DES Annurev.fluid.010908

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Detached-Eddy SimulationPhilippe R. Spalart

Boeing Commercial Airplanes, Seattle, Washington 98124; email: [email protected]

Annu. Rev. Fluid Mech. 2009. 41:181–202

First published online as a Review in Advance on August 4, 2008

The Annual Review of Fluid Mechanics is online atfluid.annualreviews.org

This article’s doi:10.1146/annurev.fluid.010908.165130

Copyright c 2009 by Annual Reviews. All rights reserved

0066-4189/09/0115-0181$20.00

Key Words

turbulence, separation, boundary layer, modeling

Abstract

Detached-eddy simulation (DES) was first proposed in 1997 and first used

1999, so its full history can be surveyed. A DES community has formed, wadepts and critics, as well as new branches. The initial motivation of higReynolds number, massively separated flows remains, for which DES is co

vincingly more capable presently than either unsteady Reynolds-averagNavier-Stokes (RANS) or large-eddy simulation (LES). This review d

cusses compelling examples, noting the visual and quantitative success DES. Its principal weakness is its response to ambiguous grids, in which t

wall-parallel grid spacing is of the order of the boundary-layer thickness.

some situations, DES on a given grid is then less accurate than RANS on tsame grid or DES on a coarser grid. Partial remedies have been found, y

dealing with thickening boundary layers and shallow separation bubbles i

central challenge. The nonmonotonic response of DES to grid refinemeis disturbing to most observers, as is the absence of a theoretical orderaccuracy. These issues also affect LES in any nontrivial flow. This revi

also covers the numerical needs of DES, gridding practices, coupling wdifferent RANS models, derivative uses such as wall modeling in LES, a

extensions such as zonal DES and delayed DES.

181



a b

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) (Mendonca et al. 2002). Figure courtesy of F. Mendonca and Ford Motor Co.

1. BASICS

Figure 1 illustrates the nature of detached-eddy simulation (DES). The aircraft geometry

complete (except for detailed surface and propulsion effects); the simulation is at flight Reynonumber; the large-eddy simulation (LES) content (resolved turbulence) in the separated reg

is rich; and the Reynolds-averaged Navier-Stokes (RANS) function plays a role on the aircranose. Furthermore, the forces and moments are accurate to within 6% (Forsythe et al. 200

This approach must still be considered experimental as a prediction method, and the accurabenefits from the thin edges on the wing; there is no marginal separation to challenge the mod

In addition, grid refinement does not indicate grid independence on the smaller components, su

as the tail surfaces. The automobile geometry is also complete, a feat of the grid generator and solver rat

than of DES (Mendonca et al. 2002). The two regions of the DES are especially well visualizsteady attached boundary layers and striking LES content around the wheels and the importan

pillar and outside mirror. The drag is dependent on the separation line near the end of the roand the accuracy of the RANS model matters. At the same time, the LES function is indispensa

to predict the aerodynamic noise and in fact the drag. These two studies reflect the broad diffusof DES.

1.1. Conceptual History

DES was created to address the challenge of high–Reynolds number, massively separated flow

which must be addressed in such fields as aerospace and ground transportation, as well as

atmospheric studies. It combined LES and RANS, spurred by the belief that each alone wpowerless to solve the problem at hand (Spalart et al. 1997). This complaint can be revisi

presently, assuming a working knowledgeof LES andRANS (Rogallo & Moin 1984, Wilcox 199 The objection to pure LES is simple and centers on computational cost. A pure LES of

airborne or ground vehicle would use well over 1011 grid points and close to 107 time steps, whis estimated to be possible in approximately 2045 (Spalart 2000). The boundary layer domina

this expense, which is necessary even if investigators solve the problem of wall modeling in LERegardless, the resolution needs in the outer region of the boundary layer are firm, with at t

least 20 points per thickness δ in each direction. No unforeseen breakthrough has occurred

