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ANRV365-FL41-11
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ANNUAL
REVIEWS
12 November 2008
14:57
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Detached-Eddy Simulation
Philippe R. Spalart
Boeing Commercial Airplanes, Seattle, Washington 98124; email: [email protected]
Annu. Rev. Fluid Mech. 2009. 41:181–202
Key Words
First published online as a Review in Advance on
August 4, 2008
turbulence, separation, boundary layer, modeling
The Annual Review of Fluid Mechanics is online at
fluid.annualreviews.org
Abstract
This article’s doi:
10.1146/annurev.fluid.010908.165130
c 2009 by Annual Reviews.
Copyright All rights reserved
0066-4189/09/0115-0181$20.00
Detached-eddy simulation (DES) was first proposed in 1997 and first used in
1999, so its full history can be surveyed. A DES community has formed, with
adepts and critics, as well as new branches. The initial motivation of high–
Reynolds number, massively separated flows remains, for which DES is convincingly more capable presently than either unsteady Reynolds-averaged
Navier-Stokes (RANS) or large-eddy simulation (LES). This review discusses compelling examples, noting the visual and quantitative success of
DES. Its principal weakness is its response to ambiguous grids, in which the
wall-parallel grid spacing is of the order of the boundary-layer thickness. In
some situations, DES on a given grid is then less accurate than RANS on the
same grid or DES on a coarser grid. Partial remedies have been found, yet
dealing with thickening boundary layers and shallow separation bubbles is a
central challenge. The nonmonotonic response of DES to grid refinement
is disturbing to most observers, as is the absence of a theoretical order of
accuracy. These issues also affect LES in any nontrivial flow. This review
also covers the numerical needs of DES, gridding practices, coupling with
different RANS models, derivative uses such as wall modeling in LES, and
extensions such as zonal DES and delayed DES.
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Figure 1
(a) Vorticity isosurfaces colored with pressure over an F-15 jet at a 65◦ angle of attack (Forsythe et al. 2004). Figure courtesy of
J. Forsythe. (b) Acoustic-source isosurface around a Ford Ka automobile (es turbo 3.1) (Mendonça et al. 2002). Figure courtesy of F.
Mendonça and Ford Motor Co.
1. BASICS
Figure 1 illustrates the nature of detached-eddy simulation (DES). The aircraft geometry is
complete (except for detailed surface and propulsion effects); the simulation is at flight Reynolds
number; the large-eddy simulation (LES) content (resolved turbulence) in the separated region
is rich; and the Reynolds-averaged Navier-Stokes (RANS) function plays a role on the aircraft’s
nose. Furthermore, the forces and moments are accurate to within 6% (Forsythe et al. 2004).
This approach must still be considered experimental as a prediction method, and the accuracy
benefits from the thin edges on the wing; there is no marginal separation to challenge the model.
In addition, grid refinement does not indicate grid independence on the smaller components, such
as the tail surfaces.
The automobile geometry is also complete, a feat of the grid generator and solver rather
than of DES (Mendonça et al. 2002). The two regions of the DES are especially well visualized:
steady attached boundary layers and striking LES content around the wheels and the important A
pillar and outside mirror. The drag is dependent on the separation line near the end of the roof,
and the accuracy of the RANS model matters. At the same time, the LES function is indispensable
to predict the aerodynamic noise and in fact the drag. These two studies reflect the broad diffusion
of DES.
1.1. Conceptual History
DES was created to address the challenge of high–Reynolds number, massively separated flows,
which must be addressed in such fields as aerospace and ground transportation, as well as in
atmospheric studies. It combined LES and RANS, spurred by the belief that each alone was
powerless to solve the problem at hand (Spalart et al. 1997). This complaint can be revisited
presently, assuming a working knowledge of LES and RANS (Rogallo & Moin 1984, Wilcox 1998).
The objection to pure LES is simple and centers on computational cost. A pure LES of an
airborne or ground vehicle would use well over 1011 grid points and close to 107 time steps, which
is estimated to be possible in approximately 2045 (Spalart 2000). The boundary layer dominates
this expense, which is necessary even if investigators solve the problem of wall modeling in LES.
Regardless, the resolution needs in the outer region of the boundary layer are firm, with at the
least 20 points per thickness δ in each direction. No unforeseen breakthrough has occurred in
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LES since 1997, and RANS is simply necessary for the large extent of thin boundary layers (the
thicker parts are discussed below).
The objection to pure RANS is not as limpid because it arises from a negative assessment of
models and the relentless attempts to build into them first-principle content and rational ideas. In
this view, RANS models can be adjusted to predict boundary layers and their separation well, but
not large separation regions, whether behind a sphere or past buildings, vehicles, in cavities, and
so on. Observers are hopeful for a new perspective that could erase this objection soon. However,
since 1997, researchers have tended to shift their effort from RANS to LES and hybrid methods.
A second motivation for DES over RANS appears in situations that, even if RANS were accurate,
would need unsteady information for engineering purposes (e.g., vibration and noise).
The original reasons to believe in DES can also be revisited. The original version of DES, which
we refer to as DES97 here, was defined as “a three-dimensional unsteady numerical solution using
a single turbulence model, which functions as a subgrid-scale model in regions where the grid
density is fine enough for a large-eddy simulation, and as a Reynolds-averaged model in regions
where it is not” (Travin et al. 2000a). A working definition is that the boundary layer is treated
by RANS, and regions of massive separation are treated with LES; the space between these areas,
known as the gray area, may be problematic unless the separation is abrupt, often fixed by the
geometry. A single model, with a RANS origin but sensitized to grid spacing via a DES limiter,
provides the desired function in both the RANS and LES limits. The mixing length then can be
limited by two constraints: the wall distance and the grid spacing. When neither constraint is felt,
the model follows its own natural RANS history; this is the case for free shear flows when they
have a grid too coarse to use LES for that particular layer.
The capability of LES in free shear flows is not in question, which does not imply that any
geometry has allowed grid convergence. Few groups have conducted grid refinement, with at
best a factor of 2 in each direction, except in homogeneous turbulence. There is only consensus
that finer grids improve the physics and that grid refinement, away from walls, has not created
bad surprises. Refinement reduces the eddy viscosity, and a plausible view of LES is that the eddy
viscosity is an error, of order 4/3 in the Kolmogorov situation. Reducing also reduces numerical
errors because the cutoff is further down the spectrum, and velocity scales like 1/3 .
RANS development has been static, as almost all the models used in DES date back to 1992.
In a natural DES, with RANS function extending to the separation line, perfection cannot be
reached, and grid refinement brings no improvement beyond the accuracy barrier of the model.
