Wall-bounded turbulenceAlexander J. Smits and Ivan Marusic Citation: Phys. Today 66(9), 25 (2013); doi: 10.1063/PT.3.2114 View online: http://dx.doi.org/10.1063/PT.3.2114 View Table of Contents: http://www.physicstoday.org/resource/1/PHTOAD/v66/i9 Published by the AIP Publishing LLC. Additional resources for Physics TodayHomepage: http://www.physicstoday.org/ Information: http://www.physicstoday.org/about_us Daily Edition: http://www.physicstoday.org/daily_edition

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In his classic 1971 text, Peter Bradshaw refers toturbulence as “the most common, the most im-portant and the most complicated kind of fluidmotion.”1 Now, more than 40 years later, theclaim remains easily justifiable. Turbulence is

likely familiar to anyone who has flown on an air-plane or watched water rush past a rock in a stream.It emerges in any number of natural and manmadesettings, from atmospheric and oceanic currents toflows in pipelines and heat exchangers. It influencesweather, pollution levels, and climate change andfigures into the design of propulsion devices, windturbines, clean rooms, artificial hearts, and irriga-tion systems.

The complexity of turbulence is evidenced bythe fact that after more than a century of concertedresearch effort, many of its seemingly simple ques-tions remain unanswered. It has been said, in fact—in a quote variously ascribed to Arnold Sommerfeld,Albert Einstein, and Richard Feynman—that “tur-bulence is the last great unsolved problem of classi-

cal physics.” Part of the complexity stems from thefact that turbulent flows are composed of recurrent,sometimes coherent flow structures, or eddyingmotions, that exhibit a range of length scales span-ning several orders of magnitude, all interactingwith one another.

The problem becomes still more complex whenthe flow is confined by one or more solid surfaces.The presence of a wall introduces new length scalesand fundamentally changes the nature of turbu-lence. Although the biggest changes are limited tothe thin layer near the surface, that layer is of out-sized practical importance. For example, the behav-ior of that near-wall region largely determines the

www.physicstoday.org September 2013 Physics Today 25

Alexander J. Smits and Ivan Marusic

Wall-bounded TURBULENCE

New experimental insights could pave the way for leaner, faster simulationsof turbulent fluid flow.

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Alexander Smits is the Eugene Higgins Professor of Mechanical andAerospace Engineering at Princeton University in Princeton, New Jersey,and a professorial fellow at Monash University in Victoria, Australia.Ivan Marusic is an Australian Research Council Laureate Fellow and aprofessor of mechanical engineering at the University of Melbourne inMelbourne, Australia.

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drag force on a plane or ship, the distribution of heatin the atmosphere, and the energy required to de-liver oil and other goods through pipelines. Con-sider that according to the US Department of Trans-portation, in 2009 the 409 000 miles of pipelines inthe US carried 21% of all ton-miles of freight. Forperspective, the US Energy Information Adminis-tration estimates that the environmental impact ofreducing transportation energy by 5% would beequivalent to that of doubling the US wind energyproduction.

Our understanding of wall-bounded turbulentflows has developed rather slowly. But recent ad-vances in computational and experimental capabil-ities have delivered new insights that may finallybreak the logjam. Experiments at a handful of newfacilities across the globe have revealed new scalingand universalities in wall-bounded flows, which inturn hold promise to vastly increase the kinds ofproblems we can solve computationally—and theefficiency with which we can solve them.

Viscosity and inertiaIn the simplest case, a turbulent flow is character-ized by a competition between viscous forces, which

damp out velocity fluctuations by dissipating ki-netic energy into heat, and inertia forces, which tendto generate and preserve velocity fluctuations. Theratio of inertia to viscous forces is known as theReynolds number, Re = UL/ν, where U and L arecharacteristic velocity and length scales and ν is thekinematic viscosity of the fluid.

If Re is less than 10 or so, inertia forces are neg-ligible and the flow is laminar and more or less per-fectly damped. The velocity field adjusts almost in-stantly to any changes in the pressure gradients thatdrive the flow. Such is the flow regime experiencedby swimming bacteria and dust particles in air.

In the intermediate range 10 < Re < 103, inertiaforces become increasingly important, though notstrong enough to give rise to persistent velocity fluc-tuations. Included in that category of laminar floware capillary and pulmonary flows in the humanbody and gliders in air or in water.