182 Spalart



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, butnot 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 canalso 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 griddensity 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 atbest 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 createdbad 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 bereached, 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 principaldifficulty 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 atbetter 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

www.annualreviews.org • Detached-Eddy Simulation 183



a

b

c

d

e

f

184 Spalart



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 foundthat drag and lift fluctuations were overpredicted by URANS, although the shedding frequency

was accurate. The term URANS here means running an unmodified (grid-insensitive) transport-equation turbulence model, in unsteady mode and with periodic spanwise conditions. Recent

findings have revealed that under fairly general conditions, these simulations in fact sustain three-dimensionality 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 3DURANS. 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, butnot 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 (whenapplicable) 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 itactually 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 complexgeometry, 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 anaircraft (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 tomodel 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 valuablefeature. 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: ReD = 5 × 104, laminar separation. Experimental dragcoefficient C d = 1.15–1.25. (a) Shear-stress transport (SST) turbulence model steady Reynolds-averagedNavier-Stokes (RANS), C d = 0.78; (b) SST 2D unsteady RANS, C d = 1.73; (c ) SST 3D unsteady RANS,C d = 1.24; (d ) Spalart-Allmaras (SA) detached-eddy simulation (DES), coarse grid, C d = 1.16; (e) SA DES,fine grid, C d = 1.26; ( f ) SST DES, fine grid, C d = 1.28. Figure courtesy of A. Travin.




a b

R e s o l v e d T

K E ( k / U 2 )

X/c

0.5

0.4

0.3

0.2

0.1

00.250 0.50 0.75 1.00

G1 (1.2 M cells)

G2 (2.7 M cells)

G3 (6.6 M cells)

G4 (10.5 M cells)Experimental peak approximately 0.5

Figure 3

(a) Flow visualizations and (b) resolved turbulent kinetic energy (TKE) for a sharp-edged delta wing at a 27◦ angle of attack, chordReynolds 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 sma

components. Grid-refinement effects were more predictable, however, on Morton’s (2003) de

wings. The simpler geometry helped, but the phenomenon of vortex breakdown is a subtle o The results are rewarding, shown visually in Figure 3 a and quantitatively in Figure 3b. Fin

grids introduce vortex shedding at the trailing edge and much finer structures in the vort The front half of the vortex is also quite different: The helical striations switch direction fr

a coarse to a fine grid. Figure 3b is especially favorable, as it suggests near-grid convergenof the resolved turbulent kinetic energy to a level that agrees with experiment both for ener

level, approximately 0.5, and location, X /c = 0.65± 0.05 (Mitchell et al. 2000). A scale-adaptsimulation also produced resolved turbulence in this flow (Egorov & Menter 2008).

Thestudyfeaturedin Figure 4 a,b also reflects thequantitative successof DES. Constantineet al. (2002) simulated the flow past a sphere with approximately 600,000 points in the baseli

grid and controlled the model in the boundary layer so that it produced laminar separationa diameter of Re = 105 and turbulent separation at Re = 1.1 × 106. The latter is somew

simplistic because in the real flow, transition and separation are not segregated (Travin et

2000a), but it is far superior to letting an untrained subgrid-scale model handle the boundalayer, effectively in RANS mode. Quite a few recent cylinder computational fluid dynamics (CF

studies even failed to select laminar separation at subcritical Reynolds numbers; Travin et a(2000a) tripless approach is needed, and Nishino et al. (2008) adopted it successfully. With t

approach, the prediction of a drag crisis is striking, and the pressure distributions are extremfavorable both when compared with experiment and when comparing baseline and fine grids.

the lower Reynolds number, DES predicts a drag coefficient of 0.41, compared with 0.40–0in experiments; at the higher Reynolds number, DES gives 0.084 and experiments give 0.12. I

tempting to extend this approach to golf balls. The drag crisis caused by dimples can be capturin a gross sense, simply by imposing turbulent separation with a smooth geometry. However,

186 Spalart



θ

C p

0 1 2

0

a

b c1.00

0.50

0.00

–0.50

–1.00

–1.500 30 60 90 120 150 180

0.5

–0.5

y / D

x /D

Re = 105 Re = 1.1 × 106

Re = 1.1 × 106

Re = 105

Figure 4

Simple bluff bodies. (a) Flow visualizations and (b) pressure distributions for a sphere. Re = 105 and 1.1× 106. Open circles anddiamonds denote experiments, whereas the dotted and dashed lines denote detached-eddy simulation (DES) on two grids. Panels a ab courtesy of K. Squires. (c ) Phase-averaged vorticity contours for a cylinder. Color gradations denote DES conducted by Mockett et(2008), and the solid line denotes experiments by the same authors.

RANS model could reproduce the dimple effect accurately, and this will require direct numericalsimulation (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, LEStreatment 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 turbulent-

separation 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




circular cylinder (Mockett & Thiele 2007); the DES visualizations are close to those shown

Figure 2e, f . The agreement on the phase-averaged flow pattern is excellent.