The computing cost of the RANS region is easily manageable, as expected, and the principal
difficulty may be to generate grids that cover all of the boundary layer well in terms of thickness.
Initially, the Spalart-Allmaras model was used, but DES now draws on several other models
(Strelets 2001) (see Section 4.1).
The gray area drew complaints as soon as 2000 in an application to an overexpanded nozzle,
although there were none for DES’s first application, which was to a thin airfoil, in 1999 (Shur
et al. 1999). Surprisingly, users quickly encountered grid spacings that disturbed the RANS model
(see Section 3.2). This motivated a relatively deep change in its formulation with shielded DES
and delayed DES (Menter & Kuntz 2002, Spalart et al. 2006) as the DES length-scale limiter now
depends on the solution, rather than on the grid only. Nonetheless, these methods are aimed at
better fulfilling the original mission of DES.
1.2. Types of Simulation for Massive Separation
Simulation for massive separation is an important field in which the differences in approach
are deep and deserve a detailed discussion. Figure 2 illustrates possible contenders for the
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simulation of flow past a circular cylinder and similar cases. The situation is not as simple as
it appeared in 1997. It was then considered obvious that unsteady RANS (URANS) solutions
suppressed three-dimensionality over two-dimensional (2D) geometries, and it had been found
that drag and lift fluctuations were overpredicted by URANS, although the shedding frequency
was accurate. The term URANS here means running an unmodified (grid-insensitive) transportequation turbulence model, in unsteady mode and with periodic spanwise conditions. Recent
findings have revealed that under fairly general conditions, these simulations in fact sustain threedimensionality and are more accurate than 2D URANS (Shur et al. 2005a). Figure 2 illustrates
the classic steady RANS (an unstable solution) and 2D URANS and includes the newer 3D
URANS. The three-dimensionality is much coarser than in DES and does not become finer
on a finer grid, which it does in DES. URANS largely suppresses three-dimensionality, but
not completely. Shur et al. (2005a) also cite and demonstrate “a troublesome sensitivity to the
spanwise period and to the turbulence model,” making 3D URANS with standard models a
weak contender for this simulation. There is no evidence that the lateral length scales in the 3D
URANS field are physical. Besides the cylinder, these authors treated an airfoil and a rounded
square.
Nishino et al. (2008) present a thorough URANS and DES study of a cylinder near a wall,
which strongly supports the idea that URANS, even if 3D, is less accurate than DES and (when
applicable) LES. More effective RANS models could be devised. Still, URANS is vulnerable to
the criticism that its partial differential equations are known, but the (Reynolds?) averaging it
actually represents is not known, in the absence of a spectral gap. A somewhat similar challenge
can be directed at DES, a point to which we return.
In spite of its failings, there are reasons to be familiar with URANS. First, some researchers do
believe in its capabilities and would dispute our conclusions from Figure 2. Second, in a complex
geometry, sometimes the DES grid and time step only allow, effectively, URANS near the smaller
components. Examples include the wiper blade on a car and the active-flow-control slot on an
aircraft (Spalart et al. 2003). It is desirable for hybrid methods to handle such situations gracefully,
even with the knowledge that the geometric detail ideally would be granted LES content on its
length scales and timescales through a finer grid and a shorter time step.
Figure 2 also vividly illustrates the response of DES to grid refinement in its LES region.
Finally, it confirms that DES solutions with different base RANS models are not sensitive to
model choice in the LES region (as opposed to the RANS region, particularly if separation occurs).
This has been verified quantitatively in many cases (e.g., a backward-facing step) and is a valuable
feature. The boundary layers being laminar, Figure 2 does not reflect DES’s value in treating
turbulent boundary layers in a manner LES cannot, but subsequent figures do.
2. STRENGTHS
This section aims to verify the soundness of DES quantitatively in the important respects of
comparison with experiment and response to grid refinement.
←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Figure 2
Vorticity isosurfaces by a circular cylinder: Re D = 5 × 104 , laminar separation. Experimental drag
coefficient Cd = 1.15–1.25. (a) Shear-stress transport (SST) turbulence model steady Reynolds-averaged
Navier-Stokes (RANS), Cd = 0.78; (b) SST 2D unsteady RANS, Cd = 1.73; (c) SST 3D unsteady RANS,
Cd = 1.24; (d ) Spalart-Allmaras (SA) detached-eddy simulation (DES), coarse grid, Cd = 1.16; (e) SA DES,
fine grid, Cd = 1.26; ( f ) SST DES, fine grid, Cd = 1.28. Figure courtesy of A. Travin.
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a
b
0.5
Experimental peak
approximately 0.5
G4 (10.5 M cells)
Resolved TKE (k/U2)
0.4
0.3
G3 (6.6 M cells)
G2 (2.7 M cells)
0.2
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0.1
G1 (1.2 M cells)
0
0
0.25
0.50
0.75
1.00
X/c
Figure 3
(a) Flow visualizations and (b) resolved turbulent kinetic energy (TKE) for a sharp-edged delta wing at a 27◦ angle of attack, chord
Reynolds number 1.56 × 106 (Morton 2003). Figure courtesy of S. Morton.
2.1. Simple Geometries
Above it was mentioned that grid refinement on the jet aircraft had nontrivial effects on the smaller
components. Grid-refinement effects were more predictable, however, on Morton’s (2003) delta
wings. The simpler geometry helped, but the phenomenon of vortex breakdown is a subtle one.
The results are rewarding, shown visually in Figure 3a and quantitatively in Figure 3b. Finer
grids introduce vortex shedding at the trailing edge and much finer structures in the vortex.
The front half of the vortex is also quite different: The helical striations switch direction from
a coarse to a fine grid. Figure 3b is especially favorable, as it suggests near-grid convergence
of the resolved turbulent kinetic energy to a level that agrees with experiment both for energy
level, approximately 0.5, and location, X/c = 0.65 ± 0.05 (Mitchell et al. 2000). A scale-adaptive
simulation also produced resolved turbulence in this flow (Egorov & Menter 2008).
The study featured in Figure 4a,b also reflects the quantitative success of DES. Constantinescu
et al. (2002) simulated the flow past a sphere with approximately 600,000 points in the baseline
grid and controlled the model in the boundary layer so that it produced laminar separation at
a diameter of Re = 105 and turbulent separation at Re = 1.1 × 106 . The latter is somewhat
simplistic because in the real flow, transition and separation are not segregated (Travin et al.