At Re > 103, however, viscous effects may not bestrong enough to damp out velocity disturbances in-

troduced into the flow field. As a result, a tiny fluc-tuation—due to, say, a small roughness element orsurface vibration—may grow to the point that itcauses the entire flow to destabilize.

Figure 1 shows one common form of distur-bance growth, a spreading patch of turbulence calleda turbulent spot, in air flowing over a flat plate. Smallvelocity fluctuations in the flow far from the plategenerate a turbulent spot near the plate’s surface.The spot grows with downstream distance until itencompasses the entire flow domain. In such a flow,bound on one side by a surface, turbulence is con-fined to a finitely thick region above the surfaceknown as a boundary layer.

The turbulent flow field is marked throughoutby irregular velocity and pressure fluctuations. Themagnitude of the velocity fluctuations may be aslarge as 10–50% of the time-averaged velocity. Thefluctuating velocity field can be viewed as a collec-tion of eddies of varying length scales.2 As we’ll see,the nature of the eddies is important in determiningthe statistical properties of the flow.

Eddies, great and smallConsider the canonical example of a flow througha pipe. If we are far downstream from the entranceto the pipe, and if the inlet conditions are steady,then the flow statistics become independent ofdownstream distance, and the flow is said to befully developed. Molecular-scale forces at thefluid–wall interface ensure that the velocity of thefluid at the wall is equal to the velocity of the wallitself; the wall is said to impose a no-slip boundarycondition. If y is the distance from the wall and thepipe is our reference frame, then both u, the instan-taneous streamwise velocity, and u‾, the time aver-age of that velocity, are zero when y = 0.

What does the velocity profile look like awayfrom the wall? If the flow is laminar, u‾ varies smoothly,as a parabola, with y. As shown in figure 2a, u‾ is great-est at the pipe’s centerline. In a turbulent flow, how-ever, eddies very effectively mix momentum—as wellas heat and mass—and significantly increase energydissipation. That tends to reduce the velocity gradientsin the bulk and, because the no-slip boundary condi-tion must still be obeyed, increase the gradients nearthe wall.

The result is a profile like the one shown in fig-ure 2b, in which the velocity gradients near the wallare much larger than they would be in a laminarflow. Viscous forces, which are proportional to ve-locity gradients, are also especially large in the near-wall region of a turbulent flow. They exert upon thewall what’s known as skin-friction drag, a key com-ponent of the resistance felt by an object—be it aship, fish, or plane—as it moves through a fluid.Skin-friction drag also influences the energy re-quired to pump fluid through a pipe. In general, themore turbulent a flow is, the more energy will belost due to the effects of the skin-friction drag.

The wall also affects the distribution of a turbu-lent flow’s energy. In the classical picture of turbu-lent flow, kinetic energy is introduced into the sys-tem in the form of large eddies. For pipe flows, thelargest eddies scale with pipe radius R and can ex-

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Turbulence

Turbulent spotLaminar flow

Figure 1. A turbulent spot develops in an initially laminar air flowacross a flat plate—a geometry known as a boundary-layer flow. Theview is from above, and the flow, from left to right, is visualized usingstreaks of smoke. Downstream, the spot grows to encompass the fulldomain of flow. (Adapted from ref. 15.)

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tend to lengths in excess of 10R. In a channel, thelargest eddies scale with the channel width; in aboundary-layer flow, they scale as the thickness ofthe turbulent layer.

In what’s known as an energy cascade, energyis transferred from large eddies to progressivelysmaller ones—usually by a mechanism called vor-tex stretching. Such cascades are thought to occurwithout significant energy loss—dissipative viscousforces are negligible at large scales—and the processis sometimes referred to as inertial transfer.

Eventually, the eddies become small enough tobe destroyed by viscous dissipation. That lengthscale sets the minimum eddy size and is known asthe Kolmogorov length scale η. Interestingly, thelength scales associated with the largest and small-est eddies give rise to an alternative definition of theReynolds number, Reη = (R/η)4/3.

For wall-bounded flows, we prefer to define theReynolds number in terms of τw, the frictional shearstress the fluid exerts on the wall, both because τwis experimentally measurable and because it is animportant parameter for many applications. At thecenterline of a pipe flow, the Kolmogorov lengthscale η can be estimated as (Rν3/uτ

3)1/4, whereuτ = (τw/ρ)1/2 is the so-called friction velocity and ρ isthe density of the fluid. Hence the Reynolds numberbecomes Reτ = uτR/ν.