2.2. Applications

DES has been applied oftenwith good results to cavities over a range of Mach numbers (Allen et

2005, Hamed et al. 2003, Langtry & Spalart 2007, Mendonca et al. 2003, Shieh & Morris 200ground vehicles (Kapadia et al. 2003, Maddox et al. 2004, Roy et al. 2004, Spalart & Squi

2004, Sreenivas et al. 2006), a simplified landing-gear truck (Hedges et al. 2002), active flcontrol 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 & Oak2006), cavitation in jets (Edge et al. 2006), buildings (Wilson et al. 2006), air inlets (Trap

et al. 2008), aircraft in a spin (Forsythe et al. 2006), high-lift devices (Cummings et al. 200 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-

systems, helicopters, combustors, and afterbodies. Chalot et al. (2007) reveal a vigorous line work in another aircraft company, Dassault. Slimon (2003) obtained positive results with (zon

DES in a turn-around duct; DES did much better than RANS with simple models, however, whmay not be expected to capture curvature effects. Publications aimed at educating users and co

writers have, appropriately, focused on grid generation (Spalart 2001) and on thorough testingthe codes (Bunge et al. 2007, Squires 2004). The terminology Euler region, RANS region, foc

region, and departure region, introduced by Spalart, may be of help. Grid adaptation in DES aLES is a future challenge.

Another promising direction is taken by Mockett et al. (2008) and Greschner et al. (200aerodynamic noise. Such studies will contribute both to interior noise in vehicles and aircr

and to community noise (airframe noise to the airline industry). We note above the industrimportance of the turbulence adjacent to the driver’s window (Figure 1b). Mockett et al. (20

studied the flow in the slat cove of an airfoil in landing configuration; the visualization w

density gradient in Figure 5 a vividly reveals much fine-scale turbulence and sound. Actual soupredictions are not included.

Greschner et al. (2008) provide sound predictions for the flow past a cylinder, placed aheof an airfoil so that its turbulent wake impinges on it (see Figure 5b). At low Mach numbe

this impingement, which causes wall-pressure fluctuations, is the dominant noise mechanis Various Ffowcs-Williams-Hawkings surfaces are used to extract far-field noise. Flow visualizati

resemble those in Figure 2, without as fine a level of resolution. This case is more onerobecause the turbulence needs to be carried all the way to the airfoil, 10 diameters downstrea

the focus region is much larger. Figure 5c compares the sound spectrum with experiment. adjustment was made in the vertical direction: In 2D geometries, there is an unsolved probl

when comparing an experiment of finite length (with some end conditions) to a simulation wperiodic boundary conditions, invariably quite narrow (in contrast, no adjustment was needed

the spectra inside the turbulence region). Once this correction is accepted, the agreement on

shape of the spectrum, over five octaves, is quite amazing.Figure 6 (Chauvetetal.2007)isofinterestfortworeasons.First,theLES-contentdevelopm

in the mixing layer is nearly immediate, which is positive, although it may be excessively 2D (sSection 3.4). Second, the simulation is simultaneously free enough of numerical dissipation

welcome LES content and robust enough to capture shocks. This result has also been achievedShur et al. (2006) in jets and by Ziefle & Kleiser (2008) in a supersonic channel with hills. Th

188 Spalart



0

20

40

60

80

Experiment

DES + FWH

θ = 90°

100

P S D ( d

B )

St = f × D/u∞

10–1 100

Y

XZ

|Δp'|

0.001 0.01 0 .1 1 10

a

b

c

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.




X /D X /D

y / D

1

0

–1

1

0

–1

1 2 3 1 2 30

a b

Figure 6

(a) Experimental schlieren (view through flow) and (b) numerical schlieren (contours in center plane) for supersonic jet. Figure taken from Chauvet et al. 2007.

studies remove the concern that LES might be barred from supersonic flows, therefore widen

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 co

ceptual nature could categorize DES as a method that is intuitively correct and often successbut dissatisfying to the purist. Below we first address these concerns and then delve into practi

issues in the remaining subsections. The criticism of URANS mentioned above (namely that the approximate PDE that is sol

is known, but the exact PDE it is meant to approximate is not) does not truly apply to DESfilter can be linked to the grid cell and to the integration implied by the CFD solver. In LE

systematic studies use filtered versions of DNS fields to steer subgrid-scale model developme