2000a), but it is far superior to letting an untrained subgrid-scale model handle the boundary
layer, effectively in RANS mode. Quite a few recent cylinder computational fluid dynamics (CFD)
studies even failed to select laminar separation at subcritical Reynolds numbers; Travin et al.’s
(2000a) tripless approach is needed, and Nishino et al. (2008) adopted it successfully. With this
approach, the prediction of a drag crisis is striking, and the pressure distributions are extremely
favorable both when compared with experiment and when comparing baseline and fine grids. At
the lower Reynolds number, DES predicts a drag coefficient of 0.41, compared with 0.40–0.51
in experiments; at the higher Reynolds number, DES gives 0.084 and experiments give 0.12. It is
tempting to extend this approach to golf balls. The drag crisis caused by dimples can be captured
in a gross sense, simply by imposing turbulent separation with a smooth geometry. However, no
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Re = 105
b
Re = 1.1 × 106
c
1.00
0.5
Re = 1.1 × 106
0.00
y/D
0.50
Cp
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a
0
–0.5
–0.50
Re = 105
0
–1.00
–1.50
1
2
x/D
0
30
60
90
120
150
180
θ
Figure 4
Simple bluff bodies. (a) Flow visualizations and (b) pressure distributions for a sphere. Re = 105 and 1.1 × 106 . Open circles and
diamonds denote experiments, whereas the dotted and dashed lines denote detached-eddy simulation (DES) on two grids. Panels a and
b courtesy of K. Squires. (c) Phase-averaged vorticity contours for a cylinder. Color gradations denote DES conducted by Mockett et al.
(2008), and the solid line denotes experiments by the same authors.
RANS model could reproduce the dimple effect accurately, and this will require direct numerical
simulation (DNS), at least of the dimple flow proper.
This is part of a general challenge stemming from the range of scales in fluid mechanics.
Compared with DNS, LES addresses the Kolmogorov viscous scale limitation, and wall modeling
addresses the similar viscous-sublayer scale. In its RANS mode, DES in addition addresses the
boundary-layer eddies of all sizes. These eddies are numerous and fairly universal. However, if
they become dependent on geometry, be it on the shape of a wiper blade or that of a dimple, LES
treatment of their scales becomes necessary for high accuracy so that many problems, in particular
active flow control, simply exceed even current grids in excess of 108 points.
Travin et al.’s (2000a) circular-cylinder study similarly included laminar- and turbulentseparation cases and a surprise-free grid-refinement study, which added confidence after Shur
et al.’s (1999) initial thin-airfoil work. Figure 4c compares DES and experiment behind a
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circular cylinder (Mockett & Thiele 2007); the DES visualizations are close to those shown in
Figure 2e, f. The agreement on the phase-averaged flow pattern is excellent.
2.2. Applications
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DES has been applied often with good results to cavities over a range of Mach numbers (Allen et al.
2005, Hamed et al. 2003, Langtry & Spalart 2007, Mendonça et al. 2003, Shieh & Morris 2001),
ground vehicles (Kapadia et al. 2003, Maddox et al. 2004, Roy et al. 2004, Spalart & Squires
2004, Sreenivas et al. 2006), a simplified landing-gear truck (Hedges et al. 2002), active flow
control by suction/blowing (Krishnan et al. 2004, Spalart et al. 2003), space launchers (Deck &
Thorigny 2007, Forsythe et al. 2002), vibrating cylinders with strakes (Constantinides & Oakley
2006), cavitation in jets (Edge et al. 2006), buildings (Wilson et al. 2006), air inlets (Trapier
et al. 2008), aircraft in a spin (Forsythe et al. 2006), high-lift devices (Cummings et al. 2004),
jet-fighter tail buffet (Morton et al. 2004), and wing-wall junctions (Fu et al. 2007). Peng &
Haase (2008) report on many promising applications at various stages of maturity: wing high-lift
systems, helicopters, combustors, and afterbodies. Chalot et al. (2007) reveal a vigorous line of
work in another aircraft company, Dassault. Slimon (2003) obtained positive results with (zonal)
DES in a turn-around duct; DES did much better than RANS with simple models, however, which
may not be expected to capture curvature effects. Publications aimed at educating users and code
writers have, appropriately, focused on grid generation (Spalart 2001) and on thorough testing of
the codes (Bunge et al. 2007, Squires 2004). The terminology Euler region, RANS region, focus
region, and departure region, introduced by Spalart, may be of help. Grid adaptation in DES and
LES is a future challenge.
Another promising direction is taken by Mockett et al. (2008) and Greschner et al. (2008):
aerodynamic noise. Such studies will contribute both to interior noise in vehicles and aircraft
and to community noise (airframe noise to the airline industry). We note above the industrial
importance of the turbulence adjacent to the driver’s window (Figure 1b). Mockett et al. (2008)
studied the flow in the slat cove of an airfoil in landing configuration; the visualization with
density gradient in Figure 5a vividly reveals much fine-scale turbulence and sound. Actual sound
predictions are not included.
Greschner et al. (2008) provide sound predictions for the flow past a cylinder, placed ahead
of an airfoil so that its turbulent wake impinges on it (see Figure 5b). At low Mach numbers,
this impingement, which causes wall-pressure fluctuations, is the dominant noise mechanism.
Various Ffowcs-Williams-Hawkings surfaces are used to extract far-field noise. Flow visualizations
resemble those in Figure 2, without as fine a level of resolution. This case is more onerous
because the turbulence needs to be carried all the way to the airfoil, 10 diameters downstream;
the focus region is much larger. Figure 5c compares the sound spectrum with experiment. An
adjustment was made in the vertical direction: In 2D geometries, there is an unsolved problem
when comparing an experiment of finite length (with some end conditions) to a simulation with
periodic boundary conditions, invariably quite narrow (in contrast, no adjustment was needed for
the spectra inside the turbulence region). Once this correction is accepted, the agreement on the
shape of the spectrum, over five octaves, is quite amazing.
Figure 6 (Chauvet et al. 2007) is of interest for two reasons. First, the LES-content development
in the mixing layer is nearly immediate, which is positive, although it may be excessively 2D (see
Section 3.4). Second, the simulation is simultaneously free enough of numerical dissipation to
welcome LES content and robust enough to capture shocks. This result has also been achieved by
Shur et al. (2006) in jets and by Ziefle & Kleiser (2008) in a supersonic channel with hills. These
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0.001 0.01 0.1
b
Y
c
100
1
10
X
Z
PSD (dB)
80
60
40
Experiment
DES + FWH
20
0
θ = 90°
10–1
100
St = f × D/u∞
Figure 5
Complex bluff bodies. (a) Schlieren picture near a slat. Panel a courtesy of C. Mockett. (b) Vorticity
isosurfaces for a rod-cylinder case. (c) Far-field spectrum. PSD, power spectral density. Panels b and c
courtesy of B. Greschner.