The energy-cascade description assumes thatthe most energetic eddies are much larger than theleast energetic ones—that is, that Reη and Reτ arelarge. Although that assumption fairly describes thebulk portion of most wall-bounded flows, it breaksdown in the near-wall region. There, even the mostenergetic eddies are relatively small, on the order ofη. In fact, because the turbulent energy in a wall-bounded flow is typically supplied by the wall,there can exist a reverse energy cascade, from smallto large scales.

The laws of the wallDirectly solving or computing the flow field of tur-bulent flows is a difficult endeavor. However, scal-ing arguments can yield valuable insights into thebehavior of canonical flows—or at least certain re-gions of those flows—and allow one to identify im-portant flow characteristics using a relatively smallnumber of nondimensional parameters. Through-out much of the 20th century, the focus of scaling ar-guments was on understanding the behavior of ve-locity and turbulent momentum transport in thenear-wall region.

Turbulent momentum transport can be ex-pressed in terms of Reynolds stresses, mathematicalproducts of velocity fluctuations. In the pipe flow,for example, the quantity ρu′u′―, where u′ = u − u‾, corresponds to a streamwise Reynolds stress.(Other important Reynolds stresses in a pipe floware ρv′v′―, ρw′w′―, and ρu′v′―, where v′ and w′ are ve-locity fluctuations in the wall-normal and cross-stream directions.)

Near the wall, Reynolds stresses must go tozero and thus momentum transport is dominatedby viscous forces. In that so-called inner region, therelevant velocity and length scales are uτ and ν/uτ;

the nondimensional wall distance is y+ = yuτ/ν; and the nondimensional streamwise velocity is u+ = u‾/uτ. In the outer region of the flow, we expectmomentum transport to be dominated by Reynoldsstresses (although viscous forces do remain impor-tant for energy dissipation at very small scales). Therelevant length scale becomes that of the largest ed-dies, R. Velocity, meanwhile, can still be scaled byuτ, which is related to the wall stress and thereforeaffects the entire profile.

In the 1930s Clark Millikan proposed that if theinner region is much thinner than the outer one,there may exist an overlap region—a “(possibly)small but finite region near the wall”3 where boththe inner and outer scalings apply. Using dimen-sional and overlap arguments, he showed that themean velocity profile in the overlap region shouldfollow a logarithmic law: u+ = κ−1 ln y+ + C, where theconstant C depends on details of the flow, but κ,known as von Kármán’s constant, is thought to beuniversal. The result is the same one that was de-duced by Ludwig Prandtl in 1925. (Here, we’venondimensionalized according to the inner-regionlength scale; an expression nondimensionalized ac-cording to the outer-region length scale takes a sim-ilar form.) Millikan’s result has received widespreadexperimental support.4

In the 1970s, Alan Townsend recognized that inthe overlap region the Reynolds stress ρu′u′― behavesin a similar way, in the sense that both the inner andouter scalings apply. Townsend argued that the ed-dies in the overlap region are essentially attached tothe wall—their size scales accordingly with theirdistance y.5 Summing over the contributions fromthose attached eddies, Townsend predicted that the

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Figure 2. Pipe-flow velocity profiles. (a) In a laminar flow, velocityvaries smoothly—as a parabola—across the pipe cross section. (Here,u‾ is the time-averaged streamwise velocity at a given distance y fromthe wall.) (b) In a turbulent flow, mixing driven by turbulent eddiesresults in a time-averaged velocity profile that’s nearly uniform in thebulk flow but falls sharply in the thin region near the wall.

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Reynolds stress should follow its own logarithmiclaw: u′u′―+ = u′u′―/uτ

2 = B − A ln(y/R), where the con-stant B depends on the details of the flow, but A isexpected to be universal. (Again, although we’venondimensionalized according to the outer-regionlength scale, an expression nondimensionalized ac-cording to the inner-region length scale takes a sim-ilar form.)

Unlike the log law of velocity, the early evidencein support of Townsend’s law of Reynolds stresses

was rather circumstantial. Both derivations rest onthe assumption of a large separation of lengthscales—that is, large Reτ. Indeed, for many realflows—including flows past ships and planes and atmospheric flows across the Earth’s surface—Reτ ison the order of 104–106 (see figure 3). Until recently,however, even state-of-the-art experimental facilitiescould achieve at most Reτ ~ 5 × 103—sufficient to seehints of Townsend’s log law but not to definitivelyconfirm it. (Some atmospheric boundary-layer exper-iments can achieve Reτ of order 106, but the experi-mental challenges are numerous and the data tend tobe of poorer quality.) In the 1990s there was a majorpush to establish facilities capable of achievinghigher Re flows. And that is where the story of wall-bounded turbulence takes a new and exciting turn.