This is known as the Clark test or a priori study and could be performed with DES but has nin LES and DES practice, models are adjusted based on results rather than explicit tests. T

new difficulty beyond those in LES is that, in the gray area, the model has a strong impact, buconvincing calibration is simply out of reach: There are far too few degrees of freedom (in DES

only C DES ). A similar problem is present even in simple LES; simply put, one would adjust tsingle Smagorinsky constant to ensure that all six subgrid stresses are correct. The problem

more severe in wall-modeled LES (WMLES) and more severe again in DES. Clear statemeare much more difficult to make, especially in view of the wide variety of anisotropies possible

the grid cell and time step, and also because of history effects, which are strong especially in all-important situation of a separating boundary layer (see Section 3.4). The essential difficulty

that the model has much more impact on WMLES and DES than it does on the notional Lsituation, namely away from walls and with a grid spacing in the inertial range. In that situati

one can arbitrarily lessen the influence of the Smagorinsky constant and similar constants with g

refinement. WMLES has been exposed to this issue less than DES, possibly because it sometimseems unable to escape channel flow.

The literature reflects a desire for an approach that is somehow more justified and mathemacally definedthan DES. Several hybridproposalsrest on theidea of splitting theturbulentenergy

a specified ratio (e.g., 70%resolved and30% modeled). This is finein simple flows, butthe strenof DES (and WMLES) is precisely that the split is different in different parts of the same soluti

Theenergysplitcanbeadjustedindifferentregions,butthisincreasesthedecisionloadfortheus

190 Spalart



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 scaleof 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 LESbehavior 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 ahigher 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 andturbulence-resolving RANS work is motivated by the disruptive effects of in DES with ambigu-

ous grids (see Section 3.2). This stimulating controversy is not over. It echoes the one in RANSmodeling 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 manageableand 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 formonotonic 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 certaindegree of grid convergence” (Nishino et al. 2008).

An order of accuracy has not even been proposed for a simulation using both modes withinDES. 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 inthe 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 com-

pounded by the residual variations owing to finite time samples; some flows have severe modu-lations 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 coefficientis 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 themodulations 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 significantpractical issues and have been worse to deal with than initially anticipated (Spalart et al. 1997).

Figure 8 a 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




a b0.6

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–0.6

0 40 80 120 160 200 0 40 80 120 160 20

1.30

1.25

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1.10

1.05

1.00

0.95

C ι , C ι

C d , C d

tU /D tU /D

Instantaneous

Time-averaged

Instantaneous

Time-averaged

Figure 7

Instantaneous ( solid line) and time-averaged (dashed line) values of force coefficients on a cylinder: (a) lift a(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 DESa wall model (see Section 3.3): The grid spacing in all directions is much smaller than δ. Tsecond level is the troublesome one: small enough for the eddy viscosity to be affected by the D

limiter but not small enough to support accurate LES content (slow LES development adds

this difficulty; see Section 3.4). Spalart et al. (2006) coined the term MSD, well after the issue wdetected by S. Deck (personal communication) and by Menter & Kuntz (2002), who pointed o

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 exam

of gradual grid refinement degrading a solution that was rather good when the RANS model wfully active (S. Deck, personal communication; see also Caruelle & Ducros 2003). Separation

a nozzle is premature and induces unsteadiness. DES users promptly explored the effects of g

spacing and sought high accuracy, with disturbing outcomes.Figure 9 is a visualization of GIS, this time on an airfoil (Menter & Kuntz 2002). Where

the RANS solution is steady and quite accurate, even in this case of incipient separation, the D

solution suffers from early separation. It also is unsteady, but in a shedding mode rather thana sound turbulence-resolving mode. The flow field then obeys the URANS equations, but wit

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 which the DES limiter is disabled as long as the flow is recognized as a boundary layer, usi

the SST F 2 function. Spalart et al. (2006) introduced DDES, which is applicable to most modEither modification successfully prevents GIS by extending the RANS region, exploiting a hist

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 conduct

long before DES; wall modeling near the walls of an LES draws on RANS technology, and eachannel LES studies even used wall functions. A new feature of DES is that the entire bound

layer can be handled by RANS. However, DES also naturally provides a simple wall model, wh

192 Spalart



0 10 20

0

N o r m a l i z e d w a l l p r e s s u r e ( P W

/ P C

)

