cyprus_paper

Simulation of Separation from Curved Surfaces
with Combined LES and RANS Schemes
F. Tessicini, N. Li and M. A. Leschziner
Department of Aeronautics, Imperial College, London, UK
mike.leschziner@imperial.ac.uk
Key words: LES; RANS-LES hybrid schemes; Near-wall modelling; Zonal
two-layer modelling.
Summary. The focus of the paper is on the performance of approximate RANS-
type near-wall treatments applied within LES strategies to the simulation of flow
separation from curved surfaces at high Reynolds numbers. Two types of combina-
tion are considered: a hybrid RANS-LES scheme in which the LES field is interfaced,
dynamically, with a full RANS solution in the near-wall layer; and a zonal scheme
in which the state of the near-wall layer is described by parabolized Navier-Stokes
equations which only return the wall shear stress to the LES domain as a wall bound-
ary condition. In both cases, the location of the interface can be chosen freely. The
two methods are applied to a flow separating from the trailing edge of a hydrofoil. A
second flow considered is one separating from a three-dimensional hill, for which the
performance of the zonal method is contrasted with a fine-grid LES and simulations
in which the near-wall layer is treated with log-law-based wall functions.
1 Introduction
Separation from curved surfaces is characterized by intermittent, rapidly vary-
ing patches of reverse flow and the ejection of large-scale vortices over an area
that can extend to several boundary-layer thicknesses in the streamwise di-
rection. The dynamics of this process are extremely complex, and attempts
to represent it with RANS schemes, even those based on elaborate second-
moment closure, have been found to almost always result in predictive fail-
ure [1]. While LES naturally captures the dynamics of the separation process,
in principle, its predictive accuracy greatly depends, in practice, on the details
of the numerical mesh, in general, and the near-wall resolution, in particular.
As found in [2], for the case of separation from a ducted two-dimensional hill,
a 1% error in the prediction of the time-mean separation line results, approxi-
mately, in a 7% error in the length of the recirculation region. This sensitivity,

2 F. Tessicini, N. Li and M. A. Leschziner
associated with the representation of the near-wall physics, is a serious ob-
stacle to the effective utilization of LES in the prediction of separation from
curved surfaces, because the demands of near-wall resolution rise roughly in
proportion to Re2
.
Approaches that aim to bypass the above exorbitant requirements are based
on wall functions and hybrid or zonal RANS-LES schemes. The use of
equilibrium-flow wall functions goes back to early proposals of Deardorff [3]
and Schumann [4], and a number of versions have subsequently been investi-
gated, 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. [5–7]). 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 held to offer a more realistic repre-
sentation 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 DES method [8], in which the interface location, yint, is dictated by the
grid parameters, through the switching condition yint = min(ywall, CDES ×
max(∆x∆y∆z)). A single turbulence model - the one-equation Spalart-
Allmaras model - is used, both as a RANS model in the inner region and
a subgrid-scale model in the outer LES region. This is thus a ’seemless’ ap-
proach. A feature of the method is that it enforces the location of the RANS-
LES switch at the y-location dictated by the streamwise and spanwise grid
density. In general flows, this density often needs to be high to achieve ade-
quate resolution of complex geometric and flow-dynamic features, both close
to the wall (e.g. separation and reattachment) and away from the wall. Thus,
the interface can be forced to be close to the wall, often as near as y+
= O(20),
defeating the rationale and objective of DES. Also, it has been repeatedly ob-
served, especially at high Reynolds numbers and coarse grids and with the
interface location being around y+
= O(100 − 200), 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 damp-
ing in the resolved motion and a misrepresentation of the resolved structure
as well as the time-mean properties.
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. [9]. With any such method, one im-
portant issue is compatibility 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

LES/RANS Modelling of Separated Flows 3
resolved turbulence. This approach, outlined in Section 2, is one of those used
below for simulating separation from curved surfaces.
Zonal schemes are similar to hybrid strategies, but 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 by Balaras and Benocci [10], Balaras et al. [11], Cabot
and Moin [12] and Wang & Moin [13]. In all these, unsteady forms of the
boundary-layer (or thin shear-flow) equations are solved across an inner-layer
of prescribed thickness, which is covered with a fine wall-normal mesh, with
a mixing-length-type algebraic model providing the eddy viscosity. Compu-
tationally, 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 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 interface. The prin-
cipal information extracted from the RANS computation is the wall shear
stress, which is fed into the LES solution as an unsteady boundary condition.
In this paper, the effectiveness of the hybrid LES-RANS scheme of Temmer-
man et al. [9] and the zonal scheme of Balaras et al. [11] is investigated in the
context of simulating separation from curved surfaces at high Reynolds num-
ber. The emphasis is on two particular configurations: a statistically spanwise-
homogeneous flow over the rear portion of a hydrofoil, separating from the
upper suction surface, and a three-dimensional flow, separating from the lee-
ward side of a circular hill in a duct.
2 Outline of Methods
2.1 The Hybrid Scheme
The principles of the hybrid scheme are conveyed in Fig. 1(a). The thickness of
the near-wall layer may be chosen freely, although in applications to follow, the
layer is simply bounded by a particular wall-parallel grid surface. The LES
and RANS regions are bridged at the interface by interchanging velocities,
modelled turbulence energy and turbulent viscosity, the last being subject to
the continuity constraint across the interface:
νLES
SGS = νRANS
t . (1)
The turbulent viscosity can be determined, in principle, from any turbulence
model. In the case of a two-equation model,
νRANS
t = Cµfµ
k2
ǫ
, (2)

