented in Spalart *et al*. (2006). It has also been repeatedly observed, especially at high Reynolds numbers and coarse grids, with the interface location being around *y*^{+}=100–200, that the high turbulent viscosity generated by the turbulence model in the inner region extends, as subgrid-scale viscosity, deep into the outer LES region, causing severe damping in the resolved motion and a misrepresentation of the resolved structure as well as of the time-mean properties. A method by Batten *et al*. (2002), referred to as the ‘Limited Numerical Scales’ approach, is not strictly a seamless method, but is similar to DES, in principle, the major difference being that switching between the LES and RANS regions is controlled by the relative values of the RANS and LES viscosities returned by separate subgrid-scale and RANS models, rather than by a comparison of the length scales . The method of Abe (2005) combines a modified quadratic eddy-viscosity model with the DES-type switching strategy to determine the interface location. The ‘Partially Integrated Transport Model’ approach of Dejoan & Schiestel (2001) and Chaouat & Schiestel (2005), the former using an eddy-viscosity model and the latter a second-moment (Reynolds stress transport) closure, rests on the concept of partitioning the turbulence spectrum into resolved and modelled parts, and the use of a single set of turbulence-model equations, in which some constants are sensitized to grid dimensions and the distance from the wall, to control the relative contribution of the resolved and modelled parts. Finally, the ‘Partially Averaged NS Method’ of Girimaji *et al*. (2003) is a simple form of the method by Dejoan and Schiestel, in so far as it uses blending functions that depend on the ratio of grid distance and turbulent length scale to gradually change the RANS model near the wall to a subgrid-scale model away from it.

In common with seamless schemes, hybrid schemes are implemented within a single numerical strategy right down to the wall. However, different models are used in the outer LES region and in the inner near-wall layer, with matching conditions imposed at the interface. Schemes of this type have been proposed by Davidson & Peng (2003), Temmerman *et al*. (2004) and Davidson & Billson (2006), the last entailing the injection of synthesized turbulence at the RANS–LES interface to combat the effect of small-scale and shear-stress depletion caused by excessive turbulence damping around the interface. The variants differ in respect of the choice of models and the precise approach to interfacing them. The method by Temmerman *et al*. will be explained in some detail below, and this will provide a better flavour of the nature of schemes within this category.

## 3. Specific LES–RANS combinations

Recent efforts by the author and his associates have focused on two particular methods, one being a hybrid LES–RANS scheme and the other being a two-layer zonal scheme. Both are documented in detail by Temmerman *et al*. (2004) and Tessicini *et al*. (2006). The difference between them is explained by reference to figure 2, which conveys the manner by which the LES and RANS regions communicate numerically.

The hybrid method uses a single computational domain. Within a predefined layer near the wall, which can be prescribed in terms of *y*^{+}, RANS equations are solved using one-equation or two-equation eddy-viscosity models that are dynamically adjusted so as to comply with continuity of the modelled (eddy/subgrid-scale) viscosity across the interface, , beyond which a LES subgrid-scale model is used. If, for example, a two-equation *k*−*ϵ* model is used in the RANS layer,(3.1)Next, the model coefficient *C*_{μ} at the interface is determined by comparison of the RANS viscosity, containing the RANS-determined turbulence energy and dissipation rate at the interface, with the LES viscosity at the interface as follows:(3.2)where ‘int’ denotes the interface and the bracket indicates the possibility of performing some averaging in a homogeneous direction to smooth the very large variation in time and space otherwise encountered. The RANS turbulence energy and, possibly, dissipation are determined from the solution of a related transport equation. The variation of *C*_{μ}(*y*) from the interface to the wall is then prescribed analytically, based on observations derived from *a priori* wall-resolved LES performed in channel flows. Among several forms investigated, the following one has been used in most simulations performed by the authors:(3.3)Details may be found in Temmerman *et al*. (2004).

The zonal method uses two overlapping grids across the near-wall layer. The LES grid extends to the wall, but is relatively coarse, maintaining cell aspect-ratio constraints appropriate to LES. Within the near-wall layer, a separate grid is inserted, which is refined towards the wall, typically to a near-wall node located at *y*^{+}*O*(1). Within that layer, parabolized RANS equations are solved for the wall-parallel velocity components, using a simple algebraic turbulence model, for example a mixing-length model,(3.4)Importantly, the wall-parallel pressure gradient in the near-wall layer is imposed on the layer from the LES solution, evaluated from the LES nodes closest to the interface. Thus, the pressure field need not be determined in that layer. In the simplest implementation, advection in the sublayer is also ignored, rendering the near-wall model one-dimensional and very cheap to solve. The reverse RANS-to-LES coupling involved only the extraction of the wall-shear stress from the parabolized equations and feeding it into the LES solution as a wall boundary condition.