0.05

0.04

0.03

0.02

0.01

X /r t

DES computation PR40

SA-URANScomputation PR40

LEA steady experimentaldata PR41.3

1.0

0.5

00 0.5 1.0 1.5 2.0

0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0

2.5 3.0 3.5 4.0

y / δ

1.0

0.5

0

1.0

0.5

0

y / δ

x /δ

aa

b

x /δ x /δ

Figure 8

(a) Types of grids in boundary layers. The dashed line represents the velocity profile. (b) Pressuredistribution in a supersonic nozzle. Figure courtesy of S. Deck. DES, detached-eddy simulation; LEA,Laboratoire d’Etudes A erodynamiques; 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. LEScontent 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 10 a 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). TheReynolds shear stress comprises modeled stress and resolved stress, which trade places as the grid

is varied (Figure 10b).




V e l o c i t y

0

–5.00

–10.00

a b

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 veloc

U +. The probability that this log-layer mismatch would be zero was nil because this study usthe pure DES97 model, adjusted for other purposes. (The study was also marked by deliber

constraints, such as equal grid spacing in the wall-parallel directions, to ensure the findings woutranslate into practice.) All other wall-modeling approaches have required adjustments to al

their log layers. Nikitin et al. (2000) mentioned the ensuing error of the order of 15% for tskin-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

mismatch closer to the wall. This is different from MSD in a near-RANS boundary layer, wh

40

35

30

25

20

15

10

5

101

102

103

104

105

y +

U +

1.0

0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 y /h

τ +

a b

Figure 10

Channel-flow, wall-modeled large-eddy simulation. (a) Velocity: Reτ = 2000 and 20,000. Each profile isshifted by five U + units. The lower two curves use approximately 140,000 grid points, and the upper curvuses approximately 1,000,000 points. The dashed line represents the log law. (b) Modeled and resolved shstress: coarser grid (dashed line) and finer grid ( solid line). Reτ = 20,000. Figure adapted from Nikitin et al2000.

194 Spalart



0 2 4 6 8

–1

0

1

x

y

0 2 4 6 8

–1

0

1

x

y

a b

Figure 11

Vorticity in a jet: (a) standard detached-eddy simulation and (b) implicit large-eddy simulation, eddy viscosity disabled. Figure courteof M. Shur.

becomes more severe as the grid is refined. Follow-on work by Piomelli and his group also showedthat 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 thesense of Figure 8 a, 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 ontypical 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 (Shuret 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 adaptationof 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 thicknessreaches 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, mak-ing 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. Thisproblem has received and deserved attention, but unlike the two problems discussed in the pre-

ceding 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 adaptationto two-equation models, in particular the SST model, which has been smooth. Recent work in-

cludes, for instance, Greschner et al.’s (2008) cubic explicit algebraic stress models. Themotivationfor 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.




4.2. Zonal Detached-Eddy Simulation

In zonal DES, the user explicitly marks different regions as RANS or as DES (Deck 2005).effect, in RANS regions, is made infinite (as opposed to zero in implicit LES). This is proba

the strongest departure from the original concept of DES, in which the use of a single but versaequation set is central, and creates most of the conceptual and practical challenges. The motivat

is to be fully safe from MSD and GIS (see Section 3.2) and to clarify the role of each region. ZoDES worked well for Brunet & Deck (2008) in the important problem of wing buffet, Chauv

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 f

question to propose to zonal DES proponents concerns complex flows, in which decisions

neededfor numerous regions (including the thickness of regions meant to contain RANS boundlayers). This is similar to issues with zonal control of laminar-turbulent transition. Which mo

will be the default, and which will be the exception? S. Deck (personal communication) is in favof RANS as the default mode; the author may disagree,and, more importantly, there is the conc

that smooth-wall separation is normally not known at the time the zones are set. Compared wDES, ZDES appears simultaneously more powerful and less self-sufficient.

4.3. Delayed Detached-Eddy Simulation and Improved DelayedDetached-Eddy Simulation

A key motivation here is precisely to avoid zonal measures, thus leaving it to the solution procto determine separation, while addressing the MSD issue that affects DES97 (see Section 3.