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LES
RANS
prescribed
y+ interface
LES−−>RANS data
transfera)
b)
LES
prescribed
y+ interfaceTurbulent
boundary
layer eq.
wall shear stress
to LES
sublayer fed back
Fig. 1. Two types of near-wall treatment: (a)The hybrid LES-RANS scheme; (b)The
two-layer zonal scheme.
and matching the subgrid-scale viscosity to the RANS viscosity at the interface
is effected by:
Cµ =
< fµ(k2
/ǫ)νSGS >
< (fµ(k2/ǫ))2 >
, (3)
where < ... > denotes averaging across any homogeneous direction, or over
a predefined patch, in case no such direction exists. An analogous approach
may be taken with any other eddy-viscosity model. With the interface Cµ
determined, the distribution across the RANS layer is needed. Temmerman
et al. [9] investigate several sensible possibilities, and the one adopted here,
based on arguments provided in the aforementioned study, is:
Cµ(d) = 0.09 + (Cµ,int − 0.09)
(1 − e−d/∆
)
(1 − e−dint/∆int )
. (4)
The numerical implementation of the coupling is straightforward and indi-
cated in Fig. 1. The numerical solution within the near-wall layer is identical
to that of the outer LES domain. The LES field at nodes nearest to the in-
terface provides the ’boundary conditions’ for the inner layer. Also, at these
nodes, Cµ is computed from (3). ’Boundary conditions’ at the interface needed
for solving the k-equation in the RANS layer are provided by the subgrid-scale
energy in the LES domain, while the interface dissipation rate is evaluated
from the subgrid-scale energy as k1.5
/(C∆), where ∆ = (∆x∆y∆z)1/3
. In
the case of two-equation RANS modelling, the turbulence equations in the

sublayer are solved by a coupled, implicit strategy, replacing an earlier se-
quential, explicit solution applied to one-equation models, which was found
to cause stability problems with two-equation models.
2.2 The Zonal Two-Layer Strategy
The objective of the zonal strategy is to provide the LES region with the
wall-shear stress, extracted from a separate modelling process applied to the
near-wall layer. The wall-shear stress can be determined from an algebraic
law-of-the-wall model or from differential equations solved on a near-wall-
layer grid refined in the wall-normal direction - an approach referred to as
”two-layer wall modelling”. The method, shown schematically in Fig. 1(b),
was originally proposed by Balaras and Benocci [10] and tested by Balaras et
al. [11] and Wang and Moin [13] to calculate the flow over the trailing edge
of a hydrofoil.
At solid boundaries, the LES equations are solved up to a near-wall node
which is located, typically, at y+
= 50. From this node to the wall, a refined
mesh is embedded into the main flow, and the following simplified turbulent
boundary-layer equations are solved:
∂ρ ˜Ui
∂t
+
∂ρ ˜Ui
˜Uj
∂xj
+
∂ ˜P
∂xi
Fi
=
∂
∂y
[(µ + µt)
∂ ˜Ui
∂y
] i = 1, 3 (5)
where y denotes the direction normal to the wall and i identify the wall-
parallel directions (1 and 3). The left-hand-side terms are collectively referred
to as Fi. In the present study, either none of the terms or only the pressure-
gradient term has been included in the near-wall approximation. The effects of
including the remaining terms are being investigated and will be reported in
future accounts. The eddy viscosity µt is here obtained from a mixing-length
model with near-wall damping, as done by Wang and Moin [13]:
µt
µ
= κy+
w (1 − e−y+
w /A
)2
(6)
The boundary conditions for equation (5) are given by the unsteady outer-
layer solution at the first grid node outside the wall layer and the no-slip
condition at y = 0. Since the friction velocity is required in equation (6)
to determine y+
(which depends, in turn, on the wall-shear stress given by
equation (5)), an iterative procedure had to be implemented wherein µt is
calculated from equation (6), followed by an integration of equation (5).
3 The Computational LES Framework
The computational method rests on a general multiblock finite-volume scheme
with non-orthogonal-mesh capabilities allowing the mesh to be body-fitted.