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a
1
b1
y/D
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Figure 6
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(a) Experimental schlieren (view through flow) and (b) numerical schlieren (contours in center plane) for a
supersonic jet. Figure taken from Chauvet et al. 2007.
studies remove the concern that LES might be barred from supersonic flows, therefore widening
the range expected for DES, given a powerful numerical method.
3. WEAKNESSES
3.1. Conceptual Issues
The need to predict turbulence numerically is far-reaching. Yet continuing concerns of a conceptual nature could categorize DES as a method that is intuitively correct and often successful
but dissatisfying to the purist. Below we first address these concerns and then delve into practical
issues in the remaining subsections.
The criticism of URANS mentioned above (namely that the approximate PDE that is solved
is known, but the exact PDE it is meant to approximate is not) does not truly apply to DES. A
filter can be linked to the grid cell and to the integration implied by the CFD solver. In LES,
systematic studies use filtered versions of DNS fields to steer subgrid-scale model development.
This is known as the Clark test or a priori study and could be performed with DES but has not;
in LES and DES practice, models are adjusted based on results rather than explicit tests. The
new difficulty beyond those in LES is that, in the gray area, the model has a strong impact, but a
convincing calibration is simply out of reach: There are far too few degrees of freedom (in DES97,
only CDES ). A similar problem is present even in simple LES; simply put, one would adjust the
single Smagorinsky constant to ensure that all six subgrid stresses are correct. The problem is
more severe in wall-modeled LES (WMLES) and more severe again in DES. Clear statements
are much more difficult to make, especially in view of the wide variety of anisotropies possible for
the grid cell and time step, and also because of history effects, which are strong especially in the
all-important situation of a separating boundary layer (see Section 3.4). The essential difficulty is
that the model has much more impact on WMLES and DES than it does on the notional LES
situation, namely away from walls and with a grid spacing in the inertial range. In that situation,
one can arbitrarily lessen the influence of the Smagorinsky constant and similar constants with grid
refinement. WMLES has been exposed to this issue less than DES, possibly because it sometimes
seems unable to escape channel flow.
The literature reflects a desire for an approach that is somehow more justified and mathematically defined than DES. Several hybrid proposals rest on the idea of splitting the turbulent energy in
a specified ratio (e.g., 70% resolved and 30% modeled). This is fine in simple flows, but the strength
of DES (and WMLES) is precisely that the split is different in different parts of the same solution.
The energy split can be adjusted in different regions, but this increases the decision load for the user.
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A separate line of critical thought regards the use of the grid spacing in the model. In LES,
of course has been standard, although it has been proposed to dissociate the filter size and grid
spacing. With RANS-LES hybrids, it has even been proposed to dispose with any length scale
of the nature of a filter width or grid scale. This led to scale-adaptive simulation (SAS). Menter
et al. (2003) use an SAS model that appears to have a pure RANS nature but achieves LES
behavior unlike any traditional RANS model. For instance, visualizations over a cylinder look just
like those in Figure 2e, f. Menter et al.’s (2003) model differs from traditional ones in its use of a
higher derivative of the velocity field, which is highly active on short scales. Travin et al.’s (2004)
turbulence-resolving RANS approach has similar features but uses the ratio of strain to vorticity
rather than a high derivative.
Besides a philosophical interest in the true nature of turbulence models, the SAS and
turbulence-resolving RANS work is motivated by the disruptive effects of in DES with ambiguous grids (see Section 3.2). This stimulating controversy is not over. It echoes the one in RANS
modeling over the use of the wall distance [as in the Spalart-Allmaras and shear-stress transport
(SST) models]. Wall distance can be expensive to calculate and has unphysical effects (e.g., with
a thin wire); however, the sustained wide use of these two models suggests that it is manageable
and has a substantial accuracy payoff. Equally active are controversies over the definition of in
noncubic grid cells (see Section 4.4). Nonuniqueness issues are most intense with delayed DES
(DDES), as discussed in Section 3.2 and Section 4.3, because even the RANS or LES nature of
the solution is in some cases dependent on initial or inflow conditions.
Finally, the issue of an order of accuracy is clear; careful users are justified in asking for one
because it is, in principle, a key step in CFD quality control; this is related to the desire for
monotonic grid convergence. A typical observation after analyzing a grid-refinement study even
in a simple geometry is the honest but vague statement that the findings are “suggesting a certain
degree of grid convergence” (Nishino et al. 2008).
An order of accuracy has not even been proposed for a simulation using both modes within
DES. In a pure LES, this order exists but depends on the quantity in question, for instance, the
resolved or total turbulent kinetic energy or the dissipation. WMLES does not deal with this
problem much better than DES does. Recent efforts at organizing the quality control of CFD in
the RANS field, in which the differential equation does not depend on the grid, would be defeated
by precisely this dependence in LES and DES.
Whether in DNS, LES, or DES, the difficulty in demonstrating grid convergence is compounded by the residual variations owing to finite time samples; some flows have severe modulations and drift. Figure 7 uses Travin et al.’s (2000a) LS1 cylinder case; the simulation covered
a generous 40 cycles of shedding, after an initial transient. The time-averaged drag coefficient
is 1.083 over the first half of the sample, but 1.033 over the second half; the lift excursions are
also noticeably less intense over the second half. Although the sample is sufficient to capture the
modulations of the lift signal, the drag’s drift is not mastered to better than several percent and
went unnoticed at the time. There is no theory that would extrapolate to infinite sample length.
As a result, searching for grid convergence to 1%, for example, is not possible.
3.2. Modeled-Stress Depletion and Grid-Induced Separation
Modeled-stress depletion (MSD) and grid-induced separation have been the most significant
practical issues and have been worse to deal with than initially anticipated (Spalart et al. 1997).
Figure 8a shows the roots of these problems, with three levels of grid density in a boundary layer.
The first level matches the initial vision of DES; it is a boundary-layer grid, with the wall-parallel
spacing in excess of the boundary-layer thickness δ, which allows full RANS function. The
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1.30
Instantaneous
1.25
0.4
Time-averaged
1.20
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Cι, Cι
0.2
0
1.15
1.10
Time-averaged
–0.2
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–0.4
1.00
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0
40
80
120
160
tU/D
200
0.95
0
40
80
120
160
200
tU/D
Figure 7
Instantaneous (solid line) and time-averaged (dashed line) values of force coefficients on a cylinder: (a) lift and
(b) drag. Re = 5 × 104 . Figure courtesy of A. Travin.
third level matches the needs of LES in the outer layer and thus of the extended use of DES as
a wall model (see Section 3.3): The grid spacing in all directions is much smaller than δ. The
second level is the troublesome one: small enough for the eddy viscosity to be affected by the DES
limiter but not small enough to support accurate LES content (slow LES development adds to
this difficulty; see Section 3.4). Spalart et al. (2006) coined the term MSD, well after the issue was
detected by S. Deck (personal communication) and by Menter & Kuntz (2002), who pointed out
a consequence of MSD called grid-induced separation (GIS).
