Large-eddy simulation of three-dimensional flow around a hill-shaped obstruction with a zonal near-wall approximation

Abstract

The focus of the paper is on the performance of an approximate ‘zonal’ near-wall treatment applied within a LES strategy to the simulation of flow separating from a three-dimensional hill at high Reynolds numbers. In the zonal scheme, the state of the near-wall layer of the flow is described by parabolized Navier–Stokes equations solved on a sub-grid embedded within a global LES mesh. The solution of the boundary-layer equations returns the wall shear stress to the LES domain as a wall boundary condition. Simulations are presented for grids containing between 1.5 and 9.6-million-nodes, the one on the finest grid being a pure LES. The comparisons included demonstrate that the zonal scheme provides a satisfactory representation of most flow properties, even on the coarsest grid, whereas the pure LES on the coarsest grid completely fails to capture the separation process.

Introduction

Three-dimensional separation from curved surfaces frequently occurs in external aerodynamics, ship hydrodynamics, turbo-machinery and all manner of curved ducts, curved aero-engine intakes being one example. Unlike separation from a sharp edge, that from a curved surface is always characterised by a highly convoluted and patchy separation area, which moves rapidly in time and space as a consequence of upstream turbulence and Kelvin–Helmholtz instability provoked by the separation process. Separation may also be intermittent and even periodic, being associated with von-Karman vortex shedding and/or Taylor–Goertler vortices. Because the separation location is not fixed by a specific geometric feature – say, a sharp corner or edge – its characteristics depend sensitively on the outer flow and also the reattachment process, if occurring at all. In circumstances in which geometric three-dimensionality is relatively weak, a closed recirculation region may arise following separation. This is the case, for example, in a spanwise uniform, unswept cylinder or surface bump, which is confined in the spanwise direction by walls that are perpendicular to the cylinder or the bump. In more complex conditions, the surface of the body interacting with the flow will be highly three-dimensional, as is the case with highly-loaded swept wings and fan blades, strongly curved circular ducts and three-dimensional smooth (hill-shaped) constrictions in conduits. In such cases, the separation pattern tends to be much more convoluted, featuring, in the mean, a wide range of topological entities such as curved detachment and attachment lines and nodes, focal points and saddles (Perry and Chong, 1987, Helman and Hesselink, 1990). Large vortical structures are shed from the surface over a substantial surface area around the mean separation line. Hence, the turbulence is distinctly non-local, and its dynamics are important. The boundary layer approaching the separation region is subjected to strong skewing and normal straining, with consequent major changes to the turbulence structure. Finally, strong streamwise vorticity and associated flow curvature within and downstream of the separated region provoke further complex interactions between the mean strain and the turbulence field.

A generic laboratory flow that combines all above features is that around a hill placed in a duct, as shown in Fig. 1. This flow has been examined extensively over several years by Simpson et al., 2002, Byun and Simpson, 2005, using elaborate LDV techniques, as well as HWA, and it is increasingly viewed as a key three-dimensional test case for prediction procedures. The hill is subjected to a boundary layer of thickness roughly one half of the hill height, one consequence of this thickness being that the structure of the boundary layer can be expected to be highly influential to the downstream evolution of the flow. The Reynolds number, based on hill height and free-stream velocity, is 130,000. As the boundary layer interacts with the hill, it is subjected to strong skewing prior to separation on the leeward side of the hill. The flow detaches, in the mean, along a separation line, roughly half-way between the hill crest and the hill foot. This merges into focal points on the leeward hill surface. Streamwise-oriented vortices are shed from the focal points, and these evolve alongside the legs of a strong horseshoe vortex formed at the upstream foot of the hill. Hugging the hill’s leeward side is a closed thin recirculation region, which reattaches close to the leeward foot of the hill.

Attempts to compute this flow with RANS methods, whether undertaken in a steady or an unsteady mode, have not been successful. For example, Wang et al. (2004) report an extensive study with various non-linear eddy-viscosity and second-moment-closure models, all giving seriously excessive separation, insufficient rate of post-reattachment recovery and wrong flow structure downstream of the hill. Attempts to induce shedding-like behaviour, within the RANS framework, through the introduction of periodic excitation in the inlet flow, invariably led to a steady flow after the excitation ceased. Similarly unsuccessful RANS results were also reported by Persson et al. (2006) in a recent study. The defects noted above are not entirely surprising, as none of the models accounts for the dynamics of the large-scale, highly energetic motions unavoidably accompanying unsteady separation.

Large-eddy simulation naturally captures, at least in principle, the dynamics of the separation process. However, the simulation of wall-bounded flows at practical Reynolds numbers faces almost untenable resource challenges, because the near-wall grid density required for the near-wall structure to be resolved rises roughly in proportion to Re². When gross features of the resolved flow are substantially affected by near-wall shear and turbulence, as may be the case herein, the quality demands from the numerical mesh, in terms of density, skewness, cell aspect ratio and inter-nodal expansion ratio, are especially stringent and further increase the computational costs.