The performance of both methods for channel and several separated flows is discussed in Temmerman *et al*. (2004) and Tessicini *et al*. (2006, 2007). Some channel-flow solutions for *Re*=42 000 (corresponding to a friction Reynolds number of *Re*_{τ}=2000) are shown in figures 3 and 4. The former compares solutions derived from the hybrid and zonal methods with a highly resolved LES and a coarse-grid LES without any wall treatment. The message conveyed by the figure is that the use of either wall model is extremely advantageous, although the upward inflections in the hybrid-scheme solution indicate that the level of turbulence, and hence turbulent shear stress, are too low around the interface. Figure 4 illustrates that some advantage can be derived from *not* averaging the constant *C*_{μ} in equation (3.2), thereby introducing at the interface small-scale fluctuations that would otherwise be removed by the spanwise averaging implied by the bracket in equation (3.2). Specifically, the figure illustrates, on the one hand, the response of the hybrid scheme with spatially averaged *C*_{μ} to an increase in the thickness of the RANS layer from *y*^{+}=120 to 280 and, on the other hand, the consequence of introducing the instantaneous *C*_{μ} as the thickness increases from *y*^{+}=280 to 610.

The performance of the zonal scheme in three separated flows is illustrated in figures 5–9. As an aside, it is noted that the first two cases required precursor boundary-layer simulations to be undertaken in order to enable the full time-dependent inflow conditions to be prescribed. This is clearly a requirement that can pose serious limitations on LES when it is used as a routine technique in a fair number of practical applications. The first case is a flow, at *Re*=2.16×10^{6} based on the chord, which separates from the upper side of a zero-camber aerofoil. In this simulation, the near-wall layer was placed at *y*^{+}=40 (evaluated just upstream of the curved section). A wall-resolved computation had been performed by Wang & Moin (2000) with 7 million nodes, while the present solution was obtained with a mesh of 1.5 million nodes. Figure 5 shows distributions of the time-mean skin-friction coefficient, predicted with and without the pressure gradient included in the parabolized equations in the RANS layer (*h* is the aerofoil thickness). Unsurprisingly, the wall-function implementation yields a result similar to that of the zonal scheme when this is applied without the pressure gradient being taken into account in equation (3.4). The skin friction is a sensitive indicator of the realism of the near-wall solution, and the results are certainly encouragingly close to the benchmark solution. Comparisons of velocity and turbulence intensity profiles, reported by Tessicini *et al*. (2006), also show good agreement, especially for the zonal scheme with the pressure gradient included.

The second example is the flow around a three-dimensional, axially symmetric hill placed on the lower wall of a rectangular duct. This flow, at a Reynolds number of 130 000 (based on the hill height and the velocity in the irrotational central portion of the duct flow), has been the subject of extensive experimental studies by Simpson *et al*. (2002) and Byun & Simpson (2005). Results reported herein are taken from a broader exposition given in a recent paper by Tessicini *et al*. (2007). The size of the computational domain is 16*H*×3.205*H*×11.67*H*, with *H* being the hill height. The hill crest is 4*H* downstream of the inlet plane. The inlet boundary layer, at −4*H*, was generated by a combination of RANS and LES precursor calculations, the former matching the experimental mean-flow data and the latter providing the spectral content. Simulations were undertaken with meshes containing between 1.5 and 36.7 million nodes, the finest-grid simulation being fully wall-resolving and reported in Li & Leschziner (2007*a*). Coarser grids were used in conjunction with the zonal scheme, wherein the interface was placed within *y*^{+}=20–40 and 40–60, using 3.5 and 1.5 million nodes, respectively.

Figure 6 shows predicted hill-surface topology maps, identifying the separation patterns that are returned by the various simulations. These were constructed from the wall-parallel velocity component closest to the wall. The experimental field is shown in figure 6*c*. An important point to make first, by reference to the wall-resolving 36.7-mio-cell simulation, is that the topology derived from the near-wall velocity field varies greatly with a wall-normal distance below *y*^{+}≈40. In fact, the separation point on the hill centre-line changes from *x*/*H*=0.8 to 0.3 as *y*^{+} reduces from 40 to 2. This is a substantial variation, also confirmed by another fine-grid simulation, reported together with the present simulation in García-Villalba *et al*. (2009), and suggests, on its own, that approximate near-wall formulations of the type considered herein are subjected to highly challenging conditions in the presence of three-dimensional separation from curved surfaces. The experimental topology is seen to be close to that derived from the fine-grid simulation at *y*^{+}=40, and this suggests lack of experimental resolution very close to the wall. As noted earlier, the near-wall velocity layer resolved with the zonal scheme lies at around *y*^{+}=20–40 or 40–60, depending on the coarseness of the grid, and it seems entirely defensible, in view of the above observations, to compare the solutions derived on these meshes with the experimental topology, as shown in figure 6. The designation ‘LES’ attached to the topology obtained from the 9.6-mio-cell simulation is to signify that no wall model was used in that case. The designation ‘no wall model’ indicates that the simulation in question used a coarse grid that was meant for the zonal scheme, but with the zonal scheme disabled, i.e. this is again a pure LES computation, although one that is far from being wall-resolving. Perhaps the most important fact conveyed by figure 6 is that no separation at all is predicted with the 1.5 million node grid when a no-slip condition is applied, while a fair representation of the separation pattern is returned with the zonal scheme. Extensive comparisons included in Tessicini *et al*. (2007) demonstrate fairly good agreement with the experiments in respect of all flow properties examined, agreement that is incomparably better than that achieved with even the most elaborate second-moment-closure RANS models (Wang *et al*. 2004).