Created only one year after Shur et al. (1999) fully defined DES, Figure 8b is an early example
of gradual grid refinement degrading a solution that was rather good when the RANS model was
fully active (S. Deck, personal communication; see also Caruelle & Ducros 2003). Separation in
a nozzle is premature and induces unsteadiness. DES users promptly explored the effects of grid
spacing and sought high accuracy, with disturbing outcomes.
Figure 9 is a visualization of GIS, this time on an airfoil (Menter & Kuntz 2002). Whereas
the RANS solution is steady and quite accurate, even in this case of incipient separation, the DES
solution suffers from early separation. It also is unsteady, but in a shedding mode rather than in
a sound turbulence-resolving mode. The flow field then obeys the URANS equations, but with a
model that has become grid dependent in an obscure and unintended manner.
Menter & Kuntz (2002) proposed a solution applicable to the SST model called shielding, in
which the DES limiter is disabled as long as the flow is recognized as a boundary layer, using
the SST F2 function. Spalart et al. (2006) introduced DDES, which is applicable to most models.
Either modification successfully prevents GIS by extending the RANS region, exploiting a history
effect. Secondary effects are covered in Section 4.3.
3.3. Logarithmic-Layer Mismatch
Simulations with an LES nature in one region and a RANS nature in another were conducted
long before DES; wall modeling near the walls of an LES draws on RANS technology, and early
channel LES studies even used wall functions. A new feature of DES is that the entire boundary
layer can be handled by RANS. However, DES also naturally provides a simple wall model, which
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0.5
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
y/δ
x/δ
1.0
1.0
0.5
0.5
0
0.5
1.0
1.5
2.0
0
x/δ
0.05
Normalized wall pressure (PW/PC)
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0
0
0.5
1.0
1.5
2.0
x/δ
b
0.04
0.03
DES computation PR40
0.02
SA-URANS
computation PR40
0.01
LEA steady experimental
data PR41.3
0
0
20
10
X/rt
Figure 8
(a) Types of grids in boundary layers. The dashed line represents the velocity profile. (b) Pressure
distribution in a supersonic nozzle. Figure courtesy of S. Deck. DES, detached-eddy simulation; LEA,
Laboratoire d’Etudes Aérodynamiques; SA-URANS, Spalart-Allmaras unsteady RANS.
Nikitin et al. (2000) attempted. The results were not perfect, but the study was successful in key
respects. The model was robust, with no need for averaging or danger of negative values. LES
content was sustained even with coarse grids, because = h/10 in most runs, where h is the
half-width of the channel. Very high Reynolds numbers were reached at little additional cost.
Figure 10a illustrates the response of Nikitin et al.’s method to Reynolds number and grid
spacing. An increase in Reynolds number on a fixed grid (same but refinement in y to retain a
first y + near 1) lengthens the modeled part of the profile, which blends into the modeled log layer
( y + roughly from 70 to 700). Grid refinement, conversely, lengthens the resolved-turbulence part
of the profile, which blends into the resolved log layer ( y + roughly from 3000 to 15,000). The
Reynolds shear stress comprises modeled stress and resolved stress, which trade places as the grid
is varied (Figure 10b).
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a
b
Velocity
0
–5.00
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–10.00
Figure 9
Vorticity contours over an airfoil: (a) Reynolds-averaged Navier-Stokes and (b) detached-eddy simulation.
Arrows indicate separation. Figure taken from Menter & Kuntz 2002.
The imperfection is that the two log layers are misaligned, by almost three wall units of velocity
U + . The probability that this log-layer mismatch would be zero was nil because this study used
the pure DES97 model, adjusted for other purposes. (The study was also marked by deliberate
constraints, such as equal grid spacing in the wall-parallel directions, to ensure the findings would
translate into practice.) All other wall-modeling approaches have required adjustments to align
their log layers. Nikitin et al. (2000) mentioned the ensuing error of the order of 15% for the
skin-friction coefficient but did not mention that the slope dU/d y is too high by 65% at y = .
Locally, this is highly inaccurate. In addition, grid refinement merely moves the same amount of
mismatch closer to the wall. This is different from MSD in a near-RANS boundary layer, which
40
1.0
a
0.8
35
30
0.6
25
τ+
U+
b
0.4
20
15
0.2
10
5
101
0
102
103
y+
104
105
0
0.2
0.4
0.6
0.8
1.0
y/h
Figure 10
Channel-flow, wall-modeled large-eddy simulation. (a) Velocity: Reτ = 2000 and 20,000. Each profile is
shifted by five U + units. The lower two curves use approximately 140,000 grid points, and the upper curve
uses approximately 1,000,000 points. The dashed line represents the log law. (b) Modeled and resolved shear
stress: coarser grid (dashed line) and finer grid (solid line). Reτ = 20,000. Figure adapted from Nikitin et al.
2000.
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b
1
0
0
y
y
a
1
–1
–1
0
2
4
x
6
8
0
2
4
x
6
8
Figure 11
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Vorticity in a jet: (a) standard detached-eddy simulation and (b) implicit large-eddy simulation, eddy viscosity disabled. Figure courtesy
of M. Shur.
becomes more severe as the grid is refined. Follow-on work by Piomelli and his group also showed
that the near-wall solution has poor LES content. The practical advantages of wall modeling by
DES, and the understanding that in practice thick wall-bounded layers lead to LES grids in the
sense of Figure 8a, motivate efforts to resolve log-layer mismatch (Piomelli & Balaras 2002,
Travin et al. 2006).
3.4. Slow Large-Eddy Simulation Development in Mixing Layers
Separation is the essential flow feature motivating DES, with the expectation that the boundary
layer is treated with RANS and is quasi-steady, but the free shear layer it feeds develops LES
content. By consensus, the sooner this takes place, the better. Unfortunately, standard DES on
typical grids does not achieve this switch very fast at all (Figure 11); a zonal approach that disables
the model in the mixing layer and activates implicit LES is visually far more successful (Shur
et al. 2005b,c). This is the case with the book-shaped grid cells typical of such regions, with one
dimension much smaller than the other two, and may be a perverse effect of the careful adaptation
of the grid to the shear layer. The DES model fails to sense the opportunity because the lateral
grid spacing is loose (here, 10% of the diameter D, with 64 points around) and the standard
definition of is used (see Section 4.4). The model defaults to RANS until the layer thickness
reaches approximately 40% of D because the mixing length in a RANS-treated mixing layer is
approximately one-tenth the vorticity thickness, much smaller than the lateral grid spacing, making the DES limiter inoperative. Other definitions are then more successful (see Section 4.4),
but in a manner dependent on the alignment and shape (book or pencil) of the grid cells. This
problem has received and deserved attention, but unlike the two problems discussed in the preceding subsections, it is remediable with grid refinement.