Following Menter & Kuntz (2002), DDES detects boundary layers and prolongs the full RANmode, evenif thewall-parallel grid spacing would normally activate theDES limiter. This detect

device depends on the eddy viscosity, so that the limiter now depends on the solution (Spalet al. 2006). This is a formal deviation from DES97 but not a different mission. DDES w

shown to resolve GIS, without impeding LES function after separation. For instance, it hand

a backward-facing-step flow well, even with grids that would cause severe MSD both upstreof the step and all along the opposite wall. DDES is likely to be the new standard version

DES.Improved delayed DES (IDDES) is more ambitious yet (Shur et al. 2008). The approach

also nonzonal and aims at resolving log-layer mismatch in addition to MSD. One basis is a ndefinition of , which includes the wall distance and not only the local characteristics of the gr

The modification tends to depress near the wall and give it a steep variation, which stimulainstabilities, boosting the resolved Reynolds stress. Other components of IDDES include n

empirical functions, some involving the cell Reynolds number, which address log-layer mismatand the bridge between wall-resolved and wall-modeled DES (grids with moderate values of

spacing in wall units, + ). These functions make the formulation less readable than that of DES

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

uniqueness of solutions. For instance, in a channel flow with periodicity and a grid and tistep capable of LES (as in Nikitin et al. 2000 and Figure 10), the solution has two branch

depending on the initial condition. If the flow is in a RANS state, with high eddy viscosity a weak perturbations, it remains in that mode and finds a steady state. If the flow starts in

LES state with low eddy viscosity and sufficient LES content, it settles into a statistically steLES. Both solutions are valid, but this situation perplexes some observers (Fr olich & von Te

2008).

196 Spalart



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 forcapturing the drag crisis of smooth bluff bodies. The mature solution depends on the level of the

turbulence variables in the initial field.

4.4. ModifiedΔ 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 successdefinitions 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 DDESmodification (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 Kelvin-

Helmholtz instability to grow. Some use the time-honored definition in LES = ( x y z)1/3, which of course reduces , but its physical justification is thin. Chauvet et al.’s (2007) length scale

≡

N 2

x y z+ N 2

y x z+ N 2

z x y , where N is the unit vector aligned with vorticity, isaimed 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 lateralgrid 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 reducedlength 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 upwindin 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 1 a 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 geome-

tries, 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 shock-

capturing capability (Hou & Mahesh 2004). The priority was given to high orders of accuracy.




An advantage of DES is the ease of programming and application. Potentially, it is activa

directly from the menu of turbulence models in many of the vendor CFD codes. This is a doub

edged advantage, as users not invested in turbulence and/or too trusting of the experts could accresults without verifying LES content, grid resolution, time step, time sample, and so on. An ea

example of this was an entry in the LESfoil workshop (Mellen et al. 2003). The simulation wformally a DES, and the results were fine. However, there is every indication from the grid th

the simulation was actually in RANS mode, even in the key region. In contrast, the genuine Lstudies struggled with all the issues of lateral domain size, resolution, and initiation of LES cont

in attached flows.

6. OUTLOOK

It is certain that DES has a future and therefore deserves a critique. Greschner et al. (200deem that “DES is still in its infancy and undergoes continuing improvements.” Under one nam

or another, a form of a RANS-LES hybrid that is capable of full RANS function in bound

layers will be in use for the foreseeable future in many industries. It will also remain conceptuadifficult, and efforts toward more predictable behavior under grid variations and better wa

modeling performance will continue. LES-content creation in attached flows will flourish, athe numerical quality of the codes will receive sustained attention. A clear need in practice is

organize and facilitate grid generation and to set guidelines for systematic refinement. Prograsuch as DESider and focused workshops will be most beneficial to the progress of DES and oth

hybrids (Peng & Haase 2008). An unfortunate trend is that models have moved away from the simplicity of DES97 in ter

of the equations and nonuniqueness of solutions (in DDES and IDDES) and in terms of the udecision load and need to mark regions (in ZDES). Users by now have identified situations

which DES gives too little eddy viscosity and others in which it gives too much. Even in DESlarge steps in the grid spacing can be used to steer the solution toward one mode or the oth

so that grid design can become involved, especially now that the dangers of ambiguous grids

known. What may be an ideal of CFD, namely that grid refinement will do no harm (in oth words, be monotonic) and follow a known power of the grid size, will remain elusive in DES a

LES (without explicit filtering), except in the simplest of flows. There are signs that a productive DES user community has formed. We must recogn

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 besid

DES and SAS (e.g., limited numerical scales, very large eddy simulation, flow simulation methoology, nonlinear disturbance equations, extra-large eddy simulation, lattice Boltzmann meth

transient RANS, partially averaged Navier-Stokes, semideterministic method, organized edsimulation, partially integrated transport model, and the self-adapting model) (some are found