The scheme is second-order accurate in space, using central differencing for
advection and diffusion. Time-marching is based on a fractional-step method,
with the time derivative being discretized by a second-order backward-biased
approximation. The flux terms are advanced explicitly using the Adams-
Bashforth method. The provisional velocity field is then corrected via the
pressure gradient by a projection onto a divergence-free velocity field. To this
end, the pressure is computed as a solution to the pressure-Poisson problem
by means of a partial-diagonalisation technique and a V-cycle multigrid algo-
rithm operating in conjunction with a successive line over-relaxation scheme.
The code is fully parallelised and was run on several multi-processor comput-
ers with up to 128 processors.
4 Application
4.1 The Configurations Simulated
In earlier papers of Temmerman et al. [7,9]) and Tessicini et al. [14], results
obtained with wall functions, hybrid RANS-LES and zonal schemes are pre-
sented for channel flow, at a Reynolds number of up to 42000, and separated
flow from a two-dimensional hill, at the relatively low Reynolds number of
21000. In both flows, the hybrid scheme was shown to perform well, and, im-
portantly, to display only weak dependence to the interface location. Here,
attention focuses on the flow around a hydrofoil with separation from its
trailing edge, and on a flow separating from a three-dimensional hill. The
configuration of the hydrofoil flow is shown in Fig. 2.
Fig. 2. One instantaneous realisation of flow over the trailing edge of a hydrofoil.
The five marked streamwise locations B to F are at x = −3.125, −2.125, −1.625,
−1.125 and −0.625 upstream of the trailing edge, where flow statistics are examined.
The flow separates from the upper side of the asymmetric trailing edge. The
Reynolds number, based on free stream velocity U∞ and the hydrofoil chord,

is 2.15 × 106
. The corresponding Reynolds number, based on hydrofoil thick-
ness, is 1.01 × 105
. Simulations were performed over the rear-most 38% of
the hydrofoil chord. The flow was previously investigated experimentally by
Blake [15] and numerically by Wang and Moin [13]. The computational do-
main is 16.5H ×41H ×0.5H, where H denotes the hydrofoil thickness. Table 1
lists the simulations performed, including the grids and the interface locations,
in terms of y+
, in the boundary layer upstream of the curved trailing edge.
The present ’coarse-grid’ results are compared to those obtained by Wang and
Moin [13] on a C-grid of 1536×96×48 nodes1
, claimed to be well-resolved. The
inter-nodal distance in both streamwise and spanwise directions is ∆+
x = 120
for both wall-approximate methods. However, the wall-normal distributions
are different, the mesh used for the hybrid scheme being considerably finer
close to the interface (see later discussion) than that for the two-layer scheme.
It is noted that the grid for both approximate methods contains only one
quarter of the number of nodes used for the reference simulation. Inflow con-
ditions were taken from Wang and Moin [13]. These had been generated in
two parts: first, an auxiliary RANS calculation was performed over the full
hydrofoil, using the v2f turbulence model by Durbin [16]; the unsteady inflow
data were then generated from two separate LES computations for flat-plate
boundary layers at zero pressure gradient. Discrete time-series of the three
velocity components at an appropriate spanwise (y-z) plane were then saved.
These data, appropriately interpolated, were fed into the inflow boundaries
of the present simulations. The upper and lower boundaries are located at
20 hydrofoil thicknesses away from the wall, to minimize numerical blocking
effects. At the downstream boundary, convective outflow conditions are ap-
plied.
Table 1. Grids, modelling practices and interface locations for hydrofoil simulations.
Case Grid SGS Model Interface y+
Reference LES 768 × 192 × 48 Dynamic -
Two-layer Fi = ∂p
∂xi
512 × 128 × 24 Dynamic 40
Two-layer Fi = 0 512 × 128 × 24 Dynamic 40
Log-law WF 512 × 128 × 24 Dynamic 40
Hybrid (2EQ-j12) 512 × 128 × 24 Yoshizawa [17] 60
Hybrid (2EQ-j19) 512 × 128 × 24 Yoshizawa [17] 120
The three-dimensional circular hill, of height-to-base ratio of 4, is located on
one wall of a duct, as shown in Fig. 3. This flow, at a Reynolds number of
130000 (based on hill height and duct velocity) has been the subject of ex-
1
The reference LES was performed on a C-type mesh. It can be approximately
translated into a 768×192×48 H-type mesh, which is directly comparable to the
wall-model meshes.