The final flow considered here combines physical with geometric complexity. In this particular case, a log-law-based wall model has been used, rather than a more elaborate hybrid or zonal scheme. The geometry, shown in figure 7, is a highly swept, high-incidence wing, giving rise to flow separation from the suction side. The wing is subjected to an incoming flow at 8.8° incidence and the Reynolds number, based on the root-chord length *C* and free-stream velocity *U*_{∞}, is approximately 210 000. The domain size is 5*C*×4*C*×6.13*C*. The numerical mesh, organized in 256 blocks, contains 23.6 million (384×192×320) nodes and was generated with careful attention to resolution around the leading-edge, trailing-edge and wing-tip regions. While this grid may be regarded fine, it is far from wall-resolving, the wall distance at some high-shear regions above the wing being of the order of *y*^{+}=20. Hence, a representation of the wall layer through a wall model is important. The results included herein have been taken from Li & Leschziner (2007*b*).

A major feature of this flow is the formation of two strong leading-edge vortices, illustrated in figure 7 by iso-surfaces of the ‘Q-criterion’, , which brings out the mainly rotational part of the flow. In the inset, the highlighted areas are those low-pressure regions that naturally accompany the high-speed flow within the core of the leading-edge vortices. The simulation is assessed by reference to a complementary experimental study at University of Manchester, in which flow visualizations and extensive laser Doppler anemometer/particle image velocimetry measurements are emerging at the time of writing. Statistical results presented below are based on averaging over 25*U*_{∞}/*C*, or five flow-through times. Mean surface streaklines (equivalent to the experimental oil-film visualization) are compared in figure 8. Both experiment and simulation feature bifurcation lines at which the leading-edge vortices separate from the mainly attached-flow region above the central portion of the wing. Both the position and size of the leading-edge vortices are well predicted, except for the fact that the predicted vortices appear to break down when approaching the trailing edge, as indicated by the sharp spanwise-turning streaklines and the highly turbulent region over the wing tips in figure 7, while the experiment indicates more stable vortical structures. The simulation also reveals secondary vortices at the rear part of the wing, identified by a pair of focal points, which do not arise in the experiment. An even more important difference relates to the flow close to the wing tips, where the simulation fails to resolve a separation line formed by the collision of outward-moving fluid along the main wing surface and the inward-moving fluid from the wing tip (highlighted by the curved arrows inserted into the experimental topology plot). However, this difference may well be due to insufficient resolution away from the wall, rather than being indicative of specific defects in the near-wall representation (figure 1). Greater clarity can only emerge from finer grid simulations, but these would inevitably be very resource intensive.

Figure 8*c* compares the predicted and experimental distances of the leading-edge vortex from the wing surface. Again, agreement is fairly close, except for the rear region in which early vortex breakdown is predicted. Finally, figure 9 presents sets of streamwise and spanwise velocity profiles normal to the suction side at 50 per cent span. At this position, the leading-edge vortex is still stable, in both the experiment and the simulation, and the agreement is good. The streamwise velocity profiles, in particular, indicate a progressive thickening of the near-wall layer associated with the progressive turning of the flow away from the trailing-edge direction towards the leading edge. Agreement between experiment and prediction is, unfortunately, not as good in the wing-tip region, where grid resolution may be insufficient, not only close to the wall, but also in outer regions.