Sagaut et al. 2006; Fr olich & von Terzi 2008). I do not believe that any of these methods provida clear remedy to the difficulties discussed here, but this could change in the future. The princi

concerns are GIS and in general the potentially poor knowledge of the nature of the simulation

each region of a complex flow: driven URANS, spontaneous URANS, or LES. This nature cchange under grid refinement and become ambiguous, and therefore it is not the case that a

grid refinement improves the solution. The nominally universal character of DES makes thobservers justifiably dubious that a sufficiently error-proof approach results, or that the user co

munity is being properly informed. Such comments are encountered more often in conversatioand anonymous reviews than in publications. It does not detract from their value, and the ta

of resolving them is an inspiring one. Locally ambiguous grids may be a permanent feature

198 Spalart



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 withgrid adaptation.

2. The link between the DES flow field and the exact or DNS flow field should be estab-lished.

3. The choice between zonal and nonzonal treatments of the turbulence needs to be ad-dressed.

4. The generation of resolved turbulence in attached boundary layers needs to becomeroutine and efficient.

DISCLOSURE STATEMENT

The author is not aware of any biases that might be perceived as affecting the objectivity of thisreview.

ACKNOWLEDGMENTS

I am grateful to Drs. Allmaras, Deck, Mockett, Strelets, Shur, and Travin for their comments onthis manuscript and their partnership over the years.

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to DES for hybrid

RANS-LES.

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Presents a wide

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work on DES and oth

hybrid approaches (bu

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ShurML, Spalart PR, StreletsMKh. 2005c.Noise predictionfor increasingly complex jets. Part II: applications.

Int J. Aeroacoust. 4:247–66

First true 3D

application, calibratio

of C DES , and successfu

prediction of airfoil

forces at all angles.

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wall-modeled LES capabilities. Int. J. Heat Fluid Flow. In press

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layer past an axisymmetric trailing edge. J. Fluid Mech. 591:215–53

Slimon S. 2003. Computation of internal separated flows using a zonal detached eddy simulation approach.

Proc. ASME Int. Mech. Eng. Congr., Pap. No. IMECE2003-43881. New York: ASME Int.

Spalart PR. 2000. Strategies for turbulence modelling and simulations. Int. J. Heat Fluid Flow 21:252–63Spalart PR. 2001. Young person’s guide to detached-eddy simulation grids. Tech. Rep. NASA CR-2001-211032.

Langley Res.Center, Hampton, Va. http://techreports.larc.nasa.gov/ltrs/PDF/2001/cr/NASA-2001-

cr211032.pdf Introduced delayed

DES to combat

grid-induced

separation.

Spalart PR, Deck S, Shur ML, Squires KD, Strelets MKh, Travin A. 2006. A new version of detached-

eddy simulation, resistant to ambiguous grid densities. Theor. Comp. Fluid Dyn. 20:181–95

Spalart PR, Hedges L, Shur M, Travin A. 2003. Simulation of active flow control on a stalled airfoil. Flow

Turbul. Combust. 71:361–73

Motivation for DES,

basic equations (with

C DES constant

undetermined), and

two-dimensional

examples.

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and on a hybrid RANS/LES approach. In Advances in DNS/LES , ed. C Liu, Z Liu, pp. 137–47.

Columbus, OH: Greyden Press

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2004, pp. 29–45Squires KD. 2004. Detached-eddy simulation: current status and perspectives. In Direct and Large-Eddy Sim-

ulation V , ed R Friedrich, BJ Geurts, O Metais, pp. 465–80. Dordrecht: Kluwer

Sreenivas K, Pankajakshan R, Nichols DS, Mitchell BCJ, Taylor LK, Whitfield DL. 2006. Aerodynamic simu-

lation of heavy trucks with rotating wheels . Presented at AIAA Aerosp. Sci. Meet. Exhib., 44th, Reno, Pap.

No. AIAA-2006-1394 Presents a wide range

applications, includin

models other than

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202 Spalart



Annual Review

Fluid Mechani

Volume 41, 200

Contents

Von K arman’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 p 17

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 p 35

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 p 53

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 p 73

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 p 91

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 Kanedap p p p p p p p p p p p 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



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

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

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

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

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

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

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

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

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

Cumulative Index of Chapter Titles, Volumes 1–41p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

Errata

An online log of corrections to Annual Review of Fluid Mechanics articles may be f

at http://fluid.annualreviews.org/errata.shtml

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