tensive experimental studies by Simpson et al. [18] and Byun & Simpson [19].
The size of the computational domain is 16H × 3.205H × 11.67H, with H
being the hill height. The hill crest is 4H downstream of the inlet plane. The
inlet conditions required particularly careful attention in this flow, because
the inlet boundary layer is thick, roughly 50% of the hill height. As indicated
in Fig. 4, the mean flow was taken from a RANS simulation that accurately
matched the experimental conditions (Wang, et al. [1]). The spectral content
was then generated separately by superposing onto the mean profile fluctu-
ations taken from a separate precursor boundary-layer simulation performed
with a quasi-periodic recycling method and rescaling the fluctuations by ref-
erence to the ratio of the friction velocity values of the simulated boundary
layer, at Reθ = 1700, and the actual boundary layer ahead of the hill, at
Reθ = 7000. Although the fluctuations only roughly match the experimental
conditions at the inlet - as can be seen from the turbulence-energy profiles in
Fig. 4 - specifying this reasonably realistic spectral representation proved to
be decisively superior to simply using uncorrelated fluctuations, even if the
latter could be matched better to the experimental profile of the turbulence
energy. Because the upper and side walls of the domain were far away from
the hill, the spectral state of the boundary layers along these walls was ignored.
x/H
-4
-2
0
2
4
6
8
Y0
1
z/H
0
2
4
6
X
Y
Z
Fig. 3. Flow over a three-dimensional hill.
Table 2 summarizes the simulations performed for the three-dimensional hill.
Only log-law-based wall functions and the zonal two-layer method will be re-
ported in this paper. To provide a reference point or yard-stick, pure LES
computations, without wall modelling, were performed on a mesh of 9.6 mil-
lion nodes using both the constant-coefficient, van-Driest damped Smagorin-
sky model and its dynamic variant. Despite this rather fine resolution, this
simulation cannot be claimed to be fully wall-resolving, as the y+
values at the
wall-nearest nodes upstream of the hill were of order 5. The two simulations
also display a non-negligible sensitivity to subgrid-scale modelling, which re-
inforces the observation that resolution is insufficient. With wall models, the
aspect ratio of the near-wall grid is (supposedly) no longer a crucial constraint

y/H
U/Uref
10-3
10-2
10-1
100
0
0.2
0.4
0.6
0.8
1
1.2
Simulation
Exp.
y/H
k/U2
ref
10-3
10-2
10-1
100
0
0.003
0.006
0.009
RANS
Exp.
LES
Fig. 4. The mean-velocity and turbulence-energy profiles at the inlet plane of the
three-dimensional hill domain.
for LES, and major savings in computational costs can be achieved by reducing
the grid resolution in the streamwise and spanwise directions. This resulted
in a much coarser 3.5-million-point mesh for the wall model simulations. Two
computations for an even coarser grid of 1.5 million nodes are also included,
one a pure LES with no-slip wall conditions imposed and another performed
with the two-layer near-wall scheme. Attention is drawn to the fact that the
grid disposition in the wall-normal direction depends greatly on the wall model
used. In the case of simple wall laws or the zonal method, the wall-distance
of the first grid point (of the LES mesh) away from the wall can be placed
at around y+
= 40 - although the separate 1d sub-layer grid over which the
parabolized RANS equations are solved is much finer, of course. In contrast,
the hybrid RANS/LES approach requires a high wall-normal refinement to be
maintained, with the wall-closest node being at y+
= 1.
Table 2. Grids, modelling practices and interface locations for three-dimensional-
hill simulations.
Case Grid SGS Model y+
of interface or
near-wall-node location
Fine-grid LES 448 × 112 × 192 Dynamic 5-10
Fine-grid LES 448 × 112 × 192 Smagorinsky 5-10
Log-law WF 192 × 96 × 192 Dynamic 20-40
Two-layer scheme 192 × 96 × 192 Dynamic 20-40
Coarse-grid LES 192 × 96 × 192 Dynamic 20-40
Two-layer scheme 192 × 64 × 128 Dynamic 40-60
Coarsest-grid LES 192 × 64 × 128 Dynamic 40-60

4.2 Results for the Hydrofoil
Results for the hydrofoil are presented in Figs. 5 to 10. First, solutions ob-
tained with the two-layer strategy and the wall-function method are con-
trasted against the highly-resolved LES in Figs. 5 and 6. The figures contain
profiles of streamwise velocity and turbulence intensity at the five stream-
wise sections along the upper hydrofoil surface, denoted B to F, respectively,
and velocity profiles at a further five sections in the wake region. Fig. 6 also
contains distributions of skin friction along the upper surface.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
0
0.1
0.2
0.3
0.4
0.5
0.6
two layer Fi
=0
two layer Fi
=dp/dxi
wall function
full LES
EFDCB
EB C D F
(y−yw)/h
(U2+V 2)1/2/Ue
0 0.15 0.3 0.45 0.6 0.75
0
0.1
0.2
0.3
0.4
0.5
0.6
two layer Fi
=0
two layer Fi
=dp/dxi
wall function
full LES
(y−yw)/h
rms(u
′
)/U∞
Fig. 5. Mean magnitude velocity (l.h.s.), rms streamwise velocity (r.h.s.) at x/H =
−3.125 (B),−2.125 (C),−1.625 (D),−1.125 (E) ,−0.625 (F).
0 1 2 3 4 5 6
-1
-0.5
0
0.5
1
two layer Fi
=0
two layer Fi
=dp/dxi
wall function
full LES
y/H
U/U∞
-8 -6 -4 -2 0
0
0.001
0.002
0.003
0.004
0.005
0.006
two layer Fi
=0
two layer Fi
=dp/dxi
wall function
full les
Cf
x/H
Fig. 6. Profiles of mean streamwise velocity in the wake at x/H = 0, 0.5, 1.0, 2.0
and 4.0 (l.h.s) and mean skin-friction coefficient Cf (r.h.s.)
With respect to the skin-friction data, it is noted first that the abrupt change
in the distribution of the reference LES solution reflects the sudden change
in surface curvature at the junction of the plane and the curved parts of the
surface. At this junction, there is also a fairly abrupt change in the pressure
gradient, and this explains the sensitivity of the results to the inclusion or ex-
clusion of the gradient in the two-layer formulation: as seen, the inclusion of
the pressure gradient leads to the peak in Cf being reproduced. With this dif-
ference aside, the overall agreement achieved between all the approximations,
particularly that including the pressure gradient, and the LES solution is