Approaches that aim to bypass the above exorbitant requirements are based either on wall functions or hybrid or zonal RANS-LES schemes. The use of equilibrium-flow wall functions goes back to early proposals by Deardorff, 1970, Schumann, 1975, and a number of versions have subsequently been investigated, which are either designed to satisfy the logarithmic velocity law (referred to as ‘log-law’, henceforth) 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 and Wengle, 1991, Hoffmann and Benocci, 1995, Temmerman et al., 2002). These can provide useful approximations in conditions not far from equilibrium, but cannot be expected to give a faithful representation of the near-wall layer in separated flow. The alternative of adopting a RANS-type turbulence-model solution for the inner near-wall layer is assumed to offer a more realistic representation of the near-wall flow in complex flow conditions at cell-aspect ratios much higher than those demanded by wall-resolved simulations.

The best-known realization of the combined RANS-LES concept is Spalart et al.’s (1997) DES method. This is one of a class of ‘seamless’ methods, the most elaborate forms of which being based on a spectral RANS-LES partitioning (Schiestel and Dejoan, 2005, Chaouat and Schiestel, 2005). The DES scheme is designed to return a RANS solution in attached flow regions and revert to LES once separation is predicted. This is done by arranging the wall-parallel cell dimensions Δx and/or Δz to be much larger than the wall-normal distance Δy, the consequence being an outward shift of the RANS-LES switching position y_int = min(y_wall, C_DES × max(Δx,Δy,Δz)) away from the wall and a dominance of the RANS scheme. This concept of extensive steady patches co-existing, seamlessly, with unsteady resolved portions raises important question marks against physical realism in areas in which separated regions border boundary layers and in post-reattachment recovery. Also, in general flows, the streamwise grid density often needs to be high to achieve adequate resolution of complex geometric and flow features, both close to the wall (e.g. in separation and reattachment) and away from the wall. Thus, another problem with DES is that the interface can be forced to move close to the wall, often as near as y⁺ ∼ O(50 − 100), in which case RANS and LES regions co-exist even in fully attached flow. In such circumstances, it has been repeatedly observed, especially at high Reynolds numbers, that the high turbulent viscosity generated by the turbulence model in the inner region extends, as subgrid-scale viscosity, deeply into the outer LES region, causing severe damping in the resolved motion and a misrepresentation of the resolved structure as well as the time-mean properties. The DES method has recently been applied to the 3d-hill flow considered in the present paper by Persson et al. (2006) with some measure of success, in so far as the DES solutions were found to be materially closer to the measured data than those obtained with RANS.

A hybrid method allowing the RANS near-wall layer to be pre-defined and to be interfaced with the LES field across a prescribed boundary has recently been proposed by Temmerman et al. (2005). With such a method, one important issue is compatibility of turbulence conditions across the interface; another (related one) is the avoidance of ‘double-counting’ of turbulence effects – that is, the over-estimation of turbulence activity due to the combined effects of modelled and resolved turbulence. A general problem often observed with this type of hybrid scheme is an insufficient level of turbulence activity just beyond the interface, as a consequence of the near-wall RANS model misrepresenting the near-wall (streaky) structure and the fact that the turbulence in the LES region close to the interface is not sufficiently vigorous, because this region is subjected to wrong or distorted structural information at the interface. Several attempts have thus been made to inject synthetic turbulence into the interface in an effort to at least partially recover the influence of the small-scale structures lost by the application of the RANS model. Alternative approaches have been proposed by Piomelli et al., 2003, Davidson and Dahlstrom, 2004, Davidson and Billson, 2006. While these measures have some beneficial effects, in terms of reducing mean-velocity anomalies, they do not and cannot cure most of the defects arising from the inevitable misrepresentation of the turbulence structure near the wall. They are also not practically usable in a general computational environment.

It is arguable that any near-wall approximation that circumvents a detailed resolution of the near-wall structure cannot be expected to return a physically correct spectral state and cannot, therefore, provide the correct ‘boundary conditions’ for the LES portion above the approximated near-wall layer. It can further be claimed, with some justification, that the most that a near-wall model can be expected to provide is a realistic representation of the wall-shear stress, and that this should be the only quantity that is fed into the LES procedure. This is the basis of a second group of approaches termed ‘zonal schemes’. Like hybrid strategies, zonal schemes involve the application of a RANS model in the near-wall layer. However, they involve a more distinct division, both in terms of modelling and numerical treatment, between the near-wall layer and the outer LES region. Such schemes have been proposed and/or investigated in the context of LES by Balaras and Benocci, 1994, Balaras et al., 1996, Cabot and Moin, 1999, Wang and Moin, 2002, Tessicini et al., 2006. The same concepts underpin methodologies formulated by Ng et al., 2002, Craft et al., 2004 for RANS computations. In all the above zonal schemes applied within LES, unsteady forms of the boundary-layer (or thin-shear-flow) equations are solved across an inner-layer of a prescribed thickness, which is covered with a fine wall-normal mesh, with a turbulence model providing the eddy viscosity. Computationally, this layer is partially decoupled from the LES region, in so far as the pressure field just outside the inner layer is imposed across the layer, i.e. the pressure is not computed in the layer. The principal information extracted from the RANS computation is the wall shear stress, which is fed into the LES solution as an unsteady boundary condition.