4. RECENT PROPOSALS
4.1. Alternate Reynolds-Averaged Navier-Stokes Models
The original formulation of DES rested on the simple Spalart-Allmaras model, and no CFD
system should ever be confined to one model. Travin et al. (2000b) pioneered the adaptation
to two-equation models, in particular the SST model, which has been smooth. Recent work includes, for instance, Greschner et al.’s (2008) cubic explicit algebraic stress models. The motivation
for complex models is debated because the RANS region normally comprises thin shear layers;
relatively thick and curved boundary layers could make using complex models worthwhile.
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4.2. Zonal Detached-Eddy Simulation
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In zonal DES, the user explicitly marks different regions as RANS or as DES (Deck 2005). In
effect, in RANS regions, is made infinite (as opposed to zero in implicit LES). This is probably
the strongest departure from the original concept of DES, in which the use of a single but versatile
equation set is central, and creates most of the conceptual and practical challenges. The motivation
is to be fully safe from MSD and GIS (see Section 3.2) and to clarify the role of each region. Zonal
DES worked well for Brunet & Deck (2008) in the important problem of wing buffet, Chauvet
et al. (2007) in jets, Simon et al. (2007) for a base flow, and Slimon (2003) in a duct.
The geometries in these studies were simple, such as the jet featured in Figure 11. A fair
question to propose to zonal DES proponents concerns complex flows, in which decisions are
needed for numerous regions (including the thickness of regions meant to contain RANS boundary
layers). This is similar to issues with zonal control of laminar-turbulent transition. Which mode
will be the default, and which will be the exception? S. Deck (personal communication) is in favor
of RANS as the default mode; the author may disagree, and, more importantly, there is the concern
that smooth-wall separation is normally not known at the time the zones are set. Compared with
DES, ZDES appears simultaneously more powerful and less self-sufficient.
4.3. Delayed Detached-Eddy Simulation and Improved Delayed
Detached-Eddy Simulation
A key motivation here is precisely to avoid zonal measures, thus leaving it to the solution process
to determine separation, while addressing the MSD issue that affects DES97 (see Section 3.2).
Following Menter & Kuntz (2002), DDES detects boundary layers and prolongs the full RANS
mode, even if the wall-parallel grid spacing would normally activate the DES limiter. This detection
device depends on the eddy viscosity, so that the limiter now depends on the solution (Spalart
et al. 2006). This is a formal deviation from DES97 but not a different mission. DDES was
shown to resolve GIS, without impeding LES function after separation. For instance, it handled
a backward-facing-step flow well, even with grids that would cause severe MSD both upstream
of the step and all along the opposite wall. DDES is likely to be the new standard version of
DES.
Improved delayed DES (IDDES) is more ambitious yet (Shur et al. 2008). The approach is
also nonzonal and aims at resolving log-layer mismatch in addition to MSD. One basis is a new
definition of , which includes the wall distance and not only the local characteristics of the grid.
The modification tends to depress near the wall and give it a steep variation, which stimulates
instabilities, boosting the resolved Reynolds stress. Other components of IDDES include new
empirical functions, some involving the cell Reynolds number, which address log-layer mismatch
and the bridge between wall-resolved and wall-modeled DES (grids with moderate values of the
spacing in wall units, +
). These functions make the formulation less readable than that of DES97.
Yet many groups have had success with IDDES in practice (Mockett & Thiele 2007).
The history effect introduced by shielding or by DDES has consequences in terms of the
uniqueness of solutions. For instance, in a channel flow with periodicity and a grid and time
step capable of LES (as in Nikitin et al. 2000 and Figure 10), the solution has two branches,
depending on the initial condition. If the flow is in a RANS state, with high eddy viscosity and
weak perturbations, it remains in that mode and finds a steady state. If the flow starts in an
LES state with low eddy viscosity and sufficient LES content, it settles into a statistically steady
LES. Both solutions are valid, but this situation perplexes some observers (Frölich & von Terzi
2008).
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Nonuniqueness, however, is not unknown in RANS practice. Some flows, such as airfoils
near maximum lift, have hysteresis both in real-world situations and in CFD. More striking is
the behavior of models in the tripless mode (Travin et al. 2000a), which is an essential tool for
capturing the drag crisis of smooth bluff bodies. The mature solution depends on the level of the
turbulence variables in the initial field.
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4.4. Modified Δ Length Scales
The IDDES length scale’s principal motivation is in a fully turbulent wall layer in the LES mode.
Other proposals relate instead to transition, more precisely the growth of LES content. Several
groups (Breuer et al. 2003, Chauvet et al. 2007, Yan et al. 2007) have tested with some success
definitions radically different from the standard one in DES, namely the maximum dimension of
the grid cell; if it is aligned with the axes, then = max(x, y, z). In contrast with the DDES
modification (which raises eddy viscosity in specific situations), all these definitions tend to reduce
it, therefore worsening the MSD tendencies. They all appear to be responses to the problem of
LES development in mixing layers (see Section 3.4) with the purpose of allowing the KelvinHelmholtz instability to grow. Some use the time-honored definition in LES = (xyz)1/3 ,
which
of course reduces , but its physical justification is thin. Chauvet et al.’s (2007) length scale
is the unit vector aligned with vorticity, is
≡ Nx2 yz + Ny2 xz + Nz2 xy, where N
aimed at the situation in which the vorticity is closely aligned with one of the grid lines.
The debate is whether promoting the 2D Kelvin-Helmholtz instability, knowing that the true
switch to 3D turbulence occurs only once the mixing-layer thickness has caught up with the lateral
grid spacing, is far superior to letting the mixing layer thicken in the RANS mode. For instance, the
RANS mode creates no sound, but the near-2D LES mode could create too much. The reduced
length scales have an advantage over the implicit LES approach shown in Figure 11 as they are
not zonal and can reverse to the normal scale when the grid is not strongly anisotropic.
5. NUMERICAL REQUIREMENTS
DES codes need qualities that are absent in many RANS codes and others that are absent in many
LES codes. Considering the filiation of the model, it is more common to start from a RANS code.