0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
2EQ-j12
2EQ-j19
full LES
(y−yw)/h
(U2+V 2)1/2/Ue
0 0.2 0.4 0.6 0.8
0
0.1
0.2
0.3
0.4
0.5
2EQ-j12
2EQ-j19
full LES
(y−yw)/h
rms(u
′
)/U∞
Fig. 7. Mean magnitude velocity (l.h.s.), rms streamwise velocity (r.h.s.) at x/H =
−3.125,−2.125,−1.625,−1.125 and −0.625.
0 1 2 3 4 5 6
-1
-0.5
0
0.5
1
2EQ-j12
2EQ-j19
full-LES
y/H
U/U∞
-8 -6 -4 -2 0
0
0.001
0.002
0.003
0.004
0.005
0.006
2EQ-j12
2EQ-j19
full LES
Cf
x/H
Fig. 8. Profiles of mean streamwise velocity in the wake at x/H = 0, 0.5, 1.0, 2.0
and 4.0 (l.h.s) and mean skin-friction coefficient Cf (r.h.s.)
encouraging - indeed, surprisingly good, considering the relative simplicity of
the methods. The simplest two-layer variant, from which the pressure-gradient
term is omitted, gives distributions for all quantities, which are close to those
derived from the wall-function approach. This is the expected behaviour, for
the former is constrained to return a solution in the near-wall layer that com-
plies with the log law, which is imposed explicitly within the wall-function
formulation. The inclusion of the pressure gradient causes early separation
relative to the LES result, as observed by reference to the skin-friction dis-
tributions. However, the velocity profiles at E and F agree closely with the
reference LES. Hence, there is a degree of inconsistency in the predictive qual-
ity of the skin friction and the flow field, implying that the recirculation zone
predicted by the approximate schemes develops at a different rate than that
predicted by the LES and recovers more quickly. As a consequence, when the
flow reaches location E, the premature separation predicted with the pressure
gradient included is not detectable and agreement is close. Conversely, when
a broadly correct separation location is predicted, the recirculation zone is
thicker than it should be at locations E and F.

j=19
j=12
x/H
y/H
-6.9 -6.8 -6.7 -6.6
1
1.02
1.04
1.06
1.08
<νt>/ν
y/H
0 5 10
1
1.02
1.04
1.06
1.08
a)
<νt>/ν
y/H
0 5 10
0.74
0.76
0.78
0.8
0.82
b)
<νt>/ν
y/H 0 2 4 6 8 10
0.54
0.56
0.58
0.6
0.62
c)
Fig. 9. Ratio of turbulent viscosity above the upper side of the hydrofoil at the
location x/H = −3.125(a), −1.625(b) and −1.125(c). Solid line: two-equation hybrid
model j = 12; dashed line: two-equation hybrid model j = 19. Interface locations
j = 12, 19 are identified, relative to a zoom of the grid around x/H = −6.9, on
upper l.h.s. insert.
Fig. 7 and 8 present results for the hybrid LES-RANS scheme, corresponding
to Figs. 5 and 6, again in comparison with the highly-resolved LES solution.
Results are here included for two hybrid simulations, both obtained with a
two-equation eddy-viscosity model applied in the near-wall layer. The simula-
tions only differ in respect of the location of the interface. Fig. 9 contains, in
the upper l.h.s. plot, a magnified view of the hybrid RANS-LES grid above the
flat portion of the upper surface preceding the curved trailing edge. This iden-
tifies the two prescribed interface locations, denoted by ’j = 12’ and ’j = 19’,
respectively. The other three plots in the figure show variations of the tur-
bulent viscosity at the three streamwise locations identified by B, C and D
in Fig. 10, and these illustrate, as expected, that shifting the interface out-
wards increases the proportion of turbulence that is represented by the RANS
model within the thickening near-wall layer. However, the comparisons given
in Figs. 7 and 8 convey a rather disconcerting sensitivity to the interface lo-
cation, not seen previously in channel-flow investigations [9]. Reference to the
skin-friction distributions reveals that, while the shift in the interface location
does not affect the result upstream of the separation point or the ability of the
scheme to resolve the sharp peak in Cf , it has a marked effect on the structure