At the frontiers of turbulenceIn the march toward higher Reynolds numbers,three experimental facilities were especially note-worthy: the Princeton University Superpipe, whichuses compressed air as the working fluid; the largeboundary-layer wind tunnel at the University ofMelbourne in Australia; and the Large CavitationChannel, an immense water tunnel established bythe US Navy in Memphis, Tennessee. Around theturn of the century, those facilities began yieldinghigh-precision measurements of near-wall velocityand stress profiles at Reτ of order 105.

Results from a Princeton experiment are shownin figure 4 and clearly exhibit the logarithmic profileof streamwise Reynolds stresses predicted byTownsend. Moreover, the logarithmic portions ofthe velocity and turbulent-stress profiles coincide inspace. That universality was previously suspected,but has only now been confirmed.

A few features of the plot deserve a closer look.First, the inner region is remarkably thin, less than1% of the pipe radius. Second, Reynolds stresses

are larger in the inner regionthan in the outer one—an indi-cation that turbulent momen-tum transport is strongest nearthe wall. Because those stressesmust equal zero at the wall, thedata suggest that a peak in tur-bulent momentum transportlies in the inner region. Otherexperiments,4 not shown, indi-cate that the peak remains fixedat y+ ≈ 12 as Reτ increases. At thesame time, the scaling of theReynolds stress indicates thatthe logarithmic region makesan increasingly important (andeventually dominant) contri -bution to the overall energy balance.

Viscous forces, on the otherhand, are always strongest nearthe wall, but they decay quicklywith increasing distance fromit. As Reτ grows, the viscosity-dominated region becomes in-creasingly thin and increas-

ingly difficult to resolve in experiments andcomputations.

New physics, new computationsUltimately, scaling laws like those derived by Millikan and Townsend tell only part of the story ofturbulent wall-bounded flow. To obtain a more de-tailed picture and to describe flows with more complicated geometries, one typically must turn tocomputation.

The ideal approach is to numerically solve thesystem’s momentum and mass balances, the Navier–Stokes and continuity equations, respectively. In di-rect numerical simulation (DNS), one solves thoseequations at grid points on a mesh; reliable resultsrequire a mesh size comparable to the smallestlength scale of motion in the system. Thus, capturingthe spatial fluctuations of a three-dimensional tur-bulent flow requires on the order of Reτ

9/4 grid points.Capturing the fluctuating field in time and account-ing for additional computing overhead6 requirescomputer resources that scale nominally as Reτ

4.Considering many real-world flows have Reτ > 106,the computing costs associated with DNS canquickly become prohibitive.

At the other end of the computing spectrum arewhat’s known as Reynolds-averaged Navier–Stokes(RANS) methods, which solve time-averaged ver-sions of the Navier–Stokes and continuity equa-

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Turbulence

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Figure 3. A turbulent flow can be characterized by a Reynolds number Reτ, commonly interpreted as the ratio of the flow’s largest and smallest length scales of motion. The real-world systems represented here—(a) human blood flows, (b) wind farms, (c) engines, (d) cargo ships, (e) hurricanes, and (f) atmospheric winds—span several orders of magni-tude in Reτ. (Figure courtesy of Olivier Cabrit.)

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tions. The Reynolds stresses due to velocity fluctu-ations are then calculated based on empirical mod-els. Fast and cheap, RANS methods are the work-horse of industry. But they can be unreliable whenapplied to flows for which the Reynolds-stress mod-els haven’t been calibrated.

Large eddy simulation (LES) is a middle way:It solves the complete Navier–Stokes and continuityequations for large-scale motions and then invokesan empirically determined effective viscosity to de-scribe forces at the smallest scales. Because LES cap-tures much of the unsteady and 3D nature of turbu-lence, it can describe many practical flows with agreater fidelity than RANS.