## 4. Prospects

Although the examples above, and others documented in the literature, demonstrate that LES–RANS hybrid schemes can operate successfully, it is unclear to what extent the questions and uncertainties raised earlier impact on the generality and prospects of any one of the many methods that have emerged over the past few years. It is undoubtedly true that all approaches lead to an intentional filtering-out of the fine-scale structure near the wall. An example of this is given in figure 10, which shows contours of streamwise vorticity across a fully developed channel flow computed with the hybrid scheme of Temmerman *et al*. (2004) with two different interface locations. As seen, shifting the interface away from the wall leads to a progressive loss of structure, i.e. the near-wall streaks are lost and unphysical ‘super streaks’ are formed. On the other hand, this is precisely the behaviour expected of a LES–RANS scheme, for the whole rationale of the approach is that a substantial part of the turbulence activity should be represented statistically, rather than in resolved form, by the near-wall RANS model.

In the face of this inevitable loss in physical fidelity and the substantial uncertainties arising from the use of highly elongated cells and conventional turbulence models in the highly unsteady flow below and in the vicinity of the interface, the question may justifiably be posed as to whether there is any real merit in trying to resolve, in detail, the unsteady features of the near-wall region, as is done by hybrid RANS–LES schemes over highly ill-disposed (extremely elongated) cells that are unsuited to unsteady conditions. In a sense, the use of the zonal scheme reflects the view that it is preferable to use relatively coarse, though low-aspect-ratio (LES-compatible), cells, with wall-normal dimension of order *y*^{+}=30 near the wall, and then only try to extract the wall condition from some auxiliary method that resolves the wall-normal processes. This approach, taken to its logical endpoint, should arguably involve the *steady-state solution* of the parabolized Navier–Stokes equations in the near-wall region. The method of Uribe *et al*. (2007), already mentioned in §2, goes part-way towards recognizing the rationale of confining the RANS element to the steady state. This approach involves an on-the-run integration of the LES solution, followed by the insertion of the time-mean solution into the turbulence model, from which the RANS stresses are extracted. These are then blended with the subgrid-scale stresses to give the total stress tensor,(4.1)where *S*_{ij} is the strain tensor and ‘〈…〉’ denotes time averaging. This stress tensor is then inserted into the LES solution in the manner normally done with the subgrid-scale stresses. The empirical blending function, *f*, is so chosen as to approach 0 at the wall, resulting in the total stresses being identical to the RANS stresses. However, apart from the arbitrary nature of the blending function, problems are posed by the high computational cost of the method and the fact that the near-wall flow is still unsteady, hence yielding a significant resolved stress tensor, in addition to the RANS stresses. More generally, there is no control over the sum of the constituents of the total stress across the near-wall region.

In the context of a zonal scheme, on-the-fly time averaging of the LES solution would be confined to the interface region. The steady solution would then be imposed on the near-wall parabolized NS equations. However, as the outcome of this process is the steady wall shear stress, a major obstacle to closing the RANS–LES loop is that the RANS model no longer yields the *unsteady* wall condition required by the LES. The question is then: can the unsteady component be recovered or reconstructed in some way?

In a study by Temmerman *et al*. (2002), the results of a wall-resolving channel-flow simulation were used to determine the correlation between the wall-friction velocity and the velocity at any distance *y*^{+} from the wall, in terms of both magnitude and direction. Some results are shown in figure 11. These indicate a fair correlation of the wall conditions with the velocity well away from the wall, at least in terms of the instantaneous magnitude of the wall shear stress. This suggests the possibility that the fluctuations of the wall shear stress, around the mean that is derived from the steady solution of the parabolized NS equation, could be extracted, at least to a first approximation, from correlations linking the wall stress to the velocity at the first computational LES node away from the wall, and derived from *a priori* studies. In an attempt to compensate for the loss of correlation shown in figure 11*b*,*c*, it might be possible to impose onto the unsteady wall shear stress an additional second-order perturbation, in terms of direction and magnitude, derived from related Gaussian probability density functions having respective variances that are related to the loss in correlation, i.e. the deviation of the correlations from 1. This is an approach currently being investigated by the writer.

## 5. Concluding remarks

This paper has demonstrated that LES–RANS approximations replacing wall-resolving LES are ‘viable’, in the sense that they provide credible solutions for complex flow problems in which pure, wall-resolving LES would be far more (often too) costly. However, current methodologies give rise to several important questions as to whether this symbiosis, in the manner implemented, is physically sensible. One key issue is the use of RANS models within a highly unsteady framework in which the time scale of the motion clearly infringes upon the time scale implicit in the time averaging on which the RANS framework rests. The discussion provided at the end of §4 suggests a possible route to addressing this lack of physical realism by exploiting the RANS element in a steady form, but this is acknowledged to raise other problems arising from the need of the RANS solution to provide the LES domain with realistic, unsteady wall-boundary conditions. As yet, there is thus no truly satisfactory LES–RANS combination for near-wall flows.

## Footnotes

One contribution of 16 to a Discussion Meeting Issue ‘Applied large eddy simulation’.

- © 2009 The Royal Society