within the separation zone. The lack of sensitivity upstream of separation is
consistent with the behaviour observed in earlier channel-flow studies already
referred to above. With the interface located at j = 12, it is observed that the
skin friction initially drops below zero at x/H = −1.5, indicating premature
separation at the start of the curved surface, but then recovers to a positive
or near-zero level. In contrast, with the interface placed at j = 19, the skin-
friction maintains its negative value following the initial separation, indicating
a fully-separated flow. Corresponding to the above differences in skin-friction
behaviour, there arise significant structural differences in the flow, as is shown
in Fig. 10. Thus, with the interface at j = 12, no distinct recirculation zone
is observed, while for j = 19, a well-formed recirculation zone covers the rear
portion of the suction side - although, as noted earlier, separation occurs too
early.
x/H
y/H
-3 -2 -1 0
0
0.5
1
1.5a)
A B C
x/H
y/H
-3 -2 -1 0
0
0.5
1
1.5b)
x/H
y/H
-3 -2 -1 0
0
0.5
1
1.5c)
Fig. 10. Streamline patterns close to the trailing edge of the hydrofoil; (a) two-
layer model Fi = ∂p
∂xi
; (b) two-equation hybrid model j = 12; (c) two-equation
hybrid model j = 19.
A possible source of the sensitivity observed above is the grid disposition
close to the interface. Fig. 9 shows that, at j = 12, the cell-aspect ratio is
around 10. While this is an entirely appropriate level for LES in an attached
boundary layer or a channel flow, the present flow in the trailing-edge region
is much more complex, and this aspect ratio may well be inappropriately high
for the LES solution just above the interface and the near-LES (URANS)
solution just below it. With the interface shifted outwards to the location
j = 19, the aspect ratio drops to around 5, and this is a much more benign
level. Hence, a tentative conclusion emerging from this application is that
caution is required when placing the interface in a region in which wall-normal

refinement, effected to ensure appropriate RANS resolution, coincides with a
coarse streamwise distribution. In practice, this constraint can be adhered to
by sensitising the interface location, locally, to the cell aspect ration, thus
ensuring that the LES solution above the interface is supported by a mesh
of low aspect ratio. It is noted here that this is not an issue in applying
the zonal two-layer scheme, as the LES mesh is divorced from the wall-layer
mesh and has a low aspect ratio even in the cells closest to the wall. As seen
from the velocity and turbulence-intensity profiles in Fig. 7, a disadvantageous
consequence of the outward shift in the interface location is, unfortunately,
an excessive outward displacement of the shear layer above the separation
zone. This is, essentially, a consequence of the premature separation, already
observed by reference to the skin-friction distributions, and the elevation of
the turbulence that is associated with the longer stretch of separated shear
layer from the point of separation to the streamwise locations ’E’ and ’F’.
Whether this defect is directly linked to the near-wall approximation is not
clear at this stage. As noted in Table 1, the subgrid-scale model used in the
hybrid RANS-LES simulation is the one-equation formulation of Yoshizawa.
Experience suggests that this model tends to be excessively dissipative, even in
channel flow and box turbulence. Hence, further simulations, with the dynamic
model, need to be performed in an effort to separate effects arising from near-
wall and subgrid-scale modelling. Additional uncertainties arise from the well-
known weaknesses of the two-equation (k − ǫ) turbulence model used in the
near-wall region, especially in separated-flow conditions, and this may well
be a contributory factor in the observed sensitivity. Finally, the manner in
which the dissipation rate is prescribed at the interface may be an influential
issue. The present practice is to extract the interface dissipation rate from the
length scale Cµ
−3/4
κy, while a dynamic process may be more appropriate,
based on the subgrid-dissipation rate. At this stage, all that can be said is
that the insensitivity to the interface location, emerging as an advantageous
property of the hybrid method in channel-flow investigations, may not carry
over to separated flow conditions. When the main flow feature to be resolved
is ’weak’ – in this case, incipient separation over a gently curved surface –
the grid characteristics may play an important role, and this needs to be
quantified.
4.3 Results for the Three-dimensional Hill
Prior to a consideration of results obtained with the near-wall approximations,
attention is directed briefly to the pure LES solution on the 9.6-million-node
mesh, some of which have already been reported by Tessicini et al. [20]. Al-
though this mesh may be regarded as fine for the Reynolds number in ques-
tion, it is, in fact, too coarse and one that compromises the accuracy of the
simulation. As noted previously, the nodal plane closest to the wall is at a
distance of y+
= 5 − 10, while the streamwise-to-wall-normal cell-aspect ratio
is of order 80 near the wall. This grid is thus found to render the simula-