However, when a wall is present the LES gridis typically constrained by the near-wall region,where turbulent and viscous momentum transportoccur on small scales (see figure 5a). As a result, forwall-bounded flows the computational costs of LESare estimated to scale as Reτ

1.8, only a modest savingsover DNS.7 If, instead, one could empirically modelthe flow in the near-wall layer (see figure 5b), thegrid requirements could be greatly relaxed, whichwould lead to a significant payoff; it’s estimated thatthe computational costs would scale as Reτ

0.2.In the 1970s James Deardorff 8 and Ulrich Schu-

mann9 independently attempted to develop suchwall-layer models. They approximated the bound-ary conditions based on time-averaged properties ofthe logarithmic region. That general approach is stillin use today in various forms, but because it doesn’taccount for the time-dependent nature of turbulentinteractions, it has achieved limited success in de-scribing high-Re flows (Reτ > 104).

Here insights from high-Re experiments—inPrinceton and Melbourne and at the University ofIllinois at Urbana-Champaign—should be helpful.Those studies all revealed unexpected flow struc-tures known as very large scale motions2 or super-structures,10 which extend as far as 10–30 R in thestreamwise direction in pipe flow. At high Re, thosesuperstructures can contribute up to 20% of the tur-bulent kinetic energy of a pipe flow.

What can those motions tell us about the muchsmaller eddies near the wall? We have postulatedthat the near-wall motions follow a universal behav-ior, independent of the flow geometry, and thatthose motions are modulated by the superstruc-tures, such that the actual velocity field at the wallis a superposition of the near-wall-eddy and super-structure velocity fields.11 Indeed, measurementssuggest that the superstructures so impose a verylow frequency modulation on the near-wall velocityfluctuations. Crucially, the parameters of the super-position and modulation can likely be derived fromthe velocity signature in the logarithmic layer.

Efforts toward implementing the wall-layermodel into LES are ongoing, but preliminary resultsare encouraging and have shown trends that are inline with experimental evidence.12 The approachholds promise for extending the computationally ac-cessible Re range up to that of atmospheric boundary-layer experiments.

Another valuable aspect of such wall-layermodels is that they elucidate the connection between

superstructures and the intensity of the local wallstress. Such information could plausibly reveal newstrategies to reduce drag by manipulating the super-structures, which are more accessible inputs for con-trol schemes than the near-wall motions that havebeen targeted in the past.

Zettaflops and yottabytesAdvances in camera, laser, and computing tech-nologies have vastly improved our ability to obtaintime- resolved velocity fields and other informationabout the dynamics of high-Re flows. However,DNS remains an important tool for generating high-fidelity, fully 3D data that are difficult to obtain ex-perimentally. Moreover, DNS will continue to serveas a benchmark for new measurement techniquesand new LES models.

To date, DNS of wall-bounded turbulence hasbeen limited to relatively low Reτ, at most around4 × 103. (A simulation at Reτ = 5 × 103 by RobertMoser and colleagues, currently ongoing, promisesto set a new record.) A tentative consensus is thatsimulations at Reτ = 104 would probably be suffi-cient to answer many of the field’s open questions—and that threshold could conceivably be reached be-fore the end of this decade. Still, it is interesting toconsider the computing resources required to carryout DNS at the high values of Reτ currently achiev-able in experiments.

According to calculations by Javier Jiménez13

that were adjusted to reflect the decreasing memory-

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Figure 4. Experimental measurements of the nondimensionalmean streamwise velocity u+ (red circles) and dimensionless stream-wise turbulent stress u′u′―+ (blue squares) in a pipe flow withReynolds number Reτ = 98 000. (Overbars indicate a time average;u′ represents a velocity fluctuation.) In a thin region (log layer,green) near the wall, both quantities scale logarithmically with dimensionless distance y+ from the wall, as indicated by the fittedcurves (black). The log layer represents the overlap of an inner region dominated by viscous forces and an outer region dominatedby inertia forces. (Adapted from ref. 16.)

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to-computing speed ratio in high-performance com-puters, one would need more than 10 zettaflops(10 × 1021 floating-point operations per second) ofcomputing power to complete a DNS at Reτ = 105 ina typical time. According to the TOP500 project,14 theworld’s fastest computer can perform at just one -millionth that speed, at about 33 petaflops. Assum-ing computing speed continues to grow at its currentexponential rate, DNS at Reτ = 105 won’t become feasible until roughly 2035.