These codes often have placed a high priority on convergence to a steady state, complex-geometry
compatibility, and shock capturing. The unsteady capability, with resolution of high frequencies
and short waves, has been neglected, and the other demands all benefit from numerical dissipation.
As a result, an extensive testing campaign and modifications to reduce dispersion, dissipation, and
time-integration errors are key (Caruelle & Ducros 2003, Mockett & Thiele 2007, Strelets 2001,
Temmerman & Hirsch 2008). The most effective schemes are structured and hybrid, not only
in their treatment of turbulence, but also in their numerics. The differencing scheme is centered
(nondissipative) or nearly so in the LES region and is more strongly upwind in the Euler and RANS
regions. This hybridization was introduced by Travin et al. (2000b) and is now widely used (e.g.,
Mockett & Thiele 2007). Conversely, the code used in Figure 1a is unstructured and uniformly
based on second-order upwind differencing, but it displays generous LES content. Therefore, it
is best to avoid blanket statements.
If the starting code is an LES code, common obstacles include the limitation to simple geometries, without implicit time integration or multiblock capabilities, let alone unstructured grids.
The addition of a transport-equation turbulence model is not trivial, and few codes have shockcapturing capability (Hou & Mahesh 2004). The priority was given to high orders of accuracy.
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An advantage of DES is the ease of programming and application. Potentially, it is activated
directly from the menu of turbulence models in many of the vendor CFD codes. This is a doubleedged advantage, as users not invested in turbulence and/or too trusting of the experts could accept
results without verifying LES content, grid resolution, time step, time sample, and so on. An early
example of this was an entry in the LESfoil workshop (Mellen et al. 2003). The simulation was
formally a DES, and the results were fine. However, there is every indication from the grid that
the simulation was actually in RANS mode, even in the key region. In contrast, the genuine LES
studies struggled with all the issues of lateral domain size, resolution, and initiation of LES content
in attached flows.
6. OUTLOOK
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It is certain that DES has a future and therefore deserves a critique. Greschner et al. (2008)
deem that “DES is still in its infancy and undergoes continuing improvements.” Under one name
or another, a form of a RANS-LES hybrid that is capable of full RANS function in boundary
layers will be in use for the foreseeable future in many industries. It will also remain conceptually
difficult, and efforts toward more predictable behavior under grid variations and better wallmodeling performance will continue. LES-content creation in attached flows will flourish, and
the numerical quality of the codes will receive sustained attention. A clear need in practice is to
organize and facilitate grid generation and to set guidelines for systematic refinement. Programs
such as DESider and focused workshops will be most beneficial to the progress of DES and other
hybrids (Peng & Haase 2008).
An unfortunate trend is that models have moved away from the simplicity of DES97 in terms
of the equations and nonuniqueness of solutions (in DDES and IDDES) and in terms of the user
decision load and need to mark regions (in ZDES). Users by now have identified situations in
which DES gives too little eddy viscosity and others in which it gives too much. Even in DES97,
large steps in the grid spacing can be used to steer the solution toward one mode or the other,
so that grid design can become involved, especially now that the dangers of ambiguous grids are
known. What may be an ideal of CFD, namely that grid refinement will do no harm (in other
words, be monotonic) and follow a known power of the grid size, will remain elusive in DES and
LES (without explicit filtering), except in the simplest of flows.
There are signs that a productive DES user community has formed. We must recognize,
however, a school of thought that considers DES to be a somewhat unsafe activity.
Owing to space limitations, this review does not discuss hybrid RANS-LES methods besides
DES and SAS (e.g., limited numerical scales, very large eddy simulation, flow simulation methodology, nonlinear disturbance equations, extra-large eddy simulation, lattice Boltzmann method,
transient RANS, partially averaged Navier-Stokes, semideterministic method, organized eddy
simulation, partially integrated transport model, and the self-adapting model) (some are found in
Sagaut et al. 2006; Frölich & von Terzi 2008). I do not believe that any of these methods provides
a clear remedy to the difficulties discussed here, but this could change in the future. The principal
concerns are GIS and in general the potentially poor knowledge of the nature of the simulation in
each region of a complex flow: driven URANS, spontaneous URANS, or LES. This nature can
change under grid refinement and become ambiguous, and therefore it is not the case that any
grid refinement improves the solution. The nominally universal character of DES makes these
observers justifiably dubious that a sufficiently error-proof approach results, or that the user community is being properly informed. Such comments are encountered more often in conversations
and anonymous reviews than in publications. It does not detract from their value, and the task
of resolving them is an inspiring one. Locally ambiguous grids may be a permanent feature of
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practical DES. One might ask, is it justified to simulate the flow past a car, when the wiper and
door handle are not well resolved? The answer depends on the purpose of the simulation.
FUTURE ISSUES
1. The numerical resolution over relevant geometries needs improvement, ultimately with
grid adaptation.
2. The link between the DES flow field and the exact or DNS flow field should be established.
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3. The choice between zonal and nonzonal treatments of the turbulence needs to be addressed.
4. The generation of resolved turbulence in attached boundary layers needs to become
routine and efficient.
DISCLOSURE STATEMENT
The author is not aware of any biases that might be perceived as affecting the objectivity of this
review.
ACKNOWLEDGMENTS
I am grateful to Drs. Allmaras, Deck, Mockett, Strelets, Shur, and Travin for their comments on
this manuscript and their partnership over the years.
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high alpha. J. Aircraft 41:193–200
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RANS/LES methods. Int. J. Heat Fluid Flow 28:1379–90
Greschner B, Jacob MC, Casalino D, Thiele F. 2008. Prediction of sound generated by a rod-airfoil configuration using EASM DES and the generalised Lighthill/FW-H analogy. Comp. Fluids 37:402–13
Hamed A, Basu D, Das K. 2003. Detached eddy simulation of supersonic flow over cavity. Presented at AIAA Aerosp.