tion sensitive to sub-grid-scale modelling, especially very close to the wall,
where the asymptotic variation of the subgrid-scale viscosity and stresses can
be very important. This sensitivity is illustrated in Figs. 11, which shows the
mean velocity-vector fields across the hill centre-plane. With the Smagorinsky
model, separation occurs too early and gives rise to a more extensive recircu-
lation zone, which results in a slower pressure recovery in the wake following
the separation and consequent differences in the flow fields downstream of
reattachment. The dynamic model gives a shorter and thinner recirculation
zone, in better agreement with the experimental observations.
x/H
y/H
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1
Fine-grid LES
(Smagorinsky)
x/H
y/H
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1
Fine-grid LES
(Dynamic)
Fig. 11. Velocity fields across the centre-plane in the leeward of the hill - comparison
between fine-grid LES solutions using different SGS models, and the experiment.
Despite the broadly satisfactory results derived with the dynamic model, some
caution is called for when assessing the physical fidelity of the results. The
use of the dynamic model poses uncertainties when it is applied on an under-
resolving grid, because the near-wall variation of the Smagorinsky constant,
following spatial averaging, is quite sensitive to the near-wall grid, and that
grid is too coarse in the present LES computation. The fact that the dynamic
model nevertheless performs better than the constant-coefficient variant is due
to the former returning a better representation of the required wall-asymptotic
variation of the Smagorinsky viscosity (O(y3
)). An estimate of the grid den-
sity required to yield a sufficiently well wall-resolved near-hill representation
suggests the need for a grid of 30 − 50 million nodes, an extremely expensive
proposition in view of the modest Reynolds number.
Results obtained with the wall-functions and zonal two-layer near-wall approx-
imations are given in Figs. 12 to 15. These show, respectively, distributions of
the pressure coefficient on the hill surface at the centre-plane, velocity-vector
fields across the centre-plane, flow topology maps on the leeward side of the
hill and velocity profiles in a cross-flow plane at the downstream location,
x/H = 3.63, for which experimental results are available. As seen, the wall-
model solutions all give quite encouraging agreement with the experiment.
Fig. 12(a) shows that, except for the coarsest-mesh (1.5 million nodes) pure
LES, all simulations predict the pressure-coefficient distribution reasonably
well. The magnified views provided in Fig. 12(b) and (c) reveal, in particular,
that the inflexion in the Cp curves, associated with the weak separation on

x/H
Cp
1 2 3
-0.6
-0.4
-0.2
0
0.2
0.4
Experiment
Fine-grid (Smagorinsky)
Fine-grid (Dynamic)
Coarse-grid, Log Law
Coarse-grid, Two-layer
(b)
x/H
Cp
1 2 3
-0.6
-0.4
-0.2
0
0.2
0.4
Experiment
Coarse-grid, pure LES
Coarsest-grid, Two-layer
Coarsest-grid, pure LES
(c)
x/H
Cp
-2 0 2 4 6 8 10
-1
-0.5
0
0.5 (a)
Fig. 12. Pressure coefficient along the hill surface at the centre-plane: (a) full view;
(b) and (c) zoomed-in view around the region where separation occurs.
x/H
y/H
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1
Fine-grid LES
(Dynamic)
x/H
y/H
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1
Coarse-grid
Log Law
x/H
y/H
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1
Coarse-grid
Two-layer
x/H
y/H
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1
Coarse-grid
pure LES
x/H
y/H
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1
Coarsest-grid
pure LES
Fig. 13. Velocity field across the centre-plane in the leeward of the hill - comparison
between pure LES solutions, wall-model solutions and the experiment. The zero-U-
velocity lines are good indications of the recirculation zone size.
the leeward side of the hill, is well captured. The benefit of using wall models
becomes especially evident in the case of extremely poor spatial resolution,
on the 1.5 million-node mesh, where the use of no-slip conditions results in
a grossly erroneous prediction of the separation process. In fact, as will be
shown below, an attached flow is predicted, and an excessively fast pressure
recovery after the hill crest is returned. In contrast, applying a wall model on
the coarsest grid results in the resolution of the separation process and thus
a better representation of the pressure-recovery process.
In the fine-grid simulations and all cases using wall models, the size and extent
of the recirculation zone on the leeward side of the hill agree fairly well with
the experimental results, as shown in Fig. 13. With poor spatial resolution
and no-slip conditions, the recirculation zone predicted is either too small - as
is the case for the coarse-grid LES on the 3.5 million-node mesh - or entirely
absent - as is the case for the 1.5-million-node mesh. Fig. 14 demonstrates

x/H
z/H
0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
Fine-grid LES
(Dynamic)
X
Z
0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
x/H
z/H
0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
Coarse-grid
Log Law
X
Z
0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
x/H
z/H
0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
Coarse-grid
Two-layer
X
Z
0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
x/H
z/H
0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
Coarsest-grid
Two-layer
Fig. 14. Topology maps predicted by the wall models, relative to the pure LES and
experiment.
0 0.4 0.8
U/Uref
0
0.2
0.4
0.6
0.8
1
y/H
0 0.4 0.8
U/Uref
|z/H|=0.33z/H=0.00 |z/H|=0.65
Exp. (LDV)
Exp. (LDV)
Log Law
Two-layer Zonal
LES (Dynamic)
LES (Smag.)
Exp. (HWA)
Coarse-grid LES
0 0.4 0.8
U/Uref
-0.1 0 0.1
W/Uref
0
0.2
0.4
0.6
0.8
1
y/H
|z/H|=0.33 |z/H|=0.65
-0.1 0 0.1
W/Uref
Exp. (LDV)
Log Law
Two-layer Zonal
LES (Dynamic)
LES (Smag.)
Fig. 15. Mean streamwise- and spanwise-velocity profiles at various spanwise loca-
tions on the downstream plane x/H = 3.63.
that both wall approximations also give a broadly faithful representation of
the flow topology, reflecting the presence of a single pair of vortices detach-
ing from the surface. With the combination of a very coarse grid and no-slip
conditions, the topology is seen to be characterised by smooth attached-flow
streaklines. In contrast, use of the two-layer model allows the correct vorti-
cal structures to be recovered. Finally, the downstream velocity profiles at
x/H = 3.63, included in Fig. 15, are generally in fairly close agreement with
the experimental data, although the computed flow appears to recover slightly
more quickly in the wake than suggested by the experiment, because all mod-
els predict slightly early reattachment of the flow at the foot of the hill. In

the region far away from the wall, beyond y/H = 0.4, the predicted velocity
profiles are noticeable different from the experimental LDA data [18]. How-
ever, they agree very well with hotwire measurements [21] made in the same
flow facility, and this discrepancy remains to be resolved.
Overall, the level of agreement with the experimental observations achieved
with the wall-function and zonal schemes is far closer than that reported in
Wang et al. [1] for RANS closures, and also closer than for the pure LES on
the same coarse grids for which the wall approximations were used. The fact
that both approximate schemes yield very similar solutions is not surprising,
for if neither the pressure gradient nor advection is included in the solution
of the RANS equations (5) in the near-wall layer, the zonal scheme inevitably
returns instantaneous representations of the log law as the solution to the
one-dimensional RANS equations in the near-wall layer. What is surprising,
however, is how good the fidelity of the solution is with experiment, in view
of the exceedingly simple approximations employed.
5 Conclusions
Two drastically different approaches to approximating the near-wall region
within a combined LES-RANS strategy have been examined by reference to
high-Reynolds-number flows involving separation from curved surfaces. The
hybrid RANS-LES scheme, applied only to the separated hydrofoil flow, has
been found to display some weaknesses not encountered previously in channel-
flow computations. Thus, the solutions in the separated zone have been found
to be sensitive to the grid disposition at the interface – specifically, the cell-
aspect ratio at the interface – although other factors may play a contributory
role, including the subgrid-scale model, the quality of the turbulence model in
the near-wall layer and the manner in which the dissipation rate is prescribed
at the LES-RANS interface, so as to serve as a ’boundary condition’ for the
dissipation-rate equation solved in the near-wall layer. The nature of the flow
may also be an issue: in this case, the separation is a relatively weak feature
ensuing from a gently-curved surface, and this can therefore be expected to
be quite sensitive to near-wall modelling.
Two forms of the zonal two-layer model have been tested for the hydrofoil
flow, one excluding and the other including the pressure-gradient term in the
solutions of the thin-shear-flow equations in the near-wall layer. The simplest
form, with the pressure gradient and advection ignored, returns a solution
that is, in effect, an instantaneous representation of the log law. Thus, as
expected, the simulations display a behaviour very close to one in which log-
law-based wall functions are imposed explicitly in the course of the simulation.
Both the two-layer formulations and the log-law-based wall-function approach
have been found to return surprisingly good results for the hydrofoil flow. In

particular, the separation behaviour is broadly correctly predicted, and the
consequence is a generally realistic representation of other flow features.
The simple zonal scheme and the wall-function approach have also yielded
pleasingly realistic solutions for the flow around the three-dimensional hill.
On coarse grids, the solutions returned by both approximate methods have
been observed to be definitively superior to the pure LES solutions on the
same grids. Most dramatically, no separation could be resolved on the coarsest
grid by the LES, while the zonal scheme still gave a credible representation
of the major features associated with the separation process. A task that
remains is to determine whether the inclusion of advection leads to a further
improvement of the performance of the zonal scheme.
6 Acknowledgements
This work was undertaken, in part, within the DESider project (Detached
Eddy Simulation for Industrial Aerodynamics). The project is funded by
the European Union and administrated by the CEC, Research Directorate-
General, Growth Programme, under Contract No. AST3-CT-2003-502842.
N. Li and M.A. Leschziner gratefully acknowledge the financial support pro-
vided by BAE Systems and EPSRC through the DARP project ”Highly Swept
Leading Edge Separation”.
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