Artmospheric broundary layer (B and W)

Designing large-eddy simulation of the turbulent boundary layer to capturelaw-of-the-wall scalingJames G. Brasseur and Tie Wei

Citation: Phys. Fluids 22, 021303 (2010); doi: 10.1063/1.3319073 View online: http://dx.doi.org/10.1063/1.3319073 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v22/i2 Published by the American Institute of Physics.

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Designing large-eddy simulation of the turbulent boundary layerto capture law-of-the-wall scalinga

James G. Brasseurb and Tie Wei (Department of Mechanical Engineering, The Pennsylvania State University, University Park,Pennsylvania 16802, USA

Received 18 September 2009; accepted 5 January 2010; published online 25 February 2010

Law-of-the-wall LOTW scaling implies that at sufficiently high Reynolds numbers the meanvelocity gradient U /z in the turbulent boundary layer should scale on u /z in theinertia-dominated surface layer, where u is the friction velocity and z is the distance from thesurface. In 1992, Mason and Thomson pointed out that large-eddy simulation LES of theatmospheric boundary layer ABL creates a systematic peak in zU /z / u /z in thesurface layer. This overshoot is particularly evident when the first grid level is within the inertialsurface layer and in hybrid LES/Reynolds-averaged NavierStokes methods such as detached-eddysimulation, where the overshoot is identified as a logarithmic layer mismatch. Negativeconsequences of the overshootspurious streamwise coherence, large-eddy structure, and verticaltransportare enhanced by buoyancy. Several studies have shown that adjustments to the modelingof the subfilter scale SFS stress tensor can alter the degree of the overshoot. A comparison amongsimulations indicates a lack of grid independence in the prediction of mean velocity that originatesin surface layer deviations from LOTW. Here we analyze the broader framework of LES predictionof LOTW scalingincluding, but extending beyond, the overshoot. Our theory includes acriterion that is necessary to remove the overshoot but is insufficient for LES to produce constantz1 / through the surface layer, and fully satisfy the LOTW. For mean shear to scale on u /zin the surface layer, we show that two additional criteria must be satisfied. These criteria can beframed in terms of three nondimensional variables that define a parameter space in which systematicadjustments can be made to the simulation to achieve LOTW scaling. This occurs when the threeparameters exceed critical values that we estimate from basic scaling arguments. The essentialdifficulty can be traced to a spurious numerical LES viscous length scale that interferes with thedimensional analysis underlying LOTW. When this spurious scale is suppressed sufficiently toretrieve LOTW scaling, the LES has been moved into the supercritical high accuracy zone HAZof our parameter space. Using eddy viscosity closures for SFS stress, we show that to move thesimulation into the HAZ, the model constant must be adjusted together with grid aspect ratio incoordinated fashion while guaranteeing that the surface layer is sufficiently well resolved in thevertical by the grid. We argue that, in principle, both the critical values that define the HAZ and thesurface layer constant when LOTW scaling is achieved can depend on details of the SFS andsurface stress models applied in the LES. We carried out over 110 simulations of the neutralrough-surface ABL to cover a wide portion of the parameter space using a low dissipation spectralcode, the Smagorinsky SFS stress model and a standard model for fluctuating surface stress. Theovershoot was found to systematically reduce and z was found to systematically approach aconstant value in the surface layer as the three parameters exceeded critical values and the LESmoved into the HAZ, consistent with the theory. However, whereas constant z was achieved overnearly the entire surface layer as the LES is moved into the HAZ, the model for surface shear stresscontinues to disrupt LOTW scaling at the first couple grid levels, and the predicted von Krmnconstant is lower than traditional values. In a comprehensive discussion, we summarize theprimary results of subsequent studies where we minimize the spurious influence of the surface stressmodel and show that the surface stress model and the SFS stress model constant influence thepredicted value of the von Krmn constant for LES in the HAZ. 2010 American Institute ofPhysics. doi:10.1063/1.3319073

aThis paper is based on an invited lecture, which was presented by the firstauthor at the 61st Annual Meeting of the Division of Fluid Dynamics of theAmerican Physical Society, held 2325 November 2008 in San Antonio,TX.

bAuthor to whom correspondence should be addressed. Electronic mail:[email protected]. Telephone: 1 814 865-3159.

PHYSICS OF FLUIDS 22, 021303 2010

1070-6631/2010/222/021303/21/$30.00 2010 American Institute of Physics22, 021303-1


I. INTRODUCTION

Accurate prediction of mean velocity and mean velocitygradient are prerequisites to large-eddy simulation LESpredictions of wall-bounded shear flows at the highest levelsof accuracy within the constraints of LES. Mean velocitygradient, in particular, enters the Reynolds stress as a sourcein streamwise velocity variance and affects directly the large-eddy structure and corresponding turbulent transport. We askwhat basic characteristics of a subfilter scale SFS stressmodel together with the grid and the numerical algorithm arerequired for a LES to capture the law-of-the-wall LOTW inthe inertia-dominated surface layer with the least alterationto the true NavierStokes physics. Alterations could include,for example, the addition of numerical dissipation or algo-rithmic modifications made in the computational domain orat its boundaries to force a logarithmic mean velocity profileto exist.

At its root, LOTW follows from scaling arguments withthe assumption that at sufficiently high outer Reynolds num-bers, only one characteristic length and velocity scale arerelevant to the vertical gradient of mean velocity, U /z,within a surface layer adjacent to a flat solid boundary thatis directly influenced by surface blockage U is the meanhorizontal velocity magnitude and z is the vertical coordinatefrom the surface. If the surface is aerodynamically smooth,LOTW assumes the existence of a viscous surface layer di-rectly adjacent to the surface that is characterized by thesingle length scale = /u, where u

2 is the mean surfaceshear stress and is the fluid kinematic viscosity. If the sur-face is homogeneously rough, the characteristic roughnessscale z0 must enter the scaling immediately adjacent to thesurface. Under LOTW, an inertial surface layer is assumed toexist outside the viscous surface layer that is sufficiently farfrom the surface that neither nor z0 are relevant scales andthat the only length scale relevant to U /z is the distancefrom the surface z, characterizing a dominant integral scaleof underlying turbulence fluctuations. LOTW assumes thatu is the only characteristic velocity scale across the entiresurface layer, so that the transition between and z as domi-nant length scales is the primary distinction between the vis-cous and inertial surface layers. Consequently, according toLOTW U /zu / in the viscous surface layer andU /zu /z in the inertial surface layer. If any other lengthor velocity scales significantly affect the mean velocity field,these scales will disrupt the LOTW scalings for U /z.

We consider a boundary layer where the Reynolds num-ber is so high that the viscous layer is unresolvable in thevertical, or that roughness elements have a scale well belowthe vertical dimension of the first grid cell adjacent to thesurface, that is z and/or z0z, where z is the verticalgrid size assuming a uniform grid. Thus, neither nor z0enter the boundary layer dynamics and scalings that are re-solved on the grid and the viscous force term in the resolvedNavierStokes equation is negligible; viscous dissipation oc-curs at scales far below the grid and viscous effects enteronly indirectly through the inertial transfer of momentumand energy between resolved and SFS motions.

In 1992, Mason and Thomson1 pointed out a fundamen-

tal flaw in LES predictions of high Reynolds number turbu-lent boundary layers: the mean velocity gradient scaled ac-cording to inertial LOTW scaling produces a well-definedpeak within the surface layer; the LES does not predict theLOTW. This overshoot has since been pointed out andstudied by many researchers. In Fig. 1 we show several ex-amples of the overshoot from the literature, where the verti-cal gradient of mean streamwise velocity is plotted normal-ized by inertial LOTW surface-layer velocity and lengthscales

z z

u

Uz

. 1

In Fig. 1 mzz is plotted against z, where is thevon Krmn constant, assumed to be 0.4. LOTW scalingimplies that z should be constant through the inertial sur-face layer, the lower 15%20% of the boundary layer depth.Estimation of the value that makes m=1 through thesurface layer in the high Reynolds number limit is difficultfrom laboratory experiment, in part because the high Rey-nolds number limit is difficult to know, u is difficult tomeasure accurately, and pressure and other corrections aresometimes applied to the data. In the atmospheric boundarylayer ABL where Reynolds number is not a major concern,global stability and lack of stationarity are. Recent compila-tions of experimental data by Nagib and Chauhan6 concludethat the von Krmn coefficient is not universal and exhibitsdependence on not only the pressure gradient but also on theflow geometry They estimate over the range from0.37 in channel flow to 0.41 in pipe flow, with a flatplate boundary layer estimate in between at 0.384. In thenear-neutral rough ABL Andreas et al.7 recently estimated0.387; however, they quote experimental values from theliterature as low as 0.33 Ref. 8 and as high as 0.40 Ref. 9with several studies reporting in the range of0.350.37.1013

In Fig. 1 m is plotted against z nondimensionalized bya boundary layer depth or, in Fig. 1b, by a Coriolis scale.The overshoot shows up clearly in many of these plots as apeak in m in the surface layer, highlighted in gray. Figure 1indicates that the overshoot is the most obvious manifesta-tion of a broader problemthe apparent inability for LES topredict LOTW scaling within the high Reynolds number tur-bulent surface layer. Similarly, a near-surface overshoot hasalso been observed in the mean gradient of temperature.2,14

Methods to reduce or eliminate the degree of overshoothave universally centered on modifying the closure for theSFS stress tensor e.g., Fig. 1; however, these have beendeveloped in the absence of a clear understanding of itssou-rce. In this study we develop a theory to explain the funda-mental characteristics underlying both the overshoot and theinability for LES to capture scaling for mean velocity gradi-ent consistent with LOTW in the inertial surface layer thatis, constant z through the lower 15%20% of the bound-ary layer. From this theory we develop requirements for theLES to predict LOTW scaling that integrate the design ofbasic features of the SFS stress model with the design of thesimulation, particularly the grid. We further show that it is

021303-2 J. G. Brasseur and T. Wei Phys. Fluids 22, 021303 2010


possible to eliminate the overshoot without the LES predict-ing LOTW scaling; eliminating the overshoot is a necessarybut not a sufficient condition to predict LOTW. From thetheory we design a three-variable parameter space in whichthe SFS model parameters and LES grid can be adjusted tosystematically weaken the influence of a spurious numericalLES viscous length scale until the LES predicts LOTWscaling in the surface layer. Although the basic theory isindependent of any particular SFS stress closure, we useeddy viscosity models to gain insight into the theory and todevelop a practical approach to designing LES within theparameter space.

We then present numerical experiments to show that thetheory accounts for primary characteristics underlyingLOTW scaling with LES. However we also learn that whenLOTW scaling is achieved, two previously hidden issues ap-pear: the surface shear stress model interferes with LOTWscaling at the first couple grid levels, and both the surfacestress model and an interaction between SFS stressmodel and aspect ratio influence the predicted value of the1 / in the surface layer when constant z is achieved.

We discuss these two issues broadly within the Discussion,Sec. VII.

The framework we present here has broader implicationsto the transition between LES and Reynolds-averagedNavierStokes RANS calculations,15 and to the interplayamong modeling, grid, and algorithm. In particular, ouranalysis is relevant to hybrid LES-RANS approaches such asdetached-eddy simulation16 DES, where the overshoot isidentified as a log-layer mismatch or superbufferlayer.1719 We discuss the applicability of the current studyin Secs. II B and VII D.

II. BACKGROUND ON THE OVERSHOOT PROBLEM

Before developing our analysis into essential mecha-nisms underlying LES prediction of LOTW scaling of meanvelocity gradient, we set the stage with a discussion of theovershoot and its history. Previous studies, many of whichhave produced significant reductions in the degree of over-shoot, provided important clues that lead to the theory devel-oped here, and present results that a theory should explain.

FIG. 1. Examples of the overshoot in mean shear from previous LES studies: a Sullivan et al. Ref. 2, b Kosovic Ref. 3, c Port-Agel et al. Ref. 4,and d Chow et al. Ref. 5. In a zi is the ABL depth defined in the traditional manner as the height where vertical turbulent heat flux is a minimum. In bz is scaled on u / f , where f is the Coriolis parameter the angular velocity at the earths surface. The simulations in c and d are pseudo-ABL/channel flowsimulations which do not contain a capping inversion and instead apply fixed horizontal velocity at z=H. =0.4 was assumed in forming m. The shadedregions indicate the surface layer.

021303-3 Designing large-eddy simulation of the turbulent boundary layer Phys. Fluids 22, 021303 2010


A. Consequences of an overshoot in mean shear rate

Figure 1 shows the essential aspects of the overshootfrom several previous LESs of the neutral boundary layerthat have focused on this issue since Mason and Thomson.1Because the overshoot is associated with the shear-dominated region of the boundary layer, it is particularlyapparent in the fully shear-driven neutral boundary layerFig. 1. Piomelli and Balaras19 point out that in DES of theshear-driven boundary layer, unphysical, nearly one-dimensional wall streaks were present in the RANSregionand shorter-scale outer-layer eddies were progres-sively formed as one moved away from the wall. Similarobservations have been made in the neutral ABL.20

In the moderately convective ABL an overshoot is pro-duced in the shear-dominated surface layer while the mixedlayer is buoyancy dominated.21 In the presence of convec-tion, vertically driven thermals couple the surface and outerboundary layers. Khanna and Brasseur20 showed that theelongated structure of streamwise turbulence fluctuationsgenerated near the surface by the interaction between strongmean shear and turbulence streaks is the source of thermalsthat penetrate the outer boundary layer. These coherent elon-gated vertical motions interact with the horizontal meanwind to form highly coherent secondary rolls that extend tothe top of the boundary layer and can extend 2040 bound-ary layer thicknesses in the mean wind direction.22 Thus,there is a direct coupling between the near-surface shear-driven streaks and very large eddy structure of the boundarylayer. It is possible that the mechanisms that underlie thecreation of the longitudinal convective rolls may be relatedto the mechanisms underlying much weaker highly elongatedstructures that have been observed in the log layer of theneutral boundary layer.23

An important negative consequence of the inner-outercoupling is that the near-surface errors from the overshootare driven vertically to infect the entire boundary layer. Todemonstrate this, Khanna and Brasseur20 applied two SFSmodels to the same LES; one produced a stronger overshootthan the other. The LES with the stronger overshoot pro-duced much stronger and coherent convective thermals andboundary layer rolls with much larger horizontal integralscales that persisted to the top of the boundary layer. Further-more, these overly coherent thermals are spuriously alignedwith the mean geostrophic wind. This spuriously strong ther-mal structure adversely influences vertical transport to theupper atmosphere of momentum, thermal energy,contaminants, and humidity. Error in humidity predictionswill likely enter cloud cover predictions and produce error insolar radiative heating at the earths surface. Incorrect pre-diction of vertical transport of CO2 and other greenhousegases may affect related predictions of upper atmospherechemistry.

The negative consequences of the overshoot in mean ve-locity gradient arise essentially from the incorrect predictionof Reynolds stress anisotropy near the surface. Juneja andBrasseur24 argued that the incorrect anisotropy results from afeedback interaction between the exaggerated mean gradientand Reynolds stress production as a consequence of inherent

under-resolution at the first grid level that occurs in LES ofhigh Reynolds number turbulent boundary layers when thefirst grid level is in the inertial layer. The errors are exacer-bated by any mechanism that enhances vertical transport,including buoyancy20 and boundary layer separation.

B. Previous studies

Given the importance of the overshoot, there have been anumber of studies that have attempted either to understandthe cause of the overshoot,24 or have attempted to remove it.Mason and Thomson,1 for example, suggested that the over-shoot is particular to eddy viscosity closures where energy isremoved at each point from the resolved scales in contrastwith the known forward/backward nature of energy transferat a point. To introduce backscatter into the simulation,they added to the resolved momentum equation a Langevin-like stochastic acceleration term, in addition to a SFS stressdivergence using the Smagorinsky closure for SFS stress.However, Mason and Thomson1 made another significantmodificationthey reduced the eddy viscosity near the sur-face by making the Smagorinsky length scale proportional toz at grid nodes near the surface.

In Fig. 1 we compare the results from four other groupsof researchers between 1994 and 2005 who explicitly ad-dressed the problem of the overshoot in LES of the ABL.The surface layer is shaded to indicate the region over whichm should be predicted as a straight vertical line and fromwhich a grid-independent prediction for mean velocityshould emanate. Figure 1a shows predictions from Sullivanet al.2 who argued that as the surface is approached, the SFSstress closure should transition from an eddy viscosity modelappropriate to LES to a model more appropriate to theRANS equation. Although their mixed eddy viscosity modelwas purely dissipative no backscatter, the overshoot wassignificantly reduced compared with a pure SFS model. LikeMason and Thomson,1 their mixed model results in a reduc-tion in net SFS stress and eddy viscosity near the surface.Whereas the overshoot could be reduced, and perhaps eveneliminated, by the adjustments to the SFS model, a depen-dence of z on z persisted Fig. 1a. The Sullivan et al.2model was further modified by Ding et al.25 Recently,Lvque et al.26 modified the Smagorinsky eddy viscosity bysubtracting mean shear from the instantaneous resolved rate-of-strain tensor. Whereas they applied the model only to lowReynolds number boundary layers with partially resolvedviscous layers, the effect is again to reduce the eddy viscos-ity near the surface. However the LES predictions failed topredict constant z over the entire surface layer.

Kosovic3 added additional nonlinear terms in velocitygradient to the eddy viscosity closure. Whereas a major im-provement in the overshoot in the neutral ABL was obtainedFig. 1b, grid resolution was quite low and the improve-ment degraded at higher grid resolution. Dynamic formula-tions of the Smagorinsky eddy viscosity closure have beenapplied by Port-Agel et al.4 and Esau27 to improve the over-shoot and capture the LOTW. We show in Fig. 1c the re-sults from Port-Agel et al.4 who developed a scale-dependent formulation of the dynamic Smagorinsky model



that produces a reduction in eddy viscosity near the surfaceand removes an apparent undershoot with the standard dy-namic model solid line. Whereas the modifications signifi-cantly reduced the overshoot, constant z was not obtainedover the surface layer.

Chow et al.5 combined a number of modeling elements,including a dynamic eddy viscosity model,28 a resolvableSFS stress model component29 combined with a deconvolu-tion procedure,30 and a canopy model.31 As shown in Fig.1d, whereas overprediction of mean shear near the surfacecould be reduced with certain combinations of elements, itwas not clear which modeling elements were responsible,and a robust grid-independent solution was not obtained. Arecent calculation by Drobinski et al.,32 however, indicatedthat a suppression of the overshoot was possible using a stan-dard one-equation model but with a more refined grid. Theirresult will be discussed in Secs. VI B and VII B in contextwith the current analysis.

Figure 2 shows that what we refer to here as an over-shoot is described as a logarithmic layer mismatch or su-perbuffer layer in the DESs of Nikitin et al.17 Spalart18 andPiomelli and Balaras19 described this as a fundamental unre-solved problem in DES. Whereas the simulations of Fig. 1contain only the inertial LOTW layer due to the presence ofsurface roughness and the fluctuating surface stress is mod-eled, the DES uses RANS to model a viscous surface layerwith no slip at the wall shown in Fig. 2a below aninertial-dominated surface layer that is simulated with LES,as shown in Fig. 2b. The logarithmic layer mismatch,

shown by the circled part of the mean velocity profile, Fig.2a, is shown in Fig. 2b to be equivalent to the overshootphenomenon discussed above.

Figures 1 and 2 motivate the current research; we seekan understanding of the essential mechanisms underlying theovershoot and, from that, a method that will both eliminatethe overshoot and predict a vertical line in m versus z overthe entire surface layer with a grid-independent predictionover the entire boundary layer. Historically, the assumptionhas been that the solution to the overshoot problem is a clo-sure issue, and all attempts to modify LES to predict LOTWhave been through adjustments to the SFS stress tensormodel. Although there have been significant advances made,the fundamental mechanisms underlying the overshoot arenot understood so that a clear path to robust grid-independentLES that eliminates the overshoot and predicts LOTW scal-ing has been elusive. We show here that the fundamentalissues underlying the overshoot and its resolution are broaderthan the SFS model and involve the basic characteristics ofthe SFS stress closure integrated with the construction of theLES grid.

C. Useful clues

In the studies described above, a number of importantobservations have been made that provide clues to underly-ing issues and which require explanation:

1 The overshoot is influenced by the details of the SFSmodel. This has been discussed above in Sec. II B Fig.

FIG. 2. The logarithmic layer mismatch in the smooth wall turbulent channel flow DESs of Nikitin et al. Ref. 17 discussed by Spalart Ref. 18. Curve A4is the simulation in b; Re=20 000. a U+=U /u plotted against z+=z /. b The log layer mismatch shown by the upper oval in a is shown here tobe an overshoot in normalized mean shear m, negating LOTW scaling over the surface layer shaded area. b was kindly generated by Spalart Ref. 33. is assumed to be 0.4.



1. Whereas there have been many variants to the mod-eling process, all have employed an eddy viscosity com-ponent. A common characteristic of the more successfuladjustments to the modeling process is a reduction ineddy viscosity near the surface, as compared with un-adulterated models.

2 The overshoot is tied to the grid. Khanna and Brasseur21pointed out that increasing the resolution of the grid,keeping all other elements of the simulation unchanged,does not diminish the magnitude of the overshoot, butmoves the peak in m closer to the surface in proportionto the vertical grid spacing, z. Similarly, Spalart18pointed out that in DES grid refinement merely movesthe same amount of logarithmic layer mismatch closerto the wall. This dependence of the overshoot on gridresolution is clear, for example, in Fig. 1d. Why thelocation of the overshoot should be proportional to z,however, is not understood. Khanna and Brasseur21pointed out that because the horizontal integral scale ofvertical velocity scales on z, vertical velocity is alwaysunder-resolved at the first grid level independent of gridresolution and that this under-resolution has negativeconsequences that must be addressed to eliminate theovershoot. Specifically, they proposed that this inherentunder-resolution at the first grid level somehow ties theovershoot to the grid. The mechanisms were unclear, butwere felt to be somehow associated with the lack ofperformance of SFS models when the integral scales areunder-resolved. This shall be discussed in Sec. IV B

3 The prediction of mean velocity gradient is grid depen-dent. A requirement for any successful numerical simu-lation is grid independence in the solution for meanvariables.34 As illustrated in Fig. 1d, the solution formean velocity gradient often does not converge as thegrid is refined; each grid produces a different solutionnot only in the overshoot region but throughout theboundary layer. Grid dependence in the flow is apparentwhen different solutions in the literature are comparede.g., Andren et al.14. We shall show in Sec. VI B that agrid-independent solution is only possible when both theovershoot is suppressed and LOTW scaling is obtained.

It is apparent from these previous studies that althoughthe overshoot is influenced by the closure for SFS stress, theovershoot problem is only part of the broader issue of accu-rately predicting the LOTW, and that these are both model-ing and numerical issues.

III. AN ANALYSIS OF THE FUNDAMENTAL NATUREOF THE OVERSHOOT: THE FIRST OBSERVATION

In this section we focus specifically on the overshoot.A primary mechanism underlying the overshoot and itsresolution can be understood by comparing true inertial-versus-viscous scaling underlying the stationary fully devel-oped smooth wall channel flow in the high Reynolds numberlimit with scaling of LES of the same high Reynolds numberchannel flow with an unresolved viscous or roughness layer.We shall find that we can relate the true physics of the chan-

nel flow to the spurious physics of the simulated channelflow that is model and algorithmically dependent and cannotbe entirely eliminated.

A. Scaling high Reynolds number smooth wallturbulent channel flow

Consider a fully developed stationary smooth wall in-compressible channel flow at Reynolds numbers sufficientlyhigh to support the classical LOTW in the surface layer.The local and global mean axial momentum balances are,respectively,

Px

=

Ttzz

+Tzz

=

Ttotzz

andPx

=

T0

, 2

where T0Ttot0u2 and is the half channel width or

boundary layer depth the height where Ttot=0. The secondexpression in Eq. 2 results from a global force balance. Thecoordinates x ,y ,z are in the mean-flow, spanwise, andwall-normal directions, respectively. Px is the mean pres-sure and Ttot=Tt+T is the total mean stress, where Tt and Tare Reynolds stress and mean viscous stress, respectively,

Ttz = uw, Tz = Uz

. 3

Capital letters and primed quantities indicate mean flow andfluctuating variables, respectively, u ,v ,w are the velocitycomponents in the x ,y ,z directions, , are density andviscosity, and the angle brackets denote ensemble averaging.

Integrating Eq. 2 in z and replacing Tz withu /zmz, the exact equation for the normalized meangradient is

mz = z+1 Tt+z z , 4where Tt

+=Tt /T0 is the ratio of the Reynolds stress Ttz to

the wall stress T0, and z+=z / is the surface-normal coordi-nate nondimensionalized by the viscous surface layer scale= /u, where = /. Equation 4 states that the totalstress Ttotz, split between the viscous and inertial contribu-tions on the left-hand side LHS and right-hand side RHSof Eq. 4, decreases linearly from T0 at the surface to zero atthe channel center plane. Near the surface z /1 in thefriction-dominated layer z+5, the turbulent stress is neg-ligible Tt

+1 and Eq. 4 is well approximated by

mz z+ z+ 5 . 5

This result suggests that mz exceeds 1 when z+1 /2.5 for 0.4, which is well within the friction-dominated portion of the surface layer.

Equation 5 indicates that an overshoot in mz existsin the smooth-surface high Reynolds number channel flow ina region bounded from below by z+2.5 and from above bythe lower margin of the inertial LOTW layer and upper mar-gin of the buffer layer. In Fig. 3 we show that this is indeedthe case. We replot, in this figure, the data from direct nu-merical simulations DNSs of the smooth wall channel atdifferent Reynolds numbers that has been graciously madeavailable to the scientific community by Iwamoto et al.35,36



at Re=300, 395, and 642, and at Re=2003 by Hoyas andJimenez,37 where Re=u /. Figure 3a shows that mz+exceeds 1 when z+ exceeds 2.5 and peaks at z+10 with amaximum value of about 2.3 independent of Reynolds num-ber. m approaches 1 at z+4050. Comparing Fig. 3cwith Fig. 3d, it is significant that the peak in mz occursnearly coincident with the crossover between turbulent stressTtz and viscous stress Tz. Both the location of overshootand the location of the crossover occur at 10. Since theposition of this overshoot scales on the viscous scale andcorresponds to the transition between the dominance of tur-bulent stress above and viscous stress below, this overshootreflects the application of an inertial length scale z in theportion of the surface layer where the appropriate lengthscale is the viscous surface scale . Figures 3c and 3dshow that the overshoot and the crossover in Tz and Ttzmove physically closer to the surface with increasingReynolds number without a reduction in maximum m.

B. LES of high Reynolds number turbulent channelflow

We compare the previous analysis with LES of the samehigh Reynolds number channel flow analyzed in Sec. III A,but in which either a viscous layer exists that is fully unre-

solved z or the surface is rough with fully unresolvedroughness scale z0z. The momentum equation for theresolved velocity ui

r is

uir

t+ui

rujr

xj=

1

pr

xi

ijSFS

xj, 6

where the viscous force has been scaled out of the momen-tum balance on account of the high local Reynolds numberson all grid nodes within the computational domain surfaceviscous layers are unresolved or nonexistent. The SFS stresstensor ij

SFS is modeled. We apply a superscript r to indicate avariable that is carried forward in the simulation as a re-solved variablethat is, after the process of explicit and im-plicit filtering in the algorithmic advancement of the mod-eled discretized version of Eq. 6. Explicit filtering isgenerally carried out algorithmically as a dealiasing step al-gorithmically, urur in Eq. 6 should be written as ururr toindicate the common application of a second explicit filter ron the nonlinear term, typically in pseudospectral LES. Im-plicit filtering arises from the dissipative nature of the modelfor ij

SFS, from numerical dissipation within the discretized

version of Eq. 6, and from any dissipative elements intro-duced algorithmically as the discretized dynamical system isadvanced in time.

The ensemble mean of Eq. 6 for stationary fully devel-oped high Reynolds number channel flow is

Px

=

TRzz

+TSzz

=

Ttotzz

withPx

=

u2

.

7

The second equation is the global force balance. Total stressTtot=TR+TS is now the summation of a turbulent stress TRformed from the fluctuating resolved velocity componentsand the shear component of the mean SFS stress tensor TS,

TRz = urwr, TSz = 13SFS . 8

Whereas the divergence of the SFS stress is physically aninertial contribution to the force balance, in application allSFS models are structured so as to contain a frictional con-tribution to the equation of motion. This is explicitly true ofeddy viscosity models and mixed models which include aneddy viscosity term.

In what follows, we do not restrict our theoretical treat-ment to any particular SFS stress model. We do, however,use eddy viscosity closures for guidance and to carry outnumerical experiments.

We seek a mechanism to extract the frictional compo-nent of the complete modeled tensor ij

SFS; that is, we seek anestimate of a scalar viscosity that extracts the part of ij

SFS thatis parallel to the resolved strain-rate tensor sij

r in the mean.For the purposes of determining the underlying causes of theovershoot in the highly shear-dominated part of the boundarylayer, we define a LES viscosity lesz as the proportion-ality between 13

SFS and 2s13r =U /z,

0.0

20.0

40.0

60.0

0 0.5 1 1.5 2 2.5

z+

m

(a) Re=300Re=395Re=642.5Re=2003

0.0

20.0

40.0

60.0

0 0.2 0.4 0.6 0.8 1T+, T

+t, T

++T

+t

(b)

0.0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

z/

m

(c)

0.0

0.1

0.2

0.3

0.4

0 0.2 0.4 0.6 0.8 1T+, T

+t, T

++T

+t

(d)

FIG. 3. DNS of the smooth wall channel flow showing the physically realovershoot in the viscous region. a Normalized mean shear m vs z+. bWall-normalized Reynolds shear stress Tt

+ filled symbols and mean viscousshear stress T

+ open symbols vs z+. The dotted, dashed, and thin straightlines are the total shear stress profiles for Reynolds number of 300, 395, and642, respectively. c m vs z /, where is the half-channel height. d Tt+filled symbols and T

+ open symbols vs z /. Total shear stress profiles forthree different Reynolds number collapse onto a single linear line. =0.4 isassumed in forming m. Re=300, 395, and 642 DNS data are from Iwa-moto et al. Ref. 35 and Iwamoto Ref. 36. In c we have added theRe=2003 DNS data from Hoyas and Jimenez Ref. 37. The horizontaldotted lines in c and d indicated the upper margin of the surface layerwhere LOTW should be predicted.



lesz TSz

2s13r

closure independent , 9

where sijr is the resolved strain-rate tensor. Furthermore, since

we are focused on the first few grid points adjacent to thesurface, we parametrize LES viscosity with its value at thefirst grid level, LESz1, and we define the normalized LESviscosity as follows:

LES lesz1, LES+ z

leszLES

. 10

We emphasize that the definitions in Eqs. 9 and 10 do notassume an eddy viscosity model and can be made for anyclosure of the SFS stress tensor. However, this estimate offrictional content is only valid with strong mean shear at thefirst grid level, and is therefore appropriate to the neutralboundary layer, and the stable and moderately convectiveABLs with shear-dominated turbulence at z1.

Replacing TS in Eq. 7 with LESLES+ zU /z, inte-

grating in z, and using inertial LOTW scaling Eq. 1, pro-duces the expression for mz,

mz =zLES

+

les+ z

1 TR+z z , 11where

zLES+

z

LES, LES

LESu

and TR+

TRT0

. 12

As in the exact channel flow equations, T0=Ttot0u2, so

that TR+z is the ratio of the resolved part of the Reynolds

shear stress to the total wall shear stress. Unlike the viscouslayer, where we could argue that the Reynolds stress is asmall percentage of wall stress and approximate Eq. 4 byEq. 2, we cannot a priori argue that TRT0. However, atthe first few grid points, z /1 and LES

+ zO1, so thatEq. 11 can be approximated by

mz zLES+ 1 TR

+ first few grid points . 13

The magnitude of TR+ depends on the relative content of re-

solved to SFS stress in total stress Ttot=TR+TS. When theseparation between resolved and SFSs is in the inertial range,implying that all integral scales are well resolved, then TSTR, and TR

+ cannot be neglected in Eq. 13. However, asdiscussed in Sec. II C, some integral scales are inherentlyunder-resolved at the first few grid levels in high Reynoldsnumber LES of wall-bounded flows. Thus, it may be the casethat TR

+1 close to the surface so that Eq. 13 is approxi-mated by mzzLES

+. If this were the case, then we would

reach the same conclusion as when Eq. 4 was reduced toEq. 5 in the viscous sublayer of the smooth wall channelflow: an overshoot would occur when zLES

+ 1 /2.5, thatis when z2.5LES.

Figure 4 shows that indeed, TSTR near the surface in atypical LES of the neutral ABL we describe the details ofour LESs of the ABL in Appendix B using the Smagorinskymodel, so that mzzLES

+ and an overshoot is produced atzLES

+ 2.5. In fact, Figs. 4a and 4b are very similar to thecurves in Figs. 3a and 3b of the real overshoot in the

smooth wall channel flow. The spurious overshoot initiatesbetween the first and second grid levels and peaks nearlycoincident with the crossover between TR and TS, very simi-lar to Fig. 3, where the real overshoot is found to be coinci-dent with the crossover between Tt and T.

The overshoot in LES appears to arise from physicssimilar to the true overshoot in smooth wall channel flow.However, in LES the frictional layer that causes the over-shoot is a numerical LES frictional layer near the surface thatarises from the frictional nature of the modeled SFS stressand any numerical and algorithmic additions to dissipation.This conclusion is analyzed further in the following sections.

C. The first criterion

The observation from Fig. 4 that the overshoot is asso-ciated with a reduction in the resolved Reynolds stress tobelow the mean SFS stress suggests that the spurious fric-tional content of the model for SFS stress has introduced aspurious length scale LES that interferes with the inertialscale z that should dominate the surface layer. The spuriousinterference of LES with inertial scaling and the consequentspurious overshoot are essentially the same physics that un-derlie the production of the real overshoot in the friction-dominated part of the LOTW layer of the smooth wall chan-nel flow. However, unlike the true viscous layer, which is anecessary consequence of frictional force with no slip, thespurious LES frictional layer must be controlled to eliminatethe frictionally induced spuriously large mean gradients nearthe surface. If it were possible to maintain the dominance ofTR over TS to the surface, then the spurious frictional contri-bution from the SFS stress model would remain suppressed.This observation suggests a criterion for elimination of theovershoot that

RTR1TS1

R O1 closure independent , 14

where the subscripts 1 mean at the first grid level. Thecritical value R, to be determined experimentally, is an or-der 1 quantity but may depend on the model for SFS stress,the lower boundary condition, the stability of the ABL, thenumerical algorithm, etc.

0.0

20.0

40.0

60.0

0 0.5 1 1.5 2 2.5

z+LES

m

(a) m

0.0

20.0

40.0

60.0

0 0.2 0.4 0.6 0.8 1T+S, T

+R, T

+R+T

+S

(b) T+RT+S

T+R+T+S

FIG. 4. Overshoot in LES of the neutral ABL using the Smagorinsky modelCs=0.2 and a 128128128 grid. a m vs zLES+ . b Wall normalizedmean resolved and SFS shear stress TR

+ and TS+ plotted against zLES+ and its

sum. =0.4 is assumed in forming m. =0.4 is assumed in forming m.



D. Understanding the ratio R and the first criterionOn what does R depend and how can it be controlled in

a LES? To gain insight into the nature of R and find ananswer to this question, we develop expressions for R basedon eddy viscosity representations for the deviatoric part ofthe SFS stress, ij

SFSdevijSFS

kkSFS /3ij,

ijSFSdev = 2tsij

r, t = tut, 15

where t and ut2 characterize length and energy at the small-

est resolved scales. t is often taken as a fixed length scalet=Ct, where = xyz1/3 is the grid size x ,y ,z arethe grid spacings in x ,y ,z and Ct is a model constant. Somemodels embed z-dependence into t in the form tz=Cttz, where tz is specified to decrease toward the sur-face and approaches away from the surface.1 Dynamicmodels take t=Ct, but determine Ct dynamically with theresult that Ct=Ctz reduces near the surface.4Thus, in each case the modeled length scale tz decreasestoward the surface. In the current work we develop scalingfrom eddy viscosity closures with constant t=Ct, butshall discuss our results in context with z-dependent t in theDiscussion, Sec.VII B.

The space-time variability in tx , t is modeledthrough a fluctuating velocity scale utx , t, given for theSmagorinsky closure by ut,Smagx , t=2sij

r sijr 1/2. In the ba-

sic one-equation model, ut,1-eqx , t=ex , t, where ex , trepresents the SFS kinetic energy, modeled through a prog-nostic equation.32,38 In both models t=Ct. Here we de-velop expressions using the Smagorinsky closure with Ctreplaced by Cs

2, as is traditional. However we shall present

results for both the Smagorinsky and one-equation eddy vis-cosity models.

Independent of the model, 13SFSdev=13

SFS and

TS1 = 13SFS1 2LESs13

r 1 = 21t1s13r 1, 16

where

LES = 1t1 with 1 = 1 + ts13r 1t1s13r 1 . 17We have found from LES of the neutral ABL that 1 is typi-cally 1.05. The ensemble mean of the eddy viscosity at thefirst grid level with the Smagorinsky closure is

t1 = Cs222sij

r sijr 1/21 2Cs

22s13r 1 = Cs

22 Uz

1,

18

where

sijr sij

r 1/21 2s13r 11 + 18 sijrsijr1s13r 12 is well approximated by 2s13r 1 since the strain-rate fluc-tuations are nearly all SFS this is especially true at the firstgrid level where the integral scales are minimally or poorlyresolved. We write the mean streamwise velocity gradient ina form appropriate for inertial scaling in the surface layer:

2s13r 1 = Uz 1 u1z1 . 19

In Eq. 19 1 is defined as the value required to make theLHS equal to the RHS at the first grid level. Only if LOTWis predicted by the LES, so that 1 is constant through thesurface layer, will 1 be the predicted value of the vonKrmn constant. When the Coriolis force is present, as inLES of the ABL, the velocity at the first grid level is at anangle 1 to the geostrophic wind velocity above the boundarylayer. In this case, Eqs. 1921, 23, 24, 27, and 28contain an additional factor cos 1, as given in Appendix A.

Inserting Eq. 19 into Eqs. 18 and 17 leads to thefollowing expression for the LES viscosity:

LES =1

1Dsuz, where

20Ds Cs

2AR4/3 Smagorinsky .

AR=x /z=y /z is the aspect ratio of the grid and z=z1 isthe vertical grid spacing. As will be discussed at length, Eq.20 indicates that the LES viscosity is proportional to thecombination Cs

2AR4/3

, and therefore is altered both by theconstant in the SFS model and by the aspect ratio of the grid,as well as by the vertical grid spacing.

Applying Kolmogorov scaling39 and LOTW scaling tothe one-equation model produces a result similar to Eq. 20,but with Ds replaced by Dk=CkAR

8/9, where Ck is the model

constant, and with 1 replaced by a different order one con-stant . The main point is that with the eddy viscosity clo-sure, the LES viscosity is proportional to the product ofmodel constant and grid aspect ratio, each raised to a powerthat depends on the closure.

In order to develop an expression for R, we note thatsince the total shear stress is Ttot=TR+TS, R is given by

R =Ttot1TS1

1 =2u

2TS1

1 closure independent ,

21

where

2 Ttot1Ttot0

N 1

N

. 22

In Eqs. 21 and 22 Ttot0 =u2 and N= /z is the number

of grid points from the surface to the top of the boundarylayer. 2 is generally very close to 1.

Inserting Eq. 19 for s13r 1 and Eq. 20 for LES intoTS1 =2LESs13

r 1, and then inserting this result for TS1 intoEq. 21, produces

R =1

Dt 1 eddy viscosity , 23

where 2 /1 is generally very close to 1. For theSmagorinsky closure, = 1 and DtDs=Cs2AR4/3, whereasfor the one-equation model DtDk=CkAR8/9 and is a dif-ferent order one constant.



The eddy viscosity closure was used in the derivation ofEq. 23 to provide insight into the mechanisms underlyingR and how it can be systematically adjusted in LES in orderto move the LES into the supercritical regime RR. Welearn that the ratio of resolved to SFS stress at the first gridlevel can be increased either by reducing the model constantor by reducing the grid aspect ratio in the combination

Dt = CtaAR

b, 24

The model constant Ct Eq. 15 and AR enter in this com-bination through the LES viscosity, Eq. 20, and the powersa and b are model dependent. Thus, LES viscosity can bedecreased and R increased by reducing either or both Ct andAR to reduce Dt.

The explanation behind the effects of model constantand AR on R is in the manner that each reduction affects thebalance between resolved and SFS stress within the totalstress Ttotz that is fixed by the global momentum balance.Reducing the model constant directly reduces the averageSFS stress TS. Reducing the aspect ratio corresponds to anincrease in resolution in the horizontal, which moves verticalvelocity variance and Reynolds stress from SFSs to resolvedscales in the horizontal. The consequence of both effects is toreduce TS relative to TR, and therefore to increase the ratioR=TR1 /TS1. Similarly, both effects reduce LES viscosity.

IV. THE BALANCE BETWEEN NUMERICAL FRICTIONAND INERTIA: A SECOND OBSERVATION

The above discussion suggests that the mechanisms thatunderlie the generation of a mean gradient overshoot and itsconsequences are associated with numerical LES frictionthat, in the modeled dynamical system, forces a physicalresponse similar to that underlying the overshoot in Newton-ian turbulent channel flow Fig. 3 that results from molecu-lar friction. Like the real viscous layer in smooth wall chan-nel flow that arises from a change in scaling from z to = /u, the overshoot in LES reflects a transition in domi-nance from the inertial surface scale z to a numerical LESviscous length scale LES=LES /u that dominates near thesurface. LES produces an overshoot in mz as a result ofthe spurious dominance of the length scale LES in a regionwhere the integral scales should scale on z.

However, for the LOTW to exist in an inertia-dominatedsurface layer, inertial effects must be sufficiently strong rela-tive to viscous forces, as measured by the ratio of the bound-ary layer depth to the viscous wall scale, the Reynolds num-ber Re= /. This suggests the existence of a LESReynolds number ReLES= /LES. Since friction in thediscretized LES dynamical system is at the core of theovershoot, we must consider all consequences of friction,including the requirement that inertial effects dominate vis-cous effects sufficiently to produce the LOTW scaling,U /zu /z.

In particular, in the smooth wall channel flow the inertiallayer will only reflect the LOTW when inertia dominates theviscous force within the surface layer sufficiently that Reexceeds a critical value Re

. The DNS data of Fig. 3 show

that the LOTW is not captured in the viscous channel floweven at Re=2003, when the viscous layer defined by thepeak in m is only 2% of the surface layer.

Similarly, one can expect that LES of the high Reynoldsnumber boundary layer can only produce LOTW scalingwhen inertia in the discretized dynamical system dominatesfriction sufficiently that the LES Reynolds number ReLESexceeds some critical value, ReLES

. Indeed one can also pos-

tulate a transitional ReLES which must be exceeded to supportturbulence in the discretized LES dynamical system. In Fig.5 we show four LESs of the ABL at increasing values ofReLES. Similar to Fig. 3 for DNS of channel flow where thepeak in the overshoot scales on the viscous scale , with theexception of the lowest ReLES thin solid curve with smalldots, the overshoot in LES scales on the numerical LESviscous scale, LES Fig. 5a. In fact, both the true channelflow overshoot and the spurious LES overshoot peak at tencorresponding viscous units. Thus, the LES overshoot movescloser to the surface along with LES as ReLES= /LES in-creases Fig. 5c and the numerical LES viscous layer oc-cupies a correspondingly smaller percentage of the surfacelayer. Similarly, the peak in mz coincides with the cross-over between mean resolved and SFS stress TR and TS Fig.5d so that the crossover also scales on LES Fig. 5b.

The thin solid curve with small dots in Fig. 5c is

0.0

20.0

40.0

60.0

0 0.5 1 1.5 2 2.5

z+LES

m

(a) ReLES=110ReLES=166ReLES=276

0.0

20.0

40.0

60.0

0 0.2 0.4 0.6 0.8 1T+S, T

+R, T

+S+T

+R

(b)

0.0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5

z/

m

(c)

0.0

0.1

0.2

0.3

0.4

0 0.5 1T+S, T

+R, T

+S+T

+R

(d)

FIG. 5. Overshoots in LES of the neutral ABL. a m vs zLES+ . b Normal-ized resolved Reynolds stress TR

+ filled symbols and mean SFS shear stressTS

+ open symbols vs zLES+ . Dotted, dashed, and thin lines are the sum ofresolved and SFS stress from low to high LES Reynolds number. c m vsz /, where is defined as the height where m=0. d TR

+ filled symbolsand TS

+ open symbols vs z /. The LES Reynolds numbers of the simula-tions are shown in a. In order of ReLES, the Smagorinsky constants andgrids were Cs=0.1, 424296, Cs=0.2, 192192128, and Cs=0.1, 128128256. The thin black line in c is a simulation with suchlow ReLES that turbulence is barely sustained Cs=0.2, 424232 andReLES=38. The horizontal dotted lines in c and d indicated the uppermargin of the surface layer where LOTW should be predicted. =0.4 isassumed in forming m.



included to illustrate the consequence of a LES Reynoldsnumber that is too low to support turbulence. The mean ve-locity profile is qualitatively similar to the parabolic profilecharacteristic of laminar Newtonian channel flow. Thus, al-though the LES equation contains no true frictional term, thenumerical LES friction inherent in the model and adjusted bythe grid as described by LES in Eq. 20 can, like real fric-tion, dampen inertial motions and prevent turbulence withinthe discretized dynamical system advanced in the LES.

The frictional content of the simulation embodied byLES should, in principle, be extended to include the numeri-cal dissipation within the specific discretization that is ap-plied to advance the LES equations in time with a SFSmodel. The LES presented in Sec. VI applies the pseu-dospectral method in the horizontal and finite difference inthe vertical on a staggered mesh and is minimally dissipa-tive. Significant frictional content within the numerical algo-rithm might strengthen the overshoot beyond what is de-scribed here.

A. The second and third criteria

The first criterion discussed in Sec. III C is a necessarycondition to eliminate the overshoot but is not a sufficientcondition to correctly predict LOTW scaling in the high Rey-nolds number surface layerthat is, constant z is the sur-face layer. A second criterion is necessary: that ReLES exceeda critical ReLES

to achieve LOTW scaling in the simulateddynamical system. In the presence of the overshoot, onewould expect that to predict the LOTW in the part of thesurface layer that is not directly affected by numerical LESfriction, ReLES

would have to achieve values in thousandssimilar to Re

in real frictional channel flow. However, un-like the DNS of channel flow, where the overshoot is real andcannot be eliminated, in LES of high Reynolds numberboundary layers the overshoot and its frictional sources arespurious. Therefore, the critical Reynolds number ReLES

re-

quired to satisfy the second criterion turns out to be muchlower when RR and the overshoot is eliminated thanwhen the overshoot is maintained as in Fig. 5.

To show this we use Eq. 19 to write

TS1 = 2LESs13r 1 = LES

u

1z1closure independent .

25

Inserting Eq. 25 into Eq. 21 yields an expression forReLES that is valid independent of the SFS model

ReLES =N

21

R + 1 closure independent . 26

Thus ReLES depends both on R and on the vertical grid res-olution N. Note that in Fig. 5c the LES given by the solidcurve has low ReLES because of low vertical grid resolution.The critical LES Reynolds number ReLES

is therefore asso-ciated not only with the critical ratio of resolved to SFSstress R, but also with a critical vertical resolution N

,

where

ReLES

=

N

21R + 1 closure independent . 27

The third criterion, that N exceed a critical value N

, follows

from the second criterion, ReLESReLES when RR.

One way to understand the requirement for a minimumvertical resolution to produce high accuracy LES of theboundary layer is simply as a manifestation of the standardcomputational requirement that all special regions with theirown characteristic dynamics be well resolved for accuratenumerical simulation. The surface layer is an example of aregion with special dynamics that requires good resolution.The surface layer occupies 15%20% of the boundary layerdepth at high Reynolds numbers. Resolving this layer with,say, ten grid points in the vertical therefore leads to an esti-mate for N

of 5065. Recognizing that R and ReLES aremodel dependent, we can estimate ReLES

roughly for R1 to be ReLES

250325. This estimate assumes that theovershoot has been eliminated and that the LOTW has beencaptured with 1=0.4. That is, we assume that there areno additional confounding influences that cause the LES todeviate from the LOTW. In Sec. VII we shall discuss con-founding influences. This estimate for ReLES

is well over anorder of magnitude lower than what we would estimate byanalogy with Re

in DNS of channel flow if the overshootwere retained, but confined to a sufficiently thin, numericallyviscous layer adjacent to the surface.

We shall find Sec. VI that for our current LES of theABL with the Smagorinsky eddy viscosity closure, 4550 isa reasonable estimate for N

and 350 is a reasonable estimatefor ReLES

. The good news is that a vertical grid resolution

Nz2N

90100 is not overly severe and doable on cur-

rent mainframes. The bad news is that most calculations inthe literature are of LES with vertical resolutions below criti-cal. Interestingly, we shall show in Sec. V B that in additionto subcritical resolution in the vertical, there are practicallimitations to the maximum vertical resolution in LES of thehigh Reynolds number boundary layer.

B. Understanding the LES Reynolds numberand the three criteria

To develop greater insight into the LES Reynolds num-ber and its control, we evaluate ReLES= /LES using theSmagorinsky eddy viscosity representation for the SFSstress, as was done to understand R in Sec. III D. There wederived an expression for the LES viscosity LES Eq. 20.Using this expression, the numerical LES viscous lengthscale is given by

LES LESu

=

1

Dtz eddy viscosity , 28

where = 1 for the Smagorinsky closure. Dividing by Eq.28, or replacing R in Eq. 26 by Eq. 23, produces thefollowing expression for the LES Reynolds number:

ReLES =

1

N

Dt

eddy viscosity . 29



Several interesting observations can be extracted fromEqs. 28 and 29. Since the overshoot scales on LES andpeaks at 10LES Fig. 5, the observation made in Sec. II Cfrom previous studiesthat the overshoot is tied to thegridcan now be explained. Equation 28 shows that ifneither the model constant nor the grid aspect ratio is alteredwhile the grid is refined, the LES viscous scale LES,and therefore the peak in m, will move closer to the surfacein proportion to the grid spacing z. However, sinceDt=Ct

aARb is left unchanged during the grid refinement, R

and the magnitude of the overshoot will not changetheovershoot simply moves closer to the surface, as shown inFigs. 3 and 5. In particular, Fig. 5 shows that as the over-shoot moves toward the surface in proportion to the gridspacing at constant AR and Ct, the ratio TR1 /TS1 =R and mag-nitude of peak m remain unchanged, and the locations ofpeak m remain attached to the crossover between TR andTS.

A second interpretation follows by replacing ReLES with

/LES in Eq. 27. The criterion RR1 is then equiva-

lent to

LESz

LES

z=

21R + 1

0.2 closure independent .

30

Equation 30 states that the spurious length scale LES aris-ing from friction within the SFS model and grid under-resolution in a LES must be confined sufficiently well withinthe first grid cell for numerical LES friction to not adverselyaffect the LES. Note that Eq. 30 is equivalent to the re-quirement that z1LES

+ 5: the first grid level must be at leastfive times larger than the spurious viscous length scale.

The inequality 30 is satisfied when RR, so that thefirst and second criteria are met when the spurious viscouslength scale is sufficiently small relative to the grid spacing.However Eq. 30 does not guarantee the third criterionN

N

. This is because the overshoot peaks at 10LES, so

that the condition given by Eq. 30 still allows partial reso-lution of the overshoot e.g., when 10LES /z2. Onecan therefore interpret the addition of the third criterionN

N

as demanding that the spurious frictional lengthscale LES be buried both sufficiently far within the first gridlevel z and within the boundary layer that the grid isgiven no opportunity to either create an overshoot or alterLOTW scaling through the influence of friction within thesurface layer.

V. A FRAMEWORK FOR HIGH-ACCURACYLARGE-EDDY SIMULATION: THE HIGH-ACCURACYZONE

By comparing Eq. 23 for R with Eq. 29 for ReLES welearn that reducing the model constant and/or the grid aspectratio in the combination Dt=Ct

aARb causes both R and ReLES

to increase and, correspondingly, LES /z to decrease.However, increasing only the vertical resolution increasesthe LES Reynolds number but has no effect on the ratio ofresolved to SFS stress R. This observation leads to the con-

cept of a RReLES parameter space in which high-accuracy LES of the high Reynolds number boundary layercould be developed. Within this framework, one can system-atically adjust the LES of the boundary layer so that, in thehigh Reynolds number limit, the overshoot is suppressed andthe LOTW is captured. The RReLES parameter space isillustrated in Fig. 6; a LES of the boundary layer is identifiedas a point on a plot of R against ReLES. In subsequent simu-lations, the LES is adjusted to move the point within theRReLES parameter space relative to the critical parametersR, ReLES

, and N

.

For the LES to capture the LOTW while resolving theovershoot, the simulation must live in the rectangular spaceRR, ReLESReLES

. We have roughly estimated R1

and ReLES 350. However in addition to the criteria

RR and ReLESReLES

, we have argued for a third crite-rion N

N

. R and ReLES are linearly related by Eq. 26,

R = 21N ReLES 1 closure independent . 31Thus, N enters in the slope of R versus ReLES. In Fig. 6 weplot Eq. 31 as a series of lines with constant slope 21 /N.Since LOTW is only captured in the supercritical region ofthe RReLES parameter space, in general 1 will vary frompoint to point on lines of constant slope 21 /N. Howeverthe variation in 1 is not so great as to obscure the stronginverse relationship between the slopes of the RReLES linesand the vertical grid resolution N. It can be shown from Eq.27 that the third criterion N

N

is met when the simula-tion lies to the right of the RReLES line with slope 21 /N

that passes through the intersection between R=R andReLES=ReLES

, as illustrated in Fig. 6. We call the wedge-

shaped region that defines supercritical LES satisfying allthree criteria RR, ReLESReLES

, and N

N

the highaccuracy zone HAZ. The LES must reside within the HAZto meet the three criteria required to both eliminate the over-shoot and capture the LOTW.

FIG. 6. Schematic of the structure of the RReLES parameter space. Thisparameter space underlies the framework we propose for designing a LESthat is capable of predicting LOTW scaling for U /z.



A. Designing LES to capture law-of-the-wall scalingThe objective is to systematically move the LES into the

HAZ of the RReLES parameter space by combining Eq.31 with knowledge gained from Eqs. 23 and 29. Al-though we have applied eddy viscosity closures to gain in-sight into the process of adjusting N, R, and ReLES to sys-tematically move the LES within the RReLES parameterspace, the basic method can be applied with any SFS stressmodel. TR1 and TS1 and therefore R and LES and thereforeReLES can be quantified independently of the SFS model,the only requirement being that all contributions to the SFSstress are included in defining ij

SFS before calculating TS1.This relatively simple process may be described in two basicsteps.

1 Adjust, and hold fixed, the vertical resolution of the gridNz so that when the simulation is fully developed, N

will exceed N

typically, Nz1.5N2N to minimizethe influence of the upper boundary condition. We shallpoint out that it is possible to over-resolve in the verti-cal.

2 Then systematically adjust the aspect ratio of the gridtogether with the model constant to move the simulationroughly along a straight line in Fig. 6 from the subcriti-cal region of the RReLES parameter space into theHAZ. With eddy viscosity closures, the model constantand aspect ratio appear in the combination Dt=Ct

aARb.

However, to move into the HAZ with any other closure,one would adjust the model constant for that closure tosystematically reduce the SFS stress together with asystematic increase in the horizontal resolution of thegrid to systematically reduce the aspect ratio.

We presume that the closure for SFS stress relies on adissipative mechanism to model the net transfer of resolvedturbulence energy to SFS motions. With this methodologyfor designing LES one can analyze systematically whatworks better or worse depending on the choice of modeltype, model details, model constant, grid resolution, gridstructure, algorithm, geometry, etc., with some understandingof underlying mechanisms. This framework provides theLES community with both physical understanding and struc-ture upon which a systematic procedure for LES design maybe based. Once the researcher has become experienced withthe method, she/he will be able to design high-accuracy LESmore rapidly using her/his favorite SFS model, algorithm,code, etc.

Whereas the model constant Ct and the model lengthscale t=Ct are uniform in the above discussions, in thegeneral eddy viscosity model the length scale t is specifiedas varying with z. The Mason and Thomson1 modification ofthe length scale t near the surface and the dynamic proce-dure that adjusts the Smagorinsky constant Cs with z e.g.,Port-Agel et al.4 are examples. The primary issue is thatthe level of eddy viscosity be adjusted in concert with thegrid aspect ratio i.e., the horizontal resolution of verticalmotions within the first few grid levels from the surface,where under-resolution is of primary concern and mean SFSstress competes with resolved stress.

B. Grid-independent LES and practical limitson grid resolution

As discussed in Sec. II C, a problem with current LES ofthe ABL is grid dependence in the mean flow. We shall showin Sec. VI that as the simulation moves systematically intothe HAZ within the RReLES parameter space, a grid-independent solution for the mean velocity is achieved.However, one cannot move the simulation infinitely far intothe HAZ along lines of fixed vertical grid resolution N,since that would require that either the model constant bedriven to zero removing the model from the dynamical sys-tem or the grid aspect ratio would be taken to zero creatinginfinitesimally thin grid cells and large computational ex-pense. Either of these limits will cause numerical problemsand simulation error regardless of computational expense.Thus, for both accuracy and practical reasons, the optimallocation for the simulation within the HAZ is near the apexof the wedge in Fig. 6 where RR, ReLESReLES

, and

NN

.

Similarly there are practical limits on vertical resolutionthat surprisingly confine the growth of LES grids on anygiven computer. Although a minimum vertical resolution isrequired to move the simulation into the HAZ, progressiveincreases in vertical resolution will force the simulation ontolines in the RReLES parameter space that have progres-sively smaller slopes, as illustrated by the dashed line in Fig.6. Increases in vertical grid resolution in the absence of otheradjustments will increase the grid aspect ratio, driving R tosubcritical values and restoring the overshoot along with itsrelated errors. With eddy viscosity models, for example, inorder that RR as N is increased, Dt=Ct

aARb must be held

constant Eq. 23. However, since the minimum model con-stant and maximum grid aspect ratio are bounded, grid as-pect ratio must, at some point, be maintained roughly fixedwith increasing vertical resolution, and horizontal grid reso-lution must increase proportionally with N. The number ofgrid points will therefore increase approximately as N

3, se-

verely limiting the vertical resolution to modest values. Thisdilemma is reminiscent of DNS where the highest Reynoldsnumber that can be simulated accurately grows slowly withincreasing computer size due to the rapid increase in reso-lution requirements with increasing Reynolds number.

VI. NUMERICAL EXPERIMENTS

To evaluate the theory and further explore the applica-tion of the RReLES framework to the development of wallbounded LES, we have carried out over 110 LESs of theneutral shear-driven ABL capped with an inversion layer tosuppress boundary layer growth and produce a quasistation-ary long-time solution. The Coriolis force is included at arelatively high level to reduce the time to reach quasistation-ary, so the mean wind is skewed relative to the geostrophicwind x direction at the first grid level see Appendix A.The code is pseudospectral in the horizontal and finite differ-ence in the vertical, so numerical dissipation is minimal. Inthe horizontal statistically homogeneous directions we applyperiodic boundary conditions; in the vertical we apply theboundary conditions, as described by Moeng38 and Sullivan



et al.40 We report here on simulations with the Smagorinskyclosure and uniform grid spacing. In particular, we apply thenonlinear Moeng38 model for total fluctuating shear stress atthe lower surface and the friction velocity is made propor-tional to the mean wind at the first grid level with a propor-tionality constant that can be related to the surface roughnesslength scale z0. A summary of the numerical algorithm andsimulations is given in Appendix B and in severalpublications.2,21,38,40

It should be noted that in our code, dealiasing in thehorizontal directions is carried out by padding rather thantruncation. These are equivalent except in the interpretationof the grid and grid length scale: the grid resolutions andaspect ratios quoted in the figures are before padding and thegrid length scale = xyz1/3 used in the eddy viscositymodel Eq. 15 is also based on the prepadded grid. In allplots we define the boundary layer thickness at the heightwhere the mean velocity gradient crosses zero. Whereas thenumber of grid points in the vertical Nz is defined before thesimulation, the number of grid points within the boundarylayer N is determined after a simulation is analyzed in thequasistationary state. We were careful to determine the qua-sistationary state consistently in all simulations see Appen-dix B. In all plots of mz=z we use a value of 0.40for .

A. The RReLES parameter space

In Fig. 7 we identify in the RReLES parameter space allthe simulations carried out for the current study. As will bediscussed below, from the predictions of mz we estimatedthe critical lines R=R, ReLES=ReLES

, and N=N

drawn onthe figures. In Fig. 7a different symbols are used accordingto the number of grid points in the vertical Nz 1.5N.These show approximate correspondence to the straight linerepresentations of Eq. 31 in Fig. 6 at constant slope21 /N. Note that the points in Fig. 7a become notice-ably more linear as the simulations enter the triangular HAZregion of the RReLES parameter space and 1 better repre-sents the von Krmn constant. To demonstrate, as per Eq.23, that R increases with decreasing Ds=Cs

2AR4/3

, in Fig.7b different symbols are used according to Ds creatinghorizontal bands of distinct symbol type consistent with in-creasing R. Note that the bands also shift to higher ReLESwith increasing R and Cs

2AR4/3 as per Eqs. 26 and 29.

B. The high accuracy zoneTo show the transition from subcritical regions of the

RReLES parameter space to the supercritical HAZ, considerthe 16 simulations on the four paths shown in Fig. 8 with thepoint symbols. Each path has a fixed vertical resolution Nzand progresses from lower to higher R and ReLES along linesof constant Nz, roughly lines of constant slope 21 /N inEq. 31.

In Fig. 9 we plot mz against z / for each of thegroups of four simulations at fixed Nz in Fig. 8. In Sec. IV Awe argued for the existence of a critical vertical resolution N

below which the surface layer is insufficiently resolved andthe LOTW can therefore not be properly captured. The four

groups of curves in Fig. 9 show the consequences of poorresolution of the surface layer in the vertical and provide anestimate for N

. In Fig. 9a the surface layer is resolved

with, at best, three grid points. This clearly insufficient ver-tical resolution is associated with mz curves that incor-rectly bend to the left from the surface with increasing z; thisoccurs even when R exceeds 1. Furthermore, the under-resolution of the surface layer in the vertical affects the meanvelocity gradient and mean velocity profiles throughout theboundary layer; the mean gradient decreases too rapidly in z.It is not until Fig. 9c, Nz=96, that this spurious drop in mwith z transitions to the profile mz extending verticallyfrom the surface as is required by LOTW. The number ofgrid points in the surface layer up to z /=0.2 is 89, sug-gesting that this is the critical resolution for the surface layerand that N

for these LESs is 4045. We have drawn a lineat constant slope /N

=0.4 /45 in Figs. 7 and 8 showing how

paths a and b remain outside the HAZ and paths c andd enter the HAZ at higher R and ReLES.

By comparing the groups of m curves c and d inFig. 9, we note that as the simulations progress to higher R

0

0.5

1

1.5

2

0 100 200 300 400 500 600 700

ReLES

(a)

*

Re*LES

N*

0

0.5

1

1.5

2

0 100 200 300 400 500 600 700

ReLES

(b)

*

Re*LES

N*

Ds

and ReLES along lines of constant Nz, in both groups the finaltwo curves approach grid independence and the LOTW iscaptured as the overshoot is suppressed. To study the transi-tion in the surface layer in detail, and to estimate R andReLES

, we compare the six simulations shown in Fig. 8 with

the open circles which lie along each of the two paths,Nz=96 and Nz=128. The variations in mz in the lower40% of the boundary layer for each point along the two pathsare shown separately in Fig. 10. Comparing the upper andlower sets of figures, we observe nearly identical transitionsalong each of the two paths into the HAZ. At the lowest Rand ReLES an overshoot obscures the surface layer and theLOTW is not predicted by the LES. As R and ReLES increaseby decreasing Cs

2AR4/3, the same transition in z takes

place along each path: the overshoot progressively decreasesand a region of constant mz progressively strengthens un-til a well-defined region of constant mz appears in thesurface layer that is maintained when R and ReLES

exceedabout 0.85 and 350, respectively, using the Smagorinsky clo-sure. The oscillations near the surface will be discussed inSec. VII. The estimates for the lines that define the HAZfrom the critical Smagorinsky model parameters R0.85,ReLES

350, and N

45 are drawn in Figs. 7, 8, and 11a.

Figure 10 shows that as Nz, Cs, and AR are adjusted so asto progressively move the LES from the subcritical part ofthe parameter space into the HAZ, the mean velocity gradi-ent approaches a fixed point independent of the path takeninto the HAZ. In Fig. 11 we plot the normalized mean ve-locity gradient distributions for five of our computed LESswithin the HAZ at different vertical grid resolutions N andcombinations Cs

2AR4/3

. Figure 11a shows the locations ofthese five simulations on the RReLES parameter space allbut the + symbol. Figure 11b shows that these simulationswithin the HAZ are approximately grid independent over theboundary layer depth. In Fig. 11c we expand the surface

layer to show how the overshoot is suppressed and theLOTW is captured by the simulation all but the dashedline.

In Fig. 7a we also show simulations from the literature.All but the simulation by Drobinski et al.32 is well outsidethe HAZ. In two simulations, one from Andren et al.14 andone from Port-Agel et al.,4 R was sufficiently high to re-move the overshoot but the vertical resolution of the gridwas insufficient for the LES to produce LOTW scaling.

In Fig. 11a the Drobinski et al.32 simulation is indi-cated with the plus symbol and in Fig. 11c with the dashedline. Drobinski et al.32 used a different numerical algorithmand a one-equation eddy viscosity model. Their model forsurface shear stress was not indicated. Whereas their criticalparameters R and ReLES

are not necessarily the same sincethey used a different SFS stress closure, their simulationappears to be well within the HAZ Fig. 11a, their over-shoot is suppressed and they have captured LOTW scalingFig. 11c. The Drobinski et al.32 simulation predicts a dif-ferent numerical value for z in the surface layer than doour simulations with the classical Smagorinsky closure andMoeng38 surface stress model. Whereas the Smagorinsky

0

0.5

1

1.5

2

0 100 200 300 400 500 600 700

ReLES

(9a) (9b)

(9c,10a)(9d)

(10b)

*

Re*LES

N*

FIG. 8. The systematic variations on the RReLES parameter space for thesimulations plotted in Figs. 9 and 10. Figure 9a shows four simulationswith the same vertical resolution, Nz=32, but decreasing Cs

2AR4/3 represented

by +, , , and . The vertical resolutions in Figs. 9b9d are Nz=64,96, and 128, respectively. The open circles show the locations of the sixsimulations in Figs. 9a and 9b. The vertical resolutions in Figs. 10a and10b are Nz=96 and 128, respectively.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

z/

m

(a)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

z/

m

(b)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

z/

m

(c)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

z/

m

(d)

(b)(a)

(c) (d)

FIG. 9. Effect of vertical grid resolution on the LES predictions of meanshear. Refer to Fig. 8 for the locations of the individual simulations on theRReLES parameter space. R and ReLES progressively increase in eachfigure in this order: +, , , . a Nz=32, Nx=Ny increased from 32+ to128 . b Nz=64, Nx=Ny increased from 64+ to 240 . c Nz=96,Nx=Ny increased from 64+ to 288 . d Nz=128, Nx=Ny increased from64+ to 360 . =0.4 is assumed in forming m. is defined as theheight where m=0. The dashed horizontal lines indicated the upper marginof the surface layer.



model simulations in Fig. 11 predicted a von Krmn con-stant 0.33, the Drobinski et al.32 simulation produces a0.35. Factors that affect the prediction of the vonKrmn constant are discussed in Sec. VII C

It turns out that the von Krmn constant issue is asso-ciated with the development of oscillations in mz at gridcells adjacent to the surface Fig. 10. The oscillations growas the simulations move farther into the HAZ along a line ofconstant Nz. As will be discussed in Sec. VII, the oscillationsoriginate from the spurious nature of the lower boundarycondition for fluctuating surface shear stress. This observa-tion is interesting, in part, because it has been previouslyreported that the LES of the ABL is insensitive to the detailsof the lower stress boundary condition.2,41 These simulations,however, were outside the HAZ and in the presence of theovershoot and its frictional source.

VII. DISCUSSION

We consider a LES in which one models only what isrequired in order to close the numerical problem, whereLOTW is a prediction and not a forced result. We restrict theproblem to Reynolds numbers where viscous or surfaceroughness scales cannot be resolved by the grid, which weregard as highly practical. In this scenario, to close the dis-cretized dynamical system, it is necessary to produce a clo-sure for SFS stress and a model for total horizontal stressfluctuations at the surface. Anything beyond this is out of thescope of our analysis.

We have developed a theory to explain the basic issuesunderlying deviations from LOTW scaling for mean velocitygradient and we laid out a strategy for the design of LES ofhigh Reynolds number wall bounded turbulent flows basedon the theoretical arguments. These deviations extend be-yond the overshoot in scale mean velocity gradient to en-compass LOTW of mean velocity in general. In this sectionwe organize this understanding and briefly refer to new is-sues that have arisen from this study and we discuss theapplication of the current framework to the logarithmic layermismatch problem in hybrid RANS/LES methods such asDES.18

A. Breakdown in law-of-the-wall scaling:Interior versus surface

Deviations from the LOTW in mean velocity and veloc-ity gradient in the inertial surface layer arise from a compe-tition between the characteristic velocity length scales u andz and other velocity or length scales that enter either from the

0

0.1

0.2

0.3

0.4

0.5 1 1.5m

(141,0.15)

1 1.5m

(230,0.38)

1 1.5m

(257,0.48)

1 1.5m

(370,0.93)

1 1.5m

(401,1.04)

1 1.5m

(454,1.27)

0

0.1

0.2

0.3

0.4

0.5 1 1.5m

(155,0.11)

1 1.5m

(243,0.26)

1 1.5m

(336,0.46)

1 1.5m

(479,0.90)

1 1.5m

(577,1.16)

1 1.5m

(669,1.39)

z/

z/

FIG. 10. Change in structure of mz as the LES moves toward and intothe HAZ. Refer to Fig. 8 for the locations of the individual simulations onthe RReLES parameter space; R and ReLES progressively increase in eachpanel from left to right. Top panel: LES with Nz=96. Bottom panel: LESwith Nz=128. ReLES and R are given on top of each figure as ReLES,R.=0.4 is assumed in forming m. is defined as the height where m=0.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

z/

m

(b)

0

0.05

0.1

0.15

0.2

1 1.5 2

z/

m

(c)

0

0.5

1

1.5

2

0 100 200 300 400 500 600 700

(a)

*

Re*LES

N*

(b)

(a)

(c)

FIG. 11. Five simulations in the HAZ at different N and DsCs2AR

4/3,

demonstrating the convergence of the predictions of mz to a relativelygrid-independent solution without the overshoot and capturing the LOTW.a gives the locations of the five simulations in HAZ. mz from Drobinskiet al. Ref. 32 are given in c with the dashed curve and in a with the plussymbol +. =0.4 is assumed in forming m. is defined as the heightwhere m=0.



true fluid physics or during the conversion to the discretizeddynamical system that is ultimately advanced on the com-puter. Turbulence motions in the surface layer at the charac-teristic length scale z, for example, may compete with mo-tions at the boundary layer scale Khanna and Brasseur21 orwith the influences of surface friction or roughness that arecharacterized by the length scales = /u and z0. If suffi-ciently strong, these confounding scales will alter the scalingU /zu /z in the inertial surface layer. LOTW assumesthat this is not the case and experiments in the laboratory andatmosphere have generally supported this assumption whenthere are no confounding scales and the ratio of the outer toviscous length and roughness scales is sufficiently large.However, when the influence of confounding characteristicscales is sufficiently strong to compete with u and z in theinertial surface layer, deviations from LOTW result.

What we have shown here is that in the conversion fromthe exact continuous dynamical system to the LES dis-cretized dynamical system that is actually advanced on thecomputer, additional characteristic scales are introduced intothe simulated dynamics that can interfere with LOTW scal-ing in the surface layer when sufficiently strong. Several el-ements in the simulation might introduce spurious character-istic scales: i models of existing terms and new terms thatare introduced into the governing equation, ii models forunknown boundary conditions, iii the type and order of thediscretization of derivatives together with grid geometry, andiv algorithmic additions, for example, to maintain numeri-cal stability. The current study has focused primarily on theinfluences of a spurious viscous length scale that is intro-duced within the computational domain by the model for theSFS stress tensor. This spurious scale is a reflection of themanner in which the net transfer of turbulence energy fromresolved to SFSs is modeled. Whereas the interscale interac-tions that underlie the transfer of energy are purely inertial inreality, all practical closures for the SFS stress tensor modelthe net transfer of energy from the resolved scales by a dis-sipative mechanism that removes energy at the smallest re-solved scales.

However, the SFS stress is often not the only term that ismodeled in the conversion from the continuous to the dis-crete dynamical system. At the very high Reynolds numbersof practical interest, and at which LOTW is valid, the surfaceviscous layer, if it exists, cannot be resolved in a LES. Thevertical derivatives in resolved and SFS stress therefore re-quire that a model be supplied for the total stress at thesurface. Hybrid schemes couple a RANS SFS stress modelon a very high aspect ratio RANS grid in an extremely thinsurface viscous layer with LES and SFS closure on a LESgrid beginning in the lower inertial surface layer, also verynear the surface. Thus, the LES part of the simulation obtainsthe lower boundary condition on stress from the RANS partof the simulation, which is far from exact. Nonhybrids

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