Sci. Meet. Exhib., 41st, Reno, Pap. No. AIAA-2003-0549
Hedges LS, Travin A, Spalart PR. 2002. Detached-eddy simulations over a simplified landing gear. J. Fluids
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Hou Y, Mahesh K. 2004. A robust, colocated, implicit algorithm for direct numerical simulation of compressible, turbulent flows. J. Comp. Phys. 205:205–21
Kapadia S, Roy S, Wurtzler K. 2003. Detached-eddy simulation over a reference Ahmed car model. Presented at
Thermophys. Conf., 36th, Orlando, Pap. No. AIAA-2003-0857
Krishnan V, Squires KD, Forsythe JR. 2004. Prediction of separated flow characteristics over a hump using RANS
and DES. Presented at AIAA Flow Control Conf., 2nd, Portland, Pap. No. AIAA-2004-2224
Langtry RB, Spalart PR. 2007. Detached-eddy simulation of a nose landing-gear cavity. Presented at. IUTAM
Symp. “Unsteady Separated Flows and Their Control,” Corfu, Greece
Maddox S, Squires KD, Wurtzler KE, Forsythe JR. 2004. Detached-eddy simulation of the ground transportation system. See McCallen et al. 2004, pp. 89–104
McCallen R, Browand F, Ross J, eds. 2004. The Aerodynamics of Heavy Vehicles: Trucks, Buses, and Trains. New
York: Springer
Mellen CP, Frölich J, Rodi W. 2003. Lessons from LESFOIL project on large-eddy simulation of flow around
an airfoil. AIAA J. 41:573–81
Mendonça F, Allen R, de Charentenay J, Kirkham D. 2003. CFD prediction of narrowband and broadband cavity
acoustics at M = 0.85. Presented at AIAA/CEAS Aeroacoust. Conf. Exhib., Hilton Head, South Carolina,
Pap. No. AIAA-2003-3303
Mendonça F, Allen R, de Charentenay J, Lewis M. 2002. Towards understanding LES and DES for industrial
aeroacoustic predictions. Presented at Int. Workshop LES Acoust., Göttingen
Menter FR, Kuntz M. 2002. Adaptation of eddy-viscosity turbulence models to unsteady separated
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Menter FR, Kuntz M, Bender R. 2003. A scale-adaptive simulation model for turbulent flow predictions.
Presented at AIAA Aerosp. Sci. Meet. Exhib., 41st, Reno, Pap. No. AIAA-2003-0767
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the time-averaged location by blowing. AIAA J. 38:793–803
Mockett C, Greschner B, Knacke T, Perrin R, Yan J, Thiele F. 2008. Demonstration of improved DES
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Morton SA, Cummings RM, Kholodar DB. 2004. High resolution turbulence treatment of F/A-18 tail buffet.
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Nikitin NV, Nicoud F, Wasistho B, Squires KD, Spalart PR. 2000. An approach to wall modeling in
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www.annualreviews.org • Detached-Eddy Simulation
Concise exploration of
DES for wall modeling
inside LES.
Presents a wide
collection of recent
work on DES and other
hybrid approaches (but
not all).
First true 3D
application, calibration
of CDES , and successful
prediction of airfoil
forces at all angles.
Introduced delayed
DES to combat
grid-induced
separation.
Motivation for DES,
basic equations (with
CDES constant
undetermined), and
two-dimensional
examples.
Presents a wide range of
applications, including
models other than
Spalart-Allmaras.
201
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First DES with grid
refinement, fair
agreement on the drag
crisis, and refined
definition of DES in
appendix.
202
14:57
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at Eur. Cong. Comput. Methods Appl. Sci. Eng., Jyväskylä, Finland
Travin AK, Shur ML, Spalart PR, Strelets MKh. 2006. Improvement of Delayed Detached-Eddy Simulation for
LES with wall modelling. Presented at Eur. Conf. CFD, ECCOMAS CDF 2006. Delft, Neth.
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AIAA/CEAS Aeroacoust. Conf., 13th, Pap. No. AIAA-2007-3621
Ziefle J, Kleiser L. 2008. Compressibility effects on turbulent separated flow in streamwise-periodic hill
channel, part 2. See Peng & Haase 2008, pp. 316–25
Spalart
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Contents
Annual Review of
Fluid Mechanics
Volume 41, 2009
Von Kármán’s Work: The Later Years (1952 to 1963) and Legacy
S.S. Penner, F.A. Williams, P.A. Libby, and S. Nemat-Nasser p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 1
Optimal Vortex Formation as a Unifying Principle
in Biological Propulsion
John O. Dabiri p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p17
Uncertainty Quantification and Polynomial Chaos Techniques
in Computational Fluid Dynamics
Habib N. Najm p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p35
Fluid Dynamic Mechanism Responsible for Breaking the Left-Right
Symmetry of the Human Body: The Nodal Flow
Nobutaka Hirokawa, Yasushi Okada, and Yosuke Tanaka p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p53
The Hydrodynamics of Chemical Cues Among Aquatic Organisms
D.R. Webster and M.J. Weissburg p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p73
Hemodynamics of Cerebral Aneurysms
Daniel M. Sforza, Christopher M. Putman, and Juan Raul Cebral p p p p p p p p p p p p p p p p p p p p p p p91
The 3D Navier-Stokes Problem
Charles R. Doering p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 109
Boger Fluids
David F. James p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 129
Laboratory Modeling of Geophysical Vortices
G.J.F. van Heijst and H.J.H. Clercx p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 143
Study of High–Reynolds Number Isotropic Turbulence by Direct
Numerical Simulation
Takashi Ishihara, Toshiyuki Gotoh, and Yukio Kaneda p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 165
Detached-Eddy Simulation
Philippe R. Spalart p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 181
Morphodynamics of Tidal Inlet Systems
H.E. de Swart and J.T.F. Zimmerman p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 203
v
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Microelectromechanical Systems–Based Feedback Control
of Turbulence for Skin Friction Reduction
Nobuhide Kasagi, Yuji Suzuki, and Koji Fukagata p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 231
Ocean Circulation Kinetic Energy: Reservoirs, Sources, and Sinks
Raffaele Ferrari and Carl Wunsch p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 253
Fluid Mechanics in Disks Around Young Stars
Karim Shariff p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 283
Annu. Rev. Fluid Mech. 2009.41:181-202. Downloaded from www.annualreviews.org
by RENSSELAER POLYTECHNIC INSTITUTE on 04/06/11. For personal use only.
Turbulence, Magnetism, and Shear in Stellar Interiors
Mark S. Miesch and Juri Toomre p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 317
Fluid and Solute Transport in Bone: Flow-Induced
Mechanotransduction
Susannah P. Fritton and Sheldon Weinbaum p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 347
Lagrangian Properties of Particles in Turbulence
Federico Toschi and Eberhard Bodenschatz p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 375
Two-Particle Dispersion in Isotropic Turbulent Flows
Juan P.L.C. Salazar and Lance R. Collins p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 405
Rheology of the Cytoskeleton
Mohammad R.K. Mofrad p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 433
Indexes
Cumulative Index of Contributing Authors, Volumes 1–41 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 455
Cumulative Index of Chapter Titles, Volumes 1–41 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 463
Errata
An online log of corrections to Annual Review of Fluid Mechanics articles may be found
at http://fluid.annualreviews.org/errata.shtml
vi
Contents
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