Even if high-Re simulations do become realiz-able in the coming decades, tools for processing andhandling large data sets will need to keep up. DNSsimulations at Reτ = 104 and Reτ = 105 would gener-ate 23 terabytes and 23 petabytes of data per timestep, respectively. Simulating the required 107 timesteps at Reτ = 105 would produce around 0.2 yot-tabytes—more digital content than currently existsin the entire world.

Clearly, storing all of that information would be neither feasible nor desirable. Simulating high-Re flows will require selective storage of data as theyare being generated, which in turn will require ahigh-level appreciation of the physics. The same ap-plies to laboratory and field studies, where the rap-idly increasing size of data sets generated in 3D,time- resolved experiments is becoming a majorissue. The process will likely be an iterative and

evolving one, with the ultimate goal being to formu-late high-fidelity empirical models.

Gathering momentumIn this article we have focused on relatively simple,canonical wall-bounded turbulent flows. Other chal-lenges emerge if one wants to model turbulent flowshaving additional parameters or complications suchas compressibility, shock waves, multiple phases,combustion, density jumps, body forces due to pres-sure gradients, and magnetic fields. Each problempresents specific challenges, but a common threadremains: the need to understand how various-sizededdies emerge, evolve, and interact in the flow.

In the coming years, computational, experimen-tal, and modeling efforts should continue to producenew advances. Computing resources will undoubt-edly continue to improve, although those changeswill also require new approaches and technologiesfor handling vast data sets. New experimental facil-ities will also be important, and at present severalsuch facilities are either under construction or underdevelopment, including sulfur fluoride tunnels inGöttingen, Germany; the CICLoPE pipe facility inPredappio, Italy; and a turbulent convection facilityin Ilmenau, Germany.

Turbulence has been the focus of concerted re-search efforts since the work of Osborne Reynoldsmore than a century ago, and the general study ofthe topic dates back even further. The field’s recentadvances bode well for the foreseeable future, andwe anticipate—and certainly hope—that the mys-teries of turbulence will not have to wait yet anothercentury to be solved.

We gratefully acknowledge funding from the Office of NavalResearch, NSF, and the Australian Research Council. Wealso thank Bob Moser for his helpful comments.

References1. P. Bradshaw, An Introduction to Turbulence and Its

Measurement, Pergamon Press, New York (1971), p. xi.2. R. J. Adrian, Phys. Fluids 19, 041301 (2007).3. C. B. Millikan, in Proceedings of the Fifth International

Congress for Applied Mechanics, J. P. D. Hartog, H. Peters,eds., Wiley, New York (1939), p. 386.

4. A. J. Smits, B. J. McKeon, I. Marusic, Annu. Rev. FluidMech. 43, 353 (2011).

5. A. A. Townsend, The Structure of Turbulent Shear Flow,2nd ed., Cambridge U. Press, New York (1976).

6. V. Yakhot, K. R. Sreenivasan, J. Stat. Phys. 121, 823(2005).

7. U. Piomelli, E. Balaras, Annu. Rev. Fluid Mech. 34, 349(2002).

8. J. W. Deardorff, J. Fluid Mech. 41, 453 (1970).9. U. Schumann, J. Comp. Phys. 18, 376 (1975).

10. N. Hutchins, I. Marusic, J. Fluid Mech. 579, 1 (2007).11. I. Marusic, R. Mathis, N. Hutchins, Science 329, 193

(2010).12. M. Inoue, R. Mathis, I. Marusic, D. I. Pullin, Phys. Fluids

24, 075102 (2012).13. J. Jiménez, J. Turbulence 4, N22 (2003).14. TOP500 List–June 2013, http://www.top500.org/list

/2013/06.15. M. Matsubara, P. H. Alfredsson, J. Fluid Mech. 430, 149

(2001).16. M. Hultmark, M. Vallikivi, S. C. C. Bailey, A. J. Smits,

Phys. Rev. Lett. 108, 094501 (2012). ■

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Turbulence

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Figure 5. The grid spacing in a large eddy simulation must be fineenough to resolve all but the smallest length scales in a turbulentflow. In this simulated boundary-layer flow, red and blue areas indi-cate eddies, and the arrows indicate the mean streamwise velocity u‾at various distances y from the wall. (a) For standard large eddy simu-lation, the necessary grid spacing is exceedingly fine due to the verysmall scales in the near-wall region. (b) A coarser grid can be used ifthe inner layer near the wall, shaded white, is described instead by anempirical model. (Adapted from ref. 